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Contents
Introduction 1
I. PRESENTATION OF THE WORKS
1. The walk 4
1.1. Description of the walk 4 1.2. First model of the walker 5 1.3. Current research 6 1.4. Application fields 8
2. The prostheses 10
2.1. Definition 10 2.2. History of the prostheses 10 2.3. The Knee prostheses made by Otto- Bock 11 2.3.1. Description of the prostheses 13 2.3.2. Kinematic properties of the prostheses 14
3. The theories used 17
3.1. Lagrange’s equation 17 3.1.1. Definition of the Lagrange’s equation 18 3.1.2. Dissipation in Lagrange’s equation 20 3.1.3. Lagrange’s equation with constraints 21 3.2. Numerical resolution and time integration 23
3.2.1. Generalities about numerical schemes 23 3.2.2. Numerical scheme used 26 3.3. The unilateral constraints 30 3.3.1. Lagrange’s multipliers 30 3.3.2. The penalty method 32
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4. The program 34
4.1. Multi-body theory: characterization of the Multi-body elements 34 4.2. Matlab and Maple 41 4.3. Walker model 43 4.4. The structure of the program 45 4.5. Program of the unilateral constraints 50
4.5.1 Lagrange’s Multipliers 50 4.5.2 The penalty method 52
II. THE IMPROVEMENTS
5. Ballistic walk 55
5.1. Definition 55 5.2. The model used 56 5.3. Definition of the impact problem of the knee 59
6. Investigation about penalty parameters 61
6.1. The experiment 61 6.1.1. Presentation of the experimental setup 61 6.1.1.1. Aim of the experiments 61 6.1.1.2. Presentation of the physical model of the leg 62 6.1.1.3. Description of the experimental setup 64 6.1.2. The measurements 65
6.1.2.1. The film 65 6.1.2.2. Analysis of the results 66 6.1.2.3. Different problems 69 6.1.2.3.1. Quality of the image 69 6.1.2.3.2. Stability of the frame 69 6.1.2.3.3. Reliability of the initial position results 70
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6.2. The simulation 72 6.2.1. The Matlab model used: analysis of the Data file 72
6.2.1.1. The initial condition 73 6.2.1.2. The update of the penalty constraint 74 6.2.2. Analysis of the simulation 75 6.2.3. Comparison between the results obtained from Matlab model and
experiments 77
6.2.4. The influence of the friction coefficient of the hinge 79 6.2.5. Analysis of the dissipation energy 80 6.2.6. Evaluation of the penalty parameters together with the hinge’s
friction coefficient 82 6.3. Conclusion 86
Appendix 87
Appendix 1A: Ballistic walk model 87 Appendix 2A: Model used in the experiment 97
