Pursuit / Evasion in

Polygonal and General Regions

The Work: by LaValle et al.

The Presentation: by Geoff Hollinger and Thanasis Ke(c)hagias

The Problem:

Pursuit / Evasion in a Polygonal Region

The Assumptions:

•Region is simply connected polygon (no holes)

•The pursuer has a map

•There is one pursuer, with 360 vision

•The evader is captured as soon as seen by the pursuer

•The evader is arbitrarily fast

•The evader always knows the pursuer’s position

The Desired Solution: An algorithm which gives

•a motion plan which guarantees capture (if such a plan exists)

•a “can’t do” output (if a guaranteed capture plan doesn’t exist).

Variants

•Non-polygonal region (e.g. with curved boundary)

•Map of the region is not available

•Probabilistic search (capture is not guaranteed but has high

probability)

•Pursuit / evasion on a graph (not a region)

Key Concepts

•The polygonal region is denoted by P.

•For every point x in P, the visibility polygon is

:V x y xy P

and the invisibility set P–V(x) is the union of several disjoint

simple connected polygons.

•Some of these polygons are clean (i.e. they certainly do not

contain the evader) and some are dirty (i.e. they may contain the

evader).

•The boundary of V(x) consists of edges;

•some of these are edges of the original P;

•the remaining are gap edges (facing “free space”)

Visibility polygon

Invisibility set

Gap edges(black is clean,Red is dirty)

Given some point x, it will have a V(x), with n associated gaps

(n 0) each of which can be clean or dirty (i.e. the invisible

component behind that gap will be clean or dirty).

This information can be encoded in an n-long string (say of 0’s

and 1’s) which we denote by B(x).

Note: B(x) can also be the empty string.

Information Space

We need an appropriate state space for the problem.

We could use (x,S) where

•x is the position of the pursuer

•S is the set of dirty points

But, we would prefer a discrete state space.

Note: when we know x, we also know V(x) and so P–V(x), i.e. the

invisible components. And S P–V(x). So we don’t really need to

put S in the state, B(x) suffices (and it is discrete).

Also: we can discretize P (break it into cells) provided we do not

lose any critical information.

Critical information is how gaps change. We need a discretization

that preserves this information.

Critical Gap Events

1. A gap disappears

2. A gap appears (it gets a 0 label)

3. A gap splits into two gaps (they inherit the parents label)

4. Two gaps merge into a new one (it gets a 1 label if any of the original gaps

had a 1)

Note: gaps can also change in noncritical ways (continuous

transformation)

Assumption: we never have an event which involves three gaps

simultaneously

A gap disappears / appears A gap splits into two / two gaps merge.

Conservative Discretization

Form a discretization D={D1,…, DN} by:

•extending all edges of P (inside P),

•extending outward segments from all pairs of vertices (inside P)

and taking all resulting sub-polygons as cells Di of the

discretization.

This is a conservative discretization, i.e. no critical gap events

occur while the pursuer moves inside one of the cells.

The rulez:

Example:

Finally

instead of (x, B(x)) use as state (Di, B(Di))

(which takes values in a discrete state space, the information space).

Now that we have the state space, we need the state transition

function. It will be a state transition graph.

We actually have two graphs:

•Gc is the connectivity graph; it has N nodes (one per cell) and its

edges follow the connectivity of the cells; it is an undirected

graph.

•GI is the information graph (the state transition graph)

•nodes: for the i-th cell Di it has 2ni nodes, where ni is the

number of gaps associated with any x in Di

•edges: they respect critical gap events and information

changes.

Note: GI is a directed graph.

Example 1:

Discretized polygon

Undirected adjacency graph

Directed information graph

Example clearing sequence:1-21/1 -> 2

Now we can formulate and solve the Pursuit/Evasion problem:

In GI , find a (shortest) path which starts from a given “all-

dirty” node and ends at some “all-clean” node (provided such

a path exists).

