Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-15T21:31:44.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Global inverse optimal exponential path-tracking control of mobile robots driven by Lévy processes

Published online by Cambridge University Press:  19 April 2021

K. D. Do Open the ORCID record for K. D. Do [Opens in a new window]
K. D. Do*
Affiliation:
School of Civil and Mechanical Engineering, Curtin University, Bentley, WA 6102, Australia Email: duc@curtin.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper formulates and solves a new problem of global practical inverse optimal exponential path-tracking control of mobile robots driven by Lévy processes with unknown characteristics. The control design is based on a new inverse optimal control design for nonlinear systems driven by Lévy processes and ensures global practical exponential stability almost surely and in the pth moment for the path-tracking errors. Moreover, it minimizes cost function that penalizes tracking errors and control torques without having to solve a Hamilton–Jacobi–Bellman or Hamilton–Jaccobi–Isaacs equation.

Keywords

Mobile robotPath-trackingInverse optimalityExponential stabilityLévy processes
Type
Article
Information
Robotica , Volume 39 , Issue 12 , December 2021 , pp. 2306 - 2332
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Murray, R. and Sastry, S., “Nonholonomic motion planning: Steering using sinusoids”, IEEE Trans. Autom. Control 38(5), 700–716 (1993).Google Scholar
Kolmanovsky, I. and McClamroch, N. H., “Developments in nonholonomic control problems”, IEEE Control Syst. Mag. 15(6), 20–36 (1995).Google Scholar
dAndrea Novel, B., G. Bastin and G. Campion, “Control of nonholonomic wheeled mobile robots by state feedback linearization”, Int. J. Rob. Res. 14(6), 543–559 (1995).10.1177/027836499501400602CrossRefGoogle Scholar
Bloch, A. M. and Drakunov, S., “Stabilization and tracking in the nonholonomic integrator via sliding mode”, Syst. Control Lett. 29(2), 91–99 (1996).10.1016/S0167-6911(96)00049-7CrossRefGoogle Scholar
Brockett, R. W., “Asymptotic Stability and Feedback Stabilization”, In: Differential Geometric Control Theory (Brockett, R. W., Millman, R. S., and Sussmann, H. J., eds.) (Birkhauser, Boston, 1983) pp. 181–191.Google Scholar
Canudas, C. d. W., B. Siciliano and G. Bastin, Theory of Robot Control (Springer, London, 1996).Google Scholar
Astolfi, A., “Discontinuous control of the Brockett integrator”, Eur. J. Control 4(1), 49–63 (1998).10.1016/S0947-3580(98)70099-8CrossRefGoogle Scholar
Lee, T. C., Song, K. T., Lee, C. H. and Teng, C. C., “Tracking control of unicycle-modelled mobile robots using a saturation feedback controller”, IEEE Trans. Control Syst. Technol. 9(2), 305–318 (2001).Google Scholar
Do, K. D., Jiang, Z. P. and Pan, J.. “A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots”, IEEE Trans. Rob. Autom. 20(3), 589–594 (2004).10.1109/TRA.2004.825470CrossRefGoogle Scholar
Do, K. D., Jiang, Z. P. and Pan, J.. “Simultaneous stabilization and tracking control of mobile robots: An adaptive approach”. IEEE Trans. Autom. Control 49(7), 1147–1151 (2004).10.1109/TAC.2004.831139CrossRefGoogle Scholar
Zohar, I., Ailon, A. and Rabinovici, R., “Mobile robot characterized by dynamic and kinematic equations and actuator dynamics: Trajectory tracking and related application,” Rob. Auton. Syst. 59(6), 343–353 (2011).10.1016/j.robot.2010.12.001CrossRefGoogle Scholar
Blazie, S., “A novel trajectory-tracking control law for wheeled mobile robots”, Rob. Auton. Syst. 59(11), 1001–1007 (2011).10.1016/j.robot.2011.06.005CrossRefGoogle Scholar
Huang, J., Wen, C., Wang, W. and Jiang, Z. P., “Adaptive stabilization and tracking control of a nonholonomic mobile robot with input saturation and disturbance”, Syst. Control Lett. 62(3), 234–241 (2013).10.1016/j.sysconle.2012.11.020CrossRefGoogle Scholar
Samson, C., “Velocity and Torque Feedback Control of a Nonholonomic Cart”, In: Advanced Robot Control, Lecture Notes in Control and Information Sciences (Canudas de Wit, C., ed.) (Berlin, Heidelberg, 1991) pp. 125–151.10.1007/BFb0039269CrossRefGoogle Scholar
