Complete Path Planning for Planar Closed Chains Among Point Obstacles

1
Complete Path Planning for Planar Closed
Chains Among Point Obstacles
Guanfeng Liu and Jeff Trinkle
Rensselaer Polytechnic Institute
2

• 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

3

• 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 Cannys roadmap method to
  C-spaces with multiple coordinate charts

4

• Overview
• Exact cell decomposition---direct cylindrical
  cell decomposition
• Atlas of two coordinate charts elbow-up and
  elbow-down torii
• Common boundary
• Complexity
• Simulation results

5
C-space Analysis

• Dimension m-3 for m-link closed chains
• Algebraic variety
• Number of components

6
(No Transcript)
7
five-bar closed chain

• Types of C-spaces which are connected

• Types of C-spaces which are disconnected

disjoint union of two tori
8
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

l3
f4
f2
l1
f3
l4
l2
f1
l5
f5
9
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)
• Elbow-up and elbow-down tori, each parametrized
  by
• (f1,,fm-3) (dimension same as C-space)
• Torii connected by boundary variety

• Our approach

10
Boundary Variety
Elbow-up torus
Elbow-down torus
11
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

12
Example A Six-bar Closed Chain

• Boundary variety B(1) connects elbow-up (S1)3
  and elbow-down (S1)3
• Recursive skeleton for decomposition

B(1)
Boundary variety
skeleton
B(2)
identified
B(3)f1,1,f1,2,f1,3,f1,4
skeleton of skeleton
13
Geometric interpretation
B(1)
Boundary variety
B(2)
skeleton
B(3)
Skeleton of skeleton
14
Cell decomposition and graph representation
Elbow-down torus
Elbow-up torus
15
graph representation
Common facets on B(1)
Elbow-up torus
Elbow-down torus
16
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
  elbow-down torii which have a common facet on the
  boundary variety

17
Video
18
Complexity analysis
Theorem
Basic idea for proof

• C-space with O(nm-3)
• components in worst case

• Each component
• decomposed into
• O(nm-4) cells

14n2-11n components
19
Topologically informed sampling-based algorithms

• Sampling C-space directly
• Sampling the boundary variety and its skeleton
• Sampling the skeleton of collision variety

20
Summary

• Global structure of C-space
• Atlas with two coordinate charts
• Boundary variety and its skeleton
• Cell-decomposition algorithm
• Topologically informed sampling-based algorithms