Preliminary design of a Very-Low-Thrust Geostationary

Preliminary design of a Very-Low-Thrust Geostationary Transfer

Orbit to Sun-Synchronous Orbit Small Satellite Transfer

Author: Chris RampersadAdvisor: Prof. C. Damaren

University of TorontoInstitute for Aerospace Studies

4925 Dufferin StreetToronto, ON M3H 5T6

Canada

Abstract

Small satellites have proven their viability for conducting meaningful missions at a fraction of thecost of larger satellites. Correspondingly, the demand for small satellite missions is increasing as is theneed for more sophisticated low-cost satellites. To enable these more advanced missions, this paperanalyzes the feasibility of using a low-thrust propulsion system to transfer to a mission orbit from areadily available launch opportunity. Specifically, a geosynchronous transfer orbit (GTO) launch to asun-synchronous orbit (SSO) small satellite trajectory is examined. The transfer from GTO to SSOorbit is designed using a direct optimization method. Due to the very low thrust levels, the transfertime to obtain the operational orbit is long (∼360 days) with numerous orbit revolutions (1400).To solve this extremely large optimization problem, a new multiple-orbit thrust parameterizationstrategy was developed. This new method has proven to be very robust and capable of handling largecomplex problems. Preliminary trajectory design shows that mission feasibility for a 100 kg smallsatellite GTO-SSO transfer greatly depends on the propulsion system’s specific impulse.

1 Introduction

Due to the high cost of satellite missions, sev-eral strategies are typically used to yield low-costmissions. Some of these initiatives include usingsmaller satellites with lower development costs, us-ing low-thrust propulsion systems with decreasedfuel requirements, and using secondary payload op-portunities for launch to reduce launch costs. Acombination of these three strategies could providesubstantial cost savings but mission studies wouldfirst be needed to determine the feasibility for thesetypes of missions.

The Canadian Space Agency (CSA) has recentlyinitiated a program to support the development ofsmall satellites. Over the next ten years, a varietyof small and micro-satellite missions are expectedto be launched. A main objective of this programis to provide low-cost access for science and technol-ogy demonstration missions. One potential conceptinvolves sending a small satellite into a non-ideal

geostationary transfer orbit (GTO) and then usinga low-thrust propulsion system to reach the mis-sion orbit. Since many GTO launches take placeeach year, there should be ample secondary pay-load launch opportunities. Furthermore, due to theavailability of GTO launches, these secondary pay-load opportunities would likely be lower in cost thana dedicated secondary launch opportunity to a tar-get orbit.

In this GTO launch strategy, the satellite wouldperform an orbital transfer from GTO to obtainits operational sun-synchronous orbit (SSO). Whilesimple in concept, the mission analysis involvedis extremely complicated due to the large out-of-plane and in-plane changes required and the verylow thrust levels available for a small satellite. Todetermine the feasibility of this concept, this pa-per explores the preliminary trajectory design for aGTO to SSO small satellite with low-thrust engines.

1

2 Trajectory Optimization

There are essentially two types of strategies thatare used to design and optimize low-thrust trajec-tories. A continuous formulation of the optimalcontrol problem, known as the “indirect” method,uses the calculus of variations to derive the nec-essary conditions for optimality. This leads to atwo-point boundary-value problem, which is gener-ally difficult to solve except for simple cases. Thisapproach has the benefit of requiring only a smallnumber of variables. However, the indirect methodoften lacks robustness and is extremely sensitive tothe initial guess of the optimization variables.

Alternatively in the “direct” method, the con-trols of the optimal control problem are discretizedto yield a parameter optimization problem. Typi-cally, the thrust magnitude and direction is param-eterized at discrete points (spaced evenly in timeor angle) throughout the trajectory. Numerical op-timization routines are then used to determine theoptimal set of parameterized controls for the spe-cific transfer problem. The direct method is signif-icantly more robust than the indirect method sinceit does not require the solutions to the control andadjoint equations. Instead, the objective function isminimized through a sequence of control updates.The drawback of the direct method is that it re-quires a large number of optimization variables toparameterize the controls and generally its resultsare slightly less accurate than the indirect formula-tion.

