Integrated cooperative guidance framework and cooperative guidance law for multi-missile

Jianbo ZHAO,Shuxing YANG,*

aSchool of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China

bKey Laboratory of Dynamics and Control of Flight Vehicle,Ministry of Education,Beijing 100081,China

1.Introduction

In modern military operation,it is a challenging task for the missile to attack a land target or a surface ship that is equipped with an antiair defense system.1To penetrate the antiair defense system,the missile should be capable of terminal evasive maneuvering,which would induce increasing cost.2An alternative way is to conduct a cooperative attack,i.e.,multiple missiles coming from different directions attack a single target simultaneously,which has been regarded as a cost effective and efficient way to address the threat of the defense system.3Therefore,the research on cooperative attack has gained increasing interest in recent years.Considering that a land target or a surface ship is either stationary or moving at a relatively low speed,the target in the cooperative attack problem in this paper is considered to be stationary as is commonly done in practice.

To achieve cooperative attack,one can perform an openloop cooperative guidance,i.e.,a common impact time is generated for all member missiles in advance,and thereafter each missile tries to arrive at the target on time independently.4–7However,a suitable common impact time is difficult to be generated in advance,since some missiles may not be able to satisfy the impact time constraint due to their specific initial conditions and limited speed.In addition,the open-loop guidance is also lack of robustness to external disturbance during engagement.8Actually,the open-loop cooperative guidance simply formulates the many-to-one cooperative attack problem as multiple one-to-one attack problems considering the impact time constraints,which cannot be considered as a genuine multi-missile cooperative guidance.9

Alternatively,with the closed-loop cooperative guidance approach,the missiles communicate to each other to synchronize the impact time.10–12As a well-known closed-loop cooperative guidance method,the Finite-Time Cooperative Guidance(FTCG)has gained much attention due to its fast convergence rate and high accuracy of the time-to-go errors of missiles.To attack a maneuvering target,Song et al.13proposed a FTCG law with impact angle constraints.In Ref.1,two distributed FTCG laws are developed based on different time-to-go estimation methods.However,in addition to the normal acceleration command,the tangential acceleration was also required with both FTCG laws,causing extra difficulties in implementation.To address this problem,a more effective FTCG law employing a hierarchical framework was proposed in Ref.14,which requires only the normal acceleration command.

For the closed-loop guidance laws introduced above,either a centralized or decentralized communication topology is utilized.However,both communication topologies are far from the optimum,15and their drawbacks shown as follows will become much more prominent if more missiles are involved in the cooperative attack.With the centralized communication topology,one or a few missiles should be designated,which can communicate with all the rest missiles.Clearly,the distances between the designated missile and the rest ones are subject to their communication capability,which incurs extra difficulty in designing the commands for cooperative missiles.In addition,poor penetration capability and system reliability would be induced if only one missile is designated,while unacceptable computational burden would be induced if more than one missile are designated.In contrast,with the decentralized communication topology,missiles can communicate only with their neighbors to reduce the computational burden,while the global information of the missile cluster cannot be obtained,causing great difficulty in making optimal decision for cooperative missiles.Moreover,it takes long time to achieve the consensus of time-to-go and some necessary conditions16are always hard to be satis fied.In addition,due to the limitation of the detective capability of seekers,the saturation constraint on Field-of-View(FOV)should be considered in the closedloop cooperative guidance,which however has been rarely seen in the existing works.And the existing FOV-constraint guidance for the common one-to-one missile-target engagement scenario17–21cannot be directly applied to the closed-loop cooperative guidance.

To address the issues above,a novel integrated cooperative guidance framework is proposed in this paper,in which missiles are distributed into several groups,and then missiles within a single group communicate by means of the centralized leader-follower framework,while the communication between groups employs the nearest-neighbor topology among leaders.The contributions of this paper lie in:(A)the centralized and decentralized communication topologies are combined and integrated into the proposed framework effectively;(B)to implement the proposed integrated cooperative guidance framework,a group of FTCG laws considering the saturation constraint on FOV(FTCG-FOV for short)are designed in this paper by extending the FTCG law in Ref.21to a group of FTCG laws and introducing a bias term to satisfy the FOV constraint;(C)the sequential approach in Ref.21is improved,which is then employed to make the FTCG-FOV satisfy the requirement of communication between groups.

The rest of this paper is organized as follows.In Section 2,the many-to-one missile-target interception engagement along with the proposed integrated cooperative guidance framework is introduced.In Section 3,a group of FTCG-FOV are developed,of which the working process in the integrated cooperative guidance framework is presented in detail.Simulation results are presented and analyzed in Section 4.Conclusions are made in the last section.

