Information Technologies in the Design of Aerospace Engineering (Studies in Systems, Decision and Control, 507) [1st ed. 2024] 3031435788, 9783031435782

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Studies in Systems, Decision and Control 507

Mykola Nechyporuk Volodymyr Pavlikov Dmytro Krytskyi   Editors

Information Technologies in the Design of Aerospace Engineering

Studies in Systems, Decision and Control Volume 507

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Mykola Nechyporuk · Volodymyr Pavlikov · Dmytro Krytskyi Editors

Information Technologies in the Design of Aerospace Engineering

Editors Mykola Nechyporuk National Aerospace University Kharkiv Aviation Institute Kharkiv, Ukraine

Volodymyr Pavlikov National Aerospace University Kharkiv Aviation Institute Kharkiv, Ukraine

Dmytro Krytskyi National Aerospace University Kharkiv Aviation Institute Kharkiv, Ukraine

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-43578-2 ISBN 978-3-031-43579-9 (eBook) https://doi.org/10.1007/978-3-031-43579-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The book “Information Technologies in The Design Of Aerospace Engineering” was written on the basis of the National Aerospace University Kharkiv Aviation Institute in 2022 based on the scientific work of teams of authors participating in the scientific research of the university. The results of the work were discussed at the conference ICTM’2022 which was held in Kharkiv, Ukraine, during November 18–20, 2022. This book presents materials related to the automation of design and creation of models of aviation equipment, namely: ways to automate the process of designing paragliders; creation of a family of unmanned aerial vehicles using the similarity method; automating the creation of passenger aircraft, for example, fuselage modeling, carrying out strength and aerodynamic analysis; creation of systems based on machine learning for automatic recognition of objects in photos and videos; an algorithm for optimal control of a group of unmanned aerial vehicles and a launch vehicle was considered when launching a group of navigation satellites into orbit; describes how an infocommunication robot performs intelligent actions in automatic or semi-automatic mode to collect information and transfer it to the control center, formulates the expansion principle for complex dynamic systems with a scheme of trajectory branches in a form convenient for constructing computational algorithms containing central and lateral branches, without interaction of subsystems after separation, and also formulates the expansion principle for the simplest complex dynamic system, taking into account the interaction of subsystems. Also, we are grateful to Springer—Janusz Kacprzyk and Thomas Ditzinger—as the editors responsible for the series “Studies in Systems, Decision and Control” for their great support in publishing these selected papers. Kharkiv, Ukraine December 2022

Mykola Nechyporuk Volodymyr Pavlikov Dmytro Krytskyi

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Contents

Information Technology for Determining the Flight Performance of a Paraglider Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dmytro Krytskyi, Oleksandr Karatanov, Olga Pohudina, Volodymyr Shevel, Andrii Bykov, Mariia Pyvovar, and Tetiana Plastun Designing a Basic Model of an Unmanned Aerial Vehicle for the Subsequent Development of a Family of Samples with Different Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valeriy Cheranovskiy, Evgeniy Druzhinin, Aleksey Kornev, Dmytro Krytskyi, Sergii Stetsenko, and Alexey Dunayev

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Transport Category Aircraft Fuselage Integrated Design . . . . . . . . . . . . . . 169 Oleksandr Dveirin, Oleksandr Hrebenikov, Andriy Humennyi, Dmytro Konyshev, and Anton Chumak Blind Evaluation of Noise Characteristics in Multichannel Images . . . . . 209 Victoriya Abramova, Sergey Abramov, Klavdiy Abramov, and Benoit Vozel Directions of Using Branched Trajectories of Determined Complex Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Olena Tachinina, Oleksandr Lysenko, Igor Romanchenko, Sergiy Ponomarenko, and Valeriy Novikov Using Krotov’s Functions for the Prompt Synthesis Trajectory of Intelligent Info-communication Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Olena Tachinina, Oleksandr Lysenko, Igor Romanchenko, Valeriy Novikov, and Ihor Sushyn

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Information Technology for Determining the Flight Performance of a Paraglider Wing Dmytro Krytskyi , Oleksandr Karatanov , Olga Pohudina , Volodymyr Shevel , Andrii Bykov , Mariia Pyvovar , and Tetiana Plastun

1 Introduction Paragliding is a very young type of aircraft (LA). In literally 20 years, paragliders have undergone a rapid evolution from a slightly improved parachute to an aircraft that allows them to fly hundreds of kilometers. In the course of the development of paragliders, a wealth of experience in their creation was accumulated, the technical capabilities of the developers have significantly increased. Nevertheless, the task of developing a new aircraft did not become trivial [1]. The need for it appears, firstly, due to the obsolescence of existing models of paragliders, secondly, because of the need to improve safety, and thirdly, because of the emergence of new technical solutions that increase flight characteristics and, accordingly, toughen competition in the market, etc. Despite the fact that the paraglider is the simplest and cheapest of the existing aircraft heavier than air, it is a rather complex technical system. The most critical stage in the development of a paraglider, or any aircraft, is the general design, which includes a technical proposal and conceptual design [2]. To ensure the competitiveness of the new paraglider model in the context of fierce competition, a narrow market and limited material resources and development time, a higher degree of accuracy in predicting aircraft characteristics is needed already at the early design stages. In turn, the flight performance of an aircraft is largely determined by its aerodynamic model [3]. This has been proven many times over in the history of aviation. This work is aimed at developing a technique for constructing a paraglider model, creating an information technology that allows, on the basis of a geometric model D. Krytskyi (B) · O. Karatanov · O. Pohudina · V. Shevel · A. Bykov · M. Pyvovar · T. Plastun Kharkiv Aviation Institute, National Aerospace University, Kharkiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_1

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of a paraglider, to determine its main aerodynamic parameters, and flight simulation using certain aerodynamic parameters. A paraglider is a non-motorized aircraft, a glider with a soft two-shell wing, inflated through the air intakes by an incoming air stream. The lift force is created due to the counter-flow of air flowing around the wing profile. It is she who maintains a certain speed relative to the air (the vector of the lift force can be directed not only up, but also forward). This airspeed is limited only by the complex drag force (wing, lines, pilot), for the constant overcoming of which the stored altitude is consumed. Therefore, in order to fly, the paraglider continuously spends altitude (gliding) [4–7]. Paragliding design is almost always about finding compromises. Most often, the flight characteristics of the apparatus and flight safety are on opposite scales. Flight performance means, first of all, the aerodynamic quality of the wing—the ratio of its horizontal speed to the rate of descent. The higher the quality, the more efficiently the paraglider uses the energy of gravity, and the further the pilot can fly from a given altitude. In addition to quality, important characteristics also include the maximum and minimum flight speed, ease of launch, and maneuverability. Safety basically means the ability of the soft wing to maintain its shape or restore it if collapse did occur. Safety is assessed by a series of tests—their set is determined by independent organizations such as the German DHV or the European CEN. The test pilot provokes the paraglider to enter certain dangerous flight modes and looks at how long it will take to restore the wing and whether active human actions are needed for this. Unfortunately, flight performance and safety are inversely related. Aircraft dominated by LH are designed for experienced pilots who are able to remain calm in a critical situation, correctly classify a dangerous mode and take conscious, timely actions to get out of it. Apparatus for beginners, on the other hand, rarely fold and recover from additions in a couple of seconds without the participation of the pilot. However, the “safe wing” model will always be inferior in its qualities to the sports models.

2 Classification Let’s consider the classification, that is, the division into main types, types and classes of paragliders. Paragliders are conventionally divided into several large classes: – in terms of carrying capacity—single and double; – as applicable—for free and motorized flight; – by target audience—training, for weekend pilots, athletes, athletes of the international level; – for sport purposes—paragliders for aerial acrobatics, cross-country, record flights;

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– according to the design features—one- and two-layer, two- and three-four-row, according to the nature of the stiffnesses used in the toe and the tip of the ribs, etc.; – in the speed range—ordinary paragliders, speedflyers for flying in strong winds in dynamic flows, speedgliders for high-speed gliding from the mountains. Due to the fact that the paraglider, being a purely sports device, is becoming more and more popular, the main classification parameter, which determines certain design solutions, is safety. In order for the pilot to be able to unambiguously determine which aircraft is in front of him, as well as in order not to release an insufficiently strong and reliable aircraft to the market, there is a certification of paragliding equipment [2]. Today in the world there are two certification systems for paragliders—the German DHV and the pan-European EN. The German system is older and more subjective, that is, the decision of which class the paraglider belongs to is made directly by the test pilot. At the moment, the EN certification system is the most modern and the most objectively assessing the behavior of the paraglider. Paragliders according to the EN certification system are divided into four classes: – A—paragliders with maximum passive safety and high resistance to exit from normal flight. Designed for all pilots, including pilots at all stages of training. – B—paragliders with good passive safety and resisting exit from normal flight. Designed for all pilots, including pilots at all stages of training. – C—paragliders with moderate passive safety and potentially dynamic responses to turbulence and pilot error. Returning to normal flight may require precise pilot action. Designed for pilots who are proficient in wing deployment techniques, who fly actively and regularly and have a good understanding of the use of a reduced safety wing. – D—paragliders with potentially harsh reactions to turbulence and pilot error. Returning to normal flight may require precise action. Designed for pilots with extensive experience in wing deployment, very active flying, with significant experience in turbulence and well aware of the features of using such a wing. The flight characteristics of paragliders are inversely related to the level of safety: the more reliable the wing, the worse it flies. The elongation of modern serial paragliders is in the range of 4.5 … 8.5 (up to 10 … 13 on experimental wings). Paragliders of about 10 years ago had lower aspect ratios, which was due to the use of thin wing profiles (to reduce drag) and the inability to provide the necessary rigidity. However, the maximum elongation of a certified paraglider is 6.7, only sports prototypes have more. Elongation is interconnected with arch: as a rule, flatter wings behave worse in extreme conditions, and the designer is forced to reduce the aspect ratio of such a wing. So, modern wings have the following aspect ratios: – category EN A—4.5 … 5.1; – category EN B—4.8 … 5.8;

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Fig. 1 Paragliding structural elements

– category EN C—5.6 … 6.5; – category EN D—6.0 … 6.7; – COMPETITION (uncertified)—6.5 … 8.5. The paraglider as a product is supplied in the following basic configuration: paraglider, harness, rescue system, packaging (backpack, bag). The main structural elements of the paraglider are shown schematically in the diagram (Fig. 1). Wing. The dome is a soft wing, consisting of an upper bearing surface (VP), a lower surface (LP) and a set of ribs. Loose ends. The riser or V-line is a construction of belts and buckles designed to transfer the load from the pilot to the line system (Fig. 2). Sometimes rigid elements are also used in the design. To change the installation angles in the design of the ends, a trimmer and an accelerator are used. Suspension system (PS) necessary for secure and comfortable fixation of the pilot to the SS. It is assumed a sitting or lying position during the flight. It can house a speed system, a rescue parachute, and devices, water ballast, etc. Rescue system. The rescue parachute should be able to rescue the pilot from a minimum height in the event of a paraglider failure (i.e., if it is impossible to land on it). The simplest parachute is a flat circle in cutting with 16, 18 or 24 lines (depending on the materials of the canopy and lines), reduced to one halyard. The parachute opens according to the percussion pattern of opening, when the canopy, which is partially filled after the throw and uncoupling, pulls out the lines from the honeycomb.

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Fig. 2 An example of the design of risers: 1—load-bearing slings; 2—control line; 3—carabiners for attaching slings; 4 articulated fastening of lines of row C; 5—toggle; 6—toggle fastening to row D; 7—accelerator; 8—trimmer; 9—a carabiner for the suspension system; 10—stirrup for accelerator control

Table 1 Materials used in the manufacture of paragliders Paragliding element

Requirements

Material

Dome

Durability, airtightness, low elongation (poorly stretched), light weight

Bologna, varnish, NYLON ripstop (Carrington, Gelvenor)

Lanyard system (a) slings (b) free ends

Strength, low elongation, minimum diameter, wear resistance, strength, light weight

Aramid (kevlar), dyneema (long molecular polyethylene) Webbing HP 20 … 25 LTKP 25–1000

Suspension system

Durability, convenience, light weight

Capron, avisent, LTKP 44–1600 Webbing HS 45 … 50

The materials used in various elements of the paraglider and PS are shown in Table 1.

3 Paragliding Design Technique Designing paragliders has one significant difference from working on airplanes or gliders: the behavior of a soft wing is almost beyond computer simulation. A paraglider is an aeroelastic system: the incoming air flow not only flows around the

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wing, but also influences its shape. The change in shape affects the nature of the flow, which again changes the shape—this is an endless chain of mutual influences. There is a fundamental work [2], covering all aspects of the design and manufacture of a paraglider, allowing its development without the use of computers. The technique proposed in this work includes the following main stages of creating a paraglider: Stage 1. Determination of the wing area and its shape in plan. Stage 1.1. The choice of the specific load on the wing. Stage 1.2. Calculation of the area depending on the weight of the pilot. Stage 1.3. Wing extension selection. Stage 1.4. Span calculation. Stage 1.5. Creation of the wing geometry in plan. Stage 2. Selection of the wing profile of the paraglider. Stage 2.1. Determination of the aerodynamic quality of the system. Stage 2.2. Profile selection. Stage 2.3. Determination of the parameters of the air intake. Stage 3. Determination of the number of wing sections. Stage 4. Construction of the geometry of the system in the plane of symmetry. Stage 5. Construction of the geometry of the system in frontal projection. Stage 6. Construction of the cutting shape of the wing panels. Stage 7. Design of a sling system. Stage 8. Making a paraglider. Stage 9. Test of the paraglider. However, the techniques described even in this book are so laborious that the need for design automation is obvious. In addition, a number of the most important design procedures are not sufficiently formalized, which results in a significant share of intuitive design decisions (the designer must “guess” the optimal value of the aerodynamic quality, “successfully” choose the wing layout and the number of its sections, the optimal air intake area, etc.) [8]. These features (apparently inevitable at this stage of development) lead to a large number of design iterations “synthesis— analysis”, associated with the refinement of the initially accepted initial data. There are several methods for designing paragliders that allow the development of a paraglider with the required characteristics [2, 5, 9]. However, these techniques imply an iterative approximation to the specified characteristics through the construction of several prototypes. The number of prototypes can go up to ten, which significantly increases the labor intensity and cost of developing a new paraglider model, which increases the cost of production models and slows down the process of updating the model range and introducing design improvements. The main source of information for developers is prototype testing, however, the use of programs for calculating the aerodynamics of rigid wings can make it possible to evaluate the airfoil flow in general terms and give quantitative estimates of the wing characteristics in non-extreme modes (without violating the wing geometry) in the early stages of design, which will simplify and speed up the process development.

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Due to the above-mentioned iteration of the design process, it was decided to focus on the problems of aerodynamic design of the wing and the analysis of design solutions as the most laborious. The design methodology does not require fundamental changes, since the tasks of developing other units and assemblies of the paraglider are well developed and do not cause significant difficulties. However, the modern capabilities of CAD and modeling systems allow, at relatively low cost, to supplement the existing methodology with the capabilities of virtual simulation [10]. This will allow us to predict some of the nuances of the wing’s behavior even before the prototype stage. The widely available programs for the design of paragliders and kites are mostly focused on building a three-dimensional model of the wing and line system and generating patterns taking into account the physical properties of fabric materials: the extensibility of the fabric, the “mattress” of the wing sections. At the same time, the flying qualities of the future paraglider will be entirely determined by the designer’s experience, and taking into account some characteristics (for example, the center of profile pressure) is possible only with the help of constructions and calculations by third-party methods (including graphic constructions on paper). A significant drawback of most of the existing available systems is the lack of an aerodynamic calculation module. It should be noted that a number of paragliding companies have their own design systems, including those with aerodynamic analysis capabilities. However, these systems are not available on the market and it is impossible to talk about their characteristics with the due degree of reliability. The most advanced of the free software for designing kites and paragliders is Surfplan/Gliderplan Hobby version (Fig. 3). The program allows user to create water kites with an inflatable balloon in the leading edge, as well as paragliders and kites inflated by a stream of air (parafoils), has a user-friendly interface, many settings, it is possible to use most modern features like oblique ribs. The program can print patterns for patterns on a regular A4 printer. User can also print to a virtual printer as PDF files, open them, for example, in CorelDraw, put them together and print to a large format plotter. Another free program for designing kites (which can be used for paragliders) is Foilmaker (Fig. 4). Several good flying kites have been developed with this program, but its capabilities are not sufficient for industrial applications. Both programs described above allow user to design a kite or paraglider of any shape, however, they do not take into account aerodynamics at all. Almost all available design systems do not have the capabilities of aerodynamic analysis of the constructed models. A rare example of an available design system with the ability to conduct aerodynamic calculations of the constructed model is fwDesign + XFLR5 (Figs. 5 and 6). The free version of fwDesign allows user to design paragliders and parafoil kites and export their models to the XFLR5 aerodynamic analysis program. Export of the model and patterns in dxf format is possible in the commercial version of the program, or (in the case of non-commercial use)—with the help of the authors of the program. XFLR5 is an aerodynamic analysis tool that allows user to determine

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Fig. 3 Surfplan interface

Fig. 4 FoilMaker interface

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Fig. 5 FwDesign interface

Fig. 6 XFLR5 interface

important profile characteristics, perform virtual blowing of models, and build polars. For these purposes, XFLR5 implements CFD methods (computational fluid dynamics methods). The advantages of the fwDesign system are such that this program (its commercial version) is widely used in Sky-Country, (Kharkiv), which produces world-class paragliders and kites. However, due to a number of shortcomings of the aerodynamic part of the program (XFLR5), the results of aerodynamic calculations differ significantly from the real characteristics of the paraglider, which makes it possible to use this part of the system only for comparative and qualitative analysis of profiles. The new PARATAILOR system has appeared on the market, which is relatively recently developed, but already has many advantages over its predecessors. In particular, the design process is in many respects more intuitive and well-visualized (Fig. 7),

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Fig. 7 PARATAILOR interface

the authors have foreseen in advance various possibilities for implementing structural wing reinforcements and different options for their execution (oblique ribs, mylar stiffeners, fishing lines in the leading and trailing edges, semi-ribs). In addition, one of the newest developments in the field of paragliding aerodynamics, the “shark profile”, has been implemented. Although, due to the novelty of the system, some functions and interface elements still look unfinished, in general the system was highly appreciated by independent paraglider designers. The PARATAILOR system, like fwDesign, has an aerodynamic module, allowing to calculate the aerodynamic characteristics of the paraglider according to its geometric model (Fig. 8). The capabilities of this module and their comparison with similar capabilities of fwDesign and the real characteristics of a flying prototype are of interest for research. In addition, there are specialized software environments for aerodynamic and hydrodynamic analysis used in industrial applications. The most famous are Flow Vision and Flow Simulation for Solid Works. Thus, one of the most important directions for improving the design process of paragliders at the present time is to increase the objectivity of aerodynamic analysis at the early stages of development.

4 Simulation Tools An ideal paraglider simulation system should have the following hardware and software elements: – soft wing paraglider simulator;

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Fig. 8 Aerodynamic module PARATAILOR

– controls that copy the corresponding controls of a real paraglider; – feedback, allowing the pilot to feel the reaction of the wing to air currents on the brakes and the body through the harness; – 3D environment display system (for example, OCULUS Rift virtual reality helmet); – a system for generating a landscape and detailed 3D environment; – weather simulation system. Figure 9 shows an example of a hardware implementation of such a system (Virtual Foot Flyer project). A significant difficulty in the implementation of such a system is the development of software that provides simulation of a soft wing inflated by an incoming air flow. Most often, developers take the path of simplifying the task and consider the wing as a rigid object. Of course, in such cases all wing transients are lost. Sometimes developers add separate elements of simulation specifically for transient modes (in particular, the addition of wingtips—“ears”, which is also one of the standard operating modes of a paraglider wing), but this approach still does not provide an opportunity to fully investigate the behavior of the wing in abnormal modes. The number of paragliding simulators on the software market is very small due to the specificity of the topic and the difficulties of its implementation. Below is an overview of free flight simulators available on the market with the ability to simulate a paraglider. Paraglider Simulator (Fig. 10)—a simplified simulator of the behavior of a paraglider in the air flow, which can be used to train a novice pilot to practice

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Fig. 9 Virtual foot flyer

wing dives and throws, that is, it simulates the toggle and accelerator operation. It does not have the ability to change the parameters of the paraglider and, due to its primitiveness, is not of interest.

Fig. 10 Paraglider simulator interface

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Fig. 11 FlightClub interface

FlightClub (Fig. 11)—a simulator of paragliding route competitions with extremely simplified graphics and gameplay. Upstream visualization is implemented by drawing a cloud at the top of the stream. Custom glider model cannot be inserted. DemoFly 5 (Fig. 12) provides the most accurate simulation of a soft wing, including transient modes, but does not provide an opportunity to create your own aircraft (most likely, the behavior and characteristics of the wing are hardcoded in the code). Looks very promising, but the authors have not updated it since 2009, carried away by the task of creating a “predictor of flows”—Thermal Assistant. This simulator is best suited for integration with a hardware system, such as the one shown in Fig. 9. ParaflySim (Fig. 13)—one of the most successful paragliding simulators in terms of combining the characteristics of the interface convenience and the elaboration of the aerodynamic model. ParaflySim includes wing folding simulation elements. However, there is no way to design or upload your own glider model. MicroFlight-Hangsim (Fig. 14)—a simulator specializing in simulating weather conditions and ultralight aircraft, with open source and great capabilities, due to which it was widely developed by the efforts of the public. It is possible to create your own models, including a paraglider (there is a ready-made—Spider). Unfortunately, in the case of a paraglider, the simulation is carried out as for a rigid wing. There is no transient simulation. Condor Soaring. Separately, among the flight simulators, we can distinguish Condor Soaring-a simulator, the purpose of which was to simulate the aerodynamics of gliders and local meteorology as accurately as possible. This simulator has received wide recognition among esports players, which has led to the specifics of its development.

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Fig. 12 DemoFly 5 in ground mode

Fig. 13 Screenshot from ParaflySim

The authors denied the possibility of creating their own models of aircraft and explicitly forbade discussion about the possibility of embedding paraglider support in the Condor.

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Fig. 14 MicroFlight—Spider paraglider

4.1 Flight Computers Paragliding pilots are increasingly using so-called flight computers—specialized software that runs on car navigators or tablets with GPS, or even smartphones. This type of software provides navigation support, and also helps to solve specific problems for non-powered flight (calculation of McCready parameters). For a high-quality solution of such problems, the aerodynamic and other characteristics of the aircraft must be entered into the flight computer. In competition conditions, a pilot using a properly configured flight computer is able to accurately plan his actions, in particular to calculate the final segment of the flight (the so-called “high-speed section”) so that the paraglider arrives at the finish line with a minimum residual height. Obviously. The most widespread programs are LK8000 (Fig. 15) on the WinMobile platform, as well as XCSoar (Fig. 16) on the Android platform. The main development environment for the paraglider wing was the noncommercial version of PARATAILOR. This system will be compared with fwDesign and SolidWorks in the course of building and analyzing a model of the Discovery 4 commercially available paraglider. The PARATAILOR system has its own aerodynamic module for calculating the characteristics of the wing, the features of which will be investigated in the course

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Fig. 15 LK8000 interface in flight mode

Fig. 16 XCSoar interface in flight mode

of this work. The calculated characteristics of the wing will be compared with the experimental results and used to simulate the wing. The Discovery 4 paraglider model was originally developed by Alexey Rakov, General Designer of Sky Country. The model was provided in the fwDesign system used in the design of paragliders at Sky Country. Therefore, it will be necessary to rebuild the model in the PARATAILOR environment using the parameters from fwDesign.

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To simulate the wing of the paraglider, the MicroFlight—Hangsim simulator will be used. There is a model of the Spider paraglider in it and it is possible to change its flight characteristics. The final check of the wing characteristics calculated from the model will be carried out on the LK8000 flight computer in Replay mode (playback of the flight from a real track taken during the flight of a real Discovery 4 glider).

5 Algorithm for Determining the Performance Characteristics of a Paraglider Wing As stated earlier, the general task of designing a paraglider is largely provided with ready-made software that implements the stages of determining the main structural and geometric characteristics. Therefore, in this work, the main attention is focused on the problems of aerodynamic design and analysis. There is a preliminary determination of the geometric and aerodynamic characteristics of the airfoil in the fwDesign system and a refined aerodynamic analysis in the PARATAILOR CFD Analysis system. The analysis of the reliability of the calculated aerodynamic data is provided based on the test results in the MicroFlight simulator environment and according to the LK8000 flight computer. At the system level, the task of obtaining output data (flight characteristics) based on the specified input (geometric parameters) should be solved. The result of processing the 3D model of the paraglider is to obtain its approximate flight performance (Fig. 17). It remains to determine exactly what tasks need to be performed at the processing stage. A separate task, not directly related to processing, but necessary for the functioning of the system, is the conversion of a 3D model from fwDesign format to PARATAILOR format.

Fig. 17 Detailing the system being developed

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6 Algorithm for Converting a 3D Model to PARATAILOR Format The paraglider model developed in fwDesign is saved in a proprietary program format that is not compatible with common CAD systems. It is possible to export a 3D model in dxf format, but the final model contains a huge number of elements (surfaces used to approximate curved surfaces), which makes it unsuitable for use in aerodynamic calculations (Fig. 18). For aerodynamic analysis in PARATAILOR, it is necessary to build a 3D model of the paraglider wing in PARATAILOR according to the known dimensions from fwDesign. To build, it is necessary to export from fwDesign a file with the coordinates of the paraglider profiles (Fig. 19). The output is a *.pfl file containing two-dimensional coordinates of the profile points (Fig. 20a). To work with this profile in PARATAILOR, user need to transform it into a threedimensional system (add a third zero coordinate to each point). The resulting file must be saved as txt, having previously cleared all comments (Fig. 20b). As can be seen in Fig. 20, in sections 1–20, a profile with a relative thickness (ratio of maximum thickness to chord) of 17.8% is used, and in sections 21–25, the relative thickness of the profile smoothly decreases to 10%, so for these sections it is necessary to rework the file with profile coordinates. As a result, the files of profiles used for the paraglider were obtained: D4-17_8.txt, D4-17_3.txt, D4-16_ 5.txt, D4-15_0.txt, D4-13_0.txt, D4-10_0.txt.

Fig. 18 Model exported from fwDesign in dxf format

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Fig. 19 Exporting a profile file from fwDesign

Fig. 20 Converting a profile file for solid works

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Fig. 21 Data dialog with the main parameters of the paraglider in PARATAILOR

When constructing a wing, it is necessary to take the profile corresponding to the current section. Creation of a new project in PARATAILOR and setting the main parameters of the wing. After starting PARATAILOR, user need to open a template project from the system kit—template.prg. After that, in the dialog called by the Data button (Fig. 21), set the following parameters. Geometry tab: – Total span [m]—wingspan (parameter P1); – Pilot’s hang height [m]—the height of the pilot’s hang point (parameter P2); – Carabiner distance [m]—distance between carabiners (parameter P3). Tab Number of cells: Total number of the cells—the number of sections of the paraglider (parameter A4). The Sew.allow tab contains sewing tolerances and is not of interest for these tasks. Profile alignment tab: Profile alignment point at [%] (parameter P5). Ribs type tab: select Orto ribs—ribs perpendicular to the wing plane. External Forces tab: – Pilot + equipment weight [N]—weight of the pilot with equipment in Newtons (950 N); – Pulling force in horizontal direction [N]—pushing force (in the case of a paramotor), set to 0 N.

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Fig. 22 Data from fwDesign for the Data PARATAILOR dialog

The information for these fields is taken from the wing model in fwDesign (Fig. 22). Constructing a wing shape in a horizontal plane. PARATAILOR and fwDesign systems differ greatly in their design approach. While fwDesign allows the user to change the dimensions and other parameters of each section of the glider separately, PARATAILOR offers convenient visual tools without burdening the user with the need to enter a lot of data. In the Flat shape dialog, the first thing to do is to press the Reset button to reset the initial values (Fig. 23). In Fig. 23 the numbers along the leading edge show the length of the chords of each rib in centimeters, the numbers along the trailing edge show the width of the sections, the value on the left is the length of the central chord of the wing in millimeters, and on the right is the tip length in millimeters. Orange handles can be used to change the size and shape of the wing. In this case, its area (Flat Area [m2 ]) and aspect ratio (Aspect Ratio) are automatically recalculated. To build the wing shape in the horizontal plane, user need the following data from fwDesign (Fig. 24): – length of the central chord (parameter P6); – wing tip chord length (parameter P7);

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Fig. 23 Dialog Flat shape. Wing shape in the horizontal plane Fig. 24 Data from fwDesign for the FlatShape dialog

– position of the leading edge at the wingtip (parameter P8); – wing area (parameter P9); – wing lengthening (parameter P10). Knowing the indicated data, in the Flat shape dialog, using movable handles, it is necessary to set the required dimensions of the central chord and wingtip, as well as adjust the wing shape, ensuring the required wing area and aspect ratio (Fig. 25). Using the same markers, user can try to find a wing shape that will give maximum aspect ratio for the same wing area, span and width, but in the case of the Discovery 4, this has apparently already been achieved. Creation of the wing shape in the frontal plane. The wing shape in the frontal plane in PARATAILOR is constructed in a similar way using movable markers.

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Fig. 25 Dialog Flat shape. Discovery 4 wing shape

To build from fwDesign, user need the following data from the Flat Shape (Fig. 26) and Canopy (Fig. 27) tabs: – – – – – –

wing area (parameter P9); projection area (parameter P11); wing lengthening (parameter P10); projection elongation (parameter P12); wing span in projection (parameter P13); the height of the frontal projection of the wing (parameter P14).

Fig. 26 Data from fwDesign (Flat Shape tab) for FaceShape dialog

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Fig. 27 Data from fwDesign (Canopy tab) for FaceShape dialog

In the course of optimizing the wing shape according to the given data, the following parameters were achieved: – – – –

Proj.area = 23.84 m2 (23.36 m2 required); Proj.aspect ratio = 3.87 (3.92 required); Proj. Wingspan = 4.8 1 m (4.78 m required); Heigth = 2.73 m (2.68 m required).

The result is shown in Fig. 28. Applying Profiles to Ribs. Setting the wing profile as a whole and individual ribs is performed in the Aifoils dialog (Fig. 29). As shown in Fig. 19, the Discovery 4 has profiles with different relative thicknesses. It is necessary to repeat this construction in PARATAILOR. On the Airfoils dialog, on the Set transitional rib tab, set the number of the rib from which the profile will change, in this case it is 21. Now, on the Import airfoil tab, user need to call the file open dialog and open the file of the main wing profile D4-17_8.txt, after which the profile will be loaded and displayed in the graphic field of the Airfoils dialog. After that, on the “Apply to…” tab, successively click on buttons 1 (apply the profile as the main wing profile) and 2 (apply the profile to the transition ribs) or, equivalently, on the large button that combines their action (Fig. 30). Using the “Select from rib” tab, user can make sure that the profile has been applied to all the ribs of the wing. Next, user need to sequentially load the rest of the profiles (D4-17_3.txt, D4-16_5.txt, D4-15_0.txt, D4-13_0.txt, D4-10_0.txt) and

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Fig. 28 Face shape dialog. Discovery 4 wing shape

Fig. 29 Airfoils dialogue in PARATAILOR

apply them to ribs 21–25, respectively. To do this, use the tool on the Apply to selected rib tab (Fig. 31). To design a paraglider in PARATAILOR, the next steps should be to set the parameters of air intakes, reinforcements, oblique ribs, lines, etc. however, this is not important for our task. The model (Fig. 32) is already ready for the analysis of aerodynamic characteristics.

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Fig. 30 Dialogue airfoils. Installing the main profile

Fig. 31 Dialogue airfoils. Setting a specific profile

7 Algorithm Aerodynamic Analysis of the Paraglider Model Of particular interest from the aerodynamic characteristics is the wing polar. The classic view of the polar is shown in Fig. 32. These characteristics are usually determined during tests of models in a wind tunnel, but the advent of powerful computing technology makes it possible to reduce many tests to virtual experiments. For this, methods of computational fluid dynamics are used. The module for calculating (Fig. 33) aerodynamic characteristics PARATAILOR allows user to determine such characteristics of the wing as the coefficients of lift, resistance, moment and, on their basis, calculate the characteristics of the wing in polar coordinates (polar of the wing). The input data for the calculation is a 3D model of the wing, the total mass of the aircraft, drag coefficients of the line system and the pilot. The calculation can be performed for a simplified model and taking into account the “mattress” of the wing. More accurate results are obtained for the second case.

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Fig. 32 Discovery 4 paraglider model

Fig. 33 Settings of the module for calculating aerodynamic characteristics

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7.1 Flight Simulation Algorithm in MicroFlight Simulator The Discovery 4 will be simulated in the MicroFlight environment based on the existing Spider model. The model consists of 3D model data and a flight dynamics file. The 3D model does not affect the behavior of the aircraft in the simulator, so its adaptation will not be considered. The dynamics of the behavior of an aircraft of the “paraglider” class can be set by editing the glider.cfg file, the structure of which is shown in Table 2. For the simulation of a non-powered paraglider, of interest are, first of all, the following the following parameters of the configuration file: – S is the area of the paraglider; – Cd0—drag coefficient at zero angle of attack; – Ki is the coefficient of inductive resistance; Table 2 Glider.cfg file structure and data Glider.cfg file structure

Explanation (from MicroFlight instructions) This file is composed of two sections: 1. The glider data:

0

HasEngine; // does the glider have engine 0 or 1

30.0

VNE = 30; // never exceed speed

16.0

s = 16.0; // m^2

0.05

cd0 = 0.05; // unitless

0.025

ki = 0.025; // unitless

120.0

float m = 120.0; // kg

3.5

cla = 3.5; // unitless

0.3

al_max = 0.3; // radians

0.0

al_min = 0.0; // radians

0.15

alfa0 = 0.15; // radians

1.0

th_factor = 1.0; // thrust efficiency

650.0

th_static = 650; // static thrust newtons

6.0

g_max = 6; // max g

6.0

sink_max = 6; // max sink rate on ground

3.0

thermal_sense = 3; // thermal sensitivity factor

0.0

glider type = 0; // 0—hang glider 1—conventional 2—paraglider

@ Low performance Trainer High wing area, low speed, low glide ratio Breaks easily, Unpowered Excels in slow speed, good for ridge flights Poor penetration against head winds

This part is a simple text only section that must begin with @ and endes when the file ends

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– Cla is the coefficient of lift of a thin profile; – Alfa0—wing angle of attack. The Cd0 value is defined as the drag coefficient of the wing at the minimum angle of attack (0°). The Ki value is defined as follows: Ki = 1/(p × e × AR)

(1)

where e—the Oswald coefficient, which is set in CFD Analysis (by default it is 0.9); AR—wing lengthening. The Cla value is calculated as: Cla = 2 × p × (AR/(AR + 2))

(2)

Apparently, due to the specific shape of the paraglider wing, the projection elongation must be taken into account in the calculations. To determine the characteristics of the paraglider in flight in the simulator, the flight simulation must be performed in the absence of wind and thermal activity (it must be configured in the simulator). It is necessary to run the simulation in the scenario when the aircraft starts moving at a certain height. After establishing a stable flight mode, it is required to record the descent rate and horizontal flight speed according to the instrument data in the simulator (GPS). From these values, the resulting flight performance of the model can be calculated.

7.2 Algorithm for Analyzing the Flight Track Using the LK8000 Flight Computer To check the calculated characteristics of the paraglider on empirical data, user need to find a suitable flight track performed on a Discovery 4 paraglider in an area for which there is a relief map in the LK8000 in the Leonardo track database or in the competition reports. The preferred track will be a track with a clearly visible “fly-by” section—a final flat gliding curve before landing. This track and the map of the area must be placed in the corresponding folders of the flight computer file system. Based on the data obtained during the aerodynamic analysis of the paraglider model, it is necessary to create a paraglider polar file in the LK8000 format. The polar file has the extended WinPilot format (Table 3): Field 1: Gross weight of the aircraft without ballast; Field 2: Maximum ballast mass; Field 3–4, 5–6, 7–8 pairs of values of horizontal speed and speed of descent, km/ h and m/s (these values are used to interpolate the velocity polar curve); Field 9: Wing area.

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Table 3 Polara file example Mass dry gross [kg]

Max water ballast [liters]

Speed 1 [km/h]

Sink 1 [m/s]

Speed 2 [km/h]

Sink 2 [m/s]

Speed 3 [km/h]

Sink 3 [m/s]

Wing area [m2 ]

330

90

75.0

−0.7

93.0

−0.74

185.00

−3.1

10.6

The polara file must be placed in the _Polars folder and loaded when configuring the LK8000. Flight simulation must be performed in Replay mode. While playing the track, the LK8000 will determine the speed and direction of the wind and plot the contour of the flight zone on the map, taking into account the wind, relief, altitude and characteristics of the paraglider. During the flight on the flight segment, it will be necessary to make sure that the landing point is in the flight zone, which will indicate that the calculated characteristics of the paraglider correspond to the real characteristics of the wing. In the event of a performance mismatch, the predicted flight area can mislead the pilot, causing a competition failure or even an accident.

8 Software Architecture The software includes the Windows operating system, the standard Microsoft Office package, the fwDesign system, the PARATAILOR system, the MicroFlight simulator with the supplied Spider paraglider model, and the PC version of the LK8000 flight computer. The structure of data flows and user interaction is shown in Fig. 34. The transfer of information between the system components is carried out manually by the user. The required profiles are exported from the fwDesign model, which are then converted to a PARATAILOR-compatible format using Excel. The system user builds a paraglider model in PARATAILOR using data from fwDesign. The model recreated in PARATAILOR is analyzed in PARATAILOR CFD Analysis. The user configures the calculation settings as needed and simulates the paraglider blowing. Virtual blowdown results are saved in Excel format. If necessary, simulation is performed for several specified parameters (weight, paraglider area, drag coefficients). The resulting set of results is brought together and analyzed by the user in the Excel environment. Further, according to the data in Excel, the necessary coefficients for the configuration files MicroFlight and LK8000 are calculated. In the MicroFlight simulator, the weather parameters and the flight scenario are set, after which the flight is performed, during which the flight performance of the model is evaluated. The Leonardo database contains a suitable flight track for a Discovery 4 paraglider (if possible, the flight should have a long gliding interval before landing). This track is loaded into the LK8000 and played in repeat mode. While playing the track, the flight computer evaluates weather conditions using the

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Fig. 34 Scheme of information exchange between system components

track information and builds the configuration of the flight zone, taking into account the characteristics of the paraglider, the direction and speed of the wind, the terrain, and the flight height above the relief. In the course of playing, the correspondence of the polar configuration of the paraglider to the real characteristics of the paraglider is assessed.

9 System Testing To test the developed information technology for studying the aerodynamic characteristics of the paraglider, a model of the Discovery 4 serial paraglider was used. According to testers, this paraglider contains an extremely successful set of flight characteristics and passive safety. It has been certified for safety class EN B, which makes it highly recommended for flight school graduates and weekend pilots. However, despite its high safety, it possesses high flying qualities, which makes it interesting for athletes as well. Maximum measured quality: 8.8 at 36.8 km/h.

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Fig. 35 Pressure distribution over the wing surface

Determination of aerodynamic characteristics in PARATAILOR CFD Analysis. Traditionally, for programs of this kind, the module for calculating the aerodynamic characteristics makes it possible to visualize the pressure distribution over the wing surface. An example is shown in Fig. 35. The CFD Analysis module allows user to vary a number of paraglider parameters such as aircraft mass, pilot drag and lines. In Fig. 36 shows the results for the default parameters of the calculation module. Of Fig. 36d, it can be seen that for this model the speed of maximum quality will be 11.48 m/s (41.33 km/h) at a speed of descent of 1.33 m/s. The tangent line drawn from the center of coordinates to the graph shows the angle of inclination of the wing gliding trajectory with the maximum quality, and the point of contact shows the corresponding components of the wing speed in this flight mode. Usually, the wing of a paraglider is tuned to this speed. From the same graph, user can determine the speed of the minimum descent and the maximum wing speed. For these speeds, the quality will be significantly lower than in the scheduling mode with maximum quality. It should be noted that when constructing these graphs, the resistance of the lines and the pilot with the harness was taken into account. For lines the value Cd = 1 was used, and for the pilot Cd = 0.5, set by default by the authors of PARATAILOR. Results for the Discovery 4 model in PARATAILOR using Cd = 1 for the line system and Cd = 2.5 for the pilot are shown in Fig. 2.61.

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Fig. 36 Calculation results, with Cd = 1 for the line system and Cd = 0.5 for the pilot: a— coefficient of the wing lift depending on the angle of attack; b—the coefficient of total drag of the wing, depending on the angle of attack; c—the quality of the wing (the ratio of the lift to the drag of the wing) depending on the angle of attack; d—polar of wing velocities (speed of maximum quality is shown, and wing gliding angle)

For the Discovery 4 serial paraglider, the maximum measured quality is 8.8 at a speed of 36.8 km/h. Obviously, the first variant of the calculation with the coefficients Cd = 1 for the line system, Cd = 0.5 for the pilot gives a picture closer to reality. Thus, as a basis for further work, we will take the calculated characteristics of the paraglider shown in Fig. 37. According to these figures, the maximum quality of 8.7 is achieved at a horizontal speed of 41.6 km/h. The rate of descent is 1.33 m/s. Installation angle of attack of the wing 4 °C. The Discovery 4’s 27 m2 wing is designed for pilots weighing between 80 and 100 kg (gross weight, which includes clothing, harness, rescue system and the wing itself). Analyzed the characteristics of the wing for a pilot weight of 95 kg, which is close to the maximum wing loading. It is interesting to see the characteristics of the

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Fig. 37 Calculation results, with Cd = 1 for the line system and Cd = 2.5 for the pilot: a—wing quality depending on the angle of attack; b—polar of wing velocities

wing with a minimum load of 80 kg. In Fig. 38 shows the corresponding performance of a Discovery 4 wing: 27 m2 with a load of 80 kg. Comparing the results (Figs. 36a, c and 38b, d), one can make sure that the maximum quality of the wing practically does not change, however, with a decrease in the wing loading, its speed decreases, including the speed of maximum quality. Reducing the speed characteristics of the wing does not in the best way affect the wing’s ability to fly upwind.

Fig. 38 Calculation results, with a pilot weight of 80 kg: a—wing quality depending on the angle of attack; b—polar of wing velocities

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The aerodynamic calculation showed a fairly accurate coincidence of the calculated wing characteristics with the experimentally determined wing characteristics of the serial Discovery 4. Compared with the results obtained earlier, one can note a significant simplification of the aerodynamic analysis, an expanded range of aerodynamic characteristics available to the user, the ability to take into account the parameters of the drag lines and the pilot, as well as a rather high accuracy of the obtained characteristics compared to the known experimental data. The final conclusion about their accuracy will be made based on the results of flight simulation and analysis of a real track using the calculated aerodynamic characteristics. Flight simulation in MicroFlight simulator. The standard model of the Spider paraglider in the simulator has the following configuration file glider.cfg (Table 4). After setting up the weather conditions in the simulator (no wind and thermal activity) and running the simulation for the Spider paraglider and waiting for the end Table 4 Glider.cfg file structure and data Glider.cfg file structure

Explanation (from MicroFlight instructions) This file is composed of two sections: 1. The glider data:

0

HasEngine; // does the glider have engine 0 or 1

20.0

VNE = 20; // never exceed speed

27.5

s = 27.5; // m^2

0.032329

cd0 = 0.032329; // unitless

0.133878

ki = 0.133878; // unitless

90.0

float m = 90.0; // kg

3.5

cla = 3.5; // unitless

0.5

al_max = 0.5; // radians

0.1

al_min = 0.1; // radians

0.25

alfa0 = 0.25; // radians

0.5

th_factor = 0.5; // thrust efficiency

350.0

th_static = 350; // static thrust newtons

10.0

g_max = 10; // max g

6.0

sink_max = 6; // max sink rate on ground

0.5

thermal_sense = 0.5; // thermal sensitivity factor

2.0

glider type = 2; // 2—paraglider

@ High performance Paraglider Good gliding performance, medium penetration and very low sink rate This high performance Paraglider is made in the shape of the most popular paragliders in use today

This part is a simple text only section that must begin with @ and endes when the file ends

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of the transient processes, we fix the following values of the horizontal and vertical velocities: – Vx = 17 knots; – Vy = 236 ft/min. Knowing the horizontal (measured in knots) and vertical (measured in feet per minute (ft/min)) speed of the paraglider, user can determine the aerodynamic quality: Quality = Vx/(Vy × 60/5280). For the standard Spider, the quality is 6.3. To modify the glider.cfg configuration file in accordance with the data of the Discovery 4 model, it is necessary to perform some calculations. Figure 36b for an angle of attack of 0 °C, we determine that the drag coefficient cd0 = 0.01. The projected aspect ratio of the Discovery 4 wing (Fig. 26) is AR = 3.92, and the Oswald coefficient (see Fig. 33) e = 0.9, which gives, in accordance with formulas (1) and (2), the values ki = 0.09022 and cla = 4.16; The setting angle of attack is 4 °C, i.e. alfa0 = 0.0698 rad. Substituting the calculated values into the configuration file, as well as the paraglider area s = 27 m2 and the aircraft mass m = 95 kg, we obtain the following form of the glider.cfg configuration file (Table 5). A flight simulation with such a configuration file gives the results (Fig. 39), which shows the values of the flight speed and descent rate): – Vx = 21 knots; – Vy = 148 ft/min; – Quality = 12.4. As can be seen, the results obtained are quite different from those predicted. Apparently, the aerodynamic model of the MicroFlight simulator cannot be a reference, however, if necessary, by selecting the coefficients, the behavior of the model can be reduced to the desired one. Flight track analysis in LK8000 flight computer. Searching the Leonardo database of the Ukrainian pilots’ route league made it possible to find the most interesting track, which was filmed during a flight on the Discovery 4 paraglider—29 m2 by pilot Pavlo Jacques. The flight took place from Mount Gimba, Carpathians. After takeoff from an altitude of 1455 m, the pilot caught the updraft and gained about 250 m in it, reaching an altitude of 1700 m, after which he flew into the valley. During the flight, the pilot practically did not process the updrafts. The landing took place at a point 8 km away from the start at an altitude of 600 m above sea level. The flight track is shown in Fig. 40, track characteristics (height above the relief, absolute height, distance from the start)—in Fig. 41. As user can see, after the climb, the flight took place in a gliding mode, which makes this track convenient for checking the calculated wing characteristics.

