Fixed-time adaptive model reference sliding mode control for an air-to-ground missile

Liang ZHANG,Changzhu WEI,Rong WU,Naigang CUI

School of Astronautics,Harbin Institute of Technology,Harbin 150001,China

KEYWORDS Adaptive controller;Air-to-ground missile;Fixed-time adaptive reaching law;Fixed-time disturbance observer;Model reference sliding mode control

Abstract This paper addresses the fixed-time adaptive model reference sliding mode control for an air-to-ground missile associated with large speed ranges,mismatched disturbances and un-modeled dynamics.Firstly,a sliding mode surface is developed by the tracking error of the state equation and the model reference state equation with respect to the air-to-ground missile.More specifically,a novel fixed-time adaptive reaching law is presented.Subsequently,the mismatched disturbances and the un-modeled dynamics are treated as the model errors of the state equation.These model errors are estimated by means of a fixed-time disturbance observer,and they are also utilized to compensate the proposed controller.Therefore,the fixed-time controller is obtained by an adaptive reaching law and a fixed-time disturbance observer.Closed-loop stability of the proposed controller is established.Finally,simulation results including Monte Carlo simulations,nonlinear six-Degree-Of-Freedom(6-DOF)simulations and different ranges are presented to demonstrate the efficacy of the proposed control scheme.

1.Introduction

The autopilot design for modern missiles is one of the most challenging problems for control system engineers in the presence of highly complex nonlinear dynamics,external disturbances, un-modeled dynamics and even actuator failures.These difficulties have attracted considerable interest in the existing references,and further posed considerable complexity and diversity in the control system design.1Although classical Linear Quadratic Regulators(LQR)have already been developed based on the dynamical models linearized around the fixed flight points,most of the existing approaches may result in rate gyro saturation for sudden pitch rate demands.2Moreover,gain-scheduling techniques,as one of the most popular control laws,have been successfully utilized for the design and implementation of various systems.3,4Meanwhile, a gain-scheduling controller guarantees good performance with the control gains scheduled by Mach number,altitude or weight to cover the entire flight envelope.Besides,the design process is simple and the designed controller has acceptable performance when the flight conditions satisfy the assumptions that the parameter variations between the operating points are sufficiently slow.5However, the gain-scheduling controller requires various operating points to cover the entire flight envelope,especially for the rapid transitions of the air-toground missile assisted by a booster rocket.To improve the performance and reliability of the missile autopilot,the Linear Parameter Varying(LPV)control law is developed,and LPVbased gain-scheduling techniques are also presented to pursue high performance.6,7Unfortunately,the construction of a Lyapunov function for the LPV is difficult,and some of the implementations to the LPV control need infinite-dimensional Linear Matrix Inequalities (LMIs), which are difficult to solve.8

With the development of the modern control theory,multiple new control laws are proposed to ensure an accurate and rapid response to various guidance commands in the presence of large disturbances and uncertainties.A robust H∞controller is designed to utilize H∞loop shaping for an agile missile under the conditions of high angles of attack,aerodynamic variations and rapidly changing dynamics.The H∞controller has a simple structure and requires no complicated gain scheduling process for many flight conditions.Moreover,the H∞controller provides satisfactory performance over the entire flight envelope.9However,the proposed controller needs two H∞controllers to cover the high-speed and low-speed regions of the flight envelope which complicates the control system.Meanwhile,various feedback control laws are proposed to maintain stability and improve performance,which achieve additional control objectives such as disturbance decoupling,attenuation or global output regulation,etc.10,11A feedback control law that attains a prespecified convergence time for a given system under any initial condition is presented.12However,one of the shortcomings of the feedback control law is that it depends on the high precision of the dynamics model.To solve the problems of aerodynamic coefficient variations,large external disturbances and control surface losses, an adaptive self-learning fuzzy control law is developed for a Bank-To-Turn(BTT)missile.13-15The fuzzy control law obtains better performance based on its dynamically self-learning ability.However,the fuzzy rules are complex and difficult to design. A nonlinear autopilot for a highly maneuverable missile is designed utilizing robust backstepping control,which is applied to the Multi-Input/Multi-Output (MIMO) model to achieve both Bank-To-Turn(BTT)and Skid-To-Turn(STT)maneuvers.16The control law contains three parts:one nominal controller and two compensation controllers counteracting unmatched/matched uncertainties.However,the backstepping controller is complex and conservative.Integrated Guidance and Control(IGC)has been proposed to overcome different bandwidths between the guidance and control loops.17-19It shows better performance than the traditional two-loop autopilot.Nevertheless,many IGC laws require target acceleration or accurate time-to-go for interception.Model Reference Adaptive Control(MRAC)has been widely studied and possesses various advantages in the control of systems with uncertainties or time-varying parameters.20MRAC can force the response of the system to track the response of a reference model asymptotically by adapting controller gains.21,22Meanwhile,Sliding Mode Control(SMC)has also been extensively studied for both linear and nonlinear uncertain systems.23-27SMC has the advantage of strong robustness in the presence of internal and external disturbances.Furthermore,in recent years,finite and fixed time convergence sliding mode control has become the focus of researches,which shows better performance than classical SMC.28-40

