Adaptive Secure Control for Leader-Follower Formation of Nonholonomic Mobile Robots in the Presence of Uncertainty and Deception Attacks

Study efforts have actively proceeded to apply the cooperation technique of multi-robot systems to various areas, such as industries, commerce, and the military. For the operation of multi-robot systems, the robots are frequently connected through a network, which can be a target for attackers. Because exposure to a cyber-attack may result in critical loss of multi-robot systems, information security is essential and various methods, such as encryption techniques, intrusion detection systems, and secure control have been studied.

Cyber-attacks are largely divided into denial-of-service (DoS) disrupting services of a host connected to the network [1,2,3], a deception attack deceiving the data [4,5,6], and a replay attack delaying data transmission [7,8,9]. Among them, since a deception attack distorts the information of neighboring robots required to operate multi-robot systems, the robots that receive the wrong information cannot move according to the control objective, and finally, cause collaboration failure. Therefore, studies on secure control have been proposed. In [10], adaptive control architectures for linear dynamical systems with sensor uncertainty and attacks were presented. In [11,12], time-varying sensor and actuator attacks were estimated by the adaptive controllers. In [13], an adaptive event-triggered mechanism was proposed for the networked control systems (NCS) under deception attacks and stochastic nonlinearity. In [14], data quantization, DoS attacks, and deception attacks of the NCS were considered. In [15], the security correction control scheme based on an interconnected adaptive observer was proposed for stochastic cyber-physical systems subject to false data injection attacks. Because the above papers are limited to linear systems, studies on nonlinear systems have been conducted. In [16], a neural-network-based controller was presented for uncertain nonlinear time-delay cyber-physical systems in the presence of sensor and actuator attacks. An approximation-based event-triggered control method was proposed to compensate for unknown injection data of lower-triangular nonlinear systems in [17]. In [18], an adaptive control scheme for second-order nonlinear systems was presented to deal with time-varying parameters and an unknown control direction brought by the injection and deception attacks. However, all these papers cannot be applied to nonholonomic mobile robots due to the underactuation problem. Although an adaptive resilient event-triggered control method at the kinematic level was proposed for single autonomous vehicles with DoS attacks [19], performance degradation due to uncertainty arising from the dynamic model is inevitable. Moreover, the deception attack problem was not considered in [19].

For the operation of multi-robot systems, formation control methods have been studied using the behavior-based approach [20], virtual structure [21], and leader-follower approach [22]. Although each approach has its advantages, the leader-follower approach has been widely used because of its simplicity, scalability, and reliability. Early studies on the leader-follower approach dealt with only the kinematics of mobile robots [22,23]. To consider model uncertainties and disturbances, studies were conducted that took into account the dynamics of mobile robots. The sliding mode control and adaptive neural network control problems were addressed to compensate for the uncertainty in [24,25], respectively. In recent years, formation control research has been conducted on connectivity preservation and collision avoidance between robots. In [26,27], formation control approaches using potential-like functions were presented. In [28], a dipolar navigation function was introduced to design the formation controller. In [29], a formation tracking method was proposed to avoid obstacles while maintaining connectivity. However, those studies mentioned above did not deal with the problem against deception attacks in the formation control design. For practical applications, it is significant to deal with the formation control of multiple mobile robots under deception attacks. This problem has not been addressed yet.

Accordingly, this paper proposes an adaptive secure control methodology for leader-follower formation of uncertain nonholonomic mobile robots in the presence of deception attacks of the leader’s information. The leader’s velocity information corrupted by unknown injected data is assumed to be transmitted to the follower robots through the network. A robust and resilient control design with adaptive attack compensation mechanisms is developed to compensate for time-varying velocity attacks where the radial basis function networks (RBFNs) are employed to deal with system uncertainties and external disturbances. Furthermore, in the proposed control design, the dynamic surface design technique is applied to circumvent the problem that the time derivatives of the virtual control laws are affected by unknown deception attack signals. It is proven that all closed-loop signals are uniformly ultimately bounded in the Lyapunov stability sense, and the formation errors are ensured for converging to an adjustable neighborhood of the origin. Finally, simulation results are given to verify the performance of the proposed theoretical approach.

Compared with the existing literature, the main contributions of this paper are as follows:

The rest of this paper is organized as follows. In Section 2, the leader-follower model is introduced, and the adaptive secure control problem is formulated for achieving the desired formation of nonholonomic mobile robots with unknown deception velocity attacks. In Section 3, the proposed secure control strategy and its stability analysis are presented. In Section 4, simulation results are given. Finally, the conclusion of this paper is drawn in Section 5.

In this section, we design the controller to achieve the control objective under the assumption that the follower j receives the corrupted data $({\overline{v}}_i,{\overline{\omega }}_i)$ from the leader i. The adaptive technique and neural networks are employed to deal with model uncertainty, external disturbance, and time-varying attack signals. The dynamic surface control (DSC) method [35] is used to design the controller at the dynamic level.

