A Novel Approach to the 2D Differential Geometric ...

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Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 167, pp. 34–40, 2007

A Novel Approach to the 2D Diﬀerential Geometric Guidance Problem By Chaoyong L I,1Þ Wuxing J ING,1Þ Zhiguo Q I2Þ and Hui W ANG2Þ 1Þ

Department of Aerospace Engineering, Harbin Institute of Technology, Harbin, P.R. China 2Þ Shanghai Electro-Mechanical Engineering Institute, Shanghai, P.R. China (Received March 22nd, 2006)

This paper presents a detailed study of the two-dimensional (2D) diﬀerential geometric (DG) guidance problem, as well as its iterative solution and initial conditions. The DG guidance curvature command is transformed from an arc length system to the time domain using the classical DG theory. Subsequently, an algorithm for commanded angleof-attack is developed to formulate the DG guidance system, whose iterative solution is established based on Newton’s iterative algorithm. Moreover, a ﬂight control system is presented using the classical PID controller so as to form the DG guidance and control system. Finally, a new necessary initial condition is deduced to guarantee the capture of a high-speed target. Simulation results demonstrate that Newton’s iterative algorithm works well and accurately in DG guidance problems and the proposed DG guidance law exhibits similar performance to the proportional navigation guidance (PNG) law in the case of intercepting a non-maneuvering target. However, the proposed method performs better than PNG in the case of intercepting a maneuvering target. Key Words:

1.

Missile Guidance, Diﬀerential Geometry, Newton Iteration, Initial Condition, Proportional Navigation

Introduction

Previous analytical studies on missile guidance problems are mostly based on the assumption that the missile follows a proportional navigation (PN) guidance command.1) As a result, researchers try to develop systems that solve coupled nonlinear ordinary diﬀerential equations or apply an optimal control method to design guidance commands. However, the resultant equations of these methods are usually very complex and costly due to the large dimension of the algebraic system. Without loss of accuracy and eﬃciency, Newton’s iterative algorithm and its variants are now of importance to compute these nonlinear algebraic equations.2) In two-dimensional (2D) engagements, the diﬀerential geometric (DG) description of a smooth curve makes use of calculus to quantify how a curve deviates locally from its linear approximation. The trajectories, in this case, are determined by the curvature commands applied normal to the unit tangent vectors of the missile and target. Classical DG curvature theory is a study of the characteristics of space curves in terms of their curvature and torsion commands. Therefore, a diﬀerent approach may be given to study missile guidance problems using DG formulations. However, there haven’t been many attempts on this subject since a three-dimensional (3D) pure PN (PPN) guidance law was derived by a proper formulation, which was found in terms of the geodesic1 and normal curvature2 of the missile’s path on the surface generated by the line-of-sight (LOS).3) Ó 2007 The Japan Society for Aeronautical and Space Sciences 1

A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. 2 Normal curvature is a quantity that characterizes the deviation of the surface at a point in the direction from its tangent plane and is the same in magnitude as the curvature of the corresponding normal section.

Notable work on DG guidance problems has been done by Chiou and Kuo.4–6) In their papers, the Frenet formula3 (Frenet-Serret equations)7) was introduced for missile guidance problems. The resultant DG guidance curvature command was shown to be a generalization of the PN guidance law, and valid for a certain set of initial conditions. Li and Jing8) examined the application of the DG guidance command to realistic tactical missile interception engagement. Ariﬀ 9,10) presented a novel DG guidance algorithm using involute information of the target’s trajectory. White11) studied the application of DG formulations to 2D interception engagement, developing and expressing the kinematics equations in DG terms. The results in their papers demonstrated that the DG guidance law performs better than the conventional PN guidance law in most cases. This paper diﬀers from prior work in three main aspects. First, it focuses on the application of a 2D DG guidance curvature command in realistic missile defense engagement. Moreover, a ﬂight control system is presented using the classical PID controller so as to form the DG guidance and control system. Second, Newton’s iterative algorithm is utilized to develop an iterative solution for the 2D DG guidance system so as to facilitate easy computation of the commanded angle-of-attack (AOA). Third, a new necessary initial condition of intercepting a high-speed target is derived based on the new assumption that the target/ missile speed ratio is bigger than one.

3

The Frenet formula describes motion of the moving co-ordinate system (the rate of change of the co-ordinate system along the curve as the point moves along the curve) in terms of the arc length and the geometric terms, such as curvature command, tangent vector and torsion command.

