A full controller for a fixed-wing UAV
AA full controller for a fixed-wing UAV
Gerardo Flores, Alejandro Flores and Andr´es Montes de Oca
Abstract — This paper presents a nonlinear control law forthe stabilization of a fixed-wing UAV. Such controller solves thepath-following problem and the longitudinal control problem ina single control. Furthermore, the control design is performedconsidering aerodynamics and state information available in thecommercial autopilots with the aim of an ease implementation.It is achieved that the closed-loop system is G.A.S. and robustto external disturbances. The difference among the availablecontrollers in the literature is: 1) it depends on available states,hence it is not required extra sensors or observers; and 2) it ispossible to achieve any desired airplane state with an ease ofimplementation, since its design is performed keeping in mindthe capability of implementation in any commercial autopilot.
Index Terms — Fixed-wing; path-following; UAV; longitudinalaircraft; dynamics Lyapunov-based control.
I. I
NTRODUCTION
Fixed-wing Unmanned Aerial Vehicles (UAVs) have be-come relevant above the quadrotors in remote sensing appli-cations, precision agriculture, and surveillance. This is dueto its flight endurance, long flying times, high speeds andenergy efficiency. Due to the inherent non-linearity conditionof fixed-wing UAVs, effective control need to be applied tothis type of vehicles so this is a topic that remains beinga challenge. In this paper we propose a full controller thatachieves to stabilize the airplane in a condition required toperform inspection and surveillance missions. Full controlmeans that it can stabilize the UAV at any desired positionand velocity taking into account the optimal angle of attackand aerodynamics. This controller is developed taking inmind its applicability in autopilots such as the popularPixhawk.
A. Literature review
In the vast majority of the works relating fixed-wingaircraft control, the problem is tackled based in three fun-damental modeling-based cases: a) the longitudinal modelwhere the airspeed, angle of attack and pitch dynamics arecontrolled; b) path-following in 2D and 3D, where desired ( x , y ) and ( x , y , z ) positions must be achieved respectively;and c) considering the full 6-DOF airplane mathematicalmodel. Each of these approaches have its own pros and cons.Studying longitudinal model and path-following problemseparately, inherits the problem of not considering the fullcontrol of the plane. Whereas in the path-following control G. Flores, A. Flores and A. Montes de Oca are with the Perceptionand Robotics Laboratory, Centro de Investigaciones en ´Optica, Le´on,Guanajuato, Mexico, 37150. (email: gfl[email protected], alejandrofl@cio.mxand [email protected]). Corresponding author: Gerardo Flores.This work was supported partially by the FORDECYT-CONACYT undergrant 292399 and by the Laboratorio Nacional de ´Optica de la Visi´on ofthe CONACYT agreement 293411. y xI F p q d sv (a) Lateral view. z xI TL D mg v (b) Longitudinal view. Fig. 1: Diagram of the fixed-wing control problem.the airspeed and aerodynamics are not considered, in thelongitudinal approach the steering of the path is not takeninto account. On the other hand, investigating the 6-DOFmodel-based control usually results in complex controls thatincludes a great quantity of parameters and terms that makesthe implementation in real UAVs a complex task. Some ofthe most recent works of these three cases are shown at TableI. Apart from the aforementioned approaches to controlfixed-wing UAVs, there is the guidance law that resolves theproblem of path following by a simple lateral acceleration a r X i v : . [ c s . S Y ] M a r ABLE I: State of the art of fixed-wing UAV controllers. command; this approach can be seen for instance in [19],and in the famous paper [20] which presents the guidancelaw implemented in the Pixhawk commercial autopilot.Regarding control techniques used for solving the controlproblem there are for instance: backstepping [21], [22], [23];gain scheduled [24]; sliding mode [25]; model referencedadaptive control [26]; model predictive control [27], [28];active disturbance rejection control [29]; among others.
