YANG Biao，JING Wuxing，and GAO Changsheng

School of Astronautics，Harbin Institute of Technology，Harbin 150001，China

Abstract: This paper proposes a cooperative guidance law for attacking a ground target with the impact angle constraint based on the motion camouflage strategy in the line-of-sight (LOS)frame. A dynamic model with the impact angle constraint is established according to the relative motion between multiple missiles and the target. The process of cooperative guidance law design is divided into two stages. Firstly，based on the undirected graph theory，a new finite-time consensus protocol on the LOS direction is derived to guarantee relative distances reach consensus. And the value of acceleration command is positive，which is beneficial for engineering realization. Secondly，the acceleration command on the normal direction of the LOS is designed based on motion camouflage and finite-time convergence，which can ensure the missiles reach the target with the desired angle and satisfy the motion camouflage state. The finitetime stability analysis is proved by the Lyapunov theory. Numerical simulations for stationary and maneuver targets have demonstrated the effectiveness of the cooperative guidance law proposed.

Keywords: cooperative guidance law，motion camouflage，impact angle constraint，finite-time.

1. Introduction

With the rapid development of defense technology in recent years，the attack ability of a single missile is greatly limited. Thus，multiple missiles cooperative attack gradually gets more and more attention [1]. Under the multiple missiles cooperative attack mode，missiles can complete the mission through information interaction and cooperation [2]. This method can enhance the defense penetration ability. And multiple missiles can accomplish some important missions which are difficult for a single missile. Cooperative guidance law design is a key technology in multiple missiles cooperative attack，which can guarantee the attack success rate and have a great engineering significance [3].

Many scholars have researched on the design of the cooperative guidance law. A new hybrid cooperative guidance law which combines inertial delay control and sliding mode control was proposed in [4]. Based on the optimal capture radius of attacker and interceptor，Garcia[5]designed a cooperative guidance for active defense of air missiles. Daughtery and Qu [6]derived a cooperative guidance law that can steer a salvo of missiles to achieve simultaneous attack. Optimal and cooperative control methods were used to derive a consensus time-to-go estimate. A cooperative mid-course guidance scheme for multiple missiles to intercept a target under the condition of large detection errors was presented in [7]. Based on the traditional proportional navigation algorithm，Zhao and Zhou [8]proposed the unified cooperative strategies for the salvo attack of multiple missiles. In order to achieve a salvo attack，Kang et al. [9]designed a cooperative guidance law based on model predictive control. Besides，time constraint is a key factor which should be considered. Finite time disturbance observer was designed to estimate the system disturbance in [10]. Then a cooperative guidance law with impact time control was derived through nonsingular fast terminal sliding mode control.For the three-dimensional cooperative guidance problem，Song et al. [11]proposed a novel guidance law which can guarantee finite-time convergence and does not need maneuvers information of the target.

When the missile attacks the target，in order to ensure the damage effect，the impact angle is usually required to be within a certain range. Thus，the guidance law with the impact angle constraint receives much attention [12].This guidance process can be transformed to the problem with terminal boundary conditions. The sliding mode control was applied to satisfy the angle constraint [13，14].Based on Dubins paths，a guidance law with desired impact angle was designed to achieve the earliest intercep-tion of a moving target [15]. Wu et al. [16]proposed a terminal guidance law through a practical flight strategy.An online correction algorithm of guidance parameters was designed to enhance the guidance performance and satisfy the angle constraint. A guidance law which considered the constraints of seeker field-of-view (FOV) and impact angle was derived in [17]. A variable-gain approach was used to satisfy the limits of FOV and lateral maneuver capability. Zhang et al. [18]presented a novel adaptive fixed-time sliding mode guidance law which can intercept maneuver targets at a desired impact angle without singularity and chattering problems. A novel distributed cooperative guidance law with arbitrary impact angle constraints was designed to achieve cooperative attack [19]. Wang and Yang [20]designed the normal acceleration and tangential acceleration to achieve simultaneous attack with the desired angle.

The motion camouflage theory is used to describe the positional relation between the predator and the prey in the biosphere，which was first proposed in [21]. It mainly means that the predator，the prey and the camouflage background are in a straight line during the predation process. And the prey cannot clearly perceive the change of the predator position [22]. Because of its camouflage characteristics，it has a large military application value. It has also been applied to the design of the guidance law in recent years [23]. Justh et al. [24]designed a motion camouflage feedback guidance law under the frenet framework and compared the guidance performance through different feedback gains. Gao et al. [25]established a biquad dynamic model of the missile and the target. Based on that，a motion camouflage guidance law for intercepting the maneuvering target was designed. A motion camouflage guidance law suitable for stamping thrust missiles has been proposed in [26]，which can effectively reduce the overload of the hit point. An adaptive motion camouflage guidance law was designed for attacking a ground target and the coefficients of the normal and binormal feedback guidance law were designed in [27].

At present，scholars have much research on cooperative guidance problems with impact angle constraints.However，in most research，the guidance command value on the direction of the line-of-sight (LOS) is large and negative sometimes，which is difficult to achieve in engineering. Besides，motion camouflage has been applied with cooperative guidance rarely. This paper develops a novel cooperative guidance law with impact angle constraints. The guidance law design is divided into two parts.In the first part，guidance command on the LOS direction is derived based on the undirected graph theory to guarantee that all missiles can attack the target simultaneously. And the value of acceleration command is positive. In the second part，according to the motion camouflage and finite-time convergence theory，guidance command on the normal direction of the LOS is given to guarantee that the missiles attack the target with desired angles and satisfy the camouflage state. Finally，simulation results are given to verify the efficiency of the cooperative guidance law.

2. Dynamic model

To describe the engagement geometry and design the guidance law，the relative motion relationships of one missile and the target are shown in Fig. 1.

Fig. 1 Relative motion of missile-target

When multiple missiles attack the target，the engagement geometry is shown in Fig. 2. Different missiles have different initial conditions. The purpose of the cooperative guidance law in this paper is that all missiles can attack the target with the desired impact angle at the same time. The missiles can exchange information through distributed communication. In this paper，subscriptidenotes the index of the missile.

Fig. 2 Multiple missile-target engagement geometry

The cooperative guidance model of multiple missiles with the impact angle constraint can be described as follows:

From the above equations，the design of cooperative guidance can be divided into two stages. The first stage is to design the acceleration command on the LOS direction，which guarantees that all missiles can arrive simultaneously in finite time. The other stage is to design the acceleration command on the normal direction of the LOS，which guarantees the LOS angles of different missiles converge to desired angles in finite time.

3. Guidance law implementation

3.1 Guidance law on the LOS direction

In the process of traditional guidance law design，the acceleration command on the normal direction of the LOS is designed to guarantee that the LOS angular rate converges to zero. And the acceleration command along the LOS is generally set to zero. However，in order to let the distance between every missile and target converge to the same in finite time，the acceleration command along the LOS should be designed in the cooperative guidance law.When multiple missiles attack the target，different missiles can exchange information to achieve cooperation.The information exchange model can be described by the graph theory. In this paper，define an undirected graph:

3.2 Motion camouflage theory

Motion camouflage relies on visual characteristics to deceive the target. When the attacker has the same image characteristics as the camouflage background (reference point) in the target’s detection vision，the target cannot know about the attacker’s motion characteristics clearly.Motion camouflage requires the attacker，target，and camouflage background (reference point) to remain in a straight line，which can minimize the parallax among attacker and camouflage background. The relative motion relationship among the attacker，the target，and the reference point is shown in Fig. 3.

Fig. 3 Motion camouflage scenario

3.3 Guidance law on the normal direction of LOS

4. Simulations

In order to verify the effectiveness of motion camouflage strategy cooperative guidance with impact angle constraints，numerical simulations are performed. In the simulation scenario，four missiles will attack a ground target cooperatively. The communication topology of missiles is shown in Fig. 4，which is undirected and connected. The weight coefficient matrix can be described as

Fig. 4 Communication topology among four missiles

Two cases for different target accelerations are selected to verify the adaptability of the cooperative guidance law. In the first case，the target is stationary. The maneuver target is considered in the second case.

Case 1The initial position of the target is (110 km，0 km，0 km) and its position is stationary. Initial conditions of four missiles are shown in Table 1.

Table 1 Initial conditions of missiles

The parameters of the guidance law on the LOS direction can be selected as follows:N=10，α1=0.6. The para-meters of the guidance law on the normal direction of the LOS can be selected as follows: μ1=0.7，μ2=0.75，μ3=0.65，μ4=0.5，β /2+W=120. The available acceleration on the LOS direction is limited between 0 and 10g.The maximum available acceleration on the normal direction of the LOS is limited to 30g，wheregis the acceleration of gravity. The simulation results are shown in Fig. 5 to Fig. 7 and Table 2.

Fig. 5 Trajectories and relative distance for Case 1

Fig. 6 Acceleration command for Case 1

Fig. 7 LOS information for Case 1

Table 2 Simulation results for stationary targets

Table 2 shows that every missile can attack the stationary target with desired LOS angle and small miss distance. The guidance time is 12.710 s，which can guarantee the missiles attack the target simultaneously. From Fig. 5，it can be seen that the initial conditions of four missiles have different directions and heights. Thus，four missiles can achieve multi-level attack for the target. Under the use of the cooperative guidance law，the relative distances gradually converge from different initial values and converge to zero finally. Fig. 6(a) shows the acceleration on the LOS direction of four missiles. At the beginning，the acceleration can adjust missile-target relative distance to achieve a rapid cooperative attack. The acceleration value is always positive and less than 10g，which is easy to realize in engineering. Fig. 6(b) shows the acceleration on the normal direction of the LOS. In order to make the LOS angle converge to desired angle and satisfy the motion camouflage state，the initial acceleration is relatively large. However，as the LOS angle converges，the acceleration becomes gentle and tends to zero. Fig. 7 shows the LOS angle rate and the LOS angle of each missile. As the relative distance decreases，the LOS angle rate becomes smaller and tends to zero. The LOS angle converges to the desired value finally.

Case 2The initial position of the target is (110 km，0 km，0 km) and its velocity is 50 m/s. The maneuver accelerations areaTx= 20sin(0.5t) m/s2andaTz=30sin(0.5t) m/s2.Initial conditions of four missiles are shown in Table 1.

The parameters of the guidance law on the LOS direction can be selected as follows:N=12，α1=0.5. The parameters of the guidance law on the normal direction of LOS can be selected as follows: μ1=0.8，μ2=0.78，μ3=0.75，μ4=0.6，β/2+W=140. Other parameters are the same as Case 1. The simulation results are shown in Fig. 8 to Fig. 10 and Table 3.

Fig. 8 Trajectories and relative distance for Case 2

Fig. 9 Acceleration command for Case 2

Fig. 10 LOS information for Case 2

Table 3 shows that four missiles can attack the maneuver target with desired LOS angle and small miss distance. Their guidance time is 13.591 s，which is more than Case 1 because of the maneuver of the target. From Fig. 8，we can see that compared with straight trajectories in Case 1，the trajectories of four missiles in Case 2 are more curved in order to attack the maneuver target.Fig. 9(a) denotes the acceleration on the LOS direction.The accelerations of four missiles are curved at the initial stage and quickly converge to achieve a rapid cooperative attack. As a consequence of target maneuvering，the acceleration curves fluctuate around zero. Fig. 9(b) shows the acceleration curves on the normal direction of the LOS. Compared with Case 1，the acceleration curves continue to change after convergence to satisfy the maneuver target. Fig. 10 shows the LOS information of each missile. Under the guidance command on the normal direction of the LOS，the LOS angle rate gradually converges and fluctuates around zero. The LOS angle converges to the desired value finally.

Table 3 Simulation results for maneuver target

From the above results，it can be seen the cooperative guidance law with impact angle constraints proposed in this paper has good guidance performance for attacking a ground target. The acceleration value on the LOS direction is always positive and small，which is easy to realize in engineering. Besides，the guidance law can satisfy the maneuver target and have motion camouflage characteristic which have the effect of confusing the target. In order to achieve attacking a large maneuver target，the next study and improvements will focus on overload analysis of the guidance law and estimation of target motion information.

5. Conclusions

In this paper，a cooperative guidance law for attacking a ground target with impact angle constraints based on the motion camouflage strategy is proposed. A dynamic model with impact angle constraints is established according to the relative motion between multiple missiles and the target in the LOS frame. Contributions of this paper are mainly twofold:

(i) According to the undirected graph theory，a new finite-time consensus protocol on the LOS direction is derived，so that all missiles can attack the target simultaneously. The acceleration value on the LOS direction is always positive and small，which is easy to realize in engineering.

(ii) Based on motion camouflage and finite-time convergence theory，guidance command on the normal direction of the LOS is given to guarantee the missiles attack the target with desired angle and satisfy the camouflage state. Thus，the guidance law has better concealment characteristics and strike effects.