Intelligent Information Processing for Inertial-Based Navigation Systems (Navigation: Science and Technology, 8) 9813345152, 9789813345157

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Navigation: Science and Technology 8

Chong Shen

Intelligent Information Processing for Inertial-Based Navigation Systems

Navigation: Science and Technology Volume 8

This series Navigation: Science and Technology (NST) presents new developments and advances in various aspects of navigation - from land navigation, marine navigation, aeronautic navigation to space navigation; and from basic theories, mechanisms, to modern techniques. It publishes monographs, edited volumes, lecture notes and professional books on topics relevant to navigation - quickly, up to date and with a high quality. A special focus of the series is the technologies of the Global Navigation Satellite Systems (GNSSs), as well as the latest progress made in the existing systems (GPS, BDS, Galileo, GLONASS, etc.). To help readers keep abreast of the latest advances in the field, the key topics in NST include but are not limited to: – – – – – – – – – – –

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Chong Shen

Intelligent Information Processing for Inertial-Based Navigation Systems

123

Chong Shen North University of China Taiyuan, Shanxi, China

ISSN 2522-0454 ISSN 2522-0462 (electronic) Navigation: Science and Technology ISBN 978-981-33-4515-7 ISBN 978-981-33-4516-4 (eBook) https://doi.org/10.1007/978-981-33-4516-4 Jointly published with Publishing House of Electronics Industry The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Publishing House of Electronics Industry. ISBN of the Co-Publisher’s edition: 978-7-121-37276-6 © Publishing House of Electronics Industry 2021 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Integrated navigation is the result of the development of modern navigation theory and technology. Inertial navigation system has the advantages of strong autonomy, high short-term accuracy, all-time, all-weather, etc., so it has been applied in most integrated navigation systems. Among them, information processing is one of the core technologies of the inertial-based integrated navigation system. During use, due to the influence of the inertial device’s own principle and working environment, the output information of the inertial-based integrated navigation system has errors, including gyroscope noise, temperature drift, discontinuous observation, etc., which will reduce the navigation accuracy and robustness of the system. So it is very necessary to study the information processing technology of the inertial-based integrated navigation system. At present, the intelligent information processing technology of inertial-based integrated navigation system has become a research hotspot in the field of navigation at home and abroad. In foreign countries, the error processing for inertial devices is mainly based on the improvement of hardware circuits, which will increase the size of the device and increase the cost; for the integrated navigation method under discontinuous observation conditions, it is mainly to increase redundant sensors, which will also increase the navigation system. Additional burden, for the intelligentization of the navigation system, research on navigation methods based on brain navigation cell models has also been carried out. In China, the main work carried out for the error processing of inertial devices includes hardware circuits and software algorithms; for integrated navigation methods under discontinuous observation conditions, relevant research results have been achieved in adding redundant sensors and constructing discontinuous observation algorithms; research work has just begun on the intelligentization of navigation systems. With the development of artificial intelligence technology, advanced tools such as deep learning have received widespread attention in all walks of life and have achieved a series of breakthrough results. Combining artificial intelligence with the traditional inertial-based integrated navigation system is expected to greatly improve the accuracy and intelligence of the navigation system, thereby achieving a breakthrough in traditional navigation technology. v

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Preface

Based on the summary of domestic and foreign research status and its own years of scientific research results, this book closely follows the latest research results in the field of artificial intelligence, combines traditional navigation technology with advanced artificial intelligence methods, and condenses intelligent information processing technology for inertial-based integrated navigation systems. Compared with competing products such as traditional inertial navigation and integrated navigation reference books, it has made breakthroughs in technological frontier and innovation. In addition, the intelligent information processing technology involved in this book is equipped with simulation verification, which can be used as a reference for scientific research and engineering personnel engaged in navigation-related majors. Taiyuan, China

Chong Shen

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

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7 7 11 17 17

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4 Temperature Drift Modeling and Compensation for Gyroscope . . . . 4.1 Temperature Drift and Modeling Method of Fog . . . . . . . . . . . . . 4.2 Fiber Gyro Temperature Error Model Based on External Temperature Change Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49

2 Synopsis of Typical Inertial Sensors and System . . . . . . . . . . 2.1 Fiber Optic Gyro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 MEMS Gyro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Inertial Navigation System . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Strapdown Inertial Navigation System . . . . . . . . . . . 2.3.2 Overview of Basic Principles of Inertial Navigation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Basic Algorithm of Strapdown Inertial System . . . .

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3 Noise Analysis and Processing Technology for Gyroscope . . 3.1 Components of Gyro Noise and Allan Analysis Method . . 3.1.1 Source and Characteristic Analysis of Gyro Noise . 3.1.2 Allan Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Denoising Algorithm for Fog Signal . . . . . . . . . . . . . . . . 3.2.1 Enhance the Wavelet Transform . . . . . . . . . . . . . . 3.2.2 Forward Linear Prediction Algorithm . . . . . . . . . . 3.2.3 LWT-FLP Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Analysis of De-Noising Results of Fog Signal . . . . 3.3 The Method of Eliminating the Angular Vibration Error of Fog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Angular Vibration Experiment and Output Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Grey FLP Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.3.3 G-FLP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3 Temperature Drift Modeling and Compensation Based on Genetic Algorithm and ELMAN Neural Network . . . . . . . . . . 4.3.1 Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Elman Neural Network . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 GA-Elman-Based Fiber Optic Gyro Temperature Drift Modeling and Compensation . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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56 57 58 60

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62 66

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5 Model and Algorithm for Discontinuous Observation Integrated Navigation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Solutions and Typical Models of Discontinuous Observation Integrated Navigation System . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Application of Kalman Filter and Neural Network in Integrated Navigation of Non-continuous Observation . . . . . . . . . . . . . . . 5.2.1 Strong Tracking Kalman Filtering . . . . . . . . . . . . . . . . . 5.2.2 Wavelet Neural Network . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Experimental Results and Analysis . . . . . . . . . . . . . . . . 5.3 Application of Self-learning Cubature Kalman Filter in Combined Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Square Root Cubature Kalman Filter . . . . . . . . . . . . . . 5.3.2 Long-Short Term Memory Neural Network . . . . . . . . . . 5.3.3 Self-learning Volume Kalman Filter . . . . . . . . . . . . . . . 5.3.4 Experimental Results and Analysis . . . . . . . . . . . . . . . . 6 Brain-Like Navigation Technology Based on Inertial/Vision System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Overview of Bionic Navigation Background . . . . . . . . . . 6.2 Bionic Navigation Mechanism . . . . . . . . . . . . . . . . . . . . . 6.2.1 Positional Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Head Direction Cell . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Grid Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Velocity Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Brain Navigation System . . . . . . . . . . . . . . . . . . . 6.3 High-Speed Effective Node Matching Algorithm . . . . . . . 6.3.1 Scan Line Strength . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 GMS (Grid-Based Motion Statistics) . . . . . . . . . . . 6.3.3 Scan Line Intensity/GMS . . . . . . . . . . . . . . . . . . . 6.4 Algorithm Verification . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 Summary of Inertial-Based Navigation Intelligent Information Processing Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Research Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1

Introduction

With the development of social economy and the improvement of people’s living standard, more and more people choose cars as a means of transportation, which leads to the road traffic system becoming increasingly complex and crowded. Meanwhile, problems such as traffic congestion delay time, traffic accidents rising year by year, vehicle exhaust emissions and excessive energy consumption also appear, causing huge losses to the national economy. This situation gradually becoming the focus of scientific research and social concern. In the face of the development trend of globalization and information technology, the traditional transportation technology and methods are obviously unable to solve the increasingly serious transportation problems. Therefore, as a revolution in Transportation industry, Intelligent Transportation System (ITS) has become the inevitable choice for the development of Transportation industry. Through the effective integration and application of advanced communication technology, information technology, sensing technology, control technology, computer technology and integrated system technology, the interaction between people, vehicles and roads is presented in a new way, thus achieve the goals of accurate, real-time, efficient, energy saving and safety. As the sensor technology, communication technology, 3S technology, remote sensing technology, geographic information system, global positioning system (GPS) and the continuous development of computer technology, traffic information collection has experienced from manual collection into a single magnetic detector traffic information collection to the development process of multi-source traffic information collection, at the same time, with the deepening of the domestic and foreign research on traffic information processing method, statistical analysis technology, artificial intelligence technology, data fusion technology, parallel computing technology is gradually applied to traffic information processing, the processing of traffic information is continuous development and innovation, more meet the needs of ITS each subsystem administrators and users. At present, the main component systems of ITS

© Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_1

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1 Introduction

Fig. 1.1 Components of an intelligent transportation system

Management and control center module

Communication module

Vehicle module

Road module

include advanced traffic information service system (ATIS), advanced traffic management system (ATMS), advanced public transportation system (APTS), advanced vehicle control system (AVCS), freight management system, electronic charging system and emergency assistance system. Among them, advanced traffic information service system and advanced traffic management system are two important means of urban traffic management. ATIS is built on the basis of a complete information network. Traffic participants provide real-time traffic information to the traffic information center by equips sensors and transmission equipment on the road, on the car, in the transfer station, in the parking lot and in the weather center. ATIS obtained these information and through the processing, real-time traffic participants to provide road traffic information, public transport information, transfer information, traffic weather information, parking lot information and other information related to travel, according to these information, travelers can determine their way of travel. At the same time, the installation of automatic navigation and positioning system in the vehicle, the system can automatically help the driver to choose the route. ATM is mainly used for traffic management, used for testing, highway traffic management and control, to provide communication between road, drivers and vehicles, it can be on the road system in the transportation environment, traffic accidents and weather conditions, such as real-time monitoring, relying on advanced vehicle testing technology and computer information processing technology, to obtain the specific information of traffic conditions, and in order to control the traffic, such as release road control information, traffic induced and accident rescue etc. Although ITS research and expression forms are various, it is basically composed of four parts: road module, vehicle module, communication module and management and control center module. As shown in Fig. 1.1: 1. Management and Control Center Module: Manage and plan the traffic according to the collected vehicle and road information, realize the vehicle tracking, dispatching and management, release the charge management and service information, and achieve the arrangement and management of emergency measures. 2. Vehicle module: According to the signal collected by various sensors, the vehicle module can calculate the navigation and positioning information of the vehicle,

1 Introduction

3 Path induction

route plan

man-machine interface Wireless communication Digital map database

Navigation and positioning module

map matching

Fig. 1.2 Basic modules of vehicle navigation and positioning system

so as to ensure that the vehicle will travel according to the dispatch or path planning command. 3. Road module: The road module can provide real-time road traffic information, monitor road traffic conditions in real time, and complete functions such as electronic toll collection, vehicle detection and management. 4. Communication module: The role of the communication module is to realize the transmission and interaction of information between the above three modules. As an important part of intelligent transportation system, vehicle module navigation and positioning technology has been the focus of many researchers. It can also be seen from the above that one of the first problems to be solved by ITS is positioning. Only by obtaining accurate vehicle location information can the other service functions of ITS be further completed. Therefore, the current intelligent transportation system takes the road and vehicle positioning as the main research object to improve the capacity of the road. The functional modules of vehicle navigation and positioning system are shown in Fig. 1.2: It is the primary function of vehicle navigation system to provide vehicle position, speed and heading information. For any vehicle navigation and positioning system with good performance, accurate and reliable vehicle positioning is the premise and foundation to realize the navigation function. Considering the characteristics of the urban environment, is the most advanced and practical vehicle navigation technology development direction based on the combination of GPS navigation technology, GPS and other navigational technology, when using GPS satellite signal are available, when the satellite signal cannot be building lead to GPS positioning, the use of other high precision positioning technology for positioning. The performance characteristics of several major navigation systems that have been applied in various fields are shown in Table 1.1: It can be seen from Table 1.1 that several kinds of existing mainstream navigation and positioning systems all have their own characteristics. Due to the good

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1 Introduction

Table 1.1 Comparison of performance characteristics of several major navigation systems Inertial navigation

Radio navigation

GPS and other satellite navigation

Autonomy

Good (full autonomy) Poor (dependent on launch pad)

Poor (dependent on satellite)

Precision level

Errors accumulate over time

The navigation precision decreases with the increase of operating distance

High precision, does not change with time, place

Information comprehensiveness

Comprehensive (position, speed, attitude information available)

Incomplete (generally only provide information of position and speed)

Information real-time and continuity

Good

Worse

Capacity of resisting disturbance

Strong

Weaker

Weak

Applied characteristics

Global coverage, all weather, ground/underwater available

Used locally to receive radio signals well

Global coverage, all weather, good reception of satellite signals

Cost/price

Higher

Lower

Low

complementary characteristics of GPS and SINS, the combined GPS/SINS navigation system has been deeply studied and has been more and more widely applied in the vehicle navigation field. Inertial instruments (including gyroscope and accelerometer) are used to measure the angular motion and linear motion of the carrier relative to the inertial space in the SINS. Among them, gyro is the core component of inertial instruments, which has a great impact on the performance of SINS, so the selection of gyro plays a crucial role in vehicle navigation and positioning accuracy. Different application fields have different requirements on positioning accuracy, as shown in Table 1.2. As can be seen from Table 1.2, in order to achieve functions such as collision prevention and intersection detection, high-precision vehicle positioning has become the goal of the development of ITS technology. In order to achieve high precision vehicle positioning, high precision gyroscope must be selected as the inertial measurement component. Since the 1960s, in addition to the rapid development of the traditional rotor gyro, a number of new types of gyros and the corresponding inertial navigation system have been developed and matured, mainly including electrostatic gyro, flexible gyro, resonant gyro, laser gyro, fiber optic gyro and micro-electromechanical Systems (MEMS) gyro. Among them, electrostatic and laser gyro have higher precision. For example, electrostatic gyro’s precision can reach 10–11 (°)/h under the conditions of microgravity and vacuum of satellites. The laser-gyro strapdown inertial navigation system has been successfully tested on aircraft and missiles

1 Introduction

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Table 1.2 Requirements of positioning accuracy for different systems Application fields

Accuracy requirement (m)

Commercial vehicle management and scheduling

100

ACI

30

Automatic voice station announcement

25–30

Data acquisition

25

Navigation and route guidance

5–20

Lost vehicle tracking

10

Special vehicles such as ambulance, armored vehicle management and scheduling

10

Public safety

10

Positioning of service places (e.g. restaurants, shopping malls)

10

Anticollision

1

Intersection detection

0.1

since the 1970s, and the navigation accuracy can reach 1n mile/h. However, the structure and manufacturing process of electrostatic and laser gyro are relatively complex and high cost, so they are used in submarines, missiles, aircraft and other high-value carriers. Flexible gyros and resonant gyros are simple in structure, low in cost and easy to be produced in batches. However, due to their low precision, they can only be used for vehicle navigation and positioning in general situations. Compared with esg and laser gyro, fiber optic gyro has its own advantages in light and small size, low power consumption, long life, high reliability and mass production. Compared with the flexible gyro and resonance gyro, the precision of fiber optic gyro is higher, which can be used in the vehicle navigation which requires the high precision of navigation and positioning. Therefore, fiber optic gyro (fog) plays an important role in the navigation and positioning of large unmanned vehicles. In addition, with the development of MEMS technology, the accuracy of MEMS gyro gradually increased, such as Norway SENSONOR company manufacturing STIM210 series of MEMS gyroscope zero bias stability up to 0.5°/h, and have the characteristics of low cost, small volume, low power consumption, so in some applications can completely replace the fiber optic gyro, have extensive application prospects, especially in the field of miniature unmanned platform, has become the most mainstream of inertial navigation and positioning method.

Chapter 2

Synopsis of Typical Inertial Sensors and System

2.1 Fiber Optic Gyro In 1976, Vail and r. w. Shorthill of the university of Utah successfully developed the first fiber optic gyro, which marked the birth of fiber optic gyro (the second generation of optical gyro) (the first generation was laser gyro). Since its inception, the fog has attracted the attention of researchers all over the world because of its flexible structure and attractive prospect. So far, it has made great progress. Since 1990, the fiber optic gyro inertial navigation system has been gradually put into use and can be mass produced. Nowadays, through the continuous efforts of researchers, many key technologies have been perfectly solved, and the precision measurement has been improved from 15°/h in the past to the order of 0.001°/h now. Compared with other gyros, FOG has the following remarkable characteristics: (1) simple structure, all-solid structure, strong resistance to acceleration and impact; (2) the design of optical fiber coil increases the detection path of laser beam, improves the detection resolution and sensitivity, and effectively overcomes the blocking problem in laser gyro; (3) no mechanical transmission parts, no wear, long service life; (4) the coherent beam can be started instantaneously due to its short propagation time; (5) it is convenient to use integrated optical path technology, and the signal is stable, digital output can be realized, and it is convenient to connect with the computer interface; (6) wide dynamic range; (7) can be used together with the laser gyro, constitute a variety of inertial navigation system sensors, especially strapdown inertial navigation system sensors. From the perspective of principle and structure, the types of fog mainly include interferometric fiber gyroscope, resonator fiber gyroscope, brillouin fiber gyroscope, mode-locked fiber gyroscope and fabry-perot fiber gyroscope. From the perspective © Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_2

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2 Synopsis of Typical Inertial Sensors and System

of structure, it can be divided into two categories: open-loop fiber optic gyroscope and closed-loop fiber optic gyroscope. From the perspective of phase demodulation mode, it can also be divided into optical heterodyne fiber optic gyroscope, phase difference offset fiber optic gyroscope and delay modulated fiber optic gyroscope. Since the 1990s, foreign fiber optic gyros have gradually entered the stage of industrial development and mass production can be realized. Fiber optic gyro manufacturers with scale production capacity are mainly distributed in the United States, followed by Germany, France and so on. Northrop built a tactical integrated inertial system production line in the early 1990s, paying tens of thousands of sets of fiber optic gyro inertial system products and nearly tens of thousands of fiber optic rate gyros to different users. Honeywell was the first company to use fiber optic gyroscope in the field of commercial aviation, and developed high-precision fiber optic gyroscope for submarine inertial navigation for the U.S. navy. The test results show that the long-term zero-bias stability of fiber optic gyroscope for 14 h (1σ) is 0.00038°/h. KVH’s open-loop fiber optic gyro (fog) has been widely used in vehicle/ship stabilization system, north finder and attitude system since 1987. In 2019, KVH integrated photon-chip technology into fiber optic gyroscope products. The random walk Angle is less than 0.0097°/h, and the zero-bias stability is 0.02°/h. Litief mainly produces closed-loop fiber optic gyros, and its single-axis fiber optic gyro delivered thousands of them around 2000. French photon Ixsea company began to develop fiber optic gyros in the 1980s. From 1986 to 1999, the company has been widely used in Marine and space applications. Japan’s Hitachi cable, mitsubishi precision and javic are the three main companies developing fiber-optic gyros in Japan, among which Hitachi cable has sold tens of thousands of fiber-optic gyros for car navigation. Fiber optic gyro (fog) has been developed in China since the early 1980s. At present, the FOG level is close to the accuracy requirement of inertial navigation system. Among them, the medium-and low-precision closed-loop fiber-optic gyro developed by Beijing university of aeronautics and astronautics has reached the practical stage, and the zero-bias stability of the bait-doped fiber-optic gyro developed by it reaches 0.01°/h in the laboratory environment. In addition, 33 institutes of aerospace, 13 institutes of aerospace times electronics, 618 institutes of aviation, 707 institutes of China ships, Zhejiang University, Beijing institute of technology and other units have also made some breakthroughs in the development of fog (Fig. 2.1). Fiber optic gyro (fog) inertial system is a general term for various strapdown inertial systems with fiber optic gyro (fog) as angular motion measuring instrument. Because different fields and carriers have different requirements on the products of fog inertial system, there are many different types of fog inertial system products. According to the different accuracy levels of fog and accelerometer, it can be divided into the following categories: (1) low precision or tactical products: general gyro precision is 10 ~ 0.1°/h (1σ) level, accelerometer precision is 10 ~ 3 g (1σ) level; (2) high precision or navigation level products: general gyro precision is 0.01°/h (1σ) level, accelerometer precision is better than 10-4 g (1σ) level;

2.1 Fiber Optic Gyro

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Fig. 2.1 gxt-4b single-axis fiber optic gyroscope of space agency 16

(3) precision products: the general gyro precision is 0.003°/h (1σ) level, the accelerometer precision is better than 10-5 g (1σ) level. All kinds of foreign fiber optic gyro inertial system products have been widely used in aerospace, aviation, ships, vehicles and other military and civilian fields, mainly including the following categories: (1) applications and typical products in missiles and rockets Northrop’s ln-200 series of fiber optic gyro inertial measurement products have been successfully used in aircraft, rockets, lunar and Mars rovers, unmanned aerial vehicles, submarines and other carriers, but also widely used in missiles, tanks, attitude stabilization systems. Ln-200 series products are composed of 3 fiber optic gyros (with an accuracy of about 1°/h) and 3 silicon microaccelerometers (with an accuracy of about g), which can provide 3d acceleration and angular velocity signals. (2) applications and typical products in spacecraft The aircraft navigation and attitude stabilization fog developed by honeywell has an accuracy of 0.00038°/h (1σ) for the Hubble space telescope. Litton also developed a FOG2500 fiber optic gyro with a drift less than 0.0005°/h (1σ) suitable for the space environment. (3) applications and typical products in aircraft More than 10,000 sets of lcr-9x series airborne fiber optic gyro course attitude reference systems have been produced by litief in Germany, which have been applied in many types of aircraft. The performance indexes of lcr-92/93 are: heading accuracy 1.0° (95%), attitude accuracy 0.5° (95%), angular rate range °/s. (4) application in ships and typical products The Marine inertial navigation system PHINS was developed by IXSea company in France. The precision of fiber optic gyro used is 0.01°/h at full temperature (-40-60 2103), 0.002°/h at normal temperature, and 0.6 n mile/h at pure inertial positioning.

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2 Synopsis of Typical Inertial Sensors and System

(5) applications and typical products in land vehicles Fiber optic gyro has been more and more application in civilian vehicles, such as Japan’s Hitachi years ago has been in mass production price is relatively cheap low precision fiber optic gyro navigation products for ordinary civil cars, mainly used in automatic stabilization the posture of the running train and safety brake system, etc., can greatly improve the ride comfort and safety of the vehicle. (6) applications and typical products in the field of rate sensing Since 1990, livef has been developing a series of FORS single-axis fiber optic gyros and using them in the field of rate sensing, with measuring accuracy ranging from 1 to 36°/h. Fo-200, another rate sensor product of life company, has a zero deviation error of about 3°/h in the full temperature range, a scale factor accuracy of about 2000 ppm, and a power consumption of less than 5 W. Its high-vibration product can work in very harsh vibration environment. In recent years, China’s inertia technology has also been rapidly developed, the product development, production and other aspects of the technology has gradually become mature, and in the sea, land, air, air and various fields have been more and more widely used. Generally speaking, the characteristics of fog inertial system are as follows: (1) fiber optic gyro inertial system has the characteristics of high reliability, long life, good parameter stability, and performance not affected by the number of starts, which can achieve the goal of long-term calibration and maintenance free; (2) the mass, volume and power consumption of the fog inertial system are relatively small, which can directly output digital quantity, and the external interface is simple, without the need for ac and high-voltage power signals; (3) fiber optic gyro inertial system has a large angular velocity measurement range (1000°/s), a wide dynamic response band (up to 200 Hz), a wide precision coverage, and a fast starting speed, which is suitable for use in high-dynamic and quick-start situations; (4) the environment adaptability of the fog inertial system is good, medium and low precision products can solve the impact of high and low temperature environment on its accuracy through software compensation; For products with high precision, appropriate temperature control measures can be adopted. Fiber optic gyro inertial system has good adaptability to the mechanical environment, and basically no mechanical resonance point can be achieved for the vibration within 2000 Hz; (5) fiber optic gyro inertial system requires no precise machining, high cleanliness assembly and other links, and the production process is relatively simple, which is conducive to the mass production and low-cost production of products. With the continuous improvement of advanced navigation and guidance system in military and civilian fields, the comprehensive requirements for high-performance inertial instruments and their systems are also increasing day by day. In summary,

2.1 Fiber Optic Gyro

11

the future development trend of fog inertial system mainly includes the following aspects: (1) to high precision, high reliability, long life direction; (2) to light, small, integrated and fully digital development direction; (3) develop towards inertial integrated navigation and multi-information fusion technology; (4) develop towards the direction of redundant and fault-tolerant inertial system; (5) to serialize, low cost, shelf type, industrialization direction. The new generation of solid state inertial devices represented by fiber optic gyro inertial system is promoting the inertial technology to a new future, and will promote the application of inertial technology in the fields of aviation, aerospace, navigation, vehicle, industrial production and daily life.

