Zhao Jiang,Zhou Rui

School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

Distributed three-dimensional cooperative guidance via receding horizon control

Zhao Jiang*,Zhou Rui

School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

The paper presents a new three-dimensional(3D)cooperative guidance approach by the receding horizon control(RHC)technique.The objective is to coordinate the impact time of a group of interceptor missiles against the stationary target.Theframework of a distributed RHC scheme is developed,in which each interceptor missile is assigned its own finite-horizon optimal control problem(FHOCP)and only shares the in formation with its neighbors.The solution of the local FHOCP is obtained by the constrained particle swarm optimization(PSO)method that is integrated into the distributed RHC framework with enhanced equality and inequality constraints.The numerical simulations show that the proposed guidance approach is feasible to implement the cooperative engagement with satisfied accuracy of target capture.Finally,the computation efficiency of the distributed RHC scheme is discussed in consideration of the PSO parameters,control update period and prediction horizon.

1.Introduction

In the last decade,autonomous guidance approaches have already been developed to improve the performance of the interceptor missiles for minimum energy control,minimum time control,impact time control and impact angle control.1–4For a single interceptor missile,the above objectives have been achieved with satisfied accuracy of target capture.Recently,many researches start to focus on the design of the guidance approaches for the multiple missiles,because the cooperative engagement can have better performance than a single interceptor missile in detecting the maneuvering targets,penetrating the defense systems and surviving the threats.5–7However,it is more difficult to achieve the impact time control and impact angle control for a group of multiple missiles,because each interceptor missile may have different initial conditions as well as possible communication limit with other members.8,9

In the current literature,two typical classes of impact time control guidance approaches have been proposed for the multi-missile salvo attack.The first class integrates the impact-time constraints into the design of the control commands for interceptor missiles.In Ref.10,the closed form of the impact time control guidance law is developed based on the proportional navigation(PN),which can guide a group of interceptor missiles to intercept a stationary target at a desirable time.In Ref.11,a time-varying navigation gain is proposed to coordinate the impact time of each interceptor missile.Then,an extension of the impact time control guidance law is used to control both the impact time and impact angle.12The above algorithms require that the global in formation of the time-to-go is available to each interceptor missile in the group.There fore,the distributed control architecture is developed on the basis of consensus protocols to improve the performance of the time-constrained guidance law.13In addition,the PN-based distributed coordination algorithms are proposed to per form the cooperative engagement against both the stationary and maneuvering target.14,15

The second class uses the leader–follower model to describe the cooperative engagement of interceptor missiles.In Ref.16,a nonlinear state tracking controller is developed to the design of the leader–follower strategy in order to achieve the impact time control guidance.Then,the consensus protocol is integrated into the leader–follower model,in which the final impact time of each follower converges to the leader in the finite time.17In Ref.18,a heterogeneous leader–follower guidance approach is also proposed for a group of interceptor missiles by using the traditional PN algorithm.Furthermore,the virtual leader scheme is used to achieve cooperative engagement by trans forming the time-constrained guidance problem to the nonlinear tracking problem.19

More recently,Ghosh et al.20develop a recursive time-togo estimation method for three-dimensional(3D)engagement of a Retro-PN guided interceptor against higher speed nonmaneuvering target.They present a navigation gain scheduling algorithm to achieve the interception at a pre-specified time.Ghosh et al.21also propose a cooperative strategy for the lower speed interceptors guided by Retro-PN guidance law to per form the salvo attack against a higher speed target.These studies are the first efforts to solve the cooperative guidance problem against moving target in 3D engagement.

The purpose of this paper is to propose a new solution framework for the 3D cooperative engagement problem.The receding horizon control(RHC)technique is employed to achieve the impact time control guidance for interceptor missiles.The main contribution of the paper is delineated in the following part:(1)the distributed RHC scheme is developed to coordinate the impact time of the interceptor missiles,each of which only shares the in formation with neighbors and solves its own local finite-horizon optimal control problem(FHOCP);(2)the swarm intelligence method is integrated into the distributed RHC framework with enhanced equality and inequality constraints.The feasibility and computation efficiency are demonstrated by some numerical simulations.The rest of the paper is organized as follows:Section 2 presents the preliminaries to the constrained particle swarm optimization(PSO)algorithm.The problem formulation of cooperative engagement is described in Section 3.In Section 4,the distributed RHC framework is developed to achieve the cooperative time-constrained guidance.In Section 5,the numerical results of the proposed approach are discussed in detail.Finally,the concluding remarks are presented in Section 6.