Fierro, R. and Lewis, F. L., “Control of a Nonholonomic Mobile Robot: Backstepping Kinematics into Dynamics”, Proceedings of the 34th IEEE Conference on Decision and Control, vol. 4, New Orleans, LA, USA (1995) pp. 3805–3810.Google Scholar
Do, K. D., “Global output-feedback path-following control of unicycle-type mobile robots: A level curve approach”, Rob. Auton. Syst. 74(A), 229–242 (2015). doi: 10.1016/j.robot.2015.07.019.CrossRefGoogle Scholar
Do, K. D., “Bounded controllers for global path tracking control of unicycle-type mobile robots”, Rob. Auton. Syst. 61(8), 775–784 (2013).10.1016/j.robot.2013.04.014CrossRefGoogle Scholar
Soetanto, D., Lapierre, L. and Pascoal, A., “Adaptive, Non-singular Path-Following Control of Dynamic Wheeled Robots”. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui (2003) pp. 1765–1770.Google Scholar
Do, K. D. and Pan, J., “Global output-feedback path tracking of unicycle-type mobile robots”, Rob. Comput. Integr. Manuf. 22(2), 166–179 (2006).10.1016/j.rcim.2005.03.002CrossRefGoogle Scholar
Do, K. D. and Pan, J., Control of Ships and Underwater Vehicles (Springer, London, 2009).10.1007/978-1-84882-730-1CrossRefGoogle Scholar
Aicardi, M., Casalino, G., Bicchi, A. and Balestrino, A., “Closed loop steering of unicycle-like vehicles via Lyapunov technique”, IEEE Rob. Autom. Mag. 2(1), 27–35 (1995).10.1109/100.388294CrossRefGoogle Scholar
Egerstedt, M., Hu, X. and Stotsky, A., “Control of mobile platforms using a virtual vehicle approach”, IEEE Trans. Autom. Control 46(11), 1777–1782 (2001).10.1109/9.964690CrossRefGoogle Scholar
Sun, S. and Cui, P., “Path tracking and a practical point stabilization of mobile robot”, Rob. Comput. Integr. Manuf. 20(1), 29–34 (2004).10.1016/S0736-5845(03)00052-8CrossRefGoogle Scholar
Khalil, H., Nonlinear Systems (Prentice Hall, New Jersey, 2002).Google Scholar
Krstic, M., Kanellakopoulos, I. and Kokotovic, P. V., Nonlinear and Adaptive Control Design (Wiley, New York, 1995).Google Scholar
Do, K. D., “Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers”, Automatica 106, 242–256 (2019). doi: 10.1016/j.automatica.2019.05.020.CrossRefGoogle Scholar
Applebaum, D., Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge University Press, New York, 2009).10.1017/CBO9780511809781CrossRefGoogle Scholar
Deng, H., Krstic, M. and Williams, R., “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance”, IEEE Trans. Autom. Control 46(8), 1237–1253 (2001).10.1109/9.940927CrossRefGoogle Scholar
Do, K. D., “Global inverse optimal stabilization of stochastic nonholonomic systems”, Syst. Control Lett. 75, 41–55 (2015). doi: 10.1016/j.sysconle.2014.11.003.CrossRefGoogle Scholar
Deng, H. and Krstic, M., “Stochastic nonlinear stabilization-part I: A backstepping design”, Syst. Control Lett. 32(3), 143–150 (1997).10.1016/S0167-6911(97)00068-6CrossRefGoogle Scholar
Wang, H. and Zhu, Q., “Global stabilization of stochastic nonlinear systems via C1 and C∞ controllers”, IEEE Trans. Autom. Control 62(11), 5880–5887 (2017).10.1109/TAC.2016.2644379CrossRefGoogle Scholar
Wang, H. and Zhu, Q., “Finite-time stabilization of high-order stochastic nonlinear systems in strict-feedback form”, Automatica 54, 284–291 (2015). doi: 10.1016/j.automatica.2015.02.016.CrossRefGoogle Scholar
Barbu, V., Cordoni, F. and Di Persio, L., “Optimal control of stochastic FitzHugh-Nagumo equation”, Int. J. Control 89(4), 746–756 (2016).10.1080/00207179.2015.1096023CrossRefGoogle Scholar
Bennett, J. and Wu, J. L., “An optimal control problem associated with SDEs driven by Lévy-type processes”, Stochastic Anal. Appl. 26, 471–494 (2008).10.1080/07362990802007004CrossRefGoogle Scholar
Tang, H. and Wu, Z., “Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes”, J. Syst. Sci. Complexity 22, 122–136 (2009). doi: 10.1007/s11424-009-9151-0.CrossRefGoogle Scholar