2.Preliminary

2.1.Problem formulation

It is considered that n missiles Mi(i=1,2,...,n)are followers and one Mlis the leader,cooperatively attacking a stationary target T.The engagement scenario is shown in the inertial reference coordinate OXY in Fig.1,in which the variables with the subscripts i and l represent the states of the follower i and the leader,respectively.Furthermore,V，a，θ，q，η and r denote the speed,normal acceleration,heading angle,Line-of-Sight(LOS)angle,lead angle,and rang-to-go,respectively.

In this work,the following assumptions are made to simplify the design and analysis process of the proposed guidance method:

(1)Missiles and target are regarded as mass points in the yaw plane.

(2)Velocities of missiles are constant.

(3)Compared with the guidance loop,the dynamic lags of autopilots and seekers can be ignored.

(4)The Angle-of-Attack(AOA)is small and can be neglected.

(5)The lead angle of each missile is small when the rangeto-go of the missile is small enough.

Assumptions 1-3 are commonly employed in the derivation of guidance laws.14Assumption 4 aims to make the lead angle as FOV,which has also universally been made.17,18,21Assumption 5 is adopted to linearize the engagement model and simplify the proof process,which is reasonable because a missile has completely locked on the target when its range-to-go is small enough.

The elapsed time is denoted as t.The relative kinematics equations of the leader and followers are given as follows:

2.2.Integrated cooperative guidance framework

For the proposed integrated cooperative guidance framework shown in Fig.2,missiles are firstly distributed into several groups and each group includes three or four missiles.Then,the centralized leader-follower framework is employed within a single group and the decentralized topology is utilized among all leaders.With this integrated framework,the consolidated coordination variable obtained via the communication among leaders can be broadcasted to all followers in real time.Compared to the existing centralized communication,more missiles can be involved in the proposed integrated framework through increasing the groups of missiles,inducing only a slight increase of the computation and communication burden of leaders.Moreover,compared to the existing decentralized communication,sincethecentralized communication is applied within each single group,the proposed integrated framework inherits many advantages,such as the suboptimal coordination variable,better convergence rate of time-to-go error and simpler communication topology.

The reasons that the leader-follower framework is adopted as the communication topology within a single group instead of the hierarchical one in the proposed integrated cooperative guidance framework are as follows.Firstly,only leaders are equipped with the information transmitters to reduce the cost.Secondly,all-communication is not required within a single group,without significantly sacrificing the penetration capability and system reliability.Thirdly,since the one-way communication is employed in the leader-follower framework,better real-time communication capability can be obtained.Fourthly,the design of guidance law for the follower is much more flexible.For example,followers can directly track the position and velocity of the leader.

3.Cooperative guidance law

3.1.FTCG laws with saturation constraint on FOV

To satisfy the communication topology of missiles within a single group,a group of FTCG-FOV is developed based on the centralized leader-follower framework,i.e.,the leader and followers implement the Proportional Navigation Guidance(PNG)and bias PNG,respectively.Hence,the acceleration command can be expressed as

where N is the navigation gain;aB1（t）and aB2（t）are used for the coordination of attack time and the saturation constraint on FOV,respectively.

Define the square of the relative error of the time-to-go between the leader and follower i as

where tgo，i（t）and tgo，l（t）are the estimated time-to-go of follower i and leader separately,and both of them can be estimated by2

In order to realize FTCG,aB1（t）is devised as

where k1is a positive constant;the positive constants m and n satisfy m ≥ n ≥ 1;the index number γ satis fies γ∈ （0，1）;sgn（·）represents the signal function.

In the lateral plane,FOV is generally regarded as the angle between the longitudinal axis of a missile and its LOS.In light of Assumption 4,FOV is identical to the lead angle.To satisfy the saturation constraint on FOV,i.e.,the constraint on lead angle,aB2is designed as

where b and k2are positive constants and b is sufficiently large;ηi，maxis the lead angle constraint for follower i;sgn（ηi）is employed to calibrate the direction of aB2;sig（x，b，θ）represents the sigmoid function defined as

Lemma 122.Consider the system=f（x），x∈Rn,with initial condition f（0）=0 and f（·）:Rn→ Rnis a continuous function.If there exists a continuous positive function defined within the neighborhood of origin as V（x）:→ R for（x）+c（V（x））β≤0，where c＞0 and β ∈ （0，1）,then the origin of the system=f（x）is finite-time stable and the stable time satis fies T ≤ ［V（x（0））］1-β［c（1- β）］-1.

