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

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

Problem with Planning without Constraints Paths may not be physically realizable

Mathematical Background Nonlinear Control System Distribution

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

Physical Interpretation of the Lie Bracket

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

Classification of a Lie Algebra Construction of a Filtration

Classification of a Lie Algebra Regular

Classification of a Lie Algebra Degree of Nonholonomy

Classification of a Lie Algebra Maximally Nonholonomic Growth Vector Relative Growth Vector

Nonholonomic Systems Example 1

Nonholonomic Systems Example 2

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

Phillip Hall Basis Example 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Maximal Growth Systems

Chained Systems

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

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

Plot Vector Fields