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University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8400231 Katbab, Abdollah THREE-DIMENSIONAL TORSO MODEL WITH MUSCLE ACTUATORS The Ohio State University Ph.D. 1983 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Three-Dimensional Torso Model with Muscle Actuators DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University by Abdollah Katbab, B.S.E.E., M.S.E.E *■** + * The Ohio State University 1983 Reading Committee: Professor H. Hemami Professor A. Keyhani Professor F.C. Weimer Approved by Advi sor Department of Electrical Engineering ACKNOWLEDGEMENTS I wish to express my appreciation to those who have aided me along the way in my years as a graduate student at The Ohio State University. I especially thank my committee chairman and advisor Professor H. Hemami, who has consistently and patiently offered assistance, encouragement, advice, and criticism , and who has afforded me a unique opportunity to define as well as execute this research program. I would like to express my most sincere thanks to the chairman of the Department, Professor H.C. Ko for his continuous support and assi stance. I wish to express my thanks and gratitude to Professor F.C. Weimer and Professor A. Keyhani for reviewing this dissertation, I appreciate their understanding and patience. I am most grateful to my wife, Soheila, for her outstanding assistance, patience, understanding, and confidence in me. Without her excellent care of our children, I would not have been able to finish this work. I am also thankful to my parents for their patience and encouragement. I would like to thank Emily Baird and Jacqueline Buckner for their excellent typing of this manuscript. This work was supported by the Electrical Engineering Department of The Ohio State University and the National Science Foundation under Grant No. ECS 820 1240. VITA November 11, 1951 Born: Shiraz, Iran 1975 B.S.E.E., Pahlavi University, Shiraz, Iran 1975 - 1978 Instructor, Technical School of Electronics, Pahlavi University 1979 - 1983 Research and Teaching Associate Department of Electrical Engineering The Ohio State University, Columbus, Ohio, USA 1980 M.S., Department of Electrical Engineering The Ohio State University PUBLICATIONS Hemami, H., A. Katbab: Constrained Inverted Pendulum Model for Evaluat ing Upright Postural S tability, Journal of Dynamic Systems, Measurement, and Control, Vol. 104, No. 4, 1982, pp. 343-349. FIELDS OF STUDY Major Field: Electrical Engineering Studies in Digital Systems: Professor H. Hemami Professor K.J. Breeding Studies in Control Systems: Professor F.C. Weimer Studies in Power Systems: Professor A. Keyhani Studies in Statistics: Professor J. Klutz i i i TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii VITA 111 LIST OF TABLES vii LIST OF FIGURES viii CHAPTER I INTRODUCTION 1 1.1 Overview 1 1.2 Organization 2 II BACKGROUND AND RELATED RESEARCH 3 2.1 Introduction 3 2.2 Kinematics and Dynamics 3 2.3 Control Mechanisms 5 2.4 Neuromuscular System 7 2.5 Human Spine Modeling and Trunk Muscles 9 2.6 Summary 10 III CONSTRAINED INVERTED PENDULUM MODEL FOR EVALUATING UPRIGHT POSTURAL STABILITY 11 3.1 Introduction 11 3.2 State Space Equations of Interconnected Rigid Bodies 11 3.2.1 One Link Model 11 3.2.2 Two Link Model 16 3.2.3 Two Link Model with Further Constraints 19 iv Table of Contents, continued. CHAPTER page 3.3 Calculation of the Holonomic Forces 21 3.3.1 One Link Model 21 3.3.2 Two Link Model 24 3.4 Elimination of the Holonomic Forces of Constraint 26 3.5 Lyapunov S tability 28 3.6 Further Constraint on One Link Torso 31 3.7 Elimination of the Nonholonomic Forces of Constraint introduced in Section 3.6 35 3.8 Simulations 36 Case 1 Lyapunov Simulation 37 Case 2 Preventing Self Rotation with Soft Constraint 