# Preliminary Design of a Very-Low-Thrust Geostationary Transfer Orbit to Sun-Synchronous Orbit Small Satellite Transfer

Preliminary design of a Very-Low-Thrust Geostationary Transfer Orbit to Sun-Synchronous Orbit Small Satellite Transfer

Author: Chris Rampersad Advisor: Prof. C. Damaren

University of Toronto Institute for Aerospace Studies 4925 Duﬀerin Street Toronto, ON M3H 5T6 Canada

Abstract Small satellites have proven their viability for conducting meaningful missions at a fraction of the cost of larger satellites. Correspondingly, the demand for small satellite missions is increasing as is the need for more sophisticated low-cost satellites. To enable these more advanced missions, this paper analyzes the feasibility of using a low-thrust propulsion system to transfer to a mission orbit from a readily available launch opportunity. Speciﬁcally, a geosynchronous transfer orbit (GTO) launch to a sun-synchronous orbit (SSO) small satellite trajectory is examined. The transfer from GTO to SSO orbit is designed using a direct optimization method. Due to the very low thrust levels, the transfer time 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 parameterization strategy was developed. This new method has proven to be very robust and capable of handling large complex problems. Preliminary trajectory design shows that mission feasibility for a 100 kg small satellite GTO-SSO transfer greatly depends on the propulsion system’s speciﬁc impulse.

1 Introduction geostationary transfer orbit (GTO) and then using a low-thrust propulsion system to reach the mis- Due to the high cost of satellite missions, sev- sion orbit. Since many GTO launches take place eral strategies are typically used to yield low-cost each year, there should be ample secondary pay- missions. Some of these initiatives include using load launch opportunities. Furthermore, due to the smaller satellites with lower development costs, us- availability of GTO launches, these secondary pay- ing low-thrust propulsion systems with decreased load opportunities would likely be lower in cost than fuel requirements, and using secondary payload op- a dedicated secondary launch opportunity to a tar- portunities for launch to reduce launch costs. A get orbit. combination of these three strategies could provide In this GTO launch strategy, the satellite would substantial cost savings but mission studies would perform an orbital transfer from GTO to obtain ﬁrst be needed to determine the feasibility for these its operational sun-synchronous orbit (SSO). While types of missions. simple in concept, the mission analysis involved The Canadian Space Agency (CSA) has recently is extremely complicated due to the large out-of- initiated a program to support the development of plane and in-plane changes required and the very small satellites. Over the next ten years, a variety low thrust levels available for a small satellite. To of small and micro-satellite missions are expected determine the feasibility of this concept, this pa- to be launched. A main objective of this program per explores the preliminary trajectory design for a is to provide low-cost access for science and technol- GTO to SSO small satellite with low-thrust engines. ogy demonstration missions. One potential concept involves sending a small satellite into a non-ideal

