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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 originIt cannot be in the image of the unicycle 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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[G. Kern, « Uniform controllability of a class of linear time-varying systems », IEEE TAC, 1982]

� 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 extended state χ1 is the longitudinal velocity v1 which has to be delayed.

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, the extended state χ1 is the longitudinal acceleration which has to be delayed.

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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

� 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]

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[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.

� 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 control laws▪ Smooth feedback but slow convergence [11]

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

� Discontinuous control lawsLyapunov based analysis ([8]) (Hovercraft Stabilization)

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

([17])▪ Transverse 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

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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

� The theorem applies for the unicycle robot with

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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]

� 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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[G. Kern, « Uniform controllability of a class of linear time-varying systems », IEEE TAC, 1982]

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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 :

[H.K. Khalil, « Nonlinear systems », Prentice Hall, 1995]

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 hybrid control strategy for unicycle and slider type robots.

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• 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

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B. d’A-N, J-M. Coron, W. Perruquetti, « FTS of nonholonomic or underactuatedmechanical systems: the examples of the unicycle and the slider », in preparation[37] P. Morin, C. Samson, « Time-varying exponential stabilization of a rigid spacecraftwith two control torques », IEEE TAC, 1997

The case of the double integrator

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E. Bernuau, W. Perruquetti, D. Efimov, E. Moulay, « Robust FT ouput feedback stabilizationof the double intergrator »,

Y. Hong, « FT stabilization and stabilizability of a class of controllable systems », SCL, 2002

L. Rosier, « Etude de qqs problèmes de stabilisation », PhD thesis, 1991

The case of the slider

s. t.

2)to stabilize the double integrator

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1)

We conclude by time-periodicity.

Stabilization of dynamical systems modeled by hyperbolic PDEs coupled withBoundary Conditions modeled by non linear ODEs.

Some examples :

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Overhead crane with flexible cable :

B. d’A-N, J.-M. C. : “Exponential stabilization of an overheadcrane with flexible cable via a back-stepping approach”, Automatica, (26), 2000.

B. d’A-N, J.-M. C : “Stabilization of an overhead crane with a variable length flexible cable”, Computational andApplied Mathematics, Vol. 21 No. 1, 2002.

Irrigation canals :

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J.-M. C., B. d’A-N, G. B. : “Les équations de Saint-Venant pour lecontrôle des canaux et des voies navigables. Penser globalement, observer et agir localement”, article pour la Recherche, Wxyz, No. 417, Mars 2008, pp. 82-83.

J.-M. C., B. d’A-N, G. B. : “A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws”, in IEEE Transaction on Automatic Control, Vol. 52, No. 1, January 2007.

Wind Instruments:

B. d’ANDRÉA-NOVEL, B. FABRE, J.-M. CORON : “An acoustic model for automatic control of a slide flute”, in Acta Acustica, Journal of the European Acoustics Association (EEA), Volume 96, 2010, pp. 713-721.

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B. d’ANDRÉA-NOVEL, J.-M. CORON, T. HÉLIE: “Asymptotic observers for a simplifiedbrass instrument model”, in Acta Acustica, Journal of the European Acoustics Association(EEA), Volume 96, 2010, pp. 733-742.

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•We have considered the simpler problem of controlling a slide flute : a cylindrical recorderwithout finger holes but with a piston mechanism to modify the length.

• From a mathematical point of view : a system of conservation laws leading to hyperbolicPDEs coupled with a nonlinear ODE with delay.

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PDEs coupled with a nonlinear ODE with delay.

flue channel

labiumlabium Bernoulli equationsBernoulli equationsWW

hh

Model of the jet channel and the mouth

The ideal boundary condition at x=0 si too simple !

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mouthmouth pipepipe

-- Interaction jet/labium, jet/pipe, jet drive mechanism [Cremer, Ising 1968; Coltman 1976].Interaction jet/labium, jet/pipe, jet drive mechanism [Cremer, Ising 1968; Coltman 1976].-- Vortex shedding at the labium [Verge, Fabre et al. 1994; Verge, Hirschberg, Caussé 1996].Vortex shedding at the labium [Verge, Fabre et al. 1994; Verge, Hirschberg, Caussé 1996].-- is the jet velocity and the pressure jump responsible of the sound production, is the jet velocity and the pressure jump responsible of the sound production, mainly composed of the pressure jump due to the jet drive mechanism, and due mainly composed of the pressure jump due to the jet drive mechanism, and due to the vortex shedding. to the vortex shedding.

hh

The pressure in the mouth is related to the output flow by the radiation impedance

� The pressure term due to the dipolar jet-drive :

Jet position Ujc~0.3Uj

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� The dissipative pressure term due to vortex shedding :

Mouth cross section at the flue exit

Jet position

Spatial amplification of the jet

Vena-contracta factor of the flow

W

Labium delay

pf = 300 Papf = 55 Pa

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pf = 5 Pa

f1s = 310 Hz (close to 290.8 Hz) f2s = 925 Hz (close to 903.7 Hz)

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• The note being chosen, we solve the linearized B.C. w.r.t. L and U0. For example, to obtain Fe ideal (c/4L) = 324 Hz, this leads to :

• This is our feedforward control algorithm.

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• Feedback :

Slide

feedforward

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Slide+

Artificialmouth

+PIDmodal

analysis

feedback

Targetpitch

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HAPPY BIRTHDAY

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JEAN-MICHEL !!!