Sampling and Searching Methods for Practical Motion Planning Algorithms

Sampling and Searching

Methods for Practical Motion

Planning Algorithms

Anna Yershova

PhD Preliminary Examination

Dept. of Computer Science

University of Illinois

29 August 2007

Presentation Overview Motion Planning Problem

Basic Motion Planning Problem Extensions of Basic Motion Planning Motion Planning under Differential Constraints

State of the Art

Thesis Statement

Technical Approach Efficient Nearest Neighbor Searching Uniform Deterministic Sampling Methods Guided Sampling for Efficient Exploration Motion Primitives Generation

Conclusions and Discussion

Given: (geometric model of a robot) (space of configurations, q, that

are applicable to ) (the set of collision free

configurations) Initial and goal configurations

Task: Compute a collision free path that connects initial and

goal configurations

Basic Motion Planning Problem ”Moving Pianos”

Given:

, , (kinematic closure

constraints) Initial and goal configurations


goal configurations

Extensions of Basic Motion Planning Problem

Given: , , State space X Input space U state transition

equation Initial and goal states


goal states

Motion Planning Problemunder Differential Constraints



State of the Art

Thesis Statement



History of Motion Planning Grid Sampling, AI Search (beginning of time-1977)

Experimental mobile robotics, etc.

Problem Formalization (1977-1983) PSPACE-hardness (Reif, 1979) Configuration space (Lozano-Perez, 1981)

Combinatorial Solutions (1983-1988) Cylindrical algebraic decomposition (Schwartz, Sharir, 1983) Stratifications, roadmap (Canny, 1987)

Sampling-based Planning (1988-present) Randomized potential fields (Barraquand, Latombe, 1989) Ariadne's clew algorithm (Ahuactzin, Mazer, 1992) Probabilistic Roadmaps (PRMs) (Kavraki, Svestka, Latombe, Overmars,

1994) Rapidly-exploring Random Trees (RRTs) (LaValle, Kuffner, 1998)

Applications of Motion Planning

Manipulation Planning

Computational Chemistryand Biology

Medical applications

Computer Graphics(motions for digital actors)

Autonomous vehicles and spacecrafts



State of the Art

Thesis Statement



Sampling and Searching Framework

Build a graph over the state (configuration) space that connects initial state to the goal:

INITIALIZATION

SELECTION METHOD

LOCAL PLANNING METHOD

INSERT AN EDGE IN THE GRAPH

CHECK FOR SOLUTION

RETURN TO STEP 2

xbest

xinit

xnew

Thesis Statement

The performance of motion planning algorithms can be significantly improved by careful consideration of sampling issues.

ADDRESSED ISSUES:

STEP 2: nearest neighbor computation

STEP 2: uniform sampling over configuration space

STEPS 2,3: guided sampling for exploration

STEP 3: motion primitives generation



State of the Art

Thesis Statement



State of Progress

100% Efficient Nearest Neighbor Searching

85% Uniform Deterministic Sampling Methods

75% Guided Sampling for Efficient Exploration

20% Motion Primitives Generation

MPNN: Nearest Neighbor Library For Motion Planning

Publications: Improving Motion Planning Algorithms by Efficient Nearest Neighbor

Searching Anna Yershova and Steven M. LaValleIEEE Transactions on Robotics 23(1):151-157, February 2007

Efficient Nearest Neighbor Searching for Motion PlanningAnna Yershova and Steven M. LaValleIn Proc. IEEE International Conference on Robotics and Automation (ICRA 2002)

Software: http://msl.cs.uiuc.edu/~yershova/mpnn/mpnn.tar.gz

Problem FormulationGiven a d-dimensional manifold, T, and a set of data points in T.

Preprocess these points so that, for any query point q T, the nearest data point to q can be found quickly.

The manifolds of interest: Euclidean one-space, represented by (0,1) R . Circle, represented by [0,1], in which 0 1 by identification. P3, represented by S3 with antipodal points identified.

Examples of topological spaces:

cylinder torus projective plane

Example: a torus

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Kd-trees

The kd-tree is a powerful data structure that is based on recursively subdividing a set of points with alternating axis-aligned hyperplanes.

The classical kd-tree uses O(dn lgn) precomputation time, O(dn) space and answers queries in time logarithmic in n, but exponential in d.

