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- https://ufdc.ufl.edu/AA00004925/00001
## Material Information- Title:
- Dynamics and optimization of a human motion problem
- Creator:
- Ghosh, Tushar Kanti, 1945-
- Publication Date:
- 1974
- Language:
- English
- Physical Description:
- x, 152 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Adjoints ( jstor )
Aircraft maneuvers ( jstor ) Boundary conditions ( jstor ) Cost functions ( jstor ) Error rates ( jstor ) Inertia ( jstor ) Kinetics ( jstor ) Mathematical models ( jstor ) Mathematical variables ( jstor ) Trajectories ( jstor ) Dissertations, Academic -- Engineering Mechanics -- UF Engineering Mechanics thesis Ph. D Human mechanics ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis -- University of Florida.
- Bibliography:
- Bibliography: leaves 149-151.
- General Note:
- Typescript.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 022776042 ( ALEPH )
14079183 ( OCLC ) ADA8842 ( NOTIS )
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DYNAMICS AND OPTIMIZATION OF A HUMAN MOTION PROBLEM By TUSHAR KANTI GHOSH A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE :.UNIk.RSITY OF FLORIDA IN PARTIAL FULrILL~.I..'-I OF THE REQUIREMENT'S FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974 DYNAMICS AND OPTIMIZATION OF A HUMAN MOTION PROBLEM By TUSHAR KANTI GHOSH A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974 TO MY FATHER AND MY MOTHER ACKNOWLEDGMENTS It has been a very happy experience to work with Dr. W. H. Boykin, Jr., during my stay at the University of Florida. I am grateful to him for his help and guidance throughout this work, from suggesting the topic of the dissertation to proofreading the manuscript. I wish to express my deepest gratitude to him, both as an efficient and enthusiastic research counselor and as a human being. I wish to thank Dr. T. E. Bullock for the many discussions I had with him about the theory and numerical methods of optimization. These discussions provided me with understanding of many of the concepts that were used in this work. I chose Drs. L. E. Malvern, U. H. Kurzweg, and 0. A. Slotterbeck to be on my supervisory committee as a way of paying tribute to them as excellent teachers. It is a pleasure to thank them. Special gratitude is expressed to Professor Malvern for going through the dissertation thoroughly and making corrections. I am thankful to Drs. T. M. Khalil and R. C. Anderson for examining this dissertation when Dr. 0. A. Slotterbeck left the University. I am thankful to Mr. Tom Boone for his interest in this work and for volunteering his services as the test subject of the experiments. Thanks are also due to the National Science Foundation which provided financial support for most of this work. Finally, it is a pleasure to thank my friend Roy K. Samras for his help in the experiments and Mrs., Edna Larrick for typing the dissertation. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF FIGURES vi NOTATION vi i i ABSTRACT ix CHAPTER 1 INTRODUCTION 1 1.0. Why and What 1 1.1. Dynamics and Optimization in Human Motion 1 1.2. Previous Work 3 1.3. The Problem Statement 7 1.4. The Kip-Up Maneuver 8 1.5. Present Work . 8 CHAPTER 2 EXPERIMENTATION AND CONSTRUCTION OF THE MATHEMATICAL MODEL 10 2.0. Introduction 10 2.1. Mathematical Model of the Kip-Up 10 2.2. The Equations of Motion 13 2.3. The Equations of Motion for the Experiment and the Integration Scheme 21 2.4. Experimental Procedure 23- 2.5. Results and Discussion 26 2.6. Sources of Errors 31 2.6.1. Imperfections in the Model 31 2.6.2. Errors in Filming and Processing the Data 31 2.6.3. The Integration Scheme 32 CHAPTER 3 ANALYTIC DETERMINATION OF HIE MINIMUM-TIME KIP-UP STRATEGY 34 3.0. Introduction 34 3.1. Mathematical Formulation of the Kip-Up Problem . 35 3.2. Bounds on the Controls 36 3.3. Torsional Springs in the Shoulder and Hip Joints 38 3.4. Boundary Conditions 39 iv TABLE OF CONTENTS (Continued) Page CHAPTER 3 (Continued) 3.5. The Necessary Conditions of Time Optimal Control 40 3.6. The Solution Methods 47 3.7. A Quasilinearization Scheme for Solving the Minimum-Time Problem 51 3.7.1. Derivation of the Modified Quasilinearization Algorithm 52 3.7.2. Approximation of the Optimal Control for the Kip-Up Problem 61 3.7.3. A Simple Example Problem for the Method of Quasilinearization 69 3.7.4. The Results With the Kip-Up Problem ... 72 3.8. Steepest Descent Methods for Solving the Minimum-Time Kip-Up Problem 79 3.8.1. Derivations for Formulation 1 81 3.8.2. Derivations for Formulation 2 93 3.8.3. Derivations for Formulation 3 106 3.8.4. The Integration Scheme for the Steepest Descent Methods 108 3.8.5. Initial Guess of the Control Function . 109 3.9. Results of the Numerical Computations and Comments 110 3.10.Comparison of the Minimum-Time Solution With Experiment 127 CHAPTER 4 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK . 130 4.1. Conclusions 130 4.2. Recommendations for Future Work 134 APPENDIX A DETERMINATION OF THE INERTIA PARAMETERS OF THE KIP-UP MODEL FROM THE-HANAVAN MODEL 138 APPENDIX B AN INVESTIGATION OF A STEEPEST DESCENT SCHEME FOR FINDING OPTIMAL BANG-BANG CONTROL SOLUTION FOR THE KIP-UP PROBLEM 143 LIST OF REFERENCES 149 BIOGRAPHICAL SKETCH 152 v LIST OF FIGURES Figure Page 1. Hanavan's Mathematical Model of a Human Being 12 2. Mathematical Model for Kip-Up 14 3. The Three-Link System 15 4. Sketch of Kip-Up Apparatus Configuration 24 5. Measured and Smoothed Film Data of Angles 0 and i|r for the Kip-Up Motion 27 6. Measured and Computed Values of cp for Swinging Motion ... 29 7. Measured and Computed Values of cp for Kip-Up Motion .... 30 8. Unmodified Control Limit Functions 37 9. Approximation of Bang-Bang Control by Saturation Control 64 10. Graphs of Optimal and Nearly Optimal Solutions Obtained via Quasilinearization for Simple Example 71 11. Modification of Corner Point Between Constrained and Unconstrained Arcs After Changes in the Unconstrained Arcs 90 12. Solution of Example Problem by the Method of Steepest Descent 105 13. Initial Guess for the Control Functions 118 14. A Non-Optimal Control Which Acquires Boundary Conditions 119 15. Approximate Minimum Time Solution by Formulation 2 of the Method of Steepest Descent 122 16. Approximate Minimum Time Solution by Formulation 3 of the Method of Steepest Descent 124 vi LIST OF FIGURES (Continued) Figure Page 17. Angle Histories for Solution of Figure 16 125 18. Difference Between Measured Angle and Mathematical Angle for Human Model Due to Deformation of Torso .... 129 19. Construction of Kip-Up Model from Hanavan Model 139 vii NOTATION Usage Meaning X dx ; total derivative of the quantity x with respect to time t. X x is a column vector. T .T x (x) Transpose of x; defined only when x is a vector or a matrix. x, y 3x ; partial derivative of x with respect to y. oy 3x x, ^ - y oy Partial derivative of the column vector x with respect to the scalar y. The result is a column vector whose i^ component is the partial derivative of the i^a component of x with respect to y- 9x xv Â¥ Partial derivative of the scalar x with respect to the column vector y; also called the gradient of x with respect to y. It is a row vector, whose i^h element is the partial deriva- 9x tive of x with respect to the i^h element of y. -V 3y Partial derivative of a column vector x by a column vector y. The result is a matrix whose (i,j)th element is the partial derivative of the ith element of x with respect to the element of y. ii ill T n 2 Norm of the column vector x. Defined as either x x or E K.x. i=l 1 1 (to be specified which one) where x is a vector of dimension n are given positive numbers, x^ is the i^h element of x. viii Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMICS AND OPTIMIZATION OF A HUMAN MOTION PROBLEM By Tushar Kanti Ghosh March, 1974 Chairman: Dr. William H. Boykin, Jr. Major Department: Engineering Science, Mechanics, and Aerospace Engineering Questions about applicability of analytical mechanics and usefulness of optimal control theory in determining optimal human motions arise quite naturally, and especially, in the context of man's increased activities in outer space and under water. So far very little work has been done to answer these questions. In this disserta tion investigations to answer these questions are presented. A particular gymnastic maneuver, namely, the kip-up maneuver is examined experimentally and theoretically. A mathematical model for a human performer is constructed for this maneuver from the best per sonalized inertia and joint centers model of a human being available today. Experiments with the human performer and photographic data col lection methods are discussed. Comparisons of the observed motion with solutions of the mathematical model equations are presented. Discrepan cies between the actual motion and the computed motion are explained in terms of principles of mechanics and errors in measurements. Some changes in the model are suggested. IX An approximate analytic solution of the kip-up maneuver performed in minimum time is obtained for the model via optimal control theory. Several numerical methods are used to determine the solution, which is compared with the observed performance of the human subject. Difficul ties in solving human motion problems by existing numerical, algorithms are discussed in terms of fundamental sources of these difficulties. Finally, recommendations for immediate'future work have been made. x CHAPTER 1 INTRODUCTION 1.0. Why and What Man's increased interest in the exploration of space and the oceans was an impetus for a better understanding of the mechanics of large motion maneuvers performed by human beings. Experience in space walking and certain athletic events brought out the fact that human intuition does not always give correct answers to questions on human motion. For certain problems the solutions must be found by analytical methods such as methods of analytical mechanics and optimal control. Broadly speaking, this work deals with the application of the principles of mechanics and optimal control theory in the analytical determination of human motion descriptors. 1.1. Dynamics and Optimization in Human Motion Dynamics provides the basic foundation of the analytic problem while optimal control theory completes the formulation of the mathemat ical problem and provides means to solve the problem. In any endeavor of analytic determination of a human motion the first steps are construc tion of a workable model having the same dynamics of the motion as that of the human performer, and obtaining the equations of motion for the human model. However, the principles of mechanics alone do not give enough information for analytical determination of a desired maneuver whenever there are sufficient degrees of freedom of movement. W'ithout 1 2 knowing what goes on in the human motor system, optimal control is presently the only known analytical method which can provide the remain ing necessary information. After a workable dynamic model of the human performer has been obtained, the remaining part of the analytic determination of a physical maneuver is a problem of control of a dynamic system where the position vectors and/or orientation of the various elements of the model and their rate of change with respect to time (representing the "states" of the "dynamic system" being considered) is to be determined. The state components change from one set of (initial) values to another set of (final) values at a later time. The "control variables," that is, the independent variables whose suitable choice will bring the change are torques of the voluntary muscle forces at the joints of the various limbs. However, the problem formulation is not mathematically complete with the above statements because there would, in general, be more than one way of transferring a system from one state to another when such transfers are possible. Constraints are required in the complete for mulation. The concept of optimization of a certain physically meaning ful quantity during the maneuver arises naturally at this point. A cost functional to be maximized or minimized gives a basic and needed struc ture to the scheme for exerting the control torques. It may be expected, logically, that, unless given special orders, a human being selects its own performance criterion for optimization while doing any physical activity. Some of the physically meaningful quantities that may be optimized during a physical activity are the total time to perform the activity, and the total energy spent during the activity. 3 1.2. Previous Work Very little work has been reported in the literature so far where the optimization considerations have been used in the study of human motion. Earlier work in the study of the mechanics of motion of living beings was done primarily from the view of grossly explaining certain maneuvers modeling the applicability of principles of rigid body mechanics. Most of this work was done under either free fall or zero gravity conditions. The models were made up of coupled rigid bodies and conservation of angular momentum in the absence of external torques was the most often used principle of mechanics. The righting maneuver of a free falling cat in midair attracted the attention of several authors in the early days of studies of living * objects. Marey's [1] photographs of a falling cat evoked discussions in 1894 in the French Acadmie des Sciences on whether an initial angu lar velocity was necessary in order to perform the righting maneuver. Guyou [2] modeled the cat by two coupled rigid bodies and explained the phenomenon with the aid of the angular momentum principle with the angu lar momentum of the entire cat identically equal to zero. Later, more photographic studies were made by Magnus [3] and McDonald [4,5,6]. McDonald made an extensive study of the falling cats with a high speed (1500 frames/second) motion picture camera. His description of their motion added many details to previous explanations. McDonald found no Numbers in brackets denote reference numbers listed in the List of References. 4 evidence for the simple motion of Magnus. In addition he studied the eyes and the vestibular organ as motion sensors. Amar [7] made one of the most complete of the early studies of human motor activities in 1914. This study of the relative motion of the head, limbs and major sections of the trunk was made with a view to study the efficiency of human motion in connection with industrial labor. Fischer [8] considered the mechanics of a body made up of n links and obtained equations of motion without introducing coordinates. He made discussions of applications of his theory to models of the human body, but did not give applications for the equations of motion he had obtained. In recent years most of the analytical studies of human motion have been associated with human beings in free fall as applied to astronauts maneuvering in space with and without external devices. McDonald [9] made extensive experimental studies of human motions such as springboard diving and the "cat-drop" maneuver. McCrank and Segar [10] considered the human body to be composed of nine connected parts. They developed a procedure for the numerical solution of their very complex equation of motion. Although some numerical results were pre sented, no general conclusions were drawn. The most significant contribution to the application of rational mechanics to problems in the reorientation of a human being without the help of external torques was made by Smith and Kane [11]. Specifically, they considered a man under free fall. In this paper the authors pointed out that the number of the unknown functions exceeded the number of the 5 equations of motions that were obtained for the system and recognized the need for optimization considerations. In order to get the adequate number of equations, they introduced a cost functional to be optimized which consisted of an integral over the total time interval of some suitable functional of the undetermined generalized coordinates. Optimization of this functional became a problem of calculus of varia tions, which yielded the necessary number of additional equations (the Euler-Lagrange equations) to solve the original problem completely. The approach of Smith and Kane suffers from one major drawback it ignores the internal forces of the system. The internal forces due to muscle groups at the various joints of the body segments are mostly voluntary, have upper bounds in their magnitudes and are responsible for the partially independent movements of the various limbs. Because they are internal forces, it is possible to eliminate them completely in any equations of motion. These can be obtained, for example, if the entire system is considered as a whole. However, such equations will be limited in number (at most six, three from the consideration of translation and three from the consideration of rotation) and will, in general, be less than the number of the unknown functions. The intro duction of a cost functional will yield via the calculus of variations the proper number of equations. However, the maneuver obtained may be beyond the physical ability of the individual. It is therefore essen tial to recognize the role of all the internal voluntary forces that come into play during a physical maneuver. Ayoub [12] considered an optimal performance problem of the human arm transferring a load from one point on a table to another point. 6 The motion considered was planar. Internal forces and constraints on the stresses were considered. A two-link model was considered for the arm and numerical solutions were obtained, using the methods of Linear Programming, Geometric Programming, Dynamic Programming and simulation. The performance criterion was a mathematical expression for the total effort spent during the activity. The motion considered was simple from the point of view of dynamics. Also, it required less than ten points to describe the entire motion. This allowed consideration of many physical constraints but the dynamics of the human body motion con sidered was quite different than when large motion of the various limbs are involved. A list of references of works on human performance from the point of view of Industrial Engineering is given in Ayoub's thesis. The research in the area of free fall problems showed that analytical solutions agreed qualitatively with observations in those cases in which rational mechanics has been applied with care. Examples of such problems are the cat-drop problem and the jack-knife-diver maneuver (Smith and Kane). Attempts at analytical solutions of activ ity problems which are not in the free fall category have not been com pletely successful. Such problems are in athletic activities (such as running, part of the pole vault maneuver and part of the high jump maneuver) and in activities associated with working on earth. Simultaneously with the study of the dynamics of motion, several investigations were made for the determination of the inertia parameters of human beings at various configurations. Knowledge of inertia param eters is essential for performing any dynamic analysis. A list of the 7 research activities in this area is given in References 13, 14 and 15. However, only Hanavan [14,15] has proposed a personalized model of a human being. This inertia model has been used in the present investi gation. 1.3. The Problem Statement The present work belongs to a broader program whose objective is to investigate the basic aspects of the applicability of rational mechanics to the solutions of any human activity problem. The aspects presently being considered are: 1. Construction of an appropriate presonalized model for the individual's maneuver under consideration 2. Formulation of a well-posed mathematical problem for the analytic description of the maneuver 3. Solution of the mathematical problem .by a suitable analytical method 4. Comparison of the analytical solution with an actual motion conducted in an experiment with a human subject 5. Determination of muscle activity and comparison with computed muscle torque histories for the maneuver. If a complete analysis based on the above steps results in the correct motion as compared with the experiment, then the results can be used for training purposes and design of man-machine systems, but, more importantly, the results will establish the applicability of rational mechanics to the solution of problems of human activity. In the present investigation, a particlar gymnastic maneuver, the kip-up, has been selected for analysis as outlined above. The methods 8 developed will, of course, be valid for other maneuvers but the ana- lyti cal model will be personalized to the maneuver and individual. Among all the common physical and athletic activities, the kip-up maneuver was found to be particularly well suited for analysis. The motion involves large motions, and is continuous and smooth. Also, it is planar and needs relatively fewer generalized co-ordinates for its complete description, since a correctly executed kip-up exhibits three "rigid" links. At the same time it is not a trivial problem from the point of view of our basic objective. The physical quantity (the performance criterion) chosen for optimization (minimization, in this case) is the total time to do the maneuver. 1.4. The Kip-Up Maneuver The kip-up maneuver is an exercise that a gymnast performs on a horizontal bar. The gymnast starts from a position hanging vertically down from the horizontal bar and rises to the top of the bar by swinging his arms and legs in a proper sequence. During the maneuver the motion is symmetric and does not involve bending of the elbows and the knees. Normally, the grip on the bar is loose most of the time. For an inex perienced person, the maneuver is not easy to perform. 1.5. Present Work In Chapter 2, a mathematical model for the kip-up motion and results of laboratory experiments to test the accuracy of the model are presented. First, a mathematical model of a professional gymnast for the kip-up motion has been constructed. The dynamic equations of 9 motion for the mathematical model were then obtained. Two sets of equations of motion were derived: one for the purpose of verifying the accuracy of the mathematical model from experiments and one for the purpose of optimization of the kip-up maneuver. Next, the results of laboratory experiments with the gymnast are presented. The gymnast was told to perform symmetric maneuvers on the horizontal bar, including the kip-up. The maneuvers were photographically recorded. Two of the many records were selected, one of a simple swinging motion with relatively small oscillations, and another of his quickest kip-up maneuver. The angle measurements from these film records were then used in the equations of motion to check the accuracy of the mathematical model. An error analysis was then performed to explain the disagreement between the experimental and the computed results. In Chapter 3, an analytic solution of the minimum time kip-up for the mathematical model was obtained by numerical computations. First, the analytical problem of the determination of the kip-up in mini mum time was stated in precise mathematical terms. This involved repre senting the equations of motion in the state variable form, specifying the boundary conditions on the state variables, establishing the bounds on the control variables, and modeling the stiffness of the shoulder and the hip joints at extreme arm and leg movements. A survey of the necessary conditions for time optimality given by the optimal control theory has been presented. Finally, several numerical schemes used for solving the kip-up maneuver problem are presented and the results of the numerical computations are discussed. In Chapter 4, final conclusions of the present investigation and recommendations for future work have been presented. CHAPTER 2 EXPERIMENTATION AND CONSTRUCTION OF THE MATHEMATICAL MODEL 2.0. Introduction In this chapter, modeling of a human being for the kip-up maneuver is considered. In Section 2.1, a mathematical model for a professional gymnast is constructed, using the personalized model of Hanavan [14]. In Section 2.2, equations of motion for the mathematical model are derived for the purpose of using them in the analytic deter mination of the subject's optimal kip-up motion. An equivalent system of first-order differential equations are derived in Section 2.3 for testing the inertia properties and structure of the model. In Sec tion 2.4, the laboratory experiments performed are described. The results of the experiments are discussed in Section 2.5. An analysis of the comparisons between the observed and computed results are also presented. 2.1. Mathematical Model of the Kip-Up The equations of motion of a deformable body such as the human body are usually partial differential equations. Presently, not enough is known or measurable about the deformation of the human body under voluntary motion to determine a partial differential equation model. Also, such models are quite difficult to handle. However, for situations 10 11 where the deformation is small compared to the displacement of a body, the deformable body may be considered as a rigid body in writing the equations of motion. The rigid link assumption has been used widely in modeling human beings. The personalized model of Hanavan [14] is based on this assumption. The elements of the Hanavan model are shown in Figure 1 and consist of fifteen simple homogeneous geometric solids. This construction allows a large number of degrees of freedom for the model and minimizes the deformation of the elements without undue com plexity. As the number of degrees of freedom increases to the maximum, it is likely that a mathematical model based on the Hanavan model becomes more accurate. However, increases in the degrees of freedom increase the model's complexity and make it difficult to analyze mathe matically. Observation of a correctly performed kip-up indicates that the human performer might be modeled quite accurately as a system of only three rigid links. The two arms form one link, the head-neck-torso system forms another, and the two legs form the third link. The shoulder and hip joints may be approximated as smooth hinges where these links are joined together. Deformation of the link consisting of the head, neck, and torso during certain periods of the maneuver is detectable when observed via high speed filming. It was felt that the effect of the deformation of the torso would be no more significant than the standard deviation in the Hanavan inertia parameters because this part of the orientation of the torso is determined by the line joining the shoulder and the hip joint centers. 12 Figure 1 o 00 >> Hanavan's Mathematical Model of a Human Being. 13 The muscle forces acting at the hip and shoulder to cause link motion have been replaced with their rigid body equivalent resultant forces and couples. If the masses of the muscles causing the forces are small compared to the other portions of the links, then the net effects of the forces are the torques of the couples. The resultants do not appear in the equations of motion. The three-link kip-up model is shown in Figure 2. It is con structed with elements of the Hanavan model. Twenty-five anthropometric dimensions of the gymnast were taken and used in a computer program for calculating first the inertia properties of the Hanavan model and then those of the kip-up model. The determination of the inertia properties of the kip-up model from those of the Hanavan model is presented in Appendix A. 2.2. The Equations of Motion The mathematical model for the kip-up motion is a three-link system executing plane motion under gravity. The active forces in the system are the pull of gravity acting on each of the elements of the system and the two muscle torques. The muscle forces at the shoulder acting at the joint between links 1 and 2 are replaced by the torque u^. Likewise, u^ is the torque for the hip joint between links 2 and 3. The system is suspended from a hinge at the upper end of link 1, repre senting the fists gripping the horizontal bar which is free to rotate on its spherical bearings. All joint hinges are assumed to be friction less. The general three-link system for which the equations of motion will be derived is shown with nomenclature in Figure 3. Equations of motion of the system were obtained via Lagrange's equations as follows: 0 Center of Gravity of an Element Figure 2. Mathematical Model for Kip-Up. 15 3 CGI a 3 *1 I = length of element 2 = distance between the hinges A and B - distance between the center of gravity of element 1 and the hinge 0 = distance between the center of gravity of element 2 and the hinge A = distance between the center of gravity of element 3 and the hinge B , CG2, CG3 locations of centers of gravity of elements 1, 2, and 3, respectively 1 = angle between O-CGl and OA = angle between A-CG2 and AB , m m = mass of elements 1, 2, and 3, respectively Z o I^, I = moments of inertia of elements 1, 2, and 3, respectively, about axes perpendicular to the xz plane through their respective centers of gravity I = g moment of inertia of the horizontal bar at the hinge 0 about its longitudinal axis acceleration due to gravity Figure 3. The Three-Link System 16 If we define A = Â§^rl + h + m2r2 +I2+m2*l + I3+ m3r3 + m34 + m34 + Ir) B = m2Vl C = m3X1X2 D = m3Vl E = m3r3^2 F = ^(m2r2+I2 + m3^2+ln3r3+I3) J mgr3 + 13 M = m^g N = (m2 + m3) V = m3 V W = ra2r2g R = m3r3g (2.2.1) with Equations (2.2.1), we can express the Lagrangian of the system as: 2 "2 L = cp [A + B cos (6+3) + C cos 8 + D cos (6+ijf) + E cos i|f] + 9 [F+ E cos \jf] 1 '2 + 6cp[2F+B cos (6+3) + C cos 6+D cos (6+i|0 + 2E cos i|f] + ^ t J + 8 i| [J + E cos ijr] + cp \[r [J + D cos (9+iJr) + E cos i|r] + M cos (cp+a) + N cos cp + V cos (cp+8) + W cos (cp+9+3) + R cos (cp+6+i)i). (2.2.2) For the Hanavan model of the angles 9 and i)r respectively, we can write the equations of motion as: 17 d Bl Bl _d_ BL dt Be 3l ^0 = u. d Bl 3l dt "5F ~ U2 3^ (2.2.3) (2.2.4) (2.2.5) Let us define = 2(A + (B+C) cos 0 + D cos (0+ \|f) + E cos i|r) a = 2F + (B+C) cos 9 + D cos (9+i|f) + 2E cos ijt z a^ = J + D cos (9+t) + E cos i)i bl a2 b = 2(F + E cos f) bg = J + E cos i|r c3 = J > d^ = cp9[2(B+C) sin 0 + 2D sin (0+\|/)] + 9\|i[2D sin (9+i|0 + 2E sin \|i] + 0i|r[2D sin (9+i|r) + 2E sin i|r] + 02[(B+C) sin 0 + D sin (0+\jf)] 2 + i| [D sin (9+\J) + E sin \|r] (M+N) sin 9 (V+W) sin (9+0) - R sin (cp+04-l) *2 d = u + i|f[29 + 20 + i|r] E sin i|r 9 [(B+C) sin 0 + D sin (0+t|r)] Z X - (V+W) sin (9+9) R sin (9+0+ijf) 2 2 d = u -9 [D sin (+ \Â¡r) + E sin i|r] 9 E sin i|r -209 E sin ljf o z - R sin (9+0+i|i) . (2.2.6) 18 With Equations (2.2.6) we obtain from Equations (2.2.2) (2.2.5) the equations of motion a P + = d1 V + 29 + c2* = d2 a3'P + b39 + C3* = d3 ' (2.2.7) (2.2.8) (2.2.9) It will be helpful to express the equations of motion in normal form in formulating optimal control problems for the system. For that purpose we define the state variables as x, =9 x =9 x_=e x. = e X = i|r x- = i|r A = (2.2.10) (2.2.11) A1 = (2.2.12) A2 = (2.2.13) (2.2.14) 19 Using these definitions, the equations of motion (2.2.7) (2.2.9) can then be written in the normal form as To write down the equations of motion in more convenient forms, we shall further define the following quantities: (2.2.15) (2.2.16) (2.2.17) (2.2.18) (2.2.19) 20 A -|T r 3 l A(X) = |_X2, x4, X6, -J B(X) = (_blC3+b3Cl) ("alC3_a3Cl) (_aib3+a3bl) (blC2 _b2Cl) (_aiC2+a2Cl) (aib2a2bl) (2.2.20) J ("blC3+b3Cl) (2.2.21) (blC2"Vl) (_aiC2+ a2Cl} (aib2 a2bl} (2.22) Using the definitions(2.2.16) (2.2.22), the equations of motion (2.2.15) may now be expressed by any one of the following equations: ?1 , c. .B j i.S ,-5rn. S_,-n,: V v- Up o this point, this analysis has been very gen era": and three-dimensional witn "-id referente to a plate- like two-dimensional problem. Equation >3} witn the aid of (?) car, be applied to any three-oixensional elasticity 22 with 9 expressed in terms of p using Equation (2.3.1). Hamilton's canonic equations are then given by cp = -g (p,cp,t) (2.3.3) and P = (P,9,t). (2.3.4) From Equations (2.3.2) (2.3.4) we obtain . ^ p-0[2F+(B+C)cos 8+D cos (6+1|0+2E cos ifr] i|r[ J+D cos (8+i|f) + E cos i|f] 2[A + (B+C) cos 8 + D cos (6+i|) + E cos \|/] (2. 3.5) p = [ (M+N) sin cp + (V+W) sin (9+0) + R sin (9+0+i|r)]. (2.3.6) If we wish to introduce the effects of the friction at the hinge 0, the equation for p becomes oh p = - F '5p (2.3.7 ) where F^ is the generalized friction torque at the hinge 0. The Integration Scheme The integration scheme integrates Equations (2.3.5) and (2.3.7). The initial condition of 9 is obtained from the measured values of 9. The initial value of p is obtained from the definition of p given in Equation (2.3.1). To compute the initial value of p, 9 has been com puted for this starting point only. The 6 and i|r data are differenti- ated numerically to generate 0 and \Jr values at every step. and are computed at every step by using these values. The value of F^ is not known a priori and it requires a separate experiment for its determination. Integrations were done with F^ = 0 and F^= C sgn 9 23 with C determined experimentally. The difference between the two cases was found to be insignificant even with C=1 ft-lb which was well above the possible friction torque at the bearings. 2.4. Experimental Procedure The test rig consisted of a horizontal bar and two motion picture cameras. The horizontal bar was made of a short solid round steel bar 1-3/16 inches in diameter and 58 inches long supported by two very rigid vertical columns through a pair of self-aligning spherical bearings. The bearings allowed free rotation of the bar with the arm of the subject. One movie camera was placed with its line of sight aligning with the horizontal bar and about 30 feet away from the bar as shown in Figure 4. Alignment of the camera's line of sight with the horizontal bar would give the correct vertical projection of the two arms on the film for determining the angle cp directly. The second camera was placed in front of the horizontal bar at the same elevation as the bar. The film taken in this camera showed whether a particular motion was sym metric or not about the vertical plane of motion and also, the angle between the two arms, which is required for computing the moment of inertia of the arms. The film speed was determined from the flashes of a strobe light regulated by a square wave generator. Experiments The experiments were done during two separate periods with the same subject, a professional gymnast. An average of 15 experiments of the subject's performance on the bar were recorded on film each day. Figure 4. Sketch of Kip-Up Apparatus Configuration. 25 The subject was told to avoid bending his arms and legs and to maintain symmetric motion. In the early days of experimentation he was told to just swing on the bar by moving his stiff arms and legs relative to the torso. These experiments were done with the idea of obtaining small angle data for verifying the inertia properties of the Hanavan model. The filming of the camera was done at speeds of 32 frames per second and 62.1 frames per second (as determined from stroboscopic measurements). In later experiments the subject was told to perform the maneuver with some specific objectives. He was told to perform what he thought would be the kip-up (1) in minimum time, (2) with minimum expenditure of energy, and (3) putting "least effort." Each of these maneuvers was repeated several times. Between any two subsequent maneuvers, the subject was given adequate rest periods to avoid fatigue. This exper iment was conducted with the idea of making optimization studies as well as testing the model. Four white tapes were stuck to the subject, on the sides of his upper and lower arms, sides of his torso, and on the sides of his legs. These tapes were aligned between joint centers as suggested by the Hanavan inertia model. Processing the Data The film speed was measured with the aid of a stroboscope by running the camera with a developed film with the same speed setting. After removing the lens the shutter was exposed to the stroboscope flash. By arresting the shutter in the stroboscope light, the shutter speed was obtained. 