Failure of the landing gear is a significant safety concern, particularly for the nose landing gear, which can cause catastrophic consequences if it malfunctions. In their study, ( Jeevanantham, Vadivelu, & Manigandan, 2017) established that the capability of the landing gear to offer structural safety is very important and especially in models like Boeing 747. ( Freitas, Infante, & Baptista, 2019) also presented an extensive laboratory study on the nose landing gear axle of a commercial aircraft. Applying material analysis reinforced with optical and scanning electron microscopies it was determined that the main cause of failure resultant from overload conditions resulting from both shear and bending stresses. Studies of this kind highlights the suitability of using the micro level to decipher the dimensions and enhancing the dependability of the landing gears.

Because of these risks, researchers have directed efforts towards new autonomous landing systems. ( Santa Cruz & Shoukry, 2022) presented NNLander-VeriF, a neural network-based method to control air- craft during autonomous landing with vision-based controllers. It showed that a vision-based autonomous landing system is capable both of verifying a safe landing and of doing so in a mechanical failure condition. ( Oszust, et al., 2018) also developed a vision-based landing system based on the realistic methodology that onboard cameras could suffice to estimate position of the aircraft during landing. The authors determined that it is possible to safely steer an aircraft in conditions when a part of the landing gear is jammed or nonoperational by using specific visual signals, like the lights on the runway.

Leader-follower control methods have emerged as a promising solution for managing autonomous systems during landings. ( Dai, He, Cai, & Yang, 2022) explored leader-follower formation control in underactuated vehicles, which can be adapted to aircraft scenarios where a ground vehicle supports landing operations by synchronizing with the aircraft's motion. Their method, which is based on adaptive backstepping and Lyapunov functions, ensures that the follower (ground vehicle) maintains a safe distance and alignment with aircraft. Furthermore, ( Darling, 2014) demonstrated a decentralized leader-follower control strategy using small UAVs. His work showed how visual-based localization can be employed to maintain formation, suggesting its application to nose landing gear malfunctions when a supporting vehicle is used to ensure safe landing.

This study aims to investigate how a control system can efficiently coordinate the velocity of a mobile platform with an aircraft's velocity in order to facilitate a secure and regulated emergency landing. The research question focuses on the effectiveness of synchronizing the velocity of the mobile platform with the velocity of the aircraft during emergency situations to ensure a safe and controlled maneuver.

The successful completion of an emergency landing, where an aircraft lands on a platform, relies heavily on coordination and control between the aircraft and the supporting platform. This part outlines the challenges and goals of creating a control system for matching velocities between the aircraft and the platform.

The main hurdle involves dynamically coordinating the velocity of the moving platform with the aircraft during landing. This necessitates a control algorithm that can adjust the platform's velocity in time to align with the desired approach velocity of the aircraft. The control system must include a feedback mechanism to monitor and assess the aircraft and platform velocities continually. This feedback is crucial for adjusting the platform's velocity, ensuring it aligns optimally and stays in sync with descending aircraft.

We aim to create a framework for emergency landing procedures involving aircraft and mobile support platforms by tackling these aspects in designing a control system for matching velocities between platforms. Through simulations and theoretical analyses, we aim to validate our proposed approach and pinpoint opportunities for improvement in real-world scenarios.

This study suggests two advanced control strategies, Proportional integral derivative controller (PID) and sliding mode control (SMC), to obtain accurate velocity synchronization between the aircraft and the support platform. This section explains the mathematical expressions of the proposed control strategies.

The control architecture also transmits the aircraft's velocity to the mobile platform, which adjusts its throttle output following the aircraft's velocity. Its closed-loop configuration enables communication between the aircraft and the platform needed for a controlled landing operation.

An aircraft's 3-DOF model was simulated using MATLAB 2021b, Simulink version 9.5. Aerodynamic coefficients C _L, C _D, C _Y are interpolated from lookup tables generated by Computational Fluid Dynamics simulations. Angles of attack range between a minimum of -10° and a maximum of 20°, Mach between 0.3 and 1.2 ( Vinuesa, 2021); however, the aerodynamic coefficient lookup table is based on subsonic tests and therefore cannot capture some crucial effects during high-velocity flight, hence its limited applicability in supersonic regimes. In this work, linear interpolation was performed in order to make the movement of points smooth. The wind gusts were simulated as a short-duration disturbance within a magnitude of 10 to 20 m/s applied for a duration of 2 seconds. In the simulation of turbulence, Gaussian white noise with a standard deviation of 0.05 m/s was used on all axes: x, y, and z. The random seed was controlled in order to make sure that results are repeatable. PID controller was tuned by trial and error while SMC controller has been designed with a sliding surface defined as: s = e ̇ ( t ) + λ e ( t ) (1) where e(t) is the error and λ is the tuning parameter. The SMC switching gain was selected for chattering reduction, and boundary layer approach was adopted to smooth the control signal ( Taheri, 2019).

