Impact time control guidance against maneuvering targets based on a nonlinear virtual relative model

Yuru BIN, Hui WANG,*, Defu LIN, Yaning WANG, Xin SUN

a School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

b Beijing Key Laboratory of UAV Autonomous Control, Beijing Institute of Technology, Beijing 100081, China

c Beijing Institute Of Electronic System Engineering, Beijing 100854, China

KEYWORDS Impact time control;Maneuvering targets;Polynomial guidance;Virtual relative model;Virtual look angle constraint

Abstract Aiming at the problem of high-precision interception of air-maneuvering targets with impact time constraints,this paper proposes a novel guidance law based on a nonlinear virtual relative model in which the origin is attached to the target.In this way,the original maneuvering target is transformed into a stationary one.A polynomial function of the guidance command in the range domain with two unknown coefficients is introduced into the virtual model,one of the coefficients is determined to achieve the impact time constraint, and the other is determined to satisfy a newly defined virtual look angle constraint.For meeting the terminal constraints simultaneously,the guidance command can finally be obtained.The resulting solution is represented as a combination of proportional navigation guidance-like term which is aimed to meet the zero miss distance constraint, a bias term for impact time control by adjusting the length of the homing trajectory, and an additional term for target maneuvers.Numerous simulations demonstrate that the proposed law achieves an acceptable impact time error for various initial conditions against different types of maneuvering targets and shows more effective performance in comparison with those of other existing guidance laws.

1.Introduction

In the past few decades, Proportional Navigation Guidance(PNG) and its variations have been well studied and widely used for their easy implementation and ideal interception for various types of targets.With the iterative development of guidance technology, guidance theory is developed vigorously in new combat scenarios.The necessity of multidimensional tactical indices makes zero miss distance no longer the only guidance requirement.In order to meet new requirements of complex combat missions, advanced guidance laws such as nonlinear guidance with multiple constraints are developed rapidly.Among them, simultaneous attack based on impact time control attracts more and more attentions.

In modern warfare, multiple missiles can achieve combat efficiency with a higher performance ratio through space and time cooperation, and simultaneous attack can effectively hit vital tactical and strategic targets which are equipped with effective self-defense systems.In the framework of salvo attack, even if some missiles are intercepted by self-defense systems, the target can still be destroyed by the remaining missiles.In the previous literature, there were a number of results that focused on designing interception-time guidance strategies against stationary and non-maneuvering targets.In 2006,Jeon et al.1proposed the Impact Time Control Guidance(ITCG) law for stationary targets based on optimal control theory, using a linear model under the small-angle hypothesis,of which a closed-form solution proved to be a combination of a classical PNG term and a feedback term of impact time error.Similarly, a new virtual guided missile method was proposed in Ref.2which transformed the time-constrained guidance design into a nonlinear tracking problem, and the acceleration command obtained was similar to that in Ref.1.The impact time control guidance for stationary targets was further studied in Refs.3–5In Ref.6a two-stage guidance strategy was derived without time-to-go estimation, and then extended to Three-Dimensional (3D) engagement.The guidance law was also shown to be capable of salvo attack against stationary targets.Further, the time-constrained interception strategies for moving targets was studied in Refs.7–10On the basis of Ref.1a novel ITCG guidance law under a nonlinear model was deduced in Ref.10and a new estimation form of time-to-go for proportional guidance was presented; meanwhile, a method for intercepting low-speed moving targets was extended based on the Predictive Intercept Point (PIP)method.

In recent years, multi-constraint guidance besides the time constraint has been developed continuously11–14.In Ref.12an additional look angle limit was added on the basis of the time constraint,and an improved ITCG guidance law with a monotonic decreasing look angle was proposed,which improved the performance compared with the traditional ITCG guidance law.In Ref.13a guidance law satisfying the impact angle and time as well as Field of View (FOV) constraints was designed.For stationary targets, Han et al.14proposed an analytical guidance law which was derived to meet both the impact time and angle constraints as well as the limit of the reduced seeker FOV that was required to maintain the lock-on condition for a strap-down seeker.At the same time, Sliding Mode Control(SMC) has been widely studied in the field of nonlinear guidance15–17, and realization of multi-constraint guidance by using sliding mode control theory has attracted much attention.In Ref.15a virtual target was introduced, and the guidance process was divided into two stages.In the first stage, a nonsingular terminal sliding mode guidance law was used to intercept a virtual target and meet a specific impact angle and time.In the second stage,proportional guidance was used to intercept real targets at specific real impact angles.In Ref.16a new sliding mode guidance law was designed, which showed good performance even with a large heading angle error and an initial negative closing speed.Similarly, a novel impact time-constrained guidance law considering the FOV constraint was proposed in Ref.17

With rising interests, some novel guidance methods are developed rapidly.In 2013, Lee et al.18proposed a Time-togo Polynomial Guidance (TPG) law, whose acceleration command was expressed as a form of the time-to-go polynomial function,and determined the coefficients of the guidance command to satisfy the impact angle and terminal acceleration constraints.Similarly, in Ref.19the guidance command was innovatively designed as a function of the remaining rangeto-go in the longitudinal direction, and a guidance law was designed so that it satisfied both the impact angle and time constraints based on the small angle hypothesis.In Refs.20–23the look angle shaping theory was applied to polynomial guidance,which provided a new idea for adjusting the impact time and angle.In Ref.24a new TPG guidance law was designed for speed-varying missiles to attack stationary targets, satisfying the terminal impact angle constraint.

The guidance methods mentioned above mainly aimed at the problem of intercepting stationary and constant-speed non-maneuvering targets.However,in practice,target maneuver should be considered when intercepting air targets.How to intercept maneuvering targets with multiple constraints has become a new research hotspot in the past few years.

In Ref.25a target maneuvering model was simplified through the transformation relationships between motion vectors of a missile and a target, and the guidance strategy for achieving zero miss distance was deduced by the theory of optimal control.For a terminal angle guidance problem,many researchers have focused on this promising field26–32.Inspired by Ref.25a terminal attack angle guidance law against maneuvering targets was deduced in Ref.26and the linearized form of the proposed law held a similar structure to that of the linear optimal Trajectory Shaping Guidance (TSG).In Ref.27Li et al.investigated the problem of impact-angle-constrained guidance law design with large initial heading errors,and then extended the guidance law to 3D scenarios in Ref.28In Ref.29the 3D guidance problem with approach angle constraints under a limited FOV and time-varying velocity was addressed without involving the LOS rate information.Further, in Refs.30–31robust incremental 3D guidance laws were proposed considering the terminal angle constraint against maneuvering targets using the sliding mode control theory.In Ref.32to realize the terminal acceleration constraint,the proposed modified bias proportional navigation guidance law could intercept maneuvering targets in a desired attack angle.

Of late, salvo attack has been shown to be both promising and formidable for guidance law design33–40.In Refs.33–35the cooperative guidance laws against maneuvering targets were studied; however, strategies in these literature used the radial component of a missile to achieve the objective, which might be a difficult task for implementation in practice.In Ref.36cooperative strategies were presented for salvo attack of multiple missiles based on the classical Proportional Navigation(PN) algorithm.However, the time-to-go estimation used was similar to that in Ref.5which was deduced under the assumptions that the target was fixed and the leading angle was small enough.What’s more, according to simulation results, it could be seen that some acceleration commands obtained reached the saturation at the beginning as well as at the end.In Ref.37a guidance strategy was originally derived for moving targets.To satisfy the impact time constraint, a parameter was tuned online, which might have been relatively computationally intensive.Based on the deviated pursuit guidance method, a new guidance law using sliding mode control satisfying the time constraint was proposed in Ref.38but for the frequent usage of the switching logics in SMC, the initial guidance acceleration commands might have become discontinuous, which reduced the theoretical value, and this shortcoming was also reflected in Ref.39An impact time guidance law without explicit time-to-go information was designed in Ref.40while taking the sum of the relative distance and ideal time-to-go as a sliding mode variable, and the ability to intercept maneuvering targets were verified by stability analysis and numerical simulations.However,the guidance command could be tremendous at the end phase.