26 The films were run on an L-W Photo Optical Data Analyzer. For the purpose of testing the inertia properties, two maneuvers were selected, one from each day's filmings. The films were projected plumb line perpendicularly on paper fixed to a vertical wall. As each frame was projected, the white tapes fixed on the subject, now clearly visible in the image, were marked out on the paper of the pad by means of a pencil and a straight edge. In this way each frame was "trans ferred" on separate sheets of paper. The angles 9 and i|r were then measured from these traces. The angle 9 was measured with respect to a vertical reference. The vertical reference was obtained from a sharp window wall in the background. Of the two sets of data processed, one was smoothed before using it in the integration scheme. This was the one filmed at the higher speed for the fast kip-up motion. A plot of the raw data and those after preliminary smoothing for this set are shown in Figure 5. 2.5 Results and Discussion The results of the integration of the equations of motion for two of the data sets analyzed are shown in Figures 6 and 7. Figure 6 shows data for swinging motion, while Figure 7 is for the kip-up. Also, in Figure 7 is given the computed results corresponding to reinitiating the integration program with the measured data. Differ ent starting points of integration were selected to eliminate the errors that were generated before these points. In Figure 6, curve 2 of the computed values for the unsmoothed data agrees well with curve 1 of the measured values for a little more Angles (Degree) for the Kip-Up Motion. (Film Speed 62.1 frames/second.) to T h c* res o 1 t s o h a i n v i so fur a re applicable to p: a tes cons;stir. of any r. jiu2- eo* la. yens having ci f f e r - ent properties an ; geomec O Cut -ro nÂ¡ here on, the e n a 1 y s i s would depend n t h s t f o o O pinte c o n s i d l r e d v i z. s 4 r. gl e layer cr mult i p 1 a y e r plates, thin or moderately thick pi a tes etc. and also on the type Of dispi a cement rune- t i on s V ?,rd V ,. Habi p [41 has demonstrated the apolica- i j - Dili rv of these results to a single layer plate. In this dissertation, a thres-1ayered elate, popu larly known as a sandwich plate will be considered. San dvr i ch oate A sandwich plate consisting of three layers is shewn in figure (2). Tne face layers are much thinner than the core. All layers are uniformly thick throughout- The two faces are of the same thickness tt, the core thick ness being t . Tne present analysis is capable of treating aixed boundary value problems. On the upper surface, the line AB separates the two regions and over which displacements and stresses respectively are prescribed. On the lower surface, the line CD separates the correspond ing regions. The line CD is located exactly below the line AD. Also, at any two points locatec on the upoer and lower surfaces of the plate and havine tne same >:, and x. cp (Radian) to CD (Radian) ( Curve 1 Measured and Smoothed Values of cp Curves 2,3,4- Computed Values of cp, Started at Different Points 2.0 w o 31 2.6. Sources of Errors The following are considered to be the sources of errors responsible for the disagreement between the computed and measured values. 2.6.1. Imperfections in the Model The human being for the motion studies was modeled as a system of rigid bodies. The response of the system to be compared to that of the rigid body model was a single generalized coordinate of the system. The errors in modeling can be lumped into the overlapping categories of (1) definition of the generalized coordinates of the rigid elements of the system, (2) deformations of link lines during motions from prior joint center measurements, and (3) significant variations during motion in the inertia properties of the torso with respect to the fixed coor dinate system. In several experiments the torso deformed with signif icant movement of the shoulder joint centers. In these cases the con stant inertia properties model is obviously incorrect. These variations not only cause inaccuracies in the inertia parameters but also result in errors in the link lines or additional errors in the mass center of element 2. These errors are reflected in errors in the angles 0 and i|r, which in turn cause a time varying "phase shift" in the computed angle cp- 2.6.2. Errors in Filming and Processing the Data The primary source of errors in filming was considered to be caused by inaccuracies in knowing precisely where the link lines were in relation to the film plane. As mentioned in Section 2.4, care was 32 taken to minimize this error. That is, the cameras were aligned with the horizontal bar so that the link line of the arms projected in the plumb line plane of the film was very nearly the correct model refer ence line. Since the link lines of the torso and legs were nearly plumb line vertical, no corrections of the image data were required for these links. Filming of static thin rods which were connected to the bar at known angles to each other and the vertical plane produces overall measurement errors of about 1 degree standard deviation. Errors up to 3 degrees can be expected in data from films of the motion experiments. These errors can cause the rates of the angles to be in error by more than 30 percent. This is the primary cause of the error in the amplitude of the angle cp computed from the dynamical equations. (Second derivatives of the data can be in error by more than 100 percent. This ruled out the use of equations such as Euler's or Lagrange's.) 2.6.3. The Integration Scheme The integration scheme uses at any step the measured values of 0 and ijf and the computed values of 9 and i|f stored previously. Once a difference between the measured (actual) and the calculated values of cp has developed at a time t due to any of the sources of errors discussed above, the system configuration determined by the calculated value of cp and the measured (actual) values of 0 and ljr at the time t will be dif ferent from that of the actual system at that time. This will cause the model to have a different response after time t than that of the actual system which has a different relative configuration. This in turn will cause a further deviation between the actual and the calculated motion that follows this instant of time. To reduce this 33 effect of propagation of error via the system equations, several reinitializations of cp and p were done by restarting the integrations at different points. Various order differentiation and integraion schemes were tested with insignificant differences in the results for smoothed data. These were done with data from filming at F=62.1 frames per second with a maximum integration step size of 2/F second. As men tioned previously, the main cause of amplitude error was the errors in the derivatives of the raw, unsmoothed data. The results obtained from the experiments show that the model for the kip-up motion constructed from the Hanavan model was reason ably good considering its kinematical simplicity. In spite of imper fections in the model, its dynamic behavior was quite similar to the actual motion, so that this model could provide reasonable estimates of optimal human performance via the theory of optimal processes and numerical solution methods. However, the results obtained from the experiments indicate that the application of rational mechanics to the analysis and design of man-machine systems could prove inadequate unless the model and the data gathering techniques can be improved. This is especially true in the design of high accuracy or low tolerance systems. CHAPTER 3 ANALYTIC DETERMINATION OF THE MINIMUM-TIME KIP-UP STRATEGY 3.0. Introduction In this chapter the determination of an analytic solution of the kip-up maneuver is presented. The problem of analytical determination of the kip-up strategy in minimum time has been cast as a problem of optimal control of dynamical systems. Before the techniques of the optimal control theory may be applied to the problem, it is necessary to state the physical problem in the language of mathematics and to introduce the physical constraints that must also be considered for the solution. Thus, the first four sections of this chapter have been devoted to the formulation of the mathematical problem. In Section 3.5 a survey of the necessary conditions for optimality obtained from the optimal control theory is presented. Since the problem under consider ation cannot be solved in closed form, numerical methods were used to obtain the solution. In Sections 3.6 3.9, the choice of the numerical methods, their derivations and the results of the numerical computations are discussed. In Section 3.10, results of the numerical computations are compared with the actual motion. 34 35 3.1. ^Mathematical Formulation of the Kip-Up Problem The problem is to determine the minimum time strategy for the man model to kip-up without violating control constraints. These con straints represent the maximum torques the man's muscles can exert for any given configuration. Formulated mathematically, we have the follow ing: For the system equations X = f(X,u) = A(X) + B(X)u (3.1.1) and the boundary conditions X(0) = X o (given) (3.1.2) where and given by Â§(X(tf))= X(tf) Xf = 0 Xf = given t = final time, to be determined X(t) is the time-dependent state vector, Xx(t) =qj(t), Xg(t) =cp(t), X3(t) = 9(t) x4(t) = 0(t), x5(t) = \Kt), xg(t) I|r(t) (3.1.3) (3.1.4) (3.1.5) find a control u(t) = [u1(t),u2(t)] (3.1.6) such that simultaneously Equations (3.1.1) (3.1.3) are satisfied, t^ is minimized, and for all values of t, 0 ^ t Â£ t the inequalities S*(X) ux(t) sJ(X) s\a) S u2(t) S S2(X) (3.1.7) 36 are satisfied. S1(X) are given functions of X and represent the bounds on the control u.(t). The functions f(X,u), A(X) and B(X) were pre- J viously given in Section 2.2. Â§ is the error in meeting the terminal values of the state variables. 3.2. Bounds on the Controls The control variable u^ is the muscle torque exerted at the shoulder joint and u is that exerted at the hip. For the individual being modeled, the functions u and u will have upper and lower limits which are functions of the state X. Samras [16] experimentally determined the maximum muscle torques at the shoulder and hip joints for various limb angles at the joints. This was done for the same subject modeled in the present study. These measurements were made under static conditions and the mximums in either flexion or extension were measured for the shoulder torque for various values of 6 and the hip torque for various values of i|i. The ex perimental bounds on the shoulder torque were then fitted by polynomials in 9. The hip torque bounds were expressed in polynomials in \jf. Even though each of these bounds might be expected to depend to some degree on all four state variables X^, X^, X^, and X^, the bounds on the shoulder torque u depend primarily on X and the bounds on the 1 O hip torque u depend primarily on X The measurements of Samras do not z o include the rate dependence X and X Although the rate effect appears 4 6 to be measurable, it is a second-order effect and quite difficult to obtain. The control limit functions are given in Figure 8. These func tions are correct only for a certain range of values of the angles Control Limit (ft-lb) Figure 8. Unmodified Control Limit Functions (Samras [16]). 38 X and X The values of S. and S0 can never be positive and those of 3 5 12 O O and can never be negative. Whenever these sign conditions are violated by extreme values of the states, S'? is set equal to zero. Also, 2 from extrapolated measurement data an upper limit has been set for S at 160.0 ft-lb and a lower limit has been set for at -100.0 ft-lb. 3.3. Torsional Springs in the Shoulder and Hip Joints Our dynamical model and the control limit functions of the shoulder and the hip do not account for the stiffness of the shoulder and the hip joints at the extremities of shoulder and leg movements. It has been observed that the shoulder joints produce a resistance to raising the arm beyond an angle of 0 30. The hip joints resist move ment for i|f > 120, or for i|r < -35. The effects of these "stops are important and must be included in the model, since the film data showed that these limits were reached. There are no data available for the stiffness of these joint stops. It was observed that, although the joints were not rigid, they were quite stiff. It was therefore decided to use stiff torsional spring models at the model's shoulder and the hip joints. These would be active when the stop angles were exceeded. For the shoulder the spring is active for 9 ^ 0.5 radian. For the hip joint the spring is active for ijj S -0.6 radian and i|r 2: 2.1 radians. The springs have equal stiffnesses. One generates a 100 ft-lb torque at the shoulder for a deflection of 0.1 radian. This corresponds to a joint stop torque of the order of the maximum voluntary torque avail able at the shoulder for the deflection of 0.1 radian. This gives a spring constant of Kg = 1000 ft-lb/rad. The spring forces at the shoulders 39 would therefore be equal to -K (9-0.5) for 9 ^ 0.5 radian and those at s the hip joints would be -K (i|r-2.1) for 4 ^ 2.1 radians and -K (4+0.6) s s for 4 ^ -0.6 radian. These torques at the shoulder and the hip joints were added to the voluntary control torques u^ and u^ when the stops were activated. 3.4. Boundary Conditions The boundary conditions for the kip-up maneuver were chosen from the experimental data of Section 2.5. The initial values selected correspond to motion which has already begun. This is beyond the initial unsymmetrical motion which occurs on beginning the first swing. This motion is difficult to model and is not important in this basic research. The final values of the state variables represent the model atop the horizontal bar still moving upward and just before body con tact with the bar. (Once the torso contacts the bar, the model is no longer valid.) The actual motion in the experiment terminated shortly after this point when the gymnast used the impact of the horizontal bar with his body to stop himself. The initial and final values of the state variables for the optimization problem are listed in Table 1. 40 TABLE 1 BOUNDARY CONDITIONS FOR A MINIMUM TIME KIP-UP MOTION State Variables Initial Value Final Value V X1 = 0.340 o 1 Xf = -2. 84 2 2 X = -2.30 o Xf - -7.05 e X3 = 0.305 x3 = 2.88 O f # 4 4 e X -0.660 X-c = 0.163 o f X5 = -0.087 o 0.436 6 6 X = -1.20 o Xf = 0.108 3.5. The Necessary Conditions for Time Optimal Control In this chapter, we look into the necessary conditions for the minimum time problem formulated in the previous chapter. The necessary conditions for optimality of motion for the case when the constraints on the control are not a function of the states are given in Reference 17. For the case where control constraints depend on the states, the nec essary condition requires a modification in the adjoint equations. These are obtained through a calculus of variations approach [18]. This approach is used in the following developments. Writing the state equations of our system as X = f(X,u) = A(X) + B(X)u (3.5.1) 41 we can construct the cost function as dt (3.5.2) where t f is free. The Hamiltonian is then given by H(X,u,X) = 1 + \Tf = 1 + \TA(X) + \TB(X)u = 1 + XTA(X) + XTb1(x)u1 + \TB2(X)U2 where X(t) is the time-dependent six-dimensional column vector of adjoint variables. A, B, B1 B and f are the quantities as defined by Equations (2.2.15) (2.2.22) in Section 2.2. The minimum-time control policy u(t) will be given by the one that minimizes the Hamiltonian (3.5.3), provided no singular arcs are present. We note that in this case the Hamiltonian is a linear func tion of the control u and therefore the minimum with respect to u occurs only in the upper and the lower bounds of u if there is no singular solution. Thus, we have, recalling the definitions of S^ in Section 3.2, T (1) If X B^(X) >0, u^ = the minimum allowable value of u^ = S*(X) (3.5.4a) (3.5.3a) (3.5.3b) (3.5.3c) (2) If \TB (X) < 0, U;L (3) If XTB2(X) > 0, u2 the maximum allowable value of u^. SJ(X) (3.5,4b) the minimum allowable value of ur S2( (3.5.5a) (4) If \TB2(X) < 0, u2 the maximum allowable value of u (3.5.5b) 42 (5) If \TB (X) = 0 T " or X B (X) = 0 J , u^ u^ = possible singular control. (3.5.6) u and u will be determined by investigating whether or not there is -i- C* a singular solution with respect to these variables. The adjoint equations will be different for the portions of the o trajectories for u corresponding to constrained and unconstrained arcs. The adjoint equations are, in general, given by \T = -H, (3.5.7) X This yields (a) When neither u nor u lie on a constraint \T = -H, = -\TA, \TB u XTB u (3.5.8) 'X -X l,x 1 "2,x 2 (b) When any one or both u and u denoted by u. (i = 1 or 2), lie on 1 Z X a constraint denoted by (j = 1 or 2) the right side of Equation (3.5.8) of the adjoint variables has the additional term - \tb. SJ - -i i X and the equation can be written as T_ j X = X A, X B, U x Bn u Z 5. x B. S. - -'X -1, 1 -2,x 2 i=1 -1 i, (3.5.9) where 6=0 if X B. = 0 l -l 6. = 1 if \ B. 4 0. i -l The boundary conditions on the state and the adjoint variables are 43 X(0) given = X \(0) = free - -o - X(t ) = given = Xf \(tf) = free and H(tf) = (1 + \Tf)t ^ 0 . (3.5.10) (3.5.11) The state and adjoint equations together with the control laws and the boundary conditions written above form a two-point boundary value problem (TPBVP) in the state and adjoint variables. If these o equations can be solved, the optimal control, u will be immediately obtained from Equations (3.5.4a) (3.5.6). Investigation of Singular Solutions T o It has been noted that if Â¡V = 0, u^ cannot be determined from the requirement that the Hamiltonian is to be minimized with o T respect to u The same is true for u when\ B = 0. Since the treatment for u^ is the'same as for u^, we shall investigate a singular control for only u^. T If the quantity \ B^ = 0 only for a single instant of time, then the situation is not of much concern because the duration of the interval is not finite and we can simply choose u = u^(t ) or u^(t+) or 0, where u^(t ) = control at the instant preceding t, u^(t4) is the instant exactly after t. The situation needs special attention when T \ B^ = 0 for a finite interval of time. If t S t Â£ t is an interval for which u is singular, it is 1 Z a clear that, for our system ^ t t^ 0 for (3.5.12) 44 and therefore, d T . Ht 5^ = 0 for (3.5.13) or x\ + \TB = 0 for ti t t2 (3.5.14) or, for the interval t ^ t ^ t the following results must hold: X cj Case 1 Only u is singular. u is nonsingular. Since u is not 1 ^ 2 singular, u is on a constraint boundary and is given by 2 u = S 2 2 j = 1 for the lower constraint j = 2 for the upper constraint. The adjoint equations are given by Equation (3.5.9) T T T T i T i l = -X A X B1 u XJ, S, X B S X 1 } v 1 ~ 2 f 2 2 a y (3.5.15) X Â§i = B X 'X = Â§1, + ?1 ui + 52 ^ X (3.5.16) (3.5.17) From Equations (3.5.14), (3.5.15), and (3.5.16), we obtain 4-h\4, jSi + ^Si, <* + s2 s2> = 0 X XX It is to be observed that the necessary condition (3.5.17) is not explicit in u^. Case 2 Both u and u are singular. The value of u is no longer X 2a 2 2 T j and the term X B S in the X Equation (3.5.15) drops out in this 2 X case. Accordingly, one obtains -XT[A,x B1 A] XT[B2 B1 Bx B21 u2 = 0 (3.5.18) X XX 45 Proceeding from the assumption that u is singular, one would also get, Â£ for this case, when both u^ and u^ are singular T C*X B_ A] \T[B1 B, ^X X ' (3.5.19) From Equations (3.5.17), (3.5.18), and (3.5.19) we can see that T T only if both \ B^ and \ Bare zero simultaneously, is it possible to find a singular solution by suitable choices of u^ and u^ from the T T t condition (3.5.13). If only one of \ B^ and \ B^, say \ B^, is zero, the requirement (^ B,) = 0 does not yield an equation explicit in dt -1 d T u as observed in Equation (3.5.17). It is thus required that (\ B.) 1 i ' l dt = 0, which will be explicit in u during the interval t ^ t ^ t J_ -L Â£ together with the requirement that the relation (3.5.17) is satisfied T at t r t^. These two conditions will ensure that \ B^ = 0 in the interval t^ ^ t S t . It is to be noted that singular control computed by the above procedure has not been proved to be the minimizing control. Additional necessary conditions analogous to the convexity conditions for singular controls have been obtained by Tait [19] and Kelley, Kopp and Moyer [20] for scalar control and by Robbins [21] and Goh [22] for vector control. For the general case of vector control these conditions, summarized by Jacobson [23], may be stated as on singular subarcs: 3 - Ldt H, =0 if q is odd (3.5.20) and :> 0 . (3.5.21) 46 d2p In these equations, * H, (X,\) is the lowest order time derivative dt P - of H, in which the control u appears explicitly, and q < 2p. For a scalar control, Equation (3.5.20) is satisfied iden tically. Equations (3.5.20) and (3.5.21) also do not constitute suf ficiency conditions for minimality. A complete set of sufficiency conditions for singular arcs has not yet been established in the literature of optimal control theory for a general nonlinear system. We can see that there are quite severe restrictions on the existence of singular arcs in the human motion problem. In the numerical methods used in the present work to determine the optimal solution, only in the method of quasilinearization is it necessary to express the control (its optimal value) in terms of the state and adjoint variables, while in the gradient methods where successive improvements are made in the control variables, this is not so. In the attempts with the quasilinearization method, singular solutions were not considered in the construction of the two-point boundary value problem in the state and adjoint variables. It was decided that if a solution to the TPBVP was obtained by quasilinearization, singular arcs would be looked for later. The gradient methods exhibit singular arcs automatically if there are any. The additional necessary condi tions for singular arcs should be checked when off-constraint arcs are exhibited by the gradient method. 47 3.6. _The Solution Methods The optimal control problem formulated in the preceding section cannot be solved in closed form. Numerical methods must therefore be used to find its solution. In the optimal control theory literature several numerical methods have been proposed for solving the differ ential equations and the optimality conditions that arise out of optimal control problems such as the present one. None of these methods guaran tees that a solution will be obtained readily, while some of the methods do not guarantee that a solution may be obtained at all. The methods are all iterative, necessitating the use of high-speed computers for all nontrivial problems. A nominal guessed trajectory is improved iteratively until the improved solution satisfactorily meets all the necessary conditions. Depending on whether the method requires finding the first or both first and second derivatives of the system equations with respect to the state and control variables, these methods are called First-Order or Second-Order methods, respectively. This is so because they, in effect, make first-order or second-order approximations of the system equations with respect to the state and control variables. The first- order methods, in general, have the property that they can start from a poor guess and make fast improvements in the beginning. They need fewer computations in each iteration. But their performance is not good near the optimal solution where the convergence rate becomes very poor. The second-order methods, on the other hand, need a good ini tial guess to be able to start but have excellent convergence prop erties near the optimal solution. Because the second-order methods 48 need computation of the second derivative of the system equations, they need more computing time per iteration, which may be excessive for some problems. Apart from the first- and second-order methods mentioned above, there is another class of methods which tries to combine the advantages of both of these methods while eliminating the disadvantages of both. The Conjugate Gradient Method, Parallel Tangent Method, and the Davidon- Fletcher-Powell Method fall into this class. These methods work very much like the first-order method except that, in the first-order expansion, the coefficients of the first-order term, or the gradient term, is modified by some transformations. These transformations are generated from the modified gradient term of the previous iteration and the gradient term of the current iteration. This has the effect of using the information that is obtained from a second derivative. It is not known which of the several methods used for solving optimal control problems is good for a given problem and one may have to try more than one method in order to obtain the solution. In the published literature, most of the illustrations of these methods are simple. In these simple problems control or state variable histories do not have wide oscillations or the system equations themselves are not complicated. This makes it truly difficult for someone without previous experience to decide upon the merits of these methods. There is no preference list, and it seems certain that there cannot be one whereby a decision can be made as to which method should be tried first so that a solution of a given problem will be obtained most efficiently. In this respect, deciding upon a computing 49 method for a given problem is still an art and depends largely on the previous experience of the individual trying to solve the problem. In the attempts to solve the minimum time problem, the method of quasilinearization was taken up first. This choice was based on several factors. This is the only method where the two-point boundary value problem obtained from the necessary conditions of optimality is solved directly, and this feature was found very attractive. As a start ing guess, this method requires the time histories of the state and adjoint variables. Time histories of the state variables were available from the experiments. (It was decided that if the method was success ful for this guess, an arbitrary and less accurate initial guess would be tried later.) When it converges, the method has a quadratic conver gence rate. Also, in spite of its being a well-known method for solv ing nonlinear two-point boundary value problems since it was first introduced by Bellman and Kalaba [24] its applications in solving optimal control problems have been very few. There was thus an added incentive for using this methodto determine its usefulness in solv ing fairly complicated optimal control problems. Sylvester and Meyer [25] proposed, with demonstrations, an efficient scheme for solving a nonlinear TPBVP using the method of quasilinearization. This scheme was available in the IBM SHARE program ABS QUASI and was used by Boykin and Sierakowski [26] who reported excellent convergence properties of the scheme for some structural optimization problems. With this record of success, the program QUASI was taken up for our problem. But with our problem several difficulties were encountered 50 from the very beginning. First, the bang-bang control law obtained from the necessary conditions had to be replaced by a suitably steep saturation type control law. Second, a slight modification in computa tion scheme was necessary when it was found that the method was unable to solve a simple example problem. The example problem could be solved with these modifications. But, in spite of all these changes and sub sequently, many attempts to generate a guess of the adjoint variables, the method could not be made to work for the human motion problem. Reasons for the difficulties encountered are discussed in detail in Section 3.7. During the attempts with quasilinearization, it was found that computations of the second derivatives of the system equations were taking an exorbitant amount of time and this was the deciding factor for the next choice of a computing method. Also, the appearance of the control function linearly in the Hamiltonian put restrictions on the use of most of the other second-order methods. The next attempts were based on the first-order steepest descent method proposed by Bryson and Denham [27,28]. The most attractive feature of this method is that the various steps involved in it render themselves to clear physical understanding. This method directly reduces the cost function in a systematic way and one obtains good insight into the basic steps in the iterative computations and can make adjustments to improve convergence and/or stability with relative ease. These features of the method of steepest descent may more than offset the advantages of other methods for some complicated problems. In the attempts with this method, three different formulations of the minimum 51 time problem were tried. In the first formulation the computations were not pursued beyond a certain point due to computational difficulties. The solution was obtained by the second formulation and verified by the third formulation. These attempts are discussed in Section 3.8. 3.7. A Quasilinearization Scheme for Solving the Minimum-Time Problem In Section 3.5 the adjoint equations and the optimal control laws (Equations (3.5.9), (3.5.4a)-(3.5.6)) have been derived for the minimum time kip-up problem. The system equations and the boundary conditions on the state and the adjoint variables are given by Equa tions (3.5.1) and (3.5.10), respectively. From these equations we can readily see that if the control variables u^ and u^ appearing in the system and adjoint equations are replaced by their optimal expressions in terms of the state and the adjoint variables, one obtains a non linear TPBVP in the state and adjoint variables. If these equations are solved, the optimal state and adjoint variable trajectories will be obtained and the optimal controls can be constructed by using the state and adjoint variables and the optimal control laws. In the TPBVP in the state and the adjoint variables, the final time is not a given constant and is to be determined from the implicit relation (3.5.11). This makes the problem one with a variable end point. The method of quasilinearization is formulated primarily for a fixed-end-point TPBVP. In problems with variable end points, the adjustment of the final time is usually done by a separate scheme, not integral with the quasilinearization scheme. Long [29] proposed a 52 scheme for converting a variable end point problem into a fixed-end point problem with the adjustment of the final time built into the quasilinearization process. For the present system, however, this scheme was not practicable because the boundary condition (3.5.11) becomes too complicated to handle in this formulation. It was decided that with a separate algorithm for adjusting a the finai time, described later in this section, the nonlinear TPBVP with free final time would be converted to a sequence of nonlinear TPBVP's with fixed final times. Each of these fixed final time problems would then be solved by the modified quasilinearization algorithm until the correct final time was obtained. The derivation of the modified quasilinearization algorithm is described below. 3.7.1. Derivation of the Modified Quasilinearization Algorithm The fixed final time nonlinear TPBVP to be solved falls in the general class of problems given by dy ^=g(i:,t). (3.7.1) With the boundary condition Bj Z) + Br + c = 0 tf = given (3.7.2) y, g, and c are of dimension n, B^ and B^ are matrices of dimension (nxn). It is being assumed that the TPBVP has been defined for the interval 0 Â£ t < t for some given t > 0. In the state and adjoint equations, if the expressions for optimal control in terms of the state and the adjoint variables are used for the control variables, one obtains 53 X = f(X,u(X,\)) = F(X,\) (say) (3.7.3a) and X = -H^(X,u(X,\),X) = G(X,X) (say). (3.7.3b) In the formulations of the TPBVP given by Equations (3,7.1) and (3.7.2), it may be seen that for the kip-up system, n=12, X ~F 0 0 I 0 y = _X , g = 1 IO 1 1 i 0 II u CQ 0 0 and -o The 0 and I appearing in the matrices B. and B represent 6x6 order null J Y and unit matrices, respectively. Let z(t) be an initial guess vector for y(t) which satisfies the boundary conditions (3.7.2). If g(y,t) is approximated by its Taylor series expansion about g(z,t), keeping only the first-order term, one obtains (y-z). y=5 9g g(y,t) = g(z,t) + Let W = so that, W. . ij (3.7.4) or, W.. = partial derivative of the i ij th . j element of y, evaluated at y = z. . th element of g with respect to the 54 With the above approximation of g(y,t), Equation (3.7.2) becomes dy = g(x,t) + W(z,t)(y-z) at - - - or, de dz = -rf + g(z,t) + W(z,t) e dt dt - - where, e = y(t) z(t) = error in the guess z(t). Rearranging the above equation, one obtains de dz W(z,t)s = + g(z,t). (3.7.5) dt - dt - Since z(t) is chosen to satisfy the boundary conditions, Bj, z (0) + z(tf) + c = 0. Subtracting this equation from Equation (3.7.2), one obtains the boundary conditions on the error e(t) as Bj Â£(0) + Br e(tf) = 0. (3.7.6) Equations (3.7.5) and (3.7.6) form a linear TPBVP in e(t) which, when solved, will give the values of the error between the guessed solution z(t) and the actual solution y(t) based on the linearized expressions of the right side of Equations (3.7.1) about the guessed solution z(t). Because of using the linearized equation instead of the full nonlinear "'n equations, the values of e(t) obtained by solving Equations (3.7.5) and (3.7.6) will not be the actual error between the guess z(t) and the solution. However, a new guess of y(t) will be obtained from e(t) by z (t) = z(t) + T) e(t) , 0 < 7] < 1. (3.7.7) 55 The algorithm of Sylvester and Meyer uses T| = 1 for all the time, which is the usual quasilinearization algorithm. It was found, while solving a simple example problem, that without the incorporation of a multiplier 7] in the expression (3.7.7), i.e. using z' (t) = z(t) + e(t) the method was unstable. The convergence property of the scheme with the incorporation of the multiplier 7] can be understood for small values of 7] by comparison with the step-size adjustment procedure of the usual steepest descent algorithms. The mathematical proof for the convergence property follows the proof of Miele and Iyer [30] and is now given. The integral squared norm of the error in the guessed solution z(t) can be expressed by the integral Similarly, the error in the solution z'(t) = z(t) + 7] e(t) is given by t f If 7] is sufficiently small, one can write g(z',t) = g(z,t) + g,z(z,t) Tie where g, = g, - z y = W(z,t) (from Equation (3.7.4)). y=z Also, for all values of 7], / z = z + 7] e . 56 From these results, one obtains, for small values of 7], j' j = 27] J {z g(z,t)}T {e We} Since e(t) satisfies the differential equation (3.7.5), this finally yields: t j' J = 27] J || z g(z,t)||2dt = a negative quantity. Thus, for sufficiently small values of 7], the reduction in the cost is assured. In the quasilinearization algorithm z^(t) takes the role of z(t) as the new guess of y(t) and the process is continued until the error in satisfying the differential equations is reduced to an acceptable value. The linear TPBVP of the error Equation's (3.7.5) and (3.7.6) is solved as follows: The time interval t = 0 to t = t^ is divided into m small inter vals. This results in m+ 1 values of t at which the solution will be computed. Equation (3.7.5) can be written in a finite central differ ence scheme as rl /Z.+Z. t.+t. [-1 -1+1 1 1+1] e. + e. z. -z. 1 -i -i+l -i+l -i i /z. + z. t. + t.