Example 1:

Discretized polygon

Undirected adjacency graph

Directed information graph

Example clearing sequence:1-21/1 -> 2

Example 2:

Discretized polygon

Undirected adjacency graph

Directed information graph

Example clearing sequences:1) 5-4-3-2 5/1 -> 4/1 -> 3/10 -> 2/02) 3-4-3-2

3/11 -> 4/1 -> 3/10 -> 2/0

Example 3:Discretized polygon

Undirected adjacency graph

Directed information graph

Example clearing sequences:1) 1-2-3-4-5

1/1 -> 2/11 -> 3/1 -> 4/01 -> 5/02) 4-5-4-3 4/11 -> 5/1 -> 4/10 -> 3/0

Example 4:

Discretized polygon

Example clearing sequences:1) 1-2-3-4-5 1/1 -> 2/11 -> 3/1 -> 4/01 -> 5/02) 7-6-5-4-3 7/11 -> 6/111 -> 5/1 -> 4/01 -> 3/0

Example 5:Example clearing sequence:10-9-8-7-6-5-4-3-2-3-4-5-12-13-18-19-20-19-18-13-14-15-16

Node Info State

10 1

9 01

8 011

7 011

6 0111

5 0111

4 0111

3 011

2 01

3 001

4 0011

5 0011

12 0011

13 0011

18 0011

19 011

20 01

19 001

18 0001

13 0001

14 0001

15 001

16 00

Discretized polygon

Example 6: Any path leads to recontamination, for instance:10-9-8-7-6-5-4-3-2-1-22

Node Info State

10 1

9 01

8 011

7 011

6 0111

5 0111

4 0111

3 011

2 011

1 11

Recontaminated!!Oh no!

Example 7:

Discretized polygon

Undirected adjacency graph

Directed information graph

Not a chance…can’t clear anything

Every node of GI can transit to two other nodes. If we assign equal probabilities to trans’swe get a Markov chain.

Its states can be divided into twoclasses: •Transient•Persistent•Trapping (subset of persistent)

Furthermore, some states can be collapsed.

Some Markov Chain Connections

It might be interesting to address questions such as:

•Decompose the chain to ergodic classes (connected components)•Determine how many trapping classes exist.•Is a particular trapping class (the all-clean one) accessible from a particular node?•If the pursuer performs a random walk on the graph

•what is the probability of hitting the trapping class?•what is the expected time to hit the trapping class?•is there an equilibrium probability distribution? •what is the rate of convergence to the equilibrium?

Variant 1: Non-polygonal region

Variant 2: Map of the region is not available

The region

A sequence of gap navigation trees: tree2tree transitions take place at critical gap events.

A gap can be chased until it disappears; when it reappears it is cleared!!!

Questions, Issues etc.

•Is there an information quantity ? •If yes, how does it evolve during the pursuit?

•Does recontamination help?•Can we reduce polygon problem to graph problem?

•If not exactly, then approximately?•Conjecture: if the polygon can be cleared starting from a particular all-dirty state, then it can be cleared starting from any all-dirty state.•How to use all this for Ember?

Biblio

•S. M. LaValle, D. Lin, L. J. Guibas, J.-C. Latombe, and R. Motwani. Finding an unpredictable target in a workspace with obstacles. In Proc. IEEE Int'l Conf. on Robotics and Automation, pages 737--742, 1997. •L. J. Guibas, J.-C. Latombe, S. M. LaValle, D. Lin, and R. Motwani. Visibility-based pursuit-evasion in a polygonal environment. In F. Dehne, A. Rau-Chaplin, J.-R. Sack, and R. Tamassia, editors, WADS '97 Algorithms and Data Structures (Lecture Notes in Computer Science, 1272), pages 17--30. Springer-Verlag, Berlin, 1997. •L. J. Guibas, J.-C. Latombe, S. M. LaValle, D. Lin, and R. Motwani. Visibility-based pursuit-evasion in a polygonal environment. International Journal of Computational Geometry and Applications, 9(5):471--494, 1999.

•L. Guilamo, B. Tovar, and S. M. LaValle. Pursuit-evasion in an unknown environment using gap navigation graphs. In Proc. IEEE International Conference on Robotics and Automation, 2004. Under review. •B. Tovar, S. M. LaValle, and R. Murrieta. Locally-optimal navigation in multiply-connected environments without geometric maps. In IEEE/RSJ Int'l Conf. on Intelligent Robots and Systems, 2003.

•Great Downloadable Book: Planning Algorithms (by Steven M. LaValle) at http://planning.cs.uiuc. edu/book.pdf

•Lavalle’s home page: http://msl.cs.uiuc.edu/~lavalle/

Great Downloadable Book