Design of Adaptive Controllers based on Christoffel Symbols of FirstKind

Design of Adaptive Controllers based on
Christoffel Symbols of First Kind

Hyper Redundant Manipulators

Robot assisted 3D shape acquisition by optical systems

Robot assisted 3D shape  acquisition by optical systems

Design of Adaptive Controllers based on Christoffel Symbols of FirstKind

Design of Adaptive Controllers based on
Christoffel Symbols of First Kind

C4B — Mobile Robotics

C4B — Mobile Robotics

Contents

1 Introduction and Motivation ....................................................................................................... 6

2 Introduction to Path Planning and Obstacle Avoidance .............................................................. 8

2.1 Holonomicity........................................................................................................................ 9

2.2 Configuration Space.......................................................................................................... 11

2.3 The Minkowski-Sum........................................................................................................... 13

2.4 Voronoi Methods............................................................................................................... 14

2.5 Bug Methods...................................................................................................................... 15

2.6 Potential Methods............................................................................................................. 15

• Estimation - A Quick Revision 19
  
  
  
  • Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
  
  
  • What is Estimation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
    
    
    
    • Defining the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
  
  
  • Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . 21
  
  
  • Maximum A-Posteriori - Estimation . . . . . . . . . . . . . . . . . . . . . . . 22

2

3

• Minimum Mean Squared Error Estimation . . . . . . . . . . . . . . . . . . . 24


• Recursive Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 25

• Least Squares Estimation 28
  
  
  
  • Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
    
    
    
    • A Geometric Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
    
    
    • LSQ Via Minimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 29
  
  
  • Weighted Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   30
    
    
    
    • Non-linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2           Long Baseline Navigation - an Example . . . . . . . . . . . . . . . . .       31
  
  
  • Kalman Filtering -Theory, Motivation and Application 35
    
    
    
    • The Linear Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . .     35
      
      
      
      • Incorporating Plant Models - Prediction . . . . . . . . . . . . . . . .     39
      
      
      • Joining Prediction to Updates . . . . . . . . . . . . . . . . . . . . . . 42
      
      
      • Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
    
    
    • Using Estimation Theory in Mobile Robotics . . . . . . . . . . . . . . . . . . 46
      
      
      
      • A Linear Navigation Problem - “Mars Lander” . . . . . . . . . . . . . 46
      
      
      • Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
    
    
    • Incorporating Non-Linear Models - The Extended Kalman Filter . . . . . . . 52
      
      
      
      • Non-linear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
      
      
      • Non-linear Observation Model . . . . . . . . . . . . . . . . . . . . . . 54

3

4

• The Extended Kalman Filter Equations . . . . . . . . . . . . . . . .     56

• Vehicle Models and Odometry 59
  
  
  
  • Velocity Steer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
  
  
  • Evolution of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
  
  
  • Using Dead-Reckoned Odometry Measurements . . . . . . . . . . . . . . . . 63 6.3.1           Composition of Transformations . . . . . . . . . . . . . . . . . . . . .         65


• Feature Based Mapping and Localisation 69
  
  
  
  • Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
  
  
  • Features and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
  
  
  • Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
  
  
  • A Probabilistic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
    
    
    
    • Probabilistic Localisation . . . . . . . . . . . . . . . . . . . . . . . . . 71
    
    
    • Probabilistic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . .         72
  
  
  • Feature Based Estimation for Mapping and Localising . . . . . . . . . . . . . 73
    
    
    
    • Feature Based Localisation . . . . . . . . . . . . . . . . . . . . . . . . 73
    
    
    • Feature Based Mapping . . . . . . . . . . . . . . . . . . . . . . . . .           74
  
  
  • Simultaneous Localisation and Mapping - SLAM . . . . . . . . . . . . . . . . 78 7.6.1           The role of Correlations  . . . . . . . . . . . . . . . . . . . . . . . . .           81


• Multi-modal and other Methods 83
  
  
  
  • Montecarlo Methods - Particle Filters . . . . . . . . . . . . . . . . . . . . . . 83

5

• Grid Based Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

• In Conclusion 89


• Miscellaneous Matters 90
  
  
  
  • Drawing Covariance Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10.2 Drawing High Dimensional Gaussians . . . . . . . . . . . . . . . . . . . . . .          93


• Example Code 94
  
  
  
  • Matlab Code For Mars Lander Example . . . . . . . . . . . . . . . . . . . .         94
  
  
  • Matlab Code For Ackerman Model Example . . . . . . . . . . . . . . . . . . 98
  
  
  • Matlab Code For EKF Localisation Example . . . . . . . . . . . . . . . . . . 100
  
  
  • Matlab Code For EKF Mapping Example . . . . . . . . . . . . . . . . . . .           103
  
  
  • Matlab Code For EKF SLAM Example . . . . . . . . . . . . . . . . . . . . . 107
  
  
  • Matlab Code For Particle Filter Example . . . . . . . . . . . . . . . . . . . . 111

The Rh-1 full-size humanoid robot: Control system design and Walkingpattern generation

The Rh-1 full-size humanoid robot: