optimal midcourse guidance algorithm for exoatmospheric ...optimal midcourse guidance algorithm for...

Research ArticleOptimal Midcourse Guidance Algorithm for ExoatmosphericInterception Using Analytical Gradients

Wenhao Du , Wanchun Chen , Liang Yang , and Hao Zhou

School of Astronautics, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing, China

Correspondence should be addressed to Liang Yang; [email protected]

Received 2 June 2019; Accepted 23 August 2019; Published 16 September 2019

Academic Editor: Maj D. Mirmirani

Copyright © 2019 Wenhao Du et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is aimed at providing a semianalytical method to solve the optimal exoatmospheric interception problem with theminimum fuel consumption. A nonlinear programming (NLP) problem with the minimum velocity increment, which involvesLambert’s problem with unspecified time-of-flight, is firstly formulated. Then, a set of Karush-Kuhn-Tucker conditions and theJacobian matrix corresponding to those conditions are derived in an analytical manner, even though the derivatives aremathematically complicated and computationally onerous. Therefore, the Newton-Raphson method can be used to efficientlysolve this problem. To further decrease computational cost, a near-optimal initialization method reducing the dimension of thesearch space is presented to provide a better initial guess. The performance of the proposed method is assessed by numericalexperiments and comparison with other methods. The results show that this method is not only of high computationalefficiency and accuracy but also applicable to onboard guidance.

1. Introduction

Exoatmospheric missile defense system is a generic term con-veying an interceptor designed to destroy any ballistic targetsdelivering nuclear, chemical, biological, or conventional war-heads outside the atmosphere. It is of great significance fornational security, and therefore, the majority of powerfulcountries such as the United States, Russia, and China havedeveloped their own missile defense systems [1–3]. In themidcourse, both interceptor and target follow Keplerianorbits and are in high-speed interception engagement withthe relative velocity greater than 10 km/s. Thus, “hit-to-kill”is the only valid way of completely destroying invadingtargets, which brings forward high requirements on theperformance of guidance system [4–6]. Actually, exoatmo-spheric midcourse guidance is a process to calculate an initialvelocity vector to ensure a successful impact with zero miss,which theoretically can be categorized into a type of Lambert’problem [7–9]. Our purpose in this paper is to develop aguidance algorithm from the point of solving Lambert’sproblem efficiently.

In 1809, Gauss [10] developed the first iterative process tosolve Lambert’s problem, but it suffers from the singularityproblem. Battin et al. [11, 12] improved Gauss’s methodand proposed an elegant algorithm that efficaciouslyremoved the singularity. Their method is based on a newtransformation and a new iteration function to achieve fastconvergence. It is mathematically elegant and practicallyimplementable, but its derivation is founded on the compli-cated geometric properties of conic sections. Avanzini [13]presented an intuitive and simplified version of Battin’smethod by parametrizing admissible orbits in terms of thetransverse eccentricity component. An iterative methodbased on Householder’s root solver was designed in [14],where the first and second derivatives of the time-of-flight(TOF) equation are analytically derived to increase the rateof convergence. In [15], Levi-Civita regularization was appliedto Lambert’s problem, and a Newton-Raphson method incombination with safety checks was adopted to achieve bothspeed and robustness. These works have successfully pro-moted the application of Lambert’s problem in the spacescience community. Additionally, Nelson and Zarchan’s

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https://doi.org/10.1155/2019/8502870

efficient method [16, 17] was popular in defense applications[18]. This method regarded the flight path angle as an itera-tive variable and employed a secant method combined withboundary values to speed up the routine. Obviously, theaforementioned researchers focused on numerical proce-dures to deal with Lambert’s problem by searching a variableiteratively, because it is impossible to derive an analyticalsolution due to transcendental equations. Therefore, it is stillof great significance to speed up the convergence of iterativealgorithms, especially for onboard autonomous scenariosrequiring computation [19–22].

Moreover, for some situations where the issue of conver-gence time is critical, two types of methods are commonlyused to improve convergence rate. The first one focuses onanalytical gradients [23]. Bate et al. [24] presented an effec-tive algorithm named as the P-iteration method, where theslope of the TOF with respect to the semilatus rectum P couldbe expressed in an analytical manner. Ahn and Lee [25] ana-lytically derived the gradient of the flight path angle withrespect to the TOF from the conditions of the two-pointorbital boundary value problem, and their method was fullyverified by comprehensive numerical experiments. Further-more, another practical way to accelerate the convergenceof iteration process is improving the initial guess. In 1990,Gooding [26] made pioneering efforts in this field and usedHalley’s iterations to set the initial guess, thereby achievingfast convergence properties. Then, Izzo [27] improved Good-ing’s method by inverting the linear approximation to theTOF curves, which significantly reduced the computationalcomplexity. Arora and Russell [28] proposed a fast androbust algorithm to solve multiple revolution Lambert’sproblem, in which a new geometry parameter was introducedto simplify the TOF equation, and therefore, an accurate ini-tial guess can be provided for rapid root-solving. In [29], Ahnand Lee also developed a method using a two-dimensionaltable involving the geometric characteristic and the normal-ized TOF to interpolate an initial guess, whose efficiencywas validated by various numerical experiments. It shouldbe noted that the aforementioned methods assume that theTOF is required to be a given constant.

The optimal exoatmospheric interception problem withthe minimum fuel consumption is to find the optimal velocityincrement to eliminate interception error and retain enoughenergy for a successful collision. This kind of problem isabbreviated as the minimum velocity increment problem(MVIP) in this paper. It is worth noting that the aforemen-tioned algorithms cannot be directly applied to the MVIP,because the TOF should be regarded as an unspecifiedparameter in the optimization process [30–32]. Commonly,an additional single-variable algorithm needs to be developedto repeatedly solve Lambert’s problem so as to determine theoptimal TOF. However, this type of method has high com-putational cost because of its overreliance on a two-leveliterative process. For onboard autonomous scenarios, thecomputation time for generating commands is very critical.Therefore, it is necessary to develop a new algorithm for solv-ing the MVIP in an inexpensive manner.

The core objective of this paper is to provide a semiana-lytical method for solving the MVIP using analytical gradi-

ents, and it has potential advantage in onboard guidance.The main contributions of this paper are stated as follows.Firstly, a typically nonlinear programming problem (NLP)with the minimum velocity increment is formulated, inwhich two transcendental equations characterizing Lam-bert’s problem and Keplerian motion are regarded as equalityconstraints in order to reduce computational difficulties.Secondly, a set of Karush-Kuhn-Tucker (KKT) conditionsfor this NLP problem is derived to determine the mini-mum velocity increment. To speed up convergence, theJacobian matrix is successfully derived on each parameterin an analytical way, even though the expressions aremathematically complicated and computationally onerous.Accordingly, root-finding algorithms such as the Newton-Raphson method can be effectively used to find the optimalsolution with a high accuracy. Thirdly, it is found that thevariable, semilatus rectum P, is insensitive to the variationof TOF. A near-optimal initialization method, which con-siders P as a constant and reduces the dimension of thesearch space, is developed to provide a better initial guess.That further accelerates the rate of convergence. In a word,the proposed method is attractive from the point of view oflow computational cost, high accuracy, and the fact that ithas potential to be used as a baseline algorithm for onboardguidance. Finally, the performance of this method is suffi-ciently verified by comprehensive experiments and compari-son with the existing methods.