2.2 MEMS Gyro In the second half of the 80 s, the birth of the MEMS gyroscope, low cost, low quality and working life is long, wide dynamic range, easy to digital, highly intelligent, and have strong overload capacity resistance, suitable for high speed, big g value, ease of integration installed in a variety of complex control system, thus realize the microelectro-mechanical integration. The excellent performance of gyroscope determines that it is bound to be widely used. MEMS gyroscope is widely used in aerospace, automobile navigation, automobile anti-lock braking system, artillery shell/grenade guidance and other fields, as shown in Fig. 2.2. Traditional gyroscope is the basic working principle of conservation of angular momentum: when sensitive element vibration in excited state, received the angular velocity and the vertical direction of vibration, the element with the inherent vibration frequency in the other direction, phase is associated with the direction of the angular velocity, amplitude is proportional to the angular velocity, through the vibration of the component can get the angular velocity gyroscope. When the gyro rotates in the direction of the axis of rotation, it will not change anything without an external force acting on it. Gyroscopes follow this principle. When the top bears the load, it starts to turn. The rate can be up to tens of thousands of revolutions per second, and work for a long time. After that, the signal is transmitted to the control terminal through various data exchange channels, and the current direction can be obtained through the control system analysis. Therefore, gyro has been applied in many key fields such as aerospace. But the principle of MEMS gyroscope is more complicated. Generally speaking, it is mainly explained as coriolis effect, that is, in the rotating system, the particle moving in a straight line still has the description of linear motion relative to the rotating system due to the action of inertia. However, due to the rotation of the system, the position of the particle relative to the system will shift after a period of time. The Koch effect theory is the theoretical basis of MEMS gyroscope. It means

12

2 Synopsis of Typical Inertial Sensors and System

According to vibration structure

Rotating vibration structure Line vibration structure

Micromachined gyroscope classification

Silicon material According to material

Non-silicon materials Piezoelectric drive

According to drive

Orthogonal line vibration structure

Non-orthogonal line vibration structure

Vibrating plate structure

Polysilicon

Vibration beam construction

quartz

Vibrating tuning fork structure

other

The accelerometer vibrates

Closed loop mode

Thin-walled hemispherical resonance gyro

Electrostatic drive

Rate gyroscope Rate integrating gyroscope

Open loop mode

Full angle mode According to detecon method

According to processing method

Rotating disc structure gyro

Monocrystalline silicon

Electromagnetic drive According to working mode

Vibrating disc structure gyro

Body micromachining

Piezoelectric detection

Surface micromachining

Capacitive detection

LIGA

Resonant cylindrical gyro Resonance ring structure gyro

Optical inspection Piezoresistive detection Tunnel effect detection

Fig. 2.2 Classification of micromachined gyros

that when an object of mass m moves with the angular velocity of w on a circle of radius r, it can generate the Koch force of size F. Compared with the traditional gyroscope, its advantages are as follows: (1) Low cost, mass production, short production cycle The fabrication process of silicon structure can be compatible with the integrated circuit process, so that the sensitive structure and the measurement and control circuit can be integrated on the same chip. At the same time, the standardized process can greatly increase the output of silicon micromechanical gyroscope (which is unmatched by traditional gyroscope), so as to reduce the cost of a single gyroscope. At present, the price of a single silicon micromechanical gyroscope on the market is about one thousandth to one millionth of that of the traditional gyroscope. At least three single-axis gyroscopes are often needed in inertial navigation system. The huge demand of the gyroscope can show the cost advantage of the silicon micromechanical gyroscope. In addition, due to the characteristics of batch processing of silicon micromachined gyroscope, once the model is formed, a large number of products can be processed in a short time and can be quickly equipped, which is of great significance to national defense construction.

2.2 MEMS Gyro

13

(2) Small size, light weight, low power consumption The sensitive structure of silicon micromachined gyroscope is machined by micromachined, and its measurement and control circuit can also be realized by integrated circuit, so the silicon micromachined gyroscope has the characteristics of small size, light weight and low power consumption. Taking the silicon micro-mechanical gyroscope ADXRS150 produced by AD (Analog Device) company as an example, its size is 7 mm × 7 mm × 3 mm, and its power consumption is about 40 mW. Therefore, silicon micromechanical gyroscope is more suitable for the occasion where there are strict requirements on volume, weight and power consumption. In practical application, a monolithic integrated triaxial micromechanical gyroscope can be used to further improve its advantages in volume, weight and power consumption. In addition, several redundant configurations of micro gyro or gyro matrix can be used in inertial navigation system to improve the accuracy and reliability of the system. (3) High reliability, long service life, good impact resistance and good dynamic performance Due to the lack of high-speed rotating rotor in the structure of silicon micromechanical gyroscope, it can be regarded as a solid device without mechanical wear, and its structure can be integrated with the electronic circuit, greatly reducing the adverse impact of external interference, so the silicon micromechanical gyroscope has high reliability and long life. In addition, due to the structure of light weight and silicon material has good elasticity, so the structure of silicon micromechanical gyroscope with inertia small, fast response, good dynamic performance, impact resistant ability, the advantages of even can bear the impact of more than 100,000 g, this makes the silicon micromechanical gyroscope can be used in the fight against impact performance and dynamic performance of high demand environment. (4) Easy to be digitized and intelligent The silicon micro-mechanical gyroscope measurement and control circuit can output analog signals, digital signals, frequency signals, etc. according to the special requirements of different inertial navigation systems. It can also be combined with the microprocessor to cooperate with peripheral sensors and related algorithms to realize self-calibration, self-detection, self-compensation, and improve the environmental adaptability. Since the first silicon micromechanical gyroscope was developed in Draper laboratory in the United States in 1988, the precision of the silicon micromechanical gyroscope has been greatly improved with the continuous optimization and improvement of the new structure and measurement and control methods. The structure proposed by Seoul national university in South Korea in 2007 is shown in Fig. 2.3. Its structure is in the form of vibration of single mass fully decoupled line. From the distribution of electrodes, it can be seen that the gyro adopts push-pull drive and differential detection, and the detection feedback electrode in the structure can realize closed-loop control of the detection loop.

14

2 Synopsis of Typical Inertial Sensors and System

Fig. 2.3 Micromachined gyroscope structure and measurement and control system, Seoul national university

German Bosch company reported a dual-mass line vibrating silicon micromechanical gyroscope drs-mm3 applied to automotive systems. Its structure adopts the vibration form of tuning fork, and the left and right mass blocks are coupled by intermediate connecting beams. In the aspect of measurement and control system, the digital integrated circuit is adopted. The digital processing part is mainly composed of driving closed-loop loop, detecting closed-loop loop (adopting the detection closed-loop control mode) and output compensation loop (Fig. 2.4). The four-mass line vibrating silicon micromechanical gyroscope proposed by the university of California, Irvine is shown in the following figure. In order to reduce the mechanical thermal noise and improve the signal-to-noise ratio of the output signal of the structure, the structure adopts a high vacuum degree (the base mechanical thermal noise in the air is about 10°/h RMS, while in the vacuum the value is increased to 0.01°/h RMS) (Fig. 2.5). Figure 2.6 presents the structure of the butterfly gyroscope developed by the National University of Defense Technology in 2011. After an initial structure two

Fig. 2.4 Micromachined gyro measurement and control system proposed by Bosch company, Germany

2.2 MEMS Gyro

15

Fig. 2.5 Micromachined gyroscope structure and photographs presented by the university of California, Irvine, USA

Fig. 2.6 The butterfly gyroscope structure of National University of Defense Technology

years ago, researchers add a quadrature correction electrode. The method of stiffness correction is used to suppress the quadrature error. The performance of the gyroscope is significantly improved after correction. Under the open-loop detection state, the drift trend is improved. The bias stability is improved from 89°/h to 17°/h, and the scale factor temperature stability is reduced from 622 to 231 ppm/°C. In 2013, Nanjing University of Science and Technology proposed a dual-mass linear vibration, dual decoupling structure and its measurement and control circuit (as shown in Fig. 2.7). With the method of vacuum packaging, the volume of this structure and overall gyroscope are 15 × 15 × 3.5 mm3 and 31 × 31 × 12 mm3 separately. In addition, the power consumption of gyroscope is 288 mW, the scale factor is nonlinear, the symmetry and repeatability are 37 ppm, 184 ppm and 155 ppm, the bias repeatability is 12°/h, the threshold and resolution are 0.008°/s similarly.

16

2 Synopsis of Typical Inertial Sensors and System

Fig. 2.7 The dual-mass gyroscope structure and its measurement and control system of Nanjing University of Science and Technology

In 2014, Southeast University proposed a dual-mass linear vibration gyroscope structure with quadrature correction combs and detection feedback teeth, which is shown in Fig. 2.8. Furthermore, based on FPGA and analog circuits, the closed-loop control loops of quadrature correction and detection are designed respectively.

Fig. 2.8 The dual-mass gyroscope structure and the measurement and control system based on analog circuits of Southeast University

2.2 MEMS Gyro

17

Fig. 2.9 STIM210 MEMS gyroscope module

Figure 2.9 shows the STIM210 utilized in this book, which is a MEMS gyroscope module dedicated to high-end application field launched by Sensonor company in Norwegian. It adopts micro-machining technology, featuring high precision, light weight, small size, etc., and has extremely strong resistance in harsh environment and excellent shock and impact resistance. STIM210 is calibrated before delivery and temperature compensated within the full temperature range. In the full working range, the bias is 10°/h and the bias instability is 0.5°/h.

2.3 Inertial Navigation System 2.3.1 Strapdown Inertial Navigation System To facilitate analysis and discussion, several frequently-used coordinate systems are defined: (1) Geocentric inertial coordinate system (I system): Leaving the revolution of the earth and relative space motion of solar system out of consideration, the center of the earth is taken as the origin of the coordinate system. Furthermore, the xi axis points to the vernal equinox point, and the z i axis points to the polar axis of the earth. The direction of yi axis and xi z i axes constitute the right-hand system. (2) Earth coordinate system (E system): This system takes the center of the earth as origin of the coordinate system. The xe axis passes through the prime meridian, and the z e axis points to the polar axis of the earth, and the direction of the ye axis is determined by the right-hand rule. The coordinate system is firmly connected with the earth, and it can be approximated that the coordinate system rotates

18

2 Synopsis of Typical Inertial Sensors and System

at the earth’s rotation angular rate, which is relative to the inertial coordinate system. (3) Geographic coordinate system (G system): Centroid of carrier is considered the origin. The ENU coordinate system and the NED coordinate system are the most common geographical coordinate systems. The x g , yg , and z g axes point to the east of the carrier (refer to the direction of the ellipsoidal unitary circle), the North (refer to the direction of the ellipsoid meridian), the Zenith (refer to the right-hand rule), and the North (refer to the direction of the ellipsoid meridian), the East (refer to the direction of the ellipsoid unitary circle), and the Earth (refer to the right-hand rule) respectively. (4) Carrier coordinate system (B system): This coordinate system is firmly connected with the carrier, and the centroid of carrier is considered its origin. The xb axis, yb axis and z b axis point to the right, upper and front of the carrier respectively. (5) Platform coordinate system (P system): In the inertial navigation system, the right-hand rectangular coordinate system which is connected with the physical platform (platform system) or the mathematical platform (strapdown system) of the inertial navigation system is called the platform coordinate system. Its coordinate origin is generally the centroid of the carrier or the centroid of inertial navigation system, and the orientation of each axis is consistent with the navigation coordinate system. The inertial navigation system can simulate the navigation coordinate system with the help of p-system.

2.3.2 Overview of Basic Principles of Inertial Navigation System Newton’s second law that laid the theoretical foundation of optical gyroscope is the rate of change (absolute rate of change) obtained with respect to the time of the inertial coordinate system, while the inertial navigation system studies the rate of change (relative rate of change) with respect to time of the vector projection on a specific moving coordinate system (such as geographic coordinate system). Supposing there is a space vector X (such  as vector,  velocity, etc.) whose magnitude and direction  and dX  represent the absolute change rate and relative change with time, dX dt i dt r change rate of the vector relative to a certain definite system and a certain moving reference system respectively. Assuming that the angular velocity of the moving reference system relative to the definite system is ω, the relationship between the absolute change rate and the relative change rate can be derived from the particle kinematic formula:   dX  dX  = +ω×X (2.1) dt i dt r

2.3 Inertial Navigation System

19

When studying the movement of a carrier in an inertial navigation system, the velocity and position of the carrier are usually determined relative to the earth. The earth coordinate system (E system) can be considered as the moving reference system, and the geocentric inertial coordinate system (I system) can be seen as the definite system. Then the angular velocity of the earth coordinate system relative to the inertial coordinate system is (i.e. the rotation speed of the earth). Assuming that the vector from the center of Earth to the carrier centroid is R, according to formula (2.1):   dR  dR  = + ωie × R dt i dt e

(2.2)

  represents the rate of change of the carrier position vector observed where dR dt e on the earth coordinate system (moving reference system), otherwise known as the velocity of the carrier relative to the earth, which is recorded as Vep . According to formula (2.1), the absolute change rate can be calculated on both sides of formula (2.2), and the relative change rate can be calculated on P system:    dVep  dωie  d 2 R  = + ωi p × Vep + ωie × (Vep + ωie × R) + ×R dt 2 i dt  p dt i Because of ωi p = ωie + ωep and



dωie  dt i

(2.3)

= 0, formula (2.2) can be written as:

  dVep  d 2 R  = + (2ωie + ωep ) × Vep + ωie × (ωie × R) dt 2 i dt  p

(2.4)

Assuming that the mass of the accelerometer’s sensitive mass block in the inertial navigation system is m, the mass m is affected by the non-gravity external force F and the earth’s gravity mG, and G is the gravitational acceleration. According to the  d2R  Newton’s second law F + mG = m dt 2  , we have: i

 d 2 R  = f +G dt 2 i

(2.5)

where f = mF indicates non-gravity external force acting on unit mass (specific force). Substituting formula (2.5) into formula (2.4) can obtain:  dVep  = f − (2ωie + ωep ) × Vep + G − ωie × (ωie × R) dt  p

(2.6)

where |ωie × R| = ωie R sin(90◦ − L) = ωie R cos L; |ωie × (ωie × R)| = ωie · 2 cos L. L is the angle between the geographic vertical (ωie R cos L) sin 90◦ = Rωie line and the equatorial plane, which is the local geographic latitude.

20

2 Synopsis of Typical Inertial Sensors and System

It can be seen that the magnitude of the centripetal acceleration ac = ωie × (ωie × 2 cos L, and the direction points to the axis of Earth, that is: R) is ac = Rωie g = G − ac Formula (2.7) can be written as:  dVep  = f − (2ωie + ωep ) × Vep + g dt  p

(2.7)

(2.8)

Formula (2.8) is the specific force equation, which is the basic formula of the inertial navigation system. The explanation of the specific force equation is as follows:  dV  (1) dtep  is change rate of carrier’s ground velocity vector Vep on platform coorp

dinate system. If formula (2.8) is projected into the P system, the specific force equation can be written as a component form:

p

p Vepp = f p − (2ωie + ωep ) × Vepp + g p p

(2.9)

p

where 2ωie + ωep represents projection of the Coriolis acceleration term in P system; p p ωie + ωep indicates centrifugal acceleration in P system, and g p is gravitational acceleration. (2) f represents the accelerometer measurement, and the specific force equation illustrates that only after the Coriolis acceleration and the centrifugal acceleration in the specific force are compensated and the gravitational acceleration is calculated, the speed of the carrier relative to the navigation coordinate system can be obtained through integration.

2.3.3 Basic Algorithm of Strapdown Inertial System 2.3.3.1

The Transformation of Coordinate System

In the process of navigation solution, the mutual transformation between coordinate systems is usually involved. The methods commonly used to describe the relationship between the two coordinate systems are directional cosine matrix method and quaternion method. (1) Directional cosine matrix method Any complex angular positional relationship between the two coordinate systems can be regarded as a basic composite of finite order, and the attitude transformation

2.3 Inertial Navigation System

21

Fig. 2.10 Carrier attitude change diagram

matrix is equal to the result of the transformation matrix determined by the basic rotation. Assuming that the attitude change of the carrier is a composite result of basic rotation around the heading axis, pitch axis, and roll axis in sequence, which can be expressed as follows (Fig. 2.10). The transformation matrix corresponding to the basic rotation around the heading axis, pitch axis and roll axis is: ⎡

⎡ ⎡ ⎤ ⎤ ⎤ cos Ψ − sin Ψ 0 1 0 0 cos γ 0 − sin γ C1n = ⎣ sin Ψ cos Ψ 0 ⎦ C2n =⎣ 0 cos θ sin θ ⎦ Cb2 = ⎣ 0 1 0 ⎦ 0 0 1 0 − sin θ cos θ sin γ 0 cos γ (2.10) The coordinate transformation matrix from O − X n Yn Z n system to O − X b Yb Z b system can be expressed as: ⎡ ⎢ Cbn = Cb2 C21 C1n = ⎣

⎤ cos γ cos Ψ + sin γ sin Ψ sin θ − cos γ sin Ψ + sin γ cos Ψ sin θ − sin γ cos θ ⎥ sin Ψ cos θ cos Ψ cos θ sin θ ⎦ sin γ cos Ψ − cos γ sin Ψ sin θ − sin γ sin Ψ − cos γ cos Ψ sin θ cos γ cos θ

(2.11)

22

2 Synopsis of Typical Inertial Sensors and System

(2) Quaternion method The definition of quaternion is: Q = q0 + q1 i + q2 j + q3 k

(2.12)

where q0 represents scalar part of quaternion, and the last three are vector part of quaternion. Using quaternion to describe the rotation of rigid body around fixed point: Q = cos

θ θ θ θ + sin cos α · i + sin cos β · j + sin cos γ · k 2 2 2 2

(2.13)

where θ is rotation angle, and cos α, cos β, cos γ represent the direction cosine between the axis of instantaneous rotation and the axis of the reference coordinate system. A quaternion can represent the direction of the rotation axis and the value of the rotation angle, and the rotation relationship can be expressed by the following operation: Rb = Q ⊗ Rn ⊗ Q∗

(2.14)

where Q∗ is conjugate quaternion of quaternion Q. According to quaternion multiplication, the formula (2.14) can be decomposed into: ⎡ 2 ⎤ 2(q1 q3 + q0 q2 ) q0 + q12 − q22 − q32 2(q1 q2 − q0 q3 ) Rb = ⎣ 2(q1 q2 + q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 − q0 q1 ) ⎦Rn (2.15) 2(q2 q3 + q0 q1 ) q02 − q12 − q22 + q32 2(q1 q3 − q0 q2 ) The attitude transformation matrix between two coordinate systems is also obtained: ⎡ 2 ⎤ 2(q1 q3 + q0 q2 ) q0 + q12 − q22 − q32 2(q1 q2 − q0 q3 ) (2.16) Cbn = ⎣ 2(q1 q2 + q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 − q0 q1 ) ⎦ 2(q2 q3 + q0 q1 ) q02 − q12 − q22 + q32 2(q1 q3 − q0 q2 ) It can be seen from formulas (2.11) and (2.16) that both the quaternion method and the direction cosine matrix method can represent the attitude transformation matrix between the two coordinate systems, that is, both can be used to describe the transformation between coordinate systems. In essence, they are equivalent.

2.3.3.2

Quaternion Algorithm for Attitude Calculation

The attitude angle of the carrier can be determined by three ordered rotations of the navigation coordinate system relative to the carrier coordinate system.

2.3 Inertial Navigation System

23

The coordinate transformation matrix Cbn from the carrier coordinate system b to the navigation coordinate system n is called the carrier attitude matrix. Since the carrier’s attitude is constantly changing, each element of the attitude matrix is a function related to time. Attitude update means that the attitude matrix is calculated according to the real-time output of the inertial device, and finally each attitude angle can be determined according to the relationship between the attitude matrix and the attitude angle. In order to solve attitude matrix, the following quaternion differential equation need to be solved first according to the theory of rigid body rotation: ˙ = 1 M∗ (ωb )Q Q 2

(2.17)

And: ⎤ by bz bx 0 −ωnb −ωnb −ωnb ⎢ bx by ⎥ bz 0 ωnb −ωnb ⎥ ⎢ω M∗ (ωb ) = ⎢ nb ⎥ by bz bx ⎣ ωnb −ωnb 0 ωnb ⎦ by bz bz ωnb ωnb −ωnb 0 ⎡

⎡

⎤ q0 ⎢ q1 ⎥ ⎥ Q=⎢ ⎣ q2 ⎦

(2.18)

q3

b where ωnb represents the projection of the rotation rate on carrier coordinate system relative to the navigation coordinate system, whose value can be obtained by the following formula: b b n n = ωib − Cbn (ωie + ωen ) ωnb

(2.19)

b is angular rate of gyros output after error compensation; Cbn represents where ωib n is rotation rate of the attitude matrix determined by lasted value of attitude update; ωie n earth; and ωen expresses the projection of the rotation rate on navigation coordinate system relative to earth coordinate system. In general, the navigation coordinate n n and ωen can be system is taken as the geographical coordinate system, then ωie expressed as:

T n = 0 ωie cos L ωie sin L ωie  n ωen = − VRNn

VE VE tan L Re Re

(2.20)

T (2.21)

where VN , V and L represent northing velocity, easting velocity and latitude calculated by the real-time measurement information of the measuring elements; Rn is radius of curvature in meridian, and Re is radius of curvature in prime vertical. Picard successive approximation method is applied to solve quaternion differential equations. According to the general solution form of differential equation, the recurrence formula of quaternion is:

24

2 Synopsis of Typical Inertial Sensors and System 1

Q(tk+1 ) = e 2

tk+1 tk

b M∗ (ωnb )dt

· Q(tk )

(2.22)

And: ⎤ 0 −θx −θ y −θz ⎢ θx 0  b  θz −θ y ⎥ ⎥ dt ≈ ⎢ M∗ ωnb ⎣ θ y −θz 0 θx ⎦ θz θ y −θx 0 ⎡

tk+1  = tk

(2.23)

where θx , θ y and θz are the angular increments of the gyroscope in the direction of x-axis, y-axis and z-axis in the carrier coordinate system within the sampling time interval from tk to tk+1 . After Taylor expansion of Eq. (2.22), we have:  Q(tk+1 ) = e

1 2 

· Q(tk ) = I +

1  2

1!

