Brigitte d’ ANDRÉA-NOVEL MINES ParisTech, PSL-Research University, Centre de robotique, 60 Bd St Michel 75006 Paris Sylvain THOREL SAGEM, 100 avenue de Paris, 91344 Massy Cedex 1
� Systems not stabilizable by means of continuous state feedback laws (Brockett condition) � Similarities concerning Controllability , Stabilizability, Flatness properties � Two different approaches for trajectory tracking of nonsingular reference trajectories and for fixed point stabilization 2
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Idem for the slider : 4
The Unicycle : the TL system is controllable if The slider : the TL system is controllable if 5
The non controllability of the tangent linearized system at a fixed point does not necessarily imply that the original nonlinear system does not satisfy the STLC property. In fact, it can be shown that the LARC is satisfied for both systems, as well as the STLC property. STLC is satisfied for the two systems at each trajectory STLC is satisfied for the two systems at each trajectory respectively BUT Let us consider a neighborhood of the origin It cannot be in the image of the unicycle dynamics at the origin. The same holds for the slider. Therefore, due to Brockett’s theorem, these two systems cannot be stabilized at fixed equilibrium points by means of continuous state feedback laws. 6
[G. Kern, « Uniform controllability of a class of linear time-varying systems », IEEE TAC, 1982] 7
� Definition found in [18] and [19]: [18] Jean lévine, Analysis and Control of Nonlinear Systems : A Flatness-basedApproach, Springer 2009, pp 143. [19] M. Fliess, J. Levine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, Int. J. Control, vol. 61, 1327-1361, 1995. 8
The unicycle robot is flat with flat outputs Y1 = x and Y2 = y : The system is dynamic feedback linearizable, the decoupling matrix being singular when the longitudinal velocity is zero. The extended state χ 1 is the longitudinal velocity v1 which has to be delayed. 9
The unicycle robot is flat with flat outputs Y1 = x and Y2 = y : The system is dynamic feedback linearizable, the decoupling matrix being singular when the longitudinal velocity is zero. 10
Tracking non singular reference trajectories for the unicycle robot using dynamic feedback linearization 11
The slider is flat with flat outputs Y1 = x and Y2 = y : The system is dynamic feedback linearizable, the decoupling matrix being singular when the longitudinal acceleration is zero, the extended state χ 1 is the longitudinal acceleration which has to be delayed. 12
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� Autonomous Indoor exploration for wheeled mobile robots � SLAM � Trajectory generation and tracking control laws � 3D reconstruction � Object recognition… � Adapt these technologies to a hybrid terrestrial and aerial quadrotor prototype 14
� Slider dynamic behavior similar to hovercraft � Tilting thrust � Hovercraft model proposed in [8] � Simplified model derived from an underactuated surface vessel modeling � Kinematic and dynamic equations [8] I. Fantoni, R. Lozano, F. Mazenc, K. Y. Pettersen, Stabilization of a nonlinear underactuated hovercraft. Conference on Decision and Control (CDC), 1999 15
� Hovercrafts belong to a more general class of marine vehicles which are known to be not asymptotically stabilizable at equilibrium points by continuous state feedback laws ([9]). by continuous state feedback laws ([9]). See also [7], [8], [10] [9] K. Y. Pettersen and O. Egeland, Exponential Stabilization of an Underactuated Surface Vessel, CDC, 1996. [10] R. W. Brockett, Asymptotic stability and feedback stabilization, Diff. Geometric Control Theory, Ed . Brockett, Millmann, Sussmann, Birkhauser, Boston, pp 181-191, 1983. [7] E. Sontag, H. Sussmann, Remarks on continous feedback, CDC, Albuquerque, 1980 . [8] C. Samson, Velocity and torque feedback control of a non holonomic cart, Advanced robot control, Springer, 1991. 16
� Trajectory tracking � Non linear control laws based on a Lyapunov analysis ▪ Surface vessel/ Position tracking/ constraint : longitudinal speed ≠ 0 ( [11] ) ▪ Surface vessel/ Posture tracking/ Exciting reference trajectory ▪ Surface vessel/ Posture tracking/ Exciting reference trajectory ( [12] ) � Flatness [13] ▪ Hovercraft stabilization/ Constraint on the reference trajectory � Sliding mode [14] ▪ Hovercraft stabilization [11] J. M. Godhavn, Nonlinear Tracking of Underactuated Surface Vessel. Decision and Control conference, 1996. [12] Pettersen and Nijmeijer. Tracking control of an underactuated surface vessel. CDC, 1998 [13] H. Sira-Ramirez and C. A. Ibanez, On the control of the Hovercraft System. Decision and control conference, 2000. [14] H. Sira-Ramirez, Dynamic second order sliding mode control of the hovercraft vessel. Control Systems technology, IEEE Transactions on, 17 2002.
