A novel cooperative guidance scenario is proposed that implements fire-and-forget attacks for seeker-less missiles with a cheap finder for stationary targets and without requiring real-time communication among missiles or precise position information. Within the proposed cooperative scenario, the classic leader-follower framework is utilized, and a two-stage cooperative guidance law is derived for the seeker-less missile. Linear-quadratic optimal control and biased proportional navigation guidance (PNG) are employed to develop this two-stage cooperative guidance law to minimize the control cost in the first stage and to reduce the maximum acceleration command in the second stage when the acceleration command is continuous. Simulations and comparisons are conducted that demonstrate the effectiveness and advantages of the proposed guidance law.

Homing guidance systems that can implement fire-and-forget attacks have been rapidly developed and widely applied [

Studies on the guidance of seeker-less missiles are mainly classified into two categories. The first category employs external guidance [

To address the aforementioned issues, a new cooperative guidance scenario is proposed to implement fire-and-forget attacks for seeker-less missiles without requiring real-time communication and precise position information. In this scenario, an onboard finder that is much cheaper than a seeker is employed for a seeker-less missile. Even though the measurement information produced by such a finder (the line-of-sight angle) is identical to that of the seeker, the finder has the following features that can clearly reduce costs, as the object of measurement would be another missile in the missile cluster: (1) the lock-on distance is remarkably reduced and (2) the requirements for identification capabilities are reduced, as information on the precise design of other missiles can be derived in advance, and there is no active disturbance or invisibility among these missiles. Therefore, this finder may be considered a cheaper version of a seeker, with a reduced lock-on distance and reduced requirements for identification capabilities. To differentiate this device from commonly used seekers, this device will be defined in this paper as a “finder.” Due to its features, a seeker-less missile with this finder will not be able to employ certain common laws of guidance, for a reduced lock-on distance and reduced identification capabilities for the finder would result in the target being undetectable.

Within this proposed cooperative guidance scenario, a missile with a seeker that can independently hit the target is located in the front of the missile cluster during the flight, while other seeker-less missiles equipped with finders can hit the target in sequence by tracking the position of the nearest missile at the front, including the missile with the seeker. In this scenario, the employment of the finder reduces costs, and the number of cooperative missiles may be conveniently increased, considering the reduced lock-on distance. In addition, to implement the aforementioned cooperative guidance scenario, a two-stage guidance law is designed for the seeker-less missile by employing linear-quadratic optimal control and biased proportional navigation guidance (PNG).

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

Within the proposed cooperative guidance scenario (as shown in Figure

Proposed cooperative guidance scenario.

In this cooperative guidance scenario, the missile

For the two-stage law of guidance for the seeker-less missile to be effective, any impact between the seeker-less missile and its leader must be avoided in the first stage, which will be studied later (Remark

Considering that the proposed cooperative guidance scenario consists of multiple groups with a leader-follower framework, a single leader-follower framework is first illustrated. During the first stage of a seeker-less missile (as a leader), its followers can be considered to be in the middle guidance stage, which is out of the scope of this work and thus not introduced here. Therefore, the missile

Geometry among the leader, follower, and target.

The speed of all missiles is considered to be constant in this work, and the relative kinematic equations between the leader and follower are given as follows:

We define the error between the position coordinate for the leader and follower relative to the

To simplify the derivation of the guidance law, a linearization process is implemented. We approximate that there is no normal acceleration for the leader, as will be illustrated in Section

As the leader flies straight, it is assumed without loss of generality that

Therefore, the linearized guidance model can be derived as

As the follower only requires the employment of the common guidance law in the second stage, it is assumed that the PNG is utilized without loss of generality. Therefore, for the first stage of guidance, the following cost function is introduced to reduce the maximum acceleration for the second stage (the first term) and to minimize the control effort (the second term).

As

Therefore, according to equation (

Substituting equation (

Then, the cost function shown in equation (

It is well-known that the solution of the optimal control problem formulated by equations (

The matrixes

Substituting equations (

Therefore, equation (

If the leader is the missile

The three criteria for the guidance mentioned in Section

For the first stage of guidance, it can be derived from equation (

Evidently, the impact between the leader and follower is avoided with the condition

In addition, by considering the limited lock-on distance of the finder, the upper limit of

Because the follower can employ the PNG during the second stage of guidance, i.e.,

Substituting equation (

Since

As the initial acceleration of the PNG is at a maximum, the term

The total control cost for the first stage can be optimized by considering the term

Clearly, it is more convenient to implement the guidance algorithm if it is reformulated with respect to variables that can be directly measured by onboard instruments. Therefore, an alternative measurable state vector for the acceleration command during the first stage is provided in this subsection. With the assumptions that

Moreover, because equation (

Then, substituting equation (

In the second stage of guidance, if the follower employs the PNG, the acceleration command would be discontinuous at the initial time, since

Both

In our analysis, three seeker-less missiles with finders, guided by a missile with a seeker, cooperatively hit a stationary target located at 12000 m and 1000 m. The measurement relationship between the missiles and their initial states of motion are presented in Figure ^{2}, and 1200 m/s^{2}, respectively.

The relationship between the missiles.

Initial states of the missiles.

Parameters | ||||
---|---|---|---|---|

Position, |
(6, 1) | (5, 0.5) | (4, 0) | (3, 0.5) |

Velocity, |
3.5 | 3.7 | 3.7 | 3.7 |

Heading angle (deg) | 10 | 0 | 0 | 0 |

The simulation results for the proposed two-stage guidance law are presented in Figure

Simulation results for the proposed cooperative guidance law.

Trajectory

Leader-follower distance

Lead angle

Normal acceleration

To verify the superiority of the proposed cost function for the first stage and the biased PNG for the second stage, two modified two-stage guidance laws using the existing formulations are introduced. For the first modified guidance law, the zero-effort miss distance is considered to produce an optimal state of initial motion for the second stage of guidance, as is commonly done in the literature; i.e., the cost function shown in equation (

With the first or second modified two-stage guidance law, all the miss distances of

Normal acceleration command for modified guidance laws.

1st modified two-stage guidance law

2nd modified two-stage guidance law

To achieve a precise hit of a seeker-less missile for a stationary target, a novel cooperative scenario for guidance is proposed in this work that is able to implement fire-and-forget attacks for seeker-less missiles without real-time communication or precise position information. Within the proposed novel cooperative guidance scenario, a two-stage cooperative guidance law is derived for a seeker-less missile. The guidance law for the first stage can produce the minimum control cost for this stage as well as the reduced maximum acceleration command for the second stage. The guidance law for the second stage can guarantee a precise hit for a seeker-less missile with a continuous acceleration command. Simulation results illustrate the effectiveness and superiority of the proposed two-stage guidance law.

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

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was supported by the National Science Foundation of China (grant number 11532002) and the Hongjian Innovation Foundation of China (grant number BQ203-HYJJ-Q2018002).