\ f -i+l -l i+l i) hi ' 2 2 / 2 h. +^' i ^ 2 2 / th where h = t t The subscript i denotes values at the i i i+1 i station, i=1,2,...,ra+l. Rearranging and simplifying the above expres sion one obtains ( + i h.w. )e. + (-1 + = h.W. )e = r. (3.7.8) 2 i i -i 2 i i -l+l -l i = 1,2,... ,m 57 where r. = z -l -l+l rui+ii i+i+^ -5i-hi 5 V 2 2 ') /z. + Z. t. + t X (3.7.9) (3.7.10) and I unit matrix of dimension n xn . The boundary conditions, Equation (3.7.2), reduce to B. e + B e = 0 i -1 r -m+1 (3.7.11) Equation (3.7.8) can be cast into the following convenient recursive expression e. + D. e. = s. -l l-i+l -i (3.7.12) where and D. = (I + i h. W.)"1 (-1+ i- h. W.) l 2 i l 2 i l - 1 -1 s. = (I + h. W. ) r. -l 2 i i -l (3.7.13) By repeated substitution, equation (3.7.12) yields the following rela tionship between e_ and e -1 -m+1 . m m e. = T + (-1) ( T D.) e -1 i -m+1 l-l where m i-1 i-1 T = s + E (-1) ( TT D.)s. i=2 j=l J 1 (3.7.14) (3.7.15) or, multiplying by on both sides of Equation (3.7.14), B. e = B. T + (-l)m B. TT D. e JL -1 l l i -m+1 1=1 m (3.7.16) Equations (3.6.16) and (3.7.11) can be solved simultaneously for and e to give -m+1 58 -m+1 where m C = (-l)m B ( TT D.) + B . I i x r 1=1 (3.7.17) (3.7.18) With e determined, e e ,. . e_ are determined in succession by -m+1 -m -m-1 -1 using the recursive relation (3.7.12). Then, the new iterate is given by zil^1(t) = zi(t) + T) e(t). The stopping condition of the algorithm is given by the fact that e should be small. When they are small, it may be seen from Equation (3.7.8) that the quantities r will also be small. From Equation (3.7.9) it is also seen that small r correspond to satisfying the central differ ence expression of the differential equations. That is, the finite dif ference equation error must be small. This does not mean that the dif ferential equation error is small unless the intervals for the differences are sufficiently small. The IBM SHARE program ABS QUASI is a program of the procedure outlined above without the provision of the multiplier T) in Equation (3.7.7), and therefore is for 7] = 1. The program was modified to intro duce and to adjust 7] to get the desired convergence. The algorithm may be described by the following steps: 1. Set up the matrices B. and B and guess a nominal trajectory Ju r z(t) that satisfies the boundary conditions (3.7.2). Set ITER = 0 (ITER for Iteration) 59 2. Do the following for i = l,2,...,m. a. Find r and W. as defined in Equations (3.7.9) and -i i (3.7.10). Find the largest element of r^, searching between the elements of each r ^, for i = l,2,...,m. Call it E2MAX. If E2MAX < specified maximum allowable error, print out z and stop the computations. b. Using Equation (3.7.13), find and s^. 3. Calculate T and C according to Equations (3.7.15) and (3.7.18). Calculate the integral norm of the error (here the norm is defined by the sum of squares of the elements of the vector r.). Set J1 = J2. -l 4. Find e using Equation (3.7.17). -mf 1 5. Decide upon a value of T], Discussions on the choice of T] will be presented in a later section. 6. Generate and store e e _,,e e. using Equation (3.7.12). -m -m-l -m-2 -l Generate the new guess z^, i = 1,2,. . ,nn-l, by doing z. = z. + T) e.. -i -i -i Set ITER = ITER + 1. 7. Find J2 and r i = l,2,...,m and find E2MAX as in step 2. 8. If J2 > Jl (unstable), go to step 11. 9. Stop if E2MAX < a prescribed value. 10. If J2 < Jl to to step 12 to continue to the next iteration. 11. If this step has been performed more than a specified number of times in this situation, go to step 13. If not, store the 60 value of the current Tj and J2. Recover the values of z of the previous iteration by doing z = z T1 e . -i -i -1 Reduce the value of T]. Generate the new by doing z. = z. + T] e. V -i -i -i Go to step 7. 12. If this step has been performed more than a specified number of times or if step 11 has been performed at least once in this iteration, go to step 13. If not, store the value of J2 and T]. Recover the new z^ as in step 11, this time by increas ing 7] to increase speed of convergence. Go to step 7. 13. Find out the minimum value of J2 that have been obtained in steps 11 and 12. If this value is greater than Jl, stop com putations and look for the cause of the instability. If this value is less than Jl, recall the T] corresponding to this J2 and regenerate the z^ for this case. Go to step 2. In this program the value of T] was selected from the consider ation that at the station m+1 the maximum value of 12 should be less than a prescribed value (j in parentheses represents the th element of the vector). This procedure limits the changes in the j existing trajectory by limiting the magnitude of the maximum fractional change in the terminal values of the variables not specified at the final time. 61 Whenever an iteration was found unstable, T] was reduced by half. When there was an improvement, a linear extrapolation formula was used to increase the value of T] so that the norm of the error J2 would decrease to a desired value. In such an attempt, however, T) was not allowed to increase beyond a certain multiple of its existing value. 3.7.2. Approximations of the Optimal Control for the Kip-Up Problem The method of quasilinearization disregards the question of singular solutions in the present investigation. It was found in Section 3.5 that there were many requirements for the existence of a singular control arq and the necessary conditions for the existence of singular controls are quite complicated. It was decided that, before going into those cases extrema without singular arc would be looked for first and if the computational method was successful, singular subarcs v/ould be looked for later. If the singular solutions are not considered, the optimal o o controls u^ and u^ become bang-bang and are given by Equations (3.5.4a) through (3.5.5b). It is convenient to rewrite these equations at this point in the following way: !f -X B (X) < 0, u = S*(X) = 0, ux = 0 > 0, u = sJ(X) (3.7.19) 62 if -X B (X) <0, u = Sg(X) =0, u = 0 (3.7.20) >0, u = S2( Let these control laws be expressed by the following expressions: u = S1 sgn (-XTÂ§1) (3.7.21) and where and i = 1 or 2 U2 = S2 Sgn (^B2) sgn (x) = sign of x = S1 when (-XTB.) < 0 1 -1 = S? when (-X^B.) > 0 i -x (3.7.22) (3.7.23) (3.7.24) With these expressions for the control laws, the state equations (3.5.1) become X = A(X) + B S1 sgn (-X11^) + Â§2 S2 sgn (-XTB2) (3.7.25) = F(X,X) (cf. Equation (3.7.3a)). In the present quasilinearization algorithm the derivatives of F(X,X) with respect to both X and X in constructing the matrix (Equation (3.7.4)) are needed. One can see that computation of the derivative of F(X,X) with respect to X will occur only as a general ized function with no numerical value for computation in the limit, because X appears only in the argument of a sign function. Instead 63 of proceeding to the limit the following approximation of the bang-bang control was made. For i = 1 and 2 o * u sa u = i i rsJ(X) if AA (-\TB ) < S*(X) AA.(-\TB.) if S2(X) > AA.(-\TB.) Â£ S1(X) 1--1 l- l -l l - S2(X) if AA.(-\TB.) > S2(X) _ 1 1 1 1 (3.7.26) * where AA^ and AA^ are two positive constants. This function of T 12 AA^(-\ B^) is called the saturation function (sat) when and are unity. The change made in the optimal control is shown graphically in o * Figure 9 for u^ and u^, i = 1 or 2. o * The controls u. and u. have been plotted against the function i i T -\ B^ near a switch point in Figures 9a and 9b, respectively. It can * o be seen that the approximation u^ differs from the optimal control u^ * only in the portion KL. By increasing the value of AA^, u^ can be made o closer to u.. i With the above approximation of the bang-bang control by a "saturation" control represented by Equation (3.7.26), one would * be able to find the derivative of the control u^ (and hence, of F) with respect to X. The derivative will be zero when the control is on a constraint (X) and will be nonzero on the arc KL which appears near a switching time. In order to represent the saturation control (3.7.26) by its linearization, it is necessary that at least one of the W , i = l,2,...,m, which contains the information of the first derivatives, be computed on the arc KL. Otherwise, this vital portion of the control would go unaccounted for in the linearized equations. This may happen (b) Approximation of Optimal Control (Saturation) Figure 9. Approximation of Bang-Bang Control by Saturation Control. o 65 if the arc KL is too steep for a given selection of integration stations. In such a case, when none of the W is computed on the switching por tions of the control variables like the arc KL,the TPBVP, Equations (3.7.5) and (3.7.6),cannot be solved as explained below. First, from a physical reasoning, it can be seen that the linear ized state equations would get decoupled from the adjoint equations if all of the W^, i = l,2,...,m, show zero derivatives of F with respect to \. This means that the first six equations of (3.7.5) would get decoupled from the last six. But the boundary conditions (3.7.6) is such that they are for the first six variables of e(t) only. Therefore, this results in a situation where there are six first-order equations with twelve boundary conditions, a situation which, in general, does not have a solution. From the point of view of computations with the present algorithm it can be shown that the matrix C in equation (3.7.17) cannot be used to solve for e -m+1 With the system equations defined as (Equations (3.7.3a) and (3.7.3b)) X = F(X,X) X = G(X,X) one obtains [W], = th where the subscript i represents the i interval. 66 Let P., Q., E., DD., R. and M., j J J 6x6 matrices. Let [W], = P P " 1 2 P P 3 4J. where P = F, = [0] for all points other than those lying on the 2 L 4J arcs such as KL in Figure 9b. Then 1 + II V'1 1 + f hi P1 I hi P2 2 hi P3 I + 77 h. P. 2 l 4 \ Q2 ^3 Q4 It may be seen that Q = [0] if P = [0]. Z Z From Equation (3.7.13), Di = [" + \ hi 'l]'1 [-; + 5 hi wi] r9! q2 rR. R7 X E_1 i 2 1 2 R E . J L 3 4J L 3 4J (say), where R1 R2 R3 R4 r-i + \ h.w. 2xi and R2 = 2 hi p2 = P9 = [0]. [0] if 67 It may again be seen that E9 = QlR9 + Q2R4 = [0-* since = Q2 = [0] when P2 = [0], Now, consider the product D. D. D. of any three succes- i l+l i+2 m sive D in the product T D.. Let this multiplication be expressed as 1 i=l 1 E11 E21 E31 E41 E12 E22 E32 E42 E13 E23 E33 E43 DD1 DD2 DD3 DD4 The expression for DD2 is DD2 = E11 E12 E23 + E21 E32 E23 + E11 E22 E43 + E21 E42 E43 If the upper right elements E ^ = E?2 = E^ = [0], it may clearly be seen that DD2 = [0]. In the product m TT D. = i=l 1 I M3 M4 one would therefore obtain M = [0] if P = [0] for all i, i = l,2,...,m. With M_ = [0], the expression for C (Equation (3.7.18)) D = (-l)m B. Tr D. + B H l r 1=1 m becomes, after using the values of B. and B , At ~!C 68 0 0 , B = I (f r I 0 0 0 "i 0 i 0 m , x ni c = (-1) = (-D _L M2_ 1 o j-1 Thus, C becomes singular if F,. = 0 for all i, i = l,2,...,m. A For a given step size or subdivision in the t-interval (0,t^) the singularity of C sets an upper limit on the steepness of the arc KL of Figure 9b whichcan be used for approximating the bang-bang control. This steepness of the arc KL depends on AA^ (or AA^) for a given X(t) and X(t). So, in selecting AA^ and AA we have to make sure that they are not too large. When the solution is obtained, however, the slope of KL does not depend on the choice of AA and AA if we decide to choose 1 z AA1 = AA = AA, say. This is because AA will be "absorbed" within X(t) , acting as a scale factor on X. In fact, if we define X as AA*X, the state and adjoint equations of our system do not change, and we could therefore select AA^ = AA^ = 1. The slope of the arc KL in the solution depends on the value of AA*X and not on AA alone. However, when the solution has not yet been obtained, the best value of AA need not be 1. If the final time, t^, selected for our problem is much larger than the minimum time, for a bang-bang optimal solution, it can be expected that the arc KL will have a relatively small slope. If the final time is gradually reduced, this slope will increase until it becomes so large that the matrix C becomes singular as explained above. At that point the solution obtained for the smallest t^ will very 69 nearly be a bang-bang control and will represent the approximate solu tion of the minimum time problem. The computation of the optimal final time via the quasilinear ization method was done according to the stopping condition outlined above. A final time would be guessed, and the TPBVP (Equations (3.7.1) and (3.7.2)) would be solved. The final time would then be reduced by reducing the integration step size, and the TPBVP would again be solved The process would be continued until the matrix C becomes singular and the TPBVP could not be solved any further. 3.7.3. A Simple Example Problem for the Method of Quasilinearization A simple problem was first taken up to explore the various features of the quasilinearization algorithm constructed above. The system was defined as X X. 2 1 X u 2 (3.7.27) (3.7.28) The constraints on u were -1 Â£ u Â£ 1 (3.7.29) The cost function to be minimized is the final time t^. 70 The adjoint equations for the system are ^ = 0 L = -x. (3.7.30) The optimal control u is u = sgn (-X ) (3.7.31) With the approximation of the optimal control u = sat (-AA\ ) Ct (3.7.32) the state and adjoint equations become X. = X x = sat (-AA X2) X, 0 (3.7.33) X0 = -X. 1 The boundary conditions are given by Equations (3.7.28). The analytical solution of the optimal control is given by t f 2 X 1 -1 X 2 t 1 u = 1 Xx = | t2 ) for 0 S t < 1 *\ u = -1 X^ = -1 + 2t t 2 > for l 2 J The solution is shown graphically in Figure 10. The problem was solved numerically by solving Equations (3.7.33) and (3.7.28) for difieren' final times t by quasilinearization. (C) (d) 1.0 0 -1.0 (e) Curve 1 - X Curve 2 X Curve 3 -* Curve 4 - \ Curve 5 - u 71 Figure 10. Graphs of Optimal and Nearly Optimal Solutions Obtained via Quasilinearization for Simple Example. 72 The solutions for t = 2.25, 2.05, and 2.005 were obtained and are shown in Figures 10a, b, and c, respectively. With t = 2.00 the matrix C became singular in iteration 10. The theoretical solution (t^ = 2.00) is shown in Figure lOd. The results for t =2.005 is a reasonably good approximation of the optimal solution. For these problems, 25 time subdivision intervals were used . The initial guess was deliberately poor, as shown in Figure lOe. The solutions were obtained in 9 to 11 iterations from this guess in all these cases. The program originally written, according to Sylvester and Meyer [25] did not have the provision for the amount of adjustment T) in the iterations. Their method was found to be unstable for some of the initial guesses. 3.7.4. The Results With the Kip-Up Problem The kip-up problem was taken up after the method of quasilinear ization was found successful in the case of the example problem. In the kip-up problem, however, many difficulties were faced from the very beginning and the problem could not finally be solved by this method. A major difference between the human motion problem and the example problem or the problem solved by Boykin and Sierakowski [26] is very prominent. In the latter problems, the control variables switched only once from one boundary to another in the entire trajectory, whereas, in the human motion problem, there were many such switchings. This made the human motion problem less amenable to iterative methods. 73 The program for the human motion problem was extremely lengthy, taking more than eleven hundred statements and requiring the use of the large core of the computer (IBM, 360-65). The subroutine (DIFEQ) which generated the right side of the state and the adjoint equations and their derivatives, turned out to be quite lengthy and required an exorbitant amount of computing time. The quasilinearization program called this subroutine at every station of the total interval and at every iteration. As a result, the program required a tremendous amount of computing timeabout 40 seconds per iteration. All computations were done in double precision. Several sources of difficulty were detected in the unsuccessful attempts to solve the human motion problem by quasilinearization. The central issue was the tremendous amount of computations with accuracy required. The total time interval was divided into 100 equal parts (seg ments). As the first (and the only one) guess of the final time, these segments were chosen to be 0.0216 second each. This made the final time (2.16 seconds) equal to the time taken by the gymnast to do the actual maneuver. With 100 segments, the program required the large core of the computer. Even though a larger number of smaller segments would have been preferred, this could not be done because it was intended not to go beyond the large core. It was later found that numerical integration of the nonlinear state equations needed time steps at least as small as one-sixth of what was taken for quasilinear ization. It was initially hoped that, since the quasilinearization algorithm solves the linearized equations instead of the nonlinear 74 equations, the central difference solution would be stable for larger integration step sizes. This was not the case. It was proved theoretically that with perfect precision the method was convergent for all initial guesses. However, it is well known for the unmodified method of quasilinearization (7] = 1) that the method is convergent only for certain initial guesses. It may therefore be fair to expect that even with the modified method, the rate of con vergence should depend on the initial guess and for some initial guesses, this convergence may be extremely poor. For this reason, several initial guesses were tried. The guess of the state variables was taken from the experimental data which were shown to agree .veil with the computed motion in Chapter 2. Different initial guesses for the adjoint variables were tried. The first attempts simply used constant values for all the adjoint variables. Convergence from this guess was nonexistent. Next, attempts were made to generate the adjoint variables from forward inte gration of the adjoint equation with the guessed values of the state variables and a guessed value of adjoint variables at time t = 0. In these cases the integrations were unstable with large numbers generated. The method was not pursued further. Lastly, the method suggested by Miele, Iyer and Well [31] was tried for generating the initial guess of the adjoint variables. In this method, an auxiliary optimization problem is constructed from the original problem. It tries to make an optimal choice of the adjoint variables such that the cumulative norm of the error in satisfying the adjoint equations for a given state variable trajectory is minimized. This is performed as follows. 75 Suppose u (X,X) is the optimal control. The state and adjoint equations may then be written as X = f (X,u (X, .)) = F(X,X) X = -f^X,u*(X,X))X = G(X,\) (3.7.34) (3.7.35) Suppose we have a guess of the state and the adjoint variables given by X(t) and X(t). Since X(t) and X(t) do not satisfy Equations (3.7.34) and (3.7.35), let e = X + f,Tx>u*(X,X))X so that e is the error in the adjoint equations (3.7.35). The above equation can be rewritten as T X = f f,x(X,uX(X,X))X (3.7.36) Now, consider the optimal control problem, where a. X(t) is a given function of time t b. X(t) is the state variable c. e is the control variable d. The system equation is given by Equation (3.7.36) e. The cost function to be minimized is 1 r> T j = I e e dt 2 2 J - o f. t^ is the final time (fixed) of the original problem g. Boundary conditions are X(0) and X(t ) are free. 76 For the optimal control problem posed above, we can construct T the Hamiltonian, using the Lagrange multipliers (6-dimensional vector): T h2 = \ It + Â£*(Â£ ~ Â£,x X) (3.7.37) The necessary conditions for optimization are T h9 ox 2,e " and e* = (H > 2 X or, after performing the differentiations H, = ^ + ?- 0T 2,f -* - and e = f, e -* X -* (3.7.38) (3.7.39) The boundary conditions are f*(0) = f*(V = 2 (3.7.40) because X(t) is free at t=0 and at t = tf. Using Equation (3.7.38) in (3.7.39) and (3.7.40), one obtains e = f, e (3.7.41) ~ ~ A. and e(0) = e(t ) 0 (3.7.42) Clearly, Equations (3.7.36), (3.7.41) and (3.7.42) form a linear TPBVP in X and e. This problem should be easier to solve, in theory, than the original TPBVP of the state and adjoint variables and should give an optimal choice for the multiples of X- 77 In several attempts this linear TPBVP could not be solved for the guessed state trajectory by either the quasilinearization program or by the IBM scientific subroutine package program DLBVP. The prime reason for the failure of the method of quasilineariza tion was finally found to be the numerical inaccuracies in the compu tations which were dominant in spite of using the double precision arithmetic. The problem was perpetuated and amplified by the large numbers in the right side of the state and adjoint equations and in the derivatives of these equations. The matrices to be inverted at the various stages, one at every step of integration and another at the end of the integration, were ill-conditioned for inversion. When a dif ferent subroutine for inversion than what came with the QUASI program was m used, different numbers resulted. The entries of the matrices T and Tf D. i=l i were very large and resulted in large numbers for some entries of C. This made the matrix C ill conditioned for inversion. Any error in inverting the matrix C would be amplified in the values of e This amplification was due to the multiplication of the inverse of C by T, a matrix used in generating s If e was in error, all other - -mfl -m+1 would be in error because they were generated from If the inversion of C was accurate, then the first six entires of computed from Equations (3.7.16,17,18) should be almost zero. But, 3 instead, large values of the order of 10 were obtained! This obviously indicated that and hence all the were being computed inaccurately. The quasilinearization program failed to solve the human motion problem due to the above three reasons, and primarily due to the last one, the numerical inaccuracies. This was felt to be rather difficult 78 to overcome since it was intimately related to the method used to solve the linear TPBVP and the structure of the original TPBVP. So far as the method was concerned, the key point was that the matrix C was becoming ill conditioned. This occurred because the recursive rela tionship between and ^ has been used to generate a relationship m between e and s which resulted in the product T D. with large I IIH-1 jL=l ^ ra entries. A look at the expression for C in terms of T D., B., i=l 1 i and B (Equation (3.7.8)) would make it clear that with such a value r m of T D C would automatically be ill conditioned. i=l 1 The other standard methods of solving linear TPBVP (Equations (3.7.5), (3.7.5)), for example, the transition matrix algorithm might have been numerically more stable. However, other difficulties arose because these methods require several forward integrations in one iteration. This means calling the subroutine DIFEQ (the subroutine to generate the right side of the differential equations and its derivatives) many more times. This increases the computing time enormously. With a step size small enough for the integration to be stable, the storage requirements, computing time, and, therefore, the cost of computing increases considerably. Even if these factors are absorbed, it may still be necessary to try several initial guesses of the adjoint variables to get the method to converge. In view of the above problems, it was concluded that the standard method of quasilinearization was not suitable for the human motion problem and so should not be pursued further. 79 , 3.8. Steepest Descent Methods for Solving the Minimum-Time Kip-Up Problem Three different formulations of first-order steepest descent methods were used after the method of quasilinearization was unsuccess ful in solving the minimum-time kip-up problem. These formulations differ from each other in the construction of the cost functional, handling of the terminal constraints, treatment of the control incre ments and in the method of adjusting the final time. The basic features of these three formulations are described below. Formulation 1 a. The cost functional is the final time. The terminal errors and the cost functional are reduced simultaneously. b. The adjustment of the final time is done by extending or truncating the final end of the trajectories. c. The control functions take the form of a sequence of constrained and unconstrained arcs. Improvements are made at the uncon strained parts only, including the junctions of the constrained and unconstrained arcs. The method is based on the works of Bryson and Denham [27,28] and Bryson and Ho [32]. Formulation 2 a. The cost functional is the sum of a scalar representing the final time and a norm of the terminal error. b. A change in the independent variable t is introduced by defin ing the transformation 80 t = a t where a is a constant, or do = 0 dT so that a is treated as an additional state variable. The final time t is directly proportional to a when is held fixed. Long [29] used this transformation of the independent variable to convert free end point TPBVP to fixed end point TPBVP for solution by the method of quasilinearization. The cost functional is reconstructed as 2 6 2 J = K O' + E K.S., K ,K. ,. . ,K > 0. o.^xi o i 6 1=1 No terminal constraints were introduced in this formulation since they (the Â§/s) are included in the cost functional, c. The control functions are assumed to be free to change in any direction while computing the gradient of the cost function. That is, the control constraints were not considered when com puting the gradient of the cost functional used to find a suit able increment in the control and o'. The control constraints were imposed in the next iteration during forward integration of the state equations. When the computed control violated a constraint in a subarc it was set equal to its limit, the constraint function on the subarc. This approach for treating constraints on the control has been used by Wong, Dressier and Luenburger [33]. 81 Formulation 3 This formulation consisted of the features (a) and (b) of Formulation 2 and (c) of Formulation 1. The derivations of the numerical algorithms for these formula tions are now presented. The basic concepts on which these algorithms are based are available in the literature [28,32]. The results are derived in a manner suitable for analysis of the outcome of the numer ical computations, and, thus, the derivations presented here are slightly different from those found in the literature for any gradient method approach to the computation of optimally controlled motion. 3.8.1. Derivations for Formulation 1 Suppose we have a continuous nominal control u^(t) and u^it) and a nominal final time t These control histories have some parts lying on the constraints S^(X), i = 1 or 2, and the remaining parts lie away from the constraints. The parts lying on constraints will be called "constrained arcs" and the parts lying off the constraints will be called "unconstrained arcs." The interesections of constrained and unconstrained arcs will be called "corner points." A nominal guess for the control variables consists of specifying the corner points and values of the control at unconstrained portions. On the constrained arcs control variables are generated from constraint functions. An initial choice of the control history and the final time will not, in general, satisfy the boundary condition and will not do so in minimum time either. One can improve upon the trajectories in the following way. At first, we establish how a particular state variable X1 (the th i component of the state vector X) changes at the final time due to 82 a small change in the control history and the final time. For that purpose, let a cost functional be defined J = X1(tf) (3.8.1) Let \1(t) be an arbitrary time-varying vector of dimension six. Since the system satisfies Equation (3.1.1), the final value of the state variable will also be given by *f T j' = XX(t ) + f X1 (f-X) dt (3.8.2) f do - - If the control variables u (t) and u (t) change by a small X j amount 6u (t) and fiu (t) there will also be a small change in the X i state variable X(t), denoted by fix(t), throughout the trajectory. It is clear that these changes in the control, denoted by fiu(t) and in the state variables denoted by fix(t), will not be independent of each other. Apart from changing the control history, an increment to the final time t by a small amount 6t^ and small increment fiX(O) to the initial state vector are also prescribed. The first-order change in J due to the changes in the control and the final time is given by Aj' = X1(tf) dtf + {Sx1 (\X)T 6x}t + (XX)T fiX(O) tf tf + f {aV f, + (x1)} fix dt + f qV f, fiu dt. (3.8.3) The fiu is chosen in the following way: For the unconstrained parts, fiu is completely free. The parts presently on the constraints will remain on the constraints for the same periods of time as before. 83 For these portions, the change in control is given by the shift of the constraint due to state changes according to the relation 6u (t) = S3 $X(t) (3.8.4) 1 " Let c} denote all those portions of the trajectory of the control u^(t), i = 1 or 2, which lie on any of the constraints s{ or 2 o S.. Also, let C. denote all those portions of the control u. which l i i do not lie on a constraint. If the expressions for on the con strained arcs given by Equations (3.8.4) are substituted in Equa tion (3.8.3) and the integration of the last term of the right side of o 1 Equation (3.8.2) is split into integrations over the intervals , o 1 , . C and C one obtains fij' = xx(t )dt + {ax1 (xV ax} + {(xV fix} 1 1 Z~Vf t=o tf + Jo {(^1)T i'X+ ('X)} dt + f (\X)T f,u dt + f (xV #, S3 6X dt o 1 j 1 X (X1)T f, u dt + f (xV f, sJ 6x dt " u2 2 1 2 X If Xx(t) is computed such that Xj(tf) = 1 for i = j = 0 for i 4 j where x{ = j element of X* and ; i X l X 1 (3.8.5) (3.8.6) (3.8.7) 84 where 6=0 on C?, 6-1 on 1 11 1 62 = 0 " 2' *2 = 1 n C2 and u1 and u used in Equation (3.8.7) are computed only when x,x 2X the controls u^ and u^ are on a constraint and u^ and u^ are replaced by the constraint expressions S^(X), one obtains 6j' = X1(t.) dt + (X1(0))T fix(0) + f (X1) f, 6U dt f f - c ux 1 + Â£'U S dt = f1 (X (t_),u(t.))dt. + aX(0))T flx(0) + f (X1)1 f, flu. - I I I o - u. 1 i.T dt + lf0(X)T f,u 6u2 dt (3.8.8) C2 2 where fi is the i^h element of the vector f(X,u). Equation (3.8.8) is the desired expression for the change of the state variable X1 at the final time t^ due to (1) a small arbitrary increment Â§u(t) given to the control variable u(t) over the unconstrained portions, (2) a small increase dt of the final time t^, and (3) an arbitrary small change 6x(0) of the initial state vector X . Similarly, one can find the expressions for the change in the terminal value of any other state variable It can be seen that if one constructs the (6 x6) matrix 85 R(t) -[V (t), 2 X (t), X3(t) 4 X (t), x5(t), x6(t) ] so that R(t) satisfies and R(t^) = I (6x6 unit matrix) R(t) = 6 f l-u. R(t) (3.8.9) (3.8.10) where the meaning of the various terms in the parentheses of right side of Equation (3.8.10) is the same as that in the same terms appear- in Equation (3.8.7), then the change in the terminal value of the state vector is given by Â§X(t )=f dt + RT(0)SX(0) f RT f, Su dt + f RT f, Â§un dt . f f ho u 1 u_o u 2 (3.8.11a) If we choose that 6x(0) =0, we obtain Sx(tf) f dt + f RT f, 5u dt + f RT f, 6u dt. - f J o -u 1 J o -u 2 C1 1 C2 2 (3.8.11b) Following the method prescribed by Bryson and Ho [32], it will now be attempted to make improvements in the terminal errors given by Â§ = X(tf) Xf and at the same time reduce the final time t^. Thus, since it is being sought to minimize t^, one would maximize -dt^, or minimize dt^ with respect to 6u_^, 6u^ subject to the constraint 6x(t ) = f(t )dt + f RT f, 6u dt + f RT f, Â§u dt -f f f v o u 1 J o u 2 C, 1 C 2 (3.8.12) 86 with 6x(t ) = 6xp(t ) - f f where OX (t^,) is a chosen decrement in the terminal error such that Â§u maintains.the first-order approximations. In this incremental minimization problem, the incremental cost functional, dt^, to be minimized, and the constraints are linear in the incremental control parameter Â§u. Such a problem does not have an extremum. However, since these are linearized relations obtained from a nonlinear system, the increments 6u Â§u and dt should be small for 1 ^ X the first-order approximation to be valid. To limit the increments 6u 6u and dt the following quadratic penalty term to the incre- X Zj X mental cost dt^ is added: I b dtf + \ So 6U1 Wl(t) dt + I Jo 6US W2(t) dt (3.8.13) where b is a positive scalar quantity and W (t) and W (t) are positive X -J scalar quantities specified as functions of time. Adding these quantities to the cost functional and adjoining the constraint relations (3.8.12) to the resulting expression by a multiplier v (a six-dimensional vector), the following problem is obtained. Minimize wrt, &u $u and X Z X dt + f 2 b dt^ + i f 6u7 W dt + i- f du^ W0dt + vT^f (t^)dt^ 'f+2Jo VU1 Tl+2Jo 2 V C1 C2 + f Rrf, 6u dt+ f RTf, *u dt 6XP(tJ C U1 1 C U2 2 f } (3.8.14) 87 If the derivatives of this functional with respect to 6u^, &u9, and dt^ are set to zero, one obtains dt = ^ [1 +vT f(t )] , f b f (3.8.15) and s = 6u = 2 W. iT *2 2 Rv on C. Rv on CL P A (3.8.16) for an extremal. Using Equations (3.8.15) and (3.8.16) in (3.8.12), one obtains Â§XP = i fil + vTf3 -IMv (3.8.17) b - M - f or where v = - 1 T I. +- f f (tj \jfiji b - f n T -IT n T -IT IM= R f, V,' f, Rdt+J R f, Wo R dt. M 'io u 1 -u o -u 2 u -i r 6*P (3.8.18) C1 1 1 C2 2 "2 (3.8.19) The value of 6xJ (t^) the desired change in the terminal values of the state variables, may be chosen as a decrement in the terminal error S. >XP(t ) = e. Â§. if li (3.8.20) where 0 < e. Â£ 1 i . th 6xP(t ) is the iL11 element of ^XP. if and 88 The steepest descent algorithm for Formulation 1 can now be described as follows: 1. Guess a nominal control history u (t) and u (t) and a final time t^. 2. Integrate the state equations forward with the nominal control and the nominal final time with the initial values of the state vector given by X(0) = X Store X (t). Compute and store - -o - Find the norm INI 3. 4. 6. 7. for some positive i = 1,...,6 (to be specified). Save the controls u^ and u^ in another variable COLD, the corner times N in the variable NOLD and the final time t^ in TFOLD. Set R(t^) = I, the (6 X6) unit matrix. Integrate backward T Equations (3.8.10). At the same time compute and store R f,^ T ^ and R f, .It is not necessary to store R(t). - u 2 Select the positive constant b and the positive quantities W (t) and W (t). Unless there is some special reason, W (t) i. Z JL and W (t) may be taken as a positive constant equal to W (to be z specified). In such a case, the storage for W^(t) and W (t) will not be needed. Select e^, i = 1,2,...,6 such that o < e. ^ l. Compute I^ using Equation Compute v, Â§u^, Su^, anc* dt and (3.8.16). Note that Â§u unconstrained parts C and (3.8.19). from Equations (3.8.18), (3.8.15), and u^ are defined for the C only, and are to be computed only on those parts. 89 Change the final time t^ to t + dt^. If the integration step size is h, this means that the total interval is to be increased or decreased at the final end (depending upon whether dt^ is positive or negative) by dt^/h, rounded to the nearest integer number of integra tion steps. In the case of a positive dt^, the controls u^(t) and u (t) in the interval TFOLD ^ t ^ t (new) will be given by extrapola- tion of the existing curves. If a control function is on a constraint at the final time, TFOLD, it should remain on the same constraint in this period and is to be generated during the forward integration in the next steps. 8. Set the control u. = UOLD. + Su., i = 1 or 2, for the uncon- l li strained portions. Integrate the state equations forward with the new control. The corner times N are to be generated again during the forward integration as described graphically in Figure 11. Figure 11 shows the different cases which may arise during the forward integration. It is seen that a new corner time is generated at the point where the unconstrained u^ + 6u^ curve, extrapolated if neces sary, meets the constraint. This part of the forward integration makes the programming complicated with many logical program statements. 9. Find the errors in the boundary conditions Â§ and find the norm j| Â§^jj as in step 2. If this norm is less than that of the previous iteration go to step 3 to continue the iterations. If it is not reduced, then do the following: a. If dt^ is too large, increase b, and go to step 9c below. |

