Iterative solution to differential geometric guidance problem Chaoyong Li and Wuxing Jing Department of Aerospace Engineering, Harbin Institute of Technology, Harbin, People’s Republic of China, and

Hui Wang and Zhiguo Qi The 8th Institute of Shanghai Academy of Spaceflight Technology, Shanghai, People’s Republic of China Abstract Purpose – To study the application of three-dimensional differential geometric (DG) guidance commands to a realistic missile defense engagement, and the application of the Newton’s iterative algorithm to DG guidance problems. Design/methodology/approach – The classical differential geometry theory is introduced firstly to transform all the variables in DG guidance commands from an arc length system to the time domain. Then, an algorithm for the angle-of-attack and the sideslip angle is developed by assuming the guidance curvature command and guidance torsion command equal to its corresponding value of current trajectory. Furthermore, Newton’s iteration is utilized to develop iterative solution of the stated algorithm and the two-dimensional DG guidance system so as to facilitate easy computation of the angle-of-attack and the sideslip angle, which are formulated to satisfy the DG guidance law. Findings – DG guidance law is viable and effective in the realistic missile defense engagement, and it is shown to be a generalization of gain-varying proportional navigation (PN) guidance law and performs better than the classical PN guidance law in the case of intercepting a maneuvering target. Moreover, Newton’s iterative algorithm has sufficient accuracy for DG guidance problem. Originality/value – Provides further study on DG guidance problem associated with its iterative solution. Keywords Aircraft navigation, Control technology Paper type Research paper

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(2001). In their papers, the Frenet formula of classical differential geometry theory (Struik, 1998) was introduced to planar and 3D missile guidance problems. The resultant DG guidance curvature command was shown to be a generalization of PN guidance law for maneuvering targets, and valid for a certain set of initial conditions. Zhang et al. (2001, 2002) introduced a robust geometric guidance algorithm using the Lyapunov stability theory and the classical differential geometry formulations, the derived guidance algorithm performs better than the PN guidance law, especially in the final stage of the engagement. Ariff et al. (2004, 2005) presented a novel DG guidance algorithm using the information of the involute of the target’s trajectory. The results indicate that the proposed guidance algorithm performs better than the PN guidance law in the case of intercepting a maneuvering target. However, regardless of the types of targets, it has a longer engagement time. White et al. (2005) examined the application of DG formulations to a planar interception engagement, whose kinematics are developed and expressed in DG terms. The resulting guidance law does not rely on local linearization and can be shown to produce guidance trajectories which are similar to that by the PN guidance law for the straight line interception of non-maneuvering targets. This paper differs from the prior works in three main aspects. First, it focuses on the application of 3D DG

Aircraft Engineering and Aerospace Technology: An International Journal 78/5 (2006) 415– 425 q Emerald Group Publishing Limited [ISSN 1748-8842] [DOI 10.1108/00022660610685800]

Chaoyong Li would like to express his deep appreciation to Dr George M. Siouris of Dayton, OH, USA, for his kind support, encouragement, and valuable suggestions on missile guidance problems and Frenet formulas.

1. Introduction Previous analytical studies on missile guidance problems are based on an assumption that the missile follows the proportional guidance command (Siouris, 2004). Then, the researchers try to solve a system of 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 iteration and its variants are of center importance now to computer these nonlinear algebraic equations (Ortega and Rheinboldt, 1970). In the past decades, there have not been many attempts to use differential geometric (DG) formulations for the missile guidance problems. Adler (1956) extended the planar PN guidance law to three dimensions, and the three-dimensional (3D) pure proportional navigation (PPN) guidance law was derived by a proper formulation which was found in terms of the geodesic and normal curvature of the missile’s path on the surface generated by the line-of-sight (LOS). The notable work on DG guidance problems were done by Chiou and Kuo (1998), Kuo and Chiou (2000) and Kuo et al.

