a joint mid-course and terminal course cooperative...

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

Chinese Journal of Aeronautics 28 (2015) xx-xx

Final Accepted Version

A Joint Mid-course and Terminal Course Cooperative Guidance Law for Multi-Missile Salvo Attack

Jie ZENG, Lihua DOU, Bin XIN* School of Automation, Beijing Institute of Technology, Beijing 100081, P. R. China

State Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing 100081, P. R. China

Abstract

Salvo attacking a surface target by multiple missiles is an effective tactic to enhance the lethality and penetrate the defense system. However, the existing cooperative guidance laws in mid-course or terminal course are not suita-ble for long-range and medium-range missiles or stand-off attacking. Because the initial conditions of the coopera-tive terminal guidance that are generally generated from the mid-course flight may not lead to a successful coopera-tive terminal guidance without proper mid-course flight adjustment. Meanwhile, the cooperative guidance in mid-course cannot solely guarantee the accuracy of the simultaneous arrival of multiple missiles. Therefore, a joint mid-course and terminal course cooperative guidance law is developed. By building a distinct leader-follower framework, the paper proposes an efficient coordinated Dubins path planning method to synchronize the arrival time of all engaged missiles in the mid-course flight. The planned flight can generate proper initial conditions for cooper-ative terminal guidance, and also benefit an earliest simultaneous arrival. In the terminal course, an existing cooper-ative proportional navigation guidance law guides all the engaged missiles to arrive at the target accurately and sim-ultaneously. The integrated guidance law for an intuitive application is summarized. Simulations demonstrate that the proposal can generate fast and accurate salvo attack.

Keywords： Cooperative systems; missile guidance; salvo attack; Mid-course flight; Dubins path. *Corresponding author. Tel.: +86 10 68912463. E-mail address: [email protected]

1. Introduction

Since the many-to-one engagement is advantageous to increase the lethality and the probability of tion1, cooperative guidance is a technique which is certain to be widely applied in future missile systems. In fact, persistent efforts have been made to meet the increasing need of cooperative guidance of missiles1-14, 20, 21.

The common missile engagement timeline can be functionally partitioned into four phases15: launching, midcourse guidance, acquisition, and terminal guidance. The existing cooperative guidance strategies mostly focus on the ter-minal guidance phase of missiles and they are based on the classic proportional navigation guidance16 (PNG) or the optimal guidance17. In 2006, Jeon et al. 1 derived a closed form of the impact time control guidance law (ITCG) based on a linear formulation. ITCG guides a missile to attack a stationary target at a presetting time. Lee et al. 2 ex-

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·2 · Chinese Journal of Aeronautics

tended the ITCG guidance law to control both the impact time and the impact angle. In 2010, Jeon and Lee3 proposed a cooperative proportional navigation (CPN) for many-to-one engagements which decreases the variance of the time-to-go (time left before hitting) of engaged missiles. Based on ITCG and consensus protocols, Zhao and Zhou4 introduced an effective hierarchical cooperative guidance architecture including both centralized and distributed co-ordination algorithms. Zou et al. 5 proposed a distributed adaptive cooperative guidance law for multiple missiles with heterogeneous leader-follower structure to implement the cooperative salvo attack. Similarly, Zhao et al. 6 pro-posed a virtual leader based scheme that achieves the impact time control indirectly by skillfully transforming the time-constrained guidance problem to a nonlinear tracking problem. Zhang et al. 7 designed a practical three dimen-sional impact time and impact angle control guidance law by using a two-stage control approach. Zhang and Ma et al. 8 designed a feasible biased PNG (BPNG) law to control the impact time and the impact angle. Based on ITCG, a biased term with the cosine of weighted leading angle is used by Zhang and Wang et al. 9 to guarantee that the field-of-view (FOV) constraint is not violated during the engagement. Furthermore, Zhang and Wang et al. 10 pro-posed a distributed cooperative scheme to ensure the convergence to the same impact time under either fixed or switching sensing/communication network. Zhao and Zhou et al. 11 presented unified cooperative strategies for the salvo attack of multiple missiles and developed guidance laws against both stationary and maneuvering targets. Lately, Zhao and Zhou et al. 12 proposed an effective 3-D guidance law to perform the cooperative engagements of multiple missiles against both a stationary target and a maneuvering one.

From another point of view, some scholars concentrate on the cooperative guidance in midcourse15, 18-22. Morgan15 addressed a midcourse guidance law which ensures a sufficiently small zero effort miss (ZEM) at the handover mo-ment and optimizes an energy cost function. Since a sooner attack is preferred in battlefield, Indig et al. 18 presented near-optimal solutions for minimum-time midcourse guidance of missile with angular constraint in both planar case and spatial case. As shown in the simulations of the work of Indig et al., the flight paths closely approximate the op-timal Dubins path19 which is the time-optimal path for vehicles with constant velocity. Tanil20 firstly made midcourse cooperative waypoint path planning for multi-missile salvo attack by adopting an evolutionary speciation approach. Obstacle avoidance and simultaneous arrival are equally emphasized in the work of Tanil, but the turning radius con-straint is neglected. Shima et al. 21 solved a simultaneous interception problem of multi-vehicles and proposed three path elongation algorithms. But all the elongated paths have curved turnings at the end of the flights, which is not fit for the midcourse guidance. The acquisition phase is considered in our proposal, and all the elongated paths have straight headings toward the target at the end of the flights. Yao et al. 23 elongated Dubins paths with bounded curva-ture and preset length. But the leader-follower scheme in our proposal ensures a soonest salvo attack.

The satisfactory effect of aforementioned guidance laws has been proved in either mid-course or terminal course. However, the two courses should not be considered separately in a cooperative guidance since the terminal guidance with closed-loop command is indispensable for a precise attacking. Meanwhile, the initial conditions of terminal course are generated from the midcourse flight and there are constraints on the initial conditions of the terminal course cooperation:

(1) The detection range constraint of seeker: all participant missiles should be in certain ranges from the target at the moment that the cooperative terminal guidance starts.

(2) The FOV constraint of seeker: all the included angles between line-of-sight (LOS) and missile headings should not violate the FOV constraints throughout the cooperative terminal course.

In short words, all the engaged missiles should have accomplished the acquisition and the handover process at the initial moment of the cooperative terminal guidance. Moreover, the time-to-go (TTG) differences among them should be small enough.

These initial constraints above are not innately satisfied without the mid-course cooperation, since the differences of the predicted flight time among the missiles cannot be eliminated from the launching moment to the terminal course. Therefore, the demand on a joint midcourse and terminal course cooperative guidance emerges. Besides, the joint cooperative guidance is required for long-range cruise missiles and those for stand-off attack. The joint mid-course and terminal course cooperative guidance at least has the following three advantages:

(1) Since missiles are relatively far from the target in the mid-course flight, the length adjustment for the mis-sile’s path has much wider range as compared with the terminal phase.

(2) The heading of missile is not constrained by FOV in the midcourse. (3) The terminal course flight is in the core defense area of the opponent. As compared with maneuvering in ter-

minal course, elongating the flight path in the midcourse has lower risk. Taking both the multi-missile handover conditions and the soonest salvo attack into consideration, this paper uti-

lizes Dubins paths and proposes a coordinated path planning method under a novel leader-follower framework. Un-like common leader-follower frameworks5,6, the framework built in this paper is for synchronizing the expected arri-val time of all engaged missiles by path planning, rather than simply unifying the missile speed, heading error, and the sight distance. The planned flight paths for all missiles not only follow the dynamics of these missiles but also achieve a soonest salvo attack.

