CS 326 A: Motion PlanningNonholonomic vs. Dynamic ConstraintsGeneral Dynamic EquationAerospace Robotics Lab RobotModeling of RobotControl-Based SamplingOptimality of a TrajectoryVariational Path/Trajectory OptimizationCS 326 A: Motion CS 326 A: Motion PlanningPlanninghttp://robotics.stanford.edu/~latombe/cs326/2003Kynodynamic Planning and Kynodynamic Planning and Under-Actuated SystemsUnder-Actuated SystemsNonholonomic vs. Dynamic Nonholonomic vs. Dynamic ConstraintsConstraintsNonholonomic constraint:q’ = f(q,u)where u is the control input (function of time), with dim(u) < dim(q)Dynamic constraint:s = (q,q’), the state of the systems’ = f(s,u) where u is the control input(first paper)General Dynamic EquationGeneral Dynamic EquationFor 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+ constraints on uAerospace Robotics Lab Aerospace Robotics Lab RobotRobotair bearinggas tankair thrustersobstaclesrobotModeling of RobotModeling of Robot xxyfq = (x,y)s = (q,q’)u = (f,)x” = (f/m) cosy” = (f/m) sinf fmaxs’ = f(s,u)If dim(u) < dim(s), then second-order nonholonomic constraint(second paper)Control-Based SamplingControl-Based SamplingPrevious sampling technique: Pick each milestone in some regionControl-based sampling:1. Pick control vector (at random or not)2. Integrate equation of motion over short duration (picked at random or not)3. The endpoint is the new milestoneTree-structured roadmapsNeed for endgame regionsOptimality of a TrajectoryOptimality of a TrajectoryOften one seeks a trajectory that optimizes a given criterion, e.g.:–smallest number of backup maneuvers, –minimal execution time, –minimal energy consumptionBobrow’s paper+ variational techniquesVariational Path/Trajectory Variational Path/Trajectory OptimizationOptimizationSteepest 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. No performance guarantee regarding optimality of computed
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