# A Robust Control Strategy for Landing an Unmanned Aerial Vehicle on a Vertically Moving Platform

Authors: Carlos Aguilar-Ibanez; Miguel S. Suarez-Castanon; Octavio Gutierrez-Frias; Jose de Jesus Rubio; Jesus A. Meda-Campana
Journal: Complexity
 (2020)

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## Abstract

In this work, we solve the uncertain unmanned aerial vehicle smooth landing problem over a moving platform, assuming that the aircraft position relative to the platform and its acceleration is always measurable. The landing task is carried out by an output-feedback robust controller, together with a repulsive force. The robust controller controls the nominal model, accomplishes the needed tracking trajectory, and counteracts the unknown uncertainties. To assure that the aircraft is always above the platform, we include a repulsive force that only works in a small vicinity of the platform. To estimate the relative aircraft velocity and platform acceleration, we use a supertwisting-based observer, assuring finite-time convergence of these signals. This fact allowed us to design the feedback state stabilizer independently of the observer design (in accordance with the separation principle). We confirmed the effectiveness of our control approach by convincing numerical simulations.


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## Body

## 1. Introduction

The challenge of landing unmanned aerial vehicles on moving platforms has attracted the attention of several researchers in the control community. It is a problem that, if solved, has many applications, such as landing an unmanned aerial vehicle (UAV) on the deck of a ship in high seas. This control task consists of regulating to zero the relative distance between the aircraft and the ship deck, the latter being subject to external motions, including pitch or vertical movements. This regulation task has to be accomplished smoothly, safely, precisely, and fast, with the primary concern of safeguarding the physical integrity of both the aircraft and the ship. Usually, this control task has been accomplished using identification, adaptive algorithms, sliding-mode control, robust control, and optimal control [1–4]. In the literature, we can find several control schemes to address the UAV shipboard landing problem. In [2], an optical flow based control law for landing on a moving platform is proposed. In [5], a vision-based algorithm is presented, where the ground effect experienced during the maneuver task is compensated for using by a sliding adaptive control law. In [6], another vision-based landing algorithm for an autonomous helicopter is presented. The landing algorithm is integrated with algorithms for the visual acquisition of the target and navigation to the goal, from an arbitrary initial position and orientation. In [7], a control strategy for autonomously landing a helicopter on a rocking ship (due to rough seas) is presented. This control strategy is implemented by two controllers that separate the time scales of translation and rotation. Marconi et al. [1] present a control strategy for the autonomous landing of a vertical take-off and landing vehicle (VTOL) on a ship with a deck that oscillates vertically due to high seas. There, the authors design an internal model-based error feedback dynamic regulator that is robust with respect to some uncertainties about the mechanical parameters that characterize the model and secures global convergence. Yang et al. [8] present a procedure for predicting a ship’s pitch and heave motions. This procedure is based on a predictive controller whose inputs are measurements of relative distance and whose outputs are the predictions of the pitch movements. Other exciting works based on the predictive control approach to solving this control task are [9–12]. In [13], the UAV landing problem is addressed using a three-step procedure: motion estimation, trajectory generation, and tracking control, where the latter is accomplished as an optimal control. The optimal control approach for landing UAVs is also used in two recently published works [14, 15]. A full review of the literature related to this problem is beyond the scope of this work; however, we suggest to the interested reader the following works [16–25].In this work, we approach the problems of feedback regulation and output regulation to accomplish the landing control maneuver task of successfully landing a vertical take-off and landing unmanned aerial vehicle (VTOL-UAV) over a vertically moving platform, under the assumptions that the vehicle model is partially known, and its relative position with respect to the platform and the VTOL-UAV vertical acceleration are both available. Due to the nature of this maneuver, we can formulate it as if it were a synchronization of two partially known systems, where the platform is the master, and the VTOL-UAV is the slave. To this end, we use the aircraft simplified dynamic model. That is, we neglected the dynamics of the other coordinates. That is, we assume that the aircraft horizontal and angular displacements are conveniently small, and in an actual implementation, such displacements can be eliminated by suitable robust controllers. Then, to solve this problem, we designed a control strategy were we first proposed a nonlinear control, consisting of two actions, in conjunction with a repulsive force. The first action, based on saturation, controls the nominal model, while the other action, which consists of a nonlinear integrator, compensates for the partially known nonlinearities. The controller and the compensator work together with the well-known supertwisting-based observer (STBO), which is in charge of estimates on a finite-time vehicle’s relative velocity and moving platform’s acceleration. The complementary repulsive force helps to guarantee that the aircraft is always above the platform. We use the Lyapunov method to carry out the closed-loop stability analysis. The novelty of this study is that, as far as we know, the Bézier spline-based trajectory planning in conjunction with a repulsive force has not been used before to solve the aforementioned control problem.We organized the remainder of the study as follows: in Section2, we introduce the VTOL-UAV model and establish the control problem; we develop the corresponding control strategy in Section 3, where we implement the needed STBO; in Section 4, we present the outcomes of the numerical simulations, which allow us to claim that our control strategy effectively lands the VTOL-UAV over the moving platform; finally, Section 5 gives the conclusions and final remarks.Notation: the symbolssm∗ and sgnx refer, respectively, to a linear saturation function and the signum function of a real number, that is, (1)sgnx=1ifx>0;−1ifx<0;∈−1,1ifx=0,snx=xifx≤nelsensgnx.


## 2. Problem Statement

To formally establish the control problem, we introduce a simplified model of a VTOL-UAV, described by the following system of equations:(2)mz¨=Kghu−mg,where z is the vertical aircraft position; u is the actual vertical rotor’s thrust; h=z−zd is the relative distance between the VTOL-UAV position, z, and the position of the vertical moving platform, zd; m is the VTOL-UAV mass; g is the gravity force; and Kh is a nonlinear function modeled as follows (see equation (31) in [2]):(3)Kh=11−D0/h+l02,where the positive constants D0 and l0, with l0>D0, can be identified from the physical. This nonlinear function is the thrust that the VTOL rotors generate. When the VTOL is approaching to the ground, the rotors will generate more thrust for a given power. This phenomenon is known as the ground effect [26]. These parameters are related to the aircraft physical dimensions [27]. The platform position is modeled as(4)zd=∑i=1nAisinωit+ζi,where Ai is the amplitude of the i−th component of the sinusoidal motion, ωi is the frequency, and ζi is the phase. Loosely speaking, zd models the platform vertical oscillatory movement due to the ocean waves. Having introduced the simplified VTOL-UAV system model, we present the primary concern of this work.


