- Email: [email protected]
Integrated cooperative guidance framework and cooperative guidance law for multi-missile
CJA 961 29 December 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx
No. of Pages 10
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Integrated cooperative guidance framework and cooperative guidance law for multi-missile
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Zhao JIANBO, Yang SHUXING *
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School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, Beijing 100081, China
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Received 4 January 2017; revised 7 December 2017; accepted 7 December 2017
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KEYWORDS
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Cooperative; Field-of-view constraint; Finite-time; Framework; Missile guidance; Multi-missiles
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Abstract An integrated cooperative guidance framework for multi-missile cooperatively attacking a single stationary target is proposed in this paper by combining both the centralized and decentralized communication topologies. Once missiles are distributed into several groups, missiles within a single group communicate with the centralized leader-follower framework, while the leaders from different groups communicate using the nearest-neighbor topology. To implement the integrated cooperative guidance framework, a group of Finite-Time Cooperative Guidance (FTCG) laws considering the saturation constraint on FOV (FTCG-FOV) are firstly derived within the centralized leader-follower framework to satisfy the communication topology of missiles in a single group. Then, an improved sequential approach is developed to adapt the FTCG-FOV to satisfy the communication topology between groups. The numerical simulations demonstrate the effectiveness and high efficiency of the integrated cooperative guidance framework and the cooperative guidance laws, as well as the superiority of the developed sequential approach. Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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In modern military operation, it is a challenging task for the missile to attack a land target or a surface ship that is equipped with an antiair defense system.1 To penetrate the antiair
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* Corresponding author at: School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China. E-mail address: [email protected] (Y. SHUXING). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
defense system, the missile should be capable of terminal evasive maneuvering, which would induce increasing cost.2 An alternative way is to conduct a cooperative attack, i.e., multiple missiles coming from different directions attack a single target simultaneously, which has been regarded as a costeffective and efficient way to address the threat of the defense system.3 Therefore, the research on cooperative attack has gained increasing interest in recent years. Considering that a land target or a surface ship is either stationary or moving at a relatively low speed, the target in the cooperative attack problem in this paper is considered to be stationary as is commonly done in practice. To achieve cooperative attack, one can perform an openloop cooperative guidance, i.e., a common impact time is gen-
https://doi.org/10.1016/j.cja.2017.12.013 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: JIANBO Z, SHUXING Y Integrated cooperative guidance framework and cooperative guidance law for multi-missile, Chin J Aeronaut (2017), https://doi.org/10.1016/j.cja.2017.12.013
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erated for all member missiles in advance, and thereafter each missile tries to arrive at the target on time independently.4–7 However, a suitable common impact time is difficult to be generated in advance, since some missiles may not be able to satisfy the impact time constraint due to their specific initial conditions and limited speed. In addition, the open-loop guidance is also lack of robustness to external disturbance during engagement.8 Actually, the open-loop cooperative guidance simply formulates the many-to-one cooperative attack problem as multiple one-to-one attack problems considering the impact time constraints, which cannot be considered as a genuine multi-missile cooperative guidance.9 Alternatively, with the closed-loop cooperative guidance approach, the missiles communicate to each other to synchronize the impact time.10–12 As a well-known closed-loop cooperative guidance method, the Finite-Time Cooperative Guidance (FTCG) has gained much attention due to its fast convergence rate and high accuracy of the time-to-go errors of missiles. To attack a maneuvering target, Song et al.13 proposed a FTCG law with impact angle constraints. In Ref. 1, two distributed FTCG laws are developed based on different time-to-go estimation methods. However, in addition to the normal acceleration command, the tangential acceleration was also required with both FTCG laws, causing extra difficulties in implementation. To address this problem, a more effective FTCG law employing a hierarchical framework was proposed in Ref. 14, which requires only the normal acceleration command. For the closed-loop guidance laws introduced above, either a centralized or decentralized communication topology is utilized. However, both communication topologies are far from the optimum,15 and their drawbacks shown as follows will become much more prominent if more missiles are involved in the cooperative attack. With the centralized communication topology, one or a few missiles should be designated, which can communicate with all the rest missiles. Clearly, the distances between the designated missile and the rest ones are subject to their communication capability, which incurs extra difficulty in designing the commands for cooperative missiles. In addition, poor penetration capability and system reliability would be induced if only one missile is designated, while unacceptable computational burden would be induced if more than one missile are designated. In contrast, with the decentralized communication topology, missiles can communicate only with their neighbors to reduce the computational burden, while the global information of the missile cluster cannot be obtained, causing great difficulty in making optimal decision for cooperative missiles. Moreover, it takes long time to achieve the consensus of time-to-go and some necessary conditions16 are always hard to be satisfied. In addition, due to the limitation of the detective capability of seekers, the saturation constraint on Field-of-View (FOV) should be considered in the closedloop cooperative guidance, which however has been rarely seen in the existing works. And the existing FOV-constraint guidance for the common one-to-one missile-target engagement scenario17–21 cannot be directly applied to the closed-loop cooperative guidance. To address the issues above, a novel integrated cooperative guidance framework is proposed in this paper, in which missiles are distributed into several groups, and then missiles within a single group communicate by means of the centralized leader-follower framework, while the communication between groups employs the nearest-neighbor topology among leaders.
Z. JIANBO, Y. SHUXING The contributions of this paper lie in: (A) the centralized and decentralized communication topologies are combined and integrated into the proposed framework effectively; (B) to implement the proposed integrated cooperative guidance framework, a group of FTCG laws considering the saturation constraint on FOV (FTCG-FOV for short) are designed in this paper by extending the FTCG law in Ref. 21 to a group of FTCG laws and introducing a bias term to satisfy the FOV constraint; (C) the sequential approach in Ref. 21 is improved, which is then employed to make the FTCG-FOV satisfy the requirement of communication between groups. The rest of this paper is organized as follows. In Section 2, the many-to-one missile-target interception engagement along with the proposed integrated cooperative guidance framework is introduced. In Section 3, a group of FTCG-FOV are developed, of which the working process in the integrated cooperative guidance framework is presented in detail. Simulation results are presented and analyzed in Section 4. Conclusions are made in the last section.
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2. Preliminary
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2.1. Problem formulation
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It is considered that n missiles Mi (i = 1, 2, . . . , n) are followers and one Ml is the leader, cooperatively attacking a stationary target T. The engagement scenario is shown in the inertial reference coordinate OXY in Fig. 1, in which the variables with the subscripts i and l represent the states of the follower i and the leader, respectively. Furthermore, V; a; h; q; g and r denote the speed, normal acceleration, heading angle, Line-of-Sight (LOS) angle, lead angle, and rang-to-go, respectively. In this work, the following assumptions are made to simplify the design and analysis process of the proposed guidance method: (1) Missiles and target are regarded as mass points in the yaw plane. (2) Velocities of missiles are constant. (3) Compared with the guidance loop, the dynamic lags of autopilots and seekers can be ignored. (4) The Angle-of-Attack (AOA) is small and can be neglected. (5) The lead angle of each missile is small when the rangeto-go of the missile is small enough.
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Fig. 1 Guidance scenario.
geometry
on
many-to-one
engagement
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Assumptions 1–3 are commonly employed in the derivation of guidance laws.14 Assumption 4 aims to make the lead angle as FOV, which has also universally been made.17,18,21 Assumption 5 is adopted to linearize the engagement model and simplify the proof process, which is reasonable because a missile has completely locked on the target when its range-to-go is small enough. The elapsed time is denoted as t. The relative kinematics equations of the leader and followers are given as follows: r_l ðtÞ ¼ Vl cos gl ðtÞ
ð1Þ
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rl ðtÞq_ l ðtÞ ¼ Vl sin gl ðtÞ
ð2Þ
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gl ðtÞ ¼ hl ðtÞ ql ðtÞ
ð3Þ
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al ðtÞ ¼ Vl h_ l ðtÞ
ð4Þ
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r_i ðtÞ ¼ Vi cos gi ðtÞ
ð5Þ
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ri ðtÞq_ i ðtÞ ¼ Vi sin gi ðtÞ
ð6Þ
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gi ðtÞ ¼ hi ðtÞ qi ðtÞ
ð7Þ
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ai ðtÞ ¼ Vi h_ i ðtÞ
ð8Þ
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2.2. Integrated cooperative guidance framework
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For the proposed integrated cooperative guidance framework shown in Fig. 2, missiles are firstly distributed into several groups and each group includes three or four missiles. Then, the centralized leader-follower framework is employed within a single group and the decentralized topology is utilized among all leaders. With this integrated framework, the consolidated coordination variable obtained via the communication among leaders can be broadcasted to all followers in real time. Compared to the existing centralized communication, more missiles can be involved in the proposed integrated framework through increasing the groups of missiles, inducing only a slight increase of the computation and communication burden of leaders. Moreover, compared to the existing decentralized communication, since the centralized communication is applied within each single group, the proposed integrated framework inherits many advantages, such as the suboptimal
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coordination variable, better convergence rate of time-to-go error and simpler communication topology. The reasons that the leader-follower framework is adopted as the communication topology within a single group instead of the hierarchical one in the proposed integrated cooperative guidance framework are as follows. Firstly, only leaders are equipped with the information transmitters to reduce the cost. Secondly, all-communication is not required within a single group, without significantly sacrificing the penetration capability and system reliability. Thirdly, since the one-way communication is employed in the leader-follower framework, better real-time communication capability can be obtained. Fourthly, the design of guidance law for the follower is much more flexible. For example, followers can directly track the position and velocity of the leader.
