Modified inverse-polynomial shaping approach with

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Original Article

Modified inverse-polynomial shaping approach with thrust and radius constraints

Proc IMechE Part G: J Aerospace Engineering 0(0) 1–13 ! IMechE 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954410015579473 uk.sagepub.com/jaero

Dongzhe Wang, Gang Zhang and Xibin Cao

Abstract The shape-based low-thrust trajectory approximation with modified inverse polynomials is studied by considering thrust and radius constraints, which require that the thrust-acceleration magnitude be less than a maximum allowed value and the trajectory radius be between a lower bound and an upper bound. Compared with the original inverse-polynomial method, the polynomial orders in the modified one are optimized. For the time-free transfer between circular orbits, it is proved that the radius monotonously changes, and the maximum thrust acceleration is obtained by solving a cubic polynomial which is the first-order expansion at half of the transfer angle. For a given maximum thrust acceleration, the minimum revolution number is analytically estimated. For the time-fixed rendezvous between circular orbits, the seventh parameter is solved by the secant method in its feasible range considering the radius constraints. The maximum tangentthrust-acceleration magnitude is estimated by solving real roots of a polynomial, and then the feasible solutions for a given maximum acceleration are determined. Numerical examples show that the maximum thrust acceleration by the proposed modified inverse-polynomial method is less than that by the original inverse-polynomial method for both the orbit transfer and rendezvous problems. Keywords Low-thrust, shape-based trajectory, modified inverse polynomials, constraints Date received: 7 September 2014; accepted: 5 March 2015

Introduction The problem of spacecraft trajectory under a specialdirection thrust acceleration has been studied for many years. The three thrust directions, circumferential (or called along-track), radial and tangent directions are being highly researched.1 Previous research mainly focused on approximate analytical trajectories, orbit raising and escape problems under a constant radial thrust acceleration2,3 or a constant tangent thrust acceleration.4,5 Tangent thrust is efficient because it changes the instantaneous orbital energy (as well as the instantaneous semimajor axis) at the maximum rate. According to the thrust magnitude of the engine, tangent thrust can be divided into two categories: impulse tangent thrust and continuous tangent thrust. For the impulse tangent thrust, Vallado6 discussed the orbit transfer with one tangent burn in detail. The well-known Hohmann transfer6 is the minimum-energy one among all two-impulse transfers. However, it is only feasible for coplanar circular orbits and coplanar elliptic orbits sharing the same apsidal line. For general coplanar elliptic orbits, the two-impulse cotangent transfer was

obtained in closed form by using the relationship between semilatus rectum and flight-direction angle.7 Based on the cotangent-transfer solution, the two-impulse cotangent rendezvous problem was transformed into solving a single variable function in the transfer-time equation.8,9 These approaches are valid for orbit maneuver problems only in the tangent-impulse case but not in the continuoustangent-thrust case. For the continuous tangent low thrust, the most important application is in the shape-based approximation method. The low-thrust trajectory is supposed to be a specified-form shape function, in which the parameters are obtained from initial and final boundary conditions. There are two merits for the

Research Center of Satellite Technology, Harbin Institute of Technology, Harbin, People’s Republic of China Corresponding author: Gang Zhang, Research Center of Satellite Technology, Harbin Institute of Technology, Science Park Building # B3, Mailbox 3012, Harbin 150080, People’s Republic of China. Email: [email protected]

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Proc IMechE Part G: J Aerospace Engineering 0(0)

assumption of tangent thrust in the shape-based method. One is tangent thrust changes the instantaneous orbital energy at the maximum rate. This results in the approximation being near to the ‘‘true’’ optimal low-thrust trajectory without the thrust-direction constraint, which was usually obtained by numerical optimization algorithms.10,11,12 The other is the trajectory motion equation can be simplified, then some analytical expressions can be derived. Petropoulos and Longuski13 used an exponential sinusoid function to approximate the shape of lowthrust gravity-assist trajectory. However, there were only four parameters in the exponential sinusoid function, thus the boundary conditions could not be satisfied at the same time. Wall and Conway14 proposed an inverse polynomial (IP) shape for low-thrust transfer and rendezvous trajectories. For the orbit transfer problem, all six parameters were obtained in closed form from the boundary conditions, whereas for the orbit rendezvous problem, the seventh parameter was numerically obtained from the time equation. To improve the method of Wall and Conway,14 the range of the seventh parameter was obtained from the change rate of the transfer angle, and then the rendezvous problem was solved in this range.15 In the global trajectory optimization competition there are limitations on the available maximum thrust. Recently, Fourier series were used to approximate the shape by considering the thrust magnitude constraint. The Fourier coefficients were solved in a resulting nonlinear programming problem.16 Furthermore, the Fourier series approximation method was used in the on–off low-thrust trajectories.17 All of the above shape-based methods are only for coplanar orbits. In addition, the pseudoequinoctial elements were used to shape three-dimensional rendezvous trajectories, but without the constraint on the tangent thrust acceleration.18 Besides the thrust-acceleration magnitude constraint, the trajectory should satisfy radius constraints. For example, in low-Earth orbits, the trajectory radius should be greater than the Earth’s radius plus the atmosphere altitude (e.g. 200 km), and less than the radius of the Earth’s sphere of influence (i.e. 925,000 km). Although the thrustacceleration constraint is considered by Taheri and Abdelkhalik,16 it is only viewed as an inequality constraint when solving the nonlinear programming problem. This paper presents a study on the shape-based method with modified IPs by considering thrust and radius constraints. For given thrust-acceleration magnitude and radius constraints between two fixed endpoints, the minimum revolutions for the transfer problem will be estimated by analytical expressions. For the rendezvous problem, the parameter will be solved by the secant method in its feasible range, then the feasible revolutions and the minimum-fuel shape-based trajectory will be obtained.