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C. L I et al.: A Novel Approach to the 2D Diﬀerential Geometric Guidance Problem vt

YI

#m ¼ pitch angle of missile’s body axis, and Target

G

yb

αm

X

P

o1

The corresponding equations for the above variables are m ¼ m0 Ptp =ðIs gÞ

θm

xv

# m ¼ m þ m m ¼ tan1 ðvmy =vmx Þ;

ϑm

Missile

XI

O

Fig. 1. Geometric description of the engagement.

2.

m ¼ rotation angle of missile’s trajectory.

xb

yv

35

where vmx and vmy are the x; y components of missile’s velocity in the inertial frame, P is the thrust magnitude, m0 is the initial missile mass, Is is the impulse, g is gravitational acceleration,1) and tp is the burn time. It should be noticed that the above relations could also be used to describe the motion of the target, simply by changing the corresponding subscript m to t.

Formulation of the Engagement 3.

The assumption of point mass is made for both the missile and target over the time interval in order to simplify this study. Figure 1 illustrates the proposed engagement of a realistic surface-to-air missile interception scenario. The inertial frame is deﬁned as XI OYI , whose origin is ﬁxed at the missile launch point, the axis OXI in the local horizontal plane points in the launch direction, and the axis OYI coincides with the local vertical direction of the launch point O. The body frame is deﬁned as xb o1 yb , whose origin o1 is ﬁxed at the missile’s center of mass, the axis o1 xb points forward along the missile’s longitudinal centerline, and the axis o1 yb points upward and orthogonal to the axis o1 xb . The velocity4 frame is deﬁned as xv o1 yv , with the axis xv o1 pointing forward and coinciding with the missile’s velocity, while the axis o1 yv points upward and orthogonal to the o1 xv axis. Without loss of generality, the thrust P, gravitation G, and atmospheric force R would be considered over the engagement, in which the thrust acts on the direction of the o1 xb axis of the body frame, gravitation acts on the opposite direction of the OYI axis of the inertial frame, and in the velocity frame, the atmospheric force can be simply expressed as: # " # " X ðCx0 þ Cxa m 2 Þv2 S=2 R¼ ; ¼ Y Cy a v2 Sm =2 where X and Y are the atmospheric drag and lift, respectively, Cx0 , Cx , and Cy are the atmospheric coeﬃcients, is the air density,1) v is the free-stream speed, S is the reference area, and m is the AOA of the missile. Therefore, the motion of the missile can be formulated in the inertial frame as follows: mv_mx ¼ P cos #m X cos m Y sin m ; ð1Þ mv_my ¼ P sin #m mg X sin m þ Y cos m where m = mass of missile,

4

Throughout this paper, the word velocity will only be used to designate a vector quantity; the corresponding scalar will be denoted as speed.

Diﬀerential Geometric Guidance and Control System

Previous treatment of the DG guidance problem has been considered only in the arc length system.4–6) With this restriction, the guidance command statement is not practical because the arc length cannot be measured by onboard sensors. In this section, the DG guidance system is presented in the time domain using classical DG theory. Moreover, a ﬂight control system (FCS) for DG guidance law is introduced to study the characteristics of the DG guidance and control system. The 2D DG guidance curvature command in the arc length system is:4) m ¼ N 2 t

cos t r 0 q0 þA ; cos m cos m

ð2Þ

where ð0 Þ denotes the derivative with respect to the arc length s along the missile’s trajectory, N is the target/ missile speed ratio, t is the curvature command of the target’s trajectory, t and m are the lead angles of the target and missile, respectively, q0 is the angular rate of the LOS (LOSR) with respect to s, A is the DG guidance gain, and r 0 is the closing speed with respect to s. Before applying the DG guidance curvature command to the engagement, it must be transformed from the arc length system to the time domain. This means that the derivations of all the variables in Eq. (2) must be obtained in respect to time, not the arc length. According to the classical DG curvature theory,7) we have: r_ dr dr dt r0 ¼ ¼ ¼ ð3Þ ds dt ds vm dq dq dt q_ q0 ¼ ¼ ¼ ; ð4Þ ds dt ds vm where ðÞ denotes the derivative with respect to time, and r_ and q_ are the closing speed and LOSR, respectively. Substituting Eqs. (3) and (4) into Eq. (2), we obtain the guidance curvature command in the time domain in the form of:

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Trans. Japan Soc. Aero. Space Sci.