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1974 TO MY FATHER AND MY MOTHER ACKNOWLEDGMENTS It has been a very happy experience to work with Dr. W. H. Boykin, Jr., during my stay at the University of Florida. I am grateful to him for his help and guidance throughout this work, from suggesting the topic of the dissertation to proofreading the manuscript. I wish to express my deepest gratitude to him, both as an efficient and enthusiastic research counselor and as a human being. I wish to thank Dr. T. E. Bullock for the many discussions I had with him about the theory and numerical methods of optimization. These discussions provided me with understanding of many of the concepts that were used in this work. I chose Drs. L. E. Malvern, U. H. Kurzweg, and 0. A. Slotterbeck to be on my supervisory committee as a way of paying tribute to them as excellent teachers. It is a pleasure to thank them. Special gratitude is expressed to Professor Malvern for going through the dissertation thoroughly and making corrections. I am thankful to Drs. T. M. Khalil and R. C. Anderson for examining this dissertation when Dr. 0. A. Slotterbeck left the University. I am thankful to Mr. Tom Boone for his interest in this work and for volunteering his services as the test subject of the experiments. Thanks are also due to the National Science Foundation which provided financial support for most of this work. Finally, it is a pleasure to thank my friend Roy K. Samras for his help in the experiments and Mrs., Edna Larrick for typing the dissertation. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF FIGURES vi NOTATION vi i i ABSTRACT ix CHAPTER 1 - INTRODUCTION 1 1.0. Why and What 1 1.1. Dynamics and Optimization in Human Motion 1 1.2. Previous Work 3 1.3. The Problem Statement 7 1.4. The Kip-Up Maneuver 8 1.5. Present Work â€¢. 8 CHAPTER 2 - EXPERIMENTATION AND CONSTRUCTION OF THE MATHEMATICAL MODEL 10 2.0. Introduction 10 2.1. Mathematical Model of the Kip-Up 10 2.2. The Equations of Motion 13 2.3. The Equations of Motion for the Experiment and the Integration Scheme 21 2.4. Experimental Procedure 23- 2.5. Results and Discussion 26 2.6. Sources of Errors 31 2.6.1. Imperfections in the Model 31 2.6.2. Errors in Filming and Processing the Data . 31 2.6.3. The Integration Scheme 32 CHAPTER 3 - ANALYTIC DETERMINATION OF HIE MINIMUM-TIME KIP-UP STRATEGY 34 3.0. Introduction 34 3.1. Mathematical Formulation of the Kip-Up Problem . . 35 3.2. Bounds on the Controls 36 3.3. Torsional Springs in the Shoulder and Hip Joints 38 3.4. Boundary Conditions 39 iv TABLE OF CONTENTS (Continued) Page CHAPTER 3 - (Continued) 3.5. The Necessary Conditions of Time Optimal Control 40 3.6. The Solution Methods 47 3.7. A Quasilinearization Scheme for Solving the Minimum-Time Problem 51 3.7.1. Derivation of the Modified Quasilinearization Algorithm 52 3.7.2. Approximation of the Optimal Control for the Kip-Up Problem 61 3.7.3. A Simple Example Problem for the Method of Quasilinearization 69 3.7.4. The Results With the Kip-Up Problem ... 72 3.8. Steepest Descent Methods for Solving the Minimum-Time Kip-Up Problem 79 3.8.1. Derivations for Formulation 1 81 3.8.2. Derivations for Formulation 2 . 93 3.8.3. Derivations for Formulation 3 106 3.8.4. The Integration Scheme for the Steepest Descent Methods 108 3.8.5. Initial Guess of the Control Function . . 109 3.9. Results of the Numerical Computations and Comments 110 3.10.Comparison of the Minimum-Time Solution With Experiment 127 CHAPTER 4 - CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK . . 130 4.1. Conclusions 130 4.2. Recommendations for Future Work 134 APPENDIX A - DETERMINATION OF THE INERTIA PARAMETERS OF THE KIP-UP MODEL FROM THE-HANAVAN HD DEL 138 APPENDIX B - AN INVESTIGATION OF A STEEPEST DESCENT SCHEME FOR FINDING OPTIMAL BANG-BANG CONTROL SOLUTION FOR THE KIP-UP PROBLEM 143 LIST OF REFERENCES 149 BIOGRAPHICAL SKETCH 152 v LIST OF FIGURES Figure Page 1. Hanavan's Mathematical Model of a Human Being 12 2. Mathematical Model for Kip-Up 14 3. The Three-Link System 15 4. Sketch of Kip-Up Apparatus Configuration 24 5. Measured and Smoothed Film Data of Angles 9 and i|r for the Kip-Up Motion 27 6. Measured and Computed Values of cp for Swinging Motion ... 29 7. Measured and Computed Values of cp for Kip-Up Motion .... 30 8. Unmodified Control Limit Functions 37 9. Approximation of Bang-Bang Control by Saturation Control . 64 10. Graphs of Optimal and Nearly Optimal Solutions Obtained via Quasilinearization for Simple Example 71 11. Modification of Corner Point Between Constrained and Unconstrained Arcs After Changes in the Unconstrained Arcs 90 12. Solution of Example Problem by the Method of Steepest Descent 105 13. Initial Guess for the Control Functions 118 14. A Non-Optimal Control Which Acquires Boundary Conditions . 119 15. Approximate Minimum Time Solution by Formulation 2 of the Method of Steepest Descent 122 16. Approximate Minimum Time Solution by Formulation 3 of the Method of Steepest Descent 124 vi LIST OF FIGURES (Continued) Figure Page 17. Angle Histories for Solution of Figure 16 . 125 18. Difference Between Measured Angle and Mathematical Angle for Human Model Due to Deformation of Torso .... 129 19. Construction of Kip-Up Model from Hanavan Model 139 vii NOTATION Usage Meaning X dx â€”; total derivative of the quantity x with respect to time t. X x is a column vector. T . .T x , (x) Transpose of x; defined only when x is a vector or a matrix. x, y 5x ; partial derivative of x with respect to y. Â«y 3x X, , â€¢*=â– - y oy Partial derivative of the column vector x with respect to the scalar y. The result is a column vector whose i^ component is the partial derivative of the i^a component of x with respect to y- 9x xâ€™yâ€™ Â¥ Partial derivative of the scalar x with respect to the column vector y; also called the gradient of x with respect to y. It is a row vector, whose i^h element is the partial deriva- 9x tive of x with respect to the i^-h element of y. -â€™yâ€™ 3y Partial derivative of a column vector x by a column vector y. The result is a matrix whose (i,j)th element is the partial derivative of the ith element of x with respect to the element of y. ii ill T n 2 Norm of the column vector x. Defined as either x x or Z K.x. i=l 1 1 (to be specified which one) where x is a vector of dimension n K. are given positive numbers, x^ is the i^h element of x. VIH Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMICS AND OPTIMIZATION OF A HUMAN MOTION PROBLEM By Tushar Kanti Ghosh March, 1974 Chairman: Dr. William H. Boykin, Jr. Major Department: Engineering Science, Mechanics, and Aerospace Engineering Questions about applicability of analytical mechanics and usefulness of optimal control theory in determining optimal human motions arise quite naturally, and especially, in the context of man's increased activities in outer space and under water. So far very little work has been done to answer these questions. In this dissertaÂ¬ tion investigations to answer these questions are presented. A particular gymnastic maneuver, namely, the kip-up maneuver is examined experimentally and theoretically. A mathematical model for a human performer is constructed for this maneuver from the best perÂ¬ sonalized inertia and joint centers model of a human being available today. Experiments with the human performer and photographic data colÂ¬ lection methods are discussed. Comparisons of the observed motion with solutions of the mathematical model equations are presented. DiscrepanÂ¬ cies between the actual motion and the computed motion are explained in terms of principles of mechanics and errors in measurements. Some changes in the model are suggested. IX An approximate analytic solution of the kip-up maneuver performed in minimum time is obtained for the model via optimal control theory. Several numerical methods are used to determine the solution, which is compared with the observed performance of the human subject. DifficulÂ¬ ties in solving human motion problems by existing numerical algorithms are discussed in terms of fundamental sources of these difficulties. Finally, recommendations for immediateâ€˜future work have been made. x CHAPTER 1 INTRODUCTION 1.0. Why and What Man's increased interest in the exploration of space and the oceans was an impetus for a better understanding of the mechanics of large motion maneuvers performed by human beings. Experience in space walking and certain athletic events brought out the fact that human intuition does not always give correct answers to questions on human motion. For certain problems the solutions must be found by analytical methods such as methods of analytical mechanics and optimal control. Broadly speaking, this work deals with the application of the principles of mechanics and optimal control theory in the analytical determination of human motion descriptors. 1.1. Dynamics and Optimization in Human Motion Dynamics provides the basic foundation of the analytic problem while optimal control theory completes the formulation of the mathematÂ¬ ical problem and provides means to solve the problem. In any endeavor of analytic determination of a human motion the first steps are construcÂ¬ tion of a workable model having the same dynamics of the motion as that of the human performer, and obtaining the equations of motion for the human model. However, the principles of mechanics alone do not give enough information for analytical determination of a desired maneuver whenever there are sufficient degrees of freedom of movement. W'ithout 1 2 knowing what goes on in the human motor system, optimal control is presently the only known analytical method which can provide the remainÂ¬ ing necessary information. After a workable dynamic model of the human performer has been obtained, the remaining part of the analytic determination of a physical maneuver is a problem of control of a dynamic system where the position vectors and/or orientation of the various elements of the model and their rate of change with respect to time (representing the "states" of the "dynamic system" being considered) is to be determined. The state components change from one set of (initial) values to another set of (final) values at a later time. The "control variables," that is, the independent variables whose suitable choice will bring the change are torques of the voluntary muscle forces at the joints of the various limbs. However, the problem formulation is not mathematically complete with the above statements because there would, in general, be more than one way of transferring a system from one state to another when such transfers are possible. Constraints are required in the complete forÂ¬ mulation. The concept of optimization of a certain physically meaningÂ¬ ful quantity during the maneuver arises naturally at this point. A cost functional to be maximized or minimized gives a basic and needed strucÂ¬ ture to the scheme for exerting the control torques. It may be expected, logically, that, unless given special orders, a human being selects its own performance criterion for optimization while doing any physical activity. Some of the physically meaningful quantities that may be optimized during a physical activity are the total time to perform the activity, and the total energy spent during the activity. 3 1.2. Previous Work Very little work has been reported in the literature so far where the optimization considerations have been used in the study of human motion. Earlier work in the study of the mechanics of motion of living beings was done primarily from the view of grossly explaining certain maneuvers modeling the applicability of principles of rigid body mechanics. Most of this work was done under either free fall or zero gravity conditions. The models were made up of coupled rigid bodies and conservation of angular momentum in the absence of external torques was the most often used principle of mechanics. The righting maneuver of a free falling cat in midair attracted the attention of several authors in the early days of studies of living * objects. Marey's [1] photographs of a falling cat evoked discussions in 1894 in the French AcadÃ©mie des Sciences on whether an initial anguÂ¬ lar velocity was necessary in order to perform the righting maneuver. Guyou [2] modeled the cat by two coupled rigid bodies and explained the phenomenon with the aid of the angular momentum principle with the anguÂ¬ lar momentum of the entire cat identically equal to zero. Later, more photographic studies were made by Magnus [3] and McDonald [4,5,6]. McDonald made an extensive study of the falling cats with a high speed (1500 frames/second) motion picture camera. His description of their motion added many details to previous explanations. McDonald found no Numbers in brackets denote reference numbers listed in the List of References. 4 evidence for the simple motion of Magnus. In addition he studied the eyes and the vestibular organ as motion sensors. Amar [7] made one of the most complete of the early studies of human motor activities in 1914. This study of the relative motion of the head, limbs and major sections of the trunk was made with a view to study the efficiency of human motion in connection with industrial labor. Fischer [8] considered the mechanics of a body made up of n links and obtained equations of motion without introducing coordinates. He made discussions of applications of his theory to models of the human body, but did not give applications for the equations of motion he had obtained. In recent years most of the analytical studies of human motion have been associated with human beings in free fall as applied to astronauts maneuvering in space with and without external devices. McDonald [9] made extensive experimental studies of human motions such as springboard diving and the "cat-drop" maneuver. McCrank and Segar [10] considered the human body to be composed of nine connected parts. They developed a procedure for the numerical solution of their very complex equation of motion. Although some numerical results were preÂ¬ sented, no general conclusions were drawn. The most significant contribution to the application of rational mechanics to problems in the reorientation of a human being without the help of external torques was made by Smith and Kane [11]. Specifically, they considered a man under free fall. In this paper the authors pointed out that the number of the unknown functions exceeded the number of the 5 equations of motions that were obtained for the system and recognized the need for optimization considerations. In order to get the adequate number of equations, they introduced a cost functional to be optimized which consisted of an integral over the total time interval of some suitable functional of the undetermined generalized coordinates. Optimization of this functional became a problem of calculus of variaÂ¬ tions, which yielded the necessary number of additional equations (the Euler-Lagrange equations) to solve the original problem completely. The approach of Smith and Kane suffers from one major drawbackâ€” it ignores the internal forces of the system. The internal forces due to muscle groups at the various joints of the body segments are mostly voluntary, have upper bounds in their magnitudes and are responsible for the partially independent movements of the various limbs. Because they are internal forces, it is possible to eliminate them completely in any equations of motion. These can be obtained, for example, if the entire system is considered as a whole. However, such equations will be limited in number (at most six, three from the consideration of translation and three from the consideration of rotation) and will, in general, be less than the number of the unknown functions. The introÂ¬ duction of a cost functional will yield via the calculus of variations the proper number of equations. However, the maneuver obtained may be beyond the physical ability of the individual. It is therefore essenÂ¬ tial to recognize the role of all the internal voluntary forces that come into play during a physical maneuver. Ayoub [12] considered an optimal performance problem of the human arm transferring a load from one point on a table to another point. 6 The motion considered was planar. Internal forces and constraints on the stresses were considered. A two-link model was considered for the arm and numerical solutions were obtained, using the methods of Linear Programming, Geometric Programming, Dynamic Programming and simulation. The performance criterion was a mathematical expression for the total effort spent during the activity. The motion considered was simple from the point of view of dynamics. Also, it required less than ten points to describe the entire motion. This allowed consideration of many physical constraints but the dynamics of the human body motion conÂ¬ sidered was quite different than when large motion of the various limbs are involved. A list of references of works on human performance from the point of view of Industrial Engineering is given in Ayoub's thesis. The research in the area of free fall problems showed that analytical solutions agreed qualitatively with observations in those cases in which rational mechanics has been applied with care. Examples of such problems are the cat-drop problem and the jack-knife-diver maneuver (Smith and Kane). Attempts at analytical solutions of activÂ¬ ity problems which are not in the free fall category have not been comÂ¬ pletely successful. Such problems are in athletic activities (such as running, part of the pole vault maneuver and part of the high jump maneuver) and in activities associated with working on earth. Simultaneously with the study of the dynamics of motion, several investigations were made for the determination of the inertia parameters of human beings at various configurations. Knowledge of inertia paramÂ¬ eters is essential for performing any dynamic analysis. A list of the 7 research activities in this area is given in References 13, 14 and 15. However, only Hanavan [14,15] has proposed a personalized model of a human being. This inertia model has been used in the present investiÂ¬ gation. 1.3. The Problem Statement The present work belongs to a broader program whose objective is to investigate the basic aspects of the applicability of rational mechanics to the solutions of any human activity problem. The aspects presently being considered are: 1. Construction of an appropriate presonalized model for the individual's maneuver under consideration 2. Formulation of a well-posed mathematical problem for the analytic description of the maneuver 3. Solution of the mathematical problem .by a suitable analytical method 4. Comparison of the analytical solution with an actual motion conducted in an experiment with a human subject 5. Determination of muscle activity and comparison with computed muscle torque histories for the maneuver. If a complete analysis based on the above steps results in the correct motion as compared with the experiment, then the results can be used for training purposes and design of man-machine systems, but, more importantly, the results will establish the applicability of rational mechanics to the solution of problems of human activity. In the present investigation, a particÃºlar gymnastic maneuver, the kip-up, has been selected for analysis as outlined above. The methods 8 developed will, of course, be valid for other maneuvers but the ana- lyti cal model will be personalized to the maneuver and individual. Among all the common physical and athletic activities, the kip-up maneuver was found to be particularly well suited for analysis. The motion involves large motions, and is continuous and smooth. Also, it is planar and needs relatively fewer generalized co-ordinates for its complete description, since a correctly executed kip-up exhibits three "rigidâ€ links. At the same time it is not a trivial problem from the point of view of our basic objective. The physical quantity (the performance criterion) chosen for optimization (minimization, in this case) is the total time to do the maneuver. 1.4. The Kip-Up Maneuver The kip-up maneuver is an exercise that a gymnast performs on a horizontal bar. The gymnast starts from a position hanging vertically down from the horizontal bar and rises to the top of the bar by swinging his arms and legs in a proper sequence. During the maneuver the motion is symmetric and does not involve bending of the elbows and the knees. Normally, the grip on the bar is loose most of the time. For an inexÂ¬ perienced person, the maneuver is not easy to perform. 1.5. Present Work In Chapter 2, a mathematical model for the kip-up motion and results of laboratory experiments to test the accuracy of the model are presented. First, a mathematical model of a professional gymnast for the kip-up motion has been constructed. The dynamic equations of 9 motion for the mathematical model were then obtained. Two sets of equations of motion were derived: one for the purpose of verifying the accuracy of the mathematical model from experiments and one for the purpose of optimization of the kip-up maneuver. Next, the results of laboratory experiments with the gymnast are presented. The gymnast was told to perform symmetric maneuvers on the horizontal bar, including the kip-up. The maneuvers were photographically recorded. Two of the many records were selected, one of a simple swinging motion with relatively small oscillations, and another of his quickest kip-up maneuver. The angle measurements from these film records were then used in the equations of motion to check the accuracy of the mathematical model. An error analysis was then performed to explain the disagreement between the experimental and the computed results. In Chapter 3, an analytic solution of the minimum time kip-up for the mathematical model was obtained by numerical computations. First, the analytical problem of the determination of the kip-up in miniÂ¬ mum time was stated in precise mathematical terms. This involved repreÂ¬ senting the equations of motion in the state variable form, specifying the boundary conditions on the state variables, establishing the bounds on the control variables, and modeling the stiffness of the shoulder and the hip joints at extreme arm and leg movements. A survey of the necessary conditions for time optimality given by the optimal control theory has been presented. Finally, several numerical schemes used for solving the kip-up maneuver problem are presented and the results of the numerical computations are discussed. In Chapter 4, final conclusions of the present investigation and recommendations for future work have been presented. CHAPTER 2 EXPERIMENTATION AND CONSTRUCTION OF THE MATHEMATICAL MODEL 2.0. Introduction In this chapter, modeling of a human being for the kip-up maneuver is considered. In Section 2.1, a mathematical model for a professional gymnast is constructed, using the personalized model of Hanavan [14]. In Section 2.2, equations of motion for the mathematical model are derived for the purpose of using them in the analytic deterÂ¬ mination of the subject's optimal kip-up motion. An equivalent system of first-order differential equations are derived in Section 2.3 for testing the inertia properties and structure of the model. In SecÂ¬ tion 2.4, the laboratory experiments performed are described. The results of the experiments are discussed in Section 2.5. An analysis of the comparisons between the observed and computed results are also presented. 2.1. Mathematical Model of the Kip-Up The equations of motion of a deformable body such as the human body are usually partial differential equations. Presently, not enough is known or measurable about the deformation of the human body under voluntary motion to determine a partial differential equation model. Also, such models are quite difficult to handle. However, for situations 10 11 where the deformation is small compared to the displacement of a body, the deformable body may be considered as a rigid body in writing the equations of motion. The rigid link assumption has been used widely in modeling human beings. The personalized model of Hanavan [14] is based on this assumption. The elements of the Hanavan model are shown in Figure 1 and consist of fifteen simple homogeneous geometric solids. This construction allows a large number of degrees of freedom for the model and minimizes the deformation of the elements without undue comÂ¬ plexity. As the number of degrees of freedom increases to the maximum, it is likely that a mathematical model based on the Hanavan model becomes more accurate. However, increases in the degrees of freedom increase the model's complexity and make it difficult to analyze matheÂ¬ matically. Observation of a correctly performed kip-up indicates that the human performer might be modeled quite accurately as a system of only three rigid links. The two arms form one link, the head-neck-torso system forms another, and the two legs form the third link. The shoulder and hip joints may be approximated as smooth hinges where these links are joined together. Deformation of the link consisting of the head, neck, and torso during certain periods of the maneuver is detectable when observed via high speed filming. It was felt that the effect of the deformation of the torso would be no more significant than the standard deviation in the Hanavan inertia parameters because this part of the orientation of the torso is determined by the line joining the shoulder and the hip joint centers. 12 Figure 1 Hanavan's Mathematical Model of a Human Being. 13 The muscle forces acting at the hip and shoulder to cause link motion have been replaced with their rigid body equivalent resultant forces and couples. If the masses of the muscles causing the forces are small compared to the other portions of the links, then the net effects of the forces are the torques of the couples. The resultants do not appear in the equations of motion. The three-link kip-up model is shown in Figure 2. It is conÂ¬ structed with elements of the Hanavan model. Twenty-five anthropometric dimensions of the gymnast were taken and used in a computer program for calculating first the inertia properties of the Hanavan model and then those of the kip-up model. The determination of the inertia properties of the kip-up model from those of the Hanavan model is presented in Appendix A. 2.2. The Equations of Motion The mathematical model for the kip-up motion is a three-link system executing plane motion under gravity. The active forces in the system are the pull of gravity acting on each of the elements of the system and the two muscle torques. The muscle forces at the shoulder acting at the joint between links 1 and 2 are replaced by the torque u^. Likewise, u^ is the torque for the hip joint between links 2 and 3. The system is suspended from a hinge at the upper end of link 1, repreÂ¬ senting the fists gripping the horizontal bar which is free to rotate on its spherical bearings. All joint hinges are assumed to be frictionÂ¬ less. The general three-link system for which the equations of motion will be derived is shown with nomenclature in Figure 3. Equations of motion of the system were obtained via Lagrange's equations as follows: 0 Center of Gravity of an Element Figure 2. Mathematical Model for Kip-Up. 15 3 CGI a 3 â€œl I = length of element 2 = distance between the hinges A and B - distance between the center of gravity of element 1 and the hinge 0 = distance between the center of gravity of element 2 and the hinge A = distance between the center of gravity of element 3 and the hinge B , CG2, CG3 locations of centers of gravity of elements 1, 2, and 3, respectively 1â€™ = angle between O-CGl and OA = angle between A-CG2 and AB , m , m = mass of elements 1, 2, and 3, respectively Z o I^, I = moments of inertia of elements 1, 2, and 3, respectively, about axes perpendicular to the xz plane through their respective centers of gravity I = g moment of inertia of the horizontal bar at the hinge 0 about its longitudinal axis acceleration due to gravity Figure 3. The Three-Link System 16 If we define A = Â§^rl + h + m2r2 +I2+m2*l + I3+ m3r3 + m34 + â€*3*2 + Ir) B = m2Vl C = m3Â¿1f2 D = m3Vl E = m3r3^2 F = ^(m2r2+I2 + m3^2+ln3r3+I3) J â€” mgr3 + 13 M = m^g N = (m2 + m3) Ajg V = m3 V W = ra2r2g R = m3r3g (2.2.1) with Equations (2.2.1), we can express the Lagrangian of the system as: â€¢ 2 "2 L = cp [A + B cos (6+3) + C cos 9 + D cos (6+ijf) + E cos i|f] + 9 [F+ E cos \jf] â€¢ . 1 *2 + 9cp[2F+B cos (6+3) + C cos 6+D cos (6+i|r) + 2E cos i|f] + ^ i|f J â€¢ â€¢ â€¢ â€¢ + 9 i|Ã [J + E cos i|r] + cp \Â¡r [J + D cos (9+iJr) + E cos i|r] + M cos (cp+a) + N cos cp + V cos (cp+9) + W cos (cp+9+3) + R cos (cp+6+i)i). (2.2.2) For the Hanavan model of the angles 9 and i)r respectively, we can write the equations of motion as: 17 d Bl _ Bl _d_ BL dt Be 3l ^0 = u. d Bl 3l dt â€¢ "SIâ€ ~ U2 3^ (2.2.3) (2.2.4) (2.2.5) Let us define = 2(A + (B+C) cos 0 + D cos (9+ij/) + E cos i|r) a = 2F + (BfC) cos 9 + D cos (9+i|Ã) + 2E cos ijt z a^ = J + D cos (9+t) + E cos i)i bl - a2 bâ€ž = 2(F + E cos |) bg = J + E cos i|r c3 = J > â€¢ â€¢ â€¢ â€¢ d^ = cp9[2(B+C) sin 0 + 2D sin (0+\|/)] + 9\|i[2D sin (9+i|0 + 2E sin \|i] + 0i|r[2D sin (9+i|r) + 2E sin i|r] + 02[(B+C) sin 0 + D sin (0+iji)] â€¢ 2 + i|r [D sin (0+\JÃ) + E sin \|r] - (M+N) sin 9 â€œ (V+W) sin (9+0) - R sin (9+04-j) â€¢ â€¢ â€¢ *2 d = u + i|f[29 + 20 + i|r] E sin i|r â€” 9 [(B+C) sin 0 + D sin (0+t|r)] Z X - (V+W) sin (9+9) - R sin (9+0+ijf) â€¢ 2 â€˜2 d = u -9 [D sin (Â©+ \Â¡r) + E sin i|r] - 9 E sin i|r -209 E sin ljf o z - R sin (9+0+i(f) . (2.2.6) 18 With Equations (2.2.6) we equations of motion ax9 + a29 + a39 4- obtain from Equations bl'9 + c *â™¦ = dx b2Â® + Â°2* = d2 b3Â® + C3* = d3 â€¢ (2.2.2) - (2.2.5) the (2.2.7) (2.2.8) (2.2.9) It will be helpful to express the equations of motion in normal form in formulating optimal control problems for the system. For that purpose we define the state variables as X 1 = cp X4=0 X5=t Xg=i (2.2.10) (2.2.11) (2.2.12) (2.2.13) A 3 (2.2.14) 19 Using these definitions, the equations of motion (2.2.7) - (2.2.9) can then be written in the normal form as To write down the equations of motion in more convenient forms, we shall further define the following quantities: (2.2.15) (2.2.16) (2.2.17) (2.2.18) (2.2.19) 20 Aâ€ž -|T r Ã¼3 l A(X) = |_X2, â€” , x4, â€” , X6, -J B(X) = (_blC3+b3Cl) ("alC3_a3Cl) (_aib3+a3bl) (blC2 _b2Cl) (_aiC2+a2Cl) (aib2â€œa2bl) (2.2.20) J <"blC3+b3Cl) (2.2.21) (_aiC2+ a2Cl} (aib2 â€œ a2bl} (2.22) Using the definitions(2.2.16) - (2.2.22), the equations of motion (2.2.15) may now be expressed by any one of the following equations: ?1 , f. ÃB' = j â€˜â€¢Sj.-a4.tn; - St3-n3; V d- vâ€™ : â€œ?â€¢; t Up to this point, this analysis has been very genÂ¬ era! and three-dimensional witn no referente to a p'.ate- 1 ike two-dimensional problem. Equation {3} witn the aid of (?; car, be applied to any three-dimensional elasticity 22 with 9 expressed in terms of p using Equation (2.3.1). Hamilton's canonic equations are then given by cp = -gâ€” (p,cp,t) (2.3.3) and P = - (P,9,t). (2.3.4) From Equations (2.3.2) - (2.3.4) we obtain . ^ p-0[2F+(B+C)cos 8+D cos (6+1|0+2E cos \|>] - i|r[ J+D cos (9+t|Ã) + E cos \[f] 2[A + (B+C) cos 9 + D cos (0+|) + E cos \|/] (2. 3.5) p = - [ (M+N) sin cp + (V+W) sin (9+0) + R sin (9+0+i|r)]. (2.3.6) If we wish to introduce the effects of the friction at the hinge 0, the equation for p becomes oh P = - TT â€œ F '5p (2.3.7 ) where F^ is the generalized friction torque at the hinge 0. The Integration Scheme The integration scheme integrates Equations (2.3.5) and (2.3.7). The initial condition of 9 is obtained from the measured values of 9. The initial value of p is obtained from the definition of p given in Equation (2.3.1). To compute the initial value of p, 9 has been comÂ¬ puted for this starting point only. The 6 and i|r data are differenti- ated numerically to generate 8 and \|f values at every step. and are computed at every step by using these values. The value of F^ is not known a priori and it requires a separate experiment for its determination. Integrations were done with F^ = 0 and F^= C sgn 9 23 with C determined experimentally. The difference between the two cases was found to be insignificant even with C=1 ft-lb which was well above the possible friction torque at the bearings. 2.4. Experimental Procedure The test rig consisted of a horizontal bar and two motion picture cameras. The horizontal bar was made of a short solid round steel bar 1-3/16 inches in diameter and 58 inches long supported by two very rigid vertical columns through a pair of self-aligning spherical bearings. The bearings allowed free rotation of the bar with the arm of the subject. Â» One movie camera was placed with its line of sight aligning with the horizontal bar and about 30 feet away from the bar as shown in Figure 4. Alignment of the camera's line of sight with the horizontal bar would give the correct vertical projection of the two arms on the film for determining the angle cp directly. The second camera was placed in front of the horizontal bar at the same elevation as the bar. The film taken in this camera showed whether a particular motion was symÂ¬ metric or not about the vertical plane of motion and also, the angle between the two arms, which is required for computing the moment of inertia of the arms. The film speed was determined from the flashes of a strobe light regulated by a square wave generator. Experiments The experiments were done during two separate periods with the same subject, a professional gymnast. An average of 15 experiments of the subject's performance on the bar were recorded on film each day. Figure 4. Sketch of Kip-Up Apparatus Configuration. 25 The subject was told to avoid bending his arms and legs and to maintain symmetric motion. In the early days of experimentation he was told to just swing on the bar by moving his stiff arms and legs relative to the torso. These experiments were done with the idea of obtaining small angle data for verifying the inertia properties of the Hanavan model. The filming of the camera was done at speeds of 32 frames per second and 62.1 frames per second (as determined from stroboscopic measurements). In later experiments the subject was told to perform the maneuver with some specific objectives. He was told to perform what he thought would be the kip-up (1) in minimum time, (2) with minimum expenditure of energy, and (3) putting "least effort." Each of these maneuvers was repeated several times. Between any two subsequent maneuvers, the subject was given adequate rest periods to avoid fatigue. This experÂ¬ iment was conducted with the idea of making optimization studies as well as testing the model. Four white tapes were stuck to the subject, on the sides of his upper and lower arms, sides of his torso, and on the sides of his legs. These tapes were aligned between joint centers as suggested by the Hanavan inertia model. Processing the Data The film speed was measured with the aid of a stroboscope by running the camera with a developed film with the same speed setting. After removing the lens the shutter was exposed to the stroboscope flash. By arresting the shutter in the stroboscope light, the shutter speed was obtained. 26 The films were run on an L-W Photo Optical Data Analyzer. For the purpose of testing the inertia properties, two maneuvers were selected, one from each day's filmings. The films were projected plumb line perpendicularly on paper fixed to a vertical wall. As each frame was projected, the white tapes fixed on the subject, now clearly visible in the image, were marked out on the paper of the pad by means of a pencil and a straight edge. In this way each frame was "transÂ¬ ferred" on separate sheets of paper. The angles 9 and i|r were then measured from these traces. The angle 9 was measured with respect to a vertical reference. The vertical reference was obtained from a sharp window wall in the background. Of the two sets of data processed, one was smoothed before using it in the integration scheme. This was the one filmed at the higher speed for the fast kip-up motion. A plot of the raw data and those after preliminary smoothing for this set are shown in Figure 5. 2.5 Results and Discussion The results of the integration of the equations of motion for two of the data sets analyzed are shown in Figures 6 and 7. Figure 6 shows data for swinging motion, while Figure 7 is for the kip-up. Also, in Figure 7 is given the computed results corresponding to reinitiating the integration program with the measured data. DifferÂ¬ ent starting points of integration were selected to eliminate the errors that were generated before these points. In Figure 6, curve 2 of the computed values for the unsmoothed data agrees well with curve 1 of the measured values for a little more Angles (Degree) for the Kip-Up Motion. (Film Speed 62.1 frames/second.) to The* results obtained so far :re applicable to platas consistir, of any r.vmse" o* layers having differÂ¬ ent properties an; geometry. E jt -'rom here on. the analyÂ¬ sis would depend â€¢ n the type o* pinte considered viz. sinÂ¬ gle layer or multiplayer plates, thin or moderately thick plates, etc., and also on one type of displacement funcÂ¬ tions V ?, r d V,. Ha b i p [41 â€˜ has demonstrated the apolica- j a oility of these results to a single layer plate. In this dissertation, a three-layered plate, popuÂ¬ larly known as a sandwich plate will be considered. San dvr 5 c h J* late A sandwich plate consisting of three layers is shewn in figure (2). Tne face layers are much thinner than the core. All layers are uniformly thick throughout- Th? two faces are of the same thickness t., the core thickÂ¬ ness being t . Toe present analysis is capable of treating mixed boundary value problems. On the upper surface, the line AB separates the two regions and â€žAs,y.