Feedback Linearization and Linear Observer for a Quadrotor Unmanned Aerial Vehicle

Transcription

1 Feedback Linearization and Linear Observer for a Quadrotor Unanned Aerial Vehicle Abdellah Mokhtari,2, Nacer K. M Sirdi, Kaal Meghriche, A. Belaidi 3 LRV : Versailles Laboratory of Robotics, University of Versailles avenue de l Europe, 784 Vélizy, France. sirdi@lrv.uvsq.fr 2 University of Science and Technology Oran Algérie. 3 E.N.S.E.T Oran Algérie Abstract Perforance and characteristics of a Luenberger observer, cobined with a classical polynoial controller (based on an accurate odel of the plant), are analyzed in this paper. The observer is shown efficient when dealing with bounded uncertainties, disturbances and noise. The analysis is based on the observer and tracking errors during transients and at steady state, and on the perforance and robustness with respect to plant uncertainties. Estiation of wind paraeters is added to reinforce the robustness. Siulation results are provided and output trajectories analyzed. Index Ters Feedback linearization ; Luenberger observer ; Estiation, Observer based control. I. INTRODUCTION Unanned Aerial Vehicles (UAV) are increasingly popular platfors due to their potential use in search and rescue, surveillance, law enforceent, inspection, apping, and aerial cineatography. For these applications, the ability of helicopters to take off and land vertically, to perfor hover flight as well as their agility and controllability, ake the ideal vehicles. Unanned aerial and autonoous vehicles have been under continuous developent in last decade, especially in civilian aerial applications such as bridges and buildings supervision, surveillance and/or testing. Their application field becoes very large. An analysis of control design approaches, suitable to this kind of systes, has been developed. The control objective is to be able to follow desired trajectories and allow autonoous otions. The syste ust obey soe desired dynaic behavior. This is iportant for applications where navigation onitoring is not easy to handle anually. An unanned quadrotor aerial vehicle (figure ()) is required to ove in different environents, showing good perforance and a great autonoy, under a variety of load conditions and unknown disturbances []. Developing a control syste that can achieve the aforeentioned goal is challenging for a variety of reasons - The nonlinear behavior of a vehicle subject to aerodynaic forces and oents. - The ultivariable character of a vehicle, leading to interaction between different coand channels. - The consistent aount of uncertainty in both high and low frequencies, due to unknown disturbances introduced by linearization of the nonlinear dynaics. Fig.. The quadrotor Aerial Vehicle (UAV) odel. The easureents of the pertinent syste variables are not always available, hence the syste state is to be estiated using observers. The ain difficulties of the otion control, for high perforance positioning, are paraetric uncertainties, neglected dynaics, and external disturbances. Since the original work by Luenberger [2], the use of state observers proves to be useful not only in syste onitoring and regulation but in failures detection and identification of dynaic systes as well. Alost all observer designs are based on a plant odel. However, the presence of disturbances, dynaic uncertainties and nonlinearities represent great challenges in practical applications. Furtherore, the available state inforation fro easureents, i.e., sensor outputs, usually does not contain full state inforation and ost often, it is corrupted by noise. This coplicates further the design of robust controllers for actual systes [3]. A