Fast C-obstacle Query Computation for Motion Planning

1
Fast C-obstacle Query Computationfor Motion
Planning
Liang-Jun Zhang1 Young J. Kim2
Gokul Varadhan1 Dinesh Manocha1 1
University of North Carolina - Chapel Hill,
USA 2 Ewha Womans University, Korea, http//gamma
.cs.unc.edu/cobstacle

• Liang-Jun Zhang
• 12/13/2005

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Configuration space

• Free space and C-obstacle

Is its configuration in C-obstacle or free space?
Do they intersect?
Obstacle
C-obstacle
Robot
Free space
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C-obstacle query in C-space

• C-obstacle query
• Does a primitive lie completely inside
  C-obstacle?
• The primitive usually is a cell.
• Goal
• Design an efficient C-obstacle
• query algorithm

C-obstacle
Free space
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Motivation- An important query for Motion
Planning

• Cell Decomposition Method
• Label Cells as FULL and EMPTY
• Complete motion planning
• Able to find a path or report path non-existence

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Previous work

• Computing the boundary of C-obstacle
• Exponential complexity Sacks 99, Sharir 97
• Degeneracy and floating point error
• Contact surface constraints
• Latombe 91, Zhu 91
• Complexity of contact surface enumeration
• To deal with non-linear contact surfaces

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Interpretation of C-obstacle query

• Do the robot and obstacle intersect at all
  configurations?

• Does the cell lie inside C-obstacle?

C-obstacle
Obstacle
Free space

• Can the robot escape from the obstacle at some
  moment?

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Algorithm overview
A(q)

• Penetration Depth
• How much does the robot penetrate into the
  obstacle at a configuration q ?
• Bounding Motion
• How much motion the robot can undergo, when its
  configuration changes from q but within the query
  primitive?
• Query criterion
• If Penetration Depth gt Bounding Motion the robot
  can not escape
• the query primitive lies inside C-obstacle

q
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Translational Penetration Depth PDt

• Minimum translation to separate A, B
• Dobkin 93, Agarwal 00, Bergen 01, Kim 02
• PDt not applicable
• The robot is allowed to both translate and
  rotate.
• Undergoing rotation, A may escape from B easier.

B
A
A
B
A
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Generalized Penetration Depth PDg

• Consider both translation and rotation
• Zhang, Kim, Varadhan, Manocha UNC-CS TR05
• Difficult for non-convex objects
• Convex A, B PDg(A,B)PDt(A,B)

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Algorithm-Lower bound on PDg

• Convex decomposition
• Eliminate non-overlapping pairs
• PDt over overlapping pairs
• LB(PDg) Max over all PDts

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C-obstacle Query Criterion
A(qa) set As config as qa
C
If Lower Bound (PDg(A(qa), B))gt Bounding
Motion, the cell C is in C-obstacle.
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Upper bound of Motion for line segment

• Configurations qa and qb
• Schwarzer,Saha,Latombe 04
• Max trajectory length over points on the
  moving robot

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Upper bound of Motion for cell
qa is the center of the cell C qb is any point on
the boundary of the cell.
Any diagonal line segment yields maximum bounding
motion.
C
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C-obstacle Query Criterion
A(qa) set As config as qa
C
If Lower Bound (PDg(A(qa), B))gt the cell C is in
C-obstacle.
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Application

• Star-shaped roadmap a complete motion planning
  approach
• Varadhan and Manocha 05
• To identify cells which lie inside C-obstacle
• No subdivisions are applied for them

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Results-gear 2T1R
video
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Results
Piano
World map
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Effective and Performance
Culled C-obstacle Cells

• Cell Culling Ratio

All C-obstacle Cells

• Query timing 0.04ms to 0.12 ms

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Speedup For Star-shaped roadmap method
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Conclusion

• A fast C-obstacle query algorithm for rigid
  robots
• Based on generalized penetration depth and
  bounding motion computation.
• Need not explicit computation of the boundary of
  free space.
• Robust and efficient
• Applied for accelerating a complete motion
  planning approach for 2D rigid robot.

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Ongoing and Future work

• A Simple Path Non-Existence Algorithm for low DOF
  robots
• L. Zhang, Y.J. Kim, D. Manocha WAFR2006
• Apply for 3D rigid robots
• Handle articulated robots

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Acknowledgements

• Army Research Office
• DARPA/REDCOM
• NSF
• ONR
• Intel Corporation
• KRF, STAR program of MOST, Ewha SMBA consortium,
  the ITRC program (Korea)

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• Thanks
• Any Questions?

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Appendix
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Generalized Penetration Depth PDg

• Consider both translation and rotation
• Zhang, Kim, Varadhan, Manocha et al. 05
• Trajectory length Separating path

Obstacle
Robot
Robot
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Generalized Penetration Depth PDg

• PDg
• Difficult for non-convex objects
• Convex A, B PDg(A,B)PDt(A,B)

MIN over all possible separating paths
MAX of the trajectory length
over all on the moving robot
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Upper bound of Motion for line segment

• Schwarzer,Saha,Latombe 04
• Max trajectory length over points on the
  moving robot
• The weighted sum of difference for x, y,
  components between qa and qb

R radius of the object