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### Robot Lab:Robot Path Planning

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

Slide 1

Introduction to Motion Planning

- Applications
- Overview of the Problem
- Basics – Planning for Point Robot
- Visibility Graphs
- Roadmap
- Cell Decomposition
- Potential Field

Slide 2

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

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

Example: Rigid Objects

Slide 6

Example: Articulated Robot

Slide 7

Is it easy?

Slide 8

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

Difficulty

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

Slide 10

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 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

- 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 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

- 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

Slide 16

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

- 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

- 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

Framework

Slide 21

Breadth-First Search

Slide 22

Breadth-First Search

Slide 23

Breadth-First Search

Slide 24

Breadth-First Search

Slide 25

Breadth-First Search

Slide 26

Breadth-First Search

Slide 27

Breadth-First Search

Slide 28

Breadth-First Search

Slide 29

Breadth-First Search

Slide 30

Breadth-First Search

Slide 31

Framework

Slide 33

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

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

- 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 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

- 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

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 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

- 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

Trapezoidal Decomposition

Slide 43

Computational Efficiency

- Running time O(n log n) by planar sweep
- Space O(n)
- Mostly for 2-D configuration spaces

Slide 44

Adjacency Graph

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

Slide 45

Summary

- Discretize the space by constructing an adjacency graph of the cells
- Search the adjacency graph

Slide 46

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

Quadtree Decomposition

Slide 48

Octree Decomposition

Slide 49

Algorithm Outline

Slide 50

Classic Path Planning Approaches

- Roadmap – Represent the connectivity of the free space by a network of 1-D curves

Slide 51

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

Attractive & Repulsive Fields

Slide 53

How Does It Work?

Slide 54

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

- 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

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

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

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

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

Slide 62

C-obstacle

Configuration Space: 2D TranslationWorkspace

Configuration Space

Goal

Free

Robot

y

x

Start

Slide 63

Configuration Space Computation

- [Varadhan et al, ICRA 2006]
- 2 Translation + 1 Rotation
- 215 seconds

Obstacle

y

x

Robot

Slide 64

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

Approaches

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

Slide 66

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

mixed

empty

Approximation Cell DecompositionConfiguration Space

- Full cell
- Empty cell
- Mixed cell
- Mixed
- Uncertain

Slide 68

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

Finding a Path by ACD

Slide 73

ACD for Path Non-existence

Connectivity graph is not connected

No path!

Sufficient condition for deciding path non-existence

Slide 76

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 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

Separation distance

A well studied geometric problem

Determine a volume in C-space which are completely free

Clearanced

Slide 80

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’: dual to clearance

Penetration Depth

A geometric computation problem less investigated

[Zhang et al. ACM SPM 2006]

‘Forbiddance’PD

Slide 82

Limitation of ACD

- Combinatorial complexity of cell decomposition
- Limited for low DOF problem
- 3-DOF robots

Slide 83

Approaches

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

Slide 84

Probabilistic roadmap motion planning

+ Efficient

+ Many DOFs

Narrow passages

Path non-existence

Complete Motion Planning

+ Complete

Not efficient

Hybrid PlanningCan we combine them together?

Slide 85

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

Results of Hybrid Planning

Slide 91

Results of Hybrid Planning

Slide 92

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

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