sliding ode observer, yielding insensitivity to unknown paraeter variations and noise, has been proposed by Utkin [4]. Dorling and Zinober copared the full and reduced order Luenberger observers with the Utkin observer [5]. In the literature, there exist several observer structures based on different ethods such as linearization by coordinate transforation and output injection [6], and variable structure approaches [7] to nae but a few. In [8], a nonlinear dynaic odel, for a quadrotor helicopter in a for suited for control design, is presented. A coparison between two control approaches, the exact linearization with dynaic extension and backstepping control using a sliding ode observer for the state estiation, as applied to a quadrotor helicopter, is ade [8][9]. The input-output decoupling proble is not solvable for this odel by eans

2 2 of a static state feedback control law [][][]. However, these observer structures need to include a plant odel in their equations. This situation inevitably generates soe practical burdens. Without a odel, observers cannot be constructed. Even if a odel is available, a reliable state reconstruction could not be expected, unless the odel is accurate enough. Even, in this case, the observers ay becoe too coplicated (due to odel coplexity) to be of practical use, especially in real tie applications. It is expected that the observer will provide robust state reconstruction in the presence of uncertainties and disturbances since the nonlinear syste (quadrotor) has been linearized through Lie derivatives. In this work, a cobined Luenberger observer ; feedback linearization controller and a disturbance estiator is designed in an overall closed loop syste to analyze efficiency when dealing with uncertainties, tie delay and wind disturbances. This paper is organized as follows : section II describes the quadrotor dynaics. In section III, we present the linearized Luenberger observer. Siulation results are discussed in section IV and a conclusion is drawn in section V. II. QUADROTOR DYNAMICS The quadrotor helicopter is assued to be a rigid body, having 6 degrees of freedo and subject to external efforts. The odel includes kineatics and dynaic equations [8]. The quadrotor helicopter is shown in figure (). The two diagonal otors and 3, are running in the sae counter-clockwise direction, whereas otors 2 and 4, run in the clockwise direction to eliinate the anti-torque. By varying the rotor speeds altogether with the sae quantity, the lift forces will change, affecting the altitude z of the syste and enabling vertical take-off / landing. Yaw angle is obtained by speeding up or slowing down the clockwise otors depending on the desired angle direction. Tilting around x axis (roll angle), allows the quadrotor to ove in the y axis direction. The otion direction depends on the angle value whether it is positive or negative. Tilting around y axis (pitch angle), allows the quadrotor to ove toward x direction. The rotor is the priary UAV source of control and propulsion. The Euler angle orientation of the flow allows the forces and oents to control the altitude and position of the syste. The absolute position is described by three coordinates (x, y, z ), and its altitude by Euler angles (ψ, θ, φ), under the conditions ( π ψ < π) for yaw ( π 2 < θ < π 2 ) for pitch ( π 2 < φ < π 2 ) for roll Using Newton s law, if F ext and T ext represent the external forces and oents respectively, the dynaic equations of the syste ay be represented as V = F ext () J ω = ω Jω + T ext (2) ū i represent the inputs of the syste. ū is the siultaneous 4 rotors thrust ; ū 2 is the thrust of the left and right rotors ; ū 3 is the thrust of