Problem analysis For the time-free orbit transfer problem, a modified IP shape function is chosen as 1 , c0 þ c1 þ c2 þ cl l þ cm m þ cn n where 34l 5 m 5 n

rðÞ ¼

2

ð1Þ where r is the radius, is the transfer angle in the polar coordinate, and c0 , c1 , c2 , cl , cm , cn are function parameters to be determined. From the initial boundary condition it is known that14 1 tan 1 e1 sin ’1 , c1 ¼ ¼ , r1 r1 p1 1 1 1 ¼ c2 ¼ 4 2 _ 2r 2p 2r 2r1 1 1 1 1

c0 ¼

ð2Þ

where is the gravity parameter, is the flight-path angle, p is the semilatus rectum, ’ is the true anomaly, and e is the eccentricity. The subscripts ‘‘1, 2’’ denote the initial and final points, respectively. Other parameters are obtained as 2

2

3

mn=fl ðml Þðnl Þ

cl 6 nl=fm 6 7 6 4 cm 5 ¼ 6 6 ðml ÞðnmÞ 4 cn ml=fn

ðmþn1Þ=fl1 ðml Þðnl Þ ðlþn1Þ=fm1 ðml ÞðnmÞ

1=fl2 ðml Þðnl Þ

3 7

1=fm2 7 7 ðml ÞðnmÞ 7

5

ðlþm1Þ=fn1 1=fn2 ðnl ÞðnmÞ ðnl ÞðnmÞ ðnl ÞðnmÞ 2 1 3 2 2 b1 r2 ðc0 þ c1 f þ c2 f Þ 6 e2 sin ’2 7 6 6 7 ðc1 þ 2c2 f Þ 5 ¼ A4 b2 p2 4 1 1 b3 p2 r2 2c2

3 7 5 ð3Þ

where f is the transfer angle at the final point. The derivative of ðÞ with respect to is denoted by ðÞ0 , the second-order and third-order derivatives are denoted by ðÞ00 and ðÞ000 , respectively. Define a polynomial variable

qðÞ ¼

1 ¼ c0 þ c1 þ c2 2 þ cl l þ cm m þ cn n r ð4Þ

then the fight-path angle is obtained from r_ r_ ¼ r c1 þ 2c2 þ lcl l1 þ mcm m1 þ ncn n1 q0 ¼ q ð5Þ

tan ¼




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and the time derivative _ is obtained from 1 _2 ¼ 4 ( r ð1=rÞ þ 2c2 þ l ðl 1Þcl l2

When " ¼ 0, equation (10) is equal to 0. When " ¼ 1, the coefficients of b1 , b3 in equation (10) are equal to 0, and the coefficient of b2 is equal to 1. Then it is not difficult to obtain

)

(

þmðm 1Þcm m2 þ nðn 1Þcn n2 4

q ¼ q þ q00

’1 q0 ð0Þ ¼ c1 ¼ e1 sin p1 ’2 q0 ðf Þ ¼ c1 þ 2c2 f þ b2 ¼ e2 sin p2

ð11Þ

ð6Þ With equation (5), the cosine of the flight-path angle is 1 q cos ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ tan q2 þ ðq0 Þ2

ð7Þ

Then the magnitude of the tangent-thrust acceleration is 8 9 l3 > > < l ðl 1Þðl 2Þcl = m3 þmðm 1Þðm 2Þcm > > : ; þnðn 1Þðn 2Þcn n3 tan =r 8 9 Ta ¼ 3 l2 2r cos > < ½ð1=rÞ þ 2c2 þ l ðl 1Þcl > = þmðm 1Þcm m2 > > : ; þnðn 1Þcn n2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 0 000 q ðq þ q Þ q2 þ ðq0 Þ ¼ 2 ðq þ q00 Þ2 ð8Þ

which indicates that l53 must be satisfied in equation (1); otherwise, lim!0 Ta ¼ 1 is obtained from equation (8). It should be notified that the last expressions in equations (5)–(8) are satisfied for both the sixthdegree transfer problem and the seventh-degree rendezvous problem. The angle derivative of the radius is obtained from equation (5) as r_ ¼ q2 q0 ¼ r2 c1 þ 2c2 þ lcl l1 _ þmcm m1 þ ncn n1

r0 ¼

ð9Þ

which is used to obtain the extreme points of the tra jectory radius. Let " ¼ =f 2 ½0, 1, then

which is used to judge the radius changes at the boundary points. The extreme points of Ta can be obtained by Ta0 ¼ 0. However, equation (8) is too complex to get the analytical values for Ta0 ¼ 0. The traditional numerical method for the maximum value of a single variable function is dividing the whole range into many subintervals, and the subintervals should be small enough such that each subinterval has a single extreme. Then a numerical method such as the golden search is used to get the single extreme in each subinterval. Finally, the maximum value is obtained by comparing all extreme points and two boundary points. However, the traditional method needs a large number of calculations. In this paper, an approximate method for the extreme points is provided. From equations (4,10) it is known that the derivatives of q with respect to need to be divided by f. When the transfer angle f is large enough, the derivatives of q will heavily decrease. By setting q00 ¼ q000 ¼ 0 in equation (8), an approximate expression of the thrust acceleration is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T^ a ¼ q0 q2 þ ðq0 Þ2 2

ð12Þ

whose extreme points are obtained by solving real roots of the following polynomial: h i dðT^ 2a Þ 2 ¼ 2q0 ðq0 Þ ð2q00 þ qÞ þ q2 q00 ¼ 0 d

ð13Þ

which is valid for both the transfer and rendezvous problems. The polynomial roots are obtained by computing the eigenvalues of companion matrix (the ‘‘roots’’ function in Matlab). Note that solving the roots of the polynomial equation (13) is simpler than solving the extreme values of the single-variable function equation (8).

lcl l1 þ mcm m1 þ ncn n1 mnl "l1 "m1 "n1 ¼ þ b1 f ðm l Þðn l Þ ðm l Þðn mÞ ðn l Þðn mÞ ðm þ n 1Þl"l1 ðl þ n 1Þm"m1 ðl þ m 1Þn"n1 þ þ þ b2 ðm l Þðn l Þ ðm l Þðn mÞ ðn l Þðn mÞ l"l1 m"m1 n"n1 þ f þ b3 ðm l Þðn l Þ ðm l Þðn mÞ ðn l Þðn mÞ


