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

CIC, Instituto Politecnico Nacional, Av. Juan de Dios Batiz s/n, U.P.A.L.M, Ciudad de Mexico 07738, Mexico ESCOM, Instituto Politecnico Nacional, Av. Juan de Dios Batiz s/n, U.P.A.L.M, Ciudad de Mexico 07738, Mexico UPIITA, Instituto Politecnico Nacional, Av. Instituto Politecnico Nacional 2580, Ciudad de Mexico 07340, Mexico SEPI ESIME Azcapotzalco, Instituto Politecnico Nacional, Av. de las Granjas no. 682, Ciudad de Mexico 02250, Mexico


Introduction
e 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. is 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.
is 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][2][3][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. e 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. is 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. ere, 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. is 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][10][11][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. e 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][17][18][19][20][21][22][23][24][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 takeoff 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. at is, we neglected the dynamics of the other coordinates. at 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. en, 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. e 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. e 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. e 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. e 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 Section 2, 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 symbols s m [ * ] and sgn[x] refer, respectively, to a linear saturation function and the signum function of a real number, that is, (1)

Problem Statement
To formally establish the control problem, we introduce a simplified model of a VTOL-UAV, described by the following system of equations: where z is the vertical aircraft position; u is the actual vertical rotor's thrust; h � z − z d is the relative distance between the VTOL-UAV position, z, and the position of the vertical moving platform, z d ; m is the VTOL-UAV mass; g is the gravity force; and K(h) is a nonlinear function modeled as follows (see equation (31) in [2]): where the positive constants D 0 and l 0 , with l 0 > D 0 , can be identified from the physical. is 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. is phenomenon is known as the ground effect [26]. ese parameters are related to the aircraft physical dimensions [27]. e platform position is modeled as where A i is the amplitude of the i − th component of the sinusoidal motion, ω i is the frequency, and ζ i is the phase. Loosely speaking, z d 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.

Control Problem.
Consider the problem (2), with measurable outputs h 1 � z − z d and y 2 � z the platform (z d , _ z d ), with sufficient accuracy for some T > 0, restricted to z(t) ≥ z d (t) for all t ≥ 0.
To solve the control problem mentioned above, we introduce the following assumptions: } and t > 0 A2: the signals € z(t) and h(t) are always measurable Notice that we must assure that z(t) ≥ z d (t) for all t ≥ 0 because we assume that the VTOL-UAV is always above the platform. is 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 z d (t) 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 h 1 and y 2 are noise free. However, if these signals are not noise free, we can overcome this problem filtering them using, for instance, a lowpass filter, obtaining the signals average values, which are very close to their actual values.
We finish this section by introducing a useful lemma, used in the forthcoming developments. Lemma 1. Consider the following uncertain second-order system: where x, y ∈ R are the states, _ φ(t) is a continuous bounded perturbation, and v is the system input, defined as where a > 0 is a constant and x r is the smooth reference signal with time derivatives _ x r and € x r . Suppose that |x (i) can be a suitable candidate. However, we use (7) because its stability analysis is easy to carry out.

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 outputfeedback robust controller, together with a repulsive fieldbased force. e robust controller is devoted to assuring tracking error convergence to zero of the nominal model, and it simultaneously compensates for the unmodeled nonlinearities. e repulsive force acts if h < 0.5δ 1 , where δ 1 > 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 K(h) is also partially known. Having obtained h . and ż t d , we proceed to design a robust output-feedback-based controller together with a trajectory planner. e planner allows us to render the aircraft to the buffer zone and lands it very smoothly on the platform. We accomplish the reference trajectory h r 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 function K(h) given in (3), based on saturation functions, together with a nonlinear integrator. is 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 K(h) 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.

Reconstruction of the Unknown Variables. Defining the synchronization errors as
where y 2 � € z is in fact a measurable signal. en, for estimating the missing variables h 2 and € z d , we use the following observer [28,29]: where constants α and β must be selected, such that for some ζ ∈ (0, 1) and 0 < τ ≪ 1. Defining the observation errors as e o 1 � h 1 − h 1 and e o 2 � h 2 − h 2 , we obtain, from the first two equations of (10) and (11), the following equations: As constants α and β are selected according to (12), we can assure that e o 1 and e o 2 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]. at is, e disadvantage of this sliding-mode observer is that we need to know, at least, an estimated value for the constant z 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 .

Robust Control Law.
Since we can recover variables _ h and € z d in a finite time, we propose a suitably robust control scheme for assuring that h 1 and h 2 asymptotically converge to zero. Before proceeding, we introduce the following useful proposition.

