CHEN Wenyu, SHAO Lei, and LEI Humin

Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China

Abstract: The cooperative interception trajectories generation of multiple interceptors to hypersonic targets is studied.First, to solve the problem of on-line trajectory generation of the single interceptor, a generation method based on neighborhood optimal control is adopted.Then, when intercepting the strong maneuvering targets, the single interceptor is insufficient in maneuverability, therefore, an on-line multiple trajectories generation algorithm is proposed, which uses the multiple interceptors intercept area (IIA)to cover the target’s predicted intercept area(PIA)cooperatively.Through optimizing the interceptors’ zero control terminal location, the trajectories are generated on-line by using the neighborhood optimal control method, these trajectories could make the IIA maximally cover the PIA.The simulation results show that the proposed method can greatly improve the interception probability, which provides a reference for the collaborative interception of multiple interceptors.

Keywords: cooperative interception, trajectories generation,near space.

1.Introduction

The hypersonic vehicle is a weapon that can fly continuously in the near space at five times of the speed of sound.Compared with the traditional ballistic missile, it has a variable trajectory, strong maneuverability in a wide scope and a high speed.The vehicle brings great pressure to the existing air defense system.Therefore, it is urgent to develop interception technology for hypersonic vehicles.Many scholars have studied the interception of hypersonic targets in the near space, including the tracking and prediction of target trajectory, and the generation and tracking of interceptor trajectory.For the generation of interceptor trajectory, the current mainstream idea is to generate an off-line trajectory according to the interceptor initial state and the target state before the launch of the interceptor.This trajectory usually guarantees one best performance index, which makes the interceptor launch to the best interceptor position in the terminal guidance.However, the target usually does not adopt the fixed flight mode, and will carry out a series of strong maneuvers.The original terminal constraints are no longer applicable, so a new optimal trajectory needs to be generated on-line.On-line trajectory generation must be completed in a short time.Because the offline optimal generation method needs a relatively long time, and the method is no longer applicable, it is necessary to select an on-line fast optimization method, and generate an on-line trajectory based on the original offline trajectory.The online trajectory must meet the changed terminal states.In addition, a single interceptor often fails to intercept the target because of its lack of maneuverability.Therefore,in order to increase the success rate of interception, we should pay more attention to the way that multiple interceptors intercept the target cooperatively.

In view of the on-line generation of correction trajectory, there are few existing researches dedicated to the near space interceptor.However, many scholars have carried out research on similar issues, which brings a lot of references for the on-line generation of trajectory of near space interceptors.With the characteristics of high efficiency and rapidity of the model predictive static programming algorithm, Dwivedi et al.studied the problem of trajectory optimization [1-4].Although the problem is mainly aimed at trajectory optimization, the essence of trajectory on-line generation and correction of the interceptor is also an optimization problem, and it just needs a more rapid solution.Therefore, the method also has a reference significance.Although the method can consider process constraints, it cannot guarantee the optimality of the index function.In [5-7], the problem of orbit correction of the lunar exploration vehicle is solved by using the theory of variable neighborhood optimal control, and the correction of control quantity under the change of the terminal constraint is solved.However, it is mainly aimedat the situation of the change of the remaining flight time,which is not consistent with the background of the interceptor.In [8] and [9], the neighborhood optimal control method and the pseudospectral method were combined to solve the correction of the control quantity.However,only for the situation where initial states change, it cannot fulfill the situation that the terminal constraint changes on-line.Williams et al.studied the trajectory optimization of spacecraft and the design of the optimal guidance law [10-12].Zhou et al.also applied the neighborhood optimal control method to study trajectory optimization, generation and tracking of the middle guidance phase, all of which took some explorations for the on-line generation of interceptor trajectory [13-18].

The cooperative interception problem of multiple interceptors stems from the inspiration of unmanned aerial vehicle (UAV)cooperative search technology, and scholars pay more and more attention to it based on the cooperative theory.In [19], to solve the problem of multiple interceptors intercepting multiple targets through optimizing the performance index, which is maximizing the probability that multiple interceptors intercept multiple targets, good results were achieved.In [20], the uncertainty of target trajectory states prediction was considered, and the terminal area of the target was described by the probability density distribution function.The reachable area of the interceptor can maximize the coverage of the possible area of the target.The linearization assumption of the motion model was made in [21], the intercepted area of the interceptor was analyzed by the true proportional guidance method, and a new cooperative strategy was proposed.Ma et al.studied the problem of multiple interceptors cooperating in intercepting highspeed targets in the longitudinal plane [22-24].The cooperative method of multiple interceptors is derived.There is no research on cooperative multiple interceptors in the middle guidance stage.

