Learning stable nonlinear dynamical systems with Gaussian mixture models

IEEE Transactions on Robotics

Volumn 27, Issue 5, 2011, Pages 943-957

  Khansari Zadeh, S Mohammad ^a   Billard, Aude ^a  

^a EPFL   (Switzerland)

Author keywords

Dynamical systems (DS); Gaussian mixture model; imitation learning; point to point motions; stability analysis

Indexed keywords

GAUSSIAN MIXTURE MODEL; GLOBAL ASYMPTOTIC STABILITY; IMITATION LEARNING; LEARNING METHODS; POINT-TO-POINT MOTION; ROBOT MOTION; STABILITY ANALYSIS; SUFFICIENT CONDITIONS; TIME INVARIANTS;

ASYMPTOTIC STABILITY; COMMUNICATION CHANNELS (INFORMATION THEORY); NONLINEAR DYNAMICAL SYSTEMS; OBJECT RECOGNITION; SYSTEM STABILITY;

DYNAMICAL SYSTEMS;

EID: 80053623760     PISSN: 15523098     EISSN: None     Source Type: Journal    
DOI: 10.1109/TRO.2011.2159412     Document Type: Article

References (34)

1. 1] S. Schaal, "Is imitation learning the route to humanoid robots?" Trends Cognitive Sci., vol. 3, no. 6, pp. 233-242, 1999. (Pubitemid 29261903)
2. D. Kulic,W. Takano, and Y. Nakamura, "Incremental learning, clustering and hierarchy formation of whole body motion patterns using adaptive hidden Markov chains," Int. J. Robot. Res., vol. 27, no. 7, pp. 761-784, 2008.
3. A. Billard, S. Calinon, R. Dillmann, and S. Schaal, "Robot programming by demonstration," in Handbook of Robotics, Cambridge,MA: MIT Press, 2008.
4. K. Dautenhahn and C. L. Nehaniv, The Agent-Based Perspective on Imitation. Cambridge, MA: MIT Press, 2002, pp. 1-40.
5. S.-M. Khansari-Zadeh and A. Billard, "Imitation learning of globally stable non-linear point-to-point robot motions using nonlinear programming," in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2010, pp. 2676-2683.
6. E. D. Sontag, "Input to state stability: Basic concepts and results," in Nonlinear and Optimal Control Theory, Berlin/Heidelberg, Germany: Springer-Verlag, 2008, ch. 3, pp. 163-220.
7. J.-H. Hwang, R. Arkin, and D.-S. Kwon, "Mobile robots at your fingertip: Bezier curve on-line trajectory generation for supervisory control," in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., vol. 2, 2003, pp. 1444- 1449.
8. R. Andersson, "Aggressive trajectory generator for a robot ping-pong player," IEEE Control Syst. Mag., vol. 9, no. 2, pp. 15-21, Feb. 1989.
9. J. Aleotti and S. Caselli, "Robust trajectory learning and approximation for robot programming by demonstration," Robot. Auton. Syst., vol. 54, pp. 409-413, 2006.
10. A. Ude, "Trajectory generation from noisy positions of object features for teaching robot paths," Robot. Auton. Syst., vol. 11, no. 2, pp. 113-127, 1993.
11. M. Muehlig,M. Gienger, S. Hellbach, J. Steil, and C. Goerick, "Task level imitation learning using variance-basedmovement optimization," in Proc. IEEE Int. Conf. Robot. Autom., 2009, pp. 1177-1184.
12. K. Yamane, J. J. Kuffner, and J. K. Hodgins, "Synthesizing animations of human manipulation tasks," ACM Trans. Graph., vol. 23, no. 3, pp. 532- 539, 2007.
13. A. Coates, P. Abbeel, and A. Y. Ng, "Learning for control from multiple demonstrations," in Proc. 25th Int. Conf. Mach. Learning, 2008, pp. 144- 151.
14. M. Hersch, F. Guenter, S. Calinon, and A. Billard, "Dynamical system modulation for robot learning via kinesthetic demonstrations," IEEE Trans. Robot., vol. 24, no. 6, pp. 1463-1467, Dec. 2008.
