Skip to main content

Control and disturbances compensation in underactuated robotic systems using the derivative-free nonlinear Kalman filter

  • Gerasimos G. Rigatos (a1)

The Derivative-free nonlinear Kalman Filter is used for developing a robust controller which can be applied to underactuated MIMO robotic systems. The control problem for underactuated robots is non-trivial and becomes further complicated if the robot is subjected to model uncertainties and external disturbances. Using differential flatness theory it is shown that the model of a closed-chain 2-DOF robotic manipulator can be transformed to linear canonical form. For the linearized equivalent of the robotic system it is shown that a state feedback controller can be designed. Since certain elements of the state vector of the linearized system cannot be measured directly, it is proposed to estimate them with the use of a novel filtering method, the so-called Derivative-free nonlinear Kalman Filter. Moreover, by redesigning the Kalman Filter as a disturbance observer, it is shown that one can estimate simultaneously external disturbance terms that affect the robotic model or disturbance terms which are associated with parametric uncertainty. The efficiency of the proposed Kalman Filter-based control scheme is tested in the case of a 2-DOF planar robotic manipulator that has the structure of a closed-chain mechanism.

Corresponding author
*Corresponding author. E-mail:
Hide All
1. Franch J., Agrawal S. K. and Sangwan V., “Differential flatness of a class of -DOF planar manipulators driven by 1 or 2 actuators,” IEEE Trans. Autom. Control 55 (2), 548554 (2010).
2. Zhang C., Franch J. and Agrawal S. K., “Differentially flat design of a closed-chain planar underactuated 2-DOF system,” IEEE Trans. Robot. 29 (1), 277282 (2013).
3. Xin X., Tanaka S., She J. and Yamasaki T., “New analytical results of energy-based swing-up control for the Pendubot,” Int. J. Non-Linear Mech., 52, 110118 (2013).
4. Lai X.-Z., Pan C.-Z., Wu M. and Yang S. X., “Unified control of n-link underactuated manipulator with single passive joint: A reduced order approach,” Mech. Mach. Theory, Elsevier 56, 170185 (2012).
5. Lai X.-Z., She J.-H., Yang S. X. and Wu M., “Comprehensive unified control strategy for underactuated two-link manipulators,” IEEE Trans. Syst. Man Cybern. Part B: Cybern. 39 (2), 389398 (2009).
6. Narikiyoa T., Sahashib J. and Misao K., “Control of a class of underactuated mechanical systems,” Nonlinear Anal.: Hybrid Syst., Elsevier 2, 231241 (2008).
7. Mahindrakar A. D., Rao S. and Banavar R. N., “Point-to-point control of a 2R planar horizontal underactuated manipulator,” Mech. Mach. Theory, Elsevier 41, 838844 (2006).
8. Hong K. S., “An open-loop control for underactuated manipulators using oscillatory inputs: Steering capability of an unactuated joint,” IEEE Trans. Control Syst. Technol. 10 (3), 469479 (2002).
9. Rubio J. J., “Adaptive least square control in discrete time of robotic arms,” Soft Comput. (2014). DOI: 10.1007/s00500-014-1300-2.
10. Gao H., Song X., Ding L., Xia K. and Li N., “Adaptive motion control of wheeled mobile robot with unknown slippage,” Int. J. Control 87 (8), 15131522 (2014).
11. Torres C., Rubio J. J., Aguilar-Ibañez C. and Perez-Cruz J. H., “Stable optimal control applied to a cylindrical robotic arm,” Neural Comput. Appl. 24 (3–4), 937944 (2014).
12. Zhang Y., Yu X., Yin Y., Peng C. and Fan Z., “Singularity-conquering ZG controllers of z2g1 type for tracking control of the IPC systems,” Int. J. Control 87 (9), 17291746 (2014).
13. Rubio J. J., Zamudio Z., Pacheco J. and Mujica-Vargas D., “Proportional derivative control with inverse dead-zone for pendulum systems,” Math. Problems Eng. 2013, 19 (2013).
14. Ri S. H., Huang J., Wang Y., Kim M. H. and An S., “Terminal sliding mode control of mobile wheeled inverted pendulum system with nonlinear disturbance observer,” Math. Problems Eng. 2014, 18 (2014).
15. Agrawal S. K. and Sangwan V., “Differentially flat designs of underactuated open-chain planar robots,” IEEE Trans. Robot. 24 (6), 14451451 (2008).
16. Franch J., Reyes A. and Agrawal S. K., “Differential Flatness of a Class of n-DOF Planar Manipulators Driven by An Arbitrary Number of Cctuators,” Proceedings of the 2013 European Control Conference, July 2013, Zurich, Switzerland.
