FU Shengnan,LIU Xiaodong,ZHANG Wenjie,and XIA Qunli,*

1.School of Mechatronical Engineering,Beijing Institute of Technology,Beijing 100081,China;2.Beijing Aerospace Automatic Control Institute,Beijing 100854,China;3.School of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China

Abstract:The traditional guidance law only guarantees the accuracy of attacking a target.However,the look angle and acceleration constraints are indispensable in applications.A new adaptive three-dimensional proportional navigation(PN)guidance law is proposed based on convex optimization.Decomposition of the three-dimensional space is carried out to establish threedimensional kinematic engagements.The constraints and the performance index are disposed by using the convex optimization method.PN guidance gains can be obtained by solving the optimization problem.This solution is more rapid and programmatic than the traditional method and provides a foundation for future online guidance methods,which is of great value for engineering applications.

Key words:proportional navigation(PN),adaptive guidance law,three-dimensional space,second-order cone programming(SOCP),convex optimal control.

1.Introduction

In recent years,convex optimization has been applied to aerospace problems,such as planetary landing[1–5],spacecraft rendezvous and proximity operations[6–8],and spacecraft trajectory planning[9–12].Second-order cone programming(SOCP)is the most commonly used method in convex optimization[13,14],and it can obtain a feasible solution without accurate initialization[15].If there is no feasible solution,then this method can conclude that the problem is unsolvable.

Research on the association of the guidance law and convex optimization,especially the proportional navigation(PN)guidance law[16–18],is attracting the attention of many researchers.The PN guidance law requires limited information[19,20]to attack a stationary target.However,for an expected impact path angle,the terminal constraint cannot be satisfied with a constant guidance gain.

Previous work includes two main classes of approaches to satisfy the impact angle constraint.(i)Based on the pure proportional navigation(PPN)guidance law,a biased pure proportional navigation(BPPN)guidance law[21]shapes the trajectory using a bias term.The term is integrated continuously before reaching a certain threshold and transforms the BPPN guidance law into the PPN guidance law.This law is an open-loop guidance law,with imprecision occurring if interrupted.Song and Zhang[22]designed a variable structure PN guidance law to realize the constraint of the terminal attack angle by estimating the distance between a missile and a target as well as the rate of the distance.(ii)Zhang et al.[23]designed a sliding mode controller that satisfies the constraint of the terminal attack angle by executing the sliding mode.The sliding mode function and its derivative converge to 0 near the sliding surface by backstepping.The controller can deal with the lag of high-order systems with good robustness.A new integrated guidance and control design scheme based on the variable structure control approach[24]is established to hit the target with both the desired impact angle and the desired flight attitude angle.Shashi et al.[25]proposed a guidance strategy that caters to both the impact angle and the impact time terminal constraints.

In addition to the impact path angle constraint,the look angle constraints should also be considered due to the seeker’s field-of-view constraint.Park et al.[26]proposed a multistage guidance law for stationary targets.The flight stage is divided into three stages:(i)the initial phase,where the look angle is controlled to reach the maximum;(ii)the middle phase,where the maximum look angle is maintained;and(iii)the terminal phase,where the impact angle is constrained to attack the target.Tekin and Erer[27]designed unbiased and biased compound proportional guidance laws using a switch strategy for the guidance gain to ensure different constraints;then,other modified PN guidance laws with a time-varying bias term[28,29]reduced the look angle monotonically to 0.

For engineering applications,acceleration is subject to maneuverability and physical structure. Liu et al.[30]proposed a time-varying PN guidance law based on convex optimization.The solution time is significantly shortened compared with that of the traditional optimization method.Huan et al.[31]improved the time-varying PN guidance law for cooperative guidance to realize the simultaneous attack of multiple missiles.

This paper presents a further study on the basis of[30].We develop a three-dimensional kinematic engagement model subject to multiple constraints based on convex optimization

The remainder of this paper is organized as follows.A three-dimensional kinematic engagement model and the equation of state are established in Section 2.All constraints are introduced,and the optimal convex control problem is established in Section 3.A successive process is formulated to solve the control problem in Section 4.The verification of the simulation model is presented,and the numerical results are discussed in Section 5.The most important conclusions are summarized in Section 6.

