MAC Versus PC: Determinism and Randomness as Complementary Approaches to Robotic Exploration of Continuous Unknown Domains

Three methods are described for exploring a continuous unknown planar region by a group of robots having limited sensors and no explicit communication. We formalize the problem, prove that its off-line version is NP-hard, and show a lower bound on the length of any solution. Then a deterministic mark and cover (MAC) algorithm is described for the on-line problem using short-lived navigational markers as a means of navigation and indirect communication. The convergence of the algorithm is proved, and its cover time is shown to be the asymptotically optimal O(A/a), where A is the total area and a is the area covered by the robot in a single step. The MAC algorithm is tested against an alternative randomized probabilistic covering (PC) method, which does not rely on sensors but is still able to cover an unknown region in an expected time that depends polynomially on the dimensions of the region. Both algorithms enable cooperation of several robots to achieve faster coverage. Finally, we show that the two methods can be combined to yield a third, hybrid algorithm with a better trade-off between performance and robustness.

[1]  Vladimir J. Lumelsky,et al.  A terrain-covering algorithm for an AUV , 1996, Auton. Robots.

[2]  Günther Schmidt,et al.  Path planning and guidance techniques for an autonomous mobile cleaning robot Robotics and Autonomous Systems 14 (1995) 199-212 , 1995, Robotics Auton. Syst..

[3]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[4]  R. Kershner.  The Number of Circles Covering a Set , 1939 .

[5]  Ricardo A. Baeza-Yates,et al.  Searching in the Plane , 1993, Inf. Comput..

[6]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[7]  Martin Held,et al.  On the Computational Geometry of Pocket Machining , 1991, Lecture Notes in Computer Science.

[8]  B. C. Virgil,et al.  The Aeneid: Virgil , 1953 .

[9]  Uriel Feige,et al.  Short random walks on graphs , 1993, SIAM J. Discret. Math..

[10]  László Lovász,et al.  Random Walks on Graphs: A Survey , 1993 .

[11]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[12]  Lynne E. Parker,et al.  On the design of behavior-based multi-robot teams , 1995, Adv. Robotics.

[13]  J. Seaman.  Introduction to the theory of coverage processes , 1990 .

[14]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[15]  D. Aldous.  Threshold limits for cover times , 1991 .

[16]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.

[17]  . Markov Chains and Random Walks on Graphs , .

[18]  Xiaotie Deng,et al.  Competitive robot mapping with homogeneous markers , 1996, IEEE Trans. Robotics Autom..

[19]  René Schott,et al.  Parallel Searching in the Plane , 1995, Comput. Geom..

[20]  Tucker R. Balch,et al.  Communication in reactive multiagent robotic systems , 1995, Auton. Robots.

[21]  Gheorghe Păun,et al.  Trading Space for Time , 2002 .

[22]  R.A. Russell,et al.  Heat trails as short-lived navigational markers for mobile robots , 1997, Proceedings of International Conference on Robotics and Automation.

[23]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..

[24]  Michael A. Erdmann.  Randomization in Robot Tasks , 1992 .

[25]  Allan Borodin,et al.  Bounds on Universal Sequences , 1989, SIAM J. Comput..

[26]  Steven M. LaValle,et al.  Evaluating motion strategies under nondeterministic or probabilistic uncertainties in sensing and control , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[27]  Esther M. Arkin,et al.  Approximation Algorithms for the Geometric Covering Salesman Problem , 1994, Discret. Appl. Math..

[28]  Israel A. Wagner,et al.  On-Line Graph Searching by a Smell-Oriented Vertex Process , 1997 .

[29]  Hiroshi Yaguchi.  Robot introduction to cleaning work in the East Japan Railway Company , 1995, Adv. Robotics.

[30]  P. Matthews.  Covering Problems for Brownian Motion on Spheres , 1988 .

[31]  Simeon C. Ntafos,et al.  Optimum watchman routes , 1986, SCG '86.

[32]  M. Docarmo.  Differential geometry of curves and surfaces , 1976 .

[33]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[34]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[35]  R. Andrew Russell,et al.  Laying and sensing odor markings as a strategy for assisting mobile robot navigation tasks , 1995, IEEE Robotics Autom. Mag..

[36]  Tunc Geveci,et al.  Advanced Calculus , 2014, Nature.

[37]  Eli Upfal,et al.  Trading space for time in undirected s-t connectivity , 1989, STOC '89.

[38]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[39]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[40]  Liqiang Feng,et al.  Navigating Mobile Robots: Systems and Techniques , 1996 .

[41]  Günther Schmidt,et al.  Path planning and guidance techniques for an autonomous mobile cleaning robot , 1994, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS'94).

[42]  Shimon Even,et al.  Graph Algorithms , 1979 .

[43]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[44]  Sameer Singh,et al.  Fuzzy neural computing of coffee and tainted-water data from an electronic nose , 1996 .

[45]  Eli Upfal,et al.  Trading Space for Time in Undirected s-t Connectivity , 1994, SIAM J. Comput..

[46]  Benjamin Kuipers,et al.  A robot exploration and mapping strategy based on a semantic hierarchy of spatial representations , 1991, Robotics Auton. Syst..

[47]  Simeon Ntafos,et al.  Watchman routes under limited visibility , 1992 .

[48]  El-Ghazali Talbi,et al.  The "Ariadne's clew" algorithm: global planning with local methods , 1993, Proceedings of 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '93).

[49]  Israel A. Wagner,et al.  Smell as a Computational Resource - A Lesson We Can Learn from the Ant , 1996, ISTCS.

[50]  Michael Jenkin,et al.  Robotic exploration as graph construction , 1991, IEEE Trans. Robotics Autom..

[51]  Charles D. Schaper,et al.  Communications, Computation, Control, and Signal Processing: A Tribute to Thomas Kailath , 1997 .

[52]  Marc Snir,et al.  On Parallel Searching , 2011, SIAM J. Comput..

[53]  B. Ripley,et al.  Introduction to the Theory of Coverage Processes. , 1989 .

[54]  Michael F. Bridgland.  Universal Traversal Sequences for Paths and Cycles , 1987, J. Algorithms.

[55]  Vladimir J. Lumelsky,et al.  Vision-guided exploration: a step toward general motion planning in three dimensions , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[56]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[57]  Douglas W. Gage,et al.  Randomized search strategies with imperfect sensors , 1994, Other Conferences.

[58]  Liqiang Feng,et al.  Measurement and correction of systematic odometry errors in mobile robots , 1996, IEEE Trans. Robotics Autom..

[59]  George A. Bekey,et al.  On autonomous robots , 1998, The Knowledge Engineering Review.

[60]  H. Sagan.  Space-filling curves , 1994 .

[61]  Per Enge,et al.  Wide area augmentation of the Global Positioning System , 1996, Proc. IEEE.

[62]  H. Woxniakowski.  Information-Based Complexity , 1988 .