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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 ( 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 in the sagittal plane 85 5.15 Bias input (Ut>) and control input (Uc) of the torso in the sagittal plane 86 5.16 Actual and reference angular positions of the torso in the sagittal plane 87 5.17 Actual and reference angular poistions of the torso in the coronal plane 87 5.18 Angular positions of the torso 88 5.19 Angular velocities of the torso 88 5.20 Rectus abdominis (Fr ) and sacrospinalis (Fs ) muscle forces 89 5.21 Right lateral flexor (Ffr) and left lateral (FfA) muscle forces 89 5.22 Sacrospinalis muscle spindle output 90 5.23 Left lateral flexor muscle spindle output 90 5.24 Left transverse muscle force 91 5.25 Bias input (Ub) and control input (Uc) of the torso 91 5.26 The crinput signal to the rectus abdominis and right lateral flexor muscles 92 5.27 The crinput signal to the sacrospinalis and left lateral flexor muscles 92 5.28 Lengths of rectus abdominis (Lr ), sacrospinalis (Ls), right lateral flexor (Lfr), and left lateral flexor (LfA) muscles 94 X FIGURE Page 5.29 Lengths of right transverse (Ltr) and transverse (L-tmuscles 94 A.l Coordinate axes transformation 101 xi CHAPTER I INTRODUCTION 1.1 OVERVIEW Interest in robotics has grown explosively in the past several years. Organizations such as universities, government agencies, and private companies increasingly have been focusing on the research, development, and application of robotics. The demands on the level of sophistication of the robots, i.e., greater ease of use and broader requirements on their abilities, are also increasing in parallel with their industrial use. Advances in understanding the many aspects of robotics are prerequisites of the support of sophisticated performance. Robotics is a huge interdisciplinary field comprised of such diverse topics as vision, force and tactile sensing, manipulator design, actuation, motion planning and control, path representation, locomotion, intelligence, decision making (knowledge representation), kinematics, and dynamics. This work deals only with those aspects of robotics involving dynamics, kinematics, control, trajectory planning, and actuators. Manipulator dynamics is an aspect of control whose understanding has developed rapidly in the past few years. Controversies such as the relation of dynamics to the other aspects of control, particularly feedback control, have also been clarified due to this recent progress. The importance of manipulator dynamics stems from its use in simu lation, analysis, and feedforward computation. Simulation of robot motions is a way of testing control problems of working with actual manipulators . An analysis of manipulator dynamics could aid in the mechanical design of prototype arms . The forces and torques experir enced throughout a mechanical arm in typical movements would yield information useful in designing joints, actuation, and structural stiff ness. A potential major role for manipulator dynamics is in the genera tion of nominal joint torques to derive a manipulator along some speci fied trajectory. The inverse dynamics computation is a feedforward con trol that could work in conjunction with the feedback controller (see Chapter V). If the measured trajectory does not satisfy the planned one, i t is the function of the feedback controller to drive the manipu lator back to desired trajectory. 1 Current robots have evolved from automatic cranes, automatic machine tools, and mechanical teleoperators; development of control technology has followed a parallel evolution from open- 1oop precision actuators and sensing systems, to single-joint servovalves and servomotors, to integrated adaptive force and position control using feedback from machine vision and touch sensing systems. However, more research is needed to advance the state of knowledge in robotics. There is a need for basic research on actuators since there is nothing that compares with natural muscle in terms of force, weight, inertia, and energy utilization. 1.2 ORGANIZATION In Chapter II of th is dissertation a brief background and review of pertinent literature concerning the kinematics and dynamics of biped systems and robotic manipulators is presented. Control mechanisms which are usually applied to these systems are also surveyed. In the last section, human spine modeling and trunk muscles are reviewed. In Chapter III, a state space formulation of systems of interconnected rigid bodies is applied to three dimensional one and two element linkage systems. The constrained multi-dimensional motion and the calculation of forces of constraint are also included. In Chapter IV, voluntary point-to-point motion of the human forearm with programmed gain parameters is considered. In Chapter V, voluntary point-to-point movement of the human trunk model in three dimensional space with muscular actuators is analyzed. In Chapter VI, the conclusion and some recommendations for further research are presented. 2 CHAPTER II BACKGROUND AND RELATED RESEARCH 2.1 INTRODUCTION This work is concerned with modelling the three-dimensional dynam ic control of the torso by muscle actuators and verification of the effectiveness of the control algorithms by digital computer simulation. For this reason, pertinent studies of kinematics and dynamics of multi - link manipulators and a number of biped models are reviewed in Section 2.2. Different control strategies are surveyed in Section 2.3 A brief background and a review of neuromuscular system models and approaches are presented in Section 2.4. Modelling of the vertebral column, and the role of some trunk muscles in the stability of the torso are reviewed in Section 2.5 2.2 KINEMATICS AND DYNAMICS Man has been fascinated with locomotion for a long time. It has been observed, questioned and admired. The scientific observation of locomotion began in 1872 when Muybridge used a series of electrically triggered cameras to photograph a galloping horse. He then went on to photograph the motion of various animals and humans [1]. Motion pic tures are still an important research tool [ 2 ], especially for recording sports a c tiv itie s. For many applications, television-computer systems are preferable since the data are recorded by the computer, can be immediately analyzed by the same computer which recorded it, and exten sive observations may be made. Several television-computer systems were built to measure gaits in as close to real-time as possible. A summary of this area of work is found in [3] containing 28 references. Recent ly automated motion measurement systems using a television camera in ter faced to a computer were constructed [4-6]. The kinematic studies have given physicians a tool for analyzing gait and associated structural abnormalities in patients [7,8], and training athletes [9]. For many applications, e.g., the control of locomotion, the design of prostheses, and the optimization of sports techniques, it is import ant to know the forces and moments which occur during locomotion. Since only the external ground reaction forces are available for measurement 3 [ 1 0 , 11], it is necessary to estimate internal forces and find indirect methods of measuring them. A survey of the use of mathematical models to solve this inverse dynamic problem is presented in [12-15]. As far as dynamics is concerned skeletal models that are made of a system of rigid bodies that are connected together by specific pin and jo in ts and have different degrees of freedom have been proposed [15]. A survey of the previous work in these areas is available in [16]. A sequence of more and more complicated biped models have been considered for stability and control [17-21]. Biped sway in the frontal plane with locked knees is considered in [ 2 2 ] and an algorithm is developed for the independent control of the three-link biped for the computation of the feedback gains that are ne essary for postural stability. Biped side step in the frontal plane is covered in [23], where linear state-variable feedback is designed to stabilize and decouple the three-link planar model of a biped. A simple mechanical model is used to simulate the stance phase of human locomotion in [24] and a mathematical representation of the human leg during the swing phase of gait is developed by [25-26]. A mathe matical model to simulate complex motion of a 17-segment humanoid which treats the external constraints and impact situations is presented in [27]. Locomotion and control studies have also been extended to the human upper extremity modeled as a torso. Chow and Jacobson [28] con sidered the multi-dimensional torso motion with a linear feedback law coupled with an on-off perturbation. The postural stability of the torso is studied via the Lyapunov method. The upper part of the human body, being approximately 70$ of the body weight, has considerable influence on the lower extremity behavior and in determining the actuat ing moments around the hip and knee jo in ts. Dynamics and coordination of torso motions in human locomotion are presented in [29], where a model is presented which completely expresses the magnitudes and phasings of torso motions and indicates the character of bipedal support and stabilization by foot placement. Analysis, design and identification of manipulators, are some other aspects of natural and man-made systems. Various approaches such as the Lagrangian, the Hamiltonian, the Lagrange-Euler, the Newton- Euler, the recursive Lagrange-Euler, and the generalized d'Alembert principle formulations [30-36] have been developed for deriving the equations of the system. The Lagrangian method has certain advantages over other model formulation techniques. However, it has a large computational redundancy resulting in lengthy and time-consuming derivations with a 4 large probability of error. A new method of Lagrangian formulation of open loop kinematic chains is discussed in [37]. The Lagrangian-Euler approach is well structured and can be expressed in matrix notation but is computationally impossible to use for real-time control unless the equations of motion are simplified. The Newton-Euler method results in an efficient set of recursive equations, but they are difficult to use for deriving advanced control laws. The Generalized Coordinate equa tions of motion give fairly "well-structured" equations at the expense of higher computations [30]. Chow and Jacobson use energy methods to derive the equation of motion. Abstract formulation of the equations of the inverted pendulum as a rigid body may be found in [38, 39]. A stable space model of interconnected rigid bodies is developed in [40] based on the Newton-Euler dynamics. 2.3 CONTROL MECHANISMS The least understood aspect of human locomotion is how humans con trol their motion, how the human neuromuscular system works, and what control laws humans use. Another research area is how these locomotion systems are controlled. The finite state control method is reported in [41], Summaries of the quadruped and multi-legged control strategies are given by Orin [42] and Lee [43]. The studies of biped locomotion and control have been based to a greater extent on the stability of the system. This is due to the inherent dynamic instability of bipeds as opposed to multi-legged systems that are statically stable. Some of the biped models that are used to study locomotion and stab ility studies are reviewed in [16, 17]. The three control schemes, i .e ., algor ithmic control, finite state control, and model reference control are discussed in [43], where a combination of finite state and model refer ence control is used to deal with varying ground constraints. In order to control m ultivariable systems with multiple inputs and outputs, considerable research has been done on decoupling and pole assignment. A detailed survey of decoupling theory is given in [44]. Pole placement is a powerful method for stabilizing and improving the performance of linear multivariable systems. Conditions required for pole assignment by means of state and output feedback have been studied and algorithms for pole assignment have been presented; see bibliography in [43]. In modal control [45, 46], the natural modes of the system are used for pole assignment. The use of state variable feedback to simultaneously decouple a system and assign its poles was studied in [47]. To control robotic manipulators, several control methods are available. Current industrial practice is to use conventional 5 servomechanisms to control manipulators. However, the dynamics of an n-degree-of-freedom manipulator are inherently nonlinear and can be described only by a set of n highly coupled, nonlinear, second-order ordinary differential equations. Use of a servomechanism models the varying dynamics of a manipulator inadequately and neglects the coupling effects of the joints [30, 48]. Dynamic control of manipulators with feedback method is discussed in [49]. For the control of a robot arm strategies such as resolved motion rate control (RMRC) [50],resolved acceleration control [51], resolved motion force control [52], near-minimum-time controller [33], the cerebellar model articulation controller (CMAC) [53], , the computed torque technique [4,8], and adaptive control [54-57] are studied. Adaptive control methods can be used to maintain good performance over a wide range of motions. Among various adaptive methods, model referenced adaptive control is the most widely used and relatively easy to implement [30]. In the MRAC method proposed in [54], a linear, second-order, time invariant differential equation is used as the reference model for each degree of freedom. Another adaptive control strategy is is based on designing a feedback control law for the perturbation equations in the vicinity of a desired motion trajectory [57]. The classical methods of optimal control have been applied to the problem of determining approximate minimum-time [33] and minimum-energy [21-59] trajectories. Linear control laws, for which each current input is a linear (or affine) functional of the past output measurements, are relatively easy to design and evaluate. The simplest and most widely used method today is independent joint control, a constrained structure linear control law analyzed effectively in [60]. The effectiveness of linear independent joint torque control is analyzed in [61]. Another pseudo-linear open loop feedback law with nonlinear pre- and post-processing of measurement and control signals is developed in [62]. Young applied sliding mode theory to develop a nonlinear switching control law with guaranteed tracking and stability properties [63]. Johnson has illustrated generalization of discontinuous concepts to locomotion and catching [64]. Freund uses methods of Lie algebra to develop a global nonlinear control law involving full-state feedback, with guaranteed stability in the absence of disturbances [65]. The relationship of stochastic control and learning has been described in [66]. 