In this study,a Fixed-Time Adaptive Model Reference Sliding Mode Control(FTAMRSMC)scheme is proposed by combining SMC and MRAC for an air-to-ground missile.The missile is first assisted by a booster rocket,and then it flies without thrust.This process will result in large speed ranges(50-400 m/s),large disturbances such as changes of the gravity center and the flight environment,mismatched disturbances and un-modeled dynamics.Therefore,the dynamic coefficients change dramatically and usually contain some considerable biases. A traditional PID controller requires many flight operation points to design control parameters.To avoid this process, a novel FTAMRSMC is developed based on a fixed-time adaptive reaching law and a fixed-time disturbance observer.The main contribution of this paper includes:

(1)The proposed control scheme covers the entire flight envelope only through one group of control parameters,which means that it does not require the gain scheduling process regardless of the large variations of the dynamic coefficients throughout the flight phase.

(2)FTAMRSMC guarantees that the errors,including the model tracking error and the estimation error for the fixed-time disturbance observer,converge to a small region of the origin in a fixed(pre-established)settling time.

(3)The proposed controller has a simple structure which is convenient to design for both the attitude autopilot and the lateral acceleration autopilot.The magnitude of the switching term for the fixed-time adaptive reaching law can be adjusted based on the sliding mode surface or the state tracking error during the flight phase,which avoids large control command responses and achieves chattering-free performance.

(4)A fixed-time disturbance observer can provide disturbance information,which enhances the control accuracy of the proposed control system.

The remainder of this paper is organized as follows:In Section 2,the missile state-space dynamics model and the reference model are discussed.The formulation of the fixed-time adaptive model reference sliding mode controller is presented and proved in Section 3.In addition,to estimate the disturbances,a fixed-time disturbance observer is also developed in Section 3. Simulation results are presented in Section 4.Finally,Section 5 concludes this work.

2.Model description

The air-to-ground missile will first be assisted by a booster rocket after separation from the aircraft.Then,it will fly without thrust during the gliding flight phase.Next,the missile will end the gliding flight and enter into the transition flight to create better working conditions for the missile seeker.More specifically,when the distance between the missile and the target is less than 4 km,the seeker starts working and the missile enters into the terminal guidance flight.Finally,the missile will hit the target under the action of the seeker and the terminal guidance law.The flight profile is shown in Fig.1.

Fig.1 Flight profile of air-to-ground missile.

2.1.Missile dynamics model state equation

The dynamics model of the air-to-ground missile is given as23

where P,V,θ and ψvrepresent the thrust,the velocity,the trajectory inclination angle and the trajectory deflection angle respectively;α,β and γvrepresent the angle of attack,the angle of sideslip and the bank angle respectively;XA,YAand ZAdenote the aerodynamic forces;Mx,Myand Mzdenote the aerodynamic torques;Jx,Jyand Jzrepresent the moments of inertia;ωx,ωy,ωzdenote the angular velocities;m is mass, ˙m is the fuel consumption rate and m0is the initial mass of the missile;g is the gravitational acceleration;?,ψ and γ represent the pitch angle,the yaw angle and the roll angle respectively;x,y,z denote the positions.The aerodynamic force and aerodynamic torque can be decomposed as

where ρ,Srefand l represent the air density,the cross-section area of the missile and the reference length of the missile respectively;andrepresent the aerodynamic coefficient derivatives with respect to α and β respectively;andare damping moment coefficients;Q=ρV2/2 is dynamic pressure;andare aerodynamic control force coefficients;are aerodynamic control torque coefficients;δx,δyand δzdenote the control surface deflections;mx0,my0,mz0are aerodynamic torque coefficients when α=β=δx=δy=δz=0.Further,Eq.(1)can be linearized into three independent channels as follows:

