FU Shengnan ,ZHOU Guanqun ,and XIA Qunli

1.School of Mechatronical Engineering,Beijing Institute of Technology,Beijing 100081,China;2.Beijing Aerospace Technology Institute,Beijing 100074,China;3.School of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China

Abstract: In this paper,a trajectory shaping guidance law,which considers constraints of field-of-view (FOV) angle,impact angle,and terminal lateral acceleration,is proposed for a constant speed missile against a stationary target.First,to decouple constraints of the FOV angle and the terminal lateral acceleration,the third-order polynomial with respect to the line-ofsight (LOS) angle is introduced.Based on an analysis of the relationship between the looking angle and the guidance coefficient,the boundary of the coefficient that satisfies the FOV constraint is obtained.The terminal guidance law coefficient is used to guarantee the convergence of the terminal conditions.Furthermore,the proposed law can be implemented under bearingsonly information,as the guidance command does not involve the relative range and the LOS angle rate.Finally,numerical simulations are performed based on a kinematic vehicle model to verify the effectiveness of the guidance law.Overall,the work offers an easily implementable guidance law with closed-form guidance gains,which is suitable for engineering applications.

Keywords:shaping guidance law,field-of-view (FOV) constraint,impact angle constraint,terminal lateral acceleration constraint.

1.Introduction

For the terminal guidance of a vehicle,in many cases,the guidance law is required to intercept the target precisely with the desired impact angle to maximize the warhead effectiveness [1?3].To realize a large impact angle,the guidance law may cause the trajectory to be highly curved,and increase the looking angle.If the looking angle exceeds the field-of-view (FOV) angle constraint,it may result in a mission failure [4?6].Thus,when designing an impact angle control,it is very important to consider the seeker’s FOV angle constraint.Besides,it is also necessary to ensure the convergence of the terminal lateral acceleration,so that the trajectory can be easily realized in practice.

The impact angle constraint is imposed on the terminal flight path angle to improve warhead effectiveness and lethality.For example,the penetration weapon needs to attack the target with a nearly vertical impact angle.Kim [7]studied the impact angle constraint firstly.They transformed a linearized engagement geometry problem into a linear quadratic optimization problem and introduced a suboptimal guidance law for reentry vehicles to intercept a non-maneuverable target.Ryoo [8]proposed practical and precise time-to-go calculation methods by approximating the missile trajectory as a third-order polynomial of time-to-go.The designed guidance law is the resultant command which is a linear combination of a step and a ramp acceleration response of the missile.An extension of that work was presented in [9]with a polynomial of order and the engagement geometry problem was transformed into a linear-quadratic optimal control problem,where the energy cost is weighted by a power of time-togo.Kim et al.[10]derived a backstep guidance law which required the information of the line-of-sight (LOS)angle,impact angle,and estimated time-to-go.Kumar et al.[11]proposed a sliding-mode-control-based guidance law to intercept stationary,constant-velocity,and maneuvering targets at the desired impact angle.Oza et al.[12]proposed a suboptimal guidance law through the model predictive static programming technique.The guidance law is three-dimensional which can satisfy terminal impact angle constraints in both azimuth and elevation directions.In these studies,the estimation of time-togo is important,which requires the implementation of precise time-to-go calculation methods.

Proportional navigation (PN) [13]and its variants are widely used because of its simple structure and strong robustness.A three-dimensional adaptive PN is used by Lu et al.[14]to satisfy the impact angle constraint,which is applied to design the terminal guidance for hypersonic vehicles.Ratnoo et al.[15]proposed a two-stage guidance law based on the PN guidance to intercept a stationary target with the desired impact angle.In the first stage a lower navigation gain is used to achieve the desired direction to switch to the second stage,and in the second stage a higher navigation gain is used to achieve the terminal constraint.An extension of that work was presented in [16]for non-maneuvering target interception with the desired impact angle.Erer et al.[17]proposed a guidance law with a bias term to the pure PN guidance command.Therein,a two-stage guidance scheme was designed to control the impact angle.Zhang et al.[18]designed a biased PN law for the large impact angle control problem,which satisfies constraints of impact time and impact angle.Zhang et al.[19]proposed a biased PN guidance law with impact time and impact angle constraints.To eliminate the time-to-go error,a feedback control with adjustable coefficients is introduced to control the impact time and impact angle simultaneously.In these studies,the impact angle constraint and other constraints are considered based on the PN guidance law.However,the guarantees on the FOV constraint are not discussed.

Considering the engineering application,the FOV constraint of the seeker also needs to be introduced.Kim et al.[20]presented the PN guidance law with switch supplementary time-varying bias terms considering the impact angle constraint and FOV angle constraint.Erer et al.[21]proposed the biased pure PN and the unbiased PN guidance laws,and designed the gain guidance schemes.Furthermore,the lateral acceleration constraint was introduced [22].For three typical cases,different guidance laws were designed.Park et al.[23]proposed an energy optimal guidance law.A three-stage guidance scheme was designed to realize different constraints in different stages.A range-to-go weighted performance index was introduced [24]using a similar guidance law.To achieve all impact angles with the FOV angle constraint,a two-stage PN guidance law was presented by Ratnoo [25].The impact angle for the FOV angle constraint can be obtained in an analytical form.The above multi-constraint guidance laws were designed for different stages.However,the switching of different guidance laws may cause problems in engineering applications.

The adaptive multi-constraint guidance law becomes very popular in recent years.Jeon et al.[26]proposed an impact-time-control guidance law.The closed-form timevarying navigation gain can be obtained from the missileto-target range and angle information.Based on the sliding mode control method,Kim et al.[24]proposed a new guidance law.Liu et al.[27]proposed a multi-constraint guidance law for stationary and non-maneuvering moving targets.The core idea is to transform into the limit of the relative speed perpendicular to the LOS angle between the missile and the target.Sharma et al.[28]proposed a two-gain feedback guidance law based on the bearings-only information.A feasible implementable region is deduced in impact angle-maximum looking angle design space.The guidance law has a simple form,and the guidance gain is easy to calculate.However,in some extreme cases,the terminal lateral acceleration is too large,which is difficult to realize in practice.

In this study,we propose an impact angle control guidance law that restricts the looking angle within the FOV angle constraint.The main contribution of the paper is that the proposed method introduces a third-order polynomial with respect to the LOS angle,and satisfies the convergence requirement of the terminal lateral acceleration,FOV angle constraint,and impact angle constraint.The guidance gain can be adjusted adaptively depending on the desired looking angle.Based on the analysis of the characteristics of the third-order polynomial,we transform the FOV angle constraint into the range of the coefficient for the looking angle,and give the analytical boundary of the looking angle to satisfy the FOV constraint.By the design of the coefficient for the looking angle and the terminal guidance law coefficient respectively,the proposed guidance law can satisfy multiple constraints simultaneously.Compared with the traditional optimal guidance law [29],the proposed guidance law does not need the information on the time-to-go,and the FOV angle constraint is added and satisfied.

The paper is organized as follows.In Section 2,the problem statement is given.In Section 3,the third-order polynomial shaping guidance law is designed.The characteristics of the looking angle are analyzed.In Section 4,the feasible guidance gains are obtained according to the boundary of the looking angle.In Section 5,the simulation results are presented to validate the performance of the proposed guidance laws.Finally,the conclusions are given in Section 6.

2.Problem statement

Considering a two-dimensional surface-to-surface engagement scenario as shown in Fig.1,whereMrepresents the center of mass of the missile andTrepresents the center of mass of the target.Assuming that the angle of attack is small,the lead angle is approximately equal to the looking angle.That is,the seeker’s FOV angle can be controlled by adjusting the looking angle.For a stationary target,assuming that the missile velocity is constant,nonlinear engagement kinematics in a polar coordinate system can be expressed as

Fig.1 Engagement scenario

whereRis the missile-target relative range,Vis the velocity of the missile,ais the lateral acceleration, γ is the flight path angle, λ is the LOS angle, σ is the looking angle,which is defined as the angle between the LOS and the velocity vector,andtrepresents the time as an independent variable.

For a surface-to-surface engagement scenario,the initial LOS angle is zero,and other initial conditions are given by

where the subscript 0 represents the initial states.

Similarly,considering the constraint of the desired impact angle γf,the terminal conditions are given by

where the subscriptfrepresents the terminal states.

Moreover,the terminal lateral acceleration needs to realize the convergence for engineering applications,given by

Assume that the maximum seeker’s FOV angle is denoted as σmax.the looking angle during the whole flight should be limited by

Equations (1)?(5) describe the terminal guidance problem in a two-dimensional plane considering the impact angle constraint and the FOV angle constraint of the seeker.For most terminal guidance problems with a stationary target,the looking angle is usually within the range (?π/2,π/2).