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3. Cooperative guidance law
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3.1. FTCG laws with saturation constraint on FOV
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To satisfy the communication topology of missiles within a single group, a group of FTCG-FOV is developed based on the centralized leader-follower framework, i.e., the leader and followers implement the Proportional Navigation Guidance (PNG) and bias PNG, respectively. Hence, the acceleration command can be expressed as
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al ðtÞ ¼ Nl Vl q_ l ðtÞ 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
ai ðtÞ ¼ Ni Vi q_ i ðtÞ þ aB1 ðtÞ þ aB2 ðtÞ
di ðtÞ , ðtgo;l ðtÞ tgo;i ðtÞÞ
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210 211 212 213
ð10Þ
218 220
ð11Þ
In order to realize FTCG, aB1 ðtÞ is devised as aB1 ðtÞ ¼
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ð9Þ
where tgo;i ðtÞ and tgo;l ðtÞ are the estimated time-to-go of follower i and leader separately, and both of them can be estimated by2 rðtÞ g2 ðtÞ tgo ðtÞ ¼ 1þ ð12Þ V 2ð2N 1Þ 8 k gm ðtÞ < 1rn iðtÞ jtgo;l ðtÞ tgo;i ðtÞjc sgnðtgo;l ðtÞ tgo;i ðtÞÞ
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where N is the navigation gain; aB1 ðtÞ and aB2 ðtÞ are used for the coordination of attack time and the saturation constraint on FOV, respectively. Define the square of the relative error of the time-to-go between the leader and follower i as 2
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if m is odd
i
: k1 gn mi ðtÞ sgnðg ðtÞÞ jtgo;l ðtÞ tgo;i ðtÞjc sgnðtgo;l ðtÞ tgo;i ðtÞÞ if m is even i r ðtÞ i
ð13Þ
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where k1 is a positive constant; the positive constants m and n satisfy m P n P 1; the index number c satisfies c 2 ð0; 1Þ; sgnðÞ represents the signal function. In the lateral plane, FOV is generally regarded as the angle between the longitudinal axis of a missile and its LOS. In light of Assumption 4, FOV is identical to the lead angle. To satisfy the saturation constraint on FOV, i.e., the constraint on lead angle, aB2 is designed as
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aB2
sgnðgi Þ ¼ k2 sigðjgi j gi;max ; b; 0Þ jgi j gi;max
ð14Þ
Fig. 2 Communication topology of integrated cooperative guidance framework.
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where b and k2 are positive constants and b is sufficiently large; gi;max is the lead angle constraint for follower i; sgnðgi Þ is employed to calibrate the direction of aB2 ; sigðx; b; hÞ represents the sigmoid function defined as sigðx; b; hÞ ¼ ð1 þ ebxþh Þ
1
ð15Þ
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Lemma 1 22. Consider the system x_ ¼ fðxÞ; x 2 Rn , with initial condition fð0Þ ¼ 0 and fðÞ : Rn ! Rn is a continuous function. If there exists a continuous positive function defined within the neighborhood of origin as VðxÞ : Rn ! R for b _ VðxÞ þ cðVðxÞÞ 6 0; where c > 0 and b 2 ð0; 1Þ, then the origin of the system x_ ¼ fðxÞ is finite-time stable and the stable time satisfies T 6 ½Vðxð0ÞÞ1b ½cð1 bÞ1 . Theorem 1. Considering the scenario that n missiles led by a leader cooperatively attack a stationary target, assumptions in Section 2 are valid. Then, with the proposed cooperative guidance law shown in Eq. (10), the square of time-to-go error di ðtÞ will converge to zero in finite time, and the lead angle constraint can be satisfied. In other words, multiple missiles can achieve attack time coordination in finite time satisfying the saturation constraint on FOV. Proof. Firstly, it can be demonstrated that based on Ni Vi q_ i ðtÞ þ aB1 ðtÞ, the square of time-to-go error di ðtÞ will converge to zero in finite time. For the leader, according to Eqs. (2)–(4) and (9), the derivative of the lead angle can be deduced as
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ð16Þ
gl ðt þ DtÞ gl ðtÞ ¼ ðN 1Þ
Vl gl ðtÞDt rl ðtÞ
ð17Þ
where Dt is a sufficiently small time interval. Hence, g2l ðt þ DtÞ g2l ðtÞ ¼ ðgl ðt þ DtÞ gl ðtÞÞðgl ðt þ DtÞ þ gl ðtÞÞ Vl gl ðtÞDt Vl gl ðtÞDt ¼ ðN 1Þ 2gl ðtÞ ðN 1Þ ð18Þ rl ðtÞ rl ðtÞ
tgo;l ðt þ DtÞ tgo;l ðtÞ ¼ Dt
ð23Þ
Using the similar way in deriving Eq. (19), one can obtain that
g2i ðt
þ DtÞ
g2i ðtÞ
Vi gi ðtÞDt aB1 Dt þ ¼ ðN 1Þ ri ðtÞ Vi
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2gi ðtÞ
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ð24Þ
Hence, substituting Eqs. (20) and (24) into Eq. (21) yields
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1 g2 ðtÞ 1 þ tgo;i ðt þ DtÞ tgo;i ðtÞ ¼ Vi Dt 1 i Vi 2 2ð2N 1ÞVi g2 ðtÞ g2i ðt þ DtÞ Vi Dt 1 i 2 Vi gi ðtÞDt aB1 Dt þri ðtÞ ðN 1Þ þ 2gi ðtÞ ri ðtÞ Vi
ð25Þ
Omitting some high-order items, one has
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ð26Þ
The difference equation of dðtÞ can be formulated as
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Vl Dt g2l ðt þ DtÞ g2l ðtÞ ¼ 2g2l ðtÞ ðN 1Þ rl ðtÞ
ð19Þ
In Ref. 14, the difference equations of range-to-go and time-to-go have been derived as g2 ðtÞ rðt þ DtÞ rðtÞ ¼ VDt 1 2
dðt þ DtÞ dðtÞ ¼ ðtgo;l ðt þ DtÞ tgo;i ðt þ DtÞÞ2
2 tgo;l ðtÞ tgo;i ðtÞ
¼ tgo;l ðt þ DtÞ tgo;l ðtÞ
tgo;i ðt þ DtÞ tgo;i ðtÞ
tgo;l ðt þ DtÞ tgo;l ðtÞ
tgo;i ðt þ DtÞ tgo;i ðtÞ þ 2 tgo;l ðtÞ tgo;i ðtÞ ð27Þ Substituting Eqs. (23) and (26) into Eq. (27) yields
Dtgi ðtÞri ðtÞ dðt þ DtÞ dðtÞ ¼ a 2 tgo;l ðtÞ tgo;i ðtÞ 2 B1 ð2N 1ÞVi
Omitting high-order items, one has
ð20Þ
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Omitting high-order items, one has
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Accordingly, based on Assumption 5, the difference equation of gl ðtÞ can be expressed as
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1 g2l ðtÞ 1 þ tgo;l ðt þ DtÞ tgo;l ðtÞ ¼ Vl Dt 1 Vl 2 2ð2N 1ÞVl g2 ðtÞ g2l ðt þ DtÞ Vl Dt 1 l 2 Vl Dt ð22Þ þrl ðtÞ 2g2l ðtÞ ðN 1Þ rl ðtÞ
Dtgi ðtÞri ðtÞ tgo;i ðt þ DtÞ tgo;i ðtÞ ¼ Dt þ aB1 ð2N 1ÞV2i
g_ l ðtÞ ¼ h_ l ðtÞ q_ l ðtÞ ¼ ðN 1Þq_ l ðtÞ Vl sin gl ðtÞ ¼ ðN 1Þ rl ðtÞ
Therefore, substituting Eqs. (19) and (20) into Eq. (21) yields
1 1 tgo ðt þ DtÞ tgo ðtÞ ¼ ðrðt þ DtÞ rðtÞÞ þ V 2ð2N 1ÞV g2 ðt þ DtÞ ½rðt þ DtÞ rðtÞ þ rðtÞ ½g2 ðt þ DtÞ g2 ðtÞ ð21Þ
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ð28Þ
Then, according to Eq. (13), Eq. (28) can be rewritten as
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2kDtjgi ðtÞjmþ1 dðt þ DtÞ dðtÞ ¼ jtgo;l ðtÞ tgo;i ðtÞjcþ1 ð2N 1ÞV2i rn1 ðtÞ i 6
2kDt ðmint
jtgo;l ðtÞ tgo;i ðtÞjcþ1
ð29Þ
where Ti is the estimated attack time of follower i. Accordingly, one has
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ddðtÞ dðt þ DtÞ dðtÞ ’ dt Dt cþ1 cþ1 2k ðmint
ð30Þ
By properly choosing the constants in Eq. (10), it can be guaranteed that gi ðtÞ – 0 during the flight. Thus, one can obtain that
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2k ðmint0 ð2N 1ÞV2i rn1 ð0Þ i
ð31Þ
Since c 2 ð0; 1Þ, one can obtain that ðc þ 1Þ=2 2 ð0; 1Þ. Therefore, according to Lemma 1, the square of time-to-go error di ðtÞ will converge to zero in finite time. And one has
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n on the maximum aB1 ðtÞ is almost opposite to that on T. Thus, generally m and n have negligible impact on the performance of the proposed FTCG (Eq. (13)). However, in some specific conditions, better cooperative guidance performance can be generated by properly choosing m and n, which will be demonstrated in Section 4. Fourthly, compared with n, m has larger impact on the acceleration command (Eq. (13)) during the initial guidance stage, since the relative variation of gi is greater than that of ri .
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Remark 3. Note that in Eq. (14), the sigmoid function is actually a switching function when the constant b tends to be infinite. However, compared with a switching function, it is unnecessary to carefully select a suitable switching point for the sigmoid function, which enhances its applicability.
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Remark 4. Notice that in Eq. (10), aB2 may be much greater than aB1 if jgi j is almost identical to gi;max , inducing difficulty in the coordination of attack time. Thus, a smaller gi;max results in longer coordination time.
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3.2. Realization of integrated cooperative guidance framework
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Inspired by the sequential approach in Ref. 14, a modified sequential approach is developed to distribute the FTCGFOV, and thus to realize the integrated cooperative guidance framework. With the modified approach, a segmented technique named as ‘two-stage method’ is adopted. In the first stage, the consensus algorithm is employed to calculate the feasible time-to-go, which is formulated as14 X aij sgnðtfgo;j ðtÞ tfgo;i ðtÞÞ jtfgo;j ðtÞ tfgo;i ðtÞj t_fgo;i ðtÞ ¼ j2Pi ð34Þ tfgo;i ð0Þ ¼ tgo;i ð0Þ
410
where Pi denotes the set of missiles that can communicate with missile i; aij is the element of the adjacency matrix representing the communication topology. According to Eq. (34), the feasible time-to-go of all missiles can converge to a consolidated constant tfgo ðT Þ in the finite time T*.14 In the second stage, based on the FTCG law, the time-to-go of all missiles can converge to tfgo ðT Þ tc in the finite time tc. Compared with the sequential approach in Ref. 14, the modified sequential approach implements the FTCG in two stages rather than one, resulting in a decrease of T þ tc and the acceleration command without any step. In the integrated cooperative guidance framework (shown in Fig. 3), the FTCG-FOV and modified sequential approach are employed. For the guidance, the FTCG-FOV is implemented by all missiles including the leaders. However, tgo;l ðtÞ in Eqs. (11) and (13) are adjusted as the feasible time-to-go. For the consensus algorithm, the modified sequential approach is employed for leaders. In the first stage, leaders calculate the feasible time-to-go by Eq. (34) and each leader broadcasts its feasible time-to-go to followers in the same group in real time. At the end of the first stage, the values of tfgo;i ðtÞ of all leaders, i.e., of all missiles, converge to the same value. In the second stage, all missiles can be guided with their own guidance laws without communication.
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401 402 403 404
350 1c
352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367
368 369
T0i 6 ½dð0Þ 2 ½cð1 cÞ=21
ð32Þ
T0i
where is the stable time of di ðtÞ. Then, it can be demonstrated that with Eq. (10), the lead angle constraint can be satisfied, as well as the requirement for finite-time missile coordination. In Eq. (14), clearly, the term jgi j gi;max is employed to increase the value of aB2 and reduce jgi j when jgi j approaches to gi;max . If jgi j is identical to gi;max , then aB2 tends to be infinite, which cannot happen. Therefore, the lead angle constraint can be satisfied by introducing aB2 . Moreover, the sigmoid function in Eq. (14) is essentially identical to a switching function, because the constant b in the sigmoid function is sufficiently large. And according to Assumption 5, jgi j is small when the range-to-go of missile is small enough. Therefore, aB2 is essentially identical to zero during the last stage of terminal guidance, i.e., the impact of aB2 on the finite-time consensus can be neglected. h Remark 1. Note that according to Assumption 5, the FTCG in Ref. 14 can be rewritten as
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ai ðtÞ ¼ Ni Vi q_ i ðtÞ þ 372 373 374 375 376 377 378 379
380 381 382 383 384 385 386 387 388 389 390
kNi V2i gi ðtÞ
jtgo ðtÞ ri ðtÞ
tgo;i ðtÞjc sgn tgo ðtÞ tgo;i ðtÞ
ð33Þ
where k is a positive constant; tgo ðtÞ denotes the average value of time-to-go of all missiles. Compared with the bias term in Eq. (33), some new items are introduced in Eq. (13), such as tgo;l , m and n. By employing tgo;l , Eq. (13) satisfies the leader-follower framework. Moreover, the FTCG in Ref. 14 is actually a specific condition of Eq. (13) when m = 1 and n = 1. Remark 2. In Eq. (13), m and n are introduced, and some noteworthy points should be noted. Firstly, the reasons why m P n P 1 is required are: (A) according to Eq. (29), if n < 1, it is difficult to choose a suitable upper bound for dðt þ DtÞ dðtÞ; (B) according to Eq. (13), if m < n, an unacceptable large command of terminal acceleration would be incurred. Secondly, it is unreasonable to choose extremely large m and n because a large m or n causes a large k1 , resulting in the great amplification of internal errors and external disturbances for the acceleration command. Thirdly, according to Eqs. (13) and (32), it can be deduced that the impact of m or
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Fig. 3
444 445 446 447 448
Flowchart of integrated cooperative guidance framework.
Remark 5. In the integrated cooperative guidance framework, the FTCG-FOV can still meet the saturation constraint on FOV, which can be demonstrated as Theorem 1. Note that in the second stage of the modified sequential approach, the difference equation of the feasible time-to-go is formulated as
Table 1
Initial conditions of follower.
Parameter
Initial position (m)
Velocity (m/ s)
Initial lead angle (°)
Follower
(900, 0)
300
25
449 451 452 453 454 455 456 457 458
tfgo ðt þ DtÞ tfgo ðtÞ ¼ Dt
ð35Þ
It can be found that Eq. (35) is identical to Eq. (23). Based on this, in the integrated cooperative guidance framework, the FTCG-FOV can realize the attack time coordination in finite time as well. The reason is that the first stage of the modified sequential approach can be finished in finite time, and in the second stage, the finite-time coordination can also be demonstrated as Theorem 1.
459
4. Simulations and analysis
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An engagement scenario is considered, in which multi-missile cooperatively attacks a single stationary target located at ½7000; 5000 m. In order to attain the precise attack for this engagement scenario, the miss distance and overload are required to be less than 2 m and 10, respectively. FTCG, FTCG-FOV and the integrated cooperative guidance framework are respectively analyzed through simulation as follows.
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4.1. Results of FTCG
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For different values of m, n and tgo;l in aB1 ; the developed FTCG (Eq. (13)) is applied. The initial conditions of the follower are shown in Table 1. According to Eq. (23), tgo;l ðtÞ merely depends on tgo;l ð0Þ, i.e., the expected impact time of the follower. tgo;l ð0Þ is set as 28 s and five cases with different values of m and n (m/n) are investigated, as shown in Table 2. Accordingly, the simulation results of the proposed FTCG are shown in Fig. 4. The missile-target distance shown in Fig. 4(a) indicates that, for different m/n, the miss distance of FTCG is less than 2 m, illustrating that the followers can precisely attack the target. Clearly, in Fig. 4(b), the time-to-go error quickly converges to zero, which means that multiple missiles can realize attack time coordination in finite time for different combina-
tions of m/n in Eq. (13). This demonstrates the effectiveness of Eq. (13). Moreover, from Fig. 4(b), it is observed that compared with the FTCG in Ref. 14 (Case 1), a faster convergence rate for time-to-go error can be obtained through properly choosing m and n. The overload of FTCG in terms of different combinations of m/n is illustrated in Fig. 4(c), which indicates that the constraints on overloads of the five cases are all satisfied, and that m has larger impact on the acceleration command than n, especially during the initial guidance stage. Then, different expected impact time of follower (te ¼ 28:0 s, 28.5 s, 29.0 s and 29.5 s) is considered in the proposed FTCG. Moreover, for the case with the expected impact time as 29.5 s, the cooperative guidance law in Ref. 9(E) is also employed for comparison. The simulation results are presented in Fig. 5. The missile-target distance shown in Fig. 5(a) suggests that, for different expected impact time, the miss distance of FTCG is less than 2 m, illustrating that the followers can precisely attack the target. In Fig. 5(b), the convergence rate of timeto-go error of the proposed FTCG is much larger than that of the cooperative guidance law in Ref. 9, demonstrating the effectiveness and superiority of the FTCG. Fig. 5(c) indicates that the overloads satisfy the constraint, and a longer expected impact time induces a larger maximum overload command, due to the larger curvature of the trajectory of follower.
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4.2. Results of FTCG-FOV
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In this subsection, the FTCG-FOV (Eq. (10)) is employed, considering different FOV angle (hFOV ) constraints (38°, 35°, 32°, 29°), and the initial conditions of the follower are the same as those in Section 4.1. The simulation results are presented in Fig. 6. In Fig. 6(a), the trajectories of followers are presented. The missile-target distance shown in Fig. 6(b) suggests that, for dif-
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Table 2 Case
Case 1
Case 2
Case 3
Case 4
Case 5
Condition
m = 1, n = 1
m = 2, n = 1
m = 2, n = 2
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Fig. 4
Results of FTCG for different m/n.
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ferent FOV angle constraints, the miss distance of FTCG-FOV is extremely low (about 0.5 m), which is much less than 2 m, illustrating that the followers can precisely attack the target. Fig. 6(c) and (d) demonstrate the conclusions of Theorem 1. Fig. 6(c) indicates that the saturation constraints on FOV can be satisfied with FTCG-FOV, demonstrating the effectiveness of Eq. (14). In Fig. 6(d), the time-to-go errors quickly converge to zero, which means that with FTCG-FOV, multiple missiles can realize attack time coordination in finite time. Moreover, Fig. 6(d) reveals that a smaller FOV angle constraint incurs longer coordination time. Fig. 6(e) indicates that the overloads satisfy the constraint.
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4.3. Results of integrated cooperative guidance framework
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In this subsection, the integrated cooperative guidance framework is employed, in which three Leaders (L1, L2 and L3) and
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Fig. 5
Results of FTCG for different expected impact time.
one Follower cooperatively attack a single stationary target. The initial conditions of all missiles are shown in Table 3. And if only PNG is employed, the impact time of L1, L2, L3 and Follower is about 29.6 s, 28.6 s, 29.0 s and 28.1 s, respectively. The maximum deviation in impact time is about 1.5 s. Moreover, the FOV angle is restricted to be less than 30°. A suitable communication topology of missiles is chosen as shown in Fig. 7. Therefore, the consensus algorithm (Eq. (34)) in this case can be expressed as 8 f t_ ðtÞ ¼ sgnðtfgo;3 ðtÞ tfgo;1 ðtÞÞ jtfgo;3 ðtÞ tfgo;1 ðtÞj > > > go;1 > < t_f ðtÞ ¼ sgnðtf ðtÞ tf ðtÞÞ jtf ðtÞ tf ðtÞj go;2 go;3 go;2 go;3 go;2 ð36Þ > t_fgo;3 ðtÞ ¼ 10sgnðtfgo;1 ðtÞ tfgo;3 ðtÞÞ jtfgo;1 ðtÞ tfgo;3 ðtÞj > > > : þsgnðtfgo;2 ðtÞ tfgo;3 ðtÞÞ jtfgo;2 ðtÞ tfgo;3 ðtÞj
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The simulation results of the integrated cooperative guidance framework are presented in Fig. 8. To reveal the superiority of the modified sequential approach, the sequential
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Fig. 6
Initial conditions of missiles.
Table 3
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Results of FTCG-FOV.
Parameter
L1
L2
L3
Follower
Initial position (m) Velocity (m/s) Initial lead angle (°)
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(100, 0) 300 25
(400, 0) 300 25
approach in Ref. 14 is also employed, of which the results are shown in Fig. 8(c) and (f). Fig. 8(a) displays the trajectories of missiles. The missile-target distance shown in Fig. 8(b) indicates that the miss distances of missiles are very small (less than 1 m). Fig. 8(c) and (d) demonstrate the conclusions of Remark 5. In Fig. 8(c), with two different sequential approaches, both the standard deviations of time-to-go can
Fig. 7
Communication topology of missiles.
converge to zero in finite time, which means that the employment of the integrated cooperative guidance framework can guarantee the attack time coordination in finite time. Moreover, for the modified sequential approach, the convergence rate of the standard deviation is faster. In Fig. 8(d), the lead angles of missiles are less than 30°, which illustrates that the saturation constraint on FOV can also be satisfied in the integrated cooperative guidance framework. Fig. 8(e) and (f) show that the modified sequential approach can satisfy the overload constraint, while the existing sequential approach14 cannot. Moreover, the overload required by the modified sequential approach does not exhibit a step between two stages. To investigate the computational efficiency of the integrated cooperative guidance framework, a centralized framework2 is considered for comparison, in which the feasible
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Results of integrated cooperative guidance framework.
5. Conclusions
Fig. 9
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Total calculation time of feasible time-to-go.
time-to-go of each missile is determined as the average value of time-to-go of other missiles. To obtain the required computational time of the feasible time-to-go, it is assumed that the feasible time-to-go is calculated 100,000 times and each missile group has four missiles. Therefore, the total required calculation time for all missiles with respect to the group number of missiles is shown in Fig. 9. It is demonstrated that the integrated cooperative guidance framework is more efficient, and this superiority will be more obvious if more missiles are involved in the cooperative attack.
(1) To address the issues of the existing communication topologies, a new integrated cooperative guidance framework is proposed in this paper, in which missiles are firstly distributed into several groups, and then missiles within a single group communicate by the centralized leader-follower framework, while the leaders from different groups communicate using the nearestneighbor topology. (2) To implement the proposed integrated cooperative guidance framework, a group of FTCG laws considering the saturation constraint on FOV (FTCG-FOV) are designed by introducing two bias terms in PNG, and an improved sequential approach is improved and then employed to make the FTCG-FOV satisfy the requirement of communication between groups. (3) The simulation results well demonstrate the effectiveness and high efficiency of the proposed integrated cooperative guidance framework and the cooperative guidance laws, as well as the superiority of the improved sequential approach.
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Acknowledgement
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This study was supported by the National Natural Science Foundation of China (No. 11532002).
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References
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1. Zhou JL, Yang JY. Distributed guidance law design for cooperative simultaneous attacks with multiple missiles. J Guid Control Dyn 2016;39(10):2436–44. 2. Jeon IS, Lee JI, Tahk MJ. Homing guidance law for cooperative attack of multiple missiles. J Guid Control Dyn 2010;33(1):275–80. 3. Wang XL, Zheng YA, Wu HL. Distributed cooperative guidance of multiple anti-ship missiles with arbitrary impact angle constraint. Aerosp Sci Technol 2015;46:299–311. 4. Wang XF, Zheng YY, Lin H. Integrated guidance and control law for cooperative attack of multiple missiles. Aerosp Sci Technol 2015;42:1–11. 5. Zhang YA, Ma GX, Liu AL. Guidance law with impact time and impact angle constraints. Chin J Aeronaut 2013;26(4):960–6. 6. Zhao SY, Zhou R, Wei C, Ding QX. Design of time-constrained guidance laws via virtual leader approach. Chin J Aeronaut 2010;23(1):103–8. 7. Zhao Y, Sheng YZGX, Liu XD. Trajectory reshaping based guidance with impact time and angle constraints. Chin J Aeronaut 2016;29(4):984–94. 8. Zhang YA, Wang XL, Wu HL. A distributed cooperative guidance law for salvo attack of multiple anti-ship missiles. Chin J Aeronaut 2015;28(5):1438–50. 9. Zhao EJ, Wang SY, Chao T, Yang M. Multiple missiles cooperative guidance based on leader-follower strategyProceedings of 2014 IEEE Chinese guidance, navigation and control conference; 2014 August 8–10. Piscataway: IEEE Press; 2014. p. 1163–7.