### 2.1. Control Problem

Consider the problem (2), with measurable outputs h1=z−zd and y2=z.. and constants D0 and l0 unknown. The control objective consists in designing a control u such that(5)zt−zdt≤δ1,z˙t−z˙dt≤δ2,for all t>T>0 and where δ1 and δ2 are sufficiently small. In other words, we desire to render the system variables zt,z¨t to a neighborhood of zero. That is to say, we want states z,z˙ of the VTOL-UAV to converge with that of the platform zd,z˙d, with sufficient accuracy for some T>0, restricted to zt≥zdt for all t≥0.To solve the control problem mentioned above, we introduce the following assumptions:A1:zdit≤z¯i, with i=0,1,2 and t>0A2: the signalsz¨t and ht are always measurableNotice that we must assure thatzt≥zdt for all t≥0 because we assume that the VTOL-UAV is always above the platform. This control problem resembles, for instance, the landing of a helicopter on a ship whose deck oscillates vertically due to sea waves, as depicted in Figure 1. We must note that the reference signal zdt and its first and second time derivatives have to be bounded (see z¯i, with i=0,1,2). On the contrary, we deal with the ideal case when signals h1 and y2 are noise free. However, if these signals are not noise free, we can overcome this problem filtering them using, for instance, a low-pass filter, obtaining the signals average values, which are very close to their actual values.Figure 1
The VTOL-UAV landing task over a vertically moving platform.We finish this section by introducing a useful lemma, used in the forthcoming developments.Lemma 1.
Consider the following uncertain second-order system:(6)x˙=y;y˙=v+φ˙t,where x, y∈R are the states, φ˙t is a continuous bounded perturbation, and v is the system input, defined as(7)v=−say−x˙r−sa/2x+y−xr+x˙r+x¨r,where a>0 is a constant and xr is the smooth reference signal with time derivatives x˙rand x¨r. Suppose that xri≤x¯ri with i=0,1,2, φ˙t≤φ¯˙, and φ˙t⟶0, as long as t⟶∞, with(8)∫0∞φtdt≤M<∞,then the tracking errors, ex=x−xr and ey=y−x˙r, globally and asymptotically converge to the origin. Notice that x¯ri with i=0,1,2 are strictly positive and constant bounds.The proof of this lemma can be found in Appendix.Remark 1.
We must emphasize that the signalv (refST0) is a bounded control. Consequently, it can be substituted by any other bounded controller, like a nested saturation-based controller, or by a twisting controller. For instance, the twisting controller(9)v=−a2sgny−x˙r−asgnx−xr+x¨rcan be a suitable candidate. However, we use (7) because its stability analysis is easy to carry out.


## 2.1. Control Problem

Consider the problem (2), with measurable outputs h1=z−zd and y2=z.. and constants D0 and l0 unknown. The control objective consists in designing a control u such that(5)zt−zdt≤δ1,z˙t−z˙dt≤δ2,for all t>T>0 and where δ1 and δ2 are sufficiently small. In other words, we desire to render the system variables zt,z¨t to a neighborhood of zero. That is to say, we want states z,z˙ of the VTOL-UAV to converge with that of the platform zd,z˙d, with sufficient accuracy for some T>0, restricted to zt≥zdt for all t≥0.To solve the control problem mentioned above, we introduce the following assumptions:A1:zdit≤z¯i, with i=0,1,2 and t>0A2: the signalsz¨t and ht are always measurableNotice that we must assure thatzt≥zdt for all t≥0 because we assume that the VTOL-UAV is always above the platform. This control problem resembles, for instance, the landing of a helicopter on a ship whose deck oscillates vertically due to sea waves, as depicted in Figure 1. We must note that the reference signal zdt and its first and second time derivatives have to be bounded (see z¯i, with i=0,1,2). On the contrary, we deal with the ideal case when signals h1 and y2 are noise free. However, if these signals are not noise free, we can overcome this problem filtering them using, for instance, a low-pass filter, obtaining the signals average values, which are very close to their actual values.Figure 1
The VTOL-UAV landing task over a vertically moving platform.We finish this section by introducing a useful lemma, used in the forthcoming developments.Lemma 1.
Consider the following uncertain second-order system:(6)x˙=y;y˙=v+φ˙t,where x, y∈R are the states, φ˙t is a continuous bounded perturbation, and v is the system input, defined as(7)v=−say−x˙r−sa/2x+y−xr+x˙r+x¨r,where a>0 is a constant and xr is the smooth reference signal with time derivatives x˙rand x¨r. Suppose that xri≤x¯ri with i=0,1,2, φ˙t≤φ¯˙, and φ˙t⟶0, as long as t⟶∞, with(8)∫0∞φtdt≤M<∞,then the tracking errors, ex=x−xr and ey=y−x˙r, globally and asymptotically converge to the origin. Notice that x¯ri with i=0,1,2 are strictly positive and constant bounds.The proof of this lemma can be found in Appendix.Remark 1.
We must emphasize that the signalv (refST0) is a bounded control. Consequently, it can be substituted by any other bounded controller, like a nested saturation-based controller, or by a twisting controller. For instance, the twisting controller(9)v=−a2sgny−x˙r−asgnx−xr+x¨rcan be a suitable candidate. However, we use (7) because its stability analysis is easy to carry out.


