MPC with Nonlinear H Control for Path Tracking of a Quad-Rotor Helicopter

Transcription

1 Proceedings of the 7th World Congress The International Federation of Autoatic Control Seoul, Korea, July 6-, 8 MPC with Nonlinear H Control for Path Tracking of a Quad-Rotor Helicopter Guilhere V. Raffo Manuel G. Ortega Francisco R. Rubio Departent of Systes Engineering and Autoation, University of Seville, Caino de los Descubriientos sn, 49, Seville, Spain {raffo,ortega,rubio}@cartuja.us.es. Abstract: This paper presents a predictive and nonlinear robust control strategy to solve the path tracking proble for a quadrotor helicopter. The dynaic otion equations are obtained by the Lagrange-Euler foralis. The control structure is perfored through a odel-based predictive controller (MPC) to track the reference trajectory and a nonlinear H controller to stabilize the rotational oveents. Siulations results in presence of aerodynaic disturbances and paraetric uncertainty are presented to corroborate the effectiveness and the robustness of the proposed strategy. Copyright c 8 IFAC.. INTRODUCTION This paper deals with a quadrotor UAV, in which the VTOL (Vertical Take-Off and Landing) is one of the concepts usually used to develop control laws. This kind of helicopter tries to reach a stable hovering and flight using the forces equilibriu produced by the four rotors [Castillo et al., 5b]. One of the advantages of the quadrotor configuration is its load capacity. Moreover, this helicopter is highly aneuverable, which allows take-off and landing as well as flight in hard environent. As a drawback, this type of UAV presents a weight and energy consuption augentation due to the extra otors. Many efforts have been ade to control the quadrotor helicopter and any strategies have been developed to solve the path tracking proble for this type of syste (see, for exaple, Mistler et al. [], Bouabdallah et al. [4], Bouabdallah and Siegwart [5], Castillo et al. [5a]). Several control strategies have been tested on the quadrotor helicopter, but ost of the do not consider external disturbances and paraetric uncertainty of the odel. In soe publications the quadrotor helicopter has been controlled using a linear H controller based on linearized odels. In Chen and Huzezan [3], a siplified nonlinear odel of the UAV oveents was presented. The path tracking proble was divided into two parts, the first one to achieve the angular rates and vertical velocity stabilization by a DOF H controller using the loop shaping technique. The sae technique was used to control the longitudinal and lateral velocities, the yaw angle and the height in the outer loop. A predictive control was designed to solve the path tracking proble. In this paper a predictive and nonlinear robust control strategy to solve the path tracking proble of the quadrotor helicopter is proposed. A state space predictive controller based on the variant tie error odel is used to track the reference trajectory. A nonlinear H controller is synthesized to stabilize the helicopter rotational oveents. The objective of MPC is to copute a future control sequence in a defined horizon in such a way that the prediction of the plant output is driven close to the reference. This is accoplished by iniizing a ulti-stage cost function in respect to the future control actions. An analytical solution can be obtained for a quadratic cost if the odel is linear and there are no constraints, otherwise an iterative ethod of optiization should be used [Caacho and Bordons, 998]. Because of its forulation MPC also allows the use of previously known references for the control law calculation [Norey-Rico et al., 999]. The goal of the nonlinear H control theory, introduced by van der Schaft in his proinent article [van der Schaft, 99], is to achieve a bounded ratio between the energy of the socalled error signals and the energy of the disturbance signals. In general, the nonlinear approach of this theory considers two Hailton-Jacobi-Bellan-Isaacs partial derivative equations (HJBI PDEs), which replace the Riccati equations in the case of the linear H control forulation. The