Information Technology for Determining the Flight Performance … Table 5 Glider.cfg file structure and data Glider.cfg file structure

Explanation (from MicroFlight instructions) This file is composed of two sections: 1. The glider data:

0

HasEngine; // does the glider have engine 0 or 1

40.0

VNE = 40; // never exceed speed

27

s = 27; // m^2

0.01

cd0 = 0.01; // unitless

0.09022

ki = 0.09022; // unitless

95.0

float m = 95.0; // kg

4.16

cla = 4.16; // unitless

0.5

al_max = 0.5; // radians

0.0

al_min = 0.0; // radians

0.0698

alfa0 = 0.0698; // radians

0.5

th_factor = 0.5; // thrust efficiency

350.0

th_static = 350; // static thrust newtons

10.0

g_max = 10; // max g

6.0

sink_max = 6; // max sink rate on ground

0.5

thermal_sense = 0.5; // thermal sensitivity factor

2.0

glider type = 2; // 2—paraglider

Fig. 39 Flight simulation of a Discovery 4 paraglider model in MicroFlight

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Fig. 40 Discovery 4 flight track

Fig. 41 Discovery 4 flight track characteristics

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Ideally, the influence of the wind should be excluded, however, the flight computer has the function of determining the direction and speed of the wind from the drift of the paraglider while moving in a thermal spiral and will take these data into account in its work. The configuration file of the paraglider polar, compiled on the basis of the simulation data (presented in Fig. 36) is shown in Table 6. In Fig. 42 the start moment is fixed. Direction to the landing site is shown in bold. The landing site is 8.3 km away from the launch site. The landing point will be a reference point for checking the calculation of the flight zone. The flight zone itself is depicted by a dotted broken line, calculated in accordance with the relief, the polar file and the current height of the paraglider. At the time of takeoff, the flight computer has no data on the strength and direction of the wind. He will be able to calculate them only after a few spirals. The highest point reached during the flight is shown in Fig. 43. By this time, the flight computer has not yet calculated the strength and direction of the wind, but it is clear that the flight zone has expanded and even includes the point of the future landing. In this case, the excess over the finish point is expected to be 230 m. The flight zone configuration will change as soon as the wind is calculated. After some time, the flight computer calculated the wind (it is 9 km/h, and the direction is shown as an arrow). The flight zone stretched out in the wind. The landing point still remains in the flying zone, closer to the edge. At the moment when 1.5 km remained before landing, the frame shown in Fig. 44. The excess over the finish line was expected to be 125 m. Table 6 Example file LK8000 polar: Discovery 4 Mass dry gross [kg]

Max water ballast [liters]

Speed 1 [km/h]

Sink 1 [m/s]

Speed 2 [km/h]

Sink 2 [m/s]

Speed 3 [km/h]

Sink 3 [m/s]

Wing area [m2 ]

95

0

29.43

−1.18

41.6

−1.33

54.75

−2

27

Fig. 42 Takeoff time displayed on the flight computer screen

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Fig. 43 The moment of maximum altitude displayed on the screen of the flight computer

Fig. 44 1.5 km left before landing

As can be seen, the paraglider flew with worse quality than it could, according to the calculations of the flight computer. The following reasons are possible: the condition of the paraglider tissue is unknown (an increase in tissue permeability leads to a deterioration in the quality of the flight); the real weight of the pilot is unknown, in addition, his paraglider had an area slightly larger than the one for which the calculations were carried out; possible atmospheric inhomogeneities (downdrafts), which led to an excessive loss of altitude; incorrect actions of the pilot; with a tailwind, the optimal action is to decelerate the wing to transfer the flight to the minimum descent mode (see Fig. 36c); there may be a discrepancy between the relief map of the real area. Based on the results of testing the prototype of the system, the following conclusions can be drawn: – a method for converting the fwDesign model to PARATAILOR has been developed; – a method for transferring aerodynamic characteristics to the environment of the simulator and the flight computer has been developed; – in the first approximation, the satisfactory agreement of the calculated aerodynamic characteristics with the real characteristics of the aircraft under study is confirmed;

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– the methodology for creating a configuration file for the simulator has shown its acceptability, as well as the necessity and possibility of its refinement to ensure a more accurate simulation; – shows the importance of accurately determining the aerodynamic characteristics of an aircraft and correctly entering these data into the flight computer to ensure the accurate operation of the pilot assistance system (calculating the flight area, etc.); – the efficiency of the complex that implements the proposed design technology has been confirmed.

10 Conclusion The authors carried out a detailed classification of paragliders according to the main types, types and classes. An analysis of the work on the design of paragliders was carried out, the features of this process were explained, and the need for its automation was substantiated. An overview of automated paraglider design systems is given, their main advantages and disadvantages are described, while the main attention is focused on the problems of aerodynamic design and analysis. An analysis of the reliability of calculated aerodynamic data is given, and the main features of the use of IT in the development of paragliders are described, which allows, based on the geometric model of a paraglider, to determine its main aerodynamic parameters, as well as flight simulation using certain aerodynamic parameters.

References 1. Kritsky, D.N., Druzhinin, E.A., Pogudina, O.K., Kritskaya, O.S.: A Method for assessing the impact of technical risks on the aerospace product development projects. In: Advances in Intelligent Systems and Computing, vol. 871, pp. 504–521 (2019). https://www.scopus.com/ inward/record.uri?eid=2-s2.0-85057811342&doi=10.1007%2f978-3-030-01069-0_36&par tnerID=40&md5=beb8ccdc4b6461d10abc97d5b36ab800. https://doi.org/10.1007/978-3-03001069-0_36 2. Landell-Mills, N.: Paragliding explained by Newtonian physics. Independent Research, pp 1–29 (2022) 3. Vermeer, N.-J., Sørensen, J., Crespo, A.: Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39(6), 467–510 (2003). https://doi.org/10.1016/S0376-0421(03)00078-2 4. Landell-Mills, N.: How airplanes generate lift is disputed (2019). https://doi.org/10.13140/RG. 2.2.34380.36487 5. Chen, Y., Lin, B., Lin, J., Wang, S.: Experimental study of wake structure behind a horizontal axis tidal stream turbine. Appl. Energy 196, 82–96 (2017). ISSN 0306-2619. https://doi.org/ 10.1016/j.apenergy.2017.03.126 6. Tang, H., Lam, K.-M., Shum, K.-M., Li, Y.: Wake effect of a horizontal axis wind turbine on the performance of a downstream turbine. Energies 12, 2395 (2019). https://doi.org/10.3390/ en12122395

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7. Nakamura, Y., Fukamachi, N.: Visualization of the flow past a Frisbee. Fluid Dyn. Res. 7(1), 31–35 (1991). https://doi.org/10.1016/0169-5983(91)90004-3 8. Kritskiy, D.N., Druzhinin, E.A., Karatanov, A.V., Kritskaya, O.S. Content management method of complex technical system development projects. In: Advances in Intelligent Systems and Computing. AISC, vol. 1080, pp. 293–303 (2020). https://www.scopus.com/ inward/record.uri?eid=2-s2.0-85076981282&doi=10.1007%2f978-3-030-33695-0_21&par tnerID=40&md5=a1a433082d5210a2c2b59a8b1296ac76. https://doi.org/10.1007/978-3-03033695-0_21 9. Maiorova, K., Vorobiov, I., Boiko, M., Suponina, V., Komisarov, O.: Implementation of reengineering technology to ensure the predefined geometric accuracy of a light aircraft keel. Eastern-Eur. J. Enterp. Technol. 6(114), 6–12 (2021). https://doi.org/10.15587/1729-4061. 2021.246414 10. Kritskiy, D., Yashin, S., Koba, S.: Unmanned aerial vehicle mass model peculiarities. In: Advances in Intelligent Systems and Computing. AISC, vol. 1265, pp. 299–308 (2021). https://www.scopus.com/inward/record.uri?eid=2-s2.0-85091130983&doi=10.1007%2f9783-030-58124-4_29&partnerID=40&md5=96c9bc1235ee042b8dab641625a87998. https://doi. org/10.1007/978-3-030-58124-4_29

Designing a Basic Model of an Unmanned Aerial Vehicle for the Subsequent Development of a Family of Samples with Different Purposes Valeriy Cheranovskiy , Evgeniy Druzhinin , Aleksey Kornev , Dmytro Krytskyi , Sergii Stetsenko , and Alexey Dunayev

Abbreviations ADC AI AV AVP BL CAD CG CM CP EN FSW ID JM LS NACA NASA NE PE PP PS RAM

Erodynamic characteristics Air intake Aerial vehicle Altitude-velocity performances Boundary layer Computer-aided design Center of gravity Cruise missile Center of pressure Ejector nozzle Forward swept wing Inlet device Jet muzzle Lifting surface National Advisory Committee for Aviation National Aeronautics and Space Administration Numerical experiment Physical experiment Power plant Propulsion system Random access memory

V. Cheranovskiy (B) · E. Druzhinin · A. Kornev · D. Krytskyi · S. Stetsenko · A. Dunayev National Aerospace University, Kharkiv Aviation Institute, Kharkiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_2

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TJE UAS UAV VF WT

Turbojet engine Unmanned aerial system Unmanned aerial vehicle Vortex former Wind tunnel

Nomenclatures H V α β X Y K cx cy cp mz M Re

Altitude Speed Angle of attack Sideslip angle Drag force Lift force Lift-to-drag ratio Drag coefficient Lift coefficient Pressure coefficient Coefficient of longitudinal moment Mach number Reynolds number

1 General Provisions Based on the complex of the tasks assigned to the unmanned aerial vehicles (UAV) of operational and tactical class (by analogy with evolution of “large” aircraft), the growth of their functional capabilities according to complexity from surveillance and reconnaissance up to attack and special missions is supposed. Against the background of continuous modernization of means of counteraction and application of protective measures on the UAV, achievement of these qualities leads to rise in price of systems. In these conditions, the basis of ensuring operational effectiveness of unmanned aerial system (UAS) in general and, first of all, its basic component—the UAV, is, firstly, the providing of conditions of effective use of the equipment of payload on the corresponding flight stages: – means of surveillance, search, guidance, aiming and tracking of the target; – operational and reliable delivery of weapons to the target; and secondly: – survivability of the UAV.

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These factors cause availability of the UAV flight performances with qualities of multimode operation and maneuverability. The multimode operation is implied as ability to carry out guided flight and to carry out the corresponding tasks in wide range of speeds and altitudes. Maneuverability—ability to change the direction of the movement and flight velocity per unit of time. The maneuverability is characterized by the value of zone of admissible angular speeds of turn. In the trajectory movement it is defined by the module and the direction of vector of overload. The listed qualities, except the decision of group of the main tasks, promote realization of active component of survivability: stability of flight in turbulent atmosphere at ultra-low heights in the mode of rounding of ground relief in simple and intricate meteorological conditions; on condition of detection or program warning—evasion from weapons owing to aerodynamic properties of the carrier. Possibility of maneuvering with low speeds on the second regimes of flight is agreed with reduction of requirements concerning energy capabilities of the launchers and reserve of the launch. The passive component of survivability is determined by the maximum reserve and is ensured by compactness of aerodynamic configuration, application of the lowreflecting forms, shielding of contrast configuration elements of the airframe, usage of the radio absorbing materials. The efficiency of UAS regarding variety of solvable tasks causes multifunctionality, that is provided by usage of family of the UAVs specialized according to functions. Creation of such family supposes reduction in cost of life cycle of system, in particular, stages of research and development, mass character of production, implementation, maintainability by means of similar or restrictedly similar scaling of the UAVs onto classes of takeoff weight and unification of the UAVs according to aerodynamic configuration, basing, design with big share of interchangeable elements and structure of the equipment constructed according to the principle of “open architecture” [2]. Similar approach is accepted in the USA when developing requirements to the perspective generation (expected by 2020) of middle UAVs—programs MQX and MQ-M—where, in particular, it is defined that replacement of the necessary equipment and gears has to be provided in regular order [3]. Thus, the UAV basic model for family of multifunction UAS has to correspond and/or provide adjustment at the corresponding stages of research and development of the following qualities: – – – – – – – –

wide range of altitudes and speeds of flight; maneuverability; low observability; high invulnerability; flight regularity; universality of basing; independence of application from weather conditions and time of day; possibility of configuring of payload;

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– high rates of maintainability, repairability and resource; – high degree of commonality and fitness to mass production. The main flight performances and operational capabilities of aerial vehicle (AV) that are defining its functionality depend on its aerodynamic characteristics (ADC). The lift-to-drag ratio of AV is one of the key aerodynamic parameters and criteria, which is characterizing the level of aerodynamic perfection. In particular, values of required thrust, speed range, altitude and characteristics of maneuverability (angular speed of rotation of AV relatively the center of gravity (CG); speed-up during acceleration and braking in horizontal flight; rate of climb; angular speed and the minimum radius of turn in the horizontal plane) depend on this characteristic. Required characteristics are reached by the choice of forms and key parameters of AV assemblies and their relative positioning, i.e. by its aerodynamic configuration. Characteristics of the movement AV relatively CG depend on efficiency of controls and controllability of AV. In modern conditions, the ideas about general configuration of perspective AV with fixed wing are based on the principles of complexification of integrated configurations of the lifting system with smooth contours and realization of favorable interference which harmoniously combine aero-gas-dynamics, internal integration and the transport perfection. Throughout this paradigm, the engine represents considerable source of impulse of aerodynamic forces, and effective power plant (PP) has to not only overcome drag force, but also create favorable contribution into lift force of AV [1]. The potential of functional combination of the motional and propulsion, and the lifting systems that is low-used at this stage, is represented as the most perspective direction of improvement of AV. PP as the tool of energy control of circulation, direct control of aerodynamic forces and controllability augmentation is applied in the systems of stream flow with partial taking of mass of air from the engine compressor or autonomous compressor [4–9]. Among difficult implementable methods of jet stream control, the greatest development was gained by full-flow systems with engines of direct reaction in the systems of thrust vector control/deflection, and also the combinations of the high-lift wing blown by jet stream with generation of forced circulation, based on Coanda effect, such as at An-72 aircraft [1]. The geometry of the UAV model of bigger sizes which is previously investigated in paper [10] (Fig. 1), in comparison with the UAV of the same class of classical configuration is accepted as the initial one. In particular, the basis of the choice of similar general configuration was formed by advantages of the UAV of integrated configuration in possibility of achievement of bigger subsonic speeds due to smaller values cx , with bigger value K, of bigger values cy due to attainment of high angles of attack, the best characteristics of maneuverability, and positioning of CG favorable for placement of consumable materials and payload.

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Fig. 1 General view of the UAV of initial geometry

2 Integration of General Configuration of the UAV The general configuration of the UAV is characterized by complex of the technical solutions investigated and tested in a variety of projects including ones finished for physical implementation [1–22]: – the folding supercritical wing of variable sweep with the large wing root extension—vortex former (VF); – the propulsion system (PS) integrated into lifting system with upper submerged vortex-free air intake (AI) and the ejector nozzle (EN) of step type; – tunnel configuration of PS with passive and active gas-dynamic regulation of the inlet device (ID); – the system of jet stream control of trajectory parameters—the thrust vector control. The main aspects defining properties and parametrization of lifting system are considered further.

2.1 Supercritical Wing of Variable Sweep On the basis of spectrum of solvable tasks, the UAV has to possess wide speed range. At this stage of the conceptual project, the engine characteristic, in which maximum speed at the majority of turbojet engines (TJE) of the interesting dimension is equal to Mach number M = 0.8–0.9, acts as fundamental restriction. From the previous analysis, the border of speed range of the UAV from the direction of the maximum

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values is defined by the crisis phenomena which initiate transition to supersonic flow regimes in local zones. At the same time, overcoming sharp drag growth by increase in power of PP is connected with decrease in weight perfection of the UAV. Local shock waves arise in zones of the greatest depression and first of all the most loaded wing experiences this. From aerodynamics positions, decrease of drag and increase of M cr * of flight are reaching by applying the supercritical profiles which are specially developed for such conditions and deprived of peak depression in characteristic of distribution of air pressure. The main disadvantage of supercritical profile—the increased negative pitching moment—is eliminated by the corresponding wing planform and wing twist. Reduction of wave drag, the increase cya max owing to increase in flight angles of attack and increase in weight perfection are carried out by the standard means: by wing sweep and wing taper. At this stage of aerodynamic designing for the purpose of the solution of indeterminacy of configuration and interaction of elements of AV design, the complex of studies of variants of the lifting surface (LS) with different sweep angles of leading edges of outer panels of wing χ l.e. is provided.

2.1.1

Integrated Supercritical Profile of Wing

The minimum drag of the supercritical profiles of the second generation which have proved in practice, is provided by angle of attack, close to zero, with characteristic practically symmetrical flow around of front half. Required lifting force is realized due to considerable positive curvature of tail part. The flow accelerated on the tail part of upper surface can reach supersonic speed, closing sound pocket by rather weak shock wave. Using the appropriate profiling, the receiving of shock-free flow and significant decrease of the drag in one reference point [23, 24] is possible; however for multimode working process the similar quality has low value. Some other its qualities described below is defining the choice of this type of profile. The property of supercritical profile in comparison with conventional, that is important for maneuverable aircrafts and is expressed by growth of characteristics of M cr * and cy on the wave flow separation modes before the sharp growth of drag, is noted. On front edge of big radius the significant suction force, and also favorable pressure profile for delaying of the separation phenomena and increase of α cr of separation at all speed range, are created. In comparison with conventional airfoil profiles that are having expressed curvature of upper arch, supercritical ones allow to receive required value of lift force with the minimized drag at the increased values of relative thickness that promotes increase in weight perfection of design. The integrated supercritical profiles realizing the main share of lifting force on the tail part possess inherent shortcoming—the increased negative pitching moment which leads to the increased losses onto balancing. On the models for wind tunnel tests it has been shown that for the airplane with supercritical wing of variable sweep the ensuring of balancing by means of twist with slightly high trim drag, than in configurations with normal profiles, is possible [23].

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Folding Wing with Root Extension

Out of condition of universality of basing, including launch from the transport-launch canister, application of folding wing is provided on the UAV. Simultaneous increase in compactness of configuration, decrease in AV drag, resistance to flow separation on moderate and high angles of attack, are provided for ensuring maneuverable characteristics, and are reached by application of wing of low aspect ratio. Besides, for the purpose of simplification of design and preservation of streamlined forms, the similar wing allows applying the scheme of lateral folding with longitudinal axle of rotation in wing root. These conditions, respectively, limit wing span and proportions of folding parts. The wing root extension—vortex former, characteristic of modern maneuverable aircrafts having wing with low aspect ratio, like the fighters Su-27, MiG-29, F-16/18/ 22, etc., on moderate and high angles of attack promotes flowing without separation in the central part of the large area, and realization of high values of cy . A priori, as a result of use of VF the favorable synergetics is created: the multimode operation of the wing is reached due to expansion of ranges of operational values of α, Y, V, K, with improvement of stability of lifting system and (under condition of coordination with the tail unit) controllability of AV. The wing with root extension has the wider center-of-gravity range that gives universality at change of payload, reduces requirements to fuel use and promotes allocation of large volumes for onboard fuel reserve [1]. Configuration of similar wing allows to provide essential lifting properties for the central fuselage part of AV, provides configuration forming of folding wing by reducing sizes of outer panels. Besides, wing root extension creates balancing component of lifting force, favorably affects characteristics of the wing drag formed of supercritical profiles, partially or completely neutralizes their shortcoming—the increased negative pitching moment.

2.2 The Conformal Power Plant Integrated into Lifting System Designing of PP has complex character and proposes acceptance of configuration solutions mainly determined by set of the factors of flowing of external and internal contours. The bases for adoption of such decisions are the general requirements of minimization of drag of AV and providing stable operating of PP in all flight envelope. Standard problems of stage of aerodynamic designing are caused by difficulties of configuration of ID owing to the operational requirements listed in subsection 1. Application of jutting AI, as a rule, is connected with additional interference and also with need of overcoming specific configuration difficulties at designing of compartments of payload/expendable load, the undercarriage of launching devices,

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for container placement or catapult launch. Such dominating factors as unification of type of basing, multimode operation, maneuverability, reserve play key role in forming general configuration of PP [1]. Above-mentioned circumstances, and also some others, find the expression in configuration of conformal and integrated by PP with the air jet engines and AI, partially or completely submerged into AV body. Similar AI are attractive from positions of compactness, weight perfection and low-visibility [1].

2.2.1

Gas-Dynamic Unity of Propulsion System and Lifting System by Means of Submerged AI and Step Type EN of Upper Dislocation

Working process in more widespread configurations with the bottom AI (Fig. 2) is in conflict with LS on the modes of takeoff, climb and maneuvering: zone depression in the neighborhood of AI on the “maximal” mode prevents from increase in vertical velocity of maneuver. Influence of upper AI antithetically. The corresponding disposition of PP, that is similar to represented in Fig. 3 [11, 12], in the process of work can provide improvement of AV lifting properties at the expense of positive contribution into balance of aerodynamic forces [1]. However, if intake of air to the bottom AI happens naturally, when angle of attack increases, the ensuring of multimode operation of ID with upper AI is complicated by condition of inflow without separation on flight runs with moderate and high Fig. 2 Examples of configurations of the single-mode non-maneuverable UAVs of air basing carried out according to “stealth” technology equipped with the lifting fuselage with folding elements of lifting system and integrated power plant with the vortex air inlet of NACA

AGM-129 ACM

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Fig. 3 Experimental models of aerodynamic configuration of the conformal power plant: a maneuverable multimode GRN-1 aerial target (2009, UAE); b non-maneuverable single-mode stealth fueler UAV of the MQ-25 Stingray project (2019, USA)

positive angles of attack with possibility of emergence of aerodynamic shadow on downwind side of the AV fuselage and demands special measures [1]. At the mode of the excess available consumption that is characteristic for high velocities of flight, the deceleration of flow in upper AI that is attracting emergence of force directed down is noncritical owing to growing with velocity pressure of efficiency of aerodynamic means of generation and control of lifting force [1]. Except its normal assignment, i.e. zone aerodynamic unloading (see Sect. 2.1.2) the functionality of wing root extension—VF of the offered general arrangement is connected with ensuring working process of PP [12, 14]. Application of the dorsal submerged AI is in harmony with differential area rule that favors expansion of flight speed range into the transonic area [1]. The similarity of multiple of goal-setting qualities with sample in Fig. 3a causes acceptance of it as the nearest analog for reasoning of the made basic design decisions and as etalon reference point for further verification.

2.2.2

The Submerged Air Intake

Mass application of series of the submerged AI of NACA of the first generation in the different systems of air supply in which strict requirements to quality of flow are not imposed (Fig. 4) became practical result of extensive researches of different

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forms and modifications of ID at present. Isolated cases of serial application of AI of NACA for air supply of cruise air-jet engines for single-mode air vehicles—the cruise missiles (CM) (RGM-84 “Harpoon”, RGM/UGM-109E “Tomahawk BlockIV”, AGM-158A JASSM) are known (Fig. 2). Equipping of airplane’s main PP with similar AI was unsuccessful and was limited by flight tests of experimental, mainly non-maneuverable aircrafts (Douglas XB-43/51, Avro-707B, North America YF-93A, “Tacit Blue”, “Shark” UAV, etc.) [1]. Defects of vortex AI are eliminated by smoothing of contours. Development of the rectangular submerged AI of NACA has led to creation of AI of the next generation which differs by smooth coupling of surfaces (Fig. 5) [25]. Unlike AI of NACA evolving from samples with the rectangular cross-section of air path, in the design of “Club” CM the submerged vortex-free AI with the channel of circular cross-section which is smoothly entered into the cylindrical fuselage (Fig. 6) is initially applied [1]. The advantage of configuration of AI of 3M-14AE CM is the possibility of the organization of vortex-free flow with satisfactory characteristics of uniformity on entrance to the engine and the best parameters of compactness. However, owing to

Fig. 4 The submerged AI of NACA of the first generation, and schematic representation of vortex working process Fig. 5 The vortex-free submerged AI with the rectangular directional entrance (Boeing)

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Fig. 6 The vortex-free compact submerged AI with the curvilinear directional entrance of 3M-14AE CM

the features of working process connected with configuration decisions, issues of homogenization of flow in the curvilinear channel of vortex-free ID remain relevant [1]. In paper [1] the method of profiling according to the principle of conformity to lines of flow, and the way of increase in gas-dynamic quality of the vortex-free submerged AI [13], which effectiveness is confirmed in special researches [19–22], are offered. Compact ID with the submerged AI (Fig. 7) with the value of preservation of total pressure and homogeneity of output flow at the level of jutting sideward ID or forward ID with the made narrower entrance cross-section for high-speed AV is developed as a result. Nevertheless, having quite high characteristics on the main flight run and engine operation, ID of this kind is characterized by quite narrow range of gas-dynamic stability in the range of 20% of changing of air consumption. That is, it well meets

Fig. 7 Flow of ID with conformal AI

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the requirements to single-mode AV and for the solution of assigned task needs application of the measures expanding the range of gas-dynamic stability [1].

2.2.3

Ensuring Multimode Operation of the Inlet Device

The given circumstances form the acceptance basis for further study of similar to represented in Fig. 7 ID with the vortex-free submerged AI, that is developed by the staff of NII PFM KhAI Scientific Research Institute in initiative order for application on multimode AV, differing by presence of system of homogenization of the inhausted flow with gas-dynamic self-regulation [13]. The objective is solved by equipping of ID by the built-in homogenizer intended for creation structurally and parametrically homogeneous flow: unidirectional with uniform fields of speeds and pressure near the output cross-section of ID; warning and prevention of the separation phenomena; ensuring gas-dynamic control. The principle of gas-dynamic regulation consists in passive and active remote control of consumption of basic mass of flow and its homogenization by change of consumption and other parameters of the boundary layer (BL).

2.2.4

Tunnel Configuration of TJE with Means of Active Regulation of the Inlet Device

Venting of ID by means of the bypass channel with engine bay and the ejector nozzle (the forming the second contour) (Fig. 8) provides automatic control of consumption on all flight modes and throttlings of PS [13]. The BL having the maximum deviations of parameters of flow, having executed function of gas lubricant, is taken away through ring slot of the bypass channel. At the same time a number of additional opportunities is implemented [1]: – thrust augmentation of PS due to increase of the mass of air rejected through by-pass shutters in the direction opposite to flight course and also due to growth of m ˙ a;

Fig. 8 The scheme of working process of tunnel configuration of PS on the main mode [79]: absorption of air, including BL, from fuselage surface, separation of wall layers in ID with their subsequent extraction through EN

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– some increase of thrust and coefficient of efficiency of PS from utilization of the warmth which is output from heat exchangers and elements of the engine design through bypass contour; – decrease of unmasking signs—thermal, radar and acoustic visibility of AV. In the simplest design version of joining of the line of bypass contour with the channel of feed of unregulated EN at approach to flight runs, critical for ID, with the speeds close to Vmax , working process in homogenizer is coordinated with the growing ejection component of jet thrust during the work of PS at the maximum mode. Thus, the tunnel configuration of pure turbojet engine functions analogously to bypass engine, promoting expansion of speeds range in the vicinity to Vmax . Displacement of BL at smaller speeds and at throttlings of PS promotes maintaining of steady operation of ID even at emergence of local separations [13]. Additional feed of ID by means of by-pass shutters of bypass contour only on upper surface of AV airframe promotes expansion of speeds range in the vicinity Vmin [13]. The lack of elements of hermetization between air duct and TJE eliminates need of the constructive organization of access into this area that leads to simplification of design and operation of PS. Reduction of requirements to the accuracy of production and to deviations of form of air duct in the zone of coupling to the engine from ±0.2 to ±1–2% of diameter of inlet cross-section of TJE promotes increase in technological effectiveness of production. The lack of labor-intensive assembly operations on joining of TJE with ID, release of the scheme of fastening of PS from the excessive redefining links increases technological effectiveness of assembly and maintainability. Against the background of the general compactness, the lack of reactions in the design of ID and fastening assemblies of TJE from thermal and vibration loads and mutual movement under the influence of overloads causes high weight perfection of the design of PP [13].

2.2.5

Ejector Nozzle of Step Type and Realization of Jet Stream Control

Disadvantages of the single-mode UAV of initial configuration concerning possible insufficient controllability because of the accepted scheme of tail unit are eliminated thanks to the organization of jet stream flow. Also based on condition of reserve of the UAV from ground thermal sensors, the nozzle of TJE is shielded by AV design by means of upper dislocation. Half-open EN is profiled by ledge in the form of upper step so that the jet stream flow out TJE jet muzzle (JM) tangentially to AV surface, and stick to it owing to Coanda effect. The flattened jet stream promotes the best sticking. Improvement of propulsive properties of the PS at the same time is expected thanks to ejection of air from the second contour of PS by semi-limited jet stream, and thanks to involvement into the force interaction of additional masses of air from the environment with the step [1].

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The functionality of upper step of EN is not limited to this. The organization in this place of the controlled surface gives it the possibility of thrust vector deflection in the process of trajectory control of AV. The partition of control surface along the symmetry plane of the UAV gives the capability of pitch control by the coordinated deflection of right and left surfaces, and the capability of roll control by their differential deflection. By means of the thrust vector deflection system the problem of ensuring maneuverability with minimization of the area, weight and drag of control surfaces is solved. At the same time, minimization of control speed expands flight speed range into zone of smaller values, that will be coordinated with the requirement of multimode operation and reduces requirements concerning energy capabilities of launchers.

3 The Designing of UAV Basic Model, the Forming of Basic Geometry, Estimations of Aerodynamic Characteristics and Flight Performances of UAV 3.1 Features of Aerodynamic Designing of the Aerial Vehicle of Integrated Configuration with the Propulsion System Included into the Lifting System The stage of aerodynamic designing is critical by the importance as the general configuration of subject of development forms exactly here, and by that the level of its technical perfection is defined. In the solution of circle of the tasks connected with obtaining required aerodynamic characteristics (ADC) of the aerial vehicle the best accuracy and the largest volume of ADC under condition of the correct statement of task are provided by natural experimental methods. However, owing to big laborintensiveness and cost of comprehensive works relatively to stage of the conceptual project of the small-size UAV, these methods are difficult implementable and often irrational [1]. Implementation into practice of research and development of the predictive numerical researches instead of natural physical tests allow to obtain necessary information about subject of designing at stages of the preliminary and draft design, long before the beginning of its physical realization. Requirement of coordination of aero-gas-dynamic characteristics of the offered system and the basic importance of interaction of the external and internal flows saturated with viscous spatial effects, already at early stages of the conceptual project makes uncontested use of methods of the numerical experiment (NE) [1]. Overcoming antagonism between acceptable quality of development and its resource support are reached owing to implementation of the new combined method of aerodynamic designing with emphasis onto NE and low-cost model tests in the wind tunnel (WT) where the guarantee of reliability is understood as satisfactory

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correlation of results of NE and the physical experiment (PE). The content of stage of aerodynamic designing is given in Fig. 9 [1]. From the formal point of view, problems of stage of gas-dynamic designing come down to factor optimization. Their solution in the inverse statement owing to complexity of initial system of the equations and boundary conditions is not represented as realistic. In this regard for profiling of external and internal contours, direct statement of boundary problems in the cycle of iterative search by means of purposeful change of form of internal borders of rated area is used [1]. Necessary simplification of design models for the purpose of economy of resources, taking into account of aerodynamic interference with different extent of influence of constituent aerodynamic phenomena and effects onto investigated parameters of flows in traditions of aerodynamic designing is reached according to the principle of factor decomposition [1]. The traditional approaches to aerodynamic designing, that on the initial stage are presenting the lifting surfaces, the fuselage and other AV parts as autonomous essences, are considered as private manifestation of the principle of decomposition onto constituent elements and physical processes. Synthesis of aerodynamic configuration of AV airframe begins from taking into account of effects of interference: the spatial flows in zones of joint of components. In relation to problems of aerodynamic designing of perspective AV of integrated configuration with PS, which is included into lifting system, the approaches which are based on the principle of decomposition in classical form are inapplicable; however, in the offered approach, the isolated components are used as reference points for identification of the general models of external and internal flowing which are directly intended for providing stages of configuration forming and detailing of external and internal forms of AV airframe [1]. From technical aspect this approach is supported with usage of the integrated technologies of computer modeling: the available software products of the computeraided design (CAD) system, the standard SolidWorks and ANSYS software packages (SP), that are implementing in particular generation methods parametrically of the managed geometry of spatial objects, providing and simplifying the iterative solution of the general and private variational tasks by means of the organization of the corresponding cycles of the predictive researches by NE methods. Available modern methods and means of aerodynamic designing do not give formal unambiguous way of solution of assigned tasks. Need of carrying out iterative complex of design actions, according to Fig. 9, is dictated by the previous experience of creation of AV-analog [26]. Rather low-cost experimental definition of stand characteristics of PS, as a rule, provided by its producer, does not give ideas about its altitude-velocity performances (AVP). And receiving the last is accompanied by a number of difficulties—availability of special barometric WT or production of full-scale flight test benches and carrying out expensive tests. Besides, the ID, that used for these purposes, are affecting onto flow parameters, and even more reduce the value of AVP of the isolated PS in tasks of integration of PS into lifting system, equipping of PS with special ID and JM [1].

Fig. 9 Block diagram of intensification of process of aerodynamic designing of integrated configurations of “AV-PS” system of stage of pilot project by method of integrated physical and numerical researches

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For the specified reason, analytical recalculation of the TJE parameters in the first approximation is made according to its available test-bench characteristic; and in the UAV model containing the TJE model, gas-dynamic parameters are set by method of “sources—outflows”. After the choice of the engine, identification of its design model is carried out into configurations of bench tests which then is integrated into the UAV general model. At detailed design phase carry out definition of integrated UAVs ADC and impact assessments of external conditions onto consumption characteristics of TJE and characteristic of gas-dynamic stability in continual area of operation conditions [1]. The offered low-resource method of aerodynamic designing of AV of integrated configurations with PS included into lifting system with use of rational combination of NE and PE consists in interactive use of results of physical modeling by means of tests in small-size industrial WT, bench and other tests, as etalons for the determined identification of element-wise mathematical models of the airframe, power plant and other components; and integrated aero-gas-dynamic characteristics of AV and PS are determined with the help of the complete homogeneous nondeterministic design model synthesized on the basis of the received identified models of components. This method allows to intensify preparation of basic data for the subsequent design stage, Figs. 10 and 11, and to avoid fatal mistakes at stages of the draft proposal and draft design [26].

3.2 Verification of Design Models, Reasoning of Use of CAD Software Packages The solution of verification task is included into complex of design actions for the purpose of forming of design basis in providing the subsequent researches on models of the created UAV [1]. Numerical computation is the most universal method of the solution of difficult applied tasks that has gained predominant dissemination. Models of gas-dynamic processes are constructed on the basis of the fundamental laws of preservation (that are applied to all set of points of area of the decision) and are brought to the system of the equations closable by set of additional connections: boundary conditions, state equations, etc. [1]. The system of the equations describing gas-dynamic flows is numerically solved in rated area. The rated area is the mapping of physical area. In the tasks of computational aerohydrodynamics such mapping is carried out by means of the computational meshes consisting of elementary volumes—cells. The form of cells, their arrangement in rated area affects the accuracy of decisions, and in certain cases the unsuccessful choice of parameters of mesh can result in impossibility of execution of calculations. In areas with strong gradients of velocity, pressure, temperature, etc. it is necessary to expect the greatest computing mistakes. The way of reduction of error of

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I II III IV V

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Development periods 0 – 0.9 0.9 – 1.6 1.6 – 3.3 3.3 – 6.3 6.3 – 9.3

The used designing methods Analytical engineering design methods Preliminary designing by NE methods Preparation of experiment in WT Experimental studies in WT Detailed aerodynamic designing by NE methods

Fig. 10 Dependence of expenses and volume of information obtained at each stage of aerodynamic designing of the UAV-analog, on time of development [27]

Fig. 11 Extrapolation of dependence of expenses on time of obtaining full volume of information about AV ADC when using only of numerical methods (dashed lines) [27]

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sampling lies in mesh refinement. Reducing of cells is carried out in areas with big gradients of parameters, enlargement of cells—in other places so that to receive approximately regular distribution of error of discretization. Difficulties arise with the geometry containing fine details, edges and fairings of variable curvature including scale contrast at mapping of PP channels, scapular wreaths of turbomachines, smallsized inlets and outlets, etc. in integrated rated area of the AV model. For obtaining the reliable numerical solution of tasks of this kind it is necessary to use rated meshes with small spatial steps. Computational burden at the same time become so considerable that because of restrictions of computing facilities it is not always possible to receive sufficiently exact solution of tasks. More flexible choice has to improve mesh in local scale [1]. In general view, especially in the first iteration of creation of numerical model with unevident morphology of flows the main approach of the solution of the described problems consists in use of adaptive methods with automatically mesh refinement. However, there is problem of providing condition of conservatism of the decision in some cases resulting in inadequacy of the decision or to the divergent process [1]. In connection with the stated, any method of numerical modeling needs the prior information providing the parametrical identification adequate to object of researches. In the field of aerodynamic experiment, the information obtained as a result of tests of model elements of AV airframe in wind tunnel with the small-sized working area and generalized then on the basis of universal representations of the theory of dimensionality and similarity is the most available and reliable [79]. The wind tunnel laboratory of KhAI University has appropriate technical base for these purposes and possesses extensive experimental backlog.

3.2.1

Statement of Numerical Experiment

Traditional providing of requirements of maneuverability owing to aerodynamic properties of the airframe assumes the normal mode of output of the UAV onto high angles of attack. Along with the solution of issues of increase in the flight duration (efficiency), the raising of aerodynamic perfection of the UAV becomes one of priority tasks. The solution of these tasks at stage of forming the general configuration of the UAV is accompanied by difficulties of assessment of ADC by means of computational methods beyond of the range of linear changes and taking into account of the “thin” phenomena of local aerodynamics occasionally having considerable impact on lift-to-drag ratio. In the conditions of speculative ideas about configuration of subject of designing, as etalon reference point for verification of design model of external flowing with similar flow morphology, the scale model of the UAV analog for wind tunnel tests is accepted (Figs. 12 and 13). For the purpose of simplification and acceleration of researches, according to the principle of decomposition, LS formed by integrated configuration of variable sweep wing with the fuselage and motor-gondola is considered (Fig. 13b) [1].

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Fig. 12 General view of scale model of the UAV-analog for WT tests

Estimates of characteristics are conducted within the theory of dimensionality and similarity. In the considered task when performing physical and numerical experiments, conditions of geometrical similarity and similarity of flows by Re number are satisfied. The model for WT tests is executed in the scale 1:2.86. WT tests of scale model are conducted in the field of self-similarity, i.e. at Re numbers that are bigger than critical value. Along with this, compliance of similarity of flow through internal paths, including the corresponding consumption characteristics of PS, will be necessary, when modeling external flow. Overcoming scale factors connected with restrictions of reconstruction of the internal flows is supposed owing to execution of design geometrical models in the scale of real AV [1]. For computational researches the following CAD software packages are used: SolidWorks and ANSYS. These CAD software packages, that based on identical mathematical base of determination of parameters of flow by method of numerical calculation of system of the equations of Navier–Stokes, have different (in the nomenclature, complexity and opportunities) sets of tools for creation and debugging of mesh models. For assessment of opportunities of the NE methods, iteration cycles of the gasdynamic calculations of flow around model of UAV LS that are differing in way of adaptation of rated mesh for the purpose of increase in accuracy of approximation of surfaces of geometrical model in the environment of SolidWorks Flow Simulations with only available k−ε model of turbulence [28] are carried out. In the environment of ANSYSFLUENT except other topology of mesh mapping of rated area, a number of models of turbulence and their modified analogs which are a priori corresponding to expected flow regimes [29–31] is in addition investigated. The received results are

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Fig. 13 Natural model of AV in the subsonic WT T-4 KhAI (a) and three-dimensional design model of lifting surface (b)

compared among themselves and with results of PE that are received in WT on scale model. General approaches to creation of numerical model. The motionless model is placed into the rated area. As the accuracy of the solution of task of external flow depends on the sizes of rated area, its borders are located at such distance from model at which there is minimum perturbation of air flow caused by flow around a model, which can be neglected. Modeling of processes of the longitudinal movement is made in symmetric statement [1]. Gas flow movement is modeled by means of Navier–Stokes’s equations describing in non-stationary statement the conservation laws of mass, impulse and energy of real gas, taking into account viscosity and compressibility according to the speed and altitude of flight. On external borders of rated area the boundary conditions (parameters of undisturbed flow on infinity, and direction of the movement of flow) are set. On surfaces of model the conditions of non-percolation and adherence of flow, and also value

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Table 1 Conditions of carrying out the numerical experiment Parameters

Solidworks [1]

ANSYS

WT tests [1]

Sizes of rated area/working area of WT, m

7.6 × 2.6 × 3.4

1.5 × 1 × 1

ø1 × 1.5

Length of model, m

3.7

1.29

1.29

Wing span, m

2.31

0.81

0.81

Range of angles of attack

–4°–28°

Reynolds number Re

11.0 · 106

4.0 · 106

4.0 · 106

Pressure p0 , Pa

103,325

99,400

99,400

Temperature, °C

20

20

18–22

Initial turbulence, %

2

2

5.6°, where the deviation does not exceed 5.25%. In the range of small angles (–3.9–3.7°) the increased discrepancy of characteristics up to 10.16% is also not represented as critical, even against the background of error of measurements of PE in the range of ±5%. For characteristic cx (α) the situation is not so unambiguous. On low negative angles of attack the similar nature of deviations with the curves received in the SolidWorks environment is observed, and that confirms the assumption of influence of holders of aerodynamic scales in PE. However, the curve k–ε RNG does not look Table 5 Comparison of results of NE with results of PE in WT Angle of attack

Values of coefficients of experiment

Comparison of results of modeling of k−ε RNG with results of experiment in WT

Comparison of results of modeling of k−ε standard with results of experiment in WT

α°

Cx pe

δ Cx

δ Cx

Cy pe

δ Cy

δ Cx (%Cx pe)

δ Cy (%Cy pe)

δ Cy

δ Cx (%Cx pe)

δ Cy (%Cy pe)

−3.9

0.019

−0.119

0.005

0.009

26.3

−7.6

0.006

0.011

31.6

−9.2

−2

0.013

−0.06

0.006

0.002

46.2

−3.3

0.002

0.003

15.4

−5.0

−0.1

0.012

0.004

0

0

0

0

0.001

0

8.3

0

1.8

0.011

0.066

0

0.006

0

9.1

0

0.012

0

18.2

3.7

0.015

0.128

0.001

0.013

6.7

10.2

0.001

0.014

6.67

10.9

5.6

0.021

0.192

0.002

0.006

9.5

3.1

0.001

0.019

4.8

9.9

7.5

0.032

0.257

0.003

0.009

9.4

3.5

0

0.026

0

10.1

9.5

0.049

0.322

0.005

0

10.2

0

0.001

0.031

2.0

9.6

11.4

0.071

0.39

0.007

0.017

9.9

4.4

0.001

0.025

1.4

6.4

13.3

0.1

0.458

0.004

0.024

4.0

5.2

0.004

0.021

4

4.6

15.2

0.133

0.519

0.005

0.016

3.8

3.1

0.006

0.01

4.5

1.9

17.1

0.17

0.574

0.003

0.023

1.8

4.0

0.002

0.008

1.2

1.4

19.1

0.21

0.619

0.001

0.019

0.5

3.1

0

0.014

0

2.3

21

0.251

0.654

0.001

0.012

0.4

1.8

0.004

0.016

1.6

2.5

23

0.281

0.642

0.002

0.01

0.7

1.6

0.011

0.042

25

0.319

0.664

0.004

0.008

1.3

1.2

0.02

0.06

6.3

9.0

27

0.362

0.698

0.002

0.014

0.6

2.0

0.028

0.061

7.7

8.7

28.9

0.384

0.706

0.03

0.028

7.8

4.0

0.061

0.097

15.9

13.7

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homogeneous because of “spike” in 46% (α = 2°). For the k-ε standard model, except for the specified features at α < 0° and discrepancy in 15.9% at α = 28.9°, quite satisfactory correlation with discrepancies less than 8.4% is traced. The result is acceptable for the k−ε RNG model in all range α = −0.1–28.9°, and in the range α = 13.3°–27° with separated flow—discrepancy does not exceed 4%. The main attention has been paid to identification of numerical model by force action from flow. Nevertheless, the achieved positive result cannot be considered full, because according to the visualized fields of boundary-layer flows (Appendix 1) in the field of separated flow (α > 10°) violation of identity of flows on outer wing part of LS is observed. By analogy with verification of numerical model in the environment of SolidWorks Flow Simulation, the following revealed parameterization factors not investigated in the course of verification of the ANSYSFLUENT model because of limitation of production resources are represented as efficient in problem of identification of flow pattern: – deviation from vector method of assigning of speed of incoming airflow in favor of turn of model—the method is probably critical in areas with domination of orthogonal meshes, is connected with increase in dimensionality of the domain and mesh model and also with need of evolution of rated mesh, corresponding to quantity of considering points; – increase in regularity of cells on internal boundaries of the domain, both on the area of the approximated surface, and in proportions of cells on coordinate axes. 3.2.8

Conclusions by Results of Verification of Software Products and Statement of Numerical Experiment

During the conducted researches in the environment of SolidWorks Flow Simulation it was established that rough approximation of surfaces of model is not enough for correct modeling of flow in places with heavy gradient of parameters of flow. Fragmentation of rated mesh on surface, increases quality of definition of the distributed characteristics near surface, but has not had impact on characteristic of cy (α) on angles of attack with mainly detached (vortex) flow and, has practically not affected characteristics of cx (α) and mz (α). From here the conclusion is drawn that fragmentation of mesh on surface gives positive effect towards increase in accuracy of calculation of cy (α) for smooth flow. This effect decreases with spreading of zones of detached flow [1]. Assessment of relative error has shown its decrease, in general from 5% up to more than 13%, and locally in the sites described only by basic cells in the first iteration—up to 50% (profile nose in cross section z = 0.693) and more (profile nose in cross section z = 0.276). The analysis of the distributed parameters on perforated model for wind tunnel tests is necessary for assessment of absolute error. Local automatic adaptation of mesh in linkage to geometrical features does not display the valid picture of the distributed parameters on surface of complicated spatial model and needs correction towards increase in regularity [79].

