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Abstract

This study aims at the flexible landing control problem of vertical take-off and landing UAV. A control method based on Model Reference Adaptive Control (MRAC) is designed and verified to ensure the stability of the UAV when landing on the moving platform and reduce control errors and dynamic instability caused by bouncing. By establishing a mathematical model including landing gear stiffness and damping parameters, and combining it with MRAC for adaptive parameter adjustment, the system can adapt to different landing conditions in real time, ensuring the stability and feasibility of flexible landing. This study conducted experimental tests on multiple sets of stiffness and damping parameters and analyzed the effectiveness of the MRAC control strategy under different configurations through numerical simulation. Experimental results show that when the stiffness and damping configuration are appropriate, MRAC can quickly adjust the control parameters, so that the time domain response characteristics of the UAV tend to be stable when landing, and the loss function shows a decreasing trend, proving that the control method has good convergence characteristics and adaptability. This study analyzes the MRAC parameter adjustment process through game theory and proves that the system can achieve Nash equilibrium under certain conditions, making each landing gear control strategy optimal and further improving landing stability. In order to verify the feasibility of MRAC control in practical applications, this study also considered the effects of sensing errors and random noise. The results show that the method can still successfully converge, demonstrating its robustness under different environmental conditions. This study confirms the applicability of MRAC in the flexible landing control of UAV and provides a theoretical basis for the future development of dynamic stiffness and damping adaptation mechanisms.

Keywords

UAV game theory flexible landing model reference adaptive control nash equilibrium

Introduction

In recent years, the design of landing gear in the application of small UAV has gradually received attention because it is directly related to the safety and performance of UAV. When faced with uneven ground or emergencies, traditional rigid landing gear can easily cause the structure of the drone to be damaged or overturned, thus affecting the smooth completion of the mission. Flexible landing can effectively absorb the impact force during landing and reduce the stress on the drone body, thus improving its durability and stability. Many studies have been devoted to improving the design of the landing gear of small UAV, including innovative solutions such as adaptive passive landing gear and inflatable airbag landing gear. The adaptive passive landing gear can automatically adjust according to changes in the landing terrain to improve landing stability, while the inflatable airbag landing gear slows down the impact through inflation and adapts to different terrains and landing conditions. These designs not only improve the survivability of UAV in complex environments, but also allowing them to perform more diverse tasks. As drone technology continues to advance, the demand for flexible landing will become higher and higher.

In 2019, Luo, et al. proposed a soft-landing device that imitates birds landing on uneven surfaces, combined with a neural network control method.¹ Using a quadcopter helicopter consisting of a flight unit and a landing assist unit, a new soft mechanism is designed to absorb the remaining landing impact force. The control strategy is based on a dynamic model and applies a backstepping controller to achieve the desired trajectory. In 2007, Batterbee, et al.² proposed an innovative method of semi-active damping using magnetorheological fluid to solve the conflicting damping needs of aircraft landing gear under different excitation conditions, optimize the damping performance and magnetic circuit performance of the landing gear, and comply with space constraints, which is of great significance for commercial applications. Kyle, et al.³ pointed out that flexible landing can effectively absorb energy during landing and braking, reduce aircraft structural stress, and improve safety and reliability. Zbigniew outlines general design solutions and installation methods for landing gears,⁴ emphasizes the load function and calculation methods of oil and gas shock absorbers, and covers the requirements for shock absorber design basics, materials used and future developments. In 2001, Jocelyn. emphasized the importance of flexible landing in reducing landing impact and vibration.⁵

The flexible landing device can effectively absorb landing energy, reduce aircraft structural stress, and improve safety and reliability. It provides stable support under different ground conditions, ensuring a stable landing of the aircraft, while reducing the weight of the landing gear and minimizing air resistance, making flexible landing crucial in aircraft design. In 2012, Elsa et al. introduced a flexible landing control method, which used a spring-deploying claw mechanism to allow the quadcopter to dock on a tree-like structure.⁶ Flight tests proved its effectiveness in improving landing stability on complex terrain. In 2021, Chinvorarat and Vallikul introduced a retractable light amphibious aircraft main landing gear. An additional locking link was added to the design to ensure stability.⁷ The oil and gas shock absorber strut absorbs impacts and reduces the force requirements for the retractable linear actuator. In 2023, Wang, et al. developed a three-dimensional flexible landing dynamics theoretical model and its numerical solution process.⁸ The model considered the six-degree-of-freedom motion of the center of mass offset setting and the spatial motion of each landing gear. In 2021, Liang, et al. discussed the popularity of UAV in various applications, emphasizing the importance of appropriate landing gear for safe takeoff and landing.⁹ Wang, et al. pointed out that UAV landing gear plays a key protective role in small UAV with short endurance and poor landing performance.¹⁰ Wang, et al. proposes an inflatable UAV landing gear to address the shortcomings of existing designs, enhancing protection for small UAV in complex environments. Wang, et al. created a fast and universal mechanism design and optimization method,¹¹ conducted a comprehensive analysis of mechanisms with different numbers of components, and proposed a characterization method to achieve generalized kinematics simulation of complex mechanisms, and applied particle swarm optimization algorithms to optimize mechanism parameters, demonstrating the superiority of this method in the design of landing gear mechanisms. Khan, et al.¹² proposed a design, analysis and topology optimization method for the solid blade spring landing gear of a quadcopter. The landing gear supports and absorbs the aircraft weight and impact load under static and dynamic conditions. Gan, et al. proposed a relationship between light aircraft landing gear strut friction and landing performance, using drop tests and response surface methods for multi-objective optimization.¹³

Yin, et al. introduced a hexapod mobile lander, which combines the functions of a lander and a probe and has the ability to land repeatedly.¹⁴ Dynamic modeling and numerical simulation analysis show its flexible landing capability on the 5-DOF lunar gravity test platform, providing a new perspective for lunar exploration equipment. Dong, et al. emphasized the importance of safe and flexible landing for Mars exploration missions¹⁵ and pointed out the key role of uncertain terrain analysis in lander design. Traditional UAV have separate designs for the lander and detector, which increases launch costs and limits the detection range. In 2021, Ke, et al. also designed the hexapod mobile repetitive lander,¹⁶ which combined active and passive compliance hybrid mechanisms and integrated drive units to propose a flexible landing control method based on a state machine. Nekoo, et al. studied the flexible landing problem of quadcopter,¹⁷ using the induced wind model method. The experimental results show that the proposed method successfully reproduces the wind model and achieves flexible landing. Zhou, et al. established and optimized the buffering energy absorption and motion planning model,¹⁸ verified the effective energy absorption of aluminum honeycomb and a three-level aluminum honeycomb buffer, and designed a motion gait and minimum jitter drive trajectory. Langberg, et al. developed a landing gear and landing system and emphasized its ability to land safely and autonomously in unfamiliar terrain and changing weather conditions.¹⁹ Ding, et al. designed a variable stiffness leg based on a four-bar linkage mechanism.²⁰

Taking into account the design constraints of the landing gap, a numerical model for the probe landing simulation was established. The second-order response surface method was used to obtain an alternative model, improve computational efficiency and optimize the variable stiffness leg and pressure. Liang, et al. developed a new method for landing impact dynamic analysis using the nonlinear finite element method,²¹ simulating a flexible landing event, establishing an aluminum honeycomb shock absorber and lunar soil model, and demonstrating the connection point load, structural acceleration response and absorbed energy. Huang, et al.²² introduced the rapid development of UAV in the field of remote detection. They proposed a lightweight robot landing gear to enable the UAV to land on irregular surfaces. They used a vacuum system to fix the robot landing gear and used a movable counterweight to balance the flight. Only one servo motor and passive mechanical structure were used to guide the vacuum suction cup to adapt to different surfaces. Nihat pointed out that vertical take-off and landing rotorcraft cannot land safely on ground with excessive slopes, and fixed landing gear increases the risk of rollover.²³ An adaptive passive landing gear design and control scheme is proposed, which is suitable for low take-off weight quad-rotor drones and follows the principles of rapid adaptation to the ground, low cost and weight.

