Formation Control for Water-Jet USV Based on Bio-Inspired Method FU Ming-yu, WANG Duan-song*, WANG Cheng-long

The formation control problem for underactuated unmanned surface vehicles (USVs) is addressed by a distributed strategy based on virtual leader strategy. The control system takes account of disturbance induced by external environment. With the coordinate transformation, the advantage of the proposed scheme is that the control point can be any point of the ship instead of the center of gravity. By introducing bio-inspired model, the formation control problem is addressed with backstepping method. This avoids complicated computation, simplifies the control law, and smoothes the input signals. The system uniform ultimate boundness is proven by Lyapunov stability theory with Young inequality. Simulation results are presented to verify the effectiveness and robust of the proposed controller.


Introduction
USV control problem is focused by many researchers. Especially formation control (Cui et al., 2009;Do, 2012;Zhang and Zhang, 2014) achieve challenging and dangerous tasks such as mine clearance, patrol, investigation, transportation of strategic materials and marine science applications, which can not only decrease personal injury but also achieve tasks that single USV cannot complete. Most of the ship formation control strategies are belongs to complicated nonlinear system, which include leader-follower method (Fahimi, 2007;Meng et al., 2012), virtual structure strategy (Zhao et al., 2012), behavioral approach (Arrichiello et al., 2006;Ma and Zeng, 2015), graph theory (Almeida et al., 2012), and artificial potential function (Li and Xiao, 2016). Ghommam and Mnif (2009) and Xie and Ma (2014) addressed the problem of maneuvering a group of underactuated ships along given path with constant disturbances by a path follower and coordinated controller. However, the way of centralized control depends too much on the overall information. Liu et al. (2016) Based on lineof-sight (LOS), extended state observer (ESO) and cascaded theory to realize the task of the multiple USV along one curve. Theoretical analysis has showed that the closedloop system is ISS. Peng et al. (2013) developed an adaptive formation controller for USV by using the neural network and dynamic surface control, which avoided the calculations of the virtual signals by introducing the first-order filter. Shojaei (2016) adopted neural network combined with adaptive robust control strategy to deal with the formation problem and acquired ideal result. The limitations of the work have common styles with too many parameters to be adjusted and the complicated process. An adaptive control design method (Peng et al., 2017a) is presented for containment maneuvering of marine surface vehicles with the adaptation laws not limited to a Lyapunov-based design and modularity between the estimator and controller. An adaptive containment maneuvering controller (Peng et al., 2017b) is proposed according to neurodynamics-based output feedback control scheme and the observer-based containment maneuvering control laws is proposed at the kinematic level. The stability for closed loop network is verified based on ISS and cascade theory. Ding and Guo (2012) proposed a formation control law without external disturbances based on leader-follower. Børhaug et al. (2011) proposed a control law for underactuated surface vessels formation based on LOS path following control and a nonlinear synchronization controller without accounting the inherent limitations due to the communication and with the control law of straight line following. Liao et al. (2015) studied the path following problem of an USV by backstepping adaptive sliding mode control. The problem is transformed into a actuated problem based on Seret-Frenet frame. However, during the controller design large calculations of derivate are involved because of the virtual signals induced by backstepping.
In this work, the formation problem for USV with time-varying disturbances is investigated. The leader-follower strategy is used as it can be easily implemented. During the control law design, a virtual is assumed to be a leader. All the USVs are needed to be coordinated to a controller to keep a desired distance and orientation, as produces a formation for the task. By changing the distance and orientation, the formation can be changed in time. With the coordination transformation, the output point position (the bow position) differs from the traditional point of the center of gravity (Peng et al., 2011). This guarantees the steady and avoids the chattering of course. Then being inspired by the bioinspired method (Yang and Zhu, 2011;Hodgkin and Huxley, 1952) three neural dynamic models are introduced. It can not only smooth the input signals and simplify the control law, but also can limit the signal to a certain range and adjust the attenuation rate by paremeters. The formation control problem of underactuated USV is addressed to cooperate with the backstepping method.

Problem formulation
In this work, the frame definition, the motion model of a single USV and formation model are shown as follows.

Single USV model
The 3-DOF dynamics and kinematics equations for a single water-jet USV are defined by Fossen (2002): (1) where denotes position (surge, sway displacements) and orientation (yaw angle) of the i-th USV in the north-east frame; represents the i-th USV's surge, sway and yaw angle velocities, respectively; is considered as the mass and inertia parameters of three axes. Hydrodynamic damping coefficients in the body coordinate is given by . The signals τ are the force and torque inputs which are provided by water-jet thrusters. And denote the surge, sway and yaw external disturbances, which are the time-varying disturbances in this work.