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Coincidence of calculated and experimental values α b and α 0 is the indicator of rather exact calculation of M 0 , and the error of calculation of mz (α) is consequence of errors of determination of M Y (α) and M X (α). Irrespective of the accuracy of solution of problem local features of design models, integral ADC are defined by the general nature of dependences and physically similar flow pattern. In this regard, for the qualitative solution of task or in the absence of need of research of “thin” processes it will be rationally be limited to the simplified settings of design models provided that the error will be in area of the standard and reasonable limits for practical application of results. For example, the engineering accuracy of 15% is considered satisfactory for preliminary design calculation [1]. It is unambiguously possible to draw conclusion, that the adaptation method used in SolidWorks Flow Simulation software with generation of regular rated mesh on streamline surface at moderate expense of resources on the basis of the ordinary computer allows to consider favorable and also to reveal and identify the adverse phenomena at early stages of aerodynamic designing for the purpose of timely acceptance of necessary measures for increase in aerodynamic perfection of subject of design [1]. Calculation of force action of flow in the environment of ANSYSFLUENT on the basis of general characteristics has shown quite acceptable results of the k-ε RNG model with the greatest discrepancies of the numerical and measured characteristics cx (α) and cy (α) about 10% in the range of positive values of α. Calculated curves, with deviations, irregular in sign, nevertheless, describe WT tests dependences in comparison with the similar curves received in SolidWorks software more precisely. However, tuning of numerical model on force factor does not guarantee achievement of topological similarity of flows. In this connection, continuation of works on identification of models at the following stages NE is supposed. For final debugging, and checking calculations, in the majority of tasks, achievement of accuracy of calculations higher than the accuracy of applied experimental methods (in this case ±5%) can be found irrational both in terms of expense of resources, and due to the lack of opportunity to check this accuracy by available methods. Other way at the solution of private tasks—to try to obtain the acceptable accuracy of results in the local range of change of parameters with the limits proved in terms of practical application according to extent of influence of this accuracy onto the end result. To these conditions, for example, quite satisfies modeling of flows in inter-vortex space of the central zone of the fuselage of the ANSYSFLUENT model for interaction research with inlet and outlet devices against the background of discrepancy of morphology of separated flow in remote zones of outer wings. Similar assumption is proved by the dominating influence of adequately modeled vortex system generated by wing root extension onto forming flows in dorsal space. At comparison of the resulting volume distribution of parameters of flow at comparative estimates of results, for example, when determining Δcx and interference factors for options of configurations of PP and ID of influence on AV ADC, models with rather detailed mesh in places of manifestation of the greatest interference and approximate calculation of flow with use of more rough mesh in remote areas can prove quite acceptable. Function of automatic adaptation of rated mesh in

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areas with high gradients of parameters, available in both software packages which is used on the subsequent iterations of calculation after definition of primary field of parameters, for adjustment of morphology of flows in contrast interference zones, is applied for these purposes. Besides, fragmentation of mesh in these zones with higher parameters of discretization, than on surface, is represented as irrational, and in the forefront there is not value of absolute error, but, for example, its oscillations depending on the conditions of fragmentation of mesh in local zone corresponding to the current iteration. The allowable limit of oscillation amplitude of the solution of ±5% serves as criterion for the termination of calculation without expectation of “exact” result [1]. The conducted researches afford grounds for separation of areas of applicability of software packages for the purpose of effective use of advantages of each of them. Advantage of ANSYSFLUENT in speed and accuracy of process of calculations suppose the usage on debugged models. The increased requirements to quality and features of complicated geometrical objects suppose the realization of additional actions in the course of their preparation and creation of mesh model on their basis; that causes duration and, often, irrationality of its use at early stages of profiling of contours within declared “Method of the accelerated creation…”. In this regard the tasks of initial design stages are assigned onto SolidWorks Flow Simulation software package. The tasks are connected with the predictive iterative variational process of forming of contours of components and the UAV in general, with determination of interference effects and preparation of basic data for the subsequent series of the adjusted calculations of integral ADC in the environment of ANSYSFLUENT where interactive work on modification of geometry of the UAV and repeated evolutions of meshes is not supposed. In relation to task of design of air-gas channel of PP the design model of the first iteration quite meets requirements of accuracy for modeling of external flow for design calculation. Positive factor of application of this model is rational expenditure of computing resources at emphasis of attention on the accuracy of the solution of internal flows [1]. Same relates to received at the considered stage the k-ε RNG ANSYSFLUENT model in the subsequent task of calculation of characteristics of internal air-gas channels under the influence of external flows. The main rated case of design calculation of ID is the flight on maximum speed. The second most important rated case of flight is the launch mode: flight on minimum speed with the maximum air consumption through ID. The main modes of operation when determining influence of the functioning ID onto AV ADC will be: – – – – – –

flight on maximum speed with low α; flight on cruising speed; modes of maximum climb rate; modes of maneuvering; flight on optimal angle of attack; takeoff mode.

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The large share of the regimes falls onto area of “the first regimes of flight”, i.e. on angles of attack α ≤ α opt = 5° corresponding to Kmax . On angles of attack up to 8° the dependence cy (α) of the second iteration (SolidWorks Flow Simulation) is in error limits of aerodynamic experiment. Therefore the design model of the second iteration can be recommended for the solution of the main tasks: – iterative design calculation of air-gas channel of PP; – the predictive researches of ADC of the AV models when flying on the first regimes with modeling of work of PP. The flow created by vortex system has considerable impact onto arrangement of the ID air inlet and, as a result, onto steady operation of PP on high angles of attack. At this stage high-quality display of morphology of flows is received in the environment of SolidWorks Flow Simulation; but obtaining integral characteristics in view of duration of calculations is represented as irrational and improbable at the scheduled time. Checking and final calculation of ADC of integral model with activated PP is supposed in the environment of ANSYSFLUENT after full identification of design models.

3.3 Definition of Main Mass-Dimensional and Energy Parameters of UAV 3.3.1

Assessment of UAV Launching Weight of the First Approximation

As a first approximation, the UAV launching weight is defined by the equation of existence of the airplane on the basis of the mass of payload: m 0I =

mp , 1 − m d − m f − m ps − m acs − m r s

where mp md mf m ps m acs mr s

the mass of payload (is accepted equal to 10 kg); relative mass of AV airframe design; relative mass of fuel; relative mass of propulsion system (TJE with JM and engine mount); relative mass of automatic control system; relative mass of recovery system.

The relative values are determined from experience of works of NII PFM Scientific Research Institute for creation of UAVs with similar design, but with corrections connected with assignment and structural materials. Then launching weight is equal: m 0I =

10 10 = = 50 (kg) 1 − 0.23 − 0.4 − 0.07 − 0.06 − 0.04 0.2

Designing a Basic Model of an Unmanned Aerial Vehicle … Table 6 The mass summary of UAV components of the first approximation

83

Parameter

Relative value

Absolute value, kg

mp

0.2

10

md

0.23

11.5

mf

0.4

20

mps

0.07

3.5

macs

0.06

3

mrs

0.04

2

1

50

Total m0I

The mass summary of main assemblies and systems of UAV is provided in Table 6.

3.3.2

Assessment of the UAV Dimensional Parameters Out of Conditions of Ground Launch Using the Low-Energy Launcher

Overall dimensions of the UAV are defined by form of LS in plane projection and its area SL S . On the basis of requirements of unification of UAV for basing, the variant of ground basing system with start from the catapult launcher is the most dependent on the size of the area of UAV LS. In this case the efficiency of complex depends on balance of parameters of the catapult launcher and UAV. Decrease of energy requirements to the ground launching equipment has dictated requirements to UAV onto decrease of minimal and control speeds of flight. These requirements will quite be agreed with paradigm of maneuverable UAV. Therefore the launching mode is considered as one of the main rated cases in which the underlying lifting properties of configuration are involved. The execution of launch provides the output of the UAV onto high angles of attack with achievement of admissible value of cadm y (β), on which execution of guided flight with maintaining lateral stability (on β angle) is possible. The controllability at low speeds of flight is provided by the taken measures of jet stream flow with realization of thrust vector control (see Sect. 2.2.5). For this reason at this stage the acceptable minimum speed caused by rational parameters of the catapult launcher, on which the lifting system of the UAV creates sufficient lifting force, acts as important design parameter. Besides, as the mediate criterion of the predictive researches of the longitudinal movement for ensuring lateral stability, the flow separation moment on outer parts of wing, which limits increase in angle of attack, is considered. The minimum admissible flight velocity is determined by condition of lateral stability as: / adm Vmin

=

2Y ρ H SL S cadm y (β)

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Fig. 28 Flow separation on 2/3 of area of outer wing (α = 15◦ ) adm The equation with two unknown (Vmin and SL S ) is decided in iteration cycle of NE by calculation of ADC of scale model of initial sample of UAV in the neighborhood of expected value of launch angle of attack α L . At this stage the assumption of small dependence of c y (α) from profiling of wing in the conditions of domination of force action of vortex flow around in the range of investigated α is accepted. In Fig. 28, the advanced stall at the outer wing tip, and the preventing of flow separation in the action area of vortex from wing root that causes gradualness of process of separation are shown. The general view of the UAV in scale 1:2 from initial sample is shown in Fig. 29. The scheme “high wing aircraft” and features of profiling of the fuselage with sloping side surfaces suppose the increased stability in lateral motion; therefore adjustment of parameters of start is supposed at the following stages of researches ADC of lateral motion. The area of plane projection except for nose cone 348 mm long of rounded shape in cross-section is taken as the characteristic area, determines the following parameters:

SL S = 0.6568 m2 —lifting surface area; adm Vmin = 55 m/s—the minimum speed of flight accepted as reasonable for starting value of VL ; cadm y (β) = 0.4—admissible value of lift force coefficient for which stability of β angle remains; α L = 15°—the launch angle of attack.

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Fig. 29 General view of the UAV

3.3.3

Definition of Specific Load onto Lifting Surface of the UAV

The specific load onto LS is equal p0 =

3.3.4

50 m0 = 76 kg/m2 = SL S 0.6568

Definition of Launching Thrust-To-Weight Ratio of the UAV and Choice of the Engine

As a first approximation, the launching thrust-to-weight ratio of the UAV is determined according to the statistical data presented in Table 7. On the basis of correlation of the sizes of components of aerodynamic configuration, the UAV is close to type of cruise missiles (CM). For this reason the generalized ranges of parameters based on the analysis of statistics [21] are presented in this table. CM, mainly, are the not maneuverable AV of classical aerodynamic configuration (with tendency of increase t 0 towards products of late release). Two CM (ground/sea launched NSM and air launched JSM) are distinguished by the greatest values t 0 and the smallest weights and positioned as maneuverable. For comparison, the parameters of two aerial targets of similar ideology with integral aerodynamic configuration and of two UAV-analogs according to assignment with similar flight speed (one with the same weight) are given.

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Table 7 Level of thrust-to-weight ratio of analogs of the UAV Assignment of analog

Excerpts

m, kg

M

P, N

t0

Cruise missiles (CM)

16 samples

1020–2275

0.50–0.95

2700–6670

0.14–0.5

Kh-35

600

0.8–0.85

4400

0.75

Maneuverable CM

NSM

344

0.95

3000

0.89

JSM

370

0.95

3000

0.83

Yabhon HMD

220

0.6

1100

0.5

Yabhon GRN

200

0.76

1100

0.55

Reconnaissance UAV

Tu-143 “Reis”

1230

0.78

9600

0.52

Aerial decoy/ multi-purpose UAV

ADM-160A MALD

45.8

0.4–0.91

227.5

0.5

50

0.8

230

0.47

Aerial targets

Designed UAV

Table 8 JetCat P200-RX parameters Parameter

Olympus HP E (AMT, Netherlands)

P200-RX (JetCat, Germany)

Thrust (max), N

230.5

230

Specific fuel consumption, kg/ N*h

0.163

0.15

Weight (with the starter generator, etc. systems), kg

3.15

2.71

Specific thrust

7.5

8.7

Length, mm

342

365

Diameter, mm

130

132

Preliminarily, from the line of small-size turbo-jet engines with similar parameters the most perfect one according to the parameter of specific thrust (provides t 0 = 0.47) and more economical JetCat P200-RX engine (Table 8) was selected.

3.4 Profiling of Wing The considered principled flight conditions with maximum speed, as more energyintensive, define this flight mode as the main rated case in the task of profiling of wing.

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87

Table 9 Calculated parameters Parameter

m, kg

H , km

ρ, kg/m3

V , m/s

S, m2

cy r

Value

50

1

1.1125

269.1

0.6568

0.0185

3.4.1

Determination of Coefficient of Required Lifting Force of Horizontal Flight

According to the principle of decomposition, in the first approximation it is accepted that on the main flight run with angle of attack close to zero, the dominating share of lifting force is created by wing. The coefficient of required lifting force is defined as: cry =

2Y ρH V 2 S

where Y = mg the required lifting force of horizontal flight corresponding to the mass of the UAV; ρH air density at calculated altitude H ; V flight speed; S characteristic area (LS area is accepted as such). Calculated parameters are given in Table 9.

3.4.2

Reasoning of the Choice of Profile, and Definition of Deformation of Medial Wing Surface

Out of compactness conditions outer part of wing is formed on basis the highly effective integrated supercritical Ja10V1S [19] profile, the NASA type [23] modified out of two conditions of the main flight run (I) for the purpose of achievement of maximum speed of flight and two common (II) for the purpose of increase in the general level of aerodynamic perfection: 1 − cry ∼ 0.02; 2 − cx ∼ min;

}

3 − combination αopt ; 4 − c y ti p > c y r oot ; where cy tip lift coefficient of wing tip cross section; cy root lift coefficient of wing root cross section.

I } II

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V. Cheranovskiy et al.

Profile medial line

Profile chord

Fig. 30 Contour of the basic Ja10V1S profile and diagram of curvature of arc

As the purpose, the last standard condition sets delay in tip stall of flow of swept wing for ensuring lateral stability at approach to critical regimes of flow. The type of profile and its ADC, calculated in two-dimensional statement, are presented in Figs. 30 and 31, respectively. The reference points for conditions (I) are marked by red contour, the reference points for conditions (II)—by green contour, respectively. The basic profile of wing tip cross section corresponds to operating conditions with excess of cy r = 0.028, value of which decreases, a priori, in the conditions of wing tip spatial overflowing of air. The profile of wing root cross section is modified for the purpose of decrease of the lifting properties into two stages. On the first stage—by reduction of curvature of the medial line up to 1/3 relatively basic (Fig. 32). The profile modified thus is adjusted on the second stage according to condition of continuous change of curvature of upper arc. Results are presented in Figs. 33 and 34. Thus, geometric twist of outer part of wing is defined by angle of incidence of wing root profile γ 0 root = –1.5° and wing tip profile γ 0 tip = –3°. At change of flight angle of attack on 4°, profiles are turning relatively of incoming flow to their value α opt . And, a priori, the local angles of inleaking onto leading edge are increasing closer to the fuselage; thanks to that, reference point (II) on characteristic K(α) of Ja10V1Sf03 profile is displaced towards achievement of local value Kmax .

3.5 Parametrical Researches of Aerodynamic Characteristics of the UAV Airframe Spatial effects of overflowing of air around wing tip reduce the lifting properties of wing; the fuselage having the lifting properties, and interaction of units of integrated aerodynamic configuration cause the solution of task in the course of iterative determination of geometrical parameters by means of technologies of three-dimensional NE. In order to avoid scale discrepancies at stage of joint calculation of external and

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Fig. 31 Calculated aerodynamic characteristics of basic profile

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Modified profile

Fig. 32 The principal diagram of receiving the modified profile by deformation of rectification of basic profile

Fig. 33 Contour and diagram of curvature of the modified Ja10V1Sf03 profile

internal flows, design calculations are conducted on full-scale models. The possibility of operating by absolute dimensional calculated parameters simplifies the analysis of force action of flows in tasks of definition of aerodynamic interference with the operating PS with the restrictions based on its AVP. In other typical cases of passive flow around for possibility of the subsequent comparison of ADC of AV of other dimensions, results of the analysis are presented within the theory of dimensionality and similarity.

3.5.1

Optimization of Deformation of Medial Surface of “Fuselage-Wing” System

The task is solved by checking calculations of “fuselage-wing” system, created by coupling of wing and fuselage, for the purpose of achievement of the conditions (I) (see Sect. 3.4.2) which are reached by correction of wing angle of incidence. At the same time, the angles of approach of edges of wing root and after extensions to outer wing, according to flow angularity, are determined by lines of flow (Fig. 35). Angles are defined on condition of smooth (vortex-free) flow on the main flight run, providing minimization of cxi . According to the resulting parameters (Fig. 35) the angles of flow of outer part of wing are increased on Δγ wing = 0.5° (γ root = –1°, γ tip = –2.5°). Thus, theoretically, on the main rated regime, conditions of flow improve from K 0root = 2.2 to K root = 4, and from K 0 tip = 3.2 to K tip = 4.8 (Figs. 31 and 34).

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Fig. 34 Calculated aerodynamic characteristics of the modified profile

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Fig. 35 The resulting angles of coupling of lateral edges of the fuselage

3.5.2

Researches of Influence of Wing Sweep onto ADC of “Fuselage-Wing” System. Determination of Parameters for More Precise Choice of TJE

The optimization task is solved with the purpose of increase in aerodynamic perfection of aerodynamic configuration (see Sect. 2.1.2). At the initial stage, characteristics of longitudinal movement in symmetric statement are investigated. Basic data. Features of design models. The scheme and general view of threedimensional configurations of basic model of lifting surface are shown in Fig. 36. According to the principle of decomposition, for reduction of dimension of task and duration of calculations, features of ADC of models of the airframe with the absent internal paths are investigated. For maintaining experimental “purity” and economy of resources onto adaptation of geometrical and mesh models, modification only of outer part of wing thanks to change of one parameter—sweep of leading edge—with values χ π.k. = χ l.e. = 35°; 0°; –20° is carried out. At the same time, the impact of change of χ π.k. χ l.e. onto relocation of the center of pressure (CP), relatively of its location for basic configuration χ l.e. = 35°, is fixed. The zone of step of nozzle is closed by the streamlined fairing. Research tasks: (a) definition of ADC in launch conditions; (b) definition of ADC in flying conditions at calculated altitude; (c) determination of drag of model in the conditions of viscous transonic flow for rated case of flight with maximum speed, and more precise definition of required thrust-to-weight ratio. Initial conditions in tasks: (a) M = 0.16; H = 0 km; α ∈ {–4°…(αmax +4°); 2°}—the range of the studied angles of attack; (b) M ∈ {0.5… 0.8; 0.1}; H = 1000 m; α ∈ {–4°…(αmax +4°); 2°}; αmax ≡ c ymax .

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Fig. 36 The scheme of creation of parametrical spatial model with changeable sweep of outer part of wing (above), and general view of the modified models (below)

Analysis of results. Calculations results are presented in the form of diagrams of ADC and the fields of flows visualized by means of streamlines in Appendix 2, Figs. 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95 and 96. The key aerodynamic parameters of the UAV airframe without internal paths of PP for different angles χ π.k. χ l.e. of outer part of wing are presented in Table 10. In all configurations the value of cya max and angle αmax exceeding angle of full stall of outer part of wing are not reached in the neighborhood of α = 18°–20°. The exceeding of this potential restriction in practice represents difficulty from position of ensuring static lateral stability; nevertheless, flight on supercritical flow regimes is of interest within the actions for increase in maneuverability of the UAV; for this reason expanded range of α is taken into consideration. In the range of mainly smooth flow the coincidence of characteristics, general for all configurations, and significant discrepancy of curves on the flow separation modes for different Mach numbers is observed. Nonlinearity, general for all configurations, is noted typical for aerodynamic configurations with wing root extensions—vortex ◦ formers and expressed by growth of derivative cαya , on angles more than 6°, induced by vortex system. With decrease of sweep, the tendency to nonlinearity of characteristic

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Table 10 Key aerodynamic parameters of UAV airframe without internal paths of PP for different angles χ π.k. χ l.e. of outer part of wing ◦

α0◦

χ π.k. χ l.e

cαya

35

0.02381

α◦

−2°…6°

−0.35°

M

0.16… 0.6

0.16… 0.6

0

0.02223/0.02389

α◦

–4°…2°/2°…6°

–0.2°

M

0.16…0.8

0.16…0.8

–20

0.01986…0.02692

α◦

–2°… 8°

–0.03°…–0.08°

M

0.16…0.6

0.16…0.6

c yamax

>34

>34

>34

◦

m αza

cxamin

K max

0.0025/ 0.0035

0.01039

8.04

–2°…–0.4°/ 0.5°…4.5°

–0.1°

6.0

0.16…0.6/ 0.5…0.6

0.6

0.42

0.0024/ 0.0019

0.01020

8.0

1°…5.5°/ 0.5°…4.0°

0.1°

6.1

0.5/0.16; 0.6…0.8

0.58

0.3

0.0010/ 0.0030

0.01010

7.6

0°…2°/ 7°…8°

0.2°

5.2

0.16…0.6

0.69

0.38

c y (α) is noted in the range of small and medium angles α = –2°–8°, right up to monotonous for configuration χ l.e. = –20°. According to combination of features, configuration χ l.e. = 35° has the best lifting properties and bigger static stability in the longitudinal movement. According to level of K max , it exceeds other configurations in 3/4 of speed range (Fig. 37). Nevertheless, superiority of swept wing and, especially configurations with the forward swept wing (FSW), for level cxamin , is confirmed (Fig. 38); that can become critical in the task of achievement of the greatest value Mmax . Besides, values cxamin for configuration χ l.e. = –20° are near to value of flight angle of attack α ≈ 0.5°, corresponding to required value C ya r = 0.0185 in flight with maximum speed. For configuration χ l.e. = 0° more intensive growth of drag with growth of compressibility effect of flow at M > 0.6 is evidently revealed. As the important quality, which is characterizing adaptedness of geometry of the AV airframe to flying conditions, its self-balancing on the main flight mode is considered. Minimization thus losses onto balancing (for normal aerodynamic configuration: providing C ya r by minimum deflection of control surfaces followed by minimization of inductive and wave components of drag) is the most critical in flight with maximum speed. According to the diagram m z (α) (Fig. 80), geometry of configuration χ π.k. χ l.e. = 35° creates negative pitching moment on the angle of attack α = 0.45°, corresponding to C ya r (Fig. 39). On maximum speed this moment will make up

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Fig. 37 Level of aerodynamic perfection of configuration depending on angle of sweep of outer part of wing

Fig. 38 Influence of sweep of outer part of wing onto the minimum drag depending on flight Mach number

Mz = m z

ρH V 2 1.1125 × 269.12 SL S b M AC = 0.0027 0.7876 × 0.875 = 74.95 Nm 2 2

On the lever of control surface of the step lCS = 0.99 m, the balancing force will make up

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Fig. 39 Fragment of moment characteristic of configuration χ π.k. χ l.e. = 35° in the neighborhood of C ya r

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Yb =

97

−Mz −74.95 = −75.7 N = lcs 0.99

Force Y b reduces value of the available lifting force onto 15%; that significantly reduces the resulting lifting capability of model with sweepback wing. And, the required thrust increases onto value ΔP r =

Yb 75.5 N = = 50.5 N, K 1.5

what makes up about 19% of value of the available thrust, leveling advantages before derivative configurations. Diagram m z (α) for configuration χ π.k. χ l.e. = 0° is characterized by the greatest instability on the flow separation modes. The next can serve as the reason for that: firstly, an inadequate modeling of morphology that needs continuation of works on identification of design models on these flight modes (see Sect. 3.2); secondly, tendency of straight wing to unorganized separation of flow. However, with big degree of confidence the diagram can be used for the analysis in the range of low angles α. Model of configuration χ π.k. χ l.e. = 0° is the closest to self-balanced one. Curves m z (α) cross abscissa in value α ≈ 0.4° (Fig. 86). Correction of center-ofgravity position by means of the CG shift back onto value ΔX CG = 0.025 m, allows to balance the model on angle of attack α = 0.6°, that is corresponding for value C ya r (Fig. 40). Change thus of center of gravity position is followed by decrease onto value about 30% of static stability and its loss in the range α < –2° at M < 0.8. FSW displaces CP into even more front position. Recalculation of characteristic m z (α) for configuration χ π.k. χ l.e. = –20° onto new position ΔX CG = –0.07 m allows to balance the model with the attendant increase in stability (Figs. 41 and 92). The model with FSW has static stability in all studied range of α. Smaller inclination of curves m z (α) in comparison with configuration χ π.k. χ l.e. = 35° means smaller losses of lifting force when balancing on other flight modes and maneuvering. However, center-of-gravity position displacement forward is disadvantageously for reasons of internal configuration. At the same time, preservation of center-ofgravity position by the displacement back of outer parts of wing is advisable from the standpoint of decrease of negative interference of the tail part. Visualization of flows (Figs. 92, 93, 94, 95 and 96) reveals development of flow separation of FSW in the direction from wing root cross section to wing tip cross section, and tendency to belated flow separation, a priori promoting maintaining lateral stability and realization of bigger values c ya (α).

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Fig. 40 Reduction of configuration χ π.k. χ l.e. = 0° to the self-balanced state

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Fig. 41 Reduction of configuration χ π.k. χ l.e. = –20° to the self-balanced state

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Adjustment of the Choice of TJE Out of Condition of Required Thrust-To-Weight Ratio in Flight with Maximum Speed

Results of preliminary calculation of ADC of models of LS have found insufficiency of thrust of the engine, chosen according to the statistics, for ensuring high-speed range. For this reason adjustment of the TJE parameters is carried out. The TJE under consideration are presented in Table 11 [33]. In consideration of the possibility of augmentation of thrust by means of flow around of the profiled step by jet stream (see Sect. 2.2.5) easier, compact and economic sample of the engine—P300-PRO is chosen. The general view of TJE is presented in Appendix 2, Fig. 97. The adjusted launching thrust-to-weight ratio of the UAV makes up: t0 =

P 300 = 0.61. = g · m0 9.81 · 50

3.6 Features of Aerodynamic Designing of Air-Gas Path with Submerged Air Intake and Nozzle of Type “The Return Step” The paradigm of designing of ID non-jutting out of fuselage lines and the nozzle of upper arrangement interacting with the surface, proceeds from two contradictory conditions. Firstly: the expected positive effect is significant increase in lift-to-drag ratio of AV due to simultaneous decrease of aerodynamic drag and increase of lifting capability caused by generation of extensive depression areas on upper surface of the airframe. Table 11 TJE parameters Parameter

P300-PRO (JetCat, Germany)

P400-PRO (JetCat, Germany)

Thrust (max), N

300

397

Specific fuel consumption, kg/ N*h

0.157

0.158

Weight, kg

2.73

3.65

Specific thrust

11.2

11.1

Air consumption, kg/s

0.5

0.67

Gas temperature, °C

480–750

480–750

Length, mm

380.5

353

Diameter, mm

132

148.4

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Secondly: the strong curvature of walls of internal path, dictated by general arrangement restrictions, inevitably involves deterioration of functional properties of the system “inlet device—compressor” [1]. Thus, achievement of positive compromise between the called contradictions is supposed. Application of submerged ID and nozzle-step, demands the carrying out the designing actions providing increase in homogeneity of flow and decrease of hydraulic resistance of curvilinear diffuser channels, bypass contour including contour of cooling of step [1]. Approach to component-wise designing as the intrasystem method is part of works of the conceptual project at the stage II (Fig. 9) when developing initial iterations of aerodynamic configuration with the subsequent verification in cycles III−IV of preliminary design—or receiving etalon reference points in case of problems with scaling of physical models. The main workload falls onto the stage V of preliminary design; its task consists in ensuring feasibility of designing of the integrated units of the “AV-PS” system in the conditions of inadequacy of their research in the form of the elements isolated from the general aerodynamic configuration. Integrated approach to aerodynamic designing of internal paths of PP includes a number of subordinated and modified aerodynamic methods of lower level, and the providing method of synthesis of parametrical geometry (Figs. 42 and 43) [1]. The ways of homogenization of flow, investigated in work [34], by means of cavities in inlet part of ID are represented as ones of little use. The known method of optimization of aerodynamic and geometrical characteristics of S-shaped ID with the semi-submerged AI (such as used for the Kh-35 cruise missile [35]), is based on the rough organization of flow by means of deformation of axial line of ID without adaptation to conditions of external flowing. The geometrical conformity with airframe

Fig. 42 The structured expression of triad of methods of designing of system “airframe—engine”

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Fig. 43 Method of designing of internal paths and their coupling with external contours

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contours (inherent in all types of the submerged AI) in case of vortex-free AI at higher level is provided by profiling of contours of ID according to morphology of flows, by thin manipulation of curvature of smoothly mating surfaces [13, 23, 24, 36, 37]. Principled basis of the existing complex of methods of profiling of ID in their initial state (Fig. 43) and the possibilities of their synthetic application following from this are described in work [1]. Conforming to the general approach, process of aerodynamic designing of ID is based on the principle of factor decomposition and is carried out into two stages. For ensuring project work, the visualization methods revealing morphological regularities of flow depending on parameterization of submodel of ID are used [15]. At the I-st stage in phase of preliminary design, the forming of geometry of ID by means of purposeful change of internal boundaries of the domain is carried out during iterative cycles of NE. Process is limited by conditions of the main rated case of cruising flight and for the purpose of economy of resources—by minimum sizes of the domain of submodel of ID with imitation of the functioning TJE by means of assignment of substantive features, type of the “sources–outflows” of the corresponding intensity in the output cross section of ID. As result, air channel of ID of fixed geometry, with appropriate aero-gas-dynamic properties and passive self-regulation, is formed [1]. Closure of autonomous model of ID is carried out by complex of the standard gas-dynamic parameters: hydraulic losses, extent of recovery of total pressure and uniformity of flow in the output cross section of ID [1, 38–46], on the basis of available statistical data for the similar ID or requirements presented by the TJE developer [1]. At the II-nd stage, the coordination of working processes of ID and TJE by means of transparent calculation of external and internal flow of AV with correction of geometry of ID, borders of area of steady modes of the system “ID + TJE” in correlation with flight envelope of AV is carried out [1].

3.6.1

The Automated Generation of Geometry of ID

The gas-dynamic principle of profiling of ID is implemented by means of the automated generation in the environment of designing. The block diagram of parametrically managed generating lines and reference axes of ID generated in the environment of CAD is presented in Fig. 44 [13]. The directional entrance represents the semi-limited curvilinear diffuser of external deceleration of flow with intensive deepening of the central zone in the neighborhood of bottom generating line under level of outer surface of the airframe, with the sideward edges converging along the flow [13]. The most approximate admissible value of external deceleration of flow and angle of tilt of the normal of inlet cross-section of AI ∠τ0 n are defined by individual features of flow of specific external configuration of AI on the limit regimes [13].

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Fig. 44 Topology of internal path of ID: 1—the bottom generating line; 2—the sideward edge; 3—the inlet cross-section of the directional entrance; 4—the inlet cross-section of air duct; 5—the outlet cross-section of ID

The curvature of air duct in single-engine version of PS forms when rounding around entrance lip of AI. Execution of the confuser entrance part of air duct up to the output cross section of swivable knee with formation of throat of ID with constriction of flow up to 10–12% is preferable. Law of change of the areas of cross sections Si = {x, θ, ϑ j } implements constriction of flow—from monotonous longwise up to the displaced closer to throat cross section, including change, local on its perimeter, where θ —angular coordinate of j segment of cross section by angle ϑ (Fig. 44) [13]. Management of scalar and vector fields of flows is carried out by mutual coordinating and deformation of the parametrically generating surfaces and reference axes. The best results are achieved by smoothing with curvature, continuous up to the second order [13].

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Fig. 45 Parametrization of optimization task

3.6.2

Ways of Achievement of Multimode Operation of “AV–PS” System with Submerged Air Intake

Properties of multimode operation of “AV–PS” system can be expressed by qualitative and quantitative compliance of the aero-gas-dynamic parameters characterizing the range of gas-dynamic stability and AV flight envelope (Fig. 45) [1]. In the course of integration of PS into AV lifting system, the available during coordination of ID, TJE and JM, improvement of the varied parameters is evaluated by the corresponding criteria of K, W, cp in aspiration to interdependent expansion of boundaries of these areas up to limiting/preset values and increase in characteristics inside flight envelope [1]. Functioning of ID on trajectory is defined by condition of flow without separation in vicinities of AI by the corresponding orientation of AV in incoming airflow. Expansion of AV flight envelope is carried out at the expense of implementation of aerodynamic unloading of the airframe by means of adopted technical solutions. Thrust control of PS on different flight runs is carried out by throttling of air-jet engine. From the positions of increase in reliability, weight efficiency, manufacturability of production and service, preference is given to ID with fixed geometry [1].

3.6.3

Efficiency Indexes of Working Process of ID

Forming the set of design actions is defined by a number of the known estimates of efficiency of working process in AI [38–46]. In addition to the standard estimates of hydraulic losses—coefficient of preservation of total pressure: σ =

po , o p∞

(1)

where p o = p + ρw2 —medium-integral value of total pressure in the output cross 2 o ∞ = p∞ + ρ∞ ·w —total section (input cross section of the TJE compressor); p∞ 2 pressure of undisturbed flow, p∞ ≡ p H ; ρ∞ ≡ ρ H , w∞ ≡ wn , and pressure ratio: 2

cp =

) ( 2 p o − p∞ , ρ∞ w∞ 2

(2)

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at the current design stage tentative estimations of heterogeneity of flow on entrance to the compressor in the form of coefficient of irregularity of the field of full pressure are applied [45, 47]: Δ p o =

p

(3)

po

maxoo , min

and coefficient of irregularity of the field of speeds: Δw =

wmax − wmin w

(4)

3.7 Determination of Boundaries of Flight Envelope According to contents of works (Fig. 43), the private optimization task of definition of UAV AVP, including definition of boundary of maximum speed in the conditions of compressibility of transonic flow is solved. As criterion, achievement of the greatest value of Mach number of horizontal flight on preset angle of attack, on condition of not exceeding of aerodynamic drag of value of TJE thrust on calculated altitude is accepted. Initial conditions: – Ya = 500 N—the lifting force counterbalancing UAV weight; – α ∈ {2…14°; 2°}—the range of the studied angles of attack. According to the engine altitude performance (Fig. 97), the restrictions of calculated maximum thrust on altitude are presented in Table 12. Calculation results are presented in the form of the combined diagram (Fig. 99) for model of configuration χ π.k. χ l.e. = 0°, where crossing of characteristics H (M, Y ) showing the built up Mach number providing required value of lifting force on the preset angle of attack and H(M, Pa max ), corresponding maximum available thrust at altitude, creates approximate boundary of the limit modes H (Mlim ). The received AVP chart gives representation about UAV flight performances. In the presence of AVP of the engine, it serves as the database for parameterization of design models in the task of coordination of working processes of “AV–PS” system. Table 12 The drag force, counterbalanced by engine thrust H, km

0

1

3

5

7

9

10

X a, N

295

260

202

158

122

95

81

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3.8 Forming of Air-Gas Paths of PP The direct task is solved by method of successive approximations. Profiling of air-gas paths of PP is conducted under the terms of the main rated case. For the high-speed UAV the flight maximum speed conditions, which are most strained on flow capacity by ID, at altitude ensuring effective work of payload are chosen. In view of lack of AVP of the engine, parameters of flow are determined approximately in proportion to the thrust variation on the diagram P = f (H ) (Fig. 97). The parameters, received at bench tests (Table 11), are accepted as the reference ones. In that case, the mass air flow rate at the altitude H = 1 km is equal m˙ 1 =

260 PH × 0.5 = 4.33 kg/s, = P0 300

where PH TJE thrust on calculated altitude, P0 maximum test-bench thrust of TJE, m˙ 0 engine air flow in bench tests. Gas temperature behind the turbine T5 is determined approximately by proportional recalculation of test-bench parameter of JC P300-PRO turbojet engine using AVP of TJ-100 small-size TJE. Initial conditions: – – – – – –

M = 0.8—flight Mach number; H = 1000 m—flight altitude; a = 366.4 m/s—sonic speed; α = 0.5°—angle of attack; m˙ = 0.433 kg/s—engine air flow; T5 = 990 K.

Analysis of results. As a result of a number of iterations of NE on profiling of components the air-gas channel of PP (shown in Fig. 46) is created. The system of internal channels consists of four main segments: the inlet device with slot-hole withdrawal; the bypass channel with the ejector; the exhaust nozzle; the channel of cooling of step with the ejector. The resultant parameters of flows are presented in Fig. 100–107. In general, external flow is smooth, continuous with perturbations of small intensity, except for faceted nose cone. The fore-part of wing root extension is enough well adapted to the flow. On the after part of the wing root extension, the small return flows, including ones induced by jet-stream effect (Fig. 104), are observed. Their elimination is supposed on the subsequent iterations of designing, including application of tip aerodynamic surfaces for the organization of flows, using them as vertical tail and shielding of hot path from the lateral sides. The initial instability of flow of BL of the directional entrance of ID has no turbulization effect onto the smooth flow in kernel of flow of the first contour (Fig. 105).

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Fig. 46 Scheme of internal paths of the power plant

Flow capacity and operation of ejectors of the second contour and contour of cooling of step is sufficient for withdrawal of BL from ID. The character of change of speeds and distribution of pressure along channels can be estimated according to Figs. 106 and 107, and irregularity of flow can be estimated according to Fig. 47. The design of the TJE starter in the form of central body before entrance into the compressor brings turbulent perturbations into flow by creation of “detachable bubble”, than determines the low level of coefficient of preservation of total pressure σ and the general level of the raised pressure losses, expressed by low value of c p , as presented in Table 13. Nevertheless, deceleration of flow in ID reaches the level exceeding atmospheric pressure on 7% at altitude H = 0 m; and indices of irregularity of flow of the stream which is inhausted by the engine indicate possibility of ensuring sufficient level of homogeneity. As the first approximation, conditionally influence of pylons holders of the starter is not considered. Profiling of step with the increased thrust augmentation coefficient, i.e. with the increased curvature, in the scheme “high wing aircraft” together with increase in curvature of the after part of the wing root extension, involves increase in midship, is followed by intensification of sideward vortex formations and growth of drag. In the version represented in Fig. 46, the step of small curvature creates the compensatory axial force of 13 N which is a little exceeding friction drag of high-speed jet stream of 12.2 N.

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Fig. 47 Spectrums of irregularity of flow in the entrance cross section of the compressor

Table 13 ID gas-dynamic parameters Parameter

σ

cp

Δ p 0

Δw

Value

0.88

0.75

0.14

0.16

4 Conclusions Based on Results of Designing of the UAV Basic Model The initial stage of implementation of technical intention of the multimode maneuverable UAV is preceded by analysis of level of technology. On the basis of it the main restrictions of area of researches, the main qualities and initial parameters of UAV are defined. Special attention is paid to the technical solutions allowing to implement required properties of subject of designing. The algorithm of design actions is made up based on the principles of composition/decomposition of components

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and methods of numerical and physical modeling. Features of method of the aerodynamic designing are considered, which is providing feasibility of complex of the specified technical solutions. Complexification of general configuration is provided by preliminary NE in the environment of SolidWorks Flow Simulation with possibility of the simplified preparation of spatial solid-state and mesh models. Serial researches of arrays of ADC are conducted in the environment of ANSYSFLUENT. Similar division of area of responsibility is caused by more labor-intensive preparation of solid-state and mesh models, but such expenses are compensated by more efficient process of NE. According to the developed method of aerodynamic designing, the decrease of resources consumption of design actions is provided by rational combination of numerical and physical experiments. Any NE needs the prior information confirming its reliability. At the considered stage, the reliability of NE is based on similarity of the numerical models verified by results of experimental studies in WT T-4 KhAI of largescale model (accepted as etalon) of the maneuverable UAV of similar aerodynamic configuration with similar working process. The received similarity of morphology of flows in the environment of SolidWorks Flow Simulation, and discrepancy of results of NE from results of physical tests in WT for low and moderate angles of attack within 12%, for high angles of attack—up to 15–17%—is quite admissible at stages of preliminary design as does not affect adequacy of relative estimates [1]. Calculated curves of ANSYSFLUENT more precisely describe WT tests dependences with the greatest discrepancies of the numerical and measured characteristics about 10%. However, tuning of numerical model on force factor does not guarantee achievement of topological similarity of flows. In this connection, continuation of works on identification of models in providing the researches ADC on flow separation regimes, dominating at the following stages, is supposed. Process of profiling of external forms and internal paths and also their coupling and coordination has iterative character; and that is provided by cycles of the repeating numerical calculations of external and internal flows. Numerical calculations of ADC of the UAV basic model received by scaling of the UAV model of the bigger dimensionality accepted as initial geometry for the purpose of determination of sufficient lifting surface area out of conditions of ensuring acceptable speed of start were carried out. Adaptation of geometry of outer part of wing is carried out for the purpose of ensuring value cy r and achievement of the minimum value cx corresponding to conditions of the main rated case of flight with Mmax , with coordination of cross sections in matching with the most favorable values of angles of incidence. Based on requirements of increase in maneuverability, events of alternative designing of lifting surface were held. On specially prepared parametrical models of the airframe with different sweep of outer part of wing comparative numerical researches for the purpose of definition of configuration with the best ADC on the main flight mode, and also on moderate and high angles of attack were conducted. According to results of calculations, more precise definition of launching thrust-to-weight ratio and parameters of the engine was carried out.

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As a result of parametrical optimization by method of successive approximations the airframe geometry with the integrated air-gas channel of PP was created, including: – ID was profiled; – the step surface with the thrust gain, which exceeds value of additional drag in the field of jet stream, was profiled; – TJE JM with the flattened nozzle was profiled; – channel of cooling of thermally-tensing elements was profiled; – coordination of ID and EN with the channel of cooling was carried out. The general configuration of the UAV represents uniform lifting system “fuselagewing” with the integrated propulsion system and is determined by variable sweep wing of small aspect ratio with the large wing root extensions—vortex formers, the upper submerged air intake and the ejector nozzle of step type with jet stream control of trailing edge. Ejector nozzle has function of deflection of thrust vector of TJE. Key parameters of the UAV are presented in Table 14. Advantage of models of configuration χ π.k. χ l.e. = 35° and χ π.k. χ l.e. = –20° before configuration χ π.k. χ l.e. = 0° in the achievement of Mmax , predicted on the basis of extrapolated up to M = 0.7; 0.8 characteristics cxmin = f (M) (Fig. 38), requires adjustment. It is reasonable to carry out the final choice of configuration after modification of model with FSW and coordination of new position of CP with preferable center-of-gravity position. Against the background of achievement of superiority on Mmax , the ensuring of lateral stability on critical and supercritical flow regimes Table 14 Key parameters of the UAV

Launching weight, kg

50

Engine/thrust, N

JetCat P300-PRO/295

Thrust-to-weight ratio

0.61

Payload weight, kg

10

Fuel weight, kg

20

Area of LS, m2 Specific load onto wing,

0.6568 kg/m2

76

Length, m

2.25

Wing span, m

0.9

Height, m

0.21

Aspect ratio of LS

1.23

Lift-to-drag ratio of LS

~8

Launching speed, m/s

55

Cruising speed, m/s f (H)

100–170

Mach number

0.16–0.78

Absolute ceiling, m

10,000

Flight endurance, min

~40

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serves as the determining factor, with goal-setting realization of the greatest values cya (α), i.e. achievement of superiority on the maximum available overload. Besides, correct comparison of lifting capability and maneuverable qualities is possible only if to take into account losses onto balancing. The listed aspects create complex of research tasks onto the subsequent stages of aerodynamic designing. ADC of the AV airframe allow to realize the altitude potential of the JC P300-PRO engine declared by producer (H = 10,000 m). For determination of sufficiency of thrust-to-weight ratio according to condition of achievement of Mmax , continuation of researches of models with the integrated propulsion system and researches devoted to profiling of step for the purpose of increase in coefficient of thrust augmentation in coordination with realization of the required balancing moment are necessary. On the example of model with sweepback wing the complex of works devoted to profiling and parametrical researches of internal flows was carried out; as a result, the active capacity of system on flight run Mmax with acceptable parameters of homogeneity of flow on entrance was demonstrated.

Appendix 1: Verification of Numerical Models See Figs. 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 and 73.

Appendix 2: Parametrical Researches ADC of Components of the UAV See Figs. 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106 and 107.

