HUANG Jia ,CHANG Sijiang,* ,and CHEN Shengfu

1.School of Energy and Power Engineering,Nanjing University of Science and Technology,Nanjing 210094,China;2.The 20th Research Institute of China Electronics Technology Group Corporation,Xi’an 710068,China

Abstract: To improve applicability and adaptability of the impact time control guidance (ITCG) in practical engineering,a twostage ITCG law with simple but effective structure is proposed based on the hybrid proportional navigation,namely,the pureproportional-navigation and the retro-proportional-navigation.For the case with the impact time error less than zero,the first stage of the guided trajectory is driven by the retro-proportionalnavigation and the second one is driven by the pure-proportionalnavigation.When the impact time error is greater than zero,both of the stages are generated by the pure-proportional-navigation but using different navigation gains.It is demonstrated by twoand three-dimensional numerical simulations that the proposed guidance law at least has comparable results to existing proportional-navigation-based ITCG laws and is shown to be advantageous in certain circumstances in that the proposed guidance law alleviates its dependence on the time-to-go estimation,consumes less control energy,and adapts itself to more boundary conditions and constraints.The results of this research are expected to be supplementary to the current research literature.

Keywords:missile guidance law,impact time control,pure-proportional-navigation,retro-proportional-navigation,time-to-go estimation.

1.Introduction

The impact time control guidance (ITCG) law has been widely investigated to carry out some special missions and tasks on the modern battlefield (e.g.,salvo impact and cooperative impact) since it was first proposed in [1].In the framework of the ITCG law,the desired impact time at which the missile captures the target is designed before the launch of the missile.The study on ITCG is also the fundamental of the guidance law controlling both impact time and impact angle.In the current research literature,both from theoretical and design perspectives,various ITCG laws can be derived in terms of various theories and techniques,such as proportional navigation[1?12],sliding-mode control (SMC) theory [13?21],Lyapunov stability theory [22,23],optimal control theory[24?26],and polynomial shaping technique [27?34],to name a few.

Based on the well-known proportional navigation(PN),the impact time control can be achieved using all kinds of additional control commands as a function of the impact time error.The additional command derived from the linear formulation consisting of the PN guidance(PNG) law and the feedback of the impact time error was given by [1].A more rigorous and generalized ITCG law based on nonlinear formulation was obtained in [2].A PNbased ITCG law with a field-of-view constraint was studied in [3]by using the cosine of the weighted heading error in the additional biased term.This guidance law was further discussed with the consideration of the uncertain system lag in [4].The studies presented in [5,6]were performed to address the impact time and impact angle constraint in the PN framework,using different time-to-go formulas and additional commands.By using the accurate time-to-go solution in the incomplete Beta function,an effective guidance command consisting of a nonlinear time-varying gain modification term and a small term biased to the traditional pure PNG was obtained in [7].On the basis of the ITCG law in [1],the field-of-view constraint was studied in [8]using a switching logic.By introducing the concept of retro-proportional-navigation[35],a three-dimensional ITCG law against high-speed non-maneuvering targets was proposed in [9],which demonstrates its feasibility and effectiveness.The navigation gain of the retro-proportional-navigation relates to the time-to-go error which can be determined using a recursive approach.The ITCG laws as presented in [10,11]was obtained via designing the time-varying navigation gain.The guidance law in [10]ensures that the seeker’s look angle decreases monotonically from the initial value to zero over the engagement,while the ITCG law in [11]has its capability in adjusting the navigation gain during the flight by predicting the impact time error.In [12],a numerical time-to-go prediction law for a three-dimensional PNG trajectory was proposed using the concept of the predicted range.

