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Robotic Pursue Evasion and Graph Search

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## Robotic Pursue Evasion and Graph Search

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**Athanasios Kehagias**Aristotle University of Thessaloniki (and Geoffrey A. Hollinger, Univ. of S. California) Robotic Pursue Evasionand Graph Search Robotic Pursuit Evasion And Graph Search**What will we talk about?**• Graph Search (GS): everybody knows it • Robotic Pursuit Evasion (PE): auro10_2.mp4 (from G. Hollinger’s thesis, CMU Robotics Institute) 2**Graph Search**• Mostly invisible evader (Parsons) • Very little about visible evader (Nowakowski) 3**Robotic Pursuit Evasion**Video: auro10_2.mp4 • Indoor environment / floorplan • Three pursuers • Dirty set • Node search 4**Applications of Robotic Search / PE**• Military • Search and Rescue • Tracking • Elderly care • Robotic exploration of hazardous sites • Computer games 5**Plan of the talk**• Historical review (1940-2010) • Models, algorithms, experiments, theorems • “Dimensions” of the PE problem • Desiderata • Questions • Discussion 6**Timeline**7**Prehistory**• The search problem in Operations Research (Koopman, 1942) • PE and Differential Games (R. Isaacs, 1955) • PE and Markov Decision Processes (Eaton and Zadeh, 1962) 8**Operations Research**• Probabilistic problem • Continuous geometry (can also be discrete) 9**Koopman, B.O., “The theory of search. III. The optimum**distribution of searching effort”, Operations Research, vol. 5, pp.613--626, 1957. • Koopman, B.O., Search and screening: general principles with historical applications, 1980. • Dobbie, J.M., “A survey of search theory”, Operations Research, vol. 16, pp. 525--537, 1968. • Benkoski, S.J. and Monticino, M.G. and Weisinger, J.R., “A survey of the search theory literature”, Naval Research Logistics, vol. 38, pp. 469--494, 1991 • Washburn, A.R., “Branch and bound methods for a search problem”, Naval Research Logistics, vol. 45, pp. 243--257, 1998 • Champagne, L. and Carl, EG and Hill, R., “Search theory, agent-based simulation, and u-boats in the bay of Biscay”, Simulation Conference, 2003, vol.1, pp. 991--998, 2004. 10**PE and Differential Games**• Continuous environment • x(t), y(t) positions of pursuer and evader • u(t), v(t) control functions of pursuer and evader • Pursuer’s gain: • Evader’s gain: 11**R. Isaacs, Differential Games, Wiley, 1965**• Vidal, R. and Shakernia, O. and Kim, H.J. and Shim, D.H. and Sastry, S. “Probabilistic pursuit-evasion games: theory, implementation, and experimental evaluation”, IEEE Trans. on Robotics and Automation, vol.18, pp. 662--669, 2002. • Li, D. and Cruz, JB and Chen, G. and Kwan, C. and Chang, M.H., “A hierarchical approach to multi-player pursuit-evasion differential games”, CDC-ECC'05, pp. 5674--5679, 2006. • Cao, H. and Ertin, E. and Kulathumani, V. and Sridharan, M. and Arora, A., “Differential games in large-scale sensor-actuator networks”, IPSN 2006, pp. 77--84, 2006. • Schenato, L. and Oh, S. and Sastry, S. and Bose, P., “Swarm coordination for pursuit evasion games using sensor networks”, ICRA 2005, pp. 2493--2498, 2006. • Bopardikar, S.D. and Bullo, F. and Hespanha, J.P., “On discrete-time pursuit-evasion games with sensing limitations”, IEEE Trans. on Robotics and Automation, vol. 24, pp.1429--1439, 2008 12**Markov Decision Processes (MDP)**• Discrete environment: a graph G=(V,E) • x(t): position of the pursuer • y(t): position of the evader • Evader is a Markov chain: with i,j in V • Pursuer chooses his position: • Joint state: z(t)=(x(t), y(t)) follows a “controlled” MC: 13**Markov Decision Processes**• Goal: choose u(t) to reach a state (i,i) • If the evader is visible we have full knowledge of z(t)=(x(t), y(t)) and it is a Markov Decision Process (MDP) • If the evader is invisible we have partial knowledge of z(t)=(x(t), y(t)) and it is a Partially Observable Markov Decision Process (POMDP) 14**Eaton, JH and Zadeh, LA , Optimal Pursuit Strategies in**Discrete-State Probabilistic Systems, ASME Transactions, Series D, Journal of Basic Engineering, vol. 84, 1962. • Hsu, D. and Lee, W.S. and Rong, N., A point-based POMDP planner for target tracking, ICRA 2008, pp. 1050-4729, 2008. • Miller, S.A. and Harris, Z.A. and Chong, E.K.P., “A POMDP framework for coordinated guidance of autonomous UAVs for multitarget tracking”, EURASIP Journal on Advances in Signal Processing, 2009. • Hutchinson, S.A. et al., “Game-Theoretic Analysis of a Visibility Based Pursuit-Evasion Game in the Presence of Obstacles”. ACC'09, pp. 373--378, 2009. • Basar, T., “Pursuit-evasion games in mobile networks, Ph.D. Thesis, 2010. 