+

1 ()2 2

2!

 + · · · Q(tk )

(2.24)

Because of: ⎤⎡ ⎤ 0 −θx −θ y −θz 0 −θx −θ y −θz ⎢ ⎢ θx 0 θz −θ y ⎥ θz −θ y ⎥ ⎥⎢ θx 0 ⎥ 2 = ⎢ ⎣ ⎦ ⎣ θ y −θz 0 θx θ y −θz 0 θx ⎦ θz θ y −θx 0 θz θ y −θx 0 ⎡ ⎤ 2 −θ ⎢ ⎥ −θ 2 ⎥ − θ 2 I =⎢ (2.25) 2 ⎣ ⎦ −θ −θ 2 ⎡

Then: 3 = −θ 2  4 = θ 4 I 5 = θ 4  6 = −θ 6 I

(2.26)

where: θ 2 = θx2 + θ y2 + θz2 Substitute the above formula:

(2.27)

2.3 Inertial Navigation System

25



 sin θ θ 2 +  Q(tk+1 ) = I cos Q(tk ) 2 θ

(2.28)

In the actual calculation, Taylor series expansion and truncation approximation will be carried out for cos θ and sin θ . 2 2 In addition, the quaternion used to characterize rotation should be normalized. Due to calculation errors, the quaternion will gradually lose the normalization characteristics during the iterative update process. It is necessary to normalize the quaternion periodically: qˆi2 , i = 0, 1, 2, 3 qi =  qˆ12 + qˆ22 + qˆ32 + qˆ42

(2.29)

where qi is normalized quaternion, and qˆi is the value obtained in iterative update process. From the quaternion derivation process of the relationship between coordinate systems in the previous section, it can be seen that the attitude matrix can be determined by the quaternion of attitude updating after normalization (formula 2.16). The relationship between attitude angle and attitude matrix can be derived from the direction cosine matrix method: ⎡

⎤ cos γ cos Ψ + sin γ sin Ψ sin θ − cos γ sin Ψ + sin γ cos Ψ sin θ − sin γ cos θ ⎢ ⎥ Cbn = ⎣ sin Ψ cos θ cos Ψ cos θ sin θ ⎦ sin γ cos Ψ − cos γ sin Ψ sin θ − sin γ sin Ψ − cos γ cos Ψ sin θ cos γ cos θ

(2.30)

The pitch angle, roll angle and heading angle of the carrier can be calculated: ⎧   ⎪ C 23 ⎪ θ = arctan √ 2 2 ⎪ ⎪ ⎨  22  C21 +C C13 γ = arctan − ⎪ ⎪  C33 ⎪ ⎪ ⎩ Ψ = arctan C21 C22

2.3.3.3

(2.31)

Solution of Carrier Velocity

When the navigation coordinate system is taken as the geographical coordinate system, the specific force equation can be expressed as: n n + ωen ) × V n + gn V˙ n = Cbn f b − (2ωie

(2.32)

n × V n represents Coriolis acceleration where f b is the output of accelerometer; 2ωie n × V n means caused by the rotation of the earth relative to the earth motion; ωen

26

2 Synopsis of Typical Inertial Sensors and System

centripetal acceleration caused by the carrier to remain on the surface of the earth, also called traction acceleration, and g n is acceleration vector. The component form of the specific force equation can be expressed as: ⎡

⎤ ⎡ V˙e ⎢ ˙ ⎥ ⎢ ⎣ Vn ⎦ = ⎣ V˙u

⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ e 0 2ωie sin L + Ve Rtan L −(2ωie cos L + V fe 0 Ve Re ) ⎥ e ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ Ve tan L Vn ⎥ 0 −R f n ⎦ − ⎣ −(2ωie sin L + Re ) ⎦ × ⎣ Vn ⎦ + ⎣ 0 ⎦ e Vn −g fu V 2ωie sin L + Ve Rtan L 0 u R e

e

(2.33)

where Ve , Vn and Vu represent velocity in the direction of east, north and zenith respectively; Re is curvature radius of earth’s prime vertical; ωie is the rotation angular n is the angular velocity of the earth; Rn is curvature radius of meridional plane; and ωen velocity of the navigation coordinate system relative to the earth coordinate system. n can be expressed as: In the NEU coordinates, the component forms of ωie and ωen ⎤ ⎤ ⎡ n ⎤ ⎡ ⎤ ⎡ n − RnV+h ωiee ωene 0 ⎥ Ve ⎣ ωien ⎦ = ⎣ ωie cos L ⎦, ⎣ ωn ⎦ = ⎢ ⎦ ⎣ Re +h enn Ve tan L n ωieu ωenu ωie sin L R +h ⎡

(2.34)

e

where h means height of carrier. Substituting the result calculated above into the specific force equation and solving the differential equation, the velocity information of carrier in three directions under the navigation coordinate system can be obtained.

2.3.3.4

Solution of Carrier Position

The differential equation of the longitude, latitude and altitude of the carrier position can be expressed as: ⎧ Ve ⎪ ⎨ λ˙ = Re +h sec L n L˙ = R V+h ⎪ ⎩ h˙ = V n u

(2.35)

Integral operation is applied to obtain the updated formula of longitude, latitude and altitude:

⎧ ˙ ⎨ L = L(0) + Ldt (2.36) λ = λ(0) + λ˙ dt ⎩ ˙ h = h(0) + hdt

Chapter 3

Noise Analysis and Processing Technology for Gyroscope

3.1 Components of Gyro Noise and Allan Analysis Method 3.1.1 Source and Characteristic Analysis of Gyro Noise 3.1.1.1

Noise Source and Its Influence on Navigation System

This chapter takes fiber optic gyroscope as an example to introduce its noise analysis and processing technology. Noise is one of the important factors affecting the output precision of fog. In the actual system, Sagnac effect is very weak, and each component of the fog may produce noise, and there are various parasitic effects, which cause the output drift and scale factor instability of the fog, and ultimately affect the performance of the fog. The sources of noise mainly include the following aspects: (1) Light source noise The wavelength variation, spectral distribution variation and output power fluctuation of the light source will directly affect the effect of optical interference. In addition, the emission state will be interfered by the light returning to the source, resulting in certain fluctuations in wavelength and luminous intensity. (2) Detector noise The role of the detector is to detect interference effects. The main error sources include detector sensitivity, preamplifier noise, granular noise and modulation frequency noise. (3) Fiber ring noise The Kerr effect, Rayleigh backscattering effect, temperature effect, birefringence effect and Faraday effect of the fiber all lead to the change of the optical information © Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_3

27

28

3 Noise Analysis and Processing Technology for Gyroscope

transmitted by the fiber ring, which leads to the gyro noise, which is also the biggest noise source of the fiber optic gyroscope. (4) Optical path device noise The noise is mainly caused by the loss of the directional coupler and the variation of the beam splitting ratio deviation. (5) Other noises Such as environmental noise, electronic noise and so on. Fiber optic gyro output noise is mainly white noise, usually with Random Walk Coefficient(RWC) to represent, it reflects the angular velocity integral (Angle) of the fiber optic gyro output over time and the √ uncertainty of the random error (Angular random error), with its unit said (◦ )/ h, also can use the standard deviation equivalent rotation rate divided by the square root of detection bandwidth  ◦ of √  √  ( )/ h / H z . The relationship is as follows:  √  √ 1 ◦ √ ( )/ h 1 (◦ )/ h / H z = 60

(3.1)

According to the definition of the random walk coefficient, the relationship between the attitude Angle error standard of the inertial system caused by the white noise of the angular rate of the fog σθ (t) and the operating time of the system t is as follows: σθ (t) = RW C ·

√

t

(3.2)

The mean value E[y(t)] of the random walk process y(t) corresponding to white noise X (t) is: 

t

E[y(t)] = E 0

  x(τ )dτ =

t

E[x(τ )]dτ = 0

(3.3)

0

This shows that the attitude Angle error of inertial navigation caused by white noise in the output signal of fiber optic gyroscope in the system application is a nonstationary angular√random walk process with a mean value of 0 and a standard deviation of RW C · t. Since the random walk coefficient is independent of the detection bandwidth (or data smoothing period), the standard deviation of the attitude Angle error of the inertial system σθ (t) is also independent of the data smoothing period. Therefore, in the algorithm of inertial navigation system, by smoothing the data of fiber optic gyroscope, the value σθ (t) cannot be reduced in principle, that is, the measurement accuracy of inertial navigation system cannot be improved. Due to the short correlation time of white noise, the error term will play a role in a short time, so it has a great impact on the short-term process of the system. Therefore,

3.1 Components of Gyro Noise and Allan Analysis Method

29

Table 3.1 Relationship between operating time of inertial navigation system and attitude Angle error caused by noise Random walk coefficient √ RW C = 0.01(◦ )/ h

Working time t/s

Standard deviation of attitude Angle error σθ (t)/()

60

4.65

600

14.70

6000

46.48

Table 3.2 Initial alignment Angle error of inertial navigation caused by fog noise (L = 45◦ ) The random walk ◦ √  coefficient/ ( )/ h

At the time t /min 1

2

3

4

5

6

7

8

9

10

0.001

2.5

1.8

1.4

1.3

1.1

1.0

0.95

0.89

0.84

0.79

0.003

7.5

5.3

4.3

3.7

3.4

3.1

2.8

2.7

2.5

2.4

the influence of fog noise on the initial attitude self-alignment process of the inertial system, the north-seeking process of the north finder and various inertial attitude stabilization √ circuits is great. For example, if the random walk coefficient RW C = 0.01(◦ )/ h is used, the relationship between attitude Angle errors corresponding to different working hours can be calculated by formula (3.2), as shown in Table 3.1. It can be seen that the working hours have increased by 100 times, while the σθ (t) caused by noise has increased by about 10 times. Noise can also cause azimuth alignment error of inertial navigation system ψ RW C (t), and the size of the alignment error Angle is proportional to the random walk coefficient and inversely proportional to the square root of the alignment time, as shown in formula (3.4). Examples are shown in Table 3.2. ψ RW C (t) =

RW C √ ωie · cos L · t

(3.4)

The positioning error of inertial navigation system caused by fog noise is shown in Fig. 3.1. In a word, the random walk coefficient is an important index to measure the noise level of fog. In order to reduce the influence of the noise on the performance of the navigation system, the structure optimization design and modern signal processing methods should be used to reduce the fog noise based on the performance improvement of the fog itself.

30

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.1 Relationship between noise and positioning error

3.1.1.2

Analysis of Noise Characteristics

(1) Angle Random Walk (ARW) Angular random walk is the result of integrated wideband rate power spectral density, and its main sources are: spontaneous radiation of photons, granular noise of detectors, thermal noise of electronic devices, mechanical jitter and other high frequency noises with shorter sampling time. In general, the bandwidth of angular random walk is less than 10 Hz, so it is within the bandwidth of most attitude control systems. Therefore, if the angular random walk cannot be accurately determined, it is very likely to become the main influencing factor of the attitude control system. The correlation time in the high frequency noise term is shorter than the sampling time, which can be considered as the random walk of gyro Angle. Most of these sources can be eliminated by design. The white noise spectrum of the gyroscope output can be used to describe these noise terms. When N is the noise amplitude, the power spectrum of angular velocity white noise is the characteristic of this random process: S ( f ) = N 2 In formula (3.5): N is the Angle random walk coefficient.

(3.5)

3.1 Components of Gyro Noise and Allan Analysis Method

31

(2) Bias Instability (BI) The electron or other term sensitive to random scintillation is the main source of the zero-biased instability, which is mainly caused by the low-frequency zero-biased fluctuation in the angular rate data, with obvious low-frequency characteristics. The rate power spectrum function of the noise can be expressed as: S ( f ) =

2

B 1 ( 2π ) f , f ≤ f0 0, f > f 0

(3.6)

where, is the cut-off frequency; B is the zero partial stability coefficient. (3) Rate Random Walk (RRW) The rate random walk is formed after the power spectral density integration of broadband angular acceleration. This error may be due to the aging effect of the crystal oscillator, or it may be the limit case of exponentially correlated noise with a long correlation time. The relevant rate PSD is: S ( f ) = (

K 2 1 ) 2π f 2

(3.7)

where, K is the rate random walk coefficient. (4) Rate Ramp (Rate Ramp, RR) Essentially, the rate ramp is not random noise, but a deterministic error. The reason why it appears in the gyroscope input and output characteristics may be caused by the temperature change of the fog caused by the external environment, or the light intensity of the fog changes slowly and monotonously over a long period of time, so as to be the pure input of the fog, which can be represented by the following equation:  = Rt

(3.8)

where, R is the rate slope coefficient. Its rate PSD is: S ( f ) =

R2 (2π f )2

(3.9)

(5) Quantization Noise Quantization noise is caused by A/D conversion during sampling and represents the lowest resolution level of the sensor. Its rate PSD is:

32

3 Noise Analysis and Processing Technology for Gyroscope

S ( f ) =

≈





sin2 (π f τ0 ) (π f τ0 )2 2 τ0 Q , f < 2τ10

τ0 Q 2

(3.10)

where, Q is the quantized noise coefficient.

3.1.2 Allan Variance For a long time, in order to accurately calculate the coefficients of each early noise term, a variety of methods have been studied, and the most commonly used method is called Allan variance analysis. Allan variance is a time-domain analysis technique, which was proposed by the national bureau of standards in the 1960s and was initially used in the frequency stability study of oscillators. The Allan variance method is recognized by IEEE as the standard detection method for gyroscope output parameter analysis, which can determine the basic random process characteristics of data noise. The characteristic of Allan variance method is that it can easily and carefully characterize and identify the statistical characteristics of various error sources. If the mechanical structure of the instrument is unknown, the noise source which may exist in the instrument can be found as long as the correct testing system is established. Allan variance can be used to analyze data separately or to supplement frequency-domain analysis techniques, and can be applied to the noise study of any instrument. In Allan analysis of variance, it is assumed that the uncertainty of data is generated by the noise sources with specific characteristics, and then the covariance of each noise source is calculated according to the data. Suppose there are N gyros with a sampling period of T to output data, set up an array with a time of respectively, and then average the sum of data points in the length array at each time T, 2T, 3T, · · · , kT (k < N/2). Allan variance can be defined as a function of time groups. Specifically, Allan variance can be defined as output rate or output Angle: t (τ )dτ

θ (t) =

(3.11)

0

The average rate between time and time is shown as follows: Ω¯ t (t) = In formula (3.12): t = mT Allan variance is defined as:

θk+m − θk t

(3.12)

3.1 Components of Gyro Noise and Allan Analysis Method

σ 2 (τ ) =

  1

1

(Ω¯ k+m − Ω¯ k )2 = 2 (θk+2m − 2θk+m + θk )2 2 2τ

33

(3.13)

In formula (3.13)  represents the population average value. After expanding formula (3.13), it can be obtained: σ 2 (τ ) =

N −2m 1 (θk+2m − 2θk+m + θk )2 2τ 2 (N − 2m) k=1

(3.14)

Allan variance can be estimated according to the above formula. The relationship between Power Spectral Density (PSD) and Allan variance can be expressed as follows:  σ (τ ) = 4 2

∞

Sω ( f )

0

sin4 (π f τ ) df (π f τ )2

(3.15)

where Sω ( f ) represents the power spectral density of random noise ω(t). It should be noted that when the filter with the transfer function is sin4 (π f t)/(π f t)2 , the output noise of the gyro is proportional to the total energy of Allan variance. The above special transfer function is the result of using the method of generating and manipulating arrays. In general, the filter bandpass depends on τ , that is, checking different types of random processes can be realized by adjusting the filter bandpass, which means checking different τ . Allan variance therefore provides a means of identifying and quantifying the various noise terms present in the data. By substituting Eqs. (3.5–3.9) into Eq. (3.15), the Allan variances of different noise items can be obtained, as shown below: Allan variance of Angle random walk is: σ 2 (τ ) =

N2 τ

(3.16)

In Eq. (3.16): N is the Angle random walk. In the logarithmic graph σ (τ ) of τ , the slope of the curve is −1/2. Allan variance of zero biased instability is: σ 2 (τ ) =

2B 2 sin3 x (sin x + 4x cos x) + Ci (2x) − Ci (4x)] [ln 2 − π 2x 2

(3.17)

In formula (3.17): x is π f 0 t, Ci is the integral function of cosine, and B is the coefficient of zero partial instability. As can be seen from the above equation, in the logarithmic curve, the curve is horizontal, that is, the slope is 0. Allan variance of rate random walk is: σ 2 (τ ) =

K 2τ 3

(3.18)

34

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.2 sample diagram of Allan anova results σ (τ )

In Eq. (3.18): K is the rate random walk coefficient. In the logarithmic graph σ (τ ) of τ , the slope of the curve is +1/2. The Allan variance of the rate slope is: σ 2 (τ ) =

R2τ 2 2

(3.19)

In Eq. (3.19): R is the rate slope coefficient. In the logarithmic graph σ (τ ) of τ , the slope of the curve is +1. The Allan variance of quantified noise is: In Eq. (3.20): Q is the quantization noise coefficient. In the logarithmic graph σ (τ ) of τ , the slope of the curve is −1. In general, all of the random processes described above are likely to occur in the gyro output. Figure 3.2 shows a typical Allan variance diagram. In most cases, tests show that different noise items will appear in different τ domains, making it easy to distinguish between different random processes in the data. Assuming that the existing random processes are statistically independent, Allan variances in any given τ domain are represented as the sum of Allan variances caused by different random processes in this τ domain, i.e.: 2 2 2 (τ ) = τ A2 RW (τ ) + τquant (τ ) + τ Bias σtot I nst (τ ) + · · ·

(3.21)

Assuming that the error source is statistically independent, the Allan variance can be represented by the sum of squares of one or several noise error sources:   K 2τ N2 3Q 2 R2τ 2 2 2 + +B ln 2 + + 2 + ··· σ (τ ) = 2 3 π τ τ 2

The above equation can be simplified to:

(3.22)

3.1 Components of Gyro Noise and Allan Analysis Method

σ 2 (τ ) =

2

Cn τ n

35

(3.23)

n=−2

In formula (3.23), τ = nT , T is the sampling period. By comparing Eqs. (3.22) and (3.23), we can get: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

√

C−1 0 N = 60 ( / h 1/2 ) √ C0 0 B = 0.664 ( / h) √ K = 60 3C1 (0 / h 3/2 ) √ ⎪ ⎪ ⎪ R = 3600√ 2C2 (0 / h 2 ) ⎪ ⎪ ⎩ 106 π C−2 √ (μrad) Q = 180×3600× 3

(3.24)

3.2 Denoising Algorithm for Fog Signal Fiber optic gyro noise is a kind of signal that changes with time caused by random disturbance of uncertain nature. In this paper, the signal filtering method is used to suppress the noise of fiber optic gyroscope, so as to obtain the accurate output signal of fiber optic gyroscope.

3.2.1 Enhance the Wavelet Transform 3.2.1.1

Wavelet Transform Theory

Wavelet transform is a method of time-scale analysis in time, scale (frequency), the local characteristics of both domain can effectively characterize signals, has low time resolution in low frequency part and high frequency resolution, in the high frequency part with high temporal resolution and low frequency resolution, especially suitable for detecting abnormal entrained in normal signal instantaneous signals and display its content. So, the wavelet transform is called a microscope for analyzing signals. The so-called wavelet refers to a wavelet sequence obtained by a parent wavelet or a basic wavelet ψ(t) after expansion and translation, ψ(t) ∈ L 2 (R),L 2 (R) represents the square integrable real space, namely the energy limited signal space, and its ˆ ˆ Fourier transform is ψ(ω). ψ(ω) meets the allowable conditions:  Cψ =

+∞

−∞

2   ˆ ψ(ω) |ω|

dω < ∞

In the continuous case, the wavelet sequence is:

(3.25)

36

3 Noise Analysis and Processing Technology for Gyroscope

1 t −b ), a, b ∈ R, a = 0 ψa,b (t) = √ ψ( a |a|

(3.26)

If x(t) is a square integrable function, called x(t) ∈ L 2 (R), ψ(t) is a function called a basic wavelet or mother wavelet, then it will be: 1 (Wψ f )(a, b) = √ a

+∞ t −b )dt, a > 0 x(t)ψ( a

(3.27)

−∞

In Eq. (3.27): ψ( t−b ) represents the conjugate operation of ψ( t−b ). Equaa a tion (3.27) is called continuous wavelet transform (CWT) of x(t). In practice, especially in the upper computer, continuous wavelet must be discretized. The corresponding discrete wavelet transform is: (Wψ f )(a, b) = |a0 |−m/2



+∞ −∞

x(t)ψ(a0−m t − nb0 )dt

(3.28)

The particular, take a0 = 2, b0 = 2 and the binary wavelet can be obtained: ψm,n (t) = 2−m/2 ψ(2−m t − n), m, n ∈ Z

3.2.1.2

(3.29)

Enhance the Wavelet Transform

In 1994, Wim Swelden proposed a new wavelet construction method, which is called Lifting Scheme, also known as Lifting wavelet transform or second-generation wavelet transform. The characteristics of the lifting wavelet transform method are as follows: (1) the multi-resolution properties of the first generation wavelet transform are retained; (2) it does not depend on the Fourier transform and can complete the wavelet transform in the time domain; (3) the transformed coefficient can be an integer; (4) when image processing is carried out, the quality of image restoration has no direct relationship with the continuation method adopted in the boundary transformation. Lifting wavelet transform is based on lifting the first-generation wavelet transform. Compared with the first-generation wavelet, lifting wavelet has the following advantages: (1) the algorithm is simple, fast and suitable for parallel processing; (2) the memory requirement is small, which is easy to be realized by DSP chip;

3.2 Denoising Algorithm for Fog Signal

37

(3) the wavelet transform of images of any size can be realized by standard operation. Wavelet lifting is a method to construct compactly supported biorthogonal wavelet. The process of the second generation wavelet based on the lifting scheme is divided into the following three steps: (1) splitting   Splitting is to divide the original signal s j = s j,k into two mutually disjoint subsets, each of which is reduced to half the length of the atomic set. Normally, a sequence is divided into odd sequence o j−1 and even sequence e j−1 , that is: Split (s j ) = (e j−1 , o j−1 )

(3.30)

    In Eq. (3.30): e j−1 = e j−1,k = s j,2k , o j−1 = o j−1,k = o j,2k+1 . (2) prediction Prediction is to use the correlation between an odd sequence and an even sequence to predict another sequence (usually an odd sequence e j−1 ) through one of the sequences (usually an even sequence o j−1 ). The degree of approximation or approximation between the two is reflected by the difference d j−1 between the actual value o j−1 and the predicted value P(e j−1 ), which is called the detail coefficient or the wavelet coefficient, corresponding to the high frequency part of the original signal s j . In general, the stronger the correlation is, the smaller the amplitude of the wavelet coefficient is. If the prediction is reasonable, the difference dataset d j−1 contains far less information than the original subset o j−1 . The prediction process is as follows: d j−1 = o j−1 − P(e j−1 )

(3.31)

In formula (3.31): the prediction operator P can be expressed by the prediction function P k , and the corresponding data itself of e j−1 can be taken as the function Pk: Pk (e j−1 , k) = e j−1 , k = s j,2k

(3.32)

Or the average value of the corresponding adjacent data in e j−1 : Pk (e j−1 ) = (e j−1 , k + e j−1,k+1 )/2 = (s j,2k + s j,2k+1 )/2

(3.33)

Or some other more complicated function. (3) update Some of the overall characteristics of the subset (such as the mean value) generated by the splitting step may not be completely consistent with the original data, and an

38

3 Noise Analysis and Processing Technology for Gyroscope

update process is required to maintain the overall characteristics of the original data. The update process can be replaced by an operator U, which is expressed as follows: s j−1 = e j−1 + U (d j−1 )

(3.34)

In formula (3.34), s j−1 is the low-frequency part of s j . Same as the prediction function, the update operator can be taken as different functions, such as: Uk (d j−1 ) = d j−1,k /2

(3.35)

Uk (d j−1 ) = (d j−1,k−1 + d j−1,k )/4 + 1/2

(3.36)

Or:

Different wavelet transforms can be constructed by taking different functions of P and U. After the wavelet is promoted, the signal s j can be decomposed into the lowfrequency part s j−1 and the high-frequency part d j−1 , and the low-frequency data subset s j−1 can repeat the same process of splitting, prediction and update, and further decompose s j−1 into d j−2 and s j−2 . In this way, after n  of the original data s j can be expressed as sub-decomposition, the wavelet · · · , d . Where s j−n represents the low frequency part of s j−n , d j−n , d j−n+1, j−1  the signal, and d j−n , d j−n+1, · · · , d j−1 is the series of high frequency parts of the signal from low to high. Each decomposition corresponds to the three steps of ascension shown above, namely splitting, predicting, and updating: Split (s j ) = (e j−1 , o j−1 ), d j−1 = o j−1 − P(e j−1 ), s j−1 = e j−1 + U (d j−1 ) (3.37) Wavelet lifting is a completely reversible process, and the steps of its inverse transformation are as follows: e j−1 = s j−1 − U (d j−1 ), o j−1 = d j−1 + P(e j−1 ), s j = Merge(e j−1 , o j−1 ) (3.38) The diagram of wavelet decomposition and reconstruction by lifting method is shown in Fig. 3.3.