m ðtÞ ¼ N 2 t

cos t r_q_ þA 2 : vm cos m cos m

ð5Þ

Furthermore, we have:7) m ¼ jrm 00 j; where rm 00 ¼

2 d2 rm d2 t dt ¼ v þ am m 2 2 ds ds ds

ð6Þ

and d2 t dvm dvm ¼ vm 2 ¼ vm 3 ¼ vm 3 v_m ; 2 ds ds dt

ð7Þ

where vm and am ¼ ½v_mx v_my T are the missile’s velocity and acceleration vectors, respectively, v_m is the rate of change in missile speed, and:1) v_m ¼ ðP cos m X mg sin m Þ=m:

ð8Þ

Therefore, the current curvature of missile’s trajectory in the time domain is: m ¼ jam v_m tm j=vm 2 ;

ð9Þ

where tm is the unit tangent vector of the missile’s trajectory. Consequently, the commanded AOA, cmd , is derived by letting the DG guidance curvature command be equal to the current trajectory curvature, thus: cmd ¼ fcmd : fm ðtÞ ¼ m gg:

ð10Þ

The output of the 2D DG guidance system is designated as cmd , which can be developed by substituting Eqs. (1)– (9) into Eq. (10). Thus, the main function of the DG FCS is to make the achieved AOA, m , ‘‘track’’ the commanded AOA, cmd , as closely as possible. More speciﬁcally, since it is impossible to measure AOA onboard, the commanded pitch angle of the missile’s body axis, #cmd ¼ m þ cmd , shall be designed as the feedback angle in a practical DG FCS. The block diagram of the FCS is given in Fig. 2. It is apparent that the proposed FCS consists of a PIDtype controller of the form: Zt u ¼ kp e þ ki eðÞd kd !y ; ð11Þ 0

where kp ; ki , and kd are PID gains, and e ¼ #cmd #m ; !y ¼ #_ m :

PI compensator

ki / s

ϑcmd

kp

−

ϑm

+ +

− kd

u

Actuator

ωy Rate gyro

δ

Missile dynamics

ϑm

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It should be pointed out that the PI part of Eq. (11) also works as a compensator in the control loop, so as to eliminate the steady-state error for a step input of the angle. Although an accelerometer is not included in the proposed FCS, which is diﬀerent from the conventional FCS,1) it is still needed in the DG guidance system, as indicated in the calculation of the ﬁrst two terms on the right-hand side of Eq. (8). Moreover, the rate gyro is used to act as a damper in order to stabilize the pitch rate, !y , which is common in many high-performance command homing missiles. 4.

Iterative Solution

It is well known that Newton’s iterative algorithm plays an important role in scientiﬁc computation,2) especially for nonlinear systems. Newton’s iteration is the only way to compute a solution. Note that the solution of the commanded AOA presented in the last section (i.e., Eq. (10)) is a theoretical algorithm, which can not be implemented onboard directly. Otherwise, the huge computational burden that is involved will cause the system to collapse. To facilitate easy computation of the commanded AOA, Newton’s classical iteration method is introduced in this section to develop an iterative solution for the DG guidance system. As previously stated, in connection with Eq. (10), all the involved commands m ðtÞ and m are all functions of cmd . Therefore, the desired function of the iterative algorithm is deﬁned as:2) f ði Þ ¼ m ðtÞ m :

ð12Þ

Hence, Newton’s iterative algorithm of Eq. (9) is: iþ1 ¼ i f ði Þ= f 0 ði Þ;

ð13Þ

f ðiþ1 Þ f ði Þ : iþ1 i

ð14Þ

where f 0 ði Þ ¼

The subscript i and i þ 1 mean the i-th and ði þ 1Þth iterations, respectively, and i means the i-th iterative solution of cmd . Consequently, applying the iterative algorithm derived to the 2D DG guidance system yields the iterative DG guidance system: f ði Þ ¼ m ðtÞ m cmd ¼ cmd : : ð15Þ iþ1 ¼ i f ði Þ=f 0 ði Þ Obviously, the guidance system indicated by Eq. (15) does not require the evaluation of the nonlinear function, and maintains the converging rate and accuracy of Newton’s method. 5.

Capture Condition

ω˜ y

Fig. 2. Illustration of a DG ﬂight control system.

