Anti-windup active disturbance rejection control of tiltable-rotors fully-actuated hexarotor

Abstract

Tiltable-rotors fully-actuated unmanned aerial vehicles achieve full control by dynamically adjusting thrust vector direction through tilting mechanisms. However, physical constraints on actuator amplitudes and rates lead to noticeable actuator saturation effects. Such saturation can cause the thrust vector to deviate from desired values, compromising control system performance and potentially resulting in unmanned aerial vehicle crashes. To address this issue, this paper proposes an anti-windup control method for tiltable-rotors fully-actuated hexarotor unmanned aerial vehicles. This method applies anti-windup principles to control allocation and controllers, obtaining the optimal actuator solution within saturation constraints. It compensates for saturation-induced disturbances using an anti-windup linear active disturbance rejection control. Simulation results demonstrate that this method enables the hexarotor unmanned aerial vehicles to achieve decoupled position and attitude tracking under various saturation constraints, proving the feasibility of resisting actuator amplitude and rate saturation.

Introduction

Conventional multirotor unmanned aerial vehicles (UAVs) exhibit underactuated properties due to the coupling between rotational and translational dynamics, rendering them incapable of tracking arbitrary flight trajectories. To address the issues inherent in conventional underactuated UAVs, researchers have proposed fully actuated UAVs with decoupled position and attitude capabilities. ¹ Fully-actuated UAVs increase control degrees of freedom by altering rotor tilt angles to vectorize thrust, effectively solving the problem of underactuated UAV dynamics. Depending on whether the tilt angles are dynamically adjusted, fully-actuated UAVs can be classified into tiltable^2,3 and fixed-tilt^4,5 types. Among them, tiltable-rotors fully-actuated UAVs (TF-UAVs) achieve greater maneuverability with relatively low mechanical complexity while retaining underactuated capabilities, which has garnered widespread attention. A TF-UAV is a fully-actuated UAV with freely tiltable rotors. Its rotors can be tilted around the arms ⁶ or inward and outward by servos mounted on the arms ⁷ to adjust the thrust direction of each rotor, thereby generating control forces and moments in any direction to achieve full six-degree-of-freedom control.

For conventional underactuated UAVs and fixed-tilt fully-actuated UAVs, the control allocation matrix is a static constant matrix and can be used to calculate the optimal rotor speeds based on the resultant force and moment. In a tiltable-rotors fully-actuated UAV system, however, the control allocation matrix becomes a variable matrix dependent on the tilt angles. This means that control allocation must consider not only the resultant force and moment but also the impact of rotor tilt angles. References Kamel et al. and ² Orozco Soto et al. ⁸ introduce a linearized control allocation scheme that linearizes the nonlinear control allocation problem through variable substitution and then solves it using the pseudo-inverse. However, this method does not consider physical constraints. Reference Jiang et al. ⁹ obtains the minimum control output solution through nonlinear programming. A different reference ¹⁰ designs a model predictive controller to generate optimal rotor speeds and tilt angle expectations. Although these methods satisfy physical constraints, they require significant computation time and resources, which do not meet real-time and efficiency requirements. References^11–13 propose a composite UAV system composed of multiple drones, which expands the range of amplitude constraints by increasing the number of actuators. However, they do not specifically address how to handle amplitude constraints. These studies provide a theoretical foundation for solving the control allocation problem but still fall short in meeting real-time and efficiency requirements.

It is important to note that the aforementioned control methods neglect actuator rate limitations, typically assuming actuators can track given values in real-time. However, in actual UAV control systems, actuator inputs are subject to not only amplitude saturation but also rate saturation. For instance, the servo’s rotation angle is limited, and there is a saturation value for angular velocity, meaning tilt time cannot be ignored. Actuator amplitude and rate saturation are common in practical systems and can negatively impact control performance, potentially jeopardizing system stability. ⁷ Control methods for amplitude and rate saturation are primarily based on the MRS model, addressing these phenomena with amplitude and rate anti-saturation compensators. ¹⁴ Reference ⁸ discusses various designs for anti-saturation compensators and their triggering

The current main approach to counteract amplitude and rate saturation is based on the model of amplitude and rate saturation (MRS). ¹⁵ This involves constructing amplitude anti-windup compensators and rate anti-windup compensators to respectively counteract amplitude and rate saturation. Reference ¹⁶ discusses various structural designs and triggering mechanisms for anti-saturation compensators. References^17–19 improve controllers for amplitude-limited problems using anti-windup methods; however, these methods overlook input rate saturation. Reference ²⁰ counteracts rate saturation with a rate anti-windup compensator and designs a low control gain to avoid amplitude saturation, but experiments have shown that this method does not effectively suppress amplitude saturation. Reference ²¹ addresses both amplitude and rate saturation by adding supplementary actuators in series, but this approach requires additional actuators, making it difficult to apply to other systems.

Based on the above analysis, this article introduces the TF-UAV, a six-rotor fully-actuated unmanned aerial vehicle (UAV) with actuators that can tilt around the arms, as shown in Figure 1. This platform is capable of independent control in all six degrees of freedom and effectively addresses the control challenges posed by actuator amplitude and rate saturation constraints.

Figure 1.

We proposed prototype model.

The specific contributions of this article can be summarized as follows:

An improved control allocation method is designed, which combines force decomposition and quadratic programming to obtain the optimal solution that satisfies amplitude saturation constraints.

The feasibility of the control allocation scheme is ensured by calculating the control reachable set.

An active disturbance rejection controller (ADRC) based on an anti-saturation expanded state observer is developed, enabling real-time estimation and compensation of saturation errors.

The overall control scheme’s effectiveness and superiority are validated through experiments.

System modeling

Assume that the tiltable-rotors fully-actuated hexarotor UAV proposed in this article is regarded as a rigid body moving under forces in a three-dimensional space, with a uniform mass distribution. The center of mass of the UAV is coplanar with the centers of the six rotors. The coordinate systems are set up as shown in Figure 2, where $F_E:(O_E-X_E,Y_E,Z_E)$ is the earth coordinate system, and $F_B:(O_B-X_B,Y_B,Z_B)$ is the body coordinate system with the origin fixed at the center of mass of the UAV. Six rotor coordinate systems $F_{Ri}:(O_{Ri}-X_{Ri},Y_{Ri},Z_{Ri})$ are also defined, with $O_E$ , $O_B$ , and $O_{R_i}$ being the origins of each coordinate system, respectively.