ð10Þ


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For the transfer between circular orbits, the radius monotonously changes, which indicates that there is no solution for q0 ¼ 0 (the detailed proof will be provided in the next section), thus all extreme points can be obtained by ðq0 Þ2 ð2q00 þ qÞ þ q2 q00 ¼ 0

Circular-to-circular transfer Monotonous change of radius

c1 ¼ c2 ¼ 0,

b1 ¼

1 1 , r2 r1

b2 ¼ b3 ¼ 0 ð15Þ

Moreover, from equation (11) it is known that q0 ð0Þ ¼ q0 ðf Þ ¼ 0

mnl"l1 ¼ ð1 "Þ2 b1 2f

ð17Þ

Thus, q0 ðÞ40 is satisfied for any 2 ½0, f since b1 ¼ 1=r2 1=r1 5 0. Actually, for an arbitrary selected group of l, m, n satisfying 34l 5 m 5 n, the inequality q0 ðÞ40 is satisfied, which indicates that "l1 "m1 "n1 þ ðm l Þðn l Þ ðm l Þðn mÞ ðn l Þðn mÞ ¼ "l1 gð"Þ50 ð18Þ which is equivalent to

gð"Þ ¼

1 "ml ðm l Þðn l Þ ðm l Þðn mÞ "nl 50 þ ðn l Þðn mÞ

ð19Þ

Since gð0Þ 4 0, gð1Þ ¼ 0, and dgð"Þ "ml1 "nl1 ¼ þ d" ðn mÞ ðn mÞ "ml1 ¼ ð1 þ "nm Þ40 ðn mÞ

for any " 2 ½0, 1 ð20Þ

For the coplanar transfer from circular to circular orbits, assume that r2 4 r1 , then the radius monotonously increases, which indicates that equation (9) is positive, i.e. q0 ðÞ40 for any 2 ½0, f . For the initial circular orbit, from equations (2) and (3) it is obvious that 1 , r1

mnl 1 l1 1 lþ1 l " " þ " b1 q ðÞ ¼ f 2 2 0

ð14Þ

Then the maximum acceleration T^ a max is obtained by comparing extreme and boundary points. For the original IP shape with l ¼ 3, m ¼ 4, n ¼ 5, the maximum (single extreme) thrust acceleration occurs at nearly half of the transfer angle.14 Thus, the golden search method can provide an accurate numerical solution of Ta max . However, this paper aims to derive an analytical approximation of Ta max without numerical iterations. In addition, for the rendezvous problem between circular orbits, the extreme thrust acceleration may be not at nearly half of the transfer angle, and in some cases there exist many extreme points. Then, there is no analytical approximation for Ta max , and it will be obtained by numerically solving equation (13). When l ¼ 3, m ¼ 4, n ¼ 5, there are at most four extreme points for the trajectory radius, thus the IP shape is unfeasible for elliptic orbits with three revolutions or more.18 Actually, there are enough extreme points for the radius with large values of l, m, n; however, after a great number of numerical tests, the modified IP shape with large orders is also unfeasible for elliptic orbits with many revolutions (e.g. six revolutions) in most cases. Then, the trajectories with thrust and radius constraints will only be solved in the transfer and rendezvous problems between circular orbits.

c0 ¼

If m ¼ l þ 1, n ¼ m þ 1 are selected in equation (1), from equation (10) it is known that

ð16Þ

which indicates that the initial and final points are two extreme points of the radius.

then equation (19) is satisfied, and q0 ðÞ is non-positive for any 34l 5 m 5 n. Finally, for any 34l 5 m 5 n, the radius monotonously changes, which indicates that the trajectory must satisfy the radius constraint for the transfer between circular orbits.

Maximum thrust acceleration for the original IP method For the original IP method,14 i.e. l ¼ 3, m ¼ 4, n ¼ 5, the polynomial is qðÞ ¼ c0 þ ð10"3 15"4 þ 6"5 Þb1

ð21Þ

The derivatives of q with respect to are q0 ¼

30b1 2 30b1 2 " 2"3 þ "4 ¼ " ð1 "Þ2 f f


ð22aÞ



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q00 ¼

5

60b1 60b1 " 3"2 þ 2"3 ¼ 2 "ð1 "Þð1 2"Þ 2 f f ð22bÞ

q000 ¼

60b1 ð1 6" þ 12"2 Þ f3

ð22cÞ

Then q0 ð0Þ ¼ q00 ð0Þ ¼ q0 ðf Þ ¼ q00 ðf Þ ¼ 0 is obtained, from equation (14) it is known that " ¼ 0 and " ¼ 1 (i.e. two boundary points) are two approximate extreme points for Ta. For the transfer problem between circular orbits, there is a single extreme point for Ta and it is near to ¼ f =2, i.e., " ¼ 1=2, especially for a large f.14 Based on this property, the single extreme point is written as 1 " ¼ þ 2

ð23Þ

where is a small value to be determined. The power of equation (23) is

q0s ¼

15b1 , 8f

ð28Þ

where " is obtained by equations (23,27) with f ¼ 1, then q^ and q^ " are independent of f. The first-order derivative is obtained as q^ 0 ¼ q^ " =f from equation (22). Substituting q^0 ¼ q^ " =f into equation (12) and squaring both sides gives

ð29Þ

Multiplying both sides of equation (29) by f4 yields a quadratic equation of f2 . Then, an estimated value of f is

q00 q00s þ q000 s f ð25Þ

where the subscript ‘‘s’’ denotes " ¼ 1=2 for the corresponding function. Substituting " ¼ 1=2 into equations (21) and (22) gives 1 qs ¼ c0 þ b1 , 2

dq^ ¼ 30b1 ð" Þ2 2ð" Þ3 þ ð" Þ4 d"