Chinese Journal of Aeronautics · 3 ·

The innovations of this paper are as follows: To our best knowledge, it is the first time to propose a joint cooperative guidance law from the perspective of satisfying the constraints on the initial conditions of cooperative terminal guidance by incorporating mid-course coordinated path planning. The effectiveness and advantages of the proposed joint cooperative guidance law has been validated by comparative simulations with different configurations of combat scenarios. As for the mid-course coordination, we apply the Dubins path with terminal heading relaxation to determine the earliest arrival time of missiles and build a distinct leader-follower framework for the synchronization of arrival time. By defining feasible path elongation patterns and deriving their applicability conditions, we propose an ef-fective coordinated path planning method for a soonest salvo attack. The mid-course cooperative guidance law can be easily derived from the planned paths, and provides ideal initial conditions for the cooperative guidance in ter-minal course. The remainder of this paper is organized as follows. Section 2 presents the integrated cooperative strategy and pre-

liminaries. Section 3 proposes the coordinated Dubins path planning method for mid-course and the guidance law. Section 4 demonstrates the simulation results and analyzes the effectiveness of the proposed method. Section 5 gives the conclusion.

2. Problem statement and preliminaries

2.1. Cooperative attack scenario

In a hypothesis battle engagement, various cruise missile systems are obliged to strike a surface target. The partic-ipant missiles are cooperatively guided to attack the target simultaneously, especially when targets are under the pro-tection of the Close-in Weapon System (CIWS). As an important part of the Theatre Missile Defense (TMD) System, CIWS is used to detect and destroy incoming anti-ship missiles and enemy aircrafts at short range from the target. Fig. 1 illustrates the scenario of three different missile systems striking a stationary target. Assume that the engaged systems are scattered in different locations, and each of them launches one constant-speed cruise missile in the at-tacking. The missiles may be different in the speeds, the minimal turning radius, the missile-target ranges and the initial headings.

Fig. 1 Scenario of a cooperative attack.

2.2. Cooperative attack strategy

The cooperation strategy proposed in this paper is mainly focused on the cooperative path planning in the mid-course. The following statements sketch the cooperation strategy. Details will be presented in Section 3.

Before launching, all the missile launching platforms that are available at the current moment send the information of their locations, initial directions, minimal turning radius and missile speeds to the centralized coordination center (CCC). Then CCC allocates proper missiles for a salvo attack according to the battle plan and the collected infor-mation. Then, the minimum flight time of engaged missiles are calculated by applying Dubins path with terminal heading relaxation and the latest arriving missile is set as the leader. Subsequently, all the other missiles become fol-lowers, and the flight paths for the followers will be re-planned to hit the target at the same time as the leader. Hence, the soonest simultaneous arrival is achieved, which is preferred since the shorter the engagement time is, the better the missile survives the threats1.

The midcourse guidance command implies an open-loop control since it is expected to follow the planned path. Nevertheless, the heading error at the hand-off moment is permissible as long as the seeker lock-on is achievable. Furthermore, the small time-to-go differences at the start moment of the cooperative terminal phase will be eliminat-ed by cooperative terminal guidance. In the terminal course, the existing technique such as ITCG and CPN can be


used for all missiles including the leader, ensuring that all the participant missiles accurately hit the target at the same time. In fact, using pre-programmed guidance command in the midcourse while implementing homing guidance control in the terminal course is quite common in the practical application of single missile guidance schemes3. In order to start the cooperative terminal guidance, the cooperative hand-off moment is set to be the moment that all the missiles have locked-on the target. In other words, all the missiles switch into the cooperative homing guidance phase when the last missile locks on the target.

It is worthy of attention that effectively and rapidly solving the midcourse path planning problem of followers is crucial in the battlefield. Therefore, we take it into consideration and propose three path patterns that can monoton-ically elongate the flight path of followers and use efficient bisection algorithm to calculate the parameters.

2.3. Preliminaries

The optimal Dubins path with terminal heading relaxation is introduced in this subsection. In the midcourse flight, missiles are controlled by a pre-programed guidance law when the target is not locked-on by the seeker. Therefore, the missile-target relative movement is not involved in this phase. However, maneuvering of the missiles is con-strained by their maximum overload, due to which the 2-D Dubins model18, 21 (see Eq. (1)) is used in this paper to describe the midcourse dynamics of missiles with the constraint of minimal turning radius.

[ ], 1,1

0m

x Vcosy Vsin

V u uR

V

θθ

θ

= = = ∈ −

=

(1)

where ( , )x y and θ are the planar coordinates and the heading of a missile, respectively, V is the velocity of the missile, mR is the minimal turning radius, u is the control input, 0u < means turning left with respect to the heading, 0u > means turning right, 0u = means keeping straight flying, and a triplet ( )x, y,θ is called a con-figuration.

The shortest path from one configuration to another must be one of the six Dubins path patterns: RSL, LSR, RSR, LSL, RLR, and LRL, 19 where L means turning left with the minimal turning radius, R means turning right corre-spondingly, and S means moving along a straight line. From the conclusions, it can be seen that the shortest Dubins path between two configurations relies on both their positions and headings.

In this research, the heading of the missile at the handover moment is not fixed, so θ in the terminal configura-tion is unrestricted. It is straightforward to prove that, in the case of terminal heading relaxation, θ in the terminal configuration is unrestricted, and LS or RS path24 is the optimal path when the target is outside the turning circle of the initial configuration. In battlefield situations, the minimal turning radii of missiles are usually hundreds of meters, while the missile-target ranges in midcourse are at least several times of the minimal turning radii. Hence, the target is certainly outside of the turning circles. The optimal Dubins path pattern of RS/LS will be applied in the path plan-ning calculation. Fig. 2 shows the LS and RS Dubins paths, the initial position of the missile is located at the origin which is denoted by S , and the initial heading of the missile is along Y-axis. The target is denoted by T ′or T , while LO and RO are the centers of left and right turning circles, respectively. The coordinates of LO and RO are ( ,0)mR− and ( ,0)mR , respectively. The coordinates of T ′ and T are ( , )T Tx y′ ′ and ( , )T Tx y .

In the rest of this paper, the RS/LS-type Dubins path with [ ]1,1u∈ − is named RS/LS path for short, and the RS/LS path with 1u = or 1u = − is named optimal RS/LS path for short.

Fig. 2 RS/LS Dubins paths with terminal heading relaxation.


3. Cooperative guidance law in the midcourse

As introduced in Subsection 2.2, the cooperative guidance commands in the midcourse are gained on the basis of the planned paths for the missiles. Firstly, the path patterns of all missiles are settled after the leader is selected and the expected path lengths of followers are known. Then, the parameters in the path pattern of each missile will be calculated according to the expected path lengths. After that, explicit guidance commands can be derived.

3.1. Mid-course cooperative path planning

In this section, we will show how to calculate the length of optimal RS/LS paths for selecting the leader. Then we propose the three path patterns for followers to elongate their paths and introduce the applicability conditions. Be-cause the path lengths under the three patterns are monotonous with respect to the respective undetermined parame-ters, the bisection algorithm is used to calculate the parameters as its effectiveness and convergence rate can be guaranteed25.

3.1.1. Selection of the leader

According to the cooperation strategy, the missile with the longest minimum flight time is selected as the leader. And the leader will follow the optimal RS/LS path for the soonest arrive. To obtain the minimum flight time of each engaged missile and select the leader, the length of the optimal RS/LS path for each missile under the initial condi-tions is calculated.

Take the optimal RS path in Fig. 2 for example. The length is calculated as:

2 2RS cos m

R m m RR

RD PT SP O T R R SO T arcO T

= + = − + ⋅ ∠ −

. (2)

The minimal turning radius mR and the coordinates of S , RO , T are known, therefore the length of RO T and

the angle between RSO and RO T

can be derived. The shortest flight time of each missile can be obtained as:

ii

i

DtV

= (3)

where iD is the Dubins path length of missile i and iV is its velocity.