B. Contribution
The motivation to develop this work is twofold: a) presentan algorithm that fully controls the fixed-wing UAV con-sidering aerodynamics and the path steering problem; andb) design a controller as simple as possible that effectivelystabilizes the system in the complete flight dynamics, i.e.stabilizing aircraft ( x , y ) position, angle of attack, pitch dy-namics, flight-path angle, yaw dynamics and airspeed (and asa consequence z dynamics) to some given desired states. Thecontrolled is designed having in mind its implementation inthe Pixhawk autopilot using the sensor information availableon it. C. Paper structure
The rest of the paper is structured as follows. In Section IIthe fixed-wing mathematical model is described, then basedon desired states an error model is obtained. Section IIIpresents the full control that stabilized the complete system,a formal proof is given. Section IV contains the simulationresults that validates the effectiveness of the control. Finally,in Section V some comments and future work are presented.II. P
ROBLEM F ORMULATION
A. System Model
Consider the following mathematical model representingthe longitudinal and lateral dynamics according to Fig. 1. m ˙ v = T cos α − D − mg sin γ mv ˙ γ = T sin α + L − mg cos γ m ˙ θ = qI y ˙ q = τ (1)˙ x = v cos ψ ˙ y = v sin ψ ˙ ψ = ω (2)where v is the airspeed of the drone, α , γ and θ are the angleof attack (AoA), the bank and the pitch angle, respectively, D and L are the drag and lift forces generated by theaerodynamics of the wing, ˙ x and ˙ y are the velocities in theinertial frame ( I ), and ψ is the yaw angle. As control inputs,the dynamics can be controlled by T , τ and ω . Without loss of generality, let normalize system (1), i.e. m = I y = v = T cos α − D − sin γ v ˙ γ = T sin α + L − cos γ ˙ θ = q ˙ q = τ . (3)We need to introduce the error of the angles. So, first, wedefine the velocity, bank angle and pitch errors as˜ v = v − v d ˜ γ = γ − γ d ˜ θ = θ − θ d . (4)Using (4) in (3) and considering that ˜ q = q − q d , then wehave ˙˜ v = T cos α − D − sin ( ˜ γ + γ d ) ˙˜ γ = T sin α + L − cos ( ˜ γ + γ d ) ˜ v + v d ˙˜ θ = q − q d ˙ q = τ − ˙ q d . If we define the following coordinate change˜ θ = (cid:18) ˜ θ ˜ θ (cid:19) = (cid:18) θ − θ d q − q d (cid:19) and consider that ˜ α = α − α d . The system changes to˙˜ v = T cos ( ˜ α + α d ) − D − sin ( ˜ γ + γ d ) ˙˜ γ = T sin ( ˜ α + α d )+ L − cos ( ˜ γ + γ d ) ˜ v + v d ˙˜ θ = ˜ θ ˙˜ θ = τ − ˙ q d (5)where ˙˜ θ = ˜ τ .For the lateral model, we define orientation and positionerrors according to Fig. 1a. This path following approach hasbeen investigated in our previous work [4] where a virtualparticle that moves over the desired path according to thevelocity and heading of the UAV is used to achieve thefollowing task. So, the errors are defined as follows [4]˙ e s = v cos ( ˜ ψ ) − ˙ s + Cc ( s ) e d ˙ s ˙ e d = v sin ( ˜ ψ ) − Cc ( s ) e s ˙ s ˙˜ ψ = ω − Cc ( s ) ˙ s where ˙ e s and ˙ e d are the x and the y errors, respectively,between the virtual particle and the position of the UAV inthe inertial frame ( I ) that is expressed in the velocity particleframe ( F ) . This rotation angle is the defined as ψ f = arctan y (cid:48) s x (cid:48) s It is also introduced the path curvature of the desiredtrajectory as C c . As v = ˜ v + v d , the previous system nows ˙ e s = ( ˜ v + v d ) cos ( ˜ ψ ) − ˙ s + Cc ( s ) e d ˙ s ˙ e d = ( ˜ v + v d ) sin ( ˜ ψ ) − Cc ( s ) e s ˙ s ˙˜ ψ = ω − Cc ( s ) ˙ s (6)Gathering (5) and (6), the complete error model is given by˙˜ v = T cos ( ˜ α + α d ) − D − sin ( ˜ γ + γ d ) ˙˜ γ = T sin ( ˜ α + α d )+ L − cos ( ˜ γ + γ d ) ˜ v + v d ˙˜ θ = ˜ θ ˙˜ θ = ˜ τ ˙ e s = v cos ( ˜ ψ ) − ˙ s + Cc ( s ) e d ˙ s ˙ e d = v sin ( ˜ ψ ) − Cc ( s ) e s ˙ s ˙˜ ψ = ω − Cc ( s ) ˙ s . (7)III. M AIN R ESULT
In this section the main result is resented. For that, letconsider the following assumptions.