2.Preliminaries

The PSO is one of popular swarm intelligence methods.22–24In this paper,we use the global particle swarm because it is fast enough to find the optimal solution of the distributed RHC problem.

Assume that｛p1，p2，...，pn｝are the n unknown parameters that have their own bounds in terms of

where aiand biare the bounds of unknown parameters.The population is Nk.Each particle k has a position vector p(k)and a velocity vector v(k)as

where p(k)and v(k)refer to search space.The elements are represented by pi(k)and vi(k).According to the bounds of unknown parameters,the related position and velocity components are limited to

Suppose that the PSO terminates at the iterations NITER.In a generic iteration j,the personal best position（k）and the global best position）can be determined.The velocity vector is described

where p(j)(k)and v(j)(k)are position and velocity vectors in each iteration;w is the inertial weight;c1and c2are cognitive and social components;r1(0,1)and r2(0,1)are random numbers.The update of the position vector is determined by23

The optimal unknown parameters are contained in the position vector that relates to the objective function J.In general,the parameter optimization problem includes many equality and inequality constraints.For equality constraints,the most typical solution is to add a penalty term to the fitness function in the form of

where ζp≥ 0 (p=1,2,...,m)is weight factor;dp(x)(p=1,2,...,m)represents the m quality constraints that relate to the n unknown parameters.Note that the values of the coefficients ζpdepend on the actual problem.For inequality constraints,a simple solution is to set the fitness function to an in finite value if the particle k violates one of the inequality constraints,i.e.,=∞.The related velocity is also set to zero,i.e=0,such that the velocity update is influenced only by the social and cognitive components.

3.Problem formulation

3.1.Basic assumptions

In this paper,the 3D nonlinear dynamics with a stationary target is used to design the guidance approach.The following conditions are assumed to describe the cooperative engagement.25

(1)The total velocity of each missile is set to the constant value.

(2)The missile and target are considered as point masses in the 3D space.

(3)The seeker and autopilot dynamics of the interceptor missiles are fast enough in comparison with the guidance loop.

3.2.Guidance geometry

By using the prescribed assumptions,the guidance geometry on the one-to-one engagement is depicted in Fig.1.M denotes the interceptor missile and T the target;r is the missile-totarget range and Vmthe total velocity of interceptor missile;the terms γmand φmare the Euler angles in the inertial referenceframe;the terms θmand ψmare the look-ahead angles in the line-of -sight frame;γLand φLare the line-of -sight angles in the inertial referenceframe.

The 3D point-mass equations of motion for the interceptor missile can be derived from the classical principles of dynamics as follows:25

where˙λyand˙λzare the components of line-of -sight angular velocity vector;Aymand Azmare defined as the yaw and pitch acceleration commands of the interceptor missile,respectively.

The traditional 3D PN guidance laws against the stationary target can be given by25

where N represents the effective navigation constant of the interceptor missile.

Fig.1 Guidance geometry on one-to-one engagement.

Fig.2 Guidance geometry on many-to-one engagement.

Suppose that Nminterceptor missiles totally participate in the cooperative engagement against a stationary target.Fig.2 shows the guidance geometry on many-to-one engagement scenario,where Mi(i=1,2,...,Nm)denotes each of the ith interceptor missile and rithe range between Miand target T;the terms θiand ψiare the look-ahead angles of Miin the line-of sight frame;the terms γiand φiare Euler angles with respect to the inertial referenceframe;it is assumed that Viis the constant speed of Miwhich may be different from each other,and the acceleration command Aionly changes the direction of Vi.The components of Aiare given by Aymiand Azmi.

The purpose of this paper is to develop a new guidanceframework that can guide the group of interceptor missiles to simultaneously impact the given stationary target even if each interceptor missile has some different initial conditions.The RHC technique is applied to design the guidance algorithm by the distributed scheme.