1 2 Trajectory Optimization 3 Solution Procedure

Averaging methods have been successfully applied There are essentially two types of strategies that in the literature to produce computationally fast so- are used to design and optimize low-thrust trajec- lutions for very large trajectory problems [5], [6], [7], tories. A continuous formulation of the optimal [8]. However, a new approach is developed here to control problem, known as the “indirect” method, solve highly complex and very large trajectory prob- uses the calculus of variations to derive the nec- lems using direct methods. In this new multiple- essary conditions for optimality. This leads to a orbit thrust parameterization strategy, the thrust two-point boundary-value problem, which is gener- proﬁle is parameterized for a single orbit and is re- ally diﬃcult to solve except for simple cases. This peated for multiple orbit revolutions. Thus, instead approach has the beneﬁt of requiring only a small of parameterizing controls evenly over the whole number of variables. However, the indirect method trajectory, the controls are parameterized every sev- often lacks robustness and is extremely sensitive to eral orbits, greatly reducing the required number the initial guess of the optimization variables. of optimization variables. Since the thrust is not Alternatively in the “direct” method, the con- parameterized evenly throughout the transfer, the trols of the optimal control problem are discretized solution will not exhibit the same degree of opti- to yield a parameter optimization problem. Typi- mality as solutions with thousands of optimization cally, the thrust magnitude and direction is param- variables or continuous thrust formulations. How- eterized at discrete points (spaced evenly in time ever, due to the slowly changing nature of low- or angle) throughout the trajectory. Numerical op- thrust transfers, this strategy provides a reasonable timization routines are then used to determine the approximation to the optimal solution and greatly optimal set of parameterized controls for the spe- reduces the computational load. ciﬁc transfer problem. The direct method is signif- One beneﬁt of the multiple-orbit thrust param- icantly more robust than the indirect method since eterization strategy is that lunar and solar grav- it does not require the solutions to the control and itational eﬀects can be readily modeled. Thus, adjoint equations. Instead, the objective function is this strategy could also be used to design very- minimized through a sequence of control updates. low-thrust small satellite lunar missions. Addition- The drawback of the direct method is that it re- ally, this strategy could be used to perform high- quires a large number of optimization variables to ﬁdelity mission planning if perturbations due to parameterize the controls and generally its results the Earth’s non-sphericity were also included. An- are slightly less accurate than the indirect formula- other potential beneﬁt of this approach is that it tion. produces minimum-fuel transfers as opposed to the For the current mission study, the transfer from minimum-time transfers that are typically associ- GTO to SSO is highly complex. As a result, this ated with averaging methods. Minimum-fuel trans- transfer problem would be extremely diﬃcult to fers using averaging methods have been solved in optimize by indirect methods. Furthermore, us- the literature, but they involve optimizing a non- ing a standard direct method approach to compute linear switching function to determine the thrust this transfer would present an overwhelming com- and coast phases. This switching function increases putational burden due to the large number of op- the complexity of the trajectory optimization; thus, timization variables that are required. One of the minimum-fuel averaging methods may be incapable largest direct method low-thrust satellite transfer of solving a GTO-SSO small satellite transfer. problems found in the literature required over 578 orbit revolutions to complete and 416,000 variables 3.1 Satellite model to solve [1]. Other direct methods in the literature In the present study, the initial mass of the satellite had transfers with 100 orbit revolutions and less is taken to be 100 kg. The low-thrust propulsion [2], [3], [4]. The current small-satellite problem is system was modelled as variable thrust with the signiﬁcantly more complicated and requires many magnitude of the thrust, T , given by: more revolutions to complete the transfer. In the present study, a new approach is employed, whereby T = 2ηP/c (1) a multiple-orbit thrust parameterization strategy is used in order to solve this very-low-thrust transfer where η is the engine eﬃciency, P is the input problem. power, c = gslIsp is the exhaust velocity, gsl is the

2 gravity at sea-level. The thrust was allowed to vary tion are signiﬁcantly more complex and compu- between zero and 25 mN and the Isp was assumed tationally demanding than the cartesian counter- to be ﬁxed. parts. However, the cartesian equations are not The propellant mass ﬂow rate,m ˙ , for the low- necessarily the preferred choice for modeling a satel- thrust engines is calculated as follows: lite’s trajectory. Since the cartesian equations of motion change very quickly with time, small step T 2ηP sizes are needed to numerically propagate the equa- m˙ = = 2 (2) c c tions. Alternatively, equinoctial elements change 3.2 State Representation much more slowly with time, which allows large step sizes to be used for numerical propagation. To de- A satellite’s trajectory can be modeled with one of termine the most appropriate state representation, several state representations (i.e., classical orbital the equinoctial and cartesian coordinates were com- elements, cartesian coordinates or equinoctial ele- pared through a many-revolution transfer test case. ments). Due to the large problem size of the GTO- The test case involves a satellite which begins in SSO satellite transfer, it is imperative that the most a geostationary orbit (GEO) and uses constant cir- eﬃcient state representation be chosen in order to cumferential thrusting for a 130-day period. Due to reduce computational cost. Classical orbital ele- the low thrust, the transfer consists of many orbit ments have a convenient intuitive representation; revolutions, which makes it sensitive to step size. however, they are not suitable for all satellite simu- Tables 1 and 2 below, present the ﬁnal state (given lations due to the singularities in circular (e=0) and in orbital elements for consistency) as a function fundamental-plane (i=0) orbits. To accommodate of integration steps for equinoctial and cartesian these types of orbits, a modiﬁed set of equinoctial states, respectively. Additionally, the computation elements were developed without singularities [9]. time to numerically propagate the equations of mo- Similarly, cartesian equations of motion are singu- tion is also given. Comparison between the compu- larity free and they can be written in a very compact tation time of the equinoctial and cartesian cases form as follows: with 9,000,000 steps, indicates that the equinoc- tial equations of motion take 1.31 times longer to d2r µ compute than the cartesian equations. However, as 2 = − 3 r + a (3) dt r shown in Tables 1 and 2, 250 propagation steps in Here, µ is the gravitational constant for the Earth, r equinoctial elements can produce accuracy equiva- is the satellite position vector relative to the Earth, lent to 10,000 steps in cartesian coordinates. Thus, and a is the thrust acceleration due to the low- equinoctial elements are clearly the preferred choice thrust engines. The equinoctial equations of mo- for computationally demanding problems.

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

Integration Computation Semi-Major Eccentricity Argument of Steps Time (s) Axis (km) Perigee (Degrees) 9,000,000 46.0799 2,194,607.8 0.7081943 96.649101 6000 0.039999 2,194,608.6 0.7081944 96.649108 4000 0.029999 2,194,610.4 0.7081946 96.649119 3000 0.019999 2,194,607.8 0.7081943 96.649056 2000 0.009999 2,194,608.9 0.7081944 96.648807 1000 0.0∗ 2,194,611.9 0.7081948 96.640621 500 0.0∗ 2,194,700.1 0.7082046 96.408770 250 0.0∗ 2,196,978.4 0.7084607 90.743281 125 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 of Steps Time (s) Axis (km) Perigee (Degrees) 9,000,000 35.0499 2,194,612.1 0.7081948 96.646606 40,000 0.15999 2,194,993.1 0.70824116 96.502910 20,000 0.08999 2,194,215.7 0.70815874 97.195957 10,000 0.05000 2,189,010.4 0.70758664 101.24273 5000 0.02999 2,159,704.9 0.70428757 124.29059 2500 0.00999 1,904,025.7 0.67204244 17.61707 1250 0.0∗ -4777.187 2743.04 -79.71218 ∗ The computation time cannot be measured accurately for very short durations.

µ ¶ r 3.3 Equations of Motion √ w 2 1 p L˙ = µp + (h sin L−k cos L)∆ (16) p w µ h 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]: where ∆r, ∆h and ∆θ are the components of the disturbances (due to thrust or perturbations) in the 2 p = a(1 − e ) (4) rotating frame deﬁned by the unit vectors: f = e cos(ω + Ω) (5) r i = (17) g = e sin(ω + Ω) (6) r krk h = tan(i/2) cos Ω (7) r × v k = tan(i/2) sin Ω (8) i = (18) h kr × vk L = Ω + ω + ν (9) (10) iθ = ih × ir (19) The auxiliary variables, w and s, in Eqs. (11)-(16) The equations of motion for the modiﬁed equinoc- are deﬁned as: tial elements in an inverse-square gravity ﬁeld are [9]: r w = 1 + f cos L + g sin L (20) 2p p p˙ = ∆θ (11) w µ s2 = 1 + h2 + k2 (21)

· The only non-Keplerian disturbances considered ˙ ∆θ in the present study are due to the low-thrust f = ∆r sin L + [(w + 1) cos L + f] w propulsion system. The thrust vector, T, is deﬁned ¸ r g∆ p in terms of thrust magnitude, T , and in- and out- −(h sin L − k cos L) h (12) w µ of-plane angles, α and β, respectively, as follows: T = T sin(α) cos(β) (22) · r ∆θ T = T cos(α) cos(β) (23) g˙ = −∆ cos L + [(w + 1) sin L + g] θ r w ¸ r Th = T sin(β) (24) f∆ p −(h sin L − k cos L) h (13) w µ where m is the spacecraft mass, and Tr, Tθ and Th are the thrust components along the basis of the ro- r p s2∆ tating frame [ir iθ ih]. The orientation of the thrust h˙ = h cos L (14) µ 2w magnitude and angles are presented below in Fig- ure 1. The acceleration disturbances, ∆ , ∆ and r r θ p s2∆ ∆ , can be found by dividing T , T and T by the k˙ = h sin L (15) h r θ h µ 2w satellite’s mass, respectively.

4 In the multiple-orbit parameterization strategy, the thrust is speciﬁed for a single orbit and then re- peated for subsequent orbits. This is accomplished by using the true longitude (L) as the independent variable, rather than time. Following the method outlined by Betts [12], the time derivative of the state can be written as dy dL ˙y = (28) dL dt in order to write the equations of motion as a func- tion of true longitude: · ¸ dy dL −1 ˙y = ˙y = (29) dL dt L˙

Here, ﬁxed step sizes are taken in true longitude rather than in time. As a result, the number of or- bit revolutions is used to deﬁne the transfer instead Figure 1: Orientation of thrust magnitude and an- of the total transfer time. gles. 4 Results 3.4 Mathematical Formulation For the optimal control problems considered here, 4.1 GTO-GEO Case Study the objective is to ﬁnd the control functions, u(t), Prior to solving the GTO-SSO transfer, a test case which minimize a performance function study was solved in order to validate the present ap- J = φ[y(tf )] (25) proach. The test case is modeled after a low-thrust many-revolution transfer given by Kluever [6]. This subject to the state equations given by method diﬀers from the present study in the follow- ˙y = f[y(t), u(t)] (26) ing areas: in Kluever, averaging methods are used, the transfer time is minimized, and the eﬀects of and a set of ﬁnal conditions Earth’s oblateness are considered. In the current Ψ[y(t )] (27) approach, the fuel is minimized (not the transfer f time) and Earth’s oblateness has not been consid- Since direct methods are used, the optimal con- ered at this stage. Nonetheless, the current method trol problem is parameterized and converted into should produce similar results to that of the liter- a nonlinear programming problem (NLP), which is ature, except with longer transfer times and lower solved via numerical optimization. In order to keep fuel requirements. the problem size small, a shooting method is used For this case study, the transfer initiates from to solve the NLP problem. In comparison to other a GTO and terminates in a GEO. The satellite’s methods, the shooting method requires a relatively parameters and the initial orbital elements are pre- small number of variables and constraints [10]. In sented in Table 3. Two cases were optimized by the this method, the state is numerically propagated present approach: a 104-orbit and a 128-orbit trans- (with a fourth order Runge-Kutta) from some ini- fer. The results of the test cases are shown below in tial time, ti, to a ﬁnal time, tf , subject to the con- Table 4. The repeat cycle describes the number of trols. Using a sequential quadratic programming orbits that are repeated for a single orbit thrust pro- (SQP) method, the controls are iteratively adjusted ﬁle. As expected, the transfer times are longer and in order to minimize the performance index and to the fuel requirements are slightly lower than the re- satisfy the ﬁnal conditions for the state. The partic- sults from Kluever. Since the fuel requirements for ular SQP implementation is from the commercially the 104-orbit test cases hardly change with repeat available NPSOL software package [11]. cycle, 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/A Present Approach 82.5 419.1 104 8 Present Approach 81.3 419.0 104 13 Present Approach 80.3 419.0 104 4 Present Approach 98.5 420.5 128 8

thrust does not overly degrade the optimality of the solution. For the 128-orbit revolution case, the tra- 150 jectory and associated thrust proﬁles are shown in 100 Figure 2 and Figures 3-5, respectively. The thrust proﬁles present the parameterization of the thrust 50 direction and magnitude for the ﬁrst 8 orbits, mid- dle 65-73 orbits, and last 8 orbits of the transfer. 0 Note that the in-plane and out-of-plane thrust an- gles were assigned values of zero when the thrust −50 magnitude was zero. In−Plane Thrust Angle (degrees) −100

orbits (1−8) −150 orbits (65−72) orbits (121−128)

0 50 100 150 200 250 300 350 True Longitude (degrees)

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

80

60 4

2 40

5000 0 4 x 10 20 0

Z (km) −2 −5000 0 −4 −4 −3 −2 −1 0 Y(km) 1 2 −20 3 4 4 x 10 X (km)

Out−of−Plane Thrust Angle (degrees) −40

−60

orbits (1−8) orbits (65−72) −80 orbits (121−128)

0 50 100 150 200 250 300 350 True Longitude (degrees)

Figure 2: GTO-GEO trajectory Case Study. Figure 4: GTO-GEO out-of-plane thrust angles.

