# Methods for orbit optimization for the LISA gravitational wave observatory

**ABSTRACT** The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA mission for detecting low-frequency gravitational waves in the frequency range from 0.1 mHz to 1 Hz, by using accurate distance measurements with laser interferometry between three spacecraft, which will be launched around 2015 and one year later reach their orbits around the Sun. In order to operate successfully, it is crucial for the constellation of the three spacecraft to have extremely high stability. In this paper, several problems of the orbit optimization of the LISA constellation are discussed by using numerical and analytical methods for satisfying the requirements of accuracy. On the basis of the coorbital restricted problem, analytical expressions of the heliocentric distance and the trailing angle to the Earth of the constellation's barycenter are deduced, with the result that the approximate analytical solution of first order will meet the accuracy requirement of the spacecraft orbit design. It is proved that there is a value of the inclination of the constellation plane that will make the variation of the arm-length a minimum. The principle for selecting the optimum starting elements of orbits at any epoch is proposed. The method and programming principles of finding the optimized orbits are also presented together with examples of the optimization design.

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**ABSTRACT:**We present a general survey of heliocentric LISA orbits, hoping that it might help in the exercise of rescoping the mission. We try to semi-analytically optimize the orbital parameters in order to minimize the disturbances coming from the Earth–LISA interaction. In a set of numerical simulations, we include non-autonomous perturbations and provide an estimate of Doppler shift and breathing as a function of the trailing angle.Classical and Quantum Gravity 01/2012; 29(3). · 3.56 Impact Factor -
##### Article: The Orbit Design of ASTROD-GW

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**ABSTRACT:**ASTROD-GW (ASTROD [Astrodynamical Space Test of Relativity Using Optical Devices] Optimized for Gravitation Wave Detection) is an optimization of ASTROD to focus on the goal of detecting gravitational waves. Its detectable wavelength is 52 times that of LISA (Laser Interferometer Space Antenna). In this paper, the mission orbit design of ASTROD-GW together with the optimization methods is presented. The mission orbits of the 3 spacecraft forming a nearly equilateral triangular array are chosen to be near the Sun-Earth Lagrangian points L3, L4 and L5. The arm lengths are about 260 million kilometers. After optimization, the variations of arm length differences are less than 10-4 AU in ten years, and the Doppler velocities of these three spacecraft are less than 4 m/s, both of which meet what required by LISA. Therefore, a number of techniques developed by LISA can be applied to ASTROD-GW.Acta Astronomica Sinica. 04/2010; -
##### Article: Design of ASTROD-GW Orbit

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**ABSTRACT:**The ASTROD-GW (ASTROD [Astrodynamical Space Test of Relativity Using Optical Devices] Optimized for Gravitation Wave Detection), the mission of the laser astrodynamical gravitational wave detection, is the scheme of optimality of the gravitational wave detection on which the ASTROD is concentrated. Its spacecraft orbits form a triangular array close to an equilateral triangle in the vicinity of the solar-terrestrial Lagrangian points L3, L4 and L5. The length of the interference arm is about 2.6×108 km and the detectable wavelength of the gravitational wave is 52 times larger than that detected by the LISA (Laser Interferometer Space Antenna). In this article, the design and optimization method of the ASTROD-GW orbit are summarized. After the orbit is optimized, the variation in the arm length difference (which can be called the interference difference in laser interferometry) within 10 years is in the order of magnitude of 10−4AU. The Doppler velocities in the three arm length directions are smaller than 4m/s, and all of them are less than that required by the LISA. Therefore the laser ranging techniques developed by the LISA can be applied to the ASTROD-GW.Chinese Astronomy and Astrophysics 01/2010; 34(4):434-446.

