In recent years, multirotor unmanned aerial vehicles, commonly referred to as UAV drones, have gained widespread adoption across various fields such as search and rescue, military operations, and scientific exploration due to their high maneuverability and ease of operation. As control technologies for UAV drones advance, research focus has shifted toward improving task complexity, control precision, and robustness. For instance, precise control of rotor thrust is crucial for UAV drones operating in wind-disturbed environments or performing aggressive maneuvers, such as obstacle avoidance and high-speed trajectory tracking. However, traditional powertrain systems in multirotor UAV drones often lack the capability to accurately regulate rotor thrust, leading to performance degradation under dynamic conditions.
Traditional thrust models for UAV drones typically rely on static aerodynamic assumptions, where rotor thrust is proportional to the square of motor speed with a constant coefficient. This simplified model works well in hover or low-speed flights without wind disturbances but fails to account for effects like relative airflow velocity, which becomes significant during high-speed maneuvers or in windy conditions. Existing approaches to address this issue include aerodynamic model-based thrust estimation, machine learning-based model training, and feedback control using direct measurements. Aerodynamic models, derived from momentum theory or blade element momentum theory, often require complex parameter identification and may not adapt well to varying conditions. Learning-based methods, while promising, suffer from high computational costs and poor generalization across different environments. Direct measurement methods, such as using strain gauges or accelerometers, face challenges like noise interference or inability to isolate individual axis thrust, limiting control accuracy.
To overcome these limitations, we propose a novel thrust feedback-based control method for multirotor UAV drones. Our approach removes the traditional motor speed inversion module in the powertrain system and incorporates a thrust feedback control loop, where thrust is measured using sensors and regulated via closed-loop control. This enables precise thrust output, enhancing the overall control performance of UAV drones in challenging scenarios. In this paper, we detail the development of this method, including the modeling of the UAV drone powertrain system, design of an input-output linearized thrust controller, and comprehensive simulation validation under wind disturbances.
The dynamics of a multirotor UAV drone can be described by rigid-body equations separating position and attitude dynamics. For a quadrotor UAV drone, which serves as a representative platform, we define an inertial frame \(\{I\}\) and a body frame \(\{B\}\). The position in the inertial frame is denoted as \(p = [x, y, z]^T \in \mathbb{R}^3\), velocity as \(v = [v_x, v_y, v_z]^T \in \mathbb{R}^3\), and attitude angles as \(\Theta = [\phi, \theta, \psi]^T \in \mathbb{R}^3\), where \(\phi\), \(\theta\), and \(\psi\) are roll, pitch, and yaw angles, respectively. The rotational matrix from body to inertial frame is \(R \in \mathbb{R}^{3 \times 3}\). The dynamics are given by:
$$
\begin{align}
\dot{p} &= v, \\
\dot{v} &= g e_3 – \frac{f}{m} R e_3, \\
\dot{\Theta} &= G \omega, \\
J \dot{\omega} &= -\omega \times J \omega + M,
\end{align}
$$
where \(g\) is gravitational acceleration, \(m\) is the mass of the UAV drone, \(f\) is the total thrust, \(M = [M_x, M_y, M_z]^T \in \mathbb{R}^3\) is the total moment, \(\omega = [\omega_x, \omega_y, \omega_z]^T \in \mathbb{R}^3\) is the body angular velocity, \(J \in \mathbb{R}^{3 \times 3}\) is the inertia matrix, \(e_3 = [0, 0, 1]^T\), and \(G\) relates attitude rate to angular velocity. For a quadrotor UAV drone with four rotors, the thrust and moments are related to individual rotor thrusts \(T_i\) through a mixing matrix:
$$
\begin{bmatrix}
f \\
M_x \\
M_y \\
M_z
\end{bmatrix}
=
\begin{bmatrix}
1 & 1 & 1 & 1 \\
0 & -d & 0 & d \\
d & 0 & -d & 0 \\
-c_\tau & c_\tau & -c_\tau & c_\tau
\end{bmatrix}
\begin{bmatrix}
T_1 \\
T_2 \\
T_3 \\
T_4
\end{bmatrix},
$$
where \(d\) is the distance from the geometric center to each rotor, and \(c_\tau\) is a constant relating thrust to torque. This model forms the basis for high-level control of UAV drones.
Traditional flight control for UAV drones often employs a cascaded PID structure with outer position loop and inner attitude loop. However, the powertrain system typically uses open-loop control, where desired thrust is mapped to motor speed commands via an inverse model, such as \(T = c_T u^2 + (1 – c_T)u\), with \(u\) as the PWM signal. This open-loop approach lacks compensation for aerodynamic disturbances, leading to thrust inaccuracies. Our thrust feedback strategy replaces this with closed-loop control. We remove the speed inversion module and introduce a thrust feedback module that measures actual thrust \(T_{m,i}\) using sensors, filters it (e.g., with a Kalman filter) to obtain \(T_{k,i}\), and uses a thrust controller to compute control input \(u\) based on desired thrust \(T_{d,i}\) and measured thrust. This structure is integrated into the flight control framework, as shown in the block diagram, enhancing robustness for UAV drones.
