Anna Yershova Dept. of Computer Science University of Illinois Sampling and Searching Methods for Practical Motion Planning Algorithms Anna Yershova Dept. of Computer Science University of Illinois

Presentation Overview Motion Planning Problem Basic Motion Planning Problem Extensions of Basic Motion Planning Motion Planning under Differential Constraints State of the Art Research Statement Technical Approach Efficient Nearest Neighbor Searching Guided Sampling for Efficient Exploration Uniform Deterministic Sampling Methods Motion Primitives Generation Conclusions and Discussion

Basic Motion Planning Problem ”Moving Pianos” 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

Extensions of Basic Motion Planning Problem Given: , , (kinematic closure constraints) Initial and goal configurations Task: Compute a collision free path that connects initial and goal configurations

Motion Planning Problem under Differential Constraints Given: , , State space X Input space U state transition equation Initial and goal states Task: Compute a collision free path that connects initial and goal states

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

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 Chemistry and Biology Medical applications Computer Graphics (motions for digital actors) Autonomous vehicles and spacecrafts

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 xnew xinit

Research 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

Nearest Neighbor Searching for Motion Planning Software: http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

Problem Formulation The manifolds of interest: Given 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 4 7 6 5 1 3 2 9 8 10 11 4 6 7 q 8 5 9 10 3 1 2 11

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. l1 4 7 6 5 1 3 2 9 8 10 11 l5 l1 l9 l6 l3 l10 l7 l4 l8 l2 l2 l3 l4 l5 l7 l6 l8 2 5 4 11 l10 8 l9 1 3 9 10 6 7

Dynamic-Domain RRTs

Small Bounding Box Large Bounding Box Bug Trap In order to investigate this we consider a toy example. A 3-d bug trap is a device for catching bugs. If the food (something sweet) is put inside the trap, the bug can get easily inside, but can never leave the trap. Already by constraining the boundary would improve the performance, since…. However, let’s consider Voronoi regions more carefully. Small Bounding Box Large Bounding Box Which one will perform better?

Voronoi Bias for the Original RRT

KD-Tree Bias for the RRT

KD-Tree Bias for the RRT

KD-Tree Bias for the RRT

Library For Generating Deterministic Sequences Of Samples Over SO(3) 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 Sequence in [0, 1]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]

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

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.

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

Motion Primitives Generation Motivating example 2: Optimal, Smooth, Nonholonomic Mobile Robot Motion Planning in State Lattices M. Pivtoraiko, R.A. Knepper, and A. Kelly

Motion Primitives Generation 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!