CS 326 A: Motion PlanningExample: Car-Like RobotTangent Space/Velocity SpaceLie BracketPath TransformNonholonomic Path Planning ApproachesCS 326 A: Motion CS 326 A: Motion PlanningPlanninghttp://robotics.stanford.edu/~latombe/cs326/2002Nonholonomic RobotsNonholonomic RobotsExample: Car-Like RobotExample: Car-Like Robotyyxxdx/dt = v cos dy/dt = v sin ddt = (v/L) tan | < dy/dx = tan Configuration space is 3-dimensional: (x, y, )But control space is 2-dimensional: (v, )LA robot is nonholonomic if its motion isconstrained by a non-integrable equationof the form f(q,q’) = 0Lower-bounded turning radiusTangent Space/Velocity Tangent Space/Velocity SpaceSpacexy(x,y,)Lie BracketLie Bracketdx/dt = v cos dy/dt = v sin ddt = (v/L) tan | < dy/dx = tan X = (v cos , v sin , 0) Y = (v cos , v sin , (v/L) tan ) X (t)Y-X-Y[X,Y] t2A Lie bracket [X,Y] is a kindof “maneuver”A nonholonomic robot is controllable iff the vector space of its velocities andthe Lie brackets of these velocities has the same dimensionality as the C-spacePath TransformPath TransformHolonomic pathNonholonomic pathNonholonomic Path Nonholonomic Path Planning ApproachesPlanning ApproachesTwo-phase planning: (Laumond’s paper)Compute collision-free path ignoring nonholonomic constraintsTransform this path into a nonholonomic oneDirect planning: (Barraquand-Latombe’s paper)Build a tree of milestones until one is close enough to the goal (deterministic or
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