For the current mission study, the transfer fromGTO to SSO is highly complex. As a result, thistransfer problem would be extremely difficult tooptimize by indirect methods. Furthermore, us-ing a standard direct method approach to computethis transfer would present an overwhelming com-putational burden due to the large number of op-timization variables that are required. One of thelargest direct method low-thrust satellite transferproblems found in the literature required over 578orbit revolutions to complete and 416,000 variablesto solve [1]. Other direct methods in the literaturehad transfers with 100 orbit revolutions and less[2], [3], [4]. The current small-satellite problem issignificantly more complicated and requires manymore revolutions to complete the transfer. In thepresent study, a new approach is employed, wherebya multiple-orbit thrust parameterization strategy isused in order to solve this very-low-thrust transferproblem.

3 Solution Procedure

Averaging methods have been successfully appliedin the literature to produce computationally fast so-lutions for very large trajectory problems [5], [6], [7],[8]. However, a new approach is developed here tosolve highly complex and very large trajectory prob-lems using direct methods. In this new multiple-orbit thrust parameterization strategy, the thrustprofile is parameterized for a single orbit and is re-peated for multiple orbit revolutions. Thus, insteadof parameterizing controls evenly over the wholetrajectory, the controls are parameterized every sev-eral orbits, greatly reducing the required numberof optimization variables. Since the thrust is notparameterized evenly throughout the transfer, thesolution will not exhibit the same degree of opti-mality as solutions with thousands of optimizationvariables or continuous thrust formulations. How-ever, due to the slowly changing nature of low-thrust transfers, this strategy provides a reasonableapproximation to the optimal solution and greatlyreduces the computational load.

One benefit of the multiple-orbit thrust param-eterization strategy is that lunar and solar grav-itational effects can be readily modeled. Thus,this strategy could also be used to design very-low-thrust small satellite lunar missions. Addition-ally, this strategy could be used to perform high-fidelity mission planning if perturbations due tothe Earth’s non-sphericity were also included. An-other potential benefit of this approach is that itproduces minimum-fuel transfers as opposed to theminimum-time transfers that are typically associ-ated with averaging methods. Minimum-fuel trans-fers using averaging methods have been solved inthe literature, but they involve optimizing a non-linear switching function to determine the thrustand coast phases. This switching function increasesthe complexity of the trajectory optimization; thus,minimum-fuel averaging methods may be incapableof solving a GTO-SSO small satellite transfer.

3.1 Satellite model

In the present study, the initial mass of the satelliteis taken to be 100 kg. The low-thrust propulsionsystem was modelled as variable thrust with themagnitude of the thrust, T , given by:

T = 2ηP/c (1)

where η is the engine efficiency, P is the inputpower, c = gslIsp is the exhaust velocity, gsl is the

2

gravity at sea-level. The thrust was allowed to varybetween zero and 25 mN and the Isp was assumedto be fixed.

The propellant mass flow rate, m, for the low-thrust engines is calculated as follows:

m =T

c=

2ηP

c2(2)

3.2 State Representation

A satellite’s trajectory can be modeled with one ofseveral state representations (i.e., classical orbitalelements, cartesian coordinates or equinoctial ele-ments). Due to the large problem size of the GTO-SSO satellite transfer, it is imperative that the mostefficient state representation be chosen in order toreduce computational cost. Classical orbital ele-ments have a convenient intuitive representation;however, they are not suitable for all satellite simu-lations due to the singularities in circular (e=0) andfundamental-plane (i=0) orbits. To accommodatethese types of orbits, a modified set of equinoctialelements were developed without singularities [9].Similarly, cartesian equations of motion are singu-larity free and they can be written in a very compactform as follows:

d2rdt2

= − µ

r3r + a (3)

Here, µ is the gravitational constant for the Earth, ris the satellite position vector relative to the Earth,and a is the thrust acceleration due to the low-thrust engines. The equinoctial equations of mo-

tion are significantly more complex and compu-tationally demanding than the cartesian counter-parts. However, the cartesian equations are notnecessarily the preferred choice for modeling a satel-lite’s trajectory. Since the cartesian equations ofmotion change very quickly with time, small stepsizes are needed to numerically propagate the equa-tions. Alternatively, equinoctial elements changemuch more slowly with time, which allows large stepsizes to be used for numerical propagation. To de-termine the most appropriate state representation,the equinoctial and cartesian coordinates were com-pared through a many-revolution transfer test case.

