Aircraft Maneuver Regulation: a Receding Horizon Backstepping Approach

Transcription

1 Aircraft Maneuver Regulation: a Receding Horizon Backstepping Approach Giuseppe Notarstefano and Ruggero Frezza Abstract Coordinated flight is a nonholonomic constraint that implies no sideslip of an aircraft. The equations of motion in coordinated flight are kinematically reducible. This property simplifies the maneuver regulation problem because under such assumption it is possible to write a lateral controller for the transverse dynamics independent of velocity. Assuming coordinated flight, the maneuver regulator consists of a model predictive controller based on the kinematic model. Since, in reality the coordinated flight assumption is seldom satisfied, the kinematic control action is back-stepped into dynamics to compute the actuation of the control surfaces. The proposed control law is tested on a multi-body SW model of an aircraft on various maneuvers, including some aggressive ones. I. INTROUCTION There exists a vast literature on flight control []. Most of the research, however, deals with the regulation around equilibrium trajectories [], [] or tracking of desired angle of attack or roll rate [], [], []. Maneuver regulation has been addressed by fewer researchers and only on simplified models as the coordinated flight model [] or the PVTOL (Planar Vertical Take Off and Landing) []. More recently, an intense research activity concerned the development of Unmanned Aerial Vehicles (UAV) and maneuver regulation is a fundamental problem that must be solved for these systems to be able to operate. Another important application of maneuver regulation in flight control has appeared in Computer Aided Engineering. The state of the art in testing aircraft design with software tools is the computation of loads in static or quasi-static situations. The estimate of dynamic loads obtained by running full fledged simulations of maneuvers is of great interest and poses the problem of flying the virtual aircraft on the maneuver. This, clearly, must be done in closed loop acting on the controls as if the virtual aircraft was real. Maneuver regulation is a different control problem than the well known trajectory tracking. In the latter, the specifications are given in terms of a desired state trajectory as a function of time. In the path following or maneuver regulation problem, the trajectory and the velocity are specified as functions of arc-length. In presence of no error, the solution to the two problems may be identical by properly specifying the velocity, but, in presence of error, the trajectories differ drastically. In one case, trajectory tracking, the longitudinal error is in position and, if the aircraft is ahead of the reference, the control system tends R. Frezza and G. Notarstefano are with the epartment of Information Egineering, University of Padova, Via Gradenigo 6/B, Padova, Italy {frezza,notarste}@dei.unipd.it to slow it down in order to be caught up by the reference. In the path following with specified velocity, the longitudinal control is a velocity control and the system does not care about position. Kinematic reducibility is a concept that has been introduced recently by Lewis and Bullo [], [6]. It basically answers the following question: when is it that the trajectories of a mechanical control system can be generated by a kinematic model? For the vehicle control problem considered here, the relevant question is the following. Let q R n be the configuration vector of the aircraft model and s the arc length of a path of the system q(s): oes there exist a set of vector fields V i, for i =,..., m such that dq(s) m = V i (q(s))u i? () ds i= If this happens, then the path following control problem may be decoupled in a kinematic lateral control problem and a longitudinal control problem so that the arc-length s follows a desired trajectory s d (t). In coordinated flight the model of an aircraft is kinematically reducible. This allows us to propose a path following control law that, at the kinematic level recalls the strategy proposed in [8] by one of the authors, and at the dynamic level acts on the control surfaces so that the coordinated flight constraint is satisfied as much as possible. The kinematic control is model predictive. In the definition of the optimal control problem the aircraft equations of motion do not appear because they are differentially flat and therefore trajectory invertible. The overall architecture of the controller is of backstepping type. The kinematic control actions are fed back to the dynamic loop that acts on the ailerons, rudder and elevators in order to track it. Experiments on various simulations run with a multibody software package are illustrated. II. AIRCRAFT MOEL The aircraft is modeled as a rigid body subject to gravity, thrust and aerodynamic forces and moments. As usual in the literature on flight dynamics, the controls are the thrust force T and the angle of deflection of the moving surfaces of the aircraft, i.e. the elevator δ e, the aileron and the rudder δ r. It is customary to describe the aircraft dynamics using the following reference frames: (a) The ground frame which is on the Earth supposed flat

