Combinatorial Filters

A problem is introduced in which a moving body (robot, human, animal, vehicle, and so on) travels among obstacles and binary detection beams that connect between obstacles or barriers. Each beam can be viewed as a virtual sensor that may have many possible alternative implementations. The task is to determine the possible body paths based only on sensor observations that each simply report that a beam crossing occurred. This is a basic filtering problem encountered in many settings, under a variety of sensing modalities. Filtering methods are presented that reconstruct the set of possible paths at three levels of resolution: (1) the possible sequences of regions (bounded by beams and obstacles) visited, (2) equivalence classes of homo-topic paths, and (3) the possible numbers of times the path winds around obstacles. In the simplest case, all beams are disjoint, distinguishable, and directed. More complex cases are then considered, allowing for any amount of beams overlapping, indistinguishability, and lack of directional information. The method was implemented in simulation. An inexpensive, low-energy, easily deployable architecture was also created which implements the beam model and validates the methods of the article with experiments.

[1]  R. Ho.  Algebraic Topology , 2022 .

[2]  Steven M. LaValle,et al.  Cyber Detectives: Determining When Robots or People Misbehave , 2010, WAFR.

[3]  P. Abbeel,et al.  Identification and Representation of Homotopy Classes of Trajectories for Search-Based Path Planning in 3D , 2012 .

[4]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[5]  Jack Snoeyink,et al.  Testing Homotopy for Paths in the Plane , 2002, SCG '02.

[6]  Steven M. LaValle,et al.  Minimalist multiple target tracking using directional sensor beams , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[7]  Jie Gao,et al.  Differential forms for target tracking and aggregate queries in distributed networks , 2013, TNET.

[8]  Leonidas J. Guibas,et al.  Sensing, tracking and reasoning with relations , 2002, IEEE Signal Process. Mag..

[9]  Steven M. LaValle,et al.  Mapping and Navigation from Permutations of Landmarks , 2006 .

[10]  Steven M. LaValle,et al.  Distance-Optimal Navigation in an Unknown Environment Without Sensing Distances , 2007, IEEE Transactions on Robotics.

[11]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[12]  Wolfram Burgard,et al.  A Probabilistic Approach to Concurrent Mapping and Localization for Mobile Robots , 1998, Auton. Robots.

[13]  Gregory Dudek,et al.  Localizing a robot with minimum travel , 1995, SODA '95.

[14]  H. W. Kuhn,et al.  11. Extensive Games and the Problem of Information , 1953 .

[15]  Leonidas J. Guibas,et al.  Visibility-Based Pursuit-Evasion in a Polygonal Environment , 1997, WADS.

[16]  Upamanyu Madhow,et al.  Tracking Multiple Targets Using Binary Proximity Sensors , 2007, 2007 6th International Symposium on Information Processing in Sensor Networks.

[17]  Sebastian Thrun,et al.  FastSLAM: a factored solution to the simultaneous localization and mapping problem , 2002, AAAI/IAAI.

[18]  Masafumi Yamashita,et al.  Online polygon search by a seven-state boundary 1-searcher , 2006, IEEE Transactions on Robotics.

[19]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[20]  Masafumi Yamashita,et al.  Searching for a Mobile Intruder in a Polygonal Region , 1992, SIAM J. Comput..

[21]  Jason M. O'Kane,et al.  Sloppy motors, flaky sensors, and virtual dirt: Comparing imperfect ill-informed robots , 2007, Proceedings 2007 IEEE International Conference on Robotics and Automation.

[22]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[23]  Justin T. Czarnowski.  Minimalist Hardware Architectures for Agent Tracking and Guidance , 2011 .

[24]  Gregory Dudek,et al.  Learning Network Topology from Simple Sensor Data , 2007, Canadian Conference on AI.

[25]  Robert Morris,et al.  Link-level measurements from an 802.11b mesh network , 2004, SIGCOMM 2004.

[26]  Leonidas J. Guibas,et al.  A Visibility-Based Pursuit-Evasion Problem , 1999, Int. J. Comput. Geom. Appl..