Theorem 1.Considering the scenario that n missiles led by a leader cooperatively attack a stationary target,assumptions in Section 2 are valid.Then,with the proposed cooperative guidance law shown in Eq.(10),the square of time-to-go error δi（t）will converge to zero in finite time,and the lead angle constraint can be satis fied.In other words,multiple missiles can achieve attack time coordination in finite time satisfying the saturation constraint on FOV.

Proof.Firstly,itcan bedemonstrated thatbased on NiVi˙qi（t）+aB1（t）,the square of time-to-go error δi（t）will converge to zero in finite time.For the leader,according to Eqs.(2)–(4)and(9),the derivative of the lead angle can be deduced as

Accordingly,based on Assumption 5,the difference equation of ηl（t）can be expressed as

where Δt is a sufficiently small time interval.Hence,

Omitting high-order items,one has

In Ref.14,the difference equations of range-to-go and time-to-go have been derived as

Therefore,substituting Eqs.(19)and(20)into Eq.(21)yields

Omitting high-order items,one has

Using the similar way in deriving Eq.(19),one can obtain that

Hence,substituting Eqs.(20)and(24)into Eq.(21)yields

Omitting some high-order items,one has

The difference equation of δ（t）can be formulated as

Substituting Eqs.(23)and(26)into Eq.(27)yields

Then,according to Eq.(13),Eq.(28)can be rewritten as

where Tiis the estimated attack time of follower i.Accordingly,one has

By properly choosing the constants in Eq.(10),it can be guaranteed that ηi（t）≠ 0 during the flight.Thus,one can obtain that

Since γ∈ （0，1）,one can obtain that （γ+1）/2 ∈ （0，1）.Therefore,according to Lemma 1,the square of time-to-go error δi（t）will converge to zero in finite time.And one has

Then,it can be demonstrated that with Eq.(10),the lead angle constraint can be satis fied,as well as the requirement for finite-time missile coordination.In Eq.(14),clearly,the term|ηi|- ηi，maxis employed to increase the value of aB2and reduce|ηi|when|ηi|approaches to ηi，max.If|ηi|is identical to ηi，max,then aB2tends to be infinite,which cannot happen.Therefore,the lead angle constraint can be satis fied by introducing aB2.Moreover,the sigmoid function in Eq.(14)is essentially identical to a switching function,because the constant b in the sigmoid function is sufficiently large.And according to Assumption 5,|ηi|is small when the range-to-go of missile is small enough.Therefore,aB2is essentially identical to zero during the last stage of terminal guidance,i.e.,the impact of aB2on the finite-time consensus can be neglected.□

Remark 1.Note that according to Assumption 5,the FTCG in Ref.14can be rewritten as

where k is a positive constant;denotes the average value of time-to-go of all missiles.

Compared with the bias term in Eq.(33),some new items are introduced in Eq.(13),such as tgo，l,m and n.By employing tgo，l,Eq.(13)satis fies the leader-follower framework.Moreover,the FTCG in Ref.14is actually a specific condition of Eq.(13)when m=1 and n=1.

Remark 2.In Eq.(13),m and n are introduced,and some noteworthy points should be noted.Firstly,the reasons why m≥n≥1 is required are:(A)according to Eq.(29),if n＜1,itis difficultto choose a suitable upper bound for δ（t+ Δt）- δ（t）;(B)according to Eq.(13),if m ＜ n,an unacceptable large command of terminal acceleration would be incurred.Secondly,it is unreasonable to choose extremely large m and n because a large m or n causes a large k1,resulting in the great amplification of internal errors and external disturbances for the acceleration command.Thirdly,according to Eqs.(13)and(32),it can be deduced that the impact of m or n on the maximum aB1（t）is almost opposite to that on T.Thus,generally m and n have negligible impact on the performance of the proposed FTCG(Eq.(13)).However,in some specific conditions,better cooperative guidance performance can be generated by properly choosing m and n,which will be demonstrated in Section 4.Fourthly,compared with n,m has larger impact on the acceleration command(Eq.(13))during the initial guidance stage,since the relative variation of ηiis greater than that of ri.

Remark 3.Note that in Eq.(14),the sigmoid function is actually a switching function when the constant b tends to be in finite.However,compared with a switching function,it is unnecessary to carefully select a suitable switching point for the sigmoid function,which enhances its applicability.

Remark 4.Notice that in Eq.(10),aB2may be much greater than aB1if|ηi|is almost identical to ηi，max,inducing difficulty in the coordination of attack time.Thus,a smaller ηi，maxresults in longer coordination time.