Figure 2.

Tiltable-rotors fully-actuated hexarotor UAV.

Kinematic modeling

Let $X={\left[\begin{array}{cccccc}x& y& z& \varphi & \theta & \psi \end{array}\right]}^{\mathrm{T}}\in \mathrm{R}{}^6$ represent the position vector of the origin $O_E$ of the coordinate system $F_E$ relative to the coordinate system $F_B$ and the attitude vector expressed by Euler angles, where $\varphi$ represents the roll angle about the $X$ -axis, $\theta$ represents the pitch angle about the $Y$ -axis, and $\psi$ represents the yaw angle about the $Z$ -axis. The linear and angular velocities of the origin $O_B$ of the coordinate system $F_B$ relative to the coordinate system $F_E$ are represented as $V={\left[\begin{array}{ccc}u& v& w\end{array}\right]}^{\mathrm{T}}$ and $\mathrm{\Omega }={\left[\begin{array}{ccc}p& q& r\end{array}\right]}^{\mathrm{T}}$ , respectively.

This article uses ${}^{(\cdot )}{R}_{(\cdot )}\in SO(3)$ to denote the rotation matrix between coordinate systems, and $s(\cdot ),c(\cdot ),t(\cdot )$ to denote $sin(\cdot ),cos(\cdot ),tan(\cdot )$ , respectively. The rotation matrix from coordinate system $F_{R_i}$ to $F_B$ is

$${}^b{\mathbf{R}}_i=\left[\begin{array}{ccc}{C}_{{\delta }_i}& -S_{{\alpha }_i}{C}_{{\delta }_i}& S_{{\alpha }_i}{S}_{{\delta }_i}\\ S_{{\delta }_i}& C_{{\alpha }_i}{C}_{{\delta }_i}& -S_{{\alpha }_i}{C}_{{\delta }_i}\\ 0& S_{{\alpha }_i}& C_{{\alpha }_i}\end{array}\right]$$

(1)

where ${\delta }_i=(i-1)*\pi /3$ is the angle between the $i$ -th rotor arm after rotation about the Z-axis in the body coordinate system and the X-axis. ${\alpha }_i$ is the tilt angle of the i-th rotor.

The rotation matrix from coordinate system $F_B$ to $F_E$ is

$${}^w{\mathbf{R}}_b=\left[\begin{array}{ccc}{C}_{\theta }{C}_{\psi }& S_{\varphi }{S}_{\theta }{C}_{\psi }-C_{\varphi }{S}_{\psi }& C_{\varphi }{S}_{\theta }{C}_{\psi }+S_{\varphi }{S}_{\psi }\\ C_{\theta }{S}_{\psi }& S_{\varphi }{S}_{\theta }{S}_{\psi }+C_{\varphi }{C}_{\psi }& C_{\varphi }{S}_{\theta }{S}_{\psi }-S_{\varphi }{C}_{\psi }\\ -S_{\theta }& S_{\varphi }{C}_{\theta }& C_{\varphi }{C}_{\theta }\end{array}\right]$$

(2)

The kinematic model of the tiltable-rotors fully-actuated hexarotor UAV is

$$\begin{array}{r}\begin{array}{c}\{\begin{array}{c}\dot{P}=V\\ {}^e{\dot{R}}_b{=}^e{R}_b{\left[\mathrm{\Omega }\right]}_{\times }\end{array}\end{array}\end{array}$$

(3)

where $P=[\begin{array}{ccc}x& y& z\end{array}{]}^T$ is the position vector, and $[\mathrm{\Omega }{]}_{\times }$ is the skew-symmetric form of $\mathrm{\Omega }$ .

Dynamic modeling

This article primarily studies the control of the tiltable-rotors fully-actuated hexarotor UAV in low-speed flight conditions, thus ignoring air resistance and gyroscopic effects. Assume that the thrust and counter-torque generated by rotor $i$ are proportional to the square of the rotor $i$ ’s speed ${\omega }_i^2$ , with $k_t$ representing the rotor thrust coefficient and $k_q$ representing the rotor counter-torque coefficient. The thrust $f_i$ and counter-torque ${\tau }_i$ generated by rotor $i$ in the rotor coordinate system can be expressed as

$$\{\begin{array}{c}{f}_i=[\begin{array}{ccc}0& 0& k_t{\omega }_i^2\end{array}]\\ {\tau }_i=[\begin{array}{ccc}0& 0& {(-1)}^i{k}_q\end{array}{\omega }_i^2]\end{array}i=1,\dots ,6$$

(4)

The tiltable-rotors fully-actuated hexarotor UAV, as a six-degree-of-freedom rigid body, can derive its system dynamic model from the Newton–Euler equations.

$$\begin{array}{r}\left[\begin{array}{c}m{P}^{..}\\ J{\mathrm{\Omega }}^{.}\end{array}\right]=\left[\begin{array}{c}-mg{e}_3\\ -\mathrm{\Omega }\times J\mathrm{\Omega }\end{array}\right]+\left[\begin{array}{c}{F}_t+F_d\\ M_t+M_{\mathrm{drag}}+M_d\end{array}\right]\end{array}$$

(5)

where $m$ is the mass of the UAV, $J$ is the body inertia matrix, $F_t$ is the total thrust of the rotors in the $F_E$ coordinate system, $M_t$ is the total torque generated by the rotor thrust, $M_{\mathrm{drag}}$ is the counter-torque generated by rotor rotation, $g$ is the gravitational acceleration, taken as $g=9.8(m\cdot s^{-2})$ , and $e^3$ is $[\begin{array}{ccc}0& 0& 1\end{array}{]}^{\mathrm{T}}$ .

$$F_t=\underset{i=1}{\overset{6}{\mathrm{\Sigma }}}{}^w{f}_i=\underset{i=1}{\overset{6}{\mathrm{\Sigma }}}{}^w{R}_b^b{R}_i{f}_i{e}_3$$

(6)

$$M_{\mathrm{t}}=\underset{i=1}{\overset{6}{\mathrm{\Sigma }}}{(}^b{p}_i{\times }^b{R}_i{f}_i{e}_3)$$

(7)

$$M_{\mathrm{drag}}=\underset{i=1}{\overset{6}{\mathrm{\Sigma }}}{(}^b{R}_i{\tau }_i{e}_3)$$

(8)

$${}^b{p}_i=R_z({\delta }_i)[\begin{array}{ccc}l& 0& 0\end{array}{]}^{\mathrm{T}}$$

(9)

where ${}^b{p}_i$ is the position of the $i$ -th rotor coordinate system relative to the body coordinate system, $R_z(\cdot )$ denotes the rotation matrix about the $Z$ -axis, and $l$ is the arm length.

