Solving Complex Low Earth Orbit-to-Geostationary Earth Orbit Transfer Problems Using Uniform Trigonometrization Method

Abstract

Low-thrust orbit transfer problems are central to reducing mission costs and enabling cleaner, more efficient space travel. However, they remain difficult to solve using mathematically superior indirect methods of optimization. This is mainly due to the sensitivity to initial guesses and ill-conditioned matrices generated using traditional indirect methods. This paper applies the Uniform Trigonometrization Method (UTM), a cutting-edge indirect optimization technique, to four cases of low-thrust low Earth orbit (LEO)-to-geostationary Earth orbit (GEO) transfer problems. Using the UTM framework, including efficient numerical continuation and problem scaling strategies, smoother optimal control solutions were obtained. The convergence of standard boundary value problem solvers, like MATLAB’s bvp4c, significantly increases while using the simplicity and efficiency of the UTM. The UTM was able to solve Case 1 in a simpler manner compared to the traditional indirect method presented in the literature. In Case 2, the UTM found results for a constant thrust value of 1 N, while a direct pseudospectral method failed to converge. The results obtained using the UTM for Case 2 have 20 times longer flight duration and revolutions of spacecraft around the Earth. The UTM efficiently performs trade studies using a continuation approach that generates additional insights into all cases of this problem. In Case 4, the UTM was able to easily generate a bang–bang control structure, which traditionally requires solving a complex multi-point boundary value problem. The results generated using the UTM are very high-resolution, as it relies on the necessary conditions of optimality and guarantees locally optimal solutions. These findings position the UTM as a promising indirect approach for solving real-world long-duration orbit transfers.

1. Introduction

Low Earth orbit (LEO)-to-geostationary Earth orbit (GEO) transfers have a wide variety of applications in the world today. For example, satellite deployment, telecommunications, or climate monitoring can make use of LEO-to-GEO transfers [1,2]. Low-thrust transfers have been extensively investigated in the literature due to their higher efficiency and lower mission costs [3,4]. There are several constraints associated with these transfer problems, such as fuel efficiency and mission longevity, which make low-thrust transfers especially appealing for modern missions. The challenge of solving these transfer optimizations is that they are computationally very difficult due to long burn durations and nonlinear orbital dynamics [5,6,7].

Various issues arise when solving LEO-to-GEO problems with low thrust, one being the increased complexity due to continuous thrusting over long periods [8]. This makes it very hard to pinpoint where and for how long the engine should be thrusting for an optimal trajectory. Unlike a traditional high-thrust impulse technique like a Hohmann or bi-elliptic transfer, a low-thrust engine does not have the power required to impart enough velocity change to complete these maneuvers. Another issue is sensitivity to initial guesses, boundary conditions, and dynamic constraints.

The typical way to solve optimal control problems (OCPs) is one of two methods, indirect or direct. Indirect methods, like those seen in ref. [9], rely on necessary conditions of optimality that guarantee locally optimal solutions. Pontryagin’s Minimum Principle (PMP) [10] provides additional conditions to select the optimal control in case the optimal control law has more options. PMP ensures that the optimal trajectory’s respective controls minimize the Hamiltonian function.

The resulting equations form a two-point boundary value problem (TPBVP) that involves state equations, costate equations, Hamiltonian conditions, and boundary conditions. This TPBVP can then be solved using methods like shooting methods, collocation methods, etc. Although indirect methods can be very accurate due to explicitly satisfied Hamiltonian conditions, they rely heavily on a good initial guess and are often numerically unstable. Furthermore, traditionally, they are considered very hard to formulate, especially if the OCPs contain state and control constraints, which require solving a multi-point boundary value problem (MPBVP).

Direct methods [11] approximate the structures of the state and control solutions using polynomials. Direct methods convert the OCP to a nonlinear programming problem (NLP) using collocation or pseudospectral transcription. An NLP solver is then used to solve the discretized problem. The benefits of using direct methods are their resilience to poor initial guesses, ease of handling path and state constraints, and their suitability for large, complex, and constrained problems. Pseudospectral methods (PSMs) are currently the state-of-the-art direct methods, which are widely used by the research community in this domain. The main problem with direct methods is that they may not guarantee strict satisfaction of necessary optimality conditions like the indirect methods. Indirect formulations analytically satisfy the necessary conditions of optimality, thereby providing Hamiltonian consistency and ensuring local optimality, which is not guaranteed by direct methods.

