Radial basis function neural networks for formation control of unmanned aerial vehicles

Abstract

This paper addresses the problem of controlling multiple unmanned aerial vehicles (UAVs) cooperating in a formation to carry out a complex task such as surface inspection. We first use the virtual leader-follower model to determine the topology and trajectory of the formation. A double-loop control system combining backstepping and sliding mode control techniques is then designed for the UAVs to track the trajectory. A radial basis function neural network capable of estimating external disturbances is developed to enhance the robustness of the controller. The stability of the controller is proven by using the Lyapunov theorem. A number of comparisons and software-in-the-loop tests have been conducted to evaluate the performance of the proposed controller. The results show that our controller not only outperforms other state-of-the-art controllers but is also sufficient for complex tasks of UAVs such as collecting surface data for inspection. The source code of our controller can be found at https://github.com/duynamrcv/rbf_bsmc.

1. Introduction

Unmanned aerial vehicles, when combined with computer vision technologies, can collect visual data of structures to provide valuable information for various tasks such as inspecting structural surfaces [1, 2], reconstructing 3D models [3, 4], identifying cracks [5, 6], and detecting corrosion and rust on steel bridges [7, 8]. However, using a single UAV for these tasks is inefficient due to the large size of the structures and the limited battery capacity of the UAV. A group of UAVs flying in a formation can be used to overcome those limitations [9–12]. The formation allows the UAVs to perform collaborative inspection to increase the efficiency and accuracy of data collection. It also allows for safe operation, as the formation control can prevent collision among the UAVs.

In formation control, the leader-follower approach is commonly used to provide flexibility in topology and trajectory selection [13, 14]. In the standard leader-follower method, one UAV is assigned as the leader, and the others are followers. The leader plays the role of a reference node for the followers to determine their locations to form the desired topology. The limitation of this approach, however, is the dependence of the system on the leader. If the leader is malfunctioning, the whole system will fail. The virtual leader-follower model can be used to cope with this problem. In this approach, the leader is purely a virtual entity, serving as a reference point for the followers to determine their positions [15]. By decoupling the physical leader from the model, this method mitigates the risk of complete system failure.

In the leader-follower model, linear controllers are commonly used to control individual UAVs to form the desired topology [15–17]. In ref. [16], linear quadratic and neural networks-based controllers are combined to control a group of UAVs considering their full dynamics. In ref. [17], the receding horizon control is employed to yield a fast convergence rate of the formation tracking control. This controller also considers the orientation between the leader and the followers for accurate formation. The decentralized $H_\infty$ -PID controller is introduced in ref. [18] to maneuver a group of UAVs to deal with the external disturbance and trailing vortex coupling from their neighbor UAVs. A leader-follower formation control technique is presented in ref. [19] to address issues related to backward error and suboptimal dynamic speed tracking in PID neural network control. Linear controllers, however, have limitations in handling constraints and parameter variation, especially when applied to nonlinear systems like UAVs.

In another approach, nonlinear controllers have been used for formation control [20–22]. In ref. [23], first and second-order sliding mode controllers are deployed to assure the asymptotic stability of the formation, taking into account modeling uncertainties. In ref. [24], an adaptive controller using the dynamic estimation of the distance between the leader and the followers is introduced to address uncertainties related to positioning errors. Two finite-time observers are used in ref. [25] to deal with bounded external disturbance force and torque. In [26], a non-uniform vector field that dynamically varies in magnitude and direction is employed to deal with the influence of wind in UAV formation control. A distributed model predictive control algorithm is introduced in ref. [27] to coordinate the operation of a fleet of UAVs considering their spatial kinematics and unidirectional data transmissions. However, the convergence of these controllers depends on the characteristics of disturbances, which are hard to model due to their varying nature. A sufficient approach would be utilizing neural networks such as the radial basis function neural network (RBFNN) to estimate disturbances and use it as the feedback for control [28–31].

In this work, we present a new controller for a group of UAVs cooperating in a formation. The UAVs use the virtual leader-follower model to determine their trajectory and form the desired topology. The controller is developed using the backstepping and sliding mode control techniques. An RBFNN is then introduced to estimate external disturbances for better control performance. Our contributions to this work are as follows:

1. I. The proposal of a new controller for UAV formation that is constructed by combining backstepping and sliding mode control techniques, thereby enabling the elimination of nonlinear components and enhancing system robustness. Additionally, the adverse effects associated with these controllers, such as “explosion of term” and “chattering,” are mitigated through the approximation of unknown factors by the neural network. As the result, the developed controller not only addresses the drawbacks of the aforementioned techniques but also augments the adaptability of the UAV system.

2. II. The design of a radial basis function neural network (RBFNN) that is capable of estimating external disturbances to compensate for input force control signals, thereby enabling the controller to maintain the required control quality.

3. III. The derivation of the stability proof for the designed controller using Lyapunov’s theorem, which is essential to ensure stable operation of the UAVs under conditions affected by external forces.