Page 1

International Journal of Modern Physics D

Vol. 17, No. 7 (2008) 1021–1042

c ? World Scientific Publishing Company

METHODS FOR ORBIT OPTIMIZATION FOR THE LISA

GRAVITATIONAL WAVE OBSERVATORY

GUANGYU LI∗,?, ZHAOHUA YI∗,†, GERHARD HEINZEL‡

ALBRECHT R¨UDIGER‡,OLIVER JENNRICH§, LI WANG¶, YAN XIA∗

FEI ZENG∗and HAIBIN ZHAO∗

∗Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, P. R. China

†Department of Astronomy, Nanjing University, Nanjing 210008, P. R. China

‡Max Planck Institute for Gravitational Physics, D-30167 Hannover, Germany

§European Space Research and Technology Center, 2200 AG Noordwijk (The Netherlands)

¶China Academy of Space Science, Beijing 100094, P. R. China

?gyl@pmo.ac.cn

Received 31 December 2006

Revised 26 May 2007

Communicated by W.-T. Ni

The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA mission

for detecting low-frequency gravitational waves in the frequency range from 0.1mHz to

1Hz, by using accurate distance measurements with laser interferometry between three

spacecraft, which will be launched around 2015 and one year later reach their orbits

around the Sun. In order to operate successfully, it is crucial for the constellation of

the three spacecraft to have extremely high stability. In this paper, several problems

of the orbit optimization of the LISA constellation are discussed by using numerical

and analytical methods for satisfying the requirements of accuracy. On the basis of

the coorbital restricted problem, analytical expressions of the heliocentric distance and

the trailing angle to the Earth of the constellation’s barycenter are deduced, with the

result that the approximate analytical solution of first order will meet the accuracy

requirement of the spacecraft orbit design. It is proved that there is a value of the

inclination of the constellation plane that will make the variation of the arm-length a

minimum. The principle for selecting the optimum starting elements of orbits at any

epoch is proposed. The method and programming principles of finding the optimized

orbits are also presented together with examples of the optimization design.

Keywords: Coorbital restricted problem; LISA constellation; optimized orbit.

1. Introduction

The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA

mission for detecting low-frequency gravitational waves in the frequency range from

0.1mHz to 1Hz. Three spacecraft will be launched around 2015, and reach their

orbits around the Sun after one year. Observation will last for 5 to 10 years. In

2009, LISA Pathfinder will be launched to test the technology’s feasibility.

1021

Page 2

1022 G. Li et al.

Fig. 1. LISA constellation moves around the Sun.

Distance variations between the spacecraft are measured with laser interferom-

etry to detect gravitational waves. In order for LISA to operate successfully, it is

crucial that the constellation of the three spacecraft have extremely high stability.

As shown in Fig. 1, the three spacecraft form an equilateral triangle with an arm-

length (side length) of around 5 × 106km. The center of the constellation moves

on the Earth orbit, trailing 20 degrees behind the Earth. The constellation plane is

tilted 60◦out of the ecliptic.1

Due to the effect of the eccentricity of the spacecraft orbit and the gravity of

the other bodies in the solar system including the Earth, and non-gravitational

effects,2,3the side length, the internal angles, the relative velocities among the

spacecraft and the trailing angle between their barycenter and the Earth vary con-

tinuously. In order to ensure the precision of the distance measurements between

the spacecraft, the variation of those parameters mentioned above should be at

least within the given ranges shown in Table 1 according to the LISA requirements.

Smaller variations may make simplifications in the hardware possible and conse-

quently are highly preferred.

Thus, the problem of optimizing the orbits of the LISA constellation can be

summarized as follows: finding a set of orbital elements ai,ei,ωi,Ωi,Ii,M0i, (i =

1,2,3) to make the following cost function take its minimum:

Q(ai,ei,ωi,Ωi,Ii,M0i) = w1?D2+ w2?α2+ w3?v2

r+ w4?θ2

(1)

Table 1.

constellation.

The variation ranges of the parameters of the LISA

Parameter Average valuePermitted variation

Arm-length l

Internal angle α

Relative velocity vr

Trailing angle to Earth θ

5 × 106km

60◦

0

20◦

?l = ±5 × 104km

?α = ±1.5◦

?vr = ±15m/s

?θ small enough

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Methods for Orbit Optimization for the LISA Gravitational Observatory1023

with appropriately chosen weights wj. The orbital elements are semi-major axis,

eccentricity, argument of perihelion, ascending node, inclination and mean anomaly

at the initial epoch, respectively.4

Since Faller5et al. presented the concept of detecting gravitational waves by

using laser ranging in space in 1985, research on the LISA constellation has been

going on for many years. Vincent6et al. (1987), Folkner7et al. (1997), Cut-

ler8(1998), Hughes9(2002), Hechler10et al. (2003), Dhurandhar11et al. (2005),

Sweetser12(2005) and K. Rajesh Nayak13et al. (2006) have investigated the LISA

constellation from the viewpoint of science and spaceflight project. Because of the

complexity of the space environment in which the LISA constellation moves, and the

extremely high stability of the constellation required in the space project, further

advanced research is expected.