6 The problems of coordinated rate control and position control of multidegree-of-freedom systems, and a solution to the endpoint position control are presented in [67]. As far as motion and trajectory planning of manipulators are concerned, paths made up of straight-line segments connected together by smooth transitions with controlledacceleration are proposed in [ 6 8]. Recursive path interpolation functions having more advantages than previous work are presented in [69]. 2.4 NEUROMUSCULAR SYSTEM The human skeleton can be regarded as a system of rigid links connected by jo in ts. This mechanical system is connected by ligaments and is operated upon by muscles which appear almost exclusively in pairs. The various joints, their actuators, sensory elements and local feedback control loops form separate control units. Motions at joints, as other voluntary actions, involve some sequence of hierarchical activ ities [70, 71]. The highest level is the decision making one where the speed and direction of the motion are determined. The algorithms for executing the motion, the control law and the process of synchronizing the muscle groups for the specific motion are established in a lower level. The lowest level is the dynamic level that includes the limbs which are involved in the motion. Skeletal muscle as the propellant of animal motion presents a special challenge to the bioengineers. Many methods of muscular contraction have been proposed, some of which attempt to explain the contractive process at the molecular level. A review of these modeling attempts is presented in [72]. Attempts are also made to simulate the macroscopic behaviour of muscle in different contractive situations [73,74]. A step by step development of a 1umped-parameter muscle model and muscle nonlinear characteristics based on the Hill's equations is presented in [75]. A mathematical model of skeletal muscle which contains two physio logical control parameters -- simulation rate and motor unit recruit ment — is presented in [76]. This model is complete in the sense that it adequately describes all possible contractive states normally occur- ing in the living muscle. An EMG-level muscle model for maximally fast 1arge-amplitude arm movement to a target is presented in [77]. Some researchers have attempted to optimize bipedal locomotion via optimal programming [70]. However, the most successful model for optimal motion problems has been that of Hatze [78]. A minimum energy solution for muscle force control is presented in [ 2 1 ], where minimum muscle energy, defined as the input chemical energy, is forwarded as the 7 criterion of optimality, and a thermodynamic model of muscle is developed to allow implementation of this criterion. Current models of the skeletal, muscular, and controller subsystems of the human neuromuscloskeletal systems are surveyed in [80], where a more re a listic approach to the modeling of these systems is indicated. The implications of expanding Hill's three-dimensional element model by replacing the series and parallel elastic elements with nonlinear visco-elastic elements is explored in [80]. A mathematical model of the total human musculoskeletal system is presented in [81]. In control and modeling of neuromuscular systems, muscle spindles and Golgi Tendon Organs play an important role in understanding locomo tion. Two types of three stretch receptors are located within a highly specified receptive organ called a muscle spindle, widely scattered throughout the fleshy parts of the muscle and usually attached at both ends to the ordinary or "extrafusal" muscle fibers. Each spindle con sists of from 2 to 12 thin muscle fibers known as "intrafusal fibers", where they are innervated by axons called gamma fibers [82]. There are two types of sensory endings in most spindles -- primary endings deriv ed from group la nerve fibers and secondary endings combine from group II fibers. Primary endings measure length plus velocity of the muscle, whereas secondary ending measures only length of the muscle. A detailed analysis of the muscle spindles is given in [82-85] among others. A linear lumped-parameter mechanical model of the muscle spindle as a neurally controlled transducer of stretch is presented in [ 8 6] with a good summary of various types of responses to the various types of inputs to these feedback receptors. Golgi tendon organs, group lb fibers, discharge proportionally to the tension existing in the extrafusal fibers due to their location in the muscle tendon junctions. A mathematical model for the Golgi tendon organ can be found in [87]. The motoneurons that directly innervate the extrafusal muscle fibers are called alpha motoneurons. Primary spindle endings, which are most sensitive to the velocity or rate of change of muscle length provide a feedback signal that is proportional to the velocity of shortening or lengthening in the extrafusal muscles. This feedback signal not only fa c ilita te s the alpha motoneuron activity of synergistic muscles, but i t also inhibits the alpha motoneuron activity of antagonistic muscles. The gamma motoneuron influence the output of alpha moneurons by adjusting the sensitivity of the spindle receptors. Most studies agree that movements generally involve coactivation of o and y motoneurons [ 8 8]. The subject of motor system activity and 8 muscle innervation is covered in [89-92] among others. The most peripheral subsystem of motor control, the postural control loop is covered in detail in [93]. 2.5 HUMAN SPINE MODELING AND TRUNK MUSCLES The human spine has an extremely complex structure. Its twenty-four mobile vertebrae experience large, three-dimensional displacements and interact with other parts of the body. A number of planar models for the study of the mechanics of the vertebral column have been formulated [94-96]. These studies are restricted to the sagittal loading of the body or to small deflection analysis. The vertebral column, however, exhibits approximate symmetry only with respect to the mid-sagittal plane in motion of flexion and extension, so three dimensional studies are of importance for an under standing of the spine's behavior. Mathematical methods for the three dimensional geometric analysis of spinal geometry have been attempted in [97-101]. These mathematical models of the spine are of two kinds: the continuum and the discrete parameter type. The former considers the spine as a rod having an infinite number of degrees-of-freedom [ 102- 104]. On the other hand, a discrete parameter model considers the spine as a structure formed by various anatomic elements like vertebrae, ligaments, muscles, and articulating surfaces. Vertebrae are represent ed by rigid bodies and ligamentous structures, muscles, and articulating facets by massless springs and dashpots [96, 100, 105-107]. Multisegmental mathematical models of the human body play a very significant role in both vehicle crash victim studies and aerospace related applications [108]. An extensive review of the mathematical models of the human body is presented in [109]. The human upper extremity has been modeled as a rigid ellip tical cylinder in a number of studies [28, 110]. As far as the movement of the human spine is concerned the muscles of the trunk can be divided into two principal groups, which are located at the lower and upper parts of the trunk. There three major muscle pairs at the lower part of the trunk that can be considered as prime movers of the human torso. The principal muscles active in sagittal flexion and extension are the rectus abdominis and sacrospinalis muscles [111, 112]. The prime muscles for lateral flexion and extension are lateral and anterior fibers of the oblique (external and internal) muscles [111-113]. The main muscles for transversal motion of the human torso are the right and left lower fibers of the internal oblique muscles [113]. 9 Floyd and Silver [89] were the firs t to make an extensive EMG study of the abdominal musculature in normal people. With a grid of paired multiple electrodes on the anterior abdominal wall, they recorded simultaneously from various parts of the rectus abdominis, the external oblique and the internal oblique. However, they used needle electrodes to study the transverse abdominis muscles. They found some differences between the right and left sides of the abdominal musculature. Their findings were confirmed la ter by others [90, 114-116] concerning athletic training and exercises in the bedridden patients. 2.6 SUMMARY In this Chapter some literature pertinent to the subject matter of this Dissertation has been reviewed. The kinematics and dynamics of biped systems and robotic manipulators were covered. Control strategies of these systems were also survayed. A brief review of the human spine modeling and trunk muscles functional roles was presented, too . 10 CHAPTER III CONSTRAINED INVERTED PENDULUM MODEL FOR EVALUATING UPRIGHT POSTURAL STABILITY 3.1 INTRODUCTION In this chapter a state space model of the three dimensional one link and interconnected two link bodies is considered. The model is capable of accepting natural constraints. Holonomic and nonholonomic constraints are embodied in the model and two examples are presented in Section 3.2.3. The holonomic forces of constraints are formulated as functions of state and input for one link and two-link models in Section 3.3, while they are eliminated from the equation of motion in Section 3.4 to reduce the dimension of the system. The study of the stab ility of the one-link model of the torso via Lyapunov's Second Method is pre sented in Section 3.5. Further constraint on the one link torso is con sidered in Section 3.6, whereas the corresponding torque of constraint is eliminated from the equations of motion in Section 3.7. Section 3.8 contains simulation results of the model. 3.2 STATE SPACE EQUATIONS OF INTERCONNECTED RIGID BODIES 3.2.1 One Link Model The human torso is considered as a rigid body with three rotational degrees of freedom 41, $, and e (Fig. 3.1). The inertial frame of reference is labled as OXYZ while OX'' 'Y' 1'Z''' represents the three dimensional space spanned by the body's principal axes. The Euler angles 4,, $, and e are generated by three successive rotations of the torso about its body axes X’" , Y '", and Z "‘, respectively, and will be described here by vector 0 . © = [4*, 4, 0]T = C01, 02, 033t (3*1) The three input torques used to maintain an upright posture expressed in inertial frame of reference are described by vector M M = [M^M^.MeDT = [Mi, M2, M3]T (3.2) The angular velocity, W, expressed in the body coordinate system is derived in Appendix A. W = B- 1 (e ) 0 (3.3) 11 F ie. 3.1: Torso as an Inverted pendulum. 12 where B"1( g ) is a 3x3 matrix COS02COS03 -sin 03 0 B-l ( e) = cos 02Sin 03 cos 03 0 (3.4) sin 02 0 -1 The matrix of principal moments of inertia about the body axes is the diagonal matrix J. Jl 0 0 J = 0 J 2 0 (3.5) 0 0 J3 The Euler's rotational equations of motion are JW = -WJW + N (3.6) where, N is the set of applied torques expressed in the body coordinate system. Equation (3.6) can be put in the following form JW = -f(W) + N (3.7) where, f(W) = WJW or (J3-J2)W2W3 f(W) = (Jl-J3)wiW3 (3.8) (J2-J1 )w]W2 and wj, W2 , and W3 are nonintegrable combinations of time derivatives of angular displacements $, and 0. Vector N has two components -- externally applied torque and the torque of the ground reaction forces N = NlNPUT + nreaction (3.9) The input torque Ninput is input vector M Eq. (3.2) while expressed in body coordinate system. From Appendix A: ninput = b~i( 0) • m (3.10) where B_1 (e) is defined by Eq. (3.4). 13 When the equations of motion are written with respect to torso center of mass, NreaCTION 1S t,ie torque generated by reaction forces at the base point of the torso about its center of mass expressed in the body coordinate system. If the point of contact of the torso with the ground is vector R in the body coordinate system and the reaction forces are shown by r [yl» y2 » y3] in the inertial coordinate system, then n REACTI0N = RBODY X IBODY (3.11) where this vector product has to be resolved in the body coordinate system. The reaction forces r are transformed to body coordinate system by tbody = AT (©) r (3.12) where A( ©) is the 3x3 coordinate transformation from the body coordinate system to the inertial coordinate system and At (q ) is its inverse derived in Appendix B. The cross product of Eq. (3.11) may be written in the matrix form [117]' Nreaction = RR AT(©)r (3.13) where RR is the skew symmetric matrix 0 -r3 r 2 RR = r3 0 -ri (3.14) -T2 ri 0 having n R = r2 (3.15) r3 For the torso shown in Fig. (3.1), r^ = r 3 = 0 and r 2 = -A. Let the coordinates of the center of gravity of the torso be vector X in the inertial coordinate system X = [x,y,z]T (3.16) and let m be the mass of the torso. The equations of the translational motion are 14 mX = G + r (3.17) where G is the gravity vector "0" G = -mg 1 (3.18) 0 The state space equations of the system are X = Y (3.19) mY = G + r (3.20) © = B(0)W (3.21) JW = -f(W) + B-1 (0)*M + RRAT(©).r (3.22) These state space equations can be combined and symbolically w ritten as Z = hi(Z)Y (3.23) m'Y = h2(Y) + h3 (Z)M + h4 (Z)r (3.24) where ZT = [x T,g T] , yt = [vT,wT], m' = I mg j | > and I h i (Z) = (3.25) B(0) G h2 (Z) = (3.26) -f(W) “ 0 h3 (Z) = (3.27) _ B-l(©) _ h4 (Z) = Lr RRAT(©)1 (3.28) 15 3.2.2 State Space Equations for Two Rigid Bodies The state space formulation of Section 3.2 can be generalized for a system of interconnected ridig bodies. For clarity the expanded form is presented first and then put in the abstract form analogous to Eqs. (3.23) and (3.24) of the previous section. Here, the number of rigid bodies is limited to two, where the generalization to larger dimension is obvious. Let the two bodies be, respectively, Body 1 and Body 2 (Fig. 3.2). Body 1 is connected to the origin of the ICS (Inertial Coordinate System) at vector Ki in B]CS (Body Coordinate System) and connected to Body 2 at vector point Lj in BjCS. Body 2 is connected to Body 1 at vector point K 2 in B2CS. Let ri be reaction forces at the origin actingon Body 1. Let r 2 be reaction forces acting on Body 2 by Body 1. n and T2are expressed in ICS. The differential equations of this two body system are ©1 * Bi (01 )Wi JlWi = -f 1 (Wx) + KKiAi- 1(o i)ri - LLiAi- 1 ( 0i ) n + Bi- 1(0i)Mi *1 = Vi miVi = 61 + n - T2 62 = B2 ( 02)W2 J2W2 = -f2(W2) + KK2A2"1(02)T2 + B2"^(C2)M2 X2 = V2 mV2 = G2 + 12 (3.29) where ©iT = C6il> ©i2 » $3 ] X ^ = [x i, y-j, z-j] GiT = [0 , -mig, 0] and Aj is the transformation matrix from B-,-CS to ICS. In abstract form the nonlinear state space equations for the two body system are 16 IT Fig. 3 .2 : Two interconnected rigid bodies. 