For a complete guidance system design in practice,in order to achieve tracking and guidance of maneuvering targets, the derivation of the guidance command is based on target state information known to missiles, and therefore an estimator is designed to obtain the target acceleration information and other state information.Aiming at the purpose of target estimation, researchers have done some research in this area.In Ref.41to estimate the location of a maneuvering target to achieve high-accuracy interception, an extended Kalman filter was composed, whose state vector consisted of cascaded state vectors of missile dynamics and target dynamics.In Ref.42a new filter with Finite Impulse Response (FIR) was derived,which made the proposed approach to increase immunity against uncertainties and provide more accurate estimates.

This paper focuses on the development of a guidance law itself;it is assumed that a missile employs a well-developed filter and the estimation of a target’s motion will not be emphasized.

In view of the existing works discussed so far, previous literature has highlighted that achieving impact time control against maneuvering targets remains to be a formidable task.This paper considers three typical attack scenarios, classified according to different types of maneuvering targets and extension to a salve attack scenario,for interception of a maneuvering target at desired interception times.

The main contributions of this paper may now be summarized as follows:

(1) Firstly, we introduce the polynomial guidance into a nonlinear virtual relative model with complex constraints.Secondly, the usage of the virtual relative guidance model25helps to intercept maneuvering targets with impact time control, while previous works based on polynomial guidance have mainly focused on stationary and constant-speed non-maneuvering targets18–24.

(2) During the guidance process,the proposed law does not require an explicit time-to-go measurement,but requires the missile-target relative distance or the so called rangeto-go.In practical engagement,range-to-go information and other signals necessary for this formulation can be directly obtained from active radar seekers.Since no time-to-go estimation is required, the proposed law seems to be more practical in implementation.

(3) The range-to-go error is taken as the single parameter in the final phase,which can be calculated analytically.The concise switching logic also guarantees the guidance command to be continuous during the whole process,what’s more, to be bounded by the target maneuver at the time impact.Through numerical simulation studies,the proposed law is proven to be effective in salvo attack.

The remainder of this paper is organized as follows.Section 2 introduces the problem statement of maneuvering target interception and describes the governing equations in the relative frame.In Section 3, an impact time control guidance law with a terminal virtual look angle constraint is derived based on the formulation of a nonlinear virtual relative model.The command structure of the guidance strategy is analyzed in detail.To demonstrate the effectiveness of the proposed guidance law, in Section 4, numerical simulations are carried out under different conditions and scenarios, and comparisons with other existing guidance laws are performed and discussed.Lastly, conclusions and remarks are presented in Section 5.

2.Problem statement

In this paper, an impact time control guidance law against maneuvering targets is proposed based on a nonlinear virtual relative model.In this relative model, a maneuvering target is equivalent to a fixed one,a constant-velocity missile is equivalent to a velocity-varying one, and thus the problem of a constant-speed missile intercepting a maneuvering target is converted to one of a speed-varying missile intercepting a virtual stationary target.Considering the two constraints of the terminal virtual look angle converging to zero and the terminal impact time being equal to a desired interception time,the guidance law in a virtual coordinate system is designed,and then the guidance acceleration command in a inertial coordinate system (depicted in Fig.1) is obtained through the transformation relation of motion vectors between a missile and a target.

The missile and the target are considered as point masses moving in a Two-Dimensional (2D) plane.M and T represent the missile and the target, respectively.VM,VTrepresent the velocity vectors of the missile and the target, respectively.(xM,yM）and(xT,yT）denote the positions of missile and target in inertial coordinate system,respectively.The missile’s Line of Sight(LOS)angle,flight path angle,and the target’s flight path angle are expressed by q,θM, and θT, respectively.aM,aTare the lateral acceleration commands perpendicular to the missile and the target,respectively.VM,VT,aM,aTare the corresponding magnitudes of the above vectors, and it’s assumed that VM>VT.The two-dimensional homing engagement in the inertial coordinate system is shown in Fig.1.

According to the engagement geometry diagram shown in Fig.1, the nonlinear relative kinematics of the engagement are formulated as

Fig.1 Engagement geometry in inertial coordinate system.