## 3. Control Strategy

Roughly speaking, we can see the control problem established above as an uncertain system synchronization problem, where the platform acts as the master and the VTOL-UAV as the slave. To this end, we propose an output-feedback robust controller, together with a repulsive field-based force. The robust controller is devoted to assuring tracking error convergence to zero of the nominal model, and it simultaneously compensates for the unmodeled nonlinearities. The repulsive force acts ifh<0.5δ1, where δ1>0 is related to the constant defined buffer zone. As we only have measurements of the relative position h and the VTOL-UAV vertical acceleration y¨, we estimate the unknown variables h˙ and z¨d by using an STBO [28], assuming that Kh is also partially known. Having obtained h. and z⃛d, we proceed to design a robust output-feedback-based controller together with a trajectory planner. The planner allows us to render the aircraft to the buffer zone and lands it very smoothly on the platform. We accomplish the reference trajectory hr using a fifth-grade Bézier spline, allowing us to plan the landing time.Comment 1.
In our opinion, the advantage of our proposal is that we give a robust control-based solution, which considers the dynamic variations in functionKh given in (3), based on saturation functions, together with a nonlinear integrator. This combination has the advantage of solving the problem in a precise manner when the external perturbations have slow variations. Contrary, for instance, to the adaptive and PI control approaches where we use an approximation to solve this kind of problem, we assume that the time derivative of Kh is equal to zero. On the contrary, if we compare our strategy with other well-known strategies, it has the drawback of having a slower time response, and the control gain parameters are difficult to tune adequately.


### 3.1. Reconstruction of the Unknown Variables

Defining the synchronization errors ash1=z−zd and h2=z˙−z˙d, we have that(10)h˙1=h2,h˙2=y2−z¨d,where y2=z¨ is in fact a measurable signal. Then, for estimating the missing variables h2 and z¨d, we use the following observer [28, 29]:(11)h^1.=h^2+βh1−h^11/2sgnh1−h^1,h^2.=y2+αsgnh1−h^1,τw^.=−w^+αsgnh1−h^1,where constants α and β must be selected, such that(12)α>z¯2,β>1+ζα+z¯21−ζ2α−z¯2,for some ζ∈0,1 and 0<τ≪1. Defining the observation errors as eo1=h1−h^1 and eo2=h2−h^2, we obtain, from the first two equations of (10) and (11), the following equations:(13)e˙o1=eo2−βeo11/2sgneo1,e˙o2=−z¨d−αsgneo1.As constantsα and β are selected according to (12), we can assure that eo1 and eo2 asymptotically converge to zero in a finite time (for details see Sections III and IV of [28]). On the contrary, if we fix the step size of the numeric integration method, ς, such that 0<ς<τ≪1, then we can assure that w^⟶−z¨d in a finite time, as long as τ⟶0 [29]. That is, there exists T0>0, such that h^2t=h˙t and w^t=−z¨dt, for allt>T0.The disadvantage of this sliding-mode observer is that we need to know, at least, an estimated value for the constantz¯2. However, we can overcome this inconvenience by selecting the constant α large enough. Similarly, to reconstruct z¨d, we need to select τ and ς small enough, depending on the natural frequency of z¨d.


### 3.2. Robust Control Law

Since we can recover variablesh˙ and z¨d in a finite time, we propose a suitably robust control scheme for assuring that h1 and h2 asymptotically converge to zero. Before proceeding, we introduce the following useful proposition.Proposition 1.
Let us consider the following uncertain second-order system:(14)q˙=p,p˙=bx,tu+ϕt,where q and p∈R are the states, x=q,pT, u∈R is the system input, and bx,t and ϕt∈R are partially unknown functions. Let as assume the following:P1: there exitsb¯>0 and b¯>0, such that b¯≤bx,t≤b¯, for t≥0P2:ϕt≤ϕ¯t<∞ for all t≥0
Consider the following controller:(15)u=1b¯vd+ζ,(16)vd=−sa/2eq+ep−saep+q¨d,where eq=q−qd, ep=p−q˙d, and qd is the continuous and bounded reference trajectory, with the first and the second time derivatives being also continuous and bounded, and the auxiliary control ζ is defined as(17)ζ=−λ0ξ−λ1ξvd+ϕ¯tkexp−λ2t+ξ,with(18)ξt=pt−p0−∫0tvdsds,where the parameters λ0,λ1,λ2,k are strictly positive with λ1b¯/b¯>1. Then, the closed-loop systems (14) and (15) assure that the tracking errors, eq and ep, asymptotically converge to zero.Proof.
From (14) and (15), we can write the closed-loop system as(19)q˙=p,p˙=vd+bx,tb¯−1vd+bx,tb¯ζ+ϕx,t.
Now, from the definition of (18), it is easy to see that we can express the equation above in a short way as(20)q˙=p;p˙=vd+ξ˙.
Before continuing with the stability analysis of the above system, we first show thatξ˙ converges to zero. To this end, we must note that the variable ξ, given in (18), can also be expressed as(21)ξt=∫0tbxs,sb¯zs+bxs,sb¯−1vds+ϕxs,sds.
Indeed, it can be justified using the simple integration of the second equation of (19) and the definition (18). From assumption P3, we must emphasize that ξ can be computed online with equation (18). Having defined the new system representation (20), we proceed to show that the virtual controller z (17) makes variables ξ and ξ˙ to asymptotically converge to zero. To this end, we substitute (17) into the time derivative of ξ, given in equation (15), leading to(22)ξ˙=−λ0b+tξ−λ1b+tξvd+ϕ¯tkexp−λ2t+ξ+b+t−1vd+ϕx,t,where(23)b+t=bx,tb¯.
Hence, to analyze the behavior of the auxiliary variableξ, we use the Lyapunov function V=ξ2/2, in which the time derivative around the trajectories of (22) can be upper bounded as(24)V˙≤−λ0b¯b¯ξ2+−λ¯1+1ξ2Wtkexp−λ2t+ξ+kexp−λ2tξWtkexp−λ2t+ξ,where λ¯1=λ1b¯/b¯ and Wt=vd+ϕ¯t≤W¯. Hence, selecting λ¯1=λ1b¯/b¯>1 and using P1, we obtain that the following inequality(25)V˙≤−2λ0b¯b¯V+kW¯exp−λ2tholds. From the last differential inequality, it is easy to conclude that V exponentially converges to zero (see Appendix). Hence, the variable ξ also exponentially converges to zero. Now, as vd (16) and ϕ¯t are bounded, then ξ˙ is bounded (see (22)), implying that ξ is uniformly continuous with ξ⟶0, as long as t⟶∞. Thus, according to the lemma of Barbalat, we can conclude that ξ˙⟶0, as long as t⟶∞.
Finally, after substituting (16) into equation (20), we obtain(26)e˙q=ep;e˙p=−sa/2eq+ep−saep+ξ˙,where ξ˙ is bounded and converges to zero. Besides, as ξ exponentially converges to zero, we have that ∫0∞ξsds≤M<∞. Consequently, according to Lemma 1, we have that eq,ep⟶0, as long as t⟶∞.Remark 2.
If we substitute the auxiliary controllerζ, defined in (17), by(27)ζ=−λ0ξ−λ1sgnξvd+ϕ¯t,the variable ξ will converge in a finite time. On the contrary, the performance of system (14), in closed loop with (15) and (16), could be improved if we substituted controller vd with any other control based on the saturation function, as follows:(28)vd=−saeq+bep+q¨d,or(29)vd=−saep+saeq+q¨d,where a and b are both positive constants, conveniently tuned.