Then, according to the cooperative strategy, the expected flight time t of all missiles is: { }ˆ max it t= .

The expected path length EiD of each missile is:

ˆ=Ei iD V t⋅ . (4) And the missile with ˆ

it t= is selected as the leader to follow its own optimal RS/LS path. The next step is to plan the flight paths for the follower missiles.

3.1.2. Patterns of elongated path for follower missiles

Because the expected flight time is longer than the calculated minimum flight time of the followers, the main pur-pose of the path planning is to extend the flight time of the followers. However, the missile’s speed should not decel-erate with the scruple that improper deceleration for the follower missile’s flight might cause aerodynamic stall. So the length of flight paths of the followers will be elongated. Meanwhile, the length adjustment and the heading alignment should be accomplished at the early stage of the mid-course flight, to achieve a lock-on as early as possi-ble. Moreover, the calculation process must be fast and effective since the response time for missiles is limited. In this condition, three patterns of elongated paths for follower missiles are designed. The combination of the applica-bility conditions for the three patterns can cover the range of all the expected path lengths. For each follower, one of the three patterns will be applied to plan the path with expected length. The selection of the pattern is determined by both the applicability conditions and the priority order of the path patterns. The priority order from high to low is the circling path, the RS/LS path and the double-turning path, since small flight coverage and less alternations of accel-eration direction are preferred.

a) Pattern 1 (RS/LS path): First, the most intuitive way to elongate the flight path is magnifying the turning ra-dius in the RS/LS path. But the extended range of this pattern is relatively narrow because the magnification of turn-ing radius is subject to the missile-target range and the initial direction of the missile, as shown in Fig. 3. The arc


ST is the longest path when magnifying the turning radius in RS/LS pattern, and the maximum path length in this case is

max1=D ST . The corresponding turning radius is / (2 sin )r s⋅ where r is the sight distance and s is the

heading error. In addition, larger turning radius will result in larger flight coverage area, which could potentially cause collision or losing communication with the launching platform, especially when the heading error is relatively large. Consequently, the turning radius will be restricted in a certain range. Suppose that maxR is the maximum safe turning radius. The maximum path length under the restriction maxR R≤ is max 2 max= ( )RSD D R . Thus, the magnifying of the turning radius is limited by the two conditions above. Last, it will be proved in the appendix (see Theorem 1) that the RS/LS path length 1PD monotonically increases with respect to the turning radius R .

max1D and max 2D can be calculated as follows. It is noticeable that the calculation of max 2D is similar to that of

RSD in formula (2).

max1 , 0sinrD ST s s πs⋅

= = ≤ < (5)

2 2 maxmax 2 max max cosR R

R

RD O T R R SO T arcO T

= − + ⋅ ∠ −

. (6)

The applicability condition of this path pattern is { }max1 max 2min ,ED D D≤ . The length of this path pattern is

2 21 2 2 2 2

2 cos cos( ) ( )

TP T T T

T T T T

R x RD x Rx y R arc arcx R y x R y

− = − + + ⋅ − − + − +

. (7)

Fig. 3 RS/LS optimal Dubins paths with different turning radii.

b) Pattern 2 (Circling path): Circling near the start point is a simple solution to cope with the path elongating problem when the expected path length is relatively large. Since the circling will be executed at least once, the ex-pected length ED should be at least RS2 +mR Dπ ⋅ .

The applicability condition of this path pattern is RS 2E mD D Rπ≥ + ⋅ . (8)

The length of this path pattern is

2 22 2 2 2 2

2 cos cos 2( ) ( )

TP T T T

T T T T

R x RD x Rx y R arc arc n Rx R y x R y

π − = − + + ⋅ − + ⋅ − + − +

(9)

where the number of circles n and the turning radius R are two unknown variables. Thus, more than one solution may exist for the equation 2 ( , )P ED n R D= . Since a small flight coverage area is preferred, the missile will circle for the maximum n with small turning radius. Then, the circles number n can be determined in advance for a unique solution as follows:

RS

2E

m

D Dn

Rπ −

=

. (10)

Because n is determined and the monotonicity of 1PD with respect to R is already proved, the monotonicity of

2PD with respect to R is obvious.


Therefore, the RS/LS path will be used for the followers if { }max1 max 2min ,ED D D< . If RS 2 RE mD D π> + ⋅ , the

circling path will be used. The above two path patterns are enough for follower missiles to extend the path length if { }max1 max 2 RSmin , 2 mD D D Rπ≥ + ⋅ . Otherwise, an eclectic solution is needed if { }max1 max 2 RSmin , 2E mD D D D Rπ< < + ⋅ .

c) Pattern 3 (Double-turning path): An illustration of the double-turning path is shown in Fig. 4. The planned path is combined by two circle arcs and a straight line. A flight arc with a certain turning angle δ is executed in advance (see Fig.4). Then, a new optimal Dubins path starting from the new position rS will be calculated and ex-ecuted. The minimal turning radius mR is used in the two circle arcs. The path length is 3 RSP m rD R Dδ= ⋅ + , where

RSrD is the path length of the RS/LS path from rS to T . The monotonicity of 3PD with respect to δ , and the conclusion that 3PD covers the range from RSD to RS 2 mD Rπ+ ⋅ will be proved in appendix (see Theorem 2). The coordinate of T in the XSY coordinate system is T( , )Tx y . Then, the coordinate of T in the r r rX S Y co-ordinate system T( , )r Trx y can be calculated as follows:

T

T

coscos sin=

sinsin cosm mr T

mTr

R Rx xRy y

δδ δδδ δ

⋅ − ⋅ − ⋅−

. (11)

The applicability condition of this path pattern is { }max1 max 2 RSmin , 2E mD D D D Rπ< < + ⋅ . (12)

The length of this path pattern is:

2 2 T3 2 2 2 2

2 cos cos( ) ( )

m r mP Tr m Tr Tr m m

Tr m Tr Tr m Tr

R x RD x R x y R arc arc R

x R y x R yδ

− = − + + ⋅ − + ⋅ − + − +

. (13)

Fig. 4 A double turning path.

The completeness of the three path patterns: Based on the above three path patterns, the length of elongated paths for followers range from RSD to +∞ ,

where RSD is the shortest path length to arrive at the target. In this sense, any follower can obtain a desirable planned path from the three path patterns.

As introduced at the outset of Subsection 3.1.2, one of the three path patterns is selected for each follower. Further, the applicability conditions and the mentioned priority order of the path patterns can uniquely determine the appro-priate path pattern for each follower with expected path length. To intuitively reveal the selection of the path pattern, Table 1 enumerates all the possible cases and the reasons for the selections.

Table 1 Selection of the path pattern in all cases Conditions Selection Reason

{ }max1 max 2 RSmin , and 2E E mD D D D D Rπ≤ ≥ + ⋅ Circling path In this case, the RS/LS path will result in a rela-tively large turning radius which is not preferred

because of its large flight coverage.

{ }max1 max 2 RSmin , and 2E E mD D D D D Rπ≥ ≥ + ⋅ Circling path The only choice

{ }max1 max 2 RSmin , and 2E E mD D D D D Rπ≤ ≤ + ⋅ RS/LS path The only choice

{ }max1 max 2 RSmin , 2E mD D D D Rπ< < + ⋅ Double turning path

This path pattern is designed to cope with this case which cannot be addressed by either Pattern 1 or 2.


The rationality of the three path patterns is reflected from the following four aspects: 1) The heading error is eliminated in the soonest way while the path is elongated to the expected length, which

will benefit the locking-on later; 2) The flight coverage is as small as possible, which reduces the chance that the missile is detected in the early

stage of attacking; 3) There is only one unknown parameter in each path pattern and the path length of each path patterns is mono-

tonic with regard to each parameter respectively, which facilitates a simple and fast calculation by bisection algorithm;

4) The resulting guidance command does not change frequently, as fickle command may potentially cause in-stability in the missile’s flight.

3.1.3. Calculation of parameters for follower paths

In order to obtain the elongated path with selected pattern and expected length, only one parameter needs to be determined, that is, the turning radius R in patterns 1 and 2 and δ in pattern 3. However, the analytical solutions of these parameters are not easy to get since the equations ( 1,2,3)Pi EiD D i= = are transcendental. The bisection algorithm is used in this proposal to solve the proper parameters for paths with expected length, as long as the path length of the three patterns can be calculated with given parameters. Meanwhile the monotonicity conditions of the path length for applying the bisection algorithm are satisfied25.

The pseudo-code of the bisection algorithm is presented in Algorithm 1.

Algorithm 1: The bisection algorithm for calculation of path parameters

Input: The path length function ( )F x , the expected path length ED , the given bound [ , ]x a b∈ , and the threshold val-ue ε .

Output: The parameter x for ex-pected path length.

%Precondition:

% (a) 0EF D− ≤ , (b) 0EF D− ≥ , and

% ( )F x is monotonic in [ , ]a b

1) maxx b= ; minx a= ;

max min0 2

x xx

+=

2) If ( ) 0EF a D− =

3) x a= ; return;

4) End If

5) If (b) 0EF D− =

6) x b= ; return;

7) End If

8) While 0( ) EF x D ε− >

9) If 0( ) 0EF x D− <

10) min 0x x= ;

11) End If

12) If 0( ) 0EF x D− >

13) max 0x x= ;

14) End If

15) max min0 2

x xx

+= ;

16) End while

17) return 0x ;

a) Arguments calculation by the bisection algorithm in the RS/LS path:

{ }max 1, min / (2 sin ), ; ( )m P ER R r R D R Ds∈ ⋅ = If the RS/LS path is chosen to obtain the expected path length, the desired turning radius R is calculated by the

bisection algorithm with 1( )P ED R D= . 1( )E P mD D R> is obvious, and { }( )1 maxmin / (2 sin ), 0P ED r R Ds⋅ − ≥ is known since the applicability condition for RS/LS path is satisfied. So the given bound for R is { }max,min / (2 sin ),mR R r Rs∈ ⋅ . The output is a proper turning radius R for followers to apply the RS/LS path.

b) Arguments calculation by the bisection algorithm in the circling path:

[ ] 2, 2 ; ( )m m P ER R R D R D∈ = If the circling path is chosen to obtain the expected path length, the desired turning radius R is calculated by the

bisection algorithm with 2 ( )P ED R D= . 2 ( )E P mD D R≥ is tenable as the applicability condition for the circling path

is satisfied. 2 1(2 ) (2 ) 2 2 0P m E P m m ED R D D R n R Dπ− = + ⋅ − > , since RS

2E

m

D Dn

Rπ −

=

. So the given bound for R is

[ ], 2m mR R R∈ . The output is a proper turning radius R for follower missiles to apply the circling path.


c) Arguments calculation by the bisection algorithm in the double turning path:

[ ] 30, ; ( )P ED Dδ π δ∈ = If the double turning path is chosen to obtain the expected path length, the desired arc angle δ to satisfy 3 ( )P ED Dδ = can be calculated by the bisection algorithm. RS 3 (0)E PD D D> = is tenable as the double turning path

is the same as RS/LS path with minimal turning radius when 0δ = . It can also be proved that 3 RS( ) 2P m ED D R Dπ π> + ⋅ > (see the appendix for detail). So the given bound for δ is [ ]0,δ π∈ . The output is a

proper arc angle δ for follower missiles to apply the double turning path.

3.2 Integrated cooperative guidance command

Because the path pattern is simple, it is easy to derive the guidance commands corresponding to the three path patterns in Subsection 3.1. We take the guidance commands in Fig. 5 for example to illustrate the three patterns. Rt is the flight time of missiles with constant normal acceleration in the RS/LS path. Ct is the flight time of missile with constant normal acceleration in the circling path. 1Dt and 2Dt represent the missile’s flight time of the two arcs in sequence in the double-turning path, respectively.

Fig. 5 Control commands for generating the planned path

The parameters Rt , Ct , 1Dt and 2Dt are calculated as follows:

2 2 2 2cos cos

( ) ( )T

R

T T T T

R xR Rt arc arcV x R y x R y

− = ⋅ − − + − +

(14)

2 2 2 2

2cos cos( ) ( )

TC

T T T T

R xR R n Rt arc arcV Vx R y x R y

π − ⋅ = ⋅ − + − + − +

(15)

1m

DR

tV

δ ⋅= (16)

T2 2 2 2 2

cos cos .( ) ( )

m m r mD

Tr m Tr Tr m Tr

R R x Rt arc arc

V x R y x R y

− = ⋅ − − + − +

(17)

Denote by Ra and Ca the acceleration values in the RS/LS path and the circling path, respectively. Denote by

1Da and 2Da the accelerations corresponding to the two arcs in the double-turning path in sequence. The negative sign in 1Da means the direction deviates from the turning direction in the RS/LS path.

2 /Ra V R= (18) 2 /Ca V R= (19) 2

1 /D ma V R= − (20) 2

2 /D ma V R= (21)

To provide an integrated guidance law, the classic CPN is employed in the terminal course. The simple form of CPN is shown as below. Interested readers may refer to the work of Jeon3 for more details.


( ),ˆ1 ( ) ( ) ( )

1i i go go i i ima N Kr t t t t t V q

m = − − −

where N is a constant which is usually set to be around 3, K is the coordination coefficient, ir is the sight dis-tance, m is the number of missiles, got is the estimate of time-to-go, got is the mean value of all the estimate of time-to-go, and q is the rate of LOS.

Each missile will switch into the terminal guidance when the sight distance to the target and the heading error are in a certain range. The switching condition is as follows:

i ir S≤ and i iCη ≤ 1, 2...,i n= ,

where ir and iη are the sight distance and the heading error of the ith missile, respectively. iS and iC are the maximum sight distance of the seeker and the FOV constraint of the ith missile, respectively. st stands for the moment when the switching conditions of all engaged missiles are satisfied.

For the thi missile which executes an RS/LS path in the mid-course, the integrated guidance command is

( )

,

,

, 0 ,

0, ,

ˆ1 (t) (t) (t) , .1

R i R

i R s

i go go i i i s

a t t

a t t tmN Kr t t V q t t

m

< <= < <

− − ≥ −

(22)

For the thi missile which executes a circling path in the mid-course, the integrated guidance command is

( )

,

,

, 0 ,

0, ,

ˆ1 (t) (t) (t) , .1

C i C

i C s

i go go i i i s

a t t

a t t tmN Kr t t V q t t

m

< <= < <

− − ≥ −

(23)

For the thi missile which executes a double turning path in the mid-course, the integrated guidance command is

( )

D1, 1

D 2, 1 1 2

1 2

,

, 0 ,

, + ,

0, ,

ˆ1 (t) (t) (t) , .1

i D

i D D D

i

D D s

i go go i i i s

a t t

a t t t ta

t t t tmN Kr t t V q t t

m

< < < <

= + < < − − ≥ −

(24)

Remark 1. The convergence of the cooperative guidance law is guaranteed jointly by the coordinated path plan-ning in mid-course and the application of CPN in terminal course. The Dubins path and the path patterns for elonga-tion are calculated by geometric method with fixed initial configuration and terminal position. Therefore, the paths planned in the mid-course will definitely reach the target. Then, the planned paths for the followers are calculated by the bisection algorithm according to the expected flight time. The bisection algorithm is an established method for determining the zero of a monotonic function, and it is considered highly efficient. The path length function of each path pattern has been proved monotonous with regard to the undetermined parameter in the appendix. Therefore, the arrival time synchronization of the planned paths is guaranteed. In other words, the flight time of followers converge to the expected flight time which is the same as the minimum flight time of the leader. In this case, the headings and the time-to-go of the missiles at the switching moment satisfy the initial conditions required by CPN for the coopera-tive terminal guidance. At the terminal guidance phase, the CPN is applied and the convergence of the algorithm has been proved by Jeon3. The impacting error of the missiles and the variance of the arrival time will converge to zero. So, the convergence of the cooperative guidance law is guaranteed.

A smooth switch from mid-course into terminal course benefits the performance of cooperative terminal guidance. And the designed three path patterns have the feather that heading straight toward the target at the transition instant. Then, the accelerations of the missiles at the transition instant st are zero if the planned paths are traced. But the configurations of missiles at the transition instant are crucial to the performance of cooperative terminal guidance. So we present the analytic method to predict the condition of missiles at st .


To calculate the configurations (positions and headings) of the missiles at the transition instant, we should first

calculate the headings and the distance of the missiles from the target since the missiles are all heading straight to-ward the target at that moment. According to the assumption that the missiles switch into the cooperative terminal guidance phase at the moment every missile detects the target, the distance of each missile is calculated as follows:

1,2...min ( )jtrans

i ij nj

Sd V

V== ⋅ (25)

where jS is the detecting range of each missile. Then we predict the transition instant as 1,2...

ˆ min ( )js j n

j

St t

V== − . And the

heading of every missile at the moment is calculated as follows: 0

, , ,trans

R i i R i R ia tθ θ= − ⋅ (26) 0

, , ,trans

C i i C i C ia tθ θ= − ⋅ (27) 0

, 1, 1, 2, 2,trans

D i i D i D i D i D ia t a tθ θ= − ⋅ − ⋅ (28) (26), (27), (28) are the calculation corresponding to the RS/LS path pattern, the Circling path pattern and the Double turning path pattern. 0

iθ is the initial heading of the missile, and acceleration is position if the missile turns right by the command.

Then the position of the missile at the transition instant ( , )trans transi ix y is as follows:

*,cos( )trans trans transi T i ix x d π θ= + ⋅ + (29)

*,sin( )trans trans transi T i iy y d π θ= + ⋅ + (30)

where *,trans

iθ means the heading is one of ,trans

R iθ , ,trans

C iθ or ,trans

D iθ . Then the predict configuration of i th missile at the transition instant is *,( , , ,0)trans trans trans

i i ix y θ which stands for the position, heading and the acceleration.

4. Simulation and analysis

The simulation consists of three parts so as to demonstrate the necessity of midcourse cooperative guidance and the performance of the proposed coordination methods for multiple missiles, respectively. In each part, three scenar-ios of simultaneous arrival are performed. The scenarios describe a group of missiles attacking a stationary target with different initial conditions. There are parameters of missiles such as cruising speed, maximum acceleration, de-tecting range need to be preset in the simulation. The choices of the parameters in the three parts are made with dif-ferent of special emphasis.

In part A), the scenarios are designed to clearly represent the three path patterns and illustrate that the proposed guidance method usually has good performance while the comparison method cannot achieve salvo attack. So the parameters of missiles are chosen according to the performance indexes of common cruise missile. The missiles are with cruising speed around the sonic velocity, and the maximum overload is in between 6g and 11g, the detecting range is set to be 8km. And the initial headings of missiles are randomly set, while the initial positions are set to gen-erate a result that all three path patterns are applied in each scenario.

In part B), the scenarios are designed to verify the advantage of proposed method in the final arrival time of the salvo attack. So the parameters of missiles are similar to the missiles in part A). And the initial headings are also randomly set, but the initial positions are set to ensure that the comparison method can also achieve a salvo attack.

In part C), the scenarios are designed to illustrate the robustness of the proposed method. So the missiles are as-sumed to be various. The missile speeds vary from Mach 0.82 to 1.62, the maximum overload is in between 9g and 22g, the detection range could be 8km, 9km or 10km. The initial positions and headings are all randomly generated.

Last, the maximum turning radius maxR is set to be m2R for all the missiles when calculating the applicability condition of the path patterns.

A) Verifying the necessity of midcourse cooperative guidance

In this part, the midcourse guidance law with terminal handover constraint recently proposed by Morgan15 is com-bined with CPN in terminal course to perform cooperative guidance. This composite method is named Morgan&CPN here and used for comparison. The two constraints described in Section 1 are taken into consideration. Hence, we assume that the missiles will not join the terminal cooperative guidance if they are not yet locked-on the target at that moment. And the first several missiles that have locked-on the target activate the cooperative terminal guidance. Then, missiles that lock-on the target afterward can join the cooperation and play an equal role only if the previous ones are not about to reach the target. In the terminal course, we consider the missiles as losing the track of target and seceding from the terminal guidance if they generate heading errors that are larger than the FOV constraint on ac-


count of detouring. The FOV constraint is that the heading error is in between 75± . Fig. 6 shows the simulation re-sults of the three scenarios when applying the above-mentioned guidance scheme. In Fig. 6, the dashed lines depict the flight path of the four engaged missiles. The marks of small circle stand for the handover points that the missiles have locked-on the target. The marks of diamond stand for the moment that missiles join the cooperative terminal guidance.

Table 2 Initial conditions of engaged missiles Velocity

(m/s) Initial Posi-tion (km)

Initial Head-ing (deg)

Minimal turn-ing radius (m)

Maximum ac-celeration (m/s2)

Detection Range(km)

Scenario A1 Missile 1 320 (18,18) 60 931 110 8 Missile 2 320 (-17,17) -90 1024 100 8 Missile 3 320 (11.5,-12.5) -90 1138 90 8 Missile 4 320 (-10,-12) 157.5 1024 100 8

Scenario A2 Missile 1 300 (-10,10) 67.5 900 100 8 Missile 2 340 (11,12) -22.5 1651 70 8 Missile 3 330 (11,-11) 90 990 110 8 Missile 4 320 (-13,-15) -45 1365 75 8

Scenario A3 Missile 1 310 (-16.5,15.5) 135 1201 80 8 Missile 2 360 (15,16) -70 1178 110 8 Missile 3 330 (18,-18) 50 1815 60 8 Missile 4 320 (-14,-16) 60 1138 90 8

The flight time of missiles hitting the target is presented in Table 3. The results shown in Fig. 6 and Table 3 are further expatiated to make the scene clear. In scenario A1, missile 4

and missile 2 successively acquire the target at 47.80s and 56.41s, CPN takes over the terminal guidance of both missiles at the moment that missile 2 locks on. Missile 3 joins the cooperative terminal guidance later at 56.48s when it acquires the target. However, missile 4 lost the target at 60.79s due to the violation of the FOV constraint when detouring, though the CPN rapidly impels missile 3 to head straight to the target to shorten the flight time. After that, missile 2 and missile 3 hit the target at 81.40s and 81.52s. Missile 1 has not acquired the target yet when missile 2 and missile 3 hit the target.

(a) Scenario A1 (b) Scenario A2 (c) Scenario A3

Fig. 6 Flight paths without midcourse cooperative guidance In scenario A2, missile 3 hit the target alone at a much earlier time that is 51.09s. Missile 1 and missile 2 succes-

sively acquire the target and successfully hit the target simultaneously at 73.10s. Missile 4 acquires the target too late at 63.57s when missile 1 and missile 2 are already very close to the target that it cannot join the terminal cooperation.