In addition to PN,the SMC theory is used to design the ITCG law,as presented in [13?21].As already mentioned,the Lyapunov stability theory can be adopted to develop ITCG due to its no mean robustness and stability.As an example,two efficient ITCG laws using the Lyapunov criteria were proposed in [22]and [23],respectively.Another alternative that can be applied to the ITCG design is the optimal control theory.In the framework of optimal control,the theoretical minimum control effort can be obtained and as a consequence,an optimal trajectory with the desired impact time is achieved,as presented in [24?26].A number of recent studies on ITCG [27?31]focus on the look angle or range shaping without using time-to-go estimation.The technique used in [32?34]is called trajectory shaping.

By comparing the aforementioned theories or techniques which play an essential role in the development of the ITCG law,it is found that the PN-based ITCG law which is the most widely studied ITCG law has its unique advantage in that it,to a large extent,possesses the merits of the PNG,including pure-proportional-navigation guidance (PPNG) and retro-proportional-navigation guidance (RPNG).Although the current study on the PNbased ITCG law is relatively wide and deep,as described previously,there still exist some limitations to be overcome,such as the controllable range of the desired impact time,the higher dependence on the time-to-go estimation,singularities in certain circumstances,and other problems associated with control effort.The main motivation for the results of this paper is to improve and extend the current PN-based ITCG law,providing a beneficial reference for the development of this kind of ITCG laws in the future.

The contribution of this effort is to propose a two-stage ITCG law which is adaptable to the required impact time with a wider range.Compared with the current research literature,this guidance law alleviates dependence on time-to-go estimation and leads to less control energy.Based on the pure-proportional-navigation and the retroproportional-navigation,the proposed guidance law is divided into two stages.In the framework of this guidance law,the impact time error is supposed to converge to zero in the first stage and then the second stage of the trajectory is only driven by the pure-proportional-navigation having zero impact time error.For the case with the impact time error less than zero,the first stage of the guided trajectory is driven by the retro-proportional-navigation,completing the task of impact time control.On the contrary,when the impact time error is greater than zero,both of the stages are driven by the pure-proportionalnavigation but with different navigation gains.Despite requiring the time-to-go estimate,the proposed guidance law still works even if the accuracy of the time-to-go estimation is insufficient.Compared with [2],the proposed ITCG law alleviates dependence on time-to-go estimation accuracy and does not result in failure due to severe impact time requirement.The initial guidance command,the control effort,and the acceleration needed during the total flight are all relatively small in contrast to [7].Moreover,the presented guidance law originally derived in the two-dimensional plane can be applied to the threedimensional engagement,as will be shown in this work.Compared with an efficient three-dimensional Lyapunovbased ITCG [22],the presented ITCG results in smaller initial acceleration command and eliminates the singularity in the case of zero initial heading error.

The organization of this paper is as follows.In Section 2,both the problem statement and a preliminary analysis are given.Section 3 covers the design of the proposed ITCG law,including the preliminary formulas,the boundary of the desired impact time,and its modified expression.The proof of the convergence of this proposed ITCG law is given in Section 4.The application of the proposed ITCG law to three-dimensional engagement is presented in Section 5.A detailed numerical simulation study is presented in Section 6 and Section 7 presents concluding remarks.

2.Problem statement and preliminary analysis

2.1 Governing equations for two-dimensional engagement

Consider a two-dimensional missile-target engagement scenario with a stationary target,as shown in Fig.1.

Fig.1 Two-dimensional engagement geometry for a stationary target

It is also assumed that (i) the missile is considered as a point mass;(ii) the speed of the missile is constant;(iii)only the normal acceleration perpendicular to the velocity vector of the missile is considered;and (iv) the autopilot lag is neglected.The governing equations from the engagement geometry shown in Fig.1 are expressed as follows:

whereRis the distance between the missile and the target,VMis the speed of the missile, ? denotes the heading error, θ is the line-of-sight (LOS) angle, γ is the flight path angle of the missile,aMis the normal acceleration of the missile,(x,y) describes the trajectory of the missile in a two-dimensional plane,the dot over the variant denotes the first derivative with respect to timet;thus,is the LOS rate.