The rest of the paper is organized as follows. Section 2provides a detailed description of the MVIP, which involvesorbital motion and Lambert’s problem, respectively. Section3 introduces the method used to solve the problem in thispaper, which is composed of the KKT conditions, analyticalgradients, initialization method, and the procedure of imple-mentation. Section 4 presents the numerical results to dem-onstrate the performance of the method. Finally, conclusionsare given in Section 5.

2. Problem Formulation

The engagement geometry of the MVIP is illustrated inFigure 1(a). At the moment T1, the target performs anorbital maneuver and its trajectory is transferred from theKeplerian orbit orb1 to a new orbit orb2. The exoatmo-spheric kill vehicle (EKV), which is a representative inter-ceptor designed for exoatmospheric interception, needs tochange its velocity vector from v0 to v1 so as to score a directhit on the predicted intercept point (PIP) PIP2. Commonly,a PIP can be uniquely determined by orbital equations if theTOF is given. Then, the midcourse guidance problem canbe formulated as the determination of an orbit having aspecified transfer time and connecting two position vec-tors, which is known as Lambert’s problem. Actually, theremust be an optimal TOF corresponding to the minimumvelocity increment Δv min, where the symbol “ ” denotesthe Euclidean norm of vector. To demonstrate the MVIPmore clearly, a schematic representation for the correlationbetween Keplerian motion and Lambert’s problem is pre-sented in Figure 1(b).

2 International Journal of Aerospace Engineering

2.1. Orbital Motion of the Target. Let the orbital elements ofthe target orbit orb2 be

ELE = ΩM , iM , aM , eM , ωM , tpM , 1

where ΩM is the right ascension of the ascending node, iM isthe inclination of the orbit, aM is the semimajor axis, eM is theeccentricity, ωM is the argument of periapsis, and tpM denotesthe time of periapsis passage. The subscript M denotes theinvading target.

The true anomaly is defined as the angle from the peri-apsis to the orbiting object measured in the direction ofmotion, which is denoted as f M . The target position vector,r2, and the corresponding geocentric distance, r2, areexpressed as follows:

r2 =aM 1 − e2M1 + eM cos f M

, 2

r2 = r2h2 = r2 h2x h2y h2zT, 3

where h2 denotes the unit vector of terminal position and isdefined as follows:

h2 =

h2x

h2y

h2z

=

cos f M + ωM cos ΩM − sin f M + ωM cos iM sin ΩM

cos f M + ωM sin ΩM + sin f M + ωM cos iM cos ΩM

sin f M + ωM sin iM

4

Furthermore, the geometric relationship between f M andits corresponding eccentric anomaly EM is

tanEM

2=

1 − eM1 + eM

tanf M2, 5

and according to orbital motion equations, the transfer time,t, can be formulated as follows:

t − tpM =a3Mμ

EM − eM sin EM 6

Then, substituting equation (5) into equation (6), theorbital motion of the target can be explicitly expressed as afunction of t and f M .

g1 t, f M = 0 7

Equations (2) and (7) indicate that the position vector,r2, is uniquely determined by the TOF. However, equation (7)is transcendental in f M and unable to be solved analytically.

2.2. Lambert’s Problem. As stated above, if the terminal posi-tion vector of the target, r2, is determined, the midcourseguidance problem can be formulated as a classical Lambert’sproblem. Its solution is the initial velocity vector, whichensures the interceptor will impact the target with zero miss.As the Keplerian orbit is confined to a plane, Lagrangiancoefficients can be used to express the equations for terminalvectors, r2 and v2, in terms of initial vectors, r1 and v1; thus,

r2 = Fr1 +Gv1,v2 = Ftr1 +Gtv1

8

Clearly, the coefficients, Ft and Gt , are simply the deriva-tives of F and G with respect to the time, respectively, andthey can be expressed in terms of cross products as follows:

F G

Ft Gt

=1HE

r2 × v1 r2 × r1v2 × v1 v2 × r1

, 9

where HE is the magnitude of HE , which is the angularmomentum vector of the EKV.

Furthermore, the position and the velocity vectors areexpressed in terms of orbital plane coordinates as follows:

r = r cos f ip + r sin f iq, 10

Δv

v0

v1

EKV

orb1

orb2PIP2

PIP1

T1

Target

(a)

Keplerianmotion

vM0

r2, r2

r1, r1Center of Earth

v1ΔvTarget

EKV

rM0

TOF

v0

PIP2

�휙, TOF

orb2

Lambert’sproblem

(b)

Figure 1: Illustration of the MVIP. (a) Geometry of exoatmospheric interception. (b) The MVIP is a combination of Lambert’s problem andKeplerian motion.

3International Journal of Aerospace Engineering

v = r cos f − rf sin f ip + r sin f + rf cos f iq, 11

where ip is a unit vector in the direction of perigee, the pointof closest approach to the earth, and iq is in the orbital planeand normal to ip such that iq, ip,HE complete a right-handedcoordinate system Fp The equation of an orbit in polar coor-dinates is written as follows:

r =P

1 + e cos f, 12

where P = a 1 – e2 is the semilatus rectum, and a, e, f denotethe semimajor axis, the eccentricity, and the true anomaly ofthe EKV orbit, respectively. Differentiating equation (12) andone can have

r =μ

Pe sin f , 13

rf =μ

P1 + e cos f , 14

where μ is the standard gravitational parameter, whichequals 3.986× 1014m3/s2. Then, substituting equations (10)and (13) into equation (9), the Lagrangian coefficients canbe rewritten as follows:

F G

Ft Gt

=

1 −r2P

1 − cos ϕr1r2 sin ϕ

μP

μ

Ptan

ϕ

21 − cos ϕ

P−

1r1

−1r2

1 −r1P

1 − cos ϕ

,

15

where ϕ denotes the angle between r2 and r1; thus,

cos ϕ =r1 ⋅ r2r1r2

=r1xh2x + r1yh2y + r1zh2z

r116

It should be noted that, ϕ is a constant in the classicalLambert’s problem, and therefore, the Lagrangian coeffi-cients are uniquely determined by P.