From equations ()to(), the control allocation matrix $A(\alpha )$ for the tiltable-rotors fully-actuated hexarotor can be derived, as seen in equation (13). For a conventional under-actuated hexarotor, the control efficiency matrix is a static constant, but in the tiltable-rotors fully-actuated hexarotor studied in this article, it is a function of the tilt angle $\alpha$ . where $U=[F_x,F_y,F_z,{\tau }_x,{\tau }_y,{\tau }_z{]}^{\mathrm{T}}$ is the control input, i.e., the control forces and torques required by the body outputted by the controller, $\alpha =[{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6{]}^{\mathrm{T}}$ is the tilt angle matrix, and $W=[{\omega }_1^2,{\omega }_2^2,{\omega }_3^2,{\omega }_4^2,{\omega }_5^2,{\omega }_6^2{]}^{\mathrm{T}}$ is the rotor speed squared matrix.

$$\begin{array}{r}U=\left[\begin{array}{c}{F}_t\\ M_t+M_{\mathrm{drag}}\end{array}\right]=A(\alpha )W\end{array}$$

(10)

$$A(\alpha )=[\begin{array}{ccc}-\frac{\sqrt{3}}{2}{k}_t{s}_{{\alpha }_1}& \frac{\sqrt{3}}{2}{k}_t{s}_{{\alpha }_2}& 0\\ \frac{1}{2}{k}_t{s}_{{\alpha }_1}& \frac{1}{2}{k}_t{s}_{{\alpha }_2}& -k_t{s}_{{\alpha }_3}\\ k_t{c}_{{\alpha }_1}& k_t{c}_{{\alpha }_2}& k_t{c}_{{\alpha }_3}\\ \frac{\sqrt{3}}{2}(l{k}_t{c}_{{\alpha }_1}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_1})& \frac{\sqrt{3}}{2}(l{k}_t{c}_{{\alpha }_2}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_2})& 0\\ -\frac{1}{2}(-l{k}_t{c}_{{\alpha }_1}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_1})& \frac{1}{2}(-l{k}_t{c}_{{\alpha }_2}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_2})& (-l{k}_t{c}_{{\alpha }_3}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_3})& \\ (l{k}_t{c}_{{\alpha }_1}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_1})& -(l{k}_t{c}_{{\alpha }_2}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_2})& (l{k}_t{c}_{{\alpha }_3}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_3})\end{array}$$

(11)

$$\begin{array}{ccc}-\frac{\sqrt{3}}{2}{k}_t{s}_{{\alpha }_3}& \frac{\sqrt{3}}{2}{k}_t{s}_{{\alpha }_4}& 0\\ \frac{1}{2}{k}_t{s}_{{\alpha }_4}& \frac{1}{2}{k}_t{s}_{{\alpha }_5}& -k_t{}_{{\alpha }_6}& \\ -\frac{\sqrt{3}}{2}(l{k}_t{c}_{{\alpha }_4}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_4})& -\frac{\sqrt{3}}{2}(l{k}_t{c}_{{\alpha }_5}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_5})& 0\\ \frac{1}{2}(-l{k}_t{c}_{{\alpha }_4}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_4})& -\frac{1}{2}(-l{k}_t{c}_{{\alpha }_5}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_5})& -(-l{k}_t{c}_{{\alpha }_6}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_6})\\ -(l{k}_t{c}_{{\alpha }_4}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_4})& (l{k}_t{c}_{{\alpha }_5}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_5})& -(l{k}_t{c}_{{\alpha }_6}+{\mathrm{k}}_q{\mathrm{s}}_{{\alpha }_6})\end{array}]$$

(11)

$$\begin{array}{rl}A=& [\begin{array}{cccccc}-\frac{\sqrt{3}}{2}{K}_T& 0& \frac{\sqrt{3}}{2}{K}_T& 0& 0& 0\\ \frac{1}{2}{K}_T& 0& \frac{1}{2}{K}_T& 0& -K_T& 0\\ 0& K_T& 0& K_T& 0& K_T\\ \frac{\sqrt{3}}{2}{K}_Q& \frac{\sqrt{3}}{2}l{K}_T& \frac{\sqrt{3}}{2}{K}_Q& \frac{\sqrt{3}}{2}l{K}_T& 0& 0\\ -\frac{1}{2}{K}_Q& \frac{1}{2}l{K}_T& \frac{1}{2}{K}_Q& -\frac{1}{2}l{K}_T& K_Q& -l{K}_T\\ K_Q& l{K}_T& -K_Q& -l{K}_T& K_Q& l{K}_T\end{array}\\ & \begin{array}{cccccc}-\frac{\sqrt{3}}{2}K& 0& \frac{\sqrt{3}}{2}{K}_T& 0& 0& 0\\ \frac{1}{2}{K}_T& 0& \frac{1}{2}{K}_T& 0& -K_T& 0\\ 0& K_T& 0& K_T& 0& K_T\\ -\frac{\sqrt{3}}{2}{K}_Q& -\frac{\sqrt{3}}{2}l{K}_T& -\frac{\sqrt{3}}{2}{K}_Q& -\frac{\sqrt{3}}{2}l{K}_T& 0& 0\\ \frac{1}{2}{K}_Q& -\frac{1}{2}l{K}_T& -\frac{1}{2}{K}_Q& \frac{1}{2}l{K}_T& -K_Q& l{K}_T\\ -K_Q& -l{K}_T& K_Q& l{K}_T& -K_Q& -l{K}_T\end{array}]\end{array}$$

(12)

Actuator modeling

The actuators of the tiltable-rotors fully-actuated hexarotor UAV mainly include six tilting rotors, each comprising a rotor and a tilt mechanism. Figure 3 shows a schematic of the tiltable rotor.

