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In this paper, a new dynamic surrounding attack cooperative guidance law against highly maneuvering target based on decoupled model is proposed. First, a new dynamic surrounding guidance strategy is proposed, and virtual targets are introduced to establish the cooperative guidance model for dynamic surrounding attack. Second, a dynamic inverse method is used to decouple the cooperative guidance model, and extended state observers (ESOs) are introduced to estimate the disturbances caused by target maneuver. Then, the impact time and dynamic surrounding guidance (ITDSG) law against highly maneuvering target is designed based on a prescribed-time stable method and the decoupled model. Finally, numerical simulations are performed to illustrate the superiority and effectiveness of the proposed ITDSG.

Cooperative guidance laws for multimissiles have captured the interest of many researchers since the seminal work by I.S. Jeon first appeared [

Considering the communications between missiles, some researches have proposed cooperative guidance laws against static or constant velocity target. Zhou and Yang [

Unlike the above scenarios, some researchers have focused on the research of cooperatively intercepting maneuvering target with multimissiles in recent years [

In order to improve the effort of attacking a target, surrounding attack strategy must be considered. In the existing literatures, in order to achieve the surrounding attack, the term LOS angle constraint is added to the basic guidance law, so that the designed guidance law can ensure the missile to approach the target with a desired LOS angle. The guidance laws for single missile attacking single target with a LOS angle constraint have been extensively studied [

Motivated by the aforementioned papers, in order to achieve the dynamic surrounding attack of multiple missiles against a highly maneuvering target and further consider the couplings between the LOS and normal LOS directions in the cooperative guidance model, we proposed the ITDSG in this paper. The main contributions of this paper can be summarized as follows:

We proposed the strategy of dynamic surrounding attack and established the cooperative guidance model between multiple missiles and virtual targets. Due to the traditional cooperative guidance laws with angle constrain need a preset LOS angle [

By using the dynamic inverse method, the coupled cooperative guidance model is decoupled, and the ITDSG is designed based a prescribed-time stable method subject to the decoupled model. This is opposed to Ref [

The proposed cooperative guidance in this paper requires less maximum overload and energy consumption compared with Ref [

The remainder of this paper is organized as follows. Section

In this section, the model description and the basic knowledge of consensus protocol are introduced.

The relative motion geometry of single missile and single target can be formulated as follows:

Differentiating Eqs. (

In Eqs. (

In order to ensure that missiles hit the target simultaneously, a variable

Compute the derivation of

Combining Eqs. (

Considering the problem of

The disturbance

Guidance geometry on Missiles-Target (virtual targets) engagement.

Suppose that communication network graph between agents can be expressed as

In this paper, the communication network graph of multiple missiles is assumed to be undirected and connected.

The first-order multiagent system with

For a first-order multiagent system as Eq. (

Consider a nonlinear system defined as

Select a continue and differentiable Lyapunov candidate function

For a nonlinear system with an unknown and bounded disturbance

Firstly, a novel dynamic surrounding attack strategy is proposed in this section by introducing the virtual targets. Then, we propose the calculation formulas of the position of virtual targets using measurable real target information. Finally, considering the couplings between the LOS and normal LOS directions of the guidance model in Eq. (

The difference between dynamic and static surrounding attack is whether a preset LOS angle constraint is required. The static surrounding attack is achieved by presetting different LOS angle constraint for each missile to achieve multiple missiles attacking the target from different directions. This attack strategy is determined at the beginning of the missile terminal guidance, which will not change with the maneuvering of the target. By introducing multiple virtual targets, the dynamic surrounding attack strategy without preset LOS angle constraint is achieved. The virtual targets are calculated according to the movement of real target, and multiple virtual targets gradually approach the real target from different directions.

In the scenario of single missile intercepting a target, at the terminal phase of intercepting, the target usually maneuvers to avoid being hit by a missile. Figure

Target escape area.

Target escape area in Case

Target escape area in Case 2

Figure

Guidance geometry on Missiles-Targets engagement.

The dynamic models of the missiles and virtual targets can be established by the three steps. Firstly, calculate the coordinate of boundary points subject to the real target position and velocity information. Then, the coordinates of

Calculate the boundary point.

Figure

Relative geometrical between target boundary point and initial position of target.

However, the seeker can only obtain the relative information of the missile and the target, such as

The boundary points can be calculated as follows. When

What is more, the coordinates (

Calculate the coordinates of virtual targets.

Considering the scenario that

When

As a result, by applying Eqs. (

Establish the dynamic model.

Similar to Eq. (

In the initial stage of intercepting, the

In this section, an adaptive dynamic inverse method is used to decouple and linearize Eq. (

Applying Lemma

According to the dynamic inverse method, the controller can be computed as follows:

Substituting the above controller

The above equation is the expected linearization one;

Finally, combining with the Eqs. (

Based on the above design method of the decoupled guidance model, a nonlinear and strong coupled guidance problem can be transformed into linear control stabilization problem. By designing the controller, all state variables can converge to 0.