Previous studies on initial conditions for DG guidance curvature command are formulated based on the assumption that the missile has a speed advantage over the target (i.e.,

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C. L I et al.: A Novel Approach to the 2D Diﬀerential Geometric Guidance Problem

YI

vm vt r er

tm

Missile

tt

ηt

ηt

vt

r ηm

rt

M

r

ηm

vm

vm

M

a

XI

O

vt

vt

T

T

Target

eq rm

37

b

ηt

ηt

T

vt

T

Fig. 3. Geometry and deﬁnition of DG terms.

r

N < 1). However, in realistic missile defense engagement, the target usually has a speed advantage over the missile (i.e., N > 1). Therefore, a practical initial condition should be developed to guarantee the interception of a high-speed target, which is addressed in this section. Referring to Fig. 3, we have: rm ¼ rt r:

ð16Þ

Taking the derivative of the last equation with respect to s and expressing r0 in terms of its normal (eq ) and tangential (er ) components, we have: tm ¼ Ntt r 0 er rq0 eq ;

ð17Þ

where tt is the unit tangent vector of the target’s trajectory. The normal (eq ) and tangential (er ) components of Eq. (17) are: 0

r ¼ N cos t cos m

ð18Þ

rq0 ¼ N sin t sin m :

ð19Þ

Again by taking derivatives of Eq. (17) with respect to s, applying the Frenet formula,7) and separating the normal and tangential components, the following relationship results: 00

2

0 0

rq þ 2r q ¼ N t cos t m cos m :

ð20Þ

Therefore, the closed-form solution of LOSR can be derived by applying Eq. (2) to Eq. (20):

M c

r ηm

ηm

vm

vm

M d

Fig. 4. Possible miss regions of the engagement.

N cos t ¼ cos me ;

ð23Þ

where me is the angle between the LOS and tme . As depicted in Fig. 4, there are four possible miss regions in the proposed engagement. In scenario 4-c, we have cos t > 0, so from Eq. (23), we know that a miss would not happen in this scenario, and the same logic applies to scenario 4-d. Regarding scenario 4-a, since 0 < m < =2 and N > 1, then from Eq. (23), we have: cos1 ð1=NÞ < t < =2:

ð24Þ

From Eq. (19), it is clear that this scenario is the miss region related to q0 > 0. Using Eq. (23), we have: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin me ¼ 1 N 2 þ N 2 sin2 t and thus ðrq0 Þme ¼ N sin t þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 N 2 þ N 2 sin2 t :

ð25Þ

ð21Þ

Substituting Eq. (24) into Eq. (25), the minimum value of ðrq0 Þme can be derived as follows: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðrq0 Þme N 2 1 > 0: ð26Þ min ¼

where r0 and q0 0 are the initial value of LOS and LOSR, respectively. It is worth stressing that r 0 ¼ 0 and q0 ¼ 0 cannot occur simultaneously, otherwise, from Eq. (17), we have:

Similarly, scenario 4-b is the miss region related to q0 < 0, and the maximum value of ðrq0 Þme in this scenario is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ð27Þ ðrq0 Þme max ¼ N 1 < 0:

q0 ¼ q0 0 ðr0 =rÞA2 ;

Before a miss occurs, we have:

tm ¼ Ntt : Obviously, a contradiction results as N > 1 and tm , tt are unit vectors. Therefore, when a miss occurs, we have:

d ðrq0 Þ ¼ r 0 q0 þ rq00 : ds

ð28Þ

r 0 ¼ 0; r 6¼ 0; q0 6¼ 0:

According to Eq. (21), if q00 > 0, then q0 > 0, for A > 2; and for q00 < 0, we have from Eq. (28):

The unit tangent vector tme is deﬁned as the ﬁctitious missile-missing velocity vector, such that if the missile velocity coincides with it, a miss occurs:

d ðrq0 Þ ¼ r 0 q0 þ rq00 < 0: ds

tm ¼ tme ;

r 0 ¼ 0;

r 6¼ 0:

Substituting the last relation into Eq. (18), we have:

ð22Þ

From Eq. (26), we note that if q0 0 > 0, then: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 < r 0 q0 0 < N 2 1:

ð29Þ

ð30Þ

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Trans. Japan Soc. Aero. Space Sci.