� Point stabilization � Time-varying control laws ▪ Smooth feedback but slow convergence [11] ▪ Homogeneous continuous feedback, fast convergence, low robustness [9], [16] (Surface vessel stabilization) � Discontinuous control laws Lyapunov based analysis ( [8] ) (Hovercraft Stabilization) � Practical stabilization ▪ Hovercraft Stabilization/ Position tracking/ C 3 reference trajectory ( [17] ) ▪ Transverse functions [18] [11] J.M. Coron, Brigitte d’Andréa-Novel, Smooth Stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots. Proceedings of Nolcos Conference, Bordeaux, June 1992, pp. 649-654. [16] K. Y. Pettersen and T. I. Fossen, Underactuated Ship Stabilization using Integral Control : Experimental Results with Cybership I. IFAC NOLCOS, 1998. [17] A. P. Aguiar, L. Cremean and J. P. Hespanha, Position Tracking for a Nonlinear Underactuated Hovercraft : Controller Design and Experimental Results, Decision and Control conference, 2003. 18 [18] P. Morin and C. Samson, Practical stabilization of driftless systems on Lie groups, INRIA, Tech. Rep. 4294, 2001.
� Model � Commands 19
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� Outputs x, y & are flat outputs: 21
� Control law: � Stability and convergence are assured for the closed loop system with: 22
� Experimental platform � Motion Capture system � Drone � Identification process � Identification process � Aerodynamic forces and moments � Friction effects (static and kinetic) � Grey box identification � Trajectory tracking results 23
• software MOTIVE Infrared cameras Streaming VRPN s250e Optitrack • Remote control Communication ZigBee Drone • Embedded computer Serial port • Microcontroller MikroKopter 24
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� Trajectory tracking realized by the flatness control law � Reference trajectory constraints: � Derivatives until the second order for the state and third order for the reference for the reference � Experimental conditions � Circular trajectory tracking with radius 1.1m � Initial position :50cm from the reference trajectory � Ground : parquet 26
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[11] J.M. Coron, Brigitte d’Andréa-Novel, Smooth Stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots. Nolcos Conference, Bordeaux, June 1992 29
� The theorem applies for the unicycle robot with 30
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[J.B. Pomet, B. Thuilot, G. Bastin, G; Campion, « A hybrid strategy for the feedback stabilization of nonholonomic mechanical systems », Proc. IEEE CDC, 1992] 32
� The theorem applies for the slider with 33
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We can consider the error tracking system at an equilibrium point with 35
[G. Kern, « Uniform controllability of a class of linear time-varying systems », IEEE TAC, 1982] 36
A change of coordinates 37
A Lyapunov function candidate : Its time-derivative : 38
By duality, OK if : [H.K. Khalil, « Nonlinear systems », Prentice Hall, 1995] 39
For the complete system, the previous law u1 and the following yaw rate viewed as virtual Input (completed by backstepping technique) ensure the fixed point stabilization: Proof : Lyapunov function candidate : Time-derivative of the Lyapunov function : 40
LaSalle arguments : On this invariant subset we can prove that e2 remains constant and that the yaw rate satisfies the assumptions of proposition 3.1 if ω is sufficiently small: 41
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• A hybrid control strategy for unicycle and slider type robots. • How to extend these results in the context of finite-time stabilization ? We collaborate in a recent research project on FTS funded by ANR with colleagues from INRIA Lille (W. Perruquetti, A. Poliakov, D. Efimov, J.P. Richard …), UPMC (J.M. Coron, E. Trélat …) and L. Rosier 44
B. d’A-N, J-M. Coron, W. Perruquetti, « FTS of nonholonomic or underactuated mechanical systems: the examples of the unicycle and the slider », in preparation [37] P. Morin, C. Samson, «Time-varying exponential stabilization of a rigid spacecraft with two control torques », IEEE TAC, 1997 45
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