17 I = hi(Z)Y m'Y = h2 (Y) + h3 (Z)M + h4 (Z)r where ZT = [ziT, Z2T] = [xiT eiT, X2T 02T] TT = CyiT, y2T] 88 CviT wf, V2T W2T3 mil i _ m J l m2 1 J 2 _ and I 0 0 B1 ( 01) h1 (Z) I 0 0 B2( 02) 61 “ h2 (Z) = -fl G2 “f 2 _ 0 0 Bl~1( ©1 ) 0 h3 (Z) = 0 0 0 B2-1 ( 02) I -I KKiAr^Gi) -L L lA r^ ei) h4 (Z) = 0 I 0 KK2A2_1 (02) (3.30) 18 3.2.3 Two Link Model with Further Constraints In this section the two body system is considered and further constraints are imposed on the rotation of the second body. The additional rotational constraints are as follow. Nonholonomic constraints result if rotation of Body 2 is restricted along certain axes in B]CS [40]. d As an example, consider two interconnected rigid bodies resembling the human arm and forearm connected at the elbow joint. The motion of the forearm is constrained to have two rotational degrees of freedom, i.e., the second body of Fig. 3.3 cannot rotate along the Zi-axis of the BiCS. Let W2 be in a plane in BjCS spanned by two fixed vectors Qi and 02- Let e = Cel e 2 ]^ be a vector of proportional ity constants. Let RT be a 1x3 vector whose row is orthogonal to vectors Ql and Q2. The nonholonomic constraint equation for this case would be [40] W2 = A2 "1 (©2 )Ai(©1 )[Ql Q2]e (3.31) Ai(©1 ) transforms Qi and Q 2 from B]CS to ICS and A 2 (©2 ) transforms the result to B2CS. Equation (3.31) may alternatively be written as RtA i (Gl )A2 (02)w2 = 0 (3.32) for the example of the human elbow jo in t le t QlT = [ 1 0 0] Q2t = [0 1 0 ] RT =[0 0 1] (3.33) As a second example, le t the motion of the second body be constrained to imitate the one degree of freedom of the knee joint, i . e . , the second body of Fig. 3.3 cannot rotate along Zi and Yj of the B^CS. Let W2 be parallel to a fixed vector Q in BiCS and let e be a vector of proportionality. Then the nonholonomic constraint equation for this case would be the same as Eq.(3.31). Let R^ be a 2x3 vector whose rows are orthogonal to vector Q, then the alternative constraint equations for this example of human knee jo in t would be the same as Eq. (3.32), where for this case Q =[10 0] rT - [°J°] (3.34) Finally, notice from these two examples that the nonholonomic constraints may be symbolically written as 19 Y z2 zl ix2 xl Z Fig. 3.3: jwo *]-fnic model of the elbow and knee jo in ts. 20 H(Z)Y = 0 (3.35) The two forces of constraint a = Cai A 2 ]T needed for knee jo in t or the one force needed for elbow jo in t enter equation of motion as [118] h5(Z)a (3.36) where h5(Z) = HT(Z) (3.37) With these constraint torques,Eq. (3.29) will be modified to Z = hi(Z)Y m'Y = h2 (Y) + h3 (Z)M + h4 (Z)r + h5(Z)A (3.38) where hi, h2, h3, and 4 h are defined in Eq. (3.30) and 0 0 h5(Z) = (3.39) 0 _ A2T( ©)Ai( 0i )R _ which is a 12x1 vector for elbow example and 12x2 vector for knee example. 3.3 CALCULATION OF THE H0L0N0MIC FORCES OF CONSTRAINTS 3.3.1 One Link Model The fact that the base of the torso remains at the origin may be used to calculate the ground reaction forces r: r = r(e,w,M) (3.40) The (holonomic) constraints are X + A( g )R = 0 (3.41) where matrix A(0) transforms the vector R from the body coordinate system to the inertial coordinate system. These holonomic constraints result when the base of the torso is restricted to be stationary on the ground, then three degrees of freedom of the body are eliminated. Holonomic constraints may be conveniently summarized in a general set of implicit equations 21 G(X,0) = 0 (3.42) where it is assumed that G is an analytic function and consequently all partial derivatives of G with respect to X and 0exist. Differentiating constraint Eq. (3.41) with respect to time [40] X+A(0)R = 0 (3.43) from the identity WW = A -!(0 ) A(0 ) (3.44) where WW is a skew symmetric matrix as in Eq. (3.2.14), i .e ., ~ 0 -W3 W2 WW = W3 0 -wi _ -v i2 wi 0 Then Eq. (3.43) becomes X + A(0)WWR = 0 (3.45) but WWR = -RRW (3.46) Therefore X - A(0)RRW = 0 (3.47) Differentiating Eq. (3.47) with respect to time X - A( 0) (RRW) - A(0 )RRW = 0 (3.48) Using Eq. (3.44) and then Eq. (3.46) X - A(0)WW(RRW) - A(0)RRW = 0 (3.49) X + A(0)(WW)2R - A(0)RRW = 0 (3.50) Writing Equs. (3.20) and (3.22) in matrix form 1 1 ml 0 -<• I + 0 J W -f(W) + B( ©)"1*M RRAT(0) (3.51) 22 and solving for [Y W]T -1 Y ml 0 G I + W 0 J -f(W) + B-l(©)-M RRAT( 0) (3.52) From Eq. (3.19) Y = X So Eq. (3.50) gives Y = A(e)RRW - A( g )(WW)2R (3.53) Substituting Y in Eq. (2.52) -1 A( ©)RRW - A(©)WW)2r ml 0 G W 0 J -f(W) + b-1*©)^ I r RRAt (©) (3.54) Multiplying both sides of Eq. (3.54) by [I, -ARR] yields I -A(©)(WW)2r = [I, -ARR] | JJ1 J RRAt (0) (3.55) Let - m l 0 - i - l D = - A(0)(WW)2r L RRAt (©) _ _ 0 J J _ B -!(0)-M-f (3.56) where I- I [I, -A(e)RR] = RRAT(©) 23 Then solving for ground reaction forcer = ( y1>y2>y3)T “ ml 0 - i - l i- -l r = D _ rraT (©) _l _ o j J L rraT(0)_ (3.57) which states that r is a function of e, W, and m -- the state and input, respectively. This model of the system is shown in Fig. 3.4. 3.3.2 Two-Link Model For the system of two rigid bodies shown in Fig. 3.2, i t is desired to calculate the six forces of holonomic constraint acting on Body 1 and Body 2 at vector points and K 2 , respectively. The two holonomic constraint equations corresponding to the holonomic forces of constraints n and 12, respectively are Xi + A]Ki = 0 X2 + A2K2 - Xj - A1L1 = 0 (3.58) Differentiating Eq. (3.58) twice with respect to time and making use of the identities WWi = A r 1 - h i WW-jK-j = -KK-j W-j (3.59) one obtains Vl I -A1KK1 0 0 -Ai(WWi)2Ki Wi -I -A1LL1 I -A2KK2 V2 -A2(WW2)YK2 - Ai(WWi)2Li W2 (3.60) ■ The six forces of constraint n and r 2 may be calculated by elimination of Y from Eqs. (3.29) and (3.60). Equation (3.60) may be given symbolically as SY = T (3.61) Solving for Y and putting in second equation of (3.30) 24 © A Fig. 3 .U: Model of the dynamics of the torso. a) Eqs. (3.56), (3.57) b) Eqs. (3.21), (3.22) 25 m'LS-lT] = h2 (Y) + h3 (Z)M + h4 (Z)r h4 (Z) r = -h2 (Y) - h3 (Z)M + m'S-lT m '-1h4 (Z)r = - n r ^ t Y ) - m'- 1h3 (Z)M + S"lT Sm’- 1h4 (Z)r = -Sm'-^fY) - Sm'-^UJM + T Note that h4 (Z) = ST, and (K K xA r1 )1 = (KKiAiT)T = A1KK1T = -A1KK1 Therefore, r » [Sm'-lS1 ] - 1 {-Sm'-l[h2 (Y) + h3 (Z)M] + T} (3.62) 3.4 ELIMINATION OF THE HOLONOMIC FORCES OF CONSTRAINT Sometimes one does not wish to compute the forces of constraint. The equations of motion for this case are obtained by elimination of the forces of constraint from Eqs. (3.19) thru (3.22). First, Eqs. (3.20) and (3.22) are combined ml 0 Y I + 0 J W •f(W) + B-if ©)-M RRAT(0 ) (3.63) If both sides of Eq. (3.63) are multiplied by row vectors that are orthogonal to the columns of the matrix associated with r, the constraint forces are eliminated. Three such row vectors are the rows of the matrix: C-rra T(©) I] (3.64) multiplying both sides of Eq. (3.63) by the matrix Eq. (3.64) yields to ml 0 -Y- [-RRAT(e), I] = [-RRAT(e), I] -f + B-^e) 0 J U (3.65) The acceleration Y may further be eliminated from Equs. (3.53) and (3.65). 26 — ml 0 —| - A(0)RR “ j [-RRAT(©), I] W = _ 0 J J I _l t , . l— G —I l— 0 “ I - A(0)(WW)2R “I -RRA (0 , I] + + |_ 0 j _ 0 J (3.66) Now Eqs. (3.66) and (3.21) are the state space equations of the system where there are no forces of constraint in these equations and while 0 and W are the states. With vector R = [0, -s., 0]T as defined before in Eq. (3.15), Eq. (3.66) can be simplified. First matrix multiplication on both sides of Eq. (3.66) is performed [-m(RR)2 + J]W = -RRAT(0)G - f(W) (3.67) + B"1(©)*M - mRR(WW)2 with use of Eq. (3.5) and (3.14) it is easy to show that Jl + nu2 0 0 [-m(RR)2 + J] = 0 J2 0 (3.68) 0 0 J3 + nu2 _ But the resulting matrix is the principal moments of inertia about the inertial coordinate system J 0 J q I Jo 2 (3.69) J q 3 _ And the term mRR(WW)2R is added to f(W) to generate f 0 (W) that corresponds to J 0 " (J03 - 0o2) W2W3 ~ F0 (W) = (J01 - J03 ) W1W3 (3.70) _ (J02 - Jol) WiW2 _ Then Eq. (3.66) simplifies to JoW = -f 0 (W) + RRAT( 0)G + B-!(0)M (3.71) which is interpreted as the rotational equations of the body written 27 about a coordinate system centered at the point of contact with the ground and parallel to the body principal axes. Now the equations of motion are 0 = B(g)W J0W = -f 0 (W) + RRAt (g)G + Ninput (3.72) 3.5 LYAPUNOV STABILITY An important attribute of the state space equations of motion derived in the previous sections is the ease with which Lyapunov's second method may be utilized for stability studies. The following nonlinear feedback torque is shown to stabilize the one link torso about its upright position: N = -BT(0)Kiq - K2W - RRAt (g)G (3.73) where Ki and K2 are two 3x3 positive definite symmetric matrices and G and W are the angular powition and velocity as state variables of the system. A positive definite Lyapunov function, V, is chosen and it is shown that its time derivative function, V, will be negative, which guarantees asympotic stab ility of the system. Let the Lyapunov function, V, be defined as Kl 0 0 V(©,W) = 0 .5[©T WT] (3.74) 0 K2 W Then its time derivative would be: Kl 0 0 V(0 ,W) = 0 .5[©t Wt] 0U K K.22 _ w W (3.75) —“ * - “ Kl 0 0 O.5 [ 0T WT] 0 K2 _ _ W _ where *eT and WT are obtained by transposing Eq. (3.72) 28 0T = wT BT(0) (3.76) WT - [-f0T(W) + (RRAT(0 )G)T + NT] (j 0-1)T noting that Kj and K 2 are symmetric matrices and (Jo - 1 )1 = (Jo 1 ) - 1 = Jo -1 using Eq. (3.76) in Eq. (3.75) yields v(e, w) = o. 5(wt b t« i0 + ©tkibw) + 0.5(nTj o -1k 2W + wTK2Jo _1N) - {3>77) 0.5(foTjo-lK2W + WTK 2J 0“1f 0) + O.5[(RRAT(0)6 )TJ0-lK2W + WTK2J 0 - 1 (RRAt (©)G )] Each of the items in the parenthesis is a scalar so it is equal to its transpose, threfore equation (3.77) simplifies to V(0 ,W) = wTbTk!© + nTj0-1k2w (3.78) - f0T(W)J0"1K2W + (RRAt (©)G )tj 0 - 1k2w Let K2=J0, then it is easy to show that the term 0 f^(W)W disappears. Now, using Eq. (3.73) in Eq. (3.78) yields V(0,W) = -WTJ0W (3.79) which is clearly a negative function. Therefore the system is stable wherever B(0) is defined. The fact that V is independent of may be interpreted as follows. Matrix l< 2=Jo in Eq. (3.73) guarantees asymptotic stability of the system (V<0) and it corresponds to angular velocity feedback. Matrix Ki determines position feedback. In a broad sense, velocity feedbacks, i.e., introduction of damping, stabilizes the system. Position feedback may change the oscillatory character of the response. Q ualitatively, th is means the system behaves as a second order linear system. To demonstrate this behavior, the system equations were linearized about the vertical stance, and numerical parameters were chosen as given in Table 3.1 [same as in 28]. The poles of the linearized system were computed for several values of Kj. For ~ 100 0 0 - Ki = 0 100 0 (3.80) 0 0 100 29 the poles are: -0.5 + 18.281 -0.5 + 3.576i, -0.5 + 3.5761 -0.5 - 18.281 -0.5 - 3.5761, -0.5 - 3.5761 for 10 0 0 Ki = 0 10 0 (3.81) 0 0 10 the poles are: -0.5 + 57651, -0 .5 + 1.0261, -0.5 + 1.0261 -0.5 - 57651, -0.5 - 1-0261 , -0.5 - 1.0261 As seen above, increasing Ki doesn't change the real part of the poles. For both values of Ki one has the same settling time — 10 seconds. TABLE 3.1 Numerical Parameters of the Model Torso weight 91 lb = 405 N Mass of torso, m 2.826 slugs = 41 Kg Base point to center of mass distance (estimated) 1.415 ft = .42 m Moment of inertial along J i = J 3 = 2.03 slu g -ft 2 = 2.7 Kg-m2 principal axes, ox'" and oz 1' 1 Jqi = Jo 3 = 7.67 slug-ft 2 = 10 Kg-m2 Moment of inertia along oy ' 11 Jq2 = J 2 = 0.299 slu g -ft 2 = 0.4 Kgm2 However, increasing Ki does increase the imaginary parts of the poles. It is to be recalled that this imaginary part corresponds to the damped natural frequency of oscillation. 30 3.6 FURTHER CONSTRAINTS ON ONE LINK TORSO In Section 3.2.1, three holonomic constraints were included in the dynamic equations. Thus, the body has lost its three translational degrees of freedom and possess only three rotational degrees of freedom. It may be desirable to impose further constraints, i.e., con straints which do not allow for self-rotation of the torso along one or two of the principal axes of the body. The human hip joint has all three rotational degrees of freedom, i .e ., the movement of the thigh relative to the hip has three degrees of freedom -- another example of a system with three holonomic constraints. The human hand is connected to the forearm, and this connection can be described by three honolomic constraints. The movement of the hand is further restricted by one more hard constraint, i.e., the hand can rotate relative to the wrist only in two orthogonal directions, i.e., the x and z axis but not y axis (Fig. 3.5). The movement of the human leg relative to the thigh is restricted not only by the three holonomic constraints of connection to thigh but also by two hard constraints that allow the leg to rotate about the thigh only about one axis. In this section it is desired to maintain 02 = The soft constraint can be achieved by position (e) and velocity ( 9) feedback in the case of torso here. One mechanism for implementing a hard holonomic constraint may be envisioned here. Assume the hand is rigidly connected to a solid ellipsoid. Assume the forearm has a half ellipsoid cavity that is slightly larger than the hand ellipsoid. Now insert the hand ellipsoid into the half ellipsoid of the forearm. It is easy to visualize how rotations about z and x (plane of paper) are possible while rotation about the y-axis (out of the paper) is impossible (Fig. 3.6). In classical terms, preventing self rotation is a holonomic constraint 31 Hand Forearm Tha hand axaa:t out of tha papar; jr aManaion of tha foraarm; z parpandicuiar to tha palm Fig. 3-5 Forearm Hand Pravantlon of rotation Fig. 3.6 32 * Projection of tha ellipsoid and two restraining cavity surfaces In me 22 plane. Anticlockwise rotation is poss.ble but clockwise rotation is not possible. Fig. 3.7 33 02 = 0 In an ellipsoid type of constraint, one has to assume that about the constraint surface there is a very slight freedom, and that two hard surfaces of constraint restrict rotation at ±e. Therefore there are two surfaces of constraint. 