ð24Þ

where k is the order of ". Then the first-order approximate expressions are obtained as q0 q0s þ q00s f ,

q^" ¼

2 4 4T2a max q^ " q^ " 2 ¼ q^ þ 2 f f

k k1 1 1 k ð" Þ ¼ þk þ Oð2 Þ 2 2 k k1 1 1 þk 2 2

q qs þ q0s f ,

there are two complex roots in all three solutions, the other one is a real root which is in the range [0,1]. Once the closed-form real root is solved, the extreme point is obtained as equation (23). Then substituting " into equations (21) and (22) gives analytical approximations of q and its derivatives, which are ^ q^ 0 , q^00 , q^ 000 . Finally, for a given transfer denoted by q, angle f, an analytical approximation of Ta max is obtained from equation (12). For a given Ta max , the minimum revolution number will be estimated. Define

q00s ¼ 0,

q000 s ¼

30b1 f3

ð26Þ Substituting equation (25) into the extreme value condition equation (14), and multiplying both sides by f2 =ð30b1 Þ yields a cubic polynomial 2 15 15 1 3 b1 þ b1 c0 þ b1 2 8 4 2 " ! # 2 1 15 15 60 2 2 b1 þ c0 þ b1 2 128 8 f 15 1 b1 c0 þ b1 ¼ 0 þ 128 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q^2" q^2 þ q^ 2" q^ 4 16T2a max =2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q^2 þ q^ 4 þ 16T2a max =2 q^ " ¼ 2 2Ta max

^f ¼ 2Ta max

ð30Þ

Finally, an analytical expression of the minimum revolution number is obtained as & ’ ^f mod ð’2 ’1 , 2Þ N^ ¼ 2

ð31Þ

where the function y denotes the nearest integer greater than or equal to y.

Maximum thrust acceleration for the modified IP method ð27Þ

When f is large enough, equation (27) is almost unchangeable with f, then the extreme point is almost independent of f. Extended numerical tests show that

For the modified IP method, the polynomial orders l, m, n are arbitrarily selected with the constraint 34l 5 m 5 n. Substituting equation (25) into the extreme value condition equation (14) yields a cubic polynomial 3 3 þ 2 2 þ 1 þ 0 ¼ 0


ð32Þ


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where the coefficients are

2

0 ¼ q2s q00s þ ðq0s Þ2 ð2q00s þ qs Þ

1=r2 ðc0 þ c1 f þ c2 f2 þ c3 f3 Þ

3

6 7 4 e2 sin ’2 =p2 ðc1 þ 2c2 f þ 3c3 f2 Þ 5

ð33aÞ

1=p2 1=r2 ð2c2 þ 6c3 f Þ 0 2 000 0 0 00 00 1 ¼ ½q2s q000 s þ ðqs Þ ð2qs þ qs Þ þ 4qs qs ðqs þ qs Þf ð33bÞ

00 00 2 00 2 ¼ 2q0s q000 s ðqs þ 2qs Þ þ ðqs Þ ð2qs þ qs Þ

þ3ðq0s Þ2 q00s f2

ð33cÞ

00 2 000 0 3 3 ¼ ½ðq0s Þ2 q000 s þ ðqs Þ ð2qs þ qs Þf

ð33dÞ

Equation (32) is valid for IPs with free orders 34l 5 m 5 n. Then an analytical approximation of Ta max is obtained from equation (12). The polynomial orders l, m, n are optimized for the minimum Ta max with the constraint 34l 5 m 5 n. In addition, for a given Ta max , the minimum revolution number is also estimated for free polynomial orders. From equations (3,15) it is known that

mn nl ð" Þl þ ð" Þm ðm l Þðn l Þ ðm l Þðn mÞ ml n ð" Þ þ ðn l Þðn mÞ ð34Þ

q^ ¼ c0 þ b1

" dq^ ð" Þl1 ð" Þm1 ¼ lmnb1 þ q^ " ¼ d" ðm l Þðn l Þ ðm l Þðn mÞ ð" Þn1 þ ðn l Þðn mÞ ð35Þ

then the minimum revolution number is also estimated by equations (30) and (31) for any 34l 5 m 5 n.

Letting new parameters which are independent of c3 2

c04

2

3

30f2

10f3

f4

3

7 1 6 6 07 48f 18f2 2f3 7 4 c5 5 ¼ 6 6 4 5 2f c06 20 8f f2 2 3 1=r2 ðc0 þ c1 f þ c2 f2 Þ 6 7 4 e2 sin ’2 =p2 ðc1 þ 2c2 f Þ 5

ð37Þ

1=p2 1=r2 2c2 and substituting equation (37) into equation (36) yields c4 ¼ c04 ð3=f Þc3 ,

c5 ¼ c05 þ ð3=f2 Þc3 ,

c6 ¼ c06 ð1=f3 Þc3

ð38Þ

From _2 4 0 in equation (6) it is known that ð1=rÞ þ 2c2 þ 6c3 þ 12c4 2 þ 20c5 3 þ 30c6 4 4 0 ð39Þ for any 2 ½0, f . Substituting equation (38) into equation (39) gives H1 ðÞ þ G1 ðÞc3 4 0

ð40Þ

where H1 , G1 are two functions independent of c3 as H1 ðÞ ¼ c0 þ 2c2 þ c1 þ ðc2 þ 12c04 Þ2 þ 20c05 3 þ ðc04 þ 30c06 Þ4 þ c05 5 þ c06 6 ð41aÞ G1 ðÞ ¼ 6 ð36=f Þ2 þ ð1 þ 60=f2 Þ3

Circular-to-circular rendezvous Parameter range of radius constraints for the original IP method

ð3=f þ 30=f3 Þ4 þ ð3=f2 Þ5 ð1=f3 Þ6 ð41bÞ

For the time-fixed orbit rendezvous problem, a seventh-degree (sixth-order) IP is used in the original IP method.14 From the initial point the parameters c0 , c1 , c2 are obtained as equation (15), and c4 , c5 , c6 are obtained from the final point and c3 as14 2 3 30f2 c4 6 1 6 7 48f 4 c5 5 ¼ 6 6 2f 4 c6 20 2

10f3 18f2 8f

f4

G1 ðÞ ¼ ð f Þ½6=f þ ð30=f2 Þ

3

7 2f3 7 5 f2

Because equation (40) is satisfied for any 2 ½0, f , then H1 ðÞ must be positive at the values of satisfying G1 ðÞ ¼ 0. Note that G1 ð0Þ ¼ G1 ðf Þ ¼ 0, then G1 ðÞ can be rewritten as

ð1=f þ 30=f3 Þ2 þ ð2=f2 Þ3 ð1=f3 Þ4 ð42Þ

ð36Þ

Thus, other roots of G1 ðÞ can be obtained in closed form by solving a quartic polynomial.




