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Fast C-obstacle Query Computationfor Motion Planning

Liang-Jun Zhang12/13/2005

Liang-Jun Zhang1 Young J. Kim2 Gokul Varadhan1 Dinesh Manocha1

1: University of North Carolina - Chapel Hill, USA2: Ewha Womans University, Korea,

http://gamma.cs.unc.edu/cobstacle

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Configuration space

Free space

C-obstacle

Do they intersect? Is its configuration in C-obstacle or free space?

• Free space and C-obstacle

Robot

Obstacle

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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

Free spaceC-obstacle

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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

EMPTY

init

goal

FULL

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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

C-obstacle

Free space

•Does the cell lie inside C-obstacle?

• Do the robot and obstacle intersect at all configurations?

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

Obstacle

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Algorithm overview• 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 > Bounding

Motion the robot can not escape

♦ the query primitive lies inside C-obstacle

q

A(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’

A

B

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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

1. Convex decomposition 2. Eliminate non-overlapping pairs

3. PDt over overlapping pairs

4. LB(PDg) = Max over all PDts

A

B

A2

A1

B1

B2

A2

A1

B2

B1

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C-obstacle Query Criterion

If Lower Bound (PDg(A(qa), B))> Bounding Motion,

the cell C is in C-obstacle.

A(qa): set A’s config as qa

aq

C

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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

bqaq,( , )a bq qA

( )aA q

( )bA q

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Upper bound of Motion for cell

,( , ) max ( , )a bq qA C A

Any diagonal line segment yields maximum bounding motion.

aq

cqbqqa is the center of the cell Cqb is any point on the boundary of the cell.

C

,( , ) max ( , )a bq qA C A

,( , )a cq qA

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C-obstacle Query Criterion

If Lower Bound (PDg(A(qa), B))>

the cell C is in C-obstacle.

A(qa): set A’s config as qa

aqcqbq

C

,( , )a cq qA

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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’: 2T+1R

video

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Results

‘Piano’‘World map’

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Effective and Performance

• Query timing: 0.04ms to 0.12 ms

Culled C-obstacle Cells• Cell Culling Ratio =

All C-obstacle Cells

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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

Robot

Obstacle

Robot

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• 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

Generalized Penetration Depth:

PDg

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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

,

, , , , , ,

( , )a bq q

a x b x a y b y a r b

A

q q q q R q q

x

ybq

aq

R: ‘radius’ of the object

,( , )a bq qA