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

C-space Analysis Dimension: m-3 for m-link closed chains Algebraic variety Number of components

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)

Cell decomposition and graph representation

Elbow-up torus Elbow-down torus

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

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

Sampling the boundary variety and its skeleton

Sampling the skeleton of collision variety

C-spaceobstacles

Elbow-down torusElbow-up torus

Summary

Global structure of C-space

Atlas with two coordinate charts

Boundary variety and its skeleton

Cell-decomposition algorithm

Topologically informed sampling-based algorithms