Platoon control design for unmanned surface vehicles subject to input delay

Vessel train formation as a new trend has been raised in cooperative control for multiple vessels. This paper addresses formation control design for a group of unmanned surface vehicles platoon considering input delay. To account for connectivity-preserving and collision-avoiding, Barrier Lyapunov function is incorporated into the constraints design of line-of-sight range and bearing. To alleviate the computational burden, neural dynamic model is employed to simplify the control design and smooth the input signals. Besides, input control arising from time delay due to mechanisms and communication is considered in the marine vessels. Within the framework of the backstepping technique, distributed coordination is accomplished in finite time and the uniformly ultimately boundness of overall system is guaranteed via rigorous stability analysis. Finally, the simulation is performed to verify the effectiveness of the proposed control method.

• The LOS range and bearing angle of decentralized leader-follower formation control have been restricted by BLF to meet the safety specification and operation performance. • For the sake of computational simplicity, bioinspired neurodynamics is employed to avoid the derivatives of virtual control inputs. The output signals of the bioinspired model are bounded in a finite interval and smooth without any sharp jumps even actual inputs have sudden changes. • The marine vehicles in the platoon are subjected to input delay. Artstein model is used as a predictor-like controller to deal with input delay in linear system. However, marine vessel models consist of nonlinear dynamics and uncertain nonlinear functions. Combined with tracking errors limitations, the nonlinear system with input time delay is converted into a delay-free system based on Artstein model.
The rest of the paper is organized as follows. In "Problem description and preliminaries" section, the problem is formulated and the preliminaries are introduced. "Platoon formation control design" section presents the controller design for connectivity preservation and collision avoidance, and the stability of the closed-loop system in the presence of input delay is rigorously analyzed. "Simulation results" section are shown. Lastly, the paper is concluded.

Problem description and preliminaries
Problem description. Consider a group of marine surface vehicles consisting of a leader and N followers.
The formation architecture in a pair of leader-follower is shown in Fig. 1. The coordinate frames {I} and {B} represent the inertial frame and body-fixed frame. The kinematics and dynamics of the i-th marine surface vessel (MSV) can be modeled as follows 23 (1) u i , ν i and r i represent surge, sway and yaw velocities in {B} , M i is the inertia matrix, C i denotes the matrix of Coriolis and centripetal, D i is the damping matrix, g i represents the restoring force vector, d i is the vector of external disturbances induced by wind, wave, and ocean currents, etc, τ i (t − t d ) denotes the control vector of the MSV with time delay, t d is the delayed time, J i (ψ i ) is the Jacobian transformation matrix, is the mass of the MSV, I zi represents the inertia moment in {B} , x gi denotes the vessel center of gravity in {B} , Xu i is the added mass in surge, Yv i and Yṙ i are the added mass in sway, and Nṙ i is the added mass in yaw. The Coriolis and centripetal matrix satisfies C i = −C T i , which is described as The damping matrix D i is given by where with the hydrodynamic damping coefficients X u i ,
Assumption 2 There exists bounded constants d i for the disturbance term d i , i = 0, 1, . . . , N of each vessel.

Assumption 3
The inertia matrix M i is inverse and we assume ||M −1 In this section, the formation objective of this paper is to design control laws such that each marine vessel modeled by (1) and (2) can follow its leader and do not violate the collision and connectivity in the platoon configuration when subject to input time delay. All signals in the closed-loop system can be guaranteed to be bounded during the whole operation.
The platoon formation objectives in this paper are to ensure that • The connectivity preservation and collision prevention are satisfied on the LOS range and bearing angle schemes between two consecutive MSVs. • The effect of input delay is considered and the stability is analyzed in constrained platoon control.
• A string of MSVs can achieve the formation tracking.
Preliminaries. Lemma 1 24 For any constant x ∈ R n , there exists a constant k satisfying |x| < k such that  where C m represents the membrane capacitance, V m is the voltage of the neuron. The parameters E k , E Na , and E p are the Nernst potentials for potassium ions, sodium ions, and passive leak current in the membrane, respectively. The functions g k , g Na , and g p denote the potassium conductance, the sodium conductance, and the passive channel, respectively. Grossberg derived the biologically inspired neurodynamic model to describe an online adaptive behavior of individuals 27 . The simplified shunting equation is obtained as where V is the neural activity (membrane potential) of the neuron. Parameters A, B, and D are nonnegative constants, namely, the passive decay rate, the upper and the lower bounds of the neural activity, respectively. The variables S(t) + and S(t) − represent the excitatory and inhibitory inputs, respectively [28][29][30] . The bioinspired model can be regarded as a low-pass filter. This method can achieve satisfactory tracking performance due to shunting characteristics. The outputs are restricted to a bounded interval and the signals obtained are smooth and continuous.