In addition, the position and velocity vectors in equation(10) can be directly written in terms of the eccentric anomalyE as follows:

r = a cos E − e ip + a 1 − e2 sin Eiq, 17

v = −aE sin E ip + a 1 − e2E cos E iq 18

Taking the derivative of equation (6), one can obtain that

E =1r

μ

a19

Substituting equations (17) and (19) into equation (9),and the Lagrangian coefficients can be rewritten as follows:

F G

Ft Gt

=1 −

ar1

1 − cos ΔE t −a3

μΔE − sin ΔE

−μa

r1r2sin ΔE 1 −

ar2

1 − cos ΔE

,

20

where ΔE denotes the difference of the eccentric anomaly,which is expressed as ΔE = E2 – E1.

Combining equations (15) and (20), the transfer timet can be expressed as a function of Lagrangian coefficients

t =G +a3

μarccos 1 −

r1a

1 − F +r1r2Ft

μa, 21

where the semimajor axis a is expressed as follows:

a = μr21 1 − F 2

2μr1 1 − F − F2t r

21r

22

22

Substituting equations (22) and (15) into equation (21),Lambert’s problem can be finally expressed as a functionof t and P

g2 t, P = 0 23

It is apparent that equation (23) is a transcendental equa-tion, and therefore, its solution can only be provided by iter-ative methods.

In summary, g1 (equation (7)) and g2 (equation (23))characterize the numerical properties of Keplerian motionand Lambert’s problem, respectively. Once a TOF is deter-mined, the intercept orbit can be ascertained by solving equa-tion (23), in which equation (7) is an equal constraint used todetermine the terminal positon vector r2. Then, substitutingthe value of P into equation (15), the initial velocity vectorv1 of the EKV can be calculated by Lagrangian coefficientsas follows:

v1 =r2 − Fr1

G24

Because v1 is uniquely defined by the TOF, a single-variable unconstrained NLP problem can be formulated forthe purpose of determining the minimum velocity incre-ment; thus, the cost function is defined as follows:

minimize J t = Δv 2, 25

where the square of the velocity increment is given by

Δv 2 = v1 − v0 2

= v1x − v0x2 + v1y − v0y

2 + v1z − v0z2 26


To find a local minimum t̂, it is necessary to derive a set offirst-order necessary conditions of this problem

dJdt t=t̂

=∂J∂P

∂P∂t

+∂J∂f M

∂f M∂t t=t̂

= 0 27

It should be noted that two transcendental equations,g1 (equation (7)) and g2 (equation (23)), are involved inthe computational process, which leads to the strong cou-pling among t, P, and f M if the transfer time is consideredas an only independent variable. Therefore, the mathemat-ical complexity is sharply increased. In order to overcomethe computational difficulties yielding from transcendentalequations, the independent variable x = t is expanded toX = t, P, f M

T. Meanwhile, g1 and g2 are considered as twoequality constraints. Although the number of iterative vari-ables increases, additional items resulting from the couplingare efficiently eliminated. Subsequently, the NLP problemformulated in equation (25) is transferred to a multivariableNLP program with equality constraints. The exhaustive deri-vations of KKT conditions and analytical gradients are pre-sented in Section 3.

3. KKT Condition Solution Procedure UsingAnalytical Gradients

This section introduces the whole procedure to solve theMVIP based on the numerical computation of the NLP prob-lem. Section 3.1 presents the mathematical basis of NLP andobtains the KKT conditions corresponding to the MVIP.Then, the Newton-Raphson method is introduced to obtainthe optimal guidance command. Section 3.2 derives the Jaco-bian matrix analytically on each parameter, which increasesthe rate of convergence of the Newton-Raphson iteration.Section 3.3 develops an initialization method used to conducta near-optimal initial guess for further accelerating the con-vergence. Section 3.4 describes the strategy to update the nor-malized iterative variables.

3.1. Derivation of the KKT Conditions. Consider a NLP prob-lem with both equality and inequality constraints. The objec-tive is to minimize the cost function J X subject to a set ofequality constraints g X and inequality constraints m X .The problem is then

minimize J X

subject to g X = 0,

m X ≤ 0

28

The KKT conditions are found by defining the aug-mented cost function, L, so that

L X, λ, η = J X − 〠K

i=1λigi X − 〠

L

j=1η jmj X , 29

where λi, η j are the KKT multipliers associated with equalityand inequality constraints, respectively. The optimal solu-tion, X̂, is determined by the gradient of the augmented costfunction [33]. If J , g, andm are continuously differentiable

functions, and then there exists KKT multipliers λ and μsuch that

∂L∂x

X̂, λ , η = 0, 30

g X̂ = 0, 31

ηm X̂ = 0, where η ≤ 0,m X̂ ≤ 0 32

For the MVIP discussed in this paper, the correspondingNLP problem is finally expressed as follows:

minimize J t, P, f M = Δv 2

subject to g1 t, f M = 0,

g2 t, P, f M = 0

33

It should be emphasized that inequality constraints canbe omitted when the iterative variables are restricted to thedefinition domain. In addition, the terminal position of Lam-bert’s problem is no longer a given constant in the MVIP, butis determined on the orbit of the target as formulated in g1.Thus, f M is involved in the third equation of equation (33),which also decreases the computational difficulty. Therefore,the independent variables in the equality constraint g2 are setto be X = t, P, f M

T , which is different from that in equation(23). Furthermore, the augmented cost function is given by

L = J − λ1g1 − λ2g2 34

Taking the partial derivative of the independent vari-ables, one can obtain

∂L∂X

= 0, X = t, P, f M 35

Rearranging equation (35), then the KKT multipliers λ1and λ2 can be formulated as follows:

λ1 = −a3Mμλ2, 36

λ2 =∂J/∂P∂g2/∂P

37

Substituting equation (36) into equation (35) and com-paring with equation (30), the KKT conditions can bedenoted by a system of transcendental equations as follows:

Φ X = g1 g2 g3T, 38


where

g1 t, f M = EM − eM sin EM −μ

a3Mt − tpM , 39

g2 t, P, f M =G +a3

μarccos 1 −

r1a

1 − F

+r1r2Ft

μa− t,

40

g3 P, f M =∂J∂f M

+a3Mμ

∂J/∂P∂g2/∂P

∂g1∂f M

−∂J/∂P∂g2/∂P

∂g2∂f M

,

41

where g1 and g2 describe the mathematical models of Kep-lerian motion and Lambert’s problem in the MVIP. Mean-while, g3 relates the minimum velocity increment of theEKV with the orbital motion of the target through the KKTconditions. To solve the system of transcendental equationsin equation (38), a Newton-Raphson iterative method isdesigned as follows.

The improvements of iterative variables are formulatedas follows:

XK+1 = XK − ΦX XK −1Φ XK , 42

where the superscript K means the K-th iteration, and theJacobian matrix ΦX X is given by

ΦX X =∂Φ X∂X

=

∂g1∂t

∂g1∂P

∂g1∂f M

∂g2∂t

∂g2∂P

∂g2∂f M

∂g3∂t

∂g3∂P

∂g3∂f M

43

In spite of mathematical complication, analytical expres-sions of all the parameters in theΦX X are precisely derivedin the following subsection, thereby significantly acceleratingthe computational efficiency and improving the accuracy incomparison with finite difference.

