Disturbance observer-based autonomous landing control of unmanned helicopters on moving shipboard

In this paper, the autonomous landing control issue on moving shipboard is investigated for unmanned helicopters subject to disturbances. The issue is studied by stabilizing the error system of the helicopter and the shipboard. The landing process is divided into two phases, i.e., homing phase, where a hierarchical double-loop control scheme is developed such that the helicopter is forced to hover synchronously at a certain altitude over the shipboard, and landing phase, where a composite landing control scheme is proposed such that the helicopter lands vertically on the shipboard in synchronization with its attitudes. The velocity and acceleration information of the shipboard as well as lump disturbances is estimated through joint state and disturbance observers. The estimates are then incorporated into the baseline feedback controller, formulating composite active anti-disturbance landing control schemes. A continuous terminal sliding mode control method is proposed for the feedback controller design, which not only effectively mitigates the chattering of the control action, but also simplifies the design process of the controller. Numerical simulations demonstrate the effectiveness and superiorities of the proposed control schemes.


Introduction
Unmanned aerial vehicles (UAVs) have received a great deal of research, development, and application for several decades [1]. Owing to the unique and significant abilities of vertical takeoff and landing, sideslip, backward flight, and hovering, unmanned helicopters have been widely utilized to perform a variety of missions in various environments. On the one hand, in all flight missions, autonomous landing is an essential stage and is prone to accidents, especially landing on moving targets. On the other hand, in all operating environments, maritime operating is very precarious and challenging due to the rough sea conditions such as low visibility, strong wind and turbulence, unpredictable motions of the shipboard, and narrow deck space [1]. For these two reasons, it is of great academic significance and research value to exploit autonomous landing schemes for helicopters on moving shipboard.
In recent decades, many researches have been carried out on autonomous landing of UAVs on mov-ing targets. The focus of these researches is different, ranging from improving the algorithm of obtaining system information [2,3], optimizing control parameters [4,5], improving the accuracy of sensors [6], to improving control performances [7][8][9][10][11][12][13][14][15]. In terms of improving control performances, some linear control methods, such as proportional-integral-derivative control [7,8], are proposed to achieve autonomous landing of UAVs in linear flight regimes. Since only specific flight regimes are considered and the complex nonlinear dynamics of UAVs are ignored, the landing performance under complex and changeable operating conditions is hardly to be satisfied. For better landing performance, many nonlinear control methods have been proposed, such as robust H ∞ control [9,10], backstepping control [11,12], and sliding mode control (SMC) [13][14][15].
Among these nonlinear control methods, SMC method is an effective method due to its remarkable advantages in improving anti-disturbance performance and robustness of the system [16,17]. As an important branch of SMC method, the terminal sliding mode control (TSMC) method has more advantages because it guarantees the finite-time convergence of the closedloop system, which is vital for the shipboard landing helicopter system since a fast landing process reduces the exposure of the helicopter to environment disturbances and thus reduces their effects to a certain extent. However, due to the existence of the discontinuous sign function, the control action of traditional SMC/TSMC is discontinuous [13], which will lead to severe control chattering and large control amplitude. Generally speaking, there are two categories of feasible strategies to mitigate the chattering of the control action. The first category is to replace the sign function in the controller with continuous function [14]. For example, in [14], by replacing the sign function with fractional-power function, a continuous control is employed for the shipboard landing control such that the helicopter lands on the moving shipboard in fixed settling time. The second category is to make the sign function appear only in the derivative of the controller [16] such that the control action is continuous and the chattering is mitigated.