02 e 02 e (3.82) where e is a very small constant. For each of these constraints a torque of constraint must be centered in the equation of motion. Instead of two equations of constraint as described in Eq. (3.6.1) only one hard constraint is considered here. It means $ cannot be less than zero but it can be larger than zero. This situation corresponds to the ellipsoid of Fig. 3.6, attached to the torso, being surrounded by segments S of a cavity such that only counter-clockwise rotation is possible (Fig.-3.7). This torque of constraint can subsequently be computed as funct ions of the state (©,W) and inputs M in the same way as r is calculated. For holonomic forces, acting at a point, the directions of the forces are known (like r). Here, because the contacts between the ellipsoids are surfaces and lines rather than points the direction of the holonomic torque is very hard to determine. In living systems, bones and their cavities have even more complex shapes and determination of the direct ion of these torques is even more difficult. To carry out the analysis here it is assumed that the constraint torque x enters Eq. (3.72) along the body y axis. Therefore Eq. (3.72) becomes f0 (W) + B"1(0)-M + RRAT(g)G + bT(Q) 1 0 (3.83) , In order to compute the torque of constraint x, a vector d: d = [0 1 0]T is defined, then Eq. (3.82) can be written as d T0 = 0 (3.84) Differentiating Eq. (3.84) twice renders dT e = o (3.85) but from Eq. (2.83) 34 0 = B(©)W + B (0) W (3.86) Then Eq. (3.85) becomes dTB(0)W + dTB ( 0) W = 0 (3.87) Solving for tl W = [dTB( 0 )]-l[-d TB( 0 )W] (3.88) Substituting W from Eq. (3.83) into Eq. (3.88) yields [dTB( 0)]-l[-d TB( 0)W] = J 0‘1[f 0 (W) + B-1(0)M + RRAT(0)G + BT(o)d x] (3.89) Multiplying both sides by CdTB(e)] -JodTB( 0)W = dTB( 0)[-fo(W) + B-!( 0 )M + RRAT(0 )G] + dTB(o)BT(0)dx (3.90) Now, Eq. (3.90) is solved for x x = [dTB(©)BT( 0)d]-l {-J 0dTB(©)W - d^Bf 0) [-f 0 (W) + B"l(o)M + RRAT(e)G ]} (3.91) The computed torque x is the torque generated by the ellipsoid of contact. 3.7 ELIMINATION OF THE NONHOLONOMIC FORCES OF CONSTRAINT INTRODUCED IN SECTION 2.6 It may be desirable to eliminate x and reduce the dimension when constraint Eq. (3.82) is satisfied. The coefficient of x in Eq. (3.83) is BT( 0 )d = [-sine, cose, 0]T (3.92) by multiplying both sides of Eq. (3.83) by row vectors that are orthogonal to BTd the holonomic torque x is eliminated. Let first such a row vector be [cose, sine, 0] (3.93) then Eq. (3.83) becomes 35 [Joicose J 02sine 0]W = [cose sine 0] (Jo3 - J02)W2W3 cose -sine 0 Mi 0 + sine cose 0 _ (Jo 2 - J0l)WlW2 _ 0 0 -1 _ _ M3 _ 0 -Asintj, jicos cp “ 0 “ + 0 0 0 -mg (3.94) -Acose Asecos^ ses<{, _ 0 F rom Eq. (3.3) wi = (cose) 4< W2 = (si n e) ip W3 = -6 (3.95) Therefore (J0lcos2 e + J 02Sin 2 e) [0 0 1] (3.97) then Eq. (3.83) becomes Jo3 9 = - V 2(J03 - J 02 )sin 2 eU 2 ) + M3 + mgAsinecos^ (3.98) The two Eqs. (3.96) and (3.98) are the same as those given in [28], where they used another method (energy method) to derive them. 3.8 SIMULATIONS To verify the results of the above analysis, three separate simulations were conducted. In all these simulations the hip, i.e., the base of the torso is stationary, and only the torso moves. No gait or locomotion is involved. 36 Case 1 - - Lyapunov Simulations The motion of the torso from an initial displacement to its equilibrium upright position was simulated with feedback formula of Eqs. (3.73) and (3.80). The physical and geometrical parameters used for the torso are given in Table 3.5.1. Initial conditions of the problem are set at 9(0 ) = [-0.25, 0 . 0 , 0.3]Trad - [-14.5, 0 , 17]Tdeg 0(0 ) = [0 . 0 , 0 . 0 , 0. 0 ]Trad/sec « [0 . 0 , 0. 0 , 0 . 0 ]Tdeg/sec The trajectories of e, W, r, N, V, and V are plotted as functions of time in Figs. 3.8 - 3.15. In Figs. 3.8 and 3.9 the trajectories of the state of the system from its initial value are plotted versus time. While those trajectories confirm the stability of the torso movement, they also point out that such control strategy is not used by humans. Normal human torso movements are not oscillatory and have shorter durations, i.e., 1-2 seconds. In order to decrease the oscillatory behavior, the state position feedback gains must be reduced, and in order to decrease the response time, angular velocity feedback gains must be increased. At best the simulated trajectories may qualitatively correspond to that of a patient suffering from tremor and spasticity. In Figs. 3.10 and 3.11 the reaction forces that the torso exerts against the hip (the ground) and the total torques needed to bring the torso to the vertical stance are plotted as functions of time. In this maneuver the vertical force Fy eventually settles to that of the weight of the body and the horizontal reaction forces converge to zero as they should. The needed torque N In Fig. 3.12 the Lyapunov function and its derivatives for the previous simulations are plotted. The oscillatory behavior of the system, more apparent in V than in V, is again confirmed. These plots show that while the Lypunov analysis is effective, more work is needed to come up with better feedback and alternative Lyapunov functions where V should depend on both the angles and the angular velocities rather than only on the angular velocities. 37 OS a t 01 S .9 1 -01 -a t ■os- Trajectory ol angular diapiacement F ig . 3 .8 os u5 04 cJ5 02 •08. Trajectory ol the angular velocity W Fig. 3.9 38 Fig. 3 .10: Ground reaction forces at the base of the torso. 60 4 0 —\ jO •«O, Fig. 3.11: Trajectory of feedback torque N. 39 - , 10 2 3 Time/Seconds Fig. 3.12: Lyapunov function and its derivative as functions of time. 03 02 in co 1 -02 -03, Time/Seconds Fig. 3.13: Angular displacenent of the torso with soft constraint. 40 05 02 I o -Ol -02 05 t5 20(0 Tima/Second* Fig. 3.1^: Trajectory of angular displacements of the torso with hard constraint. t >20 »2T •SO Time/Seconds Fig. 3.15: Hard constraint torque t applied to the torso as a function of time. 41 Case 2 — Preventing Self Rotation with Soft Constraint. The motion of the torso from an initial displacement, like those of Case 1, to its equilibrium upright position was simulated with the modified versions of feedback formula applied in [28]. For implementing the soft constraint, the same control law M^, is used: M * -K«, - I V + 2C0-g + 2Co-So-sat( 0, i ] t Me - -Kg - [S22 + 2C0*g + 2C 0 -G0 -sat( ee53e (3.99) where 1 i f ee > 0 sat( ee) = 0 if otherwise The parameters Si2, S22, and K are feedback gains and chosen to be Si2 = 53.24 N-m S22 = 13.3 N-M K = 2.7 N-ms C0 = 0.52 slug-feet = 22.6 N-m G0 =3.0 ft/S 2 = 0.9 m/S 2 The trajectory of e is shown in Fig. 3.13. It is apparent from Fig. 3.13 that the system is stable, and angle 4) is relatively con strained.A comparison with Fig. 3.8, however, indicates that the previous control strategy, even though more oscillatory is preferable for keeping angle Case 3 -- Preventing Self Rotation with Hard Constraint. For hard constraint explained in Section 3.6, where only one of the two equations of constraint as described in Eq. (3.82) is considered, a constraint torque is active whenever the constraint 4 = 0 is satisfied , i .e ., whenever the ellipsoid is pressed against the restraining surface (Fig. 3.7). The torque of constraint is zero when the ellipsoid is not pressed against the surface. The situation is analogous to pressing one's foot against the floor. The initial conditions for this case are the same as in Case 2, and the feedback strategy is also the same as in Case 2. The objective is to bring the 42 torso to the upright position. The trajectory of the Euler angles and the torque of constraint are shown in Figs. 3.14 and 3.15. The system is stable and initially satisfies the constraint and is being pressed against the constraint surface with an approximate torque of 35 ft-lb. After this initial contact, the torso loses contact with the restraining surface, i . e . , At about 1 second $ reaches zero and the ellipsoid makes contact with the restraining surface for two successive short periods of time with small torques of constraint being exerted. After this time $ appears to remain in the vicinity of zero and no further contact is made with the restraining surface. The other two angles appear to oscillate in their excursion to zero. 3.9 SUMMARY The inverted pendulum model of the torso was analyzed using Newton-Euler equations. In this formulation state variables e and W are used, which simplify the equations of motion. Two approaches to the analysis were presented — keeping the forces of reaction in the equations and computing them or eliminating them by a projection method -- the principle of virtual work. This formulation is very convenient for the control of such systems. Quadratic Lyapunov functions of the Euler angles and angular velocities can be constructed which lead to global stability, at least wherever the transformation B(q ) has an inverse. Self rotation was considered and two mechanisms of preventing it were discussed -- hard constraint and soft constraint. Some of the difficulties of the hard constraint model were pointed out. Simulations were undertaken to demonstrate the effectiveness of the Lyapunov stability approach, and reaction forces were computed. Simulations of the soft and hard constraint models were contrasted to analyze the holonomic torque of constraint that prevents self rotation. This approach is very useful in analysis of multi-body problems [10 of p .l] and eliminates the need for direct sensing and measurement of reaction forces. The Lyapunov theory is very powerful in extending stability domain of such systems. Finally, the formulation and Lyapunov approach could be extended to systems consisting of more than one body and with arbitrary holonomic and nonholonomic constraints at the jo in ts. 43 CHAPTER IV GAIN PROGRAMMING IN THE VOLUNTARY POINT-TO-POINT MOVEMENT 4.1 INTRODUCTION Fundamental to understanding the human motion is the recognition that the human motion system is an adaptive, self-optimizing system. A human operator will optimize his performance in a given task if he is given the opportunity to do so. Optimization may be achieved by adjust ing neural input commands or by adjusting other system parameters. Both may be accomplished by operating on feedback information supplied to the neural control system [122]. This paper deals with some elementary studies of such feedback processes in the motion of the forearm. A number of studies deal with this problem. Hogan [123] considered the parameter adaptation problem and stated that direct centrally programmed input to a muscle simultaneously determines the contractive force and the impedance (parameter) of the muscle. The central nervous system can adjust the system parameters by co-activation of antagonist groups of muscles. This scheme does not appear to have some of the problems inherent in feedback control. Co-activation of the antagonist pairs will increase, in effect, the net stiffness of the muscles, while the net external torque about the joint is not changed. The antagonist coactivation increases metabolic energy cost as the opposing muscles consume stored energy but produces no mechanical output work. Mains and Soechting [124] studied the neuromuscular response to sudden disturbances and were mainly concerned with the postural stability and equilibrium maintenance via the reflex system. The complete optimization of human motion and a suitable model of the skeletal muscle are studied by Hatze [78,125). Fast arm movement to a target with an EMG-Level muscle model is investigated by Kilmer et a l . [77]. A detailed mathematical control model for the skeletal muscle is also elaborated by Hatze [76]. In this chapter the forearm's planar movement is analyzed with the flexor and extensor muscles put in a working unit through reciprocal innervation as suggested by Houk [84] and control scheme as suggested by Merton [126]. The muscles and their sensory feedbacks are modified with active and passive linear elements, while both a and y motoneurons are active for a desired precise movement of the forearm. The dynamic point to point motion of the forearm is simulated on the digital computer. Angular position and velocity trajecto ries of the forearm along with the muscle forces and spindle afferent signals are computed and their functional role in the regulation of the movement are exposed. 44 Fig. 4.1: Two-musclp forearm system K W A - B 3 D - — 0- oF o Fig. 4.2: Mechanical model of muscle 45 A number of the gains of the present neuromuscular system are programmed with and without feedback information. It is also shown that transportation delays that cause severe stability problems may be compensated by additional adjustment of gain parameters of the system. 