Randomized Motion Planning Jean-Claude Latombe Computer Science Department Stanford University

Randomized Motion Planning

Jean-Claude LatombeJean-Claude Latombe

Computer Science DepartmentComputer Science DepartmentStanford UniversityStanford University

Goal of Motion PlanningGoal of Motion Planning

Answer queries about Answer queries about connectivityconnectivity of a space of a space

Classical example: find a Classical example: find a collision-free pathcollision-free path in in robot configuration space robot configuration space among static obstaclesamong static obstacles

Examples of additional constraints:Examples of additional constraints:

KinodynamicKinodynamic constraints constraints VisibilityVisibility constraints constraints

OutlineOutline Bits of historyBits of history

ApproachesApproaches

Probabilistic RoadmapsProbabilistic Roadmaps

ApplicationsApplications

ConclusionConclusion

Early WorkEarly Work

Shakey (Nilsson, 1969): Visibility graphShakey (Nilsson, 1969): Visibility graph

C = S1 x S1

Mathematical FoundationsMathematical Foundations

Lozano-Perez, 1980: Configuration SpaceLozano-Perez, 1980: Configuration Space

Computational AnalysisComputational Analysis

Reif, 1979: Hardness (lower-bound results) Reif, 1979: Hardness (lower-bound results)

Exact General-Purpose Path PlannersExact General-Purpose Path Planners

- Schwarz and Sharir, 1983: - Schwarz and Sharir, 1983: Exact cell Exact cell decomposition based on Collins techniquedecomposition based on Collins technique

- Canny, 1987: - Canny, 1987: Silhouette methodSilhouette method

Heuristic PlannersHeuristic Planners

Goal

Robot

)( GoalpGoal xxkF

0

020

0

,111

if

ifxFObstacle

Khatib, 1986:Khatib, 1986:

Potential FieldsPotential Fields

Other Types of ConstraintsOther Types of Constraints

E.g., Visibility-Based Motion PlanningE.g., Visibility-Based Motion Planning Guibas, Latombe, LaValle, Lin, and Motwani, 1997Guibas, Latombe, LaValle, Lin, and Motwani, 1997






Criticality-Based Motion PlanningCriticality-Based Motion Planning Principle:Principle:

Select a property Select a property PP over the space of interest over the space of interest Compute an arrangement of cells such that Compute an arrangement of cells such that PP stays stays

constant over each cellconstant over each cell Build a search graph based on this arrangementBuild a search graph based on this arrangement

Example: Example: Wilson’s Wilson’s Non-Directional Blocking Non-Directional Blocking Graphs for assembly planningGraphs for assembly planning

Other examples:Other examples:

Schwartz-Sharir’s cell decompositionSchwartz-Sharir’s cell decomposition Canny’s roadmapCanny’s roadmap

Criticality-Based Motion PlanningCriticality-Based Motion Planning

Advantages: Advantages:

CompletenessCompleteness InsightInsight

Drawbacks:Drawbacks:

Computational complexityComputational complexity Difficult to implementDifficult to implement

Sampling-Based Motion PlanningSampling-Based Motion Planning

Principle:Principle:

Sample the space of interest Sample the space of interest Connect sampled points by simple pathsConnect sampled points by simple paths Search the resulting graphSearch the resulting graph

Example:Example:Probabilistic Roadmaps Probabilistic Roadmaps (PRM’s)(PRM’s)

Other example:Other example:Grid-based methods (deterministic sampling)Grid-based methods (deterministic sampling)

Sampling-Based Motion PlanningSampling-Based Motion Planning

Advantages:Advantages:

– Easy to implementEasy to implement– Fast, scalable to many degrees of Fast, scalable to many degrees of

freedom and complex constraintsfreedom and complex constraints Drawbacks:Drawbacks:

– Probabilistic completenessProbabilistic completeness– Limited insightLimited insight






MotivationMotivationComputing an explicit representation of the admissibleComputing an explicit representation of the admissiblespace is hard, but checking that a point lies in the space is hard, but checking that a point lies in the admissible space is fast admissible space is fast

