Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T00:43:01.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic optimal enhancement of distributed formation control using Kalman smoothers

Published online by Cambridge University Press:  31 January 2014

Ross P. Anderson  and
Dejan Milutinović
Ross P. Anderson
Affiliation:
Department of Applied Mathematics and Statistics, Mail Stop SOEGrad, University of California, Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA
Dejan Milutinović*
Affiliation:
Computer Engineering Department, Mail Stop SOE2, University of California, Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA
*
*Corresponding author. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Summary

Beginning with a deterministic distributed feedback control for nonholonomic vehicle formations, we develop a stochastic optimal control approach for agents to enhance their non-optimal controls with additive correction terms based on the Hamilton–Jacobi–Bellman equation, making them optimal and robust to uncertainties. In order to avoid discretization of the high-dimensional cost-to-go function, we exploit the stochasticity of the distributed nature of the problem to develop an equivalent Kalman smoothing problem in a continuous state space using a path integral representation. Our approach is illustrated by numerical examples in which agents achieve a formation with their neighbors using only local observations.

Keywords

Multi-robot systemsControl of robotic systemsMotion planningMobile robots
Type
Articles
Information
Robotica , Volume 32 , Issue 2: Distributed Autonomous Robotic Systems (DARS). , March 2014 , pp. 305 - 324
Copyright
Copyright © Cambridge University Press 2014 