3.2.Realization of integrated cooperative guidance framework

Inspired by the sequential approach in Ref.14,a modified sequential approach is developed to distribute the FTCGFOV,and thus to realize the integrated cooperative guidance framework.With the modified approach,a segmented technique named as ‘two-stage method’is adopted.In the first stage,the consensus algorithm is employed to calculate the feasible time-to-go,which is formulated as14

where Pidenotes the set of missiles that can communicate with missile i;aijis the element of the adjacency matrix representing the communication topology.

According to Eq.(34),the feasible time-to-go of all missiles can converge to a consolidated constantin the finite time T*.14In the second stage,based on the FTCG law,the time-to-go of all missiles can converge toin thefinite time tc.Compared with the sequential approach in Ref.14,the modified sequential approach implements the FTCG in two stages rather than one,resulting in a decrease of T*+tcand the acceleration command without any step.

In the integrated cooperative guidance framework(shown in Fig.3),the FTCG-FOV and modified sequential approach are employed.For the guidance,the FTCG-FOV is implemented by all missiles including the leaders.However,tgo，l（t）in Eqs.(11)and(13)are adjusted as the feasible time-to-go.For the consensus algorithm,the modified sequential approach is employed for leaders.In the first stage,leaders calculate the feasible time-to-go by Eq.(34)and each leader broadcasts its feasible time-to-go to followers in the same group in real time.At the end of the first stage,the values ofof all leaders,i.e.,of all missiles,converge to the same value.In the second stage,all missiles can be guided with their own guidance laws without communication.

Fig.3 Flowchart of integrated cooperative guidance framework.

Remark 5.In the integrated cooperative guidance framework,the FTCG-FOV can still meet the saturation constraint on FOV,which can be demonstrated as Theorem 1.Note that in the second stage of the modified sequential approach,the difference equation of the feasible time-to-go is formulated as

It can be found that Eq.(35)is identical to Eq.(23).Based on this,in the integrated cooperative guidance framework,the FTCG-FOV can realize the attack time coordination in finite time as well.The reason is that the first stage of the modified sequential approach can be finished in finite time,and in the second stage,the finite-time coordination can also be demonstrated as Theorem 1.

4.Simulations and analysis

An engagement scenario is considered,in which multi-missile cooperatively attacks a single stationary target located at［7000，5000］m.In order to attain the precise attack for this engagement scenario,the miss distance and overload are required to be less than 2 m and 10,respectively.FTCG,FTCG-FOV and the integrated cooperative guidance framework are respectively analyzed through simulation as follows.

4.1.Results of FTCG

For different values of m,n and tgo，lin aB1，the developed FTCG(Eq.(13))is applied.The initial conditions of the follower are shown in Table 1.

According to Eq.(23),tgo，l（t）merely depends on tgo，l（0）,i.e.,the expected impact time of the follower.tgo，l（0）is set as 28 s and five cases with different values of m and n(m/n)are investigated,as shown in Table 2.Accordingly,the simulation results of the proposed FTCG are shown in Fig.4.

The missile-target distance shown in Fig.4(a)indicates that,for different m/n,the miss distance of FTCG is less than 2 m,illustrating that the followers can precisely attack the target.Clearly,in Fig.4(b),the time-to-go error quickly converges to zero,which means that multiple missiles can realize attack time coordination in finite time for different combina-tions of m/n in Eq.(13).This demonstrates the effectiveness of Eq.(13).Moreover,from Fig.4(b),it is observed that compared with the FTCG in Ref.14(Case 1),a faster convergence rate for time-to-go error can be obtained through properly choosing m and n.The overload of FTCG in terms of different combinations of m/n is illustrated in Fig.4(c),which indicates that the constraints on overloads of the five cases are all satis fied,and that m has larger impact on the acceleration command than n,especially during the initial guidance stage.

Table 1 Initial conditions of follower.

Then,different expected impact time of follower(te=28.0 s,28.5 s,29.0 s and 29.5 s)is considered in the proposed FTCG.Moreover,for the case with the expected impact time as 29.5 s,the cooperative guidance law in Ref.9is also employed for comparison.The simulation results are presented in Fig.5.

The missile-target distance shown in Fig.5(a)suggests that,for different expected impact time,the miss distance of FTCG is less than 2 m,illustrating that the followers can precisely attack the target.In Fig.5(b),the convergence rate of timeto-go error of the proposed FTCG is much larger than that of the cooperative guidance law in Ref.9,demonstrating the effectiveness and superiority of the FTCG.Fig.5(c)indicates that the overloads satisfy the constraint,and a longer expected impact time induces a larger maximum overload command,due to the larger curvature of the trajectory of follower.