3.2. Derivation of the Analytical Gradients. This subsectionfocuses on the analytical expression of the Jacobian matrixon each parameter. The partial derivatives of g1 are firstlyformulated due to its concise structure.

3.2.1. Partial Derivatives of g1. Since P is not explicit in g1,the partial derivative of g1 with respect to P is zero.

∂g1∂P

= 0 44

The partial derivative of g1 with respect to t can be obvi-ously expressed as follows:

∂g1∂t

= −μ

a3M45

Substituting equation (5) into equation (39), the partialderivative of g1 with respect to f M is given according to trig-onometry formulas

∂g1∂f M

=1 − e2M

3/2

1 + eM cos f M2 46

Although the partial derivatives of g1 can be formulatedsuccinctly, other formulas discussed later are very complex.Nevertheless, an exhaustive derivation of analytical gradientswill be presented.

3.2.2. Partial Derivatives of g2. Observing the expression ofg2 in equation (39), the partial derivative with respect to tcan be obviously expressed as follows:

∂g2∂t

= −1 47

For simplifying the derivations, three notations are intro-duced as follows:

k = r1r2 1 − cos ϕ ,

l = r1 + r2,

m = r1r2 1 + cos ϕ

48

Substituting equation (15) into equation (22), the semi-major axis of the intercept orbit is rewritten as follows:

a =mkP

2m − l2 P2 + 2klP − k249

Then, the partial derivative of awith respect to P isexpressed as follows:

∂a∂P

=mk l2 − 2m P2 − k2

χ20

, 50

where χ0 is another notation expressed as follows:

χ0 = 2m − l2 P2 + 2klP − k2 51

Substituting equations (15) and (49) into equation (39)and partially differentiating g2 with respect to P yields

∂g2∂P

= −G2P

+32

aμ

∂a∂P

χ3 +a3

μχ4, 52


where

χ1 =1 − kaP

,

χ2 =2aPk − k2

aP,

χ3 = arccos χ1 +tan ϕ/2 k/P − l

aP,

χ4 =2kχ2

+ aPkP− l tan

ϕ

2k − Pl

mP3 −k

P2 aPtan

ϕ

253

Partially differentiating equation (2) with respect to f M ,one can obtain that

∂r2∂f M

=r2eM sin f M1 + eM cos f M

54

Substituting equation (16) into the expression of G inequation (15), it can be rewritten as follows:

G =r2μP

r21 − r1xh2x + r1yh2y + r1zh2z2 55

Partially differentiating equation (55) with respect to f M ,and introducing notations as follows:

r1h2 = r1xh2x + r1yh2y + r1zh2z ,

r1ph2 = r1x∂h2x∂f M

+ r1y∂h2y∂f M

+ r1z∂h2z∂f M

,

sr12h2 = r21 − r1xh2x + r1yh2y + r1zh2z2,

56

where

∂h2∂f M

=

∂h2x∂f M

∂h2y∂f M

∂h2z∂f M

=

−sin f M +w cos Ω − cos f M +w cos i sin Ω

−sin f M +w sin Ω + cos f M +w cos i cos Ω

cos f M +w sin i

57

Then, the partial derivative of the Lagrangian coefficientG with respect to f M is given by

∂G∂f M

=1μP

∂r2∂f M

sr12h2 −r2r1h2r1ph2

sr12h258

In order to simplify the expression, notations χ5, χ51,χ6, χ61~χ65, χ7, and χ8 are also introduced, which areexpressed as follows:

χ5 = 2r2∂r2∂f M

s2r12h2 − r2r1h2r1ph2 ,

χ51 = 2P2χ61 + 2Pχ62 − 2r2χ6χ63,

χ6 = r1 − r1h2,

χ61 = r1h2 − r2∂r2∂f M

+ r2r1ph2,

χ62 = r1 + 2r2∂r2∂f M

χ6 − r1 + r2 r2r1ph2,

χ63 = χ6∂r2∂f M

− r2r1ph2,

χ64 =mk

χ20,

χ65 = χ63 −r2χ6a

∂a∂f M

,

χ7 =χ6

sr12h2,

χ8 =r2χ6P

− r2 − r1

59

Partially differentiating equation (49) with respect to f M ,and substituting equation (54) into the result, one finds

∂a∂f M

= Pχ5χ0

− χ64 ⋅ χ51 60

Then, substituting equations (15) and (55) into equa-tion (39) and partially differentiating g2 with respect tof M yields

∂g2∂f M

=∂G∂f M

+32

aμ

∂a∂f M

arccos χ1 +aμ

χ65Pχ2

+1μP

χ7χ8∂a∂f M

+ aχ8∂χ7∂f M

+ aχ7∂χ8∂f M

,61

where

∂χ7∂f M

= −r1r1ph2

r1 + r1h2 sr12h2,

∂χ8∂f M

=χ63P

−∂r2∂f M

62


So far, the partial derivatives of g2 have been success-fully expressed by equations (47), (52), and (61) in an ana-lytical manner. It is worth noting that g3 consists of thepartial derivatives of J , g1, andg2. Therefore, the partialderivatives of g3 involve a number of second-order partialderivatives, which are much more complicated.

3.2.3. Partial Derivatives of g3. Observing the expression forg3 in equation (39), it is easy to see that t does not appearexplicitly, so the partial derivative of g3 with respect to tcan be expressed as follows:

∂g3∂t

= 0 63

Then, let us focus on the partial derivative with respect toP. The derivation is presented as follows:

At the beginning, substituting equations (2) and (24) intoequation (33) and introducing a set of notations

JPx1 =r2xG

−FGr1x − v0x,

JPy1 =r2yG

−FGr1y − v0y ,

JPz1 =r2zG

−FGr1z − v0z ,

64

then the cost function can be rewritten as follows:

J = J2Px1 + J2Py1 + J2Pz1 65

Partially differentiating equation (65) with respect to Pand introducing another set of notations as

JPx2 =r2x − r1x 2 − F

2PG,

JPy2 =r2y − r1y 2 − F

2PG,

JPz2 =r2z − r1z 2 − F

2PG,

66

one has

∂J∂P

= 2 JPx1 JPx2 + JPy1 JPy2 + JPz1 JPz2 67

∂2a∂P2 =

2mk 2m − l22P3 + 3 2m − l2 Pk2 + 2k3l

χ30

,

∂2 J∂P2 = 2 J2Px2 + J2Py2 + J2Pz2

+1 − F

P2Gr1x JPx1 + r1y JPy1 + r1z JPz1 −

12P

∂J∂P

,

∂2g2∂P2 =

34G

P2 +32

12

1aμ

∂a∂P

2+

aμ

∂2a∂P2 χ3

+ 3aμ

∂a∂P

χ4 +a3

μ

∂χ4∂P

,

68

where

∂χ4∂P

=2kχ2

+ aP2Pl − 3kmP4 −

4a k − Pl 2kχ1m2P5χ2

+tan ϕ/2mP4 aP

2k ak − aPl −mP

+k − Pl 2

2P∂a∂P

+ a

69

Partially differentiating F in equation (15) and usingequations (16) and (54), one has