Leader-follower model
As start from simple, only two USVs are considered as a group. The formation mathematical model is shown in Fig. 1. φ l x and l y The distance between the leader USV and follower USV is given by l; the angle of the follower relative to the leader is denotes by ; represent the components of l in the north-east frame on the x and y axis, respectively. The bow position of USV is denoted by P, with x Fp , y Fp , and φ Fp of the position and orientation of P; d is the distance between P and the mass center of follower. x L , y L , φ L , u L , υ L , r L , x F , represent the position, orientation and surge, sway, and yaw angle velocities of the leader and follower, respectively. From Fig. 1 we can obtain In order to achieve the desired formation structure, following conditions should be satisfied: can be confirmed, then are unique. The control problem of can be transformed to control . As can be seen from Fig. 1, Then, The desired distance between the two USVs is considered as l d , the projection weight in the north-east frame are , then Differentiating Eq. (6) yieldṡ The errors of formation model can be defined as: The formation mathematics model can be written as follows: where μ 1 =l Assumption 2: Trajectory produced by the virtual leader is smooth enough. In a summary, the formation control problem of waterjet USV can be divided into two parts, kinematics and dynamics.
For kinematics, virtual surge and sway velocities are For dynamics, the force and torque are designed to make actual velocities approximate to virtual velocities.

Controller design
To solve the formation control problem for USV, the backstepping strategy has been used. However, the backstepping method is too complicated to be employed. In order to address this problem, we combine it with the bio-inspired method by introducing three neural dynamic models.
Step 1: Lyapunov function is chosen as: Differentiating Eq. (11), we can obtaiṅ Virtual control variables are chosen where are positive constants. In order to avoid repeating derivative , we make go through a neural dynamic model, and substitute with as the virtual variable in the process of backstepping design.
The neural dynamic model is: where are the outputs of neural dynamic model, are positive constants on behalf of the neurons of attenuation rate. The attenuation rate can be adjusted by parameter .
are regarded as positive constants denoting the upper and lower bounds of neurons dynamic . It can limit the outputs to . The parameters are chosen based on the underactuate and actuator saturation characteristic, which approximate to the practice system. are linear threshold functions of x as: z u , z υ , e u , and e υ Step 2: represent error variables, which are defined as: Then, . By substituting Eq. (14) into Eq. (12), we obtaiṅ Considered the augmented Lyapunov function The time derivative of Eq. (18) yieldṡ where , being a positive constant. Eq. (19) can be rewritten as: Then the control input is selected as: υ e υ Step 3: As is also a virtual control input, r is considered as a virtual input to control . Thuṡ τ r F In this stage, the actual control input is designed by considering Lyapunov candidate function which is given by Time derivative of the Lyapunov candidate function FU Ming-yu et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 117-122 119 (23) results iṅ When the USV is moving on the sea, the surge velocity is a none zero value. So we can choose a virtual variable , where is a positive constant.
Similarly to , is also going through a neural dynamic model, Error variable are defined as . Substituting them into Eq. (24) yieldṡ Step 4: For the sake of controlling , an augmented Lyapunov function is considered.
(28) Leṫ where is a positive constant. We can obtain the control input of torque where .

Stability analysis
For the formation system, we choose a Lyapunov function where According to Assumptions 3-4, we can obtain bounded, are assumed as the maximum value of , respectively. Based on Eq. (15) with , it results iṅ , Eq. (33) can be rewritten as: (36) For the type of absolute value of the item, without the absolute value symbol all the items are assumed larger than 0. According to Young inequality (Ghommam and Saad, 2014), we can obtain .
Inequality Eq. (38) can be rewritten as: (39) To solve the inequality, we can obtain It shows that the Lyapunov function is smaller than . This means that the control error is bounded. The radius of convergence can be small enough by adjusting the value of and .

Numerical simulation
To verify the effectiveness of the proposed formation controller of USV, the mathematical results of three USVs from Do et al. (2002) In the simulation, USV1 is a virtual leader, USV2 and USV3 are followers. The system initial states are given as , . The desired position and orientation respect to USV1 are as follows: in the first 20 s, l d12 =5 m, l d13 =5 m, , during the 20 s to 150 s, l d12 =5 m, l d13 =5 m, . We The moving trajectories of three USVs under the proposed control law are shown in Fig. 2. It can be seen that in the first 20 s, the distances of USV2 and USV3 respect to USV1 are 5 m with the orientations of and , respectively. After that, the distances of USV2 and USV3 relative to USV1 are still 5 m, the orientation are and . It is verified the effectiveness of the proposed control law. Fig. 3 shows the positions and orientation of three USV. We can see from it that in the first 20 s, the positions of x (north) are same and the positions of y (east) vary in a range of ±5. Thus the distances are . Fig. 4 shows the velocities of three USV. As can be seen from Fig.  5, the forces and torques of USV2 and USV3 are smooth except to the moment of formation changed.

Concluding remarks
The development of a bio-inspired formation controller for USV has been presented in the presence of disturbances caused by external environment. Compared with the backstepping method, the bio-inspired based scheme shows some advantages to cope with the derivation repeating problem and smooth the input signal. Stability analysis verifies that the control law can guarantee the system globally asymptotically stable. The simulation results reveal that the formation can be changed at any time with satisfying performance and robustness. The effectiveness of the control law has been validated by the simulation results and theoretical analysis.    FU Ming-yu et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 117-122 121