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Sampling and Searching Methods for Practical Motion Planning Algorithms

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29 August 2007

Sampling and Searching Methods for Practical Motion Planning Algorithms

Anna Yershova

PhD Preliminary Examination

Dept. of Computer Science

University of Illinois

- 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

Basic Motion Planning Problem”Moving Pianos”

Given:

- (geometric model of a robot)
- (space of configurations, q, thatare applicable to )
- (the set of collision freeconfigurations)
- 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 Problemunder 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

- 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

- 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

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

Thesis 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

- 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

100%Efficient Nearest Neighbor Searching

85% Uniform Deterministic Sampling Methods

75% Guided Sampling for Efficient Exploration

20% Motion Primitives Generation

Publications:

- Improving Motion Planning Algorithms by Efficient Nearest Neighbor Searching Anna Yershova and Steven M. LaValleIEEE Transactions on Robotics 23(1):151-157, February 2007
- Efficient Nearest Neighbor Searching for Motion PlanningAnna Yershova and Steven M. LaValleIn Proc. IEEE International Conference on Robotics and Automation (ICRA 2002)
Software:http://msl.cs.uiuc.edu/~yershova/mpnn/mpnn.tar.gz

Given a d-dimensional manifold, T, and a set of data points in T.

Preprocess these points so that, for any query point qT, 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

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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(dnlgn) precomputation time, O(dn) space and answers queries in time logarithmic in n, but exponential in d.

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- 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

Publications:

- Deterministic sampling methods for spheres and SO(3) Anna Yershova and Steven M. LaValle,2004 IEEE International Conference on Robotics and Automation (ICRA 2004)
- Incremental Grid Sampling Strategies in Robotics Stephen R. Lindemann, Anna Yershova, and Steven M. LaValle,Sixth International Workshop on the Algorithmic Foundations of Robotics (WAFR 2004)
Software:http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

Random Samples Halton sequence

Hammersley Points Lattice Grid

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 [0, 1]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 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

Publications:

- Planning for closed chains without inverse kinematicsAnna Yershova and Steven M. LaValle, To be submitted to ICRA 2008
- Adaptive Tuning of the Sampling Domain for Dynamic-Domain RRTsL. Jaillet, A. Yershova, S. M. LaValle and T. Simeon, In Proc. IEEE International Conference on Intelligent Robots and Systems (IROS 2005)
- Dynamic-Domain RRTs: Efficient Exploration by Controlling the Sampling DomainA. Yershova, L. Jaillet, T. Simeon, and S. M. LaValle, In Proc. IEEE International Conference on Robotics and Automation (ICRA 2005)

Which one will perform better?

Small Bounding Box Large Bounding Box

- 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

Reachability graph

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

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.

Criteria:

- Hand-picked “pleasing to the eye” trajectories
- Efficient performance of the online planner

Motivating example 2:

Optimal, Smooth, Nonholonomic Mobile Robot Motion Planning in State Lattices

M. Pivtoraiko, R.A. Knepper, and A. Kelly

The controls are chosen to reach the points on the state lattice

Criteria:

- Well separated trajectories
- Efficiency in performance

- 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]

- 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

- 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?

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Proposition 1. The algorithm correctly returns the nearest neighbor.

Proof idea:The points of kd-tree not visited by an algorithm will always be further from the query point then some point already visited.

Proposition 2. For n points in dimension d, the construction time is O(dnlgn), the space is O(dn), and the query time is logarithmic in n, but exponential in d.

Proof idea:This follows directly from the well-known complexity of the basic kd-tree.

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

x

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 Sd

- 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

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

GENERATE_RRT(xinit, K, t)

- T.init(xinit);
- Fork = 1 toKdo
- xrand RANDOM_STATE();
- xnear NEAREST_NEIGHBOR(xrand, T);
- if CONNECT(T, xrand, xnear, xnew);
- T.add_vertex(xnew);
- T.add_edge(xnear, xnew, u);
- Return T;

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The result is a tree rooted at xinit

Is this always a good idea?

refinement

expansion

Where will the random sample fall? How to control the behavior of RRT?

- Let X be an n-dimensonal ball,
in which r is very large.

- The RRT will explore n+ 1 opposite directions.
- The principle directions are vertices of a regular (n+ 1)-simplex

Expansion dominates

Balanced refinement and expansion

The tradeoff depends on the size of the bounding box

- Refinement is good when multiresolution search is needed
- Expansion is good when the tree can grow and not blocked by obstacles
Main motivation:

- Voronoi bias does not take into account obstacles
- How to incorporate the obstacles into Voronoi bias?

Nice idea, but how can this be done in practice?

Even better: Voronoi diagram for obstacle-based metric

(a) Regular RRT, unbounded Voronoi region

(b) Visibility region

(c) Dynamic domain

(a) Regular RRT, unbounded Voronoi region

(b) Visibility region

(c) Dynamic domain

Tradeoff between nearest neighbor calls and collision detection calls

Adaptive tuning of the radius:

- the radius is not fixed but is increased with every extension success and is decreased with every failure
Nearest neighbor calls:

- kd-tree based implementation
- O(logn) instead of naïve O(n) query time
Uniform sampling from dynamic domain:

- Rejection-based method is not efficient for high dimensions
- Uniform distribution should be generated directly

Motivating example 2:

Real-Time Motion Planning For Agile Autonomous Vehicles (2000)

Emilio Frazzoli, Munther A. Dahleh, Eric Feron