In scenario A3, missile 4 and missile 2 successively acquire the target at 41.70s and 55.52s. Then the terminal co-operation starts at 55.52s. Missile 4 lost the target in the cooperative guidance with missile 2 at 60.12s. At last, Mis-sile 2 hit the target at 77.74s alone. Missile 1 and missile 3 have not acquired the target until the cooperative guid-ance ends up in failure.


Table 3 Simulation record of missiles without midcourse cooperative guidance

Missile 1 Missile 2 Missile 3 Missile 4 Scenario A1

Time of acquisition - 56.41s 56.48s 47.80s Time of joining terminal cooperation - 56.41s 56.48s 56.41s

Time of target lost - - - 60.79s Time of hit - 81.40s 81.52s -

Scenario A2

Time of acquisition 44.17s 49.32s 26.71s 63.57s Time of joining terminal cooperation 49.32s 49.32s - -

Time of target lost - - - - Time of hit 73.10s 73.10s 51.09s -

Scenario A3

Time of acquisition - 55.52s - 41.70s Time of joining terminal cooperation - 55.52s - 55.52s

Time of target lost - - - 60.12s Time of hit - 77.74s - -

The results shown in these simulations illustrate the necessity of the mid-course cooperative guidance when mis-siles perform a simultaneous attack. If we postpone the launching time of the missiles that have shorter flight time or alter the launching angles in order to ameliorate the initial condition for the terminal cooperation, it will excessively increase the period of occupation of the engaged platform since the initial launching conditions are rigorous. In this proposal, the launching platform is able to launch the missile right away and forget. The platforms are out of occupa-tion and ready for other subsequent tasks right after launching.

As comparison, the simulations applied with the proposed cooperative guidance law in the same scenarios are shown. First of all, the optimal RS/LS flight paths for all missiles will be calculated at the start moment of the coop-erative attack assignment. The minimum flight time of the four missiles in the three scenarios can be obtained and are shown in Table 4. Then, the missile with longest minimum flight time, that is missile 1 in scenario A1, missile 4 in scenario A2, missile 1 in scenario A3, will be selected as the leader and follow the optimal RS/LS path. The flight time of the leader will be applied as flight time constraint in the path planning of the other missiles. The expected path length of each follower missile can be calculated according to the expected flight time and its own speed.

Table 4 Simulation record of the engaged missiles RS/LS

path flight

time (s)

Variance of RS/LS path flight

time

Hand off moment

(s)

Time-to-go by PNG at hand-off

moment (s)

Variance of time-to-go at hand-off mo-

ment

Final arrival time (s)

Variance of final arri-val time

Scenario A1 Missile 1 86.88

184.82 62.14

24.99

0.7264

87.14

0.0133 Missile 2 75.39 22.78 87.03 Missile 3 59.57 24.35 86.98 Missile 4 51.94 24.68 86.82


70.1978 44.33

22.64

0.8779

68.97



91.61 62.50

22.68

0.6059

85.19




Fig. 7 Planned flight paths of the four engaged missiles

In the coordinated path planning process, the path pattern for each missile will be chosen according to Table 1. After the path pattern is confirmed, the bisection algorithm is used in this proposal to find the proper parameters for paths with expected length. Fig. 7 depicts the planned flight paths of the four engaged missiles to attack the target simultaneously. Control commands for these planned paths are shown in Fig. 8.


Fig. 8 Mid-course guidance commands The control commands shown in Fig. 8 will be used as guidance law in the midcourse of missile flight, then all the

engaged missiles will approach the target and generate paths that are similar in path length. When all the mis-sile-target distances and missile heading errors satisfy the lock-on condition, the missiles start the cooperative termi-nal guidance. The lock-on condition in this simulation is that the sight distance is less than the detection range and the heading error is less than 30 . And according to the calculation in section 3.3, the predict configurations of mis-siles at the switching instant are: (5114,6126,-129.9 º ,0), (-6440,4563,-35.3 º ,0), (4169,-6819,129.7 º ,0), (-6156,-5109,39.7º ,0) in scenario A1, (-4047,5706,-54.7º ,0), (6749,4295,-147.5º ,0), (5819,-5182,138.3º ,0), (-4194,-5733,53.8 º ,0) in scenario A2, (-4434,5328,-50.2 º ,0), (6011,5278,-138.7º ,0), (6122,-4048,146.5 º ,0), (-5772,-4013,-145.2º,0) in scenario A3.


Fig. 9 Flight paths by the proposed guidance method

Fig. 9 depicts the paths generated by the joint cooperative guidance in the midcourse and terminal course. The flight paths in the midcourse are indicated by solid lines, and the terminal course paths by terminal guidance law are indicated by dashed lines. The four rhombuses denote the points that missiles switch into cooperative terminal guid-ance at the same time. It is obvious that the ZEM at the handover points of all missiles are sufficiently small which is


preferred for acquiring the target. As compared with Fig. 6, Fig. 9 demonstrates that the proposed cooperative mid-course guidance ensures a successful simultaneous attack and the initial launching condition is relaxed.

In scenario A1, missile 1 with the longest minimum flight time 86.88s is selected as the leader. At 62.14s, all mis-siles have locked on the target and switch into the cooperative terminal course. The heading errors of all missiles and the variance of time-to-go at the hand-off moment are sufficiently small. Then, the cooperative terminal guidance is carried out successfully. In scenario A2, missile 4 is selected as the leader. At 44.33s, all missiles switch into the co-operative terminal course. The cooperative terminal guidance is also carried out successfully. In scenario A3, missile 1 is selected as the leader. All missiles switch into the cooperative terminal course at 62.50s.

At the hand-off moments, all the missiles have locked-on the target and the time-to-go is estimated if traditional PNG is carried out in the terminal course as shown in Table 4. The final arrival time of the engaged missiles are also shown. It is obvious that due to the cooperative mid-course guidance, the variance of the time-to-go at the hand-off moment is much less than the variance of the RS/LS path flight time at the initial moment. Moreover, the variance of the final arrival time almost decreases to zero when the cooperative terminal guidance course is over. The maximum difference among the impact time of engaged missiles is 0.32 seconds in scenario A1, 0.27 seconds in scenario A2, and 0.37 seconds in scenario A3. It is rational to indicate that the four missiles impact on the target almost at the same time by applying the proposed guidance law in each scenario.

Table 5 Parameters of scenarios for attack time compare

B) Simulations to illustrate the advantage of quick salvo attack

The applicable scenarios for Morgan&CPN are much fewer than those for the proposed method. The scenarios for comparison are those in which salvo attack can be realized by both methods. The parameters are listed in Table 5. The results of the two methods are presented in Table 6 and Fig.10.

(a) Scenario B1 (Morgan&CPN) (b) Scenario B2 (Morgan&CPN) (c) Scenario B3 (Morgan&CPN

Velocity (m/s)

Initial Position

(km)

Initial Head-ing

(deg)

Minimal turning radi-us

(m)

Maximum accelera-tion (m/s2)

Scenario 1 Missile 1 320 (12,12) 60 1024 100 Missile 2 320 (-17,17) -90 1024 100 Missile 3 320 (11.5,-12.5) -90 1024 100 Missile 4 320 (-12,-14) 157.5 1024 100 Scenario 2 Missile 1 300 (-10,10) 67.5 900 100 Missile 2 340 (11,12) -22.5 1156 100 Missile 3 330 (14,-14) 90 1089 100 Missile 4 300 (-12,-12) -15 900 100 Scenario 3 Missile 1 310 (-11.5,11.5) 90 961 100 Missile 2 360 (15,16) -70 1296 100 Missile 3 330 (16,-16) 80 1089 100 Missile 4 320 (-16,-18) 60 1024 100


(d) Scenario B1 (Proposed guidance) (e) Scenario B2 (Proposed guidance) (f) Scenario B3 (Proposed guidance)

Fig.10 Comparison between Morgan&CPN and the proposed method

As shown in Table 6, the final attack time of the missiles that apply the proposed method is surely much earlier than the counterpart by Morgan&CPN. So, the proposed cooperative strategy is advantageous in the final arrival time of the salvo attack. Moreover, the elongations of path length are accomplished at the early stage of the mid-course flight. As a result, the flight paths in terminal course are smooth and short, which is also advantageous in the battle-field.