4.2.Results of FTCG-FOV

In this subsection,the FTCG-FOV(Eq.(10))is employed,considering different FOV angle(θFOV)constraints(38°,35°,32°,29°),and the initial conditions of the follower are the same as those in Section 4.1.The simulation results are presented in Fig.6.

In Fig.6(a),the trajectories of followers are presented.The missile-target distance shown in Fig.6(b)suggests that,fordifferent FOV angle constraints,the miss distance of FTCGFOV is extremely low(about 0.5 m),which is much less than 2 m,illustrating that the followers can precisely attack the target.Fig.6(c)and(d)demonstrate the conclusions of Theorem 1.Fig.6(c)indicates that the saturation constraints on FOV can be satis fied with FTCG-FOV,demonstrating the effectiveness of Eq.(14).In Fig.6(d),the time-to-go errors quickly converge to zero,which means that with FTCGFOV,multiple missiles can realize attack time coordination in finite time.Moreover,Fig.6(d)reveals that a smaller FOV angle constraint incurs longer coordination time.Fig.6(e)indicates that the overloads satisfy the constraint.

Table 2 Cases of m/n.

Fig.4 Results of FTCG for different m/n.

4.3.Results of integrated cooperative guidance framework

Fig.5 Results of FTCG for different expected impact time.

In this subsection,the integrated cooperative guidance framework is employed,in which three Leaders(Leader 1,Leader 2 and Leader 3)and one Follower cooperatively attack a single stationarytarget.The initialconditionsof allmissiles are shown in Table 3.And if only PNG is employed,the impact time of Leader 1,Leader 2 and Leader 3 and Follower is about 29.6 s,28.6 s,29.0 s and 28.1 s,respectively.The maximum deviation in impact time is about 1.5 s.Moreover,the FOV angle is restricted to be less than 30°.A suitable communication topology of missiles is chosen as shown in Fig.7.Therefore,the consensus algorithm(Eq.(34))in this case can be expressed as

Fig.6 Results of FTCG-FOV.

Table 3 Initial conditions of missiles.

Fig.7 Communication topology of missiles.

The simulation results of the integrated cooperative guidance framework are presented in Fig.8.To reveal the superiority of the modified sequential approach,the sequential approach in Ref.14is also employed,of which the results are shown in Fig.8(c)and(f).Fig.8(a)displays the trajectories of missiles.The missile-target distance shown in Fig.8(b)indicates that the miss distances of missiles are very small(less than 1 m).Fig.8(c)and(d)demonstrate the conclusions of Remark 5.In Fig.8(c),with two differentsequential approaches,both the standard deviations of time-to-go can converge to zero in finite time,which means that the employment of the integrated cooperative guidance framework can guarantee the attack time coordination in finite time.Moreover,for the modified sequential approach,the convergence rate of the standard deviation is faster.In Fig.8(d),the lead angles of missiles are less than 30°,which illustrates that the saturation constraint on FOV can also be satis fied in the integrated cooperative guidance framework.Fig.8(e)and(f)show that the modified sequential approach can satisfy the overload constraint,while the existing sequential approach14cannot.Moreover,the overload required by the modified sequential approach does not exhibit a step between two stages.

Fig.8 Results of integrated cooperative guidance framework.

Fig.9 Total calculation time of feasible time-to-go.

To investigate the computational efficiency of the integrated cooperative guidance framework,a centralized framework2is considered for comparison,in which the feasible time-to-go of each missile is determined as the average value of time-to-go of other missiles.To obtain the required computational time of the feasible time-to-go,it is assumed that the feasible time-to-go is calculated 100,000 times and each missile group has four missiles.Therefore,the total required calculation time for all missiles with respect to the group number of missiles is shown in Fig.9.It is demonstrated that the integrated cooperative guidance framework is more efficient,and this superiority will be more obvious if more missiles are involved in the cooperative attack.

5.Conclusions

(1)To address the issues of the existing communication topologies,a new integrated cooperative guidance framework is proposed in this paper,in which missiles are firstly distributed into several groups,and then missiles within a single group communicate by the centralized leader-follower framework,while the leaders from different groups communicate using the nearestneighbor topology.

(2)To implement the proposed integrated cooperative guidance framework,a group of FTCG laws considering the saturation constraint on FOV (FTCG-FOV)are designed by introducing two bias terms in PNG,and an improved sequential approach is improved and then employed to make the FTCG-FOV satisfy the requirement of communication between groups.

(3)The simulation results well demonstrate the effectiveness and high efficiency of the proposed integrated cooperative guidance framework and the cooperative guidance laws,as well as the superiority of the improved sequential approach.

Acknowledgement

This study was supported by the National Natural Science Foundation of China(No.11532002).

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