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State of the Art

Thesis Statement



Library For Generating Deterministic

Sequences Of Samples Over SO(3) Publications: Deterministic sampling methods for spheres and SO(3)

Anna Yershova and Steven M. LaValle,2004 IEEE International Conference on Robotics and Automation (ICRA 2004)

Incremental Grid Sampling Strategies in Robotics Stephen R. Lindemann, Anna Yershova, and Steven M. LaValle,Sixth International Workshop on the Algorithmic Foundations of Robotics (WAFR 2004)

Software: http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

A Spectrum of Roadmaps Random Samples Halton sequence

Hammersley Points Lattice Grid

Questions

What uniformity criteria are best suited for Motion Planning

Which of the roadmaps alone the spectrum is best suited for Motion Planning?

Measuring the (Lack of) Quality Let R (range space) denote a collection of subsets of a

sphere Discrepancy: “maximum volume estimation error over

all boxes”

Measuring the (Lack of) Quality Let denote metric on a sphere Dispersion: “radius of the largest empty ball”

The Goal for Motion Planning

We want to develop sampling schemes with the following properties:

uniform (low dispersion or discrepancy) lattice structure incremental quality (it should be a sequence) on the configuration spaces with different topologies

Layered Sukharev Grid Sequencein d

Places Sukharev grids one resolution at a time

Achieves low dispersion at each resolution

Achieves low discrepancy

Has explicit neighborhoodstructure

[Lindemann, LaValle 2003]

Layered Sukharev Grid Sequence for Spheres

Take a Layered Sukharev Grid sequence inside each face Define the ordering on faces Combine these two into a sequence on the sphere

Ordering on faces +Ordering inside faces



State of the Art

Thesis Statement



Dynamic-Domain RRTs Publications: Planning for closed chains without inverse kinematics

Anna Yershova and Steven M. LaValle, To be submitted to ICRA 2008

Adaptive Tuning of the Sampling Domain for Dynamic-Domain RRTsL. Jaillet, A. Yershova, S. M. LaValle and T. Simeon, In Proc. IEEE International Conference on Intelligent Robots and Systems (IROS 2005)

Dynamic-Domain RRTs: Efficient Exploration by Controlling the Sampling DomainA. Yershova, L. Jaillet, T. Simeon, and S. M. LaValle, In Proc. IEEE International Conference on Robotics and Automation (ICRA 2005)

Bug Trap

Which one will perform better?

Small Bounding Box Large Bounding Box

Voronoi Bias for the Original RRT

KD-Tree Bias for the RRT





State of the Art

Thesis Statement



Motion Primitives Generation

Reachability graph

Dubin’s Car Reachability Graph

Motion Primitives Generation Numerical integration can be costly for complex control

models.

In several works it has been demonstrated that the performance of motion planning algorithms can be improved by orders of magnitude by having good motion primitives

Motion Primitives Generation Motivating example 1:

Autonomous Behaviors for Interactive Vehicle Animations

Jared Go, Thuc D. Vu, James J. Kuffner

Generated spacecraft trajectories in a field of moving asteroid obstacles.


Criteria: Hand-picked “pleasing to the eye” trajectories Efficient performance of the online planner


Optimal, Smooth, Nonholonomic Mobile Robot Motion Planning in State Lattices

M. Pivtoraiko, R.A. Knepper, and A. Kelly


The controls are chosen to reach the points on the state lattice

Criteria: Well separated

trajectories Efficiency in

performance

Motivational Literature

Robotics literature:

[Kehoe, Watkins, Lind 2006] [Anderson, Srinivasa 2006] [Pivtoraiko, Knepper, Kelly 2006] [Green, Kelly 2006] [Go, Vu, Kuffner 2004] [Frazzoli, Dahleh, Feron 2001]

Motion Capture literature

[Laumond, Hicheur, Berthoz 2005] [Gleicher]

Proposed problem

Formulate the criteria of “goodness” for motion primitives in the context of Motion Planning

Automatically generate the motion primitives

Propose Efficient Motion Planning algorithms using the motion primitives

Things to investigate:

Dispersion, discrepancy in state space? In trajectory space? Robustness with respect to the obstacles? Complexity of the set of trajectories? Is it extendable to second order systems?

Thank you!

Appendix

Kd-trees. Construction

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Kd-trees. Query

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Algorithm Presentation

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Analysis of the Algorithm

Proposition 1. The algorithm correctly returns the nearest neighbor.

Proof idea: The points of kd-tree not visited by an algorithm will always be further from the query point then some point already visited.

Proposition 2. For n points in dimension d, the construction time is O(dn lgn), the space is O(dn), and the query time is logarithmic in n, but exponential in d.

Proof idea: This follows directly from the well-known complexity of the basic kd-tree.

