Bridge-Ship Collision Avoidance Control Based on AFSMC with a FRNN Estimator

For collision avoidance and maneuvering control in bridge areas, an adaptive fractional sliding mode control with fractional recurrent neural network (FRNN-AFSMC) is proposed. The uncertainties are estimated by FRNN, and the fractional gradient is adopted to improve the recurrent neural network (RNN). Its convergence has been proven. The inﬂuence of fractional order on algorithm performance is analyzed, and the simulation platform of ship collision avoidance control is built. Dynamic collision avoidance of multiple ships is simulated and veriﬁed. The results show the feasibility and eﬀectiveness of dynamic autonomous collision avoidance motion control in a dynamic ocean environment.


Introduction
Collision avoidance and maneuvering control in bridge areas are important for the autonomous intelligent navigation in ships. Due to the large inertia, nonlinearity and uncertainty of ship motion, underactuated characteristics, and strong external interferences such as wind, waves, and current, automatic collision avoidance navigation and motion control in ships is a great challenge. Wu proposed a fuzzy logic approach for ship-bridge collision alerts [1]. Li discussed the unsteady hydrodynamic loads of ships passing near bridge piers [2]. Sha discussed bridge girders against ship forecastle collision loads [3]. Pedersen studied the design of bridges against ship collisions [4]. Fang studied composite bumper systems for bridge pier protection against ship collisions [5]. Zhu discussed lattice composite bumper systems for bridge protection in ship collisions [6].
Although some ships are equipped with side thrusters, they do not work at constant speed. How to design robust controllers for underactuated ships has become a key problem. Variable structure control has become an effective control method for ships. To improve the control effect of sliding mode, fractional calculus is introduced. It helps to obtain faster response speeds, lower overshoot, smaller chattering effects, and better control performance. Fei introduced a fractional-order sliding mode control method based on recursive neural network approximation [7]. To improve the output power quality of a permanent magnet synchronous generator, Xiong proposed a fractional-order sliding mode control method [8]. Sharafian proposed a new fractional-order observer based on a sliding mode method and radial basis function neural network to distinguish the uncertainty of a fractional-order human immunodeficiency virus mathematical dynamic model [9]. Yang proposed novel passive fractional-order sliding mode control for distributed generator microgrid supercapacitor energy storage systems [10]. Moezi proposed a new adaptive interval type 2 fuzzy fractional backstepping sliding mode control method [11]. Ren proposed a fractional-order sliding mode controller based on an exponential reaching law to control the pneumatic position servo system [12]. To approximate the system uncertainty at sea, a neural network is used. A recurrent neural network (RNN) is a kind of neural network that inputs sequence data, and all its nodes are connected by chain. In 1990, the first fully connected RNN, namely, the Elman network, was proposed [13]. A new observer structure for nonlinear fractionalorder systems was proposed by Sharafian to estimate the states of fractional-order nonlinear chaotic systems with unknown dynamic models [14]. A new fractional-order error back-propagation learning algorithm was proposed to adapt to the weights of neural networks by using the Lyapunov stability strategy of fractional-order system, called Mittag-Leffler stability. e main contribution was to extend the neural observer of fractional-order dynamics by satisfying the Mittag-Leffler condition. e observer design process ensured that the observer error converged to the zero point. For a class of nonlinear fractional multiagent systems, five special types of sliding surfaces were proposed by Sharafian to overcome the consistency tracking problem [15]. e disturbance suppression control input could stabilize the uniform error dynamics of fractional-order multiagent system. Integer-order and fractional-order sliding surfaces were used to improve the accuracy and convergence speed. Based on Lyapunov direct strategy and Mittag-Leffler stability of fractional derivative, the stability of various sliding surfaces was proved. In order to show that the sliding motion was generated in finite time, the upper bound of the arrival time of the sliding surface was given. Based on the dynamic memory reset principle of fractional-order operators, it was forced to be an invariant finite time sliding mode. Jesus considered an optimal fuzzy fractional PD + I controller in which the parameters were tuned by a GA [16]. e performance of the proposed fuzzy fractional control was illustrated through some application examples. Abiyev introduced the development of fractional-order controller for dynamic control [17]. e parameters of fractional-order controller were adjusted by real coded genetic algorithm.
To update and optimize the weights of neural networks, gradient descent (GD) is a basic method. e main contributions of this paper are as follows: (1) Fractional calculus is adopted to improve RNN for approximating unknown parameters of ship model and environmental disturbance (2) FRNN-AFSMC is designed, and its convergence is proven (3) e proposed algorithm is used for collision avoidance of underactuated ships through course alterations