To design the thrust controller, we first model the single-axis powertrain system of a UAV drone, which includes the battery, electronic speed controller (ESC), brushless DC motor, and rotor. The input is the control signal \(u\), and the output is the thrust \(T\). We derive a comprehensive model incorporating aerodynamic effects. The thrust generated by a rotor is influenced by motor speed \(\varpi\) and airflow velocity. Using blade element momentum theory (BEMT), thrust can be expressed as:
$$
T = C_1 \left\{ C_2 \left[ 2 + \left( \frac{\|V^a_h\|}{\varpi R} \right)^2 \right] – 2 \frac{V^a_v}{\varpi R} \right\},
$$
where \(C_1 = \frac{1}{4} N_b \rho c_{tip} R^3 C_{l\alpha}\), \(C_2 = \theta_{tip}\), \(N_b\) is the number of blades, \(\rho\) is air density, \(c_{tip}\) is tip chord length, \(R\) is rotor radius, \(C_{l\alpha}\) is lift curve slope, \(\theta_{tip}\) is tip pitch angle, \(V^a_h\) and \(V^a_v\) are horizontal and vertical components of airflow velocity relative to the rotor. The airflow velocity \(V^a = V^s + v^i\), where \(V^s = W – V\) is the freestream velocity (with \(W\) as wind velocity and \(V\) as UAV drone velocity), and \(v^i\) is induced velocity. To compute induced velocity, we combine momentum theory and BEMT, solving an optimization problem:
$$
\min_{v^i_v} |f(v^i_v)|, \quad \text{s.t. } v^i_v > 0,
$$
where \(f(v^i_v)\) is a function derived from equating thrust expressions. This yields numerical solutions for induced velocity, which are then used in the thrust model. For the motor dynamics, we model the brushless DC motor with equations for electrical and mechanical parts. The motor torque \(M_e = C_m I_m\), where \(C_m\) is torque constant, and \(I_m\) is armature current. The motor speed dynamics are:
$$
J_m \dot{\omega} = M_e – M_c – f_m \omega,
$$
where \(J_m\) is motor inertia, \(M_c = c_\tau T\) is load torque, and \(f_m\) is viscous friction coefficient. Combining with ESC model and electrical equations, we derive the powertrain system model in state-space form, with state \(x = \omega\) (motor angular velocity) and output \(y = T\). After simplifications and substituting parameters, the model is:
$$
\begin{align}
\dot{x} &= -a_2 x – a_1 x^2 – a_0 + b_0 u, \\
y &= c_2 x^2 + c_1 x + c_0,
\end{align}
$$
where coefficients \(a_2, a_1, a_0, b_0, c_2, c_1, c_0\) depend on system parameters and airflow conditions. This model captures nonlinearities and is used for controller design for UAV drones.
Based on this model, we design an input-output linearization controller to achieve precise thrust control for UAV drones. The relative degree of the system is \(\rho = 1\), as the input \(u\) appears in the first derivative of output \(y\). Differentiating \(y\):
$$
\dot{y} = -2c_2 a_1 x^3 – (2c_2 a_2 + c_1 a_1)x^2 – (2c_2 a_0 + c_1 a_2)x – c_1 a_0 + (2c_2 b_0 x + c_1 b_0)u.
$$
We define a control law:
$$
u = \frac{v – [ -2c_2 a_1 x^3 – (2c_2 a_2 + c_1 a_1)x^2 – (2c_2 a_0 + c_1 a_2)x – c_1 a_0 ]}{2c_2 b_0 x + c_1 b_0},
$$
which linearizes the input-output map to \(\dot{y} = v\), where \(v\) is a virtual control input. To ensure tracking of desired thrust \(y_d\), we set \(v = \dot{y}_d + k_0 e\), with error \(e = y_d – y\) and gain \(k_0 > 0\). This yields error dynamics \(\dot{e} = -k_0 e\), which is asymptotically stable. Thus, the thrust controller enables accurate regulation for UAV drones.