Proportional Integral Derivative Control

More control actions and a quicker reaction are offered by the PID controller approach than by the P and PI controller systems. Control lessens the impact of the disturbance, and PID cuts down on the time needed to archive the set point. Calculating the values of gain K _p, K _i, and K _d is necessary prior to starting the system's tuning procedure. In order to optimize the PID curve's responsiveness, trial and error is employed in the tuning process. Equation (2) is utilized for the computation of all the parameters. u ( t ) = K p e ( t ) + K i ∫ 0 t e ( τ ) dτ + K d de ( t ) dt (2)

Where:

u ( t ) represents the control signal (throttle setting).

e(τ) is the error between the desired value (aircraft's velocity) and the actual value (platform's velocity) at a previous time points τ.

e ( t ) indicates the error of the difference in velocity between the aircraft and the platform, defined as e ( t ) = v aircraft − v platform (3)

K _p , K _i , and K _d are the proportional, integral, and derivative gains, respectively.

PID controllers are used to operate most control systems in most industries. In order to reduce mistake, proportional mode will expedite the process, enhance the manipulating variable more quickly, and provide dynamic response. There is no steady state error when the integral mode reaches zero offset. The derivatives mode alters the pace at which the controlled variable changes. The entire system, which includes the P, I, and D terms, commits to a minimum overshoot, a quick rising time increase in stability, and a reduction in steady state error. The characteristic changes for each controller are displayed in Table 1.

When the output is measured by a precise input signal, the control system is depicted by a crude mathematical model through system description. The P, I, and D parameters are gained using an auto-tuning technique, which is often carried out using MATLAB, and a PID controller is created. The identification system's PID closed-loop control is implemented using a Simulink model.

Sliding Mode Control

The sliding mode control (SMC) provides a sliding mode to make the error dynamics reside on a sliding surface to handle the system uncertainties and disturbances. It is very well suited for systems where the controlled quantities are perturbed; moreover, the system's parameters are uncertain or change. The central concept in sliding mode control is to create a control law that forces the system's state to the desired sliding surface, where the system is simplified and more convenient to handle. This forces the system to stay on this surface, giving robustness against disturbances and uncertainties.

The sliding mode control synthesis is done in three steps ( Slotine & Li, 1991): 1.

Select of sliding surface.

We consider a nonlinear system presented in the equation as follow: x n = f ( x , t ) + g ( x , t ) u ( x , t ) x ∈ R n , u ∈ R (4)

Where f and g are two nonlinear functions continuous supposed bounded.

The goal is to guide an aircraft (Boeing 747) towards a mobile platform, ensuring that both the aircraft and the platform arrive at the landing position simultaneously. This requires precise path tracking and velocity matching between the two systems: •

Aircraft Path Tracking: The SMC strategy focuses on guiding the aircraft along a specific trajectory.

Define Sliding Surface: The sliding surface is designed to ensure that the error between the platform's velocity and the aircraft's velocity converges to zero over time. The sliding surface can be defined as Eq. (1): s ( t ) = e ̇ ( t ) + λ e ( t )

Where:

e ( t ) = v aircraft − v platform is the velocity error between the aircraft and platform.

λ > 0 is a positive constant that affects the convergence rate of the system.

e ̇ ( t ) is the time derivative of the error.

The SMC control input is designed to drive the sliding surface S(t) to zero. The control law of SMC allows to converge very rapidly to the velocity reference even in the presence of changes in the operating conditions. The objective of this control law is to limit the state trajectories of the system so that they not only reach the sliding surface but also remain on it, even in the presence of uncertainties within the system. In essence, the control law must render the sliding surface locally appealing. Consequently, the control law should be determined by verifying a condition that guarantees the sliding variable S(t) to converge to 0. This condition is referred to as the reachability condition. Since platform velocity was to be controlled relative to aircraft velocity, the stability of the mobile platform model is a mandatory check before implementing a controller. The Lyapunov stability theorem has been used to check the stability of the mobile platform model.