3.2.2 Forward Linear Prediction Algorithm The main idea of forward linear prediction (FLP) algorithm is to multiply the previous gyro signal by the corresponding weight to predict the current gyro signal. Obviously, there is an optimal weight in the prediction process, which is obtained by iterative

3.2 Denoising Algorithm for Fog Signal

39

Fig. 3.3 Decomposition and reconstruction of the promotion method

process. In this process, the initial value of the weight is set to zero, then the difference between the predicted value and the current collected gyro signal is minimized by using the least mean square theory, and finally the weight is continuously updated to obtain a stable convergence result. Set the estimated value of current gyro signal as: x(n) ˆ =

N

α p x(n − p) = A T X (n − 1)

(3.39)

p=1

In formula (3.39):X (n − 1) = {x(n − 1), x(n − 2), · · · , x(n − N )}T is the vector composed of the gyro output at the previous time; x(n − p) is the gyro signal at the previous time; α p is the weight; N is the order, the larger the order, the better the filtering effect, but the larger the order, the larger the calculation amount of the filtering process. The difference between the current value and the predicted value (i.e. forward prediction error) and the cost function are respectively: ∧

e(n) = x(n) − x(n)

(3.40)

  J (n) = E e2 (n)

(3.41)

The least mean square theory is used to minimize the forward prediction error, that is to select the most appropriate weight to minimize the cost function. The weight iterative adjustment formula can be obtained from the least mean square value theory: A(n + 1) = A(n) + εE[e(n)X (n − 1)]

(3.42)

In order to reduce the amount of calculation, the above formula can be simplified as follows: A(n + 1) = A(n) + εe(n)X (n − 1)

(3.43)

40

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.4 Adaptive FLP algorithm structure

In formula (3.43): ε is a very small normal quantity, whose function is to control the convergence rate of the whole iterative process. A larger value of ε is conducive to fast iteration, but it will increase the deviation between MMSE and MSE, and the iteration process is easy to diverge if it is too large. Therefore, in the process of FLP prediction, the prediction error is closely related to the step size. In the initial stage, the prediction error is large, and the selection of step size should make the prediction error quickly reduce to a certain extent, so the relatively large step size can be selected; when the prediction error is reduced to a certain extent, then the small step length is taken to improve the steady-state output accuracy. Since the adjustment of ε is related to the prediction error, the following adaptive variable step size algorithm can be used: ε(n) = β(1 − exp(−α|a tan e(n)|2 ))

(3.44)

In formula (3.44): e(n) is the estimation error; β is the weighting coefficient determined by the prediction process and α is the attenuation coefficient. The optimal value of β and α can be obtained by many experiments. The structure of FLP algorithm is shown in Fig. 3.4.

3.2.3 LWT-FLP Algorithm In order to get better de-noising effect, a new de-noising algorithm, lwt-flp, is proposed by combining LWT and FLP. The specific steps are as follows: (1) Lifting wavelet decomposition and single branch reconstruction The decomposition scale of lifting wavelet is chosen as N, and the lifting wavelet coefficients of each node are reconstructed in a single branch. The high frequency coefficient is d(t) and the low frequency coefficient is c(t) after reconstruction;

3.2 Denoising Algorithm for Fog Signal

41

(2) FLP de-noising of lifting wavelet coefficients The coefficients of each layer of the reconstructed lifting wavelet described in step (1) are processed by FLP. In this prediction process, the prediction error is closely related to the step size. In the initial stage, the prediction error is large, and the selection of step size should make the prediction error quickly reduce to a certain extent. At this time, the relatively large step size can be selected. when the prediction error decreases to a certain extent, the small step length is used to improve the steady-state output accuracy. Since LWT-FLP weight coefficient adjustment is based on the frequency division data block, the step size adjustment of each frequency band should be related to the absolute error of prediction error in the frequency band.  If the mean value of  the absolute error of FLP in the J-band is E j = E e j (n) , the formula (3.44) can be transformed into:   ε j = β 1 − exp(−α E 2j )

(3.45)

(3) Lifting wavelet reconstruction In step (2), the coefficients of each layer processed by FLP are reconstructed to get the signal of fog after denoising by LWT-FLP algorithm. The structure of lwt-flp algorithm is shown in Fig. 3.5.

3.2.4 Analysis of De-Noising Results of Fog Signal A group of static output signals of the interferometric closed-loop FOG are selected to verify the algorithm. The output of the fog is collected under the conditions of horizontal placement, stable start-up and working environment temperature change (±5 °C /min). The sampling frequency is 100 Hz and the acquisition time is 40 min. The 3-scale LWT-FLP algorithm is validated by the collected fog signal, and the wavelet base of lifting wavelet is selected as DB4 wavelet. Compared with the 3-scale lifting wavelet de-noising and FLP de-noising, the results are shown in Fig. 3.6. It can be seen from Fig. 3.6 that compared with the traditional LWT and FLP de-noising methods, the LWT-FLP algorithm proposed in this paper can remove the influence of noise on gyro signal more effectively due to the layered processing strategy. Allan variance is used to analyze the de-noised signal, and the analysis results are shown in Fig. 3.7 and Table 3.3. In Table 3.3, Q is the quantization noise and N is the angle random walk. These two error coefficients mainly reflect the high frequency error of the signal. It can be seen from Table 3.1 that signal de-noising can effectively reduce the above two error coefficients. The results of LWT,FLP and LWT-FLP show that the lwt-flp algorithm proposed in this paper can effectively suppress the noise coefficient of fog and has the best de-noising effect.

42

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.5 LWT-FLP algorithm flow chart

3.3 The Method of Eliminating the Angular Vibration Error of Fog 3.3.1 Angular Vibration Experiment and Output Signal Analysis Vibration is also one of the factors affecting the performance of fog. This paper focuses on the removal of the angular vibration error of fog. Firstly, the angle vibration experiment of fog is carried out. The fog is installed on the turntable to collect the output of fog under the angle vibration. The experiment process is shown in Figs. 3.8 and 3.9. The output results are shown in Fig. 3.10. It can be seen from Fig. 3.10 that in the angular vibration environment, the output signal of fog is submerged in the noise and the angular vibration information cannot

3.3 The Method of Eliminating the Angular Vibration Error of Fog

43

Fig. 3.6 Comparison of denoising results of different algorithms Fig. 3.7 Allan variance analysis results

Table 3.3 Allan variance analysis of denoising results of different algorithms Q(rad) N(0/h1/2)

Original signal

LWT

FLP

LWT-FLP

167.01

147.37

25.01

9.09

0.97

0.84

0.066

0.064

44

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.8 Three axis turntable

Fig. 3.9 Data acquisition

be obtained. In order to eliminate the noise caused by vibration, the forward linear prediction (FLP) algorithm is introduced into the signal processing of fog. The angular vibration output of fog after FLP processing is shown in Fig. 3.11. It can be seen from Fig. 3.11 that the noise term in the output of fog after FLP processing is significantly reduced, but the residual noise still hides the true angular vibration information. In order to eliminate the influence of noise on the output of fog, a G-FLP algorithm based on Grey Theory is proposed.

3.3 The Method of Eliminating the Angular Vibration Error of Fog

45

Fig. 3.10 Fog output under angular vibration

Fig. 3.11 FLP processed FOG output

3.3.2 Grey FLP Algorithm Grey Theory (GT) is a new method to study the problems of few data, poor information and uncertainty. It takes the uncertain system of “small sample” and “poor information” with some information known and some information unknown as the research object. Through the generation and development of the known information of “part”, it extracts valuable information, and realizes the correct description and effectiveness monitor of the system operation behavior and evolution law. The theory holds that any random process can be regarded as a grey process changing in a certain space-time region, and the random quantity can be regarded as a grey quantity. At the same time, the irregular sequence of system data can be transformed into regular sequence by generating transformation. GT emphasizes to extract valuable information by studying the known information in the irregular system, and then use the known information to further reveal the unknown information, so that the system becomes “white”.

46

3 Noise Analysis and Processing Technology for Gyroscope

The model established and applied in the grey system is called grey model, which is called GM model for short. The essence of GM model is the differential equation based on the original data sequence. Among them, the most representative GM model is for the modeling of time series. It can directly convert the time series data into differential equations, make full use of system information, quantify the abstract model, and then realize the accurate prediction of system output in the absence of system characteristic knowledge. First of all, GM model accumulates the original data sequence once, and the accumulated data can show certain rules, then it uses typical curve to fit the accumulated sequence. Assuming a time series: x (0) = (xt(0) |t = 1, 2, · · · , n) = (x1(0) , x2(0) , · · · , xn(0) )

(3.46)

A new data sequence x (1) is obtained by one-time accumulation x (0) . The item t of the new data sequence x (1) is the sum of the preceding items t of the original data sequence x (0) , namely: x

(1)

=

(xt(1) |t

= 1, 2, · · · , n) =

(x1(0) ,

1

xt(0) ,

t=1

2

xt(0) , · · ·

t=2

,

n

xt(0) )

(3.47)

i=n

According to the new data sequence x (1) , the whitening equation can be established, and the following results can be obtained: d x (1) + ax (1) = u dt

(3.48)

The solution of formula (3.48) is: xt∗(1) = (x1(0) − u/a)e−a(t−1) + u/a

(3.49)

xt∗(1) is the estimated value of the sequence xt(1) , and the predicted value xt∗(0) of x obtained by one successive subtraction of xt∗(1) , namely: (0)

∗(1) xt∗(0) = xt∗(1) − xt−1

t = 2, 3, · · ·

(3.50)

3.3.3 G-FLP Algorithm In order to better remove the influence of noise on the output of fog in the environment of diagonal vibration, this paper makes full use of the advantages of grey theory and FLP algorithm, and combines them together. First, grey the data sequence, then the grey data sequence is processed by FLP, and finally whitening the data sequence to

3.3 The Method of Eliminating the Angular Vibration Error of Fog

47

get the final signal processing results. The whole algorithm structure is shown in Fig. 3.12. G-FLP algorithm is used to process the output of fog shown in Fig. 3.10, and the processed result is shown in Fig. 3.13. Compared with Figs. 3.11 and 3.13, it can be seen that G-FLP algorithm can effectively remove noise. Compared with traditional FLP algorithm, G-FLP algorithm has better performance. The statistical results of mean and standard deviation of the results processed by FLP algorithm and G-FLP algorithm are shown in Table 3.4. It can be seen from the mathematical statistical characteristics that the variance is the best one to reflect the noise characteristics. It can be seen from Table 3.4 that the variance of data processed by G-FLP algorithm is significantly smaller than that processed by FLP algorithm, so it can be concluded that the de-noising performance of G-FLP algorithm is significantly better than that of traditional FLP algorithm. According to the above experimental and simulation results of angular vibration, the G-FLP algorithm can effectively remove the influence of noise caused by angular vibration on the performance of fog, and improve the performance of fog in angular vibration environment.

Fig. 3.12 G-FLP algorithm structure

48

3 Noise Analysis and Processing Technology for Gyroscope

Fig. 3.13 Output of fog processed by G-FLP algorithm

Table 3.4 Comparison of data mean and standard deviation of different methods (arcmin)

Raw data

After FLP treatment

After G-FLP treatment

Mean value

1.96e-5

−7.9e-7

1.36e-6

Standard deviation

0.0063

0.00037

0.00018

3.4 Summary Due to the strong sensitivity of fog, in addition to sensing the small changes of the measured in use, it can always detect the changes of noise and stack the noise with useful signals, so it is very important to identify the noise in the output of fog quantitatively or qualitatively. Allan variance method can effectively solve the above problems, because it can effectively calculate the noise items and the proportion of noise items in the signal, and provide a good auxiliary role for signal preprocessing and de-noising. Allan variance analysis method can effectively analyze the noise items, but how to effectively separate the corresponding noise signals, we still need to use the de-noising algorithm to process the fog signals. In this chapter, LWT-FLP algorithm is proposed on the basis of traditional LWT and FLP algorithm. This method fully combines the advantages of lifting wavelet transform and forward linear prediction algorithm. Firstly, lifting wavelet transform is used to stratify the fog signal, and then FLP process the fog signal in different frequency domain. Finally, reconstruction of each layer of fog signal after FLP processing is carried out to the final de-noising result. Then, the G-FLP algorithm based on the grey theory and FLP algorithm is proposed for the angular vibration error of fog. Firstly, the grey accumulation operation is carried out for the angular vibration output of fog, and then the forward linear prediction is carried out for the grey data sequence. Finally, the predicted results are whitened to effectively remove the angular vibration error of fog.

Chapter 4

Temperature Drift Modeling and Compensation for Gyroscope

4.1 Temperature Drift and Modeling Method of Fog Temperature, temperature change rate and temperature gradient are the main environmental factors that cause fog error. If the temperature effect of the optical fiber is considered, when the beam passes through the optical fiber length of L with the propagation constant β(z), its phase delay is:  φ = β0 n L + β0

∂n − na ∂T

 L T (z)dz

(4.1)

0

In Formula (4.1): β0 = 2π is the propagation constant of light in vacuum; n is λ0 the effective refractive index of optical fiber; ∂∂nT is the refractive index temperature coefficient of quartz material; a is the thermal expansion coefficient; T (z) is the variation along the temperature distribution of optical fiber. In Sagnac interferometer, two beams of interference light pass through the same length L fiber in the clockwise (CW) and counter clockwise (CCW) directions respectively. It is assumed that CW light wave reaches the output end of the optical fiber at time t, and CCW light wave reaches a certain coordinate point Z at time t  = t − (L − z)/cn , where cn = c0 /n is the wave velocity in the waveguide. When the strain effect and Poisson effect are mainly considered, and the photoelastic effect is not considered, the phase delays of clockwise and counter clockwise light waves can be obtained from Eq. (4.1) as follows: ⎧ ⎪ ⎪ ⎪ ⎨

φC W = β0 n L + β0 ∂∂nT

L

T (z, t − z/cn )dz

0

L ⎪ ∂n ⎪ ⎪ ⎩ φCC W = β0 n L + β0 ∂ T T (L − z, t − z/cn )dz

(4.2)

0

© Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_4

49

50

4 Temperature Drift Modeling and Compensation for Gyroscope

Thus, the detected error of the thermal non reciprocal effect of the fiber ring is: L

β0 ∂n φ E = n c0 ∂ T

2

(T (z) − T (L − z))(2z − L)dz

(4.3)

0

It can be seen from the above formula that the temperature change has a significant impact on the output signal of fog. At present, a lot of work has been done on the modeling and compensation of FOG temperature error, among which the most widely used method is to compensate the error by establishing mathematical model. In the error modeling of fiber optic gyroscope (FOG) inertial combination products, the zero deviation and scale factor of the inertial instrument are generally modeled only for the temperature of the products when they are working, and this section mainly models the temperature drift of fog. At present, there are mature modeling and application methods for FOG temperature drift modeling. The deterministic error model (i.e. without considering random term error) of fog bias can be expressed as follows: D = D0 + DT T + DT˙ T˙

(4.4)

In the above formula: D is gyro bias; D0 is constant error; DT and DT˙ is error coefficient related to temperature T and temperature change rate T˙ respectively. Formula (4.4) expresses the temperature drift error of fog in the form of linear polynomials of temperature, rate of change of temperature and other factors. Because the temperature field is a kind of distributed parameter, which continuously acts on different structures and components of the gyroscope, there are objective differences in temperature and temperature change rate in different parts of the gyroscope, and all of them have different contributions to the temperature drift of the gyroscope. Its comprehensive function is to make the output of the gyroscope deviate from the input to form the temperature drift error. Therefore, the temperature drift of fog can be regarded as the value of temperature field in several typical spatial positions and its linear combination of time derivative. The simple and practical method to establish this kind of linear model is based on multivariate linear regression statistics. Taking the temperature and temperature change rate of r ≥ 2 positions, the multiple linear regression model of gyro temperature error is obtained as follows: D = D0 +

r

i=1

DT i · Ti +

r

DT˙ i · T˙i + ε

(4.5)

i=1

In Formula (4.5), the meanings of D, D0 , DT i , DT˙ i , Ti and T˙i are the same as those in Formula (3.4); ε are random temperature errors. For a group of temperature (and its rate of change) sampling values, after  the corresponding temperature drift value is obtained, the regression coefficient D0 DT 1 DT 2 · · · DT r DT˙ 1 DT˙ 2 · · · DT˙ r of the model can be estimated by the least square regression method.

4.1 Temperature Drift and Modeling Method of Fog

51

The modeling and compensation of FOG temperature drift can be divided into two situations: one is the modeling and compensation of FOG temperature error in the process of power on and start-up of IMU; the other is the modeling and compensation of FOG temperature error in the steady-state operation of IMU. The temperature error modeling and compensation of fog in the process of power on and start-up of IMU can be divided into two cases, one is the start-up process modeling in short time and the other is the start-up process modeling in long time. When the working time of the inertial measurement device is short (e.g. within a few minutes) after power on, the work is finished because it has not reached the thermal balance after power on. At this time, because the internal temperature environment of the fog is close to the time of power on, the influence of the temperature change on the error coefficient of the fog in a short time after power on can be basically ignored, and only the zero bias change model of fog needs to be established at different temperatures during power on start-up. The temperature of a monitoring point inside the fog is used as the input, and the bias temperature error model is as follows: D = D0 +DT (T − T0 ) + ε

(4.6)

In Formula (4.6), T refers to the temperature of monitoring point, T0 is reference temperature and ε is random error. When the inertial measurement unit works for a long time, the internal temperature field of fog will change with time and gradually reach thermal balance after being electrified, which is relatively slow and directly affects the zero deviation. In order to meet the accuracy requirements of the whole process, it is necessary to model and compensate the process of gyro zero deviation after being electrified. One (or more) temperature measuring points are set on the optical path and circuit inside the gyroscope, and the measurement temperature T1 and T2 of two points, and the temperature change rate T˙1 and T˙2 of two points are taken as state variables, then the zero bias temperature error model is: D = D0 + DT 1 T1 + DT 2 T2 + DT˙ 1 T˙1 + DT˙ 2 T˙2 + ε

(4.7)

In the field of navigation, aerospace and so on, the inertial measurement system will generally have a long enough stable time after being powered on. At this time, it is not necessary to consider the measurement accuracy of the inertial system in the starting process, but only the measurement accuracy when it is affected by the ambient temperature after its stable operation. This is also a key research topic for other types of inertial systems that need long-time work. The variation law of the temperature error caused by the external temperature disturbance after the fog works stably is similar to the variation law of the power on and start-up process of the fog, but it is also quite different. The similarity lies in the drift caused by the change of the internal temperature field, and the similar model can be selected; the difference lies in that the heat source during the power on and start-up process is mainly the instrument and components inside the inertial measurement unit, while the error source after

52

4 Temperature Drift Modeling and Compensation for Gyroscope

the work is stable is mainly the external environment temperature, which is the impact from the outside to the inside, and the temperature influence mechanism of the two is different. The gyro drift caused by the ambient temperature fluctuation is essentially a dynamic process modeling under the external temperature disturbance. In order to fully stimulate the dynamic characteristics of FOG temperature drift and enhance the adaptability of the model, a variety of temperature ranges and amplitudes should be fully considered in the test. During the test, a number of test sample data were collected for multivariate regression statistical analysis, and the temperature measurement points that have no significant impact on the model were eliminated. 1 2 3 , dT , dT ) of the The temperature (T1 , T2 , T3 ) and its temperature change rate ( dT dt dt dt top and bottom end faces of the fog and the circuit components were retained. The zero bias temperature model of the fog was as follows: D = D0 +

3

i=1

DT i · Ti +

3

DT˙ i · T˙i + ε

(4.8)

i=1

4.2 Fiber Gyro Temperature Error Model Based on External Temperature Change Rate The previous section gives the temperature error model of the fiber optic gyro during power-on and after steady-state operation. The existing temperature error model can be used to model and compensate the fiber optic gyro error caused by normal temperature changes. However, at present, the mechanism analysis and modeling compensation research of the temperature drift error of the fiber optic gyroscope working in a special temperature change environment is still relatively small. For example, when working in an alpine environment, due to the large temperature difference between the foot of the mountain and the top of the mountain, when the carrier is driven from the foot to the top, the faster temperature change rate will cause a larger temperature error to the fiber optic gyroscope. In addition, the temperature error modeling method of the traditional fiber optic gyroscope in the steady-state working condition needs to obtain the temperature of the upper and lower end surfaces of the fiber ring and the circuit components of the fiber optic gyroscope. Therefore, multiple temperature sensors need to be implanted inside the fiber optic gyroscope, and the engineering implementation is more complicated. According to the temperature drift principle of fiber optic gyro, temperature has two main ways of affecting the fiber optic gyroscope: one is the drift caused by the internal temperature field change, the heat source is mainly the instrument and its components inside the inertial measurement combination; the second is the temperature change of the external environment. When the external ambient temperature changes drastically, the internal temperature field changes slowly, and the drift due to internal temperature factors is small.

4.2 Fiber Gyro Temperature Error Model Based on External …

53

Fig. 4.1 Fiber gyro temperature experiment

First, the temperature experiment of fiber optic gyro was carried out. Place the fiber optic gyroscope in the high and low temperature test chamber. When the fiber optic gyroscope is stationary, collect the output of the fiber optic gyroscope at a temperature change rate of ±1, ±5, ±8, and ±10 °C/min. As shown in Fig. 4.1. The output data of the fiber optic gyroscope collected in the experiment is divided into two groups: the first group includes the output of the fiber optic gyroscope at a temperature change rate of ±1, ±5, −8, and +10 °C/min, This group of data was used to establish the temperature error model; The second group includes the output of the fiber optic gyroscope at 8 and −10 °C/min temperature change rate, which is used to validate the established model. The first set of data is shown in Fig. 4.2, where Fig. 4.2a is the output three-dimensional of the fiber optic gyro and Fig. 4.2b is its side view.

