Computer Aided Assessment of Open Kinematics Chains

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COMPUTER AIDED ASSESSMENT OF OPEN KINEMATICS CHAINS Kaloyan Yankov, Veselina Uzunova* Medical Faculty, Thracian University, Stara Zagora 6000, Bulgaria e-mail: [email protected] *e-mail: [email protected] Abstract. Huge diversity of natural and man-made open kinematics chains (OKC) exists. The question of creating a formal system for their evaluation is important both in robotics and in medical sciences. The present work is aimed to formulate some parameters for evaluation of OKC and to show ways of their computer assessment. A formal system of criteria for measuring the motional abilities of human limbs could be helpful for physicians in evaluating normal and pathological limb movements. Key words: Biomechanics, Kinematics, Computer Aided Design, Robotics.

INTRODUCTION Open kinematics chain (OKC) is an abstraction of bodies mutually related for the purpose of motion transmission. Every vertebrate animal has organs of movement that are OKC. In technic OKC are used in automatons with ability to perform various technological operations. Huge diversity of natural and man-made OKC exists. The need of a formal system for their evaluation arises. For artificial systems the evaluation assumes a critical place at the stage of their design. Also interesting is the problem of evaluation of OKC in biological systems. They have a natural dispersion of parameters caused by individual and species variability. Aging, an irrevocable natural process causes changes in the parameters of these systems. Changes can also result from diseases: states of living organisms leading to diminished functional capacity of the locomotory and other systems. Human movement is a result of the integrity of three components – muscular, neural and skeletal. They are regarded here in their functional unity. The locomotory system (LS), as an OKC, ensures performance of work as well as movement in space of the whole body and/or its parts. All three components suffer loss or diminished function in different diseases. Pathological processes affecting human LS are numerous. The damage can be primary: when the LS is the first affected (Dushen muscular dystrophy, osteoporosis, neuritis) and secondary: when it results from a disease of another organ and system (brain stroke). Major branches of medical sciences concerning diseases of the LS are orthopaedics, physiotherapy and prosthetics. Assessment of the functional capabilities of limbs is needed in these areas. Physiotherapy uses different techniques to recover diminished functional capacity of the LS. Prosthetics deals with cases where the function is lost completely and cannot be recovered. Limb assessment in physiotherapy is important to evaluate the extent of damage and the success of therapy, to insert corrections if necessary and to make prognosis about the recovery progress. It can be used throughout the stages of prosthetic management, namely pre-amputation, prosthetic design, prosthetic fitting, and subsequent clinical follow-ups [12]. 414 а а ия а а и “ а а За а 2004” .1 и и а и. и а ии ия. а а и аии а и а

The present work is aimed to formulate some parameters for evaluation of OKC of vertebrate animals and to show ways of their computer assessment. A formal system of criteria for measuring the motional abilities of human limbs can be helpful for physicians in creation of clear and punctual criteria for evaluating normal and pathological limb motion. DEFFINITION AND DESCRIPTION OF OKC A link is defined as a rigid body having two or more pairing elements. With regard to human limbs, links are the different moving parts such as hand, finger, shoulder etc. A kinematics pair (KP) is a joint between the surfaces of two links that keeps them in contact and relatively movable. KPs are revolute or prismatic. Living organisms only contain revolute joints. KP in the human body are the shoulder joint, hip joint etc. An open kinematics chain is a system of links, which are connected pairwise to transmit force or motion (Fig.1). The last link, that performs the goal movement, is called end effector (EE). The independent possible movements in an OKC are called Figg.1 Open kinematics chain “generalized coordinates” and they are denoted by the vector q: q = [q1,q2,….,qN]T (1) The number of generalized coordinates defines the degrees of freedom (DOF) of OKC. KPs in biological systems have 2 or 3 rotations and that is why the DOF of an OKC exceed 6. If the dimensionality of the q-vector is N>6, then the kinematic chain is redundant. This increases the mobility and the opportunity to compensate the restrictions in some joints at the expense of redundancy. Redundant kinematics chains offer several potential advantages over non-redundant ones. In a space with obstacles the extra DOFs can be used to move around or between obstacles and, thereby, to manipulate in situations otherwise inaccessible. The relationship between the joint vector q and the Cartesian end-effector vector M may be expressed as follows: M = F(q) (2) where F is a continuous non-linear vector function. Eq.2 is a direct (forward) kinematics problem. Forward kinematics gets position and orientation of the last segment in a kinematics chain by defining angles for every joint. On the other hand, inverse kinematics computes joint angles of a kinematics chain using the position and orientation of last kinematics segment to a desired target point B in space: q = F-1(B) (3) The solution of (3) is not simple. One-valued function exists for (2) but for (3) there are more solutions q for one B and infinite number of solutions for redundant OKC. CRITERIA FOR EVALUATION OF OKC 1. Joint constraints. They are the physical limits, constraining the operating range а .1