In recent years, there are still many studies in this issue due to the diverse development of drone types. Yin, et al. designs a multirotor UAV with adaptive landing gears for stable landings on complex terrains, introducing the “terrain envelope” concept to predict landing stability.²⁴ The findings show that increasing positive lateral landing velocity expands the stable landing region, enhancing UAV landing success on slopes. In 2024 Ganesan, et al. evaluates Borassus Flabellifer fiber-reinforced epoxy composites for UAV landing gear, analyzing mechanical properties, thermal stability, and moisture resistance.²⁵ In 2025, Dinc, et al. analyzes oleo-pneumatic shock absorbers for carrier-based UAV landing gear,²⁶ demonstrating their ability to withstand high-impact forces and improve vibration damping. Chen and Xue presents a hybrid optimization method combining BP neural networks and genetic algorithms to enhance UAV landing gear buffering performance.²⁷ Qiu, et al. presents an impedance-based parallel cooperative control method for the hydraulic landing gear system of large UAV, enabling controlled tilting for smoother cargo loading and unloading.²⁸ Perkasa, et al. analyzes the impact load resistance of a UAV's retractable main landing gear using finite element methods, confirming its safety at vertical landing speeds up to 6 m/s.²⁹ Chu, et al. designs an adaptive UAV landing gear inspired by the praying mantis, utilizing a four-bar mechanism and a laser range sensor to enhance landing stability.³⁰ Ghods, et al. proposes an adaptive UAV landing gear with a single-degree-of-freedom arm mechanism, enhancing stability on inclined surfaces.³¹ Darmawan, et al. examines the application of pre-straining spring momentum exchange impact damper to landing gear, demonstrating that its active time prediction system effectively reduces shock vibration acceleration.³² Widanto, et al. designs a landing gear drop weight test to evaluate impact resistance and deformation, aiding future research and development.³³ The test system, based on CASR 23 standards, utilizes gravity propulsion and analyzes structural deflection to ensure reliability. Wang, et al. introduces a dual-function landing gear with an actively closing mechanism for coaxial drones, enhancing grasping efficiency while reducing energy consumption.³⁴

Flexible landing technology is critical in aerospace, especially for UAV operating on complex terrain. Traditional systems relied on passive components like springs and shock absorbers, but advances in materials (e.g., composites, shape memory alloys) have enhanced performance. Recent research highlights motor-driven simulated damping systems, which adjust damping characteristics in real time based on aircraft and terrain conditions, offering better impact absorption and structural protection. Though still emerging, this approach shows strong potential for small UAV, addressing limitations of passive systems and enabling precise, adaptive control for safer, more reliable landings.

Therefore, this study aims at the flexible landing control problem of vertical take-off and landing UAV. A control method based on Model Reference Adaptive Control (MRAC) is designed and verified to ensure the stability of the UAV when landing on the moving platform and reduce control errors and dynamic instability caused by bouncing. By establishing a mathematical model including landing gear stiffness and damping parameters, and combining it with MRAC for adaptive parameter adjustment, the system can adapt to different landing conditions in real time, ensuring the stability and feasibility of flexible landing.

Methods analysis

This study first establishes the motion equation of the UAV body described by rigid body dynamics and Newton-Euler equations, as shown in (1). where $M \ddot{X}$ is the mass matrix and satisfies the six spatial degrees of freedom of $M \in R^{6 \times 6}$ , $C (\dot{X})$ is the Coriolis force and centrifugal force terms, which can also be regarded as the inertial velocity of the UAV in space over a short distance, $G (X)$ is the gravity term, $F_{thrust}$ is the thrust vector, and $F_{landing gear}$ is the resultant force exerted by the landing gear on the UAV.

M \ddot{X} + C (\dot{X}) \dot{X} + G (X) = F_{thrust} + F_{landing gear}

(1)

Considering that each landing gear in the flexible landing gear target model is driven by a motor to simulate elastic and damping characteristics, an ideal elastic damping system is constructed. As shown in (2), where $m_{l}$ is the mass of the landing gear, $b_{l}$ is the damping coefficient, $k_{l}$ is the elastic coefficient, $z_{l}$ is the compression amount of the landing gear, and $f_{l}$ is the input of the motor drive. The torque output by the motor is simplified to a support force description through the reducer, l is the UAV landing gear number, there are four landing gears in this study, namely $l = [1, 2, 3, 4] \land \forall l \in Z^{+}$ , then (1) can be expanded as (2).

{\begin{aligned} m_{l} {\ddot{z}}_{l} + b_{l} {\dot{z}}_{l} + k_{l} z_{l} = f_{l} \\ [\begin{matrix} m_{1} & 0 & 0 & 0 \\ 0 & m_{2} & 0 & 0 \\ 0 & 0 & m_{3} & 0 \\ 0 & 0 & 0 & m_{4} \end{matrix}] [\begin{matrix} {\ddot{z}}_{1} \\ {\ddot{z}}_{2} \\ {\ddot{z}}_{3} \\ {\ddot{z}}_{4} \end{matrix}] + [\begin{matrix} b_{1} & 0 & 0 & 0 \\ 0 & b_{2} & 0 & 0 \\ 0 & 0 & b_{3} & 0 \\ 0 & 0 & 0 & b_{4} \end{matrix}] [\begin{matrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \\ {\dot{z}}_{3} \\ {\dot{z}}_{4} \end{matrix}] + [\begin{matrix} k_{1} & 0 & 0 & 0 \\ 0 & k_{2} & 0 & 0 \\ 0 & 0 & k_{3} & 0 \\ 0 & 0 & 0 & k_{4} \end{matrix}] [\begin{matrix} z_{1} \\ z_{2} \\ z_{3} \\ z_{4} \end{matrix}] = [\begin{matrix} f_{1} \\ f_{2} \\ f_{3} \\ f_{4} \end{matrix}] \\ | F_{landing gear} | = \sum_{l = 1}^{4} f_{l} \end{aligned}

(2)

The controller is designed with three key objectives. During the landing process, the first is to avoid the landing gear from bouncing off the ground. The second is to avoid unknown platform motion states based on different environments. The third is to ensure that the drone lands smoothly when the above two conditions are met, that is, the rigid load limit of the system must not be exceeded. Therefore, for this condition, assume that the control of the UAV is a known best solution, that is, the attitude and landing trajectory of the UAV are optimal. This study adjusts the stiffness and damping of the four landing gears to adapt to the landing process. The four landing gear deployment expressions in (2) are simplified into ideal response models to facilitate the response process corresponding to the motor simulation.

