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Robot Lab: Robot Path Planning William Regli Department of Computer Science (and Departments of ECE and MEM) Drexel University Introduction to Motion Planning Applications Overview of the Problem Basics – Planning for Point Robot Visibility Graphs Roadmap Cell Decomposition

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robot lab robot path planning

Robot Lab:Robot Path Planning

William RegliDepartment of Computer Science(and Departments of ECE and MEM)Drexel University

Slide 1

introduction to motion planning
Introduction to Motion Planning
  • Applications
  • Overview of the Problem
  • Basics – Planning for Point Robot
    • Visibility Graphs
    • Roadmap
    • Cell Decomposition
    • Potential Field

Slide 2

goals
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.)

Slide 3

fundamental question
Fundamental Question

Are two given points connected by a path?

Slide 4

basic problem
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

Slide 5

hardness results
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]

Slide 9

tool configuration space
Tool: Configuration Space

Difficulty

  • Number of degrees of freedom (dimension of configuration space)
  • Geometric complexity

Slide 10

extensions of the basic problem
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

Slide 11

extensions of the basic problem12
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

Slide 12

practical algorithms
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.

Slide 13

practical algorithms14
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

Slide 14

problem formulation for point robot
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

Slide 15

framework
Framework

Slide 16

visibility graph method
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.

Slide 17

visibility graph
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.

Slide 18

computational efficiency
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

Slide 20

framework21
Framework

Slide 21

other search algorithms
Other Search Algorithms
  • Depth-First Search
  • Best-First Search, A*

Slide 32

framework33
Framework

Slide 33

summary
Summary
  • Discretize the space by constructing visibility graph
  • Search the visibility graph with breadth-first search

Q: How to perform the intersection test?

Slide 34

summary35
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?

Slide 35

classic path planning approaches
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

Slide 36

classic path planning approaches37
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

Slide 37

roadmap
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.

Slide 38

voronoi diagram
Voronoi Diagram
  • Space O(n)
  • Run time O(n log n)

Slide 39

other roadmap methods
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

Slide 40

classic path planning approaches41
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

Slide 41

cell decomposition methods
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

Slide 42

computational efficiency44
Computational Efficiency
  • Running time O(n log n) by planar sweep
  • Space O(n)
  • Mostly for 2-D configuration spaces

Slide 44

adjacency graph
Adjacency Graph
  • Nodes: cells
  • Edges: There is an edge between every pair of nodes whose corresponding cells are adjacent.

Slide 45

summary46
Summary
  • Discretize the space by constructing an adjacency graph of the cells
  • Search the adjacency graph

Slide 46

cell decomposition methods47
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

Slide 47

classic path planning approaches51
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

Slide 51

potential fields
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

Slide 52

algorithm outline55
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

Slide 55

local minima
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

Slide 56

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

Slide 57

completeness
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?

Slide 58

why complete motion planning
Probabilistic roadmap motion planning

Efficient

Work for complex problems with many DOF

Difficult for narrow passages

May not terminate when no path exists

Complete motion planning

Always terminate

Not efficient

Not robust even for low DOF

Why Complete Motion Planning?

Slide 59

path non existence problem

Robot

Initial

Goal

Path Non-existence Problem

Obstacle

Obstacle

Slide 60

main challenge

Robot

Goal

Initial

Main Challenge
  • Exponential complexity: nDOF
    • Degree of freedom: DOF
    • Geometric complexity: n
  • More difficult than finding a path
    • To check all possible paths

Obstacle

Slide 61

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

Slide 62

configuration space 2d translation

Obstacle

C-obstacle

Configuration Space: 2D Translation

Workspace

Configuration Space

Goal

Free

Robot

y

x

Start

Slide 63

configuration space computation
Configuration Space Computation
  • [Varadhan et al, ICRA 2006]
  • 2 Translation + 1 Rotation
  • 215 seconds

Obstacle

y

x

Robot

Slide 64

exact motion planning
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

Slide 65

approaches66
Approaches
  • Exact Motion Planning
    • Based on exact representation of free space
  • Approximation Cell Decomposition (ACD)
  • A Hybrid Planner Combing ACD and PRM

Slide 66

a pproximation c ell d ecomposition acd
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]

Slide 67

approximation cell decomposition

full

mixed

empty

Approximation Cell Decomposition

Configuration Space

  • Full cell
  • Empty cell
  • Mixed cell
    • Mixed
    • Uncertain

Slide 68

connectivity graph

G: Connectivity Graph

Connectivity Graph

Gf : Free Connectivity Graph

Gf is a subgraph of G

Slide 69

finding a path by acd71
Finding a Path by ACD

L: Guiding Path

  • First Graph Cut Algorithm
    • Guiding path in connectivity graph G
    • Only subdivide along this path
    • Update the graphs G and Gf

Described in Latombe’s book

Slide 71

first graph cut algorithm
First Graph Cut Algorithm

L

Only subdivide along L

Slide 72

acd for path non existence
ACD for Path Non-existence

Initial

Goal

C-space

Slide 74

acd for path non existence75

Guiding Path

ACD for Path Non-existence

Connectivity Graph

Slide 75

acd for path non existence76
ACD for Path Non-existence

Connectivity graph is not connected

No path!

Sufficient condition for deciding path non-existence

Slide 76

two gear example
Two-gear Example

Video

no path!

3.356s

Initial

Cells in C-obstacle

Roadmap in F

Goal

Slide 77

cell labeling

full

mixed

empty

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?

Slide 78

free cell query a collision detection problem
Free Cell QueryA Collision Detection Problem
  • Does the cell lie inside free space?
  • Do robot and obstacle separate at all configurations?

Robot

Obstacle

?

Configuration space

Workspace

Slide 79

clearance
Separation distance

A well studied geometric problem

Determine a volume in C-space which are completely free

Clearance

d

Slide 80

c obstacle query another collision detection problem
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

Slide 81

forbiddance
‘Forbiddance’: dual to clearance

Penetration Depth

A geometric computation problem less investigated

[Zhang et al. ACM SPM 2006]

‘Forbiddance’

PD

Slide 82

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

Slide 83

approaches84
Approaches
  • Exact Motion Planning
    • Based on exact representation of free space
  • Approximation Cell Decomposition (ACD)
  • A Hybrid Planner Combing ACD and PRM

Slide 84

hybrid planning
Probabilistic roadmap motion planning

+ Efficient

+ Many DOFs

Narrow passages

Path non-existence

Complete Motion Planning

+ Complete

Not efficient

Hybrid Planning

Can we combine them together?

Slide 85

hybrid approach for complete motion planning
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

Slide 86

connectivity graph87
Connectivity Graph

Gf : Free Connectivity Graph

G: Connectivity Graph

Gf is a subgraph of G

Slide 87

pseudo free edges
Pseudo-free edges

Initial

Goal

Pseudo free edge for two adjacent cells

Slide 88

pseudo free connectivity graph g sf
Pseudo-free Connectivity Graph: Gsf

Gsf = Gf + Pseudo-edges

Initial

Goal

Slide 89

algorithm
Algorithm
  • Gf
  • Gsf
  • G

Slide 90

results of hybrid planning93
Results of Hybrid Planning
  • 2.5 - 10 times speedup

Slide 93

summary94
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

Slide 94