4.Cooperative guidance approach

4.1.Traditional RHC

In this section,the distributed MPC framework in Ref.26is used to design the time-constrained guidance approach for the multi-missile network.Considering the traditional RHC problem,the missile dynamics of Eqs.(8)–(12)can be written in the equivalent form as follows:

where z（t）∈ Rnis the system state trajectory and u（t）∈ Rmis the system control trajectory.Then,define the constant prediction horizon as Tp∈ （0，∞）and the constant control update period as δ ∈ （0，Tp］.The common receding horizon update times are given by tc=t0+ δc,c ∈ ｛0，1，2，...｝.At each time instant tc,the RHC problem can be formulated by thefollowing FHOCP.26,27

Problem 1.For each member i∈ ｛1，2，...，Nm｝ and at the update time tc=t0+ δc,c ∈ ｛0，1，2，...｝:given z（tc）,and then,we can find

where s ∈ ［tc，tc+Tp］is the prediction horizon;z（s;tc）and u（s;tc）are the predicted state trajectory and control trajectory,respectively;Z and U are the state and control input constraints.For missile guidance problem,the control variables by Eq.(19)are typically selected as the yaw acceleration Aymiand pitch acceleration Azmi.Theallowed control variable-space U can be given by the maximum acceleration value[-Amax,Amax].The optimal control trajectory is denoted as u*（s;tc）.J is the integrated cost function including a running function F and a terminal state penalty function Φ.

4.2.Distributed RHC scheme

The decoupled time-invariant nonlinear dynamics for missile Mican be written in the equivalent form as

and then,the concatenated vectors in the system Eq.(14)can be denoted as z=[z1,z2,...,zNm],u=[u1,u2,...,uNm],and f(z,u)=[f1(z1,u1),f2(z2,u2),...,fNm(zNm,uNm)],respectively.

For traditional RHC framework in Problem 1,the states of each interceptor missile are typically coupled in the integrated cost function to achieve the impact time control guidance.The common components in Eq.(16)can be defined as

where the symbol‖·‖ denotes any vector norm in Rn;V= ｛1，2，...，Nm｝ represents the set of the interceptor missiles;E is the set of the pair-wise neighbors in the multimissile network.It is assumed that if(i,j)∈E,then(j,i)?E,and(i,i)? E for missile i∈ V.The terms α and β are the weighting constants.The control is selected as ui= ［Aymi，Azmi］.The time-to-go of each interceptor missile can be estimated by thefollowing expression10,14

The main advantage of the traditional RHC framework is the design of the cost function with Eqs.(21)and(22)which takes into account the state and control trajectories of all the interceptor missiles.It can reflect the motion of the multimissile network.However,the requirement of computation load is quite high and the guidance approach would be out of work if some interceptor missile can only obtain effective in formation from neighbors.Fig.3 illustrates an example of the communication limit between each missile that can only communicate with its neighbors in the set Ni.There fore,the time-constrained guidance should be developed in the distributed framework to achieve an agreement on the impact time.

Fig.3 Communication limit between each missile.

As shown in Fig.4,the distributed RHC framework is proposed herein for the multi-missile network.The main principle is summarized as follows.At each update time,the control inputs of the group of interceptor missiles are first initialized by using the previous predicted optimal control trajectories.Then,each member in the group of missiles receives the estimated control trajectory from its neighbors,computes the neighbors’states over the current prediction horizon,and meanwhile transmits its estimated control trajectory to the neighbors.Based on the estimated state and control trajectories from neighbors,each interceptor missile evaluates the distributed cost function of its own and finds the optimal predicted control trajectory over the current prediction horizon.Finally,the optimal control trajectory over the first control update period is implemented to update the states of each interceptor missile.