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Shang et al. 15 provided a method to judge the signpof ffiffiffi G1 ðÞ for different f, the critical value is f ¼ 2 6. There are three cases. pffiffiffi 1. If f 5 2 6, there are two solutions g1 , g2 2 ð0, f Þ for G1 ðÞ ¼ 0. If H1 ð0Þ, H1 ðg1 Þ, H1 ðg2 Þ, H1 ðf Þ are all positive, and max fH1 ðÞ=G1 ðÞg the inequality 2ðg1 , g2 Þ

5 min2ð0,g1 Þ[ð0, g2 , f Þ fH1 ðÞ=G1 ðÞg is satisfied, then the range is

max fH1 ðÞ=G1 ðÞg 5 c3 min

2ð0,g1 Þ[ð0, g2 , f Þ

¼ 3 ð=f 1Þ3

ð48bÞ

Equation (48) indicates that G2 ðÞ 4 0 for any 2 ð0, f Þ and G2 ð0Þ ¼ G2 ðf Þ ¼ 0. If H2 ð0Þ, H2 ðf Þ are both positive, then the range of c3 is c3 4 max fH2 ðÞ=G2 ðÞg 2ð0,f Þ

ð49Þ

otherwise, there is no range of c3. For a lower bound rmin , it is known that

2ðg1 , g2 Þ

5

G2 ðÞ ¼ 3 ð3=f Þ4 þ ð3=f2 Þ5 ð1=f3 Þ6

ð43Þ

fH1 ðÞ=G1 ðÞg

otherwise, there is no range of c3; pffiffiffi 2. If f ¼ 2 6, there is one solution f =2 for G1 ðÞ ¼ 0 in ð0, f Þ. If H1 ð0Þ, H1 ðf =2Þ, H1 ðf Þ are all positive, then the range is

1 1 ¼ H3 ðÞ þ G2 ðÞc3 5 0 r rmin

ð50Þ

where H3 ðÞ ¼ 1=rmin þ c0 þ c1 þ c2 2 þ c04 4 þ c05 5 þ c06 6

ð51Þ

If H3 ð0Þ, H3 ðf Þ are both positive, the range of c3 is c3 4

max

2ð0,f =2Þ[ðf =2,f Þ

fH1 ðÞ=G1 ðÞg

ð44Þ c3 5 min fH3 ðÞ=G2 ðÞg 2ð0,f Þ

ð52Þ

otherwise, there is no range of c3; pffiffiffi 3. If f 4 2 6, there is no solution for G1 ðÞ ¼ 0, i.e. G1 ðÞ 4 0 for any ð0, f Þ. If H1 ð0Þ, H1 ðf Þ are both positive, then the range is

otherwise, there is no range of c3. Finally, the intersection range of c3 satisfying equations (39) and (46) is obtained as ðc3 min , c3 max Þ.

Parameter range of radius constraints for the modified IP method c3 4 max fH1 ðÞ=G1 ðÞg

ð45Þ

2ð0,f Þ

otherwise, there is no range of c3. Then, the range of c3 satisfying equation (39) is obtained for all pcases. For the multiple-revolution ffiffiffi case, f 4 2 4 2 6, thus case 3 is satisfied and equation (45) is needed. The radius constraints can be written as rmin 5 r 5 rmax ,

1 1 1 5 5 rmax r rmin

ð46Þ

q ¼ c0 þ c1 þ c2 2 þ c3 3 þ cl l þ cm m þ cn n ð53Þ Then the range of c3 will be solved from the condition _2 4 0 and the radius constraints. The first condition _2 4 0 means that q þ q00 4 0

For an upper bound rmax , it is known that 1 1 ¼ H2 ðÞ þ G2 ðÞc3 4 0 r rmax

For the orbit rendezvous problem with free polynomial orders 44l 5 m 5 n, the modified seventhdegree (nth-order) IP is

ð47Þ

ð54Þ

where the second-order derivative is q00 ¼ 2c2 þ 6c3 þ l ðl 1Þcl l2 þ mðm 1Þcm m2

where H2 ðÞ ¼ 1=rmax þ c0 þ c1 þ c2 2 þ c04 4 þ c05 5 þ c06 6

þ nðn 1Þcn n2 ð48aÞ


ð55Þ



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Proc IMechE Part G: J Aerospace Engineering 0(0) Similar to equation (38), letting 2 3 2 03 2 3 cl cl kl 6 7 6 0 7 6 7 4 cm 5 ¼ 4 cm 5 þ c3 4 km 5, c0n cn kn 2 0 3 2 3 cl b1 6 0 7¼ 6 7 where 4 cm 5 A4 b2 5, 0 cn b3 2 3 2 3 f3 kl 6 7¼ 6 7 A4 3f2 5 4 km 5 kn 6f

ð56Þ

then equation (54) is rewritten as equation (40), i.e. H1 ðÞ þ G1 ðÞc3 4 0, where H1 ðÞ ¼ c0 þ 2c2 þ c1 þ c2 2 þ c0l l þ c0m m

Although an approximate expression can be pffiffiffiffi R obtained as t 1= 0 f q3=2 d when f is large, this expression cannot be integrated in closed form. Thus, numerical integration methods (e.g. composite Simpson’s rule) are used for equation (59). For the rendezvous problem, the transfer time tf is fixed. Then the value of c3 needs to be solved in this range ðc3 min , c3 max Þ. A large number of numerical examples show that tðc3 Þ monotonously decreases, even if it cannot be proved strictly. Actually, the range of c3 limits the transfer times as well. The feasible range of transfer time is obtained as ½tðc3 max Þ, tðc3 min Þ. Then, if ½tðc3 min Þ tf ½tðc3 max Þ tf 4 0 is satisfied, there is no solution of c3 for this given transfer time tf ; otherwise, there exists a single solution in the range c3 2 ðc3 min , c3 max Þ which is obtained by the secant method as