The test case involves a satellite which begins ina geostationary orbit (GEO) and uses constant cir-cumferential thrusting for a 130-day period. Due tothe low thrust, the transfer consists of many orbitrevolutions, which makes it sensitive to step size.Tables 1 and 2 below, present the final state (givenin orbital elements for consistency) as a functionof integration steps for equinoctial and cartesianstates, respectively. Additionally, the computationtime to numerically propagate the equations of mo-tion is also given. Comparison between the compu-tation time of the equinoctial and cartesian caseswith 9,000,000 steps, indicates that the equinoc-tial equations of motion take 1.31 times longer tocompute than the cartesian equations. However, asshown in Tables 1 and 2, 250 propagation steps inequinoctial elements can produce accuracy equiva-lent to 10,000 steps in cartesian coordinates. Thus,equinoctial elements are clearly the preferred choicefor computationally demanding problems.

Table 1: Effect of integration steps on final orbit using equinoctial orbital elements

Integration Computation Semi-Major Eccentricity Argument ofSteps Time (s) Axis (km) Perigee (Degrees)

9,000,000 46.0799 2,194,607.8 0.7081943 96.6491016000 0.039999 2,194,608.6 0.7081944 96.6491084000 0.029999 2,194,610.4 0.7081946 96.6491193000 0.019999 2,194,607.8 0.7081943 96.6490562000 0.009999 2,194,608.9 0.7081944 96.6488071000 0.0∗ 2,194,611.9 0.7081948 96.640621500 0.0∗ 2,194,700.1 0.7082046 96.408770250 0.0∗ 2,196,978.4 0.7084607 90.743281125 0.0∗ 2,227,727.0 0.7113344 66.013381

∗ The computation time cannot be measured accurately for very short durations.

3

Table 2: Effect of integration steps on final orbit using a cartesian state

Integration Computation Semi-Major Eccentricity Argument ofSteps Time (s) Axis (km) Perigee (Degrees)

9,000,000 35.0499 2,194,612.1 0.7081948 96.64660640,000 0.15999 2,194,993.1 0.70824116 96.50291020,000 0.08999 2,194,215.7 0.70815874 97.19595710,000 0.05000 2,189,010.4 0.70758664 101.242735000 0.02999 2,159,704.9 0.70428757 124.290592500 0.00999 1,904,025.7 0.67204244 17.617071250 0.0∗ -4777.187 2743.04 -79.71218

∗ The computation time cannot be measured accurately for very short durations.

3.3 Equations of Motion

The modified equinoctial elements (p, f, g, h, k, L)can be defined in terms of the classical orbital ele-ments (a, e, i, Ω, ω, ν) as follows [9]:

p = a(1− e2) (4)f = e cos(ω + Ω) (5)g = e sin(ω + Ω) (6)h = tan(i/2) cosΩ (7)k = tan(i/2) sinΩ (8)L = Ω + ω + ν (9)

(10)

The equations of motion for the modified equinoc-tial elements in an inverse-square gravity field are[9]:

p =2p

w

√p

µ∆θ (11)

f =[∆r sin L + [(w + 1) cos L + f ]

∆θ

w

−(h sin L− k cosL)g∆h

w

] √p

µ(12)

g =[−∆r cos L + [(w + 1) sinL + g]

∆θ

w

−(h sin L− k cos L)f∆h

w

] √p

µ(13)

h =√

p

µ

s2∆h

2wcos L (14)

k =√

p

µ

s2∆h

2wsin L (15)

L =√

µp

(w

p

)2

+1w

√p

µ(h sin L−k cos L)∆h (16)

where ∆r, ∆h and ∆θ are the components of thedisturbances (due to thrust or perturbations) in therotating frame defined by the unit vectors:

ir =r‖r‖ (17)

ih =r× v‖r× v‖ (18)

iθ = ih × ir (19)

The auxiliary variables, w and s, in Eqs. (11)-(16)are defined as:

w = 1 + f cosL + g sin L (20)

s2 = 1 + h2 + k2 (21)

The only non-Keplerian disturbances consideredin the present study are due to the low-thrustpropulsion system. The thrust vector, T, is definedin terms of thrust magnitude, T , and in- and out-of-plane angles, α and β, respectively, as follows:

Tr = T sin(α) cos(β) (22)Tθ = T cos(α) cos(β) (23)Th = T sin(β) (24)

where m is the spacecraft mass, and Tr, Tθ and Th

are the thrust components along the basis of the ro-tating frame [ir iθ ih]. The orientation of the thrustmagnitude and angles are presented below in Fig-ure 1. The acceleration disturbances, ∆r, ∆θ and∆h, can be found by dividing Tr, Tθ and Th by thesatellite’s mass, respectively.

4

Figure 1: Orientation of thrust magnitude and an-gles.

3.4 Mathematical Formulation

For the optimal control problems considered here,the objective is to find the control functions, u(t),which minimize a performance function

J = φ[y(tf )] (25)

subject to the state equations given by

y = f [y(t),u(t)] (26)

and a set of final conditions

Ψ[y(tf )] (27)

Since direct methods are used, the optimal con-trol problem is parameterized and converted intoa nonlinear programming problem (NLP), which issolved via numerical optimization. In order to keepthe problem size small, a shooting method is usedto solve the NLP problem. In comparison to othermethods, the shooting method requires a relativelysmall number of variables and constraints [10]. Inthis method, the state is numerically propagated(with a fourth order Runge-Kutta) from some ini-tial time, ti, to a final time, tf , subject to the con-trols. Using a sequential quadratic programming(SQP) method, the controls are iteratively adjustedin order to minimize the performance index and tosatisfy the final conditions for the state. The partic-ular SQP implementation is from the commerciallyavailable NPSOL software package [11].

In the multiple-orbit parameterization strategy,the thrust is specified for a single orbit and then re-peated for subsequent orbits. This is accomplishedby using the true longitude (L) as the independentvariable, rather than time. Following the methodoutlined by Betts [12], the time derivative of thestate can be written as

y =dydL

dL

dt(28)

in order to write the equations of motion as a func-tion of true longitude:

dydL

= y[dL

dt

]−1

=yL

(29)

Here, fixed step sizes are taken in true longituderather than in time. As a result, the number of or-bit revolutions is used to define the transfer insteadof the total transfer time.

4 Results

4.1 GTO-GEO Case Study

Prior to solving the GTO-SSO transfer, a test casestudy was solved in order to validate the present ap-proach. The test case is modeled after a low-thrustmany-revolution transfer given by Kluever [6]. Thismethod differs from the present study in the follow-ing areas: in Kluever, averaging methods are used,the transfer time is minimized, and the effects ofEarth’s oblateness are considered. In the currentapproach, the fuel is minimized (not the transfertime) and Earth’s oblateness has not been consid-ered at this stage. Nonetheless, the current methodshould produce similar results to that of the liter-ature, except with longer transfer times and lowerfuel requirements.

For this case study, the transfer initiates froma GTO and terminates in a GEO. The satellite’sparameters and the initial orbital elements are pre-sented in Table 3. Two cases were optimized by thepresent approach: a 104-orbit and a 128-orbit trans-fer. The results of the test cases are shown below inTable 4. The repeat cycle describes the number oforbits that are repeated for a single orbit thrust pro-file. As expected, the transfer times are longer andthe fuel requirements are slightly lower than the re-sults from Kluever. Since the fuel requirements forthe 104-orbit test cases hardly change with repeatcycle, it is clear that the parameterization of the

5

Table 3: Satellite parameters and initial orbital elements for test case

ao eo io Ωo ωo mo Isp Po η(RE) (deg) (deg) (deg) (kg) (s) (kW) (%)3.820 0.731 27 99.0 0 450 3300 5 65

Table 4: Results for a GTO-GEO transfer

Method Transfer Time mf Number of Orbits Repeat cycle(days) (kg) (orbits)

Kluever [6] 67.0 414.2 N/A N/APresent Approach 82.5 419.1 104 8Present Approach 81.3 419.0 104 13Present Approach 80.3 419.0 104 4Present Approach 98.5 420.5 128 8

thrust does not overly degrade the optimality of thesolution. For the 128-orbit revolution case, the tra-jectory and associated thrust profiles are shown inFigure 2 and Figures 3-5, respectively. The thrustprofiles present the parameterization of the thrustdirection and magnitude for the first 8 orbits, mid-dle 65-73 orbits, and last 8 orbits of the transfer.Note that the in-plane and out-of-plane thrust an-gles were assigned values of zero when the thrustmagnitude was zero.

−4 −3 −2 −1 0 1 2 3 4x 10

4

−4

−2

0

2

4

x 104

−5000

0

5000

Y(km)

X (km)

Z (

km)

Figure 2: GTO-GEO trajectory Case Study.