∂F∂f M

=cos ϕ − 1

P∂r2∂f M

+r2 ⋅ r1ph2Pr1

70

Then, differentiating equations (50), (65), and (52)with respect to f M and introducing two sets of notationsexpressed as

J f Mx1 =∂r2∂f M

h2x + r2∂h2x∂f M

g − r2h2x∂G∂f M

,

J f Mx2 =h2xG

∂r2∂f M

+r2G∂h2x∂f M

−r2h2x − r1xF

G2∂G∂f M

−r1xG

∂F∂f M

,

J f My1 =∂r2∂f M

h2y + r2∂h2y∂f M

g − r2h2y∂G∂f M

,

J f My2 =h2yG

∂r2∂f M

+r2G

∂h2y∂f M

−r2h2y − r1yF

G2∂G∂f M

−r1yG

∂F∂f M

,

J f Mz1 =∂r2∂f M

h2z + r2∂h2z∂f M

g − r2h2z∂G∂f M

,

J f Mz2 =h2zG

∂r2∂f M

+r2G∂h2z∂f M

−r2h2z − r1z F

G2∂G∂f M

−r1zG

∂F∂f M

,

71

Then, differentiating equations (50), (67), and (52) withrespect to P, one can obtain second-order partial derivatives one can obtain the second-order partial and mixed derivatives


Substituting equations (46), (52), (61), and (67) intoequation (39), and partially differentiating with respect toP, the partial derivative of g3 with respect to P is finallyderived as follows:

∂g3∂P

=∂2 J

∂P∂f M+

∂2 J∂P2

∂g2∂P

−∂J∂P

∂2g2∂P2

⋅a3M/μ ∂g1/∂f M − ∂g2/∂f M

∂g2/∂P2

−∂J/∂P∂g2/∂g2

⋅∂2g2∂P∂f M

73

Let us focus on the last term of the partial derivative of g3.Partially differentiating equations (54), (58) ,and (70) withrespect to f M , one can obtain that

∂2r2∂f 2M

=2e2 − e2 cos2 f M + e cos f M

1 + e cos f M2 r2,

∂2G∂f 2M

=sr12h2μP

∂2r2∂f 2M

−2r1ph2r1h2s2r12h2

∂r2∂f M

−r21ph2 − r21h2 r21 + r41h2

s3r12h2r2 ,

∂2F∂f 2M

=1r1P

2∂r2∂f M

r1ph2 − r2r1h2 −∂2r2∂f 2M

r1 − r1h2

74

Partially differentiating equation (65) with respect to f Mtwice and introducing another set of notations are expressedas follows:

J f Mx3 =∂2r2∂f 2M

h2x + 2∂r2∂f M

∂h2x∂f M

− r2h2x1G

−r2h2xG2

∂2G∂f 2M

−2J f Mx1

G∂G∂f M

− r1x J f M FG,

J f My3 =∂2r2∂f 2M

⋅ h2y + 2∂r2∂f M

∂h2y∂f M

− r2h2y1G

−r2h2yG2

∂2G∂f 2M

−2J f My1

G∂G∂f M

− r1y J f M FG,

J f Mz3 =∂2r2∂f 2M

⋅ h2z + 2∂r2∂f M

∂h2z∂f M

− r2h2z1G

−r2h2zG2

∂2G∂f 2M

−2J f Mz1

G∂G∂f M

− r1z J f M FG,

75

where

J f M FG =∂2F∂f 2M

1G

−F

G2∂2G∂f 2M

−2G2

∂F∂f M

∂G∂f M

+2FG3

∂G∂f M

2

76

Then, the second-order partial derivative of the costfunction J with respect to f M is given by

∂2 J∂f 2M

= 2 J2f Mx2 + JPx1 J f Mx3 + J2f My2

+ JPy1 J f My3 + J2f Mz2 + JPz1 J f Mz3

77

Partially differentiating equation (60) with respect to f M ,then one has

∂2a∂f 2M

= P∂χ5∂f M

1χ0

− χ64∂χ51∂f M

− 2χ5χ51χ20

+ 2χ64χ

251

χ0, 78

∂2a∂P∂f M

=∂a∂f M

1P−2Pχ30

2m − l2 P + kl χ0χ5 − 2mkχ51 + 2Pχ61 + χ62 mkχ0 ,

∂2 J∂P∂f M

= 2

JPx12PG2 J f Mx1 + Gr1x

∂F∂f M

+ 2 − F r1x∂G∂f M

+ JPx2 J f Mx2 +

JPy12PG2 J f My1 +Gr1y

∂F∂f M

+ 2 − F r1y∂G∂f M

+ JPy2 J f My2 +

JPz12PG2 J f Mz1 +Gr1z

∂F∂f M

+ 2 − F r1z∂G∂f M

+ JPz2 J f Mz2

,

∂2g2∂P∂f M

= −12P

∂G∂f M

+32arccos χ1 −

r2χ6aPχ2

aμ

∂2a∂P∂f M

+k − Pl 3k − 2r2χ6

mP3aμ

∂a∂f M

−aμ

χ63P2χ2 1 + χ1

+ 12 aμ

∂a∂P

32∂a∂f M

arccos χ1 + χ65Pχ2

−r2χ6χ1

aμP2χ2 1 + χ1

72


where

χ9 =∂2r2∂f 2M

χ6 − 2∂r2∂f M

r1ph2 + r2r1h2,

∂χ5∂f M

=∂r2∂f M

2+ r2

∂2r2∂f 2M

2s2r12h2

− 8r2r1h2r1ph2∂r2∂f M

+ 2r22 r21h2 − r21ph2 ,

∂χ51∂f M

= r1 − r2∂2r2∂f 2M

− χ9 −∂r2∂f M

22P2

− 2χ263 − 2r2χ6χ9 + r1 + 2r2 χ9

− r2r1h2 + 2 ∂r2∂f M

2χ6 2P

79

Partially differentiating equation (61), one can obtain thesecond-order partial derivative of g2 with respect to f M

∂2g2∂f 2M

=∂2G∂f 2M

+32

aμ

∂2a∂f 2M

+12a

∂a∂f M

2arccos χ1

−χ65

Pχ2 aμ∂a∂f M

+aμχ91 +

1μP

∂2a∂f 2M

χ7χ8

+ a∂2χ7

∂f 2Mχ8 + aχ7

∂2χ8

∂f 2M+ 2 ∂a

∂f M

∂χ7∂f M

χ8

+ 2∂a∂f M

χ7∂χ8∂f M

+ 2a∂χ7∂f M

∂χ8∂f M

,

80

where

∂2χ7

∂f 2M= −

r1h2r1ph2

+r1ph2sr12h2

r1h2sr12h2

− χ7∂χ7∂f M

,∂2χ8

∂f 2M

=χ9P

−∂2r2∂f 2M

81

Furthermore, partially differentiating equation (46), thesecond-order partial derivative of g1 is expressed as follows:

∂2g1∂f 2M

=eM 1 − e2M sin f M1 + eM cos f M

+ eM sin EM 82

Then, the partial derivative of g3 with respect to fM can beexpressed as follows:

∂g3∂f M

=∂2 J∂f 2M

+∂2 J

∂P∂f M

∂g2∂P

−∂J∂P

∂2g2∂P∂f M

a3M/μ ∂g1/∂f M − ∂g2/∂f M∂g2/∂P

2

+∂J/∂P∂g2/∂P

a3Mμ

∂2g1∂f 2M

−∂2g2∂f 2M

83

Now, the partial derivatives of g3 have been successfullypresented by equations (63), (73), and (83); thus, all theparameters of the Jacobian matrix have been analytically for-mulated. Although the derivations are mathematically com-plicated and computationally onerous, it only takes smallfractions of a second to compute the gradient information,which is of great significance to the improvement of the con-vergence rate. Moreover, to further improve the speed ofconvergence, a near-optimal initialization method is devel-oped in the next subsection.

3.3. Initialization Method. In order to accelerate conver-gence, this section presents an efficient method to providean improved initial guess of iterative variables X = t, P, f M

T

for the MVIP. This method is based on a discovery that semi-latus rectum, P, is insensitive to the variation of TOF in thecomputation process. In fact, the semilatus rectum P is afunction of the semimajor axis and the eccentricity. It isimpossible for EKV to substantially change these two orbitalelements due to the limitation of fuel. Therefore, the semila-tus rectum P varies within a limited range. Furthermore, aninitialization method considering the semilatus rectum as aconstant can be used to provide a warm start. The concretecomputational procedure is given as follows.

At the beginning, P is calculated from initial conditionsand considered as a constant in computation process. There-fore, the dimension of the search space is reduced to two,which are the transfer time, t, and the true anomaly, f M . Itshould be noted that, according to equation (7), t can be ana-lytically expressed as a function of f M . Therefore, the dimen-sion of the search space can be further reduced to formulate asingle-variable rooting problem. LetΨ f M denote the equal-ity constraint on f M , which is expressed as follows:

Ψ f M = g2 −a3Mμg1 84

To determine the initial guess of f M , a single-variableNewton-Raphson iteration equation is formulated as follows:

f L+1M = f LM −Ψ f LM

Ψ f LM, 85


where

Ψ f M =dg2df M

−a3Mμ

dg1df M

86

It should be pointed out that Ψ f M can be calculatedanalytically from the derivation in Section 3.2.

The iteration is finished if the difference between tL andtL+1 is small enough, and the convergence criterion of the ini-tialization method is given by

tL+1 − tL < εt 87

The proposed initializationmethod is derived based on thephysical characteristic of the exoatmospheric interception. Itcan not only reduce the difficulty in evaluation by compressingthe search space to one parameter but it also provides a feasi-ble and near-optimal initial guess for the main iteration,thereby efficiently accelerating the rate of convergence.

3.4. Updating Iterative Variables. When the initial guess hasbeen determined, optimal parameters corresponding to theminimum velocity increment can be efficiently provided bythe Newton-Rapson method with analytical gradients. If thedifference between two successive values is smaller than theconvergence criterion εX , the main iteration stops and thevelocity increment of the current iteration is the optimalsolution of the MVIP. The convergence criterion is expressedas follows:

XK+1 − XK < εX 88

Otherwise, the iterative variables X = t, P, f MT should

be updated to start the next iteration.It should be pointed out that the MVIP has significant

magnitude difference among the iterative variables. Forexample, the magnitude of semilatus rectum P reaches ashigh as 106, whereas the true anomaly f M locates on theinterval –π, π . Therefore, it is essential to normalize theiterative variables to provide a better numerical conditionfor solving the NLP problem. The normalization is achievedby multiplying a special weight matrix u and the correspond-ing convergence criterion is formulated as follows:

uXK+1 − uXK < εx , 89

where u is the weight matrix.The procedure of implementing the proposed algorithm

is included in the flowchart in Figure 2. After the input ofprogram data, the initialization method is applied to providea warm start for the main iteration. Then, analytical gradientsare used to calculate the Jacobian matrix in the computa-tional process of the Newton-Rapson iteration, which is pro-pitious for the computational efficiency. It should be notedthat the initialization method is only applied at the first stepof onboard guidance. The reason is that the optimal resultgenerated from the previous guidance cycle is very close to

the optimal one in current cycle and can be considered asthe initial guess for reducing computational cost. Further-more, a comprehensive study on the charming properties ofthe algorithm is presented in Section 4.

4. Validation of the Proposed Method

This section presents the numerical results used to verify theproposed algorithm. The validation is carried out from threedifferent perspectives. Firstly, the convergence property ofthe algorithm is investigated for nine cases with different ini-tial conditions. Secondly, comprehensive experiments areconducted to assess the computational efficiency by compar-ing the number of iteration (NOI) and computation timewith other typical methods. Thirdly, the proposed methodis implemented in the framework of midcourse guidance,which is used to validate its applicability. The correspondingcomparison of fuel consumption with other guidance laws isalso provided. All algorithms are implemented in MATLABand run on personal computer with Intel Core-i7 CPU and16G RAM under the Win10 operating system. It should benoted that all the numerical experiments have different initialconditions which are divided into three groups with differenttarget orbits. Each experiment group corresponds to each ofthree different initial velocity vectors, which is challengingenough to evaluate the performance of the proposed method.Nine simulation conditions are listed in Table 1 in detail.

4.1. Convergent Characteristics. The three groups of simula-tion cases listed in Table 1 are used to demonstrate the con-vergent characteristics of the proposed method includingthe NOI, the convergent process of iterative variables, andthe magnitude of convergence criterion. These three groupshave a large difference in their initial heading angles whichrepresent typical trajectories of intercontinental ballistic mis-siles (ICBMs). Meanwhile, the interceptor in each group hasdifferent initial velocity vector, which is designed to ensurethat the optimal impact happens at the beginning, middle,and end phase of exoatmospheric ballistic trajectory. Theengagement geometry and the iterative process of the costfunction are illustrated in Figures 3–5. Obviously, it onlytakes a few cycles for the proposed method to successfullyconverge to the optimal results in all experiments. In fact,benefiting from the analytical gradients, the cost functionhas approached a quite small neighborhood of the optimalsolution after the first two iterations. Because of the limita-tion of the length of paper, Table 2 only presents the sum-mary of the convergence properties for the cases of Group 1.

