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## Robot Lab: Robot Path Planning

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Title: Robot Lab: Robot Path Planning

1
Robot Lab Robot Path Planning
• William Regli Department of Computer Science (and
Departments of ECE and MEM) Drexel University

2
Introduction to Motion Planning
• Applications
• Overview of the Problem
• Basics Planning for Point Robot
• Visibility Graphs
• Cell Decomposition
• Potential Field

3
Goals
• Compute motion strategies, e.g.,
• Geometric paths
• Time-parameterized trajectories
• Sequence of sensor-based motion commands
• Achieve high-level goals, e.g.,
• Go to the door and do not collide with obstacles
• Assemble/disassemble the engine
• Build a map of the hallway
• Find and track the target (an intruder, a missing
pet, etc.)

4
Fundamental Question
Are two given points connected by a path?
5
Basic Problem
• Problem statement
• Compute a collision-free path for a rigid or
articulated moving object among static obstacles.
• Input
• Geometry of a moving object (a robot, a digital
actor, or a molecule) and obstacles
• How does the robot move?
• Kinematics of the robot (degrees of freedom)
• Initial and goal robot configurations (positions
orientations)
• Output
• Continuous sequence of collision-free robot
configurations connecting the initial and goal
configurations

6
Example Rigid Objects
7
Example Articulated Robot
8
Is it easy?
9
Hardness Results
• Several variants of the path planning problem
have been proven to be PSPACE-hard.
• A complete algorithm may take exponential time.
• A complete algorithm finds a path if one exists
and reports no path exists otherwise.
• Examples
• Planar linkages Hopcroft et al., 1984
• Multiple rectangles Hopcroft et al., 1984

10
Tool Configuration Space
• Difficulty
• Number of degrees of freedom (dimension of
configuration space)
• Geometric complexity

11
Extensions of the Basic Problem
• More complex robots
• Multiple robots
• Movable objects
• Nonholonomic dynamic constraints
• Physical models and deformable objects
• Sensorless motions (exploiting task mechanics)
• Uncertainty in control

12
Extensions of the Basic Problem
• More complex environments
• Moving obstacles
• Uncertainty in sensing
• More complex objectives
• Optimal motion planning
• Integration of planning and control
• Assembly planning
• Sensing the environment
• Model building
• Target finding, tracking

13
Practical Algorithms
• A complete motion planner always returns a
solution when one exists and indicates that no
such solution exists otherwise.
• Most motion planning problems are hard, meaning
that complete planners take exponential time in
the number of degrees of freedom, moving objects,
etc.

14
Practical Algorithms
• Theoretical algorithms strive for completeness
and low worst-case complexity
• Difficult to implement
• Not robust
• Heuristic algorithms strive for efficiency in
commonly encountered situations.
• No performance guarantee
• Practical algorithms with performance guarantees
• Weaker forms of completeness
• Simplifying assumptions on the space
exponential time algorithms that work in
practice

15
Problem Formulation for Point Robot
• Input
• Robot represented as a point in the plane
• Obstacles represented as polygons
• Initial and goal positions
• Output
• A collision-free path between the initial and
goal positions

16
Framework
17
Visibility Graph Method
• Observation If there is a collision-free path
between two points, then there is a polygonal
path that bends only at the obstacles vertices.
• Why?
• Any collision-free path can be transformed into a
polygonal path that bends only at the obstacle
vertices.
• A polygonal path is a piecewise linear curve.

18
Visibility Graph
• A visibility graphis a graph such that
• Nodes qinit, qgoal, or an obstacle vertex.
• Edges An edge exists between nodes u and v if
the line segment between u and v is an obstacle
edge or it does not intersect the obstacles.

19
A Simple Algorithm for Building Visibility Graphs
20
Computational Efficiency
• Simple algorithm O(n3) time
• More efficient algorithms
• Rotational sweep O(n2log n) time
• Optimal algorithm O(n2) time
• Output sensitive algorithms
• O(n2) space

21
Framework
22
23
24
25
26
27
28
29
30
31
32
Other Search Algorithms
• Depth-First Search
• Best-First Search, A

33
Framework
34
Summary
• Discretize the space by constructing visibility
graph
• Search the visibility graph with breadth-first
search
• Q How to perform the intersection test?