15**Flashlight Search**• Two non-robotic papers • Sugihara, K. and Suzuki, I. and Yamashita, M., “The searchlight scheduling problem”, SIAM Journal on Computing, vol.19, pp. 1024, 1990. • Suzuki, I. and Yamashita, M., “Searching for a mobile intruder in a polygonal region”, SIAM Journal on computing, vol. 21, pp.863, 1992. • They are in the Art Gallery tradition • They have implicitly influenced the robotics literature 16**The searchlight scheduling problem**• the searchers are stationary • they search by flashlight (a ray) • the direction of the flashlight can be changed continuously. • This paper does not cite the GS literature 17**A single mobile searcher**• Searcher can have different degrees of visibility • a searcher with k flashlights whose visibility is limited to k rays emanating from his position (k-searcher) • a searcher with a point light source who can see in all directions simultaneously (-searcher). • Necessary and sufficient conditions are presented for a polygon to be searchable by various searchers. • Studies the class of polygons for which the searcher with two flashlights is as capable as the searcher with a point light source • A necessary and sufficient condition is given for such polygons to be searchable by the searcher with two flashlights. • The complexity of generating a search schedule under some of these conditions is also discussed. • This paper cites Parsons, Megiddo and Lapaugh.**Visibility-based PE (S.M. Lavalle and coworkers)**• S. M. LaValle, D. Lin, L. J. Guibas, J.-C. Latombe, and R. Motwani. “Finding an unpredictable target in a workspace with obstacles”. ICRA 1997, pp. 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”. LNCS vol. 1272, pp.17--30, 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, vol. 9, pp. 471--494, 1999. • S. M. LaValle and J. Hinrichsen. “Visibility-based pursuit-evasion: An extension to curved environments”. IEEE Trans. On Robotics and Automation, vol. 17, pp.196--202, 2002. • And many more …(http://msl.cs.uiuc.edu/~lavalle/vispe.html) 20**Visibility-based PE (S.M. Lavalle and coworkers)**• These papers not only cite but actually discuss GS ideas • Search Number • Monotonicity • Etc. • This work has inspired many other robotics researchers • Many papers have been written on visibility-based PE • This research line continues vigorously to the present. 21**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 23**Key Concepts**• The polygonal region is denoted by F. • For every point x in F, the visibility polygon is and the invisibility set F–V(x) is the union of several disjoint simple connected polygons. • Some of these polygons are clean and some are dirty The boundary of V(x) consists of edges; • some of these are edges of the original F; • the remaining are gap edges (facing “free space”) 24**Invisibility set**Visibility polygon Gap edges (black is clean, Red is dirty) 25**A point x has 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. 26**Note: when we know x, we also know V(x) and so F–V(x),**i.e. the invisible components. And S F–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 F (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. 27**Critical Gap Events**• A gap disappears • A gap appears (it gets a 0 label) • A gap splits into two gaps (they inherit the parents label) • Two gaps merge into a new one (it gets a 1 label if any of the original gaps had a 1) • Gaps can also change in noncritical ways (continuous transformation) • Assumption: we never have events which involve three gaps simultaneously 28**A gap disappears / appears**A gap splits into two / two gaps merge. 29**Conservative Discretization**• Form a discretization D={D1,…, DN} by: • extending all edges of F (inside F), • extending outward segments from all pairs of vertices (inside F) • Take 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. 30**The rules:**Example: 31**Along with the state space, we have 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 2ninodes, 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. 32**Example 1:**Undirected adjacency graph Discretized polygon Example clearing sequence: 1-2 1/1 -> 2 Directed information graph 33**Undirected adjacency graph**Example 2: Discretized polygon • Example clearing sequences: • 5-4-3-2 • 5/1 -> 4/1 -> 3/10 -> 2/0 • 3-4-3-2 • 3/11 -> 4/1 -> 3/10 -> 2/0 34 Directed information graph**Lavalle cites some of the classic graph search papers**(Parsons, Megiddo etc.) • but he does not perform GS in the classical sense • instead he performs a shortest path computation in the information graph. • He develops an algorithm to determine if a polygon is 1-searchable • He also performs a theoretical analysis • (In what follows H(F) is the search number of polygon F.) 35**37**(By Chung, Hollinger, and Isler )**Some characteristics of the approach**• Recontamination does help • The algorithm is computationally viable essentially only for 1- searchable polygons • (this is not the same as 1-searchable graphs) • The key concept in Lavalle’s formulation is visibility. • Lavalle’s work has been very influential, his papers are cited by many other works on robotic PE. • Most of these works keep the idea of visibility and visibility partition and try to find ways to make it computationally viable. 