2.Problem formulation

2.1Three-dimensional decomposition

The relationship between a missileMand a stationary targetTin three-dimensional space is established in Fig.1.

Fig.1Standard three-dimensional relative motion relationship

ris the range between the missile and the target,vMis the missile’s velocity,θis the flight path angle,ψV mis the heading angle,λθis the line-of-sight(LOS)angle of the pitch channel,andλφis the LOS angle of the yaw channel.

Because a standard three-dimensional engagement cannot be convex,the three-dimensional relationship is decomposed into a vertical planeMVTand a lateral planeMLTin Fig.2 based on the assumption that the mutual coupling between the planes can be ignored. The subscriptsVandLindicate variables in planesMVTandMLT,respectively.

Fig.2Three-dimensional relative motion relationship

The velocity of the missilevMis decomposed intovVandvLfor the planes:

where

To more accurately model the two-dimensional plane,we design relationships between a missile and a stationary target for different planes in Fig.3 wherevis the missile’s velocity,ais the lateral acceleration perpendicular to the velocity,λis the LOS angle,andεis the look angle.

Fig.3Relative motion relationship for different planes

Angles between the two-dimensional planes and the three-dimensional space need to be correlated.PlaneMCDin Fig.4 corresponds to planeMVT,and planeMBCis designed as the horizontal plane.LineMBis the projection of the velocityvMon the horizontal plane,and lineMCis the projection ofMBon planeMCD.The relationship between the decomposed three-dimensional plane in Fig.2 and the standard three-dimensional space in Fig.4 is established by the projection.

Fig.4Projection relationship

Based on Fig.4,numerical relations can be derived:∠AMB=θ,∠BMC=ψV m?λφ,∠ADM=90?,∠ABM=90?,∠BCM=90?,∠DCM=90?.

We can obtainAB⊥MBCandDC⊥MBCfrom the above relationships.Then,we haveAB//DC.AD//BCis derived using the same method.In other words,the parallelogramABCDyieldsAB=DC.The angle relationship is given as follows:

When considering only the guidance of the vertical planeMVT,the assumptionψV m≈λφis made.Similarly,for the lateral planeMLT,we supposeθ≈λθ.

The angle relationships between the decomposed threedimensional space and the three-dimensional space can be obtained:

2.2 Two-dimensional engagement kinematics with PN

Once the three-dimensional space has been decomposed,the guidance law can be designed separately for the planes to realize three-dimensional guidance that ignores the mutual coupling between planes.For example,for planeMVT,two-dimensional engagement kinematics with the PN guidance law can be described as follows:

whereNVis the guidance coefficient.

The engagement kinematics for planeMLTcan be derived using the same approach.To facilitate subsequent linearization,the independent variable of the twodimensional kinematics is changed fromttor,namely,(7)and(8)are divided by(6),which produces the following:

whereσV=tanεVto realize the linearization and the independent variableris hidden for simplicity.The lateral acceleration command is given as follows:

The following assumptions are made:

(i)The dynamics of the control system can be neglected,andaV=aV c,whereaV cdenotes the acceleration command andaVdenotes the actual acceleration.

(ii)The autopilot missile is significantly faster than the guidance loop.

2.3 Three-dimensional equation of states and linearization

The engagement kinematics of planeMVTand planeMLTare merged,and the result is rewritten as an equation of the state:

wherex=(λVσVλLσL)Tis the state vector,u=(NVNL)Tis the control vector,and

In the three-dimensional equation of the state,control vectors are decoupled from state vectors to achieve the convergence of optimization using sequential programming methods.For application to the convex optimization method,the equation of the state needs to be partially linearized.Let{x(k),u(k)}be thekth successive solution.f(x)is linearized atx(k),andB(x)is approximated byB(x(k)),thus yielding

To validate the linearization,a trust-region constraint is chosen as follows:

whereδ∈R2is a constant vector.This constraint needs to be relaxed to increase the efficiency during optimization.

3.Optimal control problem

3.1Objective function

For optimization problems,appropriate optimization objectives are necessary to realize different performance indexes.Common optimization objectives include the maximum range,the minimum range,and the minimum time.The objective in this paper is to minimize the control energy,which corresponds to the following objective function:

wherer0is the initial range between the missile and the target,rfis the terminal range between the missile and the target,εV=arctanσVandεL=arctanσL.