In the nonlinear virtual relative coordinate system25, we define the relative flight path angle as θR, the relative velocity as VR?VM-VT, and the relative lateral acceleration command as aR?aM-aT, which is perpendicular to the direction of VR.VR,aRrepresent the magnitudes of VR,aR,respectively.The geometry relation between vectors VM,VT,aM,aTin the inertial coordinate system and VR,aRin the relative virtual coordinate system is shown in Fig.2.

Define the velocity ratio coefficient as k=VT/VM, where k<1.The transformation relations of variables in the two coordinate systems are obtained as follows25.In order to show the integrity of the research work in this paper,the derivations in Eqs.(5)–(9) are detailed in Appendix A.

The new engagement geometry can be obtained in the relative coordinate system as shown in Fig.3.Define the virtual look angle as ηR?θR-q, which means the look angle in the virtual relative model.

The corresponding nonlinear relative kinematics of the engagement are presented as follows:

Fig.2 Geometry relation between vectors in the two coordinate systems.

Fig.3 Engagement geometry in the relative coordinate system.

The initial states are r0, q0, θR(0）, ηR(0）=θR(0）-q0,where r0, q0, θR(0）are selected constants.

3.Impact time control guidance law design

3.1.Design of an impact time control guidance law based on a virtual relative model

Combining Eqs.(10) and (11) into the derivative equation of the LOS angle with respect to the relative distance, Eqs.(11)and(12)are combined to get the derivative equation of the relative flight path angle with respect to the relative distance,shown as

Substituting Eq.(20)into Eq.(19),we can simplify Eq.(19)as

In this paper, we introduce a polynomial idea in the linear model into a nonlinear model, and according to the terminal constraints, the pseudo-control is obtained by the polynomial function in the range domain as

where CN,CMare the unknown coefficients for the polynomials.CNis designed for interception with a zero virtual look angle, whereas CMis designed for impact time control.In the preceding study of polynomial guidance18,19,24,values of n and m are no greater than 4.In this paper, these coefficients are taken as m = 1, n=3 respectively.

In the previous literature of polynomial guidance,the acceleration commands of missiles as well as their corresponding achievable constraints are shown in detail in Appendix B.

By solving the differential equation in Eq.(21)with the initial condition ˉη0=sin ηR(0）, we can get

where c1is a term in the solution of ˉη, which has no functional relationship with r, r0represents initial value of r.

Considering the terminal constraints

where tfdenotes the terminal moment of guidance phase,rfrepresents the final value of r, Tdis the desired interception time that we expect.

Since the terminal relative distance rfis zero when the missile intercepts the target, the value of c1in Eq.(23) can be obtained as

In order to avoid small-angle approximation for taking tan ηRas ηRdirectly26, according to the Taylor expansion of tan ηR, the tangent value of the virtual look angle can be approximately by tan ηR=μ sin ηR=μˉη, μ ?[1,1.5）.In this way, Eq.(31) can be rewritten as

Step 0.Assign the initial values of r0,q0,θM0,θT0,VM,VT;calculate θR0 from Eq.(6), i=0.Step 1.Assign m,n,h,Td,aT(t）; // h: step size Step 2.While r(i）> 1 t=t+h;Calculate VR(i）from Eq.(5); calculate r*go from Eq.(44)(detailed below), namely r*go =VR(i）(Td-t）, respectively;Calculate ˙qi according to Eq.(10), and then calculate the acceleration command aM (the solution of aM is detailed in Eq.(42) below)Calculate A˙i =[r˙ i,θ˙Ri,θ˙ Mi,θ˙Ti]T according to Eqs.(1)–(4)and Eqs.(10)–(13);Calculate Ai+1 =Ai+ ˙Aih; assign i=i+1;Step 3.end while

Rearranging Eq.(33), we can get a quadratic equation of CM, determined by

Since there is no need for guidance control for impact time when ε=0,the only solution of the quadratic equation can be obtained as

3.2.Solution and analysis of guidance command in inertial coordinate system

According to Eq.(9), the guidance command in the inertial coordinate system is expressed as

The expression of aM1is a PNG-like term, which makes heading error in the virtual model, namely the virtual look angle of the missile, converges to zero, and realizes zero miss distance.aM2can be interpreted as an impact time error feedback term multiplied by a PNG-like term similar to aM1,which is used to restrict the flight trajectory to make sure intercepting the target at desired impact times.In the formation flight scenario, for the purpose of salvo attack, multiple missiles adjust their remaining flight time errors through appropriate maneuvers during the guidance process.In aM3, aTrepresents the maneuverability of the target, and the trigonometric function term reflects the ability of the missile to respond to the change of the target’s flight path angle.This part of the command is used to intercept the maneuvering target.