( over which displacements and stresses respectively are prescribed. On the lower surface, the line CD separates the correspondÂ¬ ing regions. The line CD is located exactly below the line AD. Also, at any two points locatec or. the upoer and lower surfaces of the plate ar.d havinc tne same x. and x. I L cp (Radian) to CD (Radian) ( 31 2.6. Sources of Errors The following are considered to be the sources of errors responsible for the disagreement between the computed and measured values. 2.6.1. Imperfections in the Model The human being for the motion studies was modeled as a system of rigid bodies. The response of the system to be compared to that of the rigid body model was a single generalized coordinate of the system. The errors in modeling can be lumped into the overlapping categories of (1) definition of the generalized coordinates of the rigid elements of the system, (2) deformations of link lines during motions from prior joint center measurements, and (3) significant variations during motion in the inertia properties of the torso with respect to the fixed coorÂ¬ dinate system. In several experiments the torso deformed with signifÂ¬ icant movement of the shoulder joint centers. In these cases the conÂ¬ stant inertia properties model is obviously incorrect. These variations not only cause inaccuracies in the inertia parameters but also result in errors in the link lines or additional errors in the mass center of element 2. These errors are reflected in errors in the angles 0 and i|r, which in turn cause a time varying "phase shift" in the computed angle cp- 2.6.2. Errors in Filming and Processing the Data The primary source of errors in filming was considered to be caused by inaccuracies in knowing precisely where the link lines were in relation to the film plane. As mentioned in Section 2.4, care was 32 taken to minimize this error. That is, the cameras were aligned with the horizontal bar so that the link line of the arms projected in the plumb line plane of the film was very nearly the correct model referÂ¬ ence line. Since the link lines of the torso and legs were nearly plumb line vertical, no corrections of the image data were required for these links. Filming of static thin rods which were connected to the bar at known angles to each other and the vertical plane produces overall measurement errors of about 1 degree standard deviation. Errors up to 3 degrees can be expected in data from films of the motion experiments. These errors can cause the rates of the angles to be in error by more than 30 percent. This is the primary cause of the error in the amplitude of the angle cp computed from the dynamical equations. (Second derivatives of the data can be in error by more than 100 percent. This ruled out the use of equations such as Euler's or Lagrange's.) 2.6.3. The Integration Scheme The integration scheme uses at any step the measured values of â€¢ â€¢ 0 and ijf and the computed values of 9 and i|f stored previously. Once a difference between the measured (actual) and the calculated values of cp has developed at a time t due to any of the sources of errors discussed above, the system configuration determined by the calculated value of cp and the measured (actual) values of 0 and ljr at the time t will be difÂ¬ ferent from that of the actual system at that time. This will cause the model to have a different response after time t than that of the actual system which has a different relative configuration. This in turn will cause a further deviation between the actual and the calculated motion that follows this instant of time. To reduce this 33 effect of propagation of error via the system equations, several reinitializations of cp and p were done by restarting the integrations at different points. Various order differentiation and integraion schemes were tested with insignificant differences in the results for smoothed data. These were done with data from filming at F=62.1 frames per second with a maximum integration step size of 2/F second. As menÂ¬ tioned previously, the main cause of amplitude error was the errors in the derivatives of the raw, unsmoothed data. The results obtained from the experiments show that the model for the kip-up motion constructed from the Hanavan model was reasonÂ¬ ably good considering its kinematical simplicity. In spite of imperÂ¬ fections in the model, its dynamic behavior was quite similar to the actual motion, so that this model could provide reasonable estimates of optimal human performance via the theory of optimal processes and numerical solution methods. However, the results obtained from the experiments indicate that the application of rational mechanics to the analysis and design of man-machine systems could prove inadequate unless the model and the data gathering techniques can be improved. This is especially true in the design of high accuracy or low tolerance systems. CHAPTER 3 ANALYTIC DETERMINATION OF THE MINIMUM-TIME KIP-UP STRATEGY 3.0. Introduction In this chapter the determination of an analytic solution of the kip-up maneuver is presented. The problem of analytical determination of the kip-up strategy in minimum time has been cast as a problem of optimal control of dynamical systems. Before the techniques of the optimal control theory may be applied to the problem, it is necessary to state the physical problem in the language of mathematics and to introduce the physical constraints that must also be considered for the solution. Thus, the first four sections of this chapter have been devoted to the formulation of the mathematical problem. In Section 3.5 a survey of the necessary conditions for optimality obtained from the optimal control theory is presented. Since the problem under considerÂ¬ ation cannot be solved in closed form, numerical methods were used to obtain the solution. In Sections 3.6 - 3.9, the choice of the numerical methods, their derivations and the results of the numerical computations are discussed. In Section 3.10, results of the numerical computations are compared with the actual motion. 34 35 3.1. ^Mathematical Formulation of the Kip-Up Problem The problem is to determine the minimum time strategy for the man model to kip-up without violating control constraints. These conÂ¬ straints represent the maximum torques the man's muscles can exert for any given configuration. Formulated mathematically, we have the followÂ¬ ing: For the system equations X = f(X,u) = A(X) + B(X)u (3.1.1) and the boundary conditions X(0) = X â€” â€” o (given) (3.1.2) where and given by Â§(X(tf))â€œ X(tf) - Xf = 0 , xf = given t = final time, to be determined X(t) is the time-dependent state vector, xx(t) = q>(t), Xg(t) =cp(t), X3(t) = 9(t) X4(t) = 0(t), X5(t) = \Kt) , Xg(t) - (t) (3.1.3) (3.1.4) (3.1.5) find a control u(t) = [u1(t),u2(t)] (3.1.6) such that simultaneously Equations (3.1.1) - (3.1.3) are satisfied, t^ is minimized, and for all values of t, 0 S t S t , the inequalities S*(X) * ux(t) * sJ(X) S2(X) S u2(t) S S2(X) (3.1.7) 36 are satisfied. S1(X) are given functions of X and represent the bounds on the control u.(t). The functions f(X,u), A(X) , and B(X) were pre- J viously given in Section 2.2. Â§ is the error in meeting the terminal values of the state variables. 3.2. Bounds on the Controls The control variable u^ is the muscle torque exerted at the shoulder joint and u is that exerted at the hip. For the individual being modeled, the functions u and u will have upper and lower limits which are functions of the state X. Samras [16] experimentally determined the maximum muscle torques at the shoulder and hip joints for various limb angles at the joints. This was done for the same subject modeled in the present study. These measurements were made under static conditions and the mÃ¡ximums in either flexion or extension were measured for the shoulder torque for various values of 9 and the hip torque for various values of i|i. The exÂ¬ perimental bounds on the shoulder torque were then fitted by polynomials in 9. The hip torque bounds were expressed in polynomials in \jf. Even though each of these bounds might be expected to depend to some degree on all four state variables X^, X^, X,., and X^, the bounds on the shoulder torque u depend primarily on X and the bounds on the 1 O hip torque u depend primarily on X . The measurements of Samras do not z o include the rate dependence X and X Although the rate effect appears 4 6 to be measurable, it is a second-order effect and quite difficult to obtain. The control limit functions are given in Figure 8. These funcÂ¬ tions are correct only for a certain range of values of the angles Control Limit (ft-lb) Figure 8. Unmodified Control Limit Functions (Samras [16]). 38 Xâ€ž and X The values of S. and S0 can never be positive and those of 3 5 12 O O and can never be negative. Whenever these sign conditions are violated by extreme values of the states, S'? is set equal to zero. Also, 2 from extrapolated measurement data an upper limit has been set for S at 160.0 ft-lb and a lower limit has been set for at -100.0 ft-lb. 3.3. Torsional Springs in the Shoulder and Hip Joints Our dynamical model and the control limit functions of the shoulder and the hip do not account for the stiffness of the shoulder and the hip joints at the extremities of shoulder and leg movements. It has been observed that the shoulder joints produce a resistance to raising the arm beyond an angle of 8 Â« 30Â°. The hip joints resist moveÂ¬ ment for ijf > 120Â°, or for i|r < -35Â°. The effects of these "stopsâ€ are important and must be included in the model, since the film data showed that these limits were reached. There are no data available for the stiffness of these joint stops. It was observed that, although the joints were not rigid, they were quite stiff. It was therefore decided to use stiff torsional spring models at the model's shoulder and the hip joints. These would be active when the stop angles were exceeded. For the shoulder the spring is active for 9 ^ 0.5 radian. For the hip joint the spring is active for ijf S -0.6 radian and i|r 2: 2.1 radians. The springs have equal stiffnesses. One generates a 100 ft-lb torque at the shoulder for a deflection of 0.1 radian. This corresponds to a joint stop torque of the order of the maximum voluntary torque availÂ¬ able at the shoulder for the deflection of 0.1 radian. This gives a spring constant of Kg = 1000 ft-lb/rad. The spring forces at the shoulders 39 would therefore be equal to -K (9-0.5) for 9 Â£ 0.5 radian and those at s the hip joints would be -K (i|r-2.1) for 4' ^ 2.1 radians and -K (4+ 0.6) s s for i|f ^ -0.6 radian. These torques at the shoulder and the hip joints were added to the voluntary control torques u^ and u^ when the stops were activated. 3.4. Boundary Conditions The boundary conditions for the kip-up maneuver were chosen from the experimental data of Section 2.5. The initial values selected correspond to motion which has already begun. This is beyond the initial unsymmetrical motion which occurs on beginning the first swing. This motion is difficult to model and is not important in this basic research. The final values of the state variables represent the model atop the horizontal bar still moving upward and just before body conÂ¬ tact with the bar. (Once the torso contacts the bar, the model is no longer valid.) The actual motion in the experiment terminated shortly after this point when the gymnast used the impact of the horizontal bar with his body to stop himself. The initial and final values of the state variables for the optimization problem are listed in Table 1. 40 TABLE 1 BOUNDARY CONDITIONS FOR A MINIMUM TIME KIP-UP MOTION State Variables Initial Value Final Value V 1 X = o 0.340 Xf = -2. 84 9 x2 = O -2.30 Xf = -7.05 e x3, O 0.305 Xf = 2.88 Ã© 4 X = o -0.660 XÃ = 0.163 â™¦ x5 = o -0.087 0.436 i x6 = o -1.20 0.108 The Necessary Conditions for Time Optimal Control In this chapter, we look into the necessary conditions for the minimum time problem formulated in the previous chapter. The necessary conditions for optimality of motion for the case when the constraints on the control are not a function of the states are given in Reference 17. For the case where control constraints depend on the states, the necÂ¬ essary condition requires a modification in the adjoint equations. These are obtained through a calculus of variations approach [18]. This approach is used in the following developments. Writing the state equations of our system as X = f(X,u) = A(X) + B(X)u (3.5.1) 41 we can construct the cost function as dt (3.5.2) where t f is free. The Hamiltonian is then given by H(X,u,\) = 1 + \Tf = 1 + \TA(X) + \TB(X)u = 1 + XTA(X) + XTb1(x)u1 + \TB2(X)U2 where X(t) is the time-dependent six-dimensional column vector of adjoint variables. A, B, B1 , B , and f are the quantities as defined by Equations (2.2.15) - (2.2.22) in Section 2.2. The minimum-time control policy uÂ°(t) will be given by the one that minimizes the Hamiltonian (3.5.3), provided no singular arcs are present. We note that in this case the Hamiltonian is a linear funcÂ¬ tion of the control u and therefore the minimum with respect to u occurs only in the upper and the lower bounds of u if there is no singular solution. Thus, we have, recalling the definitions of S^ in Section 3.2, T (1) If \ B^(X) >0, u^ = the minimum allowable value of u^ = S*(X) (3.5.4a) (3.5.3a) (3.5.3b) (3.5.3c) (2) If \TB (X) < 0, U;L (3) If XTB2(X) > 0, u2 the maximum allowable value of u^. SJ(X) (3.5,4b) the minimum allowable value of ur slâ„¢ (3.5.5a) (4) If XTB2(X) < 0, u2 the maximum allowable value of u *s> (3.5.5b) 42 (5) If \TB (X) = 0 T " or X B (X) = 0 J , u^ u^ = possible singular control. (3.5.6) u and u will be determined by investigating whether or not there is -i- Ci a singular solution with respect to these variables. The adjoint equations will be different for the portions of the o trajectories for u corresponding to constrained and unconstrained arcs. The adjoint equations are, in general, given by \T = -H, . (3.5.7) X This yields (a) When neither u nor u lie on a constraint \T = -H, = -\TA, - \TB u - XTB u . (3.5.8) 'X â€œ -â€™X - â€œl,x 1 " "2,x 2 (b) When any one or both u and u , denoted by u. (i = 1 or 2), lie on 1 Z X a constraint denoted by (j = 1 or 2) the right side of Equation (3.5.8) of the adjoint variables has the additional term - \tb. SJ - -i i â€™X and the equation can be written as T_ â€žj X = - X A, - X B, U - x Bn u - Z 5. x B. S. - -'X - -1, 1 - -2,x 2 i=1 . - -1 i, (3.5.9) where 6=0 if X B. = 0 l - -l 6. = 1 if \ B. Â¿ 0. i - -i The boundary conditions on the state and the adjoint variables are 43 X(0) - given = X \(0) = free - -o - X(t ) = given = Xf \(tf) = free and H(tf) = (1 + \Tf)t ^ 0 . (3.5.10) (3.5.11) The state and adjoint equations together with the control laws and the boundary conditions written above form a two-point boundary value problem (TPBVP) in the state and adjoint variables. If these o equations can be solved, the optimal control, u , will be immediately obtained from Equations (3.5.4a) - (3.5.6). Investigation of Singular Solutions T o It has been noted that if \ = 0, u^ cannot be determined from the requirement that the Hamiltonian is to be minimized with o T respect to u . The same is true for u whenâ€¢\ B = 0. Since the treatment for u^ is the'same as for u^, we shall investigate a singular control for only u^. T If the quantity \ B^ = 0 only for a single instant of time, then the situation is not of much concern because the duration of the interval is not finite and we can simply choose u = u^(t ) or u^(t+) or 0, where u^(t ) = control at the instant preceding t, u^(t4) is the instant exactly after t. The situation needs special attention when T \ B^ = 0 for a finite interval of time. If t S t Â£ t is an interval for which uÂ° is singular, it is 1 Z -A- clear that, for our system ^ t S 0 for (3.5.12) 44 and therefore, d T . Ht 5^ = 0 for *1 * t * *2 (3.5.13) or x\ + \TB = 0 for ti - t - t2 (3.5.14) or, for the interval t ^ t ^ t the following results must hold: x Z Case 1 Only u is singular. u is nonsingular. Since u is not 1 ^ Â£ singular, u is on a constraint boundary and is given by u = Sâ€ 2 2 j = 1 for the lower constraint j = 2 for the upper constraint. The adjoint equations are given by Equation (3.5.9) â€¢ T T T T i T i l = -X A - X B1 u - XJ, S, - X B Sâ€œ (3.5.15) X Â§i = B X 'X = Â§1, + ?1 ui + 52 ^ â€¢ X (3.5.16) From Equations (3.5.14), (3.5.15), and (3.5.16), we obtain [-*TÃ,X - v\ S2-bT52S2, jSi+^Si, O + ?2 s2> = Â° â€¢ â€” X XX It is to be observed that the necessary condition (3.5.17) is not explicit in u^. Case 2 (3.5.17) Both u and u are singular. The value of u is no longer T j â€¢ and the term X B S in the X Equation (3.5.15) drops out in this â€œ â€œ2 â€™X case. Accordingly, one obtains T -X [A, Si, Â» - h CS2 5, - 5 b2i u2 = o X XX X (3.5.18) 45 Proceeding from the assumption that u is singular, one would also get, for this case, when both u^ and u^ are singular -*VX - Â§, A] - J[B B2 - B B^ = 0 â€œX XX (3.5.19) From Equations (3.5.17), (3.5.18), and (3.5.19) we can see that T T only if both X and X Bare zero simultaneously, is it possible to find a singular solution by suitable choices of u^ and u^ from the T T t condition (3.5.13). If only one of X B^ and X , say X B^, is zero, d T the requirement (X B,) = 0 does not yield an equation explicit in dt - -1 Ãº T u as observed in Equation (3.5.17). It is thus required that â€”â€” (X B1) 1 dt = 0, which will be explicit in u , during the interval t^ ^ t ^ together with the requirement that the relation (3.5.17) is satisfied T at t=t^. These two conditions will ensure that X B^ = 0 in the interval t^ ^ t Ã¡ t . It is to be noted that singular control computed by the above procedure has not been proved to be the minimizing control. Additional necessary conditions analogous to the convexity conditions for singular controls have been obtained by Tait [19] and Kelley, Kopp and Moyer [20] for scalar control and by Robbins [21] and Goh [22] for vector control. For the general case of vector control these conditions, summarized by Jacobson [23], may be stated as on singular subarcs: 3ÃT - Ldt n H, q u = 0 if q is odd (3.5.20) and (-1)' HIT r d2p â€œ2F hâ€™u Ldt P ^ 0 . (3.5.21) 46 d2p In these equations, â€”*â€” H, (X,\) is the lowest order time derivative dt P - of H, in which the control u appears explicitly, and q < 2p. For a scalar control, Equation (3.5.20) is satisfied idenÂ¬ tically. Equations (3.5.20) and (3.5.21) also do not constitute sufÂ¬ ficiency conditions for minimality. A complete set of sufficiency conditions for singular arcs has not yet been established in the literature of optimal control theory for a general nonlinear system. We can see that there are quite severe restrictions on the existence of singular arcs in the human motion problem. In the numerical methods used in the present work to determine the optimal solution, only in the method of quasilinearization is it necessary to express the control (its optimal value) in terms of the state and adjoint variables, while in the gradient methods where successive improvements are made in the control variables, this is not so. In the attempts with the quasilinearization method, singular solutions were not considered in the construction of the two-point boundary value problem in the state and adjoint variables. It was decided that if a solution to the TPBVP was obtained by quasilinearization, singular arcs would be looked for later. The gradient methods exhibit singular arcs automatically if there are any. The additional necessary condiÂ¬ tions for singular arcs should be checked when off-constraint arcs are exhibited by the gradient method. 47 3.6. _The Solution Methods The optimal control problem formulated in the preceding section cannot be solved in closed form. Numerical methods must therefore be used to find its solution. In the optimal control theory literature several numerical methods have been proposed for solving the differÂ¬ ential equations and the optimality conditions that arise out of optimal control problems such as the present one. None of these methods guaranÂ¬ tees that a solution will be obtained readily, while some of the methods do not guarantee that a solution may be obtained at all. The methods are all iterative, necessitating the use of high-speed computers for all nontrivial problems. A nominal guessed trajectory is improved iteratively until the improved solution satisfactorily meets all the necessary conditions. Depending on whether the method requires finding the first or both first and second derivatives of the system equations with respect to the state and control variables, these methods are called First-Order or Second-Order methods, respectively. This is so because they, in effect, make first-order or second-order approximations of the system equations with respect to the state and control variables. The first- order methods, in general, have the property that they can start from a poor guess and make fast improvements in the beginning. They need fewer computations in each iteration. But their performance is not good near the optimal solution where the convergence rate becomes very poor. The second-order methods, on the other hand, need a good iniÂ¬ tial guess to be able to start but have excellent convergence propÂ¬ erties near the optimal solution. Because the second-order methods 48 need computation of the second derivative of the system equations, they need more computing time per iteration, which may be excessive for some problems. Apart from the first- and second-order methods mentioned above, there is another class of methods which tries to combine the advantages of both of these methods while eliminating the disadvantages of both. The Conjugate Gradient Method, Parallel Tangent Method, and the Davidon- Fletcher-Powell Method fall into this class. These methods work very much like the first-order method except that, in the first-order expansion, the coefficients of the first-order term, or the gradient term, is modified by some transformations. These transformations are generated from the modified gradient term of the previous iteration and the gradient term of the current iteration. This has the effect of using the information that is obtained from a second derivative. It is not known which of the several methods used for solving optimal control problems is good for a given problem and one may have to try more than one method in order to obtain the solution. In the published literature, most of the illustrations of these methods are simple. In these simple problems control or state variable histories do not have wide oscillations or the system equations themselves are not complicated. This makes it truly difficult for someone without previous experience to decide upon the merits of these methods. There is no preference list, and it seems certain that there cannot be one whereby a decision can be made as to which method should be tried first so that a solution of a given problem will be obtained most efficiently. In this respect, deciding upon a computing 49 method for a given problem is still an art and depends largely on the previous experience of the individual trying to solve the problem. In the attempts to solve the minimum time problem, the method of quasilinearization was taken up first. This choice was based on several factors. This is the only method where the two-point boundary value problem obtained from the necessary conditions of optimality is solved directly, and this feature was found very attractive. As a startÂ¬ ing guess, this method requires the time histories of the state and adjoint variables. Time histories of the state variables were available from the experiments. (It was decided that if the method was successÂ¬ ful for this guess, an arbitrary and less accurate initial guess would be tried later.) When it converges, the method has a quadratic converÂ¬ gence rate. Also, in spite of its being a well-known method for solvÂ¬ ing nonlinear two-point boundary value problems since it was first introduced by Bellman and Kalaba [24] , its applications in solving optimal control problems have been very few. There was thus an added incentive for using this methodâ€”to determine its usefulness in solvÂ¬ ing fairly complicated optimal control problems. Sylvester and Meyer [25] proposed, with demonstrations, an efficient scheme for solving a nonlinear TPBVP using the method of quasilinearization. This scheme was available in the IBM SHARE program ABS QUASI and was used by Boykin and Sierakowski [26] , who reported excellent convergence properties of the scheme for some structural optimization problems. With this record of success, the program QUASI was taken up for our problem. But with our problem several difficulties were encountered 50 from the very beginning. First, the bang-bang control law obtained from the necessary conditions had to be replaced by a suitably steep saturation type control law. Second, a slight modification in computaÂ¬ tion scheme was necessary when it was found that the method was unable to solve a simple example problem. The example problem could be solved with these modifications. But, in spite of all these changes and subÂ¬ sequently, many attempts to generate a guess of the adjoint variables, the method could not be made to work for the human motion problem. Reasons for the difficulties encountered are discussed in detail in Section 3.7. During the attempts with quasilinearization, it was found that computations of the second derivatives of the system equations were taking an exorbitant amount of time and this was the deciding factor for the next choice of a computing method. Also, the appearance of the control function linearly in the Hamiltonian put restrictions on the use of most of the other second-order methods. The next attempts were based on the first-order steepest descent method proposed by Bryson and Denham [27,28]. The most attractive feature of this method is that the various steps involved in it render themselves to clear physical understanding. This method directly reduces the cost function in a systematic way and one obtains good insight into the basic steps in the iterative computations and can make adjustments to improve convergence and/or stability with relative ease. These features of the method of steepest descent may more than offset the advantages of other methods for some complicated problems. In the attempts with this method, three different formulations of the minimum 51 time problem were tried. In the first formulation the computations were not pursued beyond a certain point due to computational difficulties. The solution was obtained by the second formulation and verified by the third formulation. These attempts are discussed in Section 3.8. 3.7. A Quasilinearization Scheme for Solving the Minimum-Time Problem In Section 3.5 the adjoint equations and the optimal control laws (Equations (3.5.9), (3.5.4a)-(3.5.6)) have been derived for the minimum time kip-up problem. The system equations and the boundary conditions on the state and the adjoint variables are given by EquaÂ¬ tions (3.5.1) and (3.5.10), respectively. From these equations we can readily see that if the control variables u^ and u^ appearing in the system and adjoint equations are replaced by their optimal expressions in terms of the state and the adjoint variables, one obtains a nonÂ¬ linear TPBVP in the state and adjoint variables. If these equations are solved, the optimal state and adjoint variable trajectories will be obtained and the optimal controls can be constructed by using the state and adjoint variables and the optimal control laws. In the TPBVP in the state and the adjoint variables, the final time is not a given constant and is to be determined from the implicit relation (3.5.11). This makes the problem one with a variable end point. The method of quasilinearization is formulated primarily for a fixed-end-point TPBVP. In problems with variable end points, the adjustment of the final time is usually done by a separate scheme, not integral with the quasilinearization scheme. Long [29] proposed a 52 scheme for converting a variable end point problem into a fixed-endÂ¬ point problem with the adjustment of the final time built into the quasilinearization process. For the present system, however, this scheme was not practicable because the boundary condition (3.5.11) becomes too complicated to handle in this formulation. It was decided that with a separate algorithm for adjusting the final time, described later in this section, the nonlinear TPBVP with free final time would be converted to a sequence of nonlinear TPBVP's with fixed final times. Each of these fixed final time problems would then be solved by the modified quasilinearization algorithm until the correct final time was obtained. The derivation of the modified quasilinearization algorithm is described below. 3.7.1. Derivation of the Modified Quasilinearization Algorithm The fixed final time nonlinear TPBVP to be solved falls in the general class of problems given by dy ^ = g(jr,t) . (3.7.1) With the boundary condition BjÂ¿ Â£(Â°) + Br + c = 0 tf = given (3.7.2) y, g, and c are of dimension n, B^ and B^ are matrices of dimension (nxn). It is being assumed that the TPBVP has been defined for the interval 0 Â£ t ^ t for some given t > 0. In the state and adjoint equations, if the expressions for optimal control in terms of the state and the adjoint variables are used for the control variables, one obtains 53 X = f(X,uÂ°(X,\)) = F(X,\) (say) (3.7.3a) and X = -H^(X,uÂ°(X,\),X) = G(X,X) (say). (3.7.3b) In the formulations of the TPBVP given by Equations (3.7.1) and (3.7.2), it may be seen that for the kip-up system, n=12, X râ€” "n F 0 0 I 0 y = _X , g = 1 IO 1 1 ' BX = i 0 II u CQ 0 0 and -o The 0 and I appearing in the matrices B. and B represent 6x6 order null Xj Y and unit matrices, respectively. Let z(t) be an initial guess vector for y(t) which satisfies the boundary conditions (3.7.2). If g(y,t) is approximated by its Taylor series expansion about g(z,t), keeping only the first-order term, one obtains (y-z). y=5 9g g(y,t) = g(z,t) + Let W = so that, W. . ij (3.7.4) or, W.. = partial derivative of the i ij th . j element of y, evaluated at y = z. . th element of g with respect to the 54 With the above approximation of g(y,t), Equation (3.7.2) becomes dy â€” = g(x,t) + W(z,t)(y-z) at - - - - - or, de dz = - -rr + g(z,t) + W(z,t) e dt dt - - - where, e = y(t) - z(t) = error in the guess z(t). Rearranging the above equation, one obtains de dz â€” - W(z,t)e = - â€” + g(z,t). (3.7.5) dt - - dt - - Since z(t) is chosen to satisfy the boundary conditions, Bj, z(0) + z(tf) + c = 0. Subtracting this equation from Equation (3.7.2), one obtains the boundary conditions on the error e(t) as BÂ¿ e(0) + Br e(tf) = 0. (3.7.6) Equations (3.7.5) and (3.7.6) form a linear TPBVP in e(t) which, when solved, will give the values of the error between the guessed solution z(t) and the actual solution y(t) based on the linearized expressions of the right side of Equations (3.7.1) about the guessed solution z(t). Because of using the linearized equation instead of the full nonlinear equations, the values of e(t) obtained by solving Equations (3.7.5) and (3.7.6) will not be the actual error between the guess z(t) and the solution. However, a new guess of y(t) will be obtained from e(t) by z (t) = z(t) + T) e(t) , 0 < 7] < 1. (3.7.7) 55 The algorithm of Sylvester and Meyer uses T| = 1 for all the time, which is the usual quasilinearization algorithm. It was found, while solving a simple example problem, that without the incorporation of a multiplier 7] in the expression (3.7.7), i.e. , using z/(t) = z(t) + e(t) the method was unstable. The convergence property of the scheme with the incorporation of the multiplier 7] can be understood for small values of 7] by comparison with the step-size adjustment procedure of the usual steepest descent algorithms. The mathematical proof for the convergence property follows the proof of Miele and Iyer [30] and is now given. The integral squared norm of the error in the guessed solution z(t) can be expressed by the integral Similarly, the error in the solution z'(t) = z(t) + 7] e(t) is given by t f If 7] is sufficiently small, one can write g(z',t) = g(z,t) + g,z(z,t) 7] e where g, = g, - z y = W(z,t) (from Equation (3.7.4)). y=z Also, for all values of 7], â€¢ / z = z + 7] e . 56 From these results, one obtains, for small values of 7], j' - j = 27] J (z - g(z,t)}T {e - We} Since e(t) satisfies the differential equation (3.7.5), this finally yields: t â€ž j' - J = - 27] J || z - g(z,t)||2dt = a negative quantity. Thus, for sufficiently small values of 7], the reduction in the cost is assured. In the quasilinearization algorithm z(t) takes the role of z(t) as the new guess of y(t) and the process is continued until the error in satisfying the differential equations is reduced to an acceptable value. The linear TPBVP of the error Equation's (3.7.5) and (3.7.6) is solved as follows: The time interval t = 0 to t = t^ is divided into m small interÂ¬ vals. This results in m+ 1 values of t at which the solution will be computed. Equation (3.7.5) can be written in a finite central differÂ¬ ence scheme as rl /Z.+Z. t.+t. [-1 -1+1 1 1+1] e. + e. z. -z. 1 -i -i+l -i+l -i i /Z . . + z. t. + t.\ f -i+l -i i+l i) hi ' 2 â€™ 2 / 2 " h. +^' i ^ 2 â€™ 2 y th where h = t - t . The subscript i denotes values at the i i i+1 i station, i=1,2,...,nH-l. Rearranging and simplifying the above expresÂ¬ sion one obtains (Ã + i h.w. )e. + (-1 + =â€¢ h.W. )e â€¢ = r. (3.7.8) 2 i i -i 2 i i -l+l -l i = 1,2,... ,m 57 where r. = z -l -l+l Cz. -,+z. t. 1 + t;n /z. , + Z. t. + t . X (3.7.9) (3.7.10) and I = unit matrix of dimension n xn . The boundary conditions, Equation (3.7.2), reduce to B. e + B e = 0 jÂ£ -1 r -m+1 (3.7.11) Equation (3.7.8) can be cast into the following convenient recursive expression e. + D. e. , = s. -l l-i+l -i (3.7.12) where and D. = (I + ih. W.)"1 (-Ã+ i- h. W.) l 2 i l 2 i l - 1 -1 s. = (I + â€” h. W. ) r. -l 2 i i -l (3.7.13) By repeated substitution, equation (3.7.12) yields the following relaÂ¬ tionship between e_ and e -1 -m+1 . m m e, = T + (-1) ( TÃ D.) e -1 - . â€ž i -m+1 l-l where m i-1 i-1 T = s + E (-1) ( TT D.)s. i=2 j=l J 1 (3.7.14) (3.7.15) or, multiplying by on both sides of Equation (3.7.14), B. e = B. T + (-l)m B. TT D. e JL -1 l - l . â€ž i -m+1 1=1 m (3.7.16) Equations (3.6.16) and (3.7.11) can be solved simultaneously for and e , to give -m+1 58 -m+1 where m C = (-l)m B - ( TT D.) + B . I . i x r 1=1 (3.7.17) (3.7.18) With e determined, e , e . ,. . . , e_ are determined in succession by -m+1 -m -m-1 -1 using the recursive relation (3.7.12). Then, the new iterate is given by zih*(t) = zi(t) + T) e(t). The stopping condition of the algorithm is given by the fact that e should be small. When they are small, it may be seen from Equation (3.7.8) that the quantities r will also be small. From Equation (3.7.9) it is also seen that small r correspond to satisfying the central differÂ¬ ence expression of the differential equations. That is, the finite difÂ¬ ference equation error must be small. This does not mean that the difÂ¬ ferential equation error is small unless the intervals for the differences are sufficiently small. The IBM SHARE program ABS QUASI is a program of the procedure outlined above without the provision of the multiplier T) in Equation (3.7.7), and therefore is for 7] = 1. The program was modified to introÂ¬ duce and to adjust 7] to get the desired convergence. The algorithm may be described by the following steps: 1. Set up the matrices B. and B and guess a nominal trajectory Xj r z(t) that satisfies the boundary conditions (3.7.2). Set ITER = 0 (ITER for Iteration) 59 2. Do the following for i = l,2,...,m. a. Find r and W. as defined in Equations (3.7.9) and -i i (3.7.10). Find the largest element of r^, searching between the elements of each r ^, for i = l,2,...,m. Call it E2MAX. If E2MAX < specified maximum allowable error, print out z and stop the computations. b. Using Equation (3.7.13), find and s^. 