ABSTRAT 1
LIST OF SYMBOLS AND ABBREVIATIONS 3
CHAPTER 1 INTRODUCTION 9
1.1. THE MOTIVATION 9
1.2. THE MAIN PROBLEM AND THE MAIN THESIS 10
1.3. OUTLINE OF THIS DISSERTATION 11
1.4. THESIS CONTRIBUTION 12
CHAPTER 2 HUMAN MOTOR CONTROL 15
2.1. THE BIOLOGICAL SYSTEM 15
2.1.1. The muscles and the joints 15
2.1.2. Muscles, motor neurons, and motor units 16
2.1.3. Proprioceptors 17
2.1.4. The spinal cord 18
2.1.5. The central nervous system 18
2.1.6. Limitations of our technical description 19
2.2. FROM FEEDBACK TO ADAPTATION 20
2.2.1. Feedback control 20
2.2.2. Adaptive control 22
2.2.3. Feed forward control and the inverse controller problem 23
2.2.4. Well and ill posed problems 24
2.2.5. Artificial neural network control 24
2.3. PARAMETER ESTIMATION 28
2.3.1. The estimation problem 28
2.3.2. Linear models 29
2.3.3. An example: parameters estimation of a linear muscle model 29
2.3.4. The order of the model 33
2.4. REDUNDANCY, DEGENERACY AND PARALLELISM 35
2.4.1. Multiple feedback loops 36
2.4.2. Spatial filtering 37
2.4.3. Learning to exploit the redundancy 38
CHAPTER 3 THE GENERAL MODEL 41
3.1. THE CONTROL PROBLEM 41
3.2. THE VIRTUE OF ADAPTATION 43
3.2.1. The hierarchy of learning and adaptation 44
3.3. THE VIRTUE OF REDUNDANCY 45
3.3.1. The formal definition of redundancy 45
3.4. THE NOTION OF MULTIPLE INVERSE CONTROLLER 48
3.5. THE GENERAL MODEL 49
CHAPTER 4 SYSTEM DYNAMICS 53
4.1. DYNAMICS DETERMINE SINGLE SOLUTION 53
4.1.1. Motor units' recruitment order 54
4.1.2. Central pattern generators 54
4.1.3. The equilibrium point hypothesis (EPH) 55
4.2. BELL SHAPED SPEED PROFILE 55
4.2.1. The phenomenon 55
4.2.2. The hill type mechanical model 56
4.2.3. The pulse excitation 57
4.2.4. The simulation result 58
4.3. STEREOTYPED RELATIONSHIPS 60
4.3.1. The experimental phenomenon 60
4.3.2. The mechanical model 61
4.3.3. The step and hold excitation 62
4.3.4. The simulation result 63
4.4. CONCLUSIONS 66
CHAPTER 5 MULTIPLE CONTROLLER 67
5.1. PIECEWISE LINEAR APPROXIMATION 67
5.2. HINGING HYPERPLANES 68
5.2.1. The hinge finding algorithm (HFA) 69
5.2.2. Function approximation by hinging hyperplanes 70
5.2.3. Extension to systems with multiple outputs 71
5.3. PMLE 72
5.3.1. The architecture 72
5.3.2. The ability to approximate inverse functions 73
5.4. FROM HH TO PMLE 74
5.4.1. Parameterization via hinging hyperplanes 74
5.4.2. Learning the forward PMLE approximator 75
5.4.3. Extension to the MIMO case 77
5.4.4. A comment concerning the algorithm complexity 78
5.5. MULTIPLE INVERSE 79
5.5.1. Constructing the complete Inverse Approximation 79
5.5.2. Illustration in one dimension 80
5.6. THE CAPABILITIES OF THE MI-PMLE 81
5.7. SIMULATION EXAMPLES 83
5.7.1. Example 1: Piecewise linear function 83
5.7.2. Example 2: Smooth function approximation 85
5.7.3. Example 3: A two dimensional function approximation 88
5.8. AN EXAMPLE: PARALLEL REDUNDANT SYSTEM 90
5.9. CONCLUSIONS 92
CHAPTER 6 LEARNING FROM EXAMPLES 93
6.1. LEARNING MULTIPLE INVERSE FROM EXAMPLES 93
6.1.1. Learning theory 94
6.1.2. Learning the multiple inverse model 95
6.1.3. Basic bounds 96
6.2. LEARNING THE INVERSE CONTROLLER 100
6.2.1. The claim 101
6.2.2. A simple one dimensional system 106
6.2.3. A simple dynamical system control 108
6.2.4. Other methods 111
6.2.5. Conclusions: IBE versus BEI 113
6.3. HOW SHOULD A MULTIPLE CONTROLLER BE LEARNED? 113
6.3.1. Criteria for the multiple inverse controller performance 114
6.3.2. Justification of the MI-PMLE construction method 115
CHAPTER 7 DISCUSSION 119
7.1. LEARNING MOTOR CONTROL OF REDUNDANT SYSTEMS 119
7.1.1. Nonlinear time-varying many-to-one system 120
7.1.2. Feedback and forward control 122
7.1.3. Multiple controllers 123
7.1.4. Learning schemes 126
7.2. OPEN PROBLEMS AND FUTURE WORK 129
7.2.1. Reaching movements and the system dynamics 129
7.2.2. A comment on models and experimental results 131
7.2.3. Multiple controllers 132
7.2.4. Degeneracy redundancy and parallelism 134
7.2.5. BEI Vs IBE in the biological motor control. 134
7.2.6. The general model 135
7.2.7. Human motor control - The next challenge 136
7.3. FINAL REMARK 139
REFERENCES 141