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Fig. 48 Comparative visualization of flows α = 4◦ : WT SolidWorks Flow Simulation ANSIS FLUENT

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Fig. 49 Comparative visualization of flows α = 6◦

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Fig. 50 Comparative visualization of flows α = 10◦

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Fig. 51 Comparative visualization of flows α = 14◦

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Fig. 52 Comparative visualization of flows α = 18◦

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Fig. 53 Comparative visualization of flows α = 20◦

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Fig. 54 Comparative visualization of flows α = 22◦

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Fig. 55 Comparative visualization of flows α = 24◦

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Fig. 56 Comparative experimental and calculated, received in the environment of SolidWorks Flow Simulation, ADC of the lifting system (M = 0.144; Re = 11.0 · 106 )

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Fig. 57 The diagram of ADC of B.9 lifting system in the increased scale

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Fig. 58 Comparative polars (SolidWorks) (M = 0.144; Re = 11.0 · 106 )

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Fig. 59 Comparative characteristics of lift-to-drag ratio (M = 0.144; Re = 11.0 · 106 )

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Fig. 60 Calibration of model (ANSYS FLUENT)

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Fig. 61 Standard list of mesh zones of design model

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Fig. 62 Boundary conditions on the domain entrance

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Fig. 63 Conditions on free external boundaries

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Fig. 64 Terminal conditions on inner boundaries

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130 Fig. 65 Calculation method

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Fig. 66 Settings of management of calculation

Fig. 67 Parameters

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Fig. 68 Conditions of convergence of output parameters

Fig. 69 Conditions of convergence of discrepancies Fig. 70 Conditions of calculation of one point of a polar

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Fig. 71 Comparative experimental and calculated, received in the environment of ANSYSFLUENT, ADC of the lifting surface (M = 0.144; Re = 4.0 · 106 )

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Fig. 72 Comparative polars (ANSYSFLUENT) (M = 0.144; Re = 4.0 · 106 )

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Fig. 73 Comparative characteristics of lift-to-drag ratio (ANSYSFLUENT) (M = 0.144; Re = 4.0 · 106 )

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Fig. 74 Dependence of coefficients cxa from angle of attack α. cxa = f (α). χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 75 Dependence of coefficients cya from angle of attack α. cya = f (α). χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 76 Polar. Dependence of coefficients cya from cxa . cya = f (cxa ). χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 77 Dependence of lift-to-drag ratio K from angle of attack α. K = f (α). χ π.k. = χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 78 Dependence of coefficients m z from angle of attack α. m z = f (α). χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 79 Dependence of center-of-pressure position XCP from angle of attack α. XCP = f(α). χ π.k. = χ π.k. χ l.e. = 35°. Under different flight conditions (M, H)

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Fig. 80 Dependence of coefficients cxa from angle of attack α. cxa = f (α). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 81 Dependence of coefficients cya from angle of attack α. cya = f (α). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 82 Polar. Dependence of coefficients cya from cxa . cya = f(cxa ). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 83 Dependence of lift-to-drag ratio K from angle of attack α. K = f (α). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 84 Dependence of coefficients m z from angle of attack α. m z = f (α). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 85 Dependence of center-of-pressure position XCP from angle of attack α. XCP = f(α). χ π.k. = χ π.k. χ l.e. = 0°. Under different flight conditions (M, H)

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Fig. 86 Dependence of coefficients cxa from angle of attack α. cxa = f (α). χ π.k. = χ π.k. χ l.e. = –20°. Under different flight conditions (M, H)

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Fig. 87 Dependence of coefficients cya from angle of attack α. cya = f (α). χ π.k. =χ π.k. χ l.e. = –20°. Under different flight conditions (M, H)

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Fig. 88 Polar. Dependence of coefficients cya from cxa . cya = f(cxa ). χ π.k. = χ π.k. χ l.e. = –20°. Under different flight conditions (M, H)

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Fig. 89 Dependence of lift-to-drag ratio K from angle of attack α. K = f (α). χ π.k. = χ π.k. χ l.e. = –20°. Under different flight conditions (M, H)

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Fig. 90 Dependence of coefficients m z from angle of attack α. m z = f (α). χ π.k. = χ π.k. χ l.e. = –20°. Under different flight conditions (M, H) and for different center-of-gravity position

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Fig. 91 Dependence of center-of-pressure position XCP from angle of attack α. XCP = f (α). χ π.k. = χ π.k. χ l.e. = –20°. Under different flight conditions (M, H)

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χl.e. = –20°

χl.e. = 35°

χl.e. = 0°

χl.e. = –20°

χl.e. = 35°

χl.e. = 0°

Fig. 92 Detached flow of models with formation of wing root vortex M = 0.16; H = 0 km; α = 18°

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χl.e. = –20°

χl.e. = 35°

χl.e. = 0°

χl.e. = –20°

χl.e. = 35°

l.e.

= 0°

Fig. 93 Development of flow separation along outer wing span M = 0.16; H = 0 km; α = 18°

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Fig. 94 Full flow separation on outer wing. Configuration χ π.k. χ l.e. = 35°. α = 18°

Fig. 95 Full flow separation on outer wing. Configuration χ π.k. χ l.e. = 0°. α = 18°

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χl.e. = –20°

χl.e. = 35°

χl.e. = 0°

Fig. 96 Delay of full flow separation on outer wing tip. Configuration χ π.k. χ l.e. = 0.20°. α = 18°

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Fig. 97 General view of JetCat P300-Pro turbojet engine

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Fig. 98 Altitude characteristic of JetCat P300-Pro turbojet engine

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Fig. 99 Flight envelope. Configuration χ π.k. χ l.e. = 0°

Fig. 100 A field of pressures on the surface of model and in a symmetry plane M = 0.8; H = 1 km; α = 0.5°

Fig. 101 A textural field of speeds on the surface of model and in a symmetry plane

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Fig. 102 The combined fields of speeds and pressures over the surface of model

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Fig. 103 Visualization by means of streamlines of jet stream of the I-st contour of PP (above) and flows over the upper surface of AV airframe (below)

Fig. 104 Visualization of flow of side edges

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Fig. 105 Visualization of flows in boundary-layer of ID (above) and a flow of the I-st contour (in the center and below)

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Fig. 106 A textural field of speeds in air-gas channel of PP in a symmetry plane

Fig. 107 A field of pressures in air-gas channel of PP in a symmetry plane

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References 1. Kornev, A.V.: Method of complex aerodynamic designing of aerial vehicles of integrated configurations with use of combinations of numerical and physical experiments: Manuscript, Ph.D. Thesis: 05.07.01. National aerospace university of name of N.E. Zhukovsky “KhAI”, Kharkov, p. 159 (2018) (in Russian) 2. Kornev, A.V., Maximov, V.P., Dmitriyev, V.A.: Forming general configuration of an air platform for the all-weather automated aerial unmanned complex for reconnaissance and attack missions; and the main results of works for its creation. In: Collected Materials of the 11th Scientific and Technical Conference “Creation and Modernization of Weapon and Military Equipment in Modern Conditions”, Feodosiya, September 8–9, pp. 278–287 (2011) (in Ukrainian) 3. Belov, K.: Attack of drones. Military-industrial courier. No. 6 (327). February 16–22 (2011) (in Russian) 4. Petrov, A.V.: The flow around of wing of large curvature with tangential blowing of streams. Sci. Mem. TsAGI (Moscow: TsAGI) XXII(2), 13–22 (1991) (in Russian) 5. Englar, R.J., Jones, G.S., Allan, B.G., Lin, J.C.: 2-D circulation control airfoil benchmark experiments intended for CFD code validation. In: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA 2009-902. Orlando, Florida: AIAA. 5–8 January 2009, p. 27 6. Coppinger, R.: Ground trials start for UK vectored thrust prototype UAV 2 Oct. 2009. http://www.flightglobal.com/news/articles/ground-trials-start-for-uk-vectored-thr ust-prototype-333030/. Accessed 06 March 2015 7. Demon-demonstrator has coped with flight without ailerons. Military Review, October 1, 2010. http://topwar.ru/1589-demon-demonstrator-spravilsya-s-polyotom-bez-yeleronov.html Accessed 10 March 2015 8. Mraz, S.J.: Could future aircraft fly without traditional control surfaces such as ailerons, flaps, and elevators? Machine Design, March 15, 2011 http://machinedesign.com/defense/couldfuture-aircraft-fly-without-traditional-control-surfaces-such-ailerons-flaps-and-elevators. Accessed 02 March 2015 9. Harley, C.D., Wilde, P.I.A., Crowther, W.J.: Application of circulation control manoeuvre effectors for three axis control of a tailless flight vehicle. In: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA 2009-146, Orlando, Florida: AIAA, 5–8 January 2009, p. 14 10. The variant of general configuration of a perspective cruise missile: Engineering sentence (the code: “Korshun-2”). National aerospace university of name of N.E. Zhukovsky “KhAI”. Supervisor: Smolyakov, A.V.; executors: Yashin, S.A., Yanakayev, V.A., Kornev, A.V., Shevko, S.V., Kharkov, NII PFM KhAI, p. 44 (2014) (in Russian) 11. Patent 99971 Ukraine, MPK (2012.01) B64D 33/00. The inlet device of the submerged type of the gas-turbine engine of an aerial vehicle. Kornev A.V.; Applicant and owner: national aerospace university of name of N.E. Zhukovsky “Kharkov Aviation Institute”. No. a201100751; declared 24.01.2011; published 25.10.2012; Bulletin No. 20; 5 p.; 3 fig (in Ukrainian) 12. Patent 103196 Ukraine, MPK (2013.01) B64D 33/00. The aerial vehicle with the dorsal inlet device. Kornev, A.V.; Applicant and owner: National aerospace university of name of N.E. Zhukovsky Kharkov Aviation Institute. No. a201100762; declared 24.01.2011; published 25.09.2013; Bulletin No. 18; 7 p.; 5 fig (in Ukrainian) 13. Patent UA 115655 C2 Ukraine, MPK (2017.01) B64D 33/02; F02C 7/04, 7/057; B64C 21/02; B63H 11/00. The inlet device with the conformal air intake, the ways of production and the ways of control, the conformal power plant implementing the specified ways, and the vehicle equipped with specified inlet device or power plant. Kornev, A.V.; Applicant and owner: Kornev A.V. No. 201402926; declared 24.03.2014; published 11.12.2017; Bulletin No. 23; 31 p.; 14 fig (in Ukrainian)

166

V. Cheranovskiy et al.

14. Kornev, A.V.: Aspects of creation of the effective power plant with the air inlet of the submerged type with external vortex formers. Sci. J. Weapon Syst. Mil.Y Equip. Kharkov: KhUPS 3(31), 167–180 (256) (2012) (in Russian) 15. Kornev, A.V., Boychuk, I.P.: Integrated approach to aerodynamic designing of paths of inlet devices with vortex-free submerged air intakes. Bulletin of Samara University. Aerosp. Equip. Technol. Mech. Eng. Samara: Samara national research university of name of academician Korolev, S.P. 16(2), 47–59 (2017) (in Russian) 16. Definition of aerodynamic characteristics of the conceptual model of AV in the subsonic wind tunnel AT-4 in the longitudinal channel (the code: KMLA): Research Report. National aerospace university of name of Zhukovsky, N.E. “KhAI”. Supervisor: V.V Chmovzh, V.V.; executors: Yeremeyev, I.D., Girka, Yu.V., Kornev, A.V., Solyanik, P.N., Kharkov, p. 96, No. 101-6/2009 (2009) (in Russian) 17. Definition of aerodynamic characteristics of the conceptual model of AV in the subsonic wind tunnel AT-4 in the lateral channel (the code: KMLA): Research Report. National aerospace university of name of N.E. Zhukovsky “KhAI”. Supervisor: Chmovzh, V.V.; executors: Yeremeyev, I.D., Girka, Yu.V., Kornev, A.V., Solyanik, P.N.–Kharkov, pp. 95, No. 101-14/2009 (2009) (in Russian) 18. Theoretical bases of management of life cycle of development of the aviation engineering on the basis of intellectualization of design process, methods of mathematical and physical modeling. Volume 1: Research report. National aerospace university of name of N.E. Zhukovsky “KhAI”. Research supervisor: Druzhinin, E.A.; responsible executor: Pogudina, O.K. Kharkov: 2011. SR No. 0109U002008 19. Adjustment of general aerodynamic configuration, main characteristics and design of a cruise missile. Determination of structure and main characteristics of the control system of a cruise missile. Volume 1 (the code: “ Korshun-2”): Research report (intermediate report). National aerospace university of name of N.E. Zhukovsky “KhAI”. Supervisor: Smolyakov, A.V.; responsible executors: Yashin, S.A., Yanakayev, V.A.; executors: Betin, A., Ryzhenko, A., Kornev, A., Dunayev, A., Bondareva, N., Tutubalin, V., Shevko, S., Kharkov, NII PFM KhAI, p. 176 (2014) (in Russian) 20. Adjustment of general aerodynamic configuration, main characteristics and design of a cruise missile. Determination of structure and main characteristics of the control system of a cruise missile. Volume 1 (the code: “ Korshun-2”): Research report (final report). National aerospace university of name of N.E. Zhukovsky “KhAI”. Supervisor: Smolyakov, A.V.; responsible executors: Yashin, S.A., Yanakayev, V.A.; executors: Betin, A., Ryzhenko, A., Kornev, A., Dunayev, Bondareva, A. N., Tutubalin, V., Shevko, S. Kharkov, NII PFM KhAI, p. 123 (2014) (in Russian) 21. Adjustment of general aerodynamic configuration, main flight performances and design of a cruise missile. Volume 1 (the code: “Korshun-2 ADC”): Research report about a component of the preliminary design “Cruise missile in a transport/launch container”, (intermediate report). “Scientific and Technical Center of Pilotless Aviation” LLC. Supervisor: A.V. Smolyakov; responsible executors: S.A. Yashin, V.A. Yanakayev; executors: S. Stetsenko, A. Kornev. Kharkov, NII PFM KhAI, p. 126 (2015) (in Russian) 22. Adjustment of general aerodynamic configuration, main flight performances and design of a cruise missile. Volume 1 (the code: “Korshun-2 ADC”): Research report about a component of the preliminary design “Cruise missile in a transport/launch container”, (final report). “Scientific and Technical Center of Pilotless Aviation” LLC. Supervisor: A.V. Smolyakov; responsible executors: S.A. Yashin, V.A. Yanakayev; executors: V. Cheranovsky, S. Stetsenko, A. Kornev, S. Shevko. Kharkov, NII PFM KhAI, pp. 140 (2015) (in Russian) 23. Harris, C.D.: NASA supercritical airfoils. A matrix of family-related airfoils. NASA Technical Paper 2969. Washington, DC: NASA, March 1990, p. 72 24. Patent US 8, 186, 616 B2 B2 USA Int. Cl. (2006.01) B64C 3/14; 39/00; 39/10. Hybrid transonicsubsonic aerofoils. Inv.: M. Shepshelovich, M. Steinbuch; assignee: Israel Aerospace Industries, Ltd.; No. 11/305,345; filed: 19.12.2005; publ.: 29.05.2012. 15 p.: 18 fig

Designing a Basic Model of an Unmanned Aerial Vehicle …

167

25. Rolls L.S.: A flight comparison of a submerged inlet and a scoop inlet at transonic speeds. NACA RM № A53A06. Washington: Langley, 42 p., (19530 26. Kornev A.V., Sereda V.A., Migalin K.V.: Metod of aerodynamic designing of aerial vehicles of integrated configurations with the submerged inlet devices and the power plant configured into the lifting system of airframe. In: Proceedings of Higher Education Institutions. Aviation Engineering. Kazan: Kazan State Technical University of name of A.N. Tupolev, No. 1, pp. 17– 25 (2018) (in Russian) 27. Druzhinin, E.A., Chmovzh, V.V., Kornev, A.V.: Use of methods of aerodynamic designing in realization process of life cycle of development of a perspective sample of aviation engineering. Sci. J. Weapon Syst. Mil.Y Equip. Kharkov: KhUPS 4(28), 48–57 (130) (2011) (in Russian) 28. Alyamovsky, A.A., Sobachkin, A.A., Odintsov, E.V. et al.: SolidWorks 2007/2008: Computer Modeling in Engineering Practice. St. Petersburg: BKhV-St. Petersburg, p. 1040. (2008) (in Russian) 29. Spalartand, P., Allmaras, S.: «A one-equation turbulence model for aerodynamic flows» Technical Report AIAA-92-0439. American Institute of Aeronautics and Astronautics (1992) 30. Launderand, B.E., Spalding, D.B.: Lectures in Mathematical Models of Turbulence. Academic Press, London, England (1972) 31. Orszag, S.A., Yakhot, V., Flannery, W.S., Boysan, F., Choudhury, D., Maruzewski, J., Patel, B.: Renormalization group modeling and turbulence simulations. In: International Conference on Near-Wall Turbulent Flows, Tempe, Arizona (1993) 32. Rusanov, A.V.: Mathematical modeling of non-stationary viscous spatial flows in air-gas channels of turbo-machines: Thesis for the degree of the Doctor of Engineering: 05.05.16. Kharkov, 388 p. (2005) (in Russian) 33. JetCat P300 PRO/JetCat https://www.jetcat.de/de/productdetails/produkte/jetcat/produkte/Pro fessionell/p300%20pro. Accessed 23 December 2019 34. Taskinoglu, E.S., Knight, D.D.: Design optimization for submerged inlets–Part I. In: 41st Aerospace Sciences Meeting and Exhibit, AIAA 2003-1247. Reno, Nevada: AIAA, Inc., 6–9 January 2003, p. 10 35. Akman, O.: Subsonic-transonic submerged intake design for a cruise missile. A thesis for Master of Science in Aerospace Engineering. Middle East Technical University, Ankara, November 2014, p. 99 36. Cosentino G.B., Holst, T.L.: Numerical optimization design of advanced transonic wing configurations. NASA Technical Memorandum 85950. Boulder, CO: University of Colorado, May 1984, p. 98 37. Boxer, V.D.: The optical researches of supercritical profiles TsAGI on transonic speeds Moscow: TsAGI: Proceedings of TsAGI, p. 24 (1973) (in Russian) 38. Rademakers R.P.M., Bindl, S., Brehm, S., Muth, B., Niehuis, R.: Investigation of flow distortion in an integrated inlet of a jet engine. In: Proceedings of the 20th German Aerospace, DLRK2013-301349. Stuttgart, Germany, Sept. 2013, p. 10 39. Gas turbine engine inlet flow distortion guidelines. Society of automotive engineers, Aerospace recommended practice, SAE-ARP1420. Warrendale, PA: SAE, Ink., Mar. 1978, p. 17 40. Stocks, C.P., Bissinger, N.C.: The design and development of the Tornado engine air intake. Aerodynamics of power plant installation, AGARD-CP-301. Toulouse, France: AGARD, 11– 14 May 1981, p. 10 41. Bloch, G.S.: An assessment of inlet total-pressure distortion requirements for the compressor research facility. Final Report, WL-TR-92-2066. Wright-Patterson AFB, Ohio: Wright laboratory, p. 38 (1992) 42. Coffman, V.M.: Comparative analysis of ways of the averaging when processing parameters of non-uniform air flow on an input in GTE. Bull. UGATU (Ufa: UGATU) 2(31), 35–42 (2009) (in Russian) 43. Goryunov, A.I., Goryunov, I.M.: The accounting of effect of non-uniformity of parameters of a working substance onto characteristics of assemblies of GTE and PP. Bull. UGATU (Ufa: UGATU) 14(3) (38), 57–61 (2010) (in Russian)

168

V. Cheranovskiy et al.

44. Bissinger, N., Breuer, T.: Basic principles–gas turbine compatibility–intake aerodynamic aspects. In: Blockley, R., Shyy, W., (eds.) Chapter EAE487 in Encyclopedia of Aerospace Engineering, vol. 8, pp. 1–10. Wiley, Chichester, UK (2010) 45. Air intakes for high speed vehicles (Prises d’air pour vehicules a grande vitesse). Chap. 2.2: Definition of intake performance and description of intake flows. AGARD advisory report 270, AD-A248 270. Neuilly-Sur-Seine: AGARD, pp. 7–18 (1991) 46. Air intakes for high speed vehicles (Prises d’air pour vehicules a grande vitesse). Chap. 4.6: Measurements in three European wind tunnels at subsonic and supersonic speeds of dynamic distortion and steady state performance of an axisymmetric pitot intake. AGARD advisory report 270, AD-A248 270. Neuilly-Sur-Seine: AGARD, pp. 232–244 (1991) 47. Harrison, N.A.: Active flow control of a boundary layer ingesting serpentine diffuser. Thesis for the degree of Master of science in aerospace engineering. Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 13 July 2005, p. 142

Transport Category Aircraft Fuselage Integrated Design Oleksandr Dveirin , Oleksandr Hrebenikov , Andriy Humennyi , Dmytro Konyshev , and Anton Chumak

1 Introduction The fuselage according to its functional features is one of the most complex units of the aircraft. It serves to accommodate payloads, crew equipment, gear, sometimes power plant and fuel. The fuselage joins the most important units of the aircraft: wing, fin, stabilizer, landing gear, power plant [1, 2]. Such functional complexity causes difficulties both in the choice of parameters, size and the fuselage form in the course of designing and calculating the external operating loadings. The fuselage consists of the nose, middle and tail sections. Structurally, the fuselage of a civil aircraft, as a rule, is a thin-walled frame structure. The aim of the article is to develop a method of integrated design and computer modeling of a civil aircraft fuselage using computer integrated systems CAD/CAM/ CAE/PLM. The initial data for the design presents in the requirements specifications, such as the value of the design range Lp of the aircraft, the mass of the payload (commercial) mp l, its dimensions, speed V (maximum and cruising), flight altitude H, base conditions (aerodrome class, takeoff length runway), lift-to-drag ratio in cruising flight mode, relative weight of the fuselage structure, fuselage life, fuselage overall dimensions and weight of the content (payload, equipment and gear), used construction materials, set of efficiency criteria. According to the initial data, the scheme of the aircraft is selected (Fig. 1), the minimum takeoff weight of the aircraft is determined, the basic parameters of the aircraft are optimized, drawings of the general view of the aircraft, aerodynamic, O. Dveirin · D. Konyshev ANTONOV State Company, Kyiv, Ukraine O. Hrebenikov · A. Humennyi · A. Chumak (B) Kharkiv Aviation Institute, National Aerospace University, Kharkiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_3

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Fig. 1 Scheme of regional transport category aircraft

mass and load-carrying structure of the aircraft is developed, the aircraft alignment is calculated. The external shape of the fuselage is determined by the outlines of the side view, the plan view of the nose and tail sections, as well as the shape of cross section. The shape of fuselage is primarily based on the aerodynamics requirements, the main parameters and characteristics of the fuselage loads during its life cycle. This calculation is convenient to present as a whole in the form of iterative process of determining the aircraft takeoff mass. The method scheme of the fuselage parameters calculation is shown in Fig. 2. To avoid the wave crisis with increasing flight speed, the form of fuselage nose section usually has a pointed shape and significant elongation [3]. The shape of the cockpit glazing is also taken into account, which is characterized by the angle ϕ of the windshield. With the increasing the number M of the flight, the angle ϕ increases as well. The application of the area rule in the design of high-speed subsonic and supersonic aircraft contributes to the reduction of aircraft drag. The shape of the cross section of fuselage is chosen not only in terms of aerodynamics, but also in terms of layout, location of engines, crew, passengers, equipment, strength requirements. The optimal cross-sectional shape is considered to be round. This shape of the cross section allows to get the minimum weight of the structure, because it provides the skin smallest thickness. As a type of circular crosssection consider cross-sections formed by a combination of two or more circles, often vertically, but sometimes horizontally. The diameter of the fuselage dF is chosen from the conditions of obtaining the minimum area of midline section Sf and fulfillment of the most important layout requirements.

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Fig. 2 Scheme of integrated design method for transport category airplane fuselage, taking into account characteristics of its nose section

For passenger and cargo aircraft, the fuselage middle is formed depending on the overall dimensions of passenger compartment or cargo cabin. The fuselage midsection dimensions of passenger aircraft are determined depending on the layout option (passenger cabin class), the height of the passenger cabin and the height of the luggage compartments located below it. In terms of design, the most optimal is a round cross-section of fuselage, because in this case a high level of strength at the lowest weight of the structure can be

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obtained. However, this form of cross-section is often suboptimal because of the requirements of passenger and luggage compartments layout located above the floor. To get full use from the width of the circular fuselage, the passengers must be placed so that the middle of the backs of the seats are located on the fuselage horizontal axis. In this case, the height of the cabin becomes irrationally large, and the height of the luggage compartment—unacceptably small. If you try to raise the passenger compartment floor to obtain the required luggage compartment height, the used width of the passenger compartment will decrease, and a sharp narrowing of the side walls above the rear seats will create an impression of depression and difficulties to get them. The following design solutions allow to eliminate these shortcomings: – placement of cargo and luggage compartments not under the floor, but in the nose and tail fuselage sections, which will increase its length; – choice of the cross-section shape not round, but oval or formed of two or more intersecting circles. These design solutions allow to ensure the implementation of layout requirements and requirements for the comfort of passengers and crew, but contradict the characteristics of strength, technological, production and economic requirements. Thus, implementing the oval shape of the cross section to meet the strength requirements, it is necessary to strengthen the structure of the fuselage in the area of the pressurized cabin, that will increase the weight of the airframe structure. Design strengthening is connected with deformations of a fuselage at influence of excess pressure of pressurized cabin. The oval fuselage cross section shape complicates the process of production and increases its cost [1]. To provide the successful rescue measures in case of emergencies, in the fuselage are provided additional cuts for special hatches and doors. According to international standards [4, 5] it is necessary to ensure the emergency evacuation of all passengers and crew on the ground for 1.5–2 min with the extended or retracted landing gear. To implement these requirements it must be determined the location and number of required cutouts for standard hatches and doors. Type I hatch of 610 × 1220 mm size is located at floor level outside the wing area. Type II hatch of 510 × 1120 mm size—outside the zone and within the wing area, where the lower edge should not be higher than 250 mm from the floor level and 430 mm from the wing level. Type III hatch of 510 × 915 mm size is placed in the wing area at a height not higher than 510 mm from the floor and not higher than 690 mm from the wing. Type IV hatch of 480 × 660 mm size is located in the wing area not higher than 740 mm from the floor and not higher than 910 mm from the wing. The required number of such hatches is determined by the number of passengers in each cabin. At number of passengers 10–40 at least one I type hatch is obligatory, at 100–200 people—two hatches must be provided, at 200–280 people—three hatches, at 280–300 people—four type I hatches. Entrance doors (necessarily without thresholds) are considered as the type I hatches.

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At the upper location of the wing, type III hatches must be provided at the top of the fuselage at the rate of one hatch per 35 passengers. Automatic inflatable rubber ladders must be installed near type I and II hatches (outside the wing area). The front door is usually placed on the left side of the fuselage. On large passenger aircraft (more than 250 people) the doors can be located on both sides (thresholds are not allowed). Windows are arranged between frames with a pitch of not less than 500 mm, width of windows—200–230 mm, height—320–350 mm. The windows pitch is usually coordinated with the pitch of the chairs, which is determined by the comfort class of the salons. As mentioned above, the main parameters and characteristics of the fuselage are determined together with the parametric calculations of other parts of the aircraft based on the requirements set in the requirements specifications, solving problems of choosing the geometric characteristics at the stage of sketch design of the aircraft as well as ensuring minimum weight and parametric analysis relating to the structure [6]. As a result of such work the aircraft mathematical model of fuselage and other units, and a mathematical model of the entire aircraft are created. The choice of nose and tail sections shape is determined by the minimum resistance, loading and unloading conditions of the fuselage content. The forms of these parts are chosen on the basis of aerodynamic researches, introduction of the new conceptual decisions, received as a result of research developments, theoretical researches, experience of a design. In Fig. 3 a general view and layout of a regional passenger aircraft fuselage is shown. The next stage of design work is the development of load-carrying structure. Loadcarrying structure (LCS) of the fuselage determines the degree of participation in the taking up the loads by load-bearing elements: – – – – – – – –

Longitudinal structural members (stringers, spars, longitudinal beams). Transverse structural members (frames). Edging cutouts in the skin. Connection of load-bearing units. Local reinforcements. Elements of construction. Fasteners. Joints of structural elements [3].

At the stage of LCS development the choice of elements parameters for fuselage design—geometrical characteristics, characteristics of element materials, their design features is carried out. The frames are divided into normal and reinforced. Normal frames are used to form the shape of cross sections and are the supports of stringers and skin. Reinforced frames provide the transfer of concentrated forces to the structure of the fuselage. Normal bulkheads are a circular (or other configuration) stamped frame made of Z-shaped sheet material or channel section with perforations for stringers. The design of each frame includes a horizontal cross beam. Normal frames together with skin are loaded with excess pressure from a pressurized cabin. In addition, normal frames prevent the fuselage structure overall loss stability during the significant overloads in flight and landing [6].

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Fig. 3 Regional passenger aircraft fuselage general view and layout. 1—nose fairing, 2—cockpit bulkhead, 3—forward entrance door, 4—front service door, 5—wing fairing, 6—rear entrance door, 7—rear service door, 8—stabilizer inspection hatch, 9—rear luggage and cargo compartment bulkhead, 10—APU compartment, 11—tail mounting compartment, 12—luggage doors 13—water supply panel, 14—back luggage hatch, 15—fairing of the landing gear, 16—main landing gear bay, 17—forward luggage hatch, 18—floor of transport cabin, 19—floor of crew cabin, 20—nose landing gear bay, 21—onboard window, 22—tail compartment hatch

Transverse beams belonging to the load-bearing elements of the frame, serve as a transverse frame of the pressurized cabin floor of the aircraft. Reinforced frames are made in the form of powerful circular frames formed by the inner and outer rims and the wall. Reinforced frames in cross section are made of Z-shape, channel or T-type. A characteristic feature of the fuselage is that it is exposed to excess pressure in pressurized cabins. Pressurized cabins are a cylindrical shell closed from the ends by bottoms, which are an important element in terms of strength, weight and space. The flat bottom is irrational considering the design weight, but, sometimes, because of space lack it can be used. Most often, such bottom is made of a thin sheet (2…3 mm), supported by a load-bearing set in the form of vertical and horizontal stiffeners. The elliptical bottom (unsupported) is a thin-walled shell, has a double curvature and requires less space for placement than a spherical one. The most rational shape of the bottom is spherical. The bottoms of other forms can also be applied. The choice of the bottom shape is often dictated by layout considerations. So in the construction of the fuselage you can find all these types. Transport aircraft often use a combination of spherical bottoms

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with different curvatures. For example, a strip of a sphere of smaller curvature is adjacent to a cylindrical part, and a dome of a sphere of greater curvature is adjacent to it. And, as a rule, both ring and radial reinforcement is applied. Fuselage stringers often have a T-shaped, Z-shaped or angular cross section form. They are easy to manufacture, but subjected to some twisting under axial loads, which can create additional (small) bending deformations between the frames in the skin. Therefore, more complex symmetrical shapes of cross sections are used, striving to ensure that the main axis of inertia of the stringer profile cross section passes through the axis of the riveted seams with the skin.

2 Methods for Calculating Fuselage Mass-Inertial and Aerodynamic Characteristics At the stage of preliminary design for the analysis of aerodynamic and mass characteristics of the aircraft methods and dependencies are used. They reflect the design features of the units in a generalized form and have statistical character, which significantly limits the possibility of reasonable choice of fuselage nose section (FNS) parameters. To determine the mass of the fuselage during the preliminary design, the dependences obtained by the statistical method are known as the first approximation formulas [2, 3]. In the formulas of O. A. Badiahin and V. M. Sheinin the diameter, fuselage length, the landing gear and engines hinges’ location are taken into account. But the FNS parameters influence on the weight of the fuselage is not taken into account. Kozlovsky’s formula takes into account the surface area of fuselage and thus allows to indirectly take into account the geometric parameters of the FNS. At the stage of sketch design, the aircraft take-off mass is determined in the second and third approximations [1, 6], which are characterized by the use of dependencies that consider mission requirements (MR), manufacturing technology and operation of aircraft in expected operating conditions. The applied methods of fuselage mass calculation in the second approximation are connected with design calculations on durability of the basic elements of its design. Thus, internal force factors in them are defined on the basis of the beam design model. This approach does not allow to take into account the load from the internal pressure of cabin, limiting its consideration by further checking the thickness of the fuselage cylindrical part skin and the introduction of additional component in the general formula of the fuselage. The geometric parameters of the FNS are taken into account in the process of determining the shape coefficients by graphically integrating the mass distribution function along the fuselage length. At the final stages of aircraft sketch design, it is possible to use more complex models and methods that allow for detailed masses analysis and optimization and geometric parameters of structural elements. However, there are very few publications in the open literature on such methods.

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During the iterative design process a theoretical drawing with the location of structural elements, their geometric coordination relative to each other is created. The next stage of design is the loads calculation, acting on the fuselage for different design cases. The loads acting on the fuselage structure are plotted, load extremes are determined, possible variants of fuselage unloading are considered. In Fig. 4 a diagram of the internal force factors acting on the transport category aircraft fuselage is shown. The allowable service life of passenger and transport aircraft depends on the strength of pressurized sections. Therefore, higher requirements should be met by pressurized sections compared to other aircraft units. Cabins should be properly pressurized providing sufficient rigidity and durability of a design, the required service life [3, 7]. The complexity of the loads, presence of large cutouts, all this create significant difficulties in calculating the strength of such structures. Determining the allowable design stresses, it is taking into account the provision of the specified service life. The level of operational loads should provide the specified service life of fuselage design. Fuselage design models are developed on the basis of the created LCS. The created design models represent a design as a system of coordinated elements that takes into account their coordination under the given loads. Beam models, analytical models of structural mechanics, the theory of elasticity, the theory of plasticity, finite element models (FEM) are used as design models for fuselage load-carrying structures of structural members and units in design calculations [3, 6]. The weight of the fuselage structure is approximately distributed between its structural and load-bearing elements as follows: the frames—up to 21… 28%, the longitudinal members—up to 30… 33%, the skin—up to 37… 40% [3]. The aerodynamic characteristics of the aircraft during the preliminary design are calculated using analytical methods that use elements of empirical and statistical dependencies. The drag coefficient is considered as the sum of the coefficients of profile, inductive, wave and additional resistance from local sources. When calculating the profile drag, the elongation of the FNS is taken into account by calculating the transition point of the laminar boundary layer to turbulent. The impedance depends on the elongation and narrowing of the FNS, as well as on the shape of generators (straight, parabolic or spherical). When calculating the additional drag using empirical dependencies the additional drag of the canopy, pitot tubes and the slots of the hatch covers are taken into account [8]. The value of the inductive resistance is determined after calculating the coefficient of the fuselage lifting force, which is affected by the angle of deviation of the FNS axis from the fuselage construction horizontal. In addition, the angle of deviation of the FNS axis affects the value of the diving moment of the fuselage. Experimental methods for determining aerodynamic characteristics are the most accurate, but require significant material and time costs, especially when conducting flight experiments. The application of these methods is appropriate in the later design stages to confirm the flight characteristics of the aircraft or in the process of developing new calculation methods to verify the results. Numerical methods for calculating the aerodynamic characteristics of the aircraft have spread significantly due to the rapid development of computer systems. In contrast to analytical methods, the

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Fig. 4 View of plots of internal force factors acting in transport category aircraft fuselage

system of equations is solved by numerical methods. The aerodynamic characteristics are obtained on the basis of the values of forces and moments acting on the model surface. The application of these methods allows to take into account all the geometric parameters specified in the three-dimensional model. Despite the need to purchase software, prepare an aerodynamic model, calculate and analyze the results, the total time is less than in a field experiment. It allows to apply these methods at the stage of sketch design of the aircraft.

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2.1 Methods of Three-Dimensional Fuselage Computer Modeling At the present stage of science and technology development, the use of integrated computer-aided design systems CAD/CAM/CAE/PLM is a prerequisite for the development and maintenance of the life cycle of competitive aircraft. The use of traditional general view drawings, LCS, layout and theoretical drawings of units is possible together with three-dimensional models to provide a clear and unambiguous presentation of information. The computer design of aircraft contains the following models [7, 9, 10]: 1. Model No. 1—master geometry of the aircraft (or model of the aircraft surface, which determines all the points lying on aircraft surface); 2. Model No. 2—model of aircraft space distribution; 3. Model No. 3—models of joints and connections on structural and technological fasteners; 4. Model No. 4—model of geometry of the whole item (analytical standards of all parts, assemblies, units and the aircraft as a whole), the model of complete computer aircraft definition. The stages of preliminary and sketch design are characterized by the mainly use of models of master geometry and space distribution, with the model FNS considered as part of the corresponding models of fuselage. In the process of creating a fuselage master geometry the geometric parameters of the surfaces of its parts are determined, coordinated and described. Traditionally used methods of descriptive geometry [11] involve the construction of external contours and the coordination of form in two stages. In the first stage, the form is linked using graphical methods (for example, the method of buttock lines and horizontals), in the second stage, the obtained form is described by analytical methods and its fixation in the form of theoretical lines and tables on the theoretical drawing of unit. Theoretical drawing is a source of information for constructing a mathematical model of unit in a computer simulation environment. It contains the following information: 1. 2. 3. 4. 5.

Coordinate systems of unit in the aircraft coordinate system; Division of the unit theoretical surface into segments; Overall and reference dimensions of unit; The axis of load-bearing elements of the unit; The location of the output forming carrier lines, which will be the formation of surface segments; 6. Tables with parameters of output sections and parameters of carrier lines; 7. Tables with nodal points. Methods of three-dimensional fuselage computer modeling using computer systems CAD/CAM/CAE consist of interconnected stages used previously. The use

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of computer systems has significantly expanded the variability of design, the degree of approximation to the best result in a shorter time [7, 10]. At each design stage, specialists have the opportunity to return to the level where the changes need to be made, to obtain the necessary characteristics and values to ensure a set of requirements for the aircraft. Using analytical methods, the geometric parameters of the aircraft and its units are determined, theoretical drawings of the aircraft and its units are created. They are the basis for creating a parametric model (PM) of the aircraft master geometry. Surfaces are created by methods of analytical geometry available in CAD/CAM/ CAE systems. Having a theoretical drawing of the unit, its master geometry is created. Surfaces are created using techniques based on the methods of analytical geometry, by solving equations describing the aircraft surfaces. The result of this solution is a master geometry [7]. The master geometry of the fuselage contains the master geometry of its parts: nose, middle and tail sections. Initial data for creating a master geometry of the fuselage: Dfm —the diameter of the fuselage middle part; Lf is the fuselage length; λn is fineness ratio of FNS; λt —elongation of fuselage tail section (FTS); shape of the fuselage nose and tail part generators. In Fig. 7 a parametric model of the fuselage master geometry of civil aircraft, created using a computer integrated system CAD/CAM/CAE Siemens NX is shown. The master geometry of the fuselage is coordinated with the master geometry of other aircraft units.

3 Parametric Modeling of Fuselage Master Geometry Modeling the fuselage surface occurs in the process of creating the master geometry of aircraft model [7] at the stage of preliminary design. Let us consider the fuselage master geometry parametric modeling, taking into account the FNS features. It is expedient to carry out modeling by parametric methods based on the fuselage parameters’ matrix (Table 1), it allows to automate further changes in models and to systematize researches of separate parameters influence on fuselage mass. In Table 1 data for six transport aircraft fuselage variants for different purposes (Fig. 5) is shown. The designations of parameters in the tables and drawings are made in accordance with the syntax of the expression editor of the Siemens NX system. Generalized theoretical drawing (Fig. 6) and matrix of geometric parameters of fuselage is used to create fuselage master geometry. Conventionally, the model is divided into models of the nose, center and tail sections of the fuselage. The model of the fuselage master geometry (Fig. 7) was further divided into parts according to the structural and functional principle in order to continued associative application of loads and calculation of the masses of fuselage parts.

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Table 1 Matrix of fuselage geometric parameters Parameter, designation, units

Value

Version

1

2

3

4

5

6

Absolute Equivalent diameter of fuselage d_f, mm

2190

2820

3350

4120

5200

7723

40

45

45

45

45

45

Glazing installation angle phi_ws, deg Sighting angle phi_v, deg

16

17

20

23

30

25

Glazing height H_ws, mm

350

480

450

500

550

500

1200

1400

1350

1600

1800

1950

Relative deviation of the FNS y_n

0.22

0.25

0.27

0.3

0.2

0.24

Relative deviation of FTS y_h

0.4

0.15

0.21

0.5

0.4

0.2

Fuselage extension lam_f

7

7.65

7.8

8

7.83

9.3

Elongation of FNS lam_n

1.4

1.4

1.8

2

1.65

2

Elongation of FTS lam_h

2.8

2.8

3

3.5

3.5

3.3

Cross section width and height ratio k_hb

1.12

1

1

1

1

1.17

Section solidity factor eta_m

0.93

π/4

π/4

π/4

π/4

0.78

Length of glazing L_ws, mm Relative

Fig. 5 Considered Configuration Options of Aircraft Fuselage. 1—light multi-purpose, 2— regional freight and passenger, 3—short-haul passenger, 4—short-haul military transport, 5— medium-haul military transport, 6—long-distance passenger

3.1 Fuselage Nose Section Parametric Model Space Distribution In addition to the three-dimensional computer model of master geometry, the implementation of the proposed design method allows to develop the FNS model of space distribution. To create a model of the aircraft space distribution, it is necessary to solve the following tasks: develop structural and technological division; paneling;

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Fig. 6 Fragment of generalized theoretical drawing of TCA fuselage

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10

11

12

13

Fig. 7 Regional aircraft fuselage. Master Geometry Model. 1—nose fairing; 2—fuselage nose section; 3—cockpit glazing; 4—emergency door; 5—nose landing gear bay; 6—entrance door; 7—fuselage central part; 8—illuminators of a passenger cabin; 9—wing center section joint; 10— main landing gear bay; 11—fastening the main landing gear; 12—emergency exit (service door); 13—fuselage tail part; 14—tail compartment hatch

determine the number and location of elements of the load-carrying structure; resolve the issue of listing and arrangement of equipment, facilities and others; system layout; layout of crew cabin and passenger cabin for different number of passengers and cabin comfort. In addition, the layout and calculations the aircraft center of mass range. In Fig. 8 shows a fragment of the transport category aircraft FNS space distribution model.

Fig. 8 Transport category aircraft FNS space distribution fragment

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3.2 Analysis of Layout and Field of View from Cockpit One of the tasks to be solved using the FNS space distribution model is to determine the viewing angles from the cockpit in order to assess their compliance with flight safety requirements in accordance with current industrial standards OST1 02721-91, AC25.773-1 and requirements for ergonomics of crew workplaces. The diagram (Fig. 9) shows the analysis method of view from the cockpit, using a parametric model of the FNS space distribution, which has been implemented using an integrated design system Siemens NX.

Fig. 9 Methods of analysis of layout of crew workplaces and view from cockpit method of determining the pilot position, providing a given view at a given glazing parameters, was performed iteratively. Layout requirements are considered as constructive constraints on pilot location

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The results of the view from the cockpit analysis can be presented in the form of a viewing angles diagram (Fig. 10a), which is convenient to compare with the requirements of current standards or in the form of a spatial curve on the fuselage surface (Fig. 10b, c), which determines the minimum necessary limits of glazing under conditions of the set FNS configuration and pilots’ position. For all the considered configurations of glazing, a significant difficulty was to provide view, taking into account the pitch angle when flying on the glide path, as well as the placement of the instrument board of the required width. As a result of the analysis, the pilot position (sighting point) was determined, which provides the required viewing angles (up to 125° to the left, 30° up, 25° down), the size of glazing and windshield angle, which will provide acceptable mass and aerodynamic characteristics of fuselage. The observed distance from the dummy to the theoretical surface is not less than 300 mm.

Fig. 10 Analysis of workplaces and view from cockpit layout. a—viewing angles diagram; b—the required limits of glazing for the original geometric parameters; c—necessary limits of glazing at the recommended geometrical parameters; 1—the actual limit of view, taking into account the risers of the glazing frame; 2—areas of insufficient view; 3—the limit of minimum view according to OST1 02721-91; 4—the limit of minimum view according to AC25.773-1; 5—bypass limits of the minimum view taking into account the pitch angle, flying on the glide path; 6—projection of the bypass limits of minimum view on the FNS surface; 7—insufficient space of cabin for the instrument board accommodation; 8—insufficient view due to the meteorological radar fairing shading; 9— instrument board area; 10—excess glazing area; 11—armrest placement area; 12—required pilot’s head free space area

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3.3 Fuselage Aerodynamics Modeling The used methods to calculate the aircraft aerodynamic characteristics at the preliminary design stage, allow to take into account some FNS parameters, such as elongation, generator shape, cross-sectional shape and angle of nose section deviation from the fuselage construction horizontal. The disadvantage of the available methods is that the FNS shape and the fuselage cross section is considered as a qualitative parameter, allowing to choose only one of several possible options (conical, parabolic, elliptical, round, rectangular). The deflection angle is taken into account only when calculating this approach. That does not allow to make full parametric analysis, but it is acceptable for results verification. The necessary data for the fuselage aerodynamic characteristics, taking into account the FNS geometric parameters, can be obtained using a finite element model of the fuselage aerodynamic flow. To determine the FNS parameters influence on the nature of the fuselage aerodynamic flow, a method of creating a corresponding finite element model was developed (Fig. 12). The values and nature of the aerodynamic load distribution on the fuselage are determined using the CFX module of the ANSYS engineering analysis system. Based on the fuselage master-geometry model, a unified FEM of aerodynamic flow around the fuselage was created and calculations were performed for the considered flight modes. To ensure minimal impact of the boundaries of calculation area on the flow nature and reduce the number of elements of the model the calculation area of elliptical cross section with a vertical position of the major axis was used (area size 15 × 20 × 50 m) and grinding elements in the boundary layer area (Fig. 11). The parameters of the aerodynamic environment are set according to state standard GOST 4401-81 “International Standard Atmosphere”. The CFD-Post component of the CFX module is used to display and visualize the calculation results. In Fig. 13 is shown the nature of the flow velocity distribution in the symmetry plane of the calculation area and the nature of the pressure distribution on the fuselage surface. In the process of model verification, the obtained fuselage aerodynamic characteristics are compared with the reference values, calculated by the known method [8]. Satisfactory accuracy results were obtained with the number of elements about 5 · 105 and the standard k-ε model of turbulence. Further refinement of aerodynamic loads is advisable when using more accurate methods for determining aerodynamic characteristics. To store and transmit data on the magnitude and nature of the fuselage aerodynamic load distribution, the values of normal pressure and tangential stresses on its surface are exported to CSV open format text files.