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Aircraft Engineering and Aerospace Technology: An International Journal

Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

Volume 78 · Number 5 · 2006 · 415 –425

Figure 2 Geometry of the frames and the guidance angles

guidance commands to a realistic missile defense engagement. To achieve this, the classical differential geometry theory is introduced to transform the guidance commands from an arc length system to the time domain. Second, the classical Newton’s iteration is utilized to develop iterative solution of the two-dimensional (2D) and 3D DG guidance systems so as to facilitate easy computation of the guidance angles, i.e. angle-of-attack and the sideslip angle, which are formulated to satisfy the DG guidance law. Third, a comparison of the interception performance of the derived 3D iterative DG guidance law and the classical 3D PN guidance law is presented in the case of intercepting non-maneuvering and maneuvering targets. Furthermore, the effect of the guidance gain to the performance of the iterative DG guidance law is investigated.

3
 2 C x0 þ Cax ða2 þ b2 Þ rv2 S=2 7 6 7 6 7 Cay rv2 S a=2 7 6 Fa ¼ 6 7 4 Y 5¼6 5 4 2Cay rv2 S b=2 Z 2

2. Dynamic formulations of the engagement The formulation of the engagement is briefly presented in this section, describing the motion of the missile and target. In particular, an assumption of point mass is made for both the missile and the target in order to simplify this guidance problem. As shown in Figure 1, the proposed engagement is a realistic surface-to-air tactical missile defense scenario. Without loss of generality, the thrust P, gravitation G and the atmospheric force Fa are considered during the entire engagement, in which case, as shown in Figure 2, the thrust acts on the direction of the x-axis of the body frame, i.e. oxbybzb in Figure 2, while gravitation acts on the opposite direction of the OYI axis of the inertial frame (Zhang, 1996); in the velocity[1] frame, i.e. oxvyvzv in Figure 2, the atmospheric force can be simply expressed as:

2X

3

2

where X, Y, and Z are the atmospheric drag, lift and side force, respectively; Cx0, C ax ; and Cay are the atmospheric coefficients during the engagement; r is the air density (Siouris, 2004); v is the free-steam speed; S is the reference area; a is the angle-of-attack; b is the sideslip angle. Therefore, the motion of the missile can be formulated in the inertial frame as follows: 2 3 2 3 P 0 6 7 6 7 I I 6 7 7 m_vm ¼ m6 ð1Þ 4 2g 5 þ C v F a þ C b 4 0 5 0 0 where m is the mass of missile, and:

Figure 1 Geometry of the engagement

m¼

m0 2 Pt p ðI s gÞ

where P is the magnitude of the thrust; C Iv is the transformation matrix from the velocity frame to the inertial frame; C Ib is the transformation matrix from the body frame to the inertial frame; m0 is the initial mass of the missile; vm is the missile’s velocity in the inertial frame; Is is the impulse; tp is the burn time; g is the gravitational acceleration (Siouris, 2004). It should be pointed out that the above equations can also be used to describe the motion of the target, simply by changing the corresponding subscript m to t.

3. Application of differential geometric guidance commands Previous treatments of the DG guidance problems have been considered only in the arc length system, when so restricted, the statement of the guidance law is not practical. In this section, the classical differential geometry theory is introduced to transform the guidance commands from an arc length system to the time domain. Moreover, an algorithm for the angle-of-attack command and the sideslip angle command is developed in the time domain to form the DG 416

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Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

Volume 78 · Number 5 · 2006 · 415 –425

guidance system, the iterative solution of which is studied in the next section. The DG guidance curvature command ksm and torsion command tsm in the arc length system are (Chiou and Kuo, 1998):

ksm ¼ N 2 kt tsm ¼ km

e v ðtÞ0 ¼ ð½km ðtÞn m ðtÞ 2 N 2 kt ðtÞn t ðtÞÞ £ e r 2 ð2_rvðtÞ=v2m þ r vðtÞ0 Þe v ðtÞ
vm Þ=ðr vðtÞÞ Moreover, according to the definitions of normal vector, and the binormal vector (Struik, 1998), the following relations are resulted: ða m 2 t m v_ m Þ n m ðtÞ ¼ ja m 2 t m v_ m j

