pursuit / evasion in polygonal and general regions the work: by lavalle et al. the presentation: by...
Post on 26-Dec-2015
222 Views
Preview:
TRANSCRIPT
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
top related