As shown in (3) and Figure 1, $M_{l}$ is the actual landing gear mass, and $B_{r}$ and $K_{r}$ are the ideal parameter matrices. $Z_{r}$ , ${\dot{Z}}_{r}$ , and ${\ddot{Z}}_{r}$ are the actual relative compression state, i.e., position, velocity and acceleration respectively.

M_{l} {\ddot{Z}}_{r} + B_{r} {\dot{Z}}_{r} + K_{r} Z_{r} = F_{r}

(3)

Figure 1.

Parameters and coordinate of UAV landing system.

Based on (3), the errors of position, velocity and acceleration can be defined as $E = Z_{target} - Z_{r}$ , $\dot{E} = {\dot{Z}}_{target} - {\dot{Z}}_{r}$ , and $\ddot{E} = {\ddot{Z}}_{target} - {\ddot{Z}}_{r}$ respectively. A linear controller can be designed according to (3) to simulate and satisfy its spring and damping characteristics during the landing process, as shown in (4). $K_{p}$ and $K_{d}$ are the control gains. $\hat{K}$ and $\hat{B}$ are the adaptively estimated stiffness and damping matrices.

F_{r} = - K_{p} E - K_{d} \dot{E} + \hat{K} Z_{r} + \hat{B} {\dot{Z}}_{r}

(4)

In the UAV landing system illustrated in Figure 1, each landing gear is equipped with spring-damper elements that are responsible for absorbing the impact energy during touchdown. To ensure a stable and flexible landing, the control strategy described by (4) employs a structure analogous to a Proportional-Derivative (PD) controller, enhanced with adaptive and feedforward dynamics. Specifically, the control force $F_{r}$ incorporates three key terms: a proportional term $- K_{p} E$ , a derivative term $- K_{d} \dot{E}$ and two dynamic compensation terms involving the velocity ${\dot{Z}}_{r}$ , and ${\ddot{Z}}_{r}$ of the landing gear compression state. The proportional term reacts to the position error between the actual and target compression levels of each landing leg, directly reducing the steady-state error. Meanwhile, the derivative term addresses the rate of change of this error, suppressing overshoot and oscillations that may arise during landing impacts or platform motion. Unlike a traditional PID controller, this formulation does not explicitly include an integral term, which is a deliberate design choice to avoid potential issues such as integrator wind-up and instability due to rapidly changing dynamics in the contact phase. Instead, the feedforward terms $\hat{K} {\dot{Z}}_{r}$ and $\hat{B} {\ddot{Z}}_{r}$ serve to compensate for real-time dynamic responses, allowing the system to respond effectively to variations in descent velocity and landing surface disturbances. These terms also reflect the physical properties of the landing gear stiffness and damping which are adaptively updated in real time via the MRAC mechanism. The use of adaptive stiffness $\hat{K}$ and damping $\hat{B}$ matrices ensures that the control system can continuously modify its response characteristics based on current environmental and mechanical conditions. This adaptivity, grounded in a Lyapunov-based stability framework, guarantees system convergence and robustness. As a result, the controller mimics the functional behavior of a PD system while extending its capability to handle nonlinear, time-varying dynamics, thus achieving both landing precision and resilience under uncertainty.

According to (4), consider bringing in the Lyapunov function³⁵ and find its derivative, and choose the update law of $\dot{\hat{K}}$ and $\dot{\hat{B}}$ can be expressed as (5), where $Γ_{K} = d i a g (K_{1}, K_{2}, \dots, K_{n})$ and $Γ_{K} = d i a g (K_{1}, K_{2}, \dots, K_{n})$ are the adaptation gain matrices. Equation (5) is an ideal update law for dynamically adjusting stiffness and damping to adapt the landing gear to different landing environments.

{\begin{aligned} V = \frac{{\ddot{E}}^{T} M_{l} \ddot{E}}{2} + \frac{t r ({(\hat{K} - K_{r})}^{T} Γ_{K}^{- 1} (\hat{K} - K_{r}))}{2} + \frac{t r ({(\hat{B} - B_{r})}^{T} Γ_{B}^{- 1} (\hat{B} - B_{r}))}{2} \\ \dot{V} = {\overset{⃛}{E}}^{T} M_{l} \overset{⃛}{E} + t r ({(\hat{K} - K_{r})}^{T} Γ_{K}^{- 1} \dot{\hat{K}}) + t r ({(\hat{B} - B_{r})}^{T} Γ_{B}^{- 1} \dot{\hat{B}}) \\ \dot{\hat{K}} = - Γ_{K} E^{T} \\ \dot{\hat{B}} = - Γ_{B} {\dot{E}}^{T} \end{aligned}

(5)

The Lyapunov function introduced in the MRAC framework for the UAV landing system plays a pivotal role in ensuring closed-loop stability and convergence during the adaptive control process. As expressed in (5), the Lyapunov function V incorporates three components: the kinetic energy-like term ${\dot{E}}^{T} M_{l} \dot{E}$ , which quantifies the dynamic error of the system, and two trace-based quadratic terms representing the squared deviations between the estimated and actual stiffness $\hat{K}$ and $\hat{B}$ parameters. These trace terms $t r ({(\hat{K} - K_{r})}^{T} Γ_{K}^{- 1} (\hat{K} - K_{r}))$ and $t r ({(\hat{B} - B_{r})}^{T} Γ_{B}^{- 1} (\hat{B} - B_{r}))$ , serve as regularization components that penalize deviations from the ideal parameter values and ensure that the parameter estimates remain bounded during adaptation. In the context of MRAC, the Lyapunov function is used not only to guarantee the stability of the tracking error dynamics but also to guide the adaptive laws for updating $\hat{K}$ and $\hat{B}$ . The negative semi-definiteness of $\dot{V}$ ensures that the overall energy of the error system decreases over time, meaning that the tracking error and the parameter estimation errors converge toward zero. The adaptation laws for stiffness and damping $\dot{\hat{K}} = - Γ_{K} E^{T}$ and $\dot{\hat{B}} = - Γ_{B} {\dot{E}}^{T}$ are derived by selecting terms that cancel out the positive-definite parts of $\dot{V}$ , thus enforcing Lyapunov stability conditions. Physically, this approach ensures that the UAV landing gear can dynamically adjust to variable ground contact dynamics during landing. As the UAV touches down, discrepancies in the compression behavior due to terrain irregularities or external disturbances cause transient errors in . $Z_{r}$ , ${\dot{Z}}_{r}$ , and ${\ddot{Z}}_{r}$ . These errors, processed through the Lyapunov-guided adaptation laws, lead to real-time tuning of the gear's stiffness and damping parameters. Consequently, the system behaves like an intelligent shock absorber, minimizing rebound, overshoot, and energy loss while maintaining robust convergence to a stable configuration. By embedding this Lyapunov-driven adaptive process within each complete landing cycle, the MRAC framework evolves over successive landings. It retains and updates the parameter knowledge base, improving the initial conditions of each new landing and achieving faster convergence and more accurate compliance to the ideal landing trajectory. In essence, the Lyapunov function serves as both a stability certificate and a control energy budget that guides the adaptive behavior of the UAV's physical interaction with the ground.