Fig. 4.2 Fiber gyro output at different temperature change rates

54

4 Temperature Drift Modeling and Compensation for Gyroscope

As can be seen from Fig. 4.2, the temperature error of the fiber optic gyroscope caused by severe temperature changes contains strong noise and drift. In order to better study the temperature drift error of the fiber optic gyroscope, we first need to denoise the output signal of the fiber optic gyroscope. Using the LWT-FLP denoising algorithm proposed in Chap. 3 to process the output signal of the fiber optic gyroscope, the denoised signal is shown in Fig. 4.3. Figure 4.3 shows the temperature drift of the fiber optic gyro after denoising. It can be seen that the temperature drift trend is very clear, which is roughly linear with time, and the variation trend of drift with the rate of temperature change needs to be further explored. First establish the relationship between fiber-optic gyro temperature drift and time: D = at + b

(4.9)

In Formula (4.9): D is the gyro temperature drift, t is time, a, b is the coefficient to be determined. As can be seen from Fig. 4.3, Coefficients a and b are not constant, but change with the temperature change rate T, namely: a = f (T˙ ), b = f (T˙ )

(4.10)

At each temperature change rate, the coefficients a and b are solved using the obtained experimental data, then a high-order polynomial fitting model based on T is established for the calculated a and b. The fitting residuals of order 1–4 are shown in Table 4.1. It can be seen from Table 4.1 that when the fitting order is 1, the coefficient residuals have been reduced to a very low; when the fitting order is 4, Good fitting result of coefficient b can be obtained. comprehensive consideration of fitting accuracy and operation, the coefficient a temperature change rate model is selected as the

Fig. 4.3 Temperature drift of fog after denoising

4.2 Fiber Gyro Temperature Error Model Based on External …

55

Table 4.1 Fitting residuals based on the rate of temperature change Fitting order

1

2

3

4

Fitted

a

0.00064

0.00063

0.00058

0.00049

Residuals (°/h)

b

0.1353

0.1352

0.0327

0.0049

first-order fitting polynomial, and the coefficient b temperature change rate model is selected as the fourth-order fitting polynomial, as shown in Fig. 4.4. And combined with Formula (4.9), the final model is obtained: D = (a0 + a1 T˙ )t + (b0 + b1 T˙ + b2 T˙ 2 + b3 T˙ 3 + b4 T˙ 4 )

(4.11)

Using the data obtained from the experiment to solve the coefficients in Formula (4.11), the temperature drift model of the fiber optic gyroscope based on the rate and time of the external temperature change is obtained. The model is shown in Fig. 4.5. Comparing Figs. 4.3 and 4.5, the method proposed in this paper approximates the temperature drift error model of fiber optic gyroscope. Make a difference between Figs. 4.3 and 4.5, and the residual error is shown in Fig. 4.6. It can be seen from Fig. 4.6 that the difference between the model output and the real data has been controlled within a very small range, that is, the model built can well reflect the temperature drift of the fiber optic gyroscope with the temperature change rate and time trend, it can effectively compensate the temperature drift of the fiber optic gyro. In order to prove the applicability of the model, the proposed model was verified by using the output of the fiber optic gyroscope collected at the temperature change rate of +8 and −10 °C/min. First denoise the gyro output, then input the temperature change rate and time into the model to get the output, and compensate the gyro data after denoising. The result is shown in Fig. 4.7.

Fig. 4.4 Fitting results of coefficients a and b

56

4 Temperature Drift Modeling and Compensation for Gyroscope

Fig. 4.5 Fiber gyro temperature drift fit model

Fig. 4.6 Model fitting residuals 3d diagram and its side view

As can be seen from Fig. 4.7, after the denoise of the fiber-optic gyro signal and the drift compensation of the proposed model, the gyro error caused by the drastic change of external temperature is well suppressed, and the correctness and applicability of the model based on the rate and time of temperature change are verified.

4.3 Temperature Drift Modeling and Compensation Based on Genetic Algorithm and ELMAN Neural Network It can be seen from Fig. 4.7 that the temperature drift model of the fiber optic gyroscope established in the environment with severe temperature changes in the previous section can suppress the temperature drift of the fiber optic gyroscope well, and the

4.3 Temperature Drift Modeling and Compensation Based on Genetic …

57

Fig. 4.7 Fiber optic gyro output processing diagram a the temperature change rate is +8 °C/min; b the temperature change rate is −10 °C/min

model is simple and the reaction speed is fast, which is very suitable for the rapid compensation of the temperature drift of the fiber optic gyroscope in the environment of severe temperature changes. But at the same time, it can also be seen that due to the simple model, the compensation accuracy is low, and the temperature drift cannot be completely eliminated, which is not practical in an environment requiring high-precision compensation. Therefore, this paper proposes a method for modeling and compensating the temperature drift of fiber optic gyroscope based on genetic algorithm and Elman neural network.

4.3.1 Neural Network At present, due to its fault tolerance, self-organization, large-scale parallel processing, association function and strong adaptive ability, artificial neural networks have gradually become powerful tools for solving practical problems. The bottleneck of science and technology has played a powerful role in promoting. The full name of the neural network is Artificial Neural Network (ANN), ANN is a network system that simulates the information processing mechanism of the human brain. Its development is based on the research results of modern neurobiology. ANN not only has the general computing ability to process numerical data, but also has the thinking, memory and learning ability to deal with knowledge. Generally, the interconnection structure of neural networks is considered by many people. There are four typical structures, namely: (1) Feedforward network. The characteristic is that the neurons are arranged in layers, which in turn constitutes an input layer, a hidden layer and an output layer, each of which can only accept the input of the previous layer of neurons;

58

4 Temperature Drift Modeling and Compensation for Gyroscope

(2) Feedback network. Its characteristic is that there is feedback between the input layer and the output layer; (3) Mutually integrated network. Its characteristic is that there may be a connection between any two neurons; (4) Hybrid network. This kind of network is a combination of mesh network and hierarchical network. Currently, representative neural network models include: (1) BP neural network. BP neural network is a multi-layer feed-forward network, and its learning method adopts the minimum mean square error. BP neural network is the most widely used and can be used for language recognition, speech synthesis, adaptive control, etc. The shortcomings of BP are long training time, only tutor training, easy to fall into local minimum; (2) RBF neural network. The RBF neural network is an extremely effective multilayer feedforward network. The RBF neuron basis function has local characteristics that produce an effective non-zero response only in a small local range. Therefore, RBF can achieve high speed during the learning process. The disadvantage of RBF is that it is difficult to learn the high frequency part of the mapping; (3) Hopfield neural network. Hopfield is one of the most typical feedback neural networks. It is one of the most studied models. The network is a single-layer network, composed of the same neurons. It does not have a self-learning associative network and requires a symmetric connection. Hopfield network can also complete functions such as associative memory and control optimization; (4) Elman neural network. It is a typical local regression neural network, which is more suitable for time series processing. It is mainly composed of input layer, hidden layer, connection layer and output layer. It is an internal feedback link based on the basic BP network. This structure is sensitive to the context structure relationship, which is beneficial to the modeling of dynamic processes. BP neural network and RBF neural network have been widely used in the research of fiber optic gyro temperature drift modeling. According to the principle of Hopfield neural network, this model has poor self-learning function and is not suitable for modeling temperature drift. Therefore, this paper attempts to use Elman neural network to model the temperature drift of fiber optic gyro.

4.3.2 Elman Neural Network The Elman neural network was proposed by Elman in 1990 for the problem of speech signals. It is a typical local recursive network and is more suitable for time series processing. The basic Elman neural network is shown in Fig. 4.8. As shown in Fig. 4.8, the nonlinear state space expression of Elman neural network is:

4.3 Temperature Drift Modeling and Compensation Based on Genetic …

59

Fig. 4.8 Elman neural network structure diagram

yo (k) =

N

2 W jo x j (k)

(4.12)

i=1

 x j (k) = f Wi1j ei (k) + Wcj3 xrc (k)

(4.13)

xrc (k) = x j (k − 1)

(4.14)

2 is the connection In Formula (4.14): y0 (k) is the output of the neural network; W jo weight of the hidden layer to the output layer; Wcj3 is the connection weight of the input layer to the hidden layer; N is the number of hidden layer nodes; f (·) is the transfer function of the hidden layer neuron, using the transig function. Elman neural network uses the optimized gradient descent method as the learning algorithm, also known as the adaptive learning rate momentum gradient descent back propagation algorithm. This algorithm can effectively improve the network training rate, and at the same time can effectively prevent the network from falling into a local minimum. The purpose of the learning process is to use the difference between the output sample value and the actual output value of the network to modify the threshold and the weight so as to minimize the sum of squared errors at the output layer of the network. Suppose the actual output vector of step k system is yo (k), and the sample vector is y(k), in the time period (0, T), the error function is defined as:

1

[yo (k) − y(k)]2 2 k=1 T

E=

(4.15)

60

4 Temperature Drift Modeling and Compensation for Gyroscope

4.3.3 Genetic Algorithm The Elman neural network is a typical dynamic neuron network. It is based on the basic structure of the BP network and stores the internal state to make it have the function of mapping dynamic features, so that the system has the ability to adapt to time-varying characteristics. It can be seen from Fig. 4.8 that the choice of network structure, initial connection weights, and thresholds has a great influence on network training, but it cannot be accurately obtained. For this defect, a genetic algorithm can be used to optimize the neural network. Genetic Algorithm (GA) is a computational model that simulates the evolutionary process of Darwinian genetic selection and natural elimination. GA was first proposed by Professor J. Holland of the University of Michigan in 1975. GA has good global optimization Performance and strong macro search capabilities. Therefore, combining the neural network with the genetic algorithm, the genetic algorithm is first used to find the weight of the neural network during training, and the search range is narrowed, and then the neural network is used to accurately solve the problem, which can achieve the purpose of rapid and efficient and global search. It can also avoid the local minimum problem. GA not only has global search capabilities, but also improves local search capabilities, enhances the ability to automatically acquire and accumulate search space knowledge, and self-apply to control the search process, thereby greatly improving the nature of the results. GA first uses genotypes to represent problem solving, selects individuals who adapt to the environment, and eliminates unsatisfactory individuals, copies and reproduces the remaining individuals, and then generates new chromosome groups through genetic operators such as crossover and mutation. According to different convergence conditions, the individuals who adapt to the environment are removed from the old and new groups, and each generation is promoted to make continuous progress. Finally, they converge to the individuals who adapt to the environment, so that the optimal solution to the problem is obtained. The correspondence between concepts in biogenetics and concepts in GA is shown in Table 4.2. The genetic algorithm implementation steps are as follows: (1) Randomly generate a certain number of initial chromosomes. The above chromosomes will form a population, and the specific number of chromosomes in the population is called the population size or size (pop-size); (2) Use the evaluation function to evaluate the pros and cons of each chromosome, and use the degree of adaptation (fitness) of the chromosome to different environments as the genetic basis; (3) Selection strategy based on fitness value. In the current population, a certain number of chromosomes are selected as a new generation of chromosomes, the higher the fitness, the greater the chance of being selected; (4) Perform crossover (i.e. mating) and mutation operations on the newly generated populations. The purpose is to make each individual in the population have diversity and avoid falling into a local optimal solution. The chromosome group (population) produced by this step is called a progeny.

4.3 Temperature Drift Modeling and Compensation Based on Genetic …

61

Table 4.2 Correspondence between the concepts of biogenetics and the concepts in GA Biogenetics concept

Concepts in genetic algorithms

Survival of the fittest When the algorithm stops, the solution of the optimal target value is most likely to be retained Individual

Solution of the objective function

Chromosome

Coding (vector) of solutions

Gene

The characteristics (or values) of each component in the solution

Fitness

Fitness function value

Population

Selected set of solutions (where the number of solutions is the size of the group)

Reproduction

Select a set of solutions according to the adaptation function

Crossover

The process of generating a new set of solutions according to the mating principle

Mutation

The process by which a component of a code changes

Then repeat the operation process of selection, crossover and mutation. After a certain number of iterations, the best chromosome obtained is taken as the optimal solution of the optimization problem. The flow chart of GA is shown in Fig. 4.9. GEN=0

end

generate initial group

yes specify the result

whether the stop criterion is met

no select genetic factors with probability

calculate the fitness value of each individual

Pr i= 0 yes

CEN=GEN+1

Pc

choose an individual

Pm

choose two individuals

choose an individual

no i= N

perform copy

copy to new group

i=i+

1

perform hybridization

GEN - current algebra N - group size Insert two child strings into the new group

i=i+

Fig. 4.9 Genetic algorithm flow

1

perform mutation

insert into new group

62

4 Temperature Drift Modeling and Compensation for Gyroscope

Fig. 4.10 Variation of temperature error of fiber optic gyroscope with temperature

4.3.4 GA-Elman-Based Fiber Optic Gyro Temperature Drift Modeling and Compensation In order to establish the temperature drift model of fiber optic gyro, the temperature experiment of fiber optic gyro was first carried out. The experimental steps are as follows: first put the fiber optic gyro into the temperature control box, adjust the temperature change rate, collect the fiber optic gyro static output, the collection time is 40 min, and the collection frequency is 100 Hz. The temperature change and corresponding fiber optic gyro output are shown in Fig. 4.10. Figure 4.10a is the data collected under the condition of temperature change rate of ±5 °C/min, and Fig. 4.10b is the data collected under the condition of temperature change rate of ±8 °C/min. It can be seen that the output of the fiber optic gyro has a clear correlation with the temperature change. When the temperature change rate is different, the output temperature error is also different. The fiber optic gyro error under temperature change includes three parts: constant error, drift error and noise. It can be seen from the figure that when the temperature drops, the fiber optic gyro error mainly includes constant error and noise, and there is no obvious drift error; when the temperature increases, the fiber optic gyro error mainly includes constant error, drift error and noise. In order to establish an accurate temperature drift model of fiber optic gyro, the constant error must be removed first. The calculation method of the constant error is: n Di (4.16) D0 = i=1 n

4.3 Temperature Drift Modeling and Compensation Based on Genetic …

63

Taking the output of a fiber optic gyroscope collected at a temperature change rate of ±5 °C/min as an example, the data sequence after removing the constant error is shown in Fig. 4.11. After removing the constant error, the signal also needs to be denoised. The LWTFLP algorithm proposed in Chap. 3 of this article is used to denoise the signal shown in Fig. 4.11. The processed result is shown in Fig. 4.12. Figure 4.12 shows the output of the fiber optic gyro after removing the constant error and noise. It can be seen from the figure that the temperature drift of the fiber optic gyro has strong nonlinearity. The multiple linear regression model and the nonlinear regression model proposed by this article (Formula 4.11). It has been difficult to establish a high-precision fiber optic gyro temperature drift model, so in

Fig. 4.11 Optical fiber gyro output after removing constant error

Fig. 4.12 Fiber optic gyro output after removing noise

64

4 Temperature Drift Modeling and Compensation for Gyroscope

'

Neural Networks

T

D

(t)

Neural Networks D (t  1) Fig. 4.13 Temperature drift model of fiber optic gyroscope based on neural network

this paper, it is proposed to use the Elman neural network based on genetic algorithm to build the model. As can be seen from Fig. 4.12, the temperature drift of the fiber optic gyro has a clear correlation with the temperature change, but the temperature drift of the fiber optic gyro has little relationship with the temperature value, because different temperature drifts will occur under the same temperature value. It can also be seen that the temperature drift of the fiber optic gyroscope is closely related to the temperature change rate. When the temperature change rate is negative (temperature drop), the temperature error of the fiber optic gyroscope mainly includes constant error and noise; while the temperature change rate is positive (temperature Ascending), there is a large drift error in the temperature error of the fiber optic gyro. At the same time, in order to compensate for the high-precision temperature drift of the fiber optic gyro at the current moment, the temperature drift of the fiber optic gyro at the previous moment is also introduced into the model input as an influencing factor, so the temperature drift model of the fiber optic gyro is established, as shown in Fig. 4.13. In Fig. 4.13, T is the temperature change rate; D  (t) and D  (t − 1) are the temperature drift of the fiber optic gyroscope at the current time and the previous time respectively. That is to say, the temperature drift of the fiber optic gyro at the current moment is related to the rate of temperature change of the outside world and the temperature drift of the fiber optic gyro at the previous moment, and a model of its relationship is established through a neural network. The temperature drift of the fiber optic gyro under the temperature change of ±5 °C/min is used as the training data to train the proposed GA-Elman neural network, and the established model is used to compensate the temperature drift of the fiber optic gyro under the temperature change of ±8 °C/min. The Elman neural network is compared, and the comparison results are shown in Fig. 4.14. As can be seen from Fig. 4.14, compared with the traditional Elman neural network, the GA-Elman neural network proposed in this paper has higher modeling accuracy, can better predict the temperature drift of the fiber optic gyro, and obtains better compensation results than the traditional Elman neural network. In summary, the method proposed in this paper has effectively removed the constant error, noise and temperature drift of the fiber optic gyroscope, and effectively suppressed the temperature effect on the output of the fiber optic gyroscope, as shown in Fig. 4.15.

4.3 Temperature Drift Modeling and Compensation Based on Genetic …

65

Fig. 4.14 Effect diagram of fiber optic gyro temperature drift compensation

Fig. 4.15 Elimination of FOG temperature error

Figure 4.15a shows the original output of FOG collected at the temperature change rate of 8 °C/min, Fig. 4.15b shows the output of FOG after removing constant error and denoising, Fig. 4.15c shows the output of FOG after compensation of temperature drift. As can be seen from Fig. 4.15, the method proposed in this paper effectively eliminates the effect of temperature change on the fiber optic gyroscope, and improves the performance of the fiber optic gyroscope in the case of ambient temperature change. In order to evaluate the effect of eliminating temperature error of the fiber optic gyroscope quantitatively, Allan variance method is used to analyze the results as follows. After compensation from Table 4.3 and Fig. 4.16, we can draw a clear conclusion: The method proposed in this paper can effectively eliminate the error of FOG caused by temperature change, compared with the original output, the Allan variance coefficient of the compensated FOG output decreases completely, which verifies the effectiveness of the proposed method.

66

4 Temperature Drift Modeling and Compensation for Gyroscope

Table 4.3 Allan variance coefficient of FOG output Allan variance coefficient Raw data

Q (μ rad)

N (°/h1/2 ) B (°/h) K (°/h3/2 ) R (°/h2 )

149.37

0.94

28.09

176.34

229

Denoising and constant 129.58 value

0.79

24.52

154.15

197.57

After compensation

0.01

0.43

7.01

1.47

9.77

Fig. 4.16 Allan variance analysis graph of FOG output

4.4 Summary Temperature and the rate of temperature change are one of the main factors that cause the error of FOG, which can be compensated by establishing a mathematical model. Generally, the scale factor error and drift error caused by the temperature change of FOG are modeled and compensated. In this paper, firstly, the influence of temperature on the scale factor of FOG is analyzed, and a hyperbolic model based on the input angular rate and temperature is established to fit the scale factor. Then, the measured data to verify the established model, and the verification results prove the accuracy of the model. Then the influence of the drastically changing ambient temperature on the output of FOG is studied, the temperature drift model of FOG based on the rate of temperature change is established, which can quickly compensate the temperature drift of FOG under the condition of drastic temperature change.

4.4 Summary

67

Finally, the Elman neural network based on genetic algorithm is proposed, and the temperature drift model of FOG based on temperature change rate and prior knowledge is established. Compared with the traditional Elman neural network, the model can compensate the FOG temperature drift more accurately.

Chapter 5

Model and Algorithm for Discontinuous Observation Integrated Navigation System

5.1 Solutions and Typical Models of Discontinuous Observation Integrated Navigation System In car integrated navigation, Due to the characteristics of GPS, when the vehicles are driving in mountainous areas, forest areas or urban areas with high-rise buildings, GPS will lose its lock because the satellite signal is easily blocked, so it can’t combine with the SINS; When the SINS work continuously and independently for a long time, the navigation error will increase rapidly with the accumulation of time, resulting in the divergence of the output of the integrated system, which can not meet the requirements of high precision, high reliability, continuous operation and adaptability to different road conditions. This section will focus on this issue. Aiming at the problem that the GPS signal of vehicle-mounted integrated navigation is easy to lose lock, domestic and foreign experts and scholars put forward some practical solutions, which are mainly divided into the following two categories: 1. Add other sensors or auxiliary facilities. For example, the combination of vehicle GPS/INS integrated navigation system with visual sensor, odometer, map matching, road network assistance, vehicle coordination and other methods, these methods increase the source of information sources, and to a certain extent, it can make up for the impact caused by GPS losing lock. But also because of the addition of auxiliary facilities, it will definitely increase the cost and complexity of the system, which is not conducive to the practical application of civilian vehicles. 2. Artificial intelligence algorithm is used to model and compensate the SINS error. The most typical idea is: when the GPS signal is available, artificial intelligence algorithm is used to model the SINS error, when the GPS signal is unlock, the trained model is used to compensate the SINS error. Among them, method (2) does not need to add additional auxiliary facilities and hardware equipment, which is simple and feasible, so it is the most studied method © Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_5

69

70

5 Model and Algorithm for Discontinuous Observation …

at present. This section will also study the method to maintain the high precision of the integrated navigation system for a long period of time even when the GPS is unlocked. According to the calibration methods and the input and output of different models, the models with the most research and most application can be divided into three types, as shown below: 1. Taking SINS output and instantaneous time as the model input, the filter estimation error is taken as model output. This model is mainly used in integrated navigation systems with output correction. Its principle is shown in Fig. 5.1. As shown in the figure: figure (a) is a schematic diagram for training the artificial intelligence model when the GPS signal is effective; Figure (b) is a schematic diagram for estimating and compensating of navigation errors by using the trained model when GPS is unlocked. It can be seen from the error characteristics of SINS themselves that the navigation error of SINS will accumulate with the increase of time, so time is taken as one of the independent variables of the SINS error. It is proved that SINS navigation error can be predicted by taking SINS navigation output and time as the input, and the effect of time is fully considered, and good results are obtained; 2. The double filter model with the SINS output velocity as input and filter estimated velocity error as output. The biggest feature of this model is that two filters are applied: one is the velocity and position filter, which is used to estimate the velocity and position errors when the GPS signal is available; The other is the

Navigation Output

SINS

+ -

Artificial intelligence model

Time

Navigation Output

Navigation error

GPS

Position

Velocity

-

+ Kalman

(a) Navigation Output

SINS

Time

Navigation Output

-

Artificial intelligence model

(b) Fig. 5.1 Model (1) functional block diagram

+

5.1 Solutions and Typical Models of Discontinuous Observation …

71

Feedback correction

GPS

+

Target position error Velocity position filter

Artificial intelligence model velocity

SINS

+ Velocity filter

+

Output correction

-

Navigation Output

Fig. 5.2 Schematic block diagram of model (2)

velocity filter. When the GPS signal is unlocked, the output of the trained artificial intelligence model can be used as the observation quantity to input the velocity filter to estimate the velocity error. Its principle is shown in Fig. 5.2. The method makes a good prediction and compensation for the SINS error. The difference between this model and other models is as follows: Method (1) model directly uses the prediction of the artificial intelligence model to compensate the SINS error, and the model is the model uses the output of the model as an observation, and then input it into to the filter to get the optimal estimate, The estimated value is used to compensate the SINS error, which can effectively restrain the divergence of integrated navigation system caused by GPS loss of lock. However, due to the use of two filters, the structure of the model is more complex, which limits its application in engineering to a certain extent; 3. The model takes the output of gyro and accelerometer in SINS as input and the estimated error of filter as output. The model is shown in Fig. 5.3. Figure (a) is a schematic diagram of training the artificial intelligence model when the GPS signal is valid; Figure (b) is a schematic diagram of using the trained model to compensate for the navigation error when the GPS loses lock. When the GPS/INS integrated navigation mode is feedback correction, that is, the filter is used to estimate the navigation error of INS and the error of gyro and accelerometer, and real-time feedback correction of gyro and accelerometer is performed. Because the output of gyro and accelerometer will be corrected at each moment, the model does not consider the impact of time accumulation. In literature, the application of this model is described in detail, and the model is applied to compensate the navigation errors of INS when GPS is lost, and good results are achieved. Immanuel Kant believes that the objective material world can only give people a sense of chaos and disorganization, and the composition of knowledge is entirely

72

5 Model and Algorithm for Discontinuous Observation … Position、Velocity、Attitude

INS

Gyro output

Artificial intelligence algorithm

Acceleration output Position Velocity

Position、Velocity

GPS

-+

Filter

Position error Velocity error Attitude error

(a) Position、Velocity、Attitude

INS

Gyro output

Acceleration output

Artificial intelligence algorithm

Position error、Velocity Error、Attitude error

(b) Fig. 5.3 Model (3) functional block diagram

processed by the “innate knowledge” inherent in the human brain, Therefore, innate form and acquired experience are the fundamental elements of knowledge. Priori knowledge base is a knowledge base formed by the accumulation of previous independent experience and provides a mapping from the current state to the next local optimal state. Specifically, prior knowledge refers to all information available for learning tasks except training data, covering a wide range. For example, in the application of automatic character recognition, 0 (Arabic numeral) and O (English letter) are very difficult to distinguish, and error often occur. In this case, we can use our prior knowledge to know that only certain characters appear in a specific context. For example, the probability of 0 (Arabic numeral) appearing in a series of Arabic numeral characters is far greater than that of O (English letter). Otherwise, in a series of English characters, the probability of the O (English letter) is much greater than that of the 0 (Arabic numeral). In mathematical modeling, the information known before modeling can be called prior knowledge, which is a part of object characteristics and mechanism, or an accurate mathematical expression, such as symmetry, monotony, concavity and convexity, or empirical criteria, such as reasoning rules, etc.