0

n t · ðe v £ e r Þ Ar v 2 n m · ðe v £ e r Þ n m · ðe v £ e r Þ

ð2Þ

t m · ðe v £ e r Þ n m · ðe0v £ e r Þ nm · er 2 þv ð3Þ b m · ðe v £ e r Þ b m ðe v £ e r Þ b m · ðe v £ e r Þ

n t ðtÞ ¼

0

where ( ) means the derivative with respect to the arc length s of the missile’s trajectory; the superscript s means the corresponding value in the arc length system; N is the target/ missile speed ratio; km and kt are the current curvatures of the missile and target’s trajectories with respect to s, respectively; nt and nm are the unit normal vectors of the missile and target’s trajectories with respect to s, respectively; bm is the unit binormal vector of the missile’s trajectory with respect to s; er is the unit vector along the LOS; tm and tt are the unit tangent vectors[2] of the missile and target’s trajectory, respectively; r0 is the closing speed with respect to s; v is the angular rate of LOS (LOSR) with respect to s; A is the DG guidance gain; ev and e0v are the unit LOSR vector and it is derivative, all with respect to s, and (Chiou and Kuo, 1998): ev ¼ e0v ¼

b m ðtÞ ¼ t m ðtÞ £ n m ðtÞ Also, the curvature and the torsion of the current trajectory with respect to time is: ja t 2 t t v_ t j kt ðtÞ ¼ ð5Þ v2t

km ðtÞ ¼ tm ðtÞ ¼

½ðkm n m 2 N 2 kt n t Þ £ e r 2 ð2r 0 v þ r v0 Þe v
rv

Before applying the DG guidance commands to the proposed engagement, they must be transformed from the arc length system to the time domain, which means the derivations of all the variables in equations (2) and (3) must be taken with respect to the time, not the arc length, which is impossible to be measured or sensed directly by the onboard sensors. According to the classical differential geometry theory (Struik, 1998), for arbitrary variable R in the arc length system, we have: _ dR dR dt R R ¼ ¼ ¼ ds dt ds vR

v¼

ktm ¼ N 2 kt ðtÞ ð4Þ 2

ttm ¼ km ðtÞ

r_ dr dr dt ¼ ¼ ds dt ds vm

þ

dq dq dt vðtÞ ¼ ¼ ds dt ds vm

e v ðtÞ ¼

jvm £ a m j2

ð7Þ

n t ðtÞ · ðe v ðtÞ £ e r Þ n m ðtÞ · ðe v ðtÞ £ e r Þ

A_rvðtÞ 1 n m ðtÞ · ðe v ðtÞ £ e r Þ v2m

ð8Þ

t m · ðe v ðtÞ £ e r Þ n m ðtÞ · ðe0v ðtÞ £ e r Þ 2 b m ðtÞ · ðe v ðtÞ £ e r Þ b m ðtÞ · ðe v ðtÞ £ e r Þ

vðtÞ n m ðtÞ · e r vm b m ðtÞ · ðe v ðtÞ £ e r Þ

ð9Þ

where the superscript t means the corresponding value in the time domain. Consequently, the angle-of-attack and sideslip angle can be derived by letting curvature of the current trajectory km(t) equal to the guidance curvature command ktm ; and torsion of the current trajectory tm(t) equal to the guidance torsion ttm ; formulated as: )) " # (" # ( t km ¼ km ðtÞ a a ð10Þ ¼ : ttm ¼ tm ðtÞ b b

where (t) means the corresponding value in the time domain; q is the angle between the LOS and the reference axis. Furthermore:

v0 ¼

vm · ða m £ a_ m Þ

ð6Þ

where vmy is the y component of the missile’s velocity in the inertial frame. Combining all of the above relations in this section, then the guidance commands in the time domain are:

where (˙) denotes the derivative with respect to the time, vR is the rate of change of the referenced arc length. Therefore, we have: r0 ¼

ja m 2 t m v_ m j v2m

where am and at are the missile and target’s acceleration vectors, respectively; v_ m and v_ t are the rates of change of the missile and target’s longitudinal speed, and v_ m is given by Zhang (1996): P cos a cos b 2 X 2 mg sin um v_ m ¼ m In this expression, um is the pitch angle of the missile’s trajectory, and: arcsin vmy um ¼ vm

½ðt m 2 Nt t Þ £ e r
rv

0

ða t 2 t t v_ t Þ ja t 2 t t v_ t j

v_ðtÞ vðtÞ_vm 2 v3m v2m

½ðt m 2 Nt t Þ £ e r
vm rv 417

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Aircraft Engineering and Aerospace Technology: An International Journal

Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

Volume 78 · Number 5 · 2006 · 415 –425

The output of the DG guidance system is designed as a and b, which can be derived by applying equations (1)-(9) to equation (10). Note that the preceding guidance system consists of feedback angles, which are formulated to satisfy the DG guidance law. It is should be pointed out that the proposed guidance system is more convenient for the flight control system in a realistic 6DOF (degree-of-freedom) model than the conventional guidance systems, which are feedback accelerations.