In this paper, to achieve the maximum coverage of the target’s predicted intercept area (PIA)by the interceptor’s intercept area (IIA), through optimizing the terminal position of multiple interceptors, the trajectories of multiple interceptors are generated on-line.Based on the existing standard trajectories, the trajectories are generated on-line with the neighborhood optimal control method.Based on direct force correction of interceptors in the terminal guidance stage, the IIA is deduced, and the terminal positions of multiple interceptors are optimized.The terminal positions of multiple interceptors are fully optimized by the on-line generation method, so as to achieve the maximum coverage of the target’s PIA by the IIA.The simulation results show that when the target maneuvers and the PIA changes longitudinally and laterally, the method in this paper can achieve the cooperative coverage of the PIA with a high coverage rate.

2.On-line trajectory generation algorithm of single interceptor

The neighborhood optimization algorithm is developed from optimal control [25].Firstly, suppose that the motion differential equation of the interceptor in longitudinal plane [26] is

whereVis the interceptor velocity, Θ is the velocity path angle,his the height,Lis the range,Pis the engine thrust, the attack angle α is the control variable,mis the mass;CLandCDare the lift coefficient and drag coefficient respectively,sis the reference area,qis the dynamic pressure, andR0is the average radius of the earth.The equation could be shown as

wherefdescribes the motion law of the interceptor,xis the state vector,uis the control vector, andtis a free variable.All derivatives in this paper are derivatives of time.

The performance index function is

where φ is the terminal quantity,t0is the initial time,tfis the terminal time,Lis the Lagrange function, andVfis the interceptor terminal velocity.

The terminal constraint is

wherehfand Θfare the terminal height and velocity path angle.The Hamiltonian function is

By using the Lagrange multiple plier method, the equality constraints and terminal constraints are introduced into the index function:

where λ ∈Rnis the Lagrange multiple plier vector, andv∈Rqis the Lagrange multiple plier.When the control quantity is unconstrained, to maximize the functional indexJ, the necessary condition should satisfy:

(i)State equation and co-state equation

(ii)Boundary and cross section conditions

(iii)Coupling condition

(iv)Whentfis free, the Hamiltonian function satisfies

The above equations are the necessary conditions for the function to satisfy the first-order optimality.When satisfying the first-order optimality, the system has a small disturbance and initial statex(t0)or terminal constraint ψfchanges, the deviation of the state variable on the reference trajectory δx(t)is

wherex*(t)represents the value along the optimal trajectory.x(t)represents the filtered data after the actual measurement.In order to maintain the optimal trajectory,the deviation of the optimal control quantity is

whereu*(t)represents the optimal control variable.The first-order optimality condition formulae (7)-(10)are obtained by taking the following variations [27].

Taking (20)into (13)and (14), there are

where

Assuming that δX(tf)is known, we can calculate the inverse recurrence integral of (22):

where Δt=-dt, δX(tk)= δX(tf).Similarly,

It can be obtained by recursion in turn:

If the first-order optimality condition of the optimal control problem has been solved,u*(t)andx*(t)of (7)-(10)are satisfied, then the coefficients in (17)-(19)can be solved along the optimal trajectory, and can be sorted into the following linear equations:

In the linear equations system composed of (26)-(28),there are a total of 2n+q+1 unknowns, which arendimensions variable δx,ndimensions variable δ λ,qdimensions variable dvand one dimension variable dtf.However, there are onlyn+q+1 equations.According to the structural theorem of the solution of the linear equation system, there must benfree variables in2n+q+1 variables.Take the variable on the right side of (27)-(29)as a free variable and record it as a vector d μ [29], then

where δxq+1,δxq+2,···,δxnare the variable δxwithout terminal constraints, which cannot be solved by (27).If the equations of (27)and (28)are linearly independent, it can be solved

Whentk=tf, by taking (31)into (26), we can get

LetCFbe

where 1 :nmeans the elements from 1 ton.Then

Then (32)can be written as

Therefore,

From the above formulae,

Take (35)and (36)into (20)to get

That means the adjustment variable of the reference control variable can be expressed by the current state deviation δx(tm)and the terminal constraint adjustmentvariable d ψf.Assume that

In this paper, only the case of terminal constraints change is involved.Therefore, the on-line generation of interceptor trajectory caused by the change of state quantity is not considered.Therefore, δx(tm)=0.When the target is maneuvering, only the end constraints of the interceptor reference trajectory are changed.Based on the original reference optimal trajectory, the following control adjustments need to be generated:

According to (39), the block diagram of neighborhood optimal control applied to trajectory on-line generation is drawn as shown in Fig.1.

Fig.1 On-line trajectory generation block diagram

In order to verify the ability of on-line trajectory generation of the interceptor under the neighborhood optimal control method, the following simulation experiments are carried out.First of all, the initial height of the interceptor is 60 km, the initial velocity is 4 000 m/s, the initial angle of reentry is -4°, the terminal height required for the standard trajectory is 30 km, and the terminal angle is 0°.With the actual situation considered, when the terminal constraints are changed due to the target’s maneuvering, and with the assumption that the terminal height changes to 28 km, the corresponding simulation figures are as follows in Fig.2-Fig.5.In Fig.2,His the flight height, andLis the flight length of the interceptor.

Fig.2 Trajectories in longitudinal plane

Fig.5 Change of attack angle

From the simulation, it can be seen that for the scenario where the terminal conditions of the reference trajectory are changing, the trajectory on-line generation under the neighborhood optimal control can find a trajectory satisfying the terminal constraints.

Fig.3 Change of ballistic inclination

Fig.4 Change of flight speed

The above simulation is only for the scenario where the position constraints change.In order to verify the universality of the method and the correction ability of interceptors for different PIAs, carry out 50 Monte Carlo simulations, keep the above initial conditions unchanged, and generate the terminal position randomly within 5 km based on the original standard trajectory.The simulation results are as follows in Fig.6-Fig.10.

Fig.6 Trajectories in longitudinal plane under 50 simulations

Fig.7 Change of ballistic inclination under 50 simulations

Fig.8 Change of flight speed under 50 simulations

Fig.9 Change of attack angle under 50 simulations

Fig.10 Terminal error of on-line trajectories generation under 50 simulations

According to Fig.10, the error between the on-line corrected terminal position and the expected terminal position is within 55 m, which can well meet the terminal constraints.Because the miss distance is not the most important index of the terminal constraint of midcourse guidance, the above error can meet the midcourse and terminal shift of midcourse guidance of the interceptor completely,and it can be corrected in the terminal guidance.It is proved that the on-line correction method of neighborhood optimal control used in this paper can complete the on-line trajectory generation of the interceptor.However,it can be seen in the figure that when the terminal constraint is modified, the required control variable of the interceptor will jump to a certain extent, which means that the control variable of the interceptor will change dramatically.The maneuverability of a single interceptor is required to be high.Therefore, this paper proposes to use multiple interceptors to complete the mission.

3.On-line trajectories generation of multiple interceptors

3.1 Multiple interceptors’ IIA cooperatively covering the PIA

Definition 1PIA: the set of all possible location points of the target at the timetf1of prediction hit is called PIA,which is recorded asR.Therefore, the area where the predicted hit probability exists can be expressed [30] as

whereP(tf1)is the covariance matrix of the prediction intercept area.

By projecting PIA and IIA into the missile target encounter plane, the coverage of PIA in three-dimensional space can be transformed into the area coverage in plane,as shown in Fig.11.

Fig.11 PIA of target and IIA of multiple interceptors

In Fig.11,MandTrepresent the interceptor and the target respectively,PMis the zero control terminal position of the interceptor, planea-b-c-dis the encounter plane of the missile and the target, wherePTis the projection of the mean point of PIA in plane,andare the projections of PIA and IIA in plane respectively.After projection, the IIA and target PIA can be expressed as

whereSmaxis the maximum correction distance,(xd,yd,zd)is the zero control terminal position.

In order to intercept the target, the interceptor needs to be overloaded about three times of the target’s maneuver value.It is undeniable that interceptors are inferior in hypersonic targets maneuver.To ensure the interceptor successfully intercepts the target, it is necessary to ensure that the IIA completely covers the PIA.From the above simulations, it can be seen that the trajectory generated online by a single interceptor can meet the changed terminal constraints after the change of the terminal constraints of the satndard trajectory, but the control variable will inevitably change dramatically.This shows that under the condition of overload limitation, a single interceptor is difficult to complete effective interception.However,if multiple interceptors are used to intercept cooperatively, and the IIA is used to cover the PIA after target’s maneuver, the probability of successful interception will be greatly improved.

3.2 Mathematical description of cooperative coverage problem

Interceptors are regarded as agents in the collaborative process.Nagents need to cover the target areaR∈R2, the location set of all agents isP={p1,p2,···,pN}.Letpi=represents agentAiin the target area, the probability density distribution function of the target area is φ(ξ):R→ R+, where ξ ∈Rrepresents any point in the target areaR.In order to describe the coverage effect of multiple agents quantitatively, leth(pi,ξ)represent the cost function of agentAicovering the point ξ:

Therefore, the coverage cost function of multiple agents to the target area can be expressed as

Formula (44)is the location optimization function.From (44), we can see that each agent has the ability to cover the points ξ ∈Rin the target area.The greater distance from the point ξ to the agent, the greater cost to cover the point, and the worse coverage performance.In order to make the cost function in the area covering problemH(P,R,φ)minimum, select the agent with the smallest distance to cover the point.Therefore, by optimizing the location of multiple agents and adjusting the allocation area of each agent, the cost function could be minimized.Based on the Voronoi segmentation principle, the above statement is proved as follows:

Theorem 1In the target areaR∈R2,Nagents are randomly distributed, and their location information isP.According to the location information of the current time,the optimal region segmentation principle of multiple agents cooperative coverage is Voronoi segmentation.

ProofIn order to minimize the coverage cost function.There is

The minimum distance from the point ξ toNagents is

According to the minimum distance, the target area can be divided intoNindependent sub areas, in which the set of the closest points to the agent can be expressed as

whereRiis the sub area to be covered by the agentAi,R={R1,R2,···,RN}represents the sub area of each agent.The optimal covering cost function of the corresponding agents to the target area is

According to this lemma, in the target area, the optimal coverage area of each agent is the Voronoi area.□

3.3 Optimization of multiple interceptors’ zero control terminal location

3.3.1 Modeling of IIA with energy constraints

It is assumed that the aerodynamic control is used in the midcourse guidance and the direct force control is used in the final guide.The aerodynamic correction ability is obviously affected by the altitude of the interceptor, and the direct lateral force correction ability is limited by the energy carried.

In the final guidance, it is assumed that the maximum working time of the orbit control engine istmax, the specific impulse of the orbit control engine isIg, the maximum steady-state thrust isPmax, and the initial mass of the interceptor terminal guidance ismm0.If the interceptor is always propelled by the maximum steady-state thrust, the mass change rate of the interceptor is

The interceptor’s maximum lateral acceleration at timetcan be expressed as follows:

According to (50), the lateral velocity and the maximum lateral maneuver displacement of the interceptor at timetcan be calculated as follows:

From the beginning of the terminal guidance timethto the interception timetf, if (tf-th)≤tmax, the maximum lateral displacement of the interceptor perpendicular to the initial velocity vector direction is

If (tf-th)≥tmax,

under the condition of direct force and energy limitation,the IIA in the terminal guidance stage of the interceptor is determined bySmax.The IIA of the interceptor can be expressed as

Among them, the IIA depends on the working time of the direct force device in the terminal guidance stage.

In order to ensure that the interceptor successfully intercepts the target at the final time, it is necessary to ensure that the IIA can completely cover the PIA at the hand-off time

whereR(tf)represents the PIA.

3.3.2 Optimal solution of multiple interceptors’zero control terminal position

At the hand-off time, the projection of the PIA in the encounter plane of the missile is, there is the point(y,z)∈, and the lateral coordinate of the interceptorMizero control terminal position at the end of the shift is(yi,zi), then the zero control miss distance of the interceptorMirelative to the point (y,z)is

There is a zero control miss distance between point(y,z)and each interceptor, and the minimum one is chosen as the zero control miss distance between point(y,z)andNinterceptors.

The mathematical expectation of zero control miss distance ofNinterceptors at the hand-off time is

wherefk(y)andfk(z)are the probability density functions of the target atyaxis andzaxis.The estimated value of the target position obeys normal distribution, the mathematical expectation of target position atyaxis andzaxis is [ μyμz], and the variance is [ σyσz].There are

p1(y,yi)p1(z,zi)is the cost function of coverage, which could judge whether the point (y,z)is within the interceptor’s intercepting capability, there are

According to the probability distribution function of the target at the hand-off time, the probability of the target appearing in the IIA is

When the flight overload of the interceptor is certain,at the hand-off time,H2depends on the position, speed direction, and sight angle of the interceptor.If the interceptor’s overload is enough, the interceptor’s IIA can completely cover the PIA of the target, so thatH2=1.To determine whether the point (y,z)is in IIA ofNinterceptors, the coverage performance function is

Ifp2=1, it means that there is an interceptor satisfyingp1(y,yi)p1(z,zi)=1, and that means the point (y,z)is in the IIA.According to the probability distribution function of the target, the probability that the target appears in the IIA ofNinterceptors is

At the hand-off time, in order to maximize the performance of multiple interceptors’ IIA coverage, it is necessary to control the status of the interceptors midcourse guidance terminal through the midcourse guidance trajectory on-line generation technology, so as to control the zero control terminal positions of the interceptors.Assuming that the number of interceptors meets the requirements,Pminis the cooperative coverage requirement, the problem can be described as

By solving the optimization problem (66), the number required and the optimal zero control terminal position of multiple interceptors can be obtained, and then the optimal medium guidance terminal state constraint of multiple interceptors can be obtained.

According to the Voronoi segmentation method of the region, the set of the closest point to the interceptorMican be

whereis the sub-region to be covered by the interceptorsMiinterceptable region.

Then (59)can be described as

In order to simplify the problem, suppose that the adjustment of the interceptor terminal position atyaxis andzaxis is independent of each other, (68)can be rewritten as

If

then we can get

By using the optimality condition, the optimal zero control terminal position can be obtained by optimizingE1yandE1z.

Based on the above, the specific solution steps of the on-line trajectories generation of cooperative multiple interceptors are as follows:

Step 1At the hand-off time, by using the target trajectory prediction algorithm, the target motion information detected by multiple-source sensors is used to get the probability density function of the target position, and the PIA of the target is obtained.

Step 2With the advance of the interception task, the prediction time of the target becomes shorter and the PIA converges gradually.According to the change of PIA, under the constraint of the cooperative coverage performance index of IIA, the optimal zero control terminal position of multiple interceptors is calculated, and the zero control terminal position is transformed into the terminal position constraint of guidance in each interceptor.

Step 3Under the new terminal state constraints, each interceptor uses the neighborhood optimal control trajectory on-line generation algorithm to generate the control instruction correction variable on-line, so as to adjust the terminal position of the interceptor trajectory and realize the control of zero control terminal position of the interceptor; if the generated terminal condition cannot be executed, it needs to return to Step 2.

Step 4Determine whether the cooperative coverage condition is met.If so, repeat Step 1; if not, go to Step 2.

4.Simulation analysis

In order to verify the effectiveness of the guidance trajectories generation method for the cooperative coverage of the IIA, the following two scenarios are simulated.The simulation assumes that the maximum operating distance of the interceptor terminal guidance is 100 km, and the terminal guidance time is taken astf=20s; the maximum correction distance of the rail control engine working for 20 s is taken asSmax=5km; the probability density function distribution of the PIA follows the normal distribution.

Scenario 1: It mainly verifies the capability of on-line trajectory generation of multiple interceptors after the longitudinal change of the PIA.By using the IIA of four interceptors to cover the PIA, the PIA at the end of the middle guidance stage is obtained by predicting the target trajectory,

Assuming that the target measurement information is updated once in the 30 s of midcourse guidance, the PIA after the update is

The initial conditions of the base trajectories of the four interceptors are set as follows in Table 1.