Ship Drift in Bridge Areas.
Assume that the origin of the calculation coordinate is the center of the channel. e longitudinal axis X is perpendicular to the bridge axis and points downstream. e transverse axis Y is parallel to the bridge axis and points to the left bank. When a ship crosses a bridge, the yaw angle is α, and the flow direction angle is β.
en the drift B 1 of the ship under the influence of currents can be calculated as follows: where S is the length of the bridge, V is the speed of the ship, and U is the speed of the current. e drift B2 by wind is where K is the coefficient, B a is the wind area above the hull waterline, B W is the area below the hull waterline, V a is the relative wind speed, and α f is the angle between the wind direction and the normal of the bridge axis.

Ship Collision Avoidance.
When the ship is sailing in narrow waters and there is a risk of collision, the ship chooses course alterations to avoid collision. e optimal ship course alteration is based on the collision risk and limited feasible solution space. In the process of collision avoidance, after one or two ships take collision avoidance measures, the nearest distance DCPA (Distance of Closest Point of Approach) and the nearest time TCPA (Time to Closest Point of Approach) of the two ships must change.
For example, the true course (TC) of a ship is 011°and its speed is 10 kn. e data of incoming ships observed by radar are listed in Table 1.
A basic calculation shows that the DCPA is 1.2 n mile. Assume that the target ship maintains this direction and speed. e baseline ship wants to maintain its speed and turn right at 09 : 15, so that the target ship passes 2.0 n mile away. e turning range and time to restore the original course are to be calculated. e calculation steps are as follows: (1) Make a triangle of relative motion. When the target ship arrives at A5, the original course of the ship is restored. e relative motion vector of the target ship is restored to A1A3. e actual relative motion line is A5Y2, and it passes 2.0 n mile away. is process is shown in Figure 1. Figure 2 shows its architecture diagram. Recurrent neural network takes sequence data as input and recurses in the evolution direction of sequence, and all nodes (cyclic units) are connected by chain. e output of a sequence is also related to the previous output. e specific form is that the network will remember the previous information and apply it to the calculation of the current output. e input of the hidden layer includes not only the output of the input layer but also the output of the previous hidden layer. Each arrow represents a transformation, and the arrow connection has weight. In the expanded structure, in the standard RNN structure, the neurons in the hidden layer also have weights. With the continuous progress of the sequence, the former hidden layer will affect the later hidden layer. O represents the output, Y represents the determined value given by the sample, and L represents the loss function. e loss function is also accumulated with the recommendation of the sequence. Each input value of RNN only establishes a weighted connection with its own route. e error signal is defined as [18]

Recurrent Neural Network.
e loss function is defined as where

Fractional-Order Calculus
its Riemann-Liouville integral is as follows: where α is the fractional order, whose real part is positive. Γ(x) is the gamma function defined as

Definition 3.
For the function f(x) � (x − x 0 ) v , the following formula holds:

Motion Model of an Underactuated
Ship. e motion model is shown in Figure 1. φ � [φ 1 , φ 2 , . . . , φ P ] denotes the position and attitude, x denotes the velocity, ‖x‖ denotes the surge velocity, x denotes the sway velocity, φ i denotes the yaw velocity, x i denotes the surge position, σ j denotes the sway position, and μ i denotes the yaw angle. Figure 3 shows the ship motion model. e mathematical model is as follows: where R is the rotation matrix and it can be calculated as e dynamic models of underactuated ships can be calculated as follows:  Figure 2: e architecture diagram of RNN.

Mathematical Problems in Engineering
e velocity equation is as follows: where F wx , F wy , and N wc are the wind disturbance, F cx , F cy , and N c are the force of current disturbance, F sx , F sy , and N s are the force of wave disturbance, and τ u and τ r denote the control input.
and hydrodynamic additional inertia. (19) and (8) can be written as where C X , C Y , and C N denote the wind coefficients. A f denotes the projection area above the water line. A s denotes the projection area on the side. L oa denotes the total length of the ship. ρ a denotes the air density.

Current Disturbance
Model. e ocean current disturbance is where F cx , F cy , and N c are the current coefficients. V c represents the velocity of the current. A fw denotes the orthographic projection area of the ship underwater. A sw denotes the side projection area of the ship underwater. L denotes the length of the ship's waterline. β denotes the drift angle. ρ denotes the density of the sea water. C x , C y , C n denote the flow force coefficients.