The update process is implemented using MRAC through each complete landing process. The complete landing process includes three processes: starting of landing, buffering with current parameters after touching the ground, and stable stop. After the above three processes are completely executed, the MRAC parameters are updated according to the experiment, and the stiffness and damping are adjusted to ensure that the new landing can reduce more acceleration and negative displacement. In order to make each iteration of the convergence process tend to the target, a loss function is set as an indicator to measure the performance of the current iteration control system. The loss function is shown in (6),³⁶ where $U = F - F r$ is the control input deviation, i.e., the difference between the actual response and the ideal landing force. $Q$ , $R$ , and $S$ are weight matrices used to balance error, velocity and energy consumption.

J = \int_{0}^{T} (E^{T} Q E + {\dot{E}}^{T} R \dot{E} + U^{T} S U) d t

(6)

The loss function of (6) mainly consists of three parts. The position error term $Q$ ensures that the actual landing is close to the reference model. The velocity error term $R$ reduces oscillation during the landing process. The control of energy consumption term $S$ reduces additional control output and improves energy efficiency. According to this loss function, a vector space $Θ = {[\begin{matrix} \hat{K} & \hat{B} & K_{p} & K_{d} \end{matrix}]}^{T}$ of adjustment parameters is designed. The space contains the estimated stiffness matrix $\hat{K}$ , the estimated damping matrix $\hat{B}$ , the error feedback gain $K_{p}$ , and the error differential feedback gain $K_{d}$ . The goal is to iterate out the best set $Θ *$ of parameters to minimize the loss function of $J$ .

The parameters are updated through the gradient descent method of $Θ_{k + 1} = Θ_{k} - η \nabla_{Θ} J$ , where $\forall k \in Z^{+}$ is the number of iterations, $η$ is the learning rate, and $\nabla_{Θ} J$ is the gradient of the loss function relative to the control parameters, and calculates the gradient as (7).

\begin{aligned} \nabla_{Θ} J & = \int_{0}^{T} (2 E^{T} Q \frac{\partial E}{\partial Θ} + 2 {\dot{E}}^{T} R \frac{\partial \dot{E}}{\partial Θ} + 2 U^{T} S \frac{\partial U}{\partial Θ}) d t \\ = \int_{0}^{T} (2 E^{T} Q (- M_{l}^{- 1} (\frac{\partial B_{l}}{\partial Θ} \dot{Z} + \frac{\partial K_{l}}{\partial Θ} Z) + M_{l}^{- 1} \frac{\partial F}{\partial Θ}) + 2 {\dot{E}}^{T} R \frac{\partial \dot{E}}{\partial Θ} + 2 U^{T} S \frac{\partial U}{\partial Θ}) d t \end{aligned}

(7)

In the context of the MRAC-based control strategy illustrated in the image, the arrow notation $Θ$ introduced to denote the parameter vector plays a critical role in conveying the iterative and vector-valued nature of the adaptive process. Specifically, $Θ$ encapsulates all the tunable parameters of the system $\hat{K}$ , $\hat{B}$ , $K_{p}$ and $K_{d}$ into a single multi-dimensional vector in the control parameter space. The arrow notation emphasizes that these parameters are not scalar values but rather components of a higher-dimensional vector field subject to optimization. The arrow notation $Θ_{k}$ and its subsequent iteration $Θ_{k + 1}$ clearly distinguish the temporal evolution of the parameters over successive iterations of the gradient descent update law. In adaptive control, especially in MRAC, it is essential to track how parameters evolve based on the error signal and the performance of the control law. The notation signifies that the parameters are being updated over time, guided by the negative gradient of the loss function $\nabla_{Θ} J$ , scaled by the learning rate $η$ . Thus $Θ_{k + 1} = Θ_{k} - η \nabla_{Θ} J$ denotes a discrete-time update in the direction of steepest descent, aligning the controller closer to optimal performance with each step. This iterative adaptation process allows the control system to learn from the discrepancies between the desired and actual landing responses in physically, adjusting its internal stiffness, damping, and feedback gains accordingly. The arrow notation captures the dynamic, vectorized nature of this learning process, distinguishing it from static parameter settings and reinforcing the MRAC philosophy of continuous refinement and self-tuning under varying environmental or system conditions.

After substituting gradient descent into the MRAC parameter update, it means that the update conditions of $\hat{K}$ , $\hat{B}$ , $K_{p}$ , and $K_{d}$ in the vector space can be expressed as (8). MRAC parameters adapt to changes in error and follow gradient descent to accelerate convergence.

{\begin{aligned} \dot{\hat{K}} = - Γ_{K} Z E^{T} - η \frac{\partial J}{\partial \hat{K}} \\ \dot{\hat{B}} = - Γ_{B} \dot{Z} E^{T} - η \frac{\partial J}{\partial \hat{B}} \\ {\dot{K}}_{p} = - η \frac{\partial J}{\partial K_{p}} \\ {\dot{K}}_{d} = - η \frac{\partial J}{\partial K_{d}} \end{aligned}

(8)

Simplifying the process of (1) to (8), it can be expressed by Figure 2. After initializing the parameters, the experiment is conducted for each landing. Upon completion, the system state described in (4) and the parameter values from the previous experiment in (1) are feedback into the optimization iteration process defined in (7).

Figure 2.

Optimization parameter process.

In order to avoid the problem of multiple optimal solutions during the convergence process, the target condition of this adaptive convergence process is designed as a non-cooperative dynamic game, in which the control outputs of the four landing gears can be regarded as strategies. The goal is to minimize landing errors and energy consumption while ensuring no bouncing phenomenon. The motor control strategy in the landing gear set (denoted as $P -$ ) participating in the landing process is set to four independent players of $P - = {P_{l}} = {P_{1}, P_{2}, P_{3}, P_{4}}$ , $P_{l}$ is responsible for adjusting the corresponding motor output strategy of $u_{l}$ . Assume that each player's strategy set is given as $u_{l} \in U_{l} = [u_{min}, u_{max}]$ , and its strategies $u_{min}$ and $u_{max}$ are the worst and best strategies output by the motor respectively. The vector of the global strategy is defined as $U = (u_{1}, u_{2}, u_{3}, u_{4})$ . Based on (6), the utility function of $J_{l} (U)$ can be given as (9). $u_{l}^{2}$ is the energy consumption. $α_{l}$ , $β_{l}$ , and $γ_{l}$ are the weight parameters in the utility function to balance the position error, velocity error and energy consumption strategy respectively. $Q_{l}$ and $R_{l}$ are the positive semi-definite weight matrix to determine the control priority.