In the existing literature, many works achieve autonomous landing by stabilizing the relative motion between the UAV and the moving target [13,14,[18][19][20]. To this end, the dynamics and full state information of the shipboard should be utilized, which generally are not available in reality due to the following techni-cal constrains. Firstly, the motion of the moving shipboard is generally a six-degree-of-freedom (6-DOF) complex model, which is difficult to establish comprehensively. Secondly, even if most of the shipboard dynamics can be established through some methods, the process would be complex and expensive. In addition, it will cause additional errors due to the unmodeled shipboard dynamics. Thirdly, the measurement of shipboard's velocity and angular velocity increases the use of sensor modules, and the measured data may be not adequately accurate because of measurement noises. Based on the above analyses, this paper adopts the method of stabilizing the error system between the helicopter and shipboard to achieve autonomous landing, but does not need to establish the shipboard dynamics. What is more, the velocity and angular velocity of the shipboard are estimated based on known signals, which not only saves sensors but also improves the accuracy of the state information.
In addition, the sea condition is rough with intricate disturbances in shipboard landing process, which brings about adverse influences to the stability and landing performance of the helicopter system [1]. Although the disturbance can be suppressed through SMC method, this is a robust approach. Specifically, the gains of the controller are usually chosen to be larger than the upper bound of the disturbance, which is quite conservative and will lead to large control chattering. To this end, an effective way is to estimate the disturbance from measurable variables and to compensate the influences of them via a feedforward manner [21][22][23]. This is suitable for the controller design in this paper because the velocity and angle velocity of the shipboard as well as the lumped disturbances can be estimated together. In recent decades, some observers for estimating various kinds of disturbances have been proposed [24][25][26][27][28]. For example, in [24], a nonlinear disturbance observer is developed for disturbances generated by an exogenous system such that the estimation error system is globally exponentially stable. Levant [25] proposes a high-order sliding mode observer (HOSMO) for a general class of disturbances, which provides finite-time convergence of the disturbance estimation error system. An innovative resonant extended state observer dedicated to harmonic disturbances is designed for 3-DOF torsional system in [27] and is extended for harmonic disturbances with unknown frequency in [28].
In this paper, the autonomous landing control problem on moving shipboard of the disturbed 6-DOF heli-copter is studied by stabilizing the error system between the shipboard and the helicopter. The landing process is composed of two phases, i.e., homing phase, at which the helicopter is controlled to be hovered synchronously at a certain altitude over the shipboard from a distant location, and landing phase, at which the helicopter should land vertically on the shipboard in synchronization with its attitudes. Each phase is achieved by applying a composite control scheme based on a joint state and disturbance observer and a continuous TSMC (CTSMC) method. Rigorous stability analysis proves the stability of the closed-loop system. The main contributions of this paper are threefold.
(1) Firstly, the landing control problem is investigated by stabilizing the error system between the shipboard and the helicopter, and meanwhile, the velocity and angular velocity of the shipboard are estimated along with lump disturbances through joint state and disturbance observers. Compared with [13,14] where the motion dynamics of the shipboard is established, the proposed scheme not only achieves autonomous landing on the premise of avoiding the complicated shipboard modeling process, but also avoids the errors caused by the unmodeled shipboard dynamics and state measurements. (2) Secondly, by assembling the observer and CTSMC method, composite control schemes are proposed to achieve autonomous landing. In such case, the lump disturbances are compensated rapidly by the observer and further suppressed by CTSMC method in a robust way, which significantly improves the anti-disturbance performance of the closedloop system. (3) Thirdly, a novel dynamic terminal sliding manifold is selected and then a CTSMC method is proposed. In contrast to the literature with traditional SMC method [13], the proposed CTSMC methodbased scheme significantly mitigates the chattering phenomenon as well as simplifying the controller design process.
The paper is organized as follows. Some preliminaries are given in Sect. 2. In Sect. 3, the 6-DOF rigid helicopter model and control objective are presented. The landing control schemes are derived in Sect. 4. Section 5 introduces the numerical simulations. Finally, conclusions are provided in Sect. 6.

Mathematical notation
The abbreviations S t , C t , and T t with t ∈ R represent the trigonometric functions sin(t), cos(t), and tan(t), respectively. The notation "×" represents the cross product of two vectors. For ∀b, z ∈ R and sgn(z) is the standard sign function, namely sgn(z) = 1 for z > 0, sgn(z) = 0 for z = 0, and sgn(z) = −1 for z < 0. Particularly, Given a matrix A m×n ∈ R m×n ,Ȧ m×n ∈ R m×n represents the time derivative of matrix A where each element is the time derivative of the corresponding element in A.