17. Rudolph J., Flatness Based Control of Distributed Parameter Systems, Examples and Computer Exercises from Various Technological Domains (Shaker Verlag, Aachen, 2003).
18. Sira-Ramirez H. and Agrawal S., Differentially Flat Systems (Marcel Dekker, New York, 2004).
19. Rigatos G. G., Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotcs and Industrial Engineering (Springer, 2011).
20. Lévine J., “On necessary and sufficient conditions for differential flatness,” Appl. Algebra Eng. Commun. Comput., Springer 22 (1), 4790 (2011).
21. Fliess M. and Mounier H., “Tracking Control and π-Freeness of Infinite Dimensional Linear Systems,” In: Dynamical Systems, Control, Coding and Computer Vision (Picci G. and Gilliam D. S., eds.), vol. 258 (Birkhaüser, 1999) pp. 4168.
22. Villagra J., d'Andrea-Novel B., Mounier H. and Pengov M., “Flatness-based vehicle steering control strategy with SDRE feedback gains tuned via a sensitivity approach,” IEEE Trans. Control Syst. Technol. 15, 554565 (2007).
23. Laroche B., Martin P. and Petit N., Commande par platitude: Equations différentielles ordinaires et aux derivées partielles (Ecole Nationale Supérieure des Techniques Avancées, Paris, 2007).
24. Martin Ph. and Rouchon P., Systèmes plats: Planification et suivi des trajectoires, Journées X-UPS, École des Mines de Paris, Centre Automatique et Systèmes, Mai (1999).
25. Bououden S., Boutat D., Zheng G., Barbot J. P. and Kratz F., “A triangular canonical form for a class of 0-flat nonlinear systems,” Int. J. Control, Taylor and Francis 84 (2), 261269 (2011).
26. Rigatos G., Siano P. and Zervos N., “PMSG Sensorless Control with the Use of the Derivative-Free Nonlinear Kalman Filter,” IEEE ICCEP 2013, IEEE International Conference on Clean Electrical Power, Alghero, Sardinia Italy, Jun. 2013.
27. Rigatos G. G., “A derivative-free Kalman Filtering approach to state estimation-based control of nonlinear dynamical systems,” IEEE Trans. Ind. Electron. 59 (10), 39873997 (2012).
28. Rigatos G. G., “Nonlinear Kalman filters and particle filters for integrated navigation of unmanned aerial vehicles,” Robot. Auton. Syst. Elsevier, 2012.
29. Chen W. H., Ballance D. J., Gawthrop P. J. and Reilly J. O., “A nonlinear disturbance observer for robotic manipulators,” IEEE Trans. Ind. Electron. 47 (4), 932938 (2000).
30. Cortesao R., Park J. and Khatib O., “Real-time adaptive control for haptic telemanipulation with Kalman Active Observers,” IEEE Trans. Robot. 22 (5), 987999 (2005).
31. Cortesao R., “On Kalman active observers,” J. Intell. Robot. Syst., Springer 48 (2), 131155 (2006).
32. Gupta A. and Malley M. K. O., “Disturbance-observer-based force estimation for haptic feedback,” ASME J. Dyn. Syst. Meas. Control 133 (1), article no, 014505 (2011).
33. Ohnishi K., “Disturbance observation - Cancellation technique,” In: Control and Mechatronics (Wilamowski B. M. and Irwin J. D., eds.) (CRC Press, 2010).
34. Miklosovic R., Radke A. and Gao Z., “Discrete Implementation and Generalization of the Extended State Observer,” Proceedings of the 2006 Americal Control Conference, Minneapolis, Minnesota, USA, 2006.
35. Rigatos G., Nonlinear Control and Filtering using Differential Flatness Approaches: Applications to Electromechanical Systems (Springer, 2015).
36. Khalil H. K., Nonlinear Systems, 2nd ed. (Prentice Hall, 1996).
37. Harris C., Hong X. and Gan Q., Adaptive Modelling, Estimation and Fusion From Data (Springer, 2002).
38. Basseville M. and Nikiforov I., Detection of Abrupt Changes: Theory and Applications (Prentice-Hall, 1993).
39. Rigatos G. and Zhang Q., Fuzzy Model Validation using the Local Statistical Approach (Publication Interne IRISA No 1417, Rennes, France, 2001).
40. Kamen E. W. and Su J. K., Introduction to Optimal Estimation (Springer, 1999).
41. Toussaint G. J., Basar T. and Bullo F., “H Optimal Tracking Control Techniques for Nonlinear Underactuated Systems,” Proceedings of the IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia (Dec. 2000).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0263-5747
  • EISSN: 1469-8668
  • URL: /core/journals/robotica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 40 *
Loading metrics...

Abstract views

Total abstract views: 213 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 26th February 2018. This data will be updated every 24 hours.