0 50 100 150 200 250 300 350

−150

−100

−50

0

50

100

150

True Longitude (degrees)

In−

Pla

ne T

hrus

t Ang

le (

degr

ees)

orbits (1−8)orbits (65−72)orbits (121−128)

Figure 3: GTO-GEO in-plane thrust angles.

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80


Out

−of

−P

lane

Thr

ust A

ngle

(de

gree

s)


Figure 4: GTO-GEO out-of-plane thrust angles.

6

0 50 100 150 200 250 300 350

0

20

40

60

80

100

120

140

160

180

200


Thr

ust M

agni

tude

(m

N)


Figure 5: GTO-GEO thrust magnitude.

4.2 GTO to SSO with Moderately-High Isp Low-Thrust Engines

Since the test case produced satisfactory results,the GTO-SSO low-thrust transfer problem was thenconsidered. One small-satellite concept envisionedusing moderately-high Isp low-thrust engines (i.e.,thrust = 25 mN thrust, initial mass = 100 kg andIsp = 800 s). While the fuel efficiency may be sig-nificantly lower than high Isp propellants, this ap-proach benefits from the lower power requirements.Here, the GTO-SSO transfer is examined using theaforementioned satellite parameters in order to ad-dress the feasibility of this concept.

It is assumed that the small satellite is launchedas a secondary payload with a GTO Cape Kennedylaunch. Table 5 presents the initial orbital elementsfor the satellite after launch. The target missionorbit is a SSO; Table 6, below, displays the targetorbital elements.

Table 5: Initial orbital elements following launch

ao eo io Ωo ωo θo

(km) (deg) (deg) (deg) (deg)24,370 0.73 28.6 0.0 0.0 0.0

Table 6: Target orbital elements for SSO

ao eo io(km) (deg)7178.0 0.0 98.6

Due to the low thrust levels and large manoeu-vre needed, many orbit revolutions were required in

order to complete the transfer. A feasible solutionwas found by formulating the transfer problem with1400 orbit revolutions and 40-orbit thrust repeatcycle. If the total number of orbits was decreased,the fuel requirements were found to increase sub-stantially. Alternatively, if many more orbits wereused, the computational load would become too de-manding. The results from the GTO-SSO transferare presented below in Table 7. The final mass ofthe satellite is very low, with over a 50% propel-lant mass fraction. Thus, a SSO mission launchedby a GTO secondary payload may not be a viablemission for the current satellite parameters. Alter-natives such as a higher initial mass or propellantIsp could be considered in order to improve the on-orbit initial mass.

The GTO-SSO trajectory and portions of thethrust profiles are shown in Figure 6 and Figures7-9, respectively. The thrust profiles show the pa-rameterization of the thrust direction and magni-tude for the first 40 orbits, middle 681-720 orbits,and last 40 orbits of the transfer. Examination ofthe thrust profiles reveals that there is a significantcoast portion during the middle 40 orbits. Further-more, the thrust angles are quite large during thefirst 40 orbits, but become relatively small by themiddle and towards the end of the transfer.

Table 7: GTO-SSO transfer results for moderately-high Isp engines

Transfer Time Orbits mf mp

(days) (kg) (kg)363.5 1400 47.9 52.1

−5−4

−3−2

−10

1

x 104

−10

1

x 104

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 104

X (km)Y (km)

Z (

km)

Figure 6: GTO-SSO trajectory with moderatelyhigh Isp.

7

0 50 100 150 200 250 300 350

−150

−100

−50

0

50

100

150


In−

Pla

ne T

hrus

t Ang

le (

degr

ees)


Figure 7: In-plane thrust angles for GTO-SSOtransfer.

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80


Out

−of

−P

lane

Thr

ust A

ngle

(de

gree

s)


Figure 8: Out-of-plane thrust angles for GTO-SSOtransfer.

0 50 100 150 200 250 300 350

0

5

10

15

20

25


Thr

ust M

agni

tude

(m

N)


Figure 9: Thrust magnitude for GTO-SSO transfer.