2.2 Preliminary analysis of the heading error variation

According to (1),the closing speed between the missile and the target is a function of the heading error.Here,we intend to explore the variations of the heading error under both the pure-and the retro-proportional-navigation guidance laws.

It is well-known that the acceleration command of the PPNG can be written as

whereNdenotes the navigation gain.

Combining (1)?(4) with (6),it can be obtained as follows:

We can see from (7) that under the assumptions of ? ∈[0,π) andN>2,the heading error of the missile decreases monotonically over the engagement for the PPNG.Using the same approach presented in [7],the accurate closed-form solution of time-to-go in the framework of the PPNG can be expressed as

whereB(z;a,b) is the incomplete beta function [36]which is defined as

in whichc,a,andbare variant parameters of the incomplete beta function.

On the other hand,the acceleration command driven by the RPNG can be written as

Similarly,combining (1)?(4) with (10),it yields

In contrast to the PPNG,the heading error increases monotonically over the engagement for the RPNG.Due to the assumptions ? ∈[0,π) andN>2,the final heading error is less than 180°.Integrating (11),the timetsthat it takes for the change from the initial heading error ?0to a certain heading error ?scan be obtained by

3.Two-stage ITCG design

This section intends to provide a detailed derivation of the ITCG law.The boundary of the desired impact time in the application of this ITCG law is also discussed.Finally,the proposed ITCG law is modified to adapt itself to more boundary conditions.

3.1 Derivation of the ITCG law

First,the impact time error is defined as

wheretgoPPNGis the time-to-go driven by the PPNG;Tgod=td?t,tdis the desired impact time andtdenotes the elapsed time after the missile’s launch.When the impact time error is zero,it is indicated that the target is captured by the missile at the desired impact time.

As previously analyzed,the impact time control can be achieved through controlling the heading error rate of the missile,since the time of flight can be expressed as a function of the heading error.We can see from (7) and(11) that the heading error rate relates to the navigation gain.Under the RPNG,the heading error rate increases as the navigation gain changes.In contrast,the heading error rate decreases as the navigation varies driven by the PPNG.In other words,the heading error rate can be controlled by switching different PNG laws,making the impact time error tend to zero and finally realizing the impact time control.Based on this idea,the ITCG law can be designed as follows:

wherekis a constant that satisfiesk>1+1/NandN>2,and sgn(·) is the signum function.

It is obvious that the convergence rate of the impact time error depends on the value ofk.Considering the fact that the available acceleration provided by the missile is limited in practice,the value ofkcannot be set too large.Typical variations of the heading error driven by the proposed ITCG law are shown in Fig.2,in which ?0andR0denote the initial heading error and the initial distance between the missile and the target,respectively;?sandRsrepresent the heading error and the distance between the missile and the target at the switching point;tsdenotes the switching time andtfis the time of flight only driven by the PPNG,i.e.,tf=tgoPPNG+t.

Fig.2 Typical variations of the heading error over the engagement using (14)

According to the definition of the impact time error,when the impact time error is less than zero,the desired impact time is greater than the time of flight only driven by the PPNG.In contrast,a positive impact time error means that the desired impact time is less than the impact time of the PPNG trajectory.As shown in Fig.2,in both cases of ξ<0 and ξ>0,the engagements are divided into two stages.The first stage (fromt=0 tot=ts) pertains to the trajectory driven by the ITCG whereas the second stage (fromt=tstot=td) refers to the trajectory only driven by the PPNG.

For the case ξ<0,the engagement consists of the RPNG-based impact time control stage (as the first stage)and the PPNG stage (as the second stage),as shown in Fig.2(a).In the first and second stages,(14) reduces torespectively.For the case ξ>0,the engagement is composed of the PPNGbased impact time control stage (as the first stage) and the PPNG stage (as the second stage).In the first stage,the impact time control is implemented usingaM=and in the second stage,(14) reduces toAlthough both of the stages are based on PPNG in this case,the specific navigation gains are different.