the front and the rear rotors ; ū 4 is the oentu difference between the clockwise turning rotors and the counter clockwise ones. The MIMO nonlinear syste has the for [2] x = f( x) + 4 ḡ i ( x)ū i (3) i= y = h( x) = col(x, y, z, ψ) (4) x T = (x, y, z, ψ, θ, φ, u, v, w, ζ, ξ, p, q, r) u v w q sin φ sec θ + r cos φ sec θ q cos φ r sin φ p + q sin φ tan θ + r cos φ tan θ Ax f( x) = + g7 (ψ, θ, φ)ζ A y + g8 (ψ, θ, φ)ζ A z + g8 (ψ, θ, φ)ζ ξ I y I z I x I z I x I y I x I y I z qr + Ap Ix pr + Aq Iy pq + Ar Iz ḡ ( x) = (,,,,,,,,,,,,, ) T ḡ 2 ( x) = (,,,,,,,,,,, d I x, ) T (5) ḡ 3 ( x) = (,,,,,,,,,,,, d I y ) T (6) ḡ 4 ( x) = (,,,,,,,,,,,, I z ) T g 7 = (cos φ cos ψ sin θ + sin φ sin ψ) g 8 = (cos φ sin θ sin ψ cos ψ sin φ) (7) g 9 = (cos θ sin φ) The real control signals (ū, ū 2, ū 3, ū 4 ) are replaced by (u, u 2, u 3, u 4 ) to avoid singularity in Lie transforation atrices when using exact linearization. In this case u is delayed by a double integrator. The other control signals reain unchanged. u = ζ ζ = ξ ξ = ū (8) u 2 = ū 2 u 3 = ū 3 u 4 = ū 4 The Input-Output linearization uses full state feedback to globally linearize the nonlinear dynaics of selected controlled outputs. Each of the output channels is differentiated, a sufficient nuber of ties, until an input control coponent appears in the resulting equation. Using the Lie derivative, feedback linearization will transfor the

3 3 nonlinear syste into a linear and non-interacting syste known as the Brunovsky for [3] d 4 x dt 4 = v d 4 y dt 4 = v 2 (9) d 4 z dt 4 = v 3 d 2 ψ dt 2 = v 4 v, v 2, v 3, and v 4, represent the new control inputs. Obviously, when developing the control law, the syste requires the priary output vector (x, y, z, ψ) and its successive derivatives to be copared with the desired state trajectories. Adopting a classical polynoial control law, the closed loop syste is represented by v = x (4) d λ 3 (x (3) x (3) d ) λ 2(ẍ ẍ d ) λ (ẋ ẋ d ) λ (x x d ) v 2 = y (4) d λ 3 (y (3) y (3) d ) λ 2(ÿ ÿ d ) λ (ẏ ẏ d ) λ (y y d ) () v 3 = x (4) d λ 3 (z (3) z (3) d ) λ 2( z z d ) λ (ż ż d ) λ (z z d ) v 4 = ψ d λ 5 ( ψ ψ d ) λ 4 (ψ ψ d ) The poles placeent is based on the choice of λ i, aking its choice decisive in defining the syste dynaics. III. Luenberger State Observer When dealing with real tie dynaic systes, it is necessary to anipulate the state vector x and the coplete easure is either expensive or difficult to ipleent. In this case, an observer ay be used to obtain an estiate to replace the non easured state coponents. A reliable state estiation is ainly required not only for control purpose, but also for other applications such as spacecraft navigation, onitoring, and fault diagnosis in echanical systes as well. However, the atheatical odel (3) is only an approxiation to the physical process and the actual plant is usually affected by external disturbances. For control ipleentation, the easured quantities are state variables x, y, z and ψ representing the translational otion and rotation around z axis respectively. Non easurable signals can be obtained by successive differentiation. Unfortunately they are containated by the easureent noise to a such degree that they can no longer be used. In fact, the accuracy of the state estiation depends largely on the goodness of the actual plant physics odel and on the estiator structure [8][3]. A. Observer Model After linearization, the quadrotor odel ay be represented in a state space for where [x, x 5 ] is the easured vector x = col(x, y, z ) ẋ = x 2 = col(ẋ, ẏ, ż ) ẋ 2 = x 3 = col(ẍ, ÿ, z ) ẋ 3 = x 4 = col( x, y, z ) ẋ 4 = v v 2 () x 5 = ψ v 3 ẋ 5 = x 6 = ψ ẋ 6 = v 4 For a vector z = [x, x 2, x 3, x 4, x 5, x 6 ], a well