3. Calculate T and C according to Equations (3.7.15) and (3.7.18). Calculate the integral norm of the error (here the norm is defined by the sum of squares of the elements of the vector r.). Set J1 = J2. -l 4. Find e . using Equation (3.7.17). -m+1 5. Decide upon a value of T], Discussions on the choice of TÂ¡ will be presented in a later section. 6. Generate and store e e _,,e e. using Equation (3.7.12). -m -m-l -m-2 -l Generate the new guess z^, i = 1,2,. . . ,nn-l, by doing z. = z. + T) e.. -i -i -i Set ITER = ITER + 1. 7. Find J2 and r , i = l,2,...,m and find E2MAX as in step 2. 8. If J2 > Jl (unstable), go to step 11. 9. Stop if E2MAX < a prescribed value. 10. If J2 < Jl to to step 12 to continue to the next iteration. 11. If this step has been performed more than a specified number of times in this situation, go to step 13. If not, store the 60 value of the current T] and J2. Recover the values of z of the previous iteration by doing z ' = z - T) e . -i -i -1 Reduce the value of T). Generate the new z^ by doing z. = z. + 7] e. . V -i -i -i Go to step 7. 12. If this step has been performed more than a specified number of times or if step 11 has been performed at least once in this iteration, go to step 13. If not, store the value of J2 and 7]. Recover the new z^ as in step 11, this time by increasÂ¬ ing 7] to increase speed of convergence. Go to step 7. 13. Find out the minimum value of J2 that have been obtained in steps 11 and 12. If this value is greater than Jl, stop comÂ¬ putations and look for the cause of the instability. If this value is less than Jl, recall the 7] corresponding to this J2 and regenerate the z^ for this case. Go to step 2. In this program the value of T] was selected from the considerÂ¬ ation that at the station m+1 the maximum value of 12 should be less than a prescribed value (j in parentheses represents the th element of the vector). This procedure limits the changes in the j existing trajectory by limiting the magnitude of the maximum fractional change in the terminal values of the variables not specified at the final time. 61 Whenever an iteration was found unstable, T] was reduced by half. When there was an improvement, a linear extrapolation formula was used to increase the value of T] so that the norm of the error J2 would decrease to a desired value. In such an attempt, however, T) was not allowed to increase beyond a certain multiple of its existing value. 3.7.2. Approximations of the Optimal Control for the Kip-Up Problem The method of quasilinearization disregards the question of singular solutions in the present investigation. It was found in Section 3.5 that there were many requirements for the existence of a singular control arq and the necessary conditions for the existence of singular controls are quite complicated. It was decided that, before going into those cases extrema without singular arc would be looked for first and if the computational method was successful, singular subarcs v/ould be looked for later. If the singular solutions are not considered, the optimal o o controls u^ and u^ become bang-bang and are given by Equations (3.5.4a) through (3.5.5b). It is convenient to rewrite these equations at this point in the following way: !f B (X) < 0, uÂ° = S*(X) = 0, ux = 0 > 0, uÂ° = sJ(X) (3.7.19) 62 if -X B (X) <0, uÂ° = Sg(X) =0, uâ€ž = 0 (3.7.20) >0, uâ€ž = S2(Â« Let these control laws be expressed by the following expressions: uÂ° = S1 sgn (-XTÂ§1) (3.7.21) and where and i = 1 or 2 U2 = S2 Sgn ("^B2) sgn (x) = sign of x = S1 when (-XTB.) < 0 1 â€œ -1 = S? when (-X^B.) > 0 i - -x (3.7.22) (3.7.23) (3.7.24) With these expressions for the control laws, the state equations (3.5.1) become X = A(X) + B â€¢ S1 sgn (-X^) + Â§2 â€¢ S2 sgn (-XTB2) (3.7.25) = F(X,X) (cf. Equation (3.7.3a)). In the present quasilinearization algorithm the derivatives of F(X,X) with respect to both X and X in constructing the matrix (Equation (3.7.4)) are needed. One can see that computation of the derivative of F(X,X) with respect to X will occur only as a generalÂ¬ ized function with no numerical value for computation in the limit, because X appears only in the argument of a sign function. Instead 63 of proceeding to the limit the following approximation of the bang-bang control was made. For i = 1 and 2 o * u w u. = i i rs^(X) if AA (-\TB ) < S*(X) AA.(-\TB.) if S2(X) > AA.(-\TB.) ^ S1(X) 1--1 l- i - -l l - (3.7.26) S2(X) if aa.(-\tb.) > S2(X) _ i â€” i â€” â€”i i â€” â˜… where AA^ and AA^ are two positive constants. This function u^ of T 12 AA^(-\ B^) is called the saturation function (sat) when and are unity. The change made in the optimal control is shown graphically in o * Figure 9 for u. and u.. i = 1 or 2. li o * The controls u. and u. have been plotted against the function i i T -\ B^ near a switch point in Figures 9a and 9b, respectively. It can * o be seen that the approximation u differs from the optimal control u^ * only in the portion KL. By increasing the value of AA^, u^ can be made o closer to u.. i With the above approximation of the bang-bang control by a "saturation" control represented by Equation (3.7.26), one would * be able to find the derivative of the control u^ (and hence, of F) with respect to \. The derivative will be zero when the control is on a constraint (X) and will be nonzero on the arc KL which appears near a switching time. In order to represent the saturation control (3.7.26) by its linearization, it is necessary that at least one of the W , i = l,2,...,m, which contains the information of the first derivatives, be computed on the arc KL. Otherwise, this vital portion of the control would go unaccounted for in the linearized equations. This may happen (b) Approximation of Optimal Control (Saturation) Figure 9. Approximation of Bang-Bang Control by Saturation Control. o 65 if the arc KL is too steep for a given selection of integration stations. In such a case, when none of the W is computed on the switching porÂ¬ tions of the control variables like the arc KL,the TPBVP, Equations (3.7.5) and (3.7.6),cannot be solved as explained below. First, from a physical reasoning, it can be seen that the linearÂ¬ ized state equations would get decoupled from the adjoint equations if all of the W^, i = l,2,...,m, show zero derivatives of F with respect to \. This means that the first six equations of (3.7.5) would get decoupled from the last six. But the boundary conditions (3.7.6) is such that they are for the first six variables of e(t) only. Therefore, this results in a situation where there are six first-order equations with twelve boundary conditions, a situation which, in general, does not have a solution. From the point of view of computations with the present algorithm it can be shown that the matrix C in equation (3.7.17) cannot be used to solve for e -m+1 With the system equations defined as (Equations (3.7.3a) and (3.7.3b)) X = F(X,X) X = G(X,X) one obtains [W], = th where the subscript i represents the i interval. 66 Let P., Q., E., DD., R. and M., j J J J J J J 1,2,3, and 4, represent 6x6 matrices. Let [W]i = P P " 1 2 P P 3 4J. where P = F, = [0] for all points other than those lying on the 2 L 4J arcs such as KL in Figure 9b. Then 1 + II V'1 1 + f hi P1 I hi P2 2 hi P3 I + 77 h. P. 2 i 4 \ Q2 ^3 Q4 It may be seen that Q = [0] if P = [0]. Z Z From Equation (3.7.13), Di = [" + \ hi â€œij'1 [-! + 5 hi "i] r9! q2 LQ3 Q4. rt X E_â€l 1 2 1 2 R E . J L 3 4J L 3 4J (say), where R1 R2 R3 R4 [-1 + i h.W. 2 i i and Râ€ž = 2 hi P2 P9 = [0]. f0] if 67 It may again be seen that E9 = QlR9 + Q2R4 = [0-* since = Q2 = [0] when P2 = [0], Now, consider the product D. â€¢ D. â€¢ D. _ of any three succes- â€™ i l+l i+2 m sive D in the product TÃ D.. Let this multiplication be expressed as 1 i=l 1 E11 E21 E31 E41 E12 E22 E32 E42 E13 E23 E33 E43 DD1 DD2 DD3 DD4 The expression for DD2 is DD2 = E11 E12 E23 + E21 E32 E23 + E11 E22 E43 + E21 E42 E43â€˜ If the upper right elements = E?2 = = [0], it may clearly be seen that DD2 = [0]. In the product m TT D. = i=l 1 â€œl m3 m4 one would therefore obtain M = [0] if P = [0] for all i, i = l,2,...,m. With M = [0], the expression for C (Equation (3.7.18)) D = (-I)â„¢ B. Tr D. + B Sj . , l r 1=1 m becomes, after using the values of B. and B , At X* 68 0 0 , B = I o' r I 0 0 0 â€œi 0 â€œi 0 m c = (-1) = (-D _Â«L M2_ 1 o â– j-1 Thus, C becomes singular if F,. = 0 for all i, i = l,2,...,m. â€” A For a given step size or subdivision in the t-interval (0,t^) the singularity of C sets an upper limit on the steepness of the arc KL of Figure 9b whichcan be used for approximating the bang-bang control. This steepness of the arc KL depends on AA^ (or AA2> for a given X(t) and X(t). So, in selecting AA^ and AA , we have to make sure that they are not too large. When the solution is obtained, however, the slope of KL does not depend on the choice of AA and AA if we decide to choose 1 z AA1 = AA = AA, say. This is because AA will be "absorbed" within X(t) , acting as a scale factor on X. In fact, if we define X as AA*X, the state and adjoint equations of our system do not change, and we could therefore select AA^ = AA2 = 1. The slope of the arc KL in the solution depends on the value of AA*X and not on AA alone. However, when the solution has not yet been obtained, the best value of AA need not be 1. If the final time, t^, selected for our problem is much larger than the minimum time, for a bang-bang optimal solution, it can be expected that the arc KL will have a relatively small slope. If the final time is gradually reduced, this slope will increase until it becomes so large that the matrix C becomes singular as explained above. At that point the solution obtained for the smallest t will very 69 nearly be a bang-bang control and will represent the approximate soluÂ¬ tion of the minimum time problem. The computation of the optimal final time via the quasilinearÂ¬ ization method was done according to the stopping condition outlined above. A final time would be guessed, and the TPBVP (Equations (3.7.1) and (3.7.2)) would be solved. The final time would then be reduced by reducing the integration step size, and the TPBVP would again be solved The process would be continued until the matrix C becomes singular and the TPBVP could not be solved any further. 3.7.3. A Simple Example Problem for the Method of Quasilinearization A simple problem was first taken up to explore the various features of the quasilinearization algorithm constructed above. The system was defined as X X. 2 1 X u 2 (3.7.27) (3.7.28) The constraints on u were -1 Â£ u Â£ 1 (3.7.29) The cost function to be minimized is the final time t^. 70 The adjoint equations for the system are ^ = 0 L = -x. (3.7.30) The optimal control u is u = sgn (-X ) (3.7.31) With the approximation of the optimal control u = sat (-AAÂ»\ ) Ct (3.7.32) the state and adjoint equations become X. = Xâ€ž xâ€ž = sat (-AAâ€¢ \2) X, - 0 (3.7.33) X0 = -X. 1 â€¢ The boundary conditions are given by Equations (3.7.28). The analytical solution of the optimal control is given by t f 2 X 1 -1 X 2 t - 1 u = 1 Xx = | t2 ) for 0 S t < 1 *\ u = -1 X^ = -1 + 2t - i t 2 ) for l 2 J The solution is shown graphically in Figure 10. The problem was solved numerically by solving Equations (3.7.33) and (3.7.28) for different final times t by quasilinearization. (C) (d) 1.0 0 -1.0 (e) Curve 1 -* Curve 2 - X Â£j Curve 3 -* Curve 4 -* \ Curve 5 -â€¢ u 71 Figure 10. Graphs of Optimal and Nearly Optimal Solutions Obtained via Quasilinearization for Simple Example. 72 The solutions for t = 2.25, 2.05, and 2.005 were obtained and are shown in Figures 10a, b, and c, respectively. With t = 2.00 the matrix C became singular in iteration 10. The theoretical solution (t^ = 2.00) is shown in Figure lOd. The results for t =2.005 is a reasonably good approximation of the optimal solution. For these problems, 25 time subdivision intervals were used . The initial guess was deliberately poor, as shown in Figure lOe. The solutions were obtained in 9 to 11 iterations from this guess in all these cases. The program originally written, according to Sylvester and Meyer [25] , did not have the provision for the amount of adjustment T) in the iterations. Their method was found to be unstable for some of the initial guesses. 3.7.4. The Results With the Kip-Up Problem The kip-up problem was taken up after the method of quasilinearÂ¬ ization was found successful in the case of the example problem. In the kip-up problem, however, many difficulties were faced from the very beginning and the problem could not finally be solved by this method. A major difference between the human motion problem and the example problem or the problem solved by Boykin and Sierakowski [26] is very prominent. In the latter problems, the control variables switched only once from one boundary to another in the entire trajectory, whereas, in the human motion problem, there were many such switchings. This made the human motion problem less amenable to iterative methods. 73 The program for the human motion problem was extremely lengthy, taking more than eleven hundred statements and requiring the use of the large core of the computer (IBM, 360-65). The subroutine (DIFEQ) which generated the right side of the state and the adjoint equations and their derivatives, turned out to be quite lengthy and required an exorbitant amount of computing time. The quasilinearization program called this subroutine at every station of the total interval and at every iteration. As a result, the program required a tremendous amount of computing timeâ€”about 40 seconds per iteration. All computations were done in double precision. Several sources of difficulty were detected in the unsuccessful attempts to solve the human motion problem by quasilinearization. The central issue was the tremendous amount of computations with accuracy required. The total time interval was divided into 100 equal parts (segÂ¬ ments). As the first (and the only one) guess of the final time, these segments were chosen to be 0.0216 second each. This made the final time (2.16 seconds) equal to the time taken by the gymnast to do the actual maneuver. With 100 segments, the program required the large core of the computer. Even though a larger number of smaller segments would have been preferred, this could not be done because it was intended not to go beyond the large core. It was later found that numerical integration of the nonlinear state equations needed time steps at least as small as one-sixth of what was taken for quasilinearÂ¬ ization. It was initially hoped that, since the quasilinearization algorithm solves the linearized equations instead of the nonlinear 74 equations, the central difference solution would be stable for larger integration step sizes. This was not the case. It was proved theoretically that with perfect precision the method was convergent for all initial guesses. However, it is well known for the unmodified method of quasilinearization (7] = 1) that the method is convergent only for certain initial guesses. It may therefore be fair to expect that even with the modified method, the rate of conÂ¬ vergence should depend on the initial guess and for some initial guesses, this convergence may be extremely poor. For this reason, several initial guesses were tried. The guess of the state variables was taken from the experimental data which were shown to agree .veil with the computed motion in Chapter 2. Different initial guesses for the adjoint variables were tried. The first attempts simply used constant values for all the adjoint variables. Convergence from this guess was nonexistent. Next, attempts were made to generate the adjoint variables from forward inteÂ¬ gration of the adjoint equation with the guessed values of the state variables and a guessed value of adjoint variables at time t = 0. In these cases the integrations were unstable with large numbers generated. The method was not pursued further. Lastly, the method suggested by Miele, Iyer and Well [31] was tried for generating the initial guess of the adjoint variables. In this method, an auxiliary optimization problem is constructed from the original problem. It tries to make an optimal choice of the adjoint variables such that the cumulative norm of the error in satisfying the adjoint equations for a given state variable trajectory is minimized. This is performed as follows. 75 Suppose u (X,X) is the optimal control. The state and adjoint equations may then be written as X = f (X,u (X, â€™â€¢.)) = F(X,X) X = -f^X,u*(X,X))X = G(X,\) (3.7.34) (3.7.35) Suppose we have a guess of the state and the adjoint variables given by X(t) and X(t). Since X(t) and X(t) do not satisfy Equations (3.7.34) and (3.7.35), let e = X + f,TÂ¿x,u*(x,X))X so that e is the error in the adjoint equations (3.7.35). The above equation can be rewritten as T X = f â€œ (3.7.36) Now, consider the optimal control problem, where a. X(t) is a given function of time t b. X(t) is the state variable c. e is the control variable d. The system equation is given by Equation (3.7.36) e. The cost function to be minimized is 1 r> T j = - I e e dt 2 2 J - - o f. t^ is the final time (fixed) of the original problem g. Boundary conditions are X(0) and X(t^) are free. 76 For the optimal control problem posed above, we can construct T the Hamiltonian, using the Lagrange multipliers (6-dimensional vector): T h2 = \ It + Â£*(Â£ ~ f,x X) . (3.7.37) The necessary conditions for optimization are T h9 - ox 2,e " and e* = - (H > 2 â€™ \ or, after performing the differentiations H, = e* + ?- 0T 2,f -* - - and e = f, e -* - X -* (3.7.38) (3.7.39) The boundary conditions are f*(0) = f*(V = 2 (3.7.40) because \(t) is free at t-0 and at t = t . Using Equation (3.7.38) in (3.7.39) and (3.7.40), one obtains e = f, e (3.7.41) â€” ~ A. â€” and e(0) = e(t ) - 0 . (3.7.42) Clearly, Equations (3.7.36), (3.7.41) and (3.7.42) form a linear TPBVP in X and e. This problem should be easier to solve, in theory, than the original TPBVP of the state and adjoint variables and should give an optimal choice for the multiples of 77 In several attempts this linear TPBVP could not be solved for the guessed state trajectory by either the quasilinearization program or by the IBM scientific subroutine package program DLBVP. The prime reason for the failure of the method of quasilinearizaÂ¬ tion was finally found to be the numerical inaccuracies in the compuÂ¬ tations which were dominant in spite of using the double precision arithmetic. The problem was perpetuated and amplified by the large numbers in the right side of the state and adjoint equations and in the derivatives of these equations. The matrices to be inverted at the various stages, one at every step of integration and another at the end of the integration, were ill-conditioned for inversion. When a difÂ¬ ferent subroutine for inversion than what came with the QUASI program was m used, different numbers resulted. The entries of the matrices T and Tf D. i=l i were very large and resulted in large numbers for some entries of C. This made the matrix C ill conditioned for inversion. Any error in inverting the matrix C would be amplified in the values of e This amplification was due to the multiplication of the inverse of C by T, a matrix used in generating s . If e was in error, all other - -mfl -m+1 would be in error because they were generated from If the inversion of C was accurate, then the first six entires of computed from Equations (3.7.16,17,18) should be almost zero. But, 3 instead, large values of the order of 10 were obtained! This obviously indicated that and hence all the were being computed inaccurately. The quasilinearization program failed to solve the human motion problem due to the above three reasons, and primarily due to the last one, the numerical inaccuracies. This was felt to be rather difficult 78 to overcome since it was intimately related to the method used to solve the linear TPBVP and the structure of the original TPBVP. So far as the method was concerned, the key point was that the matrix C was becoming ill conditioned. This occurred because the recursive relaÂ¬ tionship between and ^ has been used to generate a relationship m between eâ€ž and s , which resulted in the product TÃ D. with large â– â€œI ^ m entries. A look at the expression for C in terms of TÃ D., B., i=l 1 i and B (Equation (3.7.8)) would make it clear that with such a value r m of TÃ D , C would automatically be ill conditioned. i=l 1 The other standard methods of solving linear TPBVP (Equations (3.7.5), (3.7.5)), for example, the transition matrix algorithm might have been numerically more stable. However, other difficulties arose because these methods require several forward integrations in one iteration. This means calling the subroutine DIFEQ (the subroutine to generate the right side of the differential equations and its derivatives) many more times. This increases the computing time enormously. With a step size small enough for the integration to be stable, the storage requirements, computing time, and, therefore, the cost of computing increases considerably. Even if these factors are absorbed, it may still be necessary to try several initial guesses of the adjoint variables to get the method to converge. In view of the above problems, it was concluded that the standard method of quasilinearization was not suitable for the human motion problem and so should not be pursued further. 79 , 3.8. Steepest Descent Methods for Solving the Minimum-Time Kip-Up Problem Three different formulations of first-order steepest descent methods were used after the method of quasilinearization was unsuccessÂ¬ ful in solving the minimum-time kip-up problem. These formulations differ from each other in the construction of the cost functional, handling of the terminal constraints, treatment of the control increÂ¬ ments and in the method of adjusting the final time. The basic features of these three formulations are described below. Formulation 1 a. The cost functional is the final time. The terminal errors and the cost functional are reduced simultaneously. b. The adjustment of the final time is done by extending or truncating the final end of the trajectories. c. The control functions take the form of a sequence of constrained and unconstrained arcs. Improvements are made at the unconÂ¬ strained parts only, including the junctions of the constrained and unconstrained arcs. The method is based on the works of Bryson and Denham [27,28] and Bryson and Ho [32]. Formulation 2 a. The cost functional is the sum of a scalar representing the final time and a norm of the terminal error. b. A change in the independent variable t is introduced by definÂ¬ ing the transformation 80 t = a t where a is a constant, or do â€” = 0 dT so that a is treated as an additional state variable. The final time t is directly proportional to a when is held fixed. Long [29] used this transformation of the independent variable to convert free end point TPBVP to fixed end point TPBVP for solution by the method of quasilinearization. The cost functional is reconstructed as 2 6 2 J = K O' + E K.S., K ,K. ,. . . ,K > 0. o.^xi o i 6 1=1 No terminal constraints were introduced in this formulation since they (the Â§/s) are included in the cost functional, c. The control functions are assumed to be free to change in any direction while computing the gradient of the cost function. That is, the control constraints were not considered when comÂ¬ puting the gradient of the cost functional used to find a suitÂ¬ able increment in the control and a. The control constraints were imposed in the next iteration during forward integration of the state equations. When the computed control violated a constraint in a subarc it was set equal to its limit, the constraint function on the subarc. This approach for treating constraints on the control has been used by Wong, Dressier and Luenburger [33]. 81 Formulation 3 This formulation consisted of the features (a) and (b) of Formulation 2 and (c) of Formulation 1. The derivations of the numerical algorithms for these formulaÂ¬ tions are now presented. The basic concepts on which these algorithms are based are available in the literature [28,32]. The results are derived in a manner suitable for analysis of the outcome of the numerÂ¬ ical computations, and, thus, the derivations presented here are slightly different from those found in the literature for any gradient method approach to the computation of optimally controlled motion. 3.8.1. Derivations for Formulation 1 Suppose we have a continuous nominal control u^(t) and u^(t) and a nominal final time t . These control histories have some parts lying on the constraints S^(X), i = 1 or 2, and the remaining parts lie away from the constraints. The parts lying on constraints will be called "constrained arcs" and the parts lying off the constraints will be called "unconstrained arcs." The interesections of constrained and unconstrained arcs will be called "corner points." A nominal guess for the control variables consists of specifying the corner points and values of the control at unconstrained portions. On the constrained arcs control variables are generated from constraint functions. An initial choice of the control history and the final time will not, in general, satisfy the boundary condition and will not do so in minimum time either. One can improve upon the trajectories in the following way. At first, we establish how a particular state variable X1 (the th i component of the state vector X) changes at the final time due to 82 a small change in the control history and the final time. For that purpose, let a cost functional be defined J = X1(tf) . (3.8.1) Let \1(t) be an arbitrary time-varying vector of dimension six. Since the system satisfies Equation (3.1.1), the final value of the state variable will also be given by *f .T j' = XX(t ) + f X1 (f-X) dt . (3.8.2) f do - - - If the control variables u (t) and u (t) change by a small X z amount 6u (t) and fiu (t) , there will also be a small change in the X Â¿j state variable X(t), denoted by fiX(t), throughout the trajectory. It is clear that these changes in the control, denoted by fiu(t) and in the state variables denoted by fix(t), will not be independent of each other. Apart from changing the control history, an increment to the final time t^ by a small amount 6t^ and small increment fiX(O) to the initial state vector are also prescribed. The first-order change in J due to the changes in the control and the final time is given by Aj' = X1(tf) dtf + {Sx1 - UX)T 6x}t + (XX)T fiX(O) tf tf + f {aV f, + (x1)} fix dt + / aV f fiu dt. (3.8.3) The fiu is chosen in the following way: For the unconstrained parts, fiu is completely free. The parts presently on the constraints will remain on the constraints for the same periods of time as before. 83 For these portions, the change in control is given by the shift of the constraint due to state changes according to the relation 6u (t) = S3 $X(t) . (3.8.4) 1 x,5 " Let c} denote all those portions of the trajectory of the control u^(t), i = 1 or 2, which lie on any of the constraints s{ or 2 o S.. Also, let C. denote all those portions of the control u. which l i i do not lie on a constraint. If the expressions for on the conÂ¬ strained arcs given by Equations (3.8.4) are substituted in EquaÂ¬ tion (3.8.3) and the integration of the last term of the right side of o 1 Equation (3.8.2) is split into integrations over the intervals , o , 1 , , . C , and C , one obtains fij' = xx(t )dt + {ax1 - a1)7 ax} + {aV fix} 1 1 ** Z~vf t=o tf + Jo {(^1)T Ã'X+ (i'X)} dt + f uV f,u \ dt + f aV f,u Sd 6x dt aV f, 6u dt + f (XÃ)T f. SJ Ã“X dt " U2 2 1 ~ " 2 â€™X If \1(t) is computed such that \j(tf) = 1 for i = j = 0 for i j where x{ = j element of X* and ; i X X 1 (3.8.5) (3.8.6) (3.8.7) 84 where 6=0 on C?, 6-1 on 1 11 1 62 = 0 Â°" Â°2' *2 = 1 Â°n C2 and u1 and u used in Equation (3.8.7) are computed only when x,x 2â€™X the controls u^ and u^ are on a constraint and u^ and u^ are replaced by the constraint expressions S^(X), one obtains Sj' = XX(tâ€ž) dt + (X1(0))T fix(0) + f (X1) f, 6U dt f f - - - o - - ui 1 + Â£'u S dt = f1(X (t_),u(t.))dt. + aX(0))T flx(0) + f (X1)1 f, flu. - I - I I â€œ - Â°o - - u. 1 i.T dt + lf0(XÃ)T f,u 6u2 dt (3.8.8) C2 2 where fi is the i^h element of the vector f(X,u). Equation (3.8.8) is the desired expression for the change of the state variable X1 at the final time t^ due to (1) a small arbitrary increment Â§u(t) given to the control variable u(t) over the unconstrained portions, (2) a small increase dt^ of the final time t^, and (3) an arbitrary small change 6x(0) of the initial state vector X . Similarly, one can find the expressions for the change in the terminal value of any other state variable XJ. It can be seen that if one constructs the (6 x6) matrix 85 R(t) -[V (t), 2 X (t), X3(t) 4 X (t), x5(t), x6(t) ] so that R(t) satisfies and R(t^) = I (6x6 unit matrix) R(t) = 6 f l-â€™u. R(t) (3.8.9) (3.8.10) where the meaning of the various terms in the parentheses of right side of Equation (3.8.10) is the same as that in the same terms appear- in Equation (3.8.7), then the change in the terminal value of the state vector is given by $X(t ) = f dt.+ RT(0)Ox(0) - f RT f, 4u dt + f RT f, Â§un dt . f - f - ho - uâ€ž 1 Â°_o - u 2 (3.8.11a) If we choose that 6x(0) =0, we obtain 5x(tf) f dt + f RT f, Su dt + f RT f, 6u dt. - f J o -â€™u 1 J o -â€™u 2 C1 1 C2 ^ (3.8.11b) Following the method prescribed by Bryson and Ho [32], it will now be attempted to make improvements in the terminal errors given by | = X(tf) - Xf and at the same time reduce the final time t^. Thus, since it is being sought to minimize t , one would maximize -dt^ or minimize dt^ with respect to 6u_^, ^u^ subject to the constraint 6x(t ) = f(t Jdt + f RT f, 6u dt+f RT f, Â§u dt -f - f f v o - u 1 J o - u 2 Câ€ž 1 Câ€ž 2 (3.8.12) 86 with 6x(t ) = 6xp(t ) - f f where OX (t^,) is a chosen decrement in the terminal error such that Â§u maintains.the first-order approximations. In this incremental minimization problem, the incremental cost functional, dt^, to be minimized, and the constraints are linear in the incremental control parameter Â§u. Such a problem does not have an extremum. However, since these are linearized relations obtained from a nonlinear system, the increments 6u , Â§u , and dt should be small for 1 ^ X the first-order approximation to be valid. To limit the increments 6u , 6u , and dt , the following quadratic penalty term to the incre- X Zj X mental cost dt^ is added: I b dtf + \ So 6U1 Wl(t) dt + I Jo 6US W2(t) dtâ€™ (3.8.13) where b is a positive scalar quantity and W (t) and W (t) are positive X Â¿-J scalar quantities specified as functions of time. Adding these quantities to the cost functional and adjoining the constraint relations (3.8.12) to the resulting expression by a multiplier v (a six-dimensional vector), the following problem is obtained. Minimize wrt, &u , $u , and X Z X dt + f 2 â€” b dt^ + i f 6u7 Wâ€ž dt + i- f du^ W0dt + vT^f (t^)dt^ 'f+2Jo VU1 VU+2Jo 2 V C1 C2 + f Rrf, 6u dt+ f RTf, Su dt - 6xP(tJ CÂ° " U1 1 CÂ° â€œ U2 2 f } (3.8.14) 87 If the derivatives of this functional with respect to 6u^, &u9, and dt^ are set to zero, one obtains dt = - ^ [1+vT f(t )] , f b f (3.8.15) and s = 6u = 2 W. Â± iT *2 - U2 Rv on C. Rv on CL P A (3.8.16) for an extremal. Using Equations (3.8.15) and (3.8.16) in (3.8.12), one obtains Â§XP = - i fil + vTf3 -IMv (3.8.17) b -â– - . W " or where v = - 1 T I. . +- f f (tj \Â¡njÃ b - - f -i r 6xP(tf)+if(tf) I , I = f Rxf, W 1 fl R dt + l R1 f, W 1 fT R dt. ** cf - â€œl 1 - â€œl CÂ° - u2 2 - u2 (3.8.19) (3.8.18) The value of 6xJ(t^)> the desired change in the terminal values of the state variables, may be chosen as a decrement in the terminal error S. where >XP(t ) = - e. Â§. if li 0 < e. Â£ 1 i (3.8.20) . th $X1J(t ) is the i1,11 element of ^XP. if and 88 The steepest descent algorithm for Formulation 1 can now be described as follows: 1. Guess a nominal control history u (t) and u (t) and a final time t^. 2. Integrate the state equations forward with the nominal control and the nominal final time with the initial values of the state vector given by X(0) = X . Store X (t). Compute and store - -o - Find the norm INI ={.v/J4 3. 4. 6. 7. for some positive , i = 1,...,6 (to be specified). Save the controls u^ and u^ in another variable COLD, the corner times N in the variable NOLD and the final time t^ in TFOLD. Set R(t^) = I, the (6 X6) unit matrix. Integrate backward T Equations (3.8.10). At the same time compute and store R f,^ T ^ and R f, .It is not necessary to store R(t). - u 2 Select the positive constant b and the positive quantities W (t) and W (t). Unless there is some special reason, W (t) i. Z JL and W (t) may be taken as a positive constant equal to W (to be z specified). In such a case, the storage for W (t) and W (t) will not be needed. Select e^, , i = 1,2,...,6 such that o < e. l. Compute I^ using Equation Compute v, Â§u^, Su^, and dt and (3.8.16). Note that Â§u unconstrained parts C and (3.8.19). from Equations (3.8.18), (3.8.15), and Â®u^ are defined for the CÂ° only, and are to be computed only on those parts. 89 Change the final time t^ to t + dt^. If the integration step size is h, this means that the total interval is to be increased or decreased at the final end (depending upon whether dt^ is positive or negative) by dt^/h, rounded to the nearest integer number of integraÂ¬ tion steps. In the case of a positive dt^, the controls u^(t) and u (t) in the interval TFOLD ^ t ^ t (new) will be given by extrapola- Â¿ I tion of the existing curves. If a control function is on a constraint at the final time, TFOLD, it should remain on the same constraint in this period and is to be generated during the forward integration in the next steps. 8. Set the control u. = UOLD. + Â§u., i = 1 or 2, for the uncon- l li strained portions. Integrate the state equations forward with the new control. The corner times N are to be generated again during the forward integration as described graphically in Figure 11. Figure 11 shows the different cases which may arise during the forward integration. It is seen that a new corner time is generated at the point where the unconstrained u^ + 6u^ curve, extrapolated if necesÂ¬ sary, meets the constraint. This part of the forward integration makes the programming complicated with many logical program statements. 9. Find the errors in the boundary conditions Â§ and find the norm j| Â§^J| as in step 2. If this norm is less than that of the previous iteration go to step 3 to continue the iterations. If it is not reduced, then do the following: a. If dt^ is too large, increase b, and go to step 9c below. c c /Jl/UlltUUiUU = constraint, tâ€ž = old corner time, t = new corner time 0 n uq = old control, u = new control, u+ 6u = improved control computed by gradient at the unconstrained arc. Figure 11. Modification of the Corner Point Between Constrained and Unconstrained Arcs After Changes in the Unconstrained Arcs. to o 91 What is "too large" is to be found by trials. In the present case more than five integrations steps were found to be too large for a value of dt^. b. If dt^ is not too large, decrease the e and go to step 9c below. c. Set u = UOLD, t_^ = TFOLD, N = NOLD and go to step 5 above. The stopping conditions are given by: 1. || Â§|| is a small number, meaning that the boundary conditions have been adequately met. T 2. (1+v f), is negative or zero. This means that the correc- â– tf tions to the final time which would be computed in the next iteration, using Equation (3.8.15) would increase the final time. T T 3. For i = 1 and 2 the quantities v R f, are 1 1 a. positive when u^ is given by meaning that only a violaÂ¬ tion of the constraint will improve the cost 2 b. negative when u^ is given by S c. Zero, or a small number when u^ is not on any constraint, so that computed by Equation (3.8.16) should be zero or an acceptably small number. Comments on the Algorithm The forward integration of the state equations at every iteration requires special treatment because of the presence of the constrained and unconstrained arcs. The integration formulas are slightly differÂ¬ ent for cases when the right sides of the differential equations are On an unconstrained portion u^(t) and u^(t) are time varying. 