To describe the distributed RHC scheme,wefirst define that the neighbors of each interceptor missile Mi（i∈ V）have the control vectors u-i（t）= ｛uj（t）｝，j∈ Niand state vectors z-i（t）= ｛zj（t）｝，j∈ Ni,respectively.The decoupled nonlinear dynamics for the neighbors of missile i can be formulated as

Then,we define the following notations to distinguish different kinds of the state and control trajectories for each missile Miat current time tc.

s ∈ ［tc，tc+Tp］is the given prediction horizon.Consistent with u-i（t）and z-i（t）,the estimated control and state trajectories of the neighbors of each missile Miare defined as^u-i（s;tc）and^z-i（s;tc）,respectively.The generation of the estimated control trajectory^u-i（s;tc）and the state trajectory^z-i（s;tc）at each update time tcwill be discussed in the following part.

According to the aforementioned definition u-i（t）,the estimated control trajectory^u-i（s;tc）over the prediction horizon s ∈ ［tc，tc+Tp］is given by

As shown in Fig.5,the estimated control trajectoryu^j（s;tc）over the given prediction horizon s ∈ ［tc，tc+Tp］consists of two individual parts.The first one is the same as the previous optimal control trajectoryover the prediction horizon s ∈ ［tc，tc-1+Tp）and the second one is derived from the value ofat the time instant s=tc-1+Tp.To be specific,the estimated control trajectorycan be expressed as

Fig.4 Framework of distributed RHC scheme.

Fig.5 Generation of estimated control and state trajectories.

Thus,the estimated control trajectories of the neighbors of each missile Miare obtained by Eqs.(25)and(26),and the corresponding state trajectories^z-i（s;tc）can be also computed according to the dynamics Eq.(24).

Based on the formulation of the estimated control trajectory and state trajectory,the distributed cost function for each missile Mi（i∈ V）is given in the following expression as

where the time-to-go of the neighbors of each missile Miis estimated by

where^rj（s;tc）is the predictive missile-to-target range at the time tc.Then,the nonlinear RHC problem at each time instant tccan be formulated by the distributed FHOCP as follows.

Table 1 Pseudo-code of distributed RHC scheme for missile Mi.

Problem 2.For each member i∈ ｛1，2，...，Nm｝ and at the update time tc=t0+ δc, c ∈ ｛0，1，2，...｝: given zi（tc），z-i（tc），^ui（s;tc），^u-i（s;tc），s ∈ ［tc，tc+Tp］,find

where Jiis the distributed cost function for each missile Mi,which includes a running function Fiand a terminal state penalty function Φi.The optimal control trajectory of each interceptor missile Mi（i∈ V）is denoted as u*i（s;tc）.Typically,the distributed FHOCP will be solved by the constrained PSO algorithm.The control parameters in the PSO solver are selected as the yaw acceleration Aymiand pitch acceleration

Azmi.The allowed control variable-space U is also given by the maximum acceleration limitation[-Amax,Amax].The pseudo-code of the distributed RHC scheme for missile Miis listed as Table 1.

5.Numerical simulations

5.1.Cooperative engagement

In this section,a scenario of the 3D cooperative engagement is per formed to demonstrate the effectiveness of the proposed time-constrained guidance approach.The position of missile Mi（i∈ ｛1，2，3｝）in the inertial reference frame can be obtained by

where Xi,Yiand Ziare the position components of each missile.Suppose that a group of three missiles intercept a stationary target at(0,0,0)km.As shown in Fig.6,a simple communication topology is selected for the interceptor missiles.To be specific,missiles M1and M3can only obtain the in formation from the neighbor missile M2,i.e.,N1= ｛2｝,N2= ｛1，3｝ and N3= ｛2｝.

Table 2 lists the parameters in the distributed RHC scheme which are involved in the PSO solver and the guidance loop.Typically,the selection of the population number Nkbetween 30 and 50 can guarantee that the obtained solutions are accurate enough.The iteration number NITERshould be less than 100,because a large iteration number may decrease the computation efficiency of the PSO solver.In Table 2,the inertia,cognitive and social weights are selected as a practice.To ensure satisfactory accuracy,the constant control update period,i.e.,the guidance period,should be no more than δ=0.1 s.The prediction horizon is set to Tp=0.8 s and it also demonstrates that a smaller prediction horizon Tpmay reach a compromise between performance index and computation efficiency.