4. IV. The comparison of the proposed controller with other popular methods including model predictive control (MPC), backstepping sliding mode control (BSMC), and sliding mode control (SMC) in different scenarios to confirm its superior performance. Software-in-the-loop tests were also conducted with a cooperative bridge inspection task to verify the validity of the proposed method for practical applications.

The rest of this paper is structured as follows. Section 2 presents the dynamic and formation models of the UAVs. Section 3 introduces the proposed controller. Section 4 shows evaluation results. The paper ends with conclusions described in 5.

2. Problem formulation

To control a group of UAVs, we first consider their dynamic model and formation topology with details as follows.

2.1. UAV dynamic model

Consider a group of $n$ UAVs, each is a quadrotor with two pairs of propellers rotating in opposite directions, as described in Fig. 1. Frames $BB_xB_yB_z$ and $OXYZ$ are respectively the body-fixed and inertial frames. We use Euler angles to represent the attitude of the UAV. The configuration of the UAV includes its position $\xi =\left [x,y,z\right ]^T$ and Euler angles $\Xi =\left [\phi,\theta,\psi \right ]^T$ , with $\left \vert \phi \right \vert \leq \pi/2$ , $\left \vert \theta \right \vert \leq \pi/2$ and $\left \vert \psi \right \vert \leq \pi$ . Those angles represent the roll, pitch, and yaw orientation of the UAV, respectively. Control signals of the UAV are defined as follows:

(1) \begin{equation} \left [\begin{array}{c} f_{t}\\ \tau _{\phi }\\ \tau _{\theta }\\ \tau _{\psi } \end{array}\right ]=\left [\begin{array}{c} f_{1}+f_{2}+f_{3}+f_{4}\\ l\!\left (f_{4}-f_{2}\right )\\ l\!\left (f_{3}-f_{1}\right )\\ \tau _{2}+\tau _{4}-\tau _{1}-\tau _{3} \end{array}\right ], \end{equation}

where $l$ is the arm length; $f_{t}$ is the total thrust of four propellers; $\tau _\phi$ , $\tau _\theta$ , $\tau _\psi$ are the torques in three axes; and $f_i$ and $\tau _i$ , with $i=\left \{1,2,3,4\right \}$ , are the forces and torques generated by four propellers, respectively. According to [32], the dynamic model of the UAV is described as follows:

(2) \begin{equation} \begin{aligned} \ddot{x}&=\left (\cos \phi \sin \theta \cos \psi +\sin \phi \sin \psi \right )\dfrac{f_{t}}{m}+\dfrac{d_x}{m}\\ \ddot{y}&=\left (\cos \phi \sin \theta \sin \psi -\sin \phi \cos \psi \right )\dfrac{f_{t}}{m}+\dfrac{d_y}{m}\\ \ddot{z}&=\cos \phi \cos \theta \dfrac{f_{t}}{m}-g+\dfrac{d_z}{m}\\ \ddot{\phi }&=\dfrac{\dot{\theta }\dot{\psi }\!\left (I_{y}-I_{z}\right )+\tau _{\phi }}{I_{x}}\\ \ddot{\theta }&=\dfrac{\dot{\phi }\dot{\psi }\!\left (I_{z}-I_{x}\right )+\tau _{\theta }}{I_{y}}\\ \ddot{\psi }&=\dfrac{\dot{\phi }\dot{\theta }\!\left (I_{x}-I_{y}\right )+\tau _{\psi }}{I_{z}} \end{aligned} \end{equation}

where $I_x$ , $I_y$ , $I_z$ are the moments of inertia, $m$ is the mass of the UAV, $g$ is the gravitational acceleration, and $\left [d_x,d_y,d_z\right ]^T$ is the disturbance caused by factors such as wind or turbulent flows.

Figure 1. The structure of the quadrotor UAV in the global frame.

2.2. UAV formation model

The formation model used in this work is the virtual leader-follower model with two main components:

• Virtual leader: a virtual leader is a non-physical UAV used as a reference for other UAVs to determine their position. Its trajectory represents the trajectory of the UAV group.

• Follower: a follower is a UAV that adjusts its position based on the virtual leader. Given the reference trajectory of the leader and the expected topology, the followers calculate their trajectories and then track them to form the desired formation.

Consider virtual leader $B_L$ having position $\xi _L = \left [x_L,y_L,z_L\right ]^T$ and heading angle $\psi _L$ and follower $B_i$ having position $\xi _L = \left [x_{i},y_{i},z_{i}\right ]^T$ and yaw angle $\psi _{i}$ . Let $B^{\prime}_L$ and $B^{\prime}_i$ be their projection on the $OXY$ plane, respectively, $B^{\prime\prime}_L$ be the projection of $B_i$ on $B_LB^{\prime}_L$ , and $\Delta _i=\left [\Delta x_{i}, \Delta y_{i}, \Delta z_{i}\right ]^T$ be the desired distance between follower $B_i$ and the virtual leader, as depicted in Fig. 2. Since $\Delta z_i=B_LB^{\prime\prime}_L$ , the desired position of follower $B_i$ can be computed as:

(3) \begin{equation} \begin{aligned}{}^d\xi _i&=\text{Rot}_z\!\left (\psi _L\right )\Delta _i+\xi _L\\{}^d\psi _{i}&=\psi _L, \end{aligned} \end{equation}

where $\text{Rot}_z({\cdot})\in \mathbb{R}^{3\times 3}$ is the rotation matrix around z-axis. Equation (3) allows the followers to compute their trajectory based on the trajectory of the virtual leader and the desired formation topology.