Apparently,the first stage pertains to either RPNG-or PPNG-based impact time control,depending on the sign of the impact time error,whereas the second stage merely refers to a PPNG.Once the impact time error converges to zero in the first stage,implying that the impact time control is completed,the second stage begins.The mathematical relation between the switching timetsand the desired impact timetdcan be described as

In most cases,the value of the impact time error ξ depends on the time-to-go of PPNG and the real time-to-go in the flight.Note that the impact time error may also be affected by an inaccurate time-to-go estimate,generating either positive or negative values.In this circumstance,the sign function in the acceleration command works and,to some extent,reduces the dependence on the accuracy of the time-to-go estimate.

3.2 Boundary of the desired impact time

In the case of ξ>0,by analyzing (7),when the coefficientkis sufficiently large,the ideal minimum impact time of the missile can be approximated as

Because the available acceleration provided by the missile is limited in practical engineering,the coefficientkcannot be too large.Therefore,for the case of ξ>0,the minimum desired impact time is as follows:

We can see from (17) that the impact time control of the missile is maintained over the trajectory until interception.

For the case ξ<0,the desired impact time is required to be greater than the impact time of the PPNG.Substituting (12) into (15) yields

As illustrated by (11),when the initial heading error of the missile is non-zero,the value of ?stends to π.Taking the limit of (12),it gives

As a result,the desired impact time tends to infinity in terms of (15).Therefore,the boundary of the controllable impact time for this case can be obtained as

3.3 Practical modifications of the guidance law

As mentioned in the preceding sections,it is found that when the initial heading error is zero,the initial guidance command is also zero for both PPNG and RPNG.Thus,the ITCG law presented as (14) cannot be implemented in the condition of ?0=0 and ξ ≠0.To overcome this limitation,an additional acceleration command is added to(14),and therefore,a modified ITCG law can be expressed as

whereM>0 is a positive constant,k>1+1/N,andN>2.

It is apparent that the correction term in (21) works if and only if the initial heading error is zero.Note that by using the term |?| instead of ?,the modified ITCG law c an be effectively applied to the case of ? ∈(?π,π).

4.Proof of convergence

functionVLas follows:

This section intends to prove that the ITCG law presented as (21) enables the impact time error to converge to zero.To this end,we construct a candidate Lyapunov

Differentiating the Lyapunov function with respect to timetand considering the definition of the impact time error,it yields

Obviously,the/calculation of dVL/dtrelates to the value ofConsidering the navigation gainNas a constant,it gives

Thus,/two partial derivatives,i.e.,and,should be determined.Using the Gaussian hypergeometric function [7],the accurate solution of the time-to-go in the framework of the PPNG,i.e.,(8),can be written as

where2F1(·) represents the Gaussian hypergeometric function [36].

By transforming the Gaussian hypergeometric function into the infinite series [7],(25) can be expressed as

Differentiating the time-to-go with respect to sin2?,it yields

Differentiating the time-to-go with respect to the distance between the missile and the target,it gives

By multiplying (28) by cos?,we can obtain

Note thatg(π/2,N)=0 and the following equation

Substituting (21),(28),and (29) into (24) yields

Substitution of (33) into (23) gives

K1≥0 andK2≥0,so the derivative of the Lyapunov function is negative semi-definite.When ?=0,π/2 and ξ=0,we have dVL/dt=0.When ?=0,π/2 and ξ ≠0,we have ?˙ ≠0 in the case of ξ˙=0,implying that the impact time error can converge to zero over the engagement by using the law presented as (21).

5.Expansion of the guidance law to three-dimensional engagement

In this section,the three-dimensional engagement involving a stationary target is considered,and then,a threedimensional ITCG law is proposed using the identical idea proposed in Section 3.

5.1 Coordinate systems

The three-dimensional engagement geometry for a stationary target is shown in Fig.3.The heading error ?Mis the angle between the velocity vector of the missile and the LOS vector to the target in the three-dimensional space.Note that assumptions presented in Subsection 2.1 are also used in this section.

