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Brigitte d’ ANDRÉA-NOVELMINES ParisTech, PSL-Research University, Centre de robotique, 60 Bd St Michel

75006 Paris

Sylvain THORELSAGEM, 100 avenue de Paris, 91344 Massy Cedex

� 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

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Idem for the slider :

The Unicycle : the TL system is controllable if

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The slider : the TL system is controllable if

STLC is satisfied for the two systems at each trajectory

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.

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Let us consider a neighborhood of the origin of the unicycleIt cannot be in the image of the slider’s dynamics at the origin.

STLC is satisfied for the two systems at each trajectory

respectively

BUT

The same holds for the slider. Therefore, due to Brockett’s theorem, these twosystems cannot be stabilized at fixed equilibrium points by means of continuousstate feedback laws.

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� Definition found in [18] and [19]:

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[18] Jean lévine, Analysis and Control of Nonlinear Systems : A Flatness-based Approach, Springer 2009, pp 143.[19] M. Fliess, J. Levine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory andexamples, Int. J. Control, vol. 61, 1327-1361, 1995.

The unicycle robot is flat with flat outputs Y1 = x and Y2 = y :

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The system is dynamic feedback linearizable, the decoupling matrix beingsingular when the longitudinal velocity is zero.

The unicycle robot is flat with flat outputs Y1 = x and Y2 = y :

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The system is dynamic feedback linearizable, the decoupling matrix beingsingular when the longitudinal velocity is zero.

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Tracking non singular reference trajectoriesfor the unicycle robot using dynamicfeedback linearization

The slider is flat with flat outputs Y1 = x and Y2 = y :

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The system is dynamic feedback linearizable, the decoupling matrix beingsingular when the longitudinal acceleration is zero.

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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

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� 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

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[8] I. Fantoni, R. Lozano, F. Mazenc, K. Y. Pettersen, Stabilization of a nonlinear underactuated hovercraft. Conference on Decision and Control (CDC), 1999

� Objective : trajectory tracking

and derivatives

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� Cannot be asymptotically stabilized by continuous state feedbacks for point stabilization ([9], [10])

Underactuated

vehicles

Marine vehicles ([9], [10])

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Surface vessel

hovercraft

[9] K. Y. Pettersen and O. Egeland, Exponential Stabilization of an Underactuated Surface Vessel. Conference on Decision and Control , 1996.[10] R. W. Brockett, Asymptotic stability and feedback stabilization, Diff. Geometric Control Theory, Ed . Brockett, Millmann, Sussmann, Birkhauser, Boston, pp 181-191, 1983.

� 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

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[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, 2002.

� Point stabilization� Time-varying commands▪ Smooth feedback but slow convergence [11]

▪ Homogeneous continuous feedback, fast convergence, low robustness [9], [16] (Surface vessel stabilization)

� Discontinuous commands▪ Lyapunov based analysis ([8]) (Hovercraft Stabilization)

� Practical stabilization▪ Hovercraft Stabilization/ Position tracking/ C3 reference trajectory

([17])▪ Transverses functions [18]

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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. 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] P. Morin and C. Samson, Practical stabilization of driftless systems on Lie groups, INRIA, Tech. Rep. 4294, 2001.

� Model

� Commands

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� Outputs x, y & are flat outputs:

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� Control law:

� Stability and convergence are assured for the closed loop system with:

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� 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

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• software MOTIVE

• Remote control

Streaming VRPN

Communication ZigBee

Infrared cameras s250e Optitrack

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• Embedded computer

• Microcontroller MikroKopter

Serial port

Drone

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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 referencefor the reference

� Experimental conditions� Circular trajectory tracking with radius 1.1m

� Initial position :50cm from the reference trajectory

� Ground : parquet (tiles during identification process)

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� The theorem applies for the unicycle robot with

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� The theorem applies for the slider with

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We can consider the error tracking system at an equilibrium point with

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A change of coordinates

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A Lyapunov function candidate :

Its time-derivative :

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By duality, OK if :

For the complete system, the previous law u1 and the following yaw rate viewed as virtualInput (completed by backstepping technique) ensure the fixed point stabilization:

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Proof : Lyapunov function candidate :

Time-derivative of the Lyapunov function :

LaSalle arguments :

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On this invariant subset we can prove that e2 remains constant and that the yaw ratesatisfies the assumptions of proposition 3.1 if ω is sufficiently small.:

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• A switch control strategy for unicycle and slider type robots.

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• The time-varying control should be used for trajectory tracking but in a practical sense.

• How to extend these results in the context of finite-time stabilization ?