3.3.Saturation analysis of terminal acceleration

According to the decompositions of the acceleration command in Section 3.2,this section focuses on each part of the guidance command to analyze the convergence of terminal acceleration.

Firstly, according to Eq.(7), the cosine term in aM1can be formulated as

In this way,aM1can be seen as a PNG term,so it is reasonable to consider aM1converging to zero at the terminal guidance phase.

For aM2,on one hand,ε represents the remaining flight distance error, which is characterized by the difference between the expected flight distance and its actual value.According to Eq.(37), ε should not be less than zero at any instant to ensure CM≥0.On the other hand,as the terminal relative distance tends to be zero, the magnitude of aM2may increase sharply at the end, resulting in blowing up the acceleration command of the missile.Since ε generally shows a downward trend in the guidance process, that is to say, when the relative distance tends to be zero, ε is also close to zero.Considering the above two points, a simple switching rule is proposed.We define e as the least remaining flight distance error which is sufficiently small,and when ε0, at r=r0with t0=0,we can obtain the following condition for a proper desired impact time:

See Algorithm 2 for the pseudocode of the switching logic.

Step 0.Assign e=0.001; // the minimal tolerance on ε Step 1.calculate ε from Eq.(43)Step 2.if ε ≥e Step 3.then employ ε into Eq.(42)Step 4.else Step 5.then substitute ε=0 into Eq.(42)Step 6.end

Thus, the terminal acceleration command in the inertial coordinate system aMis simplified as

This part of the acceleration is caused by the target maneuver.It is proven that the terminal acceleration command satisfies the inequality,25i.e.,

The derivation process is detailed in Appendix E, and similar proof processes can also be seen in Ref.25Therefore, the magnitude of the acceleration command of the missile at the end will not exceed the boundary of the target’s maneuvering acceleration,which effectively avoids the saturation of the terminal acceleration command,provides conditions for tracking and intercepting high maneuvering targets, and shows its convenience for engineering implementation.

4.Simulation results

In this section, numerous simulations are carried out under typical scenarios considering different desired impact times,different types of target maneuvers, salvo attack, etc.Compared with the improved ITCG guidance law and the sliding mode guidance law, detailed analysis is presented to show the guidance performance proposed in this paper.The initial parameters for each target are shown as Table 1, and all simulations are terminated when r<1 m.Simulation results demonstrate that all the impact time errors are within 0.1 s.

4.1.Typical Scenario 1:Simulations with different impact times

In this section, the performance of the proposed guidance law is analyzed with different impact times.Four different desired impact times,i.e.,53 s,56 s,59 s,and 62 s,are selected for simulation,and the guidance command of the missile is limited to 10 g where g=9.8 m/s2.Notice that the time-to-go mentioned below is defined as r/vc,in which vcdenotes the closing velocity between the missile and the target.And XR, YRmentioned in Fig.4 (a) are defined as xT-xM, yT-yM, respectively.

Simulation results are shown in Fig.4.It can be seen from Fig.4(a) that the guidance law can intercept the target at desired impact times; meanwhile, the virtual look angle can approach the terminal constraint as expected in Fig.4(b).Fig.4(a) also clearly shows that with an increase of the expected impact times, the missile takes a longer path.Moreover,if the designated impact time is smaller,the missile takes a short-cut path to meet the impact time constraint.Fig.4(c)shows that the guidance command of missiles is reduced to the boundary of aT, avoiding the saturation of the terminal acceleration command.According to the guidance command curves shown in Fig.4(d),it can be observed that the terminal relative guidance command of the missile converges to zero.The rapid convergence of aRat the final phase is due to the fact that the remaining flight distance error has reduced to a sufficiently small positive value, so the guidance law is switched to a PNG law.It is also evident from Figs.4(e) and (f) that the target is intercepted successfully at all desired interception times.The high demand on the initial lateral acceleration, as shown in Fig.4(c), shares a similar change with the initial time-to-go in Fig.4(e) and the initial virtual look angle in Fig.4(b).In essence, it may be attributed by the requirement for the look angle reaching a desired deviation value.

