CS 326 A Motion Planning CriticalityBased Motion Planning

CS 326 A: Motion Planning Criticality-Based Motion Planning: Target Finding

Trapezoidal decomposition

Criticality-Based Planning § Define a property P § Decompose the configuration space into “regular” regions (cells) over which P is constant. § Use this decomposition for planning § Issues: - What is P? It depends on the problem - How to use the decomposition? § Approach is practical only in low-dimensional spaces: - Complexity of the arrangement of cells - Sensitivity to floating point errors

Topics of this class and the next one § Target finding Information (or belief) state/space § Assembly planning Path space

Example robot’s visibility region hiding region cleared region robot 1 2 3 4 5 6 6

Problem § A target is hiding in an environment cluttered with obstacles § A robot or multiple robots with vision sensor must find the target § Compute a motion strategy with minimal number of robot(s) 7

Assumptions § Target is unpredictable and can move arbitrarily fast § Environment is polygonal § Both the target and robots are modeled as points § A robot finds the target when the straight line joining them intersects no obstacles (omni-directional vision) 8

Animated Target-Finding Strategy 9

Does a solution always exist for a single robot? No ! Easy to test: “Hole” in the workspace Hard to test: No “hole” in the workspace 10

Effect of Geometry on the Number of Robots Two robots are needed 11

Effect of Number n of Edges Minimal number of robots N = Q(log n) 12

Effect of Number h of Holes 13

Information State visibility region a = 0 or 1 c = 0 or 1 free edge (x, y) obstacle edge b = 0 or 1 0 cleared region 1 contaminated region § Example of an information state = (x, y, a=1, b=1, c=0) § An initial state is of the form (x, y, a=1, b=1, . . . , u=1) § A goal state is any state of the form (x, y, a=0, b=0, . . . , u=0) 14

Critical Line contaminated area b=1 a=0 (x, y, a=0, b=1) cleared area b=1 a=0 Information state is unchanged (x, y, a=0, b=1) b=0 a=0 (x, y, a=0, b=0) Critical line 15

Grid-Based Discretization § Ignores critical lines Visits many “equivalent” states § Many information states per grid point § Potentially very inefficient 16

Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i. e. , the robot keeps seeing the same obstacle edges 17

Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i. e. , the robot keeps seeing the same obstacle edges 18

Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i. e. , the robot keeps seeing the same obstacle edges 19

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Search Graph § {Nodes} = {Conservative Cells} X {Information States} § Node (c, i) is connected to (c’, i’) iff: • Cells c and c’ share an edge (i. e. , are adjacent) • Moving from c, with state i, into c’ yields state i’ § Initial node (c, i) is such that: • c is the cell where the robot is initially located • i = (1, 1, …, 1) § Goal node is any node where the information state is (0, 0, …, 0) § Size is exponential in the number of edges 21

Example (C, a=1, b=1) A a B (B, b=1) b (D, a=1) E C D 22

Example (C, a=1, b=1) A (B, b=1) (D, a=1) E B C D (C, a=1, b=0) (E, a=1) 23

Example (C, a=1, b=1) A (B, b=1) (D, a=1) E B C D (C, a=1, b=0) (B, b=0) (E, a=1) (D, a=1) 24

Example (C, a=1, b=1) A (B, b=1) (D, a=1) E B C D Much smaller search tree than with grid-based discretization ! (C, a=1, b=0) (B, b=0) (E, a=1) (D, a=1) 25

Example of Target-Finding Strategy Visible Cleared Contaminated 26

More Complex Example 2 1 3 27

Example with Recontaminations 1 4 2 5 3 6 28

Example with Linear Number of Recontaminations Recontaminated area 1 3 2 4 29

Example with Two Robots (Greedy algorithm) 30

Example with Two Robots 31

Example with Three Robots 32

Robot with Cone of Vision 33

Other Topics § Dimensioned targets and robots, threedimensional environments § Non-guaranteed strategies § Concurrent model construction and target finding § Planning the escape strategy of the target 34