It is easy to obtain that, as

For the multiple-missile and multiple-virtual target system Eq. (

Define the following Lyapunov candidate function:

The derivative of Eq. (

By applying Eqs. (

Since

The convergence speed of the system states is controllable with the proposed guidance law in this paper. From Eq. (

To illustrate the effectiveness and superiority of the proposed ITDSG, we design two groups of simulations in the scenario of three missiles cooperative attacking a target with constant velocity and highly maneuvering target, respectively. Simulations with the slide mode cooperative guidance law (SMCG) in [

Let us consider the scenario of three missiles attacking single target from different directions; the first group simulations are performed to verify the performances of the proposed ITDSG compared with SMCG.

Table

Simulation conditions for missiles.

Missile | Initial position (m) | ||||
---|---|---|---|---|---|

Missile 1 | (8000, 0) | 2 | ±30 g | 600 | 18 |

Missile 2 | (9000, 1500) | 6 | ±30 g | 600 | 19 |

Missile 3 | (7000, -800) | -6 | ±30 g | 600 | 17 |

Assume that the communication network graph of three missiles is connected and undirected, Figure

Communication topology for three missiles.

The ESO parameters in LOS and normal LOS direction are the same; they are set as

Since the energy of the missile is limited, so the smaller energy needs to provide by the guidance law is, the more favorable application the guidance law is. Therefore, an energy consumption evaluation indicators are defined as Eq. (

Figure

Virtual targets with constant velocity target.

Simulation curves for a constant velocity target.

Trajectories for missiles and target

Distance between missiles and target

LOS angles for missiles

Time-to-go for missiles

Accelerations on the LOS direction

Accelerations on the normal LOS direction

The effectiveness of the proposed ITDSG against a constant velocity target can be verified as follows: Figures

In terms of missile trajectories, ITDSG has a better performance than SMCG in Figure

Energy consumption.

Missile | Case 1 | Case 2 | |||
---|---|---|---|---|---|

ITDSG | SMCG | ITDSG | SMCG | ||

^{2}) | Missile 1 | 254.4 | 220.4 | 314.5 | 1025.4 |

Missile 2 | 133.9 | 86.5 | 217.9 | 167.8 | |

Missile 3 | 222.5 | 565.9 | 504.6 | 548.4 | |

^{2}) | Missile 1 | 553.9 | 1024.7 | 1021.5 | 1858.7 |

Missile 2 | 53.7 | 41.8 | 944.9 | 911.1 | |

Missile 3 | 647.3 | 4605.3 | 1076.7 | 1740.4 |

In terms of the impact time, the SMCG and ITDSG have the same performance. From the simulation curves in Figure

As a result, in the scenario of intercepting a constant velocity target, ITDSG and SMCG both show good performance. Because ITDSG uses a dynamic surrounding attack strategy without LOS angle constraint, the trajectories by ITDSG are smoother than that by SMCG; the accelerations in normal LOS directions are also smaller.

In this case, the performance of the proposed ITDSG is investigated in the scenario of three missiles attacking single highly maneuvering target from different directions.

In order to further illustrate the superiority of ITDSG over SMCG, simulations are conducted with the same cooperative guidance laws and the same missile parameters as those in Case 1, the only difference is that the target maneuver with a bigger overload ^{2}. Furthermore, the selection principle of

Real and virtual targets with high maneuvering.

Similar to Case 1, Figure

Simulation curves for a highly maneuvering target.

Trajectories for missiles and target

Distance between missiles and target

LOS angles for missiles

Time-to-go for missiles

Accelerations on the LOS direction

Accelerations on the normal LOS direction

Similar to the analysis in Case 1, Figure

As a result, the proposed guidance law ITDSG can be suitable for constant velocity speed as well as highly maneuvering target. Besides, by introducing virtual targets instead of impact LOS angle to achieve dynamic surrounding attacks, ITDSG has better performance in trajectories and energy consumption than SMCG.

In this paper, we developed a new dynamic surrounding attack cooperative guidance law against highly maneuvering target without a preset LOS angle constraint. Firstly, we proposed the strategy of dynamic surrounding attack by introducing virtual targets and then established the cooperative guidance models between multiple missiles and multiple virtual targets. Finally, by using the dynamic inverse method to decouple the coupled cooperative guidance model, and the ITDSG subject to the decoupled model is designed based a prescribed-time stable method. Additionally, ESOs are introduced to estimate the disturbances in the LOS and normal LOS directions. In order to demonstrate the effectiveness and superiority of the proposed ITDSG, two groups of comparison simulations are carried out with SMCG in the scenarios of three missiles cooperative attack a constant velocity or highly maneuvering target.

Compared with the existing works, the advantages of the proposed ITDSG are that it does not require to set desired LOS angles in advance, and the couples in the cooperative guidance model are also considered. The limitation of the proposed guidance law in this paper lies in the need for precise target information, such as

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

The authors declared that they have no conflicts of interest to this work.

Thanks are due to the financial support provided by the National Natural Science Foundation of China (Grant No. 61973253), the Aviation Science Foundation of China (20180153001), and the Foundation of National Defense Science and Technology Key Laboratory (6142219180202).