Finite-time trajectory tracking control of quadrotor UAV via adaptive RBF neural network with lumped uncertainties

1.   Introduction

Nowadays, multi-rotor unmanned aerial vehicle (UAVs) plays an important role in many commercial applications, such as air pollution monitoring, rescue missions, precision agriculture, retail delivery, as well as academic research and military action ^[1]. The quadrotor is one of the most widely used classes of UAVs, which has the advantages of hovering, vertical take-off and landing, simple structure, and low cost. The quadrotor has four actuators and six degrees of freedom, and its dynamic model has the characteristics of strong coupling, under-actuated and susceptible to external disturbances ^[2]. Therefore, the design of high-performance flight controllers for quadrotors is a challenging task, which has received extensive attention in the academic circle.

In recent years, significant research has been done on the trajectory-tracking control of quadrotors. The conventional approach to deal with the nonlinearity is simplifying the model to linear equivalents by using dynamic inversion, feedback linearization, gain-scheduling, or Taylor's approximation, etc. However, these control methods are susceptible to model uncertainties and time-varying interference ^[3]. In order to achieve better robustness and anti-interference performance, many advanced control algorithms have been used to design flight controllers of quadrotors, for instance, neural network-based optimal mixed $\rm H_2/H_\infty$ control ^[4], terminal sliding mode control ^[5], backstepping control ^[6], adaptive fuzzy quantized control ^[7] and active disturbance rejection control ^[8].

Although these advanced control strategies improved the flight performance of the quadrotor, few results are concerned with the convergence rate of the system output. In practice, the real-time performance of the control system is vital to the flight stability of the quadrotor, and the convergence speed should be one of the key indexes for the flight control system ^[9]. The control methods concerned with the convergence rate of system output include Hybrid finite-time control ^[10], finite-time adaptive sliding mode tracking control ^[11], finite-time Lyapunov theory ^[12], global fast terminal sliding mode (GFTSM) control ^[13,14], etc. The GFTSM control method can make the system state variables converge to the equilibrium point in a limited time. However, traditional GFTSM control still has some limitations and challenges. Firstly, accurate model information is required in the calculation of the equivalent control which restricts GFTSM available to applications. Especially, some parameters are difficult to obtain because of the complex dynamics of multi-rotor UAVs. Secondly, the GFTSM control needs the upper bound of the external disturbance, which is hard to estimate in real applications. Thirdly, the system chattering is inevitable because of the discontinuous item in the switch control. The chattering will affect the stability of multi-rotor UAVs, which is not suitable for practical application.

This article focuses on the GFTSM control strategy of quadrotor UAVs based on the RBF neural network (GFTSM-RBF). This strategy has three important features: 1) The chattering phenomenon of GFTSM control can be effectively reduced by using a neural network to learn the unknown dynamics model and the upper bound of disturbance of UAV online. It also reduces the work of dynamics modeling and parameter identification before controller design. 2) The weights of the neural network are adjusted online according to the designed adaptive law to ensure the stability of the closed-loop system. 3) The equivalent control that usually requires precise model information of the system is computed directly using the RBF neural network. Therefore, the structure of the controller is simplified and the online calculation is reduced, which makes it affordable for practical applications.

The rest of the paper is arranged as follows. In Section 2, the dynamics model of a quadrotor UAV is briefly described, and a novel GFTSM-RBF controller design for trajectory tracking of the UAV is proposed. Simulation results and performance analysis are shown in Section 3. The work of this paper is summarized in the last section.

2.   Method

2.1.   Model formulation

The coordinate system and schematic of the quadrotor are shown in Figure 1. The body frame of the quadrotor $\left \{ B \right \}$ is attached to the center of mass and $\left \{ E \right \}$ is the reference frame. According to the direction of rotation, the rotors of the quadrotor can be divided into two groups, namely (1, 3) and (2, 4). These two sets of rotors produce lift force $F_1, \ldots, F_4$ and neutralize the counter-torque. The rotation of the body is controlled by the speed difference between rotors. The rolling motion of the quadrotor is accomplished by changing the speed of rotor 2 or 4. The pitching motion of the quadrotor is achieved by adjusting the speed of rotor 1 or 3. The yaw rotation of the quadrotor is accomplished by the reverse moment difference between the two pairs of rotors (1, 3) and (2, 4). The vertical motion is performed by increasing or decreasing the total speed of the rotors ^[15].