Singla, R., Parthasarathy, H., Agarwal, V. and Rana, R., “Feedback optimization problem for master–slave teleoperation tracking in the presence of random noise in dynamics and feedback”, Nonlinear Dyn. 86, 559–586 (2016). doi: 10.1007/s11071-016-2908-9.CrossRefGoogle Scholar
Jagtap, P. and Zamani, M., “Backstepping design for incremental stability of stochastic Hamiltonian systems with jumps”, IEEE Trans. Autom. Control 63(1), 255–261 (2018).10.1109/TAC.2017.2720592CrossRefGoogle Scholar
Do, K. D. and Nguyen, H. L., “Almost sure exponential stability of dynamical systems driven by Lévy processes and its application to control design for magnetic bearings”, Int. J. Control (2018). doi: 10.1080/00207179.2018.1482502.Google Scholar
Do, K. D., “Stochastic control of drill-heads driven by Lévy processes”, Automatica 103(2019), 36–45 (2019).10.1016/j.automatica.2019.01.016CrossRefGoogle Scholar
Sepulchre, R., Jankovic, M. and Kokotovic, P., Constructive Nonlinear Control (Springer, New York, 1997).10.1007/978-1-4471-0967-9CrossRefGoogle Scholar
Krstic, M. and Deng, H., Stabilization of Nonlinear Uncertain Systems (Springer, London, 1998).Google Scholar
Deng, H. and Krstic, M., “Stochastic nonlinear stabilization-part II: inverse optimality”, Syst. Control Lett. 32(3), 151–159 (1997).10.1016/S0167-6911(97)00067-4CrossRefGoogle Scholar
Do, K. D., “Inverse optimal control of stochastic systems driven by Lévy processes”, Automatica 107, 539–550 (2019). doi: 10.1016/j.automatica.2019.06.016.CrossRefGoogle Scholar
Fukao, T., Nakagawa, H. and Adachi, N., “Adaptive tracking control of a nonholonomic mobile robot”, IEEE Trans. Rob. Autom. 16(5), 609–615 (2000).10.1109/70.880812CrossRefGoogle Scholar
Park, W., Liu, Y., Zhou, Y., Moses, M. and Chirikjian, G. S., “Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map”, Robotica 26(4), 419–434 (2008).10.1017/S0263574708004475CrossRefGoogle ScholarPubMed
Thrun, S., Burgard, W. and Fox, D., Probabilistic Robotics (The MIT Press, Cambridge, 2005).Google Scholar
Barfoot, T. D. and Furgale, P. T., “Associating uncertainty with three-dimensional poses for use in estimation problems”, IEEE Trans. Robot. 30(3), 679–693 (2014).10.1109/TRO.2014.2298059CrossRefGoogle Scholar
Banks, H. T., Davis, J. L., Ernstberger, S. L., Hu, S., Artimovich, E., Dhar, A. K. and Browdy, C. L., “A comparison of probabilistic and stochastic formulations in modelling growth uncertainty and variability”, J. Biol. Dyn. 3(2–3), 130–148 (2014).10.1080/17513750802304877CrossRefGoogle Scholar
Øksendal, B. and Sulem, A., Applied Stochastic Control of Jump Diffusions, 2nd edn. (Springer, Berlin, 2007).10.1007/978-3-540-69826-5CrossRefGoogle Scholar
Mao, X., Stochastic Differential Equations and Applications, 2nd edn. (Woodhead publishing, Cambridge, 2007).Google Scholar
Do, K. D., “Inverse optimal formation control of mobile agents with collision avoidance driven by Lévy processes”, IFAC J. Syst. Control 13, 100097:1–13 (2020). doi: 10.1016/j.ifacsc.2020.100097.Google Scholar
Pan, Y., Bakshi, K. and Theodorou, E. A., “Robust trajectory optimization: A cooperative stochastic game theoretic approach”. Rob. Sci. Syst. (2015). doi: 10.15607/rss.2015.xi.029.Google Scholar
Bakshi, K. and Theodorou, E. A., “Infinite Dimensional Control of Doubly Stochastic Jump Diffusions”, Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas (2016) pp. 1145–1152.Google Scholar
Theodorou, E. A. and Todorov, E., “Stochastic Optimal Control for Nonlinear Markov Jump Diffusion Processes”, Proceedings of 2012 American Control Conference, Montréal, Canada (2012) pp. 1633–1639.Google Scholar
Do, K. D. and Pan, J., “Underactuated ships follow smooth paths with integral actions and without velocity measurements for feedback: Theory and experiments”, IEEE Trans. Control Syst. Technol. 14(2), 308–322 (2006).10.1109/TCST.2005.863665CrossRefGoogle Scholar
Hardy, G., Littlewood, J. E. and Polya, G., Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1989).Google Scholar
Krstic, M. and Li, Z. H., “Inverse optimal design of input-to-state stabilizing nonlinear controllers”. IEEE Trans. Autom. Control 43(3), 336–350 (1998).CrossRefGoogle Scholar