In 1993, Ni14et al. proposed the ASTROD mission concept. Some of its science

objectives are similar to those of the LISA project, and they have been discussed at

three international symposia in 2001, 2005 and 2006. Both LISA and ASTROD use

interferometric laser ranging, and the Doppler effects on transmitted and received

frequencies need to be addressed. LISA’s strategy is to minimize arm-length varia-

tion and relative velocity of the spacecraft. For ASTROD, the arm-length changes

of the three spacecraft are of the same order as the distances between the three

spacecraft and the relative velocities go up to 70km/s with line-of-sight velocities

varying from −20 to +20km/s. For 1064 nm (532 nm) laser light, the Doppler fre-

quency change goes up to 40 (80) GHz. For ASTROD, a strategy that relies on a

different technology is used. The recent development of optical clocks and frequency

synthesizers using optical combs makes this heterodyne problem tractable.15

Orbit design and simulation are two important parts of the pre-phase A study

of ASTROD. Wei-Tou Ni, Chien-Jen Tang, Guangyu Li, and Yan Xia took part in

the research successively,16,17which accumulated a wealth of experience for us to

study the LISA orbits. In February 2006, Guangyu Li and Yan Xia were invited to

visit the Max Planck Institute for Gravitational Physics and took part in research

on the optimization of the LISA orbit design in cooperation with Gerhard Heinzel,

Oliver Jennrich and Albrecht R¨ udiger. This paper presents our results within this

work so far.

In Sec. 2 of this paper, the selection of starting orbits for LISA for optimization

is discussed. In Sec. 3, the motion of the barycenter of the constellation on the

base of the plane coorbital motion is analyzed. In Sec. 4, the optimization algo-

rithm and program structure that are used in this work are introduced briefly. In

Sec. 5, the method to select the optimization orbits at any epoch is presented with

examples.

2. Selection of the Starting Orbits for Optimization

The starting orbits is such a set of orbits that determines the starting point of the

optimization trajectory in the orbit space, along which we can get to the targeted

Page 4

1024G. Li et al.

Fig. 2.Formation of the LISA constellation.

point and obtain the optimization orbits by given optimization algorithms, which

have been discussed in detail in many papers.7,11–13A brief review is given as the

beginning of our discussion.

As is shown in Fig. 2, the Sun is located at point S while the horizontal circle is

the orbit of the Earth, and three spacecraft SC1, SC2 and SC3 compose the LISA

constellation, the barycenter of which, point C, moves along the orbit of the Earth

coorbitally.18In this paper, we assume that the masses of the three spacecraft are

identical, so the barycenter of the constellation coincides with its geometrical center.

The inclined ellipse is the osculating orbit of spacecraft SC1 at the initial epoch

t = 0, with a,e,I and f as semi-major axis, eccentricity, inclination to the ecliptic

plane and true anomaly,4,19respectively. Assuming spacecraft SC1 is located at its

aphelion at the initial epoch, we have f0 = M0 = π here, where f0 and M0 are

the true anomaly and mean anomaly respectively at the time t = 0. Defining φ

as the angle between the constellation plane in which the three spacecraft move

and the ecliptic plane, the important relationship between the eccentricity e, the

inclination I to the ecliptic plane of the spacecraft orbit, the average arm-length l

and the angle φ is expressed by the following equations:

?

e =

1 +l2

3+

2l

√3cosφ − 1,

(2)

sinI =

lsinφ

√3(1 + e),

cosI =

√3 + lcosφ

√3(1 + e)

.

(3)

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Methods for Orbit Optimization for the LISA Gravitational Observatory 1025

Putting the Sun, S, at the origin, the ecliptic plane as the base plane and the

direction from the origin to the projection of SC1 on the ecliptic plane as x-axis,

a coordinate system, which is a non-inertial rotating coordinate system, is set up.

