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Title: Robot Lab: Robot Path Planning


1
Robot LabRobot Path Planning
  • William RegliDepartment 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
  • Roadmap
  • 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
Breadth-First Search
23
Breadth-First Search
24
Breadth-First Search
25
Breadth-First Search
26
Breadth-First Search
27
Breadth-First Search
28
Breadth-First Search
29
Breadth-First Search
30
Breadth-First Search
31
Breadth-First Search
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
Roadmap
  • 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
Other Roadmap Methods
  • Silhouette
  • First complete general method that applies to
    spaces of any dimensions and is singly
    exponential in the number of dimensions Canny
    1987
  • Probabilistic roadmaps

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
Adjacency Graph
  • 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
  • Search the adjacency graph

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
Quadtree Decomposition
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
  • Probabilistic roadmap motion planning
  • 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
  • Roadmap Canny 88
  • Criticality based method Latombe 99
  • Voronoi Diagram
  • Star-shaped roadmap Varadhan et al. 06
  • Not practical
  • Due to free space computation
  • Limit for special and simple objects
  • Ladders, sphere, convex shapes
  • 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
Roadmap in F
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 QueryAnother 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
  • Probabilistic roadmap motion 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
  • Use Probabilistic Roadmap (PRM)
  • 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
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