Control of non holonomic or under-actuated mechanical systems: The examples of the unicycle robot and the slider

d’Andréa-Novel, Brigitte; Thorel, Sylvain

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 983-1016.

This paper is dedicated to the control of some classes of mechanical systems, not stabilizable by means of at least continuous state feedback laws. This is the case for non holonomic mechanical systems, an example being the unicycle robot, or for under-actuated mechanical systems, an example being the slider. The similarity between these two kinds of dynamical systems will be analyzed, with respect to controllability, stabilizability and flatness properties. The control design for slider stabilization could be viewed, to some extent, as an extension of the control law used for unicycle systems, but it is not so obvious if one wants to stabilize the slider at a chosen reference position, with a chosen fixed orientation. In fact, a new fixed point stabilization result will be given for the slider, and therefore, a switched control strategy will be proposed for both systems, based on differential flatness property to track moving non singular reference trajectories, and switching to periodic time-varying feedback laws to ensure final stabilization at rest equilibrium points. The method will be illustrated with success through simulation results and also first experimental results in the case of a terrestrial quadrotor tracking non singular reference trajectories.

Classification : 93B, 93C, 93D
Mots clés : Non holonomic mechanical systems, under-actuated systems, fixed point stabilization, trajectory tracking, persistence of excitation condition, differential flatness property, time-varying feedback

B. d’Andréa-Novel and M. De Lara, Control Theory for Engineers. Springer (2013).

B. d’Andréa-Novel, G. Bastin and G. Campion, Dynamic feedback linearization of non holonomic wheeled mobile robots. Proc. of the IEEE Conference on Robotics and Automation. Nice (1992) 2527–2532.

B. D’Andréa-Novel, G. Campion and G. Bastin, Control of Non holonomic Wheeled Mobile Robots by State Feedback Linearization. Int. J. Robot. Res. 14 (1995) 543–559. | DOI

E. Aranda-Bricaire, C.H. Moog and J.B. Pomet, A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control 40 (1995) 127–132. | DOI | Zbl

R.W. Brockett, Asymptotic stability and feedback stabilization, Differential geometric control theory (Houghton, Mich., 1982). Vol. 27 of Progr. Math. Birkhauser, Boston, Boston, MA (1983) 181–191. MR 708502 (85e:93034). | Zbl

G. Campion, B. d’Andréa-Novel and G. Bastin, Modelling and state feedback control of non holonomic mechanical systems. Proc. of the 30th IEEE CDC (1991) 1184–1189.

C. Canudas de Wit, B. Siciliano and G. Bastin, The ZODIAC, Theory of Robot Control. Springer, 1st edition (1997). | Zbl

B. Charlet, Sur quelques problèmes de stabilisation robuste des systèmes non linéaires. Ph.D. thesis, École Nationale Supérieure des Mines de Paris (1989).

B. Charlet, J. Lévine and R. Marino, On dynamic feedback linearization. Syst. Control Lett. 13 (1989) 143–151. | DOI | Zbl

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signal Syst. 5 (1992) 295–312. | DOI | Zbl

J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law. SIAM J. Control Optim. 33 (1995) 804–833. | DOI | Zbl

J.-M. Coron, Phantom tracking method, homogeneity and rapid stabilization. Math. Control Related Fields 3 (2013) 303–322. | DOI | Zbl

J.-M. Coron, Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society (2007). | Zbl

J.-M. Coron and B. d’Andréa-Novel, Smooth stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots. Proc. of the NOLCOS Conference. Bordeaux (1992) 649–654.

J. Descusse and C.H. Moog, Decoupling with dynamic compensation for strong invertible affine nonlinear systems. Int. J. Control 42 (1985) 1387–1398. | DOI | Zbl

R. Lozano, Objets volants miniatures. Hermes Science, Lavoisier (2007).

I. Fantoni, R. Lozano, F. Mazenc and K.Y. Pettersen, Stabilization of a nonlinear under-actuated hovercraft. Proc. of the 38th IEEE CDC (1999). | Zbl

M. Fliess, J. Lévine, P. Martin and P. Rouchon, On differentially flat nonlinear systems. Proc. of the 2nd IFAC NOLCOS Symposium. Bordeaux (1992) 408–412. | Zbl

M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61 (1995) 1327–1361. | DOI | Zbl

W. Hahn, Stability of motion. Springer, Berlin, Heidelberg, New-York (1967). | Zbl

A. Isidori, Nonlinear Control Systems. Springer-Verlag, 3rd Edition, London (1995). | Zbl

Y. Kanayama, Y. Kimura, F. Miyazaki and T. Noguchi, A stable tracking control method for an autonomous mobile robot. Proc. of the IEEE Int. Conf. Robot. Automat. (1990) 384–389.

