Abstract
We construct a sensor-based feedback law that provably solves the real-time collision-free robot navigation problem in a compact convex Euclidean subset cluttered with unknown but sufficiently separated and strongly convex obstacles. Our algorithm introduces a novel use of separating hyperplanes for identifying the robot’s local obstacle-free convex neighborhood, affording a reactive (online-computed) piecewise smooth and continuous closed-loop vector field whose smooth flow brings almost all configurations in the robot’s free space to a designated goal location, with the guarantee of no collisions along the way. We further extend these provable properties to practically motivated limited range sensing models.
Keywords
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References
Trautman, P., Ma, J., Murray, R.M., Krause, A.: Robot navigation in dense human crowds:Statistical models and experimental studies of humanrobot cooperation. The InternationalJournal of Robotics Research 34(3) (2015) 335-356
Henry, P., Vollmer, C., Ferris, B., Fox, D.: Learning to navigate through crowded environments. In: Robotics and Automation, IEEE International Conference on. (2010) 981-986
Karaman, S., Frazzoli, E.: High-speed flight in an ergodic forest. In: Robotics and Automation (ICRA), IEEE International Conference on. (2012) 2899-2906
Paranjape, A.A.,Meier, K.C., Shi, X., Chung, S.J.,Hutchinson, S.: Motion primitives and 3Dpath planning for fast flight through a forest. The Int. J. Robot. Res. 34(3) (2015) 357-377
Wooden, D., Malchano, M., Blankespoor, K., Howardy, A., Rizzi, A.A., Raibert, M.: Autonomous navigation for BigDog. In: IEEE Int. Conf. Robot. Autom. (2010) 4736-4741
Johnson, A.M., Hale, M.T., Haynes, G.C., Koditschek, D.E.: Autonomous legged hill and stairwell ascent. In: IEEE Int. Symp. on Safety, Security, Rescue Robotics. (2011) 134-142
Choset, H., Lynch, K.M., Hutchinson, S., Kantor, G.A., Burgard, W., Kavraki, L.E., Thrun, S.: Principles of RobotMotion: Theory, Algorithms, and Implementations. MIT Press (2005)
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge, U.K. (2006)
Koditschek, D.E., Rimon, E.: Robot navigation functions on manifolds with boundary. Advancesin Applied Mathematics 11(4) (1990) 412-442
Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. The International Journal of Robotics Research 5(1) (1986) 90-98
Koditschek, D.E.: Exact robot navigation by means of potential functions: Some topological considerations. In: Robotics and Automation, IEEE International Conference on. (1987) 1-6
Borenstein, J., Koren, Y.: The vector field histogram-fast obstacle avoidance for mobile robots. IEEE Transactions on Robotics and Automation 7(3) (1991) 278-288
Simmons, R.: The curvature-velocity method for local obstacle avoidance. In: Robotics and Automation (ICRA), IEEE International Conference on. (1996) 3375-3382
Fox, D., Burgard, W., Thrun, S.: The dynamic window approach to collision avoidance. IEEE Robotics Automation Magazine 4(1) (1997) 23-33
Fiorini, P., Shiller, Z.: Motion planning in dynamic environments using velocity obstacles. The International Journal of Robotics Research 17(7) (1998) 760-772
Rimon, E., Koditschek, D.E.: Exact robot navigation using artificial potential functions. Robotics and Automation, IEEE Transactions on 8(5) (1992) 501-518
Lionis, G., Papageorgiou, X., Kyriakopoulos, K.J.: Locally computable navigation functions for sphere worlds. In: Robotics and Automation, IEEE Int. Conf. on. (2007) 1998-2003
Filippidis, I., Kyriakopoulos, K.J.: Adjustable navigation functions for unknown sphere worlds. In: IEEE Decision and Control and European Control Conf. (2011) 4276-4281
Burridge, R.R., Rizzi, A.A., Koditschek, D.E.: Sequential composition of dynamically dexterous robot behaviors. The International Journal of Robotics Research 18(6) (1999) 535-555
Conner, D.C., Choset, H., Rizzi, A.A.: Flow-through policies for hybrid controller synthesis applied to fully actuated systems. Robotics, IEEE Transactions on 25(1) (2009) 136-146
Choset, H., Burdick, J.: Sensor-based exploration: The hierarchical generalized Voronoi graph. The International Journal of Robotics Research 19(2) (2000) 96-125
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)
Arslan, O., Koditschek, D.E.: Exact robot navigation using power diagrams. In: Robotics and Automation (ICRA), IEEE International Conference on. (2016) 1-8
Aurenhammer, F.: Power diagrams: Properties, algorithms and applications. SIAM Journal on Computing 16(1) (1987) 78-96
Webster, R.: Convexity. Oxford University Press (1995)
Arslan, O., Koditschek, D.E.: Sensor-based reactive navigation in unknown convex sphere worlds. Technical report, University of Pennsylvania (2016) Online available at: http://kodlab.seas.upenn.edu/Omur/WAFR2016.
