Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against
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Guidance Synthesis for Evasive Maneuver of
AntiShip Missiles Against CloseIn Weapon Systems Seminar Report Presented By Ms Sarika Raju st 1 Semester, M.Tech, Roll No:10GNC10 Department of Electrical Engineering College of Engineering, Trivandrum Thiruvananthapuram16 2010 Abstract Evasive maneuvers of antiship missiles are complicated since the missile should home to the target ship alive while avoiding the defensive weapons of the target ship. Recent devel opments of antiair defense systems are threatening the survivability of antiship missiles. Here a 3 dimensional evasive maneuvers of antiship missiles against CIWS are investigated. By using a parameter optimization technique, trajectory optimization is performed using 2 methods. Based on the optimization results a 3 dimensional biased proportional navigation guidance (BPNG) that generates a barrel roll maneuver that enhance the survivability of antiship missile is also proposed. Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems.pdf (Size: 577.6 KB / Downloads: 89) Contents 1 Introduction 1 2 Equations of Motion and Aiming Errors 3 2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Aiming Errors of CIWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Optimal Evasive Maneuver Against CIWS 6 3.1 Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Biased PNG for Barrel  Roll Maneuvers 12 5 Conclusions 15 iii List of Figures 2.1 Definitions of coordinate systems and angles . . . . . . . . . . . . . . . . . . . 3 3.1 (a)Optimum trajectory (method 1). (b) Trajectory project and implimentationed on the YZ plane (method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Acceleration command profile (method 1) . . . . . . . . . . . . . . . . . . . . 8 3.3 Aiming errors (method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 (a)Optimum trajectory (method 2) (b)Trajectory project and implimentationed on YZ plane (method 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Acceleration command profile (method 2) . . . . . . . . . . . . . . . . . . . . 11 3.6 Aiming errors (method 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1 Direction of uBR and uP P N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 iv Chapter 1 Introduction The capturability of the biased PNG is ensured from the Fighter aircraft needs evasive ma neuvers just to evade attacking missiles. For antiship missiles, however, evasive maneuvers are complicated since the missile should home to the heavily defended target alive, while avoiding the defensive weapons of the target ship. Recent developments of antiair defense systems are threatening the survivability of antiship missile. It is presumed that policies of evasive maneuvers are empirically determined by antiship missiles versus defense systems engagement simulation results, rather than theoretical analysis. On the other hand, evasive maneuvers to enhance the survivability of fighter aircraft have been found in many scholarly articles. Continuously changing the maneuver direction, such as weaving or barrelroll maneuvers, is believed to be effective to enhance the survivability of fighter aircraft. However these are not adequate for the antiship missiles because they cannot satisfy a zero terminal miss distance. Some evasive maneuver policies for missiles against proportional navigation (PN)guided air defense missiles has been studied. The optimal evasive maneuver pattern has been char acterized by a deformed conical barrelroll, which is obtained from a constrained trajectory optimization problem that maximizes the timevarying weighted sum of instantaneous miss distances. The results are supported by the fact that spiral motions of tactical ballistic mis siles, which are caused by mass or configurational asymmetries, impose difficulty on antiair defense systems . A 3dimensional optimal evasive maneuver problem for antiship missiles against closein weapon stytems (CIWS), which is the most common defensive weapon for naval ships, is investigated in this paper. The acceleration command is assumed to be the sum of a con ventional pure proportional navigation guidance (PNG) term and a bias term, the former to guarantee the homing capability and the latter to minimize the performance index. Sim ilarly the performance index