Optimal Robot Path Planning Using the MinimumTime Criterion

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Optimal Robot Path Planning Using the
Minimum-Time Criterion

• JAMES E. BOBROW, 1988

Presented by Frederic Mazzella
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OUTLINE

• INTRODUCTION
• OPTIMIZATION PROCEDURE
• NUMERICAL EXAMPLES
• DISCUSSION
• CONCLUSIONS

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INTRODUCTION
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• Usual problems in robot motion generation
• - Collision-free path planning
• - Time-optimal control
• - Vision-based path planning
• - Feedback control
• Our concern A collision-free path which gives
  the minimum-time motion.

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• We know
• An initial collision-free path
• The manipulator equations of motion
• The workspace
• We want
• A parameterized time-optimal path
• (i.e. an optimal trajectory)
• How?
• By optimization of the initial feasible path

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• Why?
• Because distance is a bad (but often used)
  criterion to select a path.

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OPTIMIZATION PROCEDURE
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Method

• Geometric path represented as a set of
  parameterized interpolation functions
• Parameters are varied to minimize the path
  traversal time

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3D-path representation

• Uniform cubic B-spline polynomials
• For each spatial dimension x, we have
• And then we optimize the parameters Vi
• B-spline shape

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Advantages of this representation

• Computational efficiency
• Only 4 non-zero terms in the above formulation
• Local control of the path shape
• Any position influenced by its 4 closest
  neighbours only

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Optimization

• The path of the end-effector is defined entirely
  by the parameter s
• Optimization find the B-spline vertices Vi that
  define a path that minimizes the time

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Conditions to satisfy

• Satisfy the robot equations of motion
• Admissible torques and forces (Inertia compute
  the maximum velocity and keep the robot on path)
• Satisfy the bounds of each joint
• Collision avoidance

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NUMERICAL EXAMPLES
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Two-link planar motion in the vertical plane
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First optimization

• Path broken in two uniform cubic B-spline
  intervals.
• X and Y of the center vertex are the parameters
  to be varied during minimization (30 iterations)

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Optimal velocity profiles
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Optimal paths with 1, 3, 5, and 8 intermediate B
-spline vertices

• Traversal times
• 1 Tf 0.428 sec
• 3 Tf 0.423 sec
• 5 Tf 0.420 sec
• 8 Tf 0.418 sec
• Computation time approximately linear in the
  number of vertices used

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3D path with obstacle

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DISCUSSION
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Regularization

• Regularization was used to eliminate some local
  minima problems by optimizing J(v)
• Where C(v) is the total path curvature
• Penalizes highly curved
• or irregularly shaped functions

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Optimal path?

• Has the path converged to the true optimal?
• Who knows??? The experiments showed it seemed to
  converge, but this does not mean it was
  converging towards the optimal path

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Number of vertices to use?

• The traversal time is first greatly reduced with
  only few vertices
• The addition of more vertices only slightly
  decreases the time

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CONCLUSIONS
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• Off-line path planning method which iteratively
  computes time-optimal paths
• Uses full nonlinear equations of motion
• B-spline representation of the geometric path
• Path geometry optimized with a general-purpose
  nonlinear constrained optimization program
• Obstacle avoidance included in the algorithm