Table 6 Final attack time of two methods Final attack time (s)

Proposed method Morgan&CPN Scenario 1

Missile 1 75.36 86.62 Missile 2 75.33 86.70 Missile 3 75.36 86.71 Missile 4 75.34 86.68

Scenario 2


Scenario 3


C) Simulations to illustrate the robustness of the proposed method

In this part, the performance of the proposed method is further validated by various scenarios. Parameters such as the number of the engaged missiles, the location of the missiles, and the capability of missiles are different in the scenarios. The final arrival time of the missiles, the variance of final arrival time, and the variance of time-to-go by PNG at the hand-off moment indicate the cooperative guidance performance.

Three different scenarios are presented here. The considered performance indexes of missiles are the missile speed, the maximum acceleration, and the detection range of the seeker. The parameters of missiles in every scenario are shown in Table 7. The flight paths are shown in Fig.11. The records of simulations are presented in Table 8. In sce-nario C1, missiles 1 and 2 are from one field and have better performance than missiles 3 and 4 from another field. In scenario C2, five missiles are engaged, missiles 1, 2 are from one field, and missiles 3, 4, 5 are from another field with much longer distances to the target. In scenario C3, missile 5 alone is from one field which is much closer to the target in contrast with another field, and the rest four missiles from another field have much better performances.


Table 7 Parameters of simulation scenarios

Velocity (m/s)

Initial Posi-tion (km)

Initial Heading

(deg)

Minimal turning radi-

us (m)

Maximum acceleration

(m/s2)

Detection Range (km)

Scenario C1 Missile 1 430 (-29,5) 87 1233 150 10 Missile 2 550 (-30,-3) -63 1779 170 10 Missile 3 280 (20,1) 92 871 90 9 Missile 4 340 (19,9) -82 1156 100 9

Scenario C2 Missile 1 410 (-17,6) 12 1293 130 8 Missile 2 380 (-16,-5) -21 1313 110 8 Missile 3 510 (30,9) 153 1130 230 9 Missile 4 390 (35,1) 122 1086 140 9 Missile 5 400 (32,-6) -90 1143 140 9

Scenario C3 Missile 1 460 (-28,10) 43 1058 200 10 Missile 2 500 (-29,6) 50 1316 190 10 Missile 3 470 (-27,-1) 34 1163 190 9 Missile 4 510 (-29,-7) -29 1182 220 10 Missile 5 350 (17,2) 143 1114 110 8

The results of the simulations are quite satisfying in different scenarios with various parameters and tactical char-acteristics. As illustrated by Table 8 and Fig.11, the engaged missiles in each scenario have accomplished salvo at-tack. The variances of the final arrival time are small enough. And the adjustment of flight path length is achieved at the early stage of the mid-course flight owing to the effective path patterns for elongation. At the cooperative termi-nal guidance switching point indicated by the rhombuses, all the missiles have acquired the target and switched into cooperative terminal guidance successfully. In scenario C1, the speed of missile 2 is nearly twice of missile 3. In scenario C2, the minimum flight time of missile 4 is more than twice of missile 1 or 2. Even with such speed differ-ences, the proposed method still achieves salvo attack efficiently. Moreover, it is noticeable that the final arrival time of engaged missiles in every scenario is very close to the longest one among the minimum flight time of all the mis-siles which is theoretically the earliest completion time of the salvo attack. In this sense, the proposed method also achieves a soonest salve attack.

(a) Scenario C1 (b) Scenario C2 (c) Scenario C3

Fig. 11 Flight paths by the proposed guidance method


Table 8 Simulation records of the scenarios

RS/LS path flight

time (s)

Variance of RS/LS

path flight time

Hand off moment

(s)

Time-to-go by PNG at

hand-off mo-ment (s)

Variance of time-to-go at hand-off

moment

Final arrival time (s)

Variance of final arrival time

Scenario C1 Missile 1 70.51

47.43 41.27

32.08

0.0243

73.35

0.0130 Missile 2 55.72 32.10 73.20 Missile 3 73.41 32.33 73.52 Missile 4 62.95 31.89 73.32 Scenario C2 Missile 1 44.06

373.47 67.20

23.02

0.0170

90.22

0.0053 Missile 2 44.29 22.90 90.18 Missile 3 61.58 22.95 90.17 Missile 4 90.29 23.19 90.37 Missile 5 83.68 22.80 90.20 Scenario C3 Missile 1 65.12

26.58 42.06

23.11

0.0131

65.15

0.0045 Missile 2 59.76 22.96 65.03 Missile 3 57.56 22.87 64.97 Missile 4 58.65 22.99 65.06 Missile 5 49.14 22.77 64.97

5. Conclusion

This paper proposes a joint mid-course and terminal course cooperative guidance law for supporting the salvo at-tack of multi-missile with different initial conditions. The mid-course cooperative guidance law is derived from the path planning for all missiles under a distinct leader-follower framework. Dubins path with terminal heading relaxa-tion and the three path patterns for followers ensure a soonest salvo attack. When all missiles lock on the target, the cooperative guidance will switch from the mid-course to the terminal course. The cooperative proportional naviga-tion is employed as the terminal cooperative guidance law in order to lead the missiles to impact on the target simul-taneously. The synergy of these two courses provides an effective solution for multiple missiles to salvo attack a sur-face target, especially for long range and medium range missiles or the stand-off attacking. As for the mid-course guidance, the proposed three path patterns for follower missiles and the corresponding path calculation method based on the bisection algorithm are very simple to implement. Simulations validate the effectiveness of the joint coopera-tive guidance law.

Acknowledgement

The leader-follower framework for joint mid-course and terminal course cooperative guidance of multiple missiles in salvo attack is proposed by Bin Xin. Bisection algorithms for different path patterns are built and implemented by Jie Zeng under the guide of Bin Xin and Lihua Dou. The work of Bin Xin is supported by the National Natural Sci-ence Foundation of China (No. 61304215) and the Doctoral Program Foundation of Institutions of Higher Education of China (No. 20131101120033). The work of Lihua Dou is supported by the Beijing Education Committee Cooper-ation Building Foundation Project (CSYS100070417).

References

1. Jeon IS, Lee JI, and Tahk MJ. Impact-time-control guidance law for anti-ship missiles. IEEE Trans Control Syst Technol 2006; 14(2): 260-6. 2. Lee JI, Jeon IS, Tahk MJ. Guidance law to control impact time and angle. IEEE Trans Aerosp Electron Syst 2007; 43(1): 301-10.