ain proble in the nonlinear case is the absence of a general ethod to solve these HJBI PDEs. In Ortega et al. [5] a strategy to control echanical systes considering the tracking error dynaic equation was proposed. In such strategies a nonlinear H control, forulated via gae theory, was applied. This strategy provides, by an analytical solution, a constant gain siilar to the results obtained with the feedback linearization procedures. The reainder of the paper is organized as follows: in Section II, a description of the quadrotor helicopter odelling is given. The predictive controller for the translational oveents is presented in Section III. In Section IV, the nonlinear H controller for the rotational subsyste is developed. Soe siulation results are presented in Section V. Finally, the ajor conclusions of the work are drawn in Section VI.. Description. SYSTEM MODELLING The autonoous aerial vehicle used in this paper is a iniature four rotor helicopter. The oveent of the UAV results fro changes in the velocities of the rotors. Longitudinal otions are achieved by eans of front and rear rotors velocity, while lateral displaceents are perfored using the speed of the right and left propellers. Yaw oveent is obtained fro the difference in the counter-torque between each pair of propellers, i.e., /8/$. 8 IFAC /876-5-KR-.578

2 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 accelerating the two clockwise turning rotors while decelerating the counter-clockwise turning rotors, and vice-versa. The dynaic odel of the syste is obtained under the assuption that the vehicle is a rigid body in the space subject to one ain force (thrust) and three torques. However, this type of vehicle is a flight syste of lightweight structure and, therefore, gyroscopes effects resulting fro the rotation of the rigid body and the four propellers ust be included in the dynaic odel [Bouabdallah et al., 4]. Besides, a helicopter is an underactuated echanical syste with six degrees of freedo and only four control inputs. Due to the coplexities presented, soe assuptions are ade for odelling purposes [Koo and Sastry, 999]. The oent effects caused by the rigid body on the translational dynaic are neglected, as well as the ground effect. The center of ass and the body fixed frae origin are assued coincident. Moreover, the helicopter structure is assued to be syetric, which results in a diagonal inertia atrix.. Helicopter Kineatics The helicopter as a rigid body is characterized by a frae linked to it. Let B = {B b,bb,bb 3 } be the body fixed frae, where the B b axis is the helicopter noral flight direction, Bb is orthogonal to B b and positive to starboard in the horizontal plane, whereas B b 3 is oriented in ascendant sense and orthogonal to the plane B b OBb. The inertial frae I = {E x,e y,e z } is considered fixed with respect to the earth (see Fig. ). Fig.. Quadrotor helicopter schee. ξ ψ The vector ξ = {x,y,z} represents the position of the helicopter ass center expressed in the inertial frae I. The vehicle orientation is given by a rotation atrix R I : B I, where R I SO(3) is an orthonoral rotation atrix [Fantoni and Lozano, 995]. The rotation atrix is obtained through three successive rotations around the axes of the body fixed frae. The first one is given by a rotation around the E x axis by roll angle, ( π < φ < π), followed by a rotation of pitch angle, ( π/ < θ < π/), around the E y axis fro the new axis B b. Finally, a rotation of the yaw angle, ( π < ψ < π), is carried out around the E z axis fro the new axis B b 3 to carry the helicopter to the final position. Fro these three rotations, the following rotation atrix fro B to I is obtained: CψCθ CψSθSφ SψCφ CψSθCφ+SψSφ R I = SψCθ SψSθSφ +CψCφ SψSθCφ CψSφ () Sθ CθSφ CθCφ where C = cos( ) and S = sin( ). θ φ The kineatic equations of the rotational and translational oveents are obtained by eans of the rotation atrix. The translational kineatic can be written as: v = R I V () where v = [u v w ] T and V = [u L v L w L ] T are linear velocities expressed in the inertial frae and body fixed frae, respectively. The rotational kineatic