5.1 Solutions and Typical Models of Discontinuous Observation … Position、Velocity、Attitude

SINS

+

73 Navigation output

-

Navigation error at current time Position Velocity

Artificial intelligence model

...

(t-n) (t-2)

Position Velocity GPS

+

Navigation error at previous moment(t-1)

Kalman Filter

-

(a) Position、Velocity、Attitude

SINS

Navigation output

(t-n) Navigation error at previous moment

(t-n-1)

. . .

Artificial intelligence model

Navigation error at current time

(t-1)

(b) Fig. 5.4 Model based on prior knowledge

In this paper, prior knowledge is applied to the SINS/GPS integrated navigation, and the SINS error is modeled and predicted by using the prior knowledge. The model block diagram is shown in Fig. 5.4. In Fig. 5.4, Figure (a) is the process of modeling the SINS error when GPS signals are available. The input of the model is the SINS error estimated by the Kalman filter at the previous time, and the ideal output of the model is the SINS error estimated by the Kalman filter at the current time; Figure (b) is the process of compensating for SINS error when the GPS signal is unlocked. The Kalman filter estimation error at the previous time is used to predict the SINS error at the current time, and each predicted error at the current time will be used as the input of the model to predict the error at the next moment. In the whole model, the filter and artificial intelligence model are crucial to the final compensation result. At present, the most widely used filter is the Kalman filter, and artificial intelligence algorithm mostly use neural network models. This paper proposes to use strong tracking Kalman filter (STKF) as a filter and wavelet neural network (WNN) as a model to predict and compensate the SINS error when GPS is unlocked. The specific algorithm is described in the following section.

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5 Model and Algorithm for Discontinuous Observation …

5.2 Application of Kalman Filter and Neural Network in Integrated Navigation of Non-continuous Observation 5.2.1 Strong Tracking Kalman Filtering Strong tracking Kalman filtering (STKF) is proposed on the basis of classical Kalman filtering theory, which can ensure that the filter works in the best state. Compared with the traditional Kalman filter, STKF has the following advantages: (1) The robustness of model uncertainty is better; (2) The tracking ability of the mutation state is extremely strong, even when the system reaches steady state, it still maintains the tracking ability of the slowly changing state and the mutation state; (3) Appropriate computational complexity. Based on the traditional Kalman filter, STKF introduces a time-varying fading matrix that can be calculated online into a one-step prediction error variance matrix. So, the following formula is obtained: T T + Γk−1 Q k−1 Γk−1 Pk/k−1 = λk φk,k−1 Pk−1 φk,k−1

(5.1)

In Eq. (5.1): λk ≥ 1 is the time-varying fading matrix λk = diag[λ1(k) , λ2(k) , · · · , λn(k) ]. Among them:  λi(k) =

 S0 (k) =

λ0 , λ0 ≥ 1 1, λ0 < 1

(5.2)

λ0 = tr [N (k)]/tr [M(k)]

(5.3)

k=0 r 0 r0T [ρ S0 (k − 1) + rk rkT ]/(1 + ρ) k ≥ 1

(5.4)

T N (k) = S0 (k) − Hk−1 Q k−1 Hk−1 − β Rk

(5.5)

T T M(k) = Hk−1 Ak−1 Pk−1/k−1 Ak−1 Hk−1

(5.6)

In Eq. (5.1): 0 < ρ ≤ 1 is forgetting factor; β ≥ 1 is softening coefficient. In this paper, both ρ and β are chosen by experience, It can be seen that when the movement state changes suddenly, The increase of the estimation error rk rkT will cause the increase of error variance matrix S0 (k) and the corresponding weighting coefficient λi(k) , which will enhance the tracking ability and reliability of the filter.

5.2 Application of Kalman Filter and Neural Network …

75

Fig. 5.5 Flow chart of STKF algorithm

In fact, the sufficient condition for  STKF is to adjust the filter gain matrix K k in real  time so that the formula E rk rkT = 0 (k = 0, 1, . . . , j = 1, 2, . . .) is established, In this way, the residual sequence is kept orthogonal in time, and STKF can be forced to keep track of the actual state of the system. When λi(k) ≡ 1, (i = 1, 2, . . . , n), STKF will degenerate into the traditional Kalman filter algorithm. The flow of the STKF algorithm is shown in Fig. 5.5.

5.2.2 Wavelet Neural Network Wavelet neural network is a new type of neural network model based on wavelet analysis. It combines the advantages of wavelet and neural network and has better performance. On the one hand, wavelet transform performs multi-scale analysis of the signal through scaling and translation, which can effectively extract the local information of the signal; on the other hand, the neural network itself has the advantages of self-learning, self-analysis, and strong fault tolerance, and is a general-purpose Function approximator. Therefore, wavelet neural network has stronger approximation and fault tolerance ability. Compared with other feedforward neural networks, wavelet neural networks have obvious advantages: (1) The basic element and the

76

5 Model and Algorithm for Discontinuous Observation …

Fig. 5.6 Structure of wavelet neural network

whole structure of wavelet neural network are determined according to wavelet analysis theory, which can effectively avoid the blindness of BP neural network in structure design; (2) For the same learning task, wavelet neural network has a faster convergence speed because of its simple structure; (3) Wavelet neural network has stronger learning ability and higher precision. The weight adjustment process of wavelet neural network can be divided into two stages: (1) From the input layer of the network, the output of each layer is calculated according to the input samples. Finally, the output of the output layer is calculated, which is a forward propagation process; In the correction stage, the weights are calculated and corrected from the output layer of the network, which is a backpropagation process. The two processes alternate repeatedly until the requirements are met. The wavelet neural network is divided into three layers, input layer, hidden layer and output layer. The input layer has M(m = 1, 2, . . . , M) neurons, the hidden layer has K (k = 1, 2, . . . , K ) neurons, and the output layer has N (n = 1, 2, . . . , N ) neurons. The output layer uses sigmoid output. The specific results are shown in Fig. 5.6. The neuron excitation function selected in the hidden layer is Morlet wavelet, and its expression is as follows: 

x −b h a



      x −b x −b 2 = cos 1.75 exp −0.5 a a

(5.7)

During training, the momentum term is added to the weight and threshold correction algorithm, and the correction value obtained in the previous step is used to smooth the learning path, which can effectively avoid falling into the local minimum and speed up the learning speed. In order to avoid the oscillation caused by the correction of weights and thresholds during the sample-by-sample training process, a batch training method is adopted, which is to accumulate the correction values generated by a batch of samples and then process them. For the output of the network, the output of the wavelet node of the hidden layer of the network is first weighted and summed, and then transformed by the Sigmoid () function to obtain the final network output.

5.2 Application of Kalman Filter and Neural Network …

77

This is helpful to deal with the classification problem, while effectively avoiding the divergence during training. Given the input and output samples of group P( p = 1, 2, · · · , P), the learning rate is η(η > 0), and the momentum factor is λ(0 < λ < 1). According to the basic idea of the fastest descent method, the objective error function can be defined as follows: E=

P

Ep =

p=1

P N 1 p (d − ynp ) 2P p=1 n=1 n

p

(5.8) p

In formula (5.8): dn is the expected output of the n node of the output layer; yn is the actual output of the network. The purpose of the algorithm is to adjust the parameters continuously so that E can reach the minimum value. According to the network layer structure, the hidden layer output is:  p

Ok = h

M p Ik p , Ik = wkm xmp αk m=1

(5.9)

p

In formula (5.9): h() is the Morlet wavelet function; xm is the output of the input p layer; Ok is the output of the hidden layer; wkm is the weight between the node m of the input layer and the node k of the hidden layer. The output of the output layer is: N   p ynp = f Inp , Inp = wnk Ok

(5.10)

n=1 p

In formula (5.10): f () is the Morlet wavelet function; Im is the output of the input layer; Wnk is the weight between the node k of the input layer and the node n of the hidden layer. The training algorithm of the wavelet neural network can gradually update the connection weights between neurons, the scaling factor and translation factor of the wavelet function. Their derivation formula is as follows. First, the weight adjustment formula between the hidden layer and the output layer is: new old = wnk +η wnk

P

old δnk + λ 1 wnk

(5.11)

m=1 ∂EP

p

p

p

p

old new , wnk represent the In formula (5.11): δnk = ∂wnkn = (dn − yn )yn (1 − yn ); wnk connection weights between the hidden node k and the output layer node n before old is the momentum term. and after adjustment, respectively; 1 wnk

78

5 Model and Algorithm for Discontinuous Observation …

new old wkm = wkm +η

P

old δkm + 1 wkm

(5.12)

m=1

In formula (5.11): δkm =

∂ E nP ∂wkm

=

N

∂OP

n=1

p

old new (δnk wnk ) ∂αkk xm ; wkm , wkm represent the

connection weights between the hidden node m and the output layer node k before old is the momentum term. and after adjustment, respectively; 1 wkm aknew

=

akold

+η

P

δam + λ 1 akold

(5.13)

m=1

In formula (5.13): δak =

∂ E nP ∂ak

factor before and after adjustment;

=

N

(δnk wnk )

n=1

1 akold

∂ OkP ∂ak

; akold , aknew is the expansion

is the expansion factor momentum term.

∂OP ∂ E nP = (δnk wnk ) k ∂bk ∂bk n=1 N

δbk =

(5.14)

P In formula (5.14): bknew = bkold + η m=1 δbm + λ 1 bkold ; bkold , bknew is the translaold tion factor before and after adjustment. 1 bk is the momentum term of translation factor. The specific implementation steps of the wavelet neural network learning algorithm are as follows: 1. Initialization of the network reference gives initial values to the wavelet’s scaling factor ak , translation factor bk , network connection weights wkm and wnk , learning rate η(η > 0), and momentum factor λ(0 < λ < 1), and sets the input sample calculator p = 1; p 2. Input the corresponding expected output dn of the learning sample set; 3. Calculate the output of the hidden layer and the output layer; 4. Calculate the error and gradient vector; 5. Enter the next sample, that is p = p + 1 6. Determine whether the algorithm is over: when E < ε, that is, when the cost function E is less than a certain value ε(ε > 0) set in advance, stop the network learning process; otherwise, reset p to 1 and go to step (2).

5.2.3 Experimental Results and Analysis The SINS error compensation model based on a priori knowledge and the algorithms used in the model are introduced above, which will be verified in this section. Firstly, the SINS/GPS Integrated Navigation experiment is carried out. The GPS receiver

5.2 Application of Kalman Filter and Neural Network …

79

Fig. 5.7 Schematic diagram of vehicle SINS/GPS

used is the venus 628lp single-chip receiver with the sampling frequency of 1 Hz, and the SINS used is the inertial navigation product based on FOG with the sampling frequency of 100 Hz. The vehicle test is shown in Fig. 5.7. For the convenience of observation, the driving path is converted from the geodetic coordinate system to the plane rectangular coordinate system. The driving path in the plane rectangular coordinate system is shown in Fig. 5.8. In the figure above, the blue track is the vehicle’s driving track, and the red track is the GPS unlocked road section. When the GPS loses lock, only SINS works alone in the integrated system. Due to the characteristics of its own sensors, the error of SINS will accumulate over time. The error compensation model of sins is established by using the algorithm proposed above, and compared with the traditional method. 1. Comparison between different models For SINS error compensation when the GPS loses lock, this paper proposes an error model based on a prior knowledge, that is, the estimated value of the current time is predicted by using the previous time filter estimate. In order to verify the validity and advancement of the model, under the same algorithm (STKF/RBF algorithm), the most widely used currently is to take the gyro and accelerometer output (Output) in SINS as the input, and use the filter to estimate the position error (Position Error) is compared for the output model (OP model). The model proposed in this paper uses the position error (Position Error) estimated by the filter at the previous time to predict the position error (Position Error) at the current time, so it is simply referred to as the P-P model. The comparison results are shown in Fig. 5.9.

80

5 Model and Algorithm for Discontinuous Observation …

Fig. 5.8 Vehicle SINS/GPS driving route map

Fig. 5.9 Comparison results between different models

It can be seen from the above figure that with the model proposed in this paper, the prediction results superior to the traditional O-P model can be obtained in both the X and Y directions. In order to more intuitively reflect the superiority of the P-P model, it is proposed to use Mean Absolute Error (MAE) to reflect the prediction error, as shown in Fig. 5.10. 2. Algorithm verification and comparison Subsequently, on the basis of applying the same model (P-P model), the algorithm proposed in this paper is verified and compared with the traditional algorithm. When the GPS signal is available, use the estimated error of the filter output at the previous

5.2 Application of Kalman Filter and Neural Network …

81

Fig. 5.10 Comparison of mean absolute error (MAE/m)

time as the model input, and the estimated error of the filter output at the current time as the model output to train the model; when the GPS signal is not available, use the trained model The estimated error at the current moment is predicted, and error compensation is performed for SINS. The verification and comparison results are shown in Fig. 5.11. As can be seen from Fig. 5.11, compared with the traditional algorithm, due to the performance of the Kalman filter, etc., the algorithm based on KF/RBF divergence phenomenon, and compared with the STKF/RBF method, the STKF/WNN algorithm proposed in this paper can be better predict the optimal filter estimation of position error. In order to more intuitively reflect the superiority of the algorithm, MAE is also used to reflect the prediction error, as shown in Fig. 5.12. 3. The final compensation result of the GPS loss of lock phase The model based on prior knowledge and STKF/WNN algorithm proposed in this paper are used to compensate SINS error in GPS lock loss stage. The compensation result is shown in Fig. 5.13.

Fig. 5.11 Comparison results between different algorithms (same model)

82

5 Model and Algorithm for Discontinuous Observation …

Fig. 5.12 Comparison of mean absolute error values (MAE/m)

Fig. 5.13 Effect compensation diagram

It can be seen from Fig. 5.13 that in the phase of GPS losing lock, there is a certain error in the navigation results of pure SINS. After the error is compensated, a result with good coincidence with the real trajectory is obtained, which fully validates the effectiveness of the models and algorithms proposed in this paper.

5.3 Application of Self-learning Cubature Kalman Filter in Combined Navigation 5.3.1 Square Root Cubature Kalman Filter The Cubature Kalman Filter algorithm uses the third-order spherical-radial volumetric rule to calculate the Gaussian approximate integral of the nonlinear function. The main rule of square root cubature calculation is to pass the square root of the prediction error covariance and the posterior error covariance in the filtering process

5.3 Application of Self-learning Cubature Kalman Filter in Combined Navigation

83

to ensure the symmetry and positive definiteness of the prediction covariance. First, a model of discrete Kalman filtering needs to be established 

X (k) = f (k − 1, X (k − 1)) + q(k − 1) Z (k) = h(k, X (k)) + r (k)

(5.15)

Among them, X (k) represents state vector in state space, f (·) = Fn · X (k) represents state matrix, in non-linear filter system, the state space is non-linear transition matrix of error model, q(k − 1), r (k) use zero mean and zero covariance matrix to represent process noise and measurement noise matrix. Q(k − 1), R(k); h(·) is measurement matrix fusing position and velocity, means measurement noise matrix. Square root Cubature Kalman filtering includes two steps, namely, time prediction update and measurement update. In the process of transferring calculations, several values need to be clarified. S( k − 1|k − 1) is square root of the covariance matrix P( k − 1|k − 1). The calculation relationship between the two can be expressed as P( k − 1|k − 1) = S( k − 1|k − 1)S T ( k − 1|k − 1)

(5.16)

S R (k) and S Q (k − 1) indicate the square root factor of R(k) and Q(k − 1). Their respective calculation relations can be expressed as Q(k − 1) = S Q (k − 1)S QT (k − 1)

(5.17)

R(k) = S R (k)S QT (k)

(5.18)

Step 1: time update Firstly, Calculate cubature point χ i (k − 1|k − 1); χ i ( k − 1|k − 1) = S( k − 1|k − 1) · I (i) + Xˆ ( k − 1|k − 1), i = 1, ..2n  I (i) =

(5.19)

√ n[1]i , i = 1, . . . ., n √ − n[1]i−n , i = n + 1, . . . , 2n

√ where n [1]i identity the i-th column vector of the n × n identity matrix. Then calculate the propagation cubature point χ i∗ ( k|k − 1)   χ i∗ (k|k − 1) = f k − 1, χ i (k − 1|K − 1) , i = 1, . . . 2n

(5.20)

Finally, estimate the prior state relative to the current time and the square root of corresponding covariance matrix. 2n 1 i∗ Xˆ ( k|k − 1) = χ ( k|k − 1) 2n i=1

(5.21)

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  S( k|k − 1) = T ria X ∗ ( k|k − 1), S Q (k − 1) 1 X ∗ (k|k − 1) = √ [χ 1∗ (k|k − 1) − Xˆ (k|k − 1), . . . , 2n χ 2n∗ (k|k − 1) − Xˆ (k|k − 1)], i = 1, . . . , 2n

(5.22)

(5.23)

T ria(·) represents a triangular matrix, X ∗ ( k|k − 1) can be calculated by the formula (5.23). Step 2: Measurement update Firstly, need to calculate cubature point and propagation cubature point similarly: χ i ( k|k − 1) = S( k|k − 1) · I (i) + Xˆ ( k|k − 1), i = 1, . . . , 2n   χ i∗∗ (k|k − 1) = h k, χ i (k|k − 1) , i = 1, . . . , 2n

(5.24) (5.25)

Then estimate the square root of the prior measure and the corresponding covariance matrix Szz ( k|k − 1). The updated measurement value Z ( k|k − 1) can be calculated by formula (5.28) 2n 1 i∗∗ Zˆ ( k|k − 1) = χ ( k|k − 1) 2n i=1

(5.26)

Szz ( k|k − 1) = T ria([Z ( k|k − 1), S R (k)])

(5.27)

1 Z (k|k − 1) = √ [χ 1∗∗ (k|k − 1) − Zˆ (k|k − 1) . . . , 2n χ 2n∗∗ (k|k − 1) − Zˆ (k|k − 1)], i = 1, . . . , 2n

(5.28)

Then calculate the cross covariance matrix Sx z ( k|k − 1), update the predicted state quantity at the current moment X ( k|k − 1) and it can be obtained from the formula (5.29) Sx z ( k|k − 1) = X ( k|k − 1)Z T ( k|k − 1) 1 X (k|k − 1) = √ [χ 1 (k|k − 1) − Xˆ (k|k − 1), . . . , 2n χ 2n (k|k − 1) − Xˆ (k|k − 1)], i = 1, . . . , 2n

(5.29)

(5.30)

Finally, the Kalman filter gain is calculated, and estimated the square root of the prior state quantity at the current moment and the square root of the posterior

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covariance S( k|k).   T K (k) = Sx z ( k|k − 1)/Szz ( k|k − 1) /Szz ( k|k − 1)

(5.31)



Xˆ ( k|k) = Xˆ ( k|k − 1) + K (k) Z (k) − Zˆ ( k|k − 1)

(5.32)

S( k|k) = T ria([X ( k|k − 1) − K (k)Z ( k|k − 1), K (k)S R (k)])

(5.33)

5.3.2 Long-Short Term Memory Neural Network Long-Short Term Memory Neural Network (LSTM) is a prediction algorithm based on time series information. Recursive Neural Network (Rensive Neural Network, RNN) can only remember short-distance information in information sequence. The special structure of LSTM gives the LSTM network the ability to memorize longdistance information. Similar to the memory information of the human brain, the key behavioral sentence is a chronological data, and the behavior information is long-distance information. Unlike RNN, LSTM adds a state unit (Cell State) that can maintain long-distance information on the basis of RNN. The network is mainly composed of Input Gate, Forget Gate, and Output Gate. The purpose of this design is to avoid the problem of gradient disappearance and gradient explosion caused by the chain rule in the neural network gradient calculation. Each gate has its own different purpose. The input gate mainly specifies the information added to the state unit. The self-looping connection layer ensures that the state unit information remains unchanged from one door to another. The forget gate regulates the connection layer information of the state unit and defines the state of information deleted from the cell. The output gate outputs the information output from the status unit, and the output gate can also affect the information of the status unit. The structure of the long-term and short-term memory neural network is shown in Fig. 5.14   f t = sigmoid W f xt + U f h t−1 + b f

(5.34)

C˜ t = tanh(Wc xt + Uc h t−1 + bc )

(5.35)

  i t = sigmoid Wi xt + Ui h t−1 + b f

(5.36)

Ct = f t ◦ Ct−1 + i t ◦ Cˆ t

(5.37)

ot = sigmoid(Wo xt + Uo h t−1 + bo )

(5.38)

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Fig. 5.14 Schematic diagram of long-term and short-term memory neural network

h t = ot ◦ tanh(Ct )

(5.39)

Among them, xt is the input amount of the memory cell at time t, bi , b f , bc , bo represents the offset vector; ◦ represents the element multiplication operation; Wi , W f , Wc , Wo , U f , Uc , Uo represents the weight matrix; sigmoid(·) refers to the sigmoid function by element. First, use formula (5.35) and formula (5.36) to calculate the value of the input gate at time t and the posterior value of the state unit. Then, calculated value f t and activates the forget gate unit at the same time. Through calculating the values of i t , C˜ t and f t , update the state of the memory cell at next time t. In the final calculation step, the value of the output gate can then be calculated using the updated state of the memory cell.