Figure 3.

Rotor with servo mechanism.

Since the rotor speed is sufficiently fast, the response time of the rotor speed can be ignored. The speed of the i-th rotor can be expressed as

$${\displaystyle {\omega }_i(t)={\omega }_i^d(t)\le {\omega }_{max}}$$

(13)

where ${\omega }_i^d$ is the control allocation output value, and ${\omega }_{max}$ is the maximum rotor speed.

The tilting mechanism mainly consists of electric servos, with the tilt angular velocity being significantly lower than the rotor speed, and the tilt response process cannot be ignored. Considering the i-th tilt mechanism, with the positive direction of tilt defined as tilting to the right around the arm, the tilt angle can be expressed as

$$\{\begin{array}{c}{\alpha }_i(t)={\alpha }_i(t-1)+{\dot{\alpha }}_{max}\cdot \mathrm{\Delta }t\cdot sign({\delta }_{{\alpha }_i})\\ {\delta }_{{\alpha }_i}=\{\begin{array}{cc}0& \left|\mathrm{\Delta }{\alpha }_i\right|<\epsilon \\ \mathrm{\Delta }{\alpha }_i& \left|\mathrm{\Delta }{\alpha }_i\right|\ge \epsilon \end{array}\\ \mathrm{\Delta }{\alpha }_i={\alpha }_i^d-{\alpha }_i(t-1)\end{array}$$

(14)

where ${\alpha }_i$ is the tilt angle, $\mathrm{\Delta }t$ is the sampling time, ${\dot{\alpha }}_{max}$ is the maximum angular velocity of the servo, $sign(\cdot )$ is the sign function, $\epsilon$ is the dead zone value of the servo, and ${\alpha }_i^d$ is the control allocation output value.

Control system design

This article addresses the actuator amplitude and rate saturation issues from two perspectives. On one hand, by improving the control allocation scheme, it overcomes the shortcomings of the traditional Moore–Penrose pseudo-inverse method, aiming to ensure that the control allocation solutions are within the feasible domain. On the other hand, for rotor speed output saturation and tilt angle rate saturation problems that cannot be resolved through control allocation, an anti-saturation controller is constructed. This controller treats the errors caused by saturation as disturbances, estimates the corresponding disturbances using an anti-saturation disturbance observer, and compensates for the controller output. This approach aims to keep the control input as much as possible within the non-saturation region.

The control system block diagram is shown in Figure 4. It mainly includes the linear state error feedback control law (LSEFC), saturation limiter, control allocation (A), the controlled object, and the anti-windup extended state observer (ALESO). The controller output, after saturation limitation, is allocated to the actuators of the controlled object by the control allocation module. The ALESO observes the total disturbance and compensates through the control law.

Figure 4.

Control system block diagram.

Control allocation method design

Traditional underactuated UAVs solve the allocation problem by calculating the pseudoinverse of the control allocation matrix. This method is also applicable to fixed-tilt fully actuated UAVs. However, in the control allocation of tilt-rotor fully actuated UAVs, the control allocation matrix incorporates the tilt angle, making it impossible to directly obtain the pseudoinverse. Therefore, this section first proposes a method to linearize the control allocation matrix. By decomposing the thrust into thrust vector components and transforming the coordinates, the control allocation matrix becomes a constant matrix. In this case, the control allocation problem is converted into a linear problem, which can be solved using the pseudoinverse method to obtain the minimum norm solution.

Tilt-rotor fully actuated hexarotors experience actuator amplitude saturation. If the minimum norm solution does not meet the actuator saturation constraints, the pseudoinverse method cannot provide an effective control allocation feasible solution. To address this issue, this article proposes an anti-saturation control allocation method. By introducing quadratic programming to improve upon the pseudoinverse method, this approach can obtain an optimal solution that satisfies the actuator amplitude saturation constraints.

First, the equation (14) is linearized by factoring out the sine $\sin ({\alpha }_i)$ and cosine $\cos ({\alpha }_i)$ terms from equation (11), and combining them with the rotor speed squared matrix elements to $\mathrm{\Gamma }$ form $W$ . Here, $\mathrm{\Gamma }$ consists of the sine component ${\mathrm{\Gamma }}_{si}$ and cosine component ${\mathrm{\Gamma }}_{\mathrm{c}i}$ with respect to the tilt angle $\alpha$ .

$$\begin{array}{r}\{\begin{array}{c}{\mathrm{\Gamma }}_{si}=\sin ({\alpha }_i){\omega }_i^2\\ {\mathrm{\Gamma }}_{ci}=\cos ({\alpha }_i){\omega }_i^2\end{array}\end{array}$$

(15)

By factoring out the trigonometric variables in equation (14) and combining them with the matrix $W$ from the right-hand side of equation (11), equation (11) can be rewritten as equation (20). In this form, $A(\alpha )$ (equation (14)) becomes a constant control allocation matrix $A$ (equation (15)) that no longer contains the variable $\alpha$ , thereby linearizing the control allocation matrix and making it possible to compute its pseudoinverse.

$$U=A(\alpha )W\Rightarrow U=A\mathrm{\Gamma }$$

(16)

Therefore, using the Moore–Penrose pseudoinverse method, ²² we can obtain the minimum norm solution $\mathrm{\Gamma }*$ of equation (20) and solve for the rotor speed ${\omega }_i$ and tilt angle ${\alpha }_i$ using equation (21).

$$\begin{array}{r}\{\begin{array}{c}{\omega }_i=\sqrt{\mathrm{\Gamma }{{}_{2i-1}*}^2+\mathrm{\Gamma }{{}_{2i}*}^2}\\ {\alpha }_i=\mathrm{arctg}(\frac{{\mathrm{\Gamma }}_{2i-1}*}{{\mathrm{\Gamma }}_{2i}*})\end{array}i\in (1,2,\dots ,6)\end{array}$$

(17)

When the obtained ${\omega }_i$ and ${\alpha }_i$ satisfy the actuator amplitude constraints, it is the optimal control allocation. Otherwise, let $v_1,v_2,\dots ,v_n$ be a set of orthogonal bases of the null space of $A$ , the solution of the control allocation can be expressed as

$$\mathrm{\Gamma }=\mathrm{\Gamma }*+k_1{v}_1+k_2{v}_2+\cdots +k_n{v}_n$$

(18)

where $k_1,k_2,\dots ,k_n$ are arbitrary constants.