Zhou et al. ( Zhou, Hu, Zhu, & Ma, 2021) [11] claim Hamilton energy can be the most suitable Lyapunov function. The Hamiltonian energy refers to a system's total energy (sum of kinetic and potential energy), which is a fundamental concept in classical mechanics. Potential energy can be ignored when moving a vehicle on a straight runway. The stability analysis of the proposed control is proved using the Lyapunov candidate function approach given by: V = 1 2 S 2 (5)

The derivative of Lyapunov function V ̇ must be negative and the asymptotic stability is therefore guaranteed using sliding mode control. The necessary and sufficient condition to satisfy the reachability condition is expressed as ( Edwards & Spurgeon, 1998): S ̇ S < 0 (6)

For a convergence in finite time, we replace the condition of reachability ( Eq. (6)) which only guarantees an asymptotic convergence towards the sliding surface by a more restrictive condition called η-reachability and it is defined by: S ̇ S ≤ − η | S | , η > 0 (7)

The control variable U is the sum of the equivalent control U eq , which can be interpreted as the continuous control law that would maintain S ̇ = 0 , and a discontinuous component which presents the dynamic response providing the convergence and the sliding mode. U = U eq + U s (8)

U s is the discontinuous command usually adopted as: U s = k sgn ( S ) (9)

Where:

k is a positive gain that controls the strength of the sliding action.

sgn ( S ) is the sign function that switches the control input based on the sliding surface's value sgn ( S ) = { 1 S > 0 0 S = 0 − 1 S < 0 (10)

The sliding mode control law can be expressed as ( Liu, Gao, Yunfei, Jiahui, Wensheng, & Guanghui, 2020): U ( t ) = − k 1 . sign ( S ( t ) ) − k 2 S ( t ) (11)

k ₁ and k ₂ are a positive constant, tuned to ensure robustness against disturbances.

In order to describe the motion of the aircraft and mobile platform, a fixed inertial frame is defined in a north-west-up (NWU) reference coordinate frame and a velocity frame is placed at the center of mass of each vehicle in a forward-left-up coordinate frame. This simulation model, which focuses on an aircraft's primary motion axes (pitch, roll, and yaw), provides a fundamental understanding of aircraft dynamics and behaviour. In this paper, a Boeing-747 is considered as a point mass, and its three-dimensional equations of motion in the presence of wind disturbances are shown below ( Rucco, Sujit, Pedro Aguiar, Sousa, & Lobo, 2018): x ̇ a = v a cos φ a cos γ a + w x (12) y ̇ a = v a sin φ a cos γ a + w y (13) z ̇ a = v a sin γ a + w z (14) v ̇ a = T m − D m − g sin γ a (15) γ ̇ a = L cos ∅ a m v a − g cos γ a v a (16) φ ̇ a = L sin ∅ a m v a cos γ a (17)

Where [ x a y a z a ] T defines the position of the aircraft in the inertial frame. The velocity vector is determined by the airspeed v a , flight-path angle γ a , and heading angle φ a . The wind components within the inertial frame are represented as w x , w y , and w z . In the trajectory optimization problems addressed in this study, a constant wind velocity is presumed. The method we propose seeks to create trajectories that will act as reference trajectories, enabling the design of closed-loop control systems capable of adhering to these nominal trajectories despite uncertainties and disturbances caused by wind gusts and turbulence during actual flight. The lift, drag, and side forces are computed as follows: L = 1 2 ρ V a 2 S C L (18) D = 1 2 ρ V a 2 S C D (19) Y = 1 2 ρ V a 2 S C Y (20)

Where ρ is the atmospheric density, and S is the reference surface area of the aircraft. The lift ( C L ), drag ( C D ), and side force ( C Y ) coefficients are assumed to be from Table 2, which shows the aerodynamic parameters of the Boeing-747 used to implement the aircraft mathematical model in simulation ( McLean, 1990).

The motion of the mobile platform is confined to a two-dimensional plane on the ground, and its equations of motion in this context are as follows:

Mobile Platform has been modelled as a SISO system where the throttle is the input, and Mobile Platform velocity is the output. The platform has been modelled as single order system: x ̇ p = v p cos φ p (21) y ̇ p = v p sin φ p (22) v ̇ p = a lon (23) φ ̇ p = δ p v p (24)

Where [ x p y p z p ] T defines the position of the mobile platform in the inertial frame, and z p = 0 due to the planar-motion assumption. The velocity vector is determined by the ground velocity v p and the heading angle φ p . We assume that the path of the mobile platform is predetermined by the runway, which doesn’t have curvature ( δ p = 0 ), and we control the longitudinal acceleration a lon of the mobile platform to accelerate or decelerate along the fixed path.

By choosing the velocity frame of the aircraft as the reference frame, the motion of the mobile platform can be defined with respect to the aircraft, and a coupled dynamic system is formulated below ( Wang & McDonald, 2020): e ̇ x = v a cos μ cos γ a − v p + w x (25) e ̇ y = v a sin μ cos γ a + w y (26) e ̇ z = v a sin γ a + w z (27) v ̇ a = T m − D m − g sin γ a (28) γ ̇ a = L cos ∅ a m v a − g cos γ a v a (29) μ ̇ = L sin ∅ a m v a cos γ a (30) v ̇ p = a lon (31) where r e = [ e x e y e z ] T = r a − r p = [ x a − x p y a − y p z a − z p ] T defines the position of the mobile platform relative to the position of the aircraft. The heading angle error is defined as μ = φ a − φ p .