As stated above, the proposed method contains twoparts: the initialization method, which is used to providea near-optimal initial guess, and the main iteration whichuses analytical gradients to solve the MVIP so as to obtainthe minimum velocity increment. As shown in Table 2, twoparts successfully converge to the required tolerance withinsix cycles. The maximal error for equality constraints, g1and g2, is less than 1e – 8. The error norm of iterativeparameters monotonously decreases and its maximal erroris 3 04e – 07. Therefore, the proposed method works wellfor all cases and has a high convergence rate.


In addition, let us focus on the convergent process of Plisted in the right column of Table 2, the maximal relativeerror between the initial and optimal values is barely14.25% for Case 1, 2.17% for Case 2, and 0.22% for Case 3.Therefore, it is reasonable to assume that P can be consideredas a constant in the initialization method. Another attentionshould be paid is that a relaxed convergence criterion is usedfor the initialization method, which can further reduce thecomputation time. However, a strict convergence criterion

should be employed in the main iteration for obtaining highaccuracy results, so that

εt = 1e − 02,

εX = 1e − 0690

In summary, the proposed method is able to achieveextremely high accuracy with few iterations and is worthy

Assign problem geometry:initial position and velocity vectors

of the target and the EKV

Set P as a constant

fM

Determine correspondingTOF: tL

g1 (t, fM) = 0

Initial guessX = [t, P, fM]T

N-R iteration

K = K + 1

< �휀xuXK+1 − uXK

Corresponding PIP: r2Lagrangian coefficients:

F, G

Output:

J = =

Initialization method

Main iteration

No

Yes

No

Yes

< �휀ttL+1 − tL

M M M M

N-R iterationfL+1 =fL − �훹 (fL) / �훹 (fL)

L = L + 1

dg1

dfM�훹 (fM) =

dg2

dfM−

a3M

�휇

where �훹 (fM) is analytically expressedas

and analytical Jacobian matrix

KKT conditions

ФX (X) =

�휕g1

�휕t �휕P �휕fM



�휕g1 �휕g1

�휕g2 �휕g2 �휕g2

�휕g3 �휕g3 �휕g3

Ф (X) = Tg3]g2[g1

XK+1 = XK − ФX XK Ф XK−1

v1 = (r2 − Fr1) / G

Δv2 2

v1 − v0

Figure 2: Flowchart of the proposed method.


of practical application. Moreover, the computational effi-ciency of the proposed method is discussed comprehensivelyin the next subsection by comparing with other algorithms.

4.2. Computational Efficiency. In order to assess the compu-tational efficiency of the proposed method, a comparison

with typical methods is provided in this subsection. Theinterception scenarios are the same as that listed in Table 1.The methods used for comparison include Zarchan’s methodand the P-iteration method, which are both widely used insolving Lambert’s problem. The major difference is thatZarchan’s method is based on finite difference, whereas the

orb2

EKV

Target

Case 1

Case 3

Group 2T1

Case 2

(a)

Iterations0

01 2 3 4 5 6

1000

800

600

400

200

Case 1Case 2Case 3

�㨟Δ

v�㨟, m

/s

(b)

Figure 4: The engagement geometry and the iterative process of Group 2.

Target

Case 1Case 2

Case 3

EKV

orb2

Group 1T1

(a)

Case 1Iterations

Case 2Case 3

1000

800

600

400

2000 1 2 3 4 5 6

�㨟Δ

v�㨟, m

/s

(b)

Figure 3: The engagement geometry (a) and the iterative process (b) of Group 1.

Table 1: Initial conditions for interception.

Group 1 Group 2 Group 3

rM0 5 9, 0 84, 2 8 e + 06 –7 4, 2 8, –5 9 e + 06 5 2, 3 7, –2 6 e + 06vM0 –0 57, 5 2, 4 7 e + 03 –3 1, 6 4, 1 1 e + 03 0 3, 6 2, 2 9 e + 03rE0 4 0, 4 8, 2 3 e + 06 0 22, 6 2,−2 5 e + 06 1 7, 6 3, –1 7 e + 06

vE0Case 1 3 0, –3 0, 5 0 e + 03 –5 0, 0 0, –6 0 e + 03 6 0, 2 0, 1 0 e + 03

Case 2 1 0, 2 0, 5 0 e + 03 –4 0, 3 0, –2 0 e + 03 3 0, 4 0, 3 0 e + 03

Case 3 –1 0, 5 0, 4 0 e + 03 –3 0, 5 0, 2 0 e + 03 1 0, 5 0, 3 0 e + 03


P-iteration method has the analytical gradient. Additionally,the active set method, which is a general algorithm designedto solve NLP problem, is also used for comparison. Thenumerical results of the NOI and the computation time fordifferent methods are presented in Tables 3 and 4, respectively.

It can be seen from Table 3 that the NOI of the proposedmethod is only six, which is two to three orders of magnitudesmaller than that of Zarchan’s method and the P-iterationmethod, respectively. It should be noted that the twomethods are not pertinently designed for the MVIP, and

therefore, an additional single-variable algorithm iteratingon the TOF should be developed to recurrently solve Lam-bert’s problem, which leads to high computational burden.However, the active set algorithm is used to solved the NLPproblem transferred from the MVIP. The maximal NOI is30, which is much smaller than that of Zarchan’s methodand the P-iteration method. Moreover, the active set algo-rithm uses finite difference approach to provide gradientinformation, which leads to worse numerical accuracy thananalytical gradients. Therefore, the NOI of the proposed

Table 2: Convergent property of the proposed method.

Convergent results of Group 1Initialization method Main iteration

L f M tL+1 − tL K t, s P, km f M uXK+1 − uXK

Case 1 0 2.5000 / 0 479.99 4574.4 2.8318 /

J X = 5 73e + 05 1 2.8262 3 71e + 02 1 448.06 5274.1 2.8056 7 01e – 01Δv = 7 57e + 02 2 2.8318 6 79e + 00 2 453.18 5227.7 2.8097 4 69e – 02g1 X = 4 44e – 16 3 2.8318 2 10e − 02 3 453.25 5226.1 2.8098 1 54e – 03g2 X = 6 59e – 12 4 2.8318 2 04e – 07 4 453.26 5226.1 2.8098 3 97e – 05

5 453.26 5226.1 2.8098 1 02e – 066 453.26 5226.1 2.8098 2 61e – 08

Case 2 0 3.2000 / 0 929.85 1755.9 3.1872 /

J X = 1 04e + 05 1 3.1818 2 33e + 01 1 853.69 1777.5 3.1279 9 89e – 02Δv = 3 22e + 02 2 3.1868 6 39e + 00 2 866.05 1793.6 3.1376 2 25e – 02g1 X = 4 44e – 16 3 3.1872 5 71e – 01 3 866.25 1794.0 3.1377 5 18e – 04g2 X = 7 52e – 10 4 3.1872 4 61e – 03 4 866.25 1794.0 3.1377 1 08e – 06

5 866.25 1794.0 3.1377 1 75e – 09Case 3 0 4.0000 / 0 1664.93 2531.9 3.8067 /

J X = 3 05e + 05 1 3.8488 1 45e + 02 1 1661.48 2526.6 3.8033 7 14e – 03Δv = 5 52e + 02 2 3.8090 4 05e + 01 2 1661.36 2526.4 3.8032 2 31e – 04g1 X = −7 22e – 12 3 3.8067 2 41e + 00 3 1661.36 2526.4 3.8032 8 38e – 06g2 X = −7 66e – 09 4 3.8067 7 90e – 03 4 1661.36 2526.4 3.8032 3 04e – 07

TargetT1

EKV

Case 1

Case 2Case 3

orb2

Group 3

(a)

2200

1800

1400

1000

600

200

Iterations0 1 2 3 4 5 6

Case 1Case 2Case 3

�㨟Δ

v�㨟, m

/s

(b)

Figure 5: The engagement geometry and the iterative process of Group 3.


method is two and three times smaller than that of the activeset algorithm.