Fig.3 Three-dimensional engagement geometry for a stationary target

To mathematically describe the kinematics of a threedimensional engagement against a stationary target,three coordinate systems are required,such as the reference coordinate system (RCS),denoted asMxyz,the missile body coordinate system (MBCS),denoted asMxMyMzM,and the LOS coordinate system (LOSCS),denoted asMxLyLzL.These coordinate systems,as illustrated in Fig.4,have the same originMthat is the center of the mass of the missile.ThexL-axis of the LOSCS points to the target and is collinear with the LOS vector,while the direction of the velocity vector of the missile denotes thexMaxis of the MBCS.

Fig.4 Description of several coordinate systems

Although these mentioned coordinate systems have already been defined in [22,37],this section intends to complement mutual transformations between these coordinate systems,which are described with the use of Euler angles in Fig.4.

According to Fig.4(a),the LOSCS can be obtained by using two right-handed coordinate transformations.The first transformation is from the RCSMxyzto an intermediate systemMx1yLzwith the azimuth angle ψL,followed by the LOSCSMxLyLzL,obtained through the elevation angle θL.As shown in Fig.4(b),the MBCS consists of two transformations.The first step is fromMxLyLzLto an intermediate systemMx2yMzLwith the yaw angle ψMand the second one is fromMx2yMzLto the MBCSMxMyMzMwith the pitch angle θM.

5.2 Governing equations

Based on the aforementioned assumptions and coordinate systems,the following governing equations,similar to [22,37],can be obtained as

whereayMandazMare the yaw and pitch accelerations of the missile,respectively.

5.3 Three-dimensional ITCG law

The traditional PPNG describes the proportional relationship between the acceleration of the missile and the LOS rate.In the three-dimensional case,the acceleration vectoraPPNGfor the PPNG can be written as

where ?Lrepresents the LOS rate vector andVMis the velocity vector of the missile.

The scalar form of (42) can be expressed as

Similarly,the acceleration vectoraRPNGdriven by the RPNG can be written as

Note that considering the plane constructed by the velocity vector of the missile and the LOS vector to the target,the accurate time-to-go formula presented as (8) can be extended to the three-dimensional form as follows:

where ?M=arccos(cosθM·cosψM).

Consequently,the impact time errorfor three-dimensional engagement is defined as

Using the same idea of the two-dimensional case,the three-dimensional ITCG law can be designed as follows:

Note that specific values of the parameterskandMfor two-and three-dimensional ITCG laws,presented as (21)and (48),respectively,are not necessarily the same,as will shown in the simulation.

6.Numerical simulations and comparison study

In this section,the two-dimensional ITCG law presented in Section 3 is validated in Subsection 6.1 and the expanded three-dimensional ITCG law presented in Section 5 is validated in Subsection 6.2 by numerical simulations.The performance of these ITCG laws is also compared to those obtained in earlier studies [2,7,22].For ease of comparison,the time-to-go formula used in [2]is restated as follows:

To alleviate discontinuity of guidance command driven by frequent switching between the PPNG and the RPNG,the signum function sgn(·) used in (21) and (48)is usually replaced by a sigmoid function [15]as follows:

whereais a positive constant.

In general,the time-to-go estimation formula is independent of the specific ITCG law,so any time-to-go approximation can be incorporated into a specific ITCG law(i.e.,calculating the impact time error).Therefore,to compare the performance of various ITCG laws in the same condition,the accurate time-to-go estimation (8)can be used to replace (49) for the ITCG law in [2].Similarly,the formula (49) can also be applied to the ITCG law presented in [7].In the following simulation,some parameters in (14),(21),and (50) are set ask=1.5,M=10,anda=10,while these parameters for the threedimensional case are chosen asa=10,k=2.0,andM=10.