where a22,a24,a25,a34,a35,b22,b24,b25,b34,b35,b11and b17are defined as

The lateral acceleration values for an air-to-ground missile in the pitch channel and yaw channel are proposed as

where λhdis a proportionality coefficient that is usually set to 1.nyand nzare defined in the ballistic coordinate system.23Moreover,the force contribution of the control fins is negligible,that is,a35δzand b35δyare very small.Differentiating nyand nzwith respect to time yields

where Fyis the error of the dynamics model for the pitch channel,and

The yaw channel state equation can also be obtained as

In addition,roll channel is designed as an attitude autopilot.Hence,the state variable Xx=[γ ,ωx]Tis defined and the control input is Ux=δx,which gives the following term:

where Fxis the error of the dynamics model for the roll channel and

2.2.Reference model state equation

Next,the reference model for three channels will be proposed.The pitch channel and the yaw channel are both controlled by the lateral acceleration autopilot.As is known to all,the body transfer function for the air-to-ground missile can be obtained as second-order oscillation dynamics,as shown in Eqs.(7-109)of Ref.39.Therefore,the reference model for the pitch channel can be designed using Laplace form as

where Tyand ξyare the desired response time and the damping ratio respectively;αcdenotes the command angle of attack;αmrepresents the response angle of attack.Moreover,under the assumption of known aerodynamic coefficients,Tyand ξyare two parameters that should be designed for the reference model.The response time of the system can be obtained in advance by selecting a proper Ty,and the overshoot of the system will rely on ξy.The expected acceleration nyccan be obtained as

where δzcis the expected control surface deflections.Combining Eqs.(10)and(11)yields

Eq.(12)can be further transformed into a time-domain model.

The trajectory of the air-to-ground missile is smoother as shown in Fig.1.Therefore,the angle of attack changing rate is much larger than the trajectory inclination angle changing rate,that iswhere θmrepresents the response trajectory inclination angle.Subsequently,the small perturbation linearization equationcan be simplified asThus,Eq.(13)will be simplified as

where

For the yaw channel,the angle of sideslip can be selected as second-order reference model dynamics,which also yields

where Tzand ξzare the desired time and the damping ratio for the yaw channel respectively.The roll channel is controlled by an attitude autopilot.Hence,the reference model is given by

The corresponding state equation can be written as

where Txand ξxare the desired time and the damping ratio for the roll channel respectively. Meanwhile,,and Uxm=γcis the desired roll angle.

Finally,by ignoring the subscript,the state equation and the model reference state equation for three channels are written into a uniform form as follows:

where A is the state matrix,B is the control matrix and F represents the error dynamics;X and U are the state vector and control vector respectively;Amand Bmrepresent the reference model state matrix and control matrix respectively;Xmand Umare the reference model state vector and control vector respectively.

It is worth mentioning that the aerodynamic parameter variation is considered in the reference model by utilizing the aerodynamic data sheet in the appendix,and the aerodynamic coefficients can be obtained by the method of interpolation.

3.Fixed-time adaptive model reference sliding mode control

In this section,a fixed-time adaptive model reference sliding mode controller for an air-to-ground missile is proposed.Firstly,a sliding mode surface is designed by the tracking error of the state equation and the model reference state equation in Eq.(19).Then,a novel fixed-time adaptive reaching law is presented. Meanwhile, the disturbances and un-modeled dynamics are estimated by means of a fixed-time disturbance observer to compensate the proposed controller. Finally,closed-loop stability of the proposed controller is proved by using Lyapunov methodology.

3.1.Definitions and lemmas

Consider the following system:

where f:Rn→Rnis a nonlinear function.Suppose that the origin is an equilibrium point of system(20).

Definition 2.38The origin of system(20)is said to be fixedtime stable if it is globally finite-time stable and it will converge to the origin within the bounded convergence time T(x0).Hence,there exists a bounded positive constant Tmaxsuch that T( x0)＜Tmax.