2 and non rotating. Th axis points to North, the y axis to East and the z axis downward. (b) The body frame which is fixed to the aircraft center of gravity. Th b axis points forward along the nose of the aircraft, the y b axis to the right of the pilot and the z b axis downward. (c) The wind frame which is also fixed to the center of gravity of the aircraft, but it is rotated w.r.t. the body frame so that th w axis is directed along the velocity vector. The wind axis frame can be obtained from the body frame by two successive rotations respectively along the y b and z s axes. The z s axis is the z axis of the intermediate reference frame, obtained after the first rotation, which is called stability axes frame. The two rotation angles are called respectively angle of attack α and side-slip angle β. These angles play a crucial role in the dynamics. The angle of attack, in fact, modulates the Lift amplitude, while the side-slip angle defines the lateral slip of the aircraft. If β =, the aircraft is in coordinated flight. The dynamics of the aircraft is then described by six states which take into account the position and velocity of the center of gravity and six other states for the attitude (orientation and angular velocity). The equations of motion can, of course, be written in different sets of coordinates. For control purposes, we use the wind axes frame for the force equations and the body frame for the moment equations. Finally, applying a kinematic tranformation, the absolute position of the aircraft in the ground frame is obtained. The state of the system is, therefore, composed by the absolute position of the center of gravity X g = (x, y, z) w.r.t. the ground frame, the three Roll, Pitch and Yaw angles (µ, γ, χ) which describe the orientation of the wind axis frame w.r.t. the ground frame, the norm of the velocity vector of the aircraft V t, the angle of attack α and the sideslip angle β and, finally, the angular velocity in the body frame ω b = (p, q, r). The equations of motion are given by: Ẋ g = R gw(γ, χ)v t ; () m V t = + Y sin β + T cos β cos α mg sin γ; γ = (L cos µ Y cos β sin µ mg cos γ) + T (cos α sin β sin µ + sin α cos µ); χ = cos γ + cos γ (L sin µ + Y cos β cos µ) T (sin α sin µ cos α sin β cos µ) µ = cos β (p cos α + r sin α) + f µ(v t, µ, γ, α, β, T, L, Y ); α = q tan β(p cos α + r sin α) + f α (V t, µ, γ, α, β, T, L); β = p sin α r cos α + f β (V t, µ, γ, α, β, T, Y ); () () ω b = I ˆω b Iω b + I τ; () where R gw is the first column of the rotation matrix from wind frame to ground frame, L f µ = (tan γ sin µ + tan β) + tan γ cos µ cos β + T [sin α(tan γ sin µ + tan β) cos α tan γ cos µ sin β] mg cos γ cos µ tan β, f α = cos β ( L T sin α + mg cos γ cos µ), f β = (Y cos β + T sin β cos α + mg cos γ sin µ), and ˆω b = Y r p r q q p τ = l M N (6) (7). (8) The aerodynamic forces are the drag and the lift L, expressed in the wind frame, and the side force Y expressed in the stability axes frame. The aerodynamic moments l (roll moment), M (pitching moment) and N (yaw moment) are, instead, expressed in the body frame. They all depend on the velocity of the airflow washing the surfaces, on the air density (which depends on the altitude), on the geometry of the aircraft and finally on dimensionless coefficients which, in their more general formulation, are functions of all the states. However, taking into account only the main contributions, the aerodynamic forces and moments are given by: where L = qsc L (α, q, δ e ); = qsc (α, q, δ e ); Y = qsc Y (β, p, r, δ r ); l = qsbc l (β, p, r,, δ r ); M = qscc m (α, q, δ e ); N = qsbc n (β, p, r,, δ r ); (9) q = ρ(h)v t () and S is the wing area, b the wing span, c the mean aerodynamic chord and h the altitude. We assume to control directly the angle of deflection δ e, and δ r, we neglect the actuator dynamics. Moreover, we assume that the thrust force does not depend on the velocity, but it is simply a positive force applied in the center of gravity.