can be obtained fro the relationship between the rotation atrix and its derivative with an skewsyetric atrix [Craig, 989, Olfati-Saber, ] as follows: Ṙ I = R I S(ω) (3) η = Wη ω φ sinφ tanθ cosφ tanθ p θ = cosφ sinφ q (4) ψ sinφ secθ cosφ secθ r where η = (φ,θ,ψ), ω = (p,q,r) are the angular velocities in the body fixed frae..3 Lagrange-Euler Equations The helicopter otion equations can be expressed by the Lagrange-Euler foralis based on the kinetic and potential energy concept: Γ i = d ( ) L L (5) dt q i q i L = E c E p where L is the Lagrangian, E c is the total kinetic energy, E p is the total potential energy, q i is the generalized coordinate and Γ i are the generalized forces/torques given by nonconservative forces/torques The generalized coordinates for a rigid body rotating in the three-diensional space can be written as [Castillo et al., 5a]: q = [x y z φ θ ψ] T R 6 The Lagrangian expression of the helicopter is given by: L(q, q) = E ctrans + E crot E p (6) where E ctrans is the translational energy and E crot is the rotational energy. Firstly, the translational energy ter is developed requiring the knowledge of each generalized coordinate velocity. The linear velocity is given by (), where ξ = v and the quadratic velocity is ξ (x,y,z) = (ẋ + ẏ + ż ). Thus, the translational kinetic energy can be written as: E ctrans = ξ (x,y,z)d = ξ (x,y,z) = T ξ ξ Let E crot be the rotational kinetic energy in B expressed in I, and let de crot be the kinetic energy of a particle with differential ass d in B. Then: de crot = ( I v B ) d = ( I v Bx + I v By + I v ) Bz d (7) Therefore, the rotational kinetic energy can be obtained solving (7). Furtherore, fro the hypothesis assued on the inertia atrix, the cross products can be neglected and consequently the inertia atrix becoes diagonal. Like this the rotational kinetic energy is given by: E crot = I v B d = I ( xx φ ψ sinθ ) + I ( yy θ cosφ + ψ sinφ cosθ ) + I ( zz θ sinφ ψ cosφ cosθ ) (8) 8565

3 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 or in a copact for using (4): E crot = I xx p + I yyq + I zzr = ωt Jω (9) If the Jacobian fro ω to η in (4) is naed as W η and the following atrix is defined: J = J (η) = W T η JW η () then the kinetic energy equation (9) can be rewritten as function of the generalized coordinate η as follows: E crot = ηt J η () The potential energy E p expressed in ters of the generalized coordinates is given by: E p = gz () The coplete oveent equation is obtained fro the Lagrangian expression (6), as follows: [ ] Fξ = d ( ) L L (3) τ η dt q i q i where τ η R 3 represents the roll, pitch and yaw oents and F ξ = R I ˆF + A T is the translational force applied to the helicopter due to the ain control input U in z axis direction, with R I ˆF = R IE3 U and A T the external disturbances. Since the Lagrangian does not contain kinetic energy ters cobining ξ and η, the Lagrange-Euler equations can be divided into translational and rotational dynaics, being the Lagrange-Euler equations of the translational oveent: ξ + ge 3 = F ξ (4) Then, (4) can be expressed by eans of state vector ξ, yielding: ẍ = (cosψ sinθ cosφ + sinψ sinφ)u + A x ÿ = (sinψ sinθ cosφ cosψ sinφ)u + A y (5) z = g+ (cosθ cosφ)u + A z The Lagrange-Euler equations for the coordinate η, written in the general for, are [Castillo et al., 5a]: where M(η) = J (η). M(η) η +C(η, η) η = τ η (6) Thus, the atheatical odel that describes the helicopter rotational oveent obtained fro the Lagrange-Euler foralis is given by: η = M(η) (τ η C(η, η) η) (7) 3. ERROR BASED STATE SPACE CONTROLLER (E-SSPC) FOR PATH TRACKING In this section a control law to solve the path tracking proble by translational oveents is designed. A linear state space MPC strategy based on the error odel is perfored. Fro the error odel, two predictive controllers are synthesized. The first one controls the height through of the input U, whereas the second one akes use of this signal as a tie variant paraeter in the linear x and y otions to copute the virtual inputs. Thus, the syste (5) is rewritten in state space for ˆx(t) = f( ˆx(t),û(t)) for the controller design, where