4.3 GTO to SSO with High Isp En-gines

In the previous case study, the proposed small satel-lite concept did not prove to be an overly viable mis-sion due to the low on-orbit mass. Here, the prob-lem is reexamined using low-thrust engines withhigher Isp (3500 s), but with the same maximumthrust (25 mN). Additionally, the same initial andmission orbital parameters are used as in the previ-ous section. The transfer was again posed with 1400orbit revolutions, which was found to be feasible interms of propellent usage and computation time.The results from this trajectory are presented be-low in Table 8. This transfer is significantly morepromising than the latter case as it only requires18.8 kg of fuel to reach the mission orbit. Thus, thissmall satellite concept is a potentially feasible op-tion. Launch availability and cost would also haveto be considered in order to determine whether anear-SSO direct orbital insertion would be a pre-ferred choice over a GTO insertion.

Table 8: GTO-SSO transfer results for high Isp en-gines


(days) (kg) (kg)357.7 1400 81.2 18.8

The trajectory for this transfer is shown below inFigure 10. During the transfer, the apogee is in-creased to over 60,000 km in order to efficiently per-form the plane change manoeuvre. Portions of thethrust magnitude, and in-plane and out-of-planethrust angles are shown in Figures 11, 12 and 13,respectively.

−6−5

−4−3

−2−1

01

2

x 104

−1

0

1

x 104

−3

−2

−1

0

1

2

3

x 104

X (km)

Z (

km)

Y (km)

Figure 10: GTO-SSO trajectory with high Isp.

8

0 50 100 150 200 250 300 350

−150

−100

−50

0

50

100

150


In−

Pla

ne T

hrus

t Ang

le (

degr

ees)


Figure 11: In-plane thrust angles for high Isp GTO-SSO transfer.

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80


Out

−of

−P

lane

Thr

ust A

ngle

(de

gree

s)


Figure 12: Out-of-plane thrust angles for high Isp

GTO-SSO transfer.

0 50 100 150 200 250 300 350

0

5

10

15

20

25


Thr

ust M

agni

tude

(m

N)


Figure 13: Thrust magnitude for high Isp GTO-SSO transfer.

4.4 GTO-SSO With Node Change

In the previous case, no constraints were placed onthe value of the right ascension of the ascendingnode, Ω, upon arrival to SSO. Here, it is assumedthat the Ω needs to be equal to 60 degrees (initially0 degrees) when the satellite reaches the SSO. Theinitial and terminal conditions for this transfer aresimilar to the two previous cases studied, and thehigh value of Isp is used (3500 s). Additionally, thethrust was parameterized with a 40-orbit repeat cy-cle and the transfer was modeled with 1400 orbitrevolutions in order to compare the results with theprevious case. The results for this transfer are sum-marized in Table 9. As expected, the fuel require-ments are slightly higher (i.e., 3 kg increase) thanthe previous case study. Additionally, the transfertime requires nearly 200 more days than the pre-vious case. Thus, changing the Ω by 60 degreesproduces a substantial transfer time increase. Thislong transfer time would likely be inappropriate fora small satellite. If the thrust was parameterizedwith less than a 40-orbit repeat cycle, the transfertime could potentially be decreased. Additionally,constraints could be placed on the total transfertime to keep it under a specific duration.

In this transfer, shown below in Figure 14, thesatellite migrates far from the Earth in order to effi-ciently change both the inclination and the Ω. Thismigration accounts for the long transfer time. Thethrust profiles are shown in Figures 15-17, respec-tively, for the first 40, middle 40 and last 40 orbitsof the transfer. In this transfer, there is less out-of-plane thrusting than the previous transfer, which islikely due to the large apogee of the orbit. In Figure17, the thrust magnitude profile of the middle andfinal 40 orbits overlap and cannot be distinguishedfrom each other.

Table 9: GTO-SSO transfer results for high Isp en-gines and a fixed value of Ω


(days) (kg) (kg)541.9 1400 78.2 21.8

9

−6−4

−20

2x 10

4 −6−4

−20

2

x 104

−4

−3

−2

−1

0

1

2

3

4

x 104

Y (km)X (km)

Z (

km)

Figure 14: GTO-SSO trajectory with high Isp anda fixed value of Ω.

0 50 100 150 200 250 300 350

−150

−100

−50

0

50

100

150


In−

Pla

ne T

hrus

t Ang

le (

degr

ees)


Figure 15: In-plane thrust angles for high Isp GTO-SSO transfer with fixed value of Ω.

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80


Out

−of

−P

lane

Thr

ust A

ngle

(de

gree

s)


Figure 16: Out-of-plane thrust angles for high Isp

GTO-SSO transfer with fixed value of Ω.

