The autonomous landing of quadrotor UAV drones onto mobile platforms represents a critical phase in mission execution, directly impacting operational efficiency, equipment safety, and the potential for deployment in complex scenarios like logistics and emergency response. Achieving a landing that is simultaneously fast, stable, and precise presents a significant challenge. Traditional methods often struggle to balance the competing demands of rapid trajectory execution—to minimize time-to-land—and maintaining control stability to prevent oscillations or instability during the final approach. This work addresses this dual challenge by proposing an integrated framework combining an improved time-optimal trajectory planner with a robust flight control system, specifically designed for the autonomous landing of UAV drones on differentially-driven mobile platforms.
The core of our approach lies in generating a landing trajectory that is fundamentally time-optimal. We formulate the problem of chasing and aligning with a moving platform as an optimal control problem. For a single axis, the system dynamics can be represented as:
$$ \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u $$
where \(\mathbf{x} = [p, v]^T\) represents the position and velocity, and \(u = a\) is the control input (acceleration). The objective is to minimize the total time \(J = t_f – t_0\) subject to initial and final state constraints (\(\mathbf{x}(t_0)=\mathbf{x}_0, \mathbf{x}(t_f)=\mathbf{x}_f\)) and actuator limits \(|a| \leq a_{max}\). Applying Pontryagin’s Minimum Principle (PMP), we construct the Hamiltonian \(H\):
$$ H = 1 + \boldsymbol{\lambda}^T (\mathbf{A}\mathbf{x} + \mathbf{B}u) = 1 + \lambda_1 v + \lambda_2 u $$
The optimal control law that minimizes \(H\) is a bang-bang controller, determined by the sign of the costate \(\lambda_2\):
$$ u^* = a^* = -a_{max} \cdot \text{sgn}(\lambda_2(t)) $$
Solving the costate dynamics \(\dot{\boldsymbol{\lambda}} = -\partial H / \partial \mathbf{x}\) yields \(\lambda_2(t) = c_{20} – c_{10}t\), where \(c_{10}, c_{20}\) are constants. Consequently, the optimal acceleration switches between \(+a_{max}\) and \(-a_{max}\). The switching logic is governed by a switching function \(h\), derived from the boundary conditions:
$$ h = (p_f – p) + \frac{(v_f – v)|v_f – v|}{2a_{max}} $$
The resulting optimal control law is:
$$ a^* = \begin{cases}
a_{max}, & h > 0 \\
-a_{max}, & h < 0
\end{cases} $$
While this classical time-optimal solution provides the theoretically fastest trajectory, its direct application causes severe jitter in the acceleration profile due to infinitesimally fast switching, leading to unacceptable attitude oscillations in UAV drones. To make the trajectory feasible for real-world UAV drone control, we introduce a comprehensive modification strategy:
- Adaptive Threshold & Smooth Transition: Replace the hard switch at \(h=0\) with a smooth transition region defined by a bandwidth \(\sigma\) and an adaptive threshold \(h_{adapt} = \alpha \cdot h_{base} + \beta \cdot |h| + \gamma \cdot |v_f-v|\). This reduces high-frequency chatter.
- Saturation Compensation: Account for physical actuator limits and dynamics mismatch by adding a PI compensator that acts on the difference between the ideal unsaturated control \(u_{unsat}\) and the saturated control \(u_{sat}\).
The final, improved optimal control law implemented in the planner is:
$$ a_{im}^* = u_{sat} + K_p(u_{unsat} – u_{sat}) + K_i \int (u_{unsat} – u_{sat}) d\tau $$
Where \(u_{unsat}\) is derived from the smooth switching function. This enhanced planner generates a time-optimal reference trajectory \(\mathbf{p}_r(t), \mathbf{v}_r(t), \mathbf{a}_r(t)\) that is both fast and sufficiently smooth for stable tracking by the UAV drone’s flight controller.
The overall autonomous landing system architecture is depicted in the figure above. It integrates the trajectory planner with a hierarchical flight controller and a landing state machine. The planner takes the mobile platform’s state \((\mathbf{p}_c, \mathbf{v}_c)\) and the UAV drone’s current state \((\mathbf{p}, \mathbf{v})\) to generate the time-optimal horizontal reference trajectory. A landing state machine monitors the horizontal tracking error \((e_x, e_y)\). Based on predefined thresholds, it commands the high-level behavior: “Hold” at a safe altitude, “Track” the moving platform, or initiate “Descent” for landing. The vertical reference \(z_r\) is generated according to this state.
To accurately track the demanding reference trajectories, we employ a composite MPC-PID control architecture. This leverages the strengths of both methods: Model Predictive Control (MPC) for precise, constrained tracking of position trajectories, and PID for fast, high-frequency stabilization of the attitude loop.