Figure 2. Illustration of the virtual leader-follower formation structure.

3. Controller design for UAV formation

Given the trajectory of the virtual leader, denoted as $(\xi _L,\psi _L)$ , the desired trajectory of follower $i$ , $({}^d\xi _{i},{}^d\psi _{i})$ , in the formation can be computed based on (3). To track this trajectory, we design a dual-loop control system for each follower as shown in Fig. 3. The outer loop is a position controller that regulates the altitude and horizontal position, while the inner loop is a backstepping sliding mode controller (BSMC) that handles the UAV’s attitude, including its roll, pitch, and yaw angles. To account for external disturbances, the position controller was designed with a radial basis function neural network (RBFNN). A converter block is also included to convert the desired translational control forces into roll and pitch angles. Details of each controller are described as follows.

Figure 3. The proposed controller.

3.1. Position controller design

The position controller aims to keep the UAV’s position aligned with the desired trajectory. It is designed based on BSMC with the use of RBFNN for disturbance estimation. According to (2), dynamic equations for the translational motion of UAV $i$ are given as follows:

(4) \begin{equation} \begin{aligned} \ddot{x}_i&=u_{xi}+\dfrac{d_{xi}}{m},\\ \ddot{y}_i&=u_{yi}+\dfrac{d_{yi}}{m},\\ \ddot{z}_i&=u_{zi}+\dfrac{d_{zi}}{m}, \end{aligned} \end{equation}

where

(5) \begin{equation} \begin{aligned} u_{xi}&=\left (\cos \phi _i\sin \theta _i\cos \psi _i+\sin \phi _i\sin \psi _i\right )\dfrac{f_{ti}}{m},\\ u_{yi}&=\left (\cos \phi _i\sin \theta _i\sin \psi _i-\sin \phi _i\cos \psi _i\right )\dfrac{f_{ti}}{m},\\ u_{zi}&=\cos \phi _i\cos \theta _i\dfrac{f_{ti}}{m}-g. \end{aligned} \end{equation}

Let $\xi _i=\left [x_i,y_i,z_i\right ]^T$ and $\dot{\xi }_i=\left [\dot{x}_i,\dot{y}_i,\dot{z}_i\right ]^T$ respectively be the position and velocity of the translational motion, $U_i=\left [u_{xi},u_{yi},u_{zi}\right ]^T$ be the control signal, and $D_i=\left [\dfrac{d_{xi}}{m},\dfrac{d_{yi}}{m},\dfrac{d_{zi}}{m}\right ]^T$ , with $\left \Vert D_i\right \Vert \leq \bar{d}$ , be the external disturbance affecting UAV $i$ . Equation (4) can be rewritten as:

(6) \begin{equation} \ddot{\xi }_i=U_i+D_i. \end{equation}

The BSMC is then designed as follows.

3.1.1. Backstepping sliding mode controller (BSMC) design

Let ${}^e\xi _{i}$ be the translational error, ${}^e\xi _{i}=\xi _{i}-{}^d\xi _i$ . The virtual velocity, ${}^v\xi _i$ , of the subsystem is designed as:

(7) \begin{equation}{}^v\xi _i={}^{d}\dot{\xi }_i-\lambda _{\xi }{}^{e}\xi _i, \end{equation}

where $\lambda _{\xi }\gt 0$ is a positive definite gain. The first candidate Lyapunov function is chosen as

(8) \begin{equation}{}^1V_{\xi _i} = \dfrac{1}{2}{}^e\xi _i^T{}^e\xi _i. \end{equation}

Its derivative is given by

(9) \begin{equation}{}^1\dot{V}_{\xi _i}={}^e\xi _{i}^T{}^e\dot{\xi }_{i}={}^e\xi _{i}^T\left (\dot{\xi }_{i}-{}^d\dot{\xi }_{i}\right ). \end{equation}

Substituting $\dot{\xi }_{i} ={}^v{\xi }_i$ into (9) gives

(10) \begin{equation}{}^1\dot{V}_{\xi _i}=-\lambda _{\xi } \!\left \Vert{}^e\xi _{i}\right \Vert ^2\leq 0. \end{equation}

Hence, the system is stable with the virtual velocity chosen in (7). The sliding mode control (SMC) algorithm is then utilized to design the input control signal for the position system. The sliding surface is chosen as follows:

(11) \begin{equation} s_{\xi _i}=\gamma _{\xi }{}^e\xi _{i} + \left (\dot{\xi }_{i}-{}^v{\xi }_{i}\right ), \end{equation}

where $\gamma _{\xi }\gt 0$ is a positive definite gain. Denote $\hat{D}_i$ as the disturbance estimated via an estimator such as the RBFNN in Section 3.1.2. The derivative of $s_{\xi _i}$ then can be obtained by using $\hat{D}_i$ instead of $D_i$ as follows:

(12) \begin{equation} \dot{s}_{\xi _i}=\gamma _{\xi }\!\left (\dot{\xi }_i-{}^d\dot{\xi }_{i}\right )+U_i+\hat{D}_i-{}^v\dot{\xi }_{i}. \end{equation}

The second Lyapunov function of the subsystem is chosen as follows:

(13) \begin{equation}{}^2V_{\xi _i}=\dfrac{1}{2}s_{\xi _i}^Ts_{\xi _i} \end{equation}

The control signals are designed as follows:

(14) \begin{equation} \begin{aligned} U_{ieq}&={}^v\dot{\xi }_{i}-\gamma _{\xi }\!\left (\dot{\xi }_{i}-{}^d\dot{\xi }_{i}\right )-\hat{D}_i\\ U_{isw}&=-\left (c_{\xi 1}\text{sg}\!\left (s_{\xi _i}\right )+c_{\xi 2}s_{\xi _i}\right ), \end{aligned} \end{equation}

where $c_{\xi 1}$ and $c_{\xi 2}$ are positive gains, $U_{ieq}$ is the equivalent control signal that maintains the position variables on the sliding manifold, $U_{isw}$ is the signal that leads the subsystem to the sliding surface $s_{\xi _i}$ , and $\text{sg}\!\left (\cdot \right )$ is the piece-wise continuous function defined as

(15) \begin{equation} \text{sg}\!\left (x\right )=\left \{ \begin{array}{c} 1\\ -1\\ \dfrac{x}{\epsilon } \end{array}\right .\begin{array}{c} x\gt \epsilon \\ x\lt -\epsilon \\ \text{otherwise} \end{array} \end{equation}

where $0\lt \epsilon \lt 1$ is a predefined constant.

Theorem 3.1. Consider the position control system of the UAV. If the control signal is chosen as

(16) \begin{equation} U_i=U_{ieq}+U_{isw}, \end{equation}

the system is stable.

Proof. Taking the first derivative of ${}^2V_{\xi _i}$ gives

(17) \begin{equation}{}^2\dot{V}_{\xi _i}=s_{\xi _i}^T\dot{s}_{\xi _i}. \end{equation}

Substituting (12) into (17) gives

(18) \begin{equation}{}^2\dot{V}_{\xi _i}=s_{\xi _i}^T\!\left (\gamma _{\xi }\!\left (\dot{\xi }_{i}-{}^d\dot{\xi }_{i}\right )+U_i+\hat{D}_i-{}^v\dot{\xi }_{i}\right ). \end{equation}

By substituting (14) and (16) into (18), $\dot{V}_{\xi _i}$ becomes

(19) \begin{equation}{}^2\dot{V}_{\xi _i}=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right )-c_{\xi 2}\!\left \Vert s_{\xi _i}\right \Vert ^2\leq 0. \end{equation}

According to Lyapunov’s stability theorem, the system is stable.

3.1.2. Radial basis function neural network (RBFNN) design

During operation, UAVs are subject to inevitable disturbances such as wind or turbulent flows. Those disturbances affect the system performance, but are complex to model and analyze. We address this problem by exploiting the online learning capability of neural networks to estimate the disturbances. Previous studies on universal approximation theorems for RBFNN show that RBFNN can approximate any nonlinear function on a bounded set with an arbitrary level of accuracy [31]. In this work, we design a disturbance estimator using a neural network with the radial basis function (RBF). The network has three layers including an input layer, a hidden layer, and an output layer, as shown in Fig. 4. Position vector $\xi _i$ and its derivation $\dot{\xi }_i$ are the input of the network. At the hidden layer, neurons are activated by a radial basis function. The output of neuron $j$ is computed as:

(20) \begin{equation} h_{j}=\exp{\left (-\dfrac{\left \Vert \xi _{i}-\mu _{1j}\right \Vert ^2+\left \Vert \dot{\xi }_{i}-\mu _{2j}\right \Vert ^2}{b^2}\right )}, \end{equation}

where $b$ is a parameter controlling the width of the Gaussian function, $\mu _{1j}$ and $\mu _{2j}$ are predefined center points, and $j\in \{1,2,\ldots,m\}$ is the neuron index with $m$ being the number of neurons in the hidden layer. The output layer is a weighted sum. Let $W_i$ be the optimal weight matrix, $H_i$ be the output of the hidden layer, and $\sigma _i$ be the approximation error. Disturbance $D_i$ affecting UAV $i$ then can be expressed by:

(21) \begin{equation} D_i=W_i^TH_i+\sigma _i \end{equation}

Figure 4. The RBFNN structure.