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Z. JIANBO, Y. SHUXING 10. Zhao J, Zhou R. Distributed three-dimensional cooperative guidance via receding horizon control. Chin J Aeronaut 2016;29 (4):972–83. 11. Zhao J, Zhou R. Obstacle avoidance for multi-missile network via distributed coordination algorithm. Chin J Aeronaut 2016;29 (2):441–7. 12. Zhao J, Zhou R, Dong ZN. Three-dimensional cooperative guidance laws against stationary and maneuvering targets. Chin J Aeronaut 2015;28(4):1104–20. 13. Song JH, Song SM, Xu SL. Three-dimensional cooperative guidance law for multiple missiles with finite-time convergence. Aerosp Sci Technol 2017;67:193–205. 14. Hou DL, Wang Q, Sun XJ, Dong CY. Finite-time cooperative guidance laws for multiple missiles with acceleration saturation constraints. IET Control Theory Appl 2015;9(10):1525–35. 15. Zhao JB, Yang SX. Review and prospect of multi-missile cooperative guidance. Acta Aeronaut Astronaut Sin 2017;38 (1):020256 [Chinese]. 16. Ma GX, Zhang YA. Time-cooperative distributed guidance law design for multi-missiles. Control Decision 2014;29(5):843–7 [Chinese]. 17. Wen QQ, Xia QL, Su WX. A parameter design strategy for seeker’s field-of-view constraint in impact angle guidance. Proc Inst Mech Eng Part G: J Aerosp Eng 2015;229(13):2389–96. 18. Park BG, Kim TH, Tahk MJ. Optimal impact angle control guidance law considering the seeker’s field-of-view limits. Proc Inst Mech Eng Part G: J Aerosp Eng 2013;227(8):1347–64. 19. Lee CH, Kim TH, Tahk MJ, Whang IH. Polynomial guidance laws considering terminal impact angle and acceleration constraints. IEEE Trans Aerosp Electron Syst 2013;49(1):74–92. 20. Zhang YA, Wang XL, Wu HL. Impact time control guidance law with field of view constraint. Aerosp Sci Technol 2014;39:361–9. 21. He SM, Lin DF. A robust impact angle constraint guidance law with seeker’s field-of-view limit. Trans Inst Meas Control 2015;37 (3):317–28. 22. Bhat SP, Bernstein DS. Finite-time stability of continuous autonomous systems. SIAM J Control Optim 2000;38(3):751–66.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Integrated cooperative guidance framework and cooperative guidance law for multi-missile
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Zhao JIANBO, Yang SHUXING *
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School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, Beijing 100081, China
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Received 4 January 2017; revised 7 December 2017; accepted 7 December 2017
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KEYWORDS
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Cooperative; Field-of-view constraint; Finite-time; Framework; Missile guidance; Multi-missiles
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Abstract An integrated cooperative guidance framework for multi-missile cooperatively attacking a single stationary target is proposed in this paper by combining both the centralized and decentralized communication topologies. Once missiles are distributed into several groups, missiles within a single group communicate with the centralized leader-follower framework, while the leaders from different groups communicate using the nearest-neighbor topology. To implement the integrated cooperative guidance framework, a group of Finite-Time Cooperative Guidance (FTCG) laws considering the saturation constraint on FOV (FTCG-FOV) are firstly derived within the centralized leader-follower framework to satisfy the communication topology of missiles in a single group. Then, an improved sequential approach is developed to adapt the FTCG-FOV to satisfy the communication topology between groups. The numerical simulations demonstrate the effectiveness and high efficiency of the integrated cooperative guidance framework and the cooperative guidance laws, as well as the superiority of the developed sequential approach. Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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In modern military operation, it is a challenging task for the missile to attack a land target or a surface ship that is equipped with an antiair defense system.1 To penetrate the antiair
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* Corresponding author at: School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China. E-mail address: [email protected] (Y. SHUXING). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
defense system, the missile should be capable of terminal evasive maneuvering, which would induce increasing cost.2 An alternative way is to conduct a cooperative attack, i.e., multiple missiles coming from different directions attack a single target simultaneously, which has been regarded as a costeffective and efficient way to address the threat of the defense system.3 Therefore, the research on cooperative attack has gained increasing interest in recent years. Considering that a land target or a surface ship is either stationary or moving at a relatively low speed, the target in the cooperative attack problem in this paper is considered to be stationary as is commonly done in practice. To achieve cooperative attack, one can perform an openloop cooperative guidance, i.e., a common impact time is gen-
https://doi.org/10.1016/j.cja.2017.12.013 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: JIANBO Z, SHUXING Y Integrated cooperative guidance framework and cooperative guidance law for multi-missile, Chin J Aeronaut (2017), https://doi.org/10.1016/j.cja.2017.12.013
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erated for all member missiles in advance, and thereafter each missile tries to arrive at the target on time independently.4–7 However, a suitable common impact time is difficult to be generated in advance, since some missiles may not be able to satisfy the impact time constraint due to their specific initial conditions and limited speed. In addition, the open-loop guidance is also lack of robustness to external disturbance during engagement.8 Actually, the open-loop cooperative guidance simply formulates the many-to-one cooperative attack problem as multiple one-to-one attack problems considering the impact time constraints, which cannot be considered as a genuine multi-missile cooperative guidance.9 Alternatively, with the closed-loop cooperative guidance approach, the missiles communicate to each other to synchronize the impact time.10–12 As a well-known closed-loop cooperative guidance method, the Finite-Time Cooperative Guidance (FTCG) has gained much attention due to its fast convergence rate and high accuracy of the time-to-go errors of missiles. To attack a maneuvering target, Song et al.13 proposed a FTCG law with impact angle constraints. In Ref. 1, two distributed FTCG laws are developed based on different time-to-go estimation methods. However, in addition to the normal acceleration command, the tangential acceleration was also required with both FTCG laws, causing extra difficulties in implementation. To address this problem, a more effective FTCG law employing a hierarchical framework was proposed in Ref. 14, which requires only the normal acceleration command. For the closed-loop guidance laws introduced above, either a centralized or decentralized communication topology is utilized. However, both communication topologies are far from the optimum,15 and their drawbacks shown as follows will become much more prominent if more missiles are involved in the cooperative attack. With the centralized communication topology, one or a few missiles should be designated, which can communicate with all the rest missiles. Clearly, the distances between the designated missile and the rest ones are subject to their communication capability, which incurs extra difficulty in designing the commands for cooperative missiles. In addition, poor penetration capability and system reliability would be induced if only one missile is designated, while unacceptable computational burden would be induced if more than one missile are designated. In contrast, with the decentralized communication topology, missiles can communicate only with their neighbors to reduce the computational burden, while the global information of the missile cluster cannot be obtained, causing great difficulty in making optimal decision for cooperative missiles. Moreover, it takes long time to achieve the consensus of time-to-go and some necessary conditions16 are always hard to be satisfied. In addition, due to the limitation of the detective capability of seekers, the saturation constraint on Field-of-View (FOV) should be considered in the closedloop cooperative guidance, which however has been rarely seen in the existing works. And the existing FOV-constraint guidance for the common one-to-one missile-target engagement scenario17–21 cannot be directly applied to the closed-loop cooperative guidance. To address the issues above, a novel integrated cooperative guidance framework is proposed in this paper, in which missiles are distributed into several groups, and then missiles within a single group communicate by means of the centralized leader-follower framework, while the communication between groups employs the nearest-neighbor topology among leaders.
Z. JIANBO, Y. SHUXING The contributions of this paper lie in: (A) the centralized and decentralized communication topologies are combined and integrated into the proposed framework effectively; (B) to implement the proposed integrated cooperative guidance framework, a group of FTCG laws considering the saturation constraint on FOV (FTCG-FOV for short) are designed in this paper by extending the FTCG law in Ref. 21 to a group of FTCG laws and introducing a bias term to satisfy the FOV constraint; (C) the sequential approach in Ref. 21 is improved, which is then employed to make the FTCG-FOV satisfy the requirement of communication between groups. The rest of this paper is organized as follows. In Section 2, the many-to-one missile-target interception engagement along with the proposed integrated cooperative guidance framework is introduced. In Section 3, a group of FTCG-FOV are developed, of which the working process in the integrated cooperative guidance framework is presented in detail. Simulation results are presented and analyzed in Section 4. Conclusions are made in the last section.
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2. Preliminary
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2.1. Problem formulation
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It is considered that n missiles Mi (i = 1, 2, . . . , n) are followers and one Ml is the leader, cooperatively attacking a stationary target T. The engagement scenario is shown in the inertial reference coordinate OXY in Fig. 1, in which the variables with the subscripts i and l represent the states of the follower i and the leader, respectively. Furthermore, V; a; h; q; g and r denote the speed, normal acceleration, heading angle, Line-of-Sight (LOS) angle, lead angle, and rang-to-go, respectively. In this work, the following assumptions are made to simplify the design and analysis process of the proposed guidance method: (1) Missiles and target are regarded as mass points in the yaw plane. (2) Velocities of missiles are constant. (3) Compared with the guidance loop, the dynamic lags of autopilots and seekers can be ignored. (4) The Angle-of-Attack (AOA) is small and can be neglected. (5) The lead angle of each missile is small when the rangeto-go of the missile is small enough.