The outer-loop position controller uses MPC. We define the error state for the altitude subsystem, for example, as \(\mathbf{x}_z = [z_r – z, \dot{z}_r – \dot{z}]^T\). The discretized prediction model is \(\mathbf{x}_z(k+1) = \mathbf{A}_z \mathbf{x}_z(k) + \mathbf{B}_z \Delta u_z(k)\). MPC solves a finite-horizon optimization problem at each time step \(k\):
$$ \min_{\Delta \mathbf{U}_z} \sum_{i=1}^{N_p} ||\mathbf{x}_z(k+i|k)||^2_{\mathbf{Q}_z} + \sum_{i=0}^{N_c-1} ||\Delta u_z(k+i|k)||^2_{\mathbf{R}_z} $$
subject to the model dynamics and constraints. The first optimal control increment \(\Delta u_z^*(k|k)\) is applied, and the process repeats. A similar MPC controller handles the \(x\) and \(y\) position tracking. The MPC outputs are the desired total thrust \(U_1\) and the desired roll \((\phi_d)\) and pitch \((\theta_d)\) angles. The inner-loop attitude controller is a standard PID:
$$ \tau_\phi = K_{p,\phi}e_\phi + K_{i,\phi}\int e_\phi dt + K_{d,\phi}\dot{e}_phi $$
$$ \tau_\theta = K_{p,\theta}e_\theta + K_{i,\theta}\int e_\theta dt + K_{d,\theta}\dot{e}_theta $$
$$ \tau_\psi = K_{p,\psi}e_\psi + K_{i,\psi}\int e_\psi dt + K_{d,\psi}\dot{e}_psi $$
where \(e_\phi = \phi_d – \phi\), etc. These PID controllers calculate the moments \(U_2, U_3, U_4\) to achieve fast attitude stabilization, which is crucial for the aggressive maneuvers involved in time-optimal landing of UAV drones.
We conducted extensive simulations in MATLAB/Simulink to validate the proposed method. The quadrotor UAV drone parameters are listed in Table 1. The controller and improved planner parameters were tuned for performance.
| Parameter | Value | Unit |
|---|---|---|
| Mass (\(m\)) | 1.5 | kg |
| Arm Length (\(l\)) | 0.18 | m |
| Gravity (\(g\)) | 9.8 | m/s² |
| Moment of Inertia \(I_{xx}, I_{yy}\) | 6.85e-3 | kg·m² |
| Moment of Inertia \(I_{zz}\) | 12.90e-3 | kg·m² |
| Max Acceleration (\(a_{max}\)) | 6.0 | m/s² |
| Attitude Limit | ±0.5 | rad |
The first simulation compares the trajectory profiles generated by the classical and our improved time-optimal planner for a platform moving at constant velocity. The results, summarized in Table 2, clearly demonstrate the superiority of the improved method for controlling UAV drones.
| Metric | Classical Bang-Bang | Improved Planner |
|---|---|---|
| Acceleration Peaks | ±5.0 m/s² (instant switching) | ~3.0 m/s² (smooth transition) |
| Jerk (Implicit) | Extremely High (>100 m/s³) | Significantly Reduced |
| Switch Duration | ~0.04 s | ~0.46 s |
| Time to Reach Target (x-axis) | 1.70 s | 1.88 s |
| Trajectory Smoothness | Poor (Jittery) | Excellent (Smooth) |
The improved planner sacrifices a negligible amount of time (0.18s in this case) to gain tremendous smoothness, which is essential for stable flight of UAV drones.
Next, we compared the complete landing performance of three strategies applied to the same UAV drone: 1) Baseline MPC-PID tracking a simple reference, 2) MPC-PID tracking an offline-improved time-optimal trajectory, and 3) MPC-PID tracking an online-improved time-optimal trajectory (replanned at each step). The mobile platform moved with a constant velocity. The key landing performance metrics are shown in Table 3.
| Strategy | Time to Meet Landing Condition (X) | Time to Meet Landing Condition (Y) | Total Landing Time (Z=0) | Max Attitude Angle |
|---|---|---|---|---|
| Baseline MPC-PID | 12.0 s | 10.0 s | 22.3 s | 0.27 rad |
| Offline Time-Optimal + MPC-PID | 7.4 s | 7.9 s | 14.0 s | 0.19 rad |
| Online Time-Optimal + MPC-PID | 7.5 s | 7.4 s | 13.6 s | 0.15 rad |
The results conclusively show that integrating the time-optimal planner drastically reduces the total landing time for the UAV drone—by over 8.5 seconds compared to the baseline. The online planner offers a slight advantage in final touchdown time and, more importantly, produces the smoothest attitude response, indicating better stability. This demonstrates the critical role of trajectory planning in enhancing the performance of UAV drone landing systems.
Finally, we tested the robustness of the online time-optimal MPC-PID strategy under external disturbances. The platform followed a circular path \(\mathbf{p}_c = [2+\cos(t), 3+\sin(t), 0]\). We injected band-limited white noise into the UAV drone’s dynamics to simulate wind gusts and model uncertainties. The controller parameters were not retuned. The UAV drone successfully tracked and landed on the moving platform in 13.8 seconds, compared to 16.5 seconds in the disturbance-free case for the same path. While the landing time increased due to the disturbance, the UAV drone maintained stable flight and accurate tracking throughout the maneuver, confirming the robustness of the proposed control framework for autonomous landing of UAV drones in non-ideal conditions.
In conclusion, this work presents a cohesive solution for the fast and stable autonomous landing of quadrotor UAV drones. The improved time-optimal trajectory planner effectively bridges the gap between theoretical time-optimality and practical implementability by suppressing harmful high-frequency content. Coupled with the high-performance MPC-PID composite controller and a safety-conscious state machine, the system enables UAV drones to execute rapid landing maneuvers on moving platforms while maintaining stability and robustness against disturbances. Future work will focus on extending this framework to handle more complex environments with static and dynamic obstacles, further pushing the capabilities of autonomous UAV drone operations.