39**Hespanha**• Hespanha, J.P. and Kim, H.J. and Sastry, S., “Multiple-agent probabilistic pursuit-evasion games”, 38th Decision and Control Conference, vol.3, pp. 2432--2437,1999. • Hespanha, J.P. and Prandini, M. and Sastry, S., “Probabilistic pursuit-evasion games: A one-step Nash approach”, 39th Decision and Control Conference, vol.3, pp.2272-2277, 2000. He cites both Lavalle and the classic GS papers but his approach is quite different. • unknown maps • probabilistic formulation • game theoretic point of view His approach works on any discrete environment; his examples deal with grids. 40**Isler**• V. Isler, S. Kannan, and S. Khanna. “Randomized Pursuit-Evasion with Limited Visibility”. In ACM-SIAM Symposium on Discrete Algorithms, 2004. • V. Isler, S. Kannan, and S. Khanna. “Randomized Pursuit-Evasion in a Polygonal Environment”. IEEE Trans. on Robotics and Automation, 5(21):864--875, 2005. • V. Isler, S. Kannan, and S. Khanna. “Randomized Pursuit-Evasion with Local Visibility”. SIAM Journal on Discrete Mathematics, 1(20):26--41, 2006 41**Isler**• Probabilistic search • Algorithms and theorems about capture with high probability • Search on graphs: • evader has local visibility • evader and pursuer move simultaneously • Search on polygons • The lion-and-man game (geometric version of the cops-and-robbers game) 42**PE in polygonal regions**• Two lions can capture the man in any simply-connected polygon (Isler). • A single lion can capture the man in any simply-connected polygon in finite time (Isler). • In polygons with obstacles, multiple pursuers are necessary. • (Recently, Bhadauria et al. (2010) showed that three lions can capture the man in any polygon) • polygonal environments may require recontamination to clear with the minimal number of searchers. 43**44**(By Chung, Hollinger, and Isler )**Related Work on Probabilistic Graph Search**• Adler, M. and Racke, H. and Sivadasan, N. and Sohler, C. and Vocking, B., Randomized pursuit-evasion in graphs, Combinatorics, Probability and Computing, vol. 12, pp.225--244, 2003. • Alpern, S. and Gal, S., The theory of search games and rendezvous, 2003. 46**Gerkey, Thrun and Gordon**• Brian P. Gerkey, Sebastian Thrun, and Geoff Gordon. "Visibility-based pursuit-evasion with limited field of view". Intl. Journal of Robotics Research, vol.25, pp.299-316, 2006. • Brian P. Gerkey, Sebastian Thrun, and Geoff Gordon. "Parallel stochastic hill-climbing with small teams". In Multi-Robot Systems: From Swarms to Intelligent Automata, Vol. 3, pp. 65-77, Springer, 2005. 47**Gerkey, Thrun and Gordon**PE in polygonal environments, visibility-based discretizations In the first paper: • Visibility-based PE is extended to a limited field of view • New φ –searcher (instead of the k-searcher) • Finding the minimal number of φ-searchers is NP-complete • Complete algorithm for a single φ-searcher. In the second paper: • Market-based algorithms, heuristics and stochastic action selection • They address difficult multi-robot problems • The PARISH algorithm • Benchmark: visibility-based PE 48**Hollinger and Kehagias**• Guaranteed Search (Also Efficient Search) • G. Hollinger, S. Singh, J. Djugash, and A. Kehagias, "Efficient multi-robot search for a moving target," International Journal of Robotics Research, vol. 28, pp. 201-219, 2009. • A. Kehagias, G. Hollinger, and S. Singh, "A graph search algorithm for indoor pursuit/evasion," Mathematical and Computer Modeling, vol. 50, pp. 1305-1317, 2009. • A. Kehagias, G. Hollinger, and A. Gelastopoulos, "Searching the nodes of a graph: Theory and algorithms," ArXiv Repository, Tech. Rep. 0905.3359 • G. Hollinger, A. Kehagias, and S. Singh, "GSST: Anytime guaranteed search," Autonomous Robots, vol. 29, pp. 99-118, 2010. • G. Hollinger, A. Kehagias, and S. Singh, "Improving the efficiency of clearing with multi-agent teams," International Journal of Robotics Research, vol. 29, no. 8, pp. 1088-1105, 2010. • G. Hollinger, "Search in the physical world," Ph.D. dissertation, Robotics Institute, Carnegie Mellon University. 49**Characteristics of our approach**• Mainly indoor environments (can also handle outdoor). • Simple visibility model, simple discretization of environment. • Cells nodes, evader lives in the nodes (node search). • Evader is invisible, arbitrarily fast, omniscient. • Goal: minimal clearing search • Search must be “rooted” and “internal” • Search should be monotone and connected (but we also study nonmonotone and disconnected) • Our algorithms use tree search and Barriere’s algorithm • GSST (Guaranteed Search by Spanning Trees) • G-GSST (Guardless GSST) • Our algorithms use tree search and Barriere’s algorithm 50