Ó’Dúnlaing, C., Yap, C.K.: A retraction method for planning the motion of a disc. Journal of Algorithms 6(1) (1985) 104 - 111
Cortés, J.,Martınez, S., Karatas, T., Bullo, F.: Coverage control for mobile sensing networks. Robotics and Automation, IEEE Transactions on 20(2) (2004) 243-255
Kwok, A., Martnez, S.: Deployment algorithms for a power-constrained mobile sensor network. International Journal of Robust and Nonlinear Control 20(7) (2010) 745-763
Pimenta, L.C., Kumar, V.,Mesquita, R.C., Pereira, G.A.: Sensing and coverage for a network of heterogeneous robots. In: Decision and Control, IEEE Conference on. (2008) 3947-3952
Arslan, O., Koditschek, D.E.: Voronoi-based coverage control of heterogeneous disk-shaped robots. In: Robotics and Automation, IEEE International Conference on. (2016) 4259-4266
Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. 2nd edn. Volume 501. John Wiley & Sons (2000)
Haralick, R.M., Sternberg, S.R., Zhuang, X.: Image analysis using mathematical morphology. Pattern Analysis and Machine Intelligence, IEEE Transactions on 9(4) (1987) 532-550
Munkres, J.: Topology. 2nd edn. Pearson (2000)
Kozlov, M., Tarasov, S., Khachiyan, L.: The polynomial solvability of convex quadratic programming. USSR Comp. Mathematics and Mathematical Physics 20(5) (1980) 223-228
Bullo, F., Cortés, J.,Martinez, S.: Distributed Control of Robotic Networks: AMathematical Approach to Motion Coordination Algorithms. Princeton University Press (2009)
Rockafellar, R.: Lipschitzian properties of multifunctions. Nonlinear Analysis: Theory, Methods & Applications 9(8) (1985) 867-885
Kuntz, L., Scholtes, S.: Structural analysis of nonsmooth mappings, inverse functions, and metric projections. Journal of Math. Analysis and Applications 188(2) (1994) 346-386
Shapiro, A.: Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM Journal on Control and Optimization 26(3) (1988) 628-645
Liu, J.: Sensitivity analysis in nonlinear programs and variational inequalities via continuous selections. SIAM Journal on Control and Optimization 33(4) (1995) 1040-1060
Chaney, R.W.: Piecewise \(\text{C}^{\text{ k }}\) functions in nonsmooth analysis. Nonlinear Analysis: Theory, Methods & Applications 15(7) (1990) 649 - 660
Khalil, H.K.: Nonlinear Systems. 3rd edn. Prentice Hall (2001)
Paternain, S., Koditschek, D.E., Ribeiro, A.: Navigation functions for convex potentials in a space with convex obstacles. IEEE Transactions on Automatic Control (submitted)
Holmes, R.B.: Smoothness of certain metric projections on Hilbert space. Transactions of the American Mathematical Society 184 (1973) 87-100
Fitzpatrick, S., Phelps, R.R.: Differentiability of the metric projection in Hilbert space. Transactions of the American Mathematical Society 270(2) (1982) 483-501
Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. 2nd edn. Academic Press (2003)
Stewart, J.: Calculus: Early Transcendentals. 7th edn. Cengage Learning (2012)
Kolmanovsky, I., Garone, E., Cairano, S.D.: Reference and command governors: A tutorial on their theory and automotive applications. In: American Control Conf. (2014) 226-241
Arslan, O., Koditschek, D.E.: Smooth extensions of feedback motion planners via reference governors. In: (accepted) Robotics and Automation (ICRA), IEEE Int. Conf. on. (2017)
Vasilopoulos, V., Arslan, O., De, A., Koditschek, D.E.: Minitaur bounds home through a locally sensed artificial forest. In: (submitted) IEEE/RSJ Int. Conf. Intel. Robot. Sys. (2017)
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Arslan, O., Koditschek, D.E. (2020). Sensor-Based Reactive Navigation in Unknown Convex Sphere Worlds. In: Goldberg, K., Abbeel, P., Bekris, K., Miller, L. (eds) Algorithmic Foundations of Robotics XII. Springer Proceedings in Advanced Robotics, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-43089-4_11
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