Assumption 1:
The airspeed velocity is always positive,i.e. v ∈ ( , c ] with c ∈ R + . Assumption 2: v d and γ d are considerd constants. Assumption 3:
The initial airspeed is positive, i.e., thefixed-wing UAV has taken off and is flying in the air.The main results is presented in the next
Theorem 1:
Let system (7) under controls (14), (16), (22)and (24) then the closed loop system is globally asymptoti-cally stable.
Proof:
The goal is to get v → v d and α → α d with v d , α d ∈ R + . It is important to highlight that the algebraicequation θ = γ + α must be always fulfilled. This is acondition inherent of the AoA, pitch and flight-path angle.Let consider the following C function˜ V = ˜ V ˜ v + ˜ V ˜ γ + ˜ V ˜ θ (8)where ˜ V ˜ v =
12 ˜ v ; ˜ V ˜ γ =
12 ˜ γ ; ˜ V ˜ θ = ˜ θ T P ˜ θ (9)with P = P T >
0. The time derivative of ˜ V ˜ v is˙˜ V ˜ v = ˜ v ˙˜ v = ˜ v ( T cos ( ˜ α + α d ) − D − sin ( ˜ γ + γ d )) where we have designed the control term T cos ( ˜ α + α d ) as T cos ( ˜ α + α d ) = D + sin ( ˜ γ + γ d ) − k v ˜ v (10)and then under this control the time derivative of ˜ V ˜ v resultsin ˙˜ V ˜ v = − k v ˜ v . (11)Now obtaining the time derivative of ˜ V ˜ γ we obtain˙˜ V ˜ γ = ˜ γ ( T sin ( ˜ α + α d ) + L − cos ( ˜ γ + γ d )) V + V d where we choose the control term T sin ( ˜ α + α d ) equal to T sin ( ˜ α + α d ) = − L + cos ( ˜ γ + γ d ) − ( ˜ v + v d )( k γ ˜ γ ) (12)and therefore ˙˜ V ˜ γ = − k γ ˜ γ . (13) Considering u = T cos ( ˜ α + α d ) and u = T sin ( ˜ α + α d ) astemporal control inputs, we can extract T as T = (cid:113) u + u (14) α d is obtained from u = T sin α obtaining α as follows α d = arcsin (cid:16) u T (cid:17) , (15)this α d is the commanded for θ d = α d + γ d . The pitch errorsubsystem can be represented by˙˜ θ = A ˜ θ + B ˜ τ with A = (cid:18) (cid:19) ; B = (cid:18) (cid:19) . Let ˜ τ = − k ˜ θ ˜ θ − k ˜ θ ˜ θ (16)then ˙˜ θ = ˜ A ˜ θ where ˜ A = (cid:18) − k − k (cid:19) . It follows that˙˜ V ˜ θ = ˜ θ T ( P ˜ A + ˜ A T P ) ˜ θ = − ˜ θ T Q ˜ θ (17)with the Lyapunov equation P ˜ A + ˜ A T P = − Q with Q = Q T >
0. Then, it follows that the time derivative of (8) is given by˙˜ V = − k v ˜ v − k γ ˜ γ − ˜ θ T Q ˜ θ ≤ E ( t ) = ( ˜ v ( t ) , ˜ γ ( t ) , ˜ θ ( t ) , ˜ θ ( t ) , ˜ e s ( t ) , ˜ e d ( t ) , ˜ ψ ( t )) arebounded, i.e. for each trajectory E ( t ) there is an R ∈ R suchthat || E ( t ) || ≤ R ∀ t ≥ Ω l . By construction we can say that Ω l = { E ( t ) ∈ R : ˜ V ( E ) ≤ l } (19)where ˜ V ( E ) is a Lyapunov function for (7). Let find ˜ V ( E ) by combining ˜ V , which is not dependent of every states in(7), with the positive definite