ˆx(t)=[z(t) w (t) x(t) u (t) y(t) v (t)] T stands for the state space vector of the syste. Fro (5) and the new state space vector, the syste dynaic equation can be written in the following for: w (t) g+(cosθ(t)cosφ(t)) U (t) u (t) ˆx(t) = f(ˆx(t),û(t)) = u x (t) U (t) (8) v (t) u y (t) U (t) with: u x (t) = (cosψ(t)sinθ(t)cosφ(t)+sinψ(t)sinφ(t)) (9) u y (t) = (sinψ(t)sinθ(t)cosφ(t) cosψ(t)sinφ(t)) Equations (5) show that the oveent through the x and y axes depends on the control input U. In fact, U is the designed total thrust vector to obtain the desired linear oveent, while u x and u y can be considered as the orientations of U that cause the oveent through the x and y axes, respectively. The objective of this approach is to guarantee that the UAV follows a previously defined reference trajectory without any displaceent error. However, due to the fact that the destination coordinates varies in tie, a reference virtual vehicle having the sae quadrotor helicopter atheatical odel is placed on the track: ˆx re f (t) = f(ˆx re f (t),û re f (t)) () where ˆx re f (t) = [z re f (t) w re f (t) x re f (t) u re f (t) y re f (t) v re f (t)] T and û re f (t) = [U re f u xre f u yre f ] T are the reference states and control inputs, respectively. The input control U (t) is considered a tie variant paraeter to the reference x and y otions. Moreover, because the decentralized control structure, the roll, pitch and yaw angles are also considered as paraeters that vary in tie. Thus, by subtracting the syste () fro the syste (8) and using the Euler s ethod, the proposed translational error odel, as a tie variant discrete linear odel, is given by: x(k+ ) = A x(k)+b(k) ũ(k). () where x(k) = ˆx(k) ˆx ref (k) represents the vector error and ũ(k) = û(k) û ref (k) the control input error. Therefore, the error odel () is split up in two subsystes: height error and x and y otions error as follows: [ ][ ] [ ] z(k) x z (k+ ) = + T T w (k) cos(θ(k))cos(φ(k)) Ũ (k) () T x(k) T x xy (k+ ) = ũ (k) T ỹ(k) + U (k) [ ] ũx (k) (3) ũ y (k) ṽ (k) T U (k) So that, fro the height and longitudinal-lateral error odels the control laws can be designed in such way that the syste is forced to track the reference trajectory. The first one coputes the control input U. The idea used in this controller consists in the coputation of a control law in such a way that iniizes the cost defined by: J z = [ˆ x ] ] ] ] z x zr Qz [ˆ x z x zr + [Ũ Ũ r Rz [Ũ Ũ r, (4) where Q z and R z are diagonal definite positive weighting atrices and N z and N uz are the horizons [Rossiter, 3]. The predictions of the plant output ˆ x z (k + j k) are coputed using a linearized tie variant state space odel of the vehicle by the equation (), obtaining: ˆ x z = P z (k k) x z (k k)+h z (k k) Ũ, (5) 8566

4 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 where Ũ (k) = U (k) U re f (k) and x z (k) is the height state The internal ter of the integral expression on the left-hand error, and the height reference vectors are: side of inequality (3) can be written as: ˆx zr (k+ k) ˆx zr (k k) Û r (k k) Û r (k k) z x zr =., Ũ re f = = zt z = [ h T (x) u T ] [ ] W T h(x) W u. and. the syetric positive definite atrix W T W can be partitioned as follows: [ ] ˆx zr (k+ N k) ˆx zr (k k) Û r (k+ Nu k) Û r (k k) Miniizing the equation (4) when the constraints are not W T Q S W = considered, the control law can be obtained as: S T (3) R Ũ = [ H ] ] z Q z H z + R z [H Matrices Q and R are syetric positive definite and the fact z Q z ( x zr P z x z (k))+r z Ũ re f, (6) that W T W > guarantees that Q SR S T >. although only Ũ (k) is needed at each instant k [Caacho and Bordons, 998]. In the constrained case, an optiization algorith solves (4) at each sapling tie. However, in this work constraints are not considered. So that, by (6), U (k) = Ũ (k)+u re f (k) is obtained. Then, the next step perfors the x and y otion control inputs. The sae previous procedure, using the error odel (3), to copute the control law is carried out, and is given by: ũ xy = [ H ] xy Q xy H xy + R xy [ H xy Q xy ( x xyr P xy x xy (k) ) ] + R xy ũ xyr, (7) where ũ xy = [ũ x (k) ũ y (k)] T, and [ ] [ ] [ ] ux (k) ũx (k) uxre = + f (k) u y (k) ũ y (k) u yre f (k) The error reference states and control inputs are obtained fro the sae for that for height controller case. Fro u x (k) and u y (k) the reference roll, φ re f, and pitch, θ re f, angles for the helicopter rotational loop using equation (9) are coputed. 