First,the proposed distributed RHC scheme for cooperative engagement is tested in comparison with the traditional PN guidance law.Case 1 and Case 2 represent the RHC and PN algorithm,respectively.The PN navigation constant for each missile is set to N=3.0.The initial conditions of the three interceptor missiles are shown in Table 3.

Fig.6 Communication topology between each missile.

Table 2 Parameters in distributed RHC scheme.

Table 3 Initial conditions of Case 1 and Case 2.

The 3D trajectories of the interceptor missiles are shown in Fig.7.It can befound that thefinal impact times by using the PN algorithm are 27.4 s,24.9 s and 19.6 s,respectively.The 3D trajectories by distributed RHC scheme illustrate that missile M1moves shorter round,whereas missile M3moves farther round,both to achieve a simultaneous impact on the given target.The impact times of the group of interceptor missiles finally reach an agreement at 24.5 s,which shows the feasibility of the distributed RHC scheme.

Fig.7 3D trajectories of interceptor missiles in Case 1 and Case 2.

Fig.8 Acceleration commands of interceptor missiles in Case 1 and Case 2.

Fig.9 Times-to-go of interceptor missiles in Case 1 and Case 2.

Figs.8 and 9 show the histories of the acceleration commands and the times-to-go of the interceptor missiles.It is found that the accelerations of interceptor missiles decrease to about zero as they gradually approach the given target.It should be noted that missile M2has the least deviation in the consensus interception time with respect to the pure PN interception time.However,a high maneuver is indicated for missile M2by using the distributed RHC scheme.The acceleration command is large because the coordination of the interception time and target capture may require much more control effort.The interception time of missile M2is in fluenced by both missiles M1and M3.Typically,the cost function for each missile is determined by the penalty term for time coordination when the interception time differences are large.However,the penalty term for target capture gradually increases when the interception time difference between each missile decreases.There fore,thefeature that the acceleration command for missile M2has a high maneuver is mainly determined by the PSO-based optimization technique.As shown in Fig.9,the time-to-go error of each interceptor missile Miconverges to zero be fore t=5 s by using the distributed RHC scheme.The effectiveness of the proposed guidance approach is demonstrated.

To justify the effectiveness of the proposed guidance scheme,this part presents some simulation results with some different kinds of engagement parameters.We consider the following cases where the interceptor missiles have large initial heading errors.The velocity difference between each missile is small in Case 3,whereas large in Case 4.Table 4 lists the initial conditions of interception missiles.

The numerical results by using the distributed RHC scheme are illustrated in Figs.10–12.The impact times of the group of interceptor missiles finally reach an agreement at 23.0 s and 24.8 s,respectively.It demonstrated that the proposed guidance approach is feasible to solve the cooperative engagement of interceptor missiles with large initial heading errors.As shown in Fig.10,the interceptor missiles with large velocity differences move farther trajectories than those with small velocity differences to achieve a simultaneous impact against the given target.The histories of the acceleration commands are presented in Fig.11,which are within the maximum allowed control constraints.The acceleration commands of the interceptor missiles also decrease to around zero as they gradually approach the target.Fig.12 shows the histories of the times-to-go.It can befound that large velocity differenceslead to a relative slower convergence of the impact time between each interceptor missile.

Table 4 Initial conditions of Case 3 and Case 4.

Fig.10 3D trajectories of interceptor missiles in Case 3 and Case 4.

Fig.11 Acceleration commands of interceptor missiles in Case 3 and Case 4.

Fig.12 Times-to-go of interceptor missiles in Case 3 and Case 4.

Fig.13 3D trajectories of interceptor missiles with autopilot lags.

Fig.14 Acceleration commands of interceptor missiles with autopilot lags.

Fig.15 Times-to-go of interceptor missiles with autopilot lags.

5.2.Autopilot dynamics

The autopilot dynamics may influence the performance of the cooperative guidance algorithm.In this part,the proposed distributed RHC method is tested in consideration of autopilot lags.The autopilot dynamics can be usually described by the first-order differential equations in the form of

where τyand τzare the time constants of the autopilot;and aymiand azmiare the actual accelerations.

The dynamics Eq.(37)is included in each control update period(not in the prediction model)to test the robustness of the distributed RHC scheme.The time constant of the autopilot is set to τ=0.3 and τ=0.6,respectively.The initial conditions of the interceptor missiles are the same as those in Section 5.1.