function˜ V = ˜ V ˜ e s + ˜ V ˜ e d + ˜ V ˜ ψ (20)where˜ V ˜ e s = e s ; ˜ V ˜ e d = e d ; ˜ V ˜ ψ = ( ˜ ψ − δ ( e d )) (21)where δ ( e d ) is a function which is saturated as in [31] thatsatisfies the condition e d δ ( e d ) ≤ ∀ e d . Let’s compute thetime derivative of ˜ V e s + ˜ V e d and then we get˙˜ V e s + ˙˜ V e d = e s ˙ e s + e d ˙ e d = e s [( ˜ v + v d ) cos ˜ ψ − ˙ s + Cc ( s ) e d ˙ s ]+ e d [( ˜ v + v d ) sin ˜ ψ − Cc ( s ) e s ˙ s ] . If we define ˙ s as˙ s = k s sign ( e s ) + ( ˜ v + v d ) cos ˜ ψ (22)then ˙˜ V e s + ˙˜ V e d = − k s | e s | + ( ˜ v + v d ) e d sin ˜ ψ . (23)ow if we obtain the time derivative of ˜ V ˜ ψ ˙˜ V ψ = ( ˜ ψ − δ ( e d ))( ω − Cc ( s ) ˙ s − ˙ δ [( ˜ v + v d ) sin ˜ ψ − Cc ( s ) e s ˙ s ]) then consider the following proposed control ω = Cc ( s ) ˙ s + ˙ δ [( ˜ v + v d ) sin ˜ ψ − Cc ( s ) e s ˙ s ]] − ( ˜ v + v d ) e d (cid:18) sin ˜ ψ − sin δ ˜ ψ − δ (cid:19) − k ω sign ( ˜ ψ − δ ) (24)then˙˜ V ψ = − ( ˜ v + v d ) e d sin ˜ ψ + ( ˜ v + v d ) e d sin δ − k ω | ˜ ψ − δ ( ed ) | . (25)Using (23) and (25) in (20), it follows that˙˜ V = − k s | e s | + ˜ ve d sin δ ( e d ) + v d e d sin δ ( e d ) − k ω | ˜ ψ − δ ( ed ) | . (26)Then, as ˙˜ V = ˙˜ V + ˙˜ V we substitute (18) and (26)˙˜ V = − k v ˜ v − k γ ˜ γ − ˜ θ T Q ˜ θ − k s | e s | + ˜ ve d sin δ ( e d )+ v d e d sin δ ( e d ) − k ω | ˜ ψ − δ ( e d ) | (27)From (19) it follows that | ˜ V | = √ l . (28)Then if we define a from (27) as a = − k v ˜ v − k γ ˜ γ − ˜ θ T Q ˜ θ − k s | e s | − k ω | ˜ ψ − δ ( e d ) | (29)and we substitute (29) and (28) in (27) we have˙˜ V ≤ a + √ l | e d sin δ ( e d ) | + v d e d sin δ ( e d ) and a + √ l | e d sin δ ( e d ) | + v d e d sin δ ( e d ) ≤ a − ( v d − √ l ) | e d sin δ ( e d ) | since −| e d sin δ ( e d ) | = e d sin δ ( e d ) , then ˙˜ V ≤ v d > √ l .Also, if ψ = δ ( e d ) and the remaining states are zero, ˙˜ V = V > ∀ E ( t ) (cid:54) = V ( E ( t )) ≤ ∀ E ( t ) in Ω l . Let S ⊆ Ω l the set of all points where ˙˜ V = V = = − k ω | ˜ ψ − δ ( e d ) | → ˜ ψ = δ ( e d ) . Note that the rest of ˙˜ V is equal to zero at zero. Observing theclosed loop system (7) with the controls (14), (16) and (24)when ˜ v = ˜ γ = ˜ θ = ˜ θ = e s = ψ = δ ( e d ) one obtains˙ e s = = v d cos ˜ ψ − v d cos ˜ ψ + Cc ( s ) e d [ v d cos ˜ ψ ]= Cc ( s ) v d e d cos δ ( e d ) ˙ e d = = v d sin ˜ ψ ˙˜ ψ = ˙ δ ( v d sin ˜ ψ ) (30)then e d = ψ = M in S is the origin E ( t ) = E ( t ) → t → ∞ ; as long as the initial state E ( t ) be in Ω e . And the closed loopsystem is asymptotically stable in the arbitrary large set: Ω e provided that √ l ≤ v d . Also ˜ V → ∞ as | E ( t ) | → ∞ , i.e. ˜ V is radially unbounded, then the closed loop system is G.A.S.IV. S IMULATION E XPERIMENTS
