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
: x g1 ( x)u1 g m ( x)um Distribution span g1 ( x), , g m ( x) Lie Bracket
The Lie bracket is defined to be g f [ f , g] f g
x x The Lie bracket has the properties 1.) [ f , g ] [ g , f ] 2.) [ f , [ g , h]] [ g , [ h, f ]] [h, [ f , g ]] 0 (Jacobi identity)
Physical Interpretation of the Lie Bracket Controllability A system is controllable if for any x0 , x1 U T 0 and u : [0, T ] R m
s.t. satisfies x(0) x0 and x(T ) x1 Chows Theorem If x R n for all x U then the system is controllab le on U
( is the closure of under Lie bracketing ) Classification of a Lie Algebra Construction of a Filtration If G1 span g1 ( x), , g m ( x) Gi Gi 1 [G1 , Gi 1 ]
Where [G1 , Gi 1 ] span [ g , h] : g G1 , h Gi 1 Classification of a Lie Algebra Regular
Classification of a Lie Algebra Degree of Nonholonomy Classification of a Lie Algebra Maximally Nonholonomic
Growth Vector r Z p , ri rankGi Relative Growth Vector Z p , i ri ri 1 , r0 0
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:
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 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 ml / 1 ml Hopping Robot (First Order) Applying algorithm 1 a. Steer l and to desired values by
b. Integrating over one period Hopping Robot (First Order) Nonholonomic motion for a hopping robot
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 Hattonplot vector fields and use the body velocity integral as a height function The body velocity integral provides a decent
approximation of the systems macroscopic motion Plot Vector Fields