Furthermore, the proposed Newton’s algorithm could also be utilized for 2D DG guidance problem, so as to formulate the 2D DG guidance system, formulated as follows: ( ( )) f ðai Þ ¼ ktm 2 km ðtÞ ð15Þ a¼ a: aiþ1 ¼ ai 2 f ðai Þ=f 0 ðai Þ where the variables and functions in the preceding equation is the corresponding value in the 2D case, and:

4. Iterative solution to differential geometric guidance system

f 0 ðai Þ ¼

Obviously, the guidance system indicated by equations (14) and (15) consist also of feedback angles. However, they do not require the evaluation of the nonlinear function, i.e. equation (9), yet maintain the converging rate and accuracy of Newton’s method, which is investigated in the next section.

It is acknowledged that Newton’s iterative algorithm plays an important role in scientific computation. For large linear systems with a number of unknown variables exceeding the computational costs of a direct method may become prohibitive. However, for nonlinear systems, Newton’s iteration is the only way to compute a solution. Note that the solution of the DG guidance system presented in the last section, i.e. equation (10), is a theoretical algorithm, which cannot be implemented directly by an onboard computer. Otherwise, the huge computational burden is involved to cause the system to collapse. To facilitate easy computation of the guidance angles, in this section, the classical Newton’s iteration is introduced to develop an iterative solution of the preceding guidance system. Note that, in equation (10), all the involved commands ktm , km(t), ttm and tm(t) are functions of a and b. Therefore, the desired vector and desired function of the iterative algorithm are defined as: h iT x¼ a b "

ktm 2 km ðtÞ

f ðxi Þ ¼

5. Simulations and results For 3D engagements, simulations against non-maneuvering and maneuvering targets are presented in this section, together with a comparison of the interception performance of the 3D IDG guidance law with the classical 3D PN guidance law (Sarkar et al., 2003), pure PN (PPN) guidance law and true PN (TPN) guidance law, and a 3D PPN guidance law with specified algorithm for the guidance gain. 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 and target (m): rm0 ¼ (0;0;0); rt0 ¼ (50,000;50,000;50,000); . initial velocity of missile and target (m/s): vm0 ¼ (0;0;0); vt0 ¼ 2(1,000;1,000;1,000); . initial mass of missile and target (Kg): mm0 ¼ 1,000; mt0 ¼ 500; . thrust for missile (N): P ¼ 65,000; impulse of missile(s):Is ¼ 250; burn time (s): tp ¼ 15; no thrust for target; . the reference area for both missile and target (m2): S ¼ 0.2; . atmospheric coefficients: Cx0 ¼ 0.0774; Cax ¼ 0.00084 (1/deg2) C ay ¼ 0.0333 (1/deg); . time constant of guidance system(s): t ¼ 0.5; . simulation step (s):0.01; . effective radius of active radar (Km): 40; . initial launch azimuth angle (deg): 244.6; and . initial launch pitch angle (deg): 71.8.

# ð11Þ

ttm 2 tm ðtÞ

Hence, the Newton’s iterative algorithm of equation (9) is (Ortega and Rheinboldt, 1970): xiþ1 ¼