þ c0n n þ l ðl 1Þc0l l2 þ mðm 1Þc0m m2 þ nðn 1Þc0n n2 3

l

c3,nþ1 ¼ c3,n ðc3,n c3,n1 Þ m

G1 ðÞ ¼ 6 þ þ kl þ km þ kn

tðc3,n Þ tf , tðc3,n Þ tðc3,n1 Þ

n40

ð57aÞ

ð60Þ

ð57bÞ

where the initial guesses are c30 ¼ c3 min , c31 ¼ c3 max . When c3 is obtained, the thrust-acceleration magnitude is obtained by equation (8). The polynomial orders l, m, n are optimized for the minimum Ta max with the constraint 44l 5 m 5 n. Note that the analytical method to estimate the single extreme point of Ta max using equation (27) for the transfer problem is not valid for the rendezvous problem, since the extreme thrust acceleration may not occur at nearly half of the transfer angle and there may be many extreme points. When the transfer angle f is large, all extreme points can be numerically obtained from the polynomial equation (13), and then the maximal thrust acceleration Ta max is obtained by comparing all extreme points and two boundary points. For different revolution numbers, Ta max are different. Then for a given maximal thrust-acceleration constraint, the feasible revolution numbers and the minimumfuel solution R can be obtained. The fuel consumption _ d as in Wall and Conway.14 is V ¼ 0 f Ta ðÞ=ðÞ

n

þ l ðl 1Þkl l2 þ mðm 1Þkm m2 þ nðn 1Þkn n2

For the multiple-revolution case, equation (45) is used. Similarly, for the radius constraint condition equation (46), using the following equations: H2 ðÞ ¼ 1=rmax þ c0 þ c1 þ c2 2 þ c0l l þ c0m m þ c0n n ð58aÞ G2 ðÞ ¼ 3 þ kl l þ km m þ kn n

ð58bÞ

H3 ðÞ ¼ 1=rmin þ c0 þ c1 þ c2 2 þ c0l l

ð58cÞ

þ c0m m þ c0n n

the ranges of c3 for upper and lower radius constraints are obtained by equations (49) and (52), respectively. Finally, for the modified IP method, the intersection range of c3 satisfying _2 4 0 and the radius constraints is obtained as ðc3 min , c3 max Þ. Then the value of c3 in the transfer-time equation for the rendezvous problem will be solved in this range. Note that for any 34l 5 m 5 n in equation (1), the radius monotonously changes between circular transfers, then c3 ¼ 0 satisfies the radius constraints. Thus, c3 ¼ 0 must be in the feasible range ðc3 min , c3 max Þ.

Maximum thrust acceleration for fixed time For both the original and modified IP methods, from equation (6) the transfer time is obtained as Z f sffiffiffiffiffiffiffiffiffiffiffiffiffi00 qþq t ¼ d ð59Þ q4 0

Numerical examples Two examples are given for the time-free orbit transfer and the time-fixed orbit rendezvous, which are solved by the sixth-degree and seventh-degree IPs, respectively. The thrust-acceleration magnitude constraint and the trajectory radius constraints are considered. The numerical value (viewed as ‘‘true’’ value) of Ta max is obtained via equation (8) by dividing the whole range into 10 subintervals and using the golden search in each subinterval. The modified IP method is compared with the original IP method in the following transfer and rendezvous problems.




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Orbit transfer between circular orbits In this specific orbit transfer, the initial orbit is in a circular orbit with a radius of 1 DU, the final orbit is in a coplanar circular orbit with a radius of 3 DU, and the transfer time is free. Canonical units are used such that 1 DU is 1 AU and 2 TU is 1 year. The transfer angle with zero revolution is assigned as 180 for two fixed endpoints. These orbital parameters are the same as those of Wall and Conway.14 From Section 2 it is known that the radius monotonously increases, thus the radius constraints must be satisfied. For the original IP method, if the maximum thrust level for this transfer is set at 0.02 DU/TU2, the minimum transfer angle is obtained as 22.7572 rad by equation (30), and the minimum revolution number is obtained as 4 by equation (31) for two fixed endpoints. If the maximum thrust level for this transfer is set at 0.01 DU/TU2, the minimum transfer angle is

x 10

−3

(b)

3

9

2.8

8

2.6 Trajectory radius (DU)

Thrust acceleration (DU/TU2)

(a) 10

45.4439 rad, and the minimum revolution number is 7. If the maximum thrust level for this transfer is set at 0.001 DU/TU2, the minimum transfer angle is 452.2049 rad, and the minimum revolution number is 72. When the revolution number is 7, the thrust acceleration history is plotted in Figure 2(a). The true maximum thrust acceleration is Ta max ¼ 9:60e 3 DU=TU2 , which is less than 0.01 DU/TU2. For the modified IP method with N ¼ 7, the polynomial orders are obtained as l ¼ 3, m ¼ 4, n ¼ 100 using the numerical optimization algorithm with the constraint 34l 5 m 5 n4100. The true maximum thrust acceleration is Ta max ¼ 8:61e 3 DU=TU2 , which saves the maximum thrust acceleration by 10.3% compared with the original IP method. The thrust acceleration history for transfer using the modified IP method is plotted in Figure 1(a), which shows that the estimated value of Ta max analytically solved

7 6 5 4 3 Original IP, true Modified IP, true Modified IP, estimated

2 1

Original IP Modified IP

2.4 2.2 2 1.8 1.6 1.4 1.2

0

1

0

0.2 0.4 0.6 0.8 Normalized transfer angle ε

1

0

0.2 0.4 0.6 0.8 Normalized transfer angle ε

1

Figure 1. Transfer result comparisons for N ¼ 7 using the original and modified IP methods: (a) thrust acceleration; (b) trajectory radius.

(a)

(b)

Figure 2. Feasible rendezvous solutions for different revolutions using the original and modified IP methods: (a) maximum thrust acceleration Ta max ; (b) energy cost V.