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Fig. 11 Fragment of FEM fuselage aerodynamic flow

3.4 Analysis of Fuselage Structure Mass-Inertial Characteristics The fuselage mass-inertial characteristics have a significant impact on the aircraft airborne performance, and thus on the characteristics of its efficiency. To determine the mass-inertial characteristics, a method is proposed. Its scheme is shown in Fig. 14. To calculate the fuselage mass in the second approximation, the methods of engineering analysis are used which directly take into account the operating loads acting on fuselage. To calculate these loads, it is necessary to consider the typical flight profile of the designed aircraft and determine the modes that are characterized by the maximum values of loads, then build a flight condition envelope and determine the parameters of the calculated flight modes in accordance with AP-25 (pp. 25.321–25.373). Estimated overloads, angles of attack and excess pressure are determined in accordance with the requirements of CS-25, AP-25, literature recommendations [3, 5, 6], aerodynamic characteristics of the aircraft and the international standard atmosphere parameters (GOST 4401-81). The design pressure inside the cabin is not less than the equivalent height of 2400 m (0.6 atm). Thus, the obtained calculated flight modes allow to estimate the maximum static flight and ground loads on the fuselage at symmetrical and asymmetrical loading. It is expedient to take into account dynamic loads in the course of further design on the basis of the specified fuselage mass-centering characteristics and its parts. AP-25 requirements for bird resistance are taken into account as additional restrictions on the minimum thickness of the respective parts of fuselage structure. To calculate the mass-inertial characteristics of fuselage structure, taking into account the FNS shape, it is necessary to perform:

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Fig. 12 Scheme of fuselage aerodynamic flow modeling method

– formation of initial data and requirements on the basis of mission requirements for aircraft design, construction of a typical flight profile and flight condition envelope, determination of design flight modes parameters; – creation of a master-geometry model, a space distribution model, selection of fuselage functional and technological structural parts; – creation of a fuselage aerodynamic flow finite element model and the fuselage surface air load distribution calculation for all design cases; – creation of a generalized fem of fuselage, determination and application of loads, calculation of stress–strain state for all design modes, determination of maximum

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Fig. 13 Nature of pressure distribution on fuselage surface and flow velocity in simulation domain plane of symmetry

–

–

– –

operating and design stresses, tensile, compressive and equivalent stresses σ1 , σ3 , σe , for a single casing thickness; element-by-element calculation of the minimum required conditional thickness of regular structure skin according to the maximum allowable tensile stresses, compressive stresses (limiting ones) and equivalent stresses taking into account technological limitations regarding the minimum thickness of the skin; mass calculation, skin conditional thicknesses and surface specific weight of functional and technological parts of fuselage design, taking into account constructive and technological irregularities: edgings of cutouts, joints, connections, overlays; determination of mass, of the mass center position and inertial moments of the fuselage; analysis of calculation results.

The calculation of aircraft masses in the second approximation is based on the results of engineering analysis of their generalized models using the methods of structural mechanics and strength calculation [3], which need to introduce a number of assumptions to simplify the design scheme of the unit. The fuselage of the transport category aircraft differs from other units by a number of characteristic features directly related to its functional purpose—crew placement and payload, as well as the integration of aircraft units. The fuselage takes up, joins and balances significant power flows having a large overall height and a small cross-sectional area of the load-carrying elements, which is associated with the use of internal space to accommodate the payload. In addition, the fuselage is characterized by a large number of additional structural elements and cutouts. That causes the complexity of the direct application of analytical methods in its pure form, which requires their addition by statistical dependencies. Previously used methods of the fuselage mass calculating in the second approximation [1, 3] are associated with design calculations for the strength of the primary elements of its structure, while the internal force factors are determined on the basis of the beam calculation scheme. This approach does not allow to take into account the load from the cabin internal pressure, limiting its consideration by further checking the skin thickness of the fuselage cylindrical part and the introduction of an additional component in the general formula of the fuselage mass.

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Fig. 14 Method of fuselage mass-centering characteristics analysis scheme

The proposed method focuses on earlier stages of sketch design to clarify the geometric parameters and shapes of the fuselage and its parts. In the proposed method of calculation a finite element model of the fuselage, consisting of “shell” type elements is used. This allows to take into account the perception of internal pressure by the complex shape surface in combination

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Fig. 15 Fuselage analytical model

with mass and aerodynamic loads. The fuselage master geometry is the original geometric model. The associative connection of the finite element and parametric model of master geometry allows to automate the restructuring of the model when making changes and thus assess the impact of geometric parameters on the fuselage and its parts mass. At the same time continuity with previously applied methods concerning initial data, classification of fuselage construction masses and the account of additional structural and technological factors has been kept. The proposed method is implemented using Siemens NX integrated design systems and ANSYS engineering analysis and tested during the preliminary design of a local airliner (LA) and a light civil aircraft. The model for calculating the fuselage total stress–strain state characteristics was created in accordance with the above design scheme (Fig. 15). For creating the fuselage FEM (Fig. 16), triangular shell elements (CTRIA3) were used. The stress–strain state of the generalized model is calculated for a single shell thickness. To ensure the distribution of loads according to the previously developed layout of aircraft, the master-geometry model is supplemented by a sketch in the aircraft plane of symmetry, which contains the points of load application. In the process of FEM creating, the points of loads application are connected to the corresponding parts of the fuselage master geometry surface model by elements of the type RBE3, which provide load transfer without changing the model rigidity. The FEM is constrained at a point corresponding to the aircraft mass center associated with fixation of wing center section. The proposed mass-inertial characteristics’ calculating method is based on determining the characteristics of the total stress–strain state of the fuselage model at a single skin thickness (δ = 1 mm), for the considered calculation modes (obtained values of tensile, compressive and equivalent stresses). In Fig. 17 the nature of the

Fig. 16 Regional aircraft fuselage FEM

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Fig. 17 Nature of equivalent stress distribution

equivalent Mises stresses distribution on the fuselage surface at a single skin thickness for each i-th element of model is shown. To determine the maximum operating stresses, use the tool “Envelope” of the results panel, selecting the results of calculations as the source data. This tool allows to compare the stresses in each i-th element in different calculation cases and determine their maximum value. The fuselage is characterized by the occurrence of the greatest tensile stresses in panels with significant curvature caused by the placement of cockpit glazing, as well as in places of loads application from the landing gear and wing, but the maximum values of these stresses in different design cases are different. To calculate the required thickness of the conditional skin let us determine the maximum allowable stresses in terms of strength and durability to structures [6, 7]: p

σ1max ≤ [σ al ]; σ3max ≤ [σ cr ]; σemax ≤ [σ b ] i i i where [σ al ]—allowable material stresses at a given resource T = 80,000 flights; p [σ cr ]—critical stresses of loss of stability; [σ b ]—material strength limit; σemax = i max f · σei —maximum design stresses; safety coefficient f = 1.5. The required durability of structural elements N is calculated by the formula N = T · η, where T is the specified life time, η is the reliability coefficient determined in accordance with the applicable airworthiness standards. At the stage of the sketch design, the reliability factor η is taken to be equal to 4. For most of the transport category aircraft aluminum alloy is used as the main material of the fuselage. The durability of N of metal structural elements is described by the statistical dependence of the fatigue curve

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N · σ0m = C, where σ0 is the strength of the zero cycle; m and C are experimentally defined as constants that take into account the resource properties of the material and structural irregularities. The allowable stresses of the zero cycle [σ0 ], taking into account the provision of the specified life time are equal [σ 0 ] =

√ m C/N .

At an average value of operating stresses greater than zero, Oding’s formula is valid: √ σ0 = 2σ a σmax , where σa —amplitude stresses in the considered typical flight; σmax —maximum operating stresses. Then, the allowable stresses are equal to: [σ al ] = [σmax ] =

[σ0 ]2 . 2σ a

The critical stresses of stability loss for the fuselage panels, taking into account design and technological considerations when choosing the step of the reinforcing elements, are not less than[σ cr ] ≈ 0.8 · σ B . The required thickness of the conditional skin in each element is determined based on the maximum operating stresses at a single thickness of the skin and the maximum allowable stresses. With a constant shape of the skin to ensure strength, the thickness of the conditional skin in each i-th element must be increased in proportion to the ratio of the stresses acting in it to the maximum allowable, provided that the static strength and life time: δ1i ≥

σ1max i [σ a ]

; δ3i ≥

· f σ3max i [σ cr ]

; δei ≥

σemax · f i [σ b ]

The values of the conditional skin thickness in each element of the model will be determined by three strength criteria according to the above formulas, using the tool “Reduction” of the results panel and selecting the maximum operating stresses as source data. In addition, determining the required thickness of the conditional skin, it is necessary to take into account the technological limitations of the minimum thicknesses of materials. Using the tool “Envelope”, let us determine the maximum value of the required thickness of the conditional skin for each i-th element of the FEM:

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T δi ≥ max{δ 1i , δ3i , δei , δmin }.

In Fig. 17 the nature of distribution of the required thickness of the regional aircraft fuselage conditional skin, taking into account the loads in all considered design modes, which provides strength and manufacturability based on the considered criteria is shown. The calculation results coincide with the nature of material distribution: the greatest thicknesses are obtained at the locations of butt joints and panels with significant curvature. The thickness of the lower and side panels of the fuselage central part, as well as the side panels of the tail part is determined based on the maximum compressive stresses. The thickness of the upper panels of the central and tail sections is determined from the maximum tensile stresses. The FNS panels (except for the panels under glazing) and the FTS lower part are lightly loaded, their thickness is obtained for technological reasons. The obtained values of the conditional skin thickness correspond to the idealized regular skin. In fact, the fuselage contains many structural irregularities [3] (cutouts, connectors, joints), the mass of which at the stage of sketch design is determined on the basis of design experience or statistical dependencies. The presence of such irregularities and additional technological factors are taken into account when calculating the mass of fuselage structural parts. In general, the fuselage mass consists of the regular structure mass mr (which is determined based on the design calculation of the idealized model) and the additional mass mad , which takes into account structural and technological factors. Then the mass of a separate part of the fuselage mi can be given in the form m r = m ad + m i Each considered fuselage part of the regular structure mass is determined based on the obtained thickness of the conditional skin ( ) m r s.i = si · δi · ρi σb.i /σb. f , where si is the surface area; δ i is the average thickness of conditional skin; ρ i is the density of material; σ v.i is the tensile strength of material [3, 9] of this fuselage part; σ v.f is the yield strength of the material taken into account when calculating the thickness of the conditional skin. For the radar fairing, fuselage nose section panels, emergency departure hatch of the cockpit it is necessary to take into account the requirements for bird resistance. According to TsAGI research, the thickness of the skin of panels made of aluminum alloys located at an angle of more than 30° to the axis of fuselage should be at least 1.6 mm, provided the thickness of the fiberglass fairing radar then will be 3 mm. According to the manufacturers of aviation windows, the thickness of the windshields of the cockpit of transport category modern aircraft is 20…25 mm, side windows is about 10 mm. When calculating the mass of glazing an average value of

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15 mm is used. The thickness of the glass of the passenger compartment windows is 10 mm. The additional mass [2, 3] mad.i contains a mass of additional structural elements mad.s and additional mass due to structural and technological factors mkt : m ad.i = m ad.s + m st . Additional mass of cutouts mcut (their edging and “lids”), floors mfl , sealed bottoms msb, and butt joints mbj is related to the mass of additional structural elements mad.s : m ad.s = m cut + m f l + m sb + m bj . The additional weight of the cutout is calculated as the weight of the regular construction multiplied by the cutout factor: m cut.i = ki m cut.i . Cutout coefficients are selected according to recommendations based on the design of similar structures. Additional weights of the floor mfl , sealed bottoms msb , and butt joints mbj are determined by statistical dependences [3]: m f l = 4.48 · d 2f · λ f , m sb = 1.6 · ( pin f + 1) · d 3f ; m bj = 0.01275 · m 0 . The mass of the floor is distributed between the nose, center and tail sections in proportion to their surface area. The mass of pressurized parts is distributed between the landing gear bay, nose and tail sections. The mass of the butt joints is distributed between the load-bearing members of the fuselage mid-section, landing gear bay and the tail part. The additional mass due to structural and technological factors [3] mad includes the mass of mj joints, mfast fasteners, the additional mass due to the inaccuracy of the manufacture of parts mman and the limited range of semi-finished products mprod . m st = m j + m f ast + m man + m pr od . Fasteners mass include units’ parts connections mass. For the fuselage, this is the connection of the nose and tail sections with the central one, their additional weight is mj = 0.0667 mrs . The additional weight of the joints is caused by the need to make parts thicker and their overhang, it is approximately taken as mj = 0.1 mrs . The additional mass due to inaccuracy of the manufacture of parts mman and the limited range of semi-finished products mprod is determined by the statistical dependences: mman = 0.05 mrs , mprod = 0.05 mrk . Additional mass due to structural and technological factors is distributed between the fuselage nose, center and tail sections in proportion to their surface areas.

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Based on the obtained values of the fuselage parts masses, the thicknesses of the conditional skin δ i are calculated taking into account structural and technological factors. δ i = mi /(si ρ i ) and the surface density of the structure qi = mi /si of each part. The results of calculations are given in Table 2 and shown in Figs. 18, 19 and 20. The surface density of the structure qk.i is a criterion of mass efficiency of the structure, which allows to estimate the intensity of the loads taken up by the considered part of the fuselage, and the possibility of further reduction of its mass. Table 2 Fuselage structure parts mass analysis i

Fuselage part

mrs.i, kg

mad.s.i, kg

mi, kg

δsi, mm

1

Cowling

12.4 (62%)

7.5 (38%)

19.9 (2%)

4.82

8.68

2

FNS

46.3(36%)

84 (64%)

130.3 (11%)

3.66

9.89

3

Cockpit glazing

69 (44%)

86.3 (56%)

155.3 (13%)

4

Landing gear bay

20 (26%)

55 (74%)

75.5 (6%)

5

Windows

36.5 (47%)

41.9 (53%)

78.4 (6%)

6

FCS

190 (55%)

155 (45%)

345 (28%)

3.64

9.82

7

FCS load-bearing members

49.9 (54%)

42.5 (46%)

92.4 (7%)

6.94

18.70

8

Doors

6.62 (12%)

50.4 (88%)

57 (5%)

9.27

25.00

9

FTS

131 (46%)

155 (54%)

286 (13%)

3.45

9.32

33.8 7.35 21.5

qsi, kg/m2

84.4 19.80 53.8

Fuselage as a whole

562 (45%)

678 (55%)

1240

4.83

13.05

FNS

133 (42%)

187 (58%)

320 (26%)

6.58

17.78

FCS

286 (47%)

324 (53%)

611 (49%)

4.89

13.22

FTS

142 (46%)

166 (54%)

309 (25%)

3.64

9.83

Fig. 18 Fuselage generalized skin thickness

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Fig. 19 Comparative analysis of mass of regular structure, additional mass and surface density of fuselage (column numbers correspond to the number of the fuselage section i according to Table 2) Fig. 20 Comparative analysis of fuselage masses. 1—random, 2—FNS, 3—cockpit glazing, 4—landing gear bays, 5—windows, 6FCS, 7—FCS load-bearing members, 8—doors, 9– FTS

1 9 FTS

2 FNS

3

8 7

FCS

4 5

6

For parts of the fuselage with a high surface density (cockpit glazing, load-bearing members of the fuselage central part, doors, landing gear bay) it is advisable to further refine the design loads and optimize the design. Thus, the calculation of the mass of fuselage parts in accordance with the proposed method allows to determine the magnitude and spatial distribution of the fuselage mass in the second approximation. The obtained value of the mass of the fuselage and its components differs little from those calculated by the method of V. A. Kiselyov [3] and lies in the range of masses determined by the formulas of the first approximation [3] (V. M. Sheinin, O. A. Badyagin and V. I Kozlovsky), which indicates sufficient accuracy for the methods of the second approximation.

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Fig. 21 Determination of fuselage mass center and moments of inertia position

In the presence of more accurate data on the masses of prototypes, it is advisable to clarify the coefficients of cutouts and statistical dependencies. The obtained values of the fuselage parts masses were used to design its structure. The mass-inertial characteristics of the generalized fuselage FEM taking into account the spatial configuration and properties of the structural materials of its parts are determined using the integrated design system Siemens NX. The calculation is made element by element according to known dependencies: xm.c = ∑xi m i /m f ; ym.c = ∑yi m i /m f . In Fig. 21 the position of the fuselage mass center is shown. The output window contains the values of the inertia moments and the mass center coordinates, with help of color fuselage parts materials are marked. The obtained fuselage mass-inertial characteristics allow to specify the inertial loads in transient flight modes in the process of further design of the aircraft.

3.5 Features of Fuselage Middle and Tail Section Design The initial data for the construction of the transport aircraft fuselage theory are the required dimensions of cargo cabin, which are determined on the mission requirements basis for the developed aircraft, depending on its class and purpose. Parameters that affect the fuselage geometry are related to the flight and take-off characteristics of the aircraft, the scheme of the landing gear, wing, tail and cargo door.

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In this regard, the development of the fuselage geometric shape at the stage of sketch design is carried out in conjunction with the development of the layout of the aircraft, its wings, tail, landing gear, fairing and coordinated with their load-carrying structures and theoretical contours. The result of this development is a preliminary theoretical drawing of the fuselage. In Fig. 22 shown the fuselage theoretical contours in side and in plan views, as well as the layout of wing, landing gear, fin and stabilizer. FRP is the fuselage overall plane. FSP is the plane of the aircraft symmetry. Line 1 is the position of the ground in the parking for empty equipped aircraft. Line 2 is the position of the ground at landing. Line 3 is the position of the ground during the aircraft takeoff with the maximum takeoff weight. Line M is the line of maximum width. Line A is the line connecting ambiguously given fuselage surfaces. Lcd is the cargo cabin length. hth is the height of the threshold of cargo cabin. The line of intersection of FRP and PBF is the fuselage axis of symmetry and coincides with the X axis in the coordinate system of aircraft. TFL is theoretical floor line. Line T is the top line, line B is the bottom line of the fuselage theoretical contour. A-A (Fig. 23) is the initial section of the fuselage, which specifies its geometric shape in accordance with contour G, which is the bypass line of the combined maximum cross sections of the cargo range intended for placement in the aircraft cargo cabin. The minimum allowable clearances from the fuselage design bmin and hmin dictate the internal dimensions of the cargo cabin. Modern transport aircraft have a pressurized fuselage, and therefore inside the cargo cabin, at high altitudes, it is pressurized. In this case, the fuselage is designed for excess pressure, comparable to the pressure in the passenger compartment. This requirement determines the shape of the F-2 cross section in the form of a circle. A closed cylindrical skin with a circular cross-section is known to best absorb the air load acting from the inside, as the skin when cabin is pressurized carries only tensile

Fig. 22 Transport category aircraft fuselage. theoretical contours

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Fig. 23 Fuselage initial section

stresses and does not reload normal frames designed to maintain the fuselage shape and take up forces acting on fuselage. Although the F-2 cannot be made entirely in the form of a cylinder (in the F-2-toF-3 transition zone, the landing gear bay and the wing mounting compartment), the circle in cross section F-2 contributes to the requirements specifications of designing a fuselage with minimum weight and the highest life time. Figure 23 shows the cross section of the fuselage middle section made in the form of a round cylinder. Point O is the projection of the fuselage axis of symmetry (the line of intersection of FRP and FSP). The circle described around contour G from point O is a normal section F-2 with a fuselage diameter of 2Rf. Determining the value of Rf must be guided by its minimum to ensure the F-2 smallest cross section (to reduce weight and drag of fuselage) and the sufficiency of hb —overall height of the strong bulkhead that carries and transmits loads from wing and landing gear. As a result of the intersection of the FRP with the theoretical contour in the cross section of the fuselage middle part we obtain point M, which is maximum width point. The line of maximum width is the geometric location of points M. In the horizontal projection the line of maximum width coincides with the side line of theoretical contour F-2. In the vertical projection the line of maximum width completely or partially coincides with the fuselage overall plane. Points T and B at the intersection of theoretical contour with the plane of symmetry form the top and bottom lines of the fuselage theoretical contour. The internal dimensions of the cargo cabin in the F-2 normal section are determined by the overall height of the normal frames hb and the value hgp set in section A-A. However, the functionality of the cargo cabin of the aircraft to accommodate

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cargo depends not only on sections A-A and B-B, but on the scheme and design of the cargo door, which determine the geometric shape of the fuselage tail part and the required doorway for airdroping tasks. Lateral, upper and lower lines of the fuselage theoretical contour are given by a combination of lines and curves of the second order. The theoretical contour of the fuselage in cross section is given by a combination of second-order curves. Figure 24 shows a section of the fuselage middle section, made in the form of a circle with radius Rf . The curve bounded by points M and B, as known, can be given by two tangents and the discriminant F = ab/av. BB and MB are tangents to the curve at points B and M. Point A divides the segment of line MB in half. The condition of smooth connection of sections of curves MV and MN is the total tangent VMN passing through point M. The curve consists of four connected sections of curves MV, VM, MN and NM with discriminants F = 0.4142 provided that MO = VO = NO, is a closed arc of a circle with radius Rf = OM = OB = OH. When the value of discriminant F changes, the curvature of the CF line decreases or increases. The limit value of the discriminant is F = 0 and F = 1, which correspond to the segment of the direct CF and the combination of the segments of the direct CF and BB. The geometric shape of F-3 is given by lines B, H, M, A transforming the original section at the point of line rise B and is determined in accordance with the aerodynamic characteristics, the scheme of the cargo door and the landing gear. If the normal part of F-2 to intersection B-B is set, then further in F-3 the area of construction of the line points H will be limited to three tangents, one of which is a continuation of the line points H of the normal part of F-2. And the other two—by straight, drawn parallel to lines 2 and 3 at a distance K. Similarly, line 2 limits the theoretical circuit construction area of the landing gear fairing. The angle f is the angle of lower line rise of the fuselage tail part theoretical contour, that is the angle between the horizontal and tangent to the point of maximum curvature, in case if the line of points H is given by one curve, or the angle between horizontal and tangent (or straight) connects two curves. The geometry of the line Fig. 24 Fuselage cross section of mid-section in regular zone Fig. 25 shows the vertical and horizontal projections of F-2 and F-3. Line 2 is ground level at the aircraft landing, line 3 is ground level at takeoff with maximum weight. K is ground clearance

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of points H and the magnitude of angle f of the transport aircraft depend, on the requirement to provide the required slot in the fuselage tail section, or on the limit values of angle f at which shock stall on the lower surface of F-3 causes increase in fuselage induced drag. The upper line of the fuselage theoretical contour can be limited by two tangents, one of which is drawn through point B3 at angle b, and the other is a horizontal line at size h1 of the FRP. The angle b and the size h1 have their limit values, as their increase leads to an increase the fuselage midsection and its drag. The size h2 sets the upper position of the line of maximum width, which may be the center of a circle with radius R1, smoothly connecting with the lines of points T and B. On the horizontal plane of projection, the fuselage theoretical contour (side line), coinciding with the projection of the line of points M in the fuselage tail, can be given, (for example, a curve with discriminant Fm bounded by two tangents, the first of which comes from point M3 is elongation of points M of the fuselage middle section, and the second, connecting with the radius R2, intersects with the first one in size L1. The transition of the line H in the middle part of the fuselage from a straight line to a second-order curve leads to the cross sections transformation in the section B2–B3 due to the impossibility of their formation with one radius. If the surface F-2 is built in the form of a cylinder with radius Rf (Fig. 26) to the intersection with FTL, the lower surface should be placed by inscribed radius Rn between points H and P for taking up the floor lateral force from pressurization at the junction of floor and surface formed by radii of different sizes. The cheekbone formed at point P, due to the fact that the upper and lower surfaces in it have a separate tangent transformations should be provided ed in the F-3 into a smooth connection of the upper and lower surfaces of the fuselage. The smoothness of the junction of the fuselage lower and upper surfaces along the line A can be provided if they have a common tangent at point A in any intermediate section. Since the section of the VMA curve (section GG, Fig. 26) is an arc of a circle with radius Rm and center O, the tangent at point A will be perpendicular to the segment of line AT. The value of the discriminant of the arcs between points A and H can be a constant value F = 0.4142, or correspond to the graph of the values of Fn, given along the X axis by linear or curvilinear law. The overall height of the bottom of threshold frame h3 in section B-B at the junction of F-2 and F-3 is selected from the conditions of strength and structural layout of the components of the cargo door. The line of points M on the side view is given by a smooth connection with the FRP (Fig. 1.4). The line A is given by a smooth connection with TLP and a line of points M. Provided the minimum surface area of F-3, the narrowed fuselage in the plan view begins at the threshold of the cargo cabin. In this case, if the section of line B is a straight line, part of the curve of line A to a given point I is a derivative, i.e. the geometric place of origin of radii M. The tangent at point I is one of the initial data for constructing a line of points M.

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Fig. 25 F2 and F3 vertical and horizontal projections

Thus, the surface in the fuselage tail, bounded by lines M and A, is given by the arcs of circles with radius Rm. The surface bounded by line B and line M to point I is defined by the arcs of circles with radius Rm. The surface bounded by the lines H, A and M is given by second-order curves with a discriminant F = 0.4142. The surface bounded by line B and line M from point I is given by second-order curves with a discriminant F = 0.4142. The theory in Figs. 25, 26 and 27 is given on the condition of minimum cross sections and is not suitable for schemes of cargo doors with large mobile units, which are removed when opened inside the fuselage. Using cargo door schemes with a movable pressure bulkhead (pressurized panel, sealed fairings, pressure ladder) inside a cargo cabin such theory allows to set geometrical surface F-3 in a pressurized zone with sections in the form of a circle or close to it. It also contributes to better perception of the fuselage design overpressure and improved aerodynamic performance. In unpressurized zone the necessary space for placement of mobile units of the cargo door when it is opened is formed by setting the corresponding schedules of discriminants Ft and Fb. In this case, the increase in the surface of fuselage is compensated by simplifying the compartment of cargo door fairings and improving its functionality (for example, placing a single fairing inside the tail of the fuselage when opening the cargo door). Such a cargo door scheme and, accordingly, the theory of the tail section is the most common today and is very often used on modern transport aircraft due to the relative simplicity of its design (C-17, A-400M, KC-390, etc.).

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Fig. 26 Cheekbone transformation in area F3

Fig. 27 Transition in zone F3 from the curves given by circles with radius Rm to the curves of the second order, given by the discriminant F = 0.4142

Figures 28 and 29 show the theory of F-3 with the possibility of transforming its intersections by setting the pattern of discriminants Ft and Fb change. In area L1, upper surface F-3 can be defined by the radii Rf. The required surface F-3, which meets the specified mission requirements and the selected scheme of the cargo door, is formed by selecting the configuration of theoretical lines B, H, A, M and lines Ft and Fb of graph of cross sections’ discriminants.

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Fig. 28 Fuselage tail theory for cargo door scheme with solid sealed fairing

Fig. 29 Zone F3 sections transformation for scheme of cargo door with solid sealed fairing

From Figs. 28 and 29 it is seen that surface F-3, bounded by lines H, A and M, in cross sections is formed by second-order curves with discriminant Fb. The surface bounded by lines A and M is formed by radii RM, and in the area L1 RM = Rf. The surface bounded by lines M and B in the section L1 in cross sections is formed by the radius Rf and then by the second-order curves with discriminant Rb.

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The implementation of the method is an example of constructing the theory of the fuselage tail part of a light transport aircraft shown in Fig. 31. Namely: lines B, H, A and M, then plots of cross-sections on each frame are represented (Fig. 30). This is done in order to pre-estimate the cargo door opening in the tail, as well as to obtain initial data to create a master-geometry. As a rule, this stage ends with the release of a theoretical drawing (TD) which indicates the minimum necessary information to create a master geometry. The peculiarity of this drawing is that in addition to the graphics and technical requirements, it must specify the parameters of the main lines and contours, which usually look like parametric tables in the drawing field.

Fig. 30 The light transport aircraft fuselage. Tail section theory drawing

Fig. 31 Light transport aircraft fuselage. Tail section master-geometry

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Creating a master geometry (Fig. 31), the source surfaces are built from baselines in accordance with TD. But, as a rule, it is not possible to set all the parameters of the created surface, only specific reference lines. Therefore, such a surface usually has small distortions in the form of depressions and bulges. In this case, the theory requires editing and adjustment, with possible changes to the PM. Otherwise, further research using this model will have a large error and inaccuracy.

4 Conclusions The method of integrated design of the transport category aircraft fuselage sections using computer systems is proposed. The analysis of requirements of normative and technical documentation, features of design and design methods of transport category planes’ fuselage sections is carried out and necessity of actualization of fuselage characteristics design and calculation methods with use of parametric models and systems of integrated design CAD/CAM/ CAE/PLM is revealed. The purpose and tasks of the research are formulated. 3. The method of integrated design of TCA is developed and theoretically substantiated. Within the framework of the proposed method, parametric models of master geometry, aerodynamic flow and mass-inertial characteristics of the fuselage were created, taking into account the design features of TCA. The proposed method is used to study the influence of FNS geometric parameters on the aerodynamic and mass characteristics of the TCA fuselage, the efficiency of working with parametric models is approved. The rational configuration of TCA is determined taking into account the requirements for layout and limits of viewing from the cockpit. The choice of FNS parameters in the preliminary and sketch design of a promising aircraft for local airlines is justified, which allowed to implement and test the suitability of the proposed method for use in creation of new competitive aircraft. The used method of integrated fuselage design allowed to determine the FNS rational configuration and increase the fuel efficiency of LA by 6.4%, reduce the aerodynamic drag of the fuselage by 10%, increase the viewing angle from the cockpit by 10% compared to the previous design and provide the compliance with the requirements of the current NTD, as well as to determine the mass-inertial characteristics of the fuselage and its parts taking into account the FNS features and form a list of cockpit equipment that will meet the requirements of NTD on flight safety LA taking into account the operating conditions and modifications. The configuration of the aircraft fuselage nose section for local airlines has been developed, which allows to provide modern requirements for cockpit equipment and layout, low fuselage impedance and high aerodynamic quality and fuel efficiency in cruising mode at speeds up to 850 km/h (M = 0.8). As a result of testing using other methods and parameters of existing aircraft, the accuracy of the results obtained using the proposed method at the level of 5% was confirmed.

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References 1. Yeher, S.M., Mishyn, V.F.: Aircraft Designing: Course Book for Universities, Lyseitsev N.K. et al.: edited by S.M. Yeher. 3rd ed., revised and added M.: Mechanical Engineering, 616 p. (1983) 2. Torenbik, E.: Designing of Subsonic Aircraft. (Torenbik, E., Holubkov, E.P. Trans. from English). Mechanical Engineering, 648 p. (1983) 3. Sheynin, V.M.: Weight Designing and Efficiency of Passenger Planes [Text]: reference book / V.M. Sheinin, V. I. Kozlovskyi. – 2nd ed., revised and added M.: Mechanical Engineering, 552 p. (1984) 4. Certification Specifications and Acceptable Means of Compliance for Large Airplanes. EASA, 1135 p. (2019) 5. Kiva, D.S.: Scientific Basis of Integrated Designing of Transport Category Aircraft: Monograph. In 3 parts / Kiva, D.S., Hrebenikov, O.G. Kharkiv: KhAI, Part 2, 326 p. (2014) 6. Novozhylov, G.V.: Theory and Practice of Passenger Aircraft Designing. Nauka, 439 p. (1976) 7. Hrebenikov, O.G.: Methodology of Integrated Designing and Modeling of Assembled Aircraft Structures. Kharkiv: KhAI, 532 p. (2006) 8. Kholiavko, V.I.: Calculation of Aerodynamic Characteristics of Airplane. Course Book in 2 Parts. Part 1. Kharkiv: KhAI, 72 p. (1991) 9. Hrebenikov, O.G., Donets, O.D., Trubaiev, S.V., Chumak, A.S.: Method of General Designing of Regional Passenger Aircraft. Open Information and Computer Technologies: coll. of science works. National Aerospace University “Kharkiv Aviation Institute”, Issue 85. Kharkiv, pp. 4– 31 (2019). https://doi.org/10.32620/oikit.2019.85.01. Airworthiness standards for transport category aircraft (AP-25). M.: MAK, 322 p. (2009) 10. Bratukhina, A.G. (ed.): Information Technologies in Science-Intensive Mechanical Engineering: Computer Support of Industrial Business. Kyiv: Technika, 728 p. (2001) 11. Balabuiev, P.V., Bychkov, S.A., Hrebenikov, O.G., et al.: Fundamentals of General Designing of Aircraft with Gas Turbine Engines: Tutorial. Kharkiv: KhAI, 815 p. (2015)

Blind Evaluation of Noise Characteristics in Multichannel Images Victoriya Abramova , Sergey Abramov , Klavdiy Abramov, and Benoit Vozel

1 Introduction High resolution and the ability to cover large areas have led to the widespread use of remote sensing in agriculture, forestry, environmental monitoring, hydrology, oceanography, geology, mapping, sub-surface probing, meteorology, etc. [1–3]. There are quite many types of remote sensing systems which can be classified according to the used spectral range, the type of radiation detector, or the sounding method (active or passive). In addition, there are spaceborne and airborne remote sensing systems [1, 3]. However, regardless of the type of remote sensing system, a common feature of the obtained images is the presence of noise and distortions which may worsen both the visual quality of these images and the results of their processing. Characteristics of noise should be taken into account at different stages of image processing, such as compression [4, 5], classification [6], edge and object detection [7], etc. In some cases, it is advisable to preliminarily suppress the noise using special filters [8], however, to select a proper filter and adjust its parameters, it is necessary V. Abramova · S. Abramov · K. Abramov (B) Kharkiv Aviation Institute, National Aerospace University, Kharkiv, Ukraine e-mail: [email protected] V. Abramova e-mail: [email protected] S. Abramov e-mail: [email protected] V. Abramova Center for Physical Sciences and Technology, Vilnius, Lithuania B. Vozel University of Rennes 1, IETR UMR CNRS 6164, 22305 Lannion Cedex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_4

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to know some statistical characteristics of the noise. For real-life systems, it is often impossible to predict the characteristics of noise a priori, since they are a function of a large number of random factors [1, 3]. Therefore, information about the noise characteristics is usually extracted directly from a processed image using special blind (automatic) methods [1, 9, 10]. There are several nuances that complicate the task of blind evaluation of noise parameters. Firstly, noise often has a complex structure, characterized by the presence of signal-independent and signal-dependent components with a clear predominance of the latter. Secondly, noise can be spatially correlated to a large extent [10–12]. Although a fairly large number of blind methods for noise parameters evaluation have been designed to date [9–21], none of them is capable of providing an acceptable estimation accuracy [22] and performance in all practical situations. Thus, the task of developing new methods and improving the existing solutions does not lose its relevance. The chapter is organized as follows. First, we give the considered noise models and the accuracy criteria used. Next, we present the basic method and its performance analysis. Then, we introduce the modification of this method for multichannel images. Finally, we discuss the peculiarities of application of this modification to hyperspectral images and present the results obtained for test and real-life remote sensing images.

2 Noise Models and Accuracy Criteria A remote sensing image can be described by the following simplified model g(m, l) = n sd (m, l; s(m, l)) + n a (m, l),

(1)

where s(m, l) is an original (noise-free) image; n a (m, l) is a normally distributed additive noise with zero mean and variance σa2 ; n sd (m, l) is a functional describing the influence of signal-dependent noise. This model can be clarified depending on the imaging system used. Radar images are usually corrupted by the mixture of multiplicative and additive noise, so the model will be g(m, l) = n μ (m, l; s(m, l)) + n a (m, l),

(2)

where n μ (k, l) is a normally distributed multiplicative noise with unity mean and relative variance σμ2 . Optical and hyperspectral images are typically distorted by a mixture of additive and quasi-Poisson noise, so the model given below can be used g(m, l) = n p (m, l; s(m, l)) + n a (m, l),

(3)

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where n p (k, l) is quasi-Poisson noise with the amplification factor k. A commonly used approach to mixed noise characteristics evaluation is to build a scatter-plot of local variance and mean estimates obtained for a certain set of image blocks and fit a regression polynomial into it [18, 19]. The polynomial parameters correspond to the estimates of noise characteristics. For the considered noise models, the first-order polynomial should be used. In this case, the first-order polynomial coefficient corresponds to the signal-dependent noise parameter estimate, and the zero-order coefficient corresponds to the additive noise variance estimate. The difference between the models (2) and (3) is in the used scale along the horizontal axis. For the model (3), the regression polynomial is fitted into the scatter-plot of local estimates of variance and mean, whereas for model (2) the scatter-plot of local estimates of variance and squared mean is used. In this chapter, we will focus on hyperspectral images; so, the model (3) will be used to describe noisy images. To quantitatively characterize the accuracy of the obtained noise parameters estimates, the following criteria have been used in our earlier studies. 1. Estimation bias a = σˆ a2 − σa2 ,

(4)

where σˆ a2 is the additive noise variance estimate and σa2 is the additive noise variance true value; k = kˆ − k,

(5)

where kˆ is quasi-Poisson noise amplification factor estimate and k is the true value of quasi-Poisson noise parameter. 2. Relative estimation error δa = |a |/σa2 ,

(6)

δk = |k |/k,

(7)

Number of “acceptable” estimates. By “acceptable” we understand the estimates allow 2allow allow located within the required range σa2allow min · · · σa max (kmin · · · kmax ) which is set depending on the noise type and the application where the estimates are supposed to be used. It has been shown in [17] that if relative estimation errors of both noise components parameters do not exceed 0.2, the difference in filtering efficiency compared to the case when the true values of noise parameters are used is visually unnoticeable. That is why, further we will consider acceptable all the estimates in the range σa2 ± 0.2σa2 and k ± 0.2k.

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3 Description and Performance Analysis of the Basic Method As it has been mentioned earlier, nowadays there are quite many known blind noise characteristic evaluation methods. Most of them can be conditionally divided into three groups: (1) methods operating in spatial domain [9, 13, 15]; (2) methods operating in spectral domain [10, 11, 20–22]; (3) methods based on maximum likelihood estimation of image and noise characteristics [18, 19]. The first group of methods is characterized by high operating speed and low sensitivity to possible noise spatial correlation. Yet, such methods are significantly influenced by the image content, which often leads to essential overestimation of noise parameters’ values for highly textured images. Methods from the second group are also fast enough, but they are able to provide a noticeably better estimation accuracy of noise parameters for highly textured images compared to the methods from the first group. However, such methods tend to essentially underestimate noise parameters’ values if noise is spatially correlated. Methods from the third group demonstrate the highest estimation accuracy among the existing alternatives, but they require complex computations which result in their low operation speed. Figure 1 presents the estimates of noise parameters obtained for the test images from TID2008 [23] database using several known methods belonging to the second group [20–22] and to the third group [18]. The image indices in the database (DI) are shown along the horizontal axis (there are 25 test images in the database). There are three estimates for each image index, corresponding to red, green and blue color components. The solid bold line shows the true noise parameters values (k = 1, σa2 = 30, the noise is spatially uncorrelated), the dashed bold lines determine the boundaries of the range of acceptable estimates. All the aforementioned methods have been applied component-wise. As it is seen, the method [22] provides very unstable results, only a small part of estimates is within the required limits, other estimates are significantly overestimated or underestimated. In addition, the values of some estimates are negative, which makes no physical sense. The results provided by the method [20] are more stable, however there are quite many estimates out of the range as well. An interesting observation is that this method mostly overestimates the additive noise variance and underestimates the quasi-Poisson noise parameter correspondingly. The best accuracy is provided by the method [18] based on maximum likelihood estimation of image and noise characteristics. Only a few estimates are out of the required limits while the rest are quite close to the true values. The accuracy of the method [21] is also rather high and only a bit lower in comparison to the method [18]. Since the operating speed of method [21] is several times higher in comparison to the method [18], it seems to be more suitable for practical applications, but it is desirable to increase the accuracy of this method.

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Fig. 1 Additive (a) and quasi-Poisson (b) noise parameter estimates obtained by different methods for TID2008 test color images

The method [21] contains the following three main stages. 1. Image segmentation and cluster mean estimates ( Iˆcl c ) obtaining. An image is divided into overlapping blocks of size 8 × 8 pixels and the mean estimates Iˆ are calculated for each block. Then the maximal Iˆ and minimal Iˆ locm

locmax

locmin

mean estimates are determined and the range Iˆ locmin ... Iˆ locmax is divided into n within the c-th (c = 1, n) sub-intervals (by default n = 10); the blocks with Iˆ locm

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sub-interval are assumed to belong to the c-th cluster. Cluster mean estimates Iˆcl c are calculated as average intensity values within a cluster: Iˆcl c = Iˆ locmin + Iˆ

− Iˆ

(c − 0.5) locmax n locmin . 2. Obtaining cluster variance estimates (σˆ cl2 c ) using the method [24]. Briefly, this process consists in the following. At the first stage, the modified kurtosis [24] values are calculated for each image block and the blocks belonging to quasihomogeneous image regions are separated based on the analysis of these values. At the second stage, the statistics of 2D discrete cosine transform (DCT) coefficients are analyzed for different spatial frequencies according to the technique proposed in [25] and the final value of noise variance within the cluster is determined. 3. Line fitting through the obtained cluster centers. The fitting is carried out using double weighted least mean squares (DWLMSC) method with restrictions imposed on non-negativity of both estimates [26]. The cluster weights are calculated proportionally to the final cluster sizes. If we study the estimation results presented in Fig. 1 more in detail, we can notice that the biased estimates of noise parameters are observed for highly and medium textured images (## 1, 5, 13, 21). This happens due to the influence of image content that is not completely eliminated because of the imperfection of the homogeneous regions detector used.

4 Modification of the Basic Method for Color Images Modern imaging systems are usually multi-channel, where one snapshot is formed in different frequency sub-bands. The easiest example of a multi-channel image is a color image containing three channels corresponding to different color components (usually red, green and blue). The signal components of multi-channel images are usually characterized by a high level of inter-channel correlation. Cross-correlation factor is about 0.7 … 0.8 for color images and it can be even closer to unity for hyperspectral images [1, 2, 8, 9]. Nevertheless, noise in image components is usually uncorrelated. So, it is possible to decrease the influence of image content and, thus, increase the estimation accuracy of the method [21] by carrying out joint processing of images obtained in different channels (bands). The main idea of using inter-channel correlation for increasing the estimation accuracy of noise parameters consists in the following. Let us consider two channels of a multi-channel image corrupted by the additive noise. These channel images can be described as follows (the subscripts correspond to the channel index): g1 (m, l) = s1 (m, l) + n a1 (m, l) g2 (m, l) = s2 (m, l) + n a2 (m, l),

(8)

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Let us subtract one channel image from the other one: g1 (m, l) − g2 (m, l) = s1 (m, l) + n a1 (m, l) − s2 (m, l) − n a2 (m, l).

(9)

Taking into account high level of inter-channel correlation, we can assume s1 (m, l) ≈ s2 (m, l) and (9) may be transformed into g1 (m, l) − g2 (m, l) = s(m, l) + n a1 (m, l) − n a2 (m, l)

(10)

where s(m, l) is residual information component which is considerably less intensive than s1 (m, l) and s2 (m, l), and its influence can be neglected. This means that after subtraction, we obtain almost “clean” noise and it is possible to evaluate its characteristics more accurately due to less influence of image content. When random variables are subtracted, their variances should be added, so 2 2 var(g1 (m, l) − g2 (m, l)) = var(n a1 (m, l) − n a2 (m, l)) = σa1 + σa2 .