The numerical results of the 3D trajectories with autopilot lags are shown in Fig.13.It can befound that the interceptor missiles with the time constant τ=0.6 requires longer time to respond to the guidance commands than those with the time constant τ=0.3.The interceptor missiles also move longer trajectories in consideration of a larger time constant of the autopilot.The final impact times of the group of interceptor missile are 24.6 s and 25.2 s,respectively.Figs.14 and 15 illustrate the histories of the acceleration commands and the timesto-go with different autopilot lags.As shown in Fig.14,an explicit lag can befound in the acceleration commands with the time constant τ=0.6.In Fig.15,it also demonstrates that a larger time constant of the autopilot results in a slower convergence of the times-to-go.

5.3.Computation efficiency

Typically,the PSO parameters,control update period,and prediction horizon determine the success rate of the timeconstrained guidance approach.There fore,this part will discuss the computation efficiency of the proposed RHC scheme.

Using the standard C++,the simulations are run for several cases with different PSO parameters,control update period,and prediction horizon.In detail,the population and iteration number of the PSO solver are set to(Nk=30,NITER=50)and(Nk=50,NITER=80),respectively.The selection of the control update periods includes δ=0.05 s and δ =0.1 s.The prediction horizon Tp=4δ and Tp=8δ are used in the tests.For each case,the simulation is repeated 30 times with the same PSO parameters.The initial engagement geometries are divided into three classes by the missile-to-target range,look-ahead angle and velocity.The missile-to-target range of each interceptor missile is set to 3–6 km.The initial look-ahead angle is chosen at an interval of 10°.We also assign different velocities to interceptor missiles from 180 m/s to 220 m/s in the test.

Table 5 presents the statistical results of the PSO-based RHC scheme for comparison.It can befound that a large control update period δ typically raises the success rate of the distributed RHC scheme.The success rate for the case of δ=0.1 s reaches 100%,because the average running time of the PSO solver in each control update period δ is about 0.023 s(Nk=30,NITER=50)and 0.042 s(Nk=50,NITER=80),respectively.The proper selection of the particle population Nkand iteration number NITERmay also reach a compromise between the performance index and computation efficiency.In addition,the numerical results of the success rate demonstrate that a reasonable reduction of the prediction horizon Tpcan improve the stability of the distributed RHC scheme.

6.Conclusions

This paper proposes a new solution for the cooperative guidance problem in 3D situation.

(1)The coordination of the impact time between each interceptor missile is achieved by the design of the distributed RHC scheme.The time-constrained guidance problem is transmitted to the local FHOCP in which the group member only exchanges the in formation with its neighbors.

(2)The constrained PSO method is integrated into the distributed RHC framework and solves the local FHOCP with satisfied computation efficiency.Typically,the control update period δ=0.1 s and the prediction horizon Tp=4δ can result in a faster convergence of the timesto-go of interceptor missiles.

The future work will focus on the cooperative guidance against maneuvering targets and high-speed targets.The environment disturbance and model uncertainty should also be taken into account to per form the cooperative engagement missions.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(Nos.61273349 and 61573043).

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Zhao Jiang received the Ph.D.degree in guidance,navigation and control from Beihang University,Beijing,China.His areas of expertise include flight control and guidance,constrained trajectory optimization,cooperative control and swarm intelligence.

Zhou Rui is currently a prof essor in guidance,navigation and control from Beihang University,Beijing,China.His areas of expertise include flight control and guidance,constrained trajectory optimization,cooperative control and flight mission planning.

24 September 2015;revised 3 December 2015;accepted 8 March 2016

Available online 22 June 2016

Distributed algorithms;

Impact time;

Missile guidance;

Multiple missiles;

Particle swarm optimization(PSO);

Receding horizon control(RHC);

Three-dimensional(3D)

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.Thisisan open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 10 82316849.

E-mail address:jzhao@buaa.edu.cn(J.Zhao).

Peer review under responsibility of Editorial Committee of CJA.

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http://dx.doi.org/10.1016/j.cja.2016.06.011

1000-9361?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).