By performing quadratic convex optimization under inequality constraints, we can obtain the unique global optimal solution. ¹¹

$$\{\begin{array}{c}\begin{array}{c}\underset{k_1,k_2,\dots ,k_n}{min}\Vert \mathrm{\Gamma }\Vert \end{array}\\ \begin{array}{c}s.t.{\mathrm{\Gamma }}_{2i-1}^2+{\mathrm{\Gamma }}_{2i}^2\le {\omega }_{max}^4\end{array}\end{array}$$

(19)

Let the optimal solution be $k_1*,k_2*,\dots ,k_n*$ , The equation (24) can be solved using Sequential Quadratic Programming (SQP), ²³ then the optimal control allocation that satisfies the amplitude constraints is

$$\begin{array}{r}\begin{array}{c}\tilde{\mathrm{\Gamma }}=\mathrm{\Gamma }*+k_1*v_1+k_2*v_2+\cdots +k_n*v_n\end{array}\end{array}$$

(20)

$$\begin{array}{r}\begin{array}{c}\{\begin{array}{c}{\omega }_i=\sqrt{{\tilde{\mathrm{\Gamma }}}_{2i-1}^2+{\tilde{\mathrm{\Gamma }}}_{2i}^2}\\ \begin{array}{cc}{\alpha }_i=\mathrm{arctg}(\frac{{\tilde{\mathrm{\Gamma }}}_{2i-1}}{{\tilde{\mathrm{\Gamma }}}_{2i}})& i\in (1,2,\end{array}\cdots ,6)\end{array}\end{array}\end{array}$$

(21)

Using the above method, the optimal solution that satisfies the saturation constraints for a tilt-rotor fully actuated hexarotor can be obtained. However, it should be noted that the quadratic programming problem only has a solution for a certain range of control inputs. Section 3.2 will provide the specific range of control inputs and the proof of the existence of feasible solutions for control allocation.

Control reachable set

The actuator settings are obtained through control input via control allocation. When there are amplitude saturation constraints on the actuators, the control inputs will also have corresponding amplitude saturation constraints. For the quadratic programming problem in Section 3.1, overly large control inputs will make the problem infeasible. Therefore, this section calculates the control reachable set for the tilt-rotor fully actuated hexarotor to obtain the reference range of constrained control inputs that ensure the quadratic programming problem has feasible solutions.

Considering the amplitude saturation constraints on the control inputs, equation (19) becomes:

$$\overline{U}=A(\alpha )W=A\mathrm{\Gamma }$$

(22)

where $\overline{U}\in \mathrm{\Phi }$ represents the actual control input.

Through vector transformation, equation (27) is converted into the thrust vector form as follows:

$$\begin{array}{r}\overline{U}=A(\alpha )W=A\left[\begin{array}{c}\sin {\alpha }_1{w_1}^2\\ \cos {\alpha }_1{w_1}^2\\ \cdots \\ \sin {\alpha }_6{w_6}^2\\ \cos {\alpha }_6{w_6}^2\end{array}\right]=A\mathrm{\Gamma }\end{array}$$

(23)

where $A$ is a constant matrix that does not contain the variable $\alpha$ , and $\mathrm{\Gamma }$ is the rotor thrust vector matrix. Let:

$$\begin{array}{r}{\mathrm{\Gamma }}^{\prime }=\left[\begin{array}{c}\sin {\alpha }_1{\varpi }_1\\ \cos {\alpha }_1{\varpi }_2\\ \cdots \\ \sin {\alpha }_6{\varpi }_6\\ \cos {\alpha }_6{\varpi }_6\end{array}\right]\end{array}$$

(24)

where ${\varpi }_1\cdots {\varpi }_6$ are scalars within the range $[0,{\omega }_{max}^2]$ .

Using the UAV parameters shown in Table 1, the Monte Carlo method ²⁴ can be employed to obtain the reachable set of the thrust matrix ${\mathrm{\Gamma }}^{\prime }$ . After projection through $A$ , the control reachable set $\mathrm{\Phi }$ is obtained.

Table 1.

Parameters of the UAV.

Parameters	Definition	Numerical (Range)
$m$	Mass	$2.86kg$
$g$	Gravitational acceleration	$9.8m/s^2$
${\omega }_{max}$	Maximum rotor speed	$865rad/s$
${\alpha }_{max}$	Maximum tilt angle	$\pm \pi /2$
$J_{xx}$	Roll inertia	$4.11\times 10^{-2}kg\cdot m^2$
$J_{yy}$	Pitch inertia	$4.78\times 10^{-2}kg\cdot m^2$
$J_{zz}$	Yaw inertia	$5.99\times 10^{-2}kg\cdot m^2$
$K_T$	Thrust coefficient	$1.201\times 10^{-5}$
$K_Q$	Anti-torque coefficient	$1.57\times 10^{-7}$
$d$	Length of arm	$0.275m$
$\dot{\alpha }$	Tilting angular velocity	$[-\pi /2,\pi /2]$

Define the combined forces and torques of the six rotors in the UAV body coordinate system as follows:

$$\begin{array}{r}\{\begin{array}{c}{t}^B=\left[\begin{array}{ccc}{T}_x& T_y& T_z\end{array}\right]\\ {\tau }^B=\left[\begin{array}{ccc}{\tau }_x& {\tau }_y& {\tau }_z\end{array}\right]\end{array}\end{array}$$

(25)