is defined in terms of the timetogo weighted aiming errors of the CIWS. To simplify the numerical analysis, the bias term is constrained to be normal to the usual guidance plane of PNG, which is a plane including the line of sight (LOS) vector and the missile velocity vector. Again the optimal trajectory turns out to be the type of barrelroll. Based on the numerical optimization results, a 3dimensional biased PNG law to induce a barrelroll maneuver is also proposed. As in the numerical optimization problem, the pro posed guidance law consists of a conventional pure PNG term to guarantee homing to the target and a timevarying bias term to produce evasive barrelroll maneuvers. To define a barrelroll maneuver, the barrelroll axis and the barrelroll frequency are introduced as follows. The barrelroll axis is chosen as the LOS vector from the missile to the target, and the barrelroll frequency is a userdefined parameter that determines the angular velocity of 1 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems the barrelroll. The acceleration command for the barrelroll maneuver is computed from the cross prod uct of the missile velocity vector and the desired angular velocity vector of the barrelroll. Since the barrelroll command is always normal to the velocity vector, the proposed guidance law can be easily implemented to a missile system by using the pure PNG law. Note that the altitude of the antiship missile should be bounded in order to prevent the antiship missile from crashing into the sea. An additional altitude controller may be required to control the altitude, but such a method reduces the evasive performance of the proposed guidance law. Note that the altitude of the antiship missile should be bounded in order to prevent the antiship missile from crashing into the sea. An additional altitude controller may be required to control the altitude, but such a method reduces the evasive performance of the proposed guidance law. Fortunately the radius of the curvature of the barrelroll is maximal when the barrelroll maneuver is initiated, and it monotonically decreases as the antiship missile approaches the target. Hence by a proper choice of the direction of the barrelroll ma neuver, we can guarantee that the antiship missile does not go down below a specified height. In this paper the capturability of the proposed 3dimensional biased PNG with barrel roll maneuver against a stationary target is proved by using the Lyapunovlike approach. In our capturability analysis the nonlinear dynamics of a 3dimensional engagement are taken into accountfact that the barrelroll command does not affect the capturability of PN if the direction of the barrelroll command is carefully chosen. Department of Electrical Engineering, College of Engineering, Trivandrum 2 Chapter 2 Equations of Motion and Aiming Errors 2.1 Equations of Motion We consider the engagement geometry between the antiship missile and the target, as shown in Fig1. To simplify the equations of motion of the missile, we assume that the antiship missile is a point mass and that its speed is constant. We further assume that the autopilot and the seeker dynamics of the missile are fast enough to be neglected. Gravity is also assumed to be compensated for by autopilot. The target ship is modeled as being stationary since the maneuverability and the speed of ships are not comparable with those of antiship missiles. Figure 2.1: Definitions of coordinate systems and angles Three coordinate frames are used to describe the motion of the missile: the inertial coordinate frame, the LOS coordinate frame, and the missile velocity coordinate frame, denoted by the subscripts I, L, and M, respectively. The relationship between theIframe and the Lframe is obtained by the coordinate transformation given by iL cθL cψL cθL sψL sθL iI jL = −sψL cψL 0 jI (2.1) −sθL cψL −sθL sψL cθL kL kI Similarly the relationship between the Mframe and the Lframe is as follows. 