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Differentially Constrained Planning Mihail Pivtoraiko 1 Motion Planning The Challenge Reliable Autonomous Robots 2 NavLab 1985 Boss 2007 MER 2004 Crusher 2006 ALV 1988 XUV 1998 ID: 373005

state amp local space amp state space local search kelly tree global motion computing lattice fixed rrt pre latombe

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Slide1

A Survey ofDifferentially Constrained Planning

Mihail Pivtoraiko

1Slide2

Motion Planning

The Challenge:

Reliable Autonomous Robots

NavLab

, 1985

Boss, 2007

MER, 2004

Crusher, 2006

ALV, 1988

XUV, 1998

Stanford Cart, 1979Slide3

Agenda3

Deterministic planners

Path smoothing

Control sampling

State sampling

Randomized planners

Probabilistic roadmaps

Rapidly exploring Random TreesSlide4

Motivation4

Local

Global

ALV (Daily et al., 1988)Slide5

Unstructured Environments

Local

Global

Structure

imposed

Regular, fine grid

Standard search (A*)Slide6

Global

Unstructured and

Uncertain

Uncertain terrain

Potentially changing

LocalSlide7

Global

Unstructured and

Uncertain

Unseen obstacles

Detected up close

Invalidate the plan

LocalSlide8

Global

Unstructured and

Uncertain

Efficient

replanning

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(Simmons et al., 1996)

Gestalt (

Maimone

et al., 2002)

LocalSlide9

9Mobility Constraints

2D global planners lead to nonconvergence in difficult environments

Robot will

fail

to make the turn into the

corridor

Global planner

must

understand the need to swing

wideIssues:Passage missed, orPoint-turn is necessary…

Plan Step n

Plan Step n+1

Plan Step n+2Slide10

In the Field…10

PerceptOR/UPI, 2005Rover Navigation, 2008Slide11

11Mobility Constraints

Problem:

Mal-informed global planner

Vehicle constraints ignored:

Heading

Potential solutions:

Rapidly-Exploring Random Tree (RRT)

[

LaValle

Kuffner

, 2001]

PDST-EXPLORE

[Ladd &

Kavraki

, 2004]

Deterministic motion sampling

[

Barraquand

Latombe

, 1993]

LaValle

Kuffner

, 2001Slide12

Mobility Constraints12

Bruntingthorpe Proving Grounds Leicestershire, UK

April 1999Slide13

Extreme Maneuvering13

Kolter et al., 2010Slide14

Arbitrary…14Slide15

Dynamics Planning15Slide16

Definitions

State space

, , 

z, , , 

vSlide17

Definitions

State space

Control space

Accelerator

Steering

, , 

x ,

z, , , 

vSlide18

DefinitionsState space

Control spaceFeasibilitySatisfaction of differential constraintsGeneral formulation

x = f

(

x, u, t

)Slide19

DefinitionsState space

Control spaceFeasibilityPartially-known environmentSampled perception map

19Slide20

DefinitionsState space

Control spaceFeasibilityPartially-known environmentSampled perception mapLocal, changing info

20Slide21

DefinitionsMotion PlanningGiven two states, compute control sequence

QualitiesFeasibilityOptimalityRuntimeCompleteness

21Slide22

DefinitionsMotion PlanningDynamic

ReplanningCapacity to “repair” the planImproves reaction time22Slide23

DefinitionsMotion Planning

Dynamic ReplanningSearch SpaceSet of motion alternativesUnstructured environments SamplingState space

Control space

23Slide24

DefinitionsMotion Planning

Dynamic ReplanningSearch SpaceDeterministic samplingFixed pattern, predictable

“Curse of dimensionality”

24Slide25

DefinitionsSearch space designInput: robot properties

Output: state, control samplingMaximize planner qualities (F, O, R, C)Design principleSampling rule

25Slide26

OutlineIntroduction

Deterministic PlanningHierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMsRRTs

Derandomized

Planners

Some applications

26Slide27

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Blocked motions

Free motions

Perception horizon

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide28

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide29

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide30

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide31

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide32

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide33

Local/Global

Pros:

Local motion evaluation is fast

In sparse obstacles, works very well

Cons:

Search space differences

ALV (Daily et al., 1988)

D*/Smarty (

Stentz

& Hebert, 1994)

Ranger (Kelly, 1995)

Morphin

(

Krotkov

et al., 1996)

Gestalt (Goldberg,

Maimone

Matthies

, 2002)Slide34

EgographLocal/Global arrangement

Imposing DiscretizationPre-computed search spaceTree depth: 517 state samples per level7 segments4 velocities

19 curvatures

Lacaze

et al., 1998Slide35

OutlineIntroduction

Deterministic PlanningHierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMs

RRTs

Derandomized

Planners

Some applications

35Slide36

Path Post-Processing36

Lamiraux et al., 2002

Laumond

, Jacobs,

Taix

, Murray, 1994

Khatib

Jaouni

Chatila

Laumond

, 1997Slide37

Topological Property37

Slide38

OutlineIntroduction

Deterministic PlanningHierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMs

RRTs

Derandomized

Planners

Some applications

38Slide39

Control Space Sampling

Barraquand

Latombe

, 1993

Lindemann

LaValle

, 2006

Kammel

et al., 2008



Barraquand

Latombe

3 arcs (+ reverse) at



max

Discontinuous curvature

Cost = number of reversals

Dijkstra’s

searchSlide40

Robot-Fixed Search Space

Moves with the robotDense samplingPositionSymmetric sampling

Heading

Velocity

Steering angle

…

Tree depth

1: Local (arcs) + Global (D*) (

Stentz

& Hebert, 1994)5: Egograph (Lacaze

et al., 1998)∞: Barraquand & Latombe

(1993)40

40Slide41

OutlineIntroduction

Deterministic PlanningHierarchicalPath SmoothingControl SamplingState SamplingRandomized Planning

PRMs

RRTs

Derandomized

Planners

Some applications

41Slide42

World-Fixed Search Space

Fixed to the world

Dense sampling

(none)

Symmetric sampling

Position

Heading

Velocity

Steering angle…

DependencyBoundary value problem

Pivtoraiko

& Kelly, 2005

Examples of BVP solvers

: Dubins

, 1957 Reeds & Shepp, 1990

Lamiraux & Laumond

, 2001 Kelly & Nagy, 2002 Pancanti et al., 2004

Kelly & Howard, 2005Slide43

Robot-Fixed vs. World-Fixed43

Barraquand & Latombe

CONTROL

STATE

CONTROL

STATE

State LatticeSlide44

State Lattice Benefits44

State LatticeRegularity in state samplingPosition invariance

Pivtoraiko

& Kelly, 2005Slide45

Path Swaths45

Pivtoraiko

& Kelly, 2007

State Lattice

Regularity

Position invariance

Benefits

Pre-computing path swathsSlide46

World Fixed State Lattice46

HLUT

Pivtoraiko

& Kelly, 2005

Knepper

& Kelly, 2006

State Lattice

Regularity

Position invariance

Benefits

Pre-computing path swaths

Pre-computing heuristicsSlide47

World Fixed State Lattice47

State Lattice

Regularity

Position invariance

Benefits

Pre-computing path swaths

Pre-computing heuristics

Dynamic

replanningSlide48

World Fixed State Lattice48

State Lattice

Regularity

Position invariance

Benefits

Pre-computing path swaths

Pre-computing heuristics

Dynamic

replanningSlide49

Nonholonomic D*

Expanded States

Motion Plan

Perception Horizon

Graphics: Thomas Howard

49Slide50

Nonholonomic D*50

Pivtoraiko & Kelly, 2007Graphics: Thomas HowardSlide51

Boss“Parking lot” plannerRegular 4D state sampling

Pre-computed search spaceDepth: unlimitedMulti-resolution32 (16) headings2 velocitiesNo curvature

LIkhachev

et al., 2008Slide52

World Fixed State LatticeState LatticeRegularity

Position invarianceBenefitsPre-computing path swathsPre-computing heuristicsDynamic replanning

Dynamic search space

Search graph





…



Pivtoraiko

& Kelly, 2008Slide53

Dynamic Search Space53

Pivtoraiko & Kelly, 2008Graphics: Thomas HowardSlide54

Dynamic Search Space54Slide55

World Fixed State Lattice

START

GOAL

State Lattice

Regularity

Position invariance

Benefits

Pre-computing path swaths

Pre-computing heuristics

Dynamic

replanning

Dynamic search space

Parallelized search

Pivtoraiko

& Kelly, 2010Slide56

World Fixed State Lattice

START

GOAL

Pivtoraiko

& Kelly, 2010Slide57

START

GOAL

World Fixed State Lattice

Pivtoraiko

& Kelly, 2010

epsilon

tree size ratioSlide58

Search Space Comparison58

Robot-FixedPros: Any motion generation schemeCons:

NO Pre-computing path swaths

NO Pre-computing heuristics

NO Parallelized search

NO Dynamic

replanning

NO Dynamic search space

World-Fixed

Pros:

Pre-computing path swaths Pre-computing heuristics

Parallelized search Dynamic replanning Dynamic search space

Cons: Boundary value problemSlide59

OutlineIntroductionDeterministic Planning

HierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMsRRTs

Derandomized

Planners

Some applications

59Slide60

A Few Randomized PlannersProbabilistic Roadmaps (PRM)Kavraki

, Svestka, Latombe & Overmars, 1996Expansive

Space Tree (EST)