H.K. Khalil, Nonlinear systems. Prentice Hall, 2nd Edition (1995). | Zbl

G. Kern, Uniform controllability of a class of linear time-varying systems. IEEE Trans. Automat. Control 27 (1982) 208–210. | DOI | Zbl

P. Kokotović and H. Sussmann, A positive real condition for global stabilization of nonlinear systems. Syst. Control Lett. 13 (1989) 125–133. | DOI | Zbl

I. Kolmanovsky and N.H. Mcclamroch, Developments in non holonomic control problems. IEEE Control Syst. 15 (1995) 20-36. | DOI

E. Lefeber, K.Y. Pettersen and H. Nijmeijer, Tracking control of an under-actuated ship. IEEE Trans. Control System Technol. 11 (2003) 52–61. | DOI

D.A. Lizárraga. Obstructions to the Existence of Universal Stabilizers for Smooth Control Systems. Math. Control Signals Syst 16 (2004) 255–277. | DOI | Zbl

A. Micaelli, B. d’Andréa-Novel and B. Thuilot, Modeling and asymptotic stabilization of mobile robots equipped with two or more steering wheels. Proc. of ICARCV’92. Singapour (1992).

P. Morin and C. Samson, Control of nonlinear chained systems. From the Routh-Hurwitz stability criterion to time-varying exponential stabilizers. Rapport INRIA 3126 (1997). | Zbl

P. Morin and C. Samson, Control of nonlinear chained systems. From the Routh-Hurwitz stability criterion to time-varying exponential stabilizers. Proc. of the 36th IEEE CDC 1 (1997) 618–623. | Zbl

P. Morin and C. Samson, Control of under-actuated mechanical systems by the transverse function approach. In 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05 (2005) 7508–7513.

P. Morin and C. Samson,Transverse functions on special orthogonal groups for vector fields satisfying the LARC at the order one. In Proc. of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC (2009) 7472–7477.

R.M. Murray and S.S. Sastry, Steering non holonomic systems in chained form. In vol. 2, Proc. of the 30th IEEE CDC (1991) 1121–1126.

R.M. Murray, Z. Li, S. Shankar Sastry and S. Shankara Sastry, A mathematical introduction to robotic manipulation. CRC press (1994). | Zbl

K.Y. Pettersen and O. Egeland, Exponential stabilization of an under-actuated surface vessel. Proc. of the 35th IEEE CDC (1996) 967–972. | Zbl

K.Y. Pettersen and O. Egeland, Robust control of an under-actuated surface vessel with thruster dynamics. Proc. of the ACC (1997).

K.Y. Pettersen and H. Nijmeijer, Global practical stabilization and tracking for an under-actuated ship – a combined averaging and backstepping approach. Model. Identif. Control 20 (1999) 189–200. | DOI

J.-B. Pomet, B. Thuilot, G. Bastin and G. Campion, A hybrid strategy for the feedback stabilization of non holonomic mobile robots, Proc. of the IEEE Conf. on Robotics and Automation. Nice (1992) 129–134.

L. Praly, B. d’Andréa-Novel and J.-M. Coron, Lyapunov design of stabilizing controllers for cascaded systems. In vol. 36, IEEE Trans. Automat. Control (1991) 10.

M. Reyhanoglu. Control and stabilization of an under-actuated surface vessel. In vol. 3, Proc. of the 35th IEEE Conf. Decision Control (1996) 2371–2376.

P. Rouchon, Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. Control (1994) 345–358. | Zbl

C. Samson, Path following and time-varying feedback stabilization of a wheeled mobile robot. In vol. 13, Proc. of the International Conference on Advanced Robotics and Computer Vision. Singapour (1992) 1.1–1.5.

C. Samson, Control of chained systems. Application to path following and time-varying feedback stabilization of mobile robots. IEEE Trans. Automat. Control 40 (1995) 64–77. | DOI | Zbl

C. Samson, K. Ait-Abderrahim, Mobile robot control. Part 1: feedback control of non holonomic wheeled cart in cartesian space. Rapport de recherche RR-1288, INRIA (1990).

R. Sepulchre, M. Janković, P. Kokotović, Constructive Nonlinear Control. Springer (1997). | Zbl

S. Thorel, Conception et réalisation d’un drone hybride sol/air autonome. Ph.D. thesis, MINES ParisTech (2014).

S. Thorel and B. d’Andréa-Novel, Hybrid terrestrial and aerial quadrotor control. Proc. of the 19th IFAC World Congress. South Africa (2014).

S. Thorel and B. d’Andréa-Novel, Practical identification and flatness based control of a terrestrial quadrotor. Proc. of the IROS conference. Chicago (2014).

B. Thuilot, B. D’Andréa-Novel and A. Micaelli, Modeling and Feedback Control of Mobile Robots Equipped with Several Steering Wheels. IEEE Trans. Robot. Automat. 12 (1996) 375–390. | DOI

J. Tsinias, Sufficient Lyapunov-like conditions for stabilization. Math. Control Signals Syst. 2 (1989) 343–357. | DOI | Zbl