known result fro linear syste theory is that, for a linear tieinvariant (LTI) syste () with the dynaics ż(t) = Az(t) + Bv(t) (2) y(t) = Cz(t) + Dv(t) A A = 4 4 A A 4 2 (3) A B 3 4 B = B (4) B 3 4 B 2 C = [ ] C 4 3 C C C 4 4 C = A = ; A 2 [ B = [ ] ] (5) B 2 = [ ] (6) B 3 = [ ] B 4 = [ ] ; C 2 = ; C 3 = ; C 4 = and an observable (A,C) pair, a stable linear Luenberger observer given by ẑ(t) = Aẑ(t) + Bv(t) + L(y(t) ŷ(t)) (7) ŷ(t) = Cẑ(t) + Dv(t) ; ẑ() = ẑ can be designed by placing the poles of the observer at any desired location such that the error signals exhibit the desired dynaics [4]. ẑ(t) = (A LC) ẑ(t) + (B LD)v(t) + Ly(t) (8)

4 4 The estiation error is e = z ẑ, its dynaic equation is given by ė = (A LC)e = Âe (9) The estiation error e will converge to zero if all eigenvalues of  = (A LC) are in the left half plane. The observer design refers to selection of the gain atrix L, using the pole placeent ethod. The ain challenge in these applications is the extensive observer dependance on the accuracy of the plant atheatical odel (A, B, and C atrices). Therefore, the convergence rate of z(t) = z ẑ to zero, can arbitrarily be chosen by appropriate design of L. It follows that ẑ(t) converges exponentially to z(t) as t with a rate that depending on atrix Â. This result is valid for any atrix A and any initial condition z as long as (C,A) is an observable pair and A, C are known. B. Output States Reconstruction The observer, described in the previous section, is a state estiator with partial state (x, y, z, ψ) taken as a easured output. The observer akes an estiation of the state needed by the control law to calculate the tracking error between the desired trajectories (x d, x 2d, x 3d, x 4d, x 5d, x 6d ) and the estiated trajectories (x, x 2, x 3, x 4, x 5, x 6 ). Unfortunately, the estiated states do not involve all the output states. To obtain the entire state output, the issing variables (θ, φ, p, q, r) fro x vector(3) are calculated through the estiated values fro the nonlinear syste (3), without taking the perturbation into account. Fro (2), θ and φ can be deduced ( (sin( φ e = arcsin ˆψ)ˆx cos( ˆψ)ŷ) ) (2) u C. Wind Paraeters Estiation The final odel, obtained using feedback linearization, differs fro (4) in presence of perturbation since linearization is not exact. Considering perturbation, the syste is represented by ẋ = Ax + B η + B 2 η 2 (24) with x being the easured state vector and its successive derivatives, η and η 2 are the wind disturbances vector (Äx, Ä y,äz) and (A p, A q, A r ) respectively. A = x = col(x, x 2, x 3, x 4, x 5, x 6) I I I λ λ λ 2 λ 3 λ 4 λ 5 λ i are the control gains. B = M ; B 2 = M = ( (cos( θ e = arcsin ˆψ)ˆx + sin( ˆψ)ŷ) ) M 3 = [ ] a 45 a 46 (27) cos φ e u a (p, q, r) can be deterined ( ) fro the transforation atrix 4 = (ζsφcψsθ ζcφsψ)/(i x ); a 34 = (ζsφcθ)/(i x ) [7] which needs ψ, θ, φ. These paraeters can be evaluated fro (2) and the third derivatives ( x ˆ a 5 = (ζcψcθ)/(i y ); a 35 = (ζsθ)(i y ), y ˆ a 24 = (ζsφsψsθ + ζcφcψ)/(i x ); a 45 = Sφ/(I y Cθ ) i.e., a x ˆ (sin(φ e )sin(θ e )sin( ˆψ) + cos( ˆψ)cos(φ 25 = (ζsψcθ)/(i y ); a 46 = Cφ/(I z Cθ) (28) e ))+ y ˆ (cos(φ e )sin( ˆψ) sin(φ e )cos( ˆψ)sin(θ e ))+ cos(φ e ) ˆ ψu sin(φ e )cos θ 2 (θ e ) sin(θ e )ζ e = M 2 M 3 ; M 2 = a 4 a 5 a 24 a 25 a 34 a 35 (25) (26) φ e = cos(θ e )cos 2 (φ e )u (2) x ˆ sin( ˆψ) + ˆψu cos(φ e )sin(θ e )+ ζsin(φ e ) + cos( ˆψ) ˆ u cos(φ e ) y (22) Then (p, q, r) are deduced fro the following atrix equation : p q = sin(φ e) sec(θ e ) cos(φ e ) sec(θ e ) cos(φ e ) sin(φ e ) ˆψ θ e r sin(φ e ) tan(θ e ) cos(φ e ) tan(θ e ) φ e (23) Fig. 2. The overall closed loop syste. A proble of priary iportance is the selection of the quadratic Lyapunov function. Fro (2), the Lyapunov function which ensures convergence is chosen as V = x T P x + 2 ηt Γ η + 2 ηt 2 Γ η 2 (29)