Lemma 1.12,31For system(20),suppose a Lyapunov function V(x), parameters α1,β1,p,q, pk ＜1,qk ＞1, k ∈R+and 0 ＜Ω ＜∞, such thatThen,the system is fixed-time stable.Furthermore,the residual of the solution of system(20)can be given by

where Θ satisfies 0 ＜Θ ≤1.The convergence time is bounded as

Lemma 3.If v ∈R+and v ＞1,for any x,y ∈R,there exists

Notation 1.12In this paper,‖ ·‖is denoted as the Euclidean norm of vectors and the induced norm of matrices.For a given vectorsign(x2),...,|xn|psign(xn)]T,sig(x)p=|x|psign(x),where sign(.)denotes the sign function.

3.2.Fixed-time adaptive reaching law

According to Eq.(19),the tracking error of the state equation and the model reference state equation as well as its derivatives are shown as

The sliding mode surface is constructed as

In order to achieve the fixed-time convergence of model tracking errors and avoid large control gains in the presence of variations of aerodynamic coefficients during the whole flight phase,a novel fixed-time adaptive reaching law is presented as follows:

where 0 ＜p ＜1,q ＞1,k1＞0,k2＞0;andare adaptive gains.As shown in Eq.(26),it is obvious that the magnitude of the switching termcould be adjusted on the basis of the adaptive gainsandor the state tracking error e2.Moreover,the large control gain can result in the system instability or chattering,and thus a novel adaptive reaching law is designed.Adaptive gains can be updated by

where θ0∈(0,1)andThe initial value ofsatisfies∈ (0,1 ),^Φ ＞0 andTherefore,all of these adaptive gains will gradually decrease.c1,c2and c3satisfy the following equations:

where θ1＞1/2,θ2＞1/2,θ3＞1/2;σ1,σ2and σ3are controller parameters which are usually adjusted by simulation results.According to Eq.(27),these parameters are required to be greater than the initial value of ‖S‖ such thatandAs S approaches zero,andhold.Moreover,c1,c2and c3are designed as Eq.(28)to make the selection of parameters more convenient.

Based on Eqs.(25)and(26),the equation can be transformed into

which yields the control law

Theorem 1.If the controller is developed as Eq.(30)and the adaptive law in Eq.(27),then the control system is fixed-time convergent.

Proof.Define the error function

Consider the following Lyapunov function candidate as

Taking the derivative of VSyields

Eq.(33)can be simplified as

Moreover,the following expressions can be obtained:

Therefore,Eq.(34)can be transformed into

Substituting Eqs.(39)and(41)into Eq.(36)and making further simplification yield

Finally,Eq.(42)can be simplified as

From Lemma 1,the system tends to be practically fixedtime stable.It is worth noting that the mismatched disturbance and the un-modeled dynamics in Eq.(19)as well as the proposed controller in Eq.(30)are unknown in the real flight.Therefore,for the system=AX+BU+F,the following second-order fixed-time disturbance observer is designed30:

where Z1is the estimation of X,and Z2is the estimation of the disturbance F.The stability analysis of the proposed disturbance observer can be found in Ref.30.Thus,the disturbance F will be replaced by Z2for the proposed controller in Eq.(30),which yields

Furthermore, the switching function sign(x) can be replaced by a hyperbolic tangent function tanh(x)= (ex-e-x)/(ex+e-x)to reduce the chattering of the control system.

Finally,the close-loop stability analysis of the composite controller in Eq.(45)is presented.Consider the following Lyapunov function candidate as

Taking the derivative of V1and substituting Eq.(45)into Eq.(25),we have

According to the stability analysis of the proposed disturbance observer in Ref.30,it is obvious that‖S-Z2‖is small enough.Therefore,the following equation can be obtained:

By employing Eq.(27),we haveandSimplifying Eq.(48)yields

From Lemma 1,the composite controller also tends to be practically fixed-time stable.This completes the design of the Fixed-Time Adaptive Model Reference Sliding Mode Control(FTAMRSMC)for the air-to-ground missile.

Remark 1.It is obvious that the proposed controller has a switching termsign(S)whose magnitudedepends on the adaptive laws in Eq.(27).The magnitude of the switching term can be adjusted based on the sliding mode surface S or the state tracking error e during the flight phase, which avoids large control command responses and achieves chattering-free performance. By employing the fixed-time disturbance observer,the disturbance information can be provided, which enhances the control accuracy of the proposed control system.