To verify the stability in a computational environment, anumerical simulation of the systems (1) and (2) under con-trols (14), (16), (22) and (24) is performed. The parametersused in the simulation are: ψ a = π . , m = g = . I y = k ˜ θ = k ˜ θ = k v = k s = k γ = k ω = . γ d = ◦ , V d = (cid:113) m · g . ∗ c , where c = pS is a constant that describes theair density an the area of the wind. The initial conditionsfor the simulation are: s = x = y = − ψ = ◦ , γ = ◦ , θ = ◦ , V =
1. The induced disturbance to the ψ state is a sinusoidal signal with 20 units of amplitude and afrequency of 0 . rads . The airspeed is depicted at Fig. 2. Thecontrol T applied to the model is shown in Fig. 3. Time [s]0 1 2 3 4 5 v
123 v d v Fig. 2: Airspeed v through the UAV control for the optimalAoA considering aerodynamics of [32]. Time [s]0 1 2 3 4 5 T Fig. 3: Thrust control T applied to the fixed-wing UAV.Another important fact that determines an optimal flightof the UAV is the AoA. According to the lift coefficientof the wing profile used for this simulation, which is asymmetric profile, the best AoA in relation with the lift-drag coefficients is around 6 ◦ . Then, it is expected that thisangle is reached during stabilization process. This AoA andhe other aerodynamic terms have been taken from the UAVpresented in our previous work [32]. In Fig. 4 it is shownthe angles evolution including the AoA convergence to thedesired value equal to 6 ◦ . The control input τ is shown at Fig. Time [s]0 1 2 3 4 5 D e g r ees , . 3 Fig. 4: UAV angles α , γ and θ keeping the property of θ = α + γ .5. With this control it is possible to reach the θ d accordingto γ d and α d . Time [s]0 1 2 3 4 5 = -10010 Fig. 5: Control τ for the UAV pitch angle.In Fig.6, the control ω is plotted. With this control, ψ isguided to the desired heading for the path following. Finally,Figs. 7 and 8 depict the UAV’s position w.r.t. the desired path,which is in this case a circle of radius 20 with and withoutdisturbance in ψ , respectively.V. C ONCLUSIONS
We have presented a full controller for a fixed-wing UAVwhich presents robust capabilities for disturbances presentedin the UAV dynamics. A stability proof is presented whichdemonstrated that the closed-loop system is GAS.As future work we intend to implement this controller usedthe approach known as hardware in the loop (HIL) and finallybe implemented in the real fixed-wing UAV in a Pixhawkautopilot. R
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