The output $\hat{D}_i$ of the RBFNN approximates $D_i$ as:

(22) \begin{equation} \hat{D}_i=\hat{W}_i^TH_i, \end{equation}

where $\hat{W}_i$ is a trained weight matrix. This matrix is updated based on the following rule:

(23) \begin{equation} \dot{\hat{W}}_i=a\!\left (H_is_{\xi _i}^T-\eta \!\left \Vert s_{\xi _i}\right \Vert \hat{W}_i\right ), \end{equation}

where $a$ is a positive definite gain matrix. With this structure, the estimation of disturbance $D_i$ can be described as in Algorithm 1, where $\beta \in (0,1)$ is the momentum factor.

Algorithm 1: Pseudocode to estimate disturbances by RBFNN for UAV $i$

3.1.3. Stability of the position controller

The stability of this controller is addressed in Theorem 3.2 as follows.

Theorem 3.2. Consider UAV $i$ affected by external disturbance $D_i$ as described in (6), the control signals designed in (14), the bounded weight $\left \Vert W_i\right \Vert \leq \bar{W}_i$ , and the update rule for RBFNN in (23). If the following inequality condition is satisfied:

(24) \begin{equation} \left \Vert s_{\xi _i}\right \Vert \geq \dfrac{\bar{\sigma }+\dfrac{1}{4}\eta \bar{W}_i^{2}}{c_{\xi 2}}, \end{equation}

the position control system is stable.

Proof. Choose the candidate Lyapunov function as follows:

(25) \begin{equation} V_{\xi _i}={}^2V_{\xi _i}+\dfrac{1}{2}\text{tr}\!\left (\tilde{W}_i^Ta^{-1}\tilde{W}_i\right ), \end{equation}

where $\tilde{W}_i=W_i-\hat{W}_i$ is the error weight matrix. Taking the first derivative of $V_{\xi _i}$ gives:

(26) \begin{equation} \begin{aligned} \dot{V}_{\xi _i}&=s_{\xi _i}^{T}\dot{s}_{\xi _i}+\text{tr}\!\left (\tilde{W}_i^{T}a^{-1}\dot{\tilde{W}_i}\right )\\ &=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right ) -c_{\xi 2}\!\left \Vert s_{\xi _i}\right \Vert ^{2}-s_{\xi _i}^{T}\left (\hat{D}_i-D_{i}\right )-\text{tr}\!\left (\tilde{W}_i^{T}a^{-1}\dot{\hat{W}}_i\right )\\ &=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right ) -c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+s_{\xi _i}^{T}\sigma _i+s_{\xi _i}^{T}\tilde{W}_i^{T}H_i-\text{tr}\!\left (\tilde{W}_i^{T}a^{-1}\dot{\hat{W}}_i\right )\\ &=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right ) -c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+s_{\xi _i}^{T}\sigma _i+\text{tr}\!\left (-\tilde{W}_i^{T}\left (a^{-1}\dot{\hat{W}}_i-H_is_{\xi _i}^{T}\right )\right ) \end{aligned} \end{equation}

With the updated rule of the neural network, $\dot{V}_{\xi _i}$ can be rewritten as follows:

(27) \begin{equation} \begin{aligned} \dot{V}_{\xi _i}&=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right )-c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+s_{\xi _i}^{T}\sigma _i+\text{tr}\!\left (\tilde{W}_i^{T}\eta \!\left \Vert s_{\xi _i}\right \Vert \hat{w}_i\right )\\ &=-c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right ) -c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+s_{\xi _i}^{T}\sigma _i+\eta \!\left \Vert s_{\xi _i}\right \Vert \text{tr}\!\left (\tilde{W}_i^{T}\left (W_i-\tilde{W}_i\right )\right ) \end{aligned} \end{equation}

According to the Cauchy-Schwarz inequality, the following inequality equation can be satisfied:

(28) \begin{equation} \text{tr}\left (\tilde{W}_i^{T}\left (W_i-\tilde{W}_i\right )\right )\leq \left \Vert \tilde{W}_i\right \Vert \left \Vert W_i\right \Vert -\left \Vert \tilde{W}_i\right \Vert ^{2} \end{equation}

Thus,

(29) \begin{equation} \begin{aligned} \dot{V}_{\xi _i}&\leq -c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right )-c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+\left \Vert s_{\xi _i}\right \Vert \bar{\sigma }+\eta \!\left \Vert s_{\xi _i}\right \Vert \left (\left \Vert \tilde{W}_i\right \Vert \left \Vert W_i\right \Vert -\left \Vert \tilde{W}_i\right \Vert ^{2}\right )\\ &\leq -c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right )-c_{\xi 2}\left \Vert s_{\xi _i}\right \Vert ^{2}+\left \Vert s_{\xi _i}\right \Vert \bar{\sigma }+\dfrac{1}{4}\eta \!\left \Vert s_{\xi _i}\right \Vert \bar{W}_i^{2}-\eta \!\left \Vert s_{\xi _i}\right \Vert \left (\dfrac{1}{2}\left \Vert W_i\right \Vert -\left \Vert \tilde{W}_i\right \Vert \right )^{2} \end{aligned} \end{equation}

Based on an extension of the Lyapunov theorem [33], $\left \Vert s_{\xi _i}\right \Vert$ is bounded. Moreover, the control gain $c_{\xi 2}$ can be selected large enough so that