It is worth mentioning that across simulations, different values of Tdare chosen,which verify the efficiency of the proposed guidance law for various choices of expected impact times.

4.2.Typical Scenario 2:Simulations with different maneuvering targets

This section is aimed to investigate the performance of the proposed guidance law for different maneuvering targets.Simulation results obtained are shown in Fig.5, for a desired interception time Tdof 50 s.From the interception trajectory curves shown in Figs.5(a) and (b) and the time-to-go curve in Fig.5(c), it is confirmed that for two targets with different maneuvers and different initial positions, the interceptiontime errors converge to acceptable desired values within finite time.Meanwhile,Fig.5(d)confirms that the virtual look angle decays to zero at the end as expected.From the curves of the guidance command in Figs.5(e)and(f),the proposed guidance law shows strong sensitivity to a target maneuver.The guidance command of the missile changes in a sine form, which also indicates strong tracking capability against maneuvering targets.Furthermore, comparing the magnitudes of guidance commands in Fig.5(e), a greater missile maneuver is required when the target’s maneuver is greater.

4.3.Typical Scenario 3: Extended to a salvo attack scenario

In the previous sections, simulations verify the impact time control capability of the guidance law proposed under a oneto-one interception scenario and the guidance performance of intercepting targets with different maneuver forms.This section will extend from a one-to-one scenario to a simultaneous attack scenario to ensure a salvo interception by multiple interceptors.The initial parameters of the missiles are given in Table 2.

To testify the effectiveness of the salvo attack strategy,missiles are set to launch from four different initial positions and orientations relative to the maneuvering target.The expected impact time is chosen as 50 s, and the magnitude of the guidance command is limited to 10g.Results obtained for this implementation are shown in Fig.6.

The dotted line in Fig.6(a) demonstrates the interception curves with Augmented Proportional Navigation Guidance(APNG).The impact times of Missiles 1, 2, 3, and 4 underAPNG are 39.86 s, 28.56 s, 24.54 s, and 36.81 s, respectively,according to the simulation results.As can be observed from Fig.6(a),when the expected impact time is 50 s,the trajectory of the proposed guidance law is more curved than that of APNG.The reason is that the given impact time is longer than that of APNG, so the missiles need additional maneuvers to adjust the remaining flight time.Fig.6(b)confirms the convergence of the virtual look angle, successfully meeting the requirement of the terminal constraint.It is evident from Fig.6(c) that all the missiles do not exceed the lateral acceleration limits and finally separately converge to be within the magnitude of aTas expected.Fig.6(d) shows that for the maneuvering target, four interceptors with different initial relative positions can intercept it at the same desired time,which shows capability for salvo attack guidance.

Table 1 Simulation parameters for each target.

Fig.4 Simulation results of different desired impact times.

Fig.5 Simulation results of different maneuvering targets.

Table 2 Salvo attack initial parameters setting.

Fig.6 Simulation results of salvo attack.

4.4.Simulations with target measurement uncertainty

To evaluate the guidance performance with target measurement data errors, simulations are carried out with measurement uncertainty.Simulations results for the proposed guidance law with ten percent of uncertainty of aT,θTare shown in Fig.7.It is clearly shown that the differences between the ideal cases and the cases with measurement uncertainty are mainly reflected in the acceleration command curves from Figs.7(e) and (f).Other subfigures are almost the same as those in the ideal cases.Therefore, simulation results demonstrate that the proposed law still performs well with measurement errors existing.

4.5.Comparison with other guidance laws

The following two subsections compare the performance of the proposed guidance law with another two existing guidance laws for both stationary and maneuvering targets that aim to satisfy impact time control.

4.5.1.Comparison with classical ITCG

This subsection studies the interception performance for stationary targets, compares the guidance law in this paper with the improved ITCG guidance law,10and verifies the performance of the proposed law.

