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 Du®erin 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. Speci¯cally, 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 speci¯c 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 ¯rst 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- pro¯le is parameterized for a single orbit and is re- ally di±cult to solve except for simple cases. This peated for multiple orbit revolutions. Thus, instead approach has the bene¯t 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. ci¯c transfer problem. The direct method is signif- One bene¯t 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 e®ects 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 ¯delity 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 bene¯t 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 di±cult 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 signi¯cantly 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 = 2´P=c (1) a multiple-orbit thrust parameterization strategy is used in order to solve this very-low-thrust transfer where ´ is the engine e±ciency, 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 signi¯cantly more complex and compu- between zero and 25 mN and the Isp was assumed tationally demanding than the cartesian counter- to be ¯xed. 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 2´P 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- e±cient 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 ¯nal 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 modi¯ed set of equinoctial states, respectively.