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Fig. 1 Guidance scenario.
geometry
on
many-to-one
engagement
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Assumptions 1–3 are commonly employed in the derivation of guidance laws.14 Assumption 4 aims to make the lead angle as FOV, which has also universally been made.17,18,21 Assumption 5 is adopted to linearize the engagement model and simplify the proof process, which is reasonable because a missile has completely locked on the target when its range-to-go is small enough. The elapsed time is denoted as t. The relative kinematics equations of the leader and followers are given as follows: r_l ðtÞ ¼ Vl cos gl ðtÞ
ð1Þ
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rl ðtÞq_ l ðtÞ ¼ Vl sin gl ðtÞ
ð2Þ
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gl ðtÞ ¼ hl ðtÞ ql ðtÞ
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al ðtÞ ¼ Vl h_ l ðtÞ
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r_i ðtÞ ¼ Vi cos gi ðtÞ
ð5Þ
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ri ðtÞq_ i ðtÞ ¼ Vi sin gi ðtÞ
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gi ðtÞ ¼ hi ðtÞ qi ðtÞ
ð7Þ
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ai ðtÞ ¼ Vi h_ i ðtÞ
ð8Þ
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2.2. Integrated cooperative guidance framework
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For the proposed integrated cooperative guidance framework shown in Fig. 2, missiles are firstly distributed into several groups and each group includes three or four missiles. Then, the centralized leader-follower framework is employed within a single group and the decentralized topology is utilized among all leaders. With this integrated framework, the consolidated coordination variable obtained via the communication among leaders can be broadcasted to all followers in real time. Compared to the existing centralized communication, more missiles can be involved in the proposed integrated framework through increasing the groups of missiles, inducing only a slight increase of the computation and communication burden of leaders. Moreover, compared to the existing decentralized communication, since the centralized communication is applied within each single group, the proposed integrated framework inherits many advantages, such as the suboptimal
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coordination variable, better convergence rate of time-to-go error and simpler communication topology. The reasons that the leader-follower framework is adopted as the communication topology within a single group instead of the hierarchical one in the proposed integrated cooperative guidance framework are as follows. Firstly, only leaders are equipped with the information transmitters to reduce the cost. Secondly, all-communication is not required within a single group, without significantly sacrificing the penetration capability and system reliability. Thirdly, since the one-way communication is employed in the leader-follower framework, better real-time communication capability can be obtained. Fourthly, the design of guidance law for the follower is much more flexible. For example, followers can directly track the position and velocity of the leader.
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3. Cooperative guidance law
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3.1. FTCG laws with saturation constraint on FOV
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To satisfy the communication topology of missiles within a single group, a group of FTCG-FOV is developed based on the centralized leader-follower framework, i.e., the leader and followers implement the Proportional Navigation Guidance (PNG) and bias PNG, respectively. Hence, the acceleration command can be expressed as
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al ðtÞ ¼ Nl Vl q_ l ðtÞ 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
ai ðtÞ ¼ Ni Vi q_ i ðtÞ þ aB1 ðtÞ þ aB2 ðtÞ
di ðtÞ , ðtgo;l ðtÞ tgo;i ðtÞÞ
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ð10Þ
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ð11Þ
In order to realize FTCG, aB1 ðtÞ is devised as aB1 ðtÞ ¼
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ð9Þ
where tgo;i ðtÞ and tgo;l ðtÞ are the estimated time-to-go of follower i and leader separately, and both of them can be estimated by2 rðtÞ g2 ðtÞ tgo ðtÞ ¼ 1þ ð12Þ V 2ð2N 1Þ 8 k gm ðtÞ < 1rn iðtÞ jtgo;l ðtÞ tgo;i ðtÞjc sgnðtgo;l ðtÞ tgo;i ðtÞÞ
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where N is the navigation gain; aB1 ðtÞ and aB2 ðtÞ are used for the coordination of attack time and the saturation constraint on FOV, respectively. Define the square of the relative error of the time-to-go between the leader and follower i as 2
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if m is odd
i
: k1 gn mi ðtÞ sgnðg ðtÞÞ jtgo;l ðtÞ tgo;i ðtÞjc sgnðtgo;l ðtÞ tgo;i ðtÞÞ if m is even i r ðtÞ i
ð13Þ
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where k1 is a positive constant; the positive constants m and n satisfy m P n P 1; the index number c satisfies c 2 ð0; 1Þ; sgnðÞ represents the signal function. In the lateral plane, FOV is generally regarded as the angle between the longitudinal axis of a missile and its LOS. In light of Assumption 4, FOV is identical to the lead angle. To satisfy the saturation constraint on FOV, i.e., the constraint on lead angle, aB2 is designed as
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aB2
sgnðgi Þ ¼ k2 sigðjgi j gi;max ; b; 0Þ jgi j gi;max
ð14Þ
Fig. 2 Communication topology of integrated cooperative guidance framework.
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where b and k2 are positive constants and b is sufficiently large; gi;max is the lead angle constraint for follower i; sgnðgi Þ is employed to calibrate the direction of aB2 ; sigðx; b; hÞ represents the sigmoid function defined as sigðx; b; hÞ ¼ ð1 þ ebxþh Þ
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ð15Þ
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Lemma 1 22. Consider the system x_ ¼ fðxÞ; x 2 Rn , with initial condition fð0Þ ¼ 0 and fðÞ : Rn ! Rn is a continuous function. If there exists a continuous positive function defined within the neighborhood of origin as VðxÞ : Rn ! R for b _ VðxÞ þ cðVðxÞÞ 6 0; where c > 0 and b 2 ð0; 1Þ, then the origin of the system x_ ¼ fðxÞ is finite-time stable and the stable time satisfies T 6 ½Vðxð0ÞÞ1b ½cð1 bÞ1 . Theorem 1. Considering the scenario that n missiles led by a leader cooperatively attack a stationary target, assumptions in Section 2 are valid. Then, with the proposed cooperative guidance law shown in Eq. (10), the square of time-to-go error di ðtÞ will converge to zero in finite time, and the lead angle constraint can be satisfied. In other words, multiple missiles can achieve attack time coordination in finite time satisfying the saturation constraint on FOV. Proof. Firstly, it can be demonstrated that based on Ni Vi q_ i ðtÞ þ aB1 ðtÞ, the square of time-to-go error di ðtÞ will converge to zero in finite time. For the leader, according to Eqs. (2)–(4) and (9), the derivative of the lead angle can be deduced as
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289 290 291 293 294 295 296 298
ð16Þ
gl ðt þ DtÞ gl ðtÞ ¼ ðN 1Þ
Vl gl ðtÞDt rl ðtÞ
ð17Þ
where Dt is a sufficiently small time interval. Hence, g2l ðt þ DtÞ g2l ðtÞ ¼ ðgl ðt þ DtÞ gl ðtÞÞðgl ðt þ DtÞ þ gl ðtÞÞ Vl gl ðtÞDt Vl gl ðtÞDt ¼ ðN 1Þ 2gl ðtÞ ðN 1Þ ð18Þ rl ðtÞ rl ðtÞ
tgo;l ðt þ DtÞ tgo;l ðtÞ ¼ Dt
ð23Þ
Using the similar way in deriving Eq. (19), one can obtain that
g2i ðt
þ DtÞ
g2i ðtÞ
Vi gi ðtÞDt aB1 Dt þ ¼ ðN 1Þ ri ðtÞ Vi
304
306 307
2gi ðtÞ
308 310 311 312 313
ð24Þ
Hence, substituting Eqs. (20) and (24) into Eq. (21) yields
315 316 317
1 g2 ðtÞ 1 þ tgo;i ðt þ DtÞ tgo;i ðtÞ ¼ Vi Dt 1 i Vi 2 2ð2N 1ÞVi g2 ðtÞ g2i ðt þ DtÞ Vi Dt 1 i 2 Vi gi ðtÞDt aB1 Dt þri ðtÞ ðN 1Þ þ 2gi ðtÞ ri ðtÞ Vi
ð25Þ
Omitting some high-order items, one has
319 320 321
ð26Þ
The difference equation of dðtÞ can be formulated as
323 324
Vl Dt g2l ðt þ DtÞ g2l ðtÞ ¼ 2g2l ðtÞ ðN 1Þ rl ðtÞ
ð19Þ
