Motion Planning

1
Motion Planning

• Path Planning for a Point Robot
• (This slides are based on Latombe and Hwang
  slides)

2
Configuration SpaceTool to Map a Robot to a
Point
3
Problem
free space
free path
g
4
Problem
semi-free path
5
Types of Path Constraints

• Local constraints lie in free space
• Differential constraints have bounded
  curvature
• Global constraints have minimal length

6
Motion-Planning Framework
Continuous representation
Discretization
Graph searching (blind, best-first, A)
7
Path-Planning Approaches

• RoadmapRepresent the connectivity of the free
  space by a network of 1-D curves
• Cell decompositionDecompose the free space into
  simple cells and represent the connectivity of
  the free space by the adjacency graph of these
  cells
• Potential fieldDefine a function over the free
  space that has a global minimum at the goal
  configuration and follow its steepest descent

8
Roadmap Methods

• Visibility graphIntroduced in the Shakey project
  at SRI in the late 60s. Can produce shortest
  paths in 2-D configuration spaces

9
Simple Algorithm

• Install all obstacles vertices in VG, plus the
  start and goal positions
• For every pair of nodes u, v in VG
• If segment(u,v) is an obstacle edge then
• insert (u,v) into VG
• else
• for every obstacle edge e
• if segment(u,v) intersects e
• then goto 2
• insert (u,v) into VG
• Search VG using A

10
Complexity

• Simple algorithm O(n3) time
• Rotational sweep O(n2 log n)
• Optimal algorithm O(n2)
• Space O(n2)

11
Rotational Sweep
Check the following page http//www.cs.hut.fi/Res
earch/TRAKLA2/exercises/VisibleVertices.html
12
Rotational Sweep
13
Rotational Sweep
14
Rotational Sweep
15
Rotational Sweep
16
Reduced Visibility Graph
tangent segments
? Eliminate concave obstacle vertices
17
Generalized (Reduced) Visibility Graph
tangency point
18
Three-Dimensional Space
Computing the shortest collision-free path in a
polyhedral space is NP-hard
19
Roadmap Methods

• Voronoi diagram Introduced by Computational
  Geometry researchers. Generate paths that
  maximizes clearance. O(n log n) timeO(n) space

20
Path-Planning Approaches

• RoadmapRepresent the connectivity of the free
  space by a network of 1-D curves
• Cell decompositionDecompose the free space into
  simple cells and represent the connectivity of
  the free space by the adjacency graph of these
  cells
• Potential fieldDefine a function over the free
  space that has a global minimum at the goal
  configuration and follow its steepest descent

21
Cell-Decomposition Methods

• Two classes of methods
• Exact cell decompositionThe free space F is
  represented by a collection of non-overlapping
  cells whose union is exactly FExamples
• Polygonal decomposition
• Trapezoidal decomposition

22
Polygonal Configuration Space
23
Polygonal Configuration Space

• cells form channels instead of line paths
• channels give the robot more information to avoid
  obstacles encountered at run time

24
Polygonal Configuration Space
25
(No Transcript)
26
Trapezoidal decomposition
27
Trapezoidal decomposition
28
Trapezoidal decomposition
29
Trapezoidal decomposition
30
Planar sweep ? O(n log n) time, O(n) space
31
Generalization to 3 Dimensions
32
Cell-Decomposition Methods

• Two classes of methods
• Exact cell decomposition
• Approximate cell decompositionF is represented
  by a collection of non-overlapping cells whose
  union is contained in FExamples quadtree,
  octree, 2n-tree

33
Octree Decomposition
34
Sketch of Algorithm

• Compute cell decomposition down to some
  resolution
• Identify start and goal cells
• Search for sequence of empty/mixed cells between
  start and goal cells
• If no sequence, then exit with no path
• If sequence of empty cells, then exit with
  solution
• If resolution threshold achieved, then exit with
  failure
• Decompose further the mixed cells
• Return to 2

35
Path-Planning Approaches

• RoadmapRepresent the connectivity of the free
  space by a network of 1-D curves
• Cell decompositionDecompose the free space into
  simple cells and represent the connectivity of
  the free space by the adjacency graph of these
  cells
• Potential fieldDefine a function over the free
  space that has a global minimum at the goal
  configuration and follow its steepest descent

36
Potential Field Methods

• Approach initially proposed for real-time
  collision avoidance Khatib, 86. Hundreds of
  papers published on it.

Goal
Robot
37
Attractive and Repulsive fields
38
Local-Minimum Issue

• Perform best-first search (possibility of
  combining with approximate cell decomposition)
• Alternate descents and random walks
• Use local-minimum-free potential (navigation
  function)

39
Completeness of Planner

• A motion planner is complete if it finds a
  collision-free path whenever one exists and
  return failure otherwise.
• Visibility graph, Voronoi diagram, exact cell
  decomposition, navigation function provide
  complete planners
• Weaker notions of completeness, e.g.-
  resolution completeness (PF with best-first
  search)- probabilistic completeness (PF with
  random walks)

40

• A probabilistically complete planner returns a
  path with high probability if a path exists. It
  may not terminate if no path exists.
• A resolution complete planner discretizes the
  space and returns a path whenever one exists in
  this representation.

41
Preprocessing / Query Processing

• Preprocessing Compute visibility graph, Voronoi
  diagram, cell decomposition, navigation function
• Query processing- Connect start/goal
  configurations to visibility graph, Voronoi
  diagram- Identify start/goal cell- Search graph

42
Summary Exact Cell Decomposition
43
Summary Approximate Cell Decomposition
44
Summary Potential Fields