4.2. PHYSICAL MODEL The functional forearm considered here consists of three components: the forearm link, the mechanical model of the muscle, and the neural feedback system. The force exerted by the muscle relates to its extension, extensional velocity, and a control input that depends on the neural characteristics of the system. The third part, the neural feedback system relates the position and velocity of the forearm to the control variable governing the activity of the involved muscle pair. All synergistic muscles acting around the elbow jo in t are repre sented by a pair of identical muscles; i.e., the flexor biceps and the extensor triceps. This hypothetical two-muscle system of elbow jo in t is shown in Fig. 4.1. In this figure ae, and af are the respective moment arms, and are taken to be the same. Assume the arm is kept horizontal. The angular displacement e fully describes the motion of the forearm. The value e = 0; i.e., when the forearm is vertical is defined as the reference forearm position. The motion of the forearm is described by Newton's law a . Fe - a . Ff + mgl sin e =Ie (4.1) where I, m, g, 1 and a are given in Table 4.1, and Fe and Ff are the forces generated by the extensor and flexor muscles, respectively. Natural muscle is a nonlinear element and is governed by the iso metric force, neural excitation, force-displacement and force-velocity relationships. The isometric force is assumed to be linearly dependent on the neural excitation [122]. The force-displ acement and force- velocity relationships are respectively approximated by f(l) < 2 ln - lc (4.2) g ( l ) 0 < v < vm 46 where, In is the muscle natural length, vm is the maximum velocity of contraction, and b is a constant. The generated force vanishes at length lc and 21n - lc. Under physiological conditions that length 1 stays close to ln and v is small, both f(l) and g(l) can be approximated as unity leaving the force generator to be a linear function of neural excitation, a- The coefficient of linearity is a DC gain corresponding to the recruitment of the motoneurons. The muscle force depends on the extension and extensional velocity of the muscle and also on its activ ity , which is under neural control. When the muscle is called upon to exert a greater force, the neural excitation to individual motor units increases and more motor units are recruited [122, 128, 131]. The isotonic and isometric conditions [75,127] suggest a parallel elastic element, a parallel dashpot, and an ideal force generator (Fig. 4.2) for the muscle: Fe = Kx + Bx + oF0 (4.3) where o is the measure of muscle input activity [124], x is the muscle extension, • x is the velocity of extension, F is the force and F 0 is a DC gain. The contractile force aFo 1S due to reflex signals, central signals or both [128]. The main muscle sensory receptors are in the spindles which are sensitive to both extension and rate of change of extension. The spindle afferent fibers are, excitatory to the motoneurons of their own muscle and inhibitory to their antagonists. An increase in a-motoneurons output signals increases the contractile force exerted by the muscle fibers. The presence of y-fiber activity provides a mechanical bias to the spindles, and alters their sensitivity to stretch (static sensitivity) and rate of stretch (dynamic sensitivity). Some control may also be affected by the Golgi Tendon Organs, that is inhibitory to the homonymous motoneurons. A mathematical model for the Golgi Tendon Organ can be found in [87]. The effect of this sensory feedback is not clear on the reflex control for motion [122,129] and so it is neglected in the present analysis. The neural control system is based on the theory of neural feed back control of Merton [126, 130,131]. The control system is a loop consisting of the muscle spindles, synaptic connections from the spindl es via the dorsal roots to motoneurons in the ventral horn, and finally back to the muscle. The system acts to increase muscular activity when a muscle is stretched. This reaction is called the stretch reflex [132] ,and forms the basis of the control strategy. Houk, Cornew and Stark [133] have constructed a model for the response of amphibian spindles. Growe [134], Popple and Bowman [135], and Gottlieb and Agarwal [82] among others have presented methods for measurement of the response of mammalian muscle spindles. For small 47 stretches, x, a first order linear model for the spindle response is MS = (ksp • x + Bsp • x) H(x) (4.4) where MS is the muscle spindle output and H is the unit step function; i .e, H(x) = 1 i f x > 0 = 0 i f x < 0 The gains ksp and BSp depend on the y-efferent inputs. If the spindle is innervated by a static y-efferent, an increase in gamma firing rate increases sensitivity to stretch. This is equivalent to increasing the gain ksp. Similarly, an increase in firing rate of a dynamic y-efferent increases spindle sensitivity to rate of stretch; this corresponds to increasing the gain Bsp. 4.3. GAIN PROGRAMMING A direct input from the Central Nervous System (CNS) is also needed for voluntary motion of the forearm from the vertical stance to an extended position. Houk and Henneman [84] state that higher centers in the CNS generate a constant control signal to in itia te a motion to the desired position. However, a time-varying input signal is suggested by Inbar and Yafe [122]. In this paper, the former scheme; that is, the invariant commanding input is employed; i.e., the final desired posi tion, velocity, and acceleration are generated by the higher centers in the CNS as reference inputs to the muscular system. The muscle control system seems to be a programmable control mechanism; i . e . , the parameter gains in the muscle and muscle spindle change as the conditions of operation change. These programmed changes are transmitted by neural signals not different from the actuatinginput signals [122]. The gains of the muscle spindle are controlled through y motoneurons, while those of the muscle extrafusal fibers are controlled through a motoneurons. The schematic diagram of the muscle control system based on Fig. 4.4 is shown in Fig. 4.3. There are three phases in the voluntary movement of the arm: 1. Initially the forearm is in the vertical position. The flexor and extensor muscles are relaxed and maintain their natural (or resting) lengths. 2. Upon a command input, the extensor muscle begins to shorten. This will in itia te the motion of the forearm to the target position. Throughout this motion, the extensor muscle spindle does not produce afferent outputs. The flexor spindle, however, is active. Its output is a function of stretch and rate of stretch. 48 ( y ) 1 MUSCLE SPINDLE t. V US. £o> JC E E xft xf d VO ♦ GOLGI 2 )<- TENOOM ORGAN V xr f!S_ F GOLGI TENDON ORGAN AFFERENT (I.) ALPHA EFFERENT (u) K-) MUSCLE SPINDLE GAMMA EFFERENT ( y) FROM HIGHER CENTERS Mass . Y mn Fig. 4.4 Schematic showing anatomic connections between physiological components that participate In stretch reflex. Alpha and ganma motoneurons and Internunclal cells are denoted by a mn, y mn, and IC, respectively. E(+) and I(-) are excitatory and Inhibitory Inputs, respectively. 3. At the target position the flexor muscle generates the necessary force to hold the forearm and counteracts gravity. To evaluate the F0 parameter in Eq. (4.3) the equations of the forearm are linearized about the desired target position. For different values of F0, the natural frequencies of the linearized system are shown in Table 4.2. For an overdamped case with a physically accepted time constant, F0 = 8000. Low values of F0 correspond to weak recruitment -- a condition that occurs when one is cold and shivering. The a input is the sum of signals from the CNS, the excitatory (positive feedback) muscle afferent signal of the homonymous muscle, and the inhibitory (negative feedback) of the antagonist muscle. The target position to be reached is determined prior to the motion, and allows the gains to be programmed and be set simultaneously with the o and y si gnals. Let ep be the final equilibrium position. At bf Fe = 0 (4.5) Ff = M sin ep. (4.6) a From Fig. 3 Fe = F0 (Ue + MSe - MSf) + K xe + B xe (4.7) Ff = F0 (Uf + MSf - MSe) + K Xf + B xf (4.8) MSe and MSf represent the muscle spindle output of the extensor and the flexor muscles, respectively. At ep the following conditions hold: Fe = F0 (Ue - MSf) + Kxe (4.9) Ff = F0 (MSf) + Kxf (4.10) Adding equations (4.9) and (4.10) and noting that xe = -xf gives F0 (Ue > = Fe + Ff = JSfll Sin ep From the above equation the constant command input to the extensor muscle can be computed as a function of the terminal angle ep, i.e., Ue = (4.11) ar q Equating Eqs. (4.6) and (4.10): 51 TABLE 4.1 Numerical Parameters of a Normal Human Forearm I Moment of inertia of the forearm .2 kg-m about its axis of rotation. m Mass of the forearm 1.85 kg g gravitational acceleration 9.8 m/sec2 1 moment arm of the forearm weight .25 m a moment arm of the muscle forces .05 m TABLE 4.2 Variation of the Eigenvalues of the Forearm Model with Recruitment Gain Parameter; F0 F0 natural frequencies comment 80,000 -388, -10 overdamped - with very small time constant 8.000 -25, -15.6 overdamped with reasonable time constant 4.000 -10.3 ± 09.li damped oscillatory 1.000 -2.86 ± 05.6i damped oscillatory 100 -2.93, 1.69 unstable 52 •*"g~ sin 6p = Fq (ksp Xf) + K xf (4.12) From Fig. 5.5 of [83], the passive component, Kxf, of the muscle force is zero for the resting muscle and increases exponentially when the muscle is stretched. For the flexor muscle let: Kxf = (1 - e --1 sin 0p) (f0 MSf) (4.13) where F0 MSf is the active force component of the flexor force Ff. Substituting Eq. (4.13) into Eq. (4.12) gives sin 6f = F0 ksp Xf + (1 - e- *1 sin %) F0 ksp xf with xf = a sin of , the spindle position gain may be derived: ksp = ------(4.14) F0 a2 (2 - e ' - 1 s i n » ) By substituting (4.14) into (4.13), the muscle extrafusal fib er's position gain is -.1 sin e_ K = gaLLl^j Zl (4.is) 0 , -.1 sin e_ a2 (2 - e ^ ) Therefore the position feedback gain ksp of the muscle spindle and the extrafusal muscle fibers' elasticity K are both programmed from the knowledge of the desired final position of the forearm. The velocity feedback gain Bsp and the extrafusal muscle fiber viscosity B depend on the desired trajectories [122, 128] and the final position ep. To avoid overshoot in the motion of the forearm, viscosity B was found to assume values of about 80 N/ms_1 [136]. However, Bsp should be evaluated on-line and varies as a function of time. Through different simulations, it was found that small values of Bsp make the system unstable, whereas for large values of BSp the forearm stops before reaching the target. This led to the conclusion that initially Bsp should have small values to give the forearm enough energy to reach the target; however, i t should be increased before reaching the target so that velocity would be zero just when the target is reached. For an accurate and precise target hit, an exponentailly increasing positive BSp was needed. 4.4. SIMULATIONS Three sets of digital computer simulations of the forearm move ment were carried out to study the effect of the control strategy dis cussed: 53 1. point to point movement with no neural delay 2. point to point movement with neural delay 3. point to point movement under external force. The point-to-point motion of the forearm was simlulated on a digital computer under three cases as outlined below: Case one: From vertical position, e = 0, to an extended position of e = .2 rad. Case two: From vertical position, e = 0, to an extended position of e = .4 rad. Case three: From a flexed position of e = -.2 rad. to an extended position of e = .2 rad. All the above simulations are performed for the same period of time, i.e., 2 seconds. The muscle parameters for the above three cases are tabulated in Table 4.3. Position and velocity trajectories are shown in Fig. 4.4, and the corresponding flexor muscle spindle afferent signals are shown in Fig. 4.5. Initially, when the stretching velocity of the flexor muscle is high, the corresponding spindle is firing linearly with velocity of stretching, and then it retains a steady state value proportional to the muscle stretch in agreement with experimental results of [136]. The extensor muscle spindle is not firing. The force distributions, throughout the motion, in flexor (Ff) and extensor (Fe) muscles are shown in Fig. 4.7. Initially a sufficiently large force is generated in the extensor to initiate the motion. Subse quently this force goes to zero. The flexor is activated and sufficient force is developed to hold the body at the final position ep. The major characteristics of the above three simulations are sum marized in Table 4.4, where Fmax is the maximum force generated in the muscle, FSs is the steady state equilibrium force in the muscle, MSmax is the maximum muscle spindle afferent output, MSss. is the steady state equilibrium muscle spindle afferent output, and V,pax is the maximum , velocity of the forearm developed through the motion. The muscle spindle velocity gain, Bsp, is shown in Fig. 4.8 for the above three cases. No neural delay time was assumed in the muscle spindle paths in the above simulations. However, nerve fibers show some conduction velocity. There are multi synaptic as well as monosynaptic connections in the spinal cord. Also nerve fibers differ in size and consequently in conduction velocity. Thus there are different neural delay times for different fibers [123]. The motion of the forearm from vertical position 54 TABLE 4.3 Muscle Model Parameters Case K B Ksp 1 36 80 0.22 2 70 80 .21 3 80 80 .22 O l TABLE 4.4 on Neuro-muscuTar Variables for Point-to-Point Motion of the Forearm Under the Different Cases Case Fjmax F2max ^lss ^ s s MSlmax M$2max ^ l s s M$2ss Vmax 17.88 20 0 17.88 0 2.4 x10-3 o 2.2x10-3 0.52 36 37 0 36 0 4.6x10-3 0 4.3xl0"3 1.0 17.88 27 0 27 0 2.9x10-3 0 2.2x10-3 0.73 Case 1 tn T3 C O U 0.4 OJ tn tn c re •f* 0:2 s-*0 ctn 0.1 re ■o re 0.0 cc 0.0 1.0 2.0 Time/seconds Case 2 tn “Oc o o 0) to 0.8 toc re 0.6 re to c re 0.2 T3 re cc 0.0 0.0 1.0 2.0 Time/seconds tn Case 3 - a 1.0 so o '.8 Fig. 4.5: Angular position e and velocity 0 of the forearm. 