Defining Ω and ω as the ascending node and the argument of perihelion4,19of the

spacecraft orbit in this system, the radius vector of SC1 can be expressed as

r = R3(Ω)R1(I)R3(ω + f)(r,0,0)τ,

(4)

the superscript τ denoting the transpose operation of a matrix, Rkis the rotating

matrix around the kth axis (k = 1,3):

0sinα

cosα

R1(α) =

1

0

00

cosα

−sinα

, R3(α) =

cosα

sinα

−sinα

cosα

0

0

100

.

(5)

As I is a small quantity, 1 and I could be regarded as the approximate values of

cosI and sinI, respectively. Therefore

R1(I) = I + I

0

0

0

0

0

1

0

−1

0

(6)

where I is the unit matrix. Substituting (7) into (5) and simplifying it, we get

r = r(cos(Ω + ω + f), sin(Ω + ω + f), sin(ω + f)I)τ.

(7)

From Fig. 2, we can see that both the Ω and ω of the spacecraft SC1 equal to 3π/2.

Then at t = 0,

Ω + ω + M0= 0 (mod 2π).

(8)

To form the constellation, it is a necessary condition for all the three spacecraft

that each of their orbits keeps the relationship described in Eq. (8). Numerical

calculation indicates that the better way to select the other two orbits is:

(i) Rotating the orbit of SC1 around the z-axis by π/3 and 2π/3, respectively, to

form the orbits of SC2 and SC3;

(ii) Adjusting the mean anomalies of SC2 and SC3 to keep the relationship (8).

Thus the five intial orbit elements of the three spacecraft for optimization are shown

in Table 2. The unit of the semi-major axis a is the astronomical unit AU.

From Eq. (8), we gain

Ω + ω + f = f − M0.

(9)

Substituting it into expression (7), we get

r = r(cos(f − M0), sin(f − M0), − cos(f)I)τ.

(10)

Page 6

1026G. Li et al.

Table 2.

three spacecraft for optimization.

The 5 starting orbit elements of the

iaωΩM0

f0

SC1

SC2

SC3

1

1

1

270◦

270◦

270◦

270◦

30◦

150◦

180◦

60◦

300◦

180◦

60◦.963335

299◦.036665

It is necessary to point out that the starting formation of the constellation achieved

above is not a strictly equilateral triangle and will not become such a perfect triangle

ever in its future evolution. However, if begun from this starting formation, the

optimization formation will be achieved much more easily.

The next step is to discuss the variation of the arm-length (side length) of the

constellation in the frame of two-body motion. Although that is far from the real

motion with perturbation, it can still provide an important inspiration for our work.

For analysis convenience the three spacecraft are differentiated by subscripts 1,2,3.

So from Eq. (10), the vector from SC1 to SC2 is

(r1cosf1− r2cosf2)I

Hence the square of the arm-length is

r12≡ r2− r1=

r2cos(f2− M20) − r1cos(f1− M10)

r2sin(f2− M20) − r1sin(f1− M10)

.

(11)

r2

12= (r2− r1) · (r2− r1) = r2

−2r1r2cos[(f1− M10) − (f2− M20)] + (r1cosf1− r2cosf2)2I2. (12)

From the equation of the center4,19we have

?

We define the angle f12as the difference

?

which is a small quantity of the same order as e. With the accuracy of terms up to

e2, cosf12could be approximated as:

cosf12= 1 −f2

Noting that4,19ri= 1−ecosEi, r1−r2= e(cosE2−cosE1) ≈ e(cosM2−cosM1),

where Eiis the eccentric anomaly of the ith spacecraft, Eq. (12) becomes

1+ r2

2

fi− Mi=2e −1

4e3

?

sinMi+5

4e2sin2Mi+ ··· .

(13)

f2−M20−(f1−M10) =

2e −1

4e3

?

(sinM2−sinM1)+5

4e2(sin2M2− sin2M1)+··· ,

12

2

.

(14)

r2

12= 12e2+ 3(sin2I − 3e2) sin2

?

M1−2π

3

?

.

