Complexity Issues in Robot Motion Planning

Elif TosunMTH 353

Final Paper

Overview

Introduction & MotivationBasic DefinitionsBasic Motion PlanningManipulation PlanningAlgorithmic ApproachesConclusions

Introduction

Motion planning is aimed at providing robots with the capability of deciding automatically which motions to execute in order to achieve their tasks without colliding with other objects in their work space.

Considerations Economic Cost - time, energy, etc. Physical limitations - friction, etc.

Motivation

Robot ApplicationsManufacturingMedical SurgeryMolecular BiologyComputer GraphicsAir & Spacecraft Navigation...

Basic DefinitionsRobot: Mechanical system consisting of one or

more rigid bodies possibly connected by various joints and hinges

Configuration: Position of every point of a robot at a given instance

Degrees of Freedom(DOF): number of dimensions along which the robot can move itself.

More definitions

Workspace: Environment in which the robot moves (2D or 3D)

Configuration space: Space of all configurations of a robot

Free space: configuration space minus the space occupied by obstacles

Basic Motion Planning

Objective: To plan a collision free path of a robot with an arbitrary DOF to a goal position in 2D or 3D avoiding a set of obstacles stationary in space.

Complexity Results

3D version : Robot is a set of linked polyhedra and obstacles are fixed polyhedra in 3D

Complexity: PSPACE-hard when the robot has n links

(due to Reif, proved in 1979) P when the robot has a constant DOF

(due to Schwartz & Sharir, proved in 1983)

Complexity Results

2D version : Robot is a set of linked polygons and obstacles are fixed polygonals objects in 2D

Complexity: PSPACE-hard lower bound

(due to Schwartz & Sharir, proved in 1984)

Manipulation Planning

Objective: To have the robot move around objects in the workspace to reach a final arrangement. (Objects cannot move by themselves)

Games: SOKOBAN PushPush

Sokoban

Objective of Robot:To push boxes into their storage locations without getting himself or boxes stuck.

Rules: Cannot pull, can push only one box at a time

Sokoban

Complexity Result:

Proved to be PSPACE-hardSo all puzzles of this kind (different levels, etc.) are PSPACE-complete

(due to Culberson, 1998)

PushPush

Objective: To push blocks in order to get from an initial position to a final position

Rules: -One block at a time -Block slides the full extent

of available space

PushPush

Complexity Result PushPush is NP-hard in 2D and 3D Proof based on reduction from SAT. Open question

Is it NP-Complete (is it in NP?)OR is it PSPACE-Complete??

Algorithmic Approaches

Complete AlgorithmsProbabilistic AlgorithmsHeuristic Algorithms

Complete Algorithms

Guaranteed to find a free path between two give configurations when exists and report failure otherwise

Deal with connectivity of free space by capturing it on a graph. Cell Decomposition - partition of free space Roadmap Technique - network of curves

Not open for improvements

Probabilistic Algorithms

Trade-off exactness against running timeDon’t guarantee a solution but if exists

very likely to find it relatively quicklyExample: Probabilistic Roadmap

AlgorithmExperimental results show that

computation takes less than a secondUsed in maintenance of aircraft

Heuristic Algorithms

Many work well in practice but offer no performance guarantee

Deal with a grid on configuration space Example 1 : Potential Field Example 2 : Approximate Cell

DecompositionSpace for Improvement

Conclusions

Robot Motion Planning is DIFFICULT!!!Many open problems:

motion planning with uncertainty assembly planning approximation algorithms motion with flexible objects, and many more...

Interest moving from theoretical research to approximation algorithms and applications