4.Simulations and results

To verify the effectiveness of the proposed FTAMRSMC,the detailed results of numerical simulations for a single point are presented. Moreover, simulation results including Monte Carlo simulations,nonlinear 6-DOF simulations,and different ranges are also developed.

4.1.Single-point simulation

The proposed control scheme will be first tested with the parameters fixed for the pitch channel,and the parameters are selected to be a22=-0.0025, a24=-161.9711,a25=-147.8875,a34=0.6939,a35=0.2809 and V=178.01.The proposed FTAMRSMC control parameters are listed in Table 1.

The initial parameters of the fixed-time adaptive reaching law are set as^χ(0)=^Φ(0)=1.0 and(0)=0.5,and the model reference parameters are selected as ξy=0.707 and Ty=0.11.Meanwhile,the initial conditions of the fixed-time disturbance observer are taken as zero.Furthermore,the biases of the coefficients for a22,a24,a25,a34and a35are set to be 20%.The PID controller parameters are designed as (0 .0 05,0.6075,0.160).Moreover,the following Traditional Model Reference Sliding Mode Controller(TMRSMC)is also presented to compare with the proposed controller.

Simulation results are presented in Figs.2-5.In Fig.2,the overshoot of the model reference acceleration step response is 2.8%,and the settling time is 1.28 s,while the PID controller response performance is 4.7%and 1.83 s respectively.Correspondingly, the TMRSMC response performance is 3.6%and 1.20 s.However,there exists a large steady-state error due to the large parameter uncertainties.Meanwhile,the performance of the FTAMRSMC is 2.7%and 1.20 s.It is worth noting that the steady-state error is smaller in spite of the large parameter uncertainties.It also indicates that the proposed controller has better performance than the PID controller and TMRSMC. Furthermore, the sliding mode surface of the FTAMRSMC tends to zero within the expected settling time in Fig.3.Unfortunately,the sliding mode surface of the TMRSMC cannot approach zero.It can be seen from Fig.4 that,for the mismatched disturbances and un-modeled dynamics,the proposed disturbance observer has great estimated performance.The adaptive gains of the fixed-time reaching law are shown in Fig.5,which also indicates its strong convergence and adaptability.

Fig.2 Step response of acceleration.

Table 1 Control parameters of FTAMRSMC.

Fig.3 Response of sliding mode surface.

Fig.4 Disturbance estimated for Z2.

Fig.5 Adaptive gains of fixed-time reaching law.

Fig.6 Angle of attack α.

4.2.Nonlinear 6-DOF simulation

Next,full-time nonlinear 6-DOF simulations are carried out.The initial mass of the air-to-ground missile is m0=30 kg and the time-varying thrust is designed based on Eq.(52).We can consider the fixed target position as xT=yT=zT=0 m,and the initial conditions of the air-toground missile are set as x=-16000 m, y=5000 m,z=200 m, V=50 m/s, θ=0°, ψv=0°and ωx=ωy=ωz=0. The control parameters can be set as Tx=Ty=Tz=0.11, ξx=0.707,ξy=ξz=0.75, κ1x=10,κ1y=5,κ1z=-5,and the other parameters are the same as those in Table 1.Meanwhile,the seeker noise is considered as white Gaussian noise during the terminal guidance flight.The variance of the seeker noise is 0.4°and its mean is zero.Furthermore,the air-to-ground missile is considered to be axisymmetric. Moreover, the aerodynamic parameters and other parameters can be found in Appendix.

The guidance law for the booster rocket flight is a zero acceleration command,while the gliding flight phase will be guided by a constant angle of attack(5°)and the terminal guidance flight phase will be guided by a Proportional Navigation Guidance Law(PNGL).The simulation sampling step is set as 5 ms and the simulation results are shown in Figs.6-11.Meanwhile,the PID control parameters are shown in Table 2.

Fig.7 Angle of sideslip β.

Fig.8 Response of acceleration ny.

Fig.9 Response of acceleration nz.

Fig.10 Response of elevator.

Fig.11 Response of roll angle.