4. NONLINEAR H CONTROLLER FOR STABILIZATION Under these assuptions, an optial control signal u (x,t) ay be coputed for syste (8) if there is a sooth solution V(x,t), with V(x,t) for t, to the following HJBI equation [van der Schaft, ]: V t + T V x f(x,t)+ T V x [ γ k(x,t)kt (x,t) g(x,t)r g T (x,t) T V x g(x,t)r S T h(x)+ ht (x) ( Q SR S T) h(x) = ] V x (3) for each γ > σ ax (R), where σ ax stands for the axiu singular value. In such a case, the optial state feedback control law is derived as follows [W. Feng and I. Postlethwaite, 994]: ( u = R S T h(x)+g T (x,t) V(x,t) ). (33) x 4. Rotational Subsyste Nonlinear H Control In order to develop the nonlinear H controller the rotational oveents dynaic odel (6), obtained fro the Lagrange- Euler foralis, is used. τ η adds the control torques and external disturbances, and is redefined as: In this section the rotational subsyste control law to achieve robustness in presence of sustained disturbances and paraetric uncertainty is developed. A nonlinear H controller is able to execute this task. The controller design for echanical syste odels using Lagrange-Euler equations is carried out by a direct ethod. 4. Nonlinear H Control Theory The dynaic equation of an nth order sooth nonlinear syste which is affected by an unknown disturbance can be expressed as follows: ẋ = f(x,t)+g(x,t)u+k(x,t)ω, (8) where u R p is the vector of control inputs, ω R q is the vector of external disturbances and x R n is the vector of states. Perforance can be defined using the cost variable z R (+p) given by the expression: [ ] h(x) z = W, (9) u where h(x) R represents the error vector to be controlled and W R (+p) (+p) is a weighting atrix. If states x are assued to be available for easureent, then the optial H proble can be posed as follows [van der Schaft, 99]: Find the sallest value γ such that for any γ γ exists a state feedback u = u(x,t), such that the L gain fro ω to z is less than or equal to γ, that is: T T z dt γ ω dt. (3) τ η = τ ηa + τ ηd where τ ηa is the applied torques vector and τ ηd represents the total effect of syste odelling errors and external disturbances. As a first step to synthesize the control law, the tracking error vector is defined as follows: η η η d η η η d ˆx = = ηdt ( η η d) dt (34) where η d and η d R n are the desired trajectory and the corresponding velocity, respectively. Note that an integral ter has been included in the error vector. This ter will allow to achieve a null steady-state error when persistent disturbances are acting on the syste [Ortega et al., 5]. The following control law is proposed for the rotational subsyste: τ ηa = M(η) η +C(η, η) η T ( M(η)T ˆx+C(η, η)t ˆx ) + T u (35) The proposed control law can be split up into three different parts: the first one consists of the first three ters of that equation, which are designed in order to copensate the syste dynaics (6). The second part consists of ters including the error vector ˆx and its derivative, ˆx. Assuing τ ηd, these two ters of the control law enable perfect tracking, which eans that they represent the essential control effort needed to perfor the task. Finally, the third part includes a vector u, which represents the additional control effort needed for disturbance rejection. 8567

