UNC Chapel Hill M. C. Lin Introduction to Motion Planning Applications Overview of the Problem...

UNC Chapel Hill M. C. Lin

Introduction to Motion Planning

Applications Overview of the Problem Basics – Planning for Point Robot

– Visibility Graphs– Roadmap– Cell Decomposition– Potential Field


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.)


Fundamental Question

Are two given points connected by a path?


Basic Problem

Problem statement: Compute a collision-free pathfor 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


Example: Rigid Objects


Example: Articulated Robot


Is it easy?


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 [Hopcroftet al., 1984]– Multiple rectangles [Hopcroftet al., 1984]


Tool: Configuration Space

Difficulty– Number of degrees of freedom (dimension

of configuration space)– Geometric complexity


Extensions of the Basic Problem

More complex robots– Multiple robots– Movable objects– Nonholonomic& dynamic constraints– Physical models and deformable objects– Sensorlessmotions (exploiting task mechanics)– Uncertainty in control


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


Next Few Lectures

Present a coherent framework for motion planning problems:– Configuration space and related concepts– Algorithms based on random sampling and

algorithms based on processing critical geometric events

Emphasize “practical” algorithms with some guarantees of performance over “theoretical” or purely “heuristic” 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.


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


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


Framework


Visibility Graph Method

Observation: If there is a 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.


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.


A Simple Algorithm for Building Visibility Graphs


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


Framework


Breadth-First Search


Other Search Algorithms

Depth-First SearchBest-First Search, A*


Framework


Summary

Discretize the space by constructing visibility graph

Search the visibility graph with breadth-first search

Q: How to perform the intersection test?


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?


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


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.


Voronoi Diagram

Space O(n)Run time O(n log n)


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


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


Trapezoidal Decomposition


Computational Efficiency

Running time O(n log n) by planar sweep

Space O(n)Mostly for 2-D configuration spaces


Adjacency Graph

Nodes: cells Edges: There is an edge between every pair of

nodes whose corresponding cells are adjacent.


Summary

Discretize the space by constructing an adjacency graph of the cells

Search the adjacency graph


Cell-decomposition Methods

Exact cell decompositionApproximate 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


Quadtree Decomposition


Octree Decomposition


Algorithm Outline


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


Attractive & Repulsive Fields


How Does It Work?


Algorithm Outline

Place a regular grid G over the configuration space

Compute the potential field over GSearch G using a best-first algorithm

with potential field as the heuristic function


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


Question

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


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?

UNC Chapel Hill M. C. Lin Introduction to Motion Planning Applications Overview of the Problem...

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