15. C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning. New York: Springer-Verlag, 2006.
16. S. Schaal, C. Atkeson, and S. Vijayakumar, "Scalable locally weighted statistical techniques for real time robot learning," Appl. Intell.-Spec. Issue Scalable Robot. Appl. Neural Netw., vol. 17, no. 1, pp. 49-60, 2002. (Pubitemid 34789897)
17. A. Dempster and N. L. D. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Statist. Soc. B, vol. 39, no. 1, pp. 1-38, 1977.
18. S.-M. Khansari-Zadeh andA.Billard, "BM: An iterative algorithm to learn stable non-linear dynamical systems with Gaussian Mixture Models," in Proc. Int. Conf. Robot. Autom., 2010, pp. 2381-2388.
19. J. Slotine andW. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.
20. P. Pastor, H. Hoffmann, T. Asfour, and S. Schaal, "Learning and generalization of motor skills by learning from demonstration," in Proc. Int. Conf. Robot. Autom., 2009, pp. 1293-1298.
21. E. Gribovskaya, S. M. Khansari-Zadeh, and A. Billard, "Learning nonlinear multivariate dynamics of motion in robotic manipulators," Int. J. Robot. Res., vol. 30, pp. 1-37, 2010.
22. S. Calinon, F. D'halluin, E. Sauser, D. Caldwell, and A. Billard, "Learning and reproduction of gestures by imitation: An approach based on hidden Markov model and Gaussian mixture regression," IEEE Robot. Autom. Mag., vol. 17, no. 2, pp. 44-54, Jun. 2010.
23. E. Gribovskaya and A. Billard, "Learning nonlinear multi-variate motion dynamics for real-time position and orientation control of robotic manipulators," in Proc. 9th IEEE-RAS Int. Conf. Humanoid Robots, 2009, pp. 472-477.
24. G. McLachlan and D. Peel, Finite Mixture Models. New York: Wiley, 2000.
25. G. Schwarz, "Estimating the dimension of a model," Ann. Statist., vol. 6, pp. 461-464, 1978.
26. H. Akaike, "A new look at the statistical model identification, " IEEE Trans. Automat. Control, vol. AC-19, no. 6, pp. 716-723, Dec. 1974.
27. D. J. Spiegelhalter, N. G. Best, B. P. Carlin, and A. van der Linde, "Bayesian measures of model complexity and fit," J. R. Statist. Soc., Series B (Statist. Methodol.), vol. 64, pp. 583-639, 2002.
28. D. Nguyen-Tuong, M. Seeger, and J. Peters, "Local Gaussian process regression for real time online model learning and control," in Proc. Adv. Neural Inf. Process. Syst. 21, 2009, pp. 1193-1200.
29. E. Snelson and Z. Ghahramani, "Sparse Gaussian processes using pseudoinputs," in Proc. Adv. Neural Inf. Process. Syst. 18, 2006, pp. 1257-1264.
30. M. Seeger, C. K. I. Williams, and N. D. Lawrence, "Fast forward selection to speed up sparse Gaussian Process Regression," in Proc. 9th Int. Workshop Artif. Intell. Statist., 2003.
31. D. Cohn and Z. Ghahramani, "Active learning with statistical models," Artif. Intell. Res., vol. 4, pp. 129-145, 1996. (Pubitemid 126646151)
32. M. S. Bazaraa, H. Sherali, and C. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd ed. New York: Wiley, 2006.
33. S.-M. Khansari-Zadeh and A. Billard. (2010). "The derivatives of the SEDS optimization cost function and constraints with respect to its optimization parameters," Ecole Polytech. Fed. Lausanne, Lausanne, Switzerland, Tech. Rep., [Online]. Available: http://lasa.epfl.ch/publications/ publications.php
34. B. D. Argall, E. Sauser, and A. Billard, "Tactile guidance for policy refinement and reuse," in Proc. 9th IEEE Int. Conf.Development Learning, 2010, pp. 7-12.