A Spectrum of Planners Grid-Based Roadmaps (grids, Sukharev grids) []

optimal dispersion; poor discrepancy; explicit neighborhood structure

Lattice-Based Roadmaps (lattices, extensible lattices) optimal dispersion; near-optimal discrepancy; explicit neighborhood

structure

Low-Discrepancy/Low-Dispersion (Quasi-Random) Roadmaps (Halton sequence, Hammersley point set) optimal dispersion and discrepancy; irregular neighborhood structure

Probabilistic (Pseudo-Random) Roadmaps non-optimal dispersion and discrepancy; irregular neighborhood structure

Literature: 1916 Weyl; 1930 van der Corput; 1951 Metropolis; 1959 Korobov; 1960 Halton, Hammersley; 1967 Sobol'; 1971 Sukharev; 1982 Faure; 1987 Niederreiter; 1992 Niederreiter; 1998 Niederreiter, Xing; 1998 Owen, Matousek;2000 Wang, Hickernell

Connecting Sample Quality to Problem Difficulty

Problem Quality Measure

Difficulty Measure

Theoretical Bound

integration discrepancy bounded Hardy-Krause variation

Koksma-Hlawka inequality

optimization dispersion modulus of continuity

[N92]

motion planning dispersion corridor thickness

our analysis

Decidability of Configuration Spaces

x

Undecidability Results

Comparing to Random Sequences

Sequences for SO(3)Important points: Uniformity depends on the parameterization.

Haar measure defines the volumes of the sets in the space, so that they are invariant up to a rotation

The parameterization of SO(3) with quaternions respects the unique (up to scalar multiple) Haar measure for SO(3)

Quaternions can be viewed as all the points lying on S 3 with the antipodal points identified

Notions of dispersion and discrepancy can be extended to the surface of the sphere

Close relationship between sampling on spheres and SO(3)

Sukharev Grid on S d

Take a cube in Rd+1

Place Sukharev grid on each face Project the faces of the cube outwards to form spherical tiling Place a Sukharev grid on each spherical face

Conclusions

Random sampling in the PRMs seems to offer no advantages over the deterministic sequences

Deterministic sequences can offer advantages in terms of dispersion, discrepancy and neighborhood structure for motion planning

The RRT Construction Algorithm

GENERATE_RRT(xinit, K, t)

1. T.init(xinit);

2. For k = 1 to K do

3. xrand RANDOM_STATE();

4. xnear NEAREST_NEIGHBOR(xrand, T);

5. if CONNECT(T, xrand, xnear, xnew);

6. T.add_vertex(xnew);

7. T.add_edge(xnear, xnew, u);

8. Return T;

xnear

xinit

xnew

The result is a tree rooted at xinit

A Rapidly-exploring Random Tree (RRT)

Voronoi Biased Exploration

Is this always a good idea?

Voronoi Diagram in R 2



Refinement vs. Expansion

refinement expansion

Where will the random sample fall? How to control the behavior of RRT?

Limit Case: Pure Expansion

Let X be an n-dimensonal ball,

in which r is very large.

The RRT will explore n 1 opposite directions.

The principle directions are vertices of a regular n 1-simplex

Determining the Boundary

Expansion dominates Balanced refinement and expansion

The tradeoff depends on the size of the bounding box

Controlling the Voronoi Bias

Refinement is good when multiresolution search is needed

Expansion is good when the tree can grow and not blocked by obstacles

Main motivation:

Voronoi bias does not take into account obstacles

How to incorporate the obstacles into Voronoi bias?

Voronoi Bias for the Original RRT

Visibility-Based Clipping of the Voronoi Regions

Nice idea, but how can this be done in practice?Even better: Voronoi diagram for obstacle-based metric

(a) Regular RRT, unbounded Voronoi region

(b) Visibility region

(c) Dynamic domain

A Boundary Node

A Non-Boundary Node

(a) Regular RRT, unbounded Voronoi region

(b) Visibility region

(c) Dynamic domain

Dynamic-Domain RRT Bias

Dynamic-Domain RRT Construction

Dynamic-Domain RRT Bias

Tradeoff between nearest neighbor calls and collision detection calls

Recent Efforts

Adaptive tuning of the radius: the radius is not fixed but is increased with every extension

success and is decreased with every failure

Nearest neighbor calls: kd-tree based implementation O(log n) instead of naïve O(n) query time

Uniform sampling from dynamic domain: Rejection-based method is not efficient for high dimensions Uniform distribution should be generated directly

Adaptive Tuning of Parameter

Adaptive Tuning of Parameter


Real-Time Motion Planning For Agile Autonomous Vehicles (2000)

Emilio Frazzoli, Munther A. Dahleh, Eric Feron

Sampling and Searching Methods for Practical Motion Planning Algorithms

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