5.3.3 Self-learning Volume Kalman Filter Faced with the intelligent vehicle’s demand for intelligent navigation and positioning technology, to solve the problem of GPS/INS seamless navigation under the condition of satellite signal loss of lock, this paper proposes a new GPS/INS integrated navigation method based on self-learning volume Kalman filtering. This method divides the operation phase of the INS/GPS integrated navigation system into a training phase and an error compensation phase. The training phase is the phase where the GPS signal is valid. When the GPS signal is good, the LSTM (Long Short Term Memory) network is used to establish the navigation system. The prediction model and the optimal estimation error model use a self-learning Kalman filter composed of two cyclic filtering subsystems. The speed error and position error of

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INS are measured with the difference between the speed and position of INS and GPS. Optimal estimation to realize the self-learning function; the error compensation stage is the GPS signal loss-locking stage in the face of complex environments. At this time, the Kalman filter already has the function of self-learning, which can fully trust the prediction results of the long-term and short-term memory neural network. The navigation system compensates to improve the accuracy of the traditional intelligent vehicle navigation and positioning method under the GPS lock-free environment. It can be used for long-distance high-precision navigation and positioning in complex closed satellite signal rejection environments such as cities, indoors, underground mines, etc. The self-learning volume Kalman filter mainly includes the following parts, the specific block diagram is shown in Fig. 5.15 1. Training step: When GPS is working normally, two loop filter networks are used to optimally estimate the speed error and position error of INS with the difference between the speed and position of INS and GPS as observations; Each of the cyclic filter networks includes an LSTM neural network and a CKF filter, which are used to implement self-learning functions; In Fig. 5.15, when the GPS signal is available, the first cyclic filter network uses CKF1 to optimally estimate the INS error, its system state quantity is the position and velocity PI N S (k), VI N S (k) at the k time of the INS output, and the system observations are (Dp, Dv) the difference of the speed and position between INS and GPS. the optimal estimate of the output is (δp1 , δv1 ); and use LSTM1 to learn

Fig. 5.15 Schematic diagram of the self-learning volume Kalman filter training stage

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the relationship between the CKF1 gain at the past time and the system observation at the current time, thereby predicting the system observation, the input is the gain   (K t−n ( p, ν), . . . , K t−1 ( p, ν)) of CKF1 at time t − n, t − (n − 1), . . . , t − 1, the output is the system observation (Dp  , Dv ) at time t; The LSTM1 prediction (Dp  , Dv ) is used as the system observation of CKF2 in the second cyclic filter network. The system state quantity of CKF2 is the same as CKF1, and the optimal estimate of the output is (δp  , δν  ); The relationship between the gain and the current CKF2 optimal estimated error value, and predict the optimal estimated error value (δp  , δv ) output by CKF2: 

δp  = δp1 − δp  δv = δv1 − δv

(5.40)

training process of LSTM1 is: input the set filter gains xt =  The   ( p, ν), . . . , K t−1 K t−n ( p, ν) of the first n moments into the input gate (i t ) of the storage unit in the LSTM1 neural network for training. After the forget gate ( f t ) and output gate (ot ) are calculated, the optimal training result is obtained, output  ot = Dt−n ( p, ν), . . . , Dνt−1 ( p, ν) . Wherein, the filter gains of the first n times are obtained by processing the Cubature Kalman filter using the difference between the position and velocity output by the INS and GPS as observation information.  The training process of LSTM2 is: input the gain of CKF2 xt = K t ( p) , K t (v) into the input gate (i t ) of the storage unit of LSTM2, after the ( f t ) and  forget gate the output gate (ot ) are solved, the optimal training result ot = δp  , δv is obtained and output. (1) Error prediction and compensation step: When the GPS signal is out of lock, the system observation is predicted based on LSTM1 in the first loop filter network, and the predicted system observation is provided to CKF2 in the second loop filter network, Realize seamless navigation when the GPS signal is out of lock; meanwhile, predict and compensate the optimal estimated error value of CKF2 based on LSTM2 in the second cyclic filter network. In Fig. 5.16, when the GPS signal is out of lock, CKF1 stops working, and CKF2, LSTM1, and LSTM2 work normally. The state and prediction of CKF2 are the same as in step 1. Then get the best estimate δp  , δv . The input of LSTM1 become   K t−n ( p, ν), . . . , K t−1 ( p, ν) the gain of CKF2 at time t − n, t − (n − 1), . . . t − 1. The output is Dp  , Dv the systematic observation at the predicted time t, The input and output of LSTM2 is the same as step 1. Realize the prediction of CKF2 optimal estimation error value (δp  , δv ), Predict and compensate for the optimal estimated error value of CKF2, 

δp2 = δp  + δp  δv2 = δv + δv

(5.41)

2. Provide the optimal estimate after error compensation in step 2 to INS, and finally realize the correction of INS speed error and position error. The position

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Fig. 5.16 Schematic diagram of the self-learning volume Kalman filter prediction stage

and velocity of the inertial system are recorded as PI N S (k), VI N S (k), and the position and velocity after the optimal estimation and compensation are recorded as PI N S (k), VI N S (k). After predicting and compensating the optimal estimation error value of CKF2, it can be used as the optimal estimation of the INS error and Perform error compensation for INS. The position and speed information after the final compensation is as follows: 

PI N S (k)=PI N S (k) − δp2 VI N S (k) = VI N S (k) − δv2

(5.42)

In this prediction error compensation process, LSTM1 can generate the predicted observations as the system parameters of CKF2, and at the same time, LSTM2 can generate the predicted optimal estimated error value as the error compensation of the optimal estimated output of CKF2. In the traditional Kalman filter and neural network combined model, the optimal estimate of the Kalman filter output is generally directly compensated to the inertial system. Compared with the traditional method, in the method proposed in this paper, the training and prediction of the optimal estimation error value are added, which is equivalent to further improving the accuracy of the compensation term, thereby referring to the navigation positioning accuracy. The advantages of self-learning Kalman filtering are: (1) Self-learning Kalman filtering can learn the relationship between the filter gain and the current observation, and realize continuous seamless navigation and positioning under the condition of satellite signal rejection; (2) Compared with the single prediction model combining general Kalman filtering and neural network, the self-learning Kalman filtering method combines deep learning algorithms, adding a cyclic filtering network

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for the prediction and compensation of the optimal estimation error difference To improve navigation accuracy.

5.3.4 Experimental Results and Analysis The GPS/INS integrated navigation method based on self-learning volume Kalman filtering is introduced above. This section verifies it and carries out the vehiclemounted INS/GPS integrated navigation experiment. The inertial system is composed of STIM202 gyroscope and Model1521L accelerometer. The reference GPS uses the high-precision NovAtel ProPak6, the positioning accuracy is 1 cm (RTK), and the operating frequency is set to 100 Hz. The system has been running for 5000 s, of which 100 s are in the state of losing lock. The on-board experiment is shown in Fig. 5.17. The trajectory of the on-board experiment is shown in Fig. 5.18. The black trajectory in the figure is the reference trajectory of the vehicle. When the GPS loses lock, only SINS in the combined system works alone. Due to the characteristics of its own sensors, the error of SINS will accumulate over time. The SINS error compensation model was established using the algorithm proposed above and compared with the traditional method. 1. Verification and comparison of different models In order to compare and analyze the effectiveness of this experimental method, this paper also made a comparison of different integrated navigation system error compensation models. Different error compensation models include: pure inertial system model, model when only the first loop filter system works and traditional

Fig. 5.17 Schematic diagram of GPS/INS experimental device

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Fig. 5.18 Vehicle INS/GPS driving route map

optimization model, the traditional optimization model refers to the use of intelligent algorithms to predict the difference between the state of the navigation system and the observation Methods. Figure 5.18 shows the orbit results calculated by different models. Figure 5.19a, b show the position errors of the east and north directions respectively, and Fig. 5.20a, b show the velocity errors of the east and west directions respectively.

(a)

(b)

Fig. 5.19 Position error when GPS is out of lock a East direction; b North direction

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(a)

(b)

Fig. 5.20 Speed error when GPS loses lock (a) east direction; (b) north

In the comparison diagram of the position trajectory, the black trajectory represents the reference trajectory. As can be seen from Fig. 5.18, the trajectory based on the self-learning Kalman filter method is the closest to the reference trajectory, that is, the navigation positioning error is minimal, and high accuracy can be obtained. Navigation information. It can be seen from Figs. 5.19 and 5.20 that the pure inertial system model results in a purely divergent error state, the traditional optimization model is less robust, and the position and velocity errors are large for the satellite’s short-term lock loss. Only the position and velocity errors calculated by the model when the first loop filter system is working are small. Compared with the model when only one loop filter system works, the self-learning volume Kalman filter method provides a more accurate filter estimation result through the additional second loop filter system, compensates to the INS, and improves the integrated navigation accuracy. 2. Verification and comparison of different algorithms The self-learning volume Kalman filtering method proposed in this paper is a combination of square root volume Kalman filtering and long-short-term memory neural network. In order to verify the generalization ability and effectiveness of the longterm and short-term memory neural network, this paper changes the algorithm of the neural network based on the proposed method, and calculates the position and speed error values of different algorithms in the case of GPS loss of lock. The main comparison algorithms are: random forest regression algorithm and radial basis neural network algorithm. When GPS signals are available, the first deep learning model is trained with the CKF1 gain at the past n time in the first loop filter network as the model input, and the filter observation at the current time as the model output; the second loop filter In the network, the current CKF2 gain is used as the model input, and the current CKF2 optimal estimated error value is used as the model output

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to train the second deep learning model, and the optimal estimated error value output by CKF2 is predicted. When GPS signals are unavailable, the trained model is used to predict the estimated error at the current time, and the INS is compensated for errors. Random Forest Regression (RFR) is a classification algorithm based on the concept of integrated learning. For the training data set, this method can classify the verification results and select the verification calculation set with the most final voting results as the output. In the regression algorithm, you can train according to the input and output of the above model, and use the output as prediction data to compensate for the INS error drift when the GPS loses lock. Radial basis neural network (Radial Basis Neural Network, RBF) was originally used to solve the multivariable difference problem, which can be used for multi-data fitting. When the GPS loses lock, the RBF algorithm can be used to learn a priori data to fit the GPS. The real data is used for combination and compensation. The algorithm verification results are shown in Figs. 5.21 and 5.22. Figure 5.22a, b show the east and north velocity errors verified by different algorithms. It can be seen from the whole of Figs. 5.21 and 5.22 that the position error of the pure inertial device drifts linearly with time, and the drift distance is large. The error of the random forest regression algorithm is the largest, followed by the radial basis neural network. The long-term and short-term memory neural network has the best effect, and the errors of the north and east directions are the smallest. Because the structure of the long-term and short-term memory neural network contains a state unit that can maintain long-distance information, this unit can independently filter and extract data features based on long-term sequences, thereby completing effective prediction of data with small errors. Other algorithms do not have this feature, so that the potential information of the data cannot be deeply explored, so effective data prediction cannot be achieved.

(a)

(b)

Fig. 5.21 Position error when GPS is out of lock a East direction; b North

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Fig. 5.22 Speed error when GPS loses lock a east direction; b north direction

Chapter 6

Brain-Like Navigation Technology Based on Inertial/Vision System

6.1 Overview of Bionic Navigation Background Navigation technology plays a very important role in our life. we can easily get our current location and reach our destination with GPS even in a strange environment. However, GPS may fail to work in many occasions, such as between high-rise buildings in cities, deep sea waters, deep mountains, old forests and other remote areas. Therefore, the inertial navigation system (INS) shows its unique advantages at this time. There is no need to rely on external information, and the position of the next point can be calculated by relying only on continuously measured information about the speed and direction of its own movement, so the application of INS is very broad. INS can work in the air, the earth’s surface and even underwater all-weather and all-time. It has good stability and is not easy to be disturbed. However, as the navigation information is generated by the integration of the speed and direction information measured by the sensor, the error will be larger and larger along with the time, which leads to the problem of poor working accuracy of INS in a long period of time. How to reduce or even eliminate these errors reasonably and effectively and improve the intelligent level of INS system, has undoubtedly become one of the research hotspots in recent years. The research shows that the navigation strategy combining GPS, radio navigation and INS can complement the advantages of various navigation means and make it better. Kalman filter algorithm is commonly used to merging data. However, the application of Kalman filter algorithm needs to accurately build the motion model and observation model of the system, and the large amount of calculation for the complex dynamic environment modeling, to a certain extent, also limits the application of Kalman filter. Moreover, this model’s degree of intelligence is not high, and it only reduces the error to a certain extent, and cannot eliminate the accumulated error. On nature, many animals have outstanding guiding ability. Whether it’s thousands of miles away, or in severe weather such as storms, these magical animals always know where the road is. There is no lack of such experts in human beings. They seem © Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_6

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to have embedded a high-resolution map in their mind, so they will not lose their way in any way. This shows that organisms only rely on their own organs to obtain the perception of external information, through a certain biological mechanism. John F. Kennedy of University College London discovered a neuron in the hippocampus of rat brain as early as 1971, which is responsible for remembering location features. They named it “location cell”. More than 30 years later, the scientist couple May-Britt Moser and her husband Edward Mozer proved through a series of experiments that there is a mechanism to establish a spatial coordinate system in the animal’s brain, through which speed cells and head orientation cells can obtain motion information to generate path integral. Imitating this mechanism, Michael Milford of Queensland University of technology, Australia, made a mathematical model of mouse brain, established ratslam bionic navigation algorithm by using vision driven navigation system, and achieved good results in outdoor navigation experiments. But ratslam is a pure vision navigation algorithm, and its core sensing information comes from vision odometer. Therefore, although ratslam algorithm achieves brain like intelligence to a certain extent, it also shows the defects of poor robustness and low navigation accuracy in complex environment.

6.2 Bionic Navigation Mechanism Some animals have outstanding guiding ability, such as ants can return to their nests directly after going out for food, and migratory birds can migrate thousands of kilometers each year without losing their way. Human beings also have the ability to remember the scene. When we face a picture, our mind will come up with the location of the picture. After years of research, the 2014 Nobel Prize winner in biomedicine discovered brain location system cells based on the mechanism of animal navigation. Like the well-known GPS system, the brain positioning system is also through the acquisition of their own movement time, location information for positioning and navigation. So far, it has been found that the main neurons related to animal environmental cognition are place cell, head direction cell, grid cell and speed cell. As shown in Fig. 6.1. Fig. 6.1 Neural cells related to environmental perception of animals

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Fig. 6.2 Location of cell distribution domain

6.2.1 Positional Cell John O’Keefe, a professor at University College London, UK, first discovered the location cells in the brain of animals and identified them as one of the key cells that make up the brain location system. In 1971, O’Keefe noted that when the mouse ran to a certain place in the experimental area, a specific nerve cell G1 in the hippocampus would be activated, while other cells around it would not be activated. When you go to other places, this cell doesn’t activate and the other cell G2 is activated. This suggests that the activated cells are the location cells that the mouse perceives its own position. These location cells are not simply receiving visual information, but building the brain map of the room where the mouse recognizes. The hippocampus generates a large number of maps according to different environments, which are formed by a large number of nerve cells in different environments. Therefore, the memory of environment can be stored by specific activation combination of neurons in hippocampus. As shown in Fig. 6.2. In Fig. 6.2, the orange dot indicates that the cell is activated at the corresponding position, while the black line indicates the movement track of the mouse. When the mouse reaches a specific location, the location cells of the specific location will be activated. When the location cells at these points are activated, it indicates the specific location of the mouse in the region. These cells discharge in different locations in the hippocampus of different mice. Studies have shown that positional cells have only one activation domain in the environment.

6.2.2 Head Direction Cell The continuous attractor model can be used in the modeling and analysis of headoriented cells. The arrangement of typical cells: the adjacent cells are strongly connected, while the far cells are weakly connected. Such a stable network is a bunch of simple active cells, and the distribution rate of active cells in different

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Fig. 6.3 Distribution rate of head direction cell in different directions

directions is peaked. The direction represented by the current activity state can be obtained in several ways (Fig. 6.3). According to network dynamics, when a new active cell is injected near the active peak, the active peak will move towards the end of the active cell. Although the technique of injecting new active cells may change, this characteristic is of universal significance for the method of path synthesis. In 1995, Skaggs、Knierim and others used two sets of rotating cell groups, one focusing on the clockwise rotation direction and the other on the counterclockwise rotation direction, as shown in Fig. 6.4. These cells have the preferred direction and angular velocity direction. When the direction and angular velocity direction are consistent with the preferred state of cells, the cells transfer energy to the adjacent cells. Clockwise cells emit energy toward the head in the clockwise direction, and vice versa. The tuning curve produced by the cell coupling attraction model of rodent’s head towards cell is very close to the phenomenon observed in two regions of rodent’s brain. In the first mock exam animal model, two cell populations were used to represent the breasts in the rodent brain (PoS) and the thalamic nucleus (ATN) in breast (Fig. 6.5). If there is no angular rotation, cells in the same direction are preferred in POS and ATN to establish a close matching connection and allow stable synchronous Gaussian distribution in their respective cell systems. The offset from the cell node in the POS to the cell in the ATN is offset, and the weight is adjusted by the size and symbol of the angular velocity. During the adjustment period, the activity of ATN cell group was better than that of POS cell group. This leading behavior is supported by the fact that ATN cells of rodents are related to head orientation cells for 20–40 ms in the laboratory.

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Fig. 6.4 Sub attraction network model of rodent head orientation system. Note The outer ring formed by the head towards the cell is the active cell coding direction. The inner two rings are vestibular cells that respond to angular velocity. Hebbian learning rules were used to correct the connection between visual cells and head direction cell. The head direction cell also has close internal connections, which makes the simple local active cell group system stable

Fig. 6.5 A pair of attractive models of rodent head orientation. Note To the right leads to increased gain connection on the right, but there is no left gain connection from POS cells to ATN cells. Strong matching connections between two individual cells with the same preferred orientation are always active

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6.2.3 Grid Cell Edward Moser, a professor at Norwegian University of science and technology, and his wife May-Britt Moser discovered the role of grid cells through experiments. When the mouse passes through a wide and complex terrain, another nerve cell called the entorhinal cortex near the hippocampus of the mouse brain is activated. These cells respond to specific spatial patterns or environments, and they form grid cells as a whole. These cells form a coordinate system, just as people draw maps to divide coordinates in different directions and positions by longitude and latitude. They put the mice in boxes and let them run. They connected them to a computer to show their direction in graphs. As a result, they formed a clear hexagonal grid shape, like a honeycomb. But there are no hexagons in the box, which are abstractly formed in the mouse brain and superimposed on the environment. This means that the mouse can divide the space into honeycomb like hexagons through the mesh cells, and record the motion track on the honeycomb like mesh. Grid cells can determine the direction of their head alignment and the boundary position of the room. They coordinate with the location cells to form a complete neural circuit. This circuit system forms a complex and precise positioning system, that is, the positioning system in the brain. The discovery of positional cells and grid cells in the brain has won the Nobel Prize in physiology or medicine in 2014. Different from the position cells, the mesh cells stimulate multiple positions of equal sides of the vertices of the embedded hexagon mesh when the environment is drawn. These grids have three characteristics: the space between the excitation spaces, the direction of the grid relative to the reference axis, and the spatial phase of the network. Adjacent cells have similar spacing and orientation, but have different spatial phases. Therefore, only a small number of cells can fully cover the whole environment of the excitation area. Moreover, when the cells in the middle part of the olfactory cortex (MEC) record the moving position from the back to the side, the grid spacing increases monotonously. The characteristics of mesh cells in different layers of MEC are also changing. Layer II cells only respond to the fixed position in the environment. However, in layer III, V and VI, there are a considerable proportion of cells, called combined grid cells, which respond to both location and direction. Due to their combined characteristics, these cells are particularly suitable for the ratslam navigation model. When the initial experimental study found that the separation of the position cell and the head toward the position and direction in the cell would cause navigation problems, the combined cell was created as an engineering solution to this problem. When it is known that the location of cells becomes directional in a narrow environment, there is no obvious cell with inherent joint characteristics. Moreover, the repeated use of the linked cells rolled back through the posture cell matrix is not supported by the observation results from the single excitation region of the position cells. However, although many theories about the mechanism of neurons need to establish a network excitation area, the grid cells stimulate the observation results of multiple locations in the environment. It is objective to code multiple locations in the environment by reusing cells.

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Fig. 6.6 Velocity cell experimental data

6.2.4 Velocity Cell 2015, After winning the Nobel Prize in physiology or medicine, Edward Moser and May-Britt Moser continue to find that some nerve cells can increase the discharge rate proportionally with the increase of movement speed. By looking at the discharge frequency of such cells, we can judge the movement speed of an animal at a given point in time. The researchers named it velocity cell. In the experiment, the researchers placed the mice in a box with a top opening, and lured them to run around with random food. The experiment was conducted in a dark environment to avoid the effect son of visual information on the experiment. At the same time, in order to avoid the effect of the mouse’s own behavioral actions on the velocity cells, when analyzing the experimental data, the mouse cell activity changes were chosen to ignore all changes in the movement speed of less than 2 cm/s. First of all, the rate of specialization of velocity cells is normalized by linear transformation, and then the cell’s distribution rate is calculated and its activity is transmitted to the cell by unbiased analysis, the activity strength of the velocity cell is determined by the non-biased estimation in the experiment, the activity strength of the velocity cell is judged by the peak voltage size, and the activity of the velocity cell is materialized by a simple linear decoder consisting of the distribution field and linear filter, and then the activity state information is transmitted to the cell’s fusion cell by the head and the cell position. As shown in Fig. 6.6.