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Table 1. Rendezvous result comparisons by the original and modified IP methods. Original IP method

N¼7 N ¼ 10 N ¼ 12 N ¼ 18 N ¼ 21

Modified IP method

Ta max (km/s2)

minfrg (km)

maxfrg (km)

V (km/s)

Ta max (km/s2)

minfrg (km)

maxfrg (km)

V (km/s)

9:62e 5 6:60e 5 8:40e 5 N/A N/A

7178 7177.7 6626.4 N/A N/A

42160 42160 42160 N/A N/A

4.3735 4.3748 4.3744 N/A N/A

8:55e 5 5:66e 5 4:49e 5 2:98e 5 3:50e 5

7178 7178 7178 7178 7177.8

55127 43049 42160 42160 42160

5.1058 4.4168 4.3758 4.3758 4.3753

via equations (12), (23), and (32) is near to its true value. Moreover, the curves of the trajectory radius versus the normalized transfer angle by both the original and modified IP methods are plotted in Figure 1(b). For the modified IP method with l ¼ 3, m ¼ 4, n ¼ 100, if the maximum thrust level for the transfer is set at 0.01 DU/TU2, the minimum transfer angle is 45.69 rad, and the minimum revolution number is 6. If the maximum thrust level is set at 0.001 DU/TU2, the minimum revolution number is 65. These revolution numbers by the modified IP method are less than those by the original IP method. In addition, if the polynomial orders are not constrained to be integers, for l ¼ 3, m ¼ 3:01, n ¼ 100 and the revolution number N ¼ 7, the true maximum thrust acceleration is Ta max ¼ 8:29e 3 DU=TU2 , which is less than that for l ¼ 3, m ¼ 4, n ¼ 100.

Orbit rendezvous from LEO to GEO In this example, the initial orbit is in a low Earth circular orbit with a radius of 7178 km, the final orbit is in a geostationary circular orbit with a radius of 42,160 km, the transfer angle with zero revolution is 210 and the transfer time is 3 days. The lower bound of radius is 6578.13 km, and the upper bound is the radius of the Earth’s sphere of influence, i.e. 925,000 km. For the original IP method, when 14N412, there exists a feasible rendezvous solution for c3 satisfying _2 4 0, the radius constraints and the rendezvous-time equation. When N ¼ 0, there is no range of c3 satisfying _2 4 0. When N > 12, there exists a range of c3 satisfying equations (39) and (46); however, no solution is for the given rendezvous time. For example, when N ¼ 13, if only the c3 range ð8:650e 9, þ 1Þ for _2 4 0 is considered, there is a solution 5:2020e 9 for c3. However, the minimum radius minfrg ¼ 6095:6 km for this solution is even less than the Earth’s radius, thus it is unfeasible. In all feasible solutions, the trajectory for N ¼ 7 is the minimum-fuel one, and that for N ¼ 10 is with the minimum Ta max . For the modified IP method, the polynomial orders are l ¼ 4, m ¼ 99, n ¼ 100 using the numerical optimization algorithm with the constraint

44l 5 m 5 n4100, then there exist solutions when 34N425. In all feasible solutions, the trajectory for N ¼ 21 is the minimum-fuel one, and that for N ¼ 18 is with the minimum Ta max . The maximum thrust acceleration Ta max and the energy cost V for all feasible revolutions by both the original and modified IP methods are plotted in Figure 2, which shows that the minimum Ta max by the modified IP method is less than that by the original IP method, and the minimum costs V by both methods are almost the same. For different N, the maximum radius maxfrg, the minimum radius minfrg, Ta max and V by both methods are listed in Table 1, where ‘‘N/A’’ denotes no solution. The minimum Ta max by the original IP method is 6:60e 5 km/s2 with N ¼ 10. However, for the modified IP method, Ta max ¼ 5:66e 5 km/s2 is obtained for N ¼ 10, and the minimum Ta max is 2:98e 5 km/s2 with N ¼ 18, which saves the maximum thrust acceleration by 54.8% compared with the original IP method. When N ¼ 10, the thrust acceleration and the trajectory radius histories for rendezvous using the original and modified IP methods are plotted in Figure 3, which shows that Ta max by the modified IP method is less than that by the original IP method. In addition, the estimated maximum thrust acceleration by equation (12) is near to the ‘‘true’’ maximum thrust acceleration. The modified IP method is compared with the UFF (unconstrained finite Fourier) method and the CFF (constrained finite Fourier) method of Taheri and Abdelkhalik.16 In the UFF and CFF methods, the numerical results are different by selecting different Nr , N (numbers of Fourier terms for r, , respectively) and Dps (discretization points).16 To improve the computational efficiency, the solution of the UFF method is set as the initial guess for the CFF method, which is different from that of Taheri and Abdelkhalik.16 All computations are carried out using Matlab 2010b M-files in the hardware environment of a Pentium(R) Dual-Core CPU 3.06 GHz, RAM 2 GB with Windows 7. The results of Ta max , V, and the computation time for the modified IP, UFF and CFF methods are listed in Table 2. For N ¼ 7, there are no solutions for any selected group of Nr N , and Dps. For N ¼ 10, the values




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(a)

(b)

Figure 3. Rendezvous result comparisons for N ¼ 10 using the original and modified IP methods: (a) thrust acceleration; (b) trajectory radius.

Table 2. Rendezvous result comparisons by the modified IP, UFF and CFF methods. Modified IP method

N¼7 N ¼ 10 N ¼ 18

UFF method

CFF method

Ta max (km/s2)

V (km/s)

(s)

Ta max (km/s2)

V (km/s)

(s)

Ta max (km/s2)

V (km/s)

(s)

8:55e 5 5:66e 5 2:98e 5

5.1058 4.4168 4.3758

0.528 0.512 0.506

N/A 5:83e 5 2:82e 5

N/A 6.6218 4.4267

N/A 0.468 0.519

N/A 5:0e 5 2:50e 5

N/A 6.9400 4.4593

N/A 1.253 1.351

(a)

(b)

Figure 4. Rendezvous result comparisons for N ¼ 18 using the modified IP, UFF and CFF methods: (a) thrust acceleration; (b) trajectory radius.