(11)

Therefore, if we evaluate noise variance in the difference image (10), the obtained noise variance estimate will approximately correspond to the sum of noise variance estimates of the channel images and the only task is to divide its parts between the channels correctly. However, the latter is not as easy as it may seem. If we have only two channels, the only our option is to divide the obtained sum between them equally. If noise variances in channel images were initially different (which is typical of real-life multichannel images), this will lead to biased variance estimates for both channels, and the more the initial difference is, the more this bias is going to be [27]. To be able to determine the contribution of each channel to the obtained sum of variances more accurately, we need at least three channel images. By obtaining difference images for all the possible pairs of channel images (without repeating) and 2 2 2 , σˆ a13 , σˆ a23 ) we can obtain the following system evaluating variances in them (σˆ a12 of linear equations: ⎧ 2 2 2 = σˆ a12 ⎨ σˆ a1 + σˆ a2 2 2 2 . σˆ + σˆ a3 = σˆ a13 ⎩ a1 2 2 2 σˆ a2 + σˆ a3 = σˆ a23

(12)

Then, after this system has been solved, we obtain the estimates of noise variances in each channel image:   2 ⎧ 2 2 2 + σˆ a13 − σˆ a23 ⎨ σˆ a1 = 0.5σˆ a12  2 2 2 . − σˆ a13 + σˆ a23 σˆ 2 = 0.5σˆ a12  ⎩ a2 2 2 2 2 σˆ a3 = 0.5 σˆ a23 − σˆ a12 + σˆ a13

(13)

If an image is corrupted by signal-dependent or mixed noise, the idea of taking into account the inter-channel correlation is the same. The only difference is that we

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should evaluate noise variance separately for each cluster, since within a cluster the noise can be considered additive. So, to implement the idea of using the inter-channel correlation for the mixed noise parameters evaluation, the proposed modification of the method [21] involves the following steps. 1. Choose three channels of a multi-channel image (in case of a color image all its channels are used). 2. Provide clustering of the channel images [21] and obtain the cluster mean estimates (I1loccl1 , I2loccl2 , I3loccl3 , where cl1, cl2, cl3 are cluster indices for each channel image; cl1 = 1…n1 , cl2 = 1…n2 , cl3 = 1…n3 , where n1 , n2 , n3 are numbers of clusters for each channel image, respectively). 3. Form the difference images for all possible pairs of channels without repetitions. 4. Apply the technique from [24] to each of the difference images to obtain the 2 2 2 , σˆ cl13i , σˆ cl23i ) (boundaries of clusters were detercluster variance estimates (σˆ cl12i mined at step 2). Since for different channel images the cluster boundaries and their centers can significantly differ, the cluster variance estimates are determined separately for each clustering map. After this stage, one obtains nine variance estimates for each cluster (three estimates obtained for three difference images calculated for three different clustering maps.) 5. To obtain the cluster variance estimates that correspond to component images initially, it is necessary to compile a system of linear equations using estimates obtained from one clustering map: ⎧ 2 2 2 = σˆ cl12i ⎨ σˆ cl1i + σˆ cl2i 2 2 2 , σˆ + σˆ cl3i = σˆ cl13i ⎩ cl1i 2 2 2 σˆ cl2i + σˆ cl3i = σˆ cl23i

(14)

where i is an index of clustering map that also corresponds to the number of channels for which it has been obtained. The solution of this system looks as follows   2 ⎧ 2 2 2 + σˆ cl13i − σˆ cl23i ⎨ σˆ cl1i = 0.5σˆ cl12i  2 2 2 . (15) − σˆ cl13i + σˆ cl23i σˆ 2 = 0.5σˆ cl12i  ⎩ cl2i 2 2 2 2 = 0.5 σˆ cl23i − σˆ cl12i + σˆ cl13i σˆ cl3i 6. The final estimates of noise characteristics for initial images are obtained as parameters of a regression line fitted into the scatter-plot of cluster variance and mean estimates. It is worth paying attention that for each channel, the cluster variance and mean estimates obtained using the corresponding clustering map are used, e.g. for the first channel the reference points with coordinates (I1loccl1 , 2 σˆ cl11 ) are used, for the second channel the coordinates of needed reference points 2 2 ), and for the third one—(I3loccl1 , σˆ cl33 ). Similarly to the basic are (I2loccl1 , σˆ cl22 method [21], fitting is implemented using the DWLMSC method.

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5 Performance Analysis of the Modified Method for Color Images Figure 2 presents the noise parameters estimation results obtained for TID2008 color images using the basic method [21] and its proposed modification. For comparison, the results for the method [18] are given as well. Visual analysis of the presented plots shows that the proposed method provides the most accurate estimates of noise parameters for many images (both for signal-independent and signal-dependent components), yet for some images it can essentially overestimate or underestimate noise parameters, whereas the other two methods provide the estimates within the required range. In general, the number of noise parameters estimates obtained using the proposed method that are outside the required limits is 18 for the additive noise component and 3 for the quasi-Poisson noise component. For the basic method [21] these numbers are 34 and 3, respectively. Thus, according to this estimation criterion, the accuracy of the modified method is considerably higher and is rather close to the accuracy of method [18], for which the numbers of outranged estimates are 14 and 1 for additive and quasi-Poisson noise components, respectively. However, the plots for all the three considered methods are located quite close to each other, so it is hard to properly estimate the difference in methods’ performances relying on visual data only. So, we have calculated mean (median) noise parameter estimates by averaging (or median finding, respectively) the results for all images from TID2008 database for each method. These results are presented in Table 1. To assess how far the values of individual estimates are located from the averaged value, standard deviation (STD) and median absolute deviation (MAD) estimates are presented as well. As we can see from analysis of data presented in Table 1, the proposed method has essentially higher accuracy of additive noise parameters estimation. This is confirmed by both lower bias of the averaged estimates and their smaller STD and MAD values that indicate a higher stability of the proposed method. The biases of averaged estimates of the quasi-Poisson noise components turned out to be a bit higher for the proposed method, yet they are still within the required range and are quite close to the true value. If we compare the estimation results for the proposed method and the method [18], we see that the results for these methods are very close although the accuracy of the proposed method is still a bit lower. But the difference in accuracy can be compensated by the operating speed of the proposed method which is by several times better.

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Fig. 2 Additive (a) and quasi-Poisson (b) noise parameter estimates obtained for TID2008 test color images by the basic method [21], its proposed modification, and method [18]

6 Application of the Modified Method for Hyperspectral Images The modified noise characteristics evaluation method described above requires three channel images to be processed jointly. In the case of a color image, all its channels are used, however, if an image has a larger number of channels, various options for selecting the three channels for joint processing are possible. Obviously, the result of noise characteristics evaluation will depend on the set of images processed jointly.

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Table 1 Averaged mixed noise parameter estimates for different methods for TID2008 database Method

Estimate

Mean

mean

STD

Median

Method [21]

σˆ a2 kˆ

32.78

2.78

10.53

31.47

Proposed

σˆ a2 kˆ

31.97

Method [18]

σˆ a2 kˆ

28.8

0.998 0.994 1.01

0.002

0.107

1.97

7.29

0.006 –1.2 0.01

0.997 28.36

median

MAD

1.47

7.26

0.003 −1.64

0.071 4.57

0.078

0.98

0.02

0.05

5.53

30.77

0.77

3.94

0.07

1.01

0.01

0.05

In other words, it is important to know exactly which components of a multi-channel image are used to form a three-channel group. Modern remote sensing imaging sensors usually have from several tens to several hundred channels [2, 28–31], which makes the task of forming the three-channel groups for joint processing quite complex. In particular, hyperspectral images obtained by AVIRIS [28, 29] contain 224 channels each. Therefore, for each channel image, there are C2223 possible options for selecting two additional images for joint processing (C is the number of combinations), which is 24753 options. Of course, enumeration of all possible options is too time and resource consuming and is unacceptable for practical applications. Hence, there should be some criteria or rules which can be used while selecting channel images for joint processing. One such criterion may be the inter-channel correlation coefficient between the jointly processed images. However, it is not clear how much this parameter affects the accuracy of noise characteristics estimation and it is needed to clarify this question before applying the modified method described in Sect. 4 to hyperspectral images. To do this, we have carried out the experiment which is described below. We took two hyperspectral images obtained by AVIRIS system: Cuprite, some channels of which are shown in Fig. 3, and Lunar Lake, some channels of which are shown in Fig. 4, respectively. As it is seen, these images have a rather complex structure and contain many edges, small details and texture areas. At the same time, it is clearly noticeable that the information components of different channels have a sufficiently high degree of correlation, while the level of noise for different channels is significantly different, and for some channels this level is quite high. According to the results of studies carried out by researchers from different scientific groups [2, 3, 18], in most cases the highest values of inter-channel correlation coefficients are observed for adjacent channels of a multichannel image. Therefore, at the first stage of our research, we decided to form the three-channel groups out of neighboring channels. That is, to obtain an estimate of noise characteristics for the n-th channel, where n = 2…223, the channels with numbers n−1, n, n 1 were used; for the channel #1, the group of images with indices 224, 1, 2 was used, and for the channel #224, the group contains component images with indices 223, 224, 1.

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Fig. 3 Channels of hyperspectral image Cuprite obtained by AVIRIS system: #1 (a), #122 (b), #161 (c), #220 (d)

At the second stage, the influence of the inter-channel correlation level of the channel images included in the group on the accuracy of noise characteristics estimation has been investigated. Two variants of group formation have been considered: “the best”, in which the group included images with the maximum cross-correlation coefficients, and “the worst”, in which the group included images with the lowest values of cross-correlation coefficients.

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Fig. 4 Channels of hyperspectral image Lunar Lake obtained by AVIRIS system: #2 (a), #121 (b), #163 (c), #204 (d)

In the previous sections, it has been mentioned that in most practical situations the method [18] based on maximum likelihood estimation of image and noise characteristics demonstrates the highest accuracy among the existing methods, while its main drawback is in low operating speed due to high computational complexity. Since for real-life images, we do not know the true values of noise parameters, we decided to use the results obtained by the method [18] as a reference when analyzing the

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results obtained by the proposed method with different variants of channel groups formation. Figure 5 shows the inter-channel correlation coefficients for all the channels of images Cuprite and Lunar Lake; channel indices are given along the horizontal axis. As it is seen from the data presented, in most cases, the inter-channel correlation coefficients are quite high, and their highest values are observed mainly for neighboring channels. However, there are some channels that are characterized by very low cross-correlation coefficients with others: these are groups of channels in the vicinity of channels #120 and 160, which are characterized by the highest levels of noise and distortion. An interesting fact is that the inter-channel correlation coefficients for the images Cuprite and Lunar Lake differ significantly, but the bands of “bad” channels are present for both images and almost coincide. Figure 6 presents the histograms showing how many times each of the channels was used to form groups for joint processing in “the best” and “the worst” variants. As it is seen, in “the best” variant, most of the channel images were included in three groups. In majority of cases one of these groups was created for evaluation of noise characteristics in the current channel and the other two were formed out of the channels located in its closest neighborhood. Channel #224 turned out to be the most “popular”. It was included in six groups, which is the maximum value for “the best” variant. Let us now consider the histogram for “the worst” case, presented in Fig. 6b. Obviously, the numbers of times each of the channels had been a part of a threechannel group are much more scattered than for “the best” case. There are two obvious “leaders”, these are channels ## 161 and 164, which were included in almost all the groups. As mentioned earlier, the images obtained in these channels are characterized by the lowest quality, which is why they have the lowest correlations with the rest of channels of the considered hyperspectral images. Figure 7 shows the estimates of noise parameters in the image Cuprite, obtained according to “the worst” variant of groups formation; for comparison, the results for the reference method are given as well. The first observation that follows from the data presented is that both plots contain gaps. For the reference method, this is due to the logarithmic scale used, which ignores zeros and negative values. For the investigated method, the gaps mostly appear for the channels where the method could not detect enough homogeneous regions to provide a trustworthy estimate of the noise parameter. Another interesting observation is that there are only about a half of channels where the investigated method provided nonzero estimates of noise parameters, which, however, differ from the ones obtained by the reference method by 1–2 orders. Let us consider the results of noise characteristics evaluation obtained according to “the best” variant of groups formation. These results are presented in Fig. 8 jointly with the results for the “adjacent channels” variant. In contrast to the results for “the worst” variant, the plots in Fig. 8 contain almost no gaps. A few ones that can be found are for the channel images where the estimates of signal-dependent noise parameter turned out to be zero.

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Fig. 5 Inter-channel correlation coefficients for hyperspectral images Cuprite (a) and Lunar Lake (b)

Comparing the results for the two variants “adjacent channels” and “the best”, we can see that their plots almost coincide and both are located quite close to the reference method plot. There are a few channels where significant differences compared to the reference method are observed, however, the slightly lower accuracy of the investigated method can be compensated by its significantly (several times) higher speed. On the one hand, the obtained results indicate that the way how a group of images to be processed jointly is formed can have a significant impact on the accuracy of the investigated noise characteristics estimation method. If the cross-correlation of the images included in the group is low, and the images have a complex structure (contain many textures and small details and a small percentage of homogeneous areas), a significant decrease in the accuracy of the method is possible, up to the total

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Fig. 6 Histograms of channel usage for the image Cuprite for “the best” (a) and “the worst” (b) group forming variants

loss of its operability. It means that before forming groups, it is needed to calculate and analyze the inter-channel correlation coefficients. On the other hand, if the number of channel images is large, the process of calculating their inter-channel correlation coefficients can take a lot of time, which will decrease the operating speed of the method. This is extremely undesirable, since high operating speed is one of its main advantages. A compromise for practical use can be seen in forming groups according to the “adjacent channels” variant without any additional analysis of cross-correlation coefficients between channels. As data presented in Fig. 8 show, there is only a slight difference in the accuracy of this variant compared to the one where the groups are formed out of the images with the highest values of inter-channel correlation. This

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7 Conclusions In this chapter, the problem of blind noise parameters evaluation in multichannel images is considered. The main attention is paid to remote sensing images obtained by modern imaging sensors. The peculiarities of these images are the large number of channels, which can reach several hundred, and complex nature of the noise, which can contain both signal-independent and signal dependent components with significant predominance of the latter. All this requires the methods used for noise parameters evaluation in these images to be robust and fast enough.

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One of the methods promising in terms of meeting these requirements is the method based on the analysis of parameters of DCT coefficients’ distributions obtained for different spatial frequencies in blocks 8 × 8. In order to increase the accuracy of this method, its modification for multichannel images is proposed. The main idea of this modification is to process three channel images jointly aiming to decrease the influence of image content. The effectiveness of the proposed modification is confirmed by the numerical simulation results obtained for a large test image database. The intricacies of the application of the modified method to hyperspectral images have been considered. The main difficulty consists in the existence of many ways to form a group of three channels for joint processing. Three possible variants of group formation have been considered: “adjacent channels”, where a group has been formed out of neighboring channels, “the worst” and “the best”, where a group has been formed out of channels with the lowest or the highest values of inter-channel correlation coefficients, respectively. It has been shown that cross-correlation level of the images processed jointly affects the result of noise parameter estimation and the higher this level is the better. For practical use, it is often enough to use the neighboring channels to form a group since adjacent channels usually have quite high levels of inter-channel correlation. Despite a significant accuracy improvement due to the proposed modification, the modified method is still not able to provide acceptable accuracy in all practical situations, which means that research in this area should be continued.

References 1. Schowengerdt, R.A.: Remote Sensing Models and Methods for Image Processing. Academic Press (2007) 2. Dubovik, O., Schuster, G.L., Xu, F., Hu, Y., Bösch, H., Landgraf, J., Li, Z.: Grand challenges in satellite remote sensing. Front. Remote Sens. 2, 619818 (2021). https://doi.org/10.3389/frsen. 2021.619818 3. Kerekes, J.P.: Optical sensor technology. In: The SAGE Handbook of Remote Sensing, pp. 95– 107. SAGE Publications, London, UK (2009) 4. Christophe, E.: Hyperspectral data compression tradeoff in optical remote sensing. In: Advances in Signal Processing and Exploitation Techniques, 8th ed., pp. 9–29. Springer, Berlin (2011) 5. Bekhtin, Y.S.: Adaptive wavelet codec for noisy image compression. In: Proceedings of the 9th East-West Design and Test Symposium, September 2011, Sevastopol, Ukraine, pp. 184–188 (2011) 6. Image Classification Techniques in Remote Sensing. https://gisgeography.com/image-classific ation-techniques-remote-sensing/ (2021). Accessed 12 June 2021 7. Hu, Y., Chen, J., Pan, D., Hao, Z.: Edge-guided image object detection in multiscale segmentation for high-resolution remotely sensed imagery. IEEE Trans. Geosci. Remote Sens. 54(8), 4702–4711 (2016) 8. Zhong, P., Wang, R.: Multiple-spectral-band CRFs for denoising junk bands of hyperspectral imagery. IEEE Trans. Geosci. Remote Sens. 51(4), 2269–2275 (2013) 9. Meola, J., Eismann, M.T., Moses, R.L., Ash, J.N.: Modeling and estimation of signal-dependent noise in hyperspectral imagery. Appl. Opt. 50(21), 3829–3846 (2011)

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10. Colom, M., Lebrun, M., Buades, A., Morel, J.M.: A non-parametric approach for the estimation of intensity-frequency dependent noise. In: IEEE International Conference on Image Processing (ICIP), 27–30 October 2014, Paris, France, pp. 4261–4265 (2014) 11. Goosens, B., Pizurica, A., Wilfried, P.: Removal of correlated noise by modeling the signal of interest in the wavelet domain. IEEE Trans. Image Process. 18(6), 1–14 (2009) 12. Cocianu, C., Stan, A.: Neural Architectures for Correlated Noise Removal in Image Processing. Mathematical Problems in Engineering, vol. 2016, p. 6153749 (2016). https://www.hin-dawi. com/journals/mpe/2016/6153749/ 13. Liu, C., Szeliski, R., Kang, S.B., Zitnick, C.L., Freeman, W.T.: Automatic estimation and removal of noise from a single image. IEEE Trans. Pattern Anal. Mach. Intell. 30(2), 299–314 (2008) 14. Gao, L., Du, Q., Zhang, B., Yang, W., Wu, Y.: A comparative study on linear regression based noise estimation for hyperspectral imagery. IEEE J. Select. Top. Appl. Earth Obs. Remote Sens. 6(2), 488–498 (2013) 15. Jin, X., Xu, Z., Hirakawa, K.: Noise parameter estimation for Poisson corrupted images using variance stabilization transforms. IEEE Trans. Image Process. 23(3), 1329–1339 (2014) 16. Savant, R.V., Pradhan, D.: Estimation of noise parameters for captured image [Electronic Resource]. In: IEEE International Conference on Engineering and Technology (ICETECH), March 17–18, Coimbature, India (2016). https://ieeexplore.ieee.org/document/7569405 17. Uss, M., Vozel, B., Lukin, V., Chehdi, K.: Image informative maps for component-wise estimating parameters of signal-dependent noise. J. Electron. Imaging 22(1), 013–019 (2013) 18. Uss, M., Lukin, V., Baryshev, I., Vozel, B., Chehdi, K.: Joint estimation of additive and impulsive noise parameters in remote sensing images with fractal structure. Mod. Probl. Radioeng. Telecommun. Comput. Sci. (TCSET). Lviv-Slavsko, Ukraine 2006, 232–235 (2006) 19. Abramova, V.V., Abramov, S.K., Lukin, V.V., Vozel, B., Chehdi, K.: Scatter-plot based estimation of mixed noise parameters for remote sensing image processing. In: Aviation in the XXI-st Century: Proceedings of the 5-th World Congress, Kiev (Ukraine), vol. 2, pp. 3.7.56–3.7.60 (2012) 20. Abramova, V.V., Abramov, S.K., Lukin, V.V.: Iterative method for blind evaluation of mixed noise characteristics on images. Inf. Telecommun. Sci. 6(1), 8–14 (2015) 21. Lebrun, M., Colom, M., Buades, A., Morel, J.M.: Secrets of image denoising cuisine. Acta Numer. 21(1), 475–576 (2012) 22. Abramova, V.V., Abramov, S.K., Lukin, V.V., Egiazarian, K.O., Astola, J.T.: On required accuracy of mixed noise parameter estimation for image enhancement via denoising. EURASIP J. Image Video Process. 2014, 3 (2014). http://jivp.eurasipjournals.com/content/2014/1/3 23. Tampere Image Database 2008 TID2008, version 1.0, from: http://ponomarenko.info/tid2008. htm. Accessed 12 June 2021 24. Abramova, V.: A blind method for additive noise variance evaluation based on homogeneous region detection using the fourth central moment analysis. Telecommun. Radio Eng. 74(18), 1651–1669 (2015) 25. Kurkin, D., Lukin, V., Abramova, V., Abramov, S., Vozel, B., Chehdi, K.: Image DCT coefficient statistics and their use in blind noise variance estimation. In: Mathematical Methods in Electromagnetic Theory (MMET 2012), Kharkov, Ukraine (2012) 26. Abramov, S.K., Zabrodina, V.V., Lukin, V.V.: Taking into account physical constraints on regression line fitting in the task of blind mixed noise variance evaluation on images. In: 20-th International Conferences on «Microwave Equipment and Telecommunication Technologies» (CriMiKo’2010), Sevastopol, September 13–17, vol. 2, pp. 1229–1230 (2010) 27. Abramova, V., Abramov, S., Lukin, V., Djurovi´c, I., Simeunovi´c, M., Vozel, B.: Blind evaluation of noise characteristics in multi-channel images exploiting inter-channel correlation. In: 6th Meditarranean Conference on Embedded Computing (MECO’2017), Bar, Montenegro, pp. 145–149 (2017) 28. Airborne Visible/Infrared Imaging Spectrometer. http://aviris.jpl.nasa.gov. Accessed 12 June 2021

Blind Evaluation of Noise Characteristics in Multichannel Images

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29. Green, R.O., Eastwood, M.L., Sarture, C.M., Chrien, T.G., Aronsson, M., Chippendale, B.J.: Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (AVIRIS). Remote Sens. Environ. 65, 227–248 (1998) 30. The EO-1 Hyperion Imaging Spectrometer. https://eo1.gsfc.nasa.gov/new/validationReport/ Technology/TRW_EO1%20Papers_Presentations/10.pdf. Accessed 12 June 2021 31. Pearlman, J.S., Barry, P.S., Segal, C.C., Shepanski, J., Beiso, D., Carman, S.L.: Hyperion, a space-based imaging spectrometer. IEEE Trans. Geosci. Remote Sens. 41(6), 1160–1173 (2003)

Directions of Using Branched Trajectories of Determined Complex Dynamic Systems Olena Tachinina , Oleksandr Lysenko , Igor Romanchenko , Sergiy Ponomarenko , and Valeriy Novikov

1 Introduction Modern theory of branched dynamical systems has significant achievements. Thus, the problem of optimization of the CDS trajectory is formulated, which consists in the search for optimal controls and the trajectory of the subsystems along the sections of the branched trajectory that minimize the given criterion. Mathematical models of motion of deterministic composite dynamical systems in the form of a branched trajectory are constructed and the conditions of optimality of the branched trajectory of a deterministic composite dynamical system are found. Algorithms for finding the optimal moments of time and phase coordinates in which the CDS is divided into subsystems, methods of transforming a composite dynamic system into a discontinuous dynamic system with variables at the time of structural transformations (changing the size of state vectors and control). The necessary conditions for the optimal control of a deterministic CDS are proved and formulated in the form of basic theorems. The main theorems derive the consequences for the most typical practical cases of branching of trajectories of composite dynamical systems that formalize computational algorithms for control synthesis. O. Tachinina (B) National Aviation University, Kyiv, Ukraine e-mail: [email protected] O. Lysenko · S. Ponomarenko · V. Novikov National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine e-mail: [email protected] V. Novikov e-mail: [email protected] I. Romanchenko Central Research Institute of the Armed Forces of Ukraine, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_5

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In this paper, the provisions of the theory of branched dynamical systems are developed. Modifications of modern methods of the theory of optimal control for optimization of control of a group of unmanned aerial vehicles and a launch vehicle that puts a group of navigation satellites into orbit are presented. The application of optimality conditions for improving the algorithms of the “intelligent hint” of the motion control system of a group of unmanned aerial vehicles (UAVs) is shown. Conditions for the optimal trajectory of the UAV group have been developed, which solves the problem of monitoring the territory in the emergency zone. In the analytical form the program of movement on branched trajectories of the rocket with the separating main part which can be used in onboard computing algorithms for the similar class of devices as the reference program of their movement is received. The optimal trajectory of the rocket with the main part of the split, which brings a group of nanosatellites into Earth orbit, is calculated.

2 Problem Statement The simplest branched trajectory is a trajectory of composite dynamic system (CDS), consisting of two subsystems, which allows no more than one division or one grouping. Schemes and time diagrams of the simplest branched trajectories with division and grouping are presented in Fig. 1a, b, respectively. This material solves the problem of optimizing the CDS trajectory, which is to find the optimal control algorithms and trajectories of subsystems in areas of branched trajectory that minimize the specified criterion, as well as to find the optimal time and phase coordinates in which structural transformations of CDS.

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Fig. 1 Schemes and time diagrams of the simplest branched trajectories of CDS: a—with separation of subsystems; b—with grouping of subsystems

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The problem is solved for a group of UAVs and a launch vehicle. An algorithm for optimal control of a group of unmanned aerial vehicles and two modified algorithms for optimal control of a group of unmanned aerial vehicles, obtained on the basis of a quadratic functional and a generalized work function, are synthesized for UAVs. Also, two algorithms of “intelligent helper” are synthesized, which allows to specify the time of dissolution of the UAV group in case of random disturbances, as well as to stabilize it on a given trajectory, taking into account possible re-targeting. The optimal trajectory for launching a group of navigation satellites into orbit is calculated for the launch vehicle. Numerical results were obtained for each case.

3 Problem Solving 3.1 Algorithm for Optimal Control of a Group of Unmanned Aerial Vehicles Natural and man-made emergencies, which are increasingly occurring in our world, lead to complete or partial failure of terrestrial infrastructure, including telecommunications facilities (cellular base stations, radio relay and satellite stations, cable lines, etc.). Operational communication in such areas is possible through the deployment of occasional radio networks and telecommunication air platforms (TA) based on unmanned aerial vehicles. Unmanned aerial vehicles in the field of civil protection are used to ensure environmental control over the degree of environmental pollution, including in emergency situations, as well as for the prompt presentation of control results. They can be used to solve a wide range of tasks in the field of aviation search and rescue, search and rescue operations on land and at sea, in difficult terrain. Thus, in rescue operations, unmanned aerial vehicles are the most reliable and safe source of information for ground groups. Unmanned aerial vehicles conduct operational reconnaissance and detailed survey of the area, which allows you to timely assess the situation and make management decisions to coordinate the actions of rescue teams. To solve such problems, the group use of unmanned aerial vehicles is promising. The advantage of using a group of unmanned aerial vehicles (UAVs) becomes obvious in tasks in which it is possible to divide one complex task into several separate tasks performed by separate aircraft. For example, when monitoring large areas in a short time; in tasks of providing communication with mobile subscribers for effective interaction of ground search and rescue services; for transportation and dumping on the command of the operator of small loads to a given point [1, 2]. However, the control of UAVs in a group is a much more difficult task than the control of a single device [3–5]. This is due to the fact that in addition to controlling the flight of a separate UAV, it is necessary to ensure a certain relationship and consistency of its actions with other members of the group, taking into account their group task.

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For group control of UAVs in this section, it is proposed to use a polyergatic motion control system of the UAV group. The scheme of functioning of the polyergatic motion control system of the UAV group is presented in Fig. 2. With this method of building a control system (Fig. 2), the implementation of the planned actions of the UAV in the process of solving the general problem is entrusted to the operator. The operator must solve the navigation task, i.e., set the UAV group program trajectory, the phase coordinate of the group division, the time interval during which group division is allowed. However, in addition to the navigating task, there is also the task of keeping (stabilizing) the group and individual UAVs on the program trajectory and determining the most favorable time for separation within a given interval. To solve the above tasks, the operator must have an interactive computer system (multifunction indicator—MFI) designed to support decision-making, which, performing the function of “intelligent assistant”, would help him to set the maneuvers of UAVs, estimate the coordinates of their current location on a given trajectory, identify optimal moments of time of group maneuvers. This unit is devoted to the development of an algorithm for an “intelligent prompter” from the multifunction indicator (MFI) of the ground command post. To develop an algorithm for optimal control of the UAV group, it is proposed to use the theory of branched trajectories [6]. The concept of application of the theory of optimization of branched trajectories to the solution of the UAV group control problem is given in Fig. 3. In Fig. 3 the following designations are accepted: SPR—the starting point of the route; IPR—intermediate point of the route; GSP—UAV group separation point; Δ—label of the current position of the group; Δi (i = 1, 2)—the label of the current position of the i-th UAV after separation; Θ—mark of the set position of the group or individual UAV; 1—trajectory of the UAV group with the route points located on it, obtained as a result of navigating calculation; 2—true trajectory of movement; 3—the true point of separation of the group; x or x i (i = 1, 2)—the distance between

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the specified and the current position of the group or individual UAV. In order to physically reproduce the flight image of the UAV group, Fig. 3 can be transferred to the screen of the multifunction indicator (MFI), where the operator monitors the air situation in the emergency zone, weather conditions, monitors the position of the landing targets. Due to the action of different perturbations, the current state of the group or individual UAVs differs from the specified. The algorithm of the “intelligent assistant” produces optimal, from the point of view of the set quality criterion, control of movement of a label of current position to the set position and calculates the optimum moment of time and a phase coordinate of distribution of group. Command values of optimal control and moments of separation time are given for testing in the onboard control system (BCS) of the UAV. The evolution of the label of the current position on the screen of the MFI near the program trajectory is usually described by deterministic or stochastic linear differential equations [7]. Note that the metric of the space in which the problem is solved is used as x or x i (i = 1, 2). The possibility of representing a group of UAVs with one label on the screen of the MFI in the area of the trajectory between the points SPR and 3 (Fig. 3) is explained by the fact that the geometric dimensions of the UAV group are several orders of magnitude smaller than the distance they travel. Therefore, for the optimization problem, this assumption is correct. It is assumed that the group of UAVs consists of two vehicles, the navigation problem has already been solved and a linear deterministic model of the dynamics of the movement of the label of the current position of the UAV relative to a given position is considered [8, 9]: q x˙ =q aq x(t)+q bq u(t)

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where q x(t) ∈ E 1 is the state vector of the CDS, qu(t) ∈ E 1 is the CDS control vector, q are the indices of the branched trajectory sections along which the CDS subsystems (q = 1, 11, 12) move. The scheme of the branched trajectory of the CDS is presented in Fig. 1a.

3.2 Modified Algorithm for Optimal Control of a Group of Unmanned Aerial Vehicles, Obtained on the Basis of the Quadratic Functional The operator must set the coordinate of the start point of the joint movement of the UAV group 1x (t 0 ), time t 1 and coordinate of the UAV group separation point 1x (t 1 ), so that the UAV “11” after t11 = 6 s after the start of the joint movement and subsequent division reaches the coordinate point 11x(t11 ) = 4, and the UAV “12” by t12 = 4 s reaches the point 12x(t12 ) = 6.92 where the coordinates are measured in conventional units of distance, while minimizing the criterion Δ

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where 1 F = 0.3; 11 F = 0.1; 12 F = 0.2; 1 Q = 2; 11 Q = 1; 12 Q = 1; 1 R = 2; 11 R = 1; 12 R = 1. To solve problem (1), (2), in which it is assumed that 1 a = –2.5; 11 a = –1; 12 a = –1.5; 1 b = 1; 11 b = 0.5; 12 b = 0.5, we use the principle of Bellman’s optimality [10]. Based on the recurrent algorithm of analytical construction according to Krasovsky [13], the expression for the optimal control value will look like ] [ qu (t) = −q R −1 q b T q P(t)q x (t)t ∈ tq ∗ , tq (q = 1, q ∗ = 0; q = 11, 12, q ∗ = 1), (3) Δ

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the limit value for the auxiliary variable 1P(t1 ) is calculated taking into account the values of auxiliary variables 11P(t) and 12P(t) at the left end of branches “1–11” and “1–12” at t = t1 . Using expressions (3)–(5) to solve the scalar problem (1)–(2), we obtain the following analytical expressions for calculating the phase coordinates and auxiliary variables included in the expression for calculating the optimal control: Δ

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The moment t 1 of division of the UAV group is found from the condition 11x (t 1 ) =12 x (t 1 ). Then, substituting t 1 in the expression for 11x (t), calculate the coordinates of the separation point and then, assuming in expression (6) t = t0 = 0 when q = 1, find 1x(t 0 ). As a result of calculations we receive 1x(t 0 ) = 1, 17, t 1 = 1.094 s, 1x(t 1 ) = 1.59. Figure 4 shows a graph of joint and separate movement of UAVs. The proposed algorithm obtained on the basis of the quadratic functional takes into account the branching effect in the problem of retention (stabilization) of the UAV group on a given branched trajectory. At the same time the moments of time and coordinates of dissolution of the UAV group are optimally selected. Δ

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Fig. 4 Graph of the branched trajectory of the UAV group: 0–1—section of the trajectory of the joint movement of the UAV group; 1–11, 1–12—respectively sections of individual movement UAV “11” and UAV “12”

the “smart assistant” is to help the dispatcher determine the coordinates of the point from which the UAV group should start moving, the coordinates of the point and the time of its separation into two groups, as well as optimal controls for UAVs (5.1) on all sections of the trajectory minimum functionality ⎡ { 1⎣ J= 2

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where 1x(t) ∈ E 1 is CDS state vector, 1 j x(t) ∈ E 1 ( j = 1, 2) are state vectors of UAV “11” and UAV “12”; 1u(t) ∈ E 1 is vector of control effects of CDS, 1x(t1 ) ( j = 1, 2) are vectors of control effects of UAV “11” and UAV “12”; 1u , 1 ju are respectively optimal controls of CDS, UAV “11” and UAV “12”; t0 is the time of the beginning of the CDS movement, 12a = 0.05 is the moment of the time of the CDS separation into the UAV “11” and the UAV “12”, t1 j ( j = 1, 2) are the moments of the end time of the UAV “11” and the UAV “12”. Problem (1), (8) for a deterministic CDS moving along a branched trajectory with an arbitrary branching pattern can be solved taking into account the necessary conditions for optimal control and control constraints developed in [21]. Optimal control is calculated by the formula Δ

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where q is the indices of the sections of the branched trajectory along which the subsystems of the CDS move, q B(t) is an auxiliary variable that is the solution of

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the equation ˙ + 2q B(t)q u + 1 = 0 (q = 1, 11, 12) q B(t)

(10)

with boundary conditions B11 (t11 ) = B12 (t12 ) = 0(q = 1, 11, 12) and boundary conditions 11x(t1 ) =12 x(t1 ) =1 x(t1 ), qu(t) ∈ E 1 . Calculated by formula (9), the optimal control qu (t) allows you to get the optimal branched trajectory Δ

) ( ) } ( A (t − tq ) aq + 2q1a + 2qqa + 4q1a 2 × q x(t) = q x(tq ) exp (q = 1, 11, 12), [ ] × exp (−2q a(t − tq )) − 1 {

where q are the indices of the sections of the branched trajectory along which the CDS subsystems move (q = 1, 11, 12); A11 = A12 = 0(q = 11, 12); A1 =

2 ∑ ] } 1 { [ exp −21 j a(ti − ti j ) − 1 (q = 1). 2 a j=1 1 j

Equating the equations for 11x(t) and 12x(t) at the time t = t1 of separation of the UAV group, we find t1 and then after substitution t1 in the equation for 11x(t) and 12x(t) find the coordinates of the separation point 1x(t1 ). Using the values found t1 and 1x(t1 ), as well as the specified time t0 of the start of the UAV group, calculate using the equation for 1x(t1 ) the coordinate of the point from which the UAV group should start to minimize criterion (5.8) and ensure the arrival of UAV “11” and UAV “12” through specified time intervals in specified points (Fig. 5). In Fig. 5 shows the results of calculations t1 , 1x(t1 ), 1x(t0 ) for the following initial data: 11a = 0.5; t11 = 4; 11x(t11 ) = 0.5; 12a = 0.05, t12 = 3, 12x(t12 ) = 2;

Fig. 5 Graph of the branched trajectory of the UAV group

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1a = 0.55, t0 = 2. The following results were obtained: t1 = 2.40; 1x(t1 ) = 2.32; 1x(t0 ) = 20.72.

3.4 Algorithm of “Intelligent Assistant”, Which Allows You to Specify the Time of Disbandment of a Group of Unmanned Aerial Vehicles in Case of Random Disturbances Mathematical models of the dynamics of the movement of labels of the current position of a group of UAVs before separation and the current position of individual UAVs after separation are described by differential equations [11, 12] aq x = (q aq x+q bq u)dt+q σ (q x, t) dq W (t), t ∈ [tq ∗ , tq ] (q = 1, q ∗ = 0; q = 11, 12, q ∗ = 1);

(11)

11x(t1 ) =12 x(t1 ) =1 x(t1 ),

(12)

where q x(t) ∈ E 1 , qu(t) ∈ E 1 , qW (t) ∈ E 2 , qa, q b(q = 1, 11, 12) are constant scalar quantities; qσ (q x, t) = [q β1 , q x, q β2 ] are matrices of dimension 1 × 2; qW T (t) = [q W1 (t), q W2 (t)](q = 1, 11, 12) are two-dimensional separable Wiener processes. The scheme of the branched trajectory is presented in Fig. 2a. The group of UAVs starts its movement at the moment t0 of time from a fixed point and 1x(t0 ), further, at the moment of time t1 separate UAVs are separated, which must reach the area of their targets at the time t11 and t12 . The task of the “intelligent assistant” is to help the manager to synthesize the control of the labels of the current position qu(q x, t)(q = 1, 11, 12) and choose the time t1 so that the criterion has a minimum value. ⎡ ⎤ 2 ∑ J (U, t1 ) = M ⎣ J1 + J1i ⎦ → in f (13) U, t1 ∈θ1

j=1

where U = [1 u(t) t ∈ [t0 , t1 ], [ { 1 2 Jq = q Fq x (tq ) + 2

1i u(t)

tq

t ∈ [t1 , t1i ] (i = 1, 2); ]

[q Q q x (t)+q Rq u (t)]dt 2

2

tq ∗

(q = 1, q ∗ = 0; q = 11, 12, q ∗ = 1).

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Applying the theorem to the linear quadratic Gaussian [12] problem (11)–(13), we obtain qu (t) = −q R −1 q bq P(t)q x (q = 1, 11, 12), Δ

(14)

where q P(τ ) (q = 1, 11, 12) is the auxiliary variable, which is the solution of the differential equation [13–15] ] [ dq P qβ12 P+q R −1 q b2 q P 2 (q = 1, 11, 12) = −q Q − 2 qa + dt 2 q q P(tq ) =q F(q = 11, 12),

1 P(t1 ) =1

F+11 P(t1 )+12 P(t1 ) (q = 1).

(15) (16)

Δ

The minimum value of the functional J is calculated by the formula {

Δ

J = in f V1 (1 x(t0 ), t0 , t1 ) = in f t1 ∈θ1

t1 ∈θ1

} ] 1[ P1 (t0 , t1 )1 x 2 (t0 )+1 L(t0 , t1 ) , 2

(17)

where 1L(t0 , t1 ) calculated using equations [16] dq L = −q P(t)q β22 (q = 1, 11, 12), dt

(18)

under boundary conditions 1L(t1 ) =11 L(t1 )+12 L(t1 ), 1i L(t1i ) = 0 (i = 1, 2). For Eqs. (15) and (18) the solution can be obtained in analytical form, which allows to write explicit expressions for calculation {[

] [√ ]} √ 1β12 + 1δth 1δ(t1 − t0 )+1 B 2 {[ ] 1β12 2 −2 1a + (t1 − t0 ) 1L(t0 , t1 ) = 1 β2 1 R1 b 2 [ (√ ) ]} + ln ch 1δ(t1 − t0 )+1 B ch −1 1 B + {[ ] 2 ∑ 1iβ12 (t1i − t1 ) 1iβ22 R1i b−2 1ia + + 2 i=1 ) ]} [ (√ 1i + ln ch 1i δ(t1i − t1 )+1i B ch −1 1i B ,

1P(t0 , t1 ) =1 R1 b−2

1a +

where q B = ar th

] } ) {[ ( qβ 2 1 − qa + 1 +q b2 q P(t1 )q R −1 +q δ − 2 , 2

(19)

(20)

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Fig. 6 Dependence of the quality indicator on the time t1 of separation of the UAV group

)2 ( qβ 2 (q = 1, 11, 12), qδ =q Q q b2 q R −1 + qa + 1 2 1P(t1 ) = F1 +

2 ∑ i=1

1i R1i b−2

{( )] [√ ]} √ 1iβ12 1ia + + 1i δth 1iδ(t1i − t1 )+1i B . 2

In Fig. 6 shows the dependence V1 (1 x(t0 ), t0 ; t1 ) on t1 , calculated by formula (17) taking into account (19) and (20), for the following parameter values: 1a = 0.02; 1b = 0.1; 1ia = 0.005; 1ib = 0.1 (i = 1.2); 1β1 = 0.02; 1β2 = 0.5; 11β1 =12 β1 = 0.01; 11β2 = 0.49; 12β2 = 0.5; 1F = 1; 1Q = 0.01; 1R = 1; 11F =12 F = 0.5; 11Q =12 Q = 0.1; 11R =12 R = 1; 1x(t0 ) = 0.88; t0 = 0; t11 = 6, 5; t12 = 6.2. Let the division of a group of UAVs on the route [be permissible according to ] technical capabilities at any time in the interval θ1 = t1∗ , t1∗∗ . The requirement of minimization V1 (1 x(t0 ), t0 ; t1 ) on the parameter t1 leads to the conclusion that the most preferred moments of the separation time are the limit points of the “technical” interval θ1 , that is t1∗ = 1 and t1∗∗ = 6. However, the solution of the problem of choosing the time of separation of a group of UAVs in a polyergatic control system requires checking the condition of sufficiency of the time interval [t0 , t1 ] for the operator to understand the recommendations of the “intelligent assistant” [17–19]. The dynamic load of the operator associated with the understanding of the recommendations of the “intelligent assistant” received at the time t0 and aimed at dividing the group of UAVs at the time t1 are calculated by the formula [17] μ(t1 ) = τ1 (t1 − t0 )−1 ,

(21)

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where τ1 is the duration of the solution by the dispatcher of the problem related to the analysis of the situation on the route and the decision to divide the group of UAVs. Taking into account (21) the probability of the decision-making by the dispatcher will have the form [17] [ ][ ]−1 ρ(μ) = 1 − exp(1 − μ−1 ) 1 − μ exp(1 − μ−1 ) .

(22)

Sometimes the time τ1 in the vortex (21) will be equal to 0.5. Then. μ∗ = μ(t1∗ = 1) = 0.5/1 = 1/2, ρ(μ∗ ) = 0.7746; μ∗∗ = μ(t1∗∗ = 6) = 0.5/6 = 1/12, ρ(μ∗∗ ) = 0.9999. The result of the calculation showed that a reliable decision by the dispatcher is possible only if the recommended time of separation of the group of UAVs will be the proposed time t1∗∗ .

3.5 Algorithm “Intelligent Assistant” that Allows You to Stabilize the Unmanned Aerial Vehicle on a Given Trajectory, Taking into Account the Possible Re-targeting at Any Time in a Given Interval The physical meaning of the problem is as follows. Assume that the manager has detected two targets, one of which may be erroneous. For example, in the process of a search operation in the emergency zone, information is received about several possible areas of objects that need immediate assistance. If to perform a search operation in the possible areas of the objects consistently direct the UAV in each of these areas, it can lead to an increase in the time of the search operation and the negative consequences. Therefore, it is advisable to choose one of the zones (for example, the most probable on the primary grounds) and direct the UAV to it, but take into account the possibility of its re-targeting when receiving operational information about the location of the object in a particular zone. The dynamics of the motion of the UAV current position relative to the trajectory specified by the dispatcher and directed to the target initially selected as true is described by a linear scalar equation [17, 20–22] x(t) ˙ = ax(t) + αu(t), t ∈ [t0 , t f ], t0 = const, t f = var, x(t0 ) = X 0 , x(t f ) = X f , where X 0 , X f are known values.

(23)

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A similar equation, but with different values of the coefficients describes the dynamics of the movement of the label of the current position of the UAV relative to the possible trajectory of its movement to the second target. x˙ 0 (η) = bx 0 (η) + βu 0 (η), η ∈ [τ, tkτ ], τ ∈ [t ' , t '' ] ⊂ [t0 , t f ], τ = const, tkτ = var, x 0 (τ ) = x(τ ), x 0 (tkτ ) = X k ,

(24)

where X k is a known value or function of τ . It is necessary to optimize the process of stabilization of the UAV on the initially selected trajectory so that the criterion {

tf

I =

γ u 2 dt →

t0

min u(t)t ∈ [t0 , t f ], u 0 (η)η ∈ [τ, tkτ ],

(25)

reached a minimum value under the condition of inequality { I0 =

tkτ τ

2

φu 0 dη → A ≤ 0.

(26)

The optimal control delivers a minimum of the Hamiltonian [20–22] H (x, u, λ, t) = γ u 2 + λ(ax + αu),

(27)

calculated by the formula u(t) = −

α λ(t), 2γ

(28)

where λ(t) satisfies the equations λ˙ (t) = −aλ(t), t ∈ [t0 , t f ]\[t ' , t '' ], λ(t) = λ(t '' ) + a

{

t '' t

{ ζ+

t '' t'

λ(t)dt + ζ −1

{

t ''

λ0 (τ )dv(τ ), t ∈ [t ' , t '' ],

(29)

(30)

t

dv(τ ) − 1, ζ > 0, dv(τ ) ≥ 0, dv(τ )I 0 = 0.