In Ref. 14, the difference equations of range-to-go and time-to-go have been derived as g2 ðtÞ rðt þ DtÞ rðtÞ ¼ VDt 1 2
dðt þ DtÞ dðtÞ ¼ ðtgo;l ðt þ DtÞ tgo;i ðt þ DtÞÞ2
2 tgo;l ðtÞ tgo;i ðtÞ
¼ tgo;l ðt þ DtÞ tgo;l ðtÞ
tgo;i ðt þ DtÞ tgo;i ðtÞ
tgo;l ðt þ DtÞ tgo;l ðtÞ
tgo;i ðt þ DtÞ tgo;i ðtÞ þ 2 tgo;l ðtÞ tgo;i ðtÞ ð27Þ Substituting Eqs. (23) and (26) into Eq. (27) yields
Dtgi ðtÞri ðtÞ dðt þ DtÞ dðtÞ ¼ a 2 tgo;l ðtÞ tgo;i ðtÞ 2 B1 ð2N 1ÞVi
Omitting high-order items, one has
ð20Þ
299
301
Omitting high-order items, one has
302 303
325
Accordingly, based on Assumption 5, the difference equation of gl ðtÞ can be expressed as
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1 g2l ðtÞ 1 þ tgo;l ðt þ DtÞ tgo;l ðtÞ ¼ Vl Dt 1 Vl 2 2ð2N 1ÞVl g2 ðtÞ g2l ðt þ DtÞ Vl Dt 1 l 2 Vl Dt ð22Þ þrl ðtÞ 2g2l ðtÞ ðN 1Þ rl ðtÞ
Dtgi ðtÞri ðtÞ tgo;i ðt þ DtÞ tgo;i ðtÞ ¼ Dt þ aB1 ð2N 1ÞV2i
g_ l ðtÞ ¼ h_ l ðtÞ q_ l ðtÞ ¼ ðN 1Þq_ l ðtÞ Vl sin gl ðtÞ ¼ ðN 1Þ rl ðtÞ
Therefore, substituting Eqs. (19) and (20) into Eq. (21) yields
1 1 tgo ðt þ DtÞ tgo ðtÞ ¼ ðrðt þ DtÞ rðtÞÞ þ V 2ð2N 1ÞV g2 ðt þ DtÞ ½rðt þ DtÞ rðtÞ þ rðtÞ ½g2 ðt þ DtÞ g2 ðtÞ ð21Þ
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ð28Þ
Then, according to Eq. (13), Eq. (28) can be rewritten as
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2kDtjgi ðtÞjmþ1 dðt þ DtÞ dðtÞ ¼ jtgo;l ðtÞ tgo;i ðtÞjcþ1 ð2N 1ÞV2i rn1 ðtÞ i 6
2kDt ðmint
jtgo;l ðtÞ tgo;i ðtÞjcþ1
ð29Þ
where Ti is the estimated attack time of follower i. Accordingly, one has
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ddðtÞ dðt þ DtÞ dðtÞ ’ dt Dt cþ1 cþ1 2k ðmint
ð30Þ
By properly choosing the constants in Eq. (10), it can be guaranteed that gi ðtÞ – 0 during the flight. Thus, one can obtain that
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2k ðmint
ð31Þ
Since c 2 ð0; 1Þ, one can obtain that ðc þ 1Þ=2 2 ð0; 1Þ. Therefore, according to Lemma 1, the square of time-to-go error di ðtÞ will converge to zero in finite time. And one has
5
n on the maximum aB1 ðtÞ is almost opposite to that on T. Thus, generally m and n have negligible impact on the performance of the proposed FTCG (Eq. (13)). However, in some specific conditions, better cooperative guidance performance can be generated by properly choosing m and n, which will be demonstrated in Section 4. Fourthly, compared with n, m has larger impact on the acceleration command (Eq. (13)) during the initial guidance stage, since the relative variation of gi is greater than that of ri .
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Remark 3. Note that in Eq. (14), the sigmoid function is actually a switching function when the constant b tends to be infinite. However, compared with a switching function, it is unnecessary to carefully select a suitable switching point for the sigmoid function, which enhances its applicability.
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Remark 4. Notice that in Eq. (10), aB2 may be much greater than aB1 if jgi j is almost identical to gi;max , inducing difficulty in the coordination of attack time. Thus, a smaller gi;max results in longer coordination time.
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3.2. Realization of integrated cooperative guidance framework
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Inspired by the sequential approach in Ref. 14, a modified sequential approach is developed to distribute the FTCGFOV, and thus to realize the integrated cooperative guidance framework. With the modified approach, a segmented technique named as ‘two-stage method’ is adopted. In the first stage, the consensus algorithm is employed to calculate the feasible time-to-go, which is formulated as14 X aij sgnðtfgo;j ðtÞ tfgo;i ðtÞÞ jtfgo;j ðtÞ tfgo;i ðtÞj t_fgo;i ðtÞ ¼ j2Pi ð34Þ tfgo;i ð0Þ ¼ tgo;i ð0Þ
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where Pi denotes the set of missiles that can communicate with missile i; aij is the element of the adjacency matrix representing the communication topology. According to Eq. (34), the feasible time-to-go of all missiles can converge to a consolidated constant tfgo ðT Þ in the finite time T*.14 In the second stage, based on the FTCG law, the time-to-go of all missiles can converge to tfgo ðT Þ tc in the finite time tc. Compared with the sequential approach in Ref. 14, the modified sequential approach implements the FTCG in two stages rather than one, resulting in a decrease of T þ tc and the acceleration command without any step. In the integrated cooperative guidance framework (shown in Fig. 3), the FTCG-FOV and modified sequential approach are employed. For the guidance, the FTCG-FOV is implemented by all missiles including the leaders. However, tgo;l ðtÞ in Eqs. (11) and (13) are adjusted as the feasible time-to-go. For the consensus algorithm, the modified sequential approach is employed for leaders. In the first stage, leaders calculate the feasible time-to-go by Eq. (34) and each leader broadcasts its feasible time-to-go to followers in the same group in real time. At the end of the first stage, the values of tfgo;i ðtÞ of all leaders, i.e., of all missiles, converge to the same value. In the second stage, all missiles can be guided with their own guidance laws without communication.
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401 402 403 404
350 1c
352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367
368 369
T0i 6 ½dð0Þ 2 ½cð1 cÞ=21
ð32Þ
T0i
where is the stable time of di ðtÞ. Then, it can be demonstrated that with Eq. (10), the lead angle constraint can be satisfied, as well as the requirement for finite-time missile coordination. In Eq. (14), clearly, the term jgi j gi;max is employed to increase the value of aB2 and reduce jgi j when jgi j approaches to gi;max . If jgi j is identical to gi;max , then aB2 tends to be infinite, which cannot happen. Therefore, the lead angle constraint can be satisfied by introducing aB2 . Moreover, the sigmoid function in Eq. (14) is essentially identical to a switching function, because the constant b in the sigmoid function is sufficiently large. And according to Assumption 5, jgi j is small when the range-to-go of missile is small enough. Therefore, aB2 is essentially identical to zero during the last stage of terminal guidance, i.e., the impact of aB2 on the finite-time consensus can be neglected. h Remark 1. Note that according to Assumption 5, the FTCG in Ref. 14 can be rewritten as
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ai ðtÞ ¼ Ni Vi q_ i ðtÞ þ 372 373 374 375 376 377 378 379
380 381 382 383 384 385 386 387 388 389 390
kNi V2i gi ðtÞ
jtgo ðtÞ ri ðtÞ
tgo;i ðtÞjc sgn tgo ðtÞ tgo;i ðtÞ
ð33Þ
where k is a positive constant; tgo ðtÞ denotes the average value of time-to-go of all missiles. Compared with the bias term in Eq. (33), some new items are introduced in Eq. (13), such as tgo;l , m and n. By employing tgo;l , Eq. (13) satisfies the leader-follower framework. Moreover, the FTCG in Ref. 14 is actually a specific condition of Eq. (13) when m = 1 and n = 1. Remark 2. In Eq. (13), m and n are introduced, and some noteworthy points should be noted. Firstly, the reasons why m P n P 1 is required are: (A) according to Eq. (29), if n < 1, it is difficult to choose a suitable upper bound for dðt þ DtÞ dðtÞ; (B) according to Eq. (13), if m < n, an unacceptable large command of terminal acceleration would be incurred. Secondly, it is unreasonable to choose extremely large m and n because a large m or n causes a large k1 , resulting in the great amplification of internal errors and external disturbances for the acceleration command. Thirdly, according to Eqs. (13) and (32), it can be deduced that the impact of m or
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Fig. 3
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Flowchart of integrated cooperative guidance framework.
Remark 5. In the integrated cooperative guidance framework, the FTCG-FOV can still meet the saturation constraint on FOV, which can be demonstrated as Theorem 1. Note that in the second stage of the modified sequential approach, the difference equation of the feasible time-to-go is formulated as
Table 1
Initial conditions of follower.