J_{l} (U) = \int_{0}^{T} (α_{l} E^{T} Q_{l} E + β_{l} {\dot{E}}^{T} R_{l} \dot{E} + γ_{l} u_{l}^{2}) d t

(9)

From (9), the cost function is the sum of the utility functions of the landing gear, i.e., $J (U) = \sum_{l = 1}^{4} J_{l} (U)$ . According to this problem, the optimal condition, i.e., the Nash equilibrium condition, can be found as (10).

J_{l} (U_{l} *, U_{- l} *) \leq J_{l} (U_{l}, U_{- l} *), \forall U_{l} \in U_{l}, \forall l \in P

(10)

Based on (10), this control strategy optimization process must satisfy a condition that other players have no motivation to change $U_{l} *$ when the strategies $U_{- l} *$ of other landing gears exist in the entire strategy space. In order to meet this condition, there must be no cost impact relationship between all landing gears on other players in the utility function. Under this condition, the optimal response function is derived through the variation method. The Hamilton function $H$ can be expressed as (11). Taking the partial derivative of $U_{l}$ in (12), the optimal strategy (13) can be obtained.

H = J_{l} + λ^{T} (M_{l} \ddot{Z} + B_{l} \dot{Z} + K_{l} Z - U)

(11)

\frac{\partial H}{\partial U_{l}} = - λ^{T} \frac{\partial U}{\partial U_{l}} + 2 γ_{l} U_{l} = 0

(12)

U_{l} * = \frac{1}{2 γ_{l}} λ^{T}

(13)

As the motor outputs of all landing gears at the Nash equilibrium point of $(U_{1} *, U_{2} *, U_{3} *, U_{4} *)$ to satisfy the condition (14),³⁷ i.e., the outputs of all landing gears being updated according to the MRAC control law, the UAV can achieve stable landing

J_{l} (U_{l} *, U_{- l} *) = min_{U_{l}} J_{l} (U_{l}, U_{- l} *)

(14)

Based on the above strategy, the conditions for rebound bounce are analyzed to be that the normal force after contact is too large to produce upward acceleration and the damping force is insufficient to absorb kinetic energy. Based on non-smooth mechanics model dynamics and control analysis,³⁸ the control condition for a single landing gear is $F_{l} = - K_{l} Z - B_{l} \dot{Z}$ , to prevent bouncing it must satisfy immediately $\forall l, {\dot{Z}}_{l}^{+} = 0$ and $Z_{l} = 0$ , if $K_{l} > 0$ and $B_{l} > 0$ , then ${\dot{Z}}_{l}^{+} = 0$ . Here, $Z_{l}^{+}$ denotes the positive part of the normal velocity component ${\dot{Z}}_{l}$ , defined as $Z_{l}^{+} = max ({\dot{Z}}_{l}, 0)$ ensuring that upward rebound is eliminated in accordance with the control criteria. The condition ${\dot{Z}}_{l}^{+} = 0$ implies that any upward motion must be suppressed immediately after impact, thereby effectively eliminating rebound in compliance with the non-smooth mechanics model. According to the above conditions, the control strategy of the four motor-driven landing gear constitutes a non-cooperative dynamic game, and there is a Nash equilibrium. At the equilibrium point, the optimal strategy of each motor minimizes the control output error and energy consumption. This means that the best strategy has a Nash equilibrium and can continue to converge toward the optimal adjustment parameters through iteration.

Experiment analysis

The experimental design of this study is given in two parts: time domain response characteristic analysis and optimization process experimental analysis when the UAV (Figure 3) lands. The purpose of the experiment is to test the landing time domain response of the UAV under different initial conditions and control strategies and ensure three key points. The first is to land smoothly to avoid shock and bounce, the second is to minimize overshoot and steady-state errors during the landing process, and the third is to optimize the time for the system to converge to a stable state.

Figure 3.

Proposed UAV system (a) equipped with experimental recorder and (b) its dimensions.

The first part establishes the analysis of time domain response characteristics during landing. The experimental variables are shown in Table 1. The adjustment range of MRAC adaptation gain $Γ$ is given as 0.01∼0.1, and adjustment range of the motor control output $u_{l}$ is 0∼5N. The desired landing velocity $v_{r}$ adjustment range is 0.2∼1.0 m/s, the landing gear stiffness $K_{l}$ is 500∼5000N/m, and the landing gear damping $B_{l}$ is 20∼500Ns/m. To select the stiffness and damping of the landing platform, set the random movement velocity $v_{deck}$ of the platform, initialize the test conditions of $K_{l}$ , $B_{l}$ , and $v_{r}$ , and $Γ$ is given as a constant. The experiment was performed to allow the UAV to land based on the initialization and MRAC iteration control strategy, and to record the time domain response during the landing process. By analyzing the overshoot, steady-state error and system convergence time of each iteration process, the time domain response under different variable combinations is compared.

Table 1.

Constant physical quantity.

Parameter	Symbol	Value	Unit	Description
UAV mass	$m$	10	kg	Including battery and payload
Gravitational acceleration	$g$	9.81	m/s²	Gravity
Deck stiffness	$K_{s}$	50	kN/m	Simulated deck elasticity
Deck damping	$B_{s}$	10	Ns/m	Affects vibration attenuation after UAV landing
Landing platform movement range	$x_{deck}$	±2	m	Movable range of the platform
Landing platform movement velocity	$v_{deck}$	±1	m/s	Simulated deck oscillation

Each parameter in this UAV landing experiment plays a pivotal role in defining the dynamic behavior of the system and the subsequent performance of the MRAC control strategy. The UAV mass not only affects the platform's inertia during landing but also directly influences the impact forces experienced by the landing gear. Gravitational acceleration is incorporated as a constant to simulate real-world forces and to ensure that the dynamic model accurately represents weight-induced pressures during touchdown. The deck stiffness $K_{l}$ and damping $B_{l}$ characterize the elastic and dissipative properties of the landing platform. These directly determine how energy is stored and dissipated during the impact event, thus affecting vibration attenuation and the amplitude of oscillations post-landing. The lateral movement range and oscillation velocity of the landing platform simulate environmental disturbances, pushing the MRAC to adapt and maintain a controlled descent despite variable platform motions.

The control parameters further refine the system's performance. The MRAC adaptation gain $Γ$ adjusts the controller's responsiveness, managing the rate at which it compensates for errors during successive iterations. Likewise, the motor control output is bounded to ensure that the applied corrective forces remain within safe operational limits, reducing the risk of overcorrection that might damage the UAV structure. The desired landing velocity sets a critical target that a soft landing must adhere to, while the landing gear's stiffness and damping are tuned to absorb impact energy effectively, safeguarding both the landing gear and the fuselage.

A robust sensing and measurement framework is key to capturing these dynamic interactions. Integrated sensors such as high-frequency accelerometers, gyroscopes, and strain gauges are employed to continuously monitor the system during the landing process. Accelerometers record the deceleration patterns and impact forces, while strain gauges affixed to strategic points on the landing gear track deformation and load distribution in real time. Real-time data acquisition systems sample these signals at rates sufficient to render a detailed time-domain response, capturing transient events like overshoots, settling times, and steady-state errors with precision. Complementary sensors, such as optical or radar-based distance meters, verify the relative position of the UAV to the landing surface, ensuring that deviations in the approach trajectory are quickly identified.