Proposition 1.
Let us consider the following uncertain second-order system: where q and p ∈ R are the states, x � [q, p] T , u ∈ R is the system input, and b(x, t) and ϕ(t) ∈ R are partially unknown functions. Let as assume the following: Consider the following controller: where e q �q − q d , e p � p − _ q d , and q d 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 with where the parameters λ 0 , λ 1 , λ 2 , k are strictly positive with en, the closed-loop systems (14) and (15) assure that the tracking errors, e q and e p , asymptotically converge to zero.
Proof. From (14) and (15), we can write the closed-loop system as Now, from the definition of (18), it is easy to see that we can express the equation above in a short way as Before continuing with the stability analysis of the above system, we first show that _ ξ converges to zero. To this end, we must note that the variable ξ, given in (18), can also be expressed as 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 4 Complexity converge to zero. To this end, we substitute (17) into the time derivative of ξ, given in equation (15), leading to Hence, to analyze the behavior of the auxiliary variable ξ, we use the Lyapunov function V � ξ 2 /2, in which the time derivative around the trajectories of (22) can be upper bounded as where Hence, selecting λ 1 � λ 1 b /b > 1 and using P1, we obtain that the following inequality holds. 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 v d (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 ⟶ ∞. us, according to the lemma of Barbalat, we can conclude that _ ξ ⟶ 0, as long as t ⟶ ∞.

Remark 2.
If we substitute the auxiliary controller ζ, defined in (17), by 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 v d with any other control based on the saturation function, as follows: or v d � −s a e p + s a e q + € q d , where a and b are both positive constants, conveniently tuned.

e Dynamical Synchronization Equation.
Substituting the equation of the VTOL-UAV system (2) into equation (10), we obtain the following synchronization equation: where Φ(t) � g − € z d (t) and K(h) � K g (h)/m. Now, to propose the control u to assure (h 1 , h 2 ) ⟶ 0, as long as t ⟶ ∞, we assume that we know k, k, and Φ such that To accomplish this, we introduce the main proposition of this work.

Proposition 2. Consider system (30) in closed loop with
where h r 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 with Φ(t) � |g + w|, and If constants λ 1 and λ 2 satisfy λ 1 k /k > 1, then for any initial condition (h 1 , h 2 ), we can assure that errors h 1 and h 2 asymptotically converge to zero.
Proof. See Appendix.

Planification of the Reference Trajectory h r .
To land the VTOL-UAV on the moving platform, we use a Bézier splinebased 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 h 0 � h(t 0 ) > 0, to the buffer zone h f � h(t f ) � δ 1 > 0 with δ 1 being sufficiently small and t f ≫ t 0 conveniently fixed. at is, we define h r (t) as where ρ(t, t 0 , t f ) is the selected Bézier spline, which satisfies the following conditions: where a finite number of the time derivatives of ρ(t, t 0 , t f ) are set to zero at the initial and final maneuver times t 0 and t f (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 It is worthy to mention that Θ acts as decision function, which allows to decide when to deactivate the control action.
at is, when the aircraft is above and close enough to the platform, the controller is set to zero; otherwise, it is on.
us, we establish the practical control strategy as follows: where v d and ζ are defined in Proposition 2 and μ is a function that acts as a switch and is defined as and u r is the repulsive force defined as with its corresponding potential defined as where δ 0 � 0.5δ 1 . We must note that the repulsive force U rep 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, δ 0 and t f being sufficiently large. However, the crashing possibility cannot be avoided by any methodology, if an unexpected and strong enough crosswind appears suddenly.

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. e 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].

Experiment
Setup. We fix the system physical parameters as m � 0.25 kg, D 0 � 1, and l 0 � 5 and the controller gains, given in (15) to (17), as and the STBO gains as

(44)
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 where η(t) is the random noise.

First Experiment.
For this experiment, we fix the initial condition as (h, _ h) � (20.5, 0) and the Bézier spline parameters as (t 0 , t f ) � (0, 30) and simulate the platform hypothetical vertical movement, z d , and the crosswind effect, Δ, as follows: We show the outcome of this experiment in Figure 2, 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 h r . is error can be explained because control v d is based on a saturation function, which has the inconvenience of having a significant time of 6 Complexity convergence. However, if time t f − t 0 is sufficiently large, then we can improve the tracking error.