35
Summary
• Represent the connectivity of the configuration
space in the visibility graph
• Running time O(n3)
• Compute the visibility graph
• Search the graph
• An optimal O(n2) time algorithm exists.
• Space O(n2)
• Can we do better?

36
Classic Path Planning Approaches
• Roadmap Represent the connectivity of the free
space by a network of 1-D curves
• Cell decomposition Decompose the free space
into simple cells and represent the connectivity
of the free space by the adjacency graph of these
cells
• Potential field Define a potential function
over the free space that has a global minimum at
the goal and follow the steepest descent of the
potential function

37
Classic Path Planning Approaches
• Roadmap Represent the connectivity of the free
space by a network of 1-D curves
• Cell decomposition Decompose the free space
into simple cells and represent the connectivity
of the free space by the adjacency graph of these
cells
• Potential field Define a potential function
over the free space that has a global minimum at
the goal and follow the steepest descent of the
potential function

38
• Visibility graph
• Shakey Project, SRI Nilsson, 1969
• Voronoi Diagram
• Introduced by computational geometry
researchers. Generate paths that maximizes
clearance. Applicable mostly to 2-D configuration
spaces.

39
Voronoi Diagram
• Space O(n)
• Run time O(n log n)

40
• Silhouette
• First complete general method that applies to
spaces of any dimensions and is singly
exponential in the number of dimensions Canny
1987

41
Classic Path Planning Approaches
• Roadmap Represent the connectivity of the free
space by a network of 1-D curves
• Cell decomposition Decompose the free space
into simple cells and represent the connectivity
of the free space by the adjacency graph of these
cells
• Potential field Define a potential function
over the free space that has a global minimum at
the goal and follow the steepest descent of the
potential function

42
Cell-decomposition Methods
• Exact cell decomposition
• The free space F is represented by a collection
of non-overlapping simple cells whose union is
exactly F
• Examples of cells trapezoids, triangles

43
Trapezoidal Decomposition
44
Computational Efficiency
• Running time O(n log n) by planar sweep
• Space O(n)
• Mostly for 2-D configuration spaces

45
• Nodes cells
• Edges There is an edge between every pair of
nodes whose corresponding cells are adjacent.

46
Summary
• Discretize the space by constructing an adjacency
graph of the cells

47
Cell-decomposition Methods
• Exact cell decomposition
• Approximate cell decomposition
• F is represented by a collection of
non-overlapping cells whose union is contained in
F.
• Cells usually have simple, regular shapes, e.g.,
rectangles, squares.
• Facilitate hierarchical space decomposition

48
49
Octree Decomposition
50
Algorithm Outline
51
Classic Path Planning Approaches
• Roadmap Represent the connectivity of the free
space by a network of 1-D curves
• Cell decomposition Decompose the free space
into simple cells and represent the connectivity
of the free space by the adjacency graph of these
cells
• Potential field Define a potential function
over the free space that has a global minimum at
the goal and follow the steepest descent of the
potential function

52
Potential Fields
• Initially proposed for real-time collision
avoidance Khatib 1986. Hundreds of papers
published.
• A potential field is a scalar function over the
free space.
• To navigate, the robot applies a force
proportional to the negated gradient of the
potential field.
• A navigation function is an ideal potential field
that
• has global minimum at the goal
• has no local minima
• grows to infinity near obstacles
• is smooth

53
Attractive Repulsive Fields
54
How Does It Work?
55
Algorithm Outline
• Place a regular grid G over the configuration
space
• Compute the potential field over G
• Search G using a best-first algorithm with
potential field as the heuristic function

56
Local Minima
• What can we do?
• Escape from local minima by taking random walks
• Build an ideal potential field navigation
function that does not have local minima

57
Question
• Can such an ideal potential field be constructed
efficiently in general?

58
Completeness
• A complete motion planner always returns a
solution when one exists and indicates that no
such solution exists otherwise.
• Is the visibility graph algorithm complete? Yes.
• How about the exact cell decomposition algorithm
and the potential field algorithm?