6 200 order to complete the transfer. A feasible solution 180 was found by formulating the transfer problem with

160 1400 orbit revolutions and 40-orbit thrust repeat cycle. If the total number of orbits was decreased, 140 the fuel requirements were found to increase sub- 120 stantially. Alternatively, if many more orbits were 100 used, the computational load would become too de- orbits (1−8) 80 orbits (65−72) manding. The results from the GTO-SSO transfer orbits (121−128) Thrust Magnitude (mN)

60 are presented below in Table 7. The ﬁnal mass of the satellite is very low, with over a 50% propel- 40 lant mass fraction. Thus, a SSO mission launched 20 by a GTO secondary payload may not be a viable 0 mission for the current satellite parameters. Alter- 0 50 100 150 200 250 300 350 True Longitude (degrees) natives such as a higher initial mass or propellant Isp could be considered in order to improve the on- Figure 5: GTO-GEO thrust magnitude. orbit initial mass. The GTO-SSO trajectory and portions of the thrust proﬁles are shown in Figure 6 and Figures 4.2 GTO to SSO with Moderately- 7-9, respectively. The thrust proﬁles show the pa- High Isp Low-Thrust Engines rameterization of the thrust direction and magni- Since the test case produced satisfactory results, tude for the ﬁrst 40 orbits, middle 681-720 orbits, the GTO-SSO low-thrust transfer problem was then and last 40 orbits of the transfer. Examination of considered. One small-satellite concept envisioned the thrust proﬁles reveals that there is a signiﬁcant coast portion during the middle 40 orbits. Further- using moderately-high Isp low-thrust engines (i.e., thrust = 25 mN thrust, initial mass = 100 kg and more, the thrust angles are quite large during the ﬁrst 40 orbits, but become relatively small by the Isp = 800 s). While the fuel eﬃciency may be sig- middle and towards the end of the transfer. niﬁcantly lower than high Isp propellants, this ap- proach beneﬁts from the lower power requirements. Table 7: GTO-SSO transfer results for moderately- Here, the GTO-SSO transfer is examined using the high I engines aforementioned satellite parameters in order to ad- sp dress the feasibility of this concept. Transfer Time Orbits m m It is assumed that the small satellite is launched f p (days) (kg) (kg) as a secondary payload with a GTO Cape Kennedy launch. Table 5 presents the initial orbital elements 363.5 1400 47.9 52.1 for the satellite after launch. The target mission orbit is a SSO; Table 6, below, displays the target

4 orbital elements. x 10

2 Table 5: Initial orbital elements following launch 1.5

1 ao eo io Ωo ωo θo (km) (deg) (deg) (deg) (deg) 0.5 0

24,370 0.73 28.6 0.0 0.0 0.0 Z (km) −0.5

−1

−1.5 1 −2 0 −1 −2 Table 6: Target orbital elements for SSO 1 4 −3 x 10 0 −4 4 −1 x 10 −5 ao eo io X (km) Y (km) (km) (deg) 7178.0 0.0 98.6 Due to the low thrust levels and large manoeu- Figure 6: GTO-SSO trajectory with moderately vre needed, many orbit revolutions were required in high Isp.