(15)

Similarly we can get the two Equations for r2

The formation of the constellation is not always a strictly perfect equilateral

triangle, so that sin2I − 3e2?= 0. Let l2= 12e2, namely l = 2√3e, where l is

23and r2

31.

Page 7

Methods for Orbit Optimization for the LISA Gravitational Observatory 1027

Fig. 3. Variation of the arm-length when φ = 60◦.4776.

Fig. 4. The relationship between the inclination φ and the variation of the arm-length ∆.

the average arm-length. From Eq. (2) we have cosφ =

0.03342292561AU) which gives φ = 60◦.4776 and the variation of the arm-length

is shown as Fig. 3. Numerical calculation indicates that it is also the best value

of the inclination of the constellation plane, which makes the arm-length variation

approach a minimum in the frame of the two-body problem.

Numerical analysis also indicates that when 60◦.47 < φ < 60◦.63, the varia-

tion of the arm-length keeps at the minimum around 4.788 × 104km, as shown in

Fig. 4. Using a different method, K. Rajech Nayak et al. drew the nearly identical

conclusion.13

1

2−

√3

8l = 0.492764 (l =

Page 8

1028 G. Li et al.

Assuming φ = 60◦.4776, from Eqs. (2) and (3), the values of the eccentricity e

and the inclination I of the spacecraft orbits are

e = 0.0096483717,

I = 0◦.95292153 = 0.016631618rad.

(16)

The other elements have been given in Table 2, determining the starting orbits for

the optimization.

3. Motion of the Barycenter of the LISA Constellation

The approximate analytical expressions of the argument θ and the heliocentric dis-

tance r of the barycenter of the LISA constellation have been presented in another

paper,18which will be sketched briefly for orbit optimization in this section.

Due to the requirement of orbit design, the barycenter C is always trailing behind

the Earth, namely the initial argument θ0< 0. Hence sin|θ0|

the expressions into different forms gives

2

= −sinθ0

2. Putting

r =

1 + 2knt −

1 + esin(∆θ + nt)

3

32kk1(nt)3

,

(17)

∆θ = −3

2k(nt)2+ 2k2(nt)3+

9

256kk1(nt)4,

(18)

where e = 5 × 10−5, n is the mean motion4,19(average angular velocity) of the

barycenter, t is the mission lifetime in days, and

?

?

where µ = me/(me+ms) = 3.04×10−6, msand meis the Sun mass and the Earth

mass, respectively.

On the base of the above equations the following conclusions about the motion

of the barycenter of the LISA constellation can be drawn:

k = µsinθ0

1 +1

8csc3θ0

?

2

?

,

(19)

k1= µ

8cosθ0− cscθ0

2

1 + 2cot2θ0

2

??

,

(20)

(i) In the classical result of celestial mechanics,4,19the Lagrange point L5 is at

θ0= −60◦, and k, ?θ, and ?r all vanish if C is at that location.

(ii) The parameter k given in Eq. (19) describes the orbit instability. The rela-

tionship between k and θ0is shown in Fig. 5. When θ0increases from −60◦,

k starts its monotonic increase slowly (note the logarithmic scale). As soon

as θ0 > −5◦, when the spacecraft are quite close to the Earth, k begins to

increase rapidly, and from Eq. (18) the absolute value of ∆θ increases even

more rapidly. Consequently the orbits are much too unstable for such a small

Page 9

Methods for Orbit Optimization for the LISA Gravitational Observatory 1029

Fig. 5.The relationship between the parameter k and the initial argument θ0.

Fig. 6.The dependence of the maximum variations ∆r and ∆θ on the initial argument θ0.

value of θ0, so henceforth we consider only cases within the parameter range

θ0∈ [−60◦,−10◦].

(iii) For r0= 1AU, in our interesting parameter range θ0∈ [−60◦,−10◦], the argu-

ment θ and the increment ∆r of the heliocentric distance are nearly monotonic

functions of time t. Thus their absolute value will reach the maximum at the

end of the mission lifetime (t = 3700days). The two diagrams in Fig. 6 show

the relation between the maximum values and the initial argument θ0.

(iv) If θ0= −20◦, ∆θmax= −7◦.85 and ∆rmax= 36 × 104km are the solutions.