To validate our thrust feedback method for UAV drones, we conduct simulations in MATLAB, comparing traditional powertrain systems (TPS-UAV) with thrust feedback-controlled systems (TFCPS-UAV). We consider wind disturbances with constant horizontal and vertical components of 5 m/s. The UAV drone parameters are based on a quadrotor with MN5008-KV340 motors and P18×6.1 propellers. Key parameters are summarized in tables below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rotor radius \(R\) | 0.2286 m | Air density \(\rho\) | 1.29 kg/m³ |
| Number of blades \(N_b\) | 2 | Tip chord \(c_{tip}\) | 0.0015 m |
| Tip pitch \(\theta_{tip}\) | 0.108 rad | Lift slope \(C_{l\alpha}\) | 5.7 |
| Motor resistance \(R_m\) | 0.055 Ω | ESC resistance \(R_e\) | 0.03 Ω |
| Torque constant \(C_m\) | 0.028 V·s/rad | Motor inertia \(J_m\) | 1.292e-4 kg·m² |
| Battery voltage \(U_e\) | 23.54 V | Thrust-torque coeff. \(c_\tau\) | 0.02156 |
The thrust-torque coefficient \(c_\tau\) is fitted from motor data, yielding a linear relationship \(M_c = 0.02156 T\) with \(R^2 = 0.9999\), confirming model accuracy for UAV drones. We first verify the aerodynamic thrust model by comparing calculated thrust with manufacturer data under no-wind conditions. The error is within ±0.2 N and ±1%, demonstrating high accuracy for UAV drone applications.
We perform trajectory tracking simulations for three maneuvering paths: vertical circle, inclined circle, and lemniscate. The trajectories are defined by parametric equations. For example, the vertical circle is:
$$
\begin{align}
x(t) &= R_1 \cos(\omega_1 t – \pi/2) + x_0, \\
z(t) &= R_1 \sin(\omega_1 t – \pi/2) + R_1,
\end{align}
$$
with \(R_1 = 1\) m, \(\omega_1 = \pi/3\) rad/s, and \(x_0 = 2\) m. The inclined circle and lemniscate have similar parametric forms. The flight controller uses cascaded PID loops for position and attitude, with our thrust feedback controller at the lowest level for UAV drones.
Simulation results show that TFCPS-UAV drones achieve significantly better tracking accuracy than TPS-UAV drones under wind disturbances. For instance, in the lemniscate trajectory, TFCPS-UAV reduces position error peak-to-peak by 84.62% and thrust error peak-to-peak by 75.81%. The tables below summarize error metrics for all trajectories.
| Trajectory | System | Position Error Peak-to-Peak (m) | Position Error Variance |
|---|---|---|---|
| Vertical Circle | TPS-UAV | 0.0681 | 0.0003 |
| Vertical Circle | TFCPS-UAV | 0.0056 | 1.9865e-6 |
| Inclined Circle | TPS-UAV | 0.0675 | 0.0003 |
| Inclined Circle | TFCPS-UAV | 0.0099 | 4.6084e-6 |
| Lemniscate | TPS-UAV | 0.3160 | 0.0044 |
| Lemniscate | TFCPS-UAV | 0.0486 | 0.0001 |
| Trajectory | System | Thrust Error Peak-to-Peak (N) | Thrust Error Variance |
|---|---|---|---|
| Vertical Circle | TPS-UAV | 4.5151 | 1.1839 |
| Vertical Circle | TFCPS-UAV | 1.3381 | 0.0350 |
| Inclined Circle | TPS-UAV | 4.2888 | 1.0032 |
| Inclined Circle | TFCPS-UAV | 0.9866 | 0.0128 |
| Lemniscate | TPS-UAV | 16.0620 | 11.6412 |
| Lemniscate | TFCPS-UAV | 3.8867 | 0.5785 |
The thrust feedback controller for UAV drones ensures that actual thrust closely follows desired thrust, reducing errors caused by wind. We also analyze the impact of lift curve slope \(C_{l\alpha}\) variations on control performance for UAV drones. \(C_{l\alpha}\) is not constant in practice; we test values from 5.6 to 5.96. Results show that position error peaks when \(C_{l\alpha}\) is too low or too high, but TFCPS-UAV drones maintain lower errors than TPS-UAV drones across the range. For example, at \(C_{l\alpha} = 5.7\), TFCPS-UAV achieves position error peak-to-peak of 0.0486 m, whereas TPS-UAV has 0.316 m. This robustness is crucial for real-world UAV drone operations.
Furthermore, we simulate time-varying wind disturbances with \(W_z = 5 \sin(\pi t / 2)\) m/s. TFCPS-UAV drones reduce position error fluctuations by 79.75% compared to TPS-UAV drones, demonstrating effective disturbance rejection for UAV drones. The thrust feedback loop continuously compensates for aerodynamic effects, enabling stable trajectory tracking.
In conclusion, we have presented a thrust feedback-based control method for multirotor UAV drones to address thrust inaccuracies in wind-disturbed or maneuvering flights. By removing the open-loop speed inversion and implementing closed-loop thrust control with an input-output linearized controller, our method significantly enhances control precision and stability for UAV drones. Simulations under various conditions validate the effectiveness, showing reduced position and thrust errors. Future work will involve building a physical UAV drone platform with thrust feedback and conducting real-world experiments to further assess performance. This approach paves the way for more reliable and high-performance UAV drones in dynamic environments.