The Recent Impact of QMC Methods on Robot Motion Planning

Steven M. LaValle Stephen R. LindemannAnna Yershova

Dept. of Computer ScienceUniversity of Illinois Urbana, IL, USA

The Goal of the Talk

Introduce the MC2QMC community to the problem of robot motion planning

Survey the state of the art of the sampling techniques in Motion Planning

Discussion on the unique challenges and open problems that arise in Robotics

Talk Overview

Motion Planning Problem

QMC Philosophy in Motion Planning

A Spectrum of Planners: from Grids to Random Roadmaps

Connecting Difficulty of Motion Planning with Sampling Quality

QMC techniques and extensible lattices in the Motion Planning Planners

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

Classical Motion Planning Problem ”Moving Pianos”

Typical Configuration Spaces Translations in 2D, 3D – 2, 3

Rotations in 2D – S1

Rotations in 3D – SO(3) Motions of chains of 2D objects – (S 1)n

Motion of 3D chains – depending on the type of joints – SE(3) x S 1 x … Motions of closed chains – algebraic variaties Motions of multiple robots – (SE(3))n

Humanoid type robot performing manipulation tasks – up to 100 dimension configuration space containing a multiple copies of all the above

Obstacles in these spaces represent collisions (with obstacles, self-collisions and collisions with other robots)

Applications of Motion Planning

(Coordinated) ManipulationPlanning

Computational Chemistryand Biology

Medical applications

Computer Graphics(motions for digital actors)

Autonomous vehicles and spacecrafts

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)

Exact 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)

Probabilistic Roadmaps (PRMs)Kavraki, Latombe, Overmars, Svestka, 1994

Developed for high-dimensional spaces Avoid pitfalls of classical grid search Random sampling of Cfree

Find neighbors of each sample(radius parameter)

Local planner attempts connections “Probabilistic completeness" achieved

Other PRM variants: Obstacle-Based PRM (Amato, Wu, 1996); Sensor-based PRM (Yu, Gupta, 1998); Gaussian PRM (Boor, Overmars, van der Stappen, 1999); Medial axis PRMs (Wilmarth, Amato, Stiller, 1999; Pisula, Ho, Lin, Manocha, 2000; Kavraki, Guibas, 2000); Contact space PRM (Ji, Xiao, 2000); Closed-chain PRMs (LaValle, Yakey, Kavraki, 1999; Han, Amato 2000); Lazy PRM (Bohlin, Kavraki, 2000); PRM for changing environments (Leven, Hutchinson, 2000); Visibility PRM (Simeon, Laumond, Nissoux, 2000).

Rapidly-Exploring Random Trees (RRTs)LaValle, Kuffner, 1998

Other RRT variants: Frazzoli, Dahleh, Feron, 2000; Toussaint, Basar, Bullo, 2000; Vallejo, Jones, Amato, 2000; Strady, Laumond, 2000; Mayeux, Simeon, 2000; Karatas, Bullo, 2001; Li, Chang, 2001; Kuner, Nishiwaki, Kagami, Inaba, Inoue, 2000, 2001; Williams, Kim, Hofbaur, How, Kennell, Loy, Ragno, Stedl, Walcott, 2001; Carpin, Pagello, 2002; Urmson, Simmons, 2003.

movie

Talk Overview

Motion Planning Problem

QMC Philosophy in Motion Planning

A Spectrum of Planners: from Grids to Random Roadmaps

Connecting Difficulty of Motion Planning with Sampling Quality

QMC techniques and extensible lattices in the Motion Planning Planners

Conclusions and Discussion

QMC Philosophy

From 1989-2000 most of the community contributed planning success to randomization

Questions: Is randomization really the reason why challenging problems have

been solved? Is random sampling in PRM advantageous?

Approach: Recognize that all machine implementations of random numbers

produce deterministic sequences View sampling as an optimization problem Define criterion, and choose samples that optimize it for an

intended application

Talk Overview

Motion Planning Problem

QMC Philosophy in Motion Planning

A Spectrum of Planners: from Grids to Random Roadmaps

Connecting Difficulty of Motion Planning with Sampling Quality

QMC techniques and extensible lattices in the Motion Planning Planners

Conclusions and Discussion

Probabilistic RoadmapsKavraki, Latombe, Overmars, Svestka, 1994

Developed for high-dimensional spaces Avoid pitfalls of classical grid search Random sampling of Cfree

Find neighbors of each sample(radius parameter)

Local planner attempts connections “Probabilistic completeness" achieved

Other PRM variants: Obstacle-Based PRM (Amato, Wu, 1996); Sensor-based PRM (Yu, Gupta, 1998); Gaussian PRM (Boor, Overmars, van der Stappen, 1999); Medial axis PRMs (Wilmarth, Amato, Stiller, 1999; Pisula, Ho, Lin, Manocha, 2000; Kavraki, Guibas, 2000); Contact space PRM (Ji, Xiao, 2000); Closed-chain PRMs (LaValle, Yakey, Kavraki, 1999; Han, Amato 2000); Lazy PRM (Bohlin, Kavraki, 2000); PRM for changing environments (Leven, Hutchinson, 2000); Visibility PRM (Simeon, Laumond, Nissoux, 2000).

A Spectrum of Roadmaps Random Samples Halton sequence

Hammersley Points Lattice Grid

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

Questions

What uniformity criteria are best suited for Motion Planning

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

Talk Overview

Motion Planning Problem

QMC Philosophy in Motion Planning

A Spectrum of Planners: from Grids to Random Roadmaps

Connecting Difficulty of Motion Planning with Sampling Quality

QMC techniques and extensible lattices in the Motion Planning Planners

Conclusions and Discussion

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

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

Talk Overview

Motion Planning Problem

QMC Philosophy in Motion Planning

A Spectrum of Planners: from Grids to Random Roadmaps

Connecting Difficulty of Motion Planning with Sampling Quality

QMC techniques and extensible lattices in the Motion Planning Planners

Conclusions and Discussion

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]

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

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

Experimental Results

Random sequence produces slightly more node in the roadmap than QMC sequences and the Layered Sukharev Grid sequence

The amount of the computation is saved in Layered Sukharev Grid sequence due to efficient generation and fast nearest neighbor search

All the improvements observed so far are not very significant

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

Discussion

Are there sequences that will give a significant superior performance for motion planning?

How to develop importance sampling sequences?

How to develop deterministic techniques for sampling over general topological spaces that arise in motion planning?

What to do in higher dimensions?

How to derandomize other motion planning algorithms?