92 prescribed as time varying functions, but on a constraint, they are expressed as functions of state variables. This difference is to be taken into account in the integration scheme. The control constraining boundaries are not known before the state trajectory is known. Since the state trajectory is determined by forward integration, the determination of the corner points is difÂ¬ ficult after the control is changed at off-the-constraint portions. For this purpose a first-order prediction of the location of the conÂ¬ straint and an extrapolation of the unconstrained u + 5u curve was made to determine if they intersect to form a corner point one step ahead in the forward, state equation integration. This was needed because the integration scheme used four consecutive stored states to generate the next state. Much more computing time would be required by going back to an earlier point and redoing the integrations. First-order predicÂ¬ tions of the states Xâ€ž and X,_ for the step K + 1 were made from the step o 5 K by the formulas K+l K K x3 = x3 + h x4 and h = integration step size (constant). Only Xâ€ž and X_ are needed to find the constraints S'?. 3 5 i It is to be noticed that the modification of a corner point as illustrated in Figure 11 has not been accounted for in the derivation of the algorithm. The errors in the expression of Ã³X(t^) caused by this derivation in the algorithm from the theoretical derivation will be small only if the constrained and unconstrained arcs are much longer than the shift of the corner point. This is a significant source of error when the unconstrained arcs are small. 93 3.8.2. Derivations for Formulation 2 In this formulation, we transform the independent variable t = cfT (3. 8. 21) where T is the new independent variable. Then the system equations become dX â€”.= X' = a f(X,u) dev ' _ â€” = a =0 dT where (') denotes differentiation with respect to T. Boundary conditions are (3. 8. 22) (3. 8.23) X(0) = X (given) - -o with X(T ) not specified and to be determined, and (3.8.24) T , the fixed final value of T, given. The cost functional to be minimized is J = i Â£ K.Â§2 + K cv2 (3.8.25) 2 . li o 1=1 with the values of the K and K. specified. Constraints on the control o 1 given by Equation (3.1.7) remain unchanged. The steepest descent algorithm for minimizing the cost functional of Equation (3.8.5) was constructed as follows. Let X(T) , a vector of dimension 6, and T)(t), a scalar, be arbiÂ¬ trary functions of T. Since the system satisfies Equations (3.8.22), the cost functional is also equal to 94 J* =i Â£ K. 2 i=l 1 - . . -i 2 f f X1(tf)-X^J +K a+/ XT[crf-x']dT - f 71c/ dT. (3.8.26) In this expression S1 has been replaced by (X1(tf) - X*) Integration by parts yields tf t t t t j' = cp + f \Taf dT - [XTX]0f + fQf \'X dt - [Tlcv]/ + /Qf I]' a dT o where * = 5 J, Ki[xl(tf> -xÃ]2 + | Koâ€œ2â€™ i=l For first-order variations in u and X t â€ž ej' = cp Â§x + cp, fia + f X [f + a f, fix + cn f, fiu] dT 3. v ~~ â€”A ~ â€”U â€” tf â€œ [\T fix] + f X/T Ox dT - [71 -T] ] Â¿a + f V' 5a dT. (3.8.27) â€” â€” t â€” - tâ€žo d 'f O f Let X(t) and T) (T) be chosen such that X (Tf) = cp,x X'(T) = - (a f,x) X (3. 8.28a) (3.8.28b) Veo = - XTf (3.8.28c) ll(Tf) = (3.8.28d) Then, fi-J7 =71 Oa + f XTa f, fiu dT n v - â€” u - (3.8.29a) 7) Â¿of + a f X [f, Su + f, fiu ] dT. o J â€” u 1 - u 2 o 1 2 (3. 8.29b) 95 Equations (3.8.29) give the gradient of the cost function (3.8.26) with respect to a and 6u(t). Let us choose So = - e T) s a o (3.8.30) 6u (T) = -S \T f 1 e1(T) (3.8.31a) (T) Z, -S xV 2 " U2 e2(T) (3.8.3lb) where e , e (T) efT), and s are positive quantities. It can be seen a 1 2 that if these quantities are chosen such that &a, Â§u (t), and Ã©u^ÃT) are sufficiently small, then the first-order approximations are valid and the reduction of the cost is guaranteed. The purpose of making e^(T) and e (t) functions of T is to have control over the amount of improvements Â§u and 6u during the time interval. They will normally be 1.0 all -L Ci over, except for special reasons. The values of e , e (T), and e (T) are to be specified independently. Once they are decided upon, s may be calculated on the basis of how much improvement to the cost is desired in one iteration while remaining within the linear approximaÂ¬ tion's valid range. Suppose, in any iteration, the cost is J, and it is desired to reduce the cost by e J in the next iteration, where 0 < e Ã¡ 1. With the choice of 6u (t), 6u (T) , and Â§a already made above, Equation (3.8.29b) yields tf -e J = -s e TI2 - sc / [e (T)aTf, )2+e(T)(\Tf, )2]dT. (3.8.32) ao d 1 â€” â€” u 2 â€” â€” u o 1 2 96 Defining tf B = oÃ J [yTHff.u )2 + e2(^)(XTf,u )2] dT (3.8.33) o 1 U2 one obtains -ej = -s[ e T] + b] a o (3.8.34) s = ej/(e j\2 + B). (3.8.35) Equation (3.8.35) gives the value of s required in the determination of 6u (t), 6u (T), and 6a via the gradient terms, which, under first- order predictions, would reduce the cost by ej. In actual computation, a different change in the cost is to be expected because e will be finite and the first-order prediction will not be exact. This is true because the system is nonlinear, and, because in some intervals of T, 6u may not be implemented due to the constraints. If the cost functional is not improved, then e is to be reduced and 6u and 6a are to be recomputed. If the cost function is reduced but the terminal error is increased (meaning thereby that the cost has decreased by reducing a but increasing the terminal error), Kq is to be reduced. The selection of e was done in the following way. The change a in the cost due to the change in a and the change in u(T) are given by -s e 7) and -sB, respectively, a o Let -s e T) = e (-sB) where e ^ 0 (3.8.36) a o a a so that the change in cost due to change in a is e times that due to cl the change in u(t). Then e is given by cl 97 e = e b/T) (3.8.37) a a o and then from Equation (3.8.34), s will be given by s = ej/fB(1+ S ~ )} . (3.8.38) a o Since it is intuitively easier to specify s than e , the 3 3 advantage of choosing e in this way is obvious. 3 The Stopping Conditions The exact stopping conditions for the computations should also correspond to the conditions for optimality and they are suggested by Equations (3.8.29). Clearly, when the following three conditions are satisfied: 1. T) =0 (or small) , o T T 2. X f, and X f, are zero (or small) when the control variables - - ul u2 u and u , respectively, are not on a constraint, T T 3. when uâ€ž or u are on a constraint, X f, or X f, , resoec- 1 2 - - Uf - - u2 tively, are of opposite signs to those of the constraint, i.e., positive on a lower constraint and negative on an upper constraint then no improvement on cost is possible without violating the constraints. These indicate stopping condition. While performing the last check for stopping, the adjoint equations must be modified according to Equation (3.8.41) of Formulation 3 when any portion of the control trajectories lies on a constraint. This is necessary in order to take into account the dependence of the control constraints on the state variables. However, there are several problems to expect from these stopping conditions. These conditions will not be satisfied even if 98 a trajectory is slightly different from the optimal. The adjoint variables are extremely sensitive with respect to changes in the control functions and a slight imperfection in the solution will yield a signifÂ¬ icantly different set of adjoint variables. They will not meet the stopping conditions described above. Also, the first-order methods show poor convergence near the optimal solution and may never go sufficiently close to the exact solution in a reasonable amount of computation. Added to these problems is the problem of computation errors (round off and truncation errors) which grow in integrations over a lengthy interÂ¬ val, and will begin to show up then the terminal errors and the gradient of the cost functional are reduced to small values. In view of these factors it is very unlikely that the stopping conditions will be preÂ¬ cisely obtained. One should therefore find some other stopÂ¬ ping criteria to decide when to stop the computations and accept an approximate solution. Since it is desired to minimize the terminal errors, one can test directly whether they are small enough or not. For time optimality, it has been seen that the optimal solution is bang-bang with the excepÂ¬ tion of singular arcs. It is very difficult to start with a continuous nominal control trajectory and improve to an exact bang-bang control in any part of the trajectory. So, a steep control function arc joining an upper and lower constraint should be an acceptable approximation of a bang-bang solution. If there are singular arcs, they must be tested for optimality by seeing if they are stable or not. Also, T] should be small or negative. A negative T| indicates that the cost functional can be reduced further by increasing a. This corresponds to increasing 99 the final time, which we wish to minimize. If we already have acceptable terminal errors, no further increment in a is needed, in this case. Lastly, with all the above conditions satisfied, the final stopping condition is the insignificant improvement in the cost funcÂ¬ tional by further computations. Choice of the K. 1 If any K_ is large compared to the others, it may be expected that the corresponding boundary error will be reduced at the expense of the others. On the other hand, not all errors may improve acceptably if all the K. are made equal. If K is too small, the minimization of a i o will be slow. If K is large, a may be reduced too much, resulting in o large boundary condition errors. A reasonable way to select Kq is to increase or decrease K with the current norm of the terminal error o 6 P-L = III I! = Â§ 2 K. . (3.8.39) i=l The values of the for i = 1,2,...,6, may need to be adjusted during the iterations to achieve faster improvement of the individual terminal errors. The algorithm developed above can be summarized as follows: 1. Decide upon the values of the quantities , K^, ... ,Kg, Â£, e (T), e (T) and e as discussed above. X Z 3. 2. Make a nominal guess of T , u^(T), u9(t) and O'. This will require a few simulation trials. 100 3. Integrate the state equations (3.8.22) forward with the nominal control and the given initial values. At each inteÂ¬ gration step, check if the control u1 (t) and u (T) are within the allowable bounds (the constraints) which are to be computed at every integration step. Bring the controls on the bounds if they exceed the bounds. Store u(T) and X(T). 4. Calculate the terminal errors and their norm P and determine the quantity K . 1 2 5. Calculate the cost J = P, + â€” K a . 1 2 o 6. Set the terminal values of the adjoint variables: X. (tf) = Â§ . , T|(tf) = KqQ' (from Equations (3. 8. 28a,d). 7. Integrate the adjoint Equations (3.8.28b,c) backwards. Compute T . T and store, at each step, the values of X f (T) and X f, (T). â€” u â€” â€” u 1 2 8. Calculate B, e , and s, using Equations (3.8.33,37,38). 9. Calculate 5U (T) and 5u (T), and 5 a from Equations (3.8.30,31). JL Â¿j 10. Update u^t), u (t) , and 5a by setting u^(T) = u^(T) + 5u1(t), u (T) = u (T) + 5U (T), and a= a+ 5a. Store the old u (T), Â£ Â£ Â£ x u^ÃT), and a in UOLD and AOLD. 11. Integrate the state equations forward as in step 3, and find the value of the cost functional J. 12. If the cost functional is not reduced from the previous value, reduce e, reset the values of u (T) , u (t), and at to the previous X z values UOLD and AOLD that were stored, go to step 8. 13. If the cost is reduced but P , the norm of the error has increased, reduce the value of K , reset the values of u (T) o 1 u (T), and a to the previous values that were stored and go to step 8. 14. If both the cost function and are reduced, check to see if the stopping conditions have been met. If not, go to step 6, which is the next iteration. 15. If the number of iterations is greater than a specified number, punch on cards the values of the current u^(T) and u^(T) and stop. Manually check to see if the program is performing well. If necessary, change the values of , i = 1,2,...,6, e. ,e i (T). eâ€ž(i). Restart the program if necessary. 1 a 1 2 A Simple Numerical Example of the Method A simple second-order system with state-dependent constraints on the control was taken for solution by this method to examine the method's features as well as its effectiveness. The system was defined as follows: with the constraints on u given by The boundary conditions were chosen as X1(0) = 0 X2(0) - 0 The system is to meet the above boundary conditions in minimum time. 102 Solution With the change of the independent variable t = GT the state equations become Xi = â€œX2 X2 = â€œU a = 0 where () denotes differentiation with respect to T. While computing the adjoint variables and the improvements in the control $u, the conÂ¬ trol is assumed to be completely free as mentioned earlier in this chapter. The constraints were imposed while doing the forward inteÂ¬ gration with the improved control u + 5u in the next iteration. The equations of the adjoint variables are given by â€¢ T X = -X of,v or = 0 = -07. 1 7] = ?1X2 V The boundary conditions of the state and the adjoint variables are 3^(0) = 0 X2(0) - 0 X1(tf) = K1(X1(tf) - 1) Xgitf) = K2(X2(tf)) 7](t ) = K O Â± o 71(0) = 0. 103 The cost functional is J = \ VW - 1)2 + I WV^ + I V2' In the attempts to solve this problem, several experiments were done with respect to the choices of the initial guess of the final time, the quantity Kq and e^(T). It was found that convergence to the boundary conditions on the state variables and Xwas quite rapid from all the initial guesses tried, which may not be surprising for this simple problem. However, to accelerate minimization of the final time, some initial guesses were better than others. It was found that if the final time, i.e., the value of a, was guessed larger than optimal, then, in the initial iterations, as the boundary errors are improved for a larger nonminimal value of a, the control trajectory takes a shape other than optimal. Once the terminal errors have been reduced and the final time is still not minimal, reduction in the final time becomes very slow. Any large change in a has a tendency to greatly increase the terminal errors from their previously small values. It was found that for this problem, if the guessed value of the final time (i.e., o) was chosen less than the actual minimum time, the convergence was more rapid. In such a case it was found that the corÂ¬ rections to the control in any iteration are computed for a final time less than the minimum time which forces the control to move to the appropriate constraint boundaries. At the same time, the correction on the a was computed positive, so that the correction value of a was approached automatically in the attempts to reduce the terminal errors. 104 Large increases in a were checked by the penalty terra KqO so that the cost functional was minimized without excessive oscillations in the values of a or u(T). It was found that in the computation of the correction to the control by the formula 6u = -s e (t)\TÃ = -s e (t) \ 1 - -u 1 2 if e^(T) is chosen such that it is small where the control is near a constraint boundary and large when off the constraint, the converÂ¬ gence to an optimal bang-bang solution is accelerated after the control has taken a shape similar to the optimal control. Figure 12 shows the control trajectories at different iterations and at the solution. The exact minimum time solution is bang-bang with only one switching and the final time can be determined by forward inteÂ¬ gration of the state equations with this control. In this way, the minimum time was found to be 3.06 seconds. In the solution obtained by the penalty function method, the terminal errors for and were -.02 and -.034, respectively, and the final time (=Gi) was 2.95. The values of Kâ€ž and Kâ€ž were selected to be 1.0 for both. K was chosen by the 12 o formula â€ž2 ,2 . 2 K = .01 + (Â§ + Â§0)/or o 12 where Â§1 = x1(Tf) â€œ 1Â» and = x2(Tf). It is seen that the final time in the solution obtained by the penalty function method is a little less than the actual solution. Such a situation is to be expected from this kind of formulation where a trade off between the terminal error and the final time is bound to 1.0 u Nearest Control Constraint Boundary 1 - Initial Guess, T = 2.00 seconds 2 - At Iteration 5, T = 2.17 seconds 3 - At Approximate Solution, T = 2.95 seconds Figure 12. Solution of Example Problem by the Method of Steepest Descent. 105 106 occur in order to reduce the composite cost function defined for this problem. From a very bad initial guess, the solution was obtained in 24 iterations, taking a total computing time of 12.88 seconds. 3.8.3. Derivations for Formulation 3 In this formulation, the transformation equation of the indeÂ¬ pendent variable, the state equations, cost functional, and the control constraint functions in this formulation are exactly the same as for Formulation 2. Therefore, Equations (3.8.21) to (3.8.27) of Formulation 2 are also valid for this formulation. The difference between the two appears in the treatment of the control increments. In Formulation 3, the control function is considered to lie partly on the constraints, and the control increments are prescribed only on the portions of the conÂ¬ trol that do not lie on the constraints, as in Formulation 1. The symbols CÂ°, CÂ° Ch , Ck for portions of the u and u history that are off the X A X X constraints or on the constraint will be used as in Formulation 1. Considering the relationship between the change in the control and the change in the state variable (see Equation (3.8.4)) as in Formulation 1, one obtains the following first-order relationship between the increment in the cost functional and the increments in u and cv: 6j' = T)(0) 6a + Â« f ^Tf, 6U dT+ot f 1Â° - - u 1 â€¢L C1 1 c (3. 8.40) The equations for X(r) and Tj(t) are the same as in Formulation 2 (Equation (3.8.28)), except that the X' equation is now given by X (3.8.41) 107 explained changes, f* The term in parentheses on the right side of this in Formulation 1, Equation (3.8.7). Proceeding as in the derivations in Formulation 2 one obtains \Tf, f T - ' U1 - U1 * Â£1 n T T dT + a J X f, f, Xe, Jo - - uâ€ž - u - Â¿ dT equation is with obvious (3.8.42) s ej ~2 eat) +B a o 6 eis (3.8.43) (3.8.44a) iu = - fT X s s (3.8.44b) Â¿ U2 ~ Â¿ da = - T) s s (3.8.45) o & The quantities J, e, e , e , and e are the same quantities as defined 12 3 in Formulation 2. The numerical algorithm and the discussions of stopping condiÂ¬ tions for Formulation 2 and the comments on integration with constrained and unconstrained arcs made in Formulation 1 are valid for this formulation. The transformation of the independent variable t = qt used in Formulations 2 and 3 was introduced for the following reason. In the usual gradient schemes for optimization of trajectories and as in Formulation 1, the adjustment of the final time is done by extending or truncating the final end of the trajectory. This was not considered effiÂ¬ cient because the final time, though being as independent as the control u in determining the optimal trajectory, does not appear in the system 108 equations. By the above transformation, a measure of the final time, o', appears in the system equations and therefore direct improvement in conÂ¬ vergence was expected. 3.8.4. The Integration Scheme for the Steepest Descent Methods The integration of the ordinary differential equations for the states was found to be unstable even for moderate to very small integraÂ¬ tion step sizes when the simple Euler integration scheme was used. Subsequently, several other integration schemes such as a fourth-order Runge-Kutta, an Euler predictor corrector, and an Adams-Bashforth preÂ¬ dictor corrector scheme were tried. Out of these trials the Adams- Bashforth predictor corrector scheme was found to be the best for our problem. It required less computation per integration step and allowed larger step size than the other methods. The integration step size was held fixed because this would significantly reduce the interpolation errors for the determination of the state and control variables from their stored values. These were needed for the backward integration of the adjoint equations. The adjoint equations (equations for the adjoint variables R, X, and 7]) required a step size for integration half that required by the state equations. The integration scheme used the fourth- order Runge-Kutta method for generating the first four points for startÂ¬ ing and Adams-Bashforth predictor-corrector formulas for generating the subsequent points. Various integration step sizes were tested to find out the largest one for which the integrations were stable. From the tests the step size 0.0036 second was found to be adequate for the independent variables t in Formulation 1 and t in Formulations 2 and 3 for the 109 integration of the state equations. It was also found that the integraÂ¬ tions were stable if at the discontinuities of the right side of the differential equations, the integration was stopped and restarted from that point. This was so because the Adams-Bashforth formulas were not valid at a discontinuity, and therefore better results were obtained when starting formulas (four-step Runge-Kutta) were used after the disconÂ¬ tinuity. This feature was therefore introduced into the program. 3.8.5. Initial Guess of the Control Function It was found necessary for the initial guess of the control function to be as nearly optimal as possible if considerable computaÂ¬ tion time was to be saved. Simulation (several trial forward inteÂ¬ grations) was necessary to generate an initial guess for the relatively complicated control function. From the principles of mechanics it was noted that for a rotating system, the angular velocity tends to increase when the radius of gyration of the system is reduced, and it tends to decrease when the radius of gyration is increased. The gymnast applied this principle and counteracted the effect of gravity by timely adjustÂ¬ ments of the radius of gyration of the body about the horizontal bar. Although the gymnast is limited in strength, he was able to adequately shorten or extend his radius of gyration. Even with somewhat of a physical understanding of the mechanÂ¬ ics of the motion and the bang-bang principle, it was not possible to devise a control for the model to perform the desired maneuver. This was primarily due to the fact that a candidate control function for an extreÂ¬ mum control could not be specified fully beforehand. The bounds on the control were dependent on the state, and, thus, were not known a priori. 110 Even if the bounds on the control could be known exactly, it is difficult to predict the response of such a coupled nonlinear system. The film data of the actual motion were carefully examined via the principles of mechanics but were of no further help in generating an initial guess. It was found that the upper limit of the shoulder torque for smaller values of 9 was insufficient to allow the model to reproduce the experimentally determined motion. From the optimality conditions it was known that the optimal control should be either purely bang-bang, or bang-bang with singular subarcs. The control was therefore guessed such that it coincided with the constraints on a major portion of the time interval. 3.9. Results of the Numerical Computations and Comments Several numerical difficulties were faced during computation by the steepest descent methods. These were caused by: 1. The system equations were quite nonlinear. 2. The kip-up motion was unstable. 3. The control function had many switches and large oscillations ocÂ¬ curred in the state trajectory. 4. The system was not locally controllable about several trajectories that were tried. 5. Numerical gradient methods have several problems with the minimum time criterion. The steepest descent algorithms make control changes according to the local gradient, and for the present system these were valid only in a very small neighborhood of the control function. Due to instability Ill of the perturbation equations about the kip-up trajectory (expressed quantitatively by large entries of the order 10 to 200, in the matrix R(t) in the earlier segment of the trajectory) any small perturbation in an earlier segment of the trajectory caused larger deviations in the state variables in segments near the terminal time. Thus, unless the changes were very small, the steepest descent formulas were not able to bring about the improvement in the control. The system equations required very small integration step size. This virtually increases the length of the total time interval. It required about 570 integraÂ¬ tion steps for the forward integration of the state equations and twice as many steps for the reverse integraion of the adjoint equations. With such a long integration interval, the numerical values of the gradients are expected to be less accurate due to accumulation of truncation and round-off errors. Such inaccuracies near the earlier parts of the traÂ¬ jectory are especially detrimental when small changes in the terminal errors are desired and when the gradients themselves are small. Compared to the present problem, optimal control problems with solutions published in the literature have solutions with few switches in the control. The multiple switches in the control function and oscillations in the state trajectory result in a number of local minima. A steepest descent algorithm can get stuck in a loop at a local minimum. For the Formulations 2 and 3 it was found that several local minima of the composite cost function existed which did not satisfy the optimality conditions of the original problem. In such cases it was necessary to exit the algorithm and make manual changes before continuing the iterations. 112 Formulation 1 was tried initially since it was a direct application of a simple gradient method to the solution of the original problem. The basic premise of the controllability of the variational system naturally arises herein. A first-order approximation of an increment in the final state due to increments Â§u and $u in JL Â¿i the control and dt^ in the final time is given by (cf., Equation (3.8.11b)) dX(t_) = f(t )dt + f RTf, dt + I RTf, Â¿U dt . - f â€œ f JCÂ° " U1 1 CÂ° â€œ U2 2 If dt = 0 dX(t ) = f RTf, 6u dt + f RTf, 6u dt - f J o - u 1 >Jo - u 2 (3.9.1) It is assumed that Su and Â§u are piecewise continuous. JL Z Let the control increments be the linear combinations 6u^(t) = Â£(t) y and &u2(t) = Y2('t^T - (3.9.2) where V (t) and V (t) are suitable vectors of dimension 6 and y is a constant vector of dimension 6. Equation (3.9.1) may be rewritten as P n t T t> T ' fÂ» t t r> i i dX(t ) = I R f, V dt + R f, Vâ€ž I Jo - uH -1 J o - uâ€ž -2 C 1 C 2 T 1 dt I v . _J - (3.9. 3) It may be seen that a small arbitrary change dX(t^) in the final state is possible if and only if the matrix r"r*T T n T T 1 J R f, V dt + R f, V0 dt (3.9.4) L ÃP - u -1 Jo - u -2 J is nonsingular. 113 Theorem There exist V' (t) and V (t) such that r> T â€ž T , n T, rI Rf, V dt + Rf, V â€¢J.O - u - Ã.O - uâ€ž - dt is nonsingular if and only it f RTf, fT R dt + f RTf, fT R dt (3.9.5) CÂ° U1 U1 CÂ° " U2 U2 is nonsingular. Proof (Extension of Herme's [34] theorem (1)) T T "if pari' is obvious if one chooses V = R f, and V_ = R f, â€” â€” u -2 â€” u 1 12 To prove the "only if part," assume that the matrix f RTf, fT, R dt + f RTf, fT R dt -Â° " U1 " U1 CÂ° â€œ U2 " U2 (3.9.6) 1 2 is singular but the expression (3.9.4) is nonsingular. If the matrix of the expression (3.9.6) is singular, then there exists a vector c ^ 0 such that T c R dt + r Â«J r R dt c 0 (3.9.7) or T ft T _T T T _ -T c R f, f, R dt c + c R f, f, R dt _cÂ° ui " ui - uâ€ž - uâ€ž 2 2 c = 0. Since both the above terms are positive semidefinite, they must be individually zero. Also, the integrands of the above expressions are positive semidefinite for all time. So, we finally have 114 0 almost everywhere on C_ J 1 ,Â° -\ u. 1 (3. 9.8) u 0 almost everywhere on C o 2 T Now, premultiplying the expression (3.9.4) by c and post- multiplying by c and using the results (3.9.8), one obtains or 1 2 is singular. This contradicts the assumption that this matrix is nonsingular and completes the proof of the above theorem. From now on the matrix (3.9.6) will be called the local conÂ¬ trollability matrix, or, simply, the controllability matrix. In all solution attempts with Formulation 1, this matrix was -16 -19 found to have determinants of the order of 10 to 10 with values of the elements ranging from 0.1 to 160.0. The computed values of Â§u were of the order 100 ft-lb in some intervals. The large Â§u values persisted even when small changes in the terminal errors were programmed with no change in the final time allowed. The large Â§u caused instaÂ¬ bility in the iterations. This indicated that the terminal state comÂ¬ ponents cannot be reached simultaneously with control increments of this method for the trajectory constructed previously. Rather than attempt to i'educe the terminal state components simultaneously, it 115 was decided to form a norm of the terminal error to improve the local controllability for reducing the terminal error. This approach was used in Formulations 2 and 3. The cost functional chosen for minimization, the final time, was another source of difficulty. At the minimum time, the terminal states are just barely reachable. The exact minimum time solution cannot be obtained by the steepest descent algorithms like Formulation 1, because the controllability matrix becomes singular for the minimum time traÂ¬ jectory. (Otherwise, a small negative dt^ may be prescribed for which suitable change in the control could be found which would bring the terminal error at t - dt^ to zero. This would contradict the minimality of t^.) Computational difficulties with inversion of this matrix may be expected to appear some time before the actual minimum time is reached. The main problem was found to be the determination of the increment in the final time. The final time itself being the cost function, it is desired that the increment in the final time be negative. On the other hand, convergence of the terminal errors to zero may require an increase in the final time. Unless proper weights are given to these tendencies, the value of the final time may become either larger or smaller than the actual, minimum time of the problem. Which one of the two has occurred cannot be determined as long as the terminal errors are not small. If the errors are acceptably small, there would remain the posÂ¬ sibility that the final time can be reduced further. On the other hand, if the final time has been unknowingly reduced to less than the minimum final time of the problem, and emphasis is greater on the reduction of the final time, the terminal errors will remain large. Thus, the 116 minimum-time solution for a complicated problem is likely to result in considerable computations with these methods. During computations with Formulations 2 and 3, it was found that local minima of the composite cost functional existed. Rapid improveÂ¬ ment of the norm of the terminal error was made in the first few iterÂ¬ ations from a poor control function guess; but the ratÃ© of convergence became extremely slow. The changes in the control and the state traÂ¬ jectories became extremely small between successive iterations once the errors became small. Formulations 2 and 3 do not guarantee reduction of all the terminal errors simultaneously for a given set of the weights K and it was therefore necessary to readjust these weights to reduce all the errors in a reasonable number of iterations. The important events during the computations are now given. Formulation 2 was used first due to the following reasons: a. It is more difficult to make a good guess for a constrained- unconstrained arc-type control than for a control which is free to change everywhere. b. In Formulation 2, the changes in the control are computed everyÂ¬ where on the trajectory, but in Formulation 3 the changes on the unconstrained parts are estimated by linear approximations for small changes in the state variables. It was found that the change in the constraints were large due to small changes in the states. Therefore, the algorithm of Formulation 3 was likely to be less stable, especially when started from a poor guess. c. Formulation 2, after it ceases to give improvement, was expected to result in a good initial guess for starting Formulation 3. 117 Improvement was measured by the trend of the control to a bang-bang control or a stable off-the-constraint arc (for singular subarc) with a simultaneous reduction in the terminal error. Formulation 2 started with a nominal guess shown in Figure 13, with the terminal error Â§ = [1.66, 7.55, -2.12, -15.3, -0.237, 22.5]T. It may be noticed that the final state in the initial guess is far away from the target point. The model has reached only half the required height. It was therefore anticipated that the nominal final time was less than the minimum final time for the problem. Thus, by putting less emphasis on the reduction of the final time (keeping Kq small) the minimum time solution would be approached monotonically with the decrease in terminal error. A local minimum of the cost function was obtained after 50 iterÂ¬ ations. This was determined by examination of the gradient, which was quite small. The quantity B (defined in Equation (3.8.33)), the integral norm of the gradient of the cost functional with respect to the controls 3 had a value of 0.954. This quantity was 1.11 X 10 for the initial guess. The terminal error was not quite acceptable and the control was off the boundary most of the time. The control was changed manually by adding another switch in the control near the final time. After 75 more iterations, computations were stopped when the terminal error Â§ was reduced to [-0.00124, -0.0188, -0.0834, 0.0993, T -0.0819, 0.391] and the convergence rate was very poor. The control at this stage is shown in Figure 14. The final time was 2.0268 seconds, a reduction of 1.23 percent from the initial guess. It was noticed Control (ft-lb) Actual Control Nearest Control Constraint Boundary- Figure 13. Initial Guess for the Control Functions. 118 Actual Control Nearest Control Constraint Boundary Figure 14. A Non-Optimal Control Which Acquires Boundary Conditions. 