3.2Constraints

Once a scenario is designed,the mission has requirements in terms of initial states and terminal states.Considering the physical constraints,the guidance law is subject to the field of view and the acceleration.To achieve convexity,these constraints are denoted as a linear form or a secondorder cone form.

(i)Initial constraints:

(ii)Look angle constraints:

whereεVmax?π/2,εLmax?π/2.

(iii)Acceleration constraint:

(iv)Terminal constraints:

To satisfy all these constraints,the objective function is transformed into a discrete form:

Thus,the optimal control problem can be expressed as follows:

3.3Discretization

The linearized dynamic equation(13)has been discretized to realize convexity.The independent variableris divided intoNsegments{r0,r1,r2,...,rN},and theith discretized point is obtained byri=r0+iΔr(i=1,2,...,N),where Δr=(rf?r0)/N.The state vectorxand the control vectoruare given byxi=x(ri)andui=u(ri).The state equation can be numerically integrated by the Euler rule:

where=A(xk(ri)),=B(xk(ri))and=b(xk(ri)).By rearranging and combining the same terms,(21)can be rewritten as follows:

Let a new vector bez=(x0,x1,...,xN,u0,u1,...,uN)T;thus,the linearized dynamic equation(13)is transformed into an equality constraint:

After linearization,convexity and discretization,all constraints and the performance index are transformed into a linear or SOCP problem.As long as the discrete step is small enough,the optimization Problem O is sufficiently close to Problem A:

4.Convex problem solution

The successive steps for solving Problem O are designed as follows.

Step 1Letk=0.Initialize state vectorx(0)and control vectoru(0)according to the scenario.Vectorsx(0)andu(0)do not have to satisfy the look angle constraint and the acceleration constraint because they are used only to generate initial values ofA(x(0)),B(x(0))andc(x(0)).

Step 2In the(k+1)th iteration,SOCP Problem O is established by vectors{x(k),u(k)}.Then,Problem O is solved to obtain the solution{x(k+1),u(k+1)}.

Step 3Check whether the tight convergence constraint is met:

Equation(24)consists of a series of component-wise inequalities,andξ∈R4is a constant vector different from the trust-region constraint.If this constraint is satisfied,then proceed to Step 4;otherwise,letk=k+1 and return to Step 2.

Step 4The iteration needs to be stopped.A converged solution{x(k+1),u(k+1)}is obtained.

The control energy is minimized by the above steps. The conventional PN guidance law cannot satisfy the acceleration constraint,whereas the convex optimal method can deal with multiconstraint problems.If the control vectorui+1cannot be obtained in the(i+1)th discretized point,it is set to be replaced byuito ensure the stability and reliability of guidance.

where

5.Numerical examples

A ground-to-ground scenario is designed to demonstrate the validity of the solution proposed in this paper.Suppose a missile has a constant velocity of 200 m/s and a range to go of 4 000 m.The constraints are set as:εVmax=50?,εLmax=50?andamax=50 m/s2.The trust-region constraint is set as:

The convergence constraint is set as follows:

The optimization problem is modeled by the software MOSEK[32].Simulations are performed on a computer configured with CPUi7-4790@3.6 GHz and 8 GB of memory.

5.1Two-dimensional guidance

First,two-dimensional simulation is performed to verify the effectiveness of the method in this paper.Compared with the traditional two-dimensional model,the model here is still the three-dimensional model,although the initial and terminal states in planeMLTare set as zero.is designed as the expected terminal flight path angle.rtgois designed as the range to go.The initial and terminal parameters are stated in Table 1.

Table 1Two-dimensional flight parameters

Because a missile attacks a target with different impact angles,the angular relations in Fig.5 of those terminal flight path angle constraints with respect to one another satisfy the look angle constraint and the initial and terminal conditions.The results show that different terminal LOS angle constraints greatly affect the guidance gain pro files and the corresponding acceleration pro files.The smaller the expected terminal LOS angle,the higher the absolute values of the initial guidance gain and the acceleration.A higher altitude of the trajectory corresponds to a smaller expected terminal LOS angle.When the impact angle is given,the proposed method converges in approximately 3.2 s to obtain the solution to Problem O.