[27]  J. M. M. Montiel,et al.  The SPmap: a probabilistic framework for simultaneous localization and map building , 1999, IEEE Trans. Robotics Autom..

[28]  Steven M. LaValle,et al.  Sensing and Filtering: A Fresh Perspective Based on Preimages and Information Spaces , 2012, Found. Trends Robotics.

[29]  J. M. Bilbao,et al.  Contributions to the Theory of Games , 2005 .

[30]  Vijay Kumar,et al.  Trajectory Planning for Systems with Homotopy Class Constraints , 2012, ARK.

[31]  R. Bertram,et al.  Stochastic Systems , 2008, Control Theory for Physicists.

[32]  Gregory J. Pottie,et al.  Wireless integrated network sensors , 2000, Commun. ACM.

[33]  Ronald Parr,et al.  DP-SLAM: Fast, Robust Simultaneous Localization and Mapping Without Predetermined Landmarks , 2003, IJCAI.

[34]  Fred Cohen,et al.  Sensor Beams, Obstacles, and Possible Paths , 2008, WAFR.

[35]  Sung Yong Shin,et al.  Visibility-based pursuit-evasion in a polygonal room with a door , 1999, SCG '99.

[36]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[37]  Matthew T. Mason,et al.  An exploration of sensorless manipulation , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[38]  Nisheeth Shrivastava,et al.  Target tracking with binary proximity sensors: fundamental limits, minimal descriptions, and algorithms , 2006, SenSys '06.

[39]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[40]  Kenneth Y. Goldberg,et al.  Orienting polygonal parts without sensors , 1993, Algorithmica.

[41]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[42]  Wolfram Burgard,et al.  Probabilistic Robotics (Intelligent Robotics and Autonomous Agents) , 2005 .

[43]  Leonidas J. Guibas,et al.  The Robot Localization Problem , 1995, SIAM J. Comput..

[44]  S. LaValle.  Sensing and Filtering : A Tutorial Based on Preimages and Information Spaces , 2011 .

[45]  Hugh F. Durrant-Whyte,et al.  A solution to the simultaneous localization and map building (SLAM) problem , 2001, IEEE Trans. Robotics Autom..

[46]  Nisheeth Shrivastava,et al.  Target tracking with binary proximity sensors , 2009, TOSN.

[47]  Steven M. LaValle,et al.  Controlling Wild Bodies Using Linear Temporal Logic , 2011, Robotics: Science and Systems.

[48]  Frédo Durand,et al.  3d visibility: analytical study and applications , 1999 .

[49]  Dima Grigoriev,et al.  Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane , 1998, ISSAC '98.

[50]  P. Giblin.  Computational geometry: algorithms and applications (2nd edn.), by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Pp. 367. £20.50. 2000. ISBN 3 540 65620 0 (Springer-Verlag). , 2001, The Mathematical Gazette.

[51]  Max Dehn,et al.  Papers on Group Theory and Topology , 1987 .

[52]  Stephen G. Kobourov,et al.  Computing homotopic shortest paths efficiently , 2002, Comput. Geom..

[53]  Steven M. LaValle,et al.  Tracking hidden agents through shadow information spaces , 2008, 2008 IEEE International Conference on Robotics and Automation.

[54]  Andrew P. Bradley,et al.  The use of the area under the ROC curve in the evaluation of machine learning algorithms , 1997, Pattern Recognit..

[55]  Vin de Silva,et al.  Coordinate-free Coverage in Sensor Networks with Controlled Boundaries via Homology , 2006, Int. J. Robotics Res..

[56]  S. Shankar Sastry,et al.  A Distributed Topological Camera Network Representation for Tracking Applications , 2010, IEEE Transactions on Image Processing.

[57]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[58]  Keiji Nagatani,et al.  Topological simultaneous localization and mapping (SLAM): toward exact localization without explicit localization , 2001, IEEE Trans. Robotics Autom..