Complete Path Planning for Planar Closed Chains Among Point Obstacles Guanfeng Liu and Jeff Trinkle...
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Transcript of Complete Path Planning for Planar Closed Chains Among Point Obstacles Guanfeng Liu and Jeff Trinkle...
Complete Path Planning for Planar Closed Chains Among Point Obstacles
Guanfeng Liu and Jeff Trinkle
Rensselaer Polytechnic Institute
Outline: Motivation and overview C-space Analysis
Number of components C-space topology Local parametrization and global atlas
Boundary variety Global cell decomposition Path Planning algorithm Simulation results
Motivation:
Many applications employ closed-chain manipulators
No complete algorithms for closed chains with obstacles
Limitation of PRM method for closed chains
Difficulty to apply Canny’s roadmap method to C-spaces with multiple coordinate charts
Overview: Exact cell decomposition---direct cylindrical
cell decomposition Atlas of two coordinate charts: elbow-up and
elbow-down torii Common boundary
Complexity
Simulation results
Theorem: C-space of a single-loop closed chain is the boundary of a union of manifolds of the form:
C-space topology
p
five-bar closed chain
Types of C-spaces which are connected
Types of C-spaces which are disconnected
disjoint union of two tori
Local and global parametrization
Any m-3 joints can be used as a local chart
More than two charts for differentiable covering Example: 2n charts required to cover (S1)n
Two charts (elbow-up and elbow-down) for capturing
connectivity
1
2
3
4
5
l1
l2
l3
l4 l5
C-space Embedding
(S1)m-1 : (1,……,m-1)
R2m-4 (coordinates of m-2 vertices)
Elbow-up and elbow-down tori, each parametrized by
(1,……,m-3) (dimension same as C-space)
Torii connected by “boundary” variety
Embedding in space of dim. greater than m-3
Our approach
Boundary Variety
glue along boundary variety
P1
P2
l1l2
l3 l4
l5
or
l1 l2 l3l4
l5
Elbow-up torus Elbow-down torusP1P2
Main steps
Boundary variety and its recursive skeletons Collision varieties
Cell decomposition for elbow-up and elbow-down torii
Identify valid cells based on boundary variety
Adjacency between cells in elbow-up and elbow-down torii
Global graph representation
Example: A Six-bar Closed Chain
Boundary variety B(1) connects elbow-up (S1)3 and elbow-down (S1)3
Recursive skeleton for decomposition
Boundary variety
skeleton
skeleton of skeleton
B(1)
B(2)
B(3)={1,1,1,2,1,3,
1,4}
identified
Geometric interpretation
1
2 3l1
l2
l3 l4
l5
l2l1
l3 l4 l5 l1l3 l4 l5
l2
l1
Boundary variety
skeleton
Skeleton of skeleton
B(1)
B(2)
B(3)
graph representation
Elbow-up torus Elbow-down torus
[B1(1),1]
[1,2]
[2,B2(1)]
[B1(1),2]
[2,B2(1)]
[B1(1),1]
[1,2]
[2,B2(1)]
[B1(1),2]
[2,B2(1)]
Common facets on B(1)
Embed C-space into two (m-3)-torii Compute boundary variety and its skeleton at
each dimension Compute collision variety and its skeleton at
each dimension Decompose elbow-up and elbow-down torii into
cells Identify valid cells and construct adjacency
graphs for each torus Connect respective cells of elbow-up and elbow-
down torii which have a common facet on the boundary variety
Algorithm
Complexity analysis
Theorem:
Basic idea for proof:a. C-space with O(nm-3) components in worst case
b. Each component decomposed into O(nm-4) cells
obstacle
14n2-11n components
Topologically informed sampling-based algorithms
Sampling C-space directly
Sampling the boundary variety and its skeleton
Sampling the skeleton of collision variety
C-spaceobstacles
Elbow-down torusElbow-up torus