Moreover, from Eq. (29), we know that rq0 < r0 q0 0 , so rq0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ would never reach N 2 1. This means that a miss would not happen during the entire engagement. In the same manner, in the case of q0 0 < 0, the following relation must be satisﬁed to guarantee an interception: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð31Þ N 2 1 < r0 q0 0 < 0:

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Table 1. Comparison of the interception performance. MD (m)

Time (s)

Gain

N-M

M

N-M

M

N-M

M

V-PPN

2.37

7.61

26.52

26.45

4

2

V-TPN

1.31

4.68

26.35

26.43

1

2

DG

1.27

1.73

29.58

30.68

4

8

Hence, the necessary initial condition for the DG guidance curvature command is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jr0 q0 0 j < N 2 1: ð32Þ We conclude this section by noting that for a high-speed target, the proposed DG guidance curvature command would be eﬀective only after Eq. (32) is satisﬁed at the beginning of the guidance envelop. Otherwise, a miss would occur. Note that the proposed initial condition also works for the other guidance schemes, which have a similar LOSR response to the DG guidance curvature command. 6.

Simulation Results

Simulations against non-maneuvering and maneuvering targets are presented in this section, together with a comparison between the interception performance obtained with the DG guidance law and the benchmark guidance law. The initial conditions and constants for all of the following simulations are speciﬁed and listed below in units of meters, degrees, and seconds. Initial position of missile (m):rm0 ¼ ð0; 0Þ Initial velocity of missile (m/s):vm0 ¼ ð0; 0Þ Initial position of target (m): rt0 ¼ ð50000; 50000Þ Initial velocity of target (m/s): vt0 ¼ ð1000; 1000Þ Initial mass of missile (kg): mm0 ¼ 1000 Thrust (N):P ¼ 65000 Burn time (s): 15; Impulse (s): 250 Mass of target (kg): mt0 ¼ 500; No thrust for target Reference area of missile and target is (m2 ): S ¼ 0:2 Time constant of the guidance system (s): ¼ 0:5 Maximum permissible load factor (g): 20 Simulation step (s): 0.01 Initial launch angle of the missile (deg): m0 ¼ 75 Atmospheric coeﬃcient: Cx0 ¼ 0:0774, Cx ¼ 0:00084 (1/deg2 ), Cy ¼ 0:0333 (1/deg) The benchmark guidance law mentioned above is deﬁned as the gain-varying true PN (V-TPN) guidance law and the gain-varying PPN (V-PPN) guidance law,12) yielding: Nvppn ¼ N 0 ½vmy q_ vmx q_T ; Nvtpn ¼ N 0 vm ½ry q_

rx q_T ;

ð33Þ

where Nvppn is the commanded acceleration produced by the V-PPN guidance law; Nvtpn is the commanded acceleration produced by the V-TPN guidance law, rx and ry are the LOS vector components in the inertial frame, N 0 is eﬀective navigation ratio, and:12)

Fig. 5. Time history of the speed ratio in both cases.

N0 ¼

k3 Tgo 3 ; k2 Tgo 2 2kTgo þ 2 21

¼ ekTgo ;

ð34Þ

Tgo ¼ r=r_ where k is the guidance gain and Tgo is the time-to-go before impact. A comparison of the diﬀerent guidance schemes is presented and listed in Table 1, where the miss distance (MD) means the closest distance between target and missile before its divergence, time denotes the engagement time, N-M denotes the non-maneuvering target case, and M denotes the maneuvering target case, of which the AOA of the target is time-variant. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 ¼ 0:7616. For the N-M case, jr0 q0 0 j ¼ 0:0905 < pN ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 0 For the M case, jr0 q0 j ¼ 0:1045 < N 2 1 ¼ 0:7623. From Table 1, it is clear that the DG guidance law has similar performance to the benchmark guidance law in the case of intercepting a non-maneuvering target. However, it performs better than the other guidance schemes in the case of intercepting a maneuvering target. Moreover, regardless of the type of target, it has a longer engagement time. The time history of the target/missile speed ratio is illustrated in Fig. 5. We can see that the assumption of constant speed is not valid in this realistic scenario, neither is the assumption that the missile has a speed advantage over the target.2–4) However, the DG guidance law can still guarantee capture. The time history of solution bias, which is deﬁned as the error between the theoretical solution and the iterative solution, is presented in Fig. 6. Note that the proposed iterative

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C. L I et al.: A Novel Approach to the 2D Diﬀerential Geometric Guidance Problem

39

Fig. 6. Time history of the solution bias in both cases.

Fig. 8. Time history of the commanded accelerations for the M case.

Fig. 7. Time history of the commanded accelerations for the N-M case.