3.Guidance design

Since the missile-target relative rangeRis monotonically decreasing,some shaping guidance laws are designed as polynomial functions with independent variablesRort:

wherekN,kN?1,···,k0are polynomial coefficients,decided by initial conditions,terminal conditions,and other constraints.Using (6),the deduced guidance law often includes time-to-gotgo，which needs precise calculation methods in consideration of a curved trajectory.In order to reduce the complexity of the guidance law,the independent variable λ is introduced.In a surface-to-surface engagement scenario,we can notice that the LOS angle λ will monotonously change.Thus,the one-to-one mapping between the state and the missile-target relative rangeRcan be transformed into the mapping between the state and the LOS angle λ.Using the form of (6),we design a polynomial function as

Differentiating the looking angle σ with respect to the flight time,can be transferred intoThus,we can get the shaping guidance law:

whereMλis an adaptive guidance law coefficient.

Equation (8) is similar to the PN guidance law.If the terminal adaptive guidance law coefficient is larger than 2,the guidance law can realize the convergence of the terminal lateral acceleration.

The second-order polynomial shaping guidance law is derived from [28].In some cases,the terminal lateral acceleration cannot converge to zero.Based on the analysis of the second-order polynomial shaping guidance law,we can notice that the maximum looking angle is related to the terminal adaptive guidance law coefficient.We may not be able to limit the maximum looking angle within the FOV angle constraint and the terminal adaptive guidance law coefficient to greater than 2 simultaneously.That is,there may be contradictions that the FOV angle constraint and the terminal lateral acceleration constraint cannot be satisfied at the same time.

To solve this problem,a third-order polynomial shaping guidance law is designed in this section.Using adjustable coefficientsc1andMf,we can control the maximum looking angle and the terminal adaptive guidance law coefficient individually,to realize the FOV angle and terminal lateral acceleration constraints separately.

3.1 Guidance law

IfN=3,(7) can be rewritten as

wherek0,k1,k2andk3are polynomial coefficients.

For convenience,the terminal guidance law coefficientis recorded asMf.That is,the terminal condition needs to be satisfied with

whereMf?1 determines the terminal looking angle rate,which can be designed as a constant to satisfy the terminal constraint.

Based on the constraint in (10),we design the following form of the looking angle rate as

wherec1(c1≠0) andc2are coefficients for the looking angle,which are designed to satisfy the FOV angle constraint of the seeker.

The integral of (11) is obtained as

wherec3is another coefficient for the looking angle.

Using (2) and (3),a relationship between coefficients for the looking angle can be deduced by rearranging(12) as

Using (8) and (14),the closed-loop guidance law can be obtained as

To analyze the FOV angle,(15) can be divided into two parts as follows:

We can notice that σc(λ) is related to the coefficientc1and σMis related to the terminal guidance law coefficient.The next section will analyze the characteristics of σc(λ) and σMseparately corresponding to the path characteristics and the terminal characteristics.

3.2 Path characteristics

To keep the analysis simple and understandable,nondimensionalization should be applied.Dimensionless states are defined asandUsing (17),we can get

Differentiating (18) with respect to the LOS angleleads to

Using (18) and (20),the maximum looking angle can be deduced as

Fig.2 shows the variation offor differentwith respect to,and the maximum values are marked with points.The coefficientc1for the looking angle affects the maximum looking angleand the LOS angleis constant as ?1/3.

Fig.2 t for different with respect to

3.3 Terminal characteristics

The dimensionless state is defined as.Using (17),we can get

Differentiating (22) with respect to the LOS angleleads to

Using (22) and (24),the maximum looking anglecorresponding to the terminal characteristics can be deduced as

Fig.3 εM for different σ0/λf with respect to

3.4 Dimensionless looking angle

From Subsection 3.2 and Subsection 3.3,we can notice that the main function of the termis to control the looking maximum angle without affecting the initial and terminal conditions.The main function of the termis to satisfy initial and terminal conditions.Once the terminal guidance law coefficientMfis determined,the maximum looking angle should only be controlled by.

Using (18) and (22),we can get the dimensionless looking angle:

Differentiating (26) with respect to the LOS angleleads to

Using (27),we can get another form of the dimensionless looking angle as

To get the maximum dimensionless looking angle,we suppose that in∈(?1,0),we can get?Using (28),we can get the maximum looking angle deduced as

4.Feasible guidance gains

4.1 The lower boundary

PropositionFor∈[?1,0],there must be afor the lower boundary.