6.1 Cases for two-dimensional engagement

We consider the following simulation conditions:N=3,R0=10 000 m,andVM=250 m/s.The initial position of the missile is (0,0),and the position of the stationary target is (10 000,0).Using the same desired impact timetd=50 s,the simulation results obtained by this proposed ITCG law and those presented in [2,7]with various initial heading errors (i.e.,?0=0?,90?) are shown in Fig.5 and Fig.6,including the variations in the impact time error and the acceleration.Note that for the case ?0=0?,the time of flight only driven by the PPNG istf=40 s,and for the case ?0=90?,tf=52.44 s.Moreover,the effect of the accuracy of time-to-go estimation on guidance performance is also validated by comparing the results from (8) and (49).The corresponding results are shown in Fig.7 and Fig.8.

Fig.5 Results obtained using various ITCG laws with heading error ?0=0°

Fig.6 Results obtained using various ITCG laws with heading error ?0=90°

Fig.7 Results obtained using various ITCG laws with (8) and td=70 s

Fig.8 Results obtained using various ITCG laws with (49) and td=70 s

We can see from Fig.5 and Fig.6 that for the case ?0=0°,all three ITCG laws perform well.In Fig.5(a),from the impact time error perspective,the ITCG law presented in [7]converges more quickly than the other two ITCG laws but has a larger maximum acceleration(i.e.,more than 60 m/s2),as shown in Fig.5(b).However,it is shown in Fig.6 that for the case where ?0=90?,the impact time control cannot be successfully achieved by using the ITCG law in [2](i.e.,the simulation stops att=37.77 s).The reason for this failure is that the ITCG law in [2]cannot adapt itself to the engagement that has the desired impact time shorter than the time of flight only driven by the PPNG,which makes the command become complex.In Fig.6(a),the impact time error driven by the proposed ITCG law converges faster than that of the ITCG law in [7].The initial acceleration induced by the proposed ITCG law is slightly larger than that of the guidance law in [7],as shown in Fig.6(b).In general,the presented ITCG law has comparable results to those presented in [7].Note that the proposed ITCG law can be implemented effectively regardless of whether the impact time error is greater than zero or not.

In Fig.7,(8) is applied to the ITCG laws in [2,7]and the presented ITCG law.We can see from the figure that the impact time control can be implemented successfully by using all these ITCG laws.In Fig.7(a),the ITCG law in [7]has the shortest convergence time compared to the other two ITCG laws.Nevertheless,the initial accelerations driven by the ITCG law in [7]are apparently saturated for the casetd=70 s.As shown in Fig.7(b),the initial accelerations driven by [7]exceed 98 m/s2,and the saturated state continues for approximately 5 s,whereas the maximum accelerations driven by the other two ITCG laws over the engagement are less than 40 m/s2.

In Fig.8,(49) is used instead of (8).For the casetd=70 s,the simulation of the ITCG law in [2]stops att=58.95 s,and the impact time error driven by the ITCG law in [7]cannot converge to zero (i.e.,the final impact time error is 0.67 s).In contrast,we can see from these figures that the proposed ITCG law performs well for both cases whether (8) or (49) is used.These results indicate that the proposed ITCG law,to a large extent,reduces its dependence on the accuracy of the time-to-go estimation.

To make a detailed comparison,the performances of these three ITCG laws under various conditions (e.g.,different time-to-go estimation formulas,different desired impact time) are listed in Table 1,including the extreme acceleration,the extreme heading error,the total control energy consumption,and the final impact time error.N/A means “note applicable”.

Table 1 Performance comparison under various conditions

The extreme accelerations,the extreme heading errors,and the final impact time errors given in Table 1 are in accordance with the results shown in Fig.7 and Fig.8.The control energy consumption driven by the ITCG law presented in [7]is always much larger than that of the other two guidance laws for different time-to-go formulas and different desired impact times,which is likely detrimental to the guidance process in practical engineering.In general,the proposed guidance law is comparable to the ITCG of [2]in controlling energy consumption.Note that for some cases (i.e.,when the desired impact time is greater than 57.90 s),the ITCG of [2]that uses (49) cannot work.As illustrated in [2],the acceleration command would become a complex when the accuracy of the timeto-go estimation is insufficient,leading to a failed engagement.We can see from the above results that through using the proposed ITCG law,the dependence on time-togo estimation is reduced and the required control energy is also smaller.

