NUS CS5247 A Visibility-Based Pursuit-Evasion Problem Leonidas J.Guibas, Jean-Claude Latombe, Steven M. LaValle, David Lin, Rajeev Motwani. Computer Science Department Stanford University Stanford, CA presented by: Michał Marzec

NUS CS52472 Motivation  Applications: robotics applications: surveillance, air traffic control, military strategy, security systems. Two decision makers have totally opposing interests.  But applications are not only limited to adversarial targets… strategies to automatically locate another robot, items in a factory or people in a search/rescue effort.  We want such strategies to be carried out without human intervention. Automation of tasks.

NUS CS52473 Problem Formulation  The free space contains two types of moving actors: one or more pursuers and one evader.  We assume that any of pursuers know the initial position of the evader. We don’t know how it moves nor how fast it moves. We cannot do any prediction with respect to the moves of the evader.  How to we approach this problem? We associate a visibility region with each of the pursuers. Those may partially overlap. Each pursuer can actually see a portion of the free space.  What do we want to do? The purpose it to catch the evader, but… What do we mean by “capture”? The evader must lie in visibility region of at least one of pursuers.  The pursuers need to make sure that the space that has already been explored remains clear. They have to prevent the evader from intruding to the portion of the space that has already been searched.  It’s interesting to perform this task with minimum number of pursuers. How to determine the minimum number of pursuers that guarantees that the solution will be found?

NUS CS52474 Simple examples

NUS CS52475 Mathematical definition of the problem.  The pursuers and the evader are represented as moving points.  The initial position of the evader is unknown (e(0) = ? ) and e(t) denotes the position of the evader at any time t > 0.  We need to distinguish different zones in the free space: contaminated – might contain the evader. cleared – no evader lying in this region. recontaminated – region that has changed it’s statuses from contaminated to cleared and to contaminated again. Position of the i-th pursuer at t >= 0 Continuous path of the i-th pursuer. Motion strategy. Visibility region of any q lying in F. Is a solution strategy H(F) represents the minimum number of pursuers.

NUS CS52476 More complex cases. How about this geometry of the space? Is one pursuer still sufficient to search the space? Response: No Reason: because of holes.

NUS CS52477 Proposed solutions.  To have as many static guards as we need to completely cover the space and guarantee that any evader is immediately visible.  To have fewer pursuers but capable to move and search for an evader.  The presence of holes involves worsening of visibility. The pursuers have to penetrate farther into the corridors to check whether an evader does not hide there.

NUS CS52478 Theoretical results.  Complexity depends on topological and geometric complexity of F. For a space with h holes and n edges: For a simply connected free-space F at worst:

NUS CS52479 Proof of H(F) = 0(lg n). E 0,0E 1,0 E 1,1 E 2,0 E 2,1 E 2,3 E 2,2 E 0,0 E 1,0 E 1,1 E 2,1E 2,0 E 2,2E 2,3 Pursuers are placed on partition edges and we specify the motion strategy for N pursuers. This strategy guarantees that the evader will be seen. Partitioning is applied till the free space is triangulated.

NUS CS Proof of H(F) = 0(lg n). E 0,0E 1,0 E 1,1 E 2,0 E 2,1 E 2,3 E 2,2 E 0,0 E 1,0 E 1,1 E 2,1E 2,0 E 2,2E 2,3 Pursuers are placed on partition edges and we specify the motion strategy for N pursuers. This strategy guarantees that the evader will be seen. Partitioning is applied till the free space is triangulated.

NUS CS Proof of H(F) = 0(lg n). E 0,0E 1,0 E 1,1 E 2,0 E 2,1 E 2,3 E 2,2 E 0,0 E 1,0E 1,1 E 2,1E 2,0 E 2,2E 2,3 Pursuers are placed on partition edges and we specify the motion strategy for N pursuers. This strategy guarantees that the evader will be seen. Partitioning is applied till the free space is triangulated.

NUS CS Proof of H(F) = 0(lg n). E 0,0E 1,0 E 1,1 E 2,0 E 2,1 E 2,3 E 2,2 E 0,0 E 1,0E 1,1 E 2,1E 2,0E 2,2 E 2,3 Pursuers are placed on partition edges and we specify the motion strategy for N pursuers. This strategy guarantees that the evader will be seen. Partitioning is applied till the free space is triangulated.

NUS CS What does it show?  N = (lg n) corresponds to the height of the balanced binary tree.  Each pursuer takes cares about one level of the binary tree to search for an evader. ..so the height of the tree determines the sufficient number of pursuers, but not the minimal one.

NUS CS Usefulness of information states  What is an information state? It is an information provided by a moving visibility polygon of pursuer and which identifies all unique situations that can occur during the execution of a motion strategy. It tells us where the contaminated regions are and takes into account the positions of the pursuers.  An information space is a set of all possible information states.  We define a function that produces the information state when a specified strategy is applied between instants t0 and t1. We pass from one information state to another one after having executed a little portion of this strategy.

NUS CS Concept of conservative cells.  We search a finite cell complex that is constructed on the basis of critical information change.  A conservative cell preserves the information invariance property. In other terms when a pursuer moves along closed paths in a cell the information state is not altered. It remains stable independently of the chosen path (path invariance).  We construct a graph whose vertices correspond to the equivalence classes of the space. (conservative cells) This graph can be subsequently searched to find a solution strategy, without regard to the particular choice of path within each conservative cell.

NUS CS An algorithm for a single pursuer.  Partitioning of the space into conservative cells.  Why? We can partition the free space into convex cells by identify critical places at which edge visibility changes. GAP EDGE We associate binary labels to each gap edge: “1” – the portion of the space bordered by the gap is contaminated. “0” – otherwise (region is clear).

NUS CS An algorithm for a single pursuer.  B(q) is binary sequence of labels assigned to gaps at different instants.  q and B(q) represent the information state.  The pursuer moves so that the information state remains stable. The gap edges don’t disappear and appear and so on… The gaps continuously change during the pursuer motion but the corresponding gap edge label will not change. He moves in his cell without crossing the critical boundary.

NUS CS Ray shooting. Extension of obstacles edges to form edge-visibility cells.

NUS CS Searching the information space.  The cells and their adjacency relationships define the cell graph that must be searched. => find any sequence of cells in the graph that leave the pursuer at some position at which all gap labels are “0”. First we need to determine the correspondences between gap edges and search the graph using Dijkstra algorithm.

NUS CS Summary  The visibility-based problem is NP-hard. In the worst cases the number of information states is exponential. We also cannot clearly say whether all information states necessarily have to be represented and searched to determine a solution.  This is a visibility-based problem, so we require the pursuers to see and not to touch the evader.  The scenarios of this type have arisen in multiple applications such as robotics or surveillance with mobile robots or even search-rescue operations.  The complexity increases with number of edges.

NUS CS Extensions  Region of capture.  Include bounded velocity for the evader.  Consider a limited viewing angle ( more realistic)  Try to find the evader in minimum time ( improve the cost ).

NUS CS THE END