1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/200 2 Dynamic Constraints and Optimal Planning

2. Nonholonomic vs. Dynamic Constraints  Nonholonomic constraint: q’ = f(q,u) where u is the control input (function of time)  Dynamic constraint:  s = (q,q’), the state of the system  s’ = f(s,u) where u is the control input

3. Aerospace Robotics Lab Robot air bearing gas tank air thrusters obstacles robot

4. Modeling of Robot x y  f q = (x,y) s = (q,q’) u = (f,  ) x” = (f/m) cos  y” = (f/m) sin  f  f max

5. Example with Moving Obstacles

6. General Case For an arbitrary mechanical linkage: u = M(q)q” + C(q,q’) + G(q) + F(q,q’) where: - M is the inertia matrix - C is the vector of centrifugal and Coriolis terms - G is the vector of gravity terms - F is the vector of friction terms

7. Optimality of a Trajectory Often one seeks a trajectory that optimizes a given criterion, e.g.: –smallest number of backup maneuvers, –minimal execution time, –minimal energy consumption

8. Path Planning Approaches  Direct planning: –Build a tree of milestones until a connection to the goal has been made LaValle and Kuffner, and Donald et al.’s papers  Two-phase planning: –Compute a collision-free path ignoring constraints –Optimize this path into a trajectory satisfying the kinodynamic constraints Bobrow’s paper

9. Path Optimization  Steepest descent technique.  Parameterize the geometry of a trajectory, e.g., by defining control points through which cubic spines are fitted.  Vary the parameters. For the new values re-compute the optimal control. If better value of criterion, vary further.

10. Two-Phase Planning  Gives good results in practice  But computationally expensive (real- time planning possible?)  No performance guarantee regarding optimality of computed trajectory

11. Direct Planning  Optimality guarantee in Donald et al.’s paper, but running time exponential in number of degrees of freedom  No optimality guarantee in LaValle and Kuffner’s paper, but provably quick convergence of algorithm, allowing for real-time planning (and re-planning) among moving obstacles