Lemmas
To develop the controller, the following lemmas are given.
Lemma 4 [31] Consider the systeṁ where f : U → R n is continuous with respect to x on an open neighborhood U ⊆ R n of the origin. Assume that there exists a continuously differentiable function V (x) : U → R such that the following conditions hold (2) There exist real numbers c > 0 and α ∈ (0, 1), and an open neighborhood U 0 ⊂ U of the origin such thatV (X ) Then, for any given initial condition V (x 0 ), system (2) is finite-time stable.

Reference frames
To establish the model of the helicopter, two reference frames are defined. The first one is the inertial frame defined as I = {O e , X e , Y e , Z e }, where O e is arbitrarily fixed to a point on the earth's surface, X e points toward the geodetic north, Y e points toward the east, and Z e points vertically down along the gravity vector. The second is the body-fixed frame defined as where the origin O b is located at the gravity center of the helicopter, X b points forward of the helicopter, Y b points to the right side of the helicopter, and Z b points downward to comply with the right-hand rule. Two reference frames are illustrated in Fig. 1. Unmanned helicopters are strongly nonlinear systems with complex dynamic couplings, and they are easily affected by complex external disturbances, especially during autonomous shipboard landing missions.
To establish a helicopter model which not only reflects the nonlinear dynamics roundly but also facilitates the controller design, a general method is to establish a simplified 6-DOF rigid-body model by treating other trivial factors affecting the helicopter as uncertainties or disturbances (which are seen as lump disturbances in this paper). According to [33], the 6-DOF rigid-body model of the unmanned helicopter is given by where P = [x, y, z] T and V = [u, v, w] T represent the positions and velocities of the helicopter in inertial frame I , respectively. Θ = [φ, θ, ψ] T and Ω = [p, q, r ] T denote, respectively, the Euler attitude angles (i.e., roll angle, pitch angle, and yaw angle) and angular velocities of the helicopter in body-fixed frame B. g is the acceleration of gravity, e 3 = [0, 0, 1] T , m is the mass of the helicopter, and J = diag{J x x , J yy , J zz } is the diagonal inertia matrix. R(Θ) stands for the rotation matrix from body-fixed frame B to inertia frame I , which is defined as The attitude kinematic matrix H (Θ) is given by F d and Γ d represent the lumped force and torque disturbances. F and Γ represent the force and torque in body-fixed frame B.
In translational dynamics, it is known that the primary force is the thrust of the main rotor along Z b axis. Hence, F is expressed as where δ col represents the collective pitch of main rotor and Z w and Z col are system parameters.
In rotation dynamics, Γ is described as where L a , L b , M a , M b , N r , N col , and N ped are system parameters, whose values can be obtained by system identification. δ ped is the collective pitch of tail rotor. a and b represent, respectively, the flapping angles of the main rotor along the longitudinal and lateral axes, which are originally driven by the lateral cyclic δ lat and the longitudinal cyclic δ lon . In steady state, the relationship between flapping angles a, b and control inputs δ lat , δ lon is approximated by [34] where τ m is the main rotor's dynamics time constant and A lon , A lat , B lon , and B lat are system parameters. Combining (3)-(6) together, the simplified 6-DOF rigid-body model of the helicopter is expressed as where U = [δ col , δ lon , δ lat , δ ped ] T denotes the control Remark 1 This paper deals with smooth landing maneuver of the helicopter. Hence, the roll and pitch angles of the helicopter system (7) are considered to satisfy |φ(t)| < π 2 and |θ(t)| < π 2 for t ≥ 0. In this case, the kinematic matrix H (Θ) is nonsingular. It is reasonable because with appropriate control strategies, the roll and pitch angles of the helicopter can be guaranteed to be within the ranges of ± π 2 during normal landing process. Similar statements are also made in [11,14] and the references therein.

Control objective
The objective of this paper is to design the controller U = [δ col , δ lon , δ lat , δ ped ] T for the unmanned helicopter system (7) such that it lands on the moving shipboard in the presence of lump disturbances. The landing process consists of two phases, i.e., homing phase and landing phase. For homing phase, the homing controller U is designed such that the positions P = [x, y, z] T and yaw angle ψ of the helicopter con-

Controller design
Before proceeding, an assumption is made.
Assumption 1 All states of the helicopter are measurable. The positions and attitude angles of the shipboard are measurable.