0 50 100 150 200 250 300 350

0

5

10

15

20

25


Thr

ust M

agni

tude

(m

N)


Figure 17: Thrust magnitude for high Isp GTO-SSO transfer with fixed value of Ω.

5 Conclusion

Mission analysis software and tools are an integralpart of mission planning for all types of satellites.Low-thrust trajectory planning is particularly chal-lenging for small Earth-based satellites due to thelow thrust levels and many-orbit revolutions. Here,a direct method trajectory optimization tool wasdeveloped, using a multiple-orbit thrust parame-terization strategy, in order to design and addressthe feasibility of these types of transfers. The op-timization routines are general in nature and arecapable of handling a variety of transfer problems.This paper examined a unique small satellite mis-sion, which seeks to reduce mission cost by using anon-ideal launch insertion.

In this study, the feasibility of using a low-thrustpropulsion system to acquire the target SSO froma readily available GTO secondary launch oppor-tunity was examined. Using a 100 kg satellite, thefuel requirements were favorable (mf = 81.2) whenthe low-thrust engines had an Isp of 3500 s. How-ever, when the originally anticipated moderately-high Isp (800 s) engines were used, the SSO satellitearrival mass was unfavorably low (47.9 kg). A thirdcase study investigated the effect of a 60 degree Ωchange in the GTO-SSO transfer when using thehigh Isp engines. The change in Ω added only 3 kgmore fuel than a transfer without the change, butit increased the transfer time by 184 days. Furthertransfer studies will be needed in order to design anacceptable compromise between time and fuel.

The present approach to trajectory optimizationhas been shown to be a valuable preliminary design

10

tool for small satellite design. Furthermore, thissoftware has the capability of assessing the feasibil-ity of future mission concepts. Although the currentwork is suited for preliminary planning, it could alsobe readily evolved to perform higher-fidelity design.In future studies, the effects of Earth’s asymmetry,solar-lunar perturbations and radiation could be in-cluded in the model, albeit with increased compu-tation time.

References

[1] J. T. Betts. Very low-thrust trajectory opti-mization using a direct SQP method. Jour-nal of Computational and Applied Mathemat-ics, (120):27–40, 2000.

[2] W. A. Scheel and B. A. Conway. Optimiza-tion of very-low-thrust, many-revolution space-craft trajectories. Journal of Guidance, Con-trol, And Dynamics, 17(6):1185–1192, 1994.

[3] A. L. Herman and D. L. Spencer. Optimal, low-thrust earth-orbit transfers using higher-ordercollocation methods. Journal of Guidance,Control, and Dynamics, 25(1):40–47, 2002.

[4] P. J. Enright and B. A. Conway. Discrete ap-proximations to optimal trajectories using di-rect transcription and nonlinear programming.Journal of Guidance, Control, and Dynamics,15(4):994–1002, 1992.

[5] S. Geffroy and R. Epenoy. Optimal low-thrust transfers with constraints - generaliza-

tion of averaging techniques. Acta Astronau-tica, 41(3):133–149, 1997.

[6] C. A. Kluever and S. R. Oleson. Direct ap-proach for computing near-optimal low-thrustearth-orbit transfers. Journal of Spacecraft andRockets, 35(4):509–515, 1998.

[7] M. R. Ilgen. A hybrid method for computingoptimal low-thrust OTV trajectories. Amer-ican Astronautical Society, AAS Paper 94-129:941–958, 1994.

[8] M. R. Ilgen. Low thrust OTV guidance usinglyapunov optimal feedback control techniques.American Astronautical Society, AAS Paper93-680:1527–1545, 1993.

[9] J. T. Betts. Optimal interplanetary orbittransfers by direct transcription. Journal of theAstronautical Sciences, 42(3):247–268, 1994.

[10] J. T. Betts. Practical Methods for OptimalControl Using Nonlinear Programming. So-ciety of Industrial and Applied Mathematics,2001.

[11] P. E. Gill, W. Murray, M. A. Saunders, andM. H. Wright. User’s Guide for NPSOL 5.0: AFortran Package for Nonlinear Programming.Tech. Report SOL 86-2, Department of Opera-tions Research, Stanford University, July 1998.

[12] J. T. Betts and S. O. Erb. Optimal low thrusttrajectories to the moon. Journal of AppliedDynamical Systems, 2(2):144–170, 2003.

11

Documents

Preliminary design of a Very-Low-Thrust Geostationary