(30) \begin{equation} \left [{\bar{\sigma }+\eta \bar{W}_i^{2}/{4}}\right ]/{c_{\xi 2}}\leq b_\xi \end{equation}

Therefore, with the inequality condition (24), $\dot{V}_{p}$ can be rewritten as follows:

(31) \begin{equation} \dot{V}_{\xi _i}\leq -c_{\xi 1}s_{\xi _i}^T\text{sg}\!\left (s_{\xi _i}\right )-\eta \!\left \Vert s_{\xi _i}\right \Vert \left (\dfrac{1}{2}\left \Vert W_i\right \Vert -\left \Vert \tilde{W}_i\right \Vert \right )^{2}\leq 0 \end{equation}

The Lyapunov stability condition is satisfied.

3.2. Attitude controller design

In our system, the position controller is the outer loop of the UAV control system, as depicted in Fig. 3. Its control signal is then fed to the converter block to calculate the desired angles and translational forces based on (5) as:

(32) \begin{equation} \begin{aligned}{}^{d}\theta _i&=\arctan \left (\dfrac{u_{xi}\cos{}^{d}\psi _i+u_{yi}\sin{}^d\psi _{i}}{u_{zi}+g}\right )\\{}^d\phi _{i}&=\arctan \left (\cos{}^d\theta _{i}\dfrac{u_{xi}\sin{}^d\psi _{i}-u_{yi}\cos{}^d\psi _{i}}{u_{zi}+g}\right )\\ f_{ti}&=\dfrac{u_{zi}+g}{\cos{}^d\phi _{i}\cos{}^d\theta _{i}} \end{aligned} \end{equation}

They are used as the reference for the attitude controller, which is designed based on the BSMC. From (2), the dynamic equation for the roll angle is given by:

(33) \begin{equation} \begin{aligned} \ddot{\phi }_{i}&=\dfrac{\dot{\theta }_{i}\dot{\psi }_{i}\left (I_{y}-I_{z}\right )+\tau _{\phi _i}}{I_{x}}. \end{aligned} \end{equation}

Denote ${}^e\phi _{i} = \phi _i -{}^d\phi _{i}$ as the roll angle error. The virtual velocity, ${}^v\phi _{i}$ , is defined as:

(34) \begin{equation}{}^v\phi _{i} ={}^d\dot{\phi }_{i} - \lambda _\phi \phi _{1e}, \end{equation}

where $\lambda _\phi \gt 0$ is a positive gain. The first candidate Lyapunov function for subsystem $\phi _{1e}$ is chosen as:

(35) \begin{equation}{}^1V_{\phi _i} = \dfrac{1}{2}{}^e\phi _{i}^2. \end{equation}

Taking the first derivative of ${}^1V_{\phi }$ gives:

(36) \begin{equation}{}^1\dot{V}_{\phi _i}={}^e\phi _{i}{}^e\dot{\phi }_{i}={}^e\phi _{i}\left (\dot{\phi }_i-{}^d\dot{\phi }_{i}\right ). \end{equation}

Substituting $\dot{\phi }_i ={}^v\phi _{i}$ into (36) gives

(37) \begin{equation}{}^1\dot{V}_{\phi _i}=-\lambda _\phi{}^e\phi _{i}^2\leq 0. \end{equation}

Thus, the Lyapunov stability is guaranteed. The sliding surface of the roll angle subsystem is expressed as:

(38) \begin{equation} s_{\phi _i}=\gamma _{\phi }{}^e\phi _{i} + \left (\dot{\phi }_i -{}^e\phi _i\right ), \end{equation}

where $\gamma _\phi \gt 0$ is a positive gain. The first derivative of $s_{\phi _i}$ is given by:

(39) \begin{equation} \dot{s}_{\phi _i}=\gamma _\phi \!\left (\dot{\phi }_i-{}^d\dot{\phi }_{i}\right )+\dfrac{\dot{\theta }_{i}\dot{\psi }_{i}\!\left (I_{y}-I_{z}\right )+\tau _{\phi _i}}{I_{x}}-{}^v\dot{\phi }_{i}. \end{equation}

The control signal is then designed with two sub-control signals, ${}^{eq}\tau _{\phi _i}$ and ${}^{sw}\tau _{\phi _i}$ . ${}^{eq}\tau _{\phi _i}$ is the equivalent control signal that maintains the roll angle on the sliding manifold and ${}^{sw}\tau _{\phi _i}$ is the signal that leads the subsystem to the sliding surface $s_{\phi _i}$ . They are chosen as follows:

(40) \begin{equation} \begin{aligned}{}^{eq}\tau _{\phi _i}&=I_x\!\left ({}^v\dot{\phi }_{i}-\gamma _\phi \!\left (\dot{\phi }_i-{}^d\dot{\phi }_{i}\right )\right )-\dot{\theta }_{i}\dot{\psi }_{i}\!\left (I_{y}-I_{z}\right )\\{}^{sw}\tau _{\phi _i}&=-I_x\!\left (c_{\phi 1}\text{sg}\!\left (s_{\phi _i}\right )+c_{\phi 2}s_{\phi _i}\right ), \end{aligned} \end{equation}

where $c_{\phi 1}$ and $c_{\phi 2}$ are positive gains.