#### 3.2.1. The Dynamical Synchronization Equation

Substituting the equation of the VTOL-UAV system (2) into equation (10), we obtain the following synchronization equation:(30)h˙1=h2;h˙2=Khu−Φt,where Φt=g−z¨dt and Kh=Kgh/m. Now, to propose the control u to assure h1,h2⟶0, as long as t⟶∞, we assume that we know k¯, k¯, and Φ¯ such that(31)Φt≤Φ¯t;k¯≤Kh<k¯.To accomplish this, we introduce the main proposition of this work.Proposition 2.
Consider system (30) in closed loop with(32)u=1k¯−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r+ζ,where hr is a Bézier spline and h^2 and w^ evolve according to the system equations in (11), restricted to (12), and ζ is proposed as(33)ζ=−λ0ξ−λ1ξvd+Φ¯kexp−λ2t+ξ,with(34)vd=−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r,Φ¯t=g+w^, and(35)ξt=h^2−∫0tvdsds.If constantsλ1 and λ2 satisfy λ1k¯/k¯>1, then for any initial condition h1,h2, we can assure that errors h1 and h2 asymptotically converge to zero.Proof.
See Appendix.


#### 3.2.2. Planification of the Reference Trajectoryhr

To land the VTOL-UAV on the moving platform, we use a Bézier spline-based trajectory [30]. We most note that the Bézier spline allows us to design translation trajectories from one equilibrium point to another equilibrium point, in a previously fixed time. We propose this trajectory to bring the aircraft, from any initial condition h0=ht0>0, to the buffer zone hf=htf=δ1>0 with δ1 being sufficiently small and tf≫t0 conveniently fixed. That is, we define hrt as(36)hrt=h0+ρt,t0,tfhf−h0,where ρt,t0,tf is the selected Bézier spline, which satisfies the following conditions:(37)ρt0,t0,tf=0,ρtf,t0,tf=1,where a finite number of the time derivatives of ρt,t0,tf are set to zero at the initial and final maneuver times t0 and tf (for details, please refer [30]). To assure that both h>0 for all times and that the controller u turns off when the VTOL-UAV reaches the buffer zone, we fixed it, for practical purpose, as(38)Θ=h=h,h˙:h2+0.2h˙21/2≤1.1δ1.It is worthy to mention thatΘ acts as decision function, which allows to decide when to deactivate the control action. That is, when the aircraft is above and close enough to the platform, the controller is set to zero; otherwise, it is on. Thus, we establish the practical control strategy as follows:(39)u=μk¯vd+ur+ζ,where vd and ζ are defined in Proposition 2 and μ is a function that acts as a switch and is defined as(40)μ=0,ifh∈Θ,1,otherwise,and ur is the repulsive force defined as(41)ur=−ddhUreph,with its corresponding potential defined as(42)Ureph=δ0321h2−1δ022,ifh<δ0andh˙<0,0,otherwise,where δ0=0.5δ1. We must note that the repulsive force Urep assures that h≥0. We omitted the stability proof of system (2) in closed loop with control (39) because it mimics Proposition 1.Comment 2.
Please note that the slower the aircraft descends, the lesser the possibility to crash it with the platform. We can expect the same result if we increase either the size of the buffer zone or the magnitude of the repulsive force, by conveniently tuning, for instance,δ0 and tf being sufficiently large. However, the crashing possibility cannot be avoided by any methodology, if an unexpected and strong enough crosswind appears suddenly.


## 3.1. Reconstruction of the Unknown Variables

Defining the synchronization errors ash1=z−zd and h2=z˙−z˙d, we have that(10)h˙1=h2,h˙2=y2−z¨d,where y2=z¨ is in fact a measurable signal. Then, for estimating the missing variables h2 and z¨d, we use the following observer [28, 29]:(11)h^1.=h^2+βh1−h^11/2sgnh1−h^1,h^2.=y2+αsgnh1−h^1,τw^.=−w^+αsgnh1−h^1,where constants α and β must be selected, such that(12)α>z¯2,β>1+ζα+z¯21−ζ2α−z¯2,for some ζ∈0,1 and 0<τ≪1. Defining the observation errors as eo1=h1−h^1 and eo2=h2−h^2, we obtain, from the first two equations of (10) and (11), the following equations:(13)e˙o1=eo2−βeo11/2sgneo1,e˙o2=−z¨d−αsgneo1.As constantsα and β are selected according to (12), we can assure that eo1 and eo2 asymptotically converge to zero in a finite time (for details see Sections III and IV of [28]). On the contrary, if we fix the step size of the numeric integration method, ς, such that 0<ς<τ≪1, then we can assure that w^⟶−z¨d in a finite time, as long as τ⟶0 [29]. That is, there exists T0>0, such that h^2t=h˙t and w^t=−z¨dt, for allt>T0.The disadvantage of this sliding-mode observer is that we need to know, at least, an estimated value for the constantz¯2. However, we can overcome this inconvenience by selecting the constant α large enough. Similarly, to reconstruct z¨d, we need to select τ and ς small enough, depending on the natural frequency of z¨d.