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

1.Parker, L. E., “Multiple Mobile Robot Systems,” In: Springer Handbook of Robotics (Sciliano, B. and Khatib, O., eds.) (Springer-Verlag, Berlin, Germany, 2008), Ch. 40, pp. 921–941.Google Scholar
2.Proud, A. W., Pachter, M. and D'Azzo, J. J., “Close Formation Flight Control,” Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Portland, OR (1999) pp. 1231–1246.Google Scholar
3.Anderson, B., Fidan, B., Yu, C. and Walle, D., “UAV Formation Control: Theory and Application,” In: Recent Advances in Learning and Control (Blondel, V., Boyd, S., and Kimura, H., eds.) (Springer-Verlag, London, 2008) pp. 15–34.Google Scholar
4.Tanner, H., Jadbabaie, A. and Pappas, G., “Coordination of Multiple Autonomous Vehicles,” Proceedings of the IEEE Mediterranean Conference on Control and Automation, Rhodes, Greece (2003) pp. 3448–3453.Google Scholar
5.Elkaim, G. and Kelbley, R., “A Lightweight Formation Control Methodology for a Swarm of Non-Holonomic Vehicles,” IEEE Aerospace Conference, Big Sky, MT (2006) pp. 1–8.Google Scholar
6.Paul, T., Krogstad, T. R. and Gravdahl, J. T., “UAV Formation Flight Using 3D Potential Field,” Proceedings of the 16th Mediterranean Conference on Control and Automation, Ajaccio, France (2008) pp. 1240–1245.Google Scholar
7.Roussos, G., Dimarogonas, D. V. and Kostas, K. J., “3D navigation and collision avoidance for nonholonomic aircraft-like vehicles,” Int. J. Adapt. Control 24, 900–920 (2010).Google Scholar
8.Zou, Y. and Pagilla, P. R., “Distributed formation flight control using constraint forces,” J Guide. Control Dynam. 32 (1), 112–120 (2009).Google Scholar
9.Bullo, F., Cortes, J. and Martinez, S., Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms (Princeton University Press, Princeton, NJ, 2009).Google Scholar
10.Singh, S. N., Chandler, P., Schumacher, C., Banda, S. and Pachter, M., “Nonlinear adaptive close formation control of unmanned aerial vehicles,” Dynam. Control 10 (2), 179–194 (2000).Google Scholar
11.Sattigeri, R., Calise, A. J. and Evers, J. H., “An Adaptive Vision-Based Approach to Decentralized Formation Control,” Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, RI (2004) pp. 2575–2798.Google Scholar
12.Galzi, D. and Shtessel, Y., “UAV Formations Control Using High Order Sliding Modes,” Proceedings of the 2006 American Control Conference, Minneapolis, MN (2006) pp. 4249–4254.Google Scholar
13.Ren, W. and Beard, R., Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications (Springer-Verlag, New York, NY, 2007).Google Scholar
14.Dimarogonas, D. V., “On the rendezvous problem for multiple nonholonomic agents,” IEEE T. Automat. Control 52 (5), 916–922 (2007).Google Scholar
15.van Kampen, N. G., Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier B.V., Amsterdam, Netherlands, 2007).Google Scholar
16.Long, A. W., Wolfe, K. C., Mashner, M. J. and Chirikjian, G. S., “The Banana Distribution is Gaussian: A Localization Study with Exponential Coordinates,” Proceedings of Robotics: Science and Systems, Syndey, Australia (2012) pp. 265–272.Google Scholar
17.Wang, M. C. and Uhlenbeck, G., “On the theory of Brownian Motion II,” Rev. Mod. Phys. 17 (2–3), 323–342 (1945).Google Scholar
18.Dunbar, W. B. and Murray, R. M., “Distributed receding horizon control for multi-vehicle formation stabilization,” Automatica 42 (5), 549–558 (2006).Google Scholar
19.Freidlin, M., Functional Integration and Partial Differential Equations (Princeton University Press, Princeton, NJ, 1985).Google Scholar
20.Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, 6th ed. (Springer-Verlag, Berlin, Germany, 2003).Google Scholar
21.Yong, J., “Relations among ODEs, PDEs, FSDEs, BDSEs, and FBSDEs,” Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA (1997) pp. 2779–2784.Google Scholar
22.Kappen, H., “Linear theory for control of nonlinear stochastic systems,” Phys. Rev. Lett. 95 (20), 1–4 (2005).Google Scholar
23.Kappen, H. J., “Path integrals and symmetry breaking for optimal control theory,” J. Stat. Mech. Theory E. 2005 (21), (2005) pp. 1–25.Google Scholar
24.Todorov, E., “General Duality Between Optimal Control and Estimation,” Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico (2008) pp. 4286–4292.Google Scholar
25.Todorov, E., “Efficient computation of optimal actions.” P. Natl. Acad. Sci. USA 106 (28), 11478–11483 (2009).Google Scholar
26.Kappen, H. J., Gómez, V. and Opper, M., “Optimal control as a graphical model inference problem,” Mach. Learn. 87 (2), 159–182 (2012).Google Scholar
27.Milutinović, D., “Utilizing Stochastic Processes for Computing Distributions of Large-Size Robot Population Optimal Centralized Control,” Proceedings of the 10th International Symposium on Distributed Autonomous Robotic Systems, Lausanne, Switzerland (2010).Google Scholar
28.Palmer, A. and Milutinović, D., “A Hamiltonian Approach Using Partial Differential Equations for Open-Loop Stochastic Optimal Control,” Proceedings of the 2011 American Control Conference, San Francisco, CA (2011) pp. 2056–2061.Google Scholar