Assuming that the fuselage structure and rotors of the quadrotor are rigid, the rotational and translation movement of the quadcopter can be obtained as follows ^[16]:

where $\phi$, $\theta$ and $\psi$ represent the Euler angles (i.e., roll, pitch, and yaw angles). $x$, $y$ and $z$ are the positions of the center of gravity of the quadrotor, $g$ is the gravity, $m$ is the total mass of the quadrotor, $I_r$ is the inertia of the rotor. $\Omega_{r} = \Omega_{4}+\Omega_{3}-\Omega_{2}-\Omega_{1}$. $a_i$ and $b_{i}(i = 1, \ldots, 6)$ are constants, which given by, $a_{1} = \left(I_{y}-I_{z}\right) / I_{x}$, $a_{2} = I_{r} / I_{x}$, $a_{3} = \left(I_{z}-I_{x}\right) / I_{y}$, $a_{4} = I_{r} / I_{y}$, $a_{5} = \left(I_{x}-I_{y}\right) / I_{z}$, $b_{2} = 1 / I_{x}$, $b_{4} = 1 / I_{y}$, $b_{6} = 1 / I_{z}$. $U_{t}$ is the total thrust, $U_{\phi}$, $U_{\theta}$, $U_{\psi}$ are the torques in the $\phi$, $\theta$ and $\psi$ direction of rotation, respectively. The control allocation is:

where $l$ is the distance of the rotors on the diagonal. $b$ is the lift coefficient. $c$ is the drag coefficient. $\Omega_{1}, \ldots, \Omega_{4}$ is the rotational speed of each propeller.

Based on the model Eq (2.1) and taking the lumped disturbance into account, the quadrotor equations are formulated as

where $X = [\phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}, z, \dot{z}, x, \dot{x}, y, \dot{y}]^{T} \in\mathbb{R}^{12}$ is the state vector, $U = \left[U_{\phi}, U_{\theta}, U_{\psi}, U_{x}, U_{y}, U_{z}\right]^{T} \in\mathbb{R}^{6}$ is the input, $f(X, U)$ is nonlinear functions, $d(X, U) = \left[d_{1}, d_{2}, \ldots, d_{6}\right]^{T}$ is the lumped disturbance vector on each degree of freedom of the quadrotor. Equation (2.2) can be expended as follows:

where $U_4, U_5, U_6$ are the position's virtual control:

To simplify the design of controller, it is assumed that the vehicle does not pass through singularities $(-\pi / 2 < \phi < \pi / 2, -\pi / 2 < \theta < \pi / 2 \; {\rm { and }}-\pi < \psi < \pi)$.

In order to track the position reference signal, the desired total thrust $U_t$ and the attitude angles $(x_{1d}, x_{3d})$ can be achieved through Eq (2.4) as follows:

2.2.   Controller design

The goal of this section is to propose the GFTSM-RBF control and use it to design a closed-loop controller for the quadrotor to track the desired trajectory$x_{1 d}, x_{3 d}, x_{5 d}, x_{7 d}, x_{9 d}, x_{11 d}$. The structure of the controller is presented in Figure 2. The altitude of the quadcopter is controlled by the total thrust $U_t$, and the rotation movement is controlled by $U_1, U_2, U_3$. The desired angles of roll $x_{1d}$, pitch $x_{3d}$ and $U_t$ is achieved through Eq (2.5). The desired yaw $x_{5d}$ is used to control the heading through the yaw controller.