Homing phase
In homing phase, the goal is to design the homing controller U such that the helicopter is hovered synchronously at the predefined altitude z d over the shipboard from a distant location. To design the controller, a hierarchical double-loop control scheme is proposed where the slower dynamics (translational dynamics) is the outer loop and the faster dynamics (rotational dynamics) is the inner loop [14,18]. Since the rotational dynamics is much faster than the translational dynamics, it is considered that the virtual desired signals φ d , θ d generated from the translational dynamics have been obtained from the rotational dynamics at any sample time. Therefore, a relative velocity and disturbance observer (RVDO) is integrated with the relative position feedback control to achieve asymptotic control in outer loop, while a HOSMO and the CTSMC method are combined to achieve finite-time control in inner loop.

Translational subsystem
where From Assumption 1, it is obtained that the first-order and second-order derivatives of x s and y s are unknown, namely,Ṗ d andP d are unknown. Hence, e 2 and d 1 in (8) are unknown. To this end, a RVDO is designed to estimate e 2 and d 1 together. The estimates are then incorporated into the relative position feedback control such that a composite position tracking scheme is proposed for the position tracking controller U 1 . The detailed design process includes two steps, i.e., RVDO design and composite position tracking controller design.
Remark 2 Assumption 2 is given based on the motion model of the shipboard in [13,14] and the construction of disturbance d V . This description fully covers a general class of disturbances, such as constant, ramp, parabolic, higher order, sinusoidal disturbances, and the arbitrary combination of them.

I. RVDO design
With Assumption 2 in mind, the translational tracking error subsystem (8) is redescribed as where Next, a RVDO is designed to estimate e 2,i and the disturbance components Taking the time derivatives of estimation errors, the estimation error dynamics is obtaineḋ where

Proposition 1
The estimation error of the designed RVDO (10) asymptotically converge to zero, namely Proof For a certain i ∈ {x, y, z}, the reduced systeṁ (11) is globally exponentially stable if Φ i is a Hurwitz matrix. Regarding Π i as the input of (11), then it follows from Lemma 1 that the estimation error dynamics (11) is ISS. Note that lim t→∞ D (n i ) i = 0, namely lim t→∞ Π i = 0. Then, it can be deduced from Lemma 2 that the state of (11) asymptotically converge to zero, that is, lim t→∞ Λ i (t) = 0. This completes the proof.

II. Composite position tracking controller design
Design the position tracking controller U 1 as where K 1 = diag{k 1,x , k 1,y , k 1,z } and K 2 = diag{k 2,x , k 2,y , k 2,z } are positive definite diagonal matrices,  observer gains a 0,i , a 1,i , . . . , a 2m i ,i , b 1 (11) is a Hurwitz matrix, and the controller gains k 1,i , k 2,i in (12) are selected such that C i in (13) is a Hurwitz matrix, then the closed-loop translational tracking error subsystem (8) and (12) is asymptotically stable, that is, the position tracking error e 1 converges to zero asymptotically.
Proof (9), the translational tracking error subsystem is then governed bẏ where h i = 0, − k 2,i ε 0,i +ε 1,i +ε 3,i + · · · +ε 2m i −1,i . Taking h i as the input of (13), the reduced systemĖ i = C i E i is globally exponentially stable if C i is chosen as a Hurwitz matrix. Then from Lemma 1, the translational tracking error subsystem (13) is ISS. Based on Proposition 1, if Φ i in (11) is a Hurwitz matrix, then lim t→∞ Λ i = 0, namely lim t→∞ h i = 0. Combining this with Lemma 2 together, it gives that the translational tracking error subsystem (13) is asymptotically stable, that is, lim t→∞ e 1 (t) = 0. This completes the proof.
By the definition of U 1 , δ col and the virtual desired angles are given by

Rotational subsystem
where Next, a composite attitude tracking scheme is proposed for the attitude tracking controller U 2 by combining HOSMO and the CTSMC method. The detailed design process consists of two steps, i.e., HOSMO design and composite attitude tracking controller design.