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of a joint of a К . The generalized coordinates q as a function of the joint constraints vectors qmin and qmax define the configuration space: (4) Q = {q: qimin ≤ qi ≤ qimax, i=1,2,...N } In this way Q is a N-dimensional hyper-cube. Join constraints can be used to evaluate systems with identical kinematics schemes. For example the joint constraints for movement of the human arm in the shoulder joint are [1,3]: • Flexion/extension in the sagittal plane:[-20..90] degrees; • Elevation in the scapular/frontal plane: [0..100] degrees; • Rotation about the long axis of the humerus:[-30..60] degrees; Individual differences in the angles exist. The standard deviation of their values, signifying the individual variability had not been found in the cited reference. Also, variability due to age, sex and disease was not defined. It is known that the maximal angles are diminished in most pathological states. Diseases exist where the angles can be increased. An example of such condition is the Marphan syndrome [2]. The maximal angles are greater than normal; the functional capabilities of the LS are not improved. 2. Workspace analysis. Workspace is the set of points in the 3D space, which can be reached by the EE. The image (5) D = { M: M = F(q), q ∈ Q } is a m-dimensional workspace of the OKC. If m=2, the workspace is planar. In 3D Euclidean space m=3. 2.1. Visualization of the workspace. This approach is easy to implement and is frequently used to evaluate OKC. Workspace is a very important characteristic in robotics, used in robot work cell design [5, 9]. The range of generalized coordinates is presented with N-dimensional grid with small step ∆qi: ∆Q = [∆q1, ∆q2, …, ∆qN]T The Cartesian product of the joint coordinate values is generated and the forward problem is solved for the set of configurations. The recursive procedure is used in order to make the algorithm independent of the number of DOF (Fig.2). Procedure workspace (Joint, delta); begin repeat Joint.JointValue:= Joint.JointValue+delta; Joint.DirectProblem; Kin_chain.Visualization; If (Joint.next nil) then workspace (Joint.next); until (Joint.JointValue>= Joint.JointMax) Joint.JointValue:= Joint.JointMin; end; Fig.2. Workspace visualization algorithm 2.2. Volume of the workspace

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Workspace in most cases is bounded by complex surfaces and its estimation using trivial mathematics formulas is impossible. For this reason alogorithms based on the method “Monte Carlo”are used [6]. The method is particularly useful for complex problems that cannot be modeled by mathematical functions using deterministic methods. The statistical sampling process is based on the selection of random numbers. The investigated space D is inscribed in a hypercube H with known dimensions and volume Ω. A number P of points is generated and p∈H with uniform distribution in H. Every point has a characteristic 1, p∈D (6) χ(p) = 0, p ∉D The condition p∈D is checked by solving the inverse problem. The volume VD of the workspace is estimated using the formula

VD = Ω ∑ χ ( pi ) P

P

i =1

(7)

It is obvious that the precision of estimation depends on the number P of generated points. 3. Normalized volume criteria They are estimated by inscribing the workspace D in a space figure with known volume Ω and calculating the ratio: (8) κ = VD/Ω This coefficient shows the relative abilities of the OKC to operate in the workspace. The type of the circumscribed figure is defined according to the structure of the OKC. 3.1 Normalized spherical coefficient For anthropomorphic structures the circumscribed figure is a sphere with radius r: (9) κsph = VD/Ωsph Ωsph = 4/3πr3 For the human hand r is the distance between the shoulder and the palm. 3.2 Normalized cylindrical coefficient The cylindrical configuration robots have one rotary joint at the base and linear joints to connect the links. The space in which this robot operates is cylindrical in shape. (10) κcyl = VD/Ωcyl The axis of the cylinder circumscribed around the workspace is collinear with the translation kinematics pair. 3.3 Normalized orthogonal coefficient It is used for robots with Cartesian configurations. A Cartesian robot consists of links connected by linear joints and has three perpendicular slides. The circumscribed solid is a block. Its axes are collinear with the axes of the kinematics pairs. (11) κorth = VD/Ω orth 4. Service coefficient This is a criterion for evaluation the flexibility and the effectiveness of service inside the workspace [4]. It expresses the ability of ЕЕ to position in point В∈D with arbitrary а .1

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orientations. If ЕЕ is fixed in point В, then a closed kinematics pair is obtained (Fig.3). If КC has redundant DOF, it is possible to move it and a chosen characteristic axis of ЕЕ describes a space angle ψB: (angle of service in point В) : ψB = S/L2 S - surface of the spherical sector L – radius equal to the length of ЕЕ The coefficient of service in point В is defined as θB = ψ B/4π (12) The integral coefficient of service is defined as the middle value in all the workspace:

Θ=

1 Θ B dv VD ∫

(13)

It expresses the probability EE of the OKC to be oriented in a casual direction and arbitrary point in the workspace. When numerically integrating in Eq.13 the number of points В∈D is important, and the main requirement for them is to be uniformly distributed in D. 5. Approach coefficient The coefficient characterizes the possibilities of the OKC to position the EE with different configurations in points of the workspace: µK is the volume of the joint coordinate used to ΑB = µK/VQ ∈[0,1] position in point В VQ – the volume of phase parallelepiped in Q. The differense between servise coefficient and approach coefficient is that the first is calculated in Euclidian space and the second – in configuration space Q. The computational algorithm is common. A sphere with radius L and center point В is built. A grid including P uniformly distributed points upon the spherical surface is generated. The inverse problem for regional joints is solved for every point of the grid. If the point can be reached, then the inverse problem is solved with regard to point В and the orientation of EE is obtained. All the points for which point В is reachable have a characteristic χ(p) (Eq.6). The surface S of the spherical sector is:

4πL2 P S = ∑ χ ( pi ) P i=1

(14)

The correspondisng goal configurations are saved and they determine joint boundaries to calculate µK. SOFTWARE REALISATION The program CINDY for kinematics analysis of OKC is used and extended for this purpose [8]. The user interface was developed for easy use not only by engineers but also by physicians. It is possible to describe kinematics chain using application oriented language [10] or graphic user interface [11]. The inverse kinematics is solved applying coordinate descent method without restrictions in number of joint coordinates [7]. Space visualization includes representation of the workspace as well as the service and approach coefficients. The workspace is visualized as a cloud of points. The two coefficients have different values (from 0 to1) for different parts of the workspace. For the purposes 418 а а ия а а и “ а а За а 2004” .1 и и а и. и а ии ия. а а и аии а и а

of better visual comprehension, the points can be given different color according to the value of the coefficient. FUTURE DEVELOPMENT Investigation of patients will be performed. Groups will be formed according to age, sex and health status. The mentioned characteristics will be measured, subjected to statistical analysis and evaluation parameters will be estimated for different groups. This will give an opportunity to create criteria for normal and pathologic limb behaviour. REFERENCES 1. Basic Biomechanics of the Musculoskeletal System, Ed.by M. Nordin et.al, Second edition, Lea&Febiger, Philadelphia, London, 1989. 2. Gelehrter, T.D., F.S.Collins, D.Ginsburg. Medical Genetics, Williams&Wilkins, Second edition, A.Wawerly Company, USA. 3. Hamill, J., Et.Al. Biomechanical Basis Of Human Movement, Williams&Wilkins, Pa, Usa, 1995. 4. Kobrinski, A.A., A.E. Kobrinski. Manipulation Systems of Robots. Moskwa, Nauka, 1985. 5. Ming-shu Hsu D.Kohli. Boundary surfaces and accessibility regions for regional structures of manipulators. Mech. Mach. Theory, 22(3):277–289, 1987. 6. Vanderbilt, D., S G.Louie, A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables. J. Comput. Phys. 56, 259-271, 1984. 7. Yankov, K., Manipulator Motion Planning. Proc. Int. Workshop on Sensorial Integration for Industrial Robots, nov.22-24, 1989, Zaragoza, Spain, pp.114-118. 8. Yankov, K., Computer Simulation of Industrial Robots. Proc. Second International Conference ACMBUL'92 "Computer Applications", okt.4-8, 1992, st.Konstantine resort, Varna, Bulgaria, pp.33.1-33.8 9. Yankov, K., Interactive Design of Robot Work Cell, Proc. 14-th Int. Conf. "Systems for Automation of Engineering and Research" SAER'2000, St.Konstantin resort, Varna, 18-20 sept. 2000, Bulgaria, pp. 28-32. 10. Yankov, К., Application-Oriented Language for Describing Open Kinematic Chains. Proc. 16-th Int.Conf. "Systems for Automation of Engineering and Research" SAER'2002, St.Konstantin resort, Varna, Bulgaria, 20-22 sept. 2002, pp.95-99. 11. Yankov, K., Graphical User Interface for Describing Open Kinematics Chains. Proc. 17-th Int. Conf. "Systems for Automation of Engineering and Research” SAER2003, St.Konstantin resort, Varna, Bulgaria, 19-21 sept, 2003, pp.126-130. 12. Zheng, YP. Arthur FT Mak, Aaron KL Leung. State-of-the-art methods for geometric and biomechanical assessments of residual limbs: a review. Journal of Rehabilitation Research and Development Vol. 38 No. 5, September/October 2001.

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