Motion Planning

CS121 – Winter 2003

Basic ProblemBasic ProblemAre two given points connected by a path?

From Robotics …From Robotics …

… … to Graphic Animation to Graphic Animation ……

… … to Biologyto Biology

… … to Biologyto Biology

How Do You Get There?How Do You Get There?

?

Configuration SpaceConfiguration Space

Problems:• Geometric complexity• Number of dimensions of space• How to discretize the free space?

Approximate the free space by random sampling

Digital CharacterDigital Character

q2

q1

q3

q0

qn

q4

Q(t)

Parts DOFL 19 68H 51 118

Configuration SpaceConfiguration Space

Problems:• Geometric complexity• Number of dimensions of space• How to discretize the free space?

Approximate the free space by random sampling

Hierarchical Collision Hierarchical Collision CheckingChecking

Example in 3DExample in 3D

Hierarchical Collision Hierarchical Collision CheckingChecking

Hierarchical Collision Hierarchical Collision CheckingChecking

Performance EvaluationPerformance Evaluation

Collision checking takes between 0.0001 and .002 seconds for 2 objects of 500,000 triangles each on a 1-GHz Pentium IIICollision checking is faster when objects collide or are far apart, and gets slower when they get closer without collidingOverall collision checking time grows roughly as the log of the number of triangles

Probabilistic Roadmap Probabilistic Roadmap (PRM)(PRM)

free space

mmbb

mmgg

milestone

local path

Why It WorksWhy It Works

Narrow Passage IssueNarrow Passage Issue

EasyEasyDifficultDifficult

Probabilistic CompletenessProbabilistic Completeness

Under the generally satisfied assumption that the free space is expansive, the probability

that a PRM finds a path when one exists goes to 1 exponentially in the number of

milestones (~ running time).

Multi-Query Sampling Multi-Query Sampling StrategiesStrategies

Multi-Query Sampling Multi-Query Sampling StrategiesStrategies

• Multi-stage strategies• Obstacle-sensitive strategies• Narrow-passage strategies

Single-Query Sampling Single-Query Sampling StrategiesStrategies

mmbb

mmgg

Single-Query Sampling Single-Query Sampling StrategiesStrategies

mmbb

mmgg

• Diffusion strategies• Adaptive-step strategies• Lazy collision checking

ExamplesExamples

Nrobot = 5,000; Nobst = 83,000 Tav = 4.42 s

Nrobot = 3,000; Nobst = 50,000 Tav = 0.17 s

Design for Design for Manufacturing/ServicingManufacturing/Servicing

General ElectricGeneral Electric

General MotorsGeneral MotorsGeneral MotorsGeneral Motors

[Hsu, 2000]

Modular Reconfigurable Modular Reconfigurable RobotsRobots

Xerox, ParcXerox, Parc

Casal and Yim, 1999

Humanoid RobotHumanoid Robot[Kuffner and Inoue, 2000] (U. Tokyo)

Stability constraints

Space RoboticsSpace Robotics

air bearing

gas tank

air thrustersobstacles

robotrobot

[Kindel, 2000]Dynamic constraintsDynamic constraints

Single-Query Sampling Single-Query Sampling StrategiesStrategies

mmbb

mmgg

Total duration : 40 sec

Autonomous HelicopterAutonomous Helicopter

[Feron, 2000] (AA Dept., MIT)

Other goalsOther goals

The goal may not be to attain a given position, but to achieve a certain condition, e.g.:- Irradiate a tumor- Build a map of an environment- Sweep an environment to find a target

Radiosurgery: Irradiate a Radiosurgery: Irradiate a TumorTumor

Mobile Robots: Map BuildingMobile Robots: Map Building

Next-Best ViewNext-Best View

ExampleExample

Information StateInformation State

Example of an information state = (1,1,0)

0 : the target does not hide beyond the edge

1 : the target may hide beyond the edge

Critical CurveCritical Curve

More Complex ExampleMore Complex Example

Example with Two RobotsExample with Two Robots(Greedy algorithm)

Surgical PlanningSurgical Planning

Half-Dome, NW Face, Summer of 2010 …

Tim Bretl

Rock-Climbing RobotRock-Climbing Robot