3 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems iM cθM cψM cθL sψM sθM iL jM = −sψM cψM 0 jL (2.2) −sθM cψM −sθL sψM kM cθM kL By considering the engagement geometry depicted in Fig.2.1, we derive the following equations from the definitions of the coordinate frame and the wellknown classical principles of dynamics. r = rT − rM = riL (2.3) d dr = (riL ) = r. iL + ΩLL = −vM iM (2.4) dt dt AM = ay jM + az kM = (ΩL + ΩM ) × vM (2.5) M M ΩL = ΨL sinθL iL + ΨL cosθL kL − θ ̇L ̇ ̇ (2.6) ΩM = ψ ̇ sinθM iM − θM jM + ψ ̇ cosθM kM ̇ (2.7) M M By using the above equations and the aforementioned assumptions, we obtain the non linear equations of motion of the antiship missile given by r = −vM cosθM cosψM ̇ (2.8) ̇ rλy = vM sinθM (2.9) ̇ rλz = −vM cosθM sinψM (2.10) ay tanθL sinθM cosψM sinθM sinψM 1 ̇ M ψM = + (−vM cosθM sinψM + (vM sinθM ) + (vM cosθM sinψM ) vM cosθM rcosθM rcosθM r (2.11) aZ tanθL sinψM cosψM ̇ θM = M + (vM cosθM sinψM ) + (vM sinθM ) (2.12) vM r r Department of Electrical Engineering, College of Engineering, Trivandrum 4 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems 2.2 Aiming Errors of CIWS A CIWS is a naval shipboard defense weapon used to destroy incoming antiship missiles or enemy aircrafts at a short range. The CIWS usually is equipped with an automatic gun system of high firing rates and radar systems for tracking antiship missiles. To hit the target traveling at a high speed, the CIWS has to fire the shells to the predicted impact point since midcourse corrections of the shells are not possible. For the prediction of the impact point of the missile and the shells fired from the CIWS, the CIWS relies on the estimated missiles states of the current time. In this paper it is assumed that the CIWS exactly measures the states of the antiship missile, rM , VM , and AM . We also assume that the shells fired from the CIWS fly straight and maintain the muzzle velocity vCIW S . Then we can predict the flight time of each shell as r(t) τ (t) ≈ (2.13) vM + vC IW S Typically the fire control system of the CIWS calculates the intercept point at τt from the current time t, assuming that the missile maintains the current acceleration. If this is the case, any timevarying maneuver of the missile can produce an aiming error of the CIWS, which is calculated as t+τ (t) s2 1 AM (s1 )ds1 ds2 − AM (t)[τ (t)]2 (t + τ (t)) = (2.14) 2 t t Equation 2.14 physically represents the miss distance of the antiship missile and the CIWS shell at the moment of encounter. The aiming error of the CIWS, given in eqn 2.14, is used to analyze the evasion performance of the guidance laws designed for antiship missiles. Department of Electrical Engineering, College of Engineering, Trivandrum 5 Chapter 3 Optimal Evasive Maneuver Against CIWS To investigate the optimal evasive maneuver of antiship missiles against the CIWS, consider the following optimal control problem. Find u, which minimizes tf τ 2 (t) J= dt (3.1) (t) t0 subject to the constraints described by eqn 2.8 to eqn 2.12, the terminal inequality constraint rM (tf ) − rT  ≤ D (3.2) and the bound on control u ≤ umax (3.3) Here D is the admissible miss distance of the missile, and umax is the limit of the ac celeration command. In order treat more realistic problems, the missile is assumed to have firstorder dynamics: 1 A ̇M = (u − AM ) (3.4) τM where u is the acceleration command and where τM represents the time constant of the autopilot. A performance index of the optimal control problem should be carefully determined since the aiming error (t), given by eqn 2.14, goes to 0 as the missile approaches to the target. For example a performance index to maximize the integral of the aiming error could produce zero aiming errors even at long ranges. On the other hand a performance index to maximize the minimum of the aiming error would not be much more meaningful since the aiming error eventually converges to 0. For these reasons τ 2 (t) is included in the performance index in order to put less emphasis on the aiming errors in the final phase. The closedform solution of this optimal control problem could not be derived due to the nonlinearities included in the performance index and the inequality constraints. To find the policy of a 3dimensional evasive maneuver of the missile against the CIWS, we rely on numerical optimization techniques. The optimal control problem is converted into a parameter optimization problem which treats the discretized control and the total flight time as an unknown parameter vector, as shown below. 