xi 2 f ðxi Þ f 0 ðxi Þ

ð12Þ

where: 2

›f 1 ðxi Þ ›x1

6 f 0 ðxi Þ ¼ 4 ›f 2 ðxi Þ ›x1

›f 1 ðxi Þ ›x2

3 7

›f 2 ðxi Þ 5 ›x2

½ f ðaiþ1 Þ 2 f ðai Þ
: ðaiþ1 2 ai Þ

ð13Þ

The subscript i and i þ 1 mean, respectively, the ith and (i þ 1)th iterations; ai and bi mean the ith iterative solutions of a and b, respectively. Consequently, applying the derived iterative algorithm to 3D DG guidance system, then the iterative DG (IDG) guidance system is established, and is formulated as: 8 99 8 h iT > > > > > > > > x¼ a b > > > > > >> > > > > > > > > > " t # >> " # > " # > > > = < a => < km 2 km ðtÞ > a ð14Þ ¼ : f ðxi Þ ¼ t tm 2 tm ðtÞ > b b > > > > > > > > > > > > > > > > > > > > > > > > >> > > > ; : ;> : xiþ1 ¼ xi 2 f ðxi Þ=f 0 ðxi Þ >

The mentioned specified guidance gain algorithm is (Joseph and Asher, 2003): N0 ¼

k3 T 3go k2 T 2go 2 2kT go þ 2 2 2j21

ð16Þ

j ¼ ekT go where k is the guidance gain; Tgo ¼ 2r/vc is the time-to-go before impact; vc is the closing speed. The comparison of the interception performance between the IDG guidance law, PPN guidance law, TPN guidance law, and the specified PPN (VPPN) guidance law in both cases is listed in Table I, where the miss distance (MD) is defined as 418

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Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

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guidance law reaches its maximum value at the beginning of the guidance envelope, then it decreases gradually, and in the final stage of the engagement, the LOSR response maintains steadily and almost approaches to zero, which is more reasonable to implement for a flight control system. Moreover, according to the trend of the time histories of the LOSR response, it should be point out that the IDG guidance law is more sensitive to the change of the LOS. As shown in Figures 6 and 7, the proposed iterative algorithm converges quickly at the beginning of the guidance envelop, and the solution bias between the theoretical solution and the iterative solution is in a negligible range in both cases, which indicate that Newton’s iterative algorithm is viable and effective for practical application in missile guidance problems. As shown in Figures 8-11, it is notable that, unlike the classical PN guidance law, the IDG guidance law and the VPPN guidance law both compensate for the target’s maneuver at the beginning of the guidance envelope in both cases, which indicates that the proposed IDG guidance law is also a generalization of gain-varying PN guidance law. However, the guidance angles produced by the PN guidance law and the VPPN guidance law undergo a huge rise and almost approach saturation in the final stage of the engagement, whereas the guidance angles produced by the IDG guidance law are still stable and hold small near the impact point. As shown in Figure 12, compared with the classical PN guidance law, the curvature of the guided trajectories produced by the IDG guidance law engages its maximum value and similar to that produced by the VPPN guidance law at the beginning of the guidance envelope. However, in the final stage of the engagement, the curvature produced by the IDG guidance law decreases gradually and steadily and almost approaches to zero near the impact point, whereas the curvatures produced by the classical PN guidance law and the VPPN guidance law indicate a huge increase in magnitude, which means that the PN guidance law has the trend to rotate the guided trajectory.

Table I Comparison of interception performance MD(m) PPN TPN VPPN IDG

N-M

M

2.3321 2.8796 0.9552 2.0715

10.5324 3.9162 1.0061 0.8843

Time(s) N-M M 33.650 33.648 33.652 33.660

33.640 33.638 33.640 33.652

Gaina N-M M 6 5 5 18

4 4 5 16

Note: aFor PN guidance law, the gain denotes the effective navigation ratio corresponding to the best performance, while for the iterative DG guidance law, it denotes the guidance gain corresponding to the best performance