The ITCG guidance law is the optimal guidance law that earlier studied the impact time constraint on stationary targets.1Based on small angle approximations, on the basis of Ref.=1,the improved ITCG guidance law was deduced based on the nonlinear model in Ref.10and a new estimation of the remaining flight time of proportional navigation guidance was given.

According to Eq.(42),in the case of attacking a stationary target,the form of the guidance law in this paper can be rewritten as

When the missile attacks the target in the form of pure proportional navigation guidance (n=3), the impact time in the simulation is 30.66 s.Reasonably selecting the expected guidance time as 37 s and 42 s respectively,and limit ting the guidance command magnitude of the missile to 5g, simulation results are shown in Fig.8.

Fig.7 Simulation results of target measurement uncertainty.

As can be seen from Fig.8, both guidance laws can intercept the target desirably.Fig.8(a) shows that as the expected impact time increases, the trajectory also extends, which is reflected in the increase of the look angle in Fig.8(c).From Figs.8(a) and (c), as the trajectory of the ITCG guidance law bends rapidly in the initial stage of guidance, the corresponding look angle increases rapidly within 10 s.Compared with ITCG, the look angle of the proposed law changes relatively smoothly during the initial guidance period,and the terminal look angle finally converges to zero.Fig.8(d)shows the changes of the acceleration commands of the two guidance laws.It is observed that although the guidance law in this paper reaches the maximum constraint value of acceleration during the process, it still does not exceed the limit boundary,and the command smoothly approaches zero in the end as the guidance law has switched to PNG.Fig.8(b)shows the process of the relative distance approaching zero at desired impact times.

4.5.2.Comparison with WTG guidance law

Kim et al40.designed and deduced the Without explicit Timeto-go estimation Guidance (WTG) law with the impact time constraint based on the sliding mode control theory, and proved the interception ability against maneuvering targets through stability analysis and simulation results.In this subsection, the proposed guidance law and the WTG guidance law are used for simulation and compared.The form of the WTG guidance law is expressed as follows:

Fig.8 Comparison of simulation results with the improved ITCG guidance law.

The acceleration command of the missile is

The initial flight path angles of the missile and the target are 45°and 0°,respectively.The values of parameters in the WTG guidance law are the same as those in Ref.40which are α0=0.01, α2=0.1, k=3, kad=-0.5, η=20, and Td=80 s,while the acceleration commands of the target are aT1=0.3g sin(πt/30）and aT2=0.5g sin(πt/30）, respectively.The initial locations of the missile and the target as well as their flight speeds are the same as those in Ref.40too.

Simulation results are shown in Fig.9.When the normal acceleration is aT1, the terminal impact time of the guidance law in this paper is 80.21 s while that of WTG is 79.69 s;when the target maneuvering form is aT2,the terminal impact time of in this paper is 80.17 s and that of WTG is 79.54 s.Obviously,the expected time of this guidance law is slightly greater than 80 s, while that of the WTG guidance law is the opposite.However, the impact time error of the proposed guidance law is less than that of WTG, which shows more accurate impact time control performance.

Fig.9 Comparison of simulation results with WTG guidance law.

In this paper, the guidance law realizes a tail-chase attack on the target at the end,while head-on for WTG.The interception trajectory is shown in Fig.9(a).Compared with the large flight path angle at the initial stage of WTG,the changing rate with the proposed law is relatively small, which is reflected in the rapid reduction of the relative distance in Fig.9(b) and the shorter trajectory in Fig.9(a).Figs.9(c) and (d) demonstrate the curves of acceleration commands in the interception process.It can be found that the initial acceleration of the proposed guidance law is much smaller than that of WTG,although the guidance command has a small increase in the final phase, it quickly converges to a small value near zero.However, the acceleration command of WTG is much larger in the initial stage of guidance,and the end diverges to a given limit value.

5.Conclusions

In this paper, a new impact time control guidance law against maneuvering targets is proposed.The nonlinear engagement is established in a virtual relative motion model which is obtained through transformation relationships between motion vectors of a missile and a target.The method of polynomial guidance is introduced into the virtual model and a pseudo-control is defined as a polynomial function in the range domain.Considering impact time and newly defined terminal virtual look angle constraints, the specific expression of the pseudo-control is derived.The expressions of the missile guidance command in both the relative coordinate system and the inertial coordinate system are demonstrated.