6.2.5 Brain Navigation System Physiological studies have shown that the activation of position cells in an animal’s active environment is determined by the path integral of velocity modulation. Positional cells express the result of path integration rather than the basis for integration. When the mice reach a familiar environment, the path integrator resets to the environment perceived by the external information. These studies show that mice are

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able to accurately navigate in different environments, blending internal and external information, and in familiar environments, using external perception information as an absolute reference for error correction, as are humans. Mimicking the navigation mechanism spree of this creature, this section presents a model of an intelligent INS system of a kind of brain. The location cell model uses a continuously attracted network model, where the positional cells at the boundary are connected to the position cells at another boundary, forming a ring. The two-dimensional continuous attraction network model is a random activity package formed by local excitability and global inhibition connected to a neural board. This attraction network is driven by the spatial cell path integration system, which is reset by image information from the current location. Using the two-dimensional Gaussian distribution to create an excitatory weight connection matrix χx,y for the positional cells, the χx,y formula is: χx,y = e−(x

2

+y 2 )/W

(6.1)

where W is the width constant of the location distribution. The amount of change in positional cell activity due to local excitability connections is: P(X, Y ) =

S X −1 S Y −1  i=0

P(i, j)χx,y

(6.2)

j=0

where, S X , SY is (X, Y ) the size of the two-dimensional matrix of the position cell in space, Represents the range of activity of the attractor model on the neural plate. i, j is the distribution coefficient of X, Y. The premise of location cell iteration and visual template matching is to find the relative position of the position cell attractor in the neural board, which is represented by the subscript of the weight matrix and can be calculated by the following formula: x = (X − i)(mod S X )

(6.3)

y = (Y − j)(mod SY )

(6.4)

Cells at each location also receive global suppression signals for the entire network. The symmetry of the excitability and inhibition connection matrices ensures proper neural network dynamics and ensures that the attractors in space do not engage without unlimited excitement. The amount of activity change in the positional cell caused by the inhibitory connectivity value is: P(X, Y ) =

SX  SY  i=0 j=0

P(i, j)Ψx,y − ξ

(6.5)

6.2 Bionic Navigation Mechanism

103

where, Ψx,y is the inhibitory connection weight, ξ is the suppression level that controls the global. The activity of all location cells is non-zero and normalized. The discharge rate P t+1 (X, Y ) of the cell in the post-path integration position cell can be expressed as: P

t+1

(X, Y ) =

+1 X +1 Y  

αi, j P t (i + X, j + Y )

(6.6)

i=X j=Y

In the Eq. (6.6), αi, j is the residual amount,X and Y is the amount of bias that is rounded down in the X–Y coordinate system, which is determined by the speed and direction information. ⎤ ⎡ ⎤ ⎡ → Ci eθ v cos θ ⎥ X ⎣ ⎦=⎢ (6.7) ⎦ ⎣ → C j eθ v sin θ Y where it represents rounding down, Ci for C j the path integral constant; v is provided → by the current velocity cell; θ for the current head orientation, by eθ the head towards the cell; and for θ the unit vector pointing toward the cell. Similar to the mechanism of biological autonomous navigation, inertial navigation system is an autonomous navigation system that does not rely on external information and does not radiate energy to the outside. The basic principle of INS is based on Newtonian law of mechanics, by measuring the acceleration of the carrier in the inertial reference system, it is integrated into time, and it is transformed into the navigation coordinate system, so that information such as speed, yaw angle and position can be obtained in the navigation coordinate system. However, there is an error in the device itself, and the position information of the INS is generated by the integration, so the error accumulates over time. For organisms, the relative position of the location cells’ discharge fields is also obtained by path integrals. But when in a familiar environment, the organism discharges all the space cells involved in the path integration. The process of coordinate update is actually the process of updating the spatial cell discharge, while the process of updating the closed-loop point is the process of resetting the space cell. Following this mechanism of living organisms, this section proposes a model scheme for intelligent INS. The system can detect in real time whether the current visual information matches the pre-stored visual template, and if successfully matched, it is considered to be a “familiar place”, and then the space cells of the whole path integration network will be reset to the discharge state of the previous closed-loop point. This method can effectively eliminate the accumulated error and improve the navigation accuracy. The algorithm flow is as follows:

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Fig. 6.7 System model schema

Algorithm 6.1: A bionic navigation algorithm for self-correction of visually positional cells 1. Camera captures one frame of RGB image 2. Collect information about the motion of an object and update the speed cells and head towards the cells 3. Perform path integrals for space cells 4. Spatial geometric coordinates obtained by geometric transformation 5. Perform an image matching algorithm and get a return value R for the matching result R 6. if R = TRUE 7. Read the coordinate information of the point at which the template image is located 8. Position cell discharge reset 9. Error correction 10. end if

As shown in Fig. 6.7, the brown dotted line is the result of the carrier’s position calculation. The true trajectory of the carrier is from A, passing in a straight line of B, C, and D. However, due to cumulative errors, the trajectory of in S calculations gradually deviates from the true trajectory. The red dots in the figure are visually corrected nodes where the positional cells are reset. It can be seen that the result track after visual correction is closer to the true trajectory of the carrier, and the navigation accuracy is also significantly improved.

6.3 High-Speed Effective Node Matching Algorithm As mentioned above, the key to realize the bionic navigation algorithm proposed in this chapter is that the position cells can discharge and reset at the right position. In order to realize this process, it is necessary for the carrier to have the function of

6.3 High-Speed Effective Node Matching Algorithm

105

“memory” and “recognition” for the scene. The process of “memory” can be simulated by storing the coordinates and scenes of key points (such as intersections, in front of characteristic buildings) in advance. “Recognition” means that the carrier can recognize the template image corresponding to the scene and return its true coordinate value when passing through the location “remembered” before, so as to reset the cell discharge and eliminate the accumulated error. The traditional image matching algorithms are generally divided into three categories: gray based matching algorithm, feature-based matching algorithm and relationship based matching algorithm. Image matching algorithms need to consider both efficiency and accuracy, most of them are proposed for specific problems. However, there is no special matching algorithm for the street view image recognition problem proposed in this section. In 2004 and 2012, Michael Milford proposed the location recognition algorithm ratslam and seqslam based on the continuous multi frame image information respectively, and used the method of comparing the gray information similarity of two images in a certain region to determine whether they are the same location. This method has the advantages of small calculation and good real-time performance, but it is easy to be affected by external environmental factors (such as weather, road conditions). If the threshold is set too small, it may result in missed judgment, but if the threshold is set too large, it is easy to result in wrong judgment, so the algorithm does not have good universality. In 2017, Bian Jiawang proposed grid-based motion statistics. Compared with the traditional SIFT feature matching method, this method can eliminate the wrong matching points more effectively. Its execution speed can reach 30 frames per second on the PC, which can basically meet the real-time requirements, but the speed of the lower computer needs to slow down a lot. In order to solve these problems, this section proposes a new image matching algorithm which is suitable for location cell correction.

6.3.1 Scan Line Strength Scanning line intensity method is used to recognize scene by comparing the similarity of scanning line intensity contour of two images. The specific method is to make the template image move in the reference image, calculate the difference between the normalized values of the two intensities and sum them. The more the difference, the more similar the two images are. For template image T and base image B, assume that T(x, y) and B T(B(x, y)) are the intensity values corresponding to the pixels on the image, respectively. In the case of base image B, the sum of the intensity values for each column is S. The strength of each column is normalized to obtain the baseline image normalizing vector I. The same is true of the normalized vector I of the template image T. The similarity of the two images can be characterized by coincident beta:

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β=

n  



Ii − Ii

(6.8)

i=1

When the device is applied to the vehicle, if the relative height and inclination of the camera and the ground remain the same, two different pictures taken at the same location can be considered to have only horizontal deviation and no vertical deviation. Therefore, we can make the ROI of the template image (area of interest) pan over the base image, calculate the similarity of the corresponding area of the two images after each move, and take the minimum of all similarity values as the final similarity of the two images.



β = min

        min abs Ii − Ii−o f f ect , abs I j − I j−o f f ect 1+o f f f ect≤i≤W 1+o f f ect≤ j≤W

min



(6.9)

where β is the final similarity of the two images, offect is the maximum amount of   translation, and W is the width of the ROI. Set a threshold t, if β meets, β ≤ t to consider the two images to match successfully. As shown in Fig. 6.8, the figure on the left and the middle are two images taken at different times at the same location, while the figure on the right is taken at another location. With the middle diagram, you can observe that the scan line outline of the template image is similar to the image on the left, and the image on the right is less similar. The advantage of this algorithm is that the computation is small and can meet the requirements of real-time processing. The disadvantage is low precision, changes in environmental factors (such as lighting, pedestrians, other vehicles) can lead to misjudgment and low reliability.

Fig. 6.8 Raw image, ROI area, image scan line strength

6.3 High-Speed Effective Node Matching Algorithm

107

6.3.2 GMS (Grid-Based Motion Statistics) GMS is a simple statistical method based on feature matching. It can quickly distinguish the correct match from the wrong match, thereby improving the stability of the match. Feature matching is a basic problem in computer vision. The current main problem for feature matching is that the matching algorithm has a slow matching speed and cannot meet the real-time requirements. Fast matching is often unstable, and it is easy to produce more wrong matching points. GMS can solve this contradiction well. Similar to traditional feature matching methods (SIFT, SURF), the GMS matching process is also divided into four steps: detection, description, matching, and geometric composition. First, search for images on all scale spaces, and use Gaussian differential functions to identify potential points of interest. These points are usually some edge points, corner points, and inflection points. Then use a set of vectors to describe the key points, that is, generate feature point descriptors. This descriptor contains not only the feature points, but also the pixels around the feature points that contribute to it. The descriptor should have a high degree of independence to ensure a matching rate. Then feature matching points are generated. For example, SIFT uses the Euclidean distance of the key point feature vector as the similarity determination metric of the key points in the two images. Among the two key points, if the closest distance divided by the next closest distance is less than a certain threshold, it is determined as a pair of matching points. Finally, the detected feature points are placed in a container, and the geometric relationship is matched to eliminate the feature points that do not meet the requirements. GMS differs from traditional feature matching methods in its faster calculation speed and higher matching accuracy. GMS solves the problem of how to use the constraint of neighborhood consistency very well, so that when the matching process is performed, the points of erroneous matching can be more effectively eliminated. The core idea of GMS is simple: the smoothness of motion leads to more matching points in the neighborhood of the matching feature points. We can judge the correctness of a matching result by counting the number of matching points in the neighborhood. The basic assumption that it is based on is that the smoothness of the motion leads to the feature points in the neighborhood near the correct matching point also corresponding to each other (Fig. 6.9). First, the probability distribution of correct and incorrect matches in the neighborhood near the correct matching point is derived from a mathematical point of view.   For a pair of matching points xi , xi , assuming that the number of other feature points that can support xi in the neighborhood of xi is Ni, it can be deduced that Ni follows a binomial distribution:  B(M, pr ), xi is the right matching point (6.10) Ni ∼ B(M, pw ), xi is the wr ong matching point

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6 Brain-Like Navigation Technology Based on Inertial/Vision System

Fig. 6.9 Grid-based motion statistics

where M is the number of feature points in the neighborhood of xi and other areas feature adjacent to it; pr is the probability of matching points corresponding   to  other  points in the neighborhood of xi in the neighborhood of xi when xi , xi is a correct feature points match; pw is the probability of matching points corresponding   to other  in the neighborhood of xi in the neighborhood of xi when xi , xi is a false match. The average and standard deviation are:

√ μr = M pr , sr = M pr (1 − pr ) xi is the wr ong matching point √ μw = M pw , sw = M pw (1 − pw ) xi is the wr ong matching point

(6.11)

The objective function is: max G =

μr − μw sr + sw

(6.12)

Turning the above theoretical analysis into an operational algorithm that can be used in practice, this process mainly has to solve four problems: 1. 2. 3. 4.

How to effectively calculate scores through grid Which neighborhood should be used How many grids should be used How to calculate the threshold S.

The solution is summarized as follows: Divide the image into a grid of G = 20 × 20. The score for each grid pair is calculated only once. Calculate 3 * 3 = 9 grids around a network, as shown in Fig. 6.10 (nine grids in the red area). For grid m and grid n, the similarity score can be calculated by the following formula: Smn =

9 

|Ωm i ni |

i=1

where |Ωm i ni | is the number of matching points between grids m i and n i

(6.13)

6.3 High-Speed Effective Node Matching Algorithm

109

Fig. 6.10 9 areas for score evaluation

The threshold Smn divides the grid pair into two parts, right and wrong:

C(m, n) =

√ 1, Smn > α ti 0, other s

(6.14)

where α = 6 is the empirical value and ti is the number of feature points. The value of C(m, n) indicates whether the grid area where m, n are located is a pair of correct matches. The advantage of GMS is that it has high matching accuracy, strong adaptability to the environment, and the misjudgment rate of the scene is almost 0; At the same time, compared with the traditional feature matching algorithm, the matching time is greatly reduced, basically meeting the requirements of real-time processing (Intel i7 CPU + GTX980 GPU configuration takes about 31 ms to process a group of pictures). The disadvantage is that when the algorithm is applied to the lower computer, the calculation speed is far less than the speed on the desktop computer, and the realtime performance still needs to be improved. And this method has the characteristic that it can be accurately matched on different scale transformations, which will have matching errors when applied to scene matching. Because the car is moving from far to near, the image is scaled. The GMS algorithm is a matching algorithm based on feature points, which results in more successful matching points for images captured within a distance before and after the template image location. As shown in Fig. 6.11.

6.3.3 Scan Line Intensity/GMS There are certain disadvantages when the above two algorithms are applied to the actual scene matching alone, so this section proposes a new matching scheme. Combining the two algorithms, on the one hand, give full play to the advantages of the scanning line intensity method in speed, on the other hand, use GMS to test the matching results of the scanning line intensity method, and give full play to its advantages of accurate matching. The implementation process of the algorithm is as follows.

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6 Brain-Like Navigation Technology Based on Inertial/Vision System

Fig. 6.11 For a specific template (left image), the images taken within a certain distance in front of it (right image) will generate a large number of matching success points, which will reduce the accuracy of positioning 

First, the similarity β of the reference image R and the template image T i is calculated by the scanning line intensity method. Where Ti ∈ T, T is the set of all template images, and T i is the template i. The threshold t can be adjusted slightly  larger to avoid missed judgments as much as possible. If β < t is satisfied, the two images are considered to be similar and the next step is entered, otherwise the next frame of the reference image is read. Next, read the latitude and longitude coordinate values calculated by the INS, and calculate the distance D between the location of the template image successfully matched and the location of the latitude and longitude coordinates returned by the INS:  2  2 Ti (x) − It (x) Ti (y) − It (y) ×πR + ×πR (6.15) D= 180 180 where {Ti (x), Ti (x)} represents the latitude and longitude coordinates where the template Ti is located,{It (y), It (y)} represents the coordinate value returned by INS at time t, R is the radius of the earth. When D ≤ σ , the data is considered valid and the next algorithm is executed. Where σ is the maximum reasonable error of the distance between two points. Finally, the matching results that pass the distance test are checked again using the GMS algorithm. If the number of successfully matched points of the two images is greater than the threshold Ω, it is considered that this is a correct set of matched points, otherwise it is discarded. After the above three steps, it basically meets the real-time matching requirements of the system and can effectively eliminate some misjudgment matching points.

6.4 Algorithm Verification

111

6.4 Algorithm Verification In order to verify the actual feasibility of the above algorithm and model, we conducted a large number of related experiments, and the experimental results fully demonstrated the reliability of our model and the superiority of the algorithm. The experimental platform used in this section of the experiment is an electric experimental vehicle equipped with a bionic navigation device. The bionic navigation device is composed of LattePanda card computer, INS system, camera, high-precision GPS equipment, etc. As shown in Fig. 6.12. LattePanda is a card computer, with Windows 10 as the operating system and equipped with a wealth of input and output interfaces, mainly in the experiment to perform the task of image matching algorithm. INS mainly collects speed information and direction information, and at the same time generates path integrated position cells. INS communicates with LattePanda through the serial port protocol, so that it can receive accurate position information fed back by LattePandain real time and correct its own position cells. The entire device is powered by USB and placed at the front of the test car. The camera is connected to LattePanda, and RGB images are collected at a rate of 12 frames per second while the car is traveling, and are matched with the template images in real time. High-precision GPS is used to provide reference coordinate information during template image acquisition. The test site was selected on a road about 500 m long in the North University of China (as shown in the red track in the figure below). The location of the location cell node used for visual matching is selected at three intersections with more

Fig. 6.12 Experimental equipment

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6 Brain-Like Navigation Technology Based on Inertial/Vision System

Fig. 6.13 Driving route and location cell node location

obvious characteristics. A total of three cell corrections were performed throughout the process (green dots in the figure) (Fig. 6.13). The intensity of the scan lines at the three nodes is shown in Fig. 6.14. The image matching result is shown in Fig. 6.15. The experimental results are shown in Fig. 6.16. As shown in Fig. 6.16, the black curve is the actual trajectory of the carrier, the blue curve is the data measured by pure INS, the purple curve is the result trajectory after using the SeqSLAM algorithm, and the green curve is the brain-like navigation algorithm proposed by this design measurement trajectory. It can be seen from the figure that the navigation error of INS will gradually accumulate with the increase of its working time. However, after position cell identification and position correction, this error can be effectively eliminated, thereby improving navigation accuracy. According to high-precision GPS to provide reference information, the actual coordinates of the end point are 38.017588 °N, 112.44490 °E, while the coordinates calculated by pure INS are 38.017921 °N, 112.44493 °E, and the distance error is about 37.2 m. At the same time, the end-point coordinates measured using the brainlike navigation scheme designed are 38.017513 °N, 1122.44489 °E, and the distance error is only 8.4 m. The results show that the navigation accuracy of INS is improved by about 28.8 m every 500 m.

6.4 Algorithm Verification

Fig. 6.14 Scanning line intensity profile of the image captured at the template position

113

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6 Brain-Like Navigation Technology Based on Inertial/Vision System

Fig. 6.15 Matching results using GMS algorithm

Fig. 6.16 Experimental results

Chapter 7

Concluding Remarks

7.1 Summary of Inertial-Based Navigation Intelligent Information Processing Technology Inertial-based integrated navigation system is one of the very active research branches in the field of navigation. It has great theoretical research and application value in the civilian vehicle navigation and positioning market. This book takes inertialbased navigation intelligent information processing technology as the research object, mainly conducts gyro noise processing, temperature error mechanism analysis and modeling compensation, inertial-based integrated navigation system under discontinuous observation conditions, and inertial-based brain navigation research. Comprehensive research content, this book has made innovations in the following aspects: (1) The static noise characteristics of the fiber optic gyroscope are analyzed, the influence of noise on the fiber optic gyroscope inertial navigation system is introduced, the Allan variance is introduced to analyze the noise of the fiber optic gyroscope, and the LWT-FLP algorithm based on the lifting wavelet transform and the FLP algorithm is proposed. The verification results show that Compared with lifting wavelet transform and FLP de-noising, the LWT-FLP algorithm proposed in this paper can more effectively remove noise in the output of fiber optic gyro. At present, the wavelet analysis method and the FLP algorithm have been widely used to process the output signal of the fiber optic gyroscope. The advantages of the two are combined. The lifting wavelet transform is first used to layer the output signal of the fiber optic gyroscope, and then the signals of each layer are processed by FLP, and finally the lifting wavelet reconstruction is performed to obtain the final de-noising result, which is innovative. (2) The influence of the angular vibration environment on the output signal of the fiber optic gyro is studied, and a G-FLP algorithm based on gray theory and FLP algorithm is proposed. Compared with the traditional FLP algorithm, the G-FLP © Publishing House of Electronics Industry 2021 C. Shen, Intelligent Information Processing for Inertial-Based Navigation Systems, Navigation: Science and Technology 8, https://doi.org/10.1007/978-981-33-4516-4_7

115

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7 Concluding Remarks

algorithm can more effectively remove the fiber optic gyro angle Vibration noise. The FLP algorithm is a widely used fiber optic gyro de-noising method. The first international paper on the use of the FLP algorithm to process fiber optic gyro signals was published in 2001. The main idea of the FLP algorithm is to multiply the previous signal by the corresponding weight to predict the current At the time of the signal, a lot of research has proved that this method can effectively remove the fiber optic gyro noise; gray theory by graying the data sequence, can change the irregular data sequence into a regular sequence, better extraction and mining Value information. Compared with the output signal of static fiber optic gyro, the output of fiber optic gyro in angular vibration environment has certain regularity. By graying out the output of fiber optic gyro angular vibration, the regularity of the data sequence itself can be more obvious, so that the FLP algorithm can the current signal is better predicted, thus obtaining a better de-noising result than the traditional FLP algorithm. (3) The influence of temperature on the fiber optic gyroscope is studied. Temperature and its rate of change are one of the main environmental factors that cause the error of the fiber optic gyroscope. Errors can be compensated by establishing a mathematical model. First, the effect of temperature on the scale factor of the fiber optic gyroscope is studied, and the hyperbolic model of the scale factor of the fiber optic gyroscope based on temperature and input angular rate is established; then the data collection test of the fiber optic gyroscope under the condition of severe temperature changes is conducted, and Temperature change rate fiber optic gyro temperature error model. The simulation results show that the model can effectively compensate for the drift error caused by rapid temperature changes. Finally, an Elman neural network model based on genetic algorithm is proposed. The input of the model is the previous moment. The temperature drift of the fiber optic gyro and the temperature change rate at the current moment. The output is the temperature drift of the fiber optic gyro at the current moment. The simulation results verify that the model can accurately compensate for the temperature drift. (4) An on-board MEMS-INS/GPS navigation and positioning experiment was carried out. The output of MEMS-INS when GPS lost lock was collected. An STKF/WNN integrated navigation method based on wavelet neural network assisted strong tracking Kalman filter was designed. The navigation method can train the WNN when GPS signals are available to establish a MEMS-INS positioning error model. When the GPS signal is out of lock, the established model can be used to compensate the MEMS-INS error to improve the combined navigation. applicability. The collected data is used to verify the algorithm. The verification results show that under the condition of the satellite signal losing lock, the accumulation of MEMS-INS errors can be effectively suppressed by the assistance of WNN. In addition, a MEMS-INS/GPS integrated navigation algorithm based on selflearning volume Kalman filtering under discontinuous observation conditions is proposed. The operation phase of the MEMS-INS/GPS integrated navigation system is divided into a training phase and an error compensation phase. The

7.1 Summary of Inertial-Based Navigation Intelligent Information …

117

training phase is In the effective stage of GPS signal, the self-learning Kalman filter composed of two cyclic filtering subsystems is used to make the optimal estimation of the speed error and position error of INS with the difference between the speed and position of INS and GPS as the observation. Realize the self-learning function; the error compensation stage is the GPS signal unlocking stage. At this time, the Kalman filter has the function of predicting the observation through self-learning, and can fully trust the prediction results of the LSTM network to realize the GPS signal unlocking. Seamless navigation in the case of discontinuous observation and compensation of the Kalman filter’s optimal estimation error value to improve the accuracy of intelligent vehicle navigation and positioning in complex environments. (5) Animals have a certain cognitive ability to the environment, so they can correct their own navigation errors. Based on the excellent navigation behavior of animals, this book proposes a kind of brain navigation method to improve the precision and intelligence of MEMS inertial navigation system (MEMS-INS). By analyzing the error correction mechanism of mouse brain navigation, this method uses visual acquisition of external perception information to correct the cumulative error of INS position. In addition, in order to improve the position matching speed and accuracy of the visual scene recognition system, a position recognition algorithm (SI-GMS) combining image scan line intensity (SI) and grid-based motion statistics (GMS) is proposed. The proposed SIGMS algorithm can effectively reduce the influence of uncertain environmental factors (such as pedestrians and vehicles) on the recognition results. It solves the problem that the matching accuracy is not high when the scan line strength (SI) algorithm is used alone, or the matching speed is slow when the grid-based motion statistics (GMS) algorithm is used alone. Finally, an outdoor vehicle experiment was conducted. By comparing with high-precision reference information, the effectiveness of the inertial-based brain navigation system proposed in this book in carrier navigation and positioning is verified.

7.2 Research Prospect This book conducts in-depth research on several key technologies of inertial-based navigation intelligent information processing, and puts forward some new research ideas and solutions. Although the methods and techniques proposed in this book can improve the performance of the existing system to a certain extent, there are still some problems that need further research. On the basis of the research work in this book, the following suggestions are proposed for future research work: (1) The non-continuous observation conditions proposed in this book, that is, the INS error compensation study when the GPS signal is out of lock, are suitable for applications in indoor navigation and urban environments. However, in actual environments, multi-path errors are also one of the main sources of error, so there are It is necessary to conduct in-depth research on the real-time monitoring

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and compensation technology of multi-path errors, so as to further improve the accuracy and reliability of the navigation system; (2) This book only studies the integrated navigation system based on INS/GPS. With the development of GLONASS, Galileo and Beidou, it is necessary to further study the combined navigation method of multiple GNSS receivers (BD/GPS/GLONASS/Galileo) and INS. The accuracy and reliability will be further improved. (3) This book studies the inertial base brain navigation system based on the animal brain navigation cell model. In the later work, it is necessary to further explore the animal brain cell navigation mechanism, condense the brain navigation model, and test and verify in a complex environment to improve the environmental adaptability of inertial base brain navigation.

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