It is obvious that the proposed control law offers satisfactory performance in spite of the variations of aerodynamic coefficients during the flight phase.Compared with the PID gain-scheduling controller, the proposed controller has a strong adaptive characteristic to varying parameters without switching controllers.It should be specially mentioned that the chattering of the curve for the terminal guidance phase is caused by the seeker noise instead of the fixed-time sliding mode controller.Furthermore,the steady-state error of the fixed-time controller is smaller than that of PID and the fixed-time controller has a faster convergence,as can be seen in Fig.6.Moreover,the simulation results in Fig.8 indicate a smaller acceleration response in the initial flight time of the missile,which is important for the safety of the aircraft and the missile.The angle of the elevator deflection is less than 9°during the whole flight phase and it follows a smooth curve as shown in Fig.10.The yaw channel and the roll channel also have good performance,as shown in Figs.7,9 and 11.

4.3.Monte Carlo simulation

Next,Monte Carlo simulations are carried out considering multiple disturbances such as mass deviation,thrust deviation,aerodynamic coefficient deviation,aerodynamic moment coefficient deviation,atmospheric density deviation,wind disturbance,rotational inertia deviation and centroid eccentricity deviation. The maximum deviation values are shown in Table 3.The control parameters are also the same as those in the nonlinear 6-DOF simulation,and the number of the Monte Carlo simulation is set as 2000.The final results are shown in Figs.12 and 13.

It is found that,although there exist multiple disturbances and seeker noise,the fixed-time controller can achieve a meanmiss distance of 0.5876 m with the variance being 0.3040 m.It is verified that the proposed control law has strong robustness and extensive adaptability.

Table 2 PID control parameters.

Table 3 Maximum values of disturbances for nonlinear simulation.

Fig.12 Monte Carlo results for miss distance.

Fig.13 Monte Carlo simulation results.

Fig.14 Results of velocity.

Fig.15 Results of height and x distance.

Fig.16 Results of angle of attack.

Fig.17 Results of angle of sideslip.

Fig.18 Result of pitch rudder δz.

Fig.19 Result of yaw rudder δy.

Fig.20 Result of position z.

Fig.21 Results of roll angle.

4.4.Different ranges simulation

Lastly,the following simulations are carried out to verify the performance of the proposed control law for different ranges,and the corresponding results are given in Figs.14-21.The initial conditions are the same as those in the nonlinear 6-DOF simulation.Meanwhile,the initial positions of the missile for different ranges are set as x0=-6000 m, -8000 m,-10000 m,-12000 m,-15000 m and-18000 m respectively.Moreover,the constant angles of attack for the gliding flight are set as-3°,1°,3°,3°,5°and 9°.The control parameters are also the same as those in the above simulations.

From the above figures and Table 4,it can be seen that,even with multiple disturbances,seeker noise and different ranges,the proposed control system still performs well and achieves a miss distance of less than 1.0 m for all ranges.Furthermore,the sliding chattering is suppressed.It is important to note that,with only one group of control parameters,the proposed control system indicates good adaptability to any initial release conditions of the air-to-ground missile.

5.Conclusions

In this work,a novel fixed-time adaptive model reference sliding mode control for an air-to-ground missile is developed.It consists of a fixed-time adaptive reaching law and a fixed-time disturbance observer.

(1)The proof of the stability for the proposed control law and the disturbance observer is presented by the development of the Lyapunov methodology.

(2)Simulation results indicate that the proposed controller has a smaller overshoot and a smaller steady-state error than the PID controller and TMRSMC.Furthermore,the proposed control law achieves strong robustness and adaptability under the conditions of multiple disturbances,seeker noise and different ranges.

(3)Moreover,the miss distance is less than 1 m in the Monte Carlo simulation and the different ranges simulation,which satisfies the requirements of autopilot design.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(No.61403100),the Open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology of China (No. HIT.KLOF.MST.201704),and the Fundamental Research Funds for the Central Universities of China(No.HIT.NSRIF.2015.037).

Appendix A.

The cross-section area of the missile is Sref=0.012,and the reference length is l=1.3.Moreover,the initial center of mass is Xcg0=0.45 and the final center of mass is Xcg1=0.54.The aerodynamic coefficients are listed as follows,where CN is normal force coefficient,CA is axial force coefficient,MZG is pitch moment coefficient and XCP is pressure center coefficient.Jx=0.3,Jy=Jz=6.2(Tables A1-A4).

Table 4 Miss distance for different ranges.

Table A1 Aerodynamic coefficients for δ=0°.

Table A2 Aerodynamic coefficients for δz=-5°.

Table A3 Aerodynamic coefficients for δz=-10°.

Table A4 Roll torque coefficient.