ProofUsing (29) and (32),we can get the maximum looking angle for

Equation (33) can be transformed into the following form:

Fig.4 f() for different σ0/λf with respect to

4.2 The upper boundary

For a surface-to-surface engagement scenario,needs to keep the symbol unchanged during the whole trajectory.Considering an extreme case,there are both maximum looking angle σmaxand minimum looking angle=0 during a trajectory.The process to obtain the minimum looking angleand maximum valueare similar.Thus,using (34),we can get

Using (35),the dimensionless LOS angle at the minimum dimensionless looking anglecan be derived as

Denote the upper boundary of.Using (33)and (36),the upper boundarycan be derived as

In conclusion,the range of the dimensionless coefficientfor the looking angle satisfying the FOV angle constraint of the seeker can be concluded as

Convert (38) to a dimensional form byand,shown as

To verify the effectiveness of the designed boundary,we design a typical case withand σ0/λf=?0.4.Fig.5 shows that if ω1=φ1L,the looking angle reaches the maximum value,and ifc1=c1U,the looking angle reaches the maximum value and the minimum value.Thus,if the coefficient satisfiesc1L≤c1≤c1U,the FOV angle constraint,initial and terminal conditions can be satisfied.

Fig.5 σ for different c1 with respect to

5.Simulation results

The third-order shaping guidance law of (16) is used for simulation.A typical case is designed as the velocityV=1 000 m/s,the initial missile-target relative rangeR0=50 km,the initial LOS angle λ0=0?,and the initial looking angle σ0=15?.The desired impact angle is γf=?80?.All simulations are terminated for a miss distanceR<1×10?3m.To determine the terminal guidance law coefficientMf,we need to study the influence ofMffirst.

5.1 Influence of the terminal guidance law coefficient

We can notice that the designed shaping guidance law is similar to the PN guidance law.Thus,the terminal guidance law coefficient is designed asMf≥2 to realize the convergence of the terminal lateral acceleration.To analyze the influence of the terminal guidance law coefficient,we design some typical cases.The FOV angle constraint is σmax=30?.Different terminal guidance law coefficients are verified asMf=2,2.5,3,3.5.The coefficient ω1for the looking angle is designed by the lower boundary to realize the maximum value.Using (24),we can get the LOS angleat the maximum looking angle.Using (39),we can get coefficientsc1for different terminal guidance law coefficients as

The simulation results are shown in Fig.6?Fig.11.The missile trajectories are plotted in Fig.6.The successful interceptions with the desired impact angle are shown in Fig.7 and Fig.8.Fig.9 shows the variation in the looking angle with respect to time.A higher value of the terminal guidance law coefficientMfcorresponds to a lower maximum height of the trajectory and a lower time to adjust terminal angles.That is,the time that the looking angle reaches the FOV angle constraint is later.

Fig.6 Missile trajectories

Fig.7 Flight path angle profiles

Fig.8 LOS angle profiles

Fig.9 Looking angle profiles

The variations in the lateral acceleration and adaptive guidance law coefficients are shown in Fig.10 and Fig.11.If the terminal guidance law coefficientMfis designed too large,the maximum lateral acceleration will be a large value.Conversely,ifMfis designed too small,the terminal lateral acceleration may not achieve the convergence.

Fig.10 Lateral acceleration profiles

Fig.11 Adaptive guidance law coefficient profiles

Considering both the terminal convergence requirement and the maximum lateral acceleration constraint,we design the terminal guidance law coefficient asMf=3 in this paper.If the dynamic lag and disturbance factors will be considered in future studies,the terminal guidance law coefficient should be increased properly.

5.2 Comparison studies

The optimal guidance law(OGL)[29]is the most widely used guidance law in engineering for the impact angle constraint.The form of OGL is shown as follows:

wherenis the guidance law coefficient,tgois the time-togo,decided byR/V.To realize the convergence of the terminal lateral acceleration,the guidance law coefficient needs to satisfyn>0.In this paper,we design the guidance law coefficient for OGL asn=0.5.

The form of the second-order polynomial guidance law in [28]is shown as follows:

wherek2is the guidance law coefficient.The derivation process for the range of the guidance law coefficient is similar to that of the third-order guidance law,which will not be discussed in this paper.And the guidance law coefficientk2is set to be ?0.581 9.

In this section,the simulation is carried out for shaping guidance laws (N=2,N=3) and OGL to verify the effectiveness of the proposed guidance law.To compare the performance of different guidance laws,we design a typical case.The terminal guidance law coefficient isMf=3.

Since OGL cannot satisfy the FOV angle constraint,it is necessary to ensure that the maximum value of the looking angle to be consistent for different guidance laws.According to the pre-verified simulation for OGL,the maximum looking angle is about 65.04°.Thus,for shaping guidance laws,the FOV angle constraint is set to σmax=65.04?.

