Aiming at the requirement that the guidance law should meet the minimum miss distance and the desired terminal angle at the same time, a sliding mode variable structure control method is introduced. In order to improve the fuzzy variable structure guidance law for maneuvering target attack effect, a neural network to the optimization design is carried out on the guidance law. The neural network is trained by the samples, which is under the condition of different error coefficient of angle, the coefficient of reaching law, and the coefficient of on-off item about target. Fuzzy neural sliding mode guidance law with terminal angle constraint can increase the performance of the large maneuvering target. In addition, on the basis of the traditional PC platform visual simulation system, a new guidance law simulation platform based on embedded system and virtual reality technology is formed. The platform can verify the validity of the guidance law.

An air-to-surface missile or guided bombs are precision weapons to attack ground targets launched from the aircraft, where the precision strike is concerned with many other factors, for example, the guidance system of the terminal guidance law design is critical, and it directly affects the final precision strike weapon capacity.

The performance of the guidance system directly affects the missile’s precise guidance capability. The entire guidance process of the missile will be divided into 3 stages: the first stage guidance, the middle stage guidance, and the last stage guidance, and the performance of the last stage guidance will directly determine whether the missile can effectively strike the target, so the study of the final guidance law is to improve the overall missile. The guidance ability of the system is of great help, and it is in this context that the research work on the terminal guidance law of the missile is carried out. The guidance law is to control the missile to intercept the target according to a certain trajectory according to the relative motion information of the missile and the target. Therefore, the problem solved by the guidance law is the flight trajectory of the missile intercepting the target [

For precision-guided weapons, the main task of the guidance system is to output appropriate commands, which ultimately makes the missile’s end miss distance as small as possible. However, under certain special circumstances, while requiring the missile to accurately hit the target, it also requires the missile to have an optimal attitude when hitting the target. It is necessary to study the guidance law with the angle-of-restriction in depth and design a guidance law that can meet the requirements of miss distance and angle-of-fall constraint at the same time.

The current guidance laws in engineering practice are mostly the classic guidance laws formed in the 1960s to the 1970s, or improved versions based on these classic guidance laws. The typical guidance law representative is proportional guidance because it has the most improved versions. Proportional guidance was initially designed only for the target to be stationary, that is, the target is not maneuvering, and under the condition that the control energy is not constrained, then proportional guidance is the optimal guidance law for zero miss. However, when targeting a maneuverable target, the proportional guidance law has a relatively large off-target volume, which simply cannot meet the accuracy index required by the missile. Therefore, it is necessary to expand the proportional guidance method for development requirements. The modern guidance law has been developed with the progress of modern control theory and gradually applied to engineering. Typical representatives include optimal guidance law, variable structure guidance law, neural network guidance law, and fuzzy logic guidance law.

Our main contribution in the present paper is that we simulate and analyze the guidance law through MATLAB software. Then BP neural network fuzzy guidance law has been optimized. Therefore, a new type of fuzzy neural network variable structure terminal guidance law is obtained. Meanwhile, in this paper, a new guidance law simulation platform based on embedded platform and PC platforms using virtual reality technology is achieved, compared with the traditional MATLAB software simulation platform, the new platform is close to the underlying algorithm engineering practice, and the effect is closer to the actual battlefield display, making it easier to verify the excellent characteristics of guidance law.

The rest of this paper is organized as follows. In Section

Both missiles and targets can be seen as two different particles in space, missile and target coordinate systems are simplified into the same coordinate system, and the coordinate system is established with the distance between the missile and the target as the

The relative motion relationship between the missile and the target is shown in Figure

The relative two-dimensional relation of the terminal missile.

Equation of relative motion for both missile and target is as follows:

Derivation of time on both sides of equation (

For equation (

Putting equation (

Derivation of time on both sides of equation (

In equation (

Let

Substituting equation (

From equations (

The equation is integrated and simplified according to formula (

Equation (

In the usual missile terminal guidance, the designers hope that the guided weapon can strike ground targets at high impact angle or even vertical angles, it is necessary to ensure that the miss distance is the smallest, and the large terminal angle control of the hit target is required, and this puts forward higher requirements for the missile’s terminal guidance. Therefore, in the design of the guidance law, it is necessary to consider the issue of miss distance and the control of the missile’s terminal angle [

According to the terminal angle requirement in the terminal guidance, the related relational expression in equation (

Let the expected terminal angle of the end of the missile be

It can be seen from equation (

Derivation of time for each variable in equation (

Substitute equation (

The joint expressions (

In the equation of state of equation (

Equation (

For the problem with the terminal angular constraint, the purpose of the guidance law design is to obtain zero miss distance and the expected terminal angle at the same time, that is, the outputs

The switching function of the sliding surface is as follows:

The physical meaning of this formula is as follows: when the relative distance

Substitute equation (

In order to ensure that the state of the system can reach the sliding mode and have excellent dynamic characteristics in the process of reaching the sliding mode, the reaching law can be used to derive the controller.