3 A. Coordinated flight model Imposing coordinated flight constraint β =, () we obtain a simplified model. This is a desired condition of flight in non-acrobatic maneuver because it minimizes drag. The coordinated flight model is at the basis of our control strategy. The constraint () implies that the velocity vector is directed along the nose of the aircraft, i.e. the y component of the velocity in the body frame is zero. This is a velocity constraint which is non integrable, i.e. nonholonomic. Bullo and Lewis have shown [], [6] that under suitable conditions a dynamic model can be reduced to a kinematic model. As we shall see these conditions are satisfied by the simplified aircraft model. Substituting the constraint () in the α and β dynamics and posing also β =, two constraints on the pitch and yaw velocities arise: q w = T sin α + L(α, q, δ e ) + mg cos γ cos µ r w V t = g cos γ sin µ () where ω w = [p w q w r w ] T is the angular velocity expressed in the wind axes frame. The aerodynamic force Y is zero due to the assumption of coordinated flight. The roll velocity, instead, can be arbitrary. Hence, since the dynamics () are constrained to generate suitable angular velocities, the aircraft state space becomes seven dimensional, R S, with a three dimensional input space consisting of T, L and p w. One can easily invert the longitudinal dynamics of V t and use thrust to control the system to a desired velocity. Moreover, we can use the lift L (or equivalently the angle of attack α) to control q w and p w to control µ and so r w. Hence the dynamic system may be reduced to a kinematic model with V t, q w and r w as control inputs. The coordinated flight model is therefore kinematically reducible according to the definition given by Bullo and Lewis in [], [6]. The kinematic equations in the wind axes frame are given by: Ẋ w (t) = ˆω w X w (t) + [ ] T V t (t) () where X w (t) = [X w (t) X w (t) X w (t)] is the position of the center of gravity in the wind axes frame and ˆω w has the same structure of (7) but in the wind axes frame. Observe the state is time-varying because of the frame motion. III. CONTROL STRATEGY Our control task is a maneuver regulation which is equivalent to path following with a specified velocity. The reference inputs are a path in space and a velocity profile given as a function of the arc length of the path. Maneuver regulation differs with respect to the trajectory tracking problem in the sense that the trajectory can be reparameterized in time. This is crucial in flight control since the dynamics impose constraints on feasible maneuvers and time reparametrization may be used to meet the constraints []. Maneuver regulation implicitly demands for a decoupling of the lateral and longitudinal control tasks. The property of kinematic reducibility allows us to design a decoupled controller. The thrust input, therefore, is used only for the longitudinal control and the surfaces deflection for the lateral. The velocity control is based on feedback linearization of the velocity dynamics: T = cos β cos α ( cos β Y sin β + mg sin γ + k V p m(v ref (t) V t ) + V ref (t)), so to obtain the linear asymptotically stable dynamics () V t = V ref (t) + k V p (V ref (t) V t ). () The lateral control strategy is more complex and is based on a predictive strategy for controlling the reduced coordinated flight model, combined with a backstepping approach which solves a model matching problem and acts on the control surfaces in order to maintain coordinated flight. A. Receding Horizon Control of the coordinated flight model The kinematic control strategy for the coordinate flight model of the aircraft was introduced by Frezza et al. in [9] for vehicles subject to nonholonomic constraints and was applied to the flight control in [8]. It is a predictive, output-feedback control strategy based on the recursive approximation of the target contour with feasible system trajectories. Locally, the path Γ is assumed to be represented, in the moving frame, as X w = γ (X w, t); X w = γ (X w, t). (6) The control strategy may be summarized as follows: Measure the position of the contour and its first m derivatives w.r.t. X at look-ahead distance. Generate a locally feasible trajectory which connects to the path at the look-ahead. The trajectory is called connecting contour and is referred as γ c. An example of connecting contour in the case is depicted in fig.. Compute a control action to follow the connecting contour and apply it for a time step. Iterate the procedure. The connecting contour must be feasible and can be chosen in order to minimize a cost function. In this paper we use polynomials (which minimize the overall curvature) of degree m =. Therefore we look for the curves γ c that, at time t, satisfy the boundary conditions [ γ c(, t) γ c X w (, t) ] = [ ] [ γ c(, t) γ c X w (, t) ] = [ ] γ(, t) γ (, t) X w (7)