With increasing time, the heliocentric distance becomes larger and larger.

Selecting θ0with a larger absolute value, for example, θ0= −30◦, would sig-

nificantly decrease ∆r and ∆θ. But due to the technical and other reasons, the

value near −20◦would be a preferable choice in the LISA mission.1Although

Page 10

1030 G. Li et al.

the value ∆r and ∆θ here are larger by about an order of magnitude than the

target value given in Table 1, taking these elements as a starting point for orbit

optimization, we will still reach our goal in the end. That will be discussed in

detail in Secs. 4 and 5.

(v) Along with the increase of the heliocentric distance, the mean motion n

decreases consequently, which results in the increase of the earth-trailing angle.

If r0is slightly smaller than 1.0AU by selection, it will gradually increase to

1.0AU in the mission lifetime, which results in an effective decrease of the

variation range of the angle θ. The equation ∆r = −2∆n/(3n) can be used

to estimate an initial parameter r0 for the numerical optimization. Figure 7

shows an optimized result where the variation range of the argument θ is below

2◦.6 when θ0= −22◦and r0= 0.9996AU. During 80 percent of the mission

lifetime, the variation ∆θ is less than 0.5◦. But the variation range of ∆r is

larger than before, about 80× 104km. In the practical optimization process a

balance between the two variations is needed. It will be discussed in Sec. 5.4.

Fig. 7.An optimization result for the initial trailing argument θ0.

(vi) The solar-radiation pressure is a significant perturbation force for the space-

craft.3,20,21Specifically, the area-to-mass ratios of the spacecraft are typically

in the range from 10−3to 10−2m2/kg. Thus, to a rough first approximation,

a spacecraft in orbit about the Sun behaves as if the Sun’s gravitational con-

stant GM were reduced by the fractional amount 10−6to 10−5. This requires,

inter alia, that the semi-major axis of the LISA spacecraft be smaller than

that of the Earth by about 1/3 of this amount, 3 × 10−7to 3 × 10−6AU, if

the mean motion of the spacecraft is to match that of the Earth. However, the

actual sizes and also the actual directions of the forces due to radiation pres-

sure depend on the specifics and are very complicated, since they depend on

the reflectivities, shapes, and orientations of the various surfaces of the space-

craft.22,23Moreover, back-reaction from blackbody radiation emitted from the

spacecraft may possibly need to be taken into account.24This effect depends

Page 11

Methods for Orbit Optimization for the LISA Gravitational Observatory1031

on differential heating and cooling of the spacecraft surfaces, which depend

in turn on the thermal properties of all parts of the spacecraft. Although all

these non-gravitational effects on the spacecraft would not influence the orbits

of LISA spacecraft due to their drag-free motion, modeling these effects a pri-

ori to the accuracy contemplated in our work would be a challenging task for

the future.

4. Optimization Algorithm and Program Structure

Now it is time to discuss the problem of orbit optimization. Based on the strategy

of separate calculation, the three orbits are optimized successively, so that only

one orbit is optimized at a time, determined exclusively by the six starting orbit

elements4a,e,ω,Ω,I,M0. The problem will be discussed in the following subspace

of the orbital space of six dimensions

{xi0− gi≤ xi≤ xi0+ gi, i = 1,2,...,6}

in which a point x = (x1,x2,...,x6)τpresents an orbit and the six coordinates xi

are the six orbital elements of which the special denotation stated before will not be

used any more. The cost function (1) becomes a function of the point x. It is easy

to prove that if |∆l| < 5×104km, the inequalities |∆α| < 1◦.5 and |∆vr| < 15m/s

are certainly valid. Therefore the cost function can be simplified to

(21)

Q(x) = w1∆l2+ w2∆θ2.

(22)

The Hybrid Reactive Tabu Search(C-RTS) algorithm25presented by R. Battiti and

G. Techiolli is applied to our orbit optimization. This section gives an outline of the

algorithm and the processes in it that could be applied to the orbit optimization

problem.

4.1. Local optimization algorithm

The Affine Shaker algorithm shown in Fig. 8 is applied to resolve the optimal local

minimum. During the initialization stage, the initial base matrix is generated as

B(0) =1

4

g1

0

···

0

0

g2

···

0

···

···

···

···

0

0

···

g6

,

(23)

its column vectors bj(t) named base vectors.