29.van den Broek, B., Wiegerinck, W. and Kappen, B., “Graphical model inference in optimal control of stochastic multi-agent systems,” J. Artif. Intell. Res. 32 (1), 95–122 (2008).Google Scholar
30.van den Broek, B., Wiegerinck, W. and Kappen, B., “Optimal Control in Large Stochastic Multi-Agent Systems,” Adaptive Agents and Multi-Agent Systems III Adaptation and Multi-Agent Learning, LNAI, vol. 4865. Springer-Verlag, Berlin, Germany, pp. 15–26 (2008).CrossRefGoogle Scholar
31.Wiegerinck, W., Broek, B. and Kappen, H., “Stochastic Optimal Control in Continuous Space-Time Multi-Agent Systems,” Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, Cambridge, MA (2006) pp. 528–535.Google Scholar
32.Wiegerinck, W., van den Broek, B. and Kappen, B., “Optimal On-Line Scheduling in Stochastic Multiagent Systems in Continuous Space-Time,” Proceedings of the 6th International Joint Conference on Autonomous Agents and Multiagent Systems (2007).Google Scholar
33.Anderson, R. P. and Milutinović, D., “A Stochastic Optimal Enhancement of Feedback Control for Unicycle Formations,” Proceedings of the 11th International Symposium on Distributed Autonomous Robotic Systems (DARS), Baltimore, MD (2012) pp. 608–615.Google Scholar
34.Anderson, R. P. and Milutinović, D., “Distributed Path Integral Feedback Control Based on Kalman Smoothing for Unicycle Formations,” Proceedings of the 2013 American Control Conference, Washington, DC (2013) pp. 4611–4616.Google Scholar
35.Jadbabaie, A. and Hauser, J., “On the Stability of Unconstrained Receding Horizon Control with a General Terminal Cost,” Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL (2001) pp. 4826–4831.Google Scholar
36.Stengel, R. F., Optimal Control and Estimation (Dover, New York, NY, 1994).Google Scholar
37.Bouilly, B., Simeon, T. and Alami, R., “A Numerical Technique for Planning Motion Strategies of a Mobile Robot in Presence of Uncertainty,” Proceedings of the 1995 IEEE International Conference on Robotics and Automation. Nagoya, Japan (1995) pp. 1327–1332.CrossRefGoogle Scholar
38.Fraichard, T. and Mermond, R., “Path Planning with Uncertainty for Car-Like Robots,” Proceedings of the 1998 IEEE International Conference on Robotics & Automation, Leuven, Belgium (1998) pp. 27–32.Google Scholar
39.Lambert, A. and Gruyer, D., “Safe Path Planning in an Uncertain-Configuration Space,” Proceedings of the 2003 International Conference on Robotics and Automation, Taipei, Taiwan (Sep. 14–19, 2003) pp. 85–90.Google Scholar
40.van den Berg, J., Abbeel, P. and Goldberg, K., “LQG-MP: Optimized path planning for robots with motion uncertainty and imperfect state information,” Int. J. Robot. Res. 30 (7), 895–913 (2011).Google Scholar
41.van den Berg, J., Patil, S., Alterovitz, R., Abbeel, P. and Goldberg, K., “LQG-Based Planning, Sensing, and Control of Steerable Needles,” Vol. 68, Springer Tracts in Advanced Robotics: Algorithmic Foundations of Robotics IX (Hsu, D., Isler, V., Latombe, J.-C., Lin, M. C., eds.) (Springer-Velag, Berlin, Germany, 2011), pp. 373–389.Google Scholar
42.Mayne, D. Q., Rawlings, J. B., Rao, C. V. and Scokaert, P. O. M., “Constrained model predictive control: Stability and optimality,” Automatica 36 (6), 789–814 (2000).Google Scholar
43.Bertsekas, D., “Dynamic programming and suboptimal control: A survey from ADP to MPC,” Eur. J. Control 11 (4–5), 310–334 (2005).Google Scholar
44.Chao, Z., Zhou, S.-L., Ming, L. and Zhang, W.-G., “UAV formation flight based on nonlinear model predictive control,” Math. Probl. Eng. 2012, (2012) pp. 1–15 (and references therein).Google Scholar
45.Nascimento, T. P., Moreira, A. P. and Conceção, A. G. S., “Multi-robot nonlinear model predictive formation control: Moving target and target absence,” Robot. Auton. Syst. 61 (12), 1502–1515 (2013).Google Scholar
46.Fleming, W. and Soner, H., “Logarithmic Transformations and Risk Sensitivity,” In: Controlled Markov Processes and Viscosity Solutions (Springer, Berlin, Germany, 1993), Ch. 6, pp. 227–259.Google Scholar
47.Goldstein, H., Poole, C. P. Jr. and Safko, J. L., Classical Mechanics, 3rd ed. (Addison-Wesley, San Francisco, CA, 1980).Google Scholar
48.Gelb, A., Applied Optimal Estimation. (MIT Press, Cambridge, MA, 1974).Google Scholar
49.Särkkä, S., “Continuous-time and continuous-discrete-time unscented Rauch-Tung-Striebel smoothers,” Signal Process. 90 (1), 225–235 (2010).Google Scholar
50.Särkkä, S., “EKF/UKF Toolbox for Matlab V1.3,” available at: http://becs.aalto.fi/en/esearch/bayes/ekfukf/. Accessed November 2011.Google Scholar
51.Wan, E. A. and van der Merwe, R., “Unscented Kalman Filter,” In: Kalman Filtering and Neural Networks (Haykin, S., ed.) (John Wiley, New York, NY, 2000), Ch. 7, pp. 221–282.Google Scholar
52.Kushner, H. J. and Dupuis, P., Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd ed. (Springer, New York, NY, 2001).Google Scholar
53.Wächter, A. and Biegler, L. T., “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106 (1), 25–57, (2006).Google Scholar