the closest distance between the missile and the target before divergence, i.e. r0 . 0 occurs. N-M denotes the nonmaneuvering target case. M denotes the maneuvering target case, where the guidance angles (angle-of-attack and sideslip angle) of the target are time-varying. From Table I, it is noted that the IDG guidance law has similar performance to the conventional PN guidance law and the VPPN guidance law against a non-maneuvering target. However, it performs better than the other guidance laws in the case of intercepting a maneuvering target. However, regardless of the types of target, it has a longer engagement time. The effect of the guidance gain to the interception performance of the IDG guidance law in both cases is listed in Table II. Moreover, the influence of the guidance gain to the guidance angles in the maneuvering case is shown graphically in Figures 3 and 4. From Table II, obviously, regardless of the types of target, the guidance gain corresponding to the best performance of the IDG guidance law is between 14 and 18. Moreover, the engagement time is usually longer as the gain increases. As shown in Figures 3 and 4, it is noted that the maximum value of the angle-of-attack or the sideslip angle comes earlier and bigger as the guidance gain increase. Moreover, it should be pointed out that the IDG guidance law becomes less sensitive to the change of the LOS as the guidance gain decreases. Therefore, the guidance gain between 14 and 18 is recommended to meet the practical application. In particular, because of the solution accuracy of the iterative algorithm, the guidance gain of the IDG guidance law is not the same as that of the theoretical DG guidance law. However, they have the same influence to the interception performance. From Figure 5, it should be noticed that, regardless of the types of target, the LOSR response of the proposed IDG

6. Conclusion The results of this paper indicate clearly that the 3D DG guidance law is viable and effective in a realistic missile defense engagement. In particular, Newton’s iterative algorithm works efficiently and accurately in DG guidance problem. The comparison of the interception performance of the proposed iterative DG guidance law and the proportional navigation (PN) guidance law is illustrated graphically, and the results indicate that the proposed guidance law is a generalization of gain-varying PN guidance law, and compared with the PN guidance law, it is more sensitive to the change of the LOS, thus leading to better performance for a maneuvering target. Furthermore, the guided trajectory produced by the proposed guidance law, as shown in Figure 12, indicates that the iterative DG guidance law can guarantee a head on interception of a maneuvering target. However, the DG guidance law needs further analysis, especially in the quantification of capture region. Also, interception performance should be tested under uncertainty in the target information and in the present of the autopilot system. Moreover, the modification of proposed iterative solution or an analytical solution to the DG guidance law is expected.

Table II Interception performance Gain 2 4 6 8 10 12 14 16 18 20

Miss distance (m) M N-M 6.5330 12.7796 16.60 17.3291 16.6763 15.2880 13.3850 3.8635 2.0715 4.3421

13.0352 10.7969 13.6852 13.8598 13.0192 11.3657 6.3443 0.8804 3.5647 7.4983

Engagement time (t) N-M M 33.6600 33.6500 33.6500 33.6500 33.6500 33.6500 33.6500 33.6580 33.6600 33.6700

33.65 33.64 33.64 33.64 33.64 33.64 33.65 33.652 33.658 33.660

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Aircraft Engineering and Aerospace Technology: An International Journal

Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

Volume 78 · Number 5 · 2006 · 415 –425

Figure 3 Angle-of-attack

Figure 4 Sideslip angle

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Volume 78 · Number 5 · 2006 · 415 –425

Figure 5 Time history of the LOSR of IDG guidance law in both cases

Figure 6 Non-maneuvering case

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Volume 78 · Number 5 · 2006 · 415 –425

Figure 7 Maneuvering case

Figure 8 Angle-of-attack of the non-maneuvering case

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Volume 78 · Number 5 · 2006 · 415 –425

Figure 9 Angle-of-attack of the maneuvering case

Figure 10 Sideslip angle of the non-maneuvering case

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Chaoyong Li, Wuxing Jing, Hui Wang and Zhiguo Qi

Volume 78 · Number 5 · 2006 · 415 –425

Figure 11 Sideslip angle of the maneuvering case

Figure 12 Time history of curvatures of the guided trajectories in the maneuvering case

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Notes

White, B.A., Zbikowski, R. and Tsourdos, A. (2005), “Direct intercept guidance using differential geometric concepts”, Proceedings of 2005 AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, USA, AIAA Paper 2005-5969. Zhang, Y. (1996), Flight Dynamics of Tactical Missile, Astronautics Book Concern, Beijing. Zhang, Y-A., Hu, Y-A. and Lin, T. (2001), “Robust geometric approach to missile guidance”, Chinese Journal of Control Theory and Application, Vol. 20 No. 1, pp. 13-20. Zhang, Y-A., Hu, Y-A. and Su, S. (2002), “Geometric approach and robust control approach to three-dimensional missile guidance”, Acta Aeronautica et Astronautica Sinica, Vol. 23 No. 1, pp. 88-90.