Table 1 Initial condition setting of medium guidance reference trajectory

Among them, the initial positions of M1, M2, M3 and M4 are 10 km, 20 km, 27.07 km and 37.07 km respectively.According to the PIA, the terminal state of the interceptor can be optimized as in Table 2 and Table 3.

Table 2 Terminal condition setting of medium guidance reference trajectory

Table 3 Terminal condition setting of medium guidance correction trajectory

The simulation results are shown in Fig.12 and Fig.13.Fig.12 and Fig.13 show the simulation results of the four interceptors IIA covering the PIA of the target cooperatively.When the terminal capability of the interceptors is limited, the interceptors can only cover 2/3 of the PIA.When the target moves longitudinally in 30 s, the PIA changes fromR1toR2.According to the updated target PIA, the terminal position and velocity direction constraints of multiple interceptors are solved, and the trajectories are generated on-line as shown in Fig.13.“stan traj” means standard trajectory, “cor traj” means correction trajectory.The revised trajectories enable the four interceptors IIA to cover the target PIAR2well, and the coverage rate can reach 0.91.The longitudinal adjustment ability of midcourse guidance and the effectiveness of the cooperative coverage of multiple interceptors are verified.

Fig.12 Generation of four interceptors’ standard trajectories

Fig.13 Generation of four interceptors’ correctional trajectories

Scenario 2: In order to verify the effectiveness of cooperative coverage after lateral adjustment of midcourse guidance trajectory of multiple interceptors, it is assumed that during midcourse guidance, the target measurement information has been updated four times in total, among which the projection of the PIA in the target encounter plane at the initial time of midcourse guidance is

In the 10th second of midcourse guidance, the target measurement information is updated for the first time,and the PIA after updating is

In the 20th second of midcourse guidance, the target measurement information is updated for the second time,and the PIA after updating is

In the 30th second of midcourse guidance, the target measurement information is updated for the third time,and the PIA after updating is

In the 40th second of midcourse guidance, the measurement information of the target is updated for the fourth time, and the PIA is

According to the initial time of midcourse guidance,the PIA information at the hand-off time is obtained and generats four standard trajectories, the standard trajectories parameters are as follows in Table 4.

Table 4 Initial and terminal condition settings of medium guidance reference trajectory

Among them, M1, M2, M3 and M4 have the same position of they-axis at the initial time and terminal time,which are 10 km, 20 km, 30 km, and 40 km.The trajectories have four corrections in terminal states as in Table 5.

Table 5 Terminal condition setting of medium guidance correction trajectory

The simulation results are shown in Fig.14 and Fig.15.

Fig.14 Generation of four interceptors’ standard trajectories

Fig.15 Generation of four interceptors’ correctional trajectories

Fig.14 shows the standard trajectories generated offline by four interceptors, and the PIA in lateral is obviously more dispersive than that on the longitudinal direction.By optimizing the terminal state of the multiple interceptors, it is found that the multiple interceptors need to be deployed in the side intensively to achieve the maximum coverage of the PIA of the target in the IIA.It can be seen from Fig.14 that the standard trajectories generated by interceptors can ensure the effective coverage of IIA to the PIA obtained at the initial time of midcourse guidance; with the continuous development of midcourse guidance of multiple interceptors, the PIA of targets has been updated four times, and the PIA obtained from the four updated target measurement information,,and.According to the target PIA obtained at different times, calculate the terminal position and velocity direction constraints of multiple interceptors, and adjust the trajectories of multiple interceptors for four times, as shown in Fig.15, Mod1-Mod4 mean four corrections in terminal states.Finally, ensure that the IIA of the four interceptors can cover the PIAwell, covering a probability integral of 0.89.It is verified that the ability of lateral continuous adjustment and the effectiveness of cooperative coverage of multiple interceptors are available.

5.Conclusions

Based on the method of on-line generation of midcourse guidance trajectory under neighborhood optimal control,through optimizing the zero control terminal position, this paper proposes a strategy of cooperatively covering the PIA.Aiming at the problem that the terminal guidance correction ability of a single interceptor is insufficient,the multiple interceptor area can be used to cover the target PIA.The hit probability of the interceptor is greatly increased.