The search starts from the starting point x(0) selected randomly and is iterated

with t as loop count variable until the cost function satisfies the preset condition, or

the loop count passes the preset upper limit. In each loop a random displacement

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1032 G. Li et al.

vector is generated first:

δ =

6

?

j=1

randjbj(t − 1),

(24)

where 0 < randj≤ 1 are random numbers.

In this way, two new points x+(t) = x(t−1)+δ(t) and x−(t) = x(t−1)−δ(t) are

generated for selection. First the point x+(t) is tested. This first shot is successful

only if this point is better than the point x(t − 1). Then, if the first shot is not

successful, the point x−(t) is tested, which is named second shot. As long as a shot

is successful in any of the two tests, the search point is moved to the test point and

the affine transformation matrix is generated:

P = I + (ρ − 1)δδτ

δτδ,

(25)

where ρ > 1, to expand the search range. If neither of the two shots was successful,

the search point is not moved any further, but the affine transformation matrix is

still generated in the case that ρ < 1 to limit the search range. At the end of each

iterative loop, the base matrix is updated as

B(t) = PB(t − 1) (26)

and the loop count variable t is increased by 1.

initializing

set initial value for loop count: t = 0;

select initial point in the orbit space randomly;

generate initial base matrix B;

searching

iteratively

(0,N)

generate displacement vector δ randomly;

second shot succeed

shot didn’t succeed

first shot succeed

(0,1)?iff(x + δ) < f(x)

move to the next point: x:= x+δ;

generate the expanding matrix P;

?

(0,1)?iff(x − δ) < f(x)

?

(0,1)

move to the next point: x:= x−δ;

generate the expanding matrix P;

?generate the compressing matrix P;

transform the base matrix B:= PB;

update the loop count: t := t + 1;

Fig. 8.Flow chart of Affine Shaker algorithm.

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Methods for Orbit Optimization for the LISA Gravitational Observatory 1033

4.2. Global optimization algorithm

Due to the complexity of the problem, there are probably many optimal local min-

ima in the orbital space (22), which makes the above local optimization algorithm

get stuck in a certain local point and never search for a global optimization point.

This problem can be solved by the global optimization algorithm, C-RTS.25

The algorithm is a kind of iterative search method. The global optimal prob-

lem then could be broken down into the local optimal problems. The key point

of the algorithm is to ensure that the search trajectory covers all of the global

region. The intensified-search, the diversified-search and the prohibition period

parameter are the three major elements of the algorithm which will be illus-

trated later. Now we sketch the algorithm in two parts: data structure and search

algorithm.

4.2.1. Data structure

First, we should search for an appropriate data structure for the problem to dis-

cretize our continuous problem of the orbit optimization and establish an appropri-

ate algorithm. Because the initial search space (22) is a six-dimension space, which

can be organized in the structure of a multiway tree of boxes with 64 children for

searching expediently.

For simplicity, a 2-dimensional example is used to describe the terminology

which will be used in our algorithm. Assuming the searching is limited within the

rectangular region (or box){xi0− gi≤ xi≤ xi0+ gi, i = 1,2} , the region is named

root-box, which is a subset of the orbital space (22). Halving the root-box on each

dimension, the space can be divided into 22= 4 equal-sized children, which are

named child boxes of the root-box. The total of the child boxes cover the root-box.

The root-box divided is named parent-box now. The position of a child box in its

parent box can be identified by a 2-bit binary number (namely a decimal integer

less than 4). The first bit stands for the position of the 1st side. If it equals to

zero, the left half divided in the parent box is selected. Otherwise the right half

is selected. Similarly, the second bit stands for the position of the second side. As

shown in the left diagram of Fig. 9, the identifiers of the 4 child boxes that are the

division of the root-box are 00, 01, 10, 11, respectively.