5 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 It can also be pointed out that, despite the preceding control law ight see a not well posed syste, it will be shown afterwards that the coputed torque does not rely on joint accelerations, but on their references. Matrix T in (35) can be partitioned as follows: T = [ T T T 3 ] with T = ρi, where ρ is a positive scalar and I is the nth-order identity atrix. Substituting the expression of the control law fro (35) into the Lagrange-Euler equation of the syste (6) and defining ω = M(η)T M (η)τ ηd, one gets: M(η)T ˆx+C(η, η)t ˆx = u+ω (36) This expression represents the dynaic equation of the syste error. Taking into account this nonlinear equation, the nonlinear H control proble can be posed as follows: Find a control law u(t) such that the ratio between the energy of the cost variable z = W [ h T ( ˆx) u T] T and the energy of the disturbance signals ω is less than a given attenuation level γ. Taking into account the definition of the vector error, ˆx, and the definition of the cost variable, z, the following structures are considered for atrices Q and S in (3): Q Q Q 3 S Q = Q Q Q 3, S = S. Q 3 Q 3 Q 3 S 3 To apply the theoretical results presented in Section 4., it is necessary to rewrite the nonlinear dynaic equation of the error (36) into the standard for of the nonlinear H proble (see (8)). This can be done by defining the following expressions: ˆx = f(ˆx,t)+g( ˆx,t)u+k( ˆx,t)ω, (37) M C O O f ( ˆx,t) = T T I T T I + T (T T 3 ) T, O I I M(η) g( ˆx,t) = k(ˆx,t) = T O O where I is the identity atrix, O the zero atrix, both of n-th order, and T T T 3 T = O I I. (38) O O I As stated in Section 4., the solution of the HJBI equation depends on the choice of the cost variable, z, and particularly on the selection of function h( ˆx) (see (9)). In this paper, this function is taken to be equal to the error vector, that is, h( ˆx) = ˆx. Once this function has been selected, coputing the control law, u, will require finding the Lyapunov function, V(ˆx,t), to the HJBI equation posed in the previous section (see (3)). The details to achieve this solution can be found in Ortega et al. [5]. Matrix T = [ T T T 3 ] can be coputed by solving soe Riccati algebraic equations (see Ortega et al. [5]). Once atrix T is coputed, substituting V ( ˆx,t) in (33), control law u corresponding to the H optial index γ is given by u = R ( S T + T ) ˆx (39) Finally, if the control law (39) is replaced into (35), and after soe anipulations, the optial control law can be written as: ) τ η a = M(η) η d +C(η, η) η M(η) (K D η + K P η K I ηdt (4) A particular case can be obtained when the coponents of weighting copound W T W verify: Q = ω I, Q = ω I, Q 3 = ω 3 I, R = ω u I, (4) Q = Q 3 = Q 3 =, S = S = S 3 =. In this case, the following analytical expressions for the gain atrices have been obtained: ω K D = + ω ( ω 3 I + M(η) C(η, η)+ ) ω ωu I, K P = ω 3 ω I + ω + ω ω 3 ω ( M(η) C(η, η)+ ) ωu I, K I = ω ( 3 M(η) C(η, η)+ ) ω ωu I. These expressions have an iportant property: they do not depend on the paraeter γ. So, we obtain an algebraic expression for coputing the general optial solution for this particular case. 5. SIMULATION RESULTS The proposed control strategy has been tested by siulations in order to check the perforance attained for the path tracking proble. Siulations has been perfored considering external disturbances and paraetric uncertainties. The following vertical helix has been defined as the reference trajectory: x d = ( t ) cos, y d = ( t ) sin, z d = + t, ψ d = π 3 The initial conditions of the helicopter are (x,y,z) = (,,.5) and (φ,θ,ψ) = (,,.5)rad. The values of the odel paraeters used for siulations are the following: =.74 kg, l =., g = 9.8/s and I xx = I yy =.4 Kg., I zz =.84 Kg.. An aount of ±% in the uncertainty of the eleents of the inertia atrix has been considered in the siulations. In the siulations external disturbances on the aerodynaic oents were considered. The following persistent steps were applied: A r =.5N at t = 5s; A p = N at t = 5s; and A q = N at t = 5s. The E-SSPC paraeters were adjusted as follows: [ ] N z = N uz = 3I nz, Q z =, R z =. 5 N xy = N uxy = 3I nxy, Q xy = 5, R xy = [ ] 9 9 The nonlinear H controller gains were tuned with the following values: ω =.5, ω =.5, ω 3 = 5 y ω u =.7. Figs. to 4 present a perfect tracking of the reference trajectory when external disturbance originated by aerodynaic oents are considered. The results illustrate the robust perforance provided by the controller in the case of paraetric uncertainty in the inertia ters. Using the E-SSPC a sooth reference tracking was perfored, ainly, in the beginning of the track where the vehicle is far fro the trajectory. This is due because the predictive controller considers the future reference in the coputation of the control signal and thus, it tries to predict the path soothing the displaceent. 8568