According to (30) for the calculation λ(t) in the interval [t ' , t '' ] it is necessary to know λ0 (τ ), τ ∈ [t ' , t '' ]. For the calculation λ0 (τ ) it is necessary to find the optimal trajectory, which is described by Eq. (24). As a result of solving the system of equations

Directions of Using Branched Trajectories of Determined Complex … 0 0 0 x˙ (η) = bx (η) + βu (η); λ˙ 0 (η) = −bλ0 (η), η ∈ [τ, tkτ ],

Δ

Δ

245

Δ

(31)

where u (η) = −β(2φ)−1 λ0 (η), η ∈ [τ, tkτ ] is control that minimizes the Hamiltonian H 0 (x 0 , u 0 , λ0 , η) = φu 02 + λ0 (bx 0 + βu 0 ), we get Δ

0

λ0 (η) = cλ0 exp[−b(η − τ )], Δ

β2 cλ exp[−b(η − τ )] + cx0 exp[b(η − τ )]. 4bφ 0

0

x (η) =

(32)

(33)

Taking into account that x (τ ) = x (τ ), τ ∈ [t ' , t '' ], x (tkτ ) = X k0 ,

Δ

0

Δ

0

Δ

Δ

0

H (x (tkτ ), u (tkτ ), λ0 (tkτ ), tkτ ) = 0, Δ

0

0

Δ

write a system of three equations β2 cλ + cx0 = x (τ ), 4bφ 0 Δ

[ ( )] [ ( )] β2 cλ0 exp −b tkτ − τ + cx0 exp b tkτ − τ = X k , 4bφ cλ0 cx0 b = 0

(34)

with three unknowns cλ0 , cx0 and tkτ from which we find cλ0 as a function of cλ0 and τ . From (32) it follows that λ0 (τ ) = cλ0 (x (τ ), τ ). Substituting λ0 (τ ) = cλ0 (x (τ ), τ ) into Eq. (30), and solving it together with equation for x˙ (t), we find the optimal trajectory and control of the system (23) in the time interval [t ' , t '' ]. Outside the interval [t ' , t '' ], Eq. (23) should be solved together with Eqs. (28), (29). Let a = 1, α = 2, γ = 1, b = 1.5, β = 1.5, φ = 1,[ t0 = 0, x(t0]) = 8, X f = '' 2 2 y (τ )]− X k2 , τ ∈ [t ' , t '' ], 0.2415, X k = 0.2231, t ' = [ 0.5, t ' ]= 2.5, A = 2φb/β [ where y(τ ) = 0.0347 exp 2(τ − t ) + 7.7434 exp −2(τ − t ' ) . Then we obtain that ζ = 0.8721, dν(τ ) = μ(τ )dτ , μ(τ ) = 0 at τ ∈ [t ' , t '' ]\[t1 , t2 ] and μ(τ ) = 0.4905 at τ ∈ [t1 , t2 ], t1 = 1.3166, t2 = 1.5773. The optimal trajectory of the system (23) consists of three sections (Fig. 7): Δ

Δ

Δ

Δ

x

(1)

(t) = 8.1313 exp(−t) − 0.1313 exp(t), t ∈ [t0 , t1 ];

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Fig. 7 Graph of the optimal trajectory with an alternative Δ

x Δ

x

(3)

(2)

(t) = y(t), t ∈ [t1 , t2 ]; Δ

(t) = 1.197 exp(t2 − t), t ∈ [t2 , t f ],

Δ

gde t f = 3.1780, each of which corresponds to optimal control (1)

Δ

u (t) = 8.1313 exp(−t), t ∈ [t0 , t1 ]; )] [ [ ( ] (2) u (t) = 11.6151 exp −2 t − t ' − 0.1735 exp 2(t − t ' ) , t ∈ [t1 , t2 ]; Δ

(3)

Δ

Δ

u (t) = 1.197 exp(t2 − t), t ∈ [t2 , t f ]. In the[ time interval][t1 , t2 ] the movement of the system is limited, since I 0 = 2 2φb/β 2 x (t) − y 2 (t) = 0. The optimal value of the moment of time to reach the point X k system (24) is calculated by the formula. Δ

τ tk Δ

[ = 1/b ln

] x (τ ) + τ = 0.6666 lnx (τ ) + τ + 1, τ ∈ [t ' , t '' ]. Xk

Δ

Δ

Δ

The optimal value of criterion (25) I = 32.1025, which is slightly larger than the value I˜ = 31.9708 of the same criterion, calculated provided that the system ˜ (Fig. 7) without (23) passes from point x(t0 ) to point x(t f ) along the trajectory x(t)

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247

regard to the restriction (26). However, the point x(t f ) is reached in this case in time t˜f = 3.5 > t f and with violation of constraint (26) in the interval [t1' , t2'' ] where t1' = 1, t2'' = 2. Consider a simplified solution of the problem when the motion of the system (23) in time intervals [t0 , t1' ] and [t2'' , t f ] occurs on a trajectory constructed without constraints, and in the interval [t1' , t2'' ] with constraints. Then criterion (25) matters I = 32.3338 > I . The relationship (I − I˜)/(I − I˜) = 2.756 shows that the use of the necessary conditions for the optimality of the trajectory with the alternative allows to reduce almost 3 times the deterioration of criterion (25), which would occur in the case of system (23) on the trajectory of a simplified solution (Fig. 7). In conclusion, we note that the examples considered in this section with linear models of dynamic objects should be taken within the concept of piecewise-linear approximation of nonlinear characteristics of aircraft and information-measurement systems. Numerical examples are given in relative units. Δ

Δ

Δ

3.6 The Optimal Trajectory of the Launch Vehicle that Launches a Group of Navigation Satellites The entry into the third millennium coincided with a new stage in the development of technologies for miniature spacecraft—micro- and nanosatellites. Nanosatellites (nanosats) are spacecraft weighing from 1 to 10 kg, size 1U (10 × 10 × 10 cm), 2U (10 × 10 × 20 cm) and 3U (10 × 10 × 30 cm), which are designed to solve simple but important tasks. In world practice, nanosatellites are used for remote sensing of the Earth, environmental monitoring, earthquake forecasting, ionosphere research, etc. The period of single breakthrough results and the first successful experiments in the creation of small satellites is over. Today’s main task is to launch nanosatellites into orbit. In this section, we consider the option of launching nanosatellites as a payload (PL) based on the aircraft An-124–100, which is used as a mobile launch pad for launching a solid-propellant booster light class. The plane with the carrier rocket in the cargo compartment will take off from a regular airfield and rise to a height of about 20 km. At this altitude, with the help of an aircraft launcher, which includes a transport-launch platform (TLP) and a propeller parachute system, the launch vehicle is launched. After carrying out preparatory operations for the launch of the booster (opening of the cargo hatch, preparation of the landing system, activation of the control system, etc.), the booster on the launch vehicle under the action of the traction force of the exhaust parachutes begins to move on the floor equipment (golgangs) in side of the cargo hatch. At the moment of physical separation from the aircraft, the belts and cables connecting the booster with the transport-launch platform are separated. Then the carrier rocket, with transport container in which the nanosatellites are placed, due to its own solid propellant engine (at the initial stage of flight), and then by inertia reaches an altitude of about 600 km, which provides a discharge of payload in the form of nanosatellites. And transport-launch platform with the

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help of a parachute lands in a given place and is ready for further (multiple) use. Nanosatellites are installed inside the transport container of the carrier rocket on a special platform and pressed by springs to the lid. An electrical impulse from the launch vehicle activates the mechanism of opening the lid, which rotates at an angle of 170°. In this case, the nanosatellites under the action of the spring mechanism on the guide rails are separated from the rocket and go into orbit. The exit velocity is determined by the characteristics of the spring mechanism and the mass of the nanosatellites. The separation of the payload with low mass is due to the use of magnetic pulse drive with a capacitive energy storage. The proposed variant of launching nanosatellites into orbit is best adapted to the conditions of air launch from flying airfields (from the aerospace transport system).

3.7 The Branched Trajectory of the Launch of the Carrier Rocket, Which Allows You to Achieve the Maximum Total Height of the Rise of the Main Parts to Be Separated The optimization of the active part of the rocket’s trajectory, which launches a group of nanosatellites, should be performed taking into account the exact model of the rocket’s motion with the main part being separated, the model of the atmosphere and the Earth’s gravitational field. This optimization can be successful only with a good first approximation to the optimal solution [23–26]. The solution of the model problem presented in this section can be taken as the first approximation (suboptimal solution) for a more detailed problem. This task will more accurately take into account all the phenomena and effects associated with the motion of the carrier rocket, its main part, which divides, as well as the peculiarities of launching a group of nanosatellites into orbit. The analysis of literature sources shows that the calculation of the optimal trajectory of the launch of a group of satellites into orbit and the synthesis of optimal control of such systems were considered in the works of Aschepkova [27, 28], Sage and White [29]. Similar issues in the stochastic formulation were considered by Pugachev and Sinitsyn [30]. And the question of optimal control of deterministic composite dynamic systems that move along branched trajectories and allow to launch into orbit a group of satellites in one launch was not considered. Consider the problem in the following form. Suppose that the main part of the launch vehicle consists of two missiles. The scheme of the branched trajectory of the rocket with the separating main part is presented in Fig. 2a. The launch vehicle, the main part of which consists of two missiles, begins controlled movement at point 0 and moves further along the branch 0–1 to point 1. At this point, the main part is divided into two missiles, which using their own control system move from point 1 to points 11 and 12. We assume that the motion occurs in a vacuum in a plane-parallel gravitational field.

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Equations describing the motion of missiles in the corresponding sections of the trajectory are as follows [31–33]: q˙x 1

= g Pq cosq v q˙x 2

= q x3

sinq v − g, (q = 1, 11, 12),

(35) (36) (37)

where q are the indices of the sections of the branched trajectory along which the CDS subsystems move (q = 1, 11, 12); q x 1 , q x 3 are components of the velocity vector, directed respectively across and along the lines of force of the gravitational field; q x 2 is current coordinate, which is calculated along the lines of gravity of the gravitational field (flight altitude); Pq is overload, which is created by the power plant of the rocket, which brings a group of nanosatellites in the sections of 0–1 trajectory (q = 1), as well as the power plants of the two missiles, which are separated in sections 1–11 (q = 11) and 1–12 (q = 12); q v is pitch angle; g is the gravitational acceleration. Consider the problem with fixed moments of time: t0 is the beginning of the carrier rocket, which brings a group of nanosatellites, t1 is the time of separation of the main part, t11 and t12 are the moments of time of two missiles to reach the end points. The initial position of the carrier rocket, which launches a group of nanosatellites 1 x 1 (t0 ), 1 x 2 (t0 ), 1 x 3 (t0 ), is considered set. The overload Pq (q = 1, 11, 12) on each section of the trajectory has a constant value. It is necessary to find such a law of change vq (q = 1, 11, 12) of value of coordinates of point 1 for which criterion I = −[11 x 2 (t11 ) + 12 x 2 (t12 )] → min

(38)

reaches a minimum provided that 11 x 1 (t11 ), 11 x 2 (t11 ); 12 x 1 (t12 ), 12 x 2 (t12 ) set. On the basis of the necessary conditions for the optimality of the branched trajectory of the CDS with an arbitrary branching scheme formulated in [34], we write down the necessary conditions for the optimality of control 1 v(t)t ∈ [t0 , t1 ], 11 v(t)t ∈ [t1 , t11 ], 12 v(t)t ∈ [t1 , t12 ] of the problem (35)–(38). To construct the optimal branched trajectory of a carrier rocket that launches a group of nanosatellites, it is necessary to find the following conjugate variables ⎧ ∂ Hq ˙ ⎪ ⎪ ⎨ q λ1 = − ∂ q x 1 , ∂ Hq q˙λ2 = − ∂ q x , 2 ⎪ ⎪ ⎩ q˙λ = − ∂ Hq , 3 ∂q x

(q = 1, 11, 12)

(39)

3

where Hq = gq λ1 (t)Pq cos vq + q λ2 (t)q x 3 + g q λ3 Pq sin vq − q λ3 (t)g

(40)

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which satisfy the conditions ∂I ∂I − 11 λ2 (t11 ) = − 12 λ2 (t12 ) = 0, ∂ 11 x 2 (t11 ) ∂ 12 x 2 (t12 )

(41)

∂I − 1 λ1 (t1 ) + 11 λ1 (t1 ) + 12 λ1 (t1 ) = 0, ∂ 1 x 1 (t1 )

(42)

∂I − 1 λ2 (t1 ) + 11 λ2 (t1 ) + 12 λ2 (t1 ) = 0, ∂ 1 x 2 (t1 )

(43)

∂I − 1 λ3 (t1 ) + 11 λ3 (t1 ) + 12 λ3 (t1 ) = 0 ∂ 1 x 3 (t1 )

(44)

to minimize the Hamiltonian (40) for control q v(q = 1, 11, 12) at arbitrary values 11 λ1 (t11 ), 12 λ1 (t12 ), 11 λ3 (t11 ), 12 λ3 (t12 ). Applying the relationship (39)–(44) to the problem (35)–(38), we obtain a solution in an explicit analytical form: √ ( ) 2 Cq Rq (t) + 2Cq t + bq g Pq ln √ + q x1 t∗ , q x 1 (t) = √ ∗ ∗ Cq 2 Cq Rq (t ) + 2Cq t + bq ⎧ 1 [( )√ )√ ( ] ⎫ ∗ ∗ g Pq ⎨ 4Cq 2Cq t + bq √ Rq (t) − 2Cq t + bq Rq (t ) + ⎬ − 2 C R (t)+2Cq t+bq 4 A2 q x 2 (t) = Aq ⎩ + √q ln √ q q ∗ − Rq (t ∗ )(t − t ∗ ) ⎭ ∗ Δ

Δ

(45)

Δ

8Cq

Cq

2

Cq Rq (t )+2Cq t +bq

( )( ) ( ) (t − t ) + q x3 t∗ t − t∗ + q x2 t∗ , 2 [ ] √ ) ( ) ( g Pq √ Rq (t) − Rq (t ∗ ) − g t − t ∗ + q x 3 t ∗ , q x 3 (t) = Aq ( ) tgq v = Aq t − t ∗ + Bq (q = 1, 11, 12), −g

∗ 2

Δ

Δ

(46)

Δ

Δ

Δ

(47) (48)

where Rq (t) = aq + bq t + Cq t 2 , aq = 1 + Bq2 , bq = 2 Aq Bq , Cq = Aq2 (q = 1, 11, 12); at [ ] q = 1, t ∈ t0 , t f , t ∗ = t0 ,

A1 = 2(11 λ1 + 12 λ1 )−1 ,

B1 = [11 λ3 (t1 ) + 12 λ3 (t1 ) − 2t1 ] (11 λ1 + 12 λ1 )−1 ; at q = 11, t ∈ [t1 , t11 ], t ∗ = t1 ; q = 12, t ∈ [t1 , t12 ], t ∗ = t1 ,

Directions of Using Branched Trajectories of Determined Complex …

Aq = q λ−1 1 , Bq =

[

q λ3 (t1 )

251

] − t1 q λ−1 1 (q = 11, 12).

The parameters 11 λ1 , 12 λ1 , 11 λ3 , 12 λ3 must be selected in such a way as to satisfy the given final conditions. In Figs. 8 and 9 show the results of calculating the optimal branched trajectory for the following initial data:

Fig. 8 Graphs of parameters of the optimal branched trajectory:—optimal controls:

Fig. 9 Graphs of parameters of the optimal branched trajectory:—height and speed

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t0 = 0c, t1 = 70c, t11 = 100c, t12 = 120c, 1 x 1 (t0 )

P1 g = 2g,

=1 x3 (t0 ) = 0 m/c,

P11 g = 3.5g,

1 x 2 (t0 )

= 0 m,

P12 g = 4g, g = 9.806 m/c2 ,

11 x 1 (t11 )

= 2000 m/c,

11 x 3 (t11 )

= 0 m/c,

(49)

12 x 1 (t12 )

= 3000 m/c,

12 x 3 (t12 )

= 0 m/c,

(50)

The values of the parameters 11 λ1 , 12 λ1 , 11 λ3 (t1 ), 12 λ3 (t1 ) were found as a result of solving the gradient method of a system consisting of four nonlinear Eqs. (45), (47), at q = 11, 12 taking into account the final conditions (49) and (50). Values are used as the first approximation. 11 λ1

= 12 λ1 = −50,

11 λ3 (t1 )

= 12 λ3 (t1 ) = −30.

Calculations were stopped at 11 λ1 = −100.06, 12 λ1 = −49.00, 11 λ3 (t1 ) = −26.60, = −33.26, when the errors of compliance with the final conditions reached values of Δ11 x 1 (t11 ) = 1.049 m/s, Δ11 x 3 (t11 ) = −0.268 m/s, Δ12 x 1 (t12 ) = −1.34 m/ s, Δ12 x 3 (t12 ) = −0.510 m/s. The optimal value of criterion (38) was I = −294629 m. 12 λ3 (t1 )

tg1 v(t0 ) = 1.34; tg1 v(t1 ) = 0.401; tg11 v(t1 ) = 0.265; tg12 v(t1 ) = 0.678; 1v(t0 ) = 53◦ 16' , 1v(t10 ) = 21◦ 51' , 11v(t1 ) = 14◦ 51' , 12v(t1 ) = 34◦ 9' . Vg =

/ 2 q x1

+q x32 (q = 1, 11, 12),1 x2 (t1 ) = 124,889 km,

11 x 2 (t11 )

= 131,834 km, 12 x2 (t12 ) = 162,795 km.

Thus, in this subdivision, the program of motion along the branched trajectory of the rocket with the combined main part, which brings a group of nanosatellites into Earth orbit, is calculated in an analytical form. This program is built taking into account the necessary conditions for the optimality of the branched trajectory of the complex dynamic system with an arbitrary branching scheme [34–37]. The control effects obtained for different stages of flight and moments of separation allow to realize optimal branched trajectories and, accordingly, to efficiently use the resources of a complex dynamic system to launch a group of navigation nanosatellites. This program of motion in the form of a computational algorithm can be used in on-board computing systems of aerospace systems of a similar class for the rapid construction of a flight reference program.

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4 Conclusions 1. In this paper, the concept of a complex dynamic system is further developed. As a result, the concept of VTS includes an object such as a group of aircraft that jointly perform search and rescue tasks in emergencies. 2. Theoretical provisions of the theory of branched trajectories are brought to the level of algorithms for optimization of branched trajectories of specific VTS: – the application of optimality conditions for improving the algorithms of the “intelligent prompter” of the polyergatic motion control system of a group of unmanned aerial vehicles is shown; – shows the application of optimality conditions to improve the stabilization algorithms of the unmanned aerial vehicle on a given trajectory, taking into account the possible re-targeting at any time in a given interval; – in analytical form, the program of motion along the branched trajectory of the rocket with the main part, which brings a group of nanosatellites into Earth orbit, is calculated. The program can be used in onboard computer systems as a computational algorithm for the rapid construction of a reference program of motion of a similar class of devices. The results of research can be recommended for use in educational and scientific institutions and industrial enterprises in the development of tactical and technical requirements for control systems of complex dynamic systems, synthesis of algorithms for their operation and calculation of optimal trajectories.

References 1. Lysenko, O.: Estimation of the state vector and trajectory control of a dynamic system. IN: Flight Control Automation, MAI, pp. 32–36 (1989) 2. Hu, C., Xin, Y.: Reentry trajectory optimization for hypersonic vehicles using fuzzy satisfactory goal programming method. Int. J. Autom. Comput. 12, 171–181 (2015) 3. Bollino Kevin, P.: High-fidelity real-time trajectory optimization for reusable launch vehicles. Ph.D. Dissertation, Naval Postgraduate School (2006) 4. Sineglazov, V., Chumachenko, O., Gorbatiuk, V.: A new approach in cluster analysis. In: IEEE International Conference, «Actual Problems of Unmanned Aerial Vehicles Developments» NAU, pp. 223–226 (2017) 5. Sineglazov, V., Ischenko, V.: Intelligent system for visual navigation. In: IEEE 4th International Conference «Methods and Systems of Navigation and Motion Control», NAU, pp. 7–11 (2016) 6. Atans, M., Falb, P.: Optimal Control. Mechanical Engineering (1968) 7. Andreev, N.: The Theory of Statistically Optimal Control Systems. Nauka 416 p. (1980) 8. Yakovleva, A.: Modeling of Systems of Semi-automatic Control of Spaceships. Mashinostroenie, 280 p. (1986) 9. Samoilenko, A., Perestyuk, N.: Differential Equations with Impulse Influence. Vyscha school, 228 p. (1987) 10. Bellman, R.: Dynamic Programming. Foreign Literature Publishing House (1960)

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O. Tachinina et al.

11. 12. 13. 14.

Okhotsimsky, D., Sikharulidze, Y.: Fundamentals of Space Flight Mechanics. Nauka (1990) Fleming, W., Richel, R.: Optimal Control of Deterministic and Stochastic Systems. Mir (1978) Krasovsky, A.: Flight and Navigation Complexes. Zhukovsky (1975) Dmitrievsky, A., Lysenko, L.: Applied Problems of the Theory of Optimal Motion Control of Unmanned Aerial Vehicles. Nauka (1978) Kazakov, I.: Statistical Dynamics of Systems with Variable Structure. Nauka (1977) Mikeladze, V., Titov, V.: Basic Geometric and Aerodynamic Characteristics of Aircraft and Missiles. Mashinostroenie (1990) Silvestrov, M., Kozirov, L., Ponomarenko, V.: Automation of Control of Aircraft Taking into Account the Human Factor. Mashinostroenie (1986) Raikunov, G.: Optimization of Ballistic Support for a Flyby of a System of Spacecraft in a Circular Orbit. Fizmatlit (2011) Rosonoer, L.: The maximum principle of L.S. Pontryagin in the theory of optimal systems. Avtom. Telemech. (1959) Lysenko, O., Tachinina, O.: Method of constructing optimal trajectories with an alternative. AMU 2(8), 69–74 (2014) Tachinina, O.: Optimality conditions for the trajectory of a group of unmanned aerial vehicles with a possible change in the target of movement at any time in a given interval. Bull. AMU 1(9), 178–184 (2015) Tachinina, O.: Optimality conditions for a branching trajectory of an information robot with the current moment of separation. Adapt. Syst. Autom. Control 2(31), 110–117 (2017) Alemasov, V., Dregalin, L., Silent, L.: The Theory of Rocket Engines. Mashinostroenie (1989) Andreev, N.: The Theory of Statistically Optimal Control Systems. Nauka (1980) Andreevsky, V.: The Dynamics of the Descent of Spacecraft to Earth. Mechanical Engineering (1970) Appazov, R., Sytin, O.: Methods for Designing Trajectories of Carriers and Satellites of the Earth. Nauka (1987) Ashchepkov, L.: Optimal Control of Discontinuous Systems. Nauka (1987) Ashchepkov, L.: Optimization of sliding motions of a discontinuous system. Avtom. Telemech. 11, 30–37 (1983) Sage, E., White, C.: Optimal Control of Systems. Radio Communication (1982) Pugachev, V.: Probability Theory and Mathematical Statistics. Nauka (1979) Sirazetdinov, T.: Methods for Solving Multicriteria Problems of the Synthesis of Technical Systems. Mashinostroenie (1988) Sikharulidze, Y.: Aircraft Ballistics. Nauka (1982) Skorodelov, V.: Multipurpose demonstrator of technologies of aerospace transport systems. Aviat. Space Eng. Technol. 3, 12–16 (1997) Lysenko, O., Tachinina, O., Nikulin, O., Chumachenko, S.: Statement of the problem setting of the theory of distribution of tractors for the definition of problems of joke and of the problem in the zone of supervised situations. Tech. Mech. Sci. J. 1, 73–78 (2015) Lysenko, O., Tachinina, O.: Mathematical formulation of the problem of optimizing the movement of a group of flying robots based on unmanned aerial vehicles. Bull. AMU 1(7), 93–99 (2014) Lysenko, O., Tachinina, O., Chumachenko, S.: Conditions for the optimal trajectory of the carrier when placing sensors in the emergency zone. Tech. Mech. Sci. J. 3, 87–93 (2016) Tachinina, O.: Optimality conditions for the launch trajectory of a group of nanosatellites by an aerospace system. Tech. Mech. Sci. J. 1, 40–46 (2017)

15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

35.

36. 37.

Using Krotov’s Functions for the Prompt Synthesis Trajectory of Intelligent Info-communication Robot Olena Tachinina , Oleksandr Lysenko , Igor Romanchenko , Valeriy Novikov , and Ihor Sushyn

1 Introduction Info-communication robot (ICR) is a wireless sensor network with mobile sensors and telecommunication aero platforms which move in concert (rationally) in space. Mobile sensors assembled into clusters can be characterized as distributed (cluster) sensors. The structure and info-communication properties of cluster sensors change in real time when info-communication functions assigned to ICR are performing (the number of sensors which are part of the situationally constructed cluster are changing, the main sensor in the cluster sensor composition and the battery power reserve of each sensor are changing, the amount of service and applied information, which need to be transmitted via telecommunication aero platforms to the control center, etc.). Changing the properties of ICR in real time requires prompt calculation of rational actions that control the trajectory of its movement. So, the total spatial motion of the ICR is a branched trajectory with an arbitrary scheme of branches [1]. System O. Tachinina (B) National Aviation University, Kyiv, Ukraine e-mail: [email protected] O. Lysenko (B) · V. Novikov · I. Sushyn National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine e-mail: [email protected] V. Novikov e-mail: [email protected] I. Sushyn e-mail: [email protected] I. Romanchenko Central Research Institute of the Armed Forces of Ukraine, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Nechyporuk et al. (eds.), Information Technologies in the Design of Aerospace Engineering, Studies in Systems, Decision and Control 507, https://doi.org/10.1007/978-3-031-43579-9_6

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approach to ICR management requires apply compound dynamic systems (CDS) control methods [1–3], which would admissible rational and efficient coordination of the movement of all ICR elements (both mobile sensors and telecommunications aero platforms). Analysis of recent research and publications. Today, the wireless sensor network (mobile or fixed) is considered separately from the telecommunications platform [4– 8]. Considered that telecommunication aero platform performs an ancillary function to maintain the connectivity of the sensor network or increase throughput, or functional survivability or stability, or perform some ancillary functions to help more accurately determine the coordinates of sensors, or extend network life, or create new or more productive routes of information transfer. In addition, the telecommunications aero platform can be used to collect information from nodes of the sensor network. A holistic (system) approach to search rational control of the movement of all elements of the sensor network and telecommunication aero platform in realtime as a single system, with consideration of all kinds of restrictions, including telecommunication restrictions, which never have been applied. This approach is absolutely necessary in a situation when need the accurate prompt information about the victims in the emergency zone in conditions of almost complete destruction of infrastructure (fires, earthquakes, tsunamis, tornadoes, etc.). This information can be obtained through the use the sensors that placed on UAVs (mobile sensors), which form a “flying sensor network”. The task of operative optimization “group behavior” (optimization of a branched trajectory of movement) of mobile sensors in the aggressive environment which arises during emergency is actual. The prompt optimization algorithm is programmed in the on-board computer of the telecommunication platform, which controls the movement of mobile sensors. The success of search and rescue operation is primarily determined by sequence of the “group behavior” of the ICR elements, which (for example) are based on a “flying sensor network” with telecommunication aero platforms. ICR should provide up-to-date and high-quality (timely and reliable) information about victims and the urgent assistance which they need. Mismatch of “group behavior” of mobile sensors and telecommunication platforms as part of ICR can lead to the complete failure of the rescue operation.

2 Problem Statement Info-communication robot is considered as a compound dynamic system (CDS), which moves along a branched trajectory with arbitrary and typical branching schemes [1]. Functioning efficiency of CDS depends on the prompt optimal choice (in real time) of spatial coordinates and time points at which CDS structural transformations occur, also on the prompt optimal synthesis of control of CDS components

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

257

as they move along the branches of the trajectory in time intervals between structural transformations. The task is to develop conditions that admissible to quickly (in real time) design (build or synthesize) control of telecommunication aero platforms and mobile sensors which are part of the info-communication robot. ICR intelligence is the ability to solve the task of synthesizing the optimal branched trajectory of all ICR elements in real time based on current information about goals and restrictions. In the general setting, goals and restrictions are not stationary. They can change both determined and accidentally under the tentatively influence of not sufficiently defined external influences. That is, there is complete or partial preliminary uncertainty. Proposed to use the so-called “intelligence of action”, which is also called weak artificial intelligence to overcome this uncertainty [9–11]. The intelligence of the action is based on the “algorithmic” understanding of the current situation (prompt calculation of this action) based on the application of sufficient conditions for the optimality motion of a compound dynamic system [1]. Mathematical module of ICR motion is considered as a mathematical motion model of compound dynamic system. The theoretical basis of algorithm construction for the prompt calculation (synthesis) branched trajectory movement of ICR is sufficient conditions for the trajectory movement optimality of a deterministic compound dynamical systems for arbitrary and typical schemes of trajectory branches. The algorithm is informationally provided by accurate measurements of the state vector of mobile sensors and telecommunication aero platforms and accurate information about changes in goals and current restrictions. The task solution of finding sufficient conditions for the trajectory movement optimality of a deterministic compound dynamic system is performed in two ways: the first is to apply a modified method of Krotov’s functions (expansion principle) [12, 13]; the second is to use the method of invariant immersion [14, 15] in conjunction with the method of Krotov’s functions (the expansion principle) [12, 13]. For both methods, task solution is proposed to be sought in the form of a minimizing sequence in condition of CDS subsystems movement trajectories can be piecewise continuous.

3 The Expansion Principle for a Complex Dynamic System with an Arbitrary Scheme of Trajectory Branching In this part of the monograph, sufficient conditions for the movement trajectory optimality of a deterministic compound dynamical system with an arbitrary scheme of branching the movement trajectory are formulated in terms of the Krotov’s functions method. The conditions for branched trajectory optimality are proved in two forms: the first is a modified method of Krotov’s functions (expansion principle); the second is the modified method of Krotov’s functions (expansion principle) and the method of invariant immersion.

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3.1 Modified Expansion Principle Using the notation adopted in [1, 16–18], we formulate the general statement of the task of optimizing the CDS branched trajectory in the following form: I = S(t0l , t1l , . . . , tNl ; 1X l (t0l+ ), 2X l (t1l+ ), . . . , NX l (tNl+−1 ); N { t l− ∑ i l l− l l− 1X (t1 ), . . . , NX (tN )) + Φi (iX l , iU l , t)dt → inf D I ; i=1

(1)

l+ ti−1

(l l ( ) ( ) ( ) ( ) ( )) t0 , t1 , . . . , tNl ; 1X l t0l+ , 2X l t1l+ , . . . , NX l tNl+−1 ; 1X l t1l− , . . . , NX l tNl− ∈ B; (2) l ti−1 < til (i = 1, N ), (iX l (t), iU l (t)) ∈ Wi (i = 1, N );

(3)

[ l+ l− ] ˙ l = iF(iX l , iU l , t), t ∈ ti−1 , ti (i = 1, N ), iX

(4)

where iX l ∈ E mΣi , iX l (i = 1, N )—continuous and piecewise differentiated functions; iU l ∈ E mΣi , iU l (i = 1, N )—piecewise continuous functions; B—given ∑N subset E N +1 × E 2 i=1 nΣi ; W : E 1 → 2EnΣi +mΣi —multivalued function, D—plural l+ ≤ t ≤ til− , i = 1, N ), which satisfy the of admissible processes (iX l (t), iU l (t), ti−1 conditions (2)–(4) (D /= φ). The main (1)–(4) is minimized { }generalizing point inl+ the task formulation ≤ t ≤ til− , i = 1, N ), which is taken as sequence υ l = (iX l (t), iU l (t), ti−1 the optimal solving task Of CDS control, rather than certain admissible process l+ (iX l (t), iU l (t), ti−1 ≤ t ≤ til− , i = 1, N ). { } l+ Theorem 1 In order for the admissible sequence υ l = (iX l (t), iU l (t), ti−1 ≤ l− t ≤ ti , i = 1, N ) could be the solution of the task (1)–(4) sufficient existence of [ l+ l− ] , ti , (i = 1, N ) continuously differentiable Krotov’s functions Ψi (iX l , t), t ∈ ti−1 on iX l (t), t such that [ l+ l− ] Ri (iX l (t), iU l (t), t)M → Ri (t)t ∈ ti−1 , ti , (i = 1, N ); Δ

(5) Δ

Λ(t0l , . . . , tNl ; 1X l (t0l+ ), . . . , NX l (tNl+−1 ); 1X l (t0l− ), . . . , NX l (tNl−−1 )) → Λ,

(6)

where the symbol M → indicates convergence in measure, ( Ri (iX l , iU l , t) =

∂Ψi ∂i X l

)T

( Fi (iX l , iU l , t) +

∂Ψi ∂t

) + Φi (iX l , iU l , t);

(7)

Δ

Ri (t) = inf (iX l ,iU l )∈Wi (t) Ri (iX l , iU l , t) = 0;

(8)

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

259

Λ(t0l , . . . , NX l (tNl− )) = S(t0l , . . . , tNl ; 1X l (t0l+ ), . . . , NX l (tNl+−1 ); 1X l (tNl− , . . . , NX l (tNl− ))+ +

N ∑ [ ] l+ l+ Ψi (iX l (ti−1 ), ti−1 − Ψi (iX l (til− ), til− ;

(9)

i=1 Δ

Λ = inf B (t0 , . . . , NX (tN− )).

(10)

Proof Let the functions Ψi (iX l (t), t), (i = 1, N ) exist and manage to { } choose a sequence ν l , that satisfies the conditions (5)–(10). Suppose that ∼l

the is reached on the sequence {ν } = { minimum of the functional (1) } l l l+ l− ~ ~ iX (t), iU (t), ˜ti−1 ≤ t ≤ ˜ti , (i = 1, N ) , that does not satisfy conditions (5)– (10). Then Δ

I − I˜ ≥ 0,

(11)

∼l

Δ

where I = lim I (ν l ), I˜ = lim I (ν ). l→∞

l→∞

Herewith [ Δ

~ l (tNl− )) + I = lim I S(˜t0l , . . . , 1X l→∞

+

N { ∑ i=1

[

= lim

l→∞

til− l+ ti−1

til− l+ ti−1

i=1

{ ( ( )) ~ l tNl− = lim S ˜t0l , . . . , 1X l→∞

N { ∑

] ~ l , iU ~l , t)dt = Φi (iX

( ( ( ) ) l l l l ~ , iU ~ , t)dt] = ~ (t), t − d Ψi iX ~ (t), t + Φi (iX [d Ψi iX

~(tNl− )) Λ(˜t0 , . . . , NX

+

N { ∑

til− l+ ti−1

i=1

] ~ , iU ~ , t)dt . Ri (iX l

l

(12)

∼l

Given that for sequence {υ } fair relations Δ

~ (tNl− )) = Λ + εΛ ; lim Λ(˜t0l , . . . , NX l

l→∞

(13)

Δ

~ l , iU ~l , t) = R(t) + εRi , lim Ri (iX

l→∞

where εΛ ≥ 0, εRi ≥ 0, can be written (12) in the form

(14)

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O. Tachinina et al. Δ

I˜ = (Λ + εΛ ) +

N { ∑

ti− + ti−1

i=1

] [ R(t) + εRi dt). Δ

(15)

Due to continuity Ri (iX , iU , t) N { ∑

ti

i=1

ti−1

inf (iX ,iU )∈Wi (t) Ri (iX , iU , t)dt = inf (ν)∈D = inf (ν)∈D

N { ∑

ti− + ti−1

i=1

N { ∑

ti

i=1

ti−1

Ri (iX , iU , t)dt =

Ri (iX , iU , t)dt.

(16)

Therefore, (15) with consideration of (16), can be written as [ I˜ = inf

− (ν)∈D Λ(t0 , . . . , NX (tN ))

+

N { ∑ i=1

ti− + ti−1

] Δ

Ri (iX , iU , t)dt + ε = I + ε, (17)

∑ { t− where ε = εΛ + Ni=1 ti+ εRi dt ≥ 0. i−1 However, relation (17) contradicts inequality (11), which denies the assertion about existence of a sequence. Theorem proved. + − If there is an admissible process (iX (t), iU (t), t i−1 ≤ t ≤ t i , i = 1, N ), that delivers the minimum of the functional (1), i.e. the point is about the minima, which are the minimizing sequences of the form iX l (t) = iX (t); iU l (t) = iU (t); t0l = + − t 0 ; tll = t i ; t i−1 ≤ t ≤ t i ; i = 1, N , then following corollary follows from the Theorem 1. Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

+

−

Δ

Theorem 2 In order for the admissible process iX (t), iU (t), t i−1 ≤ t ≤ t i , (i = 1, N ) of task (1)–(4) will be optimal, sufficient existence of continuously differentiable on iX , t Krotov’s functions Ψi (iX , t)(i = 1, N ), which meet the conditions of Δ

Δ

Ri (iX (t), iU (t), t) = Ri (t) = 0,

(18)

] [ − + almost everywhere on t ∈ ti−1 , t i (i = 1, N ), Δ

Δ

Δ

Δ

+

Δ

Δ

+

Δ

Δ

Δ

Δ

Λ(t 0 , . . . , t N ; 1X (t 0 ), . . . , NX (t N −1 ); 1X (t1− ), . . . , NX (tN− )) = Λ.

(19)

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261

3.2 The Expansion Principle and Invariant Immersion Method The main idea of the method is that the initial task is included in some family of optimization tasks (invariant immersion). In doing so, it may turn out that there are between the individual tasks have simple relationships and there is one among the tasks of the family that is easily solved by the Krotov’s method. Then get the solution of the original task using the solution of the latter and the relations, that linking the individual tasks.{ As} in the previous point, for task solution, the minimizing sequence is considered ν l , which is selecting so as to minimize the criterion for task solution I = S0 (1X l (t0l+ ), t0l+ ) +

N ∑

Jj → inf D ,

(20)

i=1

where { Ji = Si (iX l (til− ), i + 1X l (t0l+ ), til ) + { JN = SN (NX l (tNl− ), tNl− ) +

tNl− tNl+

tNl− tNl+

Φi (iX l , iU l , t), (i = 1, N ),

ΦN (NX l , NU l , t)dt,

under conditions (3), (4) and (1X l (t0l+ ), t0l+ ) ∈ B0 , (NX l (tNl− ), tNl ) ∈ BN ,

(21)

(iX l (til− ), i + 1X l (t0l+ ), til ) ∈ Bi , (i = 1, N − 1),

(22)

where B0 , BN , Bi , (i = 1, N − 1)—given subsets according to E nΣi × E 1 , E nΣN × E 1 , E nΣi × E nΣi+1 × E 1 (i = 1, N − 1). The procedure for finding the minimum value of the functional (20) by the method of invariant immersion is written as follows [12–14] Δ

I = inf D I = inf D1 (J1 + inf D2 (J2 + · · · + inf DN (JN ) . . .)),

(23)

{ } l+ where D—the plural of admissible processes υ l = (iX l (t), iU l (t), ti−1 ≤ t ≤ til− , (i = 1, N ), that satisfy the conditions (3), (4), (21), (22) (D /= φ); Di (i = [l ] , til , i.e. 1, N )—a subset of the plural D, that considered in the interval ti−1 l+ (iX l (t), iU l (t), ti−1 ≤ t ≤ til− ). Let’s mark

262

O. Tachinina et al. Δ

Δ

I i = inf Di (Ji + I i+1 ) =

{

ti− + ti−1

Ri dt,

(24)

Δ

where i = N , N − 1, . . . , 2, 1; I N +1 = 0; Δ

S Ψ,i = inf Bi SΨ,i , i = N , N − 1, . . . , 2;

(25)

[ ] S Ψ,1 = inf B0 ,B1 SΨ,i + S0 (1X (t0+ ), t0 ) ;

(26)

Δ

Δ

Ri (t) = inf (iX ,iU )∈Wi (t) Ri (iX , iU , t) = 0; SΨ,N = SN (NX (tN− ), tN ) + ΨN− (NX (tN+−1 ), tN −1 ) − ΨN (NX (tN−1 ), tN );

(27) (28)

( ( ) ( ) ) ( (+ ) ) ( ( ) ) , ti−1 − Ψi iX ti− , ti + SΨ,i = Si iX ti− , i + 1X ti+ , ti + Ψi iX ti−1 Δ

+I i+1 (i + 1X (ti+ ), ti )

(29)

( ) (i = N , N − 1, . . . , 2, 1), Ri (iX l , iU l , t), i = 1, N —satisfies the Eq. (7), in which scalar continuously functions (iX l (t), t) → Ψi (iX l (t), t), are defined ] [ ldifferentiable l l η 1 for iX ∈ {E ,}t ∈ ti−1 , ti ⊂ E , i = N , N − 1, . . . , 2, 1, wanted for an admissible sequence ν l of relations [ ] lim SΨ,N (NX l (tNl− ), NX l (tNl+−1 ), tNl −1 ) = S Ψ,N , Δ

l→∞

(30)

[ ( )] l+ l lim SΨ,i iX l (til− ), i + 1X l (til+ ), iX l (ti−1 ), til , ti−1

l→∞

Δ

= S Ψ,i , (i = N , N − 1, . . . , 1),

(31)

Δ

lim Ri (iX l , iU l , t) = Ri , (i = N , N − 1, . . . , 1).

l→∞

(32)

{ } Theorem 3 In order for admissible sequence ν l will be the solution to the task (3), (4), (20)–(22) sufficient existence of such Krotov’s functions, Ψi (iX l (t), t) for which the relations are fulfilled (30)–(32). { } { } ∼l Proof Let there be a sequence ν , that is different from ν l and for which the condition is met ∼ lim I (ν ) = inf D I = I˜ , l

l→∞

(33)

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263

{ } but does not follow the relation (30)–(32), fair for the sequence ν l . Then [ (∼) [ ( ∼) [ (∼)] ]] (∼) I˜ = inf D I ν = inf D1 J1 ν + inf D2 J2 ν + · · · + inf DN JN ν . . . . (34) However, [ ( ) ] [ (∼) ] ∼l I˜i = inf D J1 ν + I˜i+1 = lim J1 ν + I˜i+1 = l→∞ [ ] { ˜til− ( l− ) ( l+ ) l l l l l = Si (i X˜ ˜ti , i+1 X˜ ˜ti , ˜ti ) + Φi (i X˜ , i U˜ , t)dt + I˜i+1 = ˜til+

[ ( ) ( ) = Si (i X˜ l ˜til− , i+1 X˜ l ˜til+ , ˜til )+ ] { ˜til− [ ] l l l l d Ψi (i X˜ (t), t) − d Ψi (i X˜ (t), t) + Φi (i X˜ ,i U˜ , t)dt + I˜i+1 = + ˜til+

] [[ ( ) ( ) l+ ), ˜ti−1 ) − Ψi (i X˜ l (˜til− ), ˜til ) + I˜i+1 = Si (i X˜ l ˜til− , i+1 X˜ l ˜til+ , ˜til ) + Ψi (i X˜ l (˜ti−1 ] { ˜til− l l ˜ ˜ Ri (i X , i U , t)dt = + ˜til+

Δ

= S Ψ,i +

{

˜ti

] [ Ri (t)dt + ε ≥ inf Di J1 (ν) + I i+1 , Δ

Δ

+ ˜ti−1

(35)

( ) where ε ≥ 0 for i = N , I N +1 = 0, SN = SN (N X tN− , tN ). Considering inequality (35) sequentially for i = N , N − 1, N − 2, . . . , 2, 1, come to the conclusion that [ ( ∼) ] [ ] (∼) (36) I˜ = inf D I ν = inf D1 J1 ν + I˜1 ≥ inf D1 J1 (ν) + I 1 = I . Δ

Δ

Δ

∼l

From expression (36) follows that the sequence {ν }, coincides with the sequence {ν l } and satisfies the relation (30)–(32), and minimizes the functional (20). Theorem proved. As a corollary of Theorem 3, formulate the result for the minimum, i.e. for the case when all members of the sequence are equal to the optimal process. ( ) Theorem 4 In order to admissible process ν = (i X (t), i U , t), t i−1 ≤ t ≤ t i , t 0 , t i i = 1, N ) of task (3), (4), (20)–(22) will be optimal, sufficient existence of continuously differentiable on i X , t functions Ψi (i X , t)i = 1, N , that meeting the conditions Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

−

Δ

Δ

Δ

+

Δ

Δ

Δ

+

Δ

Δ

Δ

SN (N X (t N ), t N ) + ΨN (N X (t N ), t N −1 ) − ΨN (N X (t N ), t N ) = S ψ,N ,

(37)

264

O. Tachinina et al. Δ

−

Δ

Δ

+

Δ

Δ

Δ

Δ

+

Δ

Si (i X (t i ), i+1 X (t i ), t i ) + Ψi (i X (t i−1 ), t i−1 )− Δ

−

Δ

Δ

Δ

Δ

+

Δ

Δ

Δ

− Ψi (i X (t i ), t i ) + I i+1 (i+1 X (t i ), t i ) = S (i = N − 1, N − 2, . . . , 1), [

∂Ψi ∂i x

(38)

| ]T | ) ( | | | iF(iX , U , t) + ∂ψi | + Φi iX , U , t = Ri (i = N , N − 1, . . . , 1), | ∂t |∧ ∧i (39) Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

for which I i , S ψ,i , Ri are calculated by formulas (24)–(27).

4 The Expansion Principle for a Compound Dynamic System with a Branching Scheme of Trajectories, Which Contains the Central and Lateral Branches, Without the Interaction of the Subsystem After Separation Consider the task in the next statement. The dynamics of CDS subsystems motion along a branched trajectory containing central and lateral branches [1] is described by equations ˙ = βf (βx, βu, t), t ∈ [tβ ∗ , tβ ] βx

(40)

βx ∈ E n , βu ∈ E mβ (β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = 1, k; j = 1, ri ) on which the restrictions are imposed (βx(t), βu(t)) ∈ Gβ (t),

(41)

where βu(t) piecewise continuous, tβ ∗ ≤ t ≤ tβ , βu(t) = βu(t + 0) = lim u(t), τ →t+0

tβ ∗ ≤ t ≤ tβ (β = i, β ∗ = i −1; β = ij, β ∗ = i; i = 1, k; j = 1, ri ), Gβ —restrictions on the limit values of the phase coordinates of the subsystems and the time of their achievement. At points in time when the CDS is divided into subsystems, the relations are fulfilled ix(ti ) − ijx(ti ) = 0 (i = 1, k; j = 1, ri ); ix(ti ) − i + 1x(ti ) = 0 (i = 1, k − 1). Emphasize that if there is a rigid mechanical connection between the subsystems, then for the n-th component of the vector state, which describes the mass change at each time point when the CDS is divided into subsystems is the relation

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

ixn (ti ) = ξ (i)i+1 xn (ti ) +

ri ∑

265

ijxn (ti ),

j=1

i = 1, k, j = 1, ri , ξ(i){1, i = 1, k − 1, 0, i = k. Need to minimize the functionality I = I (t0 , ti . . . , tk ; t11 , . . . , tkrk ; 1x(t0 ), 1x(t1 ) . . . , kx(tk ), 11x(t11 ) . . . , krk x(tkrk ); 1x(·), 1u(·); . . . , kx(·), ku(·), 11u(·), . . . , krk x(·), krk u(·) ⎛ ⎞ ri k ∑ ∑ ⎝ Ii + = Iij ⎠ → inf , (42) i=1

j=1

where { Iβ = Sβ (βx(tβ ), tβ ) +

tβ

Φβ (βx, βu, t)dt

tβ ∗

(β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = 1, k; j = 1, ri ).