## 3.2. Robust Control Law

Since we can recover variablesh˙ and z¨d in a finite time, we propose a suitably robust control scheme for assuring that h1 and h2 asymptotically converge to zero. Before proceeding, we introduce the following useful proposition.Proposition 1.
Let us consider the following uncertain second-order system:(14)q˙=p,p˙=bx,tu+ϕt,where q and p∈R are the states, x=q,pT, u∈R is the system input, and bx,t and ϕt∈R are partially unknown functions. Let as assume the following:P1: there exitsb¯>0 and b¯>0, such that b¯≤bx,t≤b¯, for t≥0P2:ϕt≤ϕ¯t<∞ for all t≥0
Consider the following controller:(15)u=1b¯vd+ζ,(16)vd=−sa/2eq+ep−saep+q¨d,where eq=q−qd, ep=p−q˙d, and qd is the continuous and bounded reference trajectory, with the first and the second time derivatives being also continuous and bounded, and the auxiliary control ζ is defined as(17)ζ=−λ0ξ−λ1ξvd+ϕ¯tkexp−λ2t+ξ,with(18)ξt=pt−p0−∫0tvdsds,where the parameters λ0,λ1,λ2,k are strictly positive with λ1b¯/b¯>1. Then, the closed-loop systems (14) and (15) assure that the tracking errors, eq and ep, asymptotically converge to zero.Proof.
From (14) and (15), we can write the closed-loop system as(19)q˙=p,p˙=vd+bx,tb¯−1vd+bx,tb¯ζ+ϕx,t.
Now, from the definition of (18), it is easy to see that we can express the equation above in a short way as(20)q˙=p;p˙=vd+ξ˙.
Before continuing with the stability analysis of the above system, we first show thatξ˙ converges to zero. To this end, we must note that the variable ξ, given in (18), can also be expressed as(21)ξt=∫0tbxs,sb¯zs+bxs,sb¯−1vds+ϕxs,sds.
Indeed, it can be justified using the simple integration of the second equation of (19) and the definition (18). From assumption P3, we must emphasize that ξ can be computed online with equation (18). Having defined the new system representation (20), we proceed to show that the virtual controller z (17) makes variables ξ and ξ˙ to asymptotically converge to zero. To this end, we substitute (17) into the time derivative of ξ, given in equation (15), leading to(22)ξ˙=−λ0b+tξ−λ1b+tξvd+ϕ¯tkexp−λ2t+ξ+b+t−1vd+ϕx,t,where(23)b+t=bx,tb¯.
Hence, to analyze the behavior of the auxiliary variableξ, we use the Lyapunov function V=ξ2/2, in which the time derivative around the trajectories of (22) can be upper bounded as(24)V˙≤−λ0b¯b¯ξ2+−λ¯1+1ξ2Wtkexp−λ2t+ξ+kexp−λ2tξWtkexp−λ2t+ξ,where λ¯1=λ1b¯/b¯ and Wt=vd+ϕ¯t≤W¯. Hence, selecting λ¯1=λ1b¯/b¯>1 and using P1, we obtain that the following inequality(25)V˙≤−2λ0b¯b¯V+kW¯exp−λ2tholds. From the last differential inequality, it is easy to conclude that V exponentially converges to zero (see Appendix). Hence, the variable ξ also exponentially converges to zero. Now, as vd (16) and ϕ¯t are bounded, then ξ˙ is bounded (see (22)), implying that ξ is uniformly continuous with ξ⟶0, as long as t⟶∞. Thus, according to the lemma of Barbalat, we can conclude that ξ˙⟶0, as long as t⟶∞.
Finally, after substituting (16) into equation (20), we obtain(26)e˙q=ep;e˙p=−sa/2eq+ep−saep+ξ˙,where ξ˙ is bounded and converges to zero. Besides, as ξ exponentially converges to zero, we have that ∫0∞ξsds≤M<∞. Consequently, according to Lemma 1, we have that eq,ep⟶0, as long as t⟶∞.Remark 2.
If we substitute the auxiliary controllerζ, defined in (17), by(27)ζ=−λ0ξ−λ1sgnξvd+ϕ¯t,the variable ξ will converge in a finite time. On the contrary, the performance of system (14), in closed loop with (15) and (16), could be improved if we substituted controller vd with any other control based on the saturation function, as follows:(28)vd=−saeq+bep+q¨d,or(29)vd=−saep+saeq+q¨d,where a and b are both positive constants, conveniently tuned.


### 3.2.1. The Dynamical Synchronization Equation

Substituting the equation of the VTOL-UAV system (2) into equation (10), we obtain the following synchronization equation:(30)h˙1=h2;h˙2=Khu−Φt,where Φt=g−z¨dt and Kh=Kgh/m. Now, to propose the control u to assure h1,h2⟶0, as long as t⟶∞, we assume that we know k¯, k¯, and Φ¯ such that(31)Φt≤Φ¯t;k¯≤Kh<k¯.To accomplish this, we introduce the main proposition of this work.Proposition 2.
Consider system (30) in closed loop with(32)u=1k¯−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r+ζ,where hr is a Bézier spline and h^2 and w^ evolve according to the system equations in (11), restricted to (12), and ζ is proposed as(33)ζ=−λ0ξ−λ1ξvd+Φ¯kexp−λ2t+ξ,with(34)vd=−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r,Φ¯t=g+w^, and(35)ξt=h^2−∫0tvdsds.If constantsλ1 and λ2 satisfy λ1k¯/k¯>1, then for any initial condition h1,h2, we can assure that errors h1 and h2 asymptotically converge to zero.Proof.
See Appendix.


