CS 326 A: Motion Planning

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CS 326 A Motion Planning

• http//robotics.stanford.edu/latombe/cs326/2002
• Dynamic Constraintsand Optimal Planning

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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

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Aerospace Robotics Lab Robot
robot
obstacles
air thrusters
gas tank
air bearing
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Modeling of Robot
q (x,y) s (q,q) u (f,a) x (f/m) cosa y
(f/m) sina f ? fmax
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Example with Moving Obstacles
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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

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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

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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
• Bobrows paper

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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.

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Two-Phase Planning

• Gives good results in practice
• But computationally expensive (real-time planning
  possible?)
• No performance guarantee regarding optimality of
  computed trajectory

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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 Kuffners
  paper, but provably quick convergence of
  algorithm, allowing for real-time planning (and
  re-planning) among moving obstacles