The reachable force set can be obtained from the constraint in equation (32):

$$\begin{array}{r}\{\begin{array}{c}{\mathrm{\Lambda }}_T=\left\{T\in {}^3|T=T_x{e}_1+T_y{e}_2+T_z{e}_3\right\}\\ s.t.{\tau }_x={\varsigma }_1,{\tau }_y={\varsigma }_2,{\tau }_z={\varsigma }_3\end{array}\end{array}$$

(26)

The reachable torque set can be obtained from the constraint in equation (33):

$$\begin{array}{r}\{\begin{array}{c}{\mathrm{\Lambda }}_{\tau }=\left\{\tau \in {}^3|\tau ={\tau }_x{e}_1+{\tau }_y{e}_2+{\tau }_z{e}_3\right\}\\ s.t.T_x={\xi }_1,T_y={\xi }_2,T_z=-mg+{\xi }_3\end{array}\end{array}$$

(27)

where $e_1,e_2,e_3$ are the unit vectors along the X, Y, and Z axes in the body coordinate system, and $\varsigma$ and $\xi$ are bounded constants.

Let $T_{max}$ and $\tau max$ be the maximum forces and torques, respectively, that the UAV can exert in any direction within the body coordinate system under constrained conditions. Then, we have:

$$T_{max}<\mathrm{\partial }({\mathrm{\Lambda }}_T),{\tau }_{max}<\mathrm{\partial }({\mathrm{\Lambda }}_{\tau })$$

(28)

where $\mathrm{\partial }(\cdot )$ represents the boundary of the reachable set.

Based on the above analysis, the reachable force set under attitude constraints and the reachable torque set under position constraints are shown in Figure 5.

Figure 5.

The reachable force set (left) and the reachable torque set (right).

The design of the control allocation is based on the analysis results of the reachable torque set and the reachable force set to ensure that the control allocation process has feasible solutions. The existence proof of feasible solutions for control allocation is provided below.

Lemma 1

Kamel et al. ² For any $U\in R^{6\times 1}$ , there exists a non-unique set of solutions $[\omega ,\alpha ]$ such that $U=A(\alpha ){\omega }^2$ .

Theorem 1

For any $U\in \mathrm{\Phi }$ , there exists a set of feasible solutions satisfying the constraint conditions, ensuring that equation (11) holds.

Proof 1

Based on Lemma 1, for any $U\in R^{6\times 1}$ , there exists a non-unique set of feasible solutions $[\omega *,\alpha *]$ $(|\omega *|\le {\omega }_{max}*,|\alpha *|\le {\alpha }_{max}*)$ such that $U*=A(\alpha *){\omega }^{{*}^2}$ .

Assume there exists a control input $U\in \mathrm{\Phi }$ , for which all solutions $[\omega *,\alpha *]$ that satisfy equation (11) do not meet the constraint conditions.

According to the definition of the reachable set, the reachable set $\mathrm{\Phi }$ can be obtained from the inverse control allocation of the constrained $[\omega ,\alpha ]$ .

Therefore, for any set $U*\in \mathrm{\Phi }$ , there exists at least one set $[\omega *,\alpha *]$ $(|\omega *|\le {\omega }_{max}*,|\alpha *|\le {\alpha }_{max}*)$ such that equation (11) holds. Hence, the assumption does not hold. This concludes the proof of Theorem 1.

Based on Lemma 1 and Theorem 1, the fully-actuated tiltable hexarotor UAV platform exhibits redundancy, allowing for multiple control allocation solutions. For any desired control input $U\in \mathrm{\Phi }$ , when the minimum norm solution does not satisfy the constraints, it is always possible to find a feasible alternative solution that meets the actuator’s physical constraints, ensuring that equation (11) holds.

Define the saturation function $sa{t}_a()$ as follows:

$$sa{t}_a(b)=\{\begin{array}{c}a\cdot sign(b)\begin{array}{cc}& if\left|b\right|>a;\end{array}\\ b\begin{array}{c}if\left|b\right|\le a;\end{array}\end{array}$$

(29)

where $sign()$ is the sign function.

For the ideal control input $U\in \mathrm{\Phi }$ , control allocation can be performed directly. When $U\notin \mathrm{\Phi }$ , the actual control input is obtained by limiting it using equation (37), and then performing control allocation. In this case, a feasible optimal actuator solution can be obtained.

$$\overline{U}=sa{t}_{{}_{\mathrm{\Phi }}}(U)$$

(30)

Anti-windup ADRC design

Consider a tilt-rotor full-actuated UAV system with amplitude and rate constraints as follows:

$$\begin{array}{r}\{\begin{array}{c}{\dot{X}}_1=X_2\\ {\dot{X}}_2=f+b_{0}U\\ Y=X_1\end{array}\end{array}$$

(31)

$X_1$ is the system state variable, $Y$ is the output, $f$ is the total system disturbance, and $U$ is the control input.

Anti-windup LESO design

The control input of the tilt-rotor full-actuated hexarotor UAV is restricted by the reachable set, so there might be a discrepancy between the actual control input and the desired control input. The proper functioning of the LESO and the accuracy of the control input are closely related. The actual control input $U_{sat}$ is difficult to obtain under input amplitude and rate saturation. Therefore, this article employs the MRS model from the literature ²⁵ to characterize $U_{sat}$ .

The MRS model of the tilt angle $\alpha$ is shown in Figure 6:

Figure 6.

MRS model of $\alpha$ .

The relationship between the rotor tilt angle and the control input can be expressed as follows:

$$\begin{array}{r}\{\begin{array}{c}\overline{U}=\mathrm{sa}{\mathrm{t}}_{\mathrm{\Phi }}(U)\\ \alpha =A(\overline{U}){|}_{\alpha }\\ \eta =\dot{\alpha }+K(\alpha -\sigma )+k_r(\eta -\dot{\sigma })\\ \dot{\sigma }=\mathrm{sa}{\mathrm{t}}_r(\eta )\\ U_{\mathrm{sat}}=A(\sigma ){\omega }^2\end{array}\begin{array}{c}\end{array}\end{array}$$

(32)

where $\overline{U}$ is the constrained control input of the tilt-rotor full-actuated hexarotor UAV system, $r={\dot{\alpha }}_{max}$ is the rate saturation limit, $K$ and $k_r$ are adjustable parameters.