7 4.3 GTO to SSO with High Isp En- 150 gines

100 In the previous case study, the proposed small satel-

50 lite concept did not prove to be an overly viable mis- sion due to the low on-orbit mass. Here, the prob-

0 lem is reexamined using low-thrust engines with higher Isp (3500 s), but with the same maximum −50 thrust (25 mN). Additionally, the same initial and In−Plane Thrust Angle (degrees) mission orbital parameters are used as in the previ- −100 orbits (1−40) orbits (681−720) ous section. The transfer was again posed with 1400 orbits (1361−1400) orbit revolutions, which was found to be feasible in −150 terms of propellent usage and computation time. 0 50 100 150 200 250 300 350 The results from this trajectory are presented be- True Longitude (degrees) low in Table 8. This transfer is signiﬁcantly more promising than the latter case as it only requires Figure 7: In-plane thrust angles for GTO-SSO 18.8 kg of fuel to reach the mission orbit. Thus, this transfer. small satellite concept is a potentially feasible op-

80 tion. Launch availability and cost would also have to be considered in order to determine whether a 60 orbits (1−40) orbits (681−720) near-SSO direct orbital insertion would be a pre- orbits (1361−1400) 40 ferred choice over a GTO insertion.

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

0 gines

−20 Transfer Time Orbits mf mp

Out−of−Plane Thrust Angle (degrees) −40 (days) (kg) (kg)

−60 357.7 1400 81.2 18.8

−80 The trajectory for this transfer is shown below in 0 50 100 150 200 250 300 350 True Longitude (degrees) Figure 10. During the transfer, the apogee is in- creased to over 60,000 km in order to eﬃciently per- Figure 8: Out-of-plane thrust angles for GTO-SSO form the plane change manoeuvre. Portions of the transfer. thrust magnitude, and in-plane and out-of-plane thrust angles are shown in Figures 11, 12 and 13, 25 respectively.

20 4 x 10

3

2 15

1

0

10 Z (km)

Thrust Magnitude (mN) −1

orbits (1−40) −2 orbits (681−720) orbits (1361−1400) 5 −3

1 2 1 0 0 −1 4 −2 4 x 10 −3 x 10 0 −1 −4 −5 −6 0 50 100 150 200 250 300 350 Y (km) X (km) True Longitude (degrees)

Figure 9: Thrust magnitude for GTO-SSO transfer. Figure 10: GTO-SSO trajectory with high Isp.

8 orbits (1−40) 4.4 GTO-SSO With Node Change 150 orbits (681−720) orbits (1361−1400)

100 In the previous case, no constraints were placed on the value of the right ascension of the ascending 50 node, Ω, upon arrival to SSO. Here, it is assumed that the Ω needs to be equal to 60 degrees (initially 0 0 degrees) when the satellite reaches the SSO. The initial and terminal conditions for this transfer are −50

In−Plane Thrust Angle (degrees) similar to the two previous cases studied, and the

−100 high value of Isp is used (3500 s). Additionally, the thrust was parameterized with a 40-orbit repeat cy- −150 cle and the transfer was modeled with 1400 orbit revolutions in order to compare the results with the 0 50 100 150 200 250 300 350 True Longitude (degrees) previous 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) than Figure 11: In-plane thrust angles for high Isp GTO- SSO transfer. the previous case study. Additionally, the transfer time requires nearly 200 more days than the pre- 80 vious case. Thus, changing the Ω by 60 degrees

60 orbits (1−40) produces a substantial transfer time increase. This orbits (681−720) orbits (1361−1400) long transfer time would likely be inappropriate for 40 a small satellite. If the thrust was parameterized

20 with less than a 40-orbit repeat cycle, the transfer time could potentially be decreased. Additionally, 0 constraints could be placed on the total transfer −20 time to keep it under a speciﬁc duration. In this transfer, shown below in Figure 14, the

Out−of−Plane Thrust Angle (degrees) −40 satellite migrates far from the Earth in order to eﬃ- −60 ciently change both the inclination and the Ω. This

−80 migration accounts for the long transfer time. The

0 50 100 150 200 250 300 350 thrust proﬁles are shown in Figures 15-17, respec- True Longitude (degrees) tively, for the ﬁrst 40, middle 40 and last 40 orbits of the transfer. In this transfer, there is less out-of- Figure 12: Out-of-plane thrust angles for high Isp plane thrusting than the previous transfer, which is GTO-SSO transfer. likely due to the large apogee of the orbit. In Figure 17, the thrust magnitude proﬁle of the middle and 25 ﬁnal 40 orbits overlap and cannot be distinguished from each other.