56 Case 1 2.0 COI o X co 0.8 o> 0.4 0.0 0.0 •1.0 2.0 Time/seconds 5.0 Case 2 4*0 CO Io 3.0 X l/> 2.0 co 1.0 0.0 1.0 2.0 Time/seconds Case 3 3.0 4.0 m io cin o 3 Ol 0.0 0 1.0 2.0 Time/seconds Fig. 4.6: Flexor muscle spindle afferent output 57 Case 1 20 oC 4-> o; i . o 2.0 Time/seconds Case 2 01c o •*-> cus 0.0 -5 0.0 2.C Time/seconds Case 3 30 15 c01 o ■M 3 QJ 0 0.0 1.0 2.0 Time/seconds Fig. 4.7: Flexor (Ff) and extensor (F0) muscle forces 58 Case 1 1.4 and "Oto 1*2 e Case 2 o o 1.0 cu co 0.8 CO QJ 0.6 OJ E -v* 0.4 toc o 0.2 4-> 5 0> 0.0 0.0 1.0 2.02.0 Time/seconds Case 3 1.4 CO T3C U2 o o . a> 0.6 E 0.4 toc o 4 -> 0.2 £ Q) 0.0 0.0 2.0 Time/seconds Fig. 4.8: Muscle spindle velocity gain, BSp 59 Radians, radians/seconds Newtons i. 4.9: Fig. Spindle gain amplification factor, (a); flexor muscle spindle spindle muscle flexor (a); factor, amplification gain Spindle output before amplification (1) and after am plification (2), (b); (b); (2), plification am after and (1) amplification before output angular position and velocity of the forearm (3); when neural fibers fibers neural when (3); forearm the of velocity and position angular ae ie delay. time have 2.0 4.0 5.0 0.0 3.0 1.0 2.0 2,5 1.5 0.0 0.5 1.0 0.8 0.6 1.0 0.4 0.1 0.2 0.0 . 10 . Time/seconds 2.0 1.0 0.0 . 10 . Time/seconds 2.0 1.0 0.0 . 10 . Time/seconds 2.0 1.0 0.0 Newtons 30 25 20 15 10 5 i. 4.10: Fig. t rs rssi h extension. the g resistin wrist at external force of 2 Newtons is acting on the forearm forearm the on acting is Newtons 2 of force external lxr uce F ad xesr F fre, hn an when forces, ) (Fe extensor and ) (Ff muscle Flexor 0.0 61 . Time/seconds 2.0 of e = 0 to extended position of e = .2 rad. is simulated next with a delay time of 50 m sec [124] in the muscle spindle path. With the same parameters, the forearm is unstable and falls down due to insufficient reflex feedback and inadequate force in the biceps to counteract the gravity. With the neural delay time, some signal amplification is required to produce sufficient reflex signal to stabilize the forearm. This amplification could be affected by gain increase (such as presynaptic facilitation [137], or by adjusting the dynamic sensitivity of the muscle spindle fibers. Gain increase is employed here, and the increase is evaluated to produce sufficient reflex signal to hold the forearm at the target position. This gain is higher at first and then falls down to unity at the steady state final position (Fig. 4.9a). The delayed muscle spindle output signal (Curve 1) and the ampli fied delayed one (Curve 2) are shown in Fig. 4.9b. The position and velocity trajectories of the forearm are shown in Fig. 4.9c. In the third case an external force is assumed to act on the forearm at the wrist resisting the extension motion. This force has a constant magnitude and is perpendicular to the forearm. The motion of the forearm from the vertical position of e = 0.2 rad. is simulated. The angular position and velocity are the same as those of Fig. 4.9c. The biceps and triceps forces are shown on Fig. 4.10. With the forearm to resist an external force of 2 Newtons at the wrist, the triceps must generate more force to move the forearm. At the final steady state equilibrium position, both the biceps and the triceps muscles are activated; therefore, more energy is consumed with the external force opposing the motion of the forearm. 4.5. SUMMARY The dynamic behavior of a one link model of the human forearm is studied. The model reference adaptive aspect and the questions of parameter versus signal adaptation are analyzed by modeling the neural control mechanisms of the muscles. It is demonstrated that, for a precise target hit, a parameter adaptation scheme, corresponding to a central programming of system gains, is necessary. This programming of gains may shed some light on further functional aspects of the central nervous system. The gains of the control system are programmed as functions of both the final position and the trajectories of the forearm. Digital computer simulations show the effectiveness of this strategy in the point-to-point movement. This amounts to "presetting" as well as "conditioning" of the muscle control gains. Point-to-point motion of the forearm under different conditions; velocities, and initial and final positions were simulated. The effect of neural delay in the reflex feedback path of the muscle spindles was studied. This delayed feature of the reflex system calls for a programmable amplification gain in the reflex path. Motion of the 62 forearm under external force was simulated to confirm validity of the model. 63 CHAPTER V VOLUNTARY POINT-TO-POINT MOVEMENT OF THE ONE RIGID BODY MODEL OF THE HUMAN SPINE 5.1 INTRODUCTION In this chapter, the three dimensional voluntary point-to-point movement of the human upper extremity is studied. In Section 5.2, the anatomy and functional role of the human trunk muscles are presented. For three dimensional movement of the torso, at least three paris of muscles, cbrresponding to the three rotational degrees of freedom of the torso, are required. In Section 5.3, the one rigid body model of the human spine with three pairs of muscles is presented, and muscle lengths are related to the angular position of the torso . In Section 5.4 the stability analysis of the model of the human trunk with muscular actu ators is covered. The muscle parameter identification and the dynamic response analysis of this system are carried out in this section. Point-to-point voluntary movement of the torso both in a plane and three dimensional space is simulated on the digital computer and the results of the simulation are presented in Section 5.5. 5.2 ANATOMY AND FUNCTIONAL ROLE OF THE HUMAN TRUNK MUSCLES The muscles of the trunk can be divided into two principal groups. One group at the lower part of the trunk, extending from the rib cage to the pelvis, controls the motion at the waist. A second group at the upper part of the trunk acts upon the shoulder girdle and upper arm. The muscles at the lower part of the trunk are three in number, each muscle of course is paired with a similar muscle on the opposite side of the body: the sacrospinal is at the back, the rectus abdominis at the front, and the obliquus (internal and external) at the sides. The sacrospinal is arises from the sacrum, the lower lumbar vertebrae, and the posterior end of the iliac crest; and its fibers are directed upward. It serves to strengthen the spine, hold the body erect, arch the back, and extend the neck (Fig. 5.1). 64 a Fig. 5.1: Sacrospinalis (a) and obliquus externus (b) muscles. 65 Fig. 5.2: Rectus abdominis (a) and obliquus externus (b) muscles. 66 The rectus abdominis is a rather slender muscle extending vertically across the front of the abdominal wall. The right and left recti are separated by a tendinous strip about an inch wide called the lina alba (white line) (Fig. 5.2). It arises from the crest of the pubis and inserts the cartilages of the fifth, sixth, and seventh ribs. Rectus abdominis is the prime mover for spinal flexion. Contraction of one rectus abdominis alone assists with lateral flexion to the same side. The external oblique muscle covers the front the side of the abdomen from the rectus abdominis to the latissimus (Fig. 5.3). It arises from the front half of the crest of the ilium, the upper edge of the fascia of the thigh, the crest of the pubis and the linea alba, and inserts by sawtooth attachments to the lower eight ribs. The external oblique is the prime mover for flexion, lateral flexion to the same side, and rotation to the opposite side. The internal oblique is situated under the external oblique, with fibers running at nearly right angles to those of the outer muscle (Fig. 5.4). It arises from the lumber fascia, the anterior two-thirds of the crest of the ilium, and the lateral half of the inguinal ligament. It inserts to the cartiliages of the eighth, ninth, and tenth ribs and the lina alba. The internal oblique is the prime mover for flexion, lateral flexion to the same side, and rotation to the same side. The rectus and two oblique muscles of the abdomen act together in all movements of rigorous flexion of the trunk, as in rising to erect sitting position when lying on the back. In lateral flexion the abdominal muscles of one side act. In rotation the external of the opposite side acts with the internal oblique of the same side. 5.3 ONE RIGID BODY MODEL OF THE HUMAN SPINE From a mechanical viewpoint, the human spine is an extremely complex structure. Its twenty-four mobile vertebrae experience large, three-dimensional displacements and are interconnected by numerous ligaments and muscles. Furthermore the spine interacts with other parts of the body, such as the ribs, the trunk and the limbs. In this work the human torso is modeled as a rigid ellip tical cylinder, representing the upper extremity. I t is hinged to the hip jo in t, which is assumed to remain stationary. For the three-dimensional movement of this model, at least three pairs of muscles are needed: two muscles for the sagittal plane motion (flexion and extension); two muscles for the coronal plane motion (lateral flexion and extension); and two muscles for transversal plane motion (self rotation of the right or left). 67 Crest of i 11i um Fig. 5.3: Obliquus externus (a) and rectus abdominis (b) muscles. 68 b Crest of 111i um Fig. 5.4: Obliquus internus (a) and rectus abdominis (b) muscles 69 Fig. 5.5 shows the schematic of the torso with three pairs of muscles. This arrangement of the muscles is assumed under the following assumptions: 1. The torso is a symmetric ellip tical cylinder with the diameter of 0.40 meters. 2. The rectus abdominis and sacrospinal is muscles are located in the sagittal plane with the points of origin on the z-axis at the distance of 0.14 meters to the origin. However, they have different points of insertion as shown in Table 5.5. 3. The right and left flexor muscles (side benders) are located in the coronal plane with the points of origin and insertion at the distance of .14 meters to the y-axis. 4. The right and left transverse muscles are located in a transverse plane. 5. All muscles are assumed to retain a straight line shape even if they are contracting. The vertebral column, as stated before, exhibits approximate symmetry with respect to the midsagittal plane in motion of flexion and extension [99]. Therefore the lateral flexor and extensor muscles are assumed to be identical and placed in the coronal plane. The lower fibers of the interior oblique muscle are in transversal plane and they are considered as two identical muscles located on the right and left side of the torso [113]. The arrangement of the torso in the sagittal plane with one pair of muscles is shown in Fig. 5.6, where the torso is shown with base point 0 representing the hip joint. In the sagittal plane, two muscles act on the torso: Rectus abdominis muscle. -- This muscle is taken as a single muscle- ^!'ch flexes the body when contracted. It has an approximate natural resting length of 0.33 meters denoted by L r0 and its force is denoted by Fr . Sacro so in aj l s muscle. — This muscle is taken as a single muscle on the back that erects" the body.. It has an approximate natural length of 0.46 meters denoted by L s0 and its force is denoted by Fr . Both muscle forces have the same arm of about 0.14 meters, denoted by x. Fig. 5.7 shows the torso in a flexed position, when rectus abdominis muscle is contracted and the sacrospinal is muscle is 70 y A 2 '///////// / til /////.■/ (a) y A* 7/11 * 111t , < * I! i / . / / ;; Fig. 5.5: Torso in the Sagittal (a), Coronal (b), and Transversal (c) planes. 71 stretched. Let vectors a and c represent the origin points of the rectus abdominis and sacrospinal is muscles, respectively, in the inertial coordinate system (Fig. 6): a = [0,0, -x]T c = [0,0, x]T Vectors b and d represent the points of attachment of these two muscles to the torso in the body coordinate system x'y'z'* b = [0 , Lyo , - X]T d = [0 , Lso , X]T Thelengths Lr of the rectus abdominis and Ls of thesacrospinal is as a function of the angle vector q and vectors a, b, c, and d are given by Lr = ia-A(e)b i 5.1 Ls = II c-A(e)d II 5.2 Matrix A(©) is the transformation matrix from the body coordinate system to the inertial coordinate system (Appendix B). Alternatively, the approximate lengths of these muscles can be obtained as follows. Fig. 5.7 shows the torso in the upright position when both rectus abdominis and sacrospinal is muscles are at rest and retaining their natural lengths; L r0 and Lso, respectively. Let Lr and Ls be the lengths of rectus abdominis and sacrospinal is muscles, respectively and let, xr = / x2 + Lro 2 xS = / x2 + Ls0 Pr = 90-a)"- 01 Pr = 90- Of- 01 The approximate lengths of the muscles as function of the angle ei are given by Lr ( ei) = / xr 2 + x2 - 2x*xr *cospr 5.3 72 y Fig. 5.6: Torso in the Sagittal plan at a flexed position. 73 so ro 4 i N il1 / 4 i / rm r/fi////*////?////m/t Fig. 5.7: Torso in the sagittal plane with muscles at rest 74 5.4 However, the simulations presented in this chapter are carried out for small variations of lengths of muscles about their resting lengths, and therefore the nonlinear functions defined in Eq. 4.2 are approximated by unity. 