The compensation for total disturbance by ADRC is based on the premise that the total disturbance is bounded. Therefore, the system can only be stabilized if the internal disturbance caused by the constrained control input is bounded. The following is the proof process for the boundedness of the disturbance caused by the constrained control input.

Lemma 2

Galeani et al. ²⁵ The introduction of the MRS model does not affect the amplitude and rate limits, and the output of the MRS model will not exceed these limits.

Theorem 2

Let the disturbance caused by input constraints be $\lambda (\mathrm{\Delta }U)$ , where $\lambda (\cdot )$ is monotonically increasing. Then $\lambda (\mathrm{\Delta }U)$ is a bounded disturbance.

Proof 2

When $U\in \mathrm{\Phi },|\eta |\le {\dot{\alpha }}_{max}$ , the MRS model is not triggered, and at this time, $U_{sat}=U,\mathrm{\Delta }U=0$ . Therefore, $\lambda (\mathrm{\Delta }U)=0$ .

When $U\in \mathrm{\Phi },|\eta |\le {\dot{\alpha }}_{max}$ , the MRS model starts to work. It can be easily obtained that:

$$\begin{array}{r}\{\begin{array}{c}sa{t}_m(\overline{U})=\overline{U}\\ sa{t}_r(\dot{\sigma })=\dot{\sigma }\end{array}\end{array}$$

(33)

From the control allocation in section 3.1, we have ${lim\dot{\sigma }}_{\alpha \to {\alpha }_{\max }}=0$ .Therefore:

$$\begin{array}{r}\{\begin{array}{c}\sigma (t)=\sigma (t-1)+\int \dot{\sigma }dt\le {\alpha }_{max}\\ A^{-1}(\sigma ,\omega )=U_{sat}\in \mathrm{\Phi }\\ \mathrm{\Delta }{U}_{max}={(U-U_{sat})}_{max}\le \mathrm{\partial }(\mathrm{\Phi })\end{array}\end{array}$$

(34)

From equation (41), we get $\lambda (\mathrm{\Delta }U)<\mathrm{\Delta }{U}_{max}\le \mathrm{\partial }(\mathrm{\Phi })$ , proving that $\mathrm{\Delta }U$ is a bounded disturbance.

To counteract amplitude and rate saturation, the output of the amplitude anti-windup loop is introduced into the state observer. The resulting observer is named the Anti-windup Linear Extended State Observer (ALESO), represented as:

$$\begin{array}{r}\{\begin{array}{c}{U}_{sat}=A^{-1}(\sigma ,\omega )\\ e=y-z_1\\ {\dot{z}}_1=z_2+3{\omega }_{0}e\\ {\dot{z}}_2=z_3+U+(b_0-1)U_{sat}+3{\omega }_0^{2}e\\ {\dot{z}}_3={\omega }_0^{3}e\end{array}\end{array}$$

(35)

where $U_{sat}$ is the actual control input, $z_1,z_2,z_3$ are the state estimates output by the observer, $y$ is the system output of this channel, and ${\omega }_0$ is the observer bandwidth.

Control law design

Based on the ADRC framework, the difference between the constrained controller and the nominal controller is treated as a disturbance. The goal of robust control is achieved through the observation and compensation of the total disturbance. Considering the second-order system described by equation (38), the total disturbance is expanded as the system state $X_3$ and assumed to be $X_3=h(X)$ . Equation (38) can be represented as the following integral cascade system:

$$\begin{array}{r}\{\begin{array}{c}{\dot{X}}_1=X_2\\ {\dot{X}}_2=X_3+b_{0}U\\ {\dot{X}}_3=g(X)\\ Y=X_1\end{array}\end{array}$$

(36)

where $Y$ is the system output and $g$ is the derivative of $h$ .

Based on the estimation of the system state and extended state by the ALESO in section 3.3.1, a linear state error feedback control law can be designed. For the second-order tilt-rotor fully actuated hexarotor system and the full-order design under ALESO, the applicable control law and ideal control input are:

$$\begin{array}{r}\{\begin{array}{c}{u}_0=k_p(r-{\mathrm{z}}_1)-k_d{z}_2\\ U=(u_0-z_3)/b_0\end{array}\end{array}$$

(37)

where $r$ is the setpoint, $k_p={\omega }_c^2$ , $k_d=2{\omega }_c$ ,and ${\omega }_c$ is the controller bandwidth.

Stability proof

Using the Lyapunov stability criterion, the closed-loop system is proven to be globally asymptotically stable under the given control law and parameters. Substituting equation (44) into equation (43) yields:

$$\begin{array}{r}\{\begin{array}{c}{\dot{X}}_1=X_2\\ {\dot{X}}_2=f+k_p(r-z_1)-k_d{z}_2\\ Y=\chi \end{array}-z_3\end{array}$$

(38)

Lemma 3

Zhang et al. ²⁶ All poles of the extended state observer are placed at , and the observer error converges to zero over time.

$$\begin{array}{r}\{\begin{array}{c}\underset{t\to \mathrm{\infty }}{lim}{z}_1=X_1\\ \underset{t\to \mathrm{\infty }}{lim}{z}_2=X_2\\ \underset{t\to \mathrm{\infty }}{lim}{z}_3=f\end{array}\end{array}$$

(39)

Substituting equation (46) into equation (45) yields:

$$\begin{array}{r}\{\begin{array}{c}{\dot{X}}_1=X_2\\ {\dot{X}}_2=-k_p{z}_1-k_d{z}_2\\ Y=X\end{array}\end{array}$$

(40)

Define the Lyapunov function as:

$$V_X=\frac{{\omega }_c^2+5}{4{\omega }_c}{X}_1^2+\frac{1}{{\omega }_c^2}{X}_1{X}_2+\frac{{\omega }_c^2+1}{4{\omega }_c^3}{X}_2^2$$

(41)

It is evident that when $X\ne 0$ , $V_X>0$ . Taking the derivative of $V_X$ and substituting equation (47) yields:

$${\dot{V}}_X=-(X_1^2+X_2^2)$$

(42)

From equation (49), it follows that ${\dot{V}}_X\le 0$ , and ${\dot{V}}_X=0$ if and only if $X_1=0$ and $X_2=0$ . By the Lyapunov criterion, the system is globally asymptotically stable.

