Sampling Methods in Robot Motion Planning Steven M. LaValle Stephen R. Lindemann Anna Yershova Dept. of Computer Science University of Illinois Urbana, IL, USA

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”

History of Motion Planning Grid Sampling, AI Search (beginning of time-1977)  Experimental mobile robotics, etc. Problem Formalization ( )  PSPACE-hardness (Reif, 1979)  Configuration space (Lozano-Perez, 1981) Complete Solutions ( )  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 C free 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, 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 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

QMC Applications Optimization problem (finding a maximum of a function):  given: continuous real function, f, defined on [0, 1] d  solution: take a point sequence ( x n )  [0, 1] d, define m 1 = f(x 1 ), and recursively set: Integration problem in higher dimensions (finding average):  given: continuous real function, f, defined on [0, 1] d  solution: QMC methods proved to be very successful in Computer Graphics:  mental images was awarded a Technical Achievement Academy Award (Oscar) for developing a rendering software in such movies as “The Matrix”, “Spider Man”, “Harry Potter”….

Basic Definitions Sample types over : Literature landmarks: 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

Measuring the (Lack of) Quality Global quality measure, used for integration:

Measuring the (Lack of) Quality Local quality measure, used for optimization:

Optimal Sequences and Point Sets Low discrepancy sequence: Low discrepancy point set: Low dispersion sequence/point set: Implied constants may be big, for example for dispersion: Low discrepancy implies good dispersion, but not necessarily optimal

Sukharev Sampling Criterion

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 Roadmaps Kavraki, Latombe, Overmars, Svestka, 1994 Developed for high-dimensional spaces Avoid pitfalls of classical grid search Random sampling of C free 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 ProblemQuality Measure Difficulty Measure Theoretical Bound integrationdiscrepancybounded Hardy- Krause variation Koksma-Hlawka inequality optimizationdispersionmodulus of continuity [N92] motion planningdispersioncorridor 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 Sequence in  d Places Sukharev grids one resolution at a time Achieves low dispersion at each resolution Achieves low discrepancy Has explicit neighborhood structure [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 R d+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 for PRMs

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

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, movie

What is the Role of Sampling in RRTs? [Lindemann, LaValle 2004] Random samples induce Voronoi bias exploration in RRTs Is this the best way to approximate the Voronoi regions? Attempts to design other sampling techniques:  use k samples at each iteration to estimate the vertex with the biggest Voronoi region  reuse these k samples for some number of iterations  deterministic samples can be used

What is the Role of Sampling in RRTs? Produces less nodes, less collision checks Not numerically robust Computations are still expensive

Discussion s Are there sequences that will give a significant superior performance for motion planning? How to develop deterministic techniques for sampling over general topological spaces that arise in motion planning? What to do in higher dimensions? Are there advantages in derandomizing other motion planning algorithms?

Discussions How to develop stratified and adaptive sampling for motion planning?