6 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems 3.1 Method 1 X = [upitch (i), uyaw (i), tf ]T, i = 0, ....., N − m. Note that the number of unknown parameters of method 1 is 2(Nm+1)+1. Since the antiship missile should hit the target within an admissible miss distance, we assume that the acceleration command during the final m intervals is void of evasive maneuvers. Let us consider the following acceleration command: 0 1 0 0 0 0 0 −1 1 0 0 0 −1/τM 0 A = 0 (3.5) −1/τT 1/τT 0 0 0 0 0 0 0 0 = u(i) = uP P N f ori = N − m − 1..., N Figure 3.1: (a)Optimum trajectory (method 1). (b) Trajectory project and implimentationed on the YZ plane (method 1) Finding u(i) without uP P N may be a natural approach. However we cannot obtain a con sistent solution from this approach. Optimizing the 2dimensional maneuver of the missile with the intercept constraint seems to have many local minima. To circumvent this difficulty Department of Electrical Engineering, College of Engineering, Trivandrum 7 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems we propose method 1, which assumes a biased PNG for homing to the target and optimizes the bias term for effective evasion. This approach reduces the optimal control problem to a 1dimensional maneuver problem without the terminal constraint, which is a lot simpler than the original 2dimensional maneuver problem. Figure 3.2: Acceleration command profile (method 1) For this optimization problem a coevolutionary augmented Lagrangian method (CEALM) is used. This method converts a constrained optimizationproblem into a minimax problem using the augmentedLagrangian formulation, which is solved by the evolution of two pop ulation groups. The advantage of the CEALM is that it does not require the gradient information of the cost and the constraints functions and that it is very robust to the initial guess of the solution. Integration of the equations of motion to evaluate the values of the performance index and the constraints is done by the Euler method. Method 1 has not provided consistent optimization results. Nonetheless all the results show sinusoidal changes in the lateral acceleration, which are characteristic of barrelroll maneuvers. Two sample trajectories, depicted in Fig.3.1, imply that the optimal maneuver of the antiship missile against CIWS should be a barrelroll type maneuver. The acceleration profile shown in Fig 3.2 clearly supports this conclusion. Both of the two cases have sinusoidal changes in the normal and the lateral accelerations, with a phase shift of 90◦ between the two axes. It is also noted that the total acceleration is saturated to its maximum value for most of the flight. Fig 3.3 shows the CIWS aiming errors caused by the sinusoidal change of the acceleration commands. Note that the aiming errors of the two cases have very similar time histories, although the trajectories are somewhat different. This may be the reason for the difficulties in obtaining a unique optimal solution; the cost function in the solution area may be fairly flat or may have numerous local minima. The costs of two cases are 124.526 and 129.820, respectively. This is a big improvement over the PNG, for which the cost turns Department of Electrical Engineering, College of Engineering, Trivandrum 8 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems Figure 3.3: Aiming errors (method 1) out to be 3038.471. To obtain more consistent solutions, the missiles evasive maneuver is now constrained to the direction perpendicular to the plane composed by the LOS vector and the missiles velocity vector. This plane is called the PN guidance plane in this paper since the missile trajectory is confined to this plane if the PN guidance is applied. (The gravity effect is ignored here.) The usual PN guidance command is given in the PN guidance plane so that homing to the target is guaranteed. This approach, referred to as method 2, significantly simplifies the optimization problem, and the number of unknown parameters is reduced to (Nm+1)+1. 