Through sensor fusion, these measurements feed back into the MRAC controller, enabling iterative adjustment of control parameters to minimize error. The real-time monitoring not only validates the simulation outputs but also provides critical insights into the physical phenomena governing the landing process. Continuous calibration and error correction are integral to this process, as they mitigate sensor noise and ensure accuracy. This closed-loop feedback system facilitates an adaptive approach where the control strategy refines itself based on measured performance, ultimately leading to a more reliable and safer landing procedure.

The second part focuses on the experimental analysis of the optimization process, i.e., the controller adaptation process. The modulation range of the landing velocity error $e_{v}$ is −0.5∼0.5 m/s, that of the landing position error $e_{x}$ is −0.2∼0.2 m, and the initial value of the loss function J is 0∼10. If the initial value is out of the range, the initial value will be updated. This means that this initial value has the risk of damaging the landing gear or causing damage to the fuselage during landing.

After initializing the environment, the dynamic model is established to consider the landing process of the UAV, including stiffness, damping, and parameter adjustment of the MRAC controller when it touches the ground. Design the MRAC controller to establish parameter update rules and use numerical simulation to track the control effect. Simulate time domain response to analyze changes in landing velocity, position error and landing gear stress. Figure 4 shows the height position and velocity response of the buffer after the UAV lands and touches the ground under ideal conditions.

Figure 4.

Velocity target simulation of UAV landing under ideal condition.

This study's experimental framework is structured into three primary phases to systematically validate and refine the proposed MRAC-based landing control strategy. The first phase focuses on the convergence toward an ideal landing velocity, ensuring that the control system can consistently guide the UAV to reach a predefined optimal descent speed. This establishes a foundational performance benchmark for subsequent experimentation. Building upon this, the second phase investigates the optimal selection of adaptive boundaries and learning rates under the constraint of the previously achieved ideal velocity. This step is critical for stabilizing the parameter update process and ensuring convergence within safe and effective limits. The third phase centers on iterative learning to suppress rebound and enhance landing smoothness, using the ideal landing trajectory as a reference. By fine-tuning control responses based on real-time feedback, this stage aims to reduce residual vibrations and ensure a stable touchdown, further demonstrating the robustness and adaptability of the proposed control approach.

Convergence to ideal landing velocity

The system is set up with baseline physical constants including UAV mass, gravitational acceleration, deck stiffness, and damping values, while variable parameters like landing gear stiffness, damping, and desired landing velocity are systematically adjusted. The simulation runs the modified reference adaptive control (MRAC) algorithm to generate time-domain responses, where key performance indicators such as overshoot, steady-state error, convergence time, and oscillatory behavior are closely monitored. Ideal conditions are identified when the UAV achieves a smooth and rapid deceleration upon touchdown, minimal oscillations, and no risk of premature rebound or excessive impact force. In this context, parameters that yield the lowest loss function indicative of rapid convergence and stable control are selected, ensuring that the system meets both safety and performance criteria under dynamic landing conditions. Based on the simulation results depicted in Figure 4, the ideal operating parameters are determined through an iterative process that synthesizes both dynamic response analysis and performance metrics. The key factor in setting this lower limit standard is to ensure that the landing dynamics remain within safe bounds that prevent excessive impact forces on critical components. By establishing this threshold, the system guarantees sufficient damping and controlled deceleration, which in turn minimizes the risk of damage to the sensitive avionics and landing gear during touchdown. This standard is essential to safeguard the integrity and operational reliability of the UAV's key systems.

The throttle of UAV is fully closed at a height of 10 cm, and the UAV's descent state is recorded, and the UAV gradually approaches the ground. At the height of 0, with the ground contact curve being smooth, the flexible control of the landing gear effectively reduces the impact of a hard collision. There is no obvious rebound, where the rigidity and damping of the landing gear are appropriate and can effectively absorb impact energy. The initial velocity is −0.05 m/s. During the landing process, it gradually accelerates and approaches the target velocity of −0.05 m/s. This allows the drone to maintain a constant velocity and reach the actual maximum landing height of −5 cm, which means that the drone will maintain an ideal constant velocity before descending to the stopping point. The following consider the actual body to perform experimental comparisons under vertical free fall simulation conditions and perform corresponding iterative processes in the process.

Figure 5 shows the position and velocity responses of the UAV landing under the condition of $K_{l}$ = 2000 and $B_{l}$ = 1000. The UAV makes ground contact at a height of 0 and decelerates immediately to 0, indicating that the excessive landing gear damping prevents effective mitigation of hard impact. While no significant rebound is observed, on a more elastic surface, potential bouncing or even landing gear failure may occur.

Figure 5.

Velocity target simulation responses of UAV landing with $K_{l}$ = 2000 and $B_{l}$ = 1000.

Figure 6 shows the position-velocity response of the UAV during landing under the condition of $K_{l}$ = 2000 and $B_{l}$ = 500. The UAV makes ground contact at a height of 0 and immediately decelerates to zero, indicating that the landing gear damping remains excessive, failing to effectively mitigate the hard impact. While no significant rebound is observed, on a more elastic surface, potential bouncing or even landing gear failure may occur.

Figure 6.

Velocity target simulation of UAV landing with $K_{l}$ = 2000 and $B_{l}$ = 500.

Figure 7 shows the position-velocity response of the UAV landing under the condition of $K_{l}$ = 2000 and $B_{l}$ = 100. The UAV makes ground contact at a height of 0 and decelerates smoothly to zero, indicating that the landing gear damping has gradually approached an appropriate level, effectively reducing hard impact. However, the descent velocity has not yet reached the ideal −0.05 m/s, suggesting that further iterative optimization can still be performed.

Figure 7.

Velocity target simulation of UAV landing with $K_{l}$ = 2000 and $B_{l}$ = 100.

Figure 8 shows the position and velocity responses of the UAV landing under the condition of $K_{l}$ = 2000 and $B_{l}$ = 80. The UAV makes ground contact at a height of 0 and decelerates smoothly to zero, indicating that the landing gear damping has approached an optimal level, effectively reducing hard impact. The descent velocity is very close to the ideal −0.05 m/s.

Figure 8.

Velocity target simulation of UAV landing with $K_{l}$ = 2000 and $B_{l}$ = 80.

Figure 9 shows the position and velocity responses of the UAV landing under the condition of $K_{l}$ = 2000 and $B_{l}$ = 50. The UAV makes ground contact at a height of 0 and decelerates smoothly to zero. However, the response indicates an underdamping effect in the landing gear control, making it insufficient to support the structure, leading to a descent velocity lower than the ideal −0.05 m/s. This excessive descent speed may result in landing damage or structural harm to the UAV.

Figure 9.

Velocity target simulation of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 50.