Fig. 9. Time history of the LOSR response.

algorithm converges quickly at the beginning of the guidance envelope, and the solution bias is in a negligible range in both cases, which indicates that Newton’s iterative algorithm is viable and eﬀective for practical application in DG guidance problems. The time histories of the commanded accelerations are presented in Figs. 7 and 8. It is clear that the DG guidance law compensates for target maneuvering and engages its maximum command acceleration at the beginning of the guidance envelope in both cases, similar to the benchmark guidance law. However, in the ﬁnal stage of engagement (i.e., endgame), the commanded accelerations produced by the benchmark guidance law undergo a huge rise that almost approaches saturation, whereas the commanded accelerations produced by the DG guidance law indicate a stable tendency and maintained small magnitude. The time history of the LOSR response in the M case is depicted in Fig. 9. We note that the LOSR responses produced by all the guidance schemes reach their maximum value at the beginning of the guidance process. However, the LOSR response produced by the DG guidance law decreases gradually, while remaining steady and almost approaches zero in the ﬁnal stage of the envelope, whereas

that produced by the PN guidance law increases dramatically, which indicates that the DG guidance curvature command is more eﬀective for restraining the LOSR response than the PN guidance law. 7.

Conclusion

The results of this paper clearly indicate that the twodimensional diﬀerential geometric (DG) guidance law is viable and eﬀective for realistic missile defense engagement. In particular, Newton’s iterative algorithm works eﬃciently and accurately in the DG guidance problem. According to the time histories of command accelerations and LOSR responses, we shall note that they have similar responses to each other, and it is clear that the DG guidance curvature command is more sensitive to the change in LOS than the PN guidance law, thus leading to better performance. Furthermore, according to Eq. (8), it is clear that the term t can be ignored based on the assumption that there’s no thrust for the target during the homing phase, and in this case, the DG guidance curvature is theoretically a generalization of the gain-varying PN guidance law.

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Trans. Japan Soc. Aero. Space Sci.

Acknowledgments Chaoyong Li is indebted to Dr. George M. Siouris of Dayton, OH, USA, for his valuable suggestions on missile guidance and control systems, as well as for his careful reviewing of this paper. The authors also appreciate an anonymous reviewer’s constructive comments and suggestions.

References 1) Siouris, G. M.: Missile Guidance and Control Systems, SpringerVerlag, New York, 2004. 2) Ortega, J. M. and Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. 3) Adler, F.P.: Missile Guidance by Three-Dimensional Proportional Navigation, J. Appl. Phys., 27(1956), pp. 500–507. 4) Kuo, C. Y. and Chiou, Y. C.: Geometric Analysis of Missile Guidance Command, IEE Proc-Control Theory and Applications, 147(2000), pp. 205–211. 5) Chiou, Y. C. and Kuo, C. Y.: Geometric Approach to Three-Dimensional Missile Guidance Problem, J. Guid. Control Dynam., 21(1998), pp. 335–341.

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6) Kuo, C. Y., Soetanto, D. and Chiou, Y. C.: Geometric Analysis of Flight Control Command for Tactical Missile Guidance, IEEE Trans. Control Syst. Technol., 9(2000), pp. 234–243. 7) Struik, D. J.: Lectures on Classical Diﬀerential Geometry, Dover, New York, 1998. 8) Li, C. Y., Jing, W. X., Wang, H. and Qi, Z. G.: Application of 2D Diﬀerential Geometric Guidance to Tactical Missile Interception, Proceedings of IEEE Aerospace Conference, Big Sky, MT, U.S.A., 2006. 9) Ariﬀ, O., Zbikowski, R., Tsourdos, A. and White, B. A.: Diﬀerential Geometric Guidance Based on the Involute of the Target’s Trajectory: 2-D Aspects, Proceedings of the 2004 American Control Conference, Boston, 2004, pp. 3640–3644. 10) Ariﬀ, O., Zbikowski, R., Tsourdos, A. and White, B. A.: Diﬀerential Geometric Guidance Based on the Involute of the Target’s Trajectory, J. Guid. Control Dynam., 28(2005), pp. 990–996. 11) White, B.A., Ariﬀ, O., Zbikowski, R. and Tsourdos, A.: Direct Intercept Guidance Using Diﬀerential Geometric Concepts, Proceedings of AIAA Guidance and Navigation Control Conference, August, 2005, AIAA Paper 2005-5969. 12) Joseph, Z. and Asher, B.: New Proportional Navigation Law for Ground-to-air Systems, J. Guid. Control Dynam., 26(2003), pp. 822– 825.