1 Throughout this paper, the word velocity will only be used to designate a vector quantity; the corresponding scalar will be denoted as speed. 2 Let C be an arbitrary curve in the space R3 and let x(s) be a parametric representation of C with the arc length s as parameter. Two points of C, corresponding to the values s and s þ h of the parameter, determine a chord C whose direction is given by the vector x(s þ h) 2 x(s). Hence, the vector: tðsÞ ¼ lim h!0

xðs þ hÞ 2 xðsÞ dx ¼ ¼ x_ ðsÞ h ds

is called the unit tangent vector to curve C at the point x(s).

About the authors Chaoyong Li was born in 1980 in Henan Province, People’s Republic of China. Currently, he is a PhD candidate of the Department of Aerospace Engineering, Harbin Institute of Technology, where he received his Bachelor’s and Master’s degree in 2003 and 2005, respectively, both in Aerospace Engineering. His current field of interest is nonlinear control and its applications to missile guidance and control systems. Chaoyong Li is the corresponding author and can be contacted at: [email protected]

References Adler, F.P. (1956), “Missile guidance by three-dimensional proportional navigation”, Journal of Applied Physics, Vol. 27 No. 5, pp. 500-7. Ariff, O., Zbikowski, R., Tsourdos, A. and White, B.A. (2004), “Differential geometric guidance based on the involute of the target’s trajectory: 2-D aspects”, Proceedings of the 2004 American Control Conference ACC’04, Boston, pp. 3640-4. Ariff, O., Zbikowski, R., Tsourdos, A. and White, B.A. (2005), “Differential geometric guidance based on the involute of the target’s trajectory”, Journal of Guidance, Control and Dynamics, Vol. 28 No. 5, pp. 990-6. Chiou, Y.C. and Kuo, C.Y. (1998), “Geometric approach to three-dimensional missile guidance problem”, Journal of Guidance, Control and Dynamics, Vol. 21 No. 2, pp. 335-41. Joseph, Z. and Asher, B. (2003), “New proportional navigation law for ground-to-air systems”, Journal of Guidance Control and Dynamics, Vol. 23 No. 5, pp. 822-5. Kuo, C.Y. and Chiou, Y.C. (2000), “Geometric analysis of missile guidance command”, IEE Proc-Control Theory and Applications, Vol. 147 No. 2, pp. 205-11. Kuo, C.Y., Soetanto, D. and Chiou, Y.C. (2001), “Geometric analysis of flight control command for tactical missile guidance”, IEEE Transactions on Control System Technology, Vol. 9 No. 2, pp. 234-43. Ortega, J.M. and Rheinboldt, W.C. (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY. Sarkar, A.K., Tiwari, P.K. and Srinivasan, S. (2003), “Generalized PN guidance law for a practical pursuer evader engagement”, Proceedings of 2003 AIAA Guidance, Navigation, and Control Conference and Exhibit, Austin, Texas, USA, AIAA Paper 2003-5651. Siouris, G.M. (2004), Missile Guidance and Control Systems, Springer-Verlag, New York, NY. Struik, D.J. (1998), Lectures on Classical Differential Geometry, Dover, New York, NY.

Wuxing Jing was born in 1965, in Henan province, People’s Republic of China. He received his MS degree (1989) and PhD (1994) from Harbin Institute of Technology (HIT). From 2000 to 2001, he was a Visiting Fellow of the Department of Aerospace Engineering, University of Glasgow. Since 1989 he has been employed at HIT, where he is currently Professor and the Director of the Department. His research interests are spacecraft dynamic and control, nonlinear system control, robust control of uncertain systems and system identification. E-mail: [email protected] Hui Wang was born in 1970, People’s Republic of China. He is currently a Senior Engineer in The 8th Institute of Shanghai Academy of Spaceflight Technology. His research interest is tactical missile guidance and control systems. E-mail: [email protected] Zhiguo Qi was born in 1975, in Henan province, People’s Republic of China. He is currently an Engineer in The 8th Institute of Shanghai Academy of Spaceflight Technology. His research interest is tactical missile guidance and control systems. E-mail: [email protected]

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