The general exponential reaching law and constant velocity reaching law can only be applied to linear time-invariant systems, and the system state equation (

The general expression of the sliding mode reaching law for a linear time-invariant system is given by

The general expression of the adaptive sliding mode reaching law is as follows:

In equation (

The physical meaning of equation (

Differentiating (

Substituting (

Bringing (

Equation (

The adaptive sliding mode guidance law has relatively strong robustness to changes in system parameters, and the speed change during the missile’s terminal guidance process is not very large, so it can be made equivalent processing, which is

So the law of guidance is obtained as follows:

Bring equation (

In formula (

The reasoning process of the fuzzy system is as follows: first, compare the differences between the input variables and membership functions to obtain the membership of each language; then the inference engine finds the corresponding rules in the knowledge base through inference operations; finally, all the results are superimposed for fuzzy output.

To perform fuzzy processing on

When the system is running,

Define the fuzzy language words set of input variables and output variables as follows: {negative large, negative middle, negative small, zero, positive small, positive middle, positive large}, which is expressed as characters: {NB， NM， NS， O， PS， PM， PB}.

The fuzzy universes of the input variables

Membership functions of input variable S.

Membership functions of input variable SC.

The fuzzy set universe of output variable

Membership functions of output variable

In order to ensure that each fuzzy language variable can cover the entire universe better, here each fuzzy language word set uses 7 variables, and each fuzzy set universe contains 15 quantization levels, so that the universe elements is twice the number of elements in the fuzzy language words set, to achieve full coverage of the universes.

The fuzzy control rules are shown in Table

Fuzzy logic control rule table.

NB | NM | NS | O | PS | PM | PB | |
---|---|---|---|---|---|---|---|

NB | PB | PB | PM | PM | PS | O | O |

NM | PB | PB | PM | PS | PS | O | O |

NS | PM | PM | PM | PS | O | NS | NS |

O | PM | PM | PS | O | NS | NM | NM |

PS | PS | PS | O | NS | NS | NM | NM |

PM | PB | O | NS | NM | NM | NM | NB |

PB | O | O | NM | NM | NM | NB | NB |

The output of the fuzzy surface.

Trajectories of different velocities.

Trajectories of different terminal angles.

Simulation was done in MATLAB software. Simulation conditions are as follows: an air-to-ground missile attacks an object on the ground, let the initial position of the missile be (0, 2000), missile speed

When the terminal angular constraint is

It can be seen from Figure

When the target speed is

It can be seen from Figure

From Table

Guidance effect under different velocities.

Speed (m/s) | Miss distance (m) | Falling angle deviation (°) | Time of flight (s) |
---|---|---|---|

15 | 3.3107 | 47.8968 | 8.3200 |

10 | 15.6580 | 10.1036 | 8.2500 |

5 | 2.0579 | 33.4270 | 8.2700 |

0 | 0.9258 | 0.0012 | 8.2300 |

Guidance effect under different angles.

Terminal angle constraint | Miss distance (m) | Falling angle deviation (°) | Time of flight (s) |
---|---|---|---|

−90 | 3.3107 | 47.8968 | 8.3200 |

−80 | 1.2544 | 5.7171 | 8.1500 |

−70 | 1.6046 | 15.840 | 8.0300 |

−60 | 2.2710 | 26.0932 | 7.9600 |

Parameter values are as follows: target speed = 15 m/s, terminal angle constraint =

Figures

Ballistic trajectory at different values of

Trajectory inclination angle at different values of

Missile normal acceleration at different values of

Guidance effect under different values of reaching law coefficient

Reaching law coefficient | Miss distance (m) | Falling angle deviation (°) | Time of flight (s) |
---|---|---|---|

0.1 | 0.3023 | 2.0896 | 8.4000 |

2 | 2.2023 | 5.2486 | 8.1200 |

10 | 0.3531 | 6.6584 | 7.8100 |

30 | 2.3912 | 173.0585 | 7.6900 |

It can be seen from the simulation diagram that when

Parameter values are as follows: target speed = 15 m/s, terminal angle constraint =

Figures

Ballistic trajectory at different angular error coefficients.

Trajectory inclination at different angular error coefficients.

Missile normal acceleration at different angular error coefficients.

As can be seen from Table

Guidance effect at different values of angular error coefficient

Reaching law coefficient | Miss distance (m) | Falling angle deviation (°) | Time of flight (s) |
---|---|---|---|

0.1 | 592.8644 | 149.5891 | 7.1100 |

0.5 | 0.8330 | 0.1252 | 8.3600 |

2 | 1.4469 | 6.4344 | 8.0900 |

5 | 2.6349 | 7.7193 | 7.9300 |

By setting different reaching law coefficients and angular error coefficients, different ballistic trajectories, trajectory inclination, and missile normal accelerations are obtained. Through the analysis of miss distance data and falling angle error data, it can be seen that the two problems of end-guided miss distance and terminal angle constraint are solved simultaneously. Through these characteristic curves, comparing the parameters such as miss distance and time of flight, different parameter value ranges are obtained under the conditions of terminal angle constraints and precise guidance.