4 max real part of eigenvalues Fig.. Connecting contour. w µ Fig.. Maximum real part of the eigenvalues The control action necessary to exactly track the connecting contour is given by: q w = V t γ c X w (, t); r w = V t γ c X w (, t). (8) In [8] the stability of the proposed control action was studied. It was shown that there exist values of p w such that the control strategy is unstable. In fact if we linearize the closed loop dynamics of the four moments ξ i = i γ (, t) X i w µ i = i γ (, t) i =... X i w (9) about the target trajectory, the transition matrix p w V t A = p w V t V t ( 6 + ξ) V t ξ µ Vt p w V t ξ µ V t ( 6 + µ ) p w V t () may have an unstable eigenvalue. However it can be shown that if p w < V t () the eigenvalue has a real part strictly negative as shown in fig. for V t = ms, = m and ξ = m. Observe that the preview distance cannot be augmented arbitrarily because it bounds the maximum rolling velocity. However we will see from simulations that the values of p w, needed by the proposed control strategy, are largely within the bound. B. Backstepping The kinematic controller described above assumes that q w and r w are control inputs. Adopting a backstepping control strategy, we feed back the reference angular velocities q w and r w, determined in (8), and act on the control surfaces in order to track them while maintaining coordinated flight. For simplicity, we make the assumption that the aerodynamic forces do not depend directly on deflections and angular velocities, i.e. = qsc (α); L = qsc L (α); Y = qsc Y (β). () This is a common assumption in flight control design in order to be able to apply feedback linearization (or dynamics inversion) [], [], []. In [] a formal justification of this approximation is also given. We observe that, due to the constraints (), tracking q w and r w corresponds to tracking the reference values µ and ᾱ obtained by inverting (numerically) the following relations q w (t) (t) = T (ᾱ, t) sin ᾱ + L(ᾱ) + mg cos γ(t) cos µ r w (t)v t (t) = g cos γ(t) sin µ. () Observe that we may use the current value of γ due to the boundary conditions (7) imposed on the connecting contour. The equations of motion for µ, α and β () and for the angular velocity () are in the so called strict feedback form [7]. Hence, a backstepping approach may be used to track µ and ᾱ and to control β to zero. In particular, we choose p c = p cos α + r sin α q c = q r c = p sin α + r cos α () as virtual control inputs. Using feedback linearization the controls are found to be: p c = cos β( f µ ( ) + µ + k µ ( µ µ)) q c = sin β( f µ ( ) + µ + k µ ( µ µ))+ f α ( ) + ᾱ + k α (ᾱ α) r c = f β ( ) + k β β () where µ and ᾱ are obtained differentiating () after substituting the expression (8) for q w (t) and r w (t). The resulting error dynamics are given by: ė µ = k µ e µ + cos β (p c p c ), ė α = k α e α + (q c q c ) + tan β(p c p c ), β = k β β + (r c r c ). (6)

5 Finally we can feedback linearize the dynamics () of the body angular velocity in order to track the virtual controls p c, q c and r c found in the previous step. Supposing that l(, δ r ), M(δ e ) and N(, δ r ) are invertible w.r.t. (, δ e, δ r ), we may control directly the aerodynamic moments l M N = ˆω b Iω b + I p + k p ( p p) q + k q ( q q) r + k q ( r r). (7) where p, q and r are computed by inverting (). Applying the control action (7) the virtual inputs p c, q c and r c converge to the desired values p c, q c and r c. Hence, the system (6) is linear asymptotically stable with decaying inputs and thus it converges to the desired values of µ, α and β. Remark The backstepping technique used here is also known in the flight control literature as dynamic inversion by means of time scale separation. The dynamics may be, in fact, separated in fast (angular velocity dynamics) and slow (bank angle, angle of attack and side-slip angle dynamics) ones. This implies that the fast states may be used as virtual controls supposing they track their slowing changing reference exactly. IV. SIMULATIONS The robustness of the proposed control strategy has been tested on different virtual aircraft prototypes by performing several maneuvers. We show here two examples of maneuvers: the tracking of a sinusoidal path and a more aggressive double turn on the horizontal plane. The model used in the simulation is a Cessna7 with the following parameters: m =. Kg, I xx = 8. Kg m, I yy = 8.9 Kg m, I zz = Kg m, I xy = I yz = I xz =. A linear formulation has been used for the aerodynamic forces and moments. In the first simulation the aircraft performs a changing on altitude and lateral coordinate by moving along a sinusoidal path. In fig. (b) the errors along the three axes are shown as function of the length of the path. Ignoring numerical miscalculations due to the discontinuous curvature of the path, the errors along the three axes are less than. m except for the initial transient where altitude error is larger because the aircraft starts from an untrimmed condition. The sideslip angle β is controlled to zero, while the angle of attack α and the bank angle µ are modulated in order to perform the lateral and longitudinal motions, fig. (c). The angular velocities are depicted in fig. (d) and finally the thrust force and the surfaces deflections are shown in fig. (e) and (f). The same graphs are reported for the second maneuver in fig.. This maneuver is more aggressive, in fact the errors are larger than the previous case, however they are less than.6 m, fig. (b). The peaks of the errors occur in the points of (discontinuous) changing of the curvature. V. CONCLUSIONS We have proposed a control strategy for maneuvering an autonomous aircraft. A receding horizon technique is used to control the coordinated flight model of the aircraft by means of angular velocities. Then a backstepping control is applied to the real inputs so that the aircraft reach coordinated flight and tracks the angular velocity computed with the predictive strategy. Two maneuver examples have been presented to show the controller robustness. REFERENCES [] R. J. Adams, J. M. Buffington, S. S. Banda esign of Nonlinear Control Laws for High-Angle-of-Attack Flight, Journal of Guidance, Control and ynamics, Vol. 7, No., June-August 99. [] Al-Hiddabi S.A., McClamroch N.H., Tracking and maneuver regulation control for nonlinear nonminimum phase systems: application to flight control Control, IEEE Transactions on Systems Technology, Volume, Issue 6, Nov. Page(s): [] R. Bhattacharya, G. J. Balas, M. A. Kaya, A. Packard, Nonlinear Receding Horizon Control of an F-6 Aircraft, American Control Conference, Arlington, Virginia, June. []. J. Bugajski,. F. Enns, Nonlinear Control Law with Application to High Angle-of-Attack Flight, Journal of Guidance Control adn ynamics, Vol., No, May-June 99. [] F. Bullo, K. Lynch Kinematic Controllability for ecoupled Trajectory Planning in Underactuated Mechanical Systems, IEEE Transaction on Robotics and Automation, Vol. 7, Issue, Aug. pp. -. [6] F. Bullo and A. Lewis, Low-order controllability and kinematic reduction for affine connection control systems, SIAM Journal of Control and Optimization, Jan. [7] R. Freeman, P. Kokotovic, Robust Nonlinear Control esign, Boston, Birkhauser,. [8] R. Frezza, Path Following for Air Vehicles in Coordinated Flight, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, September 9-, 999, Atlanta, USA. [9] R. Frezza, G. Picci, S. Soatto, Non-holonomic Model-based Predictive Output Tracking of an Unknown Three-dimensional Trajectory, Proceedings of 7th IEEE Conference on ecision and Control, 998. [] J. Hauser and R. Hindman, Aggressive Flight Maneuvers, Proceedings of the 6th IEEE Conference on ecision and Control, - ec. 997, pp. 86-9, vol.. [] A. Jadbabaie and J. Hauser, Control of a thrust vectored flying wing: a receding horizon-lpv approach, International Journal of Robust and Nonlinear Control, Vol., No. 9, July, pp [] C. Nielsen, M. Maggiore. Maneuver regulation via transverse feedback linearization : Theory and examples, IFAC symposium on nonlinear control systems (NOLCOS, semi-planary presentation). September [] S. N. Singh, M. Steinberg, A. B. Page, Nonlinear Adaptive and Sliding Mode Flight Path Control of F/A-8 Model, IEEE Transaction on Aerospace and Electronic Systems, vol. 9, No., October. [] S. A. Snell,. F. Enns, W. L. Garrard Jr. Nonlinear Inversion Flight Control for a Supermaneuverable Aircraft, Journal of Guidance, Control and ynamics, Vol., No., July-August 99. [] M.L.Steinberg, Comparison of Intelligent, Adaptive and Nonlinear Flight Control Laws, Journal of Guidance, Control and ynamics, Vol., No., July-August.

6 z(meter) 6 8 y(meter) x(meter) (a) reference path trajectory, e y, e z (meter)..... e y e z (b) α, β, µ (deg) (c) α β µ p, q, r (deg/sec) (d) p q r T x Thrust (e), δ e, δ r (deg) (f) δ e δ r Fig.. Following of a sinusoidal path y(meter) 6 8 reference path trajectory x(meter) (a), e y, e z (meter) e y e z (b) α, β, µ (deg) α β µ (c) p q r x 6 Thrust δ e δ r p, q, r (deg/sec) T, δ e, δ r (deg) (d) (e) (f) Fig.. Following of a double turn path at constant altitude