20

Table 9: GTO-SSO transfer results for high Isp en- 15 gines and a ﬁxed value of Ω

10 Thrust Magnitude (mN)

5 orbits (1−40) Transfer Time Orbits m m orbits (681−720) f p orbits (1361−1400) (days) (kg) (kg) 0 541.9 1400 78.2 21.8 0 50 100 150 200 250 300 350 True Longitude (degrees)

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

9 4 x 10 25

4

3 20 2

1

0 15 Z (km) −1

−2

−3 10 Thrust Magnitude (mN) −4

−6 2 5 −4 0 orbits (1−40) −2 −2 orbits (681−720) 0 −4 4 orbits (1361−1400) 4 2 −6 x 10 x 10 Y (km) X (km) 0

0 50 100 150 200 250 300 350 True Longitude (degrees)

Figure 14: GTO-SSO trajectory with high Isp and a ﬁxed value of Ω. Figure 17: Thrust magnitude for high Isp GTO- SSO transfer with ﬁxed value of Ω.

150

100 5 Conclusion

50 Mission analysis software and tools are an integral part of mission planning for all types of satellites. 0 Low-thrust trajectory planning is particularly chal- lenging for small Earth-based satellites due to the −50

In−Plane Thrust Angle (degrees) low thrust levels and many-orbit revolutions. Here, orbits (1−40) a direct method trajectory optimization tool was −100 orbits (681−720) orbits (1361−1400) developed, using a multiple-orbit thrust parame- −150 terization strategy, in order to design and address

0 50 100 150 200 250 300 350 the feasibility of these types of transfers. The op- True Longitude (degrees) timization routines are general in nature and are capable of handling a variety of transfer problems. This paper examined a unique small satellite mis- Figure 15: In-plane thrust angles for high I GTO- sp sion, which seeks to reduce mission cost by using a SSO transfer with ﬁxed value of Ω. non-ideal launch insertion. In this study, the feasibility of using a low-thrust 80 propulsion system to acquire the target SSO from orbits (1−40) 60 orbits (681−720) orbits (1361−1400) a readily available GTO secondary launch oppor- tunity was examined. Using a 100 kg satellite, the 40 fuel requirements were favorable (mf = 81.2) when 20 the low-thrust engines had an Isp of 3500 s. How-

0 ever, when the originally anticipated moderately- high Isp (800 s) engines were used, the SSO satellite −20 arrival mass was unfavorably low (47.9 kg). A third

Out−of−Plane Thrust Angle (degrees) −40 case study investigated the eﬀect of a 60 degree Ω change in the GTO-SSO transfer when using the −60 high Isp engines. The change in Ω added only 3 kg −80 more fuel than a transfer without the change, but 0 50 100 150 200 250 300 350 True Longitude (degrees) it increased the transfer time by 184 days. Further transfer studies will be needed in order to design an acceptable compromise between time and fuel. Figure 16: Out-of-plane thrust angles for high Isp The present approach to trajectory optimization GTO-SSO transfer with ﬁxed value of Ω. has been shown to be a valuable preliminary design

10 tool for small satellite design. Furthermore, this tion of averaging techniques. Acta Astronau- software has the capability of assessing the feasibil- tica, 41(3):133–149, 1997. ity of future mission concepts. Although the current work is suited for preliminary planning, it could also [6] C. A. Kluever and S. R. Oleson. Direct ap- be readily evolved to perform higher-ﬁdelity design. proach for computing near-optimal low-thrust In future studies, the eﬀects of Earth’s asymmetry, earth-orbit transfers. Journal of Spacecraft and solar-lunar perturbations and radiation could be in- Rockets, 35(4):509–515, 1998. cluded in the model, albeit with increased compu- [7] M. R. Ilgen. A hybrid method for computing tation time. optimal low-thrust OTV trajectories. Amer- ican Astronautical Society, AAS Paper 94- References 129:941–958, 1994.

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