The controller integrates the GFTSM control and the adaptive RBF neural network in the frame of Lyapunov theory, as shown in Figure 3. There are four steps to compute the control as shown in Figure 4. The design steps are as follows:

Step 1. The tracking errors of the quadrotor are defined as

Thus, the derivative of the error are

And the second time derivative of errors are as follows

Step 2. Based on the GFTSM control theory ^[18], the sliding mode surfaces of the quadrotor position and attitude tracking control system is designed as:

where, $\alpha_{i}$ and $\beta_{i}$ are positive constants. $q_i$ and $r_{i}(q_i < r_{i})$ are odd integers. The time derivative of the sliding surfaces are

Step 3. The Lyapunov candidate functions are designed as follows

The derivative of Lyapunov candidate functions are obtained

According to Eq (2.11), the control law of the quadrotor can be designed as

where,

and,

where, $U_{i}^{e q}$ is equivalent control and $U_{i}^{s w}$ is the switch control. $\lambda_{i}$ and $\delta_{i}$ are positive parameters. Take Eq (2.12) into Eq (2.10), the derivative of $s_i$ are

Step 4. Adaptive RBF design: The control law $U_i$ is difficult to calculate because $U_{i}^{e q}$ contains the dynamic model and the disturbance term of the system. To obtain the control law, the RBF neural network is used to estimate the $U_{i}^{e q}$. Suppose a neural network with hidden layers ^[17] is adopted:

where ${\boldsymbol{x}}_{i} = \left[s_{i}, x_{k d}\right]^{T}, (k = 2 j-1)$ is the input of the neural network, $\hat{{\boldsymbol{W}}}_{i} = \left[w_{i 1}, w_{i 2}, \ldots, w_{i n}\right]^{T}$ is the weights of the neural networks, ${\boldsymbol{h}}_{i}\left({\boldsymbol{x}}_{i}\right) = \left[h_{i 1}, h_{i 2}, \cdots, h_{i n}\right]^{T}$ is the radial basis function and given by:

where $\sigma_i$ is $j$ th standard deviation. ${\boldsymbol{c}}_{i j}$ is the $j$ th center vector.

In addition, the adaptive law of $\hat{{\boldsymbol{W}}}_{i}$ is

The final design of control law is

Substitute Eq (2.14) into Eq (2.10), then the derivative of the sliding surface variable is

Theorem 2.1. Considering the quadrotor UAV dynamics (Eq (2.3)) subject to lumped disturbance, there exists a set of control gains such that the proposed GFTSM (Eq (2.14)) for the quadrotor UAV can ensure that all signals in the closed-loop control system are bounded and the finite-time convergence of all the outputs to the designated trajectory can be guaranteed.

Proof. Assume that the equivalent control $U_{i}^{e q}$ can be estimated by the RBF neural network, and the minimum estimation error is $\varepsilon_{i} > 0$. The equivalent control $U_{i}^{e q}$ becomes

where ${\boldsymbol{W}}_{i}$ is an ideal value of $\hat{{\boldsymbol{W}}}_{i}$. Hence, the estimation error of $U_{i}^{e q}$ is

where $\tilde{W}_{i} = \hat{{\boldsymbol{W}}}_{i}-{\boldsymbol{W}}_{i}$ is the error of weight vector, and submit it to Eq (2.15).

The following Lyapunov function is used

The time derivative of $V_{i}^{w s}$ is

Take Eqs (2.13) and (2.16) into Eq (2.17), the derivative of the sliding surface variable can be written as (take $i = 1$ for example).

If $\delta_{1} > \left|\varepsilon_{1} / s_{1}^{q_{1} / r_{1}}\right|$, then $\dot{V}_{1}^{w s} \leq 0$, which means $s_1$ and $\tilde{{\boldsymbol{W}}}_{1}$ converge to zero.

Define a Lyapunov candidate function

Submit $s_i = 0$ and Eqs (2.9)–(2.18), the derivative of Eq (2.18) are obtained.

According to Eq (2.19), $e_i$ converges to zero too. Equation (2.16) can be rewritten as (take $i = 1$ for example):

where, $\lambda_{1}^{\prime} = \lambda_{1} b_{2}$, $\delta_{1}^{\prime} = \left(b_{2} \delta_{1}+\frac{b_{2} \varepsilon_{1}}{s_{1}^{q_{1} / r_{1}}}\right)$.