5.4 STABILITY ANALYSIS OF THE HUMAN SPINE MODEL In this section the stability of the torso with muscle actuators is studied. As mentioned before at least three pairs of muscles are required for three dimensional movement of the torso. The muscles are modeled with linear springs and dashpots and a force generator that depends linearly on the neural actuating inputs to the muscles ( a-motoneuron signals) as shown in Fig. 4.2 of Chapter 4. Each pair of muscles make an agonist-antagonist pair interconnected through reciprocal innervation as explained in Chapter 4. The control block diagram of a pair of muscles is shown in Fig. 4.3 For the present study, the muscles are assumed to have identical models with the same parameter values. This simplifies the analysis of the model. However, one could use different muscle parameters and even more muscles to represent actual musculature of the human spine. The stability of the torso is studied by linearizing the dynamic equations of the torso about an operating point. This study is carried out first for the case that the muscles are modeled by passive springs and dashpots with no muscle spindle feedbacks and secondly for the case that the muscle spindles are active, and spindle feedback is available. The equations of motion of the torso in three dimensional space were derived in Chapter 3: 0 = B(0 )W J0W = -f0 (W) + RRAt (©)G + N 5.5 " where 5.6 The muscle moments are denoted by vector M.in the inertial coordinate system consisting of three components Mj, M 2 , and M3 corresponding to the sagittal, transversal and coronal planes, respectively, 75 Mi = X.cosei [P’r_frs3 M2 = x.cosea [Ftr-F’t^] 5.7 M3 = x.cos 03 [Ffr-F fA] where, Fr = rectus abdominis muscle force Fs = sacrospinal is muscle force Ftr = right transverse muscle force Ft£ = le ft transverse muscle force Ffr = right lateral flexor muscle force Ff^ = le ft lateral flexor muscle force The force generated in a muscle Is explained in Chapter 4: Fi - K0(Uj + MS-j - MSj') + Kej + Bej 5.8 for i = 1,6 and j = 1,3 where Fj = force in muscle i K0 = a constant gain corresponding to the motorneuron recruitment Ui = open loop input to muscle i from higher centers of nervous system MS-j = muscle spindle afferent signal from the muscle i, which is excitatory to muscle i MS-j' = muscle spindle afferent signal from the antagonist muscle to the muscle i, which is inhibitory to muscle i. K = Muscle fiber's elasticity 0j = angular position variable corresponding to the plane where muscle i and its antagonist muscle lie B = Muscle fib e r's viscosity 9j = time derivative of variable 0j 76 Muscle spindle afferent output is a function of both e and e, i .e ., MS = KSp0 + Bgp0 5.9 where, Ksp = muscle spindle's fiber elasticity Bsp = muscle spindle's fiber viscousity In feedback control terminology, Ksp and Bsp are the position and velocity feedback gains, respectively, and in physiology they represent the static and dynamic sensitivity parameters [82]. These muscle model parameters are evaluated from the knowledge of the dynamic behavior and steady state position of the torso. Table 5.1 shows a set of values for these parameters for the steady state position of ei = 0 . 2 , 02 = 0 , and 03 = .2 radians. The gain K0 is given the same value as found in Chapter 4, i.e ., 8000 . Variable K is evaluated such that at this operating point the passive component of the force generated in any of the stretched muscles is about 3% of the total force in the same muscle [83]. For the evaluation of KSp the torso is assumed to be in a flexed position in the sagittal plane, where ei = 0.2 radians. Ksp should assume a value such th at i t will give rise to a sufficient force in the stretched muscle to hold the torso at this operating point. The viscous parameter B is given the same value as found in Chapter 4. The velocity gain BSp of the muscle spindle depenJs on the desired dynamic response of the torso and can be evaluated by pole placement. Table 5.2 gives pole locations as a function of Bsp. For small Bsp, i.e., when Bsp is about 0.005, the complex conjugate pairs of poles indicate an oscillatory behavior for the dynamic response of the system. The stability analysis of the torso with the muscle parameters of Table 5.1 is carried out by linearizing the equations of motion of the torso about the upright position. The linearized equations of motion are put in the following form: X = AX + BU 5.10 where, X is the 6x1 vector of the angular positions 0 = [ 01, 02, 03]^ and velocities W = [wj, W2 , W3 ] of the torso representing the state variables of the system, i.e., X = [ 0 W]T and U is the 6x1 vector of the inputs to the muscles. 77 Table 5.1 Muscle Model Parameters KO K B Ksp Bsp 8000 38.26 80 .1546 0.01 Table 5.2 Variation of the Eigenvalues of the Closed Loop System with Muscle Spindle System; Bsp Bsp Eigenvalues Comments 0 .1 -306.54, -1.456 large time -12.3, -0.0193 constants -12.3, -0.0193 0 .0 1 -46,37, 09.625 Acceptable -2.1282, -.11179 response -2.1282, -.11179 0.005 -21 ± 2.32i Oscil ato ry -1.524, -.156 behavior in -1.524, -.156 transverse modes Table 5.3a gives the values of poles of the open loop system, whe the muscle spindles are not active, i.e., all the six muscles are represented only by passive springs and dashpots. One pair of poles corresponding one each in the sagittal and coronal planes of the system are in the right half of the frequency plane, since the torso is inherently unstable due to gravity. However, the poles corresponding to the transversal plane, -27.513 and -.47308, are stable; in agreement with the fact that the torso is not under gravity effect in this plane of motion. Therefore the passive springs and dashpots are adequate to stabilize the torso in the transversal plane. With spindle feedback the unstable poles of the torso are moved to the left side of the complex plane. Table 5.3b shows the poles of the stabilized system. The torso could be stabilized about the upright position by increasing the muscle viscoelastic parameters, i.e., K and B. However, at the upright operating point the muscles are at rest so these parameters are fairly low. Moreover, physiological experiments show that the reflex control system is effective for postural stability [84, 129, 133]. The effect of the reflex control system is even more pronounced for small changes in the lengths of muscles [138]. It could be surmised that for small disturbances around the steady state operating points the reflex control system plays the major role in stabilization of the torso, but for larger displacements of the torso the viscoelastic parameters of the muscle would be adjusted properly by d irect inputs from the higher centers of the nervous system such as cr mononeuron signals [ 122]. 5.5 POINT-TO-POINT MOTION In this section the voluntary point-to-point motion of the torso is studied by digital computer simulation. In itia lly the torso is in the vertical stationary position and the muscles are at rest. To initiate any motion of the torso, the proper muscles have to be activated by the central nervous system. In addition, depending on the required final steady state position and dynamic response behavior of that motion, the gain parameters of the muscle model are preset by efferent input signals to the muscle, i.e., the a and y motoneuron signal s. The motion of the torso from the upright position to the point of [. 2 , 0, 0]T and e = [.2, 0, .ZV are considered here for the fol 1owi ng: Case 1. — In this case a command input activates the rectus abdominis muscle to shorten and in itia te the motion of the torso to = 0 .2 radians in the sagittal plane. Its antagonist muscle — sacrospinal is — starts to build up a force that would be enough to hold the torso at 79 Table 5.3 (a) Open Loop Eigenstructure of the Torso ETGSTft VECTOR WITH 6 COMPONENTS) 27.513 --4.7308 -4*7308 -.4 8 6 5 3♦6108 3 ♦ 6 i. o; AMAT (A 6 BY 6 MATRIX) 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 17.082 0 0 - 1 .1 2 0 0 0 -13.385 0 0 -28 0 0 0 17.082 0 0 - 1 .1 2 BMAT (A 6 BY 1 MATRIX) 0 0 0 112 2800 112 (b) Closed Loop Eigenstructure of the Torso EIGS (A VECTOR WITH 6 COMPONENTS) -46.374 -9.6255 -2.1282 -2.1282 -.11179 -.11179 AMAT (A 6 BY 6 MATRIX) O O O 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ♦23792 0 0 2 .24 0 0 0 -446.38 0 0 -56 0 0 0 -.2 3 7 9 2 0 0 - 2 .2 4 BMAT (A 6 BY 1 MATRIX) 0 0 0 112 2800 112 80 the planned final steady state position. The command input to the contracting muscle -- rectus abdominis -- depends on the target position and is evaluated such that at that position the net force in the rectus abdominis is zero and this muscle is relaxed; i . e . , its magnitude should be the same as the spindle feedback signal produced by the antagonist muscle-sacrospi nal i s. Since the force of gravity continuously acts to destabilize the torso, a bias input is required to maintain any position. This input which is only a function of position is computed by an approximate inverse dynamics to counteract gravity force and drive the torso to the desired position dictated by the reference input position and velocity trajectories. Whenever the actual state (position and velocity) of the torso is different from the reference ones, a feedback input is necessary to compensate the bias input. This feedback input is the re su lt of comparing the actual and reference state of the system and then weighting the error signal by linear feedback gains. The controller block diagram is shown in Fig. 5.8. The motion of the torso in the sagittal plane was simulated on the digital computer with the muscle parameter shown in Table 5.1. The muscle spindle velocity gain, Bsp, was assumed to be 0.01 and from Table 5.2 the poles of the system corresponding to the variables ej and *e are -2.1282 , -.11179 These poles give a large settling time of about 9 seconds to speed up the system response, the feedback gains of the controller shown in Fig. 5.8 were chosen to move the poles of the system to give smaller time constants. For new multiple poles of -2.1282, the feedback gain is K = [k2, 0, 0, k2, 0, 0]T where ki = 0.0383 and k 2 = 018 , corresponding to ei and ei, respectively. The actual and reference states are represented by the vectors ^actual = C 01» 0 , 0 , 01, 0 , 0 ]T Xref = [®lref» ®lref» 0]"^ The digital computer simulation results for this case are shown in Figs. 5.9 - 5.15. Fig. 5.9 shows the reference angular position and velocity. Fig. 10 shows the actual and reference angular positions, while Fig. 5.11 shows the actual and reference velocities. A delay of about .7 seconds is seen in the settling of position with respect to the reference one. Fig. 5.12 shows the distribution of force in the rectus abdominis (Fr) and sacrospinal is (Fs) muscles. The steady state value 81 Bias input (ub) Control input (uc) Inverse Plant 0 / Referense State Feedback > Gains (Xref) / V ■ K D l Actual state (Xactual) Fig. 5.8: The Controller Black Diagram. 0.6 - o lref Q) 0 .5 to (/) c re •5 re o n 0.2 co c lre f re •r— ■D re o n 0 . 1 }— 0.0 0 0.5 Time /Sec. Fig. 5.9: Reference position (0ref) and velocity <©ref) of the Torso in the sagittal plane. 0 .3 0 0 .1 8 0 .1 6 O 14 0.12 lr e f f— 0.10 •o fC oz 0 .0 8 lActual o.os 0 .0 4 0.02 0.00 ' 0 5 1.0 Time/Sec. Fig. 5.10: Actual and reference angular positions of the Torso in the sagittal plane 83 Newtons Radians/Sec i. .1 Ata ad eeec aglr eoiis f h Tro in Torso the of velocities angular reference and Actual 5.11: Fig. i. .2 Rcu admns F ad arsia s F) uce forces. muscle (Fg) is Sacrospinal and ) (F abdominis Rectus 5.12: Fig. .7 0 6 O .4 0 0 0.0 .3 0 2 h sgta plane. sagittal the 1 ref 1 actual 5 . Time/Sec. 1.0 .5 84 Time/Sec. Newtons i. .3 Msua tru atn o te os i te aitl plane. sagittal the in Torso the on acting torque Muscular 5.13: Fig. i. .4 Scopnls uce pnl otu i te aitl plane. sagittal the in output spindle muscle Sacrospinalis 5.14: Fig. 10 -1 30 -3 -ao SO -S -3 E 20 30 0 1 40 30 35 20 25 15 lO - 0 . 10 Time/Sec 1.0 0.5 85 TTT j Time/Sec 70 E -3 SO — SO 30 30 to Fig. 5.15: Bias input (Uh) and Control Input (Uc) of the Torso in the sagittal plane. 86 Radians Radians i. .6 Ata ad eeec aglr oiin o te os i the in Torso the of positions angular reference and Actual 5.16: Fig. i. .7 Ata ad eeec aglr oiin o te os in Torso the of positions angular reference and Actual 5.17: Fig. 16 .1 0 18 - 8 .1 0 o.eo 0.00 0.10 0.12 0.20 0.02 4 .0 0 6 .0 0 8 .0 0 16 .1 0 18 .1 0 14 .1 0 0.12 08 .0 0 o.io 0.00 0.02 4 .0 0 6 .0 0 aitl plane. sagittal h Crnl plane. Coronal the 3ref Ire# actual 3 1 actual 87 Time/Sec Time/Sec Radians/Sec i. .8 Aglr oiin o te Torso the of positions Angular 5.18: Fig. i. .9 Aglr eoiis f h Torso the of velorities Angular 5.19: Fig. 0 2 . 0 16 .1 0 8 .1 0 .is o 4 .1 0 o.io 06 .0 0 8 .0 0 0.00 4 .0 0 - - .5 0 .3 0 0.2 0.1 0.6 0.0 0.1 0.2 88 Time/Sec Time/Sec ■400 350 3C0 250 200 150 100 50 Time/Sec Fig. 5.20: Rectus abdominis (Fr ) and Sacrospinal is (Fg) muscle forces 350 300 250 200 150 100 50 0 .5 1.0 Time/Sec Fig. 5.21: Right lateral flexor (Ffr) and left lateral flexor (Ffl) muscle forces. 89 45 r— E-3 •40 - 35 - 30 25 20 15 lO 0 5 ' 1.0 Time/Sec Fig. 5.22: Sacrospinal is muscle spindle output •45 E-3 40 35 30 (/> c 25 o + -> 2 0 lO Time/Sec Fig. 5.23: Left lateral flexor muscle spindle output 90 Newtons force muscle Transverse Left 5.24: _ -ig. Newtons i. .5 Ba ipt U) n Cnrl nu (c o te Torso the of (Uc) input Control and (U.) input Bias 5.25: Fig. 0 1 - 0 6 50 20 30 43 10 —i —i —i —i —r i— i— i— i— i— i— i— i— i— #5 . Time/Sec 1.0 91 i i » i i i i Time/Sec Newtons Newtons i. .7 Te tipt inl o h Scopnli ad eft lateral t f le and is Sacrospinal the to signal ct-input The 5.27: Fig. g 52: h c-nu sga t te ets boii ad right and abdominis rectus the to signal ct-input The 5.26: ig. E -3 -3 E -3 E so 35 - •45 s a 30 0 3 •40 •45 15 30 35 10 20 ss 15 10 aea feo muscles flexor lateral lxr muscles flexor 92 Time/Sec Time/Sec of Fs is equal to 255.08 N. This force is sufficient to counteract the gravity and hold the torso at 0.2 rad. Fig. 5.13 shows the muscular torque which is acting on the body. Fig. 5.14 gives the sacrospinal is muscle spindle output. It increases with the velocity and then retains a steady state value which is the activation signal to the sacrospinal is muscle. In Fig. 5.15 the bias input (%) and control input (Uc) are shown. The control input is higher than the bias Input initially and then retains the same value as the bias input at the steady-state final position. Case 2. — In this case the motion of the torso from the upright position to the point of ei = 0 . 