ForN=3,the coefficient for the looking anglec1can be decided by the proposed boundaries.Using (32) and(37),we can get boundaries asc1L=?2.837 3 andc1U=5.250 8.ForN=2,we design an extreme case to prove that the second-order shaping guidance law may not be able to realize the convergence of the terminal lateral acceleration.

Fig.12?Fig.18 show the simulation results of the comparison between the second-order shaping guidance law,the third-order shaping guidance law,and OGL.In Fig.12,the trajectories forN=3(c1=?2.837 3) and OGL correspond to the maximum looking angle,and trajectories forN=2 andN=3(c1=5.250 8) correspond to the upper boundary of guidance coefficients to realize the minimum looking angle.Fig.13 and Fig.14 show that the flight path angle and LOS angle change rapidly at the end for the second-order and third-order shaping guidance laws.Fig.15 shows that for the lower boundary of the third-order shaping guidance law,the looking angle can reach the FOV angle constraint.For the upper boundary,the looking angle also changes rapidly at the end.

Fig.12 Missile trajectories

Fig.13 Flight path angle profiles with respect to time

Fig.14 LOS angle profiles with respect to time

Fig.15 Looking angle profiles with respect to time

Since the independent variable of the designed polynomial guidance law is the LOS angle,variations have been redrawn with respect to the LOS angle.Fig.16 shows that for the third-order shaping guidance law,the looking angle profile for the case ofN=3(c1=5.250 8) reaches the maximum and minimum values which are designed previously.The control range of the looking angle for the third-order shaping guidance law is obviously larger than that of the second-order shaping guidance law.Fig.17 shows that in the extreme case,the second-order shaping guidance law may not realize the convergence of the terminal lateral acceleration.As shown in Fig.17 and Fig.18,the third-order shaping guidance law can realize the convergence of the terminal lateral acceleration within the boundary.However,the maximum lateral acceleration at the end is too large,which needs to be limited in the engineering application.

Fig.16 Looking angle profiles with respect to LOS angle

Fig.17 Lateral acceleration profiles with respect to LOS angle 1

Fig.18 Lateral acceleration profiles with respect to LOS angle 2

Compared with the traditional OGL,the FOV angle constraint is introduced for the third-order shaping guidance law.In addition,the traditional OGL needs the missile-target relative range to obtain the time-to-go.The shaping guidance law does not need the distance information of the missile and the target.Compared with the second-order shaping guidance law,the proposed guidance law can realize the convergence of the terminal lateral acceleration within the guidance coefficient boundary.

5.3 Realistic simulations

In this section,simulations are carried out with a realistic missile model.Assuming flat,non-rotating Earth,equations of motion can be expressed as

wherexis the downrange of the missile;yis the altitude;Tis the thrust;Dis the drag;mis the mass of the missile;gis the gravity acceleration.These parameters are borrowed from [30].To compensate the gravity acceleration,the guidance law in (16) is augmented as

This case considers different impact angles γf=[?60?,?70?,?80?,?90?]with FOV angle constraints as σmax=[30?,45?,60?,75?].The terminal guidance law coefficient is designed asMf=3.Besides,we design the coefficientc1as the lower boundary to get the maximum looking angle.

Fig.19 shows that the missile intercepts the target with different desired impact angles and FOV angle constraints.Fig.20 shows that the desired impact angles can be satisfied by the proposed guidance law.Looking angles reach the maximum FOV angle and converge to zero at the end as shown in Fig.21.The lateral acceleration profiles are shown in Fig.22.Terminal lateral acceleration is convergent,but do not converge to zero due to the compensation of the gravity acceleration in (43).The missile speed profiles decided by aerodynamics are shown in Fig.23.

Fig.19 Missile trajectories with respect to time

Fig.20 Flight path angle profiles with respect to time

Fig.21 Looking angle profiles with respect to time

Fig.22 Lateral acceleration profiles with respect to time

Fig.23 Speed profiles with respect to time

6.Conclusions

To solve the problem that the second-order polynomial shaping guidance law may not satisfy the convergence requirement of the terminal lateral acceleration,we propose a third-order polynomial shaping guidance law.Besides,we analyze the characteristics of the looking angle,which provides adjustment references for the guidance coefficients.

We transform the range of looking angles into the range of the guidance coefficients,which can satisfy the FOV angle and terminal lateral acceleration constraints simultaneously.

The simulation results show that the third-order polynomial shaping guidance law can satisfy the convergence of the terminal lateral acceleration and the FOV angle constraint,compared with the traditional OGL and the second-order shaping guidance law.Since the proposed guidance law does not need the missile-target relative range,it has a wider application prospect in practice.