The missile uses fuzzy variable structure terminal guidance law at the end, which can better hit low-speed ground targets, such as tanks and armored vehicles with a speed of 15 m/s.

But for high-speed targets, the effect is not ideal. The main reason is to achieve the constraint of the terminal angle, and the missile needs to track the target trajectory in time by increasing the overload. But in the process of terminal guidance, the time is very short, and it is very difficult for the missile to provide a large overload in a short time. So for missiles, the guidance law is always expected to provide a straight or smooth trajectory [

Different reaching law coefficients and angular error coefficients in variable structures have an impact on the final guidance law; when tracking a large maneuvering target, these two coefficients have an optimal range, so when the target is a large maneuver, they can automatically learn and adjust these two coefficients. The neural network system formed by this training should be able to track well big maneuvering target.

In the network structure design, a single hidden layer BP neural network is used. The theory proves that a feedforward network with a single hidden layer can map all continuous functions, and only two hidden layers are needed when learning discontinuous functions. Therefore, a single hidden layer can be used to map the fuzzy guidance law. The input layer has three input variables, which are the input line-of-sight angular rate

Figure

Neural network guidance loop.

In the above formula,

Considering the number of neurons in the input and output layers and the number of training samples, the number of hidden neurons was finally determined to be 20 after several simulations.

The transfer function of the hidden neuron is a nonlinear transfer function

Based on the fuzzy logic controller, this group of samples is obtained by adjusting the sizes of

Select several typical situations are as follows:

In a neural network, to determine the weight matrix from the hidden layer to the output layer, it represents 49 regular outputs. Use the rule table in Table

Fuzzy rule table.

NB | NM | NS | O | PS | PM | PB | |
---|---|---|---|---|---|---|---|

NB | PB | PM | PM | PM | PS | O | O |

NM | PS | PB | PM | PS | PS | O | O |

NS | PM | PM | PM | PS | O | NS | NS |

O | PB | PM | PS | O | NS | NM | NS |

PS | PS | PS | O | NS | NS | NB | NM |

PM | PB | O | NS | NM | NM | NM | NB |

PB | O | O | NM | PS | NM | NB | NB |

Fuzzy neural network training results.

An air-to-surface missile attacks an object on the ground, let the initial position of the missile be (0, 2000), the speed of the missile is

Ballistic trajectory of two algorithms.

Normal acceleration of the two algorithms.

From the ballistic trajectory in Figure

Figure

Table

Guidance effect comparison.

Miss distance (m) | Falling angle deviation (°) | Time of flight (s) | |
---|---|---|---|

Fuzzy guidance law | 0.8238 | 7.2829 | 8.1100 |

Neural fuzzy guidance law | 0.4029 | 1.3893 | 8.1500 |

Virtual reality simulation first solves the mathematical model of the simulation system by a numerical analysis method and then displayed on the screen by the display technology of the computer system. This will give people an intuitive and realistic experience. The actual missile guidance law is solved by a missile-borne computer, which is a typical embedded system, which is quite different from a PC platform. If the designed guidance law can be calculated in an embedded system, the performance of the guidance law will be better verified. The above-mentioned fuzzy variable structure terminal guidance law with terminal angle constraint is combined with two technologies of embedded system and virtual reality to design a new guidance law simulation platform to intuitively show the research results of guidance law [

Figure

New style simulation platform framework.

Simulation parameters are as follows: the armed helicopter found a ground armored target and was ready to launch an air-to-ground missile to destroy it. Let the initial position of the missile be 2000 m above the ground, the speed of the missile is

In Figure

Virtual helicopter launching missile graph.

Virtual missile flying graph in the air.

Virtual missile accurate hit target graph.

Figure

Physical map of the simulation platform.

The missile’s terminal guidance not only needs to meet the miss distance but also requires that the terminal angle of the attack be restricted. In order to meet this requirement, this paper proposes a terminal guidance law based on sliding mode variable structure, and blurs the jitter problem in the guidance law by the fuzzy logic method. By setting different approach law coefficients and angular error coefficients, different ballistic trajectories, trajectory inclination, and missile normal accelerations are obtained. Through the analysis of miss distance data and terminal angular error data, it can be seen that the two problems of terminal guidance miss distance and terminal angular constraint are solved simultaneously. At the same time, these characteristic curves, by comparing graphics, missed targets, flight time, etc., summarized the different parameter value ranges that meet both the terminal angle constraint and the precise guidance conditions.

The BP neural network is used to optimize the fuzzy variable structure terminal guidance law with terminal angle constraint, which effectively solves the problem of fuzzy variable structure terminal guidance law for large maneuvering targets. By the neural network self-learning and adaptive capabilities using large maneuvering targets as sample inputs, use the best reaching law coefficient and angle error coefficient. The simulation results show that the fuzzy variable structure terminal guidance law optimized by the neural network has improved the guidance accuracy and other aspects significantly.

Based on this, a new guidance law simulation platform based on the combination of embedded system and virtual reality technology is designed. Simulation experiments verify the correctness of the guidance law and the display effect is better.

All data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.