Hsu,

Kindel

Latombe

& Rock, 2001

Rapidly-Exploring Random Tree (RRT)

LaValle & Kuffner, 2001R* Search

Likhachev & Stentz, 2008

LaValle

Kuffner, 2001Slide61

Probabilistic RoadmapStatic workspacesE.g. industrial workcells

Two phases:Learning: construct the roadmapQuery: actually planStructure: undirected graph

Originally applied to

holonomic

robots

61Slide62

Learning PhaseTwo steps:ConstructionConstructs edges and vertices to cover free

C-space uniformlyExpansionTries to detect “difficult” regions and samples them more densely62Slide63

Construction StepTwo sub-steps:Sample a random configuration and add to the graph

Select n neighbor vertices and (try to) connect to the new vertexComponentsDistance metricLocal plannerConnections between vertices

Kavraki

Svestka

Latombe

Overmars

, 1996Slide64

Query PhaseGraph is constructedApply any shortest-path graph search Smoothing:

Find “shortcuts”Local planner is reused64

Kavraki

Svestka

Latombe

Overmars

, 1996Slide65

OutlineIntroductionDeterministic Planning

HierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMs

RRTs

Derandomized

Planners

Some applications

65Slide66

Original RRTRapidly-Exploring Random TreeProposed for kino

-dynamic planningHolonomic randomized planners existed beforeProposed to meet the need for randomized planners under differential constraintsToday, a work-horse for randomized searchProbabilistically complete

No optimality guarantees

Avoids “curse of dimensionality”Slide67

RRT in a NutshellGiven a tree (initially only x_{init})Pick a sample, x, in state space, X

RandomlySample uniform distribution over XFind nearest neighbor tree node, x_{near}, to xFind a control that approaches x_{near}Unless you solve the BVP problem, won’t approach exactlyJust do your best; you’ll arrive at x_{new} near to x

Add x_{new} and the edge (x_{near}, x_{new}) to the tree

RepeatSlide68

RRT OriginsKhatib, IJRR

5(1), 1986Potential fields for obstacle avoidance, mobile robots and manipulatorsBarraquand & Latombe, IJRR 10(6), 1993

Builds and searches a graph connecting local minima of a potential field

Monte-Carlo technique to escape local minima via

Brownian

motions

Kavraki

Svestka

, Latombe & Overmars, Transactions 12(4), 1996Probabilistic roadmaps

Randomly generate a graph in a configuration spaceMulti-query method, best for fixed manipulators

Hsu, Latombe & Motwani, Int. J. of

Comput. Geometry & App., 1997

Expansive Space Tree (EST), single-query methodChoose tree node to extend via biased probability measure

Apply random controlCollision checking in state*time spaceSlide69

Rapidly-ExploringVoronoi bias

Sampling uniform distribution over state spaceLarge empty regions have higher probability of being sampledHence, tree “prefers” growing into empty regions

Naïve random tree

RRTSlide70

Voronoi BiasSlide71

AnalysisConvergence to solutionProbability of failure (to find solution)

decreases exponentially with the number of iterationsProbabilistic completenessSlide72

ExamplesSlide73

Quick overview of CBiRRTConstrained

Bi-directional RRTStart with “unconstrained” RRTAssume some constraints, e.g.:End-effector poseTorque

For each sample, x_{rand}, in X

Get x_{new} by tweaking x_{rand} until constraints satisfied

… via optimization (gradient descent)

In text, “project sample onto constraint manifold”Slide74

Growing the TreeExtending tree toward random sampleThe sample is q_{target}

Nearest neighbor: q_{near}Step from q_{near} to q_{target}In state spaceProject each step onto constraint manifoldUntil you reach q_{target}’s projectionReturn, if can’t continue, e.g. Obstacles

Can’t projectSlide75

Results

5kg

6kg

8kgSlide76

OutlineIntroductionDeterministic Planning

HierarchicalPath SmoothingControl SamplingState SamplingRandomized PlanningPRMsRRTs

Derandomized

Planners

Some Applications

76Slide77

Randomness for PlanningPros:Allows rapid exploration of state space

(via uniform sampling)Less susceptible to local minimaCons:Inability to provide performance guaranteesObscures other useful features of plannersMoreover:

There are deterministic incremental sampling methods

E.g.,

Halton

points

Van

der

Corput

sequence (1935); generalized to multiple dimensions by Halton.Slide78

Derandomized RRTLet’s implement

Voronoi bias explicitlyGiven treeCompute Voronoi diagram wrt its nodesPick the sample to extend toward:

Centroid

of largest