1. The Recent Impact of QMC Methods on Robot Motion Planning Steven M. LaValle Stephen R. Lindemann Anna Yershova Dept. of Computer Science University of Illinois Urbana, IL, USA

2. The Goal of the Talk  Introduce the MC 2 QMC 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

3. 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

4. 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”

5. Typical Configuration Spaces  Translations in 2D, 3D – 2, 3  Rotations in 2D – S 1  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)

6. Applications of Motion Planning (Coordinated) Manipulation Planning Computational Chemistry and Biology Medical applications Computer Graphics (motions for digital actors) Autonomous vehicles and spacecrafts

7. 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)

8. 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).

9. 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

10. 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

11. 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

12. 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

13. 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).

14. A Spectrum of Roadmaps Random Samples Halton sequence Hammersley Points Lattice Grid

15. 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

16. Questions  What uniformity criteria are best suited for Motion Planning  Which of the roadmaps alone the spectrum is best suited for Motion Planning?

17. 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

18. 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

19. Decidability of Configuration Spaces x

20. Undecidability Results

21. Comparing to Random Sequences

22. 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

23. 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

24. 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]

25. 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)

26. 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

27. 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

28. 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

29. 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

30. 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?