Table I. Parameters of the Hummingbird quadrotor.

Table II. Parameters of the proposed controller.

Theorem 3.3. Consider the roll angle subsystem (33). If the control signal is designed as:

(41) \begin{equation} \tau _{\phi _i}={}^{eq}\tau _{\phi _i}+{}^{sw}\tau _{\phi _i}, \end{equation}

the roll angle control system is stable.

Proof. The candidate Lyapunov function of the roll angle subsystem is chosen as follows:

(42) \begin{equation} V_{\phi _i}=\dfrac{1}{2}s_{\phi _i}^2. \end{equation}

Taking the first derivative of $V_\phi$ gives

(43) \begin{equation} \dot{V}_{\phi _i}=s_{\phi _i}\dot{s}_{\phi _i}. \end{equation}

By substituting (39) into (43), we have

(44) \begin{equation} \dot{V}_{\phi _i}=s_{\phi _i}\!\left (\gamma _\phi \!\left (\dot{\phi }_i-{}^d\dot{\phi }_{i}\right )+\dfrac{\dot{\theta }_{i}\dot{\psi }_{i}\!\left (I_{y}-I_{z}\right )+\tau _{\phi _i}}{I_{x}}-{}^v\dot{\phi }_{i}\right ). \end{equation}

Finally, substituting (40) and (41) into (44) gives

(45) \begin{equation} \dot{V}_{\phi _i}=-c_{\phi 1} s_{\phi _i}\text{sg}\!\left (s_{\phi _i}\right )-c_{\phi 2}s_{\phi _i}^2\leq 0. \end{equation}

Thus, the Lyapunov stability of the roll angle control system is guaranteed.

The control signals for the pitch and yaw angles can be obtained by applying the design process similar to the roll angle. As a result, the pitch control signals are obtained as:

(46) \begin{equation} \begin{aligned} \tau _{\theta _i}&={}^{eq}\tau _{\theta _i}+{}^{sw}\tau _{\theta _i}\\{}^{eq}\tau _{\theta _i}&=I_y\!\left ({}^v\dot{\theta }_{i}-\gamma _\theta \!\left (\dot{\theta }_i-{}^d\dot{\theta }_{i}\right )\right )-\dot{\phi }_{i}\dot{\psi }_{i}\!\left (I_{z}-I_{x}\right )\\{}^{sw}\tau _{\theta _i}&=-I_y\!\left (c_{\theta 1}\text{sg}\!\left (s_{\theta _i}\right )+c_{\theta 2}s_{\theta _i}\right ), \end{aligned} \end{equation}

and the yaw control signals are given by:

(47) \begin{equation} \begin{aligned} \tau _{\psi _i}&={}^{eq}\tau _{\psi _i}+{}^{sw}\tau _{\psi _i}\\{}^{eq}\tau _{\psi _i}&=I_z\!\left ({}^v\dot{\psi }_{i}-\gamma _\psi \!\left (\dot{\psi }_i-{}^d\dot{\psi }_{i}\right )\right )-\dot{\phi }_{i}\dot{\theta }_{i}\!\left (I_{x}-I_{y}\right )\\{}^{sw}\tau _{\psi _i}&=-I_z\!\left (c_{\psi 1}\text{sg}\!\left (s_{\psi _i}\right )+c_{\psi 2}s_{\psi _i}\right ). \end{aligned} \end{equation}

4. Results

To evaluate the performance of the proposed control system, we have conducted a number of evaluations and comparisons.1 The UAV model used is the Hummingbird quadrotors [34], whose parameters are shown in Table I. Parameters of the position and attitude controllers are chosen as shown in Table II. The desired formation is a triangular shape with $\Delta _1=\left [2,0,0\right ]^T$ , $\Delta _2=\left [0,0,2\right ]^T$ , and $\Delta _3=\left [0,0,-2\right ]^T$ , as depicted in Fig. 5. Comparisons are conducted between the proposed controller (RBF-BSMC) and three other controllers namely model predictive control (MPC) [27, 35], backstepping sliding mode control (BSMC) [36, 37], and sliding mode control (SMC) [23, 38] in different scenarios.

Figure 5. The desired topology.

4.1. Scenario 1

In this scenario, external disturbances acting on the formation are generated based on the combination of the rectangle and full wavelength “1-cosine” wind model [39], as shown in Fig. 6. The initial positions of the UAVs are set as $\xi _{1}=\left [3,2,4\right ]^T$ , $\xi _{2}=\left [2,1,4\right ]^T$ , and $\xi _{3}=\left [0,0,4\right ]^T$ . The desired trajectory of the virtual leader is a spiral with the $z$ coordinate increasing over time as expressed in (48).