3. Jeon IS, Lee JI. Homing guidance law for cooperative attack of multiple missiles. J Guidance Contr Dyn 2010; 33(1): 275-80. 4. Zhao SY, Zhou R. Cooperative guidance for multi-missile salvo attack. Chin J Aeronaut 2008; 21(6): 533-9. 5. Zou L, Kong FE, Zhou R, Wu J. Distributed adaptive cooperative guidance for multi-missile salvo attack. J Bei-jing Univ Aeronaut Astronaut 2012; 38(1): 128-132 [Chinese]. 6. Zhao SY, Zhou R, Chen W. Design of time-constrained guidance laws via virtual leader approach. Chin J Aeronaut 2010; 23(1): 103-8. 7. Zhang YG, Zhang YA. Three-dimensional guidance law to control impact time and impact angle: a two-stage con-trol approach. Control Theory Appl 2010; 27(10): 1429-34. 8. Zhang YA, Ma GX, Liu AL. Guidance law with impact time and impact angle constraints. Chin J Aeronaut 2013; 26(4): 960-6. 9. Zhang YA, Wang XL, Wu HL. Impact time control guidance law with field of view constraint. Aerosp Sci Technol 2014; 39: 361-9. 10. Zhang YA, Wang XL, Wu HL. A distributed cooperative guidance law for salvo attack of multiple anti-ship mis-siles. Chin J Aeronaut 2015; 28(5): 1438-50. 11. Zhao J, Zhou R. Unified approach to cooperative guidance laws against stationary and maneuvering targets. Non-linear Dyn 2015; 81(4): 1635-47. 12. Zhao J, Zhou R, Dong Z. Three-dimensional cooperative guidance laws against stationary and maneuvering tar-gets. Chin J Aeronaut 2015; 28(4): 1104-20. 13. Zeng J, Dou LH, Xin B. Cooperative Salvo Attack Using Guidance Law of Multiple Missiles. Journal of Ad-vanced Computational Intelligence & Intelligent Informatics 2015; 19:301-6. 14. Zhou J, Yang J, Li Z. Simultaneous attack of a stationary target using multiple missiles: a consensus-based ap-proach. Sci China Inform Sci 2017; 60(7):1-14. 15. Morgan RW. Midcourse guidance with terminal handover constraint. Proceedings of the American Control Con-ference; 2016 July 6-8; Boston, MA, USA. IEEE; 2016. p. 6006-11. 16. Guelman M. A qualitative study of proportional navigation. IEEE Trans Aerosp Electron Syst 1971; 7 (4): 637-43. 17. Ho YC, Bryson Jr AE, and Baron S. Differential games and optimal pursuit-evasion strategies. IEEE Trans. Au-tom. Control 1965; 10(4): 385-9. 18. Indig N, Benasher J Z, Sigal E. Near-optimal minimum-time guidance under spatial angular constraint in atmos-pheric flight. J Guidance Contr Dyn 2016; 39(7):1-15. 19. Dubins LE. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. Am J Math 1957; 79(3): 497-516. 20. Tanil C. Cooperative Path Planning for Multiple Missiles - An Evolutionary Speciation Approach. AIAA Guid-ance, Navigation, and Control Conference; 2012 Aug 13-16; Minneapolis, MN, USA. 2012. p. 1-13. 21. Meyer Y, Isaiah P, Shima T. On Dubins paths to intercept a moving target. Automatica 2015; 53:256-63. 22. Wang L, Yao Y, He FH, Liu K. A novel cooperative mid-course guidance scheme for multiple intercepting mis-siles. Chin J Aeronaut 2017; 30(3):1140-53. 23. Yao W, Qi N, Zhao J, Neng W. Bounded curvature path planning with expected length for Dubins vehicle enter-ing target manifold. Robot Auton Syst 2017; 97:217-29. 24. Xin B, Chen J, Xu DL, Chen YW. Hybrid encoding based differential evolution algorithms for Dubins traveling salesman problem with neighborhood. Control Theory Appl 2014; 31(7): 941-54. 25. Parida PK, Gupta DK. A cubic convergent iterative method for enclosing simple roots of nonlinear equations. Appl Math Comput 2007; 187(2):1544-51.

Appendix

Theorem 1: In the RS/LS pattern of optimal Dubins path, the path length will monotonically increase if the turn-ing radius is magnified.

Proof: Due to the symmetry, the RS path will be illustrated in the proof, and the same approach can be applied for the LS path. As shown in Fig. A1, the missile path starts from the point S , the target is located at T , the LOS dis-tance between S and T is r , the initial heading is represented by vector V , s is the initial heading error, the turning radii for two RS paths are R and R + ∆ , respectively. The path RS1 with R as turning radius is com-posed of

1SP and 1PT , α is the angle of

1SP while 1RO is the center. The path RS2 with R + ∆ ( 0∆ > ) as

turning radius is composed of 2SP and

2P T , β is the angle of

2SP while 2RO is the center. Line 1 1RO P

inter-sects with

2SP at A, and the angle of 1 2R RO AO∠ is θ .


Fig. A1 Two RS paths with different turning radii

First,

1 1 1(RS )D SP PT= +

, (A1)

2 2 2(RS )D SA AP P T= + +

, (A2)

where

2 2 2 2 1AP P T AP P T AT PT+ > + > >

, (A3)

if

1SPSA ≥ , the conclusion can be proved then.

Obviously,

( ) ,SA R β= + ∆ ⋅ (A4)

1 ,SP R α= ⋅ (A5)

When α π≥ , α β θ= − such that

1SPSA > , then the conclusion is proved.

When α π< , α β θ= + , and that

sin sinR

θ α∆ + ∆

= , (A6)

then

11

1

sin( sin ( )) (R )

sin(R )sin ( )

SPSA RR

R

αα α

αα

−

−

∆− = − ⋅ + ∆ − ⋅

+ ∆∆

= ⋅∆ − + ∆+ ∆

. (A7)

Define

1 sin( ) sin ( )FR R

α αα −∆ ⋅ ∆= −

+ ∆ + ∆. (A8)

Set R

λ ∆=

+ ∆, and 0 1λ< < , then 0 λ α π< ⋅ < . If

2πλ α⋅ > , then ( ) 0F α > , and the conclusion is proved.

When 02πλ α≤ ⋅ ≤ , we can get 1sin ( sin )λ α λ α−⋅ > ⋅ if sin( ) sinλ α λ α⋅ > ⋅ because of the monotonicity of the

sine function in [0, ]2π . Define

sin( )( ) sinG λ αα αλ⋅

= − . (A9)

Obviously, (0) 0G = and ( ) cos( ) cos 0G α λ α α= ⋅ − >

,

which leads to


( ) (0) for [0, ).G Gα α π≥ ∈ (A10)

Therefore, ( ) 0F α ≥ when 02πλ α≤ ⋅ ≤ .

So, the path length will monotonically increase with respect to the turning radius in RS/LS pattern. □ Theorem 2: In the double turning path, the path length will monotonically increase with respect to the angle δ . Proof: Recursive method is used in this proof. As shown in Fig. A2, the flight path 1 is a RS pattern path when 0δ = , while the path 2 is a double-turning path.

Both of them start from S , and end at T . The turning radii in two paths are the same. Such that path 2 is longer than path 1 because the RS pattern path is the shortest path when the initial position and direction, the final position and the turning radius are fixed.20

Consider path 3 which is a double-turning path with the first arc angle being ( 0)δ τ τ+ > . If ignoring the flight arc angle δ , the rest of path 2 is a RS pattern path starting from rS , and the rest of path 3 is a double-turning path starting from rS . Therefore, it is obvious that path 3 is longer than path 2, and path 2 is longer than path 1 as long as

0, 0δ τ> > . In other words, the length of the double turning path will monotonically increase with respect to the angle δ .

Fig. A2 Double turning paths with the first arc angle being δ and δ τ+ .

Besides, it is also obvious that the length of the double turning-path is longer than (RS) 2 RmD π+ ⋅ if δ π> . Then path length in this pattern can range from (RS)D to (RS) 2 RmD π+ ⋅ and the double turning path is a logical

choice when the expected path length is in between { }max1 max 2min ,D D and (RS) 2 RmD π+ ⋅ . □

a joint mid-course and terminal course cooperative...

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