Theorem 2 Consider the rotational tracking error subsystem (16) under the attitude tracking controller
Proof The proof consists of three steps. The first step is to prove that e 3 , e 4 of (16) and s 1 of (19) are not escape to infinity during the period before the estimation errors of HOSMO (17) converge to zero. In the second step, it is proved that the dynamic terminal sliding manifold s 1 converges to zero in finite time. Finally, the finite-time convergence of the attitude tracking error e 3 is proved in the third step.
Step 3 Substituting (19) into (16), one has Let T = max{T r , T s }. When t ≥ T , ξ 2 = 0 and s 1 = 0 are kept. Then, (30) turns tö According to Lemma 5, the attitude tracking error e 3 converges to zero in finite time. This completes the proof.

With Remark 1 in mind, H (Θ) is invertible with the inverse being
By the identity U 2 = H BU, it gives H −1 U 2 = BU . Then, by some algebraic transformations, the control inputs δ lon , δ lat , and δ ped are given as where δ col is from (14). According to Theorems 1 and 2, it can be concluded that under the homing controller (14) and (32), the positions P = [x, y, z] T and yaw angle ψ of the helicopter converge to the desired position P d = [x s , y s , z d ] T and yaw angle ψ d asymptotically. The hierarchical control structure for homing phase is shown in Fig. 3.

Remark 3
The attitude tracking controller U 2 is embedded in the dynamic terminal sliding manifold s 1 . Part of U 2 (namely U eq 2 ) is designed to cancel out the terms other than U 2 in s 1 , and the other part (namely U n 2 ) is to makeṡ 1 = −η 1 sgn(s 1 ) hold. Sinceṡ 1 =U n 2 = −η 1 sgn(s 1 ), U 2 can be obtained directly from s 1 without actually taking the mathematical derivative of s 1 . In contrast to the literature with traditional SMC method, for example [12], the proposed CTSMC method-based scheme simplifies the design process of the controller.

Remark 4
In the proposed hierarchical control scheme for homing phase, the translational dynamics (the outer loop) are controlled under the asymptotic stability control scheme, while the finite-time convergence control scheme is adopted for the rotational dynamics (the inner loop). The reason mainly lies in two aspects. On the one hand, in contrast to asymptotic stable system, finitetime convergent system has superiorities as fast convergence rate and good anti-disturbance performance [35,36], which are highly desirable for helicopter systems landing in harsh sea conditions. On other hand, due to the underactuated property of the helicopter, the desired signals φ d , θ d are generated from the position tracking controller in the outer loop and are then fed into the inner loop as given reference signals, so φ d and θ d must be at least second-order differentiable. However, finite-time control is essentially discontinuous control, which means that the controller is discontinuous (such as traditional discontinuous TSMC method), or nonsmooth control, namely the controller is continuous but non-differentiable (such as the finite-time control given in Lemma 5 and the proposed CTSMC method in this paper). With this in mind, the finite-time control is only adopted in the inner loop, while the asymptotic control is developed in the outer loop. Since the translational dynamics (the outer loop) runs at an order of magnitude slower than the rotational dynamics (the inner loop) [18], such design also meets the dynamic response speed requirements of the hierarchical control system.

Landing phase
In landing phase, the goal is to design the landing controller U such that the helicopter lands vertically on the shipboard in synchronization with its attitudes. During the landing period, only attitude angles Θ = [φ, θ, ψ] T and vertical position z are considered, while the dynamics of lateral and longitudinal positions x, y are ignored.
Let Υ = [Θ T , z] T ∈ R 4 , Ξ = [Ω T , w] T ∈ R 4 . Based on the helicopter model (7), the landing dynamic system is expressed as where , and d l = 3 , in which d V,3 denotes the third row of d V .
Take Υ s = [φ s , θ s , ψ s , z s ] T as the attitude angles and vertical position of the shipboard. Let e 5 = [e 5,φ , e 5,θ , e 5,ψ , e 5,z ] T = Υ − Υ s represent the landing tracking error, and e 6 = GΞ −Υ s . The landing tracking error system is then given by

I. HOSMO design
Based on Assumption 4, the following HOSMO is designed to estimate e 6 and d 3 in (34), where χ 1 , χ 2 , and χ 3 are the estimates of e 5 , e 6 , and d 3 , are observer gains to be determined, which are all positive definite diagonal matrices.
With the same argument, the inverse of G is G −1 = H −1 0 0 1 . By some algebraic transformations, the landing controller U = [δ col , δ lon , δ lat , δ ped ] T is computed as Proof The proof of Theorem 3 is similar to that of Theorem 2, so it is omitted here.

