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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.

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 [

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 [

To update and optimize the weights of neural networks, gradient descent (GD) is a basic method. The main contributions of this paper are as follows:

Fractional calculus is adopted to improve RNN for approximating unknown parameters of ship model and environmental disturbance

FRNN-AFSMC is designed, and its convergence is proven

The proposed algorithm is used for collision avoidance of underactuated ships through course alterations

Assume that the origin of the calculation coordinate is the center of the channel. The longitudinal axis _{1} of the ship under the influence of currents can be calculated as follows:

The drift B2 by wind is_{a} is the wind area above the hull waterline, _{W} is the area below the hull waterline, _{a} is the relative wind speed, and _{f} is the angle between the wind direction and the normal of the bridge axis.

When the ship is sailing in narrow waters and there is a risk of collision, the ship chooses course alterations to avoid collision. The 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. The data of incoming ships observed by radar are listed in Table

Data of incoming ships observed by radar.

Time | True course (°) | Bearing (°) | Distance ( |
---|---|---|---|

09 : 00 | 052 | 041 | 10.0 |

09 : 06 | 051 | 040 | 8.5 |

09 : 12 | 048 | 038 | 7.0 |

A basic calculation shows that the DCPA is 1.2

Make a triangle of relative motion.

Determine the relative position A4 of the target ship at 09 : 15.

Create a tangent of distance circle through A4 with DCPA = 2

The parallel line A3Y11 of A4Y1 is drawn through point A3.

Take point

Make a parallel line of the A1A3 tangent to the 2.0

Ship collision avoidance.

Figure

The architecture diagram of RNN.

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. The output of a sequence is also related to the previous output. The specific form is that the network will remember the previous information and apply it to the calculation of the current output. The 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.

The error signal is defined as [

The loss function is defined as

For the absolute integrable function

For the absolute integrable function

For the function

The motion model is shown in Figure

Ship motion.

The mathematical model is as follows:

The dynamic models of underactuated ships can be calculated as follows:

The velocity equation is as follows:

The wind disturbance is

The ocean current disturbance is

The wave disturbance is

The theoretical basis of gradient descent method is the concept of gradient. The 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. The weight is updated using the gradient method:

The fractional order derivative of

The derivative of

Substituting (

Substituting (

Substituting (

The loss function

From the Taylor mean value theorem with Lagrange remainder, we have

Applying (

The proof of Lemma

The set {L(t)} is convergent:

From (

Every bounded monotonic sequence of real numbers converges. From Lemma

The proof of Lemma

Assume that (

Denote

From (

Since

When

Thus, we have

Substituting (

Substituting (

Thus, we have

From Theorem

Denote

The tracking error of position and attitude is

The tracking error of velocity is

The following fractional sliding surface functions are constructed:

The derivative of (

Substituting (

The sliding mode control law is constructed:

Subtracting (

Denote

(

The adaptive control law is constructed:

The fractional sliding surface is constructed:

The derivative of (

Substituting (

The derivative of (

The derivative of (

It can be obtained by calculation that

The derivative of (

Substituting (

The control rate is designed as follows:

Subtracting (

Denote

(

The adaptive control rate is constructed:

With fractional-order sliding mode controllers (

Define the Lyapunov function:

The derivative of (

Substituting (

Substituting (

Substituting (

The derivative of (

Substituting (

Substituting (

Substituting (

Substituting (

Substituting (

Substituting (

Substituting (

The derivative of (

Substituting (

Substituting (

The following conditions are established:

We have

The system is asymptotically stable, given that the stability of the presented method cannot be sustained due to the dynamic memory of the fractional operators.

A container ship [

Sea weather condition data.

Time of observation | Significant wave height (m) | Maximum wave height (m) | Wind speed (m/s) | Maximum wind speed (m/s) | Wind direction | Wind force |
---|---|---|---|---|---|---|

07.50 PM | 1.40 | 2.20 | 9.9 | 12.6 | North-northeast | 5 |

08.00 PM | 1.60 | 2.40 | 10.40 | 12.60 | North | 5.00 |

08.10 PM | 1.60 | 2.40 | 10.80 | 14.10 | North-northeast | 6.00 |

08.20 PM | 1.60 | 2.40 | 11.50 | 13.80 | North-northeast | 6.00 |

08.30 PM | 1.70 | 2.20 | 10.60 | 13.70 | North-northeast | 5.00 |

08.40 PM | 1.70 | 2.20 | 10.60 | 13.30 | North | 5.00 |

Table

Tidal height on Dec. 30, 2019.

Tide time | Tidal height |
---|---|

12.00 AM | 447 |

01.00 AM | 516 |

02.00 AM | 549 |

02.04 AM | 549 |

03.00 AM | 524 |

04.00 AM | 434 |

05.00 AM | 319 |

Table

Ship collision avoidance parameters.

Before collision avoidance | After collision avoidance | ||
---|---|---|---|

Ship no. | Course (°) | Speed (kn) | Course |

1 | 137 | 10.5 | 118.3 |

2 | 314.6 | 7.5 | 298.3 |

Figure

Speed response curve of ship 1.

Figure

Position and attitude response of ship 1.

Figure

Roll response curve of ship 1.

Figure

Input rudder angle curve of ship 1.

Figure

Speed response curve of ship 2.

Figure

Position and attitude response curve of ship 2.

Figure

Roll response curve of ship 2.

Figure

Input rudder angle curve of ship 2.

Figure

Position of two ships.

Figures

Table

Table

Comparison of the control performances of different algorithms.

SMC | RNN-ASMC | FRNN-AFOSMC | |
---|---|---|---|

Overshoot | 19% | 13% | 9% |

Adjustment time (s) | 192 | 176 | 156 |

Figure

Performance comparison of different algorithms.

Figure

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

The data used are included within the manuscript.

The authors declare that they have no conflicts of interest.

This work was supported by Xiamen Social Science Research Project from Xiamen Federation of Social Sciences and Xiamen Academy of Social Sciences (XMSK2021C04).