Nr ¼ 3, N ¼ 3, and Dps ¼ 60 are used. For N ¼ 18, the values Nr ¼ 4, N ¼ 4, and Dps ¼ 60 are used, then the maximum thrust acceleration by the UFF method is Ta max ¼ 2:82e 5 km/s2, which is near to

Ta max ¼ 2:98e 5 km/s2 by the modified IP method. For N ¼ 10 and N ¼ 18 in the CFF method, the maximum thrust accelerations are constrained to be less than 5:0e 5 km/s2 and 2:5e 5 km/s2, respectively.



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Although the CFF method provides a smaller Ta max , longer computation time is needed. The thrust acceleration and the radius histories by three methods for N ¼ 18 are plotted in Figure 4.

Conclusions This paper studies the coplanar shape-based approximation method with modified IPs by considering the thrust-acceleration-magnitude and radius constraints. For the time-free transfer between circular orbits, the radius monotonously changes, and the minimum revolution number is analytically derived for a given maximum thrust acceleration. For the time-fixed rendezvous between circular orbits, the seventh parameter is obtained considering the radius constraints, then the feasible shape-based trajectories for a given maximum thrust are obtained. The maximum thrust acceleration by the modified IP method is less than that of the original IP method for both the transfer and rendezvous problems, even though their energy consumptions are close. In addition, the constrained finite Fourier method provides a smaller maximum thrust acceleration, but longer computation time is needed. The proposed technique for thrust and radius constraints can be used in preliminary orbit design and initial guess of constrained trajectory optimization for low-Earth orbits and interplanetary transfers. However, both the original and modified IP methods are only valid for the trajectories between circular orbits. Thus, new shape-based methods will be studied for elliptic orbits in the future. Funding This work was supported in part by the National Natural Scientific Foundation of China (grant numbers 11402062 and 61333003), the China Postdoctoral Science Foundation (grant number 2014T70354), and the Open Research Fund of State Key Laboratory of Astronautic Dynamics of China (grant number 2014ADL-DW0203).

References 1. Battin RH. An introduction to the mathematics and methods of astrodynamics, revised edition (AIAA Education Series). Reston, VA: AIAA, 1999. 2. Mengali G and Quarta AA. Escape from elliptic orbit using constant radial thrust. J Guid Control Dyn 2009; 32(3): 1018–1022. 3. Quarta AA and Mengali G. New look to the constant radial acceleration problem. J Guid Control Dyn 2012; 35(3): 919–929. 4. Benney D. Escape from a circular orbit using tangential thrust. Jet Propulsion 1958; 28: 167–169. 5. Bombardelli C, Bau` G and Pelaéz J. Asymptotic solution for the two-body problem with constant tangential thrust acceleration. Celest Mech Dyn Astr 2011; 110(3): 239–256. 6. Vallado DA. Fundamentals of astrodynamics and applications. 3rd edn. New York: McGraw-Hill Companies, Inc., 2007.

7. Zhang G, Zhou D, Mortari D, et al. Analytical study of tangent orbit and conditions for its solution existence. J Guid Control Dyn 2012; 35(1): 186–194. 8. Zhang G, Zhou D, Sun Z, et al. Optimal two-impulse cotangent rendezvous between coplanar elliptical orbits. J Guid Control Dyn 2013; 36(3): 677–685. 9. Zhang G, Cao X and Zhou D. Two-impulse cotangent rendezvous between coplanar elliptic and hyperbolic orbits. J Guid Control Dyn 2014; 37(3): 964–970. 10. Yam CH, Lorenzo DD and Izzo D. Low-thrust trajectory design as a constrained global optimization problem. Proc IMechE, Part G: J Aerospace Engineering 2011; 225(11): 1243–1251. 11. Shafieenejad I, Novinzadeh AB and Molazadeh VR. Comparing and analyzing min-time and min-effort criteria for free true anomaly of low-thrust orbital maneuvers with new optimal control algorithm. Aerospace Science and Technology 2014; 35: 116–134. 12. Shafieenejad I, Novinzadeh AB and Molazadeh VR. Introducing a novel algorithm for minimum-time lowthrust orbital transfers with free initial condition. Proc IMechE, Part G: J Aerospace Engineering 2015; 229(2): 333–351. 13. Petropoulos AE and Longuski JM. Shape-based algorithm for automated design of low-thrust, gravity-assist trajectories. J Spacecr Rockets 2004; 41(5): 787–796. 14. Wall B and Conway BA. Shape-based approach to lowthrust rendezvous trajectory design. J Guid Control Dyn 2009; 32(1): 95–101. 15. Shang HB, Cui PY and Qiao D. A shape based design approach to interplanetary low-thrust transfer trajectory. J Astronautics 2010; 31(6): 1569–1574. (in Chinese). 16. Taheri E and Abdelkhalik O. Shape based approximation of constrained low-thrust space trajectories using Fourier series. J Spacecr Rockets 2012; 49(3): 535–546. 17. Abdelkhalik O and Taheri E. Approximate on-off lowthrust space trajectories using fourier series. J Spacecr Rockets 2012; 49(5): 962–965. 18. Novak DM and Vasile M. Improved shaping approach to the preliminary design of low-thrust trajectories. J Guid Control Dyn 2011; 34(1): 128–147.

Appendix Notation A a b cj cl , cm , cn c0l , c0m , c0n e G g H

matrix for function parameters cl , cm , cn semimajor axis parameter defined by boundary conditions function parameters function parameters with free orders function parameters independent of c3 in rendezvous problems eccentricity function independent of c3 for rendezvous problem function for radius changes function independent of c3 for rendezvous problem



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13

p q N r Ta

parameters for c3 range in rendezvous problem semilatus rectum polynomial variable number of revolutions radius thrust acceleration

’

t V "

coefficients of cubic polynomial for transfer time fuel consumption small value for extreme points of Ta flight-path angle normalized transfer angle

1 2 max min f s

computation time true anomaly gravitational parameter transfer angle

Subscripts and superscripts


initial final maximum minimum terminal " ¼ 1=2 for corresponding functions