### 3.2.2. Planification of the Reference Trajectoryhr

To land the VTOL-UAV on the moving platform, we use a Bézier spline-based trajectory [30]. We most note that the Bézier spline allows us to design translation trajectories from one equilibrium point to another equilibrium point, in a previously fixed time. We propose this trajectory to bring the aircraft, from any initial condition h0=ht0>0, to the buffer zone hf=htf=δ1>0 with δ1 being sufficiently small and tf≫t0 conveniently fixed. That is, we define hrt as(36)hrt=h0+ρt,t0,tfhf−h0,where ρt,t0,tf is the selected Bézier spline, which satisfies the following conditions:(37)ρt0,t0,tf=0,ρtf,t0,tf=1,where a finite number of the time derivatives of ρt,t0,tf are set to zero at the initial and final maneuver times t0 and tf (for details, please refer [30]). To assure that both h>0 for all times and that the controller u turns off when the VTOL-UAV reaches the buffer zone, we fixed it, for practical purpose, as(38)Θ=h=h,h˙:h2+0.2h˙21/2≤1.1δ1.It is worthy to mention thatΘ acts as decision function, which allows to decide when to deactivate the control action. That is, when the aircraft is above and close enough to the platform, the controller is set to zero; otherwise, it is on. Thus, we establish the practical control strategy as follows:(39)u=μk¯vd+ur+ζ,where vd and ζ are defined in Proposition 2 and μ is a function that acts as a switch and is defined as(40)μ=0,ifh∈Θ,1,otherwise,and ur is the repulsive force defined as(41)ur=−ddhUreph,with its corresponding potential defined as(42)Ureph=δ0321h2−1δ022,ifh<δ0andh˙<0,0,otherwise,where δ0=0.5δ1. We must note that the repulsive force Urep assures that h≥0. We omitted the stability proof of system (2) in closed loop with control (39) because it mimics Proposition 1.Comment 2.
Please note that the slower the aircraft descends, the lesser the possibility to crash it with the platform. We can expect the same result if we increase either the size of the buffer zone or the magnitude of the repulsive force, by conveniently tuning, for instance,δ0 and tf being sufficiently large. However, the crashing possibility cannot be avoided by any methodology, if an unexpected and strong enough crosswind appears suddenly.


## 3.2.1. The Dynamical Synchronization Equation

Substituting the equation of the VTOL-UAV system (2) into equation (10), we obtain the following synchronization equation:(30)h˙1=h2;h˙2=Khu−Φt,where Φt=g−z¨dt and Kh=Kgh/m. Now, to propose the control u to assure h1,h2⟶0, as long as t⟶∞, we assume that we know k¯, k¯, and Φ¯ such that(31)Φt≤Φ¯t;k¯≤Kh<k¯.To accomplish this, we introduce the main proposition of this work.Proposition 2.
Consider system (30) in closed loop with(32)u=1k¯−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r+ζ,where hr is a Bézier spline and h^2 and w^ evolve according to the system equations in (11), restricted to (12), and ζ is proposed as(33)ζ=−λ0ξ−λ1ξvd+Φ¯kexp−λ2t+ξ,with(34)vd=−sa/2h1+h^2−hr−h˙r−sah^2−h˙r+h¨r,Φ¯t=g+w^, and(35)ξt=h^2−∫0tvdsds.If constantsλ1 and λ2 satisfy λ1k¯/k¯>1, then for any initial condition h1,h2, we can assure that errors h1 and h2 asymptotically converge to zero.Proof.
See Appendix.


## 3.2.2. Planification of the Reference Trajectoryhr

To land the VTOL-UAV on the moving platform, we use a Bézier spline-based trajectory [30]. We most note that the Bézier spline allows us to design translation trajectories from one equilibrium point to another equilibrium point, in a previously fixed time. We propose this trajectory to bring the aircraft, from any initial condition h0=ht0>0, to the buffer zone hf=htf=δ1>0 with δ1 being sufficiently small and tf≫t0 conveniently fixed. That is, we define hrt as(36)hrt=h0+ρt,t0,tfhf−h0,where ρt,t0,tf is the selected Bézier spline, which satisfies the following conditions:(37)ρt0,t0,tf=0,ρtf,t0,tf=1,where a finite number of the time derivatives of ρt,t0,tf are set to zero at the initial and final maneuver times t0 and tf (for details, please refer [30]). To assure that both h>0 for all times and that the controller u turns off when the VTOL-UAV reaches the buffer zone, we fixed it, for practical purpose, as(38)Θ=h=h,h˙:h2+0.2h˙21/2≤1.1δ1.It is worthy to mention thatΘ acts as decision function, which allows to decide when to deactivate the control action. That is, when the aircraft is above and close enough to the platform, the controller is set to zero; otherwise, it is on. Thus, we establish the practical control strategy as follows:(39)u=μk¯vd+ur+ζ,where vd and ζ are defined in Proposition 2 and μ is a function that acts as a switch and is defined as(40)μ=0,ifh∈Θ,1,otherwise,and ur is the repulsive force defined as(41)ur=−ddhUreph,with its corresponding potential defined as(42)Ureph=δ0321h2−1δ022,ifh<δ0andh˙<0,0,otherwise,where δ0=0.5δ1. We must note that the repulsive force Urep assures that h≥0. We omitted the stability proof of system (2) in closed loop with control (39) because it mimics Proposition 1.Comment 2.
Please note that the slower the aircraft descends, the lesser the possibility to crash it with the platform. We can expect the same result if we increase either the size of the buffer zone or the magnitude of the repulsive force, by conveniently tuning, for instance,δ0 and tf being sufficiently large. However, the crashing possibility cannot be avoided by any methodology, if an unexpected and strong enough crosswind appears suddenly.


## 4. Numerical Simulations

To assess the effectiveness of the proposed control strategy, we design and run three numerical experiments. In the first experiment, we perturb the aircraft with a crosswind. The second experiment consists of comparison among our control strategy (OCS), and the continuous and smooth PID-based strategy (PIDS), and an adaptive strategy (AS); the latter two are, respectively, proposed in [4, 13]. Finally, the third experiment is also a performance comparison, among OCS and a version of the discontinuous twisting-based strategy (TBS), found in [31].


### 4.1. Experiment Setup

We fix the system physical parameters asm=0.25kg, D0=1, and l0=5 and the controller gains, given in (15) to (17), as(43)k¯=4,k¯=4.5,a=3,λ0=3,λ1=2.2,λ2=0.5,and the STBO gains as(44)α=0.5,β=3,τ=0.1.For the experiments two and three, we suppose that the system acceleration is noisily perturbed by a normally distributed random process with zero mean and whose standard deviation is 1, defined as(45)Δst=ηtm5, where ηt is the random noise.