It is shown in this paper that with the guidance strategy proposed, it is successful to achieve interception of maneuvering targets at desired impact times or with negligible errors in the same.Simulation results demonstrate that for different types of maneuvering targets,the proposed law performs satisfactorily as we expect.Additionally, the effectiveness of the proposed law is further validated through extending to a salvo attack scenario.The results of additional simulations involving target measurement errors further demonstrate the robustness properties of the proposed strategy.

The encouraging results in this paper motivate further investigations.As the guidance command is derived on a planar plane, in future work, a 3D guidance problem may need further discussions and investigations.Moreover,if additional constraints are added, for instance, terminal acceleration and impact velocity, the proposed guidance strategy still performing successful interception could also be a promising direction of future studies.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the Beijing Key Laboratory of UAV Autonomous Control, China and the Key Project of Chinese Ministry of Education (No.2022CX02702).

Appendix A.Eqs.(5)–(9) are derived as follows.

Appendix B.In the previous polynomial guidance literature,most of the guidance commands were expressed as polynomials related to the relative distance and time-to-go information.Table B118,19,24,26lists the comparison results of guidance commands and corresponding constraints in previous different literature.

CTis unknown coefficient for the polynomials which aimed to control the impact time.

where xf,yfdenote the locations of targets.

Above all, most of previous literature utilized polynomial guidance for impact-angle control.In Ref.19the CTterm was designed for impact time control, but was assumed as a constant value, and the proposed guidance law was supposed to attack stationary targets.However, in this paper, the term for impact time control is time-varying, and the guidance law is effective for maneuvering targets as well as stationary ones.

Appendix C.In Eq.(31),the integral function can be rewritten as

This approximation is reasonable because we constrain the terminal virtual look angle ηRto zero, satisfying the small angle hypothesis,and this approximation in Eq.(C2)becomes more accurate in the final phase of guidance,which guarantees successful control for interception time.

In Fig.C1,we can clearly find that as the virtual look angle ηRapproaches zero,the difference between the two functions is narrowing.When the virtual look angle turns zero at the time of impact, the values of the above functions are the same.However, if ηRis greater than 80°, it may affect the accuracy of the interception time.

Appendix D.According to Eqs.(5)and(33),the relative speed VRand the desired range to go r*goare formulated by Eqs.(D1)and (D2), respectively.

From Eq.(D1), it’s shown that VRhas a nonlinear timevarying relationship with θM,θT.It is worth analyzing the influences of different values of VRon the guidance performance.According to Li et.al27, the average relative speed is approximated by the average values of the current relative speed VRand the terminal relative speed VRf.Motivated bythis approach, we set different weight coefficients for VRand VRf, and evaluate the guidance performance through simulation results.Note that for the proposed guidance law,VRfcannot be derived analytically,but could be solved numerically by simulation.

Table B1 Comparison with different polynomial guidance laws.

Fig.C1 Curves of two functions.

In this way, Eq.(D2) can be rewritten as

where 0.5 ≤α ≤1,0<β ≤0.5,α+β=1.

When α=0.5, β=0.5, it’s the approach similar to that in Ref.27and α=1, β=0 is the method of this paper.Simulations results for the proposed guidance law with different weight coefficients are shown in Fig.D1.It’s clearly shown from Fig.D1(a) that with different coefficients, the missiles can successfully intercept the maneuvering target.However,the differences are reflected in the acceleration command curves in Figs.D1(c)and(d)and the virtual look angle curves in Fig.D1(b).From Fig.D1(b),as α decreases and β increases,the terminal virtual look angle fails to be constrained to zero as expected, and this shortcoming is also reflected by the divergences of the terminal acceleration commands in Figs.D1(c)and (d).

Appendix E.According to Eqs.(46) and (5), we have

The right-hand side of the equation is a monotonic function with VRas the independent variable.According to Eq.(E2),the range of VRcan be obtained as

which verifies inequality Eq.(51) successfully.