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 elbowdown torii
Common boundary
 Complexity
 Simulation results
C-space Analysis
 Dimension: m-3 for m-link closed chains
 Algebraic variety
 Number of components
 C-space topology
Theorem:
C-space of a single-loop
closed chain is the
boundary of a union of
manifolds of the form:
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
l
1
f2
f1
l2
l3
f4
f3
l4
l5
f5
C-space Embedding
 Embedding in space of dim. greater than m-3

(S1)m-1 : (f1,……,fm-1)

R2m-4 (coordinates of m-2 vertices)
 Our approach


Elbow-up and elbow-down tori, each parametrized by
(f1,……,fm-3) (dimension same as C-space)
Torii connected by “boundary” variety
Boundary Variety
Elbow-up torus
P1
P2
l3
l1
l4
l2
l5
P1
or
l1
Elbow-down torus
P2
l2
l3
l4
l5
glue along boundary variety
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
1 3
elbow-down (S )
 Recursive skeleton for decomposition
Boundary variety
B(1)
skeleton
identified
B(2)
B(3)={f1,1,f1,2,f1,3,f1,4
skeleton of skeleton
}
Geometric interpretation
l1
l1
l2
f1
l3
l
f2 3 f3
l2
l4
l4
B(1)
l5
l5
l1
l3
l2
Boundary variety
B(2)
l4
l5
skeleton
B(3)
l1
Skeleton of skeleton
Cell decomposition and graph representation
Elbow-up torus
Elbow-down torus
graph representation
[B1(1),1]
[B1(1),1]
[B1(1),2]
[1,2]
[1,2]
[B1(1),2]
[2,B2(1)]
[2,B2(1)]
[2,B2(1)]
[2,B2(1)]
Common facets on B(1)
Elbow-up torus
Elbow-down torus
Algorithm
 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 elbowdown torii which have a common facet on the
boundary variety
Video
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
obstacles
C-space
 Sampling the boundary variety and its skeleton
Elbow-up torus
Elbow-down torus
 Sampling the skeleton of collision variety
Summary
 Global structure of C-space
 Atlas with two coordinate charts
 Boundary variety and its skeleton
 Cell-decomposition algorithm
 Topologically informed sampling-based algorithms