2 , 02 = 0 , and 03 = 0 .2 radians is considered. Initially the rectus abdominis and right side flexor muscles are activated, and at the final required position these muscles are relaxed and their antagonits are stretched and generate enough force to hold the body and counteract the gravity force. Since the torso is assumed to be symmetric and since the three pairs of muscles are assumed to be identical, the dynamic response of the torso projected onto the sagittal and coronal planes would be similar to that of Case 1. The feedback gain vector for this case is K = Ckl. 0, k3, k2, 0, k4]T where, ki and k3 are position gains equal to 0.0383 and k 2 and k 4 are velocity gains equal to 0.018. With this feedback gain vector, the poles of the system corresponding to 01, 03, ei, and 63 are all located at -2.128. The actual and reference state vectors are respectively represented by ^actual = [ 9l> 0 > 03» 6l» 03]^ Xref = C 01 ref* 0* 93ref> * 01ref. 0 , 93ref]T The simulation results for this case are shown in Figs. 5.16 - 5.27. Fig. 5.16 shows the actual and reference angular position of the torso in the sagittal plane. The amount of time delay in the motion of the torso is in the same order of those of Case 1. Fig. 5.17 shows the same trajectory for 03. Fig. 5.18 shows the three angular positions of the torso. It is noted that 01 and 03 both go to .2 radians, but 02 after some variations about zero stays almost at zero. Fig. 5.19 shows the angular velocities of the torso. Figs. 5.20 and 5.21 show the variations of the force in the rectus abdominis (Fr ), sacrospinal is (Fs), right lateral flexor (Ffr), and left lateral flexor (FfA) muscles. Since the motion projections onto the sagittal and coronal planes are equivalent, the corresponding muscle forces are also equivalent for the example. Figs. 5.22 and 5.23 show the muscle spindle outputs of the stretched muscles, i.e., the sacrospinal is and the left 93 i. .8 Lnts f ets boii ( ) scopnls L ), (L sacrospinalis ), (L abdominis rectus of Lengths 5.28: Fig. i. .9 Lnts f ih tases (tr) n l t transverse ft le and ) r (Lt transverse right of Lengths 5.29: Fig. Meters Meters 0 . 195 195 . 0 8 9 .1 0 7 9 .1 0 194 9 .1 0 198 8 9 .1 0 9 9 .1 0 0.200 0.201 0.202 0.202 50 0 .5 0 35 5 .3 0 40 0 .4 0 5 .4 0 0.20 0.20 5 .3 0 15 .1 0 0 .3 0 ' right lateral flexor (Lfr )» and le ft lateral flexor (L ^) ^) (L flexor lateral ft le and )» (Lfr flexor lateral right ' muscles L muscles. (L 10 . Time/Sec 2.0 1.0 ° 10 . Time/Sec 2.0 1.0 ° 94 r t Table 5.4 The Insertion and Origin Coordinates of the Torso Muscles in the Three Dimensional Space* Muscle - Origin Insertion Resting (X,Y,Z) (X,Y,Z) Length Rectus abdominis (0, 0, .14) (0, 33, .14) 0.33 Sacrospinal is (0, 0, -.14) (0, .46, -14) 0.46 Right flexor (-.14, 0, 0) (-14, .20, 0) 0.20 Left flexor (.14, 0, 0) (.14, .20, 0) 0.20 Right transverse (0, 0, .14) (-.14, 0, 0) 0.28 Left transverse (0, 0, .14) (0.14, 0, 0) 0.28 * Unit is meter lateral flexor. In Fig. 5.24, the force distribution in the left transverse muscle is shown. This force is generated to keep 02 near zero. If there were no transverse muscle force, due to the coupling between angular positions when ei and 93 are activated, 02 would be activated also. Consequently, the system would be unstable. To keep the 02 angle at zero, a transverse force of about 50 Newtons is generated. Fig. 5.25 shows the bias input (Ut>) and compensated control input (Uc) to the rectus abdominis and right lateral flexor muscles. The a-inputs to the muscles are shown in Figs. 5.26 and 5.27. In Fig. 5.26 the crinput to the rectus abdominis and right lateral flexor muscles are shown. To start the motion of the torso, a large initial a signal activates these muscles and then the activity of these muscles is inhibited afterwards by the antagonist muscle spindles. Fig. 5.26 shows the a input signals to the sacrospinal is and left lateral flexor muscles that correspond to the force that holds the body at the final steady state point. The lengths of muscles can be calculated by vector subtraction as explained in Section 5.5, where the vectors represent the origin and insertion points in the inertial and body coordinate systems, respectively. Table 5.4 shows the points of insertion, origin, and the resting lengths of the muscles. When the torso moves from the upright position, where all muscles are at their resting lengths, to a position that ©i = 0. 2, 02 = 0. 0, and 03 = 0.2 the variations in the lengths of muscles are shown in Figs. 5.28 and 5.29. Fig. 5.28 shows the lengths of rectus abdominis (Lr ), sacrospinal is (Ls), right lateral flexor (Lfr), and left lateral flexor (Lf^) muscles. Fig. 5.29 shows the lengths of the transverse muscles; right transverse (Ltr) and left transverse d-t*). 5.7 SUMMARY A short discussion of the human trunk muscles was presented. The torso was modeled as one rigid body moving in the three dimensional space with three pairs of muscles. Each muscle pair can move the torso in a plane. Muscle parameters were selected. S tability of the torso both with and without spindle feedback was considered. Voluntary point-to-point movement of the torso in a plane and three dimensional space were analyzed and the result of digital computer simulations were presented. A dynamic controller is employed to have the torso follow a set of reference state trajectories with no steady- state error. Muscle forces and muscle sensory signals during the motion of the torso were plotted as functions of time. The variations in the length of the muscles as a function of time as the torso moves in multidimensional space was also presented. 96 CHAPTER VI CONCLUSIONS 6.1 SUMMARY A state space formulation of systems of interconnected rigid bodies is applied to three dimensional one and two element linkage systems. The one-element linkage system is a human torso, and the two- element linkage system is that of the elbow or the knee joint with the appropriate nonholonomic constraints. For the one-rigid-body system both the free body movements and the constrained system are considered. Two approaches to the analysis are presented. The first approach is keeping the forces of constraint in the equation of motion and computing them as a function of state (posi tion and velocity) and input (gravity and actuating input). This gives a method for indirect measurement of the internal forces and eliminates the need of putting sensors at the joints if the actual parameters of the system are known. In the second approach these forces of constraint are eliminated from the equations of motion by a projection method to project the large dimensional state space onto a smaller manifold, since certain constraints can never be violated, and the calculation of the corresponding forces may not be needed or may be computationally expenssive. Two mechanisms of preventing self rotation of manipulator links about their long axis are discussed — hard constraint, where the constraint is never violated and soft constraint, which allows small movement of the system on both sides of the constraint surface. Some of the difficulties of the hard constraint model are also pointed out. Quadratic Lyapunov functions of the Euler angles and angular velocities can be constructed which lead to global stability. Digital computer simulations are carried out for the inverted pendulum model with a nonlinear state feedback, and the general stability of this sys tem is considered via Lyapunov's Second Method. The Lyapunov theory is found to be powerful in extending the stab ility domain of such systems. The above theory is simplified and applied to the dynamic behavior of a one-link model of the human forearm. The actuators are a pair of muscles at the elbow joint. These muscles are modeled by linear springs, dashpots and force generators. The neuromuscular control system in the lower level of the control hierarchy is considered and its feedback and feedforward sensory paths are investigated. The model 97 reference adaptive aspect and the question of parameter versus signal adaptation mechanisms are analyzed. The parameters of the control system are programmed as functions of both final position and the trajectories of the point-to-point movement of the forearm. It is demonstrated that, for good performance and zero steady-state error, a parameter adaptation scheme, corresponding to a central programming of system gains, is necessary. This programming of gains may shed some lig h t on further functional aspects of the central nervous system. Also, the effect of neural delay in the reflex feedback path of the muscle spindles are studied. This delayed feature of the reflex system c alls for a programmable amplification gain in the reflex path. The rigid body model of the torso is combined with the above muscle models to provide a more realistic model of the torso movement. A brief discussion of the human trunk muscles is presented. Anatomy and the functional role of the major muscle groups of the human trunk are considered. It is assumed that the prime mover of the human spine consists of only 3 identical pairs of muscles, where each muscle pair can move the human torso in one plane. Stability of the torso both with and without spindle feedback is considered. It is concluded that the torso is not stable without the muscle spindle feedbacks. The neuromuscular control system gain parameters are selected for the desired point-to-point movement and good dynamic response. Voluntary point-to-point movement of the torso in a plane and three dimensional space are analyzed and the result of digital computer simulations are presented. The control problem is postulated such that the dynamic system should f ir s t be stable in the vicinity of an operating point, and secondly, the system follow a set of reference state trajectories with no steady state error. A dynamic controller is employed for this purpose. The linear feedback gains of the controller are assumed to be set by higher centers of the nervous system for a desired dynamic response. This is carried out by computing the gains for a set of desired closed loop eigenvalues of the system. Muscle forces, motoneuron input signals, spindle feedback efferent signals, and muscle length variations are all recorded as the torso moves in the multidimensional space. 6.2 RECOMMENDATIONS Formulation of the equations of motion for multibody systems which is best from a computational point of view, should be further explored. Methods to eliminate the d iffic u ltie s in implementing the hard constraints in the equations of motion and explicit computation of their corresponding forces of constraint must be developed. 98 In using Lyapunov's methods for sta b ility , better feedback torque should be constructed so that the dynamic responses of the torso would be as those of normal human torso movement. The nonlinear feedback torque used in this study had some drawbacks such as oscillatory behavior, large time constant, and insensitivity of the responses to some of the feedback gains. Moveover, the power of this method should be examined with multi body and multidimensional structures. More improved muscle models are needed to consider all the non- linearities and characteristics of the natural muscle. In this work, the Hill's muscle model consisting of three linear parallel elements was used. Better models should include series viscoelastic components and nonlinearities such as force-velocity and force-displacement relationships. A more re a listic muscle model should have some dynamics in the motor recruitment process. Further work should be done on modeling the Golgi Tendon Organ and including it in the control system. This kind of transducer was not taken into consideration here. Another area of work is to identify different muscle parameters for different muscles of the human body. The muscles used in this study were assumed to be identical. Nerve fibers show some time delay in the conduction of signals. More work needs to be done in modeling and measuring these delays in different sensory fibers. For point-to-point motion of the torso only three pairs of muscles were considered with straight line shape and point connection to the body. More refined movements call for more muscles or muscles with parallel fibers with line connection to the body. Exact measurements of the parameters of the muscles and also insertion and origin points are needed. The human spine was modeled as a one link rigid body. Better models of this mechanically complicated structure would be multisegmental with more muscles connecting the segments. A multidimensional study of an improved model would reveal more realistic results for diagnostic, medical, sports and rehabilitation purposes. 99 APPENDIX A In this Appendix the input torques expressed in the inertial coordinate system will be transformed to the body coordinate system. Let M = [Mi, M2, M3]t be the vector of input torques that will give the body rotational displacements of e = [ei, 02, 63]T with respect to inertial coordinate system. Let N = [Ni, N 2, N3F be the equivalent input vector with respect to the body coordinate system. Consider Fig. A.a, where ox'y'z' is obtained by applying Mi along ox axis. In Fig. A.b ox'y'z' is the result of the application of M2 along oy' where Ml is resolved along ox' and oz'. Figure A.c shows the ox'"y‘' 'z ''', which is obtained as a result ofapplying M 3 and coincides with the body coordinate system. Vector N is the resultant input torque in the body coordinate system and can be put in the following matrix form: Ml ~ cos N = B -!(0)M (A.2) The time derivate of vector 0 = [ 01, 02, 03]T will be in the direction of M = [Mi, M 2, M3]T, where the angular velocity vector W = [Wi, W2, W3]T will be expressed in body coordinate system by the same relation as (A.2) as follows W = B-l( 0) 0 (A.3) 100 x, x' (a) Z Y', Y" (b) V Z" X" (c) Z" Fig. A: Coordinate axes transformation. 101 APPENDIX B In this Appendix the transformation matrix A(e) which transforms any matrix from body coordinate system to inertial coordinate system is derived. Consider Fig. A.l of Appendix A. OXY represents the inertial coordinate system. OX'Y'Z' coincides with the body coordinate system when 0 = [ei 0 0]T. It is transformed to OXYZ by the following matrix notation: “ X “ “ 1 0 0 “ - X' - Y = 0 costp -sintp Y* _ Z _ _0 sincj, cosc|>_ _ Z’ _ (B. 1) by the same token 0X"Y"Z" of Fig. A.2 transforms to OXYZ by 1 1 o “ X “ i—* o COS _ Z _ _ 0 S<), COScJ; _ _ -si n ip 0 cosij) _ z" _ (B.2) and OX" 'Y" Z '" of Fig. A.3 transforms to the inertial coordinate system by “ X “ “ 1 0 0 “ cosp 0 sinp Y = 0 cos()j -sin _ _-si n cos 0 sine 0 - X" ' - Y11 i -sine cose 0 0 0 1 Z". 102 Equation (B.3) may be represented by the following abstract form ICS = [Al( eX)-A2( G2)*A3( e3)]BCS (B.4) where ICS and BCS represent inertial and body coordinate systems, respectively. To simplify the notation, Eq. (B.4) can be written as ICS = A( 0) BCS (B.5) 103 REFERENCES 1. Muybridge, E., The Human Figure in Motion, Dover Publications, New York, 1955. 2. Plagenhoef, S., Patterns of Human Locomotion: A Cinematographic Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1971. 3. 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