Recent advances in indirect methods have sought to address the issues of numerical sensitivity, poor convergence, and difficulty in initializing costate variables. Traditional formulations based on PMP offer adequate solutions, but solving the resulting TPBVP remains difficult without a good initial guess. To mitigate this, ref. [12] came up with a two-step process using analytical initialization followed by continuation that allowed the solution of increasingly complex problems using simpler problems as a foundation.

More recently, Ref. [9] has used two different heuristic approaches: regularization through the hyperbolic tangent function and developing the problem using a multi-arc formulation. However, the latter was found to be the better choice. In addition, some hybrid methods have been developed that combine the strengths of both direct and indirect methods. For example, the Pontryagin–Bellman Differential Dynamic Programming algorithm [13] was developed by incorporating PMP into the Differential Dynamic Programming formulation. These developments signal a shift towards more stable, practical implementation of optimal strategies, paving the way for alternative transformations, such as the indirect method used in this paper, the Uniform Trigonometrization Method (UTM).

Furthermore, direct methods result in lower resolution results as compared to mathematically richer indirect methods for certain problems, such as singular control problems. Chattering or solutions with many jitters are obtained using direct methods that cannot be implemented in a real-world scenario. To solve the outstanding challenges surrounding indirect methods since their inception, several recent advancements [7,14,15,16,17,18,19,20,21,22,23] have given birth to the UTM, which generates optimal results with higher resolution. Recently, it was also discovered that indirect methods result in a better and different solution than direct methods for certain OCPs [24]. Several other complex OCPs from different domains have been solved using the UTM [25,26,27,28].

The UTM is an indirect optimization method that formulates OCPs using a trigonometric transformation of control inputs [7]. This allows the method to deal with bang–bang, singular, and nonlinear bounded controls, as well as state path constraints in a unified and numerically stable way. Unlike other smoothing methods, the UTM avoids trial-and-error-based error parameters and costate-dependent denominators by incorporating smooth trigonometric regularization terms directly into the cost functional. This guarantees continuity of the control while preserving the structure of the Hamiltonian. As demonstrated in this paper, the UTM was able to solve complex problems with a faster computation time than the traditional indirect approach.

The main contribution of this paper is that we apply the UTM to low-thrust LEO-to-GEO transfers at realistic thrust levels such as 1 N. This represents one of the first demonstrations of an indirect method achieving convergence for such low-thrust levels. These realistic problems are typically very difficult to solve due to poor convergence. In particular, we show that state-of-the-art PSM solvers are unable to converge or require unrealistic computational time for these scenarios. The UTM, combined with continuation strategies, is capable of achieving stable and accurate solutions. This shows the applicability of the UTM as a practical indirect approach to trajectory design for problems where direct methods have been favored in the past.

The remainder of this paper proceeds as follows. Section 2 is a summary of the UTM, followed by four different transfer problems solved in subsequent sections. Section 3 includes a problem taken from Ref. [29] that serves as a baseline problem. Section 4 presents a problem similar to Section 3, with constant thrust values reduced from 20 N to 1 N. Section 5 has a similar problem as Section 3, with higher Isp values to see how well the UTM can handle different engine parameters. Section 6 includes a minimum-fuel optimization problem, in which thrust is an added control that varies between 0 N and 20 N. Section 7 concludes this study.

2. Background on the Uniform Trigonometrization Method

OCPs often involve complex constraints on the states and controls, for example, bounds, singular arc, or path constraints. Traditional indirect methods offer high-accuracy solutions but suffer from implementation complexity, especially for problems relating to complex orbit transfers. This can result in the problems becoming numerically stiff and difficult to solve due to discontinuities and non-uniqueness, or the need to explicitly manage inequality constraints. The UTM simplified the problem formulation and solution process by using trigonometric functions to implicitly impose control bounds while using BVP solvers such as MATLAB’s bvp4c. Unlike other control-input-limiting techniques such as $ϵ$-TRIG regularization [14], hyperbolic tangent smoothing [30,31,32,33], and polynomial smooth regularization [34], the UTM introduces smooth trigonometric substitutions directly into the Hamiltonian, eliminating the need for empirical tuning parameters and ensuring that the controls remain continuous.

2.1. General Optimal Control Problem Formulation

An OCP can be formulated as follows.

$$\begin{array}{cc}\hfill & J=q(x\left(t_f\right),t_f,x\left(t_0\right),t_0)+{\int }_{t_0}^{t_f}L(x,u,t)dt\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill & \mathrm{subject}\mathrm{to}:\dot{x}=f(x,u,t)\hfill \end{array}$$

(1b)

$$\begin{array}{cc}\hfill & \psi (x\left(t_0\right),t_0)=0\hfill \end{array}$$

(1c)

$$\begin{array}{cc}\hfill & \varphi (x\left(t_f\right),t_f)=0\hfill \end{array}$$

(1d)

J is the performance index, which represents the total cost or performance of the system over time. The term q is a terminal cost that accounts for the performance at the end of the trajectory. $x\left(t\right)$ is the state vector, which is a collection of variables that describe the condition of the systems at time t. $u\left(t\right)$ is the control vector, which is a set of variables that are directly manipulated to change the behavior of the system. $f(x,u,t)$ is the dynamic function that describes the system equations of motion. $L(x,u,t)$ is the path cost; this quantifies the “cost” of being in a certain state $x\left(t\right)$ and applying a certain control $u\left(t\right)$. $t_0$ and $t_f$ are the initial and final times, respectively. $\psi$ and $\varphi$ enforce the initial and final constraints on the states.

The Hamiltonian H combines the dynamics and the cost into a single scalar quantity. It is used as a framework to derive the necessary conditions of optimality and is given below.

$$H=L(x,u,t)+{\lambda }^{\top }f(x,u,t)$$

(2)

$\lambda \left(t\right)$ is the costate vector, which is the adjoint variable associated with each state variable. The necessary conditions for optimality are stated below.

$$\begin{array}{cc}\hfill \dot{\lambda }\hfill & =-{\left({\displaystyle \frac{\partial H}{\partial x}}\right)}^{\top }\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial H}{\partial u}}& =0\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill \lambda \left(t_0\right)& \hfill ={\nu }_0{\displaystyle \frac{\partial \psi }{\partial x\left(t_0\right)}}\hfill \end{array}$$

(3c)

$$\begin{array}{cc}\hfill \lambda \left(t_f\right)& ={\displaystyle \frac{\partial q}{\partial x\left(t_f\right)}}+{\nu }_f{\displaystyle \frac{\partial \varphi }{\partial x\left(t_f\right)}}\hfill \end{array}$$

(3d)

$$\begin{array}{cc}\hfill {\left.\left(H+{\displaystyle \frac{\partial q}{\partial t}}+{\nu }_f{\displaystyle \frac{\partial \varphi }{\partial t}}\right)\right|}_{t=t_f}& =0\hfill \end{array}$$

(3e)

When there is more than one option in the optimal control law, the PMP must be satisfied as follows.

$$H(t,x^{*},u^{*},{\lambda }^{*})\le H(t,x^{*},u,{\lambda }^{*})$$

(4)

2.2. Applying the Uniform Trigonometrization Method

To automatically enforce bounds on a control, the UTM maps the original control to a new control in sine form, as shown below. This trigonometric transformation ensures that the control stays between its limits, $u_{\nu ,min}$ and $u_{\nu ,max}$, respectively. The resulting equation is shown below.

$$\begin{array}{cc}\hfill u_{\nu }& =\left({\displaystyle \frac{u_{\nu ,max}-u_{\nu ,min}}{2}}\right)sin\left(u_{\nu ,\mathrm{TRIG}}\right)+\left({\displaystyle \frac{u_{\nu ,max}+u_{\nu ,min}}{2}}\right)\hfill \end{array}$$

(5)

The sine function keeps the control smoothly bounded between its physical limits and avoids discontinuities associated with bang–bang transitions. This transformation gives a continuous and numerically stable control history. Furthermore, such a trigonometric substitution helps in finding singular controls using indirect methods without using additional conditions like Kelley’s condition [29].

The UTM modifies the Hamiltonian to account for linear and nonlinear bounded controls and path constraints. The general form assumes the following structure.

$$H=H_0(t,x,\lambda )+\sum _{i=1}^r{H}_{1i}(t,x,\lambda )u_i+\sum _{j=1}^s\sum _{k=1}^{d_j}{a}_{jk}{u}_j^k$$

(6)

To enforce bounds and eliminate discontinuities, the path cost is regularized. The new expressions for the objective function, the path cost function, and the dynamics are given below. $u_i$ and $u_j$ describe how control inputs change the time evolution of the system.

$$\begin{array}{cc}\hfill \tilde{J}& =q(x\left(t_f\right),t_f,x\left(t_0\right),t_0)+{\int }_{t_0}^{t_f}\tilde{L}(x,u,t)dt\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill \tilde{L}& =L+\sum _{i=1}^r{ϵ}_{i}cos\left(u_{i,\mathrm{TRIG}}\right)+\sum _{j=1}^q{\eta }_{j}sec\left({\displaystyle \frac{\pi }{2}}\left({\displaystyle \frac{2(s_j-s_{max,j}-s_{min,j})}{s_{max,j}-s_{min,j}}}\right)\right)\hfill \end{array}$$

(7b)

$$\begin{array}{cc}\hfill {\dot{x}}_i& =f_i(t,x)+g_i(t,x)u_i\hfill \end{array}$$

(7c)

$$\begin{array}{cc}\hfill {\dot{x}}_j& =f_j(t,x)+g_j(t,x,u_j)\hfill \end{array}$$

(7d)

2.3. Optimality Conditions in the UTM

For linearly bounded controls, the optimality conditions are as follows.

$${\displaystyle \frac{\partial H}{\partial u_{i,\mathrm{TRIG}}}}=-{ϵ}_{i}sin\left(u_{i,\mathrm{TRIG}}\right)+2a_i{H}_{1i}cos\left(u_{i,\mathrm{TRIG}}\right)=0$$

(8)

For nonlinear controls, the optimality conditions are as follows.

$${\displaystyle \frac{\partial H}{\partial u_{j,\mathrm{TRIG}}}}=\left(\sum _{k=1}^{d_j}k{a}_{jk}{(2b_j+2a_{j}sin\left(u_{j,\mathrm{TRIG}}\right))}^{k-1}\right)cos\left(u_{j,\mathrm{TRIG}}\right)=0$$

(9)

2.4. Closed-Form Control Expressions

Linear and nonlinear controls are described by Equations (10) and (11), respectively.

$$u_{i,\mathrm{TRIG}}^{*}=\left\{\begin{array}{c}arctan\left({\displaystyle \frac{2a_i{H}_{1i}}{{ϵ}_i}}\right)\hfill \\ arctan\left({\displaystyle \frac{2a_i{H}_{1i}}{{ϵ}_i}}\right)+\pi \hfill \end{array}\right.$$

(10)

$$u_{j,\mathrm{TRIG}}^{*}=\left\{\begin{array}{c}-{\displaystyle \frac{\pi }{2}}\hfill \\ arcsin\left({\zeta }_{1j}(t,x,\lambda )\right)\hfill \\ ⋮\hfill \\ arcsin\left({\zeta }_{d_j-1,j}(t,x,\lambda )\right)\hfill \\ {\displaystyle \frac{\pi }{2}}\hfill \end{array}\right.$$

(11)

The UTM simplifies the numerical solution of constrained OCPs by replacing hard constraints with smooth bounded trigonometric representations and by embedding regularization terms in the path cost. This reduces the complexity of the problem from an MPBVP to a standard TPBVP, making them solvable with classic numerical methods without loss of generality or control accuracy.

Note that the traditional indirect methods require scaling of both the states and costates, while the UTM required scaling of only the states for the problem in this paper. Although both the traditional and UTM approaches use numerical continuation strategies, the UTM can converge more quickly due to the more autonomous nature of the framework. With a more random guess, the UTM can still converge to a solution compared to traditional methods [7,15].

After going over the structure and optimality conditions of the UTM, we can now look at its application in orbit transfer scenarios. In the following sections, we consider a sequence of challenging low-thrust transfers to assess the method’s accuracy, convergence properties, and robustness compared to existing approaches. The different cases presented and solved in this study are shown in

Table 1. Four different cases of low-thrust LEO-to-GEO transfer in this study.

Case #	Thrust (N)	Isp (s)
1	20	6000
2	1	6000
3	20	10,000
4	Between 0 and 20	6000

.

3. Case 1: Minimum-Time Low-Thrust Low Earth Orbit-to-Geostationary Earth Orbit Transfer

This problem comes from Appendix C of ref. [29] authored by Longuski et al. The original problem was posed as a final radius-maximizing optimization. However, we can reformulate this problem as a minimum-time problem. This problem has never been solved using the UTM. Additionally, the traditional indirect methods required several steps to make this problem well conditioned for solving using MATLAB’s bvp4c. We used the MATLAB 2025a version for this study.

3.1. Problem Statement

Our goal is to find the minimum-time trajectory for an orbit transfer from a 300 km altitude LEO to a 35,786 km altitude GEO. The optimization problem is as follows.

$$\begin{array}{cc}\hfill \mathrm{minimize}:J& =t_f\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}:\dot{r}& =u\hfill \end{array}$$

(12b)

$$\begin{array}{cc}\hfill \dot{u}& ={\displaystyle \frac{v^2}{r}}-{\displaystyle \frac{\mu }{r^2}}+{\displaystyle \frac{Tsin\alpha }{m_0-\left|\dot{m}\right|t}}\hfill \end{array}$$

(12c)

$$\begin{array}{cc}\hfill \dot{v}& =-{\displaystyle \frac{uv}{r}}+{\displaystyle \frac{Tcos\alpha }{m_0-\left|\dot{m}\right|t}}\hfill \end{array}$$

(12d)

$$\begin{array}{cc}\hfill \dot{\theta }& ={\displaystyle \frac{v}{r}}\hfill \end{array}$$

(12e)

where r is the radius; u is the radial velocity; v is the tangential component of velocity; $\theta$ is the longitudinal angle; $\alpha$ is the thrust angle; T is thrust; $m_0$ is the initial mass; $\dot{m}$ is the mass flow rate; and $\mu$ is the gravitational parameter.

For all the cases presented in this article, the following general assumptions are used.

1.

A planar orbit transfer under unperturbed Keplerian dynamics.

2.

Initial and final orbits are assumed to be circular.

3.

The spacecraft is treated as a point mass with no attitude dynamics.

4.

The spacecraft is assumed to have continuous power availability.

5.

The LEO is equatorial.

In addition to the general assumptions, we assume a constant thrust of 20 N for this case. We have similar case-specific assumptions for the remaining cases. The constants used for this case are shown in

Table 2. Constants for Case 1 of the LEO-to-GEO problem.

Parameter	Unit	Value
g	m/s2	9.80665
$m_0$	kg	1500
Isp	s	6000
T	N	20
$\dot{m}$	kg/s	$3.3991\times {10}^{-4}$

.

GPOPS-II, based on PSMs, was used to compare with the UTM results. The parameters used to setup and solve this problem using GPOPS-II are shown in

Table 3. GPOPS-II setup parameters and their values for this study.

Parameter	Value
Solver	IPOPT
Mesh Refinement Method	hp-PattersonRao
Maximum Iterations	100
Minimum Collocation Points	2
Maximum Collocation Points	3
Mesh Refinement Tolerance	${10}^{-6}$
Linear Solver	mumps
Derivatives Supplier	sparseCD
Derivative Level	second
Method	RPM-Differentiation

.

The bounds, initial, and final values are shown in

Table 4. Boundary and bounding values for the time and states of the LEO-to-GEO problem.

Parameter	Unit	Initial Value	Final Value	Minimum Value	Maximum Value
t	days	0	Free	0	10
r	km	6678	42,164	6378	43,164
$\theta$	rad	0	Free	0	1000$\pi$
u	km/s	0	0	−1	10
v	km/s	7.7258	3.0747	0	20

. For this problem, since we are optimizing time, its final value is unknown and its maximum value is unbounded. The states u and v are functions of the final radius, which is fixed, and we want it to be circular. $u_f$ turns out to be 0 for a circular orbit. We can find $v_f$ using the following equation:

$$\begin{array}{cc}\hfill v_f& =\sqrt{{\displaystyle \frac{\mu }{r_f}}}\hfill \end{array}$$

(13)

The specific values of the minimum and maximum bounds do not matter, as long as the final value of the states is within them. Since $\theta$ depends on how many revolutions the optimized orbit transfer takes, this is found after the problem is solved. For simplicity, we chose the lower bound of $\theta$ as 0 degrees. The maximum value of $\theta$ is unbounded, but we chose it as a very high value of $1000\pi$ while using GPOPS-II. Similarly, we chose an upper limit for time as 10 days. Note that the values on bounds can be changed based on the trial-and-error method.

3.2. Solution Process Using the Uniform Trigonometrization Method

To make the original problem well conditioned to gain quicker convergence using indirect methods, a scaling strategy was employed. Along with this scaling strategy, a numerical continuation strategy, as suggested in ref. [35], was also used in this study. These strategies are discussed in the following subsections.

3.2.1. Scaling Strategy

For this problem, unit scaling was not used. Instead, the following scaling scheme was used. The new state vector for the spacecraft is given by

$$x_s=[r_s,\theta ,u_s,v_s,t_s]$$

The new scaled variables are related to the original variables through the following set of equations. In these equations, the values of $t_f$, $