5 5 x represents the tracking errors between the estiated and desired values. By coputing V and aking V < we deduce the adaptation law for paraeters tuning η = 2Γ BT P x η 2 = 2Γ 2 BT 2 P x (3) Finally, the closed loop syste with the cobined controller-observer-estiator is represented in figure (2). IV. Siulation and Results Siulation is carried out using the following quadrotor paraeters : I x = I y = I z =.24N/rad/s 2, = 2.kg, d =., and g = 9.8/s 2. Fig. 5. Tracking error for x,y,z,ψ ) Without disturbance: In this case, we use (A x = A y = A z = ); (A p = A q = A r = ). Figure 4 shows a good convergence of the yaw trajectory. Figures 5 and 6 show a sooth behavior of tracking errors and the control signals respectively. Fig. 3. Trajectories behavior without disturbances To evaluate the perforance of the proposed observer and control, the reference (desired) trajectory used to carry out siulation, is a vertical helix, see figure (3), with equation x d = 2 cos( t 2 ) y d = 2 sin( t 2 ) (3) z d = t ψ d = π 3 Fig. 6. Control signals v, v 2, v 3, v 4 2) With disturbances: The estiation gains for wind disturbance [A p ; A q ; A r ] are Γ 2 = [2. 4 ; 2. 4 ; 4] around x axis (A p ), y axis (A q ) and z axis (A r ). Using the aerodynaic disturbances (Moents) A p =.8; A q = ; A r =.8; the following results are obtained : In fig-7 we present tracking errors showing that disturbance is well rejected and fig-8 shows a good convergence of paraeters estiation. Fig. 4. Yaw trajectory (λ, λ, λ 2, λ 3 ) are the coefficients of the polynoial (s+ ) 4 and (λ 4, λ 5 ) the ones of (s + 5) 2. The desired poles for the closed loop syste used to deterine L are { } 5.5; 5.5; 5.5; 6.6; 6.6; 6.6; 7.7; P 4 = 7.7; 7.7; 8.8; 8.8; 8.8; 4.95; 4.95 To evaluate the perforance and robustness of the sliding observer, siulation is carried out considering the following cases : without disturbance, with disturbance, and ultiately with uncertainties. Fig. 7. Tracking error with wind disturbances 3) With uncertainties: Uncertainties of 2% are introduced on the ass and inertial coefficients I x, I y, I z to show the behavior of the observer-control law cobination toward error odelling. The obtained results are shown in fig-9. 4) Concluding rearks: -It is noted fro the obtained results without perturbation, (fig-3, 4, 5 and 6), that the Luenberger observer gives quite satisfactory results, especially when the poles placeent is chosen judiciously. This can be seen fro trajectories tracking error vanishing after a finite tie with a perfect convergence shape. -When wind disturbances are introduced, the results presented in fig-7 reflect the good robustness of the ixed

6 6 observer-controller. This is confired by the convergence of tracking error (fig-7) and disturbance rejection. The estiation of wind paraeters, around x axis (A p ), y axis (A q ) and z axis (A r ), exhibits a very good convergence as presented in fig-8. This shows that the syste dynaic behavior is sensitive to aerodynaic oents disturbances. -However, the estiation of translational wind forces (given by A x, A y and A z ) is not represented here since the observer-controller exhibit an efficiency to overcoe this type of perturbation without need to an estiationcopensation procedure. -The observer robustness is also tested by introducing 2% uncertainties on the syste paraeters, I x, I y, I z (fig-9). -The output state vector convergence is obtained despite the (well known) non-robustness of the exact linearization approach when syste paraeters suffer severe