By solving Eq (2.20), the convergence time of $s_i$ from $s_{1}(0) \neq 0$ to $s_1(t_s) = 0$ is obtained:

3.   Results

The simulation results given in this section verify the effectiveness and performance of the proposed controller. Table 1 lists the main physical parameters of the quadrotor used in the simulation. The proposed flight controllers parameters are set as: $\alpha_1 = 4.5, \beta_1 = 1.4, q_1 = 3, r_1 = 7, \sigma_1 = 0.2, \lambda_1 = 10,$ $\alpha_2 = 8.2, \beta_2 = 2.5, q_2 = 3.5, r_2 = 7.5, \sigma_2 = 0.2,$ $\lambda_2 = 20, \eta_1 = 10.5, \eta_2 = 14.5, n = 5, c = [-2.6, -1.2, 0, -1.2, -2.6]$

In the simulation flight, the quadrotor tracked 3d trajectory under the external interference and parameter uncertainties. The initial state values of the quadrotor are [0, 0, 0]rad and [0, 0, 0]m. The disturbance terms are set as $d_1 = d_2 = d_3 = 0.2sin(4t)$, $d_4 = d_5 = d_6 = 0.06cos(4t)$. Seven seconds after the simulation started, the quadrotor weight was suddenly reduced by $30\%$. Comparative simulations with the conventional GFTSM control method proposed in ^[19] are also given.

The simulation results of the proposed controller are shown in Figures 5–10. As shown in Figure 5, the 3D flight trajectory demonstrated that the proposed controller has succeeded following the 3D flight trajectory in finite time, but the traditional GFTSM has steady-state error since the parameter perturbation. Figure 6 shows the time evolution of position variables ($x, y$ and $z$). It can be seen that the abrupt change of $z$ variable due to the mass variation was overcome by the GFTSM-RBF controller within 2 seconds, but GFTSM failed. Figure 7 shows the tracking errors of position variables. It can be observed that the proposed GFTSM-RBF achieves better position tracking than traditional GFTSM. Figures 8 and 9 shows the trajectory tracking of attitude angles($\phi, \theta$ and $\psi$). It can be observed that both of the two control systems can track the attitude references accurately, but the GFTSM-RBF controller has a faster tracking speed and smoother yaw angle $\psi$. Figure 10 presents the control inputs of the two control approaches. As expected, GFTSM exhibits chattering in the control input, and the GFTSM-RBF nearly eliminates the chattering. As a result, the GFTSM-RBF method presents a faster tracking speed and greater robustness against sustained time-varying disturbances and parametric uncertainties.

4.   Conclusions

This paper presents an adaptive GFTSM-NN controller to realize 3D trajectory tracking for the quadrotor with unknown disturbance and dynamic uncertainty. Global fast terminal sliding surfaces are designed for finite-time convergence of all the outputs of quadrotor. The equivalent control of the GFTSM controller is estimated by the RBF-NN. Adaptive laws are developed to compute the weights of RBF-NN. The quadrotor's closed-loop stability and finite-time convergence is guaranteed through the Lyapunov theory and subsequent analysis demonstrated in this paper. Finally, a comparison of the proposed control technique is presented with the conventional GFTSM. The results demonstrate that the GFTSM-NN achieves faster response speed, more robust to dynamic uncertainty, and lower chattering than the GFTSM. In future work, the proposed GFTSM-RBF approach will be validated by a real quadrotor UAV to perform the trajectory-tracking task. Also, control input constraints will be considered.

Acknowledgments

This study was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB460020); the National Natural Science Foundation of China (Grant No.52005231); the Foundation Research Project of Jiangsu Province (the Natural Science Fund No. BK20170315); and Changzhou Sci & Tech Program of China (Grant No. CJ20179017).

Conflict of interest

All authors disclosed no relevant relationships.