6 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 5 Inertia +% Inertia % Finally, the robustness, the soothness and the predictive feature of the presented control strategy has been also corroborated by siulations. z [] y [].5.5 x [] Fig.. Path following with external disturbances..5.5 x [] y [] Inertia +% Inertia %.5 Inertia +% Inertia % 5 5 z [] Inertia +% Inertia % tie [s] Fig. 3. Position (x,y,z) with external disturbances..4.. φ [rad] θ [rad] Inertia +%. Inertia % Inertia +% Inertia %. 5 5 ψ [rad] Inertia +% Inertia % tie [s] Fig. 4. Orientation (φ,θ,ψ) with external disturbances. 6. CONCLUSIONS In this paper a predictive and robust control strategy to solve the path tracking proble for a quadrotor helicopter has been presented. The proposed strategy was designed in consideration of external disturbances like aerodynaic oents. Through the state space predictive controller for the linear oveents a good and sooth perforance in the reference tracking has been achieved. A robust control based on nonlinear H theory has been used for the helicopter stabilization, which is able to reject oent disturbances. Besides, the H controller robustness has been checked under uncertainty in the inertia ters. ACKNOWLEDGEMENTS The authors would like to thank Jörn Klaas Gruber for his help in this work. Also, CICYT for funding this work under grants DPI4-649, DPI and DPI REFERENCES S. Bouabdallah and R. Siegwart. Backstepping and Slidingode Techniques Applied to an Indoor Micro Quadrotor. In Proc. IEEE Int. Conf. on Robot. and Autoat., pages 59 64, Barcelona, Spain, 5. S. Bouabdallah, A. Noth, and R. Siegwart. PID vs LQ Control Techniques Applied to an Indoor Micro Quadrotor. In Proc. IEEE Int. Conf. on Intelligent Robots and Systes, volue 3, pages , Sendai, Japan, 4. E. F. Caacho and C. Bordons. Model Predictive Control. Springer-Verlag, New York, 998. P. Castillo, R. Lozano, and A. Dzul. Stabilization of a Mini Rotorcraft with Four Rotors. IEEE Control Systes Magazine, pages 45 55, Dic 5a. P. Castillo, R. Lozano, and A. E. Dzul. Modelling and Control of Mini-Flying Machines. Springer-Verlag, London, UK, 5b. M. Chen and M. Huzezan. A Cobined MBPC / DOF H Controller for a Quad Rotor UAV. In Proc. AIAA, 3. J. J. Craig. Introduction to Robotics - Mechanics y Control. Addison-Wesley Publishing Copany, Inc., USA, nd edn edition, 989. I. Fantoni and R. Lozano. Nonlinear Control for Underactuated Mechanical Systes. TSpringer Verlag, London, 995. T. J. Koo and S. Sastry. Output tracking control design of a helicopter odel based on approxiate linearization. In Proc. of the CDC, Florida, USA, 999. V. Mistler, A. Benallegue, and N. K. M Sirdi. Exact linearization and noninteracting control of a 4 rotors helicopter via dynaic feedback. In Proc. IEEE Int. Workshop on Robot and Huan Inter. Counic.,. J. E. Norey-Rico, J. Góez-Ortega, and E. F. Caacho. A Sith-Predictor-Based Generalised Predictive Controller for Mobile Robot Path-Tracking. In Control Engineering Practice, volue 7, pages 79 74, London, England, 999. R. Olfati-Saber. Nonlinear Control of Underactuated Mechanical Systes with Application to Robotics and Aerospace Vehicles. PhD thesis, Massachusetts Institute of Technology,. M. G. Ortega, M. Vargas, C. Vivas, and F. R. Rubio. Robustness Iproveent of a Nonlinear H Controller for Robot Manipulators via Saturation Functions. Journal of Robotic Systes, (8):4 437, 5. J. A. Rossiter. Model-Based Predictive Control: A Practical Approach. CRC Press, New York, 3. A. van der Schaft. L -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, New York,. A. van der Schaft. L -Gain Analysis of Nonlinear Systes and Nonlinear State Feedback Control. IEEE Trans. Autoat. Control, 37(6):77 784, 99. W. Feng and I. Postlethwaite. Robust Nonlinear H /Adaptative Control of Robot Manipulator Motion. Proc. Instn. Mech. Engrs., 8: 3,