uncertainties. However attention ust be paid on the choice of observer gain L to avoid noise aplification around the desired trajectories. Fig. 8. Wind paraeters estiation for A p =.8; A q =.5; A r = V. CONCLUSION In this paper, a kino-dynaic odel of a quadrotor helicopter is presented. We developed a nonlinear controller with observers. A feedback linearization using Luenberger observer is applied to the quadrotor UAV. The linear observer is used to rebuilt the non easured variables required to enhance robustness of the control law. An adaptive estiator is added to the overall syste to estiate the effect of the external disturbances such as wind. The whole control syste (observer-estiator-controller) constitutes an interesting contribution to control systes equipped with a iniu nuber of sensors. This approach shows a good robustness of the controller and perits to reduce the nuber of sensors to be used. Intensive siulations were carried out to validate the perforance and stability of the controller. The robustness study was realized in siulations taking into account uncertainties and disturbances with a noise corrupted easured state. The obtained results show good convergence of estiated values and satisfying tracking errors of desired trajectories. Also, they show that the estiator added to the control reinforces the robustness and stability of the overall syste. The adaptive part allows estiation of paraeters which ay not be well known and ay change during operation. Fig. 9. Tracking errors for % uncertainties on ; I x; I y; I z References [] V.Mislter, A. Benallegue and N. K.M Sirdi, Linéarisation exacte et découplage entrées-sorties, Coparaison entre l hélicoptère standard et l hélicoptère 4 rotors, CIFA 22. [2] D. Luenberger, Observers for Multivariable Systes, IEEE Trans. Auto. Control, Vol., pp. 9-97, 966. [3] Jong-Rae Ki, Model-Error Control Synthesis : A New Approach to Robust Control, PhD Dissertation, Texas A&M University August 22. [4] V. I. Utkin, Sliding Modes in Control and Optiization, Springer-Verlag, Berlin, 992. [5] C. M. Dorling and A.S. I. Zinober, A coparaison study of the sensitivity of observers, Proceeding of First IASTED Syposiu on Applied Control and Identification, Copenhagen, pp , 983. [6] A. J. Krener and A. Isidori, Linearization by Output Injection and Nonlinear Observers, Systes and Control Letters, Vol. 3, pp , 983. [7] B. L. Walcott and S. H. Zak, State Observation of Nonlinear Uncertain Dynaical Systes, IEEE Transactions on Autoatic Control, Vol.32, pp. 66-7, 987. [8] L. Mederreg, F. Diaz and N. K. M sirdi, Nonlinear Backstepping Control with Observer Design for a 4-Rotors Helicopter. Internal report, LRV deceber 23. [9] Erdinç Altug, Jaes P. Ostrowski, Robert Mahony. Control of a Quadrotor Helicopter Using Visual Feedback. GRASP Lab. University of Pennsylvania, Philadelphia. [] J.C. Avila-Vilchis, B. Brogliato, A. Dzul, R. Lozano, Nonlinear Modelling and Control of Helicopters, Autoatica, 23. [] H. Shi, T. J. Koo, F. Ho ann, and S. Sastry. A coprehensive study of control design for an autonoous helicopter. In Proceedings of the 37th Conference on Decision and Control, pp , Tapa, Florida, Deceber 998. [2] V. Mistler, A. Benallegue, N. K. M Sirdi, Exact Linearization and Noninteracting Control of a 4-Rotors Helicopter via Dynaic Feedback, th IEEE Int. Workshop on Robot-Huan Interactive Counication (Septeber 8-2, 2 Bordeaux and Paris). [3] S.J. Kwon, W. K. Chung, Cobined Synthesis of State Estiator and Perturbation Observer. Journal of Dynaic Systes, Measureent, and Control., Vol. 25. ASME,March 23. [4] T. Kailath, Linear Systes, Prentice-Hall, Englewood Cliffs,NJ, 98.., February, 27, 24. LRV/TRVI.Nk3