### 4.2. First Experiment

For this experiment, we fix the initial condition ash,h˙=20.5,0 and the Bézier spline parameters as t0,tf=0,30 and simulate the platform hypothetical vertical movement, zd, and the crosswind effect, Δ, as follows:(46)zdt=0.25sint2cost2;Δtm=0.2sin2t4+0.25.We show the outcome of this experiment in Figure2, where we see that the VTOL-UAV reaches the buffer zone after 27 seconds and the estimated velocity matches the actual velocity after 5 seconds. Consequently, we can conclude that our control approach effectively lands the aircraft over the platform, with a good performance. In fact, the final error is in the order of 10−2 meters. We must note that there is a small error between h and hr. This error can be explained because control vd is based on a saturation function, which has the inconvenience of having a significant time of convergence. However, if time tf−t0 is sufficiently large, then we can improve the tracking error.Figure 2
The closed-loop response of the VTOL-UAV perturbed by a crosswind when tracking a trajectory to land over a moving platform, with a programmed time of 30 seconds: (a) VTOL-UAV landing task; (b) comparison between actual and desired positions; (c) comparison between actual and desired velocities; (d) control action.
(a)(b)(c)(d)


### 4.3. Second Experiment

For this experiment, we fix the initial condition ash,h˙=10.5m.1m/s and the Bézier spline parameters as t0,tf=0,30. We simulate the platform hypothetical vertical movement as in the first experiment, while we externally perturbed the system by a crosswind, defined as Δt/m=0.1sintcost+0.1, and we fixed the OCS saturation function parameter as a=6. As we already mentioned, the experiment goal is to make a comparison among the OCS, PIDS, and AS strategies and to have an idea of how good our solution is with respect to the other two well-established approaches. In order to accomplish a fair comparison, we assume that h˙ is measurable, and we add the repulsive force to the strategies PIDS and AS and use the same control gains for both strategies. Then, we propose the PIDS as(47)u=1b¯−kph˜1−kdh˜2−ki∫0th˜1sds+h¨r+9.8+ur,and the AS as(48)u=d^−kph˜1−kdh˜2+h¨r+9.8+ur,where b¯=4, h˜1=h−hr, and h˜2=h.−hr. and d^ is the gain parameter estimation of Kgh based on the projective adaptive control law proposed in [13] and the control gains fixed as kp=1, kd=2, and ki=1, with ur proposed as in (41). We show the obtained results from the numerical simulation in Figure 3. In this figure, we show the performance index of the three control strategies, which is defined as(49)IT=1T∫0Th˜12s+h˜22sds,and their corresponding control actions. From the performance indexes, we can see that OCS has a better performance than PIDS, while AS exhibits the less efficient performance index. In the same figure, we can see that the OCS control action shows fewer overshoots; however, the chattering phenomenon is always present. Between the PIDS and AS control actions, we can say that the first one exhibits more overshoots. We can also see that the PIDS strategy exhibits chattering when it is close to the equilibrium point, that is, after 20 seconds. Finally, in Figure 4 we show the trajectory tracking position error for the three control strategies. From this figure, we can see that OCS has a slightly better performance. Please note that the overshoots that we can see after 27 seconds are due to the effect of the repulsive force.Figure 3
Comparison among the control strategies OCS, PID, and AS, when the system is perturbed by a normally distributed random process with zero mean.
(a)(b)(c)(d)Figure 4
Comparison of the tracking trajectory position errorh˜=h−hr of strategies OCS, PID, and AS, when the system is perturbed by a crosswind.
(a)(b)(c)


### 4.4. Third Experiment

The control task is the same as in the second experiment, but we fix theBézier parameters as t0,tf=0,50. In this experiment, once again, we perturbed the acceleration equation by adding it to the noisy signal Δst. Additionally, to make the experiment more realistic and illustrative, we added a noise signal to the output y. That is, ym=ya+0.1×ηt, where ym and ya are, respectively, the measurable and actual outputs, and ηt is a normally distributed random process with zero mean; we assume that h. is not available and estimated using the STBO (11). Then, we use the following TBS:(50)u=−r1signh˜1−r2signh˜2+ur,where the controller gains satisfy r1>r2>0 and the following conditions:(51)r2+Ck¯<r1<k¯−k¯r2−2Ck¯−k¯,where(52)C=maxΦ¯t+ci,and ci>0 is any constant. We most note that, for this experiment, we fix C=5. Hence, according to (51), we can select r1=10 and r2=19. Then, to make the comparison, we use the following error index:(53)hM=h˜12+h˜22.We show the comparison outcome in Figure5, where we can see that, as we expected, the TBS outperforms OCS because the closed-loop response of the TBS is almost immune to external matching perturbation, having the inconvenience of exhibiting the chattering phenomena in the controller, as we can see it in Figure 5, preventing its actual implementation. In favor of OCS, we mention that the controller has a smooth response, which, in our opinion, is due to the integration action because it is averaging the noise effect over the system. This behavior can allow us to implement OCS for real applications.Figure 5
Comparison between the control strategies OCS and TBS, when the system is perturbed by a normally distributed random process with zero mean: (a) OCS error index; (b) TBS error index; (c) OCS control action; (d) TBS control action.
(a)(b)(c)(d)Comment 3.
We must mention that we implement the sign function as(54)signx=xx+10−3,with integration step fixed as 10−4.


## 4.1. Experiment Setup

We fix the system physical parameters asm=0.25kg, D0=1, and l0=5 and the controller gains, given in (15) to (17), as(43)k¯=4,k¯=4.5,a=3,λ0=3,λ1=2.2,λ2=0.5,and the STBO gains as(44)α=0.5,β=3,τ=0.1.For the experiments two and three, we suppose that the system acceleration is noisily perturbed by a normally distributed random process with zero mean and whose standard deviation is 1, defined as(45)Δst=ηtm5, where ηt is the random noise.