Remark 5
In homing phase, all states of the helicopter are considered such that the helicopter (7) is an underactuated 6-DOF system with four control inputs. Because of the underactuated property, a hierarchical double-loop control scheme is developed. For the reasons mentioned in Remark 4, the outer loop adopts asymptotic stability control scheme by combining a RVDO with relative position feedback control, while the inner loop adopts finite-time convergence control scheme by combining a HOSMO with CTSMC method. In landing phase, since the landing process is fast, only the degrees of freedom of attitude angles and altitude are considered. Hence, the landing dynamic system (33) is a fully actuated 4-DOF system with four control inputs. To this end, the landing control scheme does not need to adopt hierarchical double-loop control, but can be the same as the inner loop scheme in homing phase, that is, a composite landing scheme with the integration of a HOSMO and CTSMC method.

Simulations studies
To show the effectiveness and the superiorities of the proposed composite landing control scheme, numerical simulations are performed in this section.
The model parameters of the helicopter are given as follows [37]:  Figure 4 depicts the three-dimensional trajectories of the helicopter and shipboard in homing phase. Figure 5 gives the positions and yaw angle response curves of the helicopter under the proposed homing control scheme. It can be observed from Figs. 4 and 5 that under the homing controller, the positions and yaw angle of the helicopter converge to the desired values asymptotically even in the presence of lump disturbances. The time history of the homing controller with four control inputs is shown in Fig. 6, which shows that the control inputs are continuous. The estimation errors on e 2 and d 1 of the RVDO (10) are provided in Fig. 7. From Fig. 7, it is shown that both the estimation errors of the system state and the disturbance converge to zero asymptotically, which indicates that the velocity and acceleration of the shipboard as well as the disturbance can be accurately estimated by the observer. Figure 8 depicts the estimation errors of d 2 from the HOSMO (17), which can be seen to converge to zero in finite time.
The simulation results are shown in Figs. 9, 10, 11, and 12. Figures 9 and 10 demonstrate the attitude angles and vertical position response curves of the helicopter

(f)
in three simulations. It is observed from the curves under CTSMC + HOSMO and CTSMC schemes that the landing error curves under the CTSMC + HOSMO scheme are much smaller than those under the CTSMC scheme, which fully demonstrates the good compensation of HOMSO. From the curves under the CTSMC + HOSMO and SMC + HOSMO schemes, it is shown that compared to asymptotic SMC + HOSMO scheme, finite-time CTSMC + HOSMO scheme achieves smaller landing errors of the closed-loop system, which clearly shows the superiorities of the finite-time control scheme. Figure 11 depicts the time histories of the landing controllers under three schemes. It is observed that the control action is discontinuous under the SMC + HOSMO scheme, while they are being continuous under the proposed CTSMC + HOSMO and the compared CTSMC schemes. This fully verifies that the proposed CTSMC method effectively mitigates the chattering problem caused by switching functions in traditional SMC method. The estimation error curves on e 6 and d 3 of the HOSMO (35) are shown in Fig. 12, which indicates that the estimation errors of the system state and the disturbance converge to zero in finite time.

Conclusions
The autonomous landing control problem on moving shipboard has been investigated for disturbed helicopters in this paper. By dividing the whole landing process into homing phase and landing phase, the helicopter has been forced to land on a moving shipboard from a distant location. The velocity and acceleration information of the shipboard has been estimated along with lump disturbances through the RVDO/HOSMO such that the errors caused by the unmodeled shipboard dynamics and inaccurate state measurements are effectively avoided, which is one of the major contribution of this paper. Based on the suppression from CTSMC method and feedforward compensation of the disturbance estimates from RVDO/HOSMO, the proposed composite landing scheme has provided double guarantees for the anti-disturbance performance and robustness of the closed-loop system. In addition, the proposed control method mitigates the chattering of the control action as well as simplifying the design process of the controller.
The limitation of this work exists on treating various factors affecting the helicopter system as lumped disturbances, which may lead to insufficient targeted compensation for these factors. Future works will be concentrated on detailed modeling and analysis of these factors, as well as the design of targeted disturbance observation and compensation schemes.