Randomized Motion Planning Jean-Claude Latombe Jean-Claude Latombe Computer Science Department Computer Science Department Stanford University Stanford University
Dec 20, 2015
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
OutlineOutline Bits of historyBits of history
ApproachesApproaches
Probabilistic RoadmapsProbabilistic Roadmaps
ApplicationsApplications
ConclusionConclusion
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
OutlineOutline Bits of historyBits of history
ApproachesApproaches
Probabilistic RoadmapsProbabilistic Roadmaps
ApplicationsApplications
ConclusionConclusion
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
OutlineOutline Bits of historyBits of history
ApproachesApproaches
Probabilistic RoadmapsProbabilistic Roadmaps
ApplicationsApplications
ConclusionConclusion
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
Map BuildingMap Building
[Gonzalez, 2000][Gonzalez, 2000]
Map BuildingMap Building
[Gonzalez, 2000][Gonzalez, 2000]
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
OutlineOutline Bits of historyBits of history
ApproachesApproaches
Probabilistic RoadmapsProbabilistic Roadmaps
ApplicationsApplications
ConclusionConclusion
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