Wave Disturbance Model. e wave disturbance is
where a denotes the average wave amplitude. χ denotes the encounter angle. C xw , C yw , and C nw denote wave coefficients. λ is the length of the wave.

FRNN Algorithm.
e theoretical basis of gradient descent method is the concept of gradient. e relationship between gradient and directional derivative is as follows: the direction of gradient is consistent with the direction of obtaining the maximum value of directional derivative, and the modulus of gradient is the maximum value of directional derivative of function at this point. e weight is updated using the gradient method: where w kj is the weight, L is the loss function of the neural network, η is the learning rate coefficient, and ɑ is the fractional order. e fractional order derivative of L(t) is where v is the weight from the hidden layer to the output layer of the network. e derivative of L(t) is Substituting (25) into the first item of (24) yields where y is the output of the training samples. Substituting (27) into the first item of (5) yields Substituting (28) into the first item of (26) yields
Proof. From the Taylor mean value theorem with Lagrange remainder, we have Applying (29) to (31), we have where c > 1/2L ″ . From (32), one can obtain that (30) will hold if the following is satisfied: e proof of Lemma 1 is completed.
(35) e proof of Lemma 2 is completed.

Neural Fractional-Order Sliding Mode Control.
Denote η d as the desired position and attitude. Denote υ d as the desired velocity.
e tracking error of position and attitude is e tracking error of velocity is e following fractional sliding surface functions are constructed: where c 1 > 0. α is the fractional order. e derivative of (49) is obtained: Substituting (20) into (50), one can obtain e sliding mode control law is constructed: where η 1 > 0. f u is the approximation of f u : where ε 1 is the approximation error. Subtracting (53) from (54), (55) can be written as e adaptive control law is constructed: e fractional sliding surface is constructed: where c 2 > 0. β is the fractional order. e derivative of (59) is obtained: Substituting (48) into (60), one can obtain and v d is constructed as follows: e derivative of (62) is obtained: and u d is constructed as follows: e derivative of (64) is obtained: It can be obtained by calculation that (66) e derivative of (63) can obtain Substituting (66) into (67), one can obtain e control rate is designed as follows: where k 2 > 0, η 2 > 0. f r is the approximation of f r : where ε 2 is the approximation error. Subtracting (70) from (71), Denote (72) can be written as e adaptive control rate is constructed:

Stability Analysis
Theorem 2. With fractional-order sliding mode controllers (52) and (69) and adaptive control laws (58) and (75), the system converges in an asymptotically stable way.

Simulation Studies
A container ship [19] is used for testing. e length of the ship is 175 m. Table 2 lists sea weather condition data. Table 3 lists the tidal height on Dec. 30, 2019. Table 4 lists the ship collision avoidance parameters between two ships. Figure 4 shows the speed response curve of ship 1. Figure 5 shows the position and attitude response curve of ship 1. Figure 6 shows the roll response curve of ship 1. Figure 7 shows the input rudder angle of ship 1. Figure 8 shows the speed response curve of ship 2. Figure 9 shows the position and attitude response curve of ship 1. Figure 10 shows the roll response curve of ship 2. Figure 11 shows the input rudder angle of ship 2. Figure 12 shows the resultant positions of ships 1 and 2 to avoid collisions. Figures 4-12 show that FRNN-AFSMC can realize the collision avoidance between two ships. Table 4 shows the performance comparison of different algorithms, including SMC, ASMC with RNN (RNN-ASMC), and FRNN-AFSMC. Table 5 shows that FRNN-AFSMC has less overshoot and a shorter system regulation time with a wider parameter selection range. Figure 13 shows the yawing response curve of the closedloop control system, where the horizontal axis represents the time in seconds and the longitudinal axis represents the yaw angle in unit of rad. Figure 13 shows that, compared with basic sliding mode control algorithm, the overshoot of FRNN-AFOSMC is smaller and the regulation time of the system is shorter.

Conclusions
In this paper, FRNN-AFSMC is proposed and shown to converge for collision avoidance in bridge areas. e uncertainties are estimated by FRNN. e influence of fractional order on the algorithm performance is analyzed. Multiple ships dynamic collision avoidance is simulated and verified. e results show the performance comparison of different algorithms, including SMC, ASMC with RNN (RNN-ASMC), and FRNN-AFSMC. e accuracy of the control algorithm will be improved in future research.
Data Availability e data used are included within the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 13