Let us focus on the computation time as shown inTable 4. The average value is barely 0.089ms for proposedmethod, but is 7.614ms for Zarchan’s method, 4.451ms forthe P-iteration method, and 19.42ms for the active set algo-rithm. It is obvious that the computation time of the pro-posed method is much smaller than that of the others. Itshould be pointed out that the active set algorithm costsmuch time due to the invocation of toolbox functions inMATLAB, whereas the proposed method is independentand only consists of basic operation.

In conclusion, the average values of NOI and computa-tion time are both improved by two to three orders of magni-tude, which makes the proposed method to be particularlysuited for onboard applications. The corresponding statistic

is shown in Figure 6. It should be noted that Zarchan’smethod and the P-iteration method have been demonstratedto be effective in solving Lambert’s problem, and these reduc-tions do not necessarily mean that the proposed method out-performs that of the conventional algorithms. In fact, thecomputational challenge of the MVIP arises from the combi-nation of Lambert’s problem and orbital motion, whichbrings more transcendental equations into the mathematicalmodel. The proposed method is developed for this specificproblem and derives a set of analytical gradients to speedup convergence, and thereby having a higher computationalefficiency than those typical methods.

4.3. Application in the Midcourse Guidance. In this subsec-tion, the proposed method is used as the baseline algorithmfor midcourse guidance, in which the interception scenarios

Table 3: Comparisons of the number of iteration.

Number of iteration (NOI)Group 1 Group 2 Group 3

Case 1 Case 2 Case 3 Case 1 Case 2 Case 3 Case 1 Case 2 Case 3

Zarchan’s method 1676 1596 1737 1616 1564 1662 1614 1554 1657

P-iteration 174 90 118 144 56 88 212 160 126

Active set 18 30 14 20 17 17 19 15 19

Proposed method 6 5 4 4 3 6 5 6 6

Table 4: Comparisons of the computation time.

Computation time (ms)Group 1 Group 2 Group 3


Zarchan’s method 4.46 7.86 7.03 8.14 6.87 4.18 10.04 10.50 9.45

P-iteration 2.78 4.01 4.07 3.37 5.83 1.86 4.13 6.81 7.20

Active set 32.42 26.86 12.50 22.60 16.59 17.48 18.10 12.37 15.86

Proposed method 0.090 0.078 0.067 0.090 0.103 0.116 0.078 0.089 0.090

1 2 3 4Methods

Number of iterationComputation time

Zarchan P-iteration Active set Proposedmethod

104

103

102

101

100

102

101

100

10−1

10−2

Aver

age n

umbe

r of i

tera

tion

(no.

of m

etho

ds)

Aver

age c

ompu

tion

time (

no. o

f met

hods

)

Figure 6: Efficiency comparison of four methods for solving the MVIP.


are the same as that listed in Table 1. In order to demon-strate its superb performance, several typical guidance laws,which have been widely implemented for exoatmosphericinterception, are also applied for comparison. Those guid-ance laws include proportional navigation (PN) [34], aug-mented proportional navigation (APN) [35], and optimalterminal guidance (OTG) [36]. It should be noted that thesethree methods are different from their gravity models. PNassumes that the gravity difference is zero, whereas APNand OTG assume that gravity differences are constant andlinear, respectively [37].

All numerical results have been recorded, and the result-ing velocity increments for various methods are listed inTable 5. Obviously, the proposed method significantly out-performs the other three guidance laws in terms of the veloc-ity increment because it regards the minimum velocityincrement as the cost function. Although the Case 1 of Group3 is the worst case, the velocity increment of the proposedmethod is only 72.4% of that of PN (27.6% reduction),73.5% of that of APN (27.5% reduction), and 73.9% of thatof OTG (27.1% reduction). For some special cases, the costreduction is even higher than 84%. Consequently, the pro-posed method is able to achieve very favorable fuel efficiencyand retain enough energy for a successful collision. It shouldbe noted that, because Keplerian motion is not taken intoaccount, PN, APN, and OTG are not suitable for tail-chaseinterception scenarios such as the Case 3 in Group 1 andGroup 2. The velocity increments for these three methodsare higher than 2264m/s and 1738m/s for Group 1 andGroup 2, respectively. However, the values of the proposedmethod are only 552m/s and 421m/s, which suggest thatthe proposed method has a more extensive applicability.Computational efficiency was fully discussed in Subsection4.2. It can be concluded that the proposed method is suffi-ciently applicable for midcourse guidance.

5. Conclusion

In this paper, a semianalytical method is proposed for solvingthe exoatmospheric midcourse guidance problem with theminimum velocity increment. A nonlinear programmingproblem relating Lambert’s problem with Keplerian motionis firstly formulated. Then, Karush-Kuhn-Tucker conditionsand its Jacobian matrix are successfully derived in an analyt-ical manner. Therefore, Newton-Raphson method can beefficiently employed to obtain the optimal result. In orderto further improve the computational efficiency, a near-

optimal initialization method is developed to provide a warmstart. Several experiments and the comparison with existingmethods are carried out to evaluate the performance of theproposed method. Numerical results show that, even forworse case scenarios, the maximal number of iterationsrequired is less than 6, the maximal computation time is lessthan 0.116ms, and the cost reduction is more than 24.1%.Apparently, the proposed method outperforms typicalmethods and is of rapid convergence, high computationalefficiency, and wide applicability.

Data Availability

The simulation data used to support the findings of this studyare available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Table 5: Comparisons of simulation results.

Δv (m/s)Group 1 Group 2 Group 3


PN 1046 996 3575 802 721 2751 557 1244 1398

APN 1030 678 2673 753 371 2089 545 1097 1027

OTG 1025 592 2264 738 284 1738 543 1058 906

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Minimum cost reduction (%) 26.1 45.6 75.6 26.3 61.3 75.8 24.1 25.3 63.1

Maximum cost reduction (%) 27.6 67.7 84.6 32.2 84.7 84.7 26.0 36.5 76.1


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