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Motion Planning for Point Robots CS 659 Kris Hauser

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Agenda Skimming through Principles Ch. 2, 5.1, 6.1

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Fundamental question of motion planning Are the two given points connected by a path? Forbidden region Feasible space collision with obstacle lack of stability lost sight of target …

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Basic Problem Statement: Compute a collision-free path for a rigid or articulated object among static obstacles Inputs: Geometry of moving object and obstacles Kinematics of moving object (degrees of freedom) Initial and goal configurations (placements) Output: Continuous sequence of collision-free robot configurations connecting the initial and goal configurations

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Why is this hard?

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Tool: Configuration Space Problems: Geometric complexity Space dimensionality

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Tool: Configuration Space

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Problem free space s g free path obstacle

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Problem g obstacle s Semi-free path

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Types of Path Constraints Local constraints: e.g., avoid collision with obstacles Differential constraints: e.g., bounded curvature Global constraints: e.g., minimal length

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Motion-Planning Framework Continuous representation Discretization Graph searching (blind, best-first, A*)

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Path-Planning Approaches Roadmap Represent the connectivity of the free space by a network of 1-D curves Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

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Roadmap Methods Visibility graph Introduced in the Shakey project at SRI in the late 60s. Can produce shortest paths in 2-D configuration spaces

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Simple (Naïve) Algorithm 1.Install all obstacles vertices in VG, plus the start and goal positions 2.For every pair of nodes u, v in VG 3. If segment(u,v) is an obstacle edge then 4. insert (u,v) into VG 5. else 6. for every obstacle edge e 7. if segment(u,v) intersects e 8. then goto 2 9. insert (u,v) into VG 10.Search VG using A*

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Complexity Simple algorithm: O(n 3 ) time Rotational sweep: O(n 2 log n) Optimal algorithm: O(n 2 ) Space: O(n 2 )

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Rotational Sweep

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Reduced Visibility Graph tangent segments Eliminate concave obstacle vertices can’t be shortest path

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Generalized (Reduced) Visibility Graph tangency point

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Three-Dimensional Space Computing the shortest collision-free path in a polyhedral space is NP-hard Shortest path passes through none of the vertices locally shortest path homotopic to globally shortest path

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Roadmap Methods Voronoi diagram Introduced by Computational Geometry researchers. Generate paths that maximizes clearance. O(n log n) time O(n) space

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Roadmap Methods Visibility graph Voronoi diagram Silhouette First complete general method that applies to spaces of any dimension and is singly exponential in # of dimensions [Canny, 87] Probabilistic roadmaps

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Path-Planning Approaches Roadmap Represent the connectivity of the free space by a network of 1-D curves Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

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Cell-Decomposition Methods Two classes of methods: Exact cell decomposition The free space F is represented by a collection of non- overlapping cells whose union is exactly F Example: trapezoidal decomposition

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Trapezoidal decomposition

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critical events criticality-based decomposition … Trapezoidal decomposition

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Planar sweep O(n log n) time, O(n) space Trapezoidal decomposition

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Cell-Decomposition Methods Two classes of methods: Exact cell decomposition Approximate cell decomposition F is represented by a collection of non-overlapping cells whose union is contained in F Examples: quadtree, octree, 2 n -tree

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Octree Decomposition

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Sketch of Algorithm 1.Compute cell decomposition down to some resolution 2.Identify start and goal cells 3.Search for sequence of empty/mixed cells between start and goal cells 4.If no sequence, then exit with no path 5.If sequence of empty cells, then exit with solution 6.If resolution threshold achieved, then exit with failure 7.Decompose further the mixed cells 8.Return to 2

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Path-Planning Approaches Roadmap Represent the connectivity of the free space by a network of 1-D curves Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

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Potential Field Methods Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on it. Goal Robot

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Attractive and Repulsive fields

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Local-Minimum Issue Perform best-first search (possibility of combining with approximate cell decomposition) Alternate descents and random walks Use local-minimum-free potential (navigation function)

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Sketch of Algorithm (with best-first search) 1.Place regular grid G over space 2.Search G using best-first search algorithm with potential as heuristic function

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Simple Navigation Function 0 11 1 2 22 23 3 3 44 5

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1 122 2 3 3 3 45 21 0 4

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1 122 2 3 3 3 45 21 0 4

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Completeness of Planner A motion planner is complete if it finds a collision-free path whenever one exists and return failure otherwise. Visibility graph, Voronoi diagram, exact cell decomposition, navigation function provide complete planners Weaker notions of completeness, e.g.: - resolution completeness (PF with best-first search) - probabilistic completeness (PF with random walks)

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A probabilistically complete planner returns a path with high probability if a path exists. It may not terminate if no path exists. A resolution complete planner discretizes the space and returns a path whenever one exists in this representation.

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Preprocessing / Query Processing Preprocessing: Compute visibility graph, Voronoi diagram, cell decomposition, navigation function Query processing: - Connect start/goal configurations to visibility graph, Voronoi diagram - Identify start/goal cell - Search graph

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Some Variants Moving obstacles Multiple robots Movable objects Assembly planning Goal is to acquire information by sensing Model building Object finding/tracking Inspection Nonholonomic constraints Dynamic constraints Stability constraints Optimal planning Uncertainty in model, control and sensing Exploiting task mechanics (sensorless motions, under- actuated systems) Physical models and deformable objects Integration of planning and control Integration with higher- level planning

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Issues for Future Lectures on Planning Space dimensionality Geometric complexity of the free space Constraints other than avoiding collision The goal is not just a position to reach Etc …

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Readings Principles Ch. 3.1-3.4 *Lozano-Perez (1986) *Lozano-Perez (1983)

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