Platoon formation control design
As shown in Fig. 1, the relative distance, ρ i , between each pair of MSVs and LOS range, ϕ i , are defined as The tracking errors of the MSVs are defined as where ρ i,des is the desired LOS range.
To avoid collision and connectivity maintenance among vehicles, the desired distance during the whole moving process must satisfy the following equation where ρ i,min col and ρ i,max com represent the minimum safety distance and maximum effective communication distance respectively. For convenience, we define the minimum and maximum distance errors as Then the constraints of the LOS range errors become The constraints of the yaw angle errors have similar property as where ē ψi and e ψi are denoted as the maximum and minimum bounds of yaw angle errors.
Step 1: Define the tracking error as Consider the symmetric barrier Lyapunov function candidate as where k ai and k bi are positive constants satisfying the inequalities |e ρi | < k ai , |e ψi | < k bi , respectively. The time derivative of V 1i yields www.nature.com/scientificreports/ According to Eq. (17), the time derivatives of e ρi and e ψi are given by Step 2: The stabilizing function α i = [α 1i , α 2i , α 3i ] T is designed as follows where k di and k ψi are positive constants.
To avoid the complicated math operations on the derivative of α i , let α i pass through a neural dynamic model and substitute α ci = [α c1i , α c2i , α c3i ] T with α i in the following backstepping design. The bioinspired neurodynamics is adopted to smooth the virtual velocity control variables and obtain their derivatives. The neural dynamic model is constructed as [28][29][30] with where α ci is the output of the neural dynamic model, A i , B i and U i are positive parameters, which can be chosen to adjust the attenuation rate. The output can be limited within the region [−U i , B i ] . Then define the error z αi as The term N ci is defined as the following expression Utilizing the mean value theorem, then we obtain where the bounding function N ci (||z si ||) is a globally positive function. z si is defined as Design the following control law (1) and (2) satisfying Assumptions 1-3, under the virtual control laws (27), (28) and (29), the filter (30) and control input (43). For any � 0 > 0 for the initial conditions V 2i (0) < � 0 , then the following properties hold

Theorem 1 Consider N+1 USVs with dynamics
If the tuning parameters are selected as www.nature.com/scientificreports/

Simulation results
To demonstrate the performance of the proposed formation control method, a platoon consisting of one leader and three followers is designed. The marine vehicle model parameters are taken from Cybership-II 31 . It is a 1:70 scale replica of a supply vessel from the marine control laboratory in Norwegian University of Science and Technology. The corresponding parameters are listed in Table 1.    www.nature.com/scientificreports/ 2s. The control parameters are chosen as k d1 = 12, k d2 = k d3 = 1, k ψ1 = 1, k ψ2 = k ψ3 = 0.5, k a1 = k a2 = k a3 = 1, [8,8,6].
Simulation results are shown in Figs. 3, 4, 5, 6 and 7. The response curves of MSV1, MSV2 and MSV3 are plotted in red dash lines, purple dash lines, and blue dash lines, respectively. Figure 3 depicts platoon formation process of the 4 MSVs. Each marine vessel follows its leader with a satisfactory tracking performance during the entire process of moving. The distances between successive vehicles shown in Fig. 4 stay within the maximum connectivity distance 6 m and the minimum collision distance 4 m. It indicates that the LOS range tracking errors are within the predefined region bound. The desired distances between each leader and follower satisfy the inequality constraints 0 < ρ i,min col < ρ i ≤ ρ i,max com , i = 1, 2, 3 . The connectivity and collision prevention among the 4 MSVs are guaranteed during the formation achievement. Figure 5 represents the bearing angle tracking errors e ψ , which does not violate the constraints [−0.5rad, 0.5rad] . It demonstrates that bearing angles are constrained effectively. Figure 6 displays the control inputs of the following MSVs. The velocities of three followers are shown in Fig. 7. The simulation results demonstrate the connectivity preservation and collision prevention are satisfied, and the string of the MSVs can achieve the formation tracking in the presence of input delay.

Conclusions
Platoon formation control for a string MSVs has been developed in the presence of input delay and output constraints in this paper. BLF has been proposed to constraint LOS range and bearing angle tracking errors to satisfy the collision avoidance and connectivity maintenance. Next, bioinspired neurodynamics has been incorporated into the kinematic design to avoid the complicated computation of the leader vessel's acceleration. Furthermore, input delay system has been converted into a delay-free system and the stability has been    www.nature.com/scientificreports/