(43)

Denote the plural of all controls βu(·) through Δ(βx, t, tβ ), defined on the segment [t, tβ ], which satisfy the conditions (41) and such that the trajectory of the system (40) is also defined on the segment [tβ ∗ , tβ ]. Plural of pairs ϑβ = (β x(t), β u(t)), that satisfy the conditions (0x(t0 ), t0 ) ∈ Q0 ,

(44)

where Q0 —range of admissible values of phase coordinates at the beginning of the system; (ix(ti ), ti ) ∈ Qi (ti−1 < ti , i = 1, k)

(45)

where Qi —range of admissible values of phase coordinates at the separation points at which the separation occurs on ri (i = 1, k) subsystems, which moving to individual endpoints (ijx(tij ), tij ) ∈ Qij (i = 1, k, j = 1, ri ),

(46)

where Qij —range of admissible values of phase coordinates at the time of completion of the movement of subsystems, and ( ) conditions (40), (41) will be called a plural of admissible processes Dβ Dβ /= ∅ β = i, ij; i = 1, kj = 1, r i . On the plural D = D1 × U r1 j=1 D1j UD2 × U r2 j=1 D2j U . . . UDk × U rk j=1 Dkj

(47)

266

O. Tachinina et al.

functional is given (42). Need to find the sequence {

} ϑ l β , β = i, ij; i = 1, k; j = 1, ri } { = βx(t), βu(t), t0 , tβl , β = 1, ij; i = 1, k; j = 1, ri ⊂ D,

In which the functional (42) aim for its smallest value on the plural D: ) ( I ϑ l β , β = i, ij; i = 1, k; j = 1, ri → inf D I = I . Δ

(48)

Theorem 5 In order to admissible sequence (ϑ l β , β = i, ij; i = 1, k; j = 1, ri ) will be the solution of the task (40)–(46), (48) sufficient existence of continuously differentiable Krotov’s functions Ψβ (β xl (t), t), t ∈ [tβ ∗ , tβ ](β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = 1, k; j = 1, ri ) such that ] [ Rβ (β xl (t), β ul (t), t)→M Rβ (t) on t ∈ tβ ∗ , tβ , Δ

(49) Δ

Λ(β xl (t), tol ), β (tβ ), t l o , t l β , β = i, ij; i = 1, k;j = 1, ri ) → Λ,

(50)

where ( Rβ (β xl , β ul , t) =

∂Ψβ ∂β x l

)T β

f (β xl , β ul , t) +

∂Ψβ + Φβ (β xl , β ul , t), ∂t

(51)

Δ

Rβ (t) = inf (β x(t),β u(t))∈Gβ , Rβ (β x, β u, t) = 0

(52)

Λ(1 xl (t0l ), β xl (t l β ), t l 0 , t l β , β = i, ij, i = 1, k;j = 1, ri ) = =

k ∑ {

l l Si (i xl (til ), til ) + Ψi (i xl (ti−1 ), ti−1 ) − Ψi (i xl (til ), til )+

i=1

⎫ ri [ ]⎬ ∑ + Sij (ij xl (tijl ), tijl ) + Ψij (ij xl (til ), til ) − Ψij (ij xl (tijl ), tijl ) , ⎭

(53)

j=1

Δ

Λ = inf Q Λ(i xl , tol ), β xl (tβl ), t0l , tβl , β = i, ij, i = 1, k;j = 1, ri ).

(54)

Task (40)–(46), (48) is a special case of task (1)–(4). Therefore, the content of Theorem 5 and relations (49)–(54) follow directly from Theorem 1 as a corollary. In addition, the proof of Theorem 5 can be performed using the same method of proof that was applied to Theorem 1. Take the method of direct proof of Theorem 5.

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

267

{ l } ∼ Suppose there is a sequence ϑ β , β = i, ij, i = 1, k;j = 1, ri , that minimizes the functional (48) but does not satisfy conditions (49)–(54). Then Δ

I − I˜ ≥ 0,

(55)

I = I (ϑ l β , β = i, ij; i = 1, k;j = 1, ri ),

(56)

where Δ

∼l

I˜ = I (ϑ β , β = i, ij; i = 1, k;j = 1, ri ),

(57)

at the same time ⎡ ⎤ ) ( l ri k ∑ ∑ ∼ ∼l ⎣I (ϑ i , i = 1, k + Iij , ϑ ij , j = 1, ri ⎦ = I˜ = j=1

j=1

= lim

k ∑ {

l→∞

( ) (l ) l ( ) Si (i x˜ l ˜til , ˜til ) + Ψi (i x˜ l ˜ti−1 , ˜ti−1 ) − Ψi (i x˜ l ˜til , ˜til )+

j=1

⎫ ( ) ( ) ]⎬ ( ) Sij (ij x˜ l ˜tijl , ˜tijl ) + Ψij (ij x˜ l ˜til , ˜til ) − Ψij (ij x˜ l ˜tijl , ˜tijl + + ⎭ j=1 ⎤ ⎡ { ti ri { tij k ∑ ∑ ⎣ lim Ri (i x˜ l , i u˜ l , t)dt + lim Rij (i x˜ l , i u˜ l , t)dt ⎦ = + k [ ∑

ti−1 l→∞

i=1 Δ

= Λ + εΛ +

⎧ k ⎨{ ∑ i=1

⎡ { k ∑ ⎣ =Λ+ Δ

⎩

ti

j=1 ti

[Ri (t) + εR,i ]dt +

ti−1

Δ

Ri (t) +

ti−1

i=1

Δ

⎤

ri { ∑

tij

j=1

ti

ti l→∞

ri { ∑

tij

j=1

ti

Δ

[Rij (t) + εR,ij ]dt

⎫ ⎬ ⎭

=

Δ

Rij (t)⎦dt + ε,

(58)

where ⎡ { k ∑ ⎣ ε = ε∧ + i=1

ti ti−1

εR,i dt +

⎤

ri { ∑

tij

j=1

ti

εR,i dt ⎦ ≥ 0, ε∧ ≥ 0, εR,i ≥ 0, εR,ij ≥ 0.

Due to continuity Rβ (β x, β u, t)(β = i, ij; i = 1, k;j = 1, ri )

268

O. Tachinina et al. k [{ ∑

ti

i=1

ti−1

+

inf (i x,i u)∈Gi (t), Ri (i x, i u, t)dt+ ⎤

ri { ∑

tij

j=1

ti

inf (ij x,ij u)∈Gij (t), Rij (ij x, ij u, t)dt ⎦ =

⎡ { k ∑ ⎣ = inf D

ti

Ri (i x, i u, t)dt +

ti−1

i=1

⎤

ri { ∑

tij

j=1

ti

Rij (ij x, ij u, t)dt ⎦.

(59)

The last inequality in (58) with consideration of relation (59) will have the form ⎧ ⎤⎫ ⎡ { ti ri { tij k ⎬ ⎨ ∑ ∑ ⎣ Ri (i x, i u, t)dt + Rij (ij x, ij u, t)dt ⎦ + I˜ = inf D Λ + ⎭ ⎩ ti−1 ti i=1 j=1 { ( )} (60) ε = inf D I ϑβ , β = i, ij; i = 1, k;j = 1, ri + ε = I + ε. Δ

After substitution (60) in inequality (55) appearing a contradiction, which is caused by an {incorrect initial assumption about } the possibility of the existence ∼l

of a sequence ϑ β , β = i, ij; i = 1, k;j = 1, ri , that does not have the properties formulated in Theorem 5. The theorem is proved. The following theorem formulates sufficient conditions that must satisfy the ) (( ϑ β , β = i, ij; i = 1, k;j = 1, ri = (β x(t), β u(t), t 0 , t β ; β = i, ij, minimum ) i = 1, k;j = 1, ri , i.e., the minimizing sequence of the partial view βxl (t) = βx(t), βul (t) = βu(t), t0l (t) = t 0 , tβl (t) = t β at all l. Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Theorem 6 In order for the admissible process (β x(t), β u(t), t 0 , t β ; β = i, ij, i = i = 1, k;j = 1, ri ) of task (40)–(46) will be optimal, sufficient existence of continuously differentiable on β x, t Krotov’s functions ψβ (β x, t) (β = i, ij; i = 1, k;j = 1, ri ), which meet the conditions of. Δ

Δ

Δ

Rβ (β x, β u(t), t) = Rβ (t),

(61)

] [ almost everywhere on t ∈ tβ ∗ , tβ , ( ) Λ(β x tβ , t 0 , t β ; β = i, ij, i = 1, k;j = 1, ri ) = Λ, Δ

Δ

Δ

Δ

(62)

Δ

where Rβ , Rβ (t), Λ, Λ are determined by equalities (51)–(54). As a corollary of Theorem 3, formulate solution of task (40)–(46), (48) by using both methods of invariant immersion and Krotov’s functions.

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

269

} { Theorem 7 In order for the admissible sequence ϑ l β , β = i, ij, i = 1, k;j = 1, ri , will be the solution of the task (40)–(46), (48) sufficient existence of continuously l differentiable Krotov’s functions Ψβ (β x (t), t), for which the relations are fair Δ

( ( ) ( ) ) ( ) lim Sψ,β βxl tβ , tβl ; βxl tβ ∗ , tβl ∗ = sψ,β , (βx tβ ∗ , tβ ∗ ), Δ

l→∞

(63)

Δ

lim Rβ (β xl , β ul , t) = Rβ (t)

l→∞

( β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = k, j = 1, rk ; ) i = k − 1, j = 1, rk−1 ; . . . ; i = 1, j = 1, ri ,

(64)

where ( ) ( ) l l Sψ,i (i xl til , til ; i xl t l i−1 , t l i−1 ) = Si (i xl (til ), til ) + Ψi (i xl (ti−1 ), ti−1 )− −Ψ i (i xl (til ), til ) +

ri ∑

( ) ( ) S ψ,ij (ij xl til , til ) + S ψ,i+1 (i+1 xl til , til ), S ψ,i+1 = 0, Δ

Δ

Δ

(65)

j=1

( ( ) ) ( ) S ψ,i i xl t l i−1 , t l i−1 = inf (i x(ti ),ti )∈Qi Sψ,i (i xl (ti ), ti ; i xl t l i−1 , t l i−1 ), Δ

( Rβ

β x ,β u l

l

)

, t = Φβ

(66)

) T ( ( ) ∂ψβ (β xl , t ) ∂ψ β xl , t l l ) βf β x ,β u , t + β x ,β u , t + ( ∂ β xl ∂t (67)

(

l

l

)

Δ

Rβ = inf (β x,β u)∈G(t), Rβ (β x(t), β u, t) = 0,

(68)

tβ∗ ≤ t ≤ tβ , β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = 1, k; 1, rk ; j = 1, ri , ( Sψ,ij

ij x

l

( ) ( ( ) ) ( ) ) ( ) tijl , tijl ; ij xl til til = Sij ij xl tijl , tijl + ψij ij xl (ti ), ti + ( ( ) ) + ψij ij xl tijl , tijl

( ( ) ) ( ) ( ) S ψ,ij ij xl t l i , t l i = inf (ij x(tij ),tij )∈Qij Sψ,ij (ij x tij , tij ; ij xl t l i , t l i )

(69)

Δ

(70)

given that (i xq (ti ) = (i+1 xq (ti )=ij xq (ti )(q = 1, n − 1; i = 1, k − 1), k x q (tk )=kj xq (tk )

) ( q = 1, n − 1 ,

(71)

270

O. Tachinina et al.

and the n-th phase coordinate describes the change in mass (i+1 xn (ti ) = ξi , i xn (ti ), ij xn (ti ) = ξij i xn (ti ), ξi +

ri ∑

) ( ξij = 1 i = 1, k , 0 < ξβ < 1, β = i, ij; i = 1, k; j = 1, ri ;

(72)

j=1

In turn, theorem about minimal follows from the Theorem 6. Δ

Δ

Δ

Δ

Theorem 8 In order to admissible process (β x(t), β u(t), t 0 , t β ; β = i, ij, i = 1, k;j = 1, ri ) of the task (40)–(46), (48) will be optimal, sufficient existence of functions. continuously( differentiable on (β x, t) Krotov’s ) Ψβ (β x, t) β = i, ij, i = 1, k;j = 1, ri , which meet the conditions of ( ) ( ) ( ) Sψ,β (β x t β , t β ; β x tβ ∗ , tβ ∗ ) = S ψ,β (β x tβ ∗ , tβ ∗ ),

(73)

] [ Rβ (β x, β u, t) = Rβ almost everywhere on t ∈ tβ ∗ , tβ

(74)

Δ

Δ

Δ

Δ

Δ

Δ

Δ

β = i, β ∗ = i − 1; β = ij, β ∗ = i; i = 1, k; 1, rk ; j = 1, ri , with consideration of relation (65)–(72).

5 The Expansion Principle for the Simplest Compound Dynamic System with Consideration Subsystems Interaction 5.1 The Simplest Branched Trajectory of a Compound Dynamic System with Its Division into Two Interacting Subsystems Consider the task definition of optimization a typical branched CDS trajectory with its division into two interacting subsystems in the following form [1, 19–23]. The dynamics of CDS subsystems motion along a branched trajectory is described by equations of the form βx = βf (βx, βu, t), tE[tβ ∗ , tβ ] (

) β = 1, β ∗ = 0; β = 11, β ∗ = 1; β = 12, β ∗ = 1 ,

(75)

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

271

t0 < t1 < t12 < t11 , where βx ∈ E n , βu ∈ E mβ , βu(t)—piecewise continuous control, tβ ∗ ≤ t ≤ tβ , βu(t) = βu(t + 0) = lim u(τ ) τ →t+1

At the time of subsystems separation, the conditions must be met (1 x(t), 1 u(t)) ∈ W1 (t), t ∈ [t0 , t1 ];

(76)

(11.12 x(t), (11.12 u(t)) ∈ W11.12 (t), (t) ∈ [t1 , t12 ];

(77)

(11 x(t), 11u(t)) ∈ W11 (t), t ∈ [t12 , t11 ];

(78)

) ( 1xr (t1 ) = 11xr (t1 ) = 12xr (t1 ) r = 1, n − 1 ; 1xn (t1 ) = 11xn (t1 ) + 12xn (t1 ),

(79)

βxn (t)—phase coordinate, which describes the change in mass in mechanical CDS; n+m 2n+m +m 2n+m W1 (t) : E 1 → 2E 1 , W11,12 (t) : E 1 → 2E 11 12 , W11 (t) : E 1 → 2E 11 — multivalued functions. Control βu(t), phase coordinates βx(t), points of time tβ ∗ , tβ (β = 1, β ∗ = 0; β = 11, β ∗ = 1, β = 12, β ∗ = 1) affect the criterion I = I1 + I11 + I12 → inf D ,

(80)

where ( ( ) ) Iβ = Sβ βx tβ , tβ +

{

tβ

Φβ (βx, βu, t)dt(β = 1, β ∗ = 0; β = 11, β ∗ = 1,

tβ ∗

β = 12, β ∗ = 1); 11.12x(t) = col(11xT (t), 12xT (t)); 11.12u(t) = col(11uT (t), 12uT (t)). Considering that (1 x(t0 ), (1 x(t1 ), (12 x(t12 ), (11 x(t11 ); t0 , t1 , t12 , t11 ) ∈ Q,

(81)

272

O. Tachinina et al.

as a solution of the task let’s look for a sequence { l} ν = {1 xl (t), 1 ul (t), 12 xl (t), 12 ul (t), 11 xl (t), 11 ul (t); l t l 0 , t l 1 , t l 12 , t l 11 ) ≤ t ≤ t11 },

Which minimizes the functional (80) on the plural of admissible processes (ν) ∈ D ( ) I ν l → inf D [I1 + I11 + I12 ] = I . Δ

(82)

Task (75)–(82) is a special case of task (1)–(4). Having estab= ({0, 1, 2, 3}, {0, 1, 12, 11}, lished correspondence of indexes I {(0, 0), phase coordinates ℵ = (1, 1), (2, 12), (3, 11)}), ({1 X , 2 X , 3 X }, {1 x, (11 xT , 12 xT )T , 12 x}, {(1 X , 1 x), (2 X , (11 xT , 12 xT )T ), (3 X , 11 x)}) and controls U = ({1 U , 2 U , 3 U }, {1 u, (11 uT , 12 uT )T , 12 u}, {(1 U , 1 u), T T T (2 U , (11 u , 12 u ) ), (3 U , 11u)}) formulate the solution of this task as a corollary of the Theorems 1–4. Corollaries of Theorems 1 and 2, to prove which uses a modified V.F. Krotov’s method. Corollary of Theorem 1 about the minimizing sequence. In order for the admissible sequence {ν l } will be the solution of the task (75)–(82) sufficient existence of continuously differentiable on βxl , t Krotov’s functions ψβ (βxl (t), t), t ∈ [tβ ∗ , tβ ](β = 1, β ∗ = 0; β = 11.12, β ∗ = 1, t11.12 = t12 ; β = 11, β ∗ = 12) such that Rβ (βxl , βul , t) M

Δ

→ Rβ (t) almost everywhere on t ∈ [tβl ∗ , tβl ]

(83)

(β = 1, β ∗ = 0; β = 11.12, β ∗ = 1; β = 11, β ∗ = 12); Δ

l l l l l Λ(1xl (t0l ), 1xl (t1l ), 12xl (t12 ), 11xl (t12 ), 11xl (t11 ), t0l , t1l , t12 , t11 ) → Λ,

(84)

where ) ∂ψβ T βf (βxl (t), βul , t) ∂β x ∂ψβ + Φβ (βxl (t), βul , t)(β = 1; 11) + ∂t ) ( ∂ψ11.12 T 11f (11xl , 11ul , t)+ R11.12 (11.12x, 11.12ul , t) = ∂11 xl ) ( ∂ψ11.12 T + 12f (12xl , βul , t)+ ∂12 xl (

Rβ (βxl (t), βul , t) =

(85)

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

+

273

∂ψ11.12 + Φ11 (11xl , 11ul , t) + Φ12 (12xl , 12ul , t), ∂t

(86)

Δ

Rβ (t) = inf (βx,βu)∈Wβ(t) Rβ (βx(t), βu, t) = 0,

(87)

l Λ(1xl (t0l ), . . . , t11 ) = S1 (1xl (t1l ), t1l ) + ψ1 (1xl (t0l ), t0l ) − ψ1 (1xl (t1l ), t1l )+ l l l l + S12 (12xl (t12 ), t12 ) + ψ11.12 (11.12xl (t1l ), t1l ) − ψ11.12 (11.12xl (t12 ), t12 )+ l l l l l l + S11 (11xl (t11 ), t11 ) + ψ11 (11xl (t12 ), t12 ) − ψ11 (11xl (t11 ), t11 ),

(88)

Δ

l Λ = inf Q Λ(1xl (t0l ), . . . , t11 ).

(89)

Corollary of Theorem 2 about the minima. In order to admissible process (βx(t), βu(t), (β = 1, 11.12, 11)t 0 , t 1 , t 12 , t 11 ; t 0 ≤ t ≤ t 1 ) of the task (75)–(82) will be optimal, sufficient existence of continuously differentiable on βx, t Krotov’s functions ψβ (βx(t), t)(β = 1, 11.12, 11), which meet the conditions of Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

( ) Rβ βx, βu, t = Rβ (t) Δ

Δ

Δ

(90)

(β = 1, β ∗ = 0; β = 11.12, β ∗ = 1; β = 11, β ∗ = 12), Almost everywhere on t ∈ [tβ ∗ , tβ ], Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Λ(1x(t 0 ), 1x(t 1 ), 12x(t 12 ), 11x(t 12 ), 11x(t 11 ), t 0 , t 1 , t 12 , t 11 ) = Λ,

(91)

Δ

where Rβ (t), Λ—are given by Eqs. (87), (89). For the proof of corollaries of Theorems 3 and 4 used the method of invariant immersion and the V.F. Krotov’s method, provided that Q = Q0 × Q1 × Q12 × Q11 ,

(92)

where the plurals Qβ (β = 0, 1, 11) consist of elements respectively (1x(t0 ), t0 ), (1x(t1 ), t1 )(11x(t11 ), t11 ), and the plural Q12 consists of elements (11x(t12 ), 12x(t12 ), t12 ). Corollary of Theorem 3 about the minimizing sequence. In order for the admissible sequence {ν l } will be the solution of the task (75)–(82) sufficient existence of such Krotov’s functions for which relations must be met (83), (85)–(87) and lim [S1 (1xl (t1l ), t1l ) + ψ1 (1xl (t0l ), t0l ) − ψ1 (1xl (t1l ), t1l )+ ( (l ) l ) ( ( ) ) + S12 12xl t12 , t12 + ψ11.12 11.12xl t1l , t1l − ( (l ) l ) ( (l ) l ) − ψ11.12 11.12xl t12 , t12 + S11 11xl t11 , t11 +

l→∞

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= inf Q0 [ψ1 (1x(t0 ), t0 ) + inf Q1 [S1 (1x(t1 ), t1 ) − ψ1 (1x(t1 ), t1 )+ + ψ11.12 (11.12x(t1 ), t1 ) + inf Q12 [S12 (12x(t12 ), t12 ) − ψ11.12 (11.12x(t12 ), t12 )+ + ψ11 (11x(t12 ), t12 ) + inf Q11 [S11 (11x(t11 ), t11 ) − ψ11 (11x(t11 ), t11 )]]]] ∑ = S ψ,β , Δ

(93)

β

where Δ

Δ

S ψ,0 = inf Q0 ψ1 (1x(t0 ), t 0 ), S ψ,1 = = inf Q1 [S1 (1x(t1 ), t 1 ) − ψ1 (1x(t1 ), t 1 ) + ψ11.12 (11.12x(t0 ), t 0 )], Δ

S ψ,12 = inf Q12 [S12 (12x(t12 ), t 12 ) − ψ11.12 (11.12x(t11.12 ), t11.12 ) +ψ11.11 (11.11x(t12 ), t12 )], Δ

S ψ,11 = inf Q11 [S 11 (11x(t11 ), t 11 ) − ψ11 (11x(t11 ), t 11 )]. Corollary of Theorem 4 about the minima. In order to admissible process (βx(t), βu(t), (β = 1, 11.12, 11)t 0 , t 1 , t 12 , t 11 ; t 0 ≤ t ≤ t 11 ) of the task (75)–(82) will be optimal, sufficient existence of continuously differentiable on βx, t Krotov’s functions ψβ (βx(t), t)(β = 1, 11.12, 11), which meet the conditions of (87), (90), Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

( ( ) ) ψ1 1x t 0 , t 0 = S ψ,0 , Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

(94) Δ

Δ

Δ

Δ

S1 (1 x(t 1 ), t 1 ) − ψ1 (1 x(t 1 ), t 1 ) + ψ11.12 (11.12 x(t 1 ), t 1 ) = S ψ,1 , Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

S12 (12 x(t 12 ), t 12 ) − ψ11.12 (11.12 x(t 12 ), t 12 ) + ψ11 (11 x(t 12 ), t 12 ) = S ψ,12 , Δ

Δ

Δ

Δ

Δ

Δ

(95) (96)

Δ

S11 (11 x(t 11 ), t 11 ) − ψ11 (11 x(t 11 ), t 11 ) = S ψ,11 , Δ

(97)

where S ψ,β (β = 0, 1, 12, 11) are given by the respective expressions (94)–(97).

5.2 The Simplest Branched Trajectory with Subsystems Grouping Consider the task definition:

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I = I1 + I11 + I12 + S0 (1x(t0 ), t0 ) + S1 (1x(t1 ), t1 ) → inf ,

(98)

where { t ∗ Iβ = tββ Φβ (βx, βu, t)dt (β ∗ = 0, β = 1; β ∗ = 1, β = 11, 12); (1x(t0 ), t0 ) ∈ Q0 ,

(99)

( ( ) ) βx tβ , tβ ∈ Qβ (β = 1, 11),

(100)

(βx(t), βu(t)) ∈ Wβ (t)(β = 1, 11, 12, 11),

(101)

1xr (t1 ) = 11xr (t1 ) = 12xr (t1 )(r = 1, n − 1), 1xn (t1 ) = 11xn (t1 ) + 12xn (t1 ),

(102)

βx = βf (βx, βu, t), tE[tβ ∗ , tβ ],

(103)

where βxEE n , βuEE mβ ; βu—piecewise continuous; tβ ≤ t ≤ tβ ∗ ; βu(t) = βu(t + 0) = lim u(τ ); (β ∗ = 0, β = 1; β ∗ = 1, β = 11, 12); t11 < t12 < t1 < t0 ; τ →t+0

11.12x(t), 11.12u(t), βxn (t), (β = 1, 11, 12), Qβ (β = 0, 1, 11, 12) and Wβ (t)(β = 1, 11, 12, 11)—notations that have the same meaning as in the task (75)–(80), and considering (81). The solution of the task (98)–(103) will be sought in the form of a sequence l l l {ν l } = {11xl (t), 11ul (t), 12xl (t), 12ul (t), 1xl (t), 1ul (t); t11 , t12 , t1l , t0l , t11 ≤ t ≤ t0l }, that minimizes the functional (98) on the plural of admissible processes (ν) ∈ D Δ

I (ν l ) → inf D [I1 + I11 + I12 + S1 (1x(t1 ), t1 ) + S0 (1x(t0 ), t0 )] = I 0 .

(104)

Let’s establish the correspondence of indices I , phase coordinates ℵ, controls U between task (1)–(4) and task (98)–(103), which is a special case of the first: I = ({0, 1, 2, 3}, {0, 1, 12, 11}, {(0, 0), (1, 1), (2, 12), (3, 11)}), ℵ = ({1 X , 2 X , 3 X }, {1x, (11x, 12x), 11x}, {(1X , 11x), (2X , (11x, 12x)), (3X , 11x)}), U = ({1U , 2U , 3U }, {1u, (11u, 12u), 11u}, {(1U , 1u), (2 U , (11u, 12u)), (3U , 11u)}).

Let’s single out separately four consequences from Theorems 1–4 for the case of the simplest branched trajectory with grouping of subsystems. The results obtained by using the modified V.F. Krotov’s method.

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Corollary of Theorem 1 about the minimizing sequence. In order for the admissible sequence {ν l } will be the solution of the task (98)–(103) sufficient existence of continuously differentiable on βxι , t Krotov’s functions ψβ (βxl , t), t ∈ [tβl , tβl ∗ ](β = 11, β ∗ = 12; β = 11.12, β ∗ = 1, t11.12 = t12 ; β = 1; β ∗ = 0) such that Δ

Rβ (βxl , βul , t)→M Rβ (t)

(105)

almost everywhere on t ∈ [tβl , tβl ∗ ](β = 11, β ∗ = 12; β = 11.12, β ∗ = 1, β = 1; β ∗ = 0); (l ) (l ) ( l ) l( l ) l( l ) l l l l ) ( , 11xl t12 , 12xl t12 , 1x t1 , 1x t0 , t11 , t12 , t1 , t0 → Λ, Λ 11xl t11 Δ

Δ

(106)

Δ

Δ

where Rβ (βx, βu, t), Rβ (t) are calculated from expressions (85)–(87), (89); ( (l ) ) l l l l Λ 11xl t11 ), t11 ) − ψ11 (11xl (t12 ), t12 ) , . . . , t0l = ψ11 (11xl (t11 l l + ψ11.12 (11.12xl (t12 ), t12 ) − ψ11.12 (11.12xl (t1l ), t1l ) + S1 (1xl (t1l ), t1l )

+ ψ1 (1xl (t1l ), t1l ) − ψ1 (1xl (t0l ), t0l ) + S0 (1xl (t0l ), t0l ).

(107)

Corollary of Theorem 2 about the minima.. In order to admissible process (βx(t), βu(t)(β = 1, 11.12, 11), t 11 , t 12 , t 1 , t 0 ; t 11 ≤ t ≤ t 0 ) of the task (98)–(103) will be optimal, sufficient existence of continuously differentiable on βx, t Krotov’s functions ψβ (βx, t)(β = 11, 11.12, 1), which meet the conditions of Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Rβ (βx, βu, t) = R(t) almost everywhere on t ∈ [tβl , tβl ∗ ]

(108)

(β = 11, β ∗ = 12; β = 11.12, β ∗ = 1, t11.12 = t11 ; β = 1; β ∗ = 0); Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Λ(11x(t 11 ), (11x(t 12 ), (12x(t 12 ), (1x(t 1 ), (1x(t 0 ), t 11 , t 12 , t 1 , t 0 ) = Λ, Δ

where Rβ (t), Λ—are calculated according to expressions (87) and (89) considering (107). Theorems obtained by using the invariant immersion method and the V.F. Krotov’s method, provided that Q = Q11 × Q12 × Q1 × Q0 ,

(109)

where elements of plurals Qβ (β = 11, 1, 0) are described in task (98)–(103), and the plural Q12 consists of elements (11x(t12 ), 12x(t12 ), t12 ). Corollary of theorem 3 about the minimizing sequence. In order for the admissible sequence {ν l } will be the solution of the task (98)–(103) sufficient existence of such Krotov’s functions for which fair relation (85)–(87), (105), (107) and

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l l l l l lim Λ(11xl (t11 ), (11xl (t12 ), (12xl (t12 ), (1xl (t1l ), (1xl (t0l ), t11 , t12 , t1l , t0l ) =

l→∞

= infQ11 [ψ11 (11x(t11 ), t11 )+inf Q12 [ψ11.12 (11.12x(t12 ), t12 ) − ψ11 (11x(t12 ), t12 )+ + inf Q1 [S1 (1x(t1 ), t1 ) − ψ11.12 (11.12x(t1 ), t1 ) + ψ1 (1x(t1 ), t1 ) ∑ + inf Q0 [S0 (1x(t0 ), t0 ) − ψ1 (1x(t0 ), t0 )]]]] = S ψ,β , Δ

(110)

β

where Δ

S ψ,11 = inf Q11 [ψ11 (11x(t11 ), t11 )],

(111)

Δ

S ψ,12 = inf Q12 [ψ11.12 (11.12x(t12 ), t12 )] − [ψ11 (11x(t12 ), t12 )],

(112)

S ψ,12 = inf Q1 [S1 (1x(t1 ), t1 ) − ψ11.12 (11.12x(t1 ), t1 ) + ψ1 (1x(t1 ), t1 )],

(113)

Δ

Δ

S ψ,0 = inf Q0 [S0 (1x(t0 ), t0 ) − ψ1 (1x(t0 ), t0 )]

(114)

(β = 11, 12, 1, 0). Corollary of Theorem 4 about the minima. In order to admissible process (βx(t), βu(t) (β = 11, 11.12, 1), t 11 t 12 t 1 t 0 ; t 11 ≤ t ≤ t 0 ) of the task (98)–(103) will be optimal, sufficient existence of continuously differentiable on βx, t Krotov’s functions ψβ (βx, t) (β = 11, 11.12, 1), which meet the conditions of (87)–(90) almost everywhere on t ∈ [tβ , tβ ∗ ](β = 11, β ∗ = 12; β = 11.12, β ∗ = 1, t11.12 = t11 ; β = 1; β ∗ = 0), also Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

ψ11 (11x(t 11 ), t 11 ) = S ψ,11 , Δ

Δ

Δ

Δ

Δ

Δ

Δ

ψ11.12 (11.12x(t 12 ), t 12 ) − ψ11 (11x(t 12 ), t 12 ) = S ψ,12 , Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

S1 (1x(t 1 ), t 1 ) − ψ11.12 (11.12x(t 1 ), t 1 ) + ψ1 (1x(t 1 ), t 1 ) = S ψ,1 , Δ

Δ

Δ

Δ

Δ

Δ

Δ

S0 (1x(t 0 ), t 0 ) − ψ1 (1x(t 0 ), t 0 ) = S ψ,0 , Δ

where S ψ,β (β = 11, 12, 1, 0)—are given by the respective expressions (111)–(114).

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6 Example of Prompt Synthesis the Branched Trajectory of Info-communication Robot (Movement Trajectory of the “Flying Sensor Network”), Which Based on the Application of a Modified Krotov’s Functions Method Task definition. The scheme of the branched movement trajectory of the compound dynamic system (info-communication robot), consisting of one telecommunication aero platform (TAP) and two mobile sensors (MS1 and MS2), which will be abbreviated as “flying sensor network”, that contains the central and lateral branches (Fig. 1). Assume that: (1) Sensors are fixed on the aero platform (we will denote the “connection” TAP + MS1 + MS2) at the start time of movement and after the separation of MS1, MS2 and TAP begin to move to a given fixed points; (2) after sensors separation from aero platform there is no interaction between TAP and MS1, MS2, which would impose restrictions on the control or phase variables TAP and MS1, MS2 after their separation; (3) the movement in the plane of the horizon is considered (the Earth is flat and does not rotate). Maneuvering characteristics comparison (radius and time of flat reversal of the cruising speed vector by 90°) of TAP + MS1 + MS2, TAP, MS1, MS2 with the distance which they must move, also with the total flight duration, allows to consider TAP + MS1 + MS2, TAP, MS1, MS2 as material points, the motion dynamics of the projections of which in the horizon plane is described by equations of the form [6, 24–29]: qx ˙ 1 =q u1 u2 ,

(115)

qx ˙ 2 =q u1 u2 ,

(116)

q = i, ij; i = 1; j = 1, 2, 3, where qx1 , qx2 —the current coordinates; they are calculating along the abscissa and ordinate axes, respectively;qu1 , qu2 —the control variables; they are modulus of the velocity vector and the angle between the abscissa axis and the velocity vector, respectively for the physical content of the task;qu1 ∈ [0, ququ21max q—indices of the sections of the branched trajectory along which the TAPs move + MS1 + MS2, TAP, MS1, MS2 (q = i, ij; i = 1; j = 1, 2, 3). It is necessary to minimize the vector criterion, the components of it are time intervals that are counted from the start time moment of movement TAP + MS1 + MS2 until the time of arrival MS1, MS2 i TAP to destinations W = [t11 − t0 t12 − t0 t13 − t0 ] → , min 1U ∈1 Ω∀t∈[t0 ,t1 ],1j U ∈1j Ω∀t∈[t1 ,t1j ],j=1,2,3;(1 x(t1 ),t1 )∈Q

(117)

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q x2

8 B(6;7) 7

11

6 5

P(8;5)

12

D(7,2468; 4,6942) 1

4

C(10;3)

3

13

2 1 0 А(0;0)

1

2

3

4

5

6

7

8

9

10

11

12

q x1

Fig. 1 Graphic image of the optimal branched trajectory of the “flying sensor network”: physical dimension of the variables plotted on the abscissa and ordinate axes - km

[ ] [ ] where qU = q u1q u2 , q = 1, 1j; j = 1, 2, 3; qΩ = 0; q u1max × [0; 2π ], Q— plural of points whose coordinates in the range from 0 to 10 on the abscissa axis and from 0 to 8 on the ordinate and are reached at time t1 (Fig. 1). In Fig. 1 the following designations are accepted: (·)A(0; 0) – the beginning of the movement of TAP + MS1 + MS2; (·)B(6; 7)− destination for MS1; ; 7+3 ) = P(8; 5)− destination for TAP; (·)C(10; 3)− destination for MS2; (·)P( 6+10 2 2 (·)D(7, 2468; 4, 6942)—the branch point of the trajectory of the CDS type TAP + MS1 + MS2; q ∈ {1, 1j, }j = 1, 2, 3. To find a compromise solution of multicriteria task, turn to the additive form of the scalar criterion: { W = α1

t1 t0

dt +

3 ∑ j=1

{ α1j

t1j

dt → min,

(118)

t1

] [ 1U ∈1 Ω∀t ∈ [t1 , t1 ], 1jU ∈1j Ω∀t ∈ t1 , t1j , = 1, 2, 3, (1 x(t1 ), t1 ) ∈ Q). To solve task (115), (116), (118) use the corollary of Theorem 2, which is stated in Sect. 3.1 of this section of the monograph. The conditions stated in this corollary allow to operatively optimize the trajectory of the CDS with the scheme of

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branching the trajectory containing the central and lateral branches, without the interaction of subsystems after separation. According to these conditions, the following optimization algorithm is as follows: 1. Construct a function R(x(t), u(t), t), the structure of which for each branch in this case will be the same: ∂ϕ(x, t) ∂ϕ T − × f (x, u, t), ∂t ∂x Φ(x, u, t) = 1; f (x, u, t) = [u1 u2 ],

R(x(t), u(t), t) = Φ(x, u, t) −

Consider that. ϕ(x, t) = ϕt (t) + ϕx (x), ϕt (t) = β × t; ] [ ∂ϕ ∂ϕ ∂ϕ ∂ϕ . = β; = ∂t ∂[x1 x2 ] ∂x1 ∂x2 Finally, have R(x(t), u(t), t) = 1 − β −

∂ϕ u ∂x1 1

−

∂ϕ ∂x2

× u2 → min ,where [u1 u2 ]∈Ω

{˙x1 = u1 =q u1 u2 , x˙ 2 = u2 =q u2 , u2 , q = i, ij; i = 1; j = 1, 2, 3. ∂φ ∂ϕ = b1 , ∂x = b2 and the Let φx (x) = b1 x1 + b2 x2 , where b1,2 > 0. Then ∂x 1 2 function (x, ) reaches a minimum at u1 = u1max , u2 = u2max . 2. Choose β from the condition 1 − β − b1 u22max1max .

Then the equation of motion on the branches of a branched trajectory will take the form {˙x1 = u1max x˙ 2 = u2max { or

dx1 u1max = . dx2 u2max ⇒ x1 = u2maxu·x1max 2 +const

(119)

Equation (119) is a straight line; In other words, the final branch of the branched trajectory in this task statement is straight. We use this property of the desired optimal branched trajectory, proved by using the Krotov’s function, for the final solution of the task. It is necessary to connect the lines AD, DC, DP, DB (i.e., find the coordinates of point D) so that the criterion W given by expression (118) has a minimum value: W = α1 (t1 − t0 ) +

3 ∑ J =1

( ) αij t1j − t1 =

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AD DB DP DB + α11 + α12 + α13 = 1u1max 11u1max 12u1max 13u1max / α1 = ∗ (xD − 1x1 (t0 ))2 + (yD − 1x2 (t0 ))2 + 1u1max / 3 ∑ ( ( ))2 ( ( ))2 α1j xD − ijx1 tij + yD − ijx2 tij → min + 1jujmax J =1 = α1

where (xD ; yD ) ∈ [0; 10] × [0; 8]. Let’s put that α1 = 0.4, α11 = α12 = α13 = 0.2; 1u1max km/h; 11u km/h1max ; 12u km/h1max ; 13u km/h1max . Then, using the computer mathematics system MATLAB get: 1x1 = 7.247 km, 1 x2 = 4.694 km, t1 = 0.8634 h, t11 = 0.5243 h, t12 − t1 = 0.813 h, t13 = 0.646 h. Thus, the application of a modified method of Krotov’s functions (developed on tasks with branched movement trajectories of compound dynamic systems) allowed at the previous stage of synthesis to substantiate the form of the optimal trajectory on each branch and reduce the initial task to a nonlinear programming task that can be solved quickly on the aero platform board for desired location points (quickly set by the operator) of the MS.

7 Conclusion 1. Obtained sufficient optimality conditions control of a deterministic compound dynamic system, which can be considered a theoretical basis for constructing algorithms of weak artificial intelligence - intelligence of action, which will allow to prompt trajectories calculating of info-communication robots in uncertainty conditions. 2. Sufficient optimality conditions are proved using the modified Krotov’s expansion principle and the invariant immersion method in combination with the Krotov’s function method and are formulated as a modified Krotov’s expansion principle for deterministic compound dynamical systems. 3. In the form of corollaries from the obtained sufficient conditions were formulated: the expansion principle for CDS with a scheme of trajectory branching, containing the central and lateral branches, without the interaction of subsystems after separation; the expansion principle for the simplest CDS with consideration of the subsystem’s interaction; the expansion principle for the simplest CDS with a subsystem grouping.

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The practical significance of the obtained conditions is possibility of developing computational procedures on their basis for the prompt calculation of the optimal branched trajectories of info-communications robots.

References 1. Lysenko, O., Tachinina, O.: Methods for the synthesis of optimal control of deterministic compound dynamical systems with branch. In: Handbook of Research on Artificial Intelligence Applications in the Aviation and Aerospace Industries, pp. 323–351. Copyright: IGI Globa (2020) 2. Tachinina, O.: Conditions for the optimality of the branching trajectory of the information robot with the current moment of separation. In: Interdepartmental Scientific and Technical Collection, pp. 110–117. “Polytechnic” Publishing House (2017) 3. Lysenko, O., Tachinina, O.: Expansion principle for compound dynamic systems with an arbitrary trajectory branching scheme. Probl. Inf. Manag. (NAU) 4(60), 51–57 (2017) 4. Alfattani, S., Wael, J., Yongacoglu, A.: Multi-UAV data collection framework for wireless sensor networks. In: IEEE Global Communications Conference (Globecom) (2019) 5. Romaniuk, V., Lysenko, O., Romaniuk, A., Zhuk, O.: Increasing the efficiency of data gathering in clustered wireless sensor networks using UAV. Inf. Telecommun. Sci. 11(1), 102–107 (2020) 6. Rahman, S., Cho, Y.: UAV positioning for throughput maximization. EURASIP J. Wirel. Commun. Netw. 20, 452–463 (2018) 7. Romaniuk, A.: MAC-protocol for data collection in wireless sensors networks with UAV. Collect. Sci. Papers MITI 4, 84–91 (2018) 8. Yeasir, M., Habib, A., Moh, S.: Routing protocols for UAV-aided wireless sensor networks. Appl. Sci. 10(12), 4077 (2020) 9. Bondarev, V., Ade, F.: Artificial Intelligence. Publisher SevNTU (2002) 10. Luger, J.: Artificial Intelligence: Strategies and Methods for Solving Compound Problems. Publishing House “Williams” (2003) 11. Stewart, R., Norvig, P.: Artificial Intelligence: Modern Approach. Publishing House “Williams” (2006) 12. Repnikova, N.: Theory of Automatic Control: Classics and Modernity. NTUU «KPI» (2011) 13. Yang, X.: Optimization Techniques and Applications with Examples. Wiley (2018) 14. Dorf, R., Bishop, R.: Modern Control Systems. Laboratory of Basic Knowledge (2002) 15. Hudvyn, G., Hrebe, S., Sylhado, M.: Design of Control Systems. Laboratory of Basic Knowledge (2004) 16. Lysenko, O., Tachinina, O., Valuiskyi, S. et al.: Application models of info-telecommunication technologies on the basis of unmanned aerial vehicles in emergencies. In: NAU (2016) 17. Tachinina, O.: Method of dynamic programming for information robot’s branching path optimization. Electron. Control Syst. (NAU) 3(53), 100–105 (2017) 18. Lysenko, O., Tachinina, O.: Method of path constructing of information robot on the basis of unmanned aerial vehicle. Proc. Natl. Aviat. Univ. (NAU) 4(73), 60–68 (2017) 19. Lysenko, O., Chumachenko, S., Shevchenko, V.: Mathematical models and information technologies for assessment and forecasting of the state of the environment in test sites, pp.43–52. Publisher: Pro Langs, language: Bulgarian (2017) 20. Lysenko, O., Valuiskyi, S.: Secured wireless sensor network for environmental monitoring. Volume of Scientific Papers, Security forum, vol. 2, pp. 528–532, Banská Bystrica (2016) 21. Tachinina, O., Lysenko, O., Chumachenko, S.: Optimality conditions for the movement trajectory of the carrier when placing the sensors in the emergency zone. Tech. Mech. Sci. J. 3, 87–93 (2016)

Using Krotov’s Functions for the Prompt Synthesis Trajectory …

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22. Lysenko, O., Tachinina, O.: Dynamic programming method for arbitrary branching trajectory optimization of a compound dynamic system. Probl. Inf. Manag. Sci. Papers Coll. (NAU) 3(59), 38–43 (2017) 23. Popescu, D., Stoican, F., Stamatescu, G.: A survey of collaborative UAV–WSN systems for efficient monitoring sensors. 19(21), 4690 (2019) 24. Qin, Z., Li, A., Dong, C.: Completion time minimization for multi-UAV information collection via trajectory planning. Sensors 19(18), 4032 (2019) 25. Liu, B., Zhu, H.: Energy-effective data gathering for UAV-aided. Sensors (Basel) 19(11), 2506 (2019) 26. Tachinina, O., Lysenko, O., Nazarenko, O.: New interpretation of the functional of generalized work in the task of optimal control of small-sized unmanned aerial vehicles. Probl. Inf. Manag. Coll. Sci. Works 4, 84–91 (2015) 27. Tachinina, O.: Optimality conditions for the branching trajectory of the information robot with the current separation moment. Adapt. Syst. Autom. Control (Publisher «Polytechnic») 2(31), 110–117 (2017) 28. Lysenko, O., Tachinina, O.: The scenario-based approach for control of multi-object dynamic system motion. In: IEEE International Conference Actual Problems of Unmanned Air Vehicles Developments (2015) 29. Lysenko, O.I., Tachinina, O.M., Alekseeva, I.V.: Path constructing method of unmanned aerial vehicle. In: IEEE International Conference Actual Problems of Unmanned Aerial (2017)