119 120 that the constraint boundaries had moved far away from their original locations. On the other hand, all the significant changes in the conÂ¬ trol had been small compared to u and occurred in the first 15 iterations. Since the optimal solution should be bang-bang, or, possibly, bang-bang with singular subarcs, the control was modified again by placing the controls on the nearer of the two boundaries. This was done in an attempt to reduce the final time by using the apparent excess control energy. These changes in the control initially increased the terminal errors but also increased the prospects of meeting the terminal condiÂ¬ tions in slightly less time. The terminal error decreased in the subÂ¬ sequent iterations. This time it was found that over several segments, especially on the second half of the control interval, the constraint boundary moved inward but the increments in the control were computed outward. As a result, the actual change in the control used when the control was not allowed to violate the constraints increased the cost functional instead of decreasing it. If the controls were allowed to violate the constraints, the cost decreased. At this point it was noted that the terminal errors were quite sensitive to small changes in the control. The final conditions could be met by adjusting the controls only in the later parts of the traÂ¬ jectory. Slight changes in the earlier parts of the control history caused two problems. First, they caused significant changes in the control boundaries which were in directions opposite to the directions for improved cost. Second, the computed control increments were less accurate near the beginning of the solution time, so that with high 121 sensitivity, improvements were less likely via control increments near the beginning. When no changes were allowed in the earlier part of the control history, the terminal errors reduced to [0.0970, -0.136, -0.0883, -0.0340, -0.0790, -0.0438] at the 17 iteration. The final time was 1.992 seconds. The convergence rate of the cost function was again extremely slow in the last several iterations. The gradient of the cost functional with respect to the control was small and the gradient with respect to a was found to oscillate between positive and negative values during the last several iterations. The control history is shown in Figure 15. This control was considered to be very nearly the minimum time control. From the solution it was clear that there were no singular subarcs and the optimal control would be bang-bang if further refineÂ¬ ment of the solution was made. However, observing the sensitivity of the final time to the control changes from Figure 14 to Figure 15 (about 1% for large control changes) it was believed that no significant reducÂ¬ tion in the final time could be found. Formulation 3 was used to attempt further improvements in the solution and to check the validity of the above assertions. The conÂ¬ trols were modified by placing them exactly on the constraints when they were close to the constraints, and off the constraint arcs were made steeper. The terminal errors initially increased because of these changes. Formulation 3 was found to converge more slowly than FormulaÂ¬ tion 2. This was due to the fact that in Formulation 3 the changes in the control were made on relatively small segments. The changes in the constraints were found to be larger than those of the control increments Control (ft-lb) Actual Control 122 123 in the unconstrained parts. This was because the constraints were very sensitive to changes in the states. The algorithm for Formulation 3 was able to reduce the terminal errors for 90 iterations until it was necessary to stop increments in the first half of the control function. Again the gradient there became th noisy and inaccurate. Computations were stopped at the 120 iteration when the errors were [0.0504, 0.869, -0.0366, 0.0154, 0.0081, -0.113], The errors for the state variables X., , Xâ€ž, X., Xc, and Xâ€ž were accept- 1 o 4 o b ably small and the error for Xwas slightly greater than what was obtained before by Formulation 2. The values of the 6u in the final ten iterations were of the order 0.01 ft-lb. Larger values resulted in divergence or an increase in cost. The control history obtained by Formulation 3 is shown in Figure 16. The final time in the trajectory was 1.992, the same as the final time of Figure 15. The final time was less in the intermediÂ¬ ate iterations, but it increased back to this value. From these results it was concluded that the solution should indeed be bang-bang, but no significant improvement in the final time could be achieved. It may be pointed out that the solution (b) of the Example Problem in Section 3.7, shows the final time 2.025 was only 1.25 percent more than the exact bang-bang solution final time of 2.000. But in the solution (b) the control was off the constraint (during the switch) for 28 percent of the total time. A preliminary investigation of a steepest descent scheme for bang-bang solution is described in Appendix B. Control (ft-lb) Actual Control 124 Angles (Radian) 3.0. 125 126 The difference between the state trajectories for the solutions of Figures 15 and 16 was insignificant. The trajectories for the angles cp, 6, and \jr are shown in Figure 17 for the solution of Figure 15. Examination of the Computed Solution by the Principles of Mechanics The computed solution can be explained qualitatively by considerÂ¬ ing the average motion of the links and the mechanics of rotating systems. We notice that positive controls tend to reduce the radius of gyration of the system at most configurations and hence they tend to increase |cp[ which is the major contributor to the angular momentum of the system. The negative controls do the opposite. The controls remain positive during most of the time in order to obtain increased Jcpj except near the end of a swing, i. e. , at the maximum amplitude. There, the controls switch to negative values in order to limit the swing prior to the final swing and thus save time. It should be noted that the first swing should be just high enough so that with his limited strength, the gymnast is able to swing to the top on the second swing. Thus, during the first two times, the angle 9 reached its maximum amplitudes. There are four switches each in the controls u and u . At the end of the last swing we should not expect JL Z any more switch. However, in order to meet the terminal conditions, the legs had to be brought down, requiring another switch in u . z This switch also helped the angles 0 and cp to reach their terminal values. After this has happened, one more switch each for the controls was needed to meet the terminal conditions on the angular rates. 127 Without any terminal conditions on the angular rates, the solution could have been obtained without the last switch in each of the controls and in slightly less final time. 3.10. Comparison of the Minimum-Time Solution With Experiment A comparison between the control histories of the computed solution and the maneuver of the gymnast recorded in the experiment with the same boundary conditions was not possible because the control forces used by the gymnast could not be determined. The photographic measurements of the angles were not accurate enough for two successive numerical differentiations needed for finding the control torques for the actual rigid body motion from the equations of motion. The time histories of the generalized coordinates cp, 9, and i|f for the experiment and the computed solution are given in Figures 5 and 17. The trajecÂ¬ tories of 9 and iji are quite different in the two cases. The differÂ¬ ences are primarily due to the differences between portions of the conÂ¬ trol limit functions used in the mathematical model and the actual strength of the gymnast. During the simulation runs it was found that the model's arm strength was insufficient for small values of 9. Thus, the actual state trajectory was not a good guess for starting. A nomÂ¬ inal trajectory had to be determined which was more nearly correct for a gymnast with less strength at extremes of his limb movement. It may be noticed that in the measurements of the angle 0 of the kip-up motion, the angle 9 remained non-negative throughout the motion, whereas in the computed minimum time solution 9 became negative for a short duration, with a minimum value of -.25 radians. This 128 discrepancy is largely due to inaccuracy in modeling. The modeling inaccuracies include modeling the head-torso system as a rigid link, and inaccuracies in the inertia parameters and the functions describÂ¬ ing the constraints on the control torques. From the physical considerations, it can be seen that raised arms can indeed bend more than .25 radians behind the torso. But then the bending is shared by the deformation of the torso, as demonstrated in Figure 18. Under such conditions there will be differences in the angles 9 and measured from the tape strips and the corresponding angles of the model based on lines between joint centers. This is illustrated in Figure 18. In the optimization procedure, an inequality constraint on the angle 9 v/ould be able to keep 9 from being less than a prespecified value. Such a constraint was not considered in this investigation for several reasons. First, it was thought that if the strength measureÂ¬ ments (the control limits) were accurate, the states would remain within acceptable bounds naturally and without requiring a constraint on the states imposed externally. Second, the exact limits and the nature (hard or soft) of the constraints on the angles need further experÂ¬ imental investigations. And, lastly, it was anticipated that the violation of the state constraint, if any, would not be significant enough to justify going into the complications associated with hard state constraints. 129 Figure 18. Difference Between Measured Angle and Mathematical Angle for Human Model Due to Deformation of Torso. CHAPTER 4 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 4.1. Conclusions The results of an investigation involving application of rational mechanics and optimal control theory to a human motion probÂ¬ lem have been presented. The kip-up maneuver of gymnastics was selected for this purpose. First, a three-link model was constructed for the human performerâ€”a professional gymnast. The dynamic propÂ¬ erties of the model were compared with those of the gymnast by means of experiments. A minimum-time strategy for the kip-up maneuver was then obtained by numerical computation for the mathematical model. The solution obtained was compared with an actual minimum time maneuÂ¬ ver performed by the gymnast. It was found that the human being may be modeled by rigid links for the purpose of dynamic analysis of human motion and man-machine systems. However, the model cannot be tested effectively unless accurÂ¬ ate measurements of the angular velocities and angular accelerations along with the angles are made. Photographic determination of the limb positions is not accurate enough for determining within the uncertainÂ¬ ties of the model the derivatives of the movements involved. 130 131 Modifications were needed for the shoulder and hip joints to account for the stiffness and the deformation of these joints during extreme arm and leg movements. An accurate knowledge of the maximum strength of the person being modeled at the junctions of the various body segments is necesÂ¬ sary for correctly determining the optimal performance. For this reason, these measurements must be taken at small intervals of the joint angles and for the entire range of the possible movements so that a polynomial representation or interpolation of these limits with respect to the angles is sufficiently accurate. In the present invesÂ¬ tigation, the control limit functions used did not represent the true strength of the gymnast at the extremities of the arm movement, and as a result, a significant difference in the actual maneuver of the gymnast and the optimal motion of the mathematical model was obtained. After an accurate mathematical model has been constructed, the optimal motion has to be determined by numerical computation. The analytical determination of the minimum time kip-up maneuver involved working with a significantly nonlinear system with control constraints which depend on the state variables. Several numerical methods were used to obtain the solution. A modified quasilinearizaÂ¬ tion method was tried first but the attempt was unsuccessful. First- order steepest descent methods were then used with three different formulations. A nearly optimal numerical solution was obtained from these formulations. The solution was compared with experimental results. 132 It was found that the method of quasilinearization was not suitable for the highly nonlinear human motion problems involving multiple switches in the control. It was also noted that the method would not be efficient for problems which may admit singular solutions. It (or, any other method which determines the optimal solution via solution of a two-point boundary value problem of the state and adjoint variables) uses the optimal expression for the control variables in terms of the state and adjoint variables. Several problems would be faced in determining the optimal control when singular arcs are to be considered. First, the equalities in the necessary conditions for singular subarcs would be difficult to verify in the limited precision of digital computations. Second, sufficiency conditions for optimality of singular arcs for a general nonlinear system have not yet been estabÂ¬ lished. Lastly, if the initial guess of the state and adjoint variables is poor, then it may happen that in an intermediate iteration the state and adjoint variable histories correspond to a singular control arc on a certain time interval, but they do not do so for the same time interval in the next iteration. In such a case the iterations are likely to be unable to improve the solutions. The method of quasilinearization, or, likewise, any second-order method requires more computation and storage than first-order methods. For the human motion, the method of quasilinearization required an excesÂ¬ sive amount of computing time per iteration and, more critically, nearly maximum storage. The second-order methods were therefore not considered further. 133 Existing first-order steepest descent methods are not efficient for solving such problems either. Though the solution was obtained by these methods, simulation and physical reasoning played an important role. A Davidon-Fletcher-Powell scheme of a modified gradient method was also used but without improvement in the results. The method of constrained and unconstrained arcs is not effecÂ¬ tive as a starting method when the control constraints of the constrained and unconstrained arcs are not known in advance. The effectiveness of the constrained-unconstrained arc method depends on the sensitivity of the control constraints to changes in the state variables. Its converÂ¬ gence is poor when the constraints are sensitive to changes in the states. On the other hand, treating the control as entirely free while computing the gradients and later imposing the constraints while impleÂ¬ menting the control is a much simpler scheme and converges more rapidly when a significant portion of the control history is away from the conÂ¬ straints. The method becomes ineffective (and theoretically incorrect) when controls are mostly near the constraints and the movement of the constraints becomes significant. Each of the steepest descent algorithms requires a good initial guess when several switches occur in the optimal control. They may not be able to produce a required switch in the control in problems with multiple control switches. In Formulations 2 and 3 of the steepest descent method, the adjustment of the final time was done by adjusting the state variable a introduced by changing the independent variable from t to T, where t=or. 134 This change was expected to improve the performance of the steepest descent method with free final time because a appeared in the state equations. This approach was not found to be more efficient than the conventional method of changing the final time, i.e., by extending or truncating the final end of the trajectory. The gradient of the norm of the terminal error with respect to This required the change Z>a be kept extremely small in iterations, using nearly optimal results. It was significant that, during the first manual change of the control after achieving a local minimum, it was necessary to increase a to its original value of 1.0 (it had dropped to 0.977) and truncate the trajectory. This adjustment is the same as the conventional method of adjustment of the final time. 5 For the relatively quick kip-up maneuver, 1.36 X 10 bytes of storage (with double precision arithmetic) and a computing time of about six seconds per iteration (for IBM 370-165 Computer, Fortran G) were required for Formulations 2 and 3 of the steepest descent method. 4.2. Recommendations for Future Work The following areas need to be considered for future work in the study of dynamics and analytic determination of human motions: 1. Improvement of the Inertia Model of Hanavan and Data Gathering Experiments: The most significant inaccuracy in the three-link inertia model for the kip-up maneuver was observed to be the deformation of the shoulder and bending of the torso during certain periods of the motion. An accurate determination of the average location of the joint centers 135 at the shoulder and hip is sufficient for modeling the middle link of the three-link kip-up model. The stiffnesses of the hip and shoulder joints during extreme arm and leg movements should also be determined by experiment. The bending of the torso might be accurately modeled by dividing the torso-head link into two links. The upper link might consist of the head and upper torso with the lower torso of the Hanavan model. These two links might be joined by a smooth hinge with a torsional spring and a voluntary torque. The introduction of the additional degree of freedom to take care of the deformation of the torso will increase the complexity of the dynamical equations of the system further. Work needs to be done to improve the data gathering techniques, that is, determination of the generalized coordinates and their rates. Photographic measurements are extremely time consuming and inaccurate. Preliminary evaluations [35] show that the angular differentiating accelerometers (ADA) will provide the desired rate and angle accurÂ¬ acies. The devices are quite small and have a low power requirement. With a light weight telemetry system the human performer is not encumÂ¬ bered by wiring. Once accurate determination of the angles and their rates is possible, the inertia model may be improved by using the experimental data and some of the system identification techniques. 2. Measurement of the Muscle Torques: An accurate knowledge of the strength of the individual is needed in the analytic determination of a given maneuver. It needs to be investigated how' the strength at a joint depends on the rate of 136 contraction of the muscle group and the time for which the force acts. The actual muscle forces or their equivalent torques during a maneuver should also be determined in order to have a better understanding of the muscle activities. If accurate information of the inertia properÂ¬ ties and the angles and their rates is available, these forces can be obtained from the equations of motion. However, determination of these forces using biomedical devices such as electromyography electrodes might give valuable information in this area. 3. Determination of Better Numerical Techniques for Human Motion Problems: It was clear from the present investigation of the two- second duration kip-up maneuver that analytic solution of more compliÂ¬ cated and longer duration motion would be extremely difficult. ConsiderÂ¬ ation of state-constraints may be necessary to ensure that the stresses generated during the motion at any part of the human body remain within certain limits. Also, constraints should be imposed on the rate of change of the control forces because a human being may not be able to execute bang-bang control. These constraints will further complicate the optimization problems and introduce interesting biomechanical questions. It therefore seems that some numerical methods should be found which would give an approximate extremum (i. e. , not give an exact extremum) but would be efficient in handling the constraints and other complications of the human motion problem. APPENDIXES APPENDIX A DETERMINATION OF THE INERTIA PARAMETERS OF THE KIP-UP MODEL FROM THE HANAVAN MODEL The three elements of the kip-up model were constructed by- combining the elements of the Hanavan model. The inertia properties of the kip-up model were obtained from geometric considerations and the parallel axes and rotation theorems of moment of inertia. Figure 19 shows the three elements of the kip-up model and their comÂ¬ ponents from the Hanavan model. The dimensions of the Hanavan model, including the location of the hinge axes A and B, could be obtained from the program of Hanavan. Input of this program were the 25 anthroÂ¬ pometric dimensions taken from the subject. The formulas for the varÂ¬ ious dimensions of the elements of the kip-up model are presented below. These formulas were added to Hanavan's program in order to obtain the inertia parameters of the kip-up model. SM(8) [R(4)+{l-T](8)}SL(8)]cos y + SM(6) [R (4) + SL (8)+ f 1-T) (8) }SL(6)]cos y Tl~ SM(4) + SM (8) + SM (6) i = [R(4) + SL(8) + SL(6) - R(6)] cos y m = 2[SM(4) + SM(8) + SM(6)] | I = SIYY(4) + SM(4)rJ + i[SIYY(8) + SIZZ(8) + (SIYY(8) - SIZZ(8)} cos 2v] + SM(8) [r -ÃR(4) + SL(8) - T] (8) SL(8) }cos y] 2 +|[SIYY(6) + SIZZ(6)+ {SIYY(6) -SIZZ(6)]cos y] + SM(6)[{R(4>+ SL(8) 2 + SL(8) + SL(6) - ~ (6)}cos y - r ] 138 139 Figure 19. Construction of Kip-Up Model from Hanavan Model. 140 - -SM(1)R(1) + SM(2)7](2)SL(2) + SM(3) [SL(2)+7] (3) SL(3) ] _ I-2 ~ SM(1) + SM(2) SM(3) ^ J t, = SL(2) + SL(3) - DELSH - R(6) m = SM(1) + SM(2) + SM(3) Zi I2 = SM(1)[R(1) + R(6) + r ]2 + SIYY(l) + SM(2)[R(6) + ^ - 11(2) SL(2)]2 + SIYY (2) + SM(3) [R (6) + *2 ~ SL(2) - T| (3)SL(3)]2 + SIYY (3) l = DELSH + |sM(10)l](10)SL(10) + SM(12) [SL(10) + T)(12)SL(12) + SM(14)[SL(10) + SL(12)+R(14)]}/{SM(10) + SM(12) + SM(14) } m = 2[SM(10) + SM(12) + SM(14)] Ã¼ i I_ = SIYY(10) + SM(10) [ i - DELSH - 7| (10) SL (10) ] 2 Z o + SIYY (12) + SM(12)[i - DELSH - SL(10) - T] (12) SL(12) ]2 Ã¼ + SIYY(14) + SM(14)[SL(10) + SL(12) + R(14) - + DELSH]2 141 Anthropometric Dimensions of the Subj ect Dimension Value (inches) 1. Ankle Circumference 9. 25 2. Auxiliary Arm Circumference 13.0 3. Buttock Depth 8.125 4. Chest Breadth 12.75 5. Chest Depth 9. 25 6. Elbow Circumference 11.0 7. Fist Circumference 11.0 8. Forearm Length (Lower arm length) 10.75 9. Foot Length 10.0 10. Knee Circumference 13.5 11. Head Circumference 22. 5 12. Hip Breadth 12.0 13. Shoulder Height (Acromial Height) 54.0 14. Sitting Height 34.75 15. Sphyrion Height 3.5 16. Stature 66. 5 17. Substernal Height 46.5 18. Thigh Circumference 20.625 19. Tibale Height 16.125 20. Trochanteric Height 34. 25 21. Upper Arm Length 12.0 22. Weight 138 lb 23. Waist Breadth 10. 25 24. Waist Depth 7.625 25. Waist Circumference 6.75 142 Inertia Properties of the Kip-Up Model 22.29 17.68 inches inches rl= 12. 68 inches ^ r2 = 2.57 inches > II ro u 12.32 inches â€l = 0.47 slug ^ 3 to II 2. 39 slug ) m3 1.42 slug J 28. 20 208.98 140.90 slug-in slug-in slug-in lengths of the elements 1 and 2 distances of the CG's of elements 1, 2, and 3 from their upper hinge points mass of the elements 1, 2, 3 moments of inertia about the centers of gravity for elements 1, 2, 3 APPENDIX B AN INVESTIGATION OF A STEEPEST DESCENT SCHEME FOR FINDING OPTIMAL BANG-BANG CONTROL SOLUTION FOR THE KIP-UP PROBLEM An investigation was made to determine the suitability of the steepest descent algorithm of Bryson and Denham [28] for determining optimal bang-bang control for the kip-up problem. The investigation, described in this section, showed that extremely small changes in the switching times were necessary for the first-order relationship between the changes in the switching time and changes in the terminal state vector to be valid. The small changes in the switching time ruled out the use of fixed integration step size as used for the continuous control. The change of the switching time by one integration step was found to be much too large for the first-order relationship to be valid. Therefore, an attempt to find a bang-bang optimal control by using a fixed step size of 0.0036 second (with a total of about 550 inteÂ¬ gration steps) was not successful. Even though a scheme for avoiding use of fixed integration step size was available, the method was not pursued further because of anticipated numerical problems. The imporÂ¬ tant results of the investigation are now presented in more detail. Suppose each of the controls u (t) and u (t) is on either the -L Z upper limit or lower limit for all time. Also, let the control u (t) , i = 1 or 2, have the switching times t^,t^,...jt1 , i.e., at these times 12 N. i the control switches from an upper to a lower limit or vice versa. 143 144 We shall first find out the change in the final state vector due to a small change in one of the switching times t*. For this we can use the results of Section 3.8, Equation (3.8.11) which is written as ^X(t ) = f(t)dt + RT(0)6X(0) + f RTf, Gu dt + f RTf, 6uâ€ž dt . (B.l) - f - 1 - JcÂ° - U1 1 JcÂ° - u2 2 In this equation CÂ° and CÂ° denote the unconstrained portions of the control trajectories u^(t) and u^it), respectively. In the present case the unconstrained arcs are the portions of the control trajectories which change from one constraint to another due to the shift of the switching times. They are therefore around the switching times and are not known a priori. But, since the shifts in the switching times will o be made small in any iteration, the presence of the integrals over and CÂ° can be ignored as second-order terms while generating R(t) by backward integration of Equations (3.8.10). The value of R(t) will therefore be the solution of the following equations: where R(tf) (6x6 unit matrix) J R - -fÃ'X + Ã'u, SÃ,x+1-â€™uâ€ž 4JTr (B.2) (B. 3) 'X j = 1 for u^ on the lower constraint j = 2 for u^ on the upper constraint K = 1 for u on the lower constraint Â¿-j (B.4) K = 2 for u^ on the upper constraint . Suppose a small change Â§x(0) at the initial condition of the state vector X(0) is prescribed so that the initial state becomes 145 X + Â§X(0). Also, let the switching times remain unchanged, and the -o - final time changes from tf to t + dtf. The resulting change in the final values of the state variables can be obtained by setting Â§u = 0 in Equation (B.l), which yields Â¿X(tf) = RT(0) ftX(O) + f(t ) dtf . (B.5) If at any other time t , 0 Ã¡ t ^ t , the state vector X(t1> is changed by the amount 6x(t ), the change in the final state vector, if the same switching times are maintained, will similarly be given by Ã©X(tf) = RT(t1) SXC^) + f(tf) dtf . (B. 6) Now consider a change in the switching time t^. Suppose the 2 1 control u switches from to at this switching time, and that this switching time will be advanced by the amount dtj. All other switching times remain unchanged. Consider the value of the state vector X(t) at the time t^ + dt^. Before the change in the switching time, we have, to the first order X(t*+6t*) X(tJ) + X(t^) â€¢j dt (t^ means just after t^) = x(t1) + X(t^) + A + f, uâ€ž + f, uâ€ž - ~ \ 1 " u2 2 A + f, S1 + f, u - - U1 1 - U2 2 1+ dtÃ 1 dtl ( B. 7 ) where the vector A(X) is defined in Equation (2.2.20). 146 After the change in the switching time, we have, similarly, X(t* + Ã“t^) = X(t^) + A + f, S2 + f, u0 " - U1 1 - U2 2 (B. 8) From (B.7) and (B.8), we obtain the change in the state vector at time t^ + dt^ due to the change in the switching time. Ã“X(t* + 6t^) = f,u (S2 - SJ) dt* (B. 9) 12 12 If the switch was from to we would have obtained (S^ - S^) 2 1 1 instead of (S^ - S^) in Equation (B.9). If dt^ was negative the relaÂ¬ tion (B. 9) would still be valid. Corresponding to the change in the state vector at t* + dt* given by Equation (B.9), the change in the final state vector given by Equation (B.6) is 6x(tf) = RT(t*+6t*)f,u (S2-S*)dt* + f(tf) dtf or, to the first order ax(tf> = f(tf)dtf + RTiUl(srsÃ> i dti (B.10) In the nominal trajectory, suppose switches from to occur at t* t* t*,... and switches from S* to S2 occur at 13 5 i i t*, t* t^,..., for i = 1 and 2. The result (b.10) can be extended 2 4 5 for the case when any of the switching times t1 is advanced by dt1. This will yield N, 6x(t ) = f(t )dt + Z (-1) i=l Nâ€ž + Z (-1) i-1 J-l T 2 1 R Ã'u (V5? 1 1 dti 1 . 1 rTÃ.u <Â£<$> 2 dt2 ,2 j (B.11) 147 The relations (B.10) and (B.ll) are exactly the same as the gradient relations given in [24] for bang-bang control, but are derived differently. This derivation brings out the fact that the change in the final state vector will be given by the linear relations only if the change in the state vector at a switching time caused by the change in switching time is small enough to make the relation (B.6) valid. To find the maximum value of )6x(t1)j valid for use in expresÂ¬ sion (B.6), the following test was performed. First, with one forward and one backward integration, with a nominal trajectory, the value of R(0) was determined. The initial time t = 0 was used for t . Several values of 6x(0) were chosen and for each initial state vector given by Xq + ^x(O), the state equations were integrated forward with the same switching times and with dt^=0. The final values of the state vector X(tf) obtained by the integration were used to find the actual changes in the final values of the state vector from the value of the nominal trajectory. These actual changes T were then compared with the predicted change R(0) Â§X(0). From these comparisons it was found that 6x(0) with elements of the order of 0.01 was too large. The relation (B.6) was valid for a much smaller (sÂ« 0.001) 6x(0). On the other hand, the elements of Â§X(t) caused by a change in the switching time by one integration step of 0.0036 second used in Sections 3.7 and 3.8 were found to be of the order of 0.1 to 6.0â€”much larger than what could be used in Equation (B.6). From the matrix R(0), which had large entries, it was clear that the change in the switching time should be much smaller. 148 In order to find an optimal bang-bang control by this method, the following features in the determination of the influence function R(t) should be incorporated. 1. A nominal guess consisting of the switching times and a final time is to be made first. The switching times specify the control completely by telling which constraints the control is on. 2. Integrate the state equations forward with the initial value X^. Generate u from the constraints. Store X(t^). It is not necessary to store X(t) and u(t) for any other t. 3. Integrate the state equations backwards and the influence Equations (B.4) simultaneously. Use final value of the state variables as those obtained from the forward integration. Generate control from the constraints. Generating the state and the control variables again during the backward integration (they were generated once in step 2) is necesÂ¬ sary to eliminate interpolation errors. This feature however will not introduce significantly more computations when the control lies always on a constraint. LIST OF REFERENCES 1. Marey, M., "Des Mouvements que Certains Animaux Executent pour Retomber sur Leurs Pieds, Lorsqu'ils Sont Precipites d1un Lieu Eleve," Comptes Rendus, Vol. 119, 1894, pp. 714-717. 2. Guyou, M., "Note Relative a la Communication de M. Marey," Comptes Rendus, Vol. 119, 1894, pp. 717-718. 3. Magnus, R. , "Wie Sich die Fallende Katze in der Luft Undreht," Arch. Neerlandaises die Physiologie, Vol. 7, 1922, pp. 218-222. 4. McDonald, D. A., "The Righting Movements of a Free Falling Cat," J. Physiol., Vol. 129, 1955, pp. 34-35. 5. McDonald, D. A., "How Does a Falling Cat Turn Over?" St. Berth- domew's Hospital J., Vol. 56, 1955, pp. 254-258. 6. McDonald, D. A., "How Does a Cat Fall on Its Feet?" New Scientist, Vol. 10, 1961, pp. 501-503. 7. Amar, J., The Human Motor, George Routledge, London, 1920. 8. Fischer, 0., Theoretische Grundlagen fur Eine Mechanik der Lebenden Korper, B. G. Teubner, Leipzig, 1906. 9. McDonald, D. A., "How Does a Man Twist in the Air?â€ New Scientist, Vol. 10, 1961, pp. 501-503. 10. McCrank, J. M. and Segar, D. R., "Torque Free Rotational Dynamics of a Variable-Configuration Body (Application to Weightless Man)," M. S. Thesis, Air Force Institute of Technology, 1964. 11. Smith, P. G. and Kane, T. R., "On the Dynamics of the Human Body in Free Fall," Journal of Applied Mechanics, Vol. 35, 1968, pp. 167- 168. 12. Ayoub, M. A., A Biomedical Model for the Upper Extremity Using Optimizing Techniques, Ph.D. Thesis, Texas Tech UniverÂ¬ sity, 1971. 13. Santschi, W. R., DuBois, J., and Omoto, C., "Moments of Inertia and Centers of Gravity of the Living Human Body," Aerospace Medical Research Laboratory, AMRL-TDR-63-36, 1963. 149 150 14. Hanavan, E. P., "A Personalized Mathematical Model of the Human Body," Aerospace Medical Research Laboratory, AMRL-TR-102, 1964. 15. Hanavan, E. P., "A Personalized Mathematical Model of the Human Body," AIAA Paper No. 65-498, 1965. 16. Samras, R. K. , Muscle Torque Measurements, Effects on Optimal Motion, and Human Performance Evaluation, M. S. Thesis, University of Florida, 1971. 17. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidge, R. V., and Mishchelenko, E. G., The Mathematical Theory of Optimal Processes, Interscience Publishers, New York, 1962. 18. Bryson, A. E. , Denham, W. F., and Dreyfus, S. E. , "Optimal ProgramÂ¬ ming Problems with Inequality Constraints. I: Necessary Conditions for External Solutions,â€ AIAA Journal, Vol. 1, 1963, pp. 2544-2550. 19. Tait, K., "Singular Problems in Optimal Control," Ph.D. Thesis, Harvard University, 1965. 20. Kelley, H. J. , Kopp, R. E. , and Moyer, A. G. , "Singular Extremals," Topics in Optimization, G. Leitmann (ed.), Vol. II, Chapter 3, Academic Press, New York, 1966. 21. Robbins, H. M. , "Optimality of Intermediate-Thrust Arcs of Rocket Trajectories," AIAA Journal, Vol. 3, 1965, pp. 1094-1098. 22. Goh, B. S., "Necessary Conditions for Singular Extremals Involving Multiple Control Variables," SIAM, Journal on Control, Vol. 2, 1964, pp. 234-240. 23. Jacobson, D. H., "Totally Singular Quadratic Minimization Problems," IEEE Transactions on Automatic Control, Vol. AC-16, No. 6, 1971 (Special Issue on Linear-Quadratic-Gaussian Problem), pp. 651-658. 24. Bellman, R. E., and Kalaba, R. E., Quasilinearization and Nonlinear Boundary Value Problems, The RAND Corporation, Report No. R-438-PR, 1965. 25. Sylvester, R. J. , and Meyer, F., "Two-Point Boundary Value Problems by Quasilinearization," SIAM, Journal on Applied Mathematics, (2) 13, 1965, pp. 586-602. 26. Boykin, W. H., Jr., and Sierakowski, R. L., "Remarks on Pontryagin's Maximum Principle Applied to a Structural Optimization Problem," The Aeronautical Journal of the Royal Aeronautical Society, 1972, pp. 175-176. 151 27. Bryson, A. E., and Denham, W. F., "A Steepest Ascent Method for Solving Optimal Programming Problems," Journal of Applied Mechanics, 1962, pp. 247-257. 28. Bryson, A. E., and Denham, W. F., "Optimal Programming Problems with Inequality Constraints. II: Solution of Steepest-Ascent,' AIAA Journal, Vol. 2, No. 1, 1964, pp. 25-34. 29. Long, R. S., "Newton Raphson Operator; Problems with Undetermined End Points," AIAA Journal, Vol. 3, No. 7, 1965, pp. 1351-1352. 30. Miele, A., and Iyer, R. R., Modified Quasilinearization Method for Solving Nonlinear Two-Point Boundary-Value Problems, Rice University Aero Astronautics Report No. 79, 1970. 31. Miele, A., Iyer, R. R., and Well, K. H., Modified Quasilinearization and Optimal Initial Choice of the Multipliers, Part 2â€”Optimal Control Problems, Rice University Aero Astronautics Report No. 77, 1970. 32. Bryson, A. E., and Bo, Y. C., Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969. 33. Wong, P. J., Dressier, R. M., and Luenburger, D. G., "A Combined Parallel-Tangents/Penalty-Function Approach to Solve Trajectory- Optimization Problems," AIAA Journal, Vol. 9, No. 12, 1971, pp. 2443-2448. 34. Hermes, H., "Controllability and the Singular Problem," SIAM, Journal on Control. Ser. A, Vol. 2, No. 2, 1965, pp. 241-260. 35. Little, J. L., The Design and Analysis of a Human Body Motion Measurement System, M. S. Thesis, University of Florida, 1972. BIOGRAPHICAL SKETCH Tushar Kanti Ghosh was born on August 17, 1945, in Calcutta, India. He was graduated in the year 1962 from Ballygunge Government High School, Calcutta. In the same year, he entered the Indian Institute of Technology, Kharagpur, to study Mechanical Engineering. He received the degrees Bachelor of Technology in Mechanical EngineerÂ¬ ing with First Class Honors, and Master of Technology in Machine Design in the years 1967 and 1969, respectively. He came to the University of Florida to pursue further studies in dynamics and controls. He received the degree of Master of Engineering from the Department of Engineering Science and Mechanics in the year 1970. Since then he has been working on his Ph.D. degree which will be awarded in March, 1974, from the same department. 152 I certify that I have read this study and that in ray opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of .Doctor of Philosophy. Associate Professor of Engineering SciÂ¬ ence, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. <Â¡Â£Â± Ctcoc/u2^T^L-Z. Lawrence E. Malvern Professor of Engineering Science, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Engineering Science, Mechanics and Aerospace Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'S/S VS Ulrich H. Kurzweg â€™ Associate Professor of Engineering SciÂ¬ ence, Mechanics and Aerospace Engineering I certify^ that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas Ã‰. Bullock Associate Professor of Electrical EngineerÂ¬ ing I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Tarek M. Khalil Assistant Professor of Industrial and Systems Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. March, 1974 ngineermg Dean, Graduate School UNIVERSITY OF FLORIDA I III III IIIÂ» inn ii iii" "â€™Â¿'Wn 3 1262 08556 7450 |