Probabilistic Roadmap (PRM)Probabilistic Roadmap (PRM)

admissible space

mmbb

mmgg

milestone

[Kavraki, Svetska, Latombe,Overmars, 95][Kavraki, Svetska, Latombe,Overmars, 95]

Sampling StrategiesSampling Strategies

Multi vs. single query strategiesMulti vs. single query strategies Multi-stage strategiesMulti-stage strategies Obstacle-sensitive strategiesObstacle-sensitive strategies Lazy collision checkingLazy collision checking Probabilistic biases (e.g., potential fields)Probabilistic biases (e.g., potential fields)

mb

mg

PRM With Dynamic Constraints in State x Time SpacePRM With Dynamic Constraints in State x Time Space

endgame region

[Hsu, Kindel, Latombe, and Rock, 2000][Hsu, Kindel, Latombe, and Rock, 2000]

m’ = f(m,u)m’ = f(m,u)

Relation to Art-Gallery ProblemsRelation to Art-Gallery Problems

[Kavraki, Latombe, Motwani, Raghavan, 95]

Narrow Passage IssueNarrow Passage Issue

Desirable Properties of a PRMDesirable Properties of a PRM

Coverage:Coverage:The milestones should see most of the admissible The milestones should see most of the admissible space to guarantee that the initial and goal space to guarantee that the initial and goal configurations can be easily connected to the configurations can be easily connected to the roadmaproadmap

Connectivity:Connectivity:There should be a 1-to-1 map between the There should be a 1-to-1 map between the components of the admissible space and those of components of the admissible space and those of the roadmap the roadmap

Complexity MeasuresComplexity Measures

-goodness-goodness[Kavraki, Latombe, Motwani, and Raghavan, 1995][Kavraki, Latombe, Motwani, and Raghavan, 1995]

Path clearancePath clearance[Kavraki, Koulountzakis, and Latombe, 1996][Kavraki, Koulountzakis, and Latombe, 1996]

-complexity-complexity[Overmars and Svetska, 1998][Overmars and Svetska, 1998]

ExpansivenessExpansiveness[Hsu, Latombe, and Motwani, 1997][Hsu, Latombe, and Motwani, 1997]

Expansiveness of Admissible SpaceExpansiveness of Admissible Space

Expansiveness of Admissible SpaceExpansiveness of Admissible Space

Lookout of Lookout of F1F1

The admissible space is expansive if each of its subsets has a large lookout

Prob[failure] = K exp(-r)

Two Very Different CasesTwo Very Different Cases

ExpansiveExpansivePoorly expansivePoorly expansive

A Few RemarksA Few Remarks

Big computational saving is achieved at the cost of Big computational saving is achieved at the cost of slightly reduced completenessslightly reduced completeness

Computational complexity is a function of the Computational complexity is a function of the shape of the admissible space, not the size needed shape of the admissible space, not the size needed to describe itto describe it

Randomization is not really needed; it is a Randomization is not really needed; it is a convenient incremental schemeconvenient incremental scheme






Design for Manufacturing and ServicingDesign for Manufacturing and Servicing

General ElectricGeneral Electric

General MotorsGeneral MotorsGeneral MotorsGeneral Motors

[Hsu, 2000][Hsu, 2000]

Robot Programming and PlacementRobot Programming and Placement

[Hsu, 2000][Hsu, 2000]

Graphic Animation of Digital ActorsGraphic Animation of Digital Actors

[Koga, Kondo, Kuffner, and Latombe, 1994][Koga, Kondo, Kuffner, and Latombe, 1994]

The MotionThe MotionFactoryFactory

Vision module imageActor camera image

Digital Actors With Visual SensingDigital Actors With Visual Sensing

Segment environmentSegment environment Render false-color scene offscreen Render false-color scene offscreen Scan pixels & record IDsScan pixels & record IDs

Simulated VisionSimulated Vision Kuffner, 1999Kuffner, 1999

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

Space RoboticsSpace Robotics

air bearingair bearing

gaz tankgaz tank

air thrustersair thrustersobstacles

robotrobot

[Kindel, Hsu, Latombe, and Rock, 2000][Kindel, Hsu, Latombe, and Rock, 2000]

