Nonholonomic Motion Planning: Steering Using Sinusoids

1. Nonholonomic Motion Planning: Steering Using Sinusoids R. M. Murray and S. S. Sastry

2. Motion Planning without Constraints • Obstacle positions are known and dynamic constrains on robot are not considered. From Planning, geometry, and complexity of robot motion By Jacob T. Schwartz, John E. Hopcroft

3. Problem with Planning without Constraints Paths may not be physically realizable

4. Mathematical Background • Nonlinear Control System • Distribution

5. Lie Bracket • The Lie bracket has the properties • The Lie bracket is defined to be 1.) (Jacobi identity) 2.)

6. Physical Interpretation of the Lie Bracket

7. Controllability • A system is controllable if for any • Chow’s Theorem

8. Classification of a Lie Algebra • Construction of a Filtration

9. Classification of a Lie Algebra • Regular

10. Classification of a Lie Algebra • Degree of Nonholonomy

11. Classification of a Lie Algebra • Maximally Nonholonomic • Growth Vector • Relative Growth Vector

12. Nonholonomic Systems • Example 1

13. Nonholonomic Systems • Example 2

14. Phillip Hall Basis The Phillip Hall basis is a clever way of imposing the skew-symmetry of Jacobi identity

15. Phillip Hall Basis • Example 1

16. Phillip Hall Basis • A Lie algebra being nilpotent is mentioned • A nilpotent Lie algebra means that all Lie brackets higher than a certain order are zero • A lie algebra being nilpotent provides a convenient way in which to determine when to terminate construction of the Lie algebra • Nilpotentcy is not a necessary condition

17. Steering Controllable Systems Using Sinusoids: First-Order Systems • Contract structures are first-order systems with growth vector • Contact structures have a constraint which can be written • Written in control system form

18. Steering Controllable Systems Using Sinusoids: First-Order Systems More general version

19. Derive the Optimal Control: First-Order Systems • To find the optimal control, define the Lagrangian • Solve the Euler-Lagrange equations

20. Derive the Optimal Control: First-Order Systems Example Lagrangian: Euler-Lagrange equations:

21. Derive the Optimal Control: First-Order Systems • Optimal control has the form where is skew symmetric • Which suggests that that the inputs are sinusoid at various frequencies

22. Steering Controllable Systems Using Sinusoids: First-Order Systems Algorithm yields

23. Hopping Robot (First Order) • Kinematic Equations • Taylor series expansion at l=0 • Change of coordinates

24. Hopping Robot (First Order) • Applying algorithm 1 a. Steer l and ψ to desired values by b. Integrating over one period

25. Hopping Robot (First Order) • Nonholonomic motion for a hopping robot

26. Steering Controllable Systems Using Sinusoids: Second-Order Systems Canonical form:

27. Front Wheel Drive Car (Second Order) • Kinematic Equations • Change of coordinates

28. Front Wheel Drive Car (Second Order) • Sample trajectories for the car applying algorithm 2

29. Maximal Growth System • Want vectorfields for which the P. Hall basis is linearly independent

30. Maximal Growth Systems

31. Chained Systems

32. Possible Extensions Canonical form associated with maximal growth 2 input systems look similar to a reconstruction equation

33. Possible Extensions • Pull a Hatton…plot vector fields and use the body velocity integral as a height function • The body velocity integral provides a decent approximation of the system’s macroscopic motion

34. Plot Vector Fields