## 1. Introduction

In recent years, enhancements to solar-electric propulsion technology [

1,

2,

3] have seen their greater usage in Earth-orbiting satellites [

1,

4,

5], for both station-keeping as well as orbital transfer purposes [

5]. These applications have largely been for relatively larger satellites, and a variety of electric propulsion technologies are currently being miniaturized for future incorporation in nano-satellites as well [

6,

7]. The primary focus of this paper is the electric orbit-raising maneuver during the deployment of satellites operating in the geosynchronous equatorial orbit (GEO). The telecommunication satellite industry has started adopting electric propulsion for performing long-time scale orbit-raising maneuvers to the GEO. Instances include the deployment of SatMex and ABS satellites using Boeing’s 702-SP all-electric architecture (2015) [

8], and that of the EutelSat 172B satellite using Airbus’ fully-electric propulsion system (2017) [

9], as well as China’s SJ-13 mission (2017) [

10]. As more future space missions begin incorporating electric orbit-raising to GEO, there is a need for efficient mission design tools to address the challenges associated with the complicated maneuver spanning several months. In particular, we are interested in tools that facilitate fast and robust analysis of numerous electric orbit-raising scenarios to help identify best technology selections for the satellite bus.

The main challenge of the long-duration electric orbit-raising maneuver arises due to the low acceleration imparted by the onboard electric propulsion systems, relative to the local gravitational acceleration due to the Earth. The multiple eclipses that the satellite encounters en route to GEO deprive the electric propulsion system of power from the solar arrays, thereby adding to the complexity of the transfer.

For instance,

Figure 1 depicts a 225-revolution low-thrust transfer trajectory from a geosynchronous transfer orbit (GTO) to the GEO, spanning more than five months. Such a multi-revolution long duration transfer means a long transit time through the Van Allen belts, causing damage to the solar arrays, and therefore impacting thrust availability during the transfer. Mission design for an all-electric satellite therefore involves identifying various variables that affect the orbit-raising maneuver: number and type of electric thrusters for the satellite’s propulsion system, sizing of the solar array and shielding requirements, battery sizing in case thrusters need to be supported during eclipses, launch vehicle and injection orbit elements. In this paper, we consider a computational framework to generate low-thrust orbit-raising trajectories in a fast, robust, and automated manner, in order to facilitate automated analysis of numerous orbit-raising trajectories as part of exploring the design space.

The computation of an all-electric orbit-raising trajectory requires the solution of a challenging optimal control problem. A number of methods have been developed in order to compute optimal low-thrust orbit-raising trajectory, broadly falling under the categories of direct and indirect optimization based methodologies. While indirect techniques use calculus of variations to determine the necessary conditions of optimality and set up a two-point boundary value problem [

11,

12,

13,

14,

15,

16], direct techniques avoid using calculus of variations and instead rely on direct transcription and collocation to set up a parameter optimization problem, often solved using commercial software such as SNOPT and IPOPT [

17,

18,

19,

20,

21]. These methodologies typically rely on good quality user-provided initial guesses in order to rely on numerical convergence of the underlying algorithm [

22]. In order to address these issues, a number of alternative approaches approaches have been developed, such as shape-based techniques, Q-law, or semi-analytical approaches [

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

37]. Furthermore, dynamic models to capture the underlying translation dynamics of the spacecraft often play an important role in the convergence of direct or indirect optimization techniques, and several such models have been used in the literature, orbital elements, equinoctial elements, Cartesian, and spherical coordinates; please see Ref. [

38] for a comparison of different dynamic models in terms of their impact on the numerical solutions of orbit-raising optimal control problems.

This paper utilizes a novel dynamic model that has proven to be effective for solving orbit-raising optimal control problems [

38,

39]. This model uses dynamical parameters as the states of the spacecraft, instead of geometrical quantities. This model also uses a non-inertial reference frame that can be obtained by a 2-1-3 rotation sequence of Euler angles instead of a traditional 3-1-3 rotation sequence used in the orbital mechanics. This dynamic model removes the singularity arising for both equatorial and circular orbits, the singularity being a major drawback with traditional orbital elements. The singularity is shifted to a special case of polar orbits when the right ascension of the ascending node is 0 or 2

$\pi $ deg. Five out of this six new orbital elements are regularized, that is, they change very slowly for a thrusting spacecraft and remain constant for pure Keplerian motion. This makes them suitable for the trajectory optimization schemes that compute trajectories for long-time-scale transfers. Additionally, the current paper utilizes a novel optimization scheme introduced in Ref. [

39]; this technique breaks the multi-revolution problem into a sequence of optimization sub-problems, computing the all-electric orbit-raising trajectory in the order of tens of seconds on a standard personal computer without the need of any user-provided initial guess. This optimization tool can be used to analyze various electric orbit-raising mission scenarios and also can compute the trajectories to the GEO due to the non-singularity of the formulation in the equatorial plane. Recently, the work in Ref. [

39] was extended in order to analyze a variety of different aspects of the orbit-raising problem: effect of

${J}_{2}$ perturbations and accurate shadow model [

40], effects of using different types of electric thrusters [

41], analysis of attitude control during the maneuver [

42], effect of planning horizon length [

43], effect of objective function weights [

44], and their adaptive modification [

45].

The contributions of the paper are two important enhancements of the optimization problem described in Ref. [

39], in which the thrust magnitude was determined by a pre-set scheme (thrust in Sun-lit segments of trajectory and coast in eclipses) and power loss during transfer was not considered. First, we allow for coasting arcs within the Sun-lit portion of the orbit-raising trajectory. This is achieved through the application of the concept of thruster efficiency originally proposed by Petropoulos [

46] for the Q-law based framework. The thruster efficiency is defined for each segment of discretized trajectory in the optimization sub-problem, and allows for deciding the thrust magnitude. This enhancement is important to identify trade-offs between deployed mass and deployment time for the satellite. Second, we also allow for the consideration of radiation damage within the optimization framework, by proposing a neural network, trained on AP-9 radiation data that computes the radiation damage over a revolution based on the osculating orbital elements. The neural network predicts the power degradation over a revolution, and updates the thrust availability for the next revolution. Prior efforts on capturing the radiation dose were an analytical model based on AP-8 data [

47] and semi-analytical approach utilizing AP-9 data and the SCREAM software [

48]; however, to the best of knowledge of the authors, there has not been any effort on utilizing a neural network based modeling of the solar array degradation. The paper is organized as follows: we first present the mathematical overview of the problem under consideration, then elaborate the two proposed extensions, and finally present numerical simulations to demonstrate our proposed methodology.