Simulation experiments

Numerical simulation experiments

To validate the effectiveness of the control scheme proposed in this article and to fine-tune the controller as well as related parameters, comprehensive numerical simulations were conducted using MATLAB/Simulink. These simulations aimed to evaluate the proposed method by comparing it against two widely used approaches: the control allocation method introduced in reference ² and the conventional ADRC.

The simulation scenario was carefully designed to replicate a realistic flight process. Specifically, the hexarotor UAV was tasked with taking off from the ground and completing a rectangular trajectory flight at an altitude of 6 m while maintaining zero attitude angles throughout the flight. The rectangular trajectory had a length of 4 m, representing a practical test case for trajectory tracking under constrained conditions. During the simulation, the system was subjected to both actuator amplitude saturation and rate saturation constraints, which closely mimic real-world limitations of UAV actuators.

The saturation constraint parameters used in the simulations are summarized in Table 2. In this context, the amplitude constraint defines the maximum allowable rotor speed, while the rate constraint specifies the maximum angular velocity permitted for the tilting mechanism of the hexarotor. To thoroughly assess the robustness of the proposed method under different operating conditions, three distinct saturation constraint settings were applied.

Saturation Constraint 1 served as the baseline configuration, representing typical actuator limitations.

The feasibility of the control allocation scheme is ensured by calculating the control reachable set.

Saturation Constraint 2: Compared to Saturation Constraint 1, the amplitude constraint is enhanced by reducing the maximum thrust output of individual rotors.

Compared to Saturation Constraint 1, the rate constraint is tightened, meaning the tilting mechanism requires more time to achieve the desired angles.

Table 2.

Parameters of saturation constraint.

	Amplitude constraint	Rate constraint
Constraint 1	826 $rad/s$	$60\circ /0.8s$
Constraint 2	750 $rad/s$	$60\circ /0.8s$
Constraint 3	826 $rad/s$	$60\circ /1.2s$

These varied constraints were introduced to evaluate the proposed method’s adaptability and robustness when operating under different levels of actuator saturation.

With both control schemes were fully tuned, the position tracking curves for the X, Y, and Z axes under different actuator saturation constraints are shown in Figures 7 to 9, respectively. From these figures, it can be observed that both the conventional ADRC method and the proposed method can achieve stable position tracking for the tilt-rotor fully-actuated hexarotor UAV under certain amplitude and rate constraints.

Figure 7.

X-axis comparison curve of conventional ADRC (left) and our method (right).

Figure 8.

Y-axis comparison curve of conventional ADRC (left) and our method (right).

Figure 9.

Z-axis comparison curve of conventional ADRC (left) and our method (right).

However, when the amplitude constraint is tightened, the conventional ADRC method causes position tracking to diverge, while the proposed anti-windup ADRC method maintains stable trajectory tracking performance. Similarly, when the rate constraint is enhanced, system performance is significantly affected. Nevertheless, the proposed method still ensures stable trajectory tracking, demonstrating its robustness under stricter actuator constraints.

Figure 10 illustrates the comparison of the rotor speed setpoint curves generated by the pseudo-inverse method and the proposed anti-windup control allocation method during the aforementioned flight process. As shown in the figure, the Moore–Penrose pseudo-inverse method fails to effectively address actuator saturation issues. In contrast, the proposed anti-windup control allocation method ensures that the motor speeds remain within the constraint range, demonstrating its capability to effectively prevent actuator saturation.

Figure 10.

Speed curve comparison of Moore–Penrose pseudo-inverse method (left) and our approach (right).

Figure 11 presents the position and attitude tracking curves obtained by the conventional active disturbance rejection control (ADRC) and the proposed anti-windup ADRC under the influence of the proposed anti-windup control allocation method. From the curves, it can be observed that the proposed anti-windup ADRC enables the tilt-rotor fully actuated hexarotor UAV to smoothly and quickly reach the designated waypoints. Compared with the conventional ADRC method, the proposed approach achieves smaller overshoot, shorter settling time, and a reduced impact of position changes on attitude angles.

Figure 11.

Position and attitude curves of conventional ADRC (left) and our approach (right).

Semi-physical simulation experiment

Semi-physical simulation is a real-time simulation method that uses a simulation model to replace part of the physical hardware. In this study, Gazebo is used as the physical environment, and a 3D model generated by SolidWorks is used to replace the real physical model for visualization.

The tilt-rotor fully actuated hexarotor is set to a height of 6m. Upon reaching the specified height, it begins a zero-attitude circular trajectory motion with an amplitude of 1m. The flight position trajectory during this process is shown in Figure 12.

Figure 12.

Flight trajectory.

During this process, the position and attitude tracking curves of the tilt-rotor fully actuated hexarotor are shown in Figures 13 and 14, respectively. From these position and attitude curves, it can be seen that the proposed method allows the hexarotor to smoothly and rapidly track the specified trajectory with minimal attitude changes, achieving decoupled position and attitude tracking control.

Figure 13.

Position curves.

Figure 14.

Attitude curves.

Figures 15 and 16 show the set values of rotor tilt angles and rotor speeds during the flight process, respectively. The desired signals of the actuators do not exceed the amplitude and rate limits, indicating that the proposed method can effectively avoid actuator saturation.

Figure 15.

Curves of $\alpha$ .

Figure 16.

Curves of $\omega$ .

Conclusions

This article applies the anti-windup concept to the design of control allocation and controllers to address the actuator saturation problem in a tilt-rotor fully actuated hexarotor. The anti-windup control allocation module is used to obtain the desired actuator values that satisfy the saturation constraints. An anti-windup ADRC is constructed, treating the error caused by saturation as part of the total disturbance, which is estimated by the anti-windup extended state observer and used to compensate for the controller output, thereby keeping the control input within the non-saturation range as much as possible. Experimental results show that this method can effectively overcome the problems caused by actuator saturation, allowing the hexarotor to achieve normal position and attitude decoupled tracking control. Future work will further optimize the mathematical model and consider actuator failures.