Total duration : 40 secTotal duration : 40 sec

Autonomous HelicopterAutonomous Helicopter

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

Interacting Nonholonomic RobotsInteracting Nonholonomic Robots

yy11

xx22

d

xx11

yy22

(Grasp Lab - U. Penn)(Grasp Lab - U. Penn)

Map BuildingMap Building

[Gonzalez, 2000][Gonzalez, 2000]

Next-Best View ComputationNext-Best View Computation





Radiosurgical PlanningRadiosurgical Planning

Cyberknife System (Accuray, Inc.) Cyberknife System (Accuray, Inc.) CARABEAMER Planner CARABEAMER Planner

[Tombropoulos, Adler, and Latombe, 1997][Tombropoulos, Adler, and Latombe, 1997]

Radiosurgical PlanningRadiosurgical Planning

• 2000 < Tumor < 22002000 < B2 + B4 < 22002000 < B4 < 22002000 < B3 + B4 < 22002000 < B3 < 22002000 < B1 + B3 + B4 < 22002000 < B1 + B4 < 22002000 < B1 + B2 + B4 < 22002000 < B1 < 22002000 < B1 + B2 < 2200

• 0 < Critical < 5000 < B2 < 500

T

C

B1

B2

B3B4

T

Sample CaseSample Case

50% Isodose Surface

80% Isodose Surface

Conventional system’s plan CARABEAMER’s plan

Reconfiguration Planning for Modular RobotsReconfiguration Planning for Modular Robots

Xerox, ParcXerox, Parc

Casal and Yim, 1999

Prediction of Molecular MotionsPrediction of Molecular Motions

[Singh, Latombe, and Brutlag, 1999][Singh, Latombe, and Brutlag, 1999]

Ligand-protein bindingLigand-protein binding Protein foldingProtein folding[Apaydin, 2000][Apaydin, 2000]

Capturing Energy LandscapeCapturing Energy Landscape[Apaydin, 2000][Apaydin, 2000]

Energy

Predicted binding site

Predicted binding site

Active site

15-20 kcal/ mol 10-12 kcal/ mol 10-12 kcal/ mol






ConclusionConclusion PRM planners have successfully solved many diverse PRM planners have successfully solved many diverse

complex motion problems with different constraints complex motion problems with different constraints (obstacles, kinematics, dynamics, stability, visibility, (obstacles, kinematics, dynamics, stability, visibility, energetic)energetic)

They are easy to implementThey are easy to implement Fast convergence has been formally proven in expansive Fast convergence has been formally proven in expansive

spaces. As computers get more powerful, PRM planners spaces. As computers get more powerful, PRM planners should allow us to solve considerably more difficult problemsshould allow us to solve considerably more difficult problems

Recent implementations solve difficult problems with many Recent implementations solve difficult problems with many degrees of freedom at quasi-interactive ratedegrees of freedom at quasi-interactive rate

IssuesIssues Relatively large standard deviation of Relatively large standard deviation of

planning timeplanning time

No rigorous termination criterion when no No rigorous termination criterion when no solution is foundsolution is found

New challenging applicationsNew challenging applications ……

Planning Minimally Invasive SurgeryPlanning Minimally Invasive SurgeryProcedures Amidst Soft-Tissue StructuresProcedures Amidst Soft-Tissue Structures

Planning Nice-Looking Motions Planning Nice-Looking Motions for Digital Actors for Digital Actors

A Bug’s Life (Pixar/Disney) Toy Story (Pixar/Disney)

Tomb Raider 3 (Eidos Interactive) Final Fantasy VIII (SquareOne)The Legend of Zelda (Nintendo)

Antz (Dreamworks)

Dealing with 1,000s of Degrees of FreedomDealing with 1,000s of Degrees of Freedom

Protein foldingProtein folding

Main Common DifficultyMain Common Difficulty

Formulating motion constraintsFormulating motion constraints

Documents

Randomized Motion Planning Jean-Claude Latombe Computer Science Department Stanford University