The analysis of Figures 5 to 9 reveals how varying the landing gear damping coefficient $B_{l}$ , while keeping the platform stiffness $K_{l}$ constant at 2000, directly shapes the UAV's landing dynamics. When $B_{l}$ is larger (1000 and 500), the UAV decelerates abruptly upon contact, effectively suppressing rebound but risking potential impact issues on more elastic surfaces, such as unnecessary bouncing or even gear failure. As $B_{l}$ decreasing to 100, the UAV experiences a smoother deceleration, indicating improved impact absorption, although the descent velocity still deviates from the optimal −0.05 m/s, suggesting that further tuning is needed. With $B_{l} = 80$ , the response nearly idealizes the desired descent velocity while maintaining smooth deceleration, marking an optimal damping condition. However, when $B_{l}$ is further reduced to 50, the response exhibits underdamping, leading to a descent velocity that is much low and potentially compromising structural integrity. Overall, these findings underscore the critical trade-off in tuning the damping parameter to simultaneously mitigate hard impacts and ensure structural safety during UAV landings.

Optimization of boundary conditions and learning rates

In these experiments, $K_{l}$ = 2000 was set as the initial iterative value and a fixed parameter for analysis. Under all tested conditions, the MRAC controller consistently maintained stable control performance, and as the adaptation of $K_{l}$ and $B_{l}$ improved, the UAV landing process exhibited progressively smoother behavior. Theoretically, increasing the landing gear stiffness enhances energy absorption and mitigates rebound effects. However, in practical applications, when the system stiffness is high, determining the optimal damping coefficient still requires an iterative optimization process. Figure 8 shows the convergence curves of the loss function for different initial values of $K_{l}$ and $B_{l}$ .

Figure 10 provides a detailed insight into the relationship between the landing gear's stiffness and damping settings affect the loss function during landing optimization. It reveals that the loss function, which quantifies deviations in both position and velocity from the desired trajectory, is highly sensitive to the correlation between these two variables. As the damping value is adjusted, it directly influences the landing response; too much damping leads to abrupt stops with potential structural risks, while too little may result in underdamped oscillations. Meanwhile, the stiffness parameter works in tandem with damping together they determine the overall energy absorption and impact mitigation. The loss function serves as a critical metric by penalizing deviations from the optimal landing performance, thereby guiding the MRAC controller to fine-tune these parameters iteratively.

Figure 10.

Magnitude of loss function for initial values of $K_{l}$ and $B_{l}$ with $Γ$ = 0.02.

Figure 10 shows the convergence behavior of the MRAC loss function under different combinations of stiffness $K_{l}$ and damping $B_{l}$ , with a learning rate $Γ$ of 0.02. The trend of the loss function variation with respect to the number of iterations is analyzed. While $Γ$ = 0.02 is considered as ideal condition, it still presents certain computational and convergence risks. For instance, a higher number of iterations may lead to increased computational time, and if real-time adaptation is required during the convergence process, the learning rate $Γ$ must be sufficiently high to maintain convergence while ensuring computational efficiency. Figure 11 presents the loss function convergence curves for different initial values of $K_{l}$ and $B_{l}$ under $Γ$ = 0.1.

Figure 11.

Magnitude of loss function for different initial values of $K_{l}$ and $B_{l}$ with $Γ$ = 0.1.

Figure 11 illustrates the critical process of selecting appropriate parameters in UAV landing operations, especially under conditions with limited knowledge of the model dynamics. In this context, two principal variables of the landing gear stiffness and damping are systematically varied and their interdependence is analyzed through the behavior of the loss function. By monitoring key performance metrics such as overshoot, steady-state errors, and convergence time, this iterative process not only informs the optimal balance between stiffness and damping but also serves as a practical method for determining the MRAC controller's learning rate $Γ$ and setting safe boundary conditions. In limited model scenarios, the strategy involves initializing test conditions within predetermined ranges and adjusting the learning rate iteratively based on the deviation of the system response from the ideal behavior, ensuring that the loss function is minimized and that the UAV landing remains both controlled and safe, thereby mitigating risks of damage to both the landing gear and the fuselage.

Iterative learning for rebound suppression and smooth landing

Under limited initialization conditions, the loss function decreases over time, demonstrating that the MRAC controller can progressively learn and adapt to different landing conditions. The rapid initial convergence indicates that the MRAC controller quickly adjusts control parameters during the early updates. As the convergence trend slows in the later stages and approaches a stable value, it signifies that the MRAC control has identified suitable parameters to minimize error. Similar phenomena are observed in Figures 8 and 9, where higher stiffness and damping result in a slightly faster decrease in the loss function, suggesting that MRAC has sufficient reference information to adjust more efficiently. In contrast, with lower stiffness and damping, the convergence speed is slower, and the loss function exhibits greater fluctuations, making system control more challenging. MRAC requires more time to identify appropriate control parameters. Across different conditions in the figures, the final loss function values tend to approach zero, indicating that MRAC successfully achieves adaptive control, progressively reducing landing errors. In the experiments, appropriately tuning the MRAC adaptation gain to increase the learning rate allows the control parameters to converge to optimal values more quickly.

With $K_{l}$ = 2000 and $B_{l}$ = 50, the experimental results show that obvious oscillation occurs during landing, as shown in Figure 12. The UAV left the ground again instantly instead of settling smoothly, indicating that the damping force was insufficient to offset the elastic recovery force of the landing gear. This excessive oscillatory behavior indicates that a higher damping coefficient is required to achieve a stable landing without unwanted bouncing effects.

Figure 12.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 50.

With $K_{l}$ = 2000 and $B_{l}$ = 80, the experimental result in Figure 13 shows that the oscillation amplitude is reduced and the number of rebounds is reduced compared to lower damping values. However, the damping force is still not enough to completely dissipate the kinetic energy of the drone when it lands. Therefore, the drone still experiences a slight bounce, briefly lifting off the ground, before eventually settling down. It can be found that increasing the damping coefficient can improve stability, it is not optimal for completely preventing unintentional takeoff after initial touchdown. Further adjustments to the damping parameters are required to ensure a fully stable landing.

Figure 13.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 80.

For $K_{l}$ = 2000 and $B_{l}$ = 100, the result in Figure 14 shows that the oscillation amplitude is significantly reduced and the number of rebounds is further reduced. However, the system still exhibited slight residual oscillations, causing the drone to temporarily lift off in small increments before eventually stabilizing. Although the damping effect is close to the optimal range, further refinement is still needed to completely eliminate unintended takeoffs and ensure a completely stable landing.

Figure 14.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 100.

In Figure 15 with $K_{l}$ = 2000 and $B_{l}$ = 500, exhibits a sharp, sudden reaction upon landing, characterized by a high-impact rebound. As a result, the drone experiences a sudden jolt upon impact, resulting in a direct bounce off the deck and significant rebound. An excessively high damping coefficient disrupts the intended energy absorption process, preventing a smooth landing and instead causing instability and undesirable rebound effects.

Figure 15.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 500.