6.2 Cases for three-dimensional engagement

In this simulation,we consider the following conditions:N=3,VM=250 m/s,θM0=10?,ψM0=10?,and (xT,yT,zT)=(6 000,6 000,0).Note that θM0and ψM0are the initial values of θMand ψM,respectively,and (xT,yT,zT) denotes the position coordinate of the target.To evaluate the effect of the accuracy of time-to-go estimation on the engagement performance,equations (8) and (49) are incorporated into the proposed ITCG law.As shown in Fig.9,the results obtained by the proposed ITCG law with various time-to-go formulas are also compared with those obtained in [22]for the cases oftd=50 s.To validate the proposed ITCG law in a more extensive manner,the special condition oftd=70 s,θM0=0?,and ψM0=0?is simulated,as shown in Fig.10.

Fig.9 Simulation results of proposed three-dimensional ITCG law with td=50 s

Fig.10 Simulation results of proposed three-dimensional ITCG law with td=70s,θM0=0°,and ψM0=0°

We can see from Fig.9 that although the engagement can be completed by the proposed three-dimensional ITCG law using (8) and (49) and the ITCG law proposed in [22],the initial accelerations in pitch and yaw shown in[22]are extremely large,exceeding 98 m/s2.For this case,the state with saturated acceleration command lasts for approximately 4 s.It can be also seen from Fig.9(d) that the three-dimensional trajectory formed by using the ITCG law in [22]requires greater acceleration than the proposed ITCG law whether (8) and (49) are used or not.Moreover,the impact time error driven by the proposed ITCG law using (8) converges faster than that driven by the proposed guidance law using (49),as shown in Fig.9(a).In Fig.9(b),the acceleration in yaw driven by the proposed guidance law using (8) appears to be smaller than that using (49) over the engagement.On the other hand,the acceleration in pitch driven by the proposed ITCG law using (8) is comparable to that driven by the ITCG law using (49),as illustrated in Fig.9(c) .

Note that the simulation condition used for Fig.10 represents a special case in which the ITCG law presented in[22]fails to accomplish the impact time control,whereas the proposed three-dimensional ITCG law performs well.In Fig.10(a),the impact time error driven by [22]cannot converge.Although the missile can fly to the target (i.e.,only driven by PPNG),as illustrated in Fig.10(b),the time of flight over the trajectory is approximately 33.94 s,far from the desired impact timetd=70 s.There exists a singularity (i.e.,?M0=0) for the guidance law in [22],resulting in a zero command over the trajectory.On the contrary,the proposed ITCG law is shown to be advantageous under this particular circumstance,overcoming singularity.

7.Conclusions

This paper proposes a two-stage ITCG law based on a hybrid proportional navigation.The presented ITCG law can be easily understood and implemented due to its simple but effective structure.The acceleration command used in the first stage of guided trajectory follows either the RPNG or the PPNG,depending on the sign of the impact time error.As a result,the task of controlling the impact time is accomplished in the first stage,making the impact time error become zero,while the second stage follows a pure-proportional-navigation-based trajectory.By using Lyapunov theory,the convergence of this guidance law has been proved in a mathematical manner.It is demonstrated by two-and three-dimensional numerical simulations that the proposed ITCG at least has comparable performance in comparison with those in earlier studies.A number of simulation results indicate that the presented guidance law alleviates its dependence on the accuracy of time-to-go estimation,consumes relatively less control energy,and adapts itself to more boundary conditions and constraints.The ITCG law reported here provides a means toward deeper understanding of how to achieve the desired impact time in the framework of PN and is expected to be supplementary to the current research literature.