Parameter
Initial position (m)
Velocity (m/ s)
Initial lead angle (°)
Follower
(900, 0)
300
25
449 451 452 453 454 455 456 457 458
tfgo ðt þ DtÞ tfgo ðtÞ ¼ Dt
ð35Þ
It can be found that Eq. (35) is identical to Eq. (23). Based on this, in the integrated cooperative guidance framework, the FTCG-FOV can realize the attack time coordination in finite time as well. The reason is that the first stage of the modified sequential approach can be finished in finite time, and in the second stage, the finite-time coordination can also be demonstrated as Theorem 1.
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4. Simulations and analysis
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An engagement scenario is considered, in which multi-missile cooperatively attacks a single stationary target located at ½7000; 5000 m. In order to attain the precise attack for this engagement scenario, the miss distance and overload are required to be less than 2 m and 10, respectively. FTCG, FTCG-FOV and the integrated cooperative guidance framework are respectively analyzed through simulation as follows.
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4.1. Results of FTCG
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For different values of m, n and tgo;l in aB1 ; the developed FTCG (Eq. (13)) is applied. The initial conditions of the follower are shown in Table 1. According to Eq. (23), tgo;l ðtÞ merely depends on tgo;l ð0Þ, i.e., the expected impact time of the follower. tgo;l ð0Þ is set as 28 s and five cases with different values of m and n (m/n) are investigated, as shown in Table 2. Accordingly, the simulation results of the proposed FTCG are shown in Fig. 4. The missile-target distance shown in Fig. 4(a) indicates that, for different m/n, the miss distance of FTCG is less than 2 m, illustrating that the followers can precisely attack the target. Clearly, in Fig. 4(b), the time-to-go error quickly converges to zero, which means that multiple missiles can realize attack time coordination in finite time for different combina-
tions of m/n in Eq. (13). This demonstrates the effectiveness of Eq. (13). Moreover, from Fig. 4(b), it is observed that compared with the FTCG in Ref. 14 (Case 1), a faster convergence rate for time-to-go error can be obtained through properly choosing m and n. The overload of FTCG in terms of different combinations of m/n is illustrated in Fig. 4(c), which indicates that the constraints on overloads of the five cases are all satisfied, and that m has larger impact on the acceleration command than n, especially during the initial guidance stage. Then, different expected impact time of follower (te ¼ 28:0 s, 28.5 s, 29.0 s and 29.5 s) is considered in the proposed FTCG. Moreover, for the case with the expected impact time as 29.5 s, the cooperative guidance law in Ref. 9(E) is also employed for comparison. The simulation results are presented in Fig. 5. The missile-target distance shown in Fig. 5(a) suggests that, for different expected impact time, the miss distance of FTCG is less than 2 m, illustrating that the followers can precisely attack the target. In Fig. 5(b), the convergence rate of timeto-go error of the proposed FTCG is much larger than that of the cooperative guidance law in Ref. 9, demonstrating the effectiveness and superiority of the FTCG. Fig. 5(c) indicates that the overloads satisfy the constraint, and a longer expected impact time induces a larger maximum overload command, due to the larger curvature of the trajectory of follower.
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4.2. Results of FTCG-FOV
507
In this subsection, the FTCG-FOV (Eq. (10)) is employed, considering different FOV angle (hFOV ) constraints (38°, 35°, 32°, 29°), and the initial conditions of the follower are the same as those in Section 4.1. The simulation results are presented in Fig. 6. In Fig. 6(a), the trajectories of followers are presented. The missile-target distance shown in Fig. 6(b) suggests that, for dif-
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Table 2 Case
Case 1
Case 2
Case 3
Case 4
Case 5
Condition
m = 1, n = 1
m = 2, n = 1
m = 2, n = 2
m = 3, n = 1
m = 3, n = 2
Fig. 4
Results of FTCG for different m/n.
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ferent FOV angle constraints, the miss distance of FTCG-FOV is extremely low (about 0.5 m), which is much less than 2 m, illustrating that the followers can precisely attack the target. Fig. 6(c) and (d) demonstrate the conclusions of Theorem 1. Fig. 6(c) indicates that the saturation constraints on FOV can be satisfied with FTCG-FOV, demonstrating the effectiveness of Eq. (14). In Fig. 6(d), the time-to-go errors quickly converge to zero, which means that with FTCG-FOV, multiple missiles can realize attack time coordination in finite time. Moreover, Fig. 6(d) reveals that a smaller FOV angle constraint incurs longer coordination time. Fig. 6(e) indicates that the overloads satisfy the constraint.
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4.3. Results of integrated cooperative guidance framework
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In this subsection, the integrated cooperative guidance framework is employed, in which three Leaders (L1, L2 and L3) and
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7
Fig. 5
Results of FTCG for different expected impact time.
one Follower cooperatively attack a single stationary target. The initial conditions of all missiles are shown in Table 3. And if only PNG is employed, the impact time of L1, L2, L3 and Follower is about 29.6 s, 28.6 s, 29.0 s and 28.1 s, respectively. The maximum deviation in impact time is about 1.5 s. Moreover, the FOV angle is restricted to be less than 30°. A suitable communication topology of missiles is chosen as shown in Fig. 7. Therefore, the consensus algorithm (Eq. (34)) in this case can be expressed as 8 f t_ ðtÞ ¼ sgnðtfgo;3 ðtÞ tfgo;1 ðtÞÞ jtfgo;3 ðtÞ tfgo;1 ðtÞj > > > go;1 > < t_f ðtÞ ¼ sgnðtf ðtÞ tf ðtÞÞ jtf ðtÞ tf ðtÞj go;2 go;3 go;2 go;3 go;2 ð36Þ > t_fgo;3 ðtÞ ¼ 10sgnðtfgo;1 ðtÞ tfgo;3 ðtÞÞ jtfgo;1 ðtÞ tfgo;3 ðtÞj > > > : þsgnðtfgo;2 ðtÞ tfgo;3 ðtÞÞ jtfgo;2 ðtÞ tfgo;3 ðtÞj
530
The simulation results of the integrated cooperative guidance framework are presented in Fig. 8. To reveal the superiority of the modified sequential approach, the sequential
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Fig. 6
Initial conditions of missiles.
Table 3
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Results of FTCG-FOV.
Parameter
L1
L2
L3
Follower
Initial position (m) Velocity (m/s) Initial lead angle (°)
(100, 0) 300 25
(200, 0) 300 25
(100, 0) 300 25
(400, 0) 300 25
approach in Ref. 14 is also employed, of which the results are shown in Fig. 8(c) and (f). Fig. 8(a) displays the trajectories of missiles. The missile-target distance shown in Fig. 8(b) indicates that the miss distances of missiles are very small (less than 1 m). Fig. 8(c) and (d) demonstrate the conclusions of Remark 5. In Fig. 8(c), with two different sequential approaches, both the standard deviations of time-to-go can
Fig. 7
Communication topology of missiles.
converge to zero in finite time, which means that the employment of the integrated cooperative guidance framework can guarantee the attack time coordination in finite time. Moreover, for the modified sequential approach, the convergence rate of the standard deviation is faster. In Fig. 8(d), the lead angles of missiles are less than 30°, which illustrates that the saturation constraint on FOV can also be satisfied in the integrated cooperative guidance framework. Fig. 8(e) and (f) show that the modified sequential approach can satisfy the overload constraint, while the existing sequential approach14 cannot. Moreover, the overload required by the modified sequential approach does not exhibit a step between two stages. To investigate the computational efficiency of the integrated cooperative guidance framework, a centralized framework2 is considered for comparison, in which the feasible
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Fig. 8
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Results of integrated cooperative guidance framework.
5. Conclusions
Fig. 9
567 568 569 570 571 572 573 574 575 576
Total calculation time of feasible time-to-go.
time-to-go of each missile is determined as the average value of time-to-go of other missiles. To obtain the required computational time of the feasible time-to-go, it is assumed that the feasible time-to-go is calculated 100,000 times and each missile group has four missiles. Therefore, the total required calculation time for all missiles with respect to the group number of missiles is shown in Fig. 9. It is demonstrated that the integrated cooperative guidance framework is more efficient, and this superiority will be more obvious if more missiles are involved in the cooperative attack.
(1) To address the issues of the existing communication topologies, a new integrated cooperative guidance framework is proposed in this paper, in which missiles are firstly distributed into several groups, and then missiles within a single group communicate by the centralized leader-follower framework, while the leaders from different groups communicate using the nearestneighbor topology. (2) To implement the proposed integrated cooperative guidance framework, a group of FTCG laws considering the saturation constraint on FOV (FTCG-FOV) are designed by introducing two bias terms in PNG, and an improved sequential approach is improved and then employed to make the FTCG-FOV satisfy the requirement of communication between groups. (3) The simulation results well demonstrate the effectiveness and high efficiency of the proposed integrated cooperative guidance framework and the cooperative guidance laws, as well as the superiority of the improved sequential approach.
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Acknowledgement
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This study was supported by the National Natural Science Foundation of China (No. 11532002).
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