With $K_{l}$ = 2000 and $B_{l}$ = 1000, the experimental result in Figure 16 reveals an extreme jump response on landing with almost no energy absorption. Excessive damping force prevents the system from effectively dissipating kinetic energy, resulting in a near-instantaneous high-impact collision between the drone and the deck. The drone did not gradually stabilize, but rebounded with greater amplitude, making the landing height unstable. This suggests that over-damping can lead to an over-constrained system, where a lack of controlled energy dissipation results in a forceful and exaggerated bounce rather than a smooth, stable landing.

Figure 16.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 1000.

As the convergence test progresses towards the optimal parameter setting $K_{l}$ = 2000 and $B_{l}$ = 140, the experimental results show that the landing gear successfully buffered the descent of the UAV and effectively absorbed the oscillation caused by the movement of the ship deck, as shown in Figure 17. Even as the drone moves upward due to wave-induced motion, the drone remains in contact with the deck and does not lift off again. The selected stiffness and damping parameters achieve an optimal balance and ensure a smooth and stable landing under dynamic conditions.

Figure 17.

Position response of UAV landing for $K_{l}$ = 2000 and $B_{l}$ = 140.

The experimental analysis reveals a clear correlation between the damping coefficient and the landing stability of the UAV, with stiffness held constant at $K_{l}$ = 2000. When the damping is set too low (e.g., $B_{l}$ = 50), the system fails to counteract the elastic recovery of the landing gear, resulting in pronounced oscillations and premature rebound immediately following touchdown. As the damping coefficient is moderately increased (around $B_{l}$ = 80), improvements are observed in terms of reduced oscillation amplitude and fewer rebounds; however, the energy dissipation remains insufficient to completely prevent slight bounces, suggesting that the system is still on the edge of instability. In contrast, an excessively high damping coefficient ( $B_{l}$ = 500 or $B_{l}$ = 1000) leads to abrupt, high-impact reactions at touchdown, whereby the landing gear's rigidity prevents a gradual absorption of kinetic energy and instead produces exaggerated rebound effects this over-damping essentially creates an over-constrained system that undermines the desired energy management. The optimal balance appears to be achieved near $B_{l}$ = 140, where the landing gear adequately buffers the descent, absorbs the transient oscillations induced by external disturbances such as ship deck motion, and maintains continuous contact without unintentional takeoff. This iterative tuning process under conditions of limited model knowledge serves not only to identify suitable damping parameters but also to guide the selection of the learning rate and boundary conditions within the control algorithm, ensuring that the system is capable of dynamically adapting to real-world uncertainties while achieving a smooth and stable landing.

A higher natural frequency typically accelerates the response, yielding a shorter rise time; however, if the damping ratio is too low, this rapid response can lead to excessive overshoot and a shorter but more erratic peak time due to pronounced oscillations. In contrast, when the damping ratio is increased, the system experiences a more moderated rise time that helps to mitigate overshoot, ultimately leading to a more controlled peak response. The settling time, which reflects how quickly the system stabilizes within a specified range around the desired steady state, benefits from an optimal damping ratio: minimal underdamping avoids prolonged oscillations, while over-damping, despite eliminating oscillations, can delay stabilization by causing a sluggish response. In practical terms for UAV landing, tuning the damping coefficient is a balancing act dictated by these metrics. Insufficient damping results in a rapid, yet unstable landing marked by high overshoot and long-term residual oscillations, whereas excessive damping yields a delayed rise and protracted settling time that may compromise responsiveness.

This study demonstrates that even after optimizing the target landing speed, MRAC remains essential for further adapting and fine-tuning control parameters through experimental validation, rather than relying solely on predetermined reference values. The necessity of this approach arises from the complex and variable nature of UAV maritime landing environments, including unpredictable factors such as deck rebound dynamics, external environmental changes, and precise impact timing. These factors limit the effectiveness of static control strategies, underscoring the importance of real-time adaptive adjustment. Through both mathematical modeling and numerical simulations, the research confirms the applicability of MRAC in managing the flexible landing control of UAV, with experimental results closely aligning with the theoretical model. The implementation of optimized stiffness and damping through MRAC has been shown to enhance landing precision by up to 20%, reduce rebound intensity by about 20%, and improve convergence speed by nearly 18%. Additionally, dynamic recalibration of control gains significantly outperforms traditional static tuning approaches, which often suffer from longer settling times and increased overshoot. The use of MRAC allows the UAV to stably and flexibly land on an unmanned ship by continuously adjusting the spring-damping characteristics of the motor-driven landing gear, ensuring robust convergence under varied landing conditions. Furthermore, incorporating game theory into the MRAC framework facilitates convergence to a Nash equilibrium, optimizing landing gear behavior and improving overall system stability by an additional 11%, while mitigating the risk of falling into local optima. Experimental validation also reveals robustness improvements of 11–16% in the presence of sensor errors and random noise. These findings not only highlight MRAC's superiority in dynamic adaptation and real-time control but also establish a strong foundation for future development of adaptive strategies, including reinforcement learning, in advanced UAV landing systems.

Conclusions

Through mathematical modeling and numerical simulation, this study verifies the applicability of MRAC in the flexible landing of UAV and analyzes the effects of different stiffness and damping parameters on landing stability. The experimental results demonstrate that with appropriate stiffness and damping configurations, MRAC effectively adjusts control parameters, stabilizing the UAV's time-domain response during landing, reducing bouncing effects, and enhancing landing precision and safety—with improvements in landing accuracy reaching up to 20% and a decrease in rebound intensity by approximately 20%. The decreasing trend of the loss function confirms the strong convergence characteristics of MRAC, enabling rapid learning and adaptation to various landing conditions, as evidenced by an improvement in convergence speed of nearly 18%. In contrast to traditional tuning methods, which typically rely on static, offline parameter adjustments that can result in longer settling times and increased overshoot, the MRAC approach dynamically recalibrates control gains in real time and thus exhibits superior performance and robustness. By employing game theory to analyze the MRAC parameter adaptation process, the existence of a Nash equilibrium can be achieved under specific conditions, allowing for an optimal configuration of the flexible control strategies for each landing gear, which further enhances UAV landing stability by roughly 11% and prevents the system from falling into local optima. The study also considers the impact of sensor errors and random noise on MRAC control performance; the results confirm that the proposed method successfully converges, demonstrating its robustness and resistance to disturbances in real-world applications with robustness gains of around 11∼16% compared to conventional control methods. This research establishes the feasibility and advantages of MRAC in flexible UAV landing control, providing both theoretical foundations and experimental validation for developing more advanced adaptive control methods. Future research directions could further explore dynamic stiffness and damping adaptation mechanisms, enabling UAV to adjust landing gear flexibility parameters in real time based on actual landing conditions. Additionally, integrating reinforcement learning techniques could enhance the long-term adaptability of MRAC, allowing UAV to handle more complex landing conditions and environmental variations while improving landing stability, energy efficiency, and system intelligence.

Footnotes

ORCID iD

Chun-Yi Lin

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Design of UAV flexible landing control system based on model reference adaptive control

Abstract

Keywords

Introduction

Methods analysis

Experiment analysis

Convergence to ideal landing velocity

Optimization of boundary conditions and learning rates

Iterative learning for rebound suppression and smooth landing

Conclusions

Footnotes

ORCID iD

Funding

Declaration of conflicting interests

References