42 Case 3 Preventing Self Rotation with Hard Constraint 42 3.9 Summary 43 IV GAIN PROGRAMMING IN THE VOLUNTARY POINT-TO-POINT MOVEMENT OF THE HUMAN FOREARM 44 4.1 Introduction 44 4.2 Physical Model 46 4.3 Gain Programming 48 4.4 Simul ations 53 4.5 Summary 62 v Table of Contents, continued. CHAPTER page V VOLUNTARY POINT-TO-POINT MOVEMENT OF THE ONE RIGID BODY MODEL OF THE HUMAN SPINE 64 5.1 Introduction 64 5.2 Anatomy and Functional Role of the Human Trunk Muscles 64 5.3 One Rigid Body Model of theHuman Spine 67 5.4 Stability Analysis of the Human SpineModel 75 5.5 Point-to-Point Motion 79 5.6 Summary 96 VI CONCLUSIONS 97 APPENDIX A 100 APPENDIX B 102 REFERENCES 104 vi LIST OF TABLES TABLE Page 3.1 Numerical parameters of the model 30 4.1 Numerical parameters of a normal human forearm 52 4.2 Variation of the Eigenvalues of the forearm model with recruitment gain parameter 52 4.3 Muscle model parameters 55 4.4 Neuromuscular variables for point-to-point motion of the forearm under the different cases 55 5.1 Muscle model parameters 78 5.2 Variation of the Eigenvalues of the closed loop system with muscle spindle velocity feedback gain; Bsp 78 5.3a Open loop Eigenstructure of the torso 80 5.3b Closed loop Eigenstructure of the torso. 80 5.4 The insertion and origin coordinates of the torso muscles in the three dimensional space 95 v ii LIST OF FIGURES Page Torso as an inverted pendulum 12 Two innerconnected rigid bodies 17 Two link model of the elbow and knee joints 20 Model of the dynamics of the torso 25 The hand axes 32 Prevention of Rotation 32 Projection of the ellipsoid and two restraining cavity surfaces in the XZ plane. 33 Trajectory of angular displacement (<p,<(»,©) 38 Trajectory of the angular velocity W 38 Ground reaction forces at the base of the torso 39 Trajectory of feedback torque N 39 Lyapunov function and its derivative as functions of time 40 Angular displacement of the torso with soft constraint 40 Trajectory of angular displacements of the torso with hard constraint 41 Hard constraint forque t applied to the torso as a function of time 41 Two-muscle forearm system 45 Mechanical Model of muscle 45 Block diagram of the stretch reflex system and model of dynamics of the human forearm 49 viii Schematic showing anatomic connections between physiological components that participate in stretch reflex 50 Angular position e and velocity e of the forearm 56 Flexor muscle spindle afferent output 57 Flexor (Ff) and extensor (Fe ) muscle forces 58 Muscle spindle velocity gain, BSp 59 a) Spindle gain amplification factor; b) flexor muscle spindle output before amplification ( 1 ) and after amplification ( 2 ); c) angular position and velocity of the forearm when neural fibers have time delay 60 Flexor muscle (Ff) and extensor (Fe ) forces, when an external force of 2 Newtons is acting on the forearm at wrist resisting the extension 61 Sacrospinal is (a) and obliquus externus (b) muscles 65 Rectus abdominis (a) and obliquus externus (b) muscles 66 Obliquus externus (a) and rectus abdominis (b) muscles 68 Obliquus internus (a) and rectus abdominis (b) muscles 69 Torso in the sagittal (a), coronal (b), and transversal (c) planes 71 Torso in the sagittal plane at a flexed position 73 Torso in the sagittal plane with muscles at rest 74 The controller block diagram 82 Reference position (eref) and velocity (eref) of the torso in the sagittal plane 83 Actual and reference angular positions of the torso in the sagittal plane 83 ix FIGURE Page 5.11 Actual and reference angular velocities of the torso in the sagittal plane 84 5.12 Rectus abdominis (Fr ) and sacrospinalis (Fs ) muscle forces 84 5.13 Muscular torque acting on the torso in the sagittal plane 85 5.14 Sacrospinalis muscle spindle output

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