3.2 Method 2 X = [k(i), tf ]T, i = 0, ....., N − m iLM f ori = 0....., N − m − 1 u(i) = uP P N + k(i) (3.6) iLM  u(i) = uP P N f ori = N − m..., N (3.7) Two sample trajectories are depicted in Fig 4.1. The result of case 1 shows a barrelroll maneuver. In case 2, however, the rotating direction of the missile changes once in the mid dle. Consequently the acceleration command of case 2 has an abrupt change, as shown in Fig 4.2, which produces large aiming errors of the CIWS observed in Fig 4.3. However these two cases have very similar costs: 214.595 and 214.245 for case 1 and 2, respectively. We have obtained various results showing the reversal of the rotating direction observed in case 2. This may be the reason for the difficulties in obtaining the unique optimal solution. From the results of the numerical analysis on optimal evasive trajectory, we see that the optimal evasive maneuver of the antiship missiles against the CIWS should have sinusoidal acceleration commands to produce a barrelroll type trajectory. Motivated by this observa tion we propose, in the next section, a guidance law, which is a biased PNG, to generate a homing trajectory with a barrelroll maneuver. Department of Electrical Engineering, College of Engineering, Trivandrum 9 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems Figure 3.4: (a)Optimum trajectory (method 2) (b)Trajectory project and implimentationed on YZ plane (method 2) Department of Electrical Engineering, College of Engineering, Trivandrum 10 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems Figure 3.5: Acceleration command profile (method 2) Figure 3.6: Aiming errors (method 2) Department of Electrical Engineering, College of Engineering, Trivandrum 11 Chapter 4 Biased PNG for Barrel  Roll Maneuvers To obtain a trajectory of the barrelroll type, consider the following 3dimensional biased PNG: uBP N = uP P N + uBR (4.1) where uP P N is the conventional pure PN(PPN) guidance command for homing and where uBR is a bias term to generate barrelroll maneuvers. Pure PNG: The conventional PPNG law generates an acceleration command proportional to the LOS rate, which is expressed as uP P N = NM ΩL × VM = NM vM (−λ ̇x sinθM cosψM − λy sinθM sinψM + λz cosθM )jM + (4.2) ̇ ̇ +NM vM (λ ̇x sinψM − λy cosψM )kM ̇ Since the onboard seeker cannot measure , the 3dimensional PPNG law is practically given by ̇ ̇ ̇ uP P N = NM vM (−λy sinθM sinψM + λz cosθM )jM + NM vM (−λy cosψM )kM (4.3) BarrelRoll Command : A barrelroll maneuver can be defined by the barrelroll axis and the barrelroll frequency. The command for a barrelroll maneuver can be defined by uBR = ωBR × VM (4.4) If the barrelroll axis is in the same direction as the velocity, the barrelroll maneuver, does not occur ωBR //VM =⇒ uBR = 0 (4.5) Therefore a forced maneuver is needed to generate a velocity component which is normal to the barrelroll axis. The command magnitude for a barrelroll is calculated as 1 − cos2 θM cos2 ψM uBR  = ωBR vM (4.6) As the missile velocity approaches the direction of the LOS vector, the command magnitude decreases. 3Dimensional Biased PNG: A 3dimensional BPNG law for the barrelroll maneuver is given by ̇ ̇ uBP N = NM vM (−λY sinθM sinψM + λz cosθM − ωBR vM sinθM cosψM jM (4.7) ̇ +NM vM (−λy cosψM ) + ωBR vM sinψM 12 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems Figure 4.1: Direction of uBR and uP P N For any antiship missile, the missile altitude should be lowerbounded properly in order to prevent a crash into the sea. An altitude controller can be used for this purpose, but it may deteriorate the evasive performance of the proposed guidance law by reducing the magnitude of the barrelroll command. In the following we investigate the time history of the radius of the barrelroll to check the possibility of a sea crash. The barrelroll maneuver occurs in the maneuver plane, and its instantaneous radius of curvature is calculated as LP 1 − cos2 θM cos2 ψM VM (t) vM ρ(t) = = (4.8) ωBR ωBR LP where VM represents the project and implimentationion of the missile velocity on the maneuver plane. The role of PPN is to align the velocity vector of the missile with the LOS vector. The PPN LP command project and implimentationed on the maneuver plane is always in the opposite direction of VM , as shown in Fig 4.1. Hence, the time rate of the radius of curvature of barrelroll is calculated as LP uLP VM (t) /dt = − PPN ρ(t) = ̇ (4.9) ωBR ωBR where uLP N is the PPN command project and implimentationed onto the maneuver plane. From eqn 4.2 the PP magnitude of uLP N is calculated as PP 2 N M vM uLP N = 1 − cos2 θM cos2 ψM cosθM cosψM (4.10) PP r By substituting eqn 4.9 into eqn 4.10, we obtain an alternative expression of ρ· (t), givenby It is noted that the radius of curvature decreases as long as the missile is heading toward the target: 2 NM vM ρ· (t) = 1 − cos2 θM cos2 ψM ≤ 0, f or − π/2 ≤ θM , ψM < π/2 (4.11) cosθM cosψM rωBR Department of Electrical Engineering, College of Engineering, Trivandrum 13 Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems In general the antiship missile attacks the target with sea skimming during the midcourse guidance phase. Hence the initial flight path angle θM (0) can be assumed zero, and then the shape of the barrelroll is determined by ψM (0). The barrelroll maneuver occurs counter clockwise in the maneuver plane. If ψM (0) > 0 and θM (0 = 0, then the missile arises from the sea surface. Since the radius of the curvature of the barrelroll is monotonously decreasing, the altitude of the missile is always higher than the initial altitude. For ψM (0)<0 the direction of the barrelroll axis should be reversed to avoid a crash into the sea. For ψM (0) = 0, the barrelroll maneuver does not occur so that a forced maneuver is required to generate some velocity component in the maneuver plane. Capturability of 3D BPNG: To treat more general situations, we consider the case that the speed of the antiship missile is time varying. In this case the acceleration of the missile, given by (5), is modified as AM = ax iM + ay jM + az kM = ax iM + (ΩL Ω) (4.12) M M M M The other equations of the motion of the antiship missile do not need correction. For capturability analysis of the proposed 3dimensional BPNG law, it is assumed that the initial missile heading satisfies cosθM (0)cosψM (0) > 0 (4.13) This condition requires that the antiship missile be launched toward the target. A missile guided by the proposed BPNG law for the barrelroll maneuver always captures a stationary target within a finite time, provided that the navigation constant is larger than one. Department of Electrical Engineering, College of Engineering, Trivandrum 14 Chapter 5 Conclusions In this paper 3dimensional evasive maneuvers of the antiship missile against the CIWS are investigated. By using a direct parameter optimization technique, trajectory optimization is performed with two methods for control parameterization: 1) the direction of the lateral acceleration is free, and 2) it is constrained to be orthogonal to the plane generated by the velocity vector and by the LOS vector. For the first method the optimal evasive maneuver of the missile turns out to be sinusoidal acceleration commands in the pitch and the yaw channel, but a consistent solution is not obtained. When the bias term of acceleration command is constrained, the solutions look more like a barrelroll, although we have not been able to obtain this unique solution again. Based on the optimization results, we propose a 3dimensional BPNG that generates a barrelroll maneuver to enhance the survivability of antiship missiles. In the proposed guidance law, a barrelroll maneuver is defined by the barrelroll frequency and the barrelroll axis. The barrelroll frequency determines the radius of the curvature of the barrelroll project and implimentationed on the maneuver plane so that it can be used as a trajectory design parameter. The target capture of the BPNG is guaranteed even for the case of timevarying missile speed. Furthermore a proper choice of the barrelroll direction enables the missile to avoid the possibility of a sea crash. By being compared with the results of the numerical optimization, the proposed guidance law has an evasion performance close to the case of the constrained maneuver treated in method 2, which is asignificant improvement over the evasion performance of PPNG. Due to the time lag of the missile dynamics, large barrelroll maneuvers could increase the miss distance in reality. Hence the barrelroll frequency should be carefully selected, and the initial heading error must not be too large. 15 References [1] YoonHwan ,K., ChangKyung ,R., and MinJea,T.,“Guidance Synthesis for Evasive Maneuver of AntiShip Missiles Against CloseIn Weapon Systems” IEEE Transactions on Aerospace and Electronic Systems, Vol 46, No.3, 2010, pp.13761387 [2] Zarchan, P “Proportional navigation and weaving targets”. Journal of Guid ance,Control and Dynamics, 18, 5 (1995),969974. [3]Imado, F. and Uehara,“S.Highg barrel roll maneuvers against proportional navigation from optimal control viewpoint”.Journal of Guidance, Control, and Dynam ics, 21, 6 (1998),876881. [4] Imado, F. and Miwa, S. “Missile guidance algorithm against highg barrel maneuvers”. Journal of Guidance, Control, and Dynamics, 17, 1 (1994), 123128. 16 



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