Let us define the errors as where $e_{4,j}={[e_{4,1,j},e_{4,2,j}]}^{\top }$, ${\alpha }_j={[{\alpha }_{1,j},{\alpha }_{2,j}]}^{\top }$ is the virtual control, ${\alpha }_{f,j}={[{\alpha }_{f,1,j},{\alpha }_{f,2,j}]}^{\top }$ is the filtered signal obtained by ${\Gamma }_j{\dot{\alpha }}_{f,j}+{\alpha }_{f,j}={\alpha }_j$, ${\alpha }_{f,j}\left(0\right)={\alpha }_j\left(0\right)$, ${\Gamma }_j$ is a diagonal matrix and positive definite, and ${\theta }_{r,j}$ is the reference orientation to be defined later. In the leader-follower formation, the orientations of the leader i and the follower j cannot be equal while the formation is turning [25]. Thus, instead of following the orientation of the leader i, the reference orientation ${\theta }_{r,j}$ is required for the follower j.

Step 1: Substituting (2) and (4) into the time derivatives of (7)–(9) yields the following error dynamics, Consider the Lyapunov function candidate as where ${\gamma }_{v,j}$ and ${\gamma }_{\omega ,j}$ are positive constants, ${\tilde{a}}_{v,j}={\overline{a}}_{v,i}-{\widehat{a}}_{v,j}$, ${\tilde{a}}_{\omega ,j}={\overline{a}}_{\omega ,i}-{\widehat{a}}_{\omega ,j}$, and ${\widehat{a}}_{v,j}$ and ${\widehat{a}}_{\omega ,j}$ are the estimates of ${\overline{a}}_{v,i}$ and ${\overline{a}}_{\omega ,i}$, respectively. By Assumption 1-(iii) and the inequality, we obtain the time derivative of (13) along (10)–(12) as follows:

The virtual controls ${\alpha }_{1,j}$ and ${\alpha }_{2,j}$ are chosen as and the reference orientation is updated by where with positive constants $k_{1,j}$, $k_{2,j}$, $k_{3,j}$, and ${ϵ}_j$. Substituting (15) and (16) into (14) and using Lemma 1 yield where from Lemma 1, the following inequalities are used:

We choose the adaptation laws as where ${\sigma }_{1,j}$ and ${\sigma }_{2,j}$ are positive constants.

Step 2: The time derivative of (10) along (3) is

Multiplying $B_j{M}_j$ both sides of (19) yields where Note that there exists the inverse of $B_j$, and the constant matrix ${\overline{M}}_j$ is positive definite.

Consider the Lyapunov function candidate as where ${\gamma }_{l,j}>0$, ${\tilde{W}}_{l,j}=W_{l,j}-{\widehat{W}}_{l,j}$, $W_{l,j}$ is the constant optimal weight vector, and ${\widehat{W}}_{l,j}$ is the estimate of $W_{l,j}$. By Assumption 1(ii), the time derivative of (21) along (20) yields where $f_j\left(X_j\right)={[f_{1,j},f_{2,j}]}^{\top }=-B_j(C_j\left({\nu }_j\right){\nu }_j+D_j{\nu }_j-M_j{\Gamma }^{-1}{\varpi }_j)+{\overline{\tau }}_{d,j}^2{e}_{4,j}/\left(2{\varsigma }_j\right)$ with $X_j={[{\nu }_j^{\top },{\varpi }_j^{\top },e_{4,j}^{\top }]}^{\top }$ and a constant ${\varsigma }_j>0$. Note that the continuous functions $f_{n,j}$ for $n=1,2$ can be approximated by RBFNs as follows: $f_{n,j}\left(X_j\right)=W_{n,j}^{\top }{\Phi }_j+{\epsilon }_{n,j}$.

We chose the actual control inputs and update laws as where $k_{4,j}$, ${\sigma }_{3,4}$, and ${\sigma }_{4,j}$ are positive constants, and Consider the Lyapunov function candidate $V_{T,j}=V_{1,j}+V_{2,j}$. Then, the following theorem gives the main result of this paper.

In this section, we simulate five robots to demonstrate the performance of the proposed control scheme. Figure 2 shows the desired formation and information flow. The model parameters of each robot follow closely [30]. The initial postures of all robots are set to $q_0={[0,0,\pi /4]}^{\top }$, $q_1={[-3,-3,\pi /2]}^{\top }$, $q_2={[3,-3,\pi /2]}^{\top }$, $q_3={[-6,-3,\pi /2]}^{\top }$, and $q_4={[6,-3,\pi /2]}^{\top }$. The external disturbances are given by ${\tau }_{d,j}=[0.7cos\left(1.5t\right),$${1.2sin\left(0.5t\right)]}^{\top }$. The time-varying attack signals are set to $a_{v,n}=sin(\pi t/5)$ and $a_{\omega ,n}=cos(\pi t/5)$, where $n=0,1,2$. The design parameters are chosen as ${\gamma }_{v,j}={\gamma }_{\omega ,j}={\gamma }_{1,j}={\gamma }_{2,j}=5$, $k_{n,j}=5$, ${\sigma }_{n,j}=0.002$, ${\Gamma }_j=\mathrm{diag}[0.01,0.01]$, and ${ϵ}_j=0.5$ where $n=1,\dots ,4$. We assume that the robot 0 moves autonomously and the trajectory is generated by (2). The velocities of the robot 0 are chosen as follows: ${\nu }_0={[0.25(1-cos(\pi t/5)),0]}^{\top }$ for $0\le t<5$, ${\nu }_0={[0.5,0]}^{\top }$ for $5\le t<25$, ${\nu }_0={[0.5,0.2cos(2\pi t/20)]}^{\top }$ for $25\le t<45$, and ${\nu }_0={[0.5,0]}^{\top }$ for $t\ge 45$.