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CS 326 A: Motion Planning

CS 326 A: Motion Planning. http://robotics.stanford.edu/~latombe/cs326/2002 Dynamic Constraints and Optimal Planning. Nonholonomic vs. Dynamic Constraints. Nonholonomic constraint: q’ = f( q ,u) where u is the control input (function of time) Dynamic constraint:

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CS 326 A: Motion Planning

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  1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/2002 Dynamic Constraintsand 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 robot obstacles air thrusters gas tank air bearing

  4. f a y x Modeling of Robot q = (x,y) s = (q,q’) u = (f,a) x” = (f/m) cosa y” = (f/m) sina f  fmax

  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

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