Fig.5Two-dimensional pro files with different impact angles in MVT

Angles,acceleration and lateral displacement in planeMLTremain at approximately zero during the flight in Fig.6.Empirically,the valuesNV?20 andNL?20 are implemented to avoid divergence and the guidance gainNLis maximized in the range from 500 m to 3 000 m.The reason for this phenomenon is detailed in the next section for the same condition under three-dimensional guidance.

Fig.6Two-dimensional pro files with different impact angles in MLT

5.2Three-dimensional guidance with a constant velocity

Three-dimensional simulations are performed in this section.The initial or terminal parameters in planesMVTandMLTare designed to be the same as those in Table 2.

Table 2Three-dimensional flight parameters

All constraints can be satisfied for both planeMVTand planeMLTin Fig.7.Parameters in planeMLTchange significantly because the initial flight path angle and the look angle in planeMLTare set to 15?,which is a different value from that of the two-dimensional simulation.When the impact angle is given,the proposed method converges in approximately 3.2 s to obtain the solution to Problem O.

Fig.7Three-dimensional angle pro files with a constant velocity

The guidance gainNLreaches the maximum near 2 500 m,which is similar to the two-dimensional simulation.We find that the look angle in theMLTplane is close to 0 in Fig.7.Thus,the performance index for planeMLTin the objective function becomes out of control.The most likely outcome is a divergent control vector.Therefore,empirically limiting the guidance law is necessary.The terminal flight path angle constraints for planeMVTaffect the parameters of planeMLTin Fig.8 according to the three-dimensional engagement kinematics in(1)and(2).

Fig.8 Three-dimensional guidance pro files with a constant velocity

Thus, a guidance law ignoring mutual coupling between planes is feasible.Fig.8 shows that all constraints are satisfied during the flight.

Pro files for the altitude and lateral displacement with different impact angles and a more indicative view of the trajectory in three dimensions are shown in Fig.9.

5.3Three-dimensional guidance with a time-varying velocity

For a missile with a time-varying velocity,the velocity can be approximated by the last guidance circle. The increment of the velocity is given by the following:

whereD=0.5ρV2SrefCDis the aerodynamic drag,mis the mass,andgis the gravitational acceleration.We setρ=1.225e?0.001h(0.108?0.4×10?5h),Sref=0.1 m2,CD=0.02,m=40 kg,andg=9.81 m/s2,wherehis the altitude.The velocityVused to calculate the aerodynamic drag is obtained by the last guidance circle,and the velocity in the new guidance circle is given byV+ΔV.

Other conditions are the same as those in Section 5.2.All constraints of the terminal angles can be satisfied for both planeMVTand planeMLTin Fig.10.

Fig.10Three-dimensional angle pro files with a time-varying velocity

When the impact angle is given,the proposed method converges in approximately 3.2 s to obtain the solution to Problem O.

For a missile with a time-varying velocity,the guidance gain and acceleration in planeMVTin Fig.11 are significantly different from those of a missile with a constant velocity in Fig.8.The guidance gain and acceleration decrease,especially in the initial phase.The total acceleration satisfies the constraint,and the velocity pro files are very close for different impact angles.

Pro files for the altitude and lateral displacement with different impact angles and a more indicative view of the trajectory in three dimensions with a time-varying velocity are shown in Fig.12.

Fig.11Three-dimensional guidance pro files with atime-varying velocity

Fig.12Three-dimensional position pro files with a time-varying velocity

6.Conclusions

In this paper,a reliable and adaptive three-dimensional guidance law subject to common initial,look angle,acceleration,and terminal constraints is developed by using convex optimization.For three-dimensional engagement kinematics with PN,the three-dimensional space is divided into two planes and the coupling between planes is ignored.The simplified model can transform the secondorder constraint into an SOCP problem for convexity.Through linearization,convexity and discretization,all constraints and the performance index are transformed into a linear or SOCP problem,and the convex optimal control problem can be obtained by the continuous solution method.The simulation results show that the adaptive three-dimensional guidance law is suitable for twodimensional and three-dimensional engagement kinematics with a constant or a time-varying velocity.This method can satisfy all the constraints,and it could possibly be applied for online guidance in future research.

Compared with the traditional guidance law, the requirement of the acceleration constraint can be realized using the proposed method.However,the method has some shortcomings in terms of accuracy. Future studies could try to find a more accurate solution to three-dimensional modeling.