59
Why Complete Motion Planning?
• Complete motion planning
• Always terminate
• Not efficient
• Not robust even for low DOF
• Efficient
• Work for complex problems with many DOF
• Difficult for narrow passages
• May not terminate when no path exists

60
Path Non-existence Problem
Obstacle
Obstacle
61
Main Challenge
• Exponential complexity nDOF
• Degree of freedom DOF
• Geometric complexity n
• More difficult than finding a path
• To check all possible paths

Obstacle
62
Approaches
• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid planner

63
Configuration Space 2D Translation
Workspace
Configuration Space
Goal
Free
Robot
y
x
Start
64
Configuration Space Computation
• Varadhan et al, ICRA 2006
• 2 Translation 1 Rotation
• 215 seconds

Obstacle
?
y
x
Robot
65
Exact Motion Planning
• Approaches
• Exact cell decomposition Schwartz et al. 83
• Criticality based method Latombe 99
• Voronoi Diagram
• Not practical
• Due to free space computation
• Limit for special and simple objects
• 3DOF

66
Approaches
• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM

67
Approximation Cell Decomposition (ACD)
• Not compute the free space exactly at once
• But compute it incrementally
• Relatively easy to implement
• Lozano-Pérez 83
• Zhu et al. 91
• Latombe 91
• Zhang et al. 06

68
Approximation Cell Decomposition
Configuration Space
• Full cell
• Empty cell
• Mixed cell
• Mixed
• Uncertain

69
Connectivity Graph
Gf Free Connectivity Graph
Gf is a subgraph of G
70
Finding a Path by ACD
Initial
Goal
71
Finding a Path by ACD
• First Graph Cut Algorithm
• Guiding path in connectivity graph G
• Only subdivide along this path
• Update the graphs G and Gf

L Guiding Path
Described in Latombes book
72
First Graph Cut Algorithm
L
Only subdivide along L
73
Finding a Path by ACD
74
ACD for Path Non-existence
Initial
Goal
C-space
75
ACD for Path Non-existence
Connectivity Graph
76
ACD for Path Non-existence
Connectivity graph is not connected
No path!
Sufficient condition for deciding path
non-existence
77
Two-gear Example
Video
no path!
3.356s
Initial
Cells in C-obstacle
Goal
78
Cell Labeling
• Free Cell Query
• Whether a cell completely lies in free space?
• C-obstacle Cell Query
• Whether a cell completely lies in C-obstacle?

79
Free Cell Query A Collision Detection Problem
• Does the cell lie inside free space?
• Do robot and obstacle separate at all
configurations?

Robot
Obstacle
?
Configuration space
Workspace
80
Clearance
• Separation distance
• A well studied geometric problem
• Determine a volume in C-space which are
completely free

d
81
C-obstacle Query Another Collision Detection
Problem
• Does the cell lie inside C-obstacle?
• Do robot and obstacle intersect at all
configurations?

Robot
?
Obstacle
Configuration space
Workspace
82
Forbiddance
• Forbiddance dual to clearance
• Penetration Depth
• A geometric computation problem less investigated
• Zhang et al. ACM SPM 2006

PD
83
Limitation of ACD
• Combinatorial complexity of cell decomposition
• Limited for low DOF problem
• 3-DOF robots

84
Approaches
• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM

85
Hybrid Planning
• Efficient
• Many DOFs
• Narrow passages
• Path non-existence
• Complete Motion Planning
• Complete
• Not efficient

Can we combine them together?
86
Hybrid Approach for Complete Motion Planning
• Capture the connectivity for mixed cells
• Avoid substantial subdivision
• Use Approximation Cell Decomposition (ACD)
• Completeness
• Improve the sampling on narrow passages

87
Connectivity Graph
Gf Free Connectivity Graph
G Connectivity Graph
Gf is a subgraph of G
88
Pseudo-free edges
Initial
Goal
Pseudo free edge for two adjacent cells
89
Pseudo-free Connectivity Graph Gsf
Gsf Gf Pseudo-edges
Initial
Goal
90
Algorithm
• Gf
• Gsf
• G

91
Results of Hybrid Planning
92
Results of Hybrid Planning
93
Results of Hybrid Planning
• 2.5 - 10 times speedup

3 DOF 3 DOF 4 DOF 4 DOF 4 DOF 4 DOF
timing cells timing cells timing cells
Hybrid 34s 50K 16s 48K 102s 164K
ACD 85s 168K ? ? ? ?
Speedup 2.5 3.3 10 ? 10 ?
94
Summary
• Difficult for Exact Motion Planning
• Due to the difficulty of free space configuration
computation
• ACD is more practical
• Explore the free space incrementally
• Hybrid Planning
• Combine the completeness of ACD and efficiency of
PRM