## 4.2. First Experiment

For this experiment, we fix the initial condition ash,h˙=20.5,0 and the Bézier spline parameters as t0,tf=0,30 and simulate the platform hypothetical vertical movement, zd, and the crosswind effect, Δ, as follows:(46)zdt=0.25sint2cost2;Δtm=0.2sin2t4+0.25.We show the outcome of this experiment in Figure2, where we see that the VTOL-UAV reaches the buffer zone after 27 seconds and the estimated velocity matches the actual velocity after 5 seconds. Consequently, we can conclude that our control approach effectively lands the aircraft over the platform, with a good performance. In fact, the final error is in the order of 10−2 meters. We must note that there is a small error between h and hr. This error can be explained because control vd is based on a saturation function, which has the inconvenience of having a significant time of convergence. However, if time tf−t0 is sufficiently large, then we can improve the tracking error.Figure 2
The closed-loop response of the VTOL-UAV perturbed by a crosswind when tracking a trajectory to land over a moving platform, with a programmed time of 30 seconds: (a) VTOL-UAV landing task; (b) comparison between actual and desired positions; (c) comparison between actual and desired velocities; (d) control action.
(a)(b)(c)(d)


## 4.3. Second Experiment

For this experiment, we fix the initial condition ash,h˙=10.5m.1m/s and the Bézier spline parameters as t0,tf=0,30. We simulate the platform hypothetical vertical movement as in the first experiment, while we externally perturbed the system by a crosswind, defined as Δt/m=0.1sintcost+0.1, and we fixed the OCS saturation function parameter as a=6. As we already mentioned, the experiment goal is to make a comparison among the OCS, PIDS, and AS strategies and to have an idea of how good our solution is with respect to the other two well-established approaches. In order to accomplish a fair comparison, we assume that h˙ is measurable, and we add the repulsive force to the strategies PIDS and AS and use the same control gains for both strategies. Then, we propose the PIDS as(47)u=1b¯−kph˜1−kdh˜2−ki∫0th˜1sds+h¨r+9.8+ur,and the AS as(48)u=d^−kph˜1−kdh˜2+h¨r+9.8+ur,where b¯=4, h˜1=h−hr, and h˜2=h.−hr. and d^ is the gain parameter estimation of Kgh based on the projective adaptive control law proposed in [13] and the control gains fixed as kp=1, kd=2, and ki=1, with ur proposed as in (41). We show the obtained results from the numerical simulation in Figure 3. In this figure, we show the performance index of the three control strategies, which is defined as(49)IT=1T∫0Th˜12s+h˜22sds,and their corresponding control actions. From the performance indexes, we can see that OCS has a better performance than PIDS, while AS exhibits the less efficient performance index. In the same figure, we can see that the OCS control action shows fewer overshoots; however, the chattering phenomenon is always present. Between the PIDS and AS control actions, we can say that the first one exhibits more overshoots. We can also see that the PIDS strategy exhibits chattering when it is close to the equilibrium point, that is, after 20 seconds. Finally, in Figure 4 we show the trajectory tracking position error for the three control strategies. From this figure, we can see that OCS has a slightly better performance. Please note that the overshoots that we can see after 27 seconds are due to the effect of the repulsive force.Figure 3
Comparison among the control strategies OCS, PID, and AS, when the system is perturbed by a normally distributed random process with zero mean.
(a)(b)(c)(d)Figure 4
Comparison of the tracking trajectory position errorh˜=h−hr of strategies OCS, PID, and AS, when the system is perturbed by a crosswind.
(a)(b)(c)


## 4.4. Third Experiment

The control task is the same as in the second experiment, but we fix theBézier parameters as t0,tf=0,50. In this experiment, once again, we perturbed the acceleration equation by adding it to the noisy signal Δst. Additionally, to make the experiment more realistic and illustrative, we added a noise signal to the output y. That is, ym=ya+0.1×ηt, where ym and ya are, respectively, the measurable and actual outputs, and ηt is a normally distributed random process with zero mean; we assume that h. is not available and estimated using the STBO (11). Then, we use the following TBS:(50)u=−r1signh˜1−r2signh˜2+ur,where the controller gains satisfy r1>r2>0 and the following conditions:(51)r2+Ck¯<r1<k¯−k¯r2−2Ck¯−k¯,where(52)C=maxΦ¯t+ci,and ci>0 is any constant. We most note that, for this experiment, we fix C=5. Hence, according to (51), we can select r1=10 and r2=19. Then, to make the comparison, we use the following error index:(53)hM=h˜12+h˜22.We show the comparison outcome in Figure5, where we can see that, as we expected, the TBS outperforms OCS because the closed-loop response of the TBS is almost immune to external matching perturbation, having the inconvenience of exhibiting the chattering phenomena in the controller, as we can see it in Figure 5, preventing its actual implementation. In favor of OCS, we mention that the controller has a smooth response, which, in our opinion, is due to the integration action because it is averaging the noise effect over the system. This behavior can allow us to implement OCS for real applications.Figure 5
Comparison between the control strategies OCS and TBS, when the system is perturbed by a normally distributed random process with zero mean: (a) OCS error index; (b) TBS error index; (c) OCS control action; (d) TBS control action.
(a)(b)(c)(d)Comment 3.
We must mention that we implement the sign function as(54)signx=xx+10−3,with integration step fixed as 10−4.


## 5. Conclusions

In this work, we propose a control strategy for landing a VTOL-UAV smoothly on a vertically moving platform. We designed this control strategy assuming that the VTOL-UAV model was partially known, while its position relative to the platform and the platform acceleration were both available. Our landing control approach is based on designing an output-feedback robust controller, in conjunction with a repulsive force. The robust controller consists of two control actions: one based on saturation functions, and the other is a nonlinear integrator. The former controls the nominal model and assures the tracking trajectory task, and the latter compensates for the partially known uncertainties. Additionally, we introduce a repulsive force controller to guarantee that the aircraft is always above the platform, and it only works within a very small vicinity (buffer zone). We use an STBO to estimate the nonavailable aircraft relative velocity and the moving platform acceleration. We select the aircraft reference trajectory as a Bézier spline, devoted to bringing the system from any initial condition towards the buffer zone in a fixed time. To assess the effectiveness of our control method, we carry out two numerical simulations, the first of which performs the landing task when a crosswind perturbs the aircraft. The second experiment consists of a comparison among the strategies OCS and AS. From the results obtained in this simulation, shown in Figure3, we can see that our control strategy has a better performance than the other two. Finally, we carry out the corresponding stability analysis based on the Lyapunov method.



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*Source: 2917684-2020-07-15.xml*