One of the aforementioned four boxes is selected stochastically as the starting

point for the global optimum search. Once two different local minima are found in

one child-box during the search, the child-box is then subdivided in the aforesaid

way immediately. Assuming the child-box with the identifier of 10 is divided, 4

smaller child-boxes will be generated. Their identifiers relative to their parent-box

10 are still 00, 01, 10, 11, respectively, while their absolute identifiers, namely the

position relative to the root-box are 1000, 1001, 1010 and 1011. The higher 2 bits

are the identifer of the parent-box and the lower 2 bits are the identifers of the

child-boxs, shown as the right diagram of Fig. 9.

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1034 G. Li et al.

Fig. 9. Dividing a 2-dimension region, boxes and their identifiers.

Hereafter, the global optimum is searched within the set of 7 undivided boxes

with identifier 00, 01, 11, 1000, 1001, 1010, 1011, respectively, which is named

leaf-box. The depths of the leaf-boxes are different. The depth of the first three

leaf-boxes are 0, while the others are 1. The further the searching and dividing is

going, the larger is the depth of the leaf-box. Therefore, the data structure of the

multiway tree of boxes with 4 children is built up now.

By inverting any bit of the identifier of a box, the box obtained is adjacent to the

primary box. For example, the neighbors of the box 00 are 10 and 01. At each step,

the search moves to an appropriate neighboring box, which is named the node of

the search trajectory. If the current node is box 10, which no longer exists because

of the division, one of the 4 child-box of the box 10 is selected randomly as the node

replacement. Vice versa, if the current node is box 1001 and the next node will be

0001 which is also not existing because of the non-division of box 00, the new node

replacement will be box 00. All the nodes generated during the successive search

can be linked into a trajectory according to their generation order.

As soon as a leaf-box emerges during the search, a point in it is selected stochas-

tically where the value of cost function is calculated by the method in Sec. 5.3 and

is regarded as the evaluation of the box. When the box emerges again, another

random point in it is selected and the value of the cost function at this point is also

calculated. If this value is better than the earlier evaluation of the box, it will be

regarded as the new evaluation. Otherwise, the evaluation will remain unchanged.

The problem is discretized in this way.

Then back to the orbit optimization problem itself, the difference is merely

that the orbital space (21) is a 6-dimension space. 26= 64 child-boxes will be

produced by the division of a box. The position of a child-box in its parent-box can

be identified with a 6-bit binary number (namely a decimal integer less than 64).

The position of the ith side is determined by the ith bit of the binary number. If it

equals to zero, the left half of the parent box is selected. Otherwise the right half is

selected. The successive search will build up a data structure of the multiway tree

of boxes with 64 children.

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Methods for Orbit Optimization for the LISA Gravitational Observatory 1035

4.2.2. Searching algorithm

The flow chart of the global optimization algorithm is shown in Fig. 10. The algo-

rithm is composed of two parts: initializing and searching iteratively. The tasks of

initializing are to set up the initial values of the relative parameters and to select

the starting point of the search trajectory stochastically. The search is iterated until

the cost function satisfies the preset condition, or the loop count overruns the preset

upper limit. In each of the iterations, the parameters, such as prohibition period etc,

are modified according to the condition of the time at first, and the searching mode

is determined subsequently. The prohibition period is an important parameter for

this algorithm, as it helps the searching trajectory to avoid running into endless

loops. This algorithm has two search modes: the intensified-search which moves

to the leaf-box in the neighborhood with optimum evaluation to find the global

minimum, and the diversified-search which moves to a random neighbor leaf-box to

avoid running into a local minimum. The algorithm follows the three rules shown

below:

(i) If the evaluation of the current leaf-box is better than the evaluations of all

the neighbor leaf-boxes in the intensified-search, the Affine Shaker algorithm

in Sec. 4.1 is activated to find a local minimum.

(ii) Once two different local minima are found in one leaf-box, the box is divided

by the method in the previous subsection.

(iii) Moving from a leaf-box to another visited one in the prohibition period is

prohibited. And the prohibition period is decided based upon the condition of

that time.

A detailed description of this algorithm is presented in Ref. 25.

initializing

searching

iteratively

(1,N)

modifying parameters, such as prohibition period etc;

determining the search mode;

intensified-search

(0,1)

move to the optimum in the neighborhood;

Affine Shaker;

(0,1)

divide the box;

?

diversified-search

(0,1)

move stochastically;

Flow chart of the global optimization algorithm.Fig. 10.