Second Experiment.
For this experiment, we fix the initial condition as (h, _ h) � (10.5 m.1 m/s) and the Bézier spline parameters as (t 0 , t f ) � (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 Δ(t)/m � 0.1 sin(t)cos(t) + 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. en, we propose the PIDS as and the AS as and d is the gain parameter estimation of K g (h) based on the projective adaptive control law proposed in [13] and the control gains fixed as k p � 1, k d � 2, and k i � 1, with u r 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 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.

ird Experiment.
e control task is the same as in the second experiment, but we fix the Bézier parameters as (t 0 , t f ) � (0, 50). In this experiment, once again, we perturbed the acceleration equation by adding it to the noisy signal Δ s (t). Additionally, to make the experiment more realistic and illustrative, we added a noise signal to the output y. at is, y m � y a + 0.1 × η(t), where y m and y a are, respectively, the measurable and actual outputs, and η(t) is a normally distributed random process with zero mean; we assume that h . is not available and estimated using the STBO (11). en, we use the following TBS: where the controller gains satisfy r 1 > r 2 > 0 and the following conditions: where and c i > 0 is any constant. We most note that, for this experiment, we fix C � 5. Hence, according to (51), we can select r 1 � 10 and r 2 � 19. en, to make the comparison, we use the following error index: We show the comparison outcome in Figure 5, 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. is behavior can allow us to implement OCS for real applications.

Complexity 9
Comment 3. We must mention that we implement the sign function as with integration step fixed as 10 − 4 .

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. e robust controller consists of two control actions: one based on saturation functions, and the other is a nonlinear integrator. e 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. e second experiment consists of a comparison among the strategies OCS and AS. From the results obtained in this simulation, shown in Figure 3, 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.

Appendix
Proof. Proof of Inequality (25): first of all, multiplying both sides of (25) by exp(λt), we obtain d dt where λ � 2λ 0 b /b. For simplicity, we suppose that λ 2 ≠ λ. Now, integrating both sides of the above inequality, we obtain which leads us, after using some simple algebra, to the following inequality: From the above, we conclude that V exponentially converges to zero. We can probe the case when λ 2 � λ is in a similar fashion.  Proof of Lemma 1: after some manipulation, it is easy to see that system (6), in closed loop with (7), can be written as _ e x � e y , _ e y � −s a e y − s a/2 e x + e y + _ φ(t), where e x � x − x r and e y � y − _ x r . Notice that the right-hand side of the equation above is Lipchitz, then states e x and e y remain bounded in a finite time. at is, the system is forward complete. Now, introducing the linear transformations x 1 � e x − e y and x 2 � e y into system (A.4), we obtain Now, since _ φ(t) ⟶ 0, as long as t ⟶ ∞, we can select t 0 > 0, such that | _ φ(t)| < a/4, for all t > t 0 . us, selecting the positive definite function V (2) � x 2 2 /2, we obtain that its time derivative, around the trajectories of the second equation of (A.5), leads to the following inequality: Because system (A.5) is Lipchitz with respect to the states (x 1 , x 2 ), they cannot exhibit finite time of escape. Notice that, if |x 2 | > a, then from (A.6), we have that _ V (2) < 0. is implies that there exists t 1 > t 0 , such that |x 2 (t)| < a, for all t > t 1 . When this condition is satisfied, we evidently have that s a [x 2 ] � x 2 . Consequently, the first equation of (A.5) reads as for all t > t 1 . Using the positive definite function V (1) � x 2 1 /2 and recalling that | _ φ(t)| < a/4, for all t > t 1 , we have that the time derivative of V (1) , around the trajectories of the system above, leads to the following inequality: Observe that, if |x 1 | > a/2, then _ V (1) < 0. us, there exists a finite time t 2 > t 1 such that |x 1 (t)| ≤ a/2 for all t ≥ t 2 . After these conditions hold, system (A.5) reads as where β 0 � ‖B‖, β 1 � β‖x(t 2 )‖, and β 2 � β‖B‖. Notice that Now, using the Cauchy mean value theorem, we can upper bound the first term of the right-hand side of (A.15) as follows (the fact that Similarly, the second term of the right-hand side of (A.15) satisfies the following equality: erefore, from inequalities (A.14)-(A.19), we conclude that the state x converges to zero for any initial condition. at is, system (A.4) converges globally and asymptotically to the origin.
Proof of Proposition 2. For simplicity, we carry out the proof assuming that h r � 0. As h 2 and w are estimated by the STBO (11), with its corresponding gains fulfilling condition (12), we can conclude that (h 2 , w) converges in finite time to Complexity with η h (t) and η w (t) bounded, such that η h (t) ⟶ 0 and η w (t) ⟶ 0, as long as t ≥ T * , with T * being finite. Under these conditions, it is easy to see that the trajectories of the closed-loop system (32) with (30) does not escape to infinity for all t ≥ T * . Moreover, after t ≥ T * , signals u, v d , ξ(t), and Φ(t) are given as follows: where z is defined in (33). As the above signals coincide with the expression given in Proposition 1, with λ 1 k /k > 1, then we can assure that h 1 and h 2 converge to zero, as long as t ⟶ ∞.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.