sampling and searching methods for practical motion planning algorithms anna yershova dept. of...
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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
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
Task: Compute a collision free path that connects initial and
goal configurations
Extensions of Basic Motion Planning Problem
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
Motion Planning Problemunder Differential Constraints
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 Chemistryand 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
xinit
xnew
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 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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Library For Generating Deterministic
Sequences Of Samples Over SO(3)
Software: http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz
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
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?