(48) \begin{equation} \begin{aligned} x_{L}&=5\cos \!\left (\dfrac{2\pi }{20}t\right )\text{ (m)},\\ y_{L}&=5\sin \!\left (\dfrac{2\pi }{20}t\right )\text{ (m)},\\ z_{L}&=0.5t+5\text{ (m)},\\ \psi _{L}&=\dfrac{2\pi }{20}t+\dfrac{\pi }{2}\text{ (rad)}. \end{aligned} \end{equation}

Figure 6. The external disturbance acting on the formation generated based on the rectangle and full wavelength “1-cosine” wind model in scenario 1.

Fig. 7 shows the 3D views of the trajectory tracking results of the UAV formation. It can be seen that the UAVs quickly reach the initial positions to form the desired shape. They then maintain the shape while following the reference trajectory. However, the trajectory of the proposed method is smoother and more accurate than the others due to its capability to estimate the disturbance via the RBFNN and use it as feedback to adjust the control signals. This result can be further verified via the tracking errors as shown in Fig. 8a. It can be seen that the average tracking errors of the proposed controller quickly converge to zeros, whereas those errors of the other controllers largely fluctuate due to disturbances. In addition, the maximum and minimum tracking errors of the UAVs are also very small with our method, which confirm its stability for formation control.

Figure 7. Trajectories of the UAV formation generated by the four controllers in scenario 1.

Figure 8. The tracking errors and estimated disturbances of the controllers in scenario 1.

Fig. 8b shows the disturbances estimated for each UAV by the proposed controller. After the transition period, the estimation starts to converge to the real disturbance. This provides feedback for the controller to adjust the control signal for better tracking performance.

4.2. Scenario 2

In this scenario, the rectangle wind model is used to generate external disturbances as shown in Fig. 9. The initial positions of the UAVs are set as $\xi _{1}=\left [2,0,9\right ]^T$ , $\xi _{2}=\left [5,2,7\right ]^T$ , and $\xi _{3}=\left [-3,-2,8\right ]^T$ . The desired trajectory of the virtual leader is generated based on an inspection path used to collect surface data [40].

Figure 9. The external disturbance acting on the formation generated based on the rectangle wind model in scenario 2.

The 3D of the UAV formation tracking results are shown in Fig. 10. It can be seen that all controllers are able to drive the UAVs to reach their reference positions and then track them to form the desired shape. The proposed controller, however, introduces smaller tracking errors than the other controllers, as shown in Fig. 11a, due to its disturbance estimator. As shown in Fig. 11b, the estimation closely follows the actual disturbances except for the positions where step changes happen. At those positions, a transition period is needed for the estimator to converge to the new steady state. However, the settling time of the estimator is short allowing it to provide timely feedback to the controller.

Figure 10. Trajectories of the UAV formation generated by the four controllers in scenario 2.

Figure 11. The tracking errors and estimated disturbances of the controllers in scenario 2.

4.3. Validation with software-in-the-loop tests

To further validate the proposed control system, we have carried out software-in-the-loop (SIL) tests that involve the inspection of a scaled-down 3D model of a real bridge with 5 columns, as shown in Fig. 12. The UAV model used is a Hummingbird quadrotor2 developed based on Gazebo-based RotorS simulator [32], as depicted in Fig. 13. The formation used includes two topologies, vertical and triangular shapes, as shown in Fig. 14. According to our previous work [41], the generated path to inspect the bridge includes two stages. The first stage covers all columns of the bridge using a vertical formation. The second one uses a triangular formation to cover the side and top surfaces of the bridge, as depicted in Fig. 12.

Figure 12. The planned paths to inspect the bridge.

Figure 13. The Hummingbird drone model used in validation [32, 34].

Given the planned paths, the UAVs start to fly from positions $\left [-8,-8,0\right ]^T$ , $\left [-4,-8,0\right ]^T$ , $\left [4,-8,0\right ]^T$ , and reach their initial positions to form a vertical formation as shown in Fig. 12a. The formation then tracks the planned path to acquire surface images of the bridge.3 Fig. 15 shows the tracking errors of the UAVs during operation. It can be seen that the errors quickly converge to small values in both inspection stages except between time steps 1070 and 1120, where there is a change in the formation topology. The errors in the first stage are slightly larger than in the second one as the UAVs frequently change their direction to navigate around each column of the bridge. Nevertheless, the average tracking error of less than 5 cm is sufficient for most UAV-related applications and thus confirms the validity of our approach.

Figure 14. The vertical and triangular formation topologies used in SIL tests.

Figure 15. The tracking errors of UAV formation in the Gazebo SIL.

5. Conclusion

In this paper, we have presented a robust control system using RBFNN for a group of UAVs flying in a formation. By combining BSC with SMC, the controller can handle nonlinearity to increase its control performance. The use of RBFNN enables the system to estimate external disturbances to enhance its control robustness. By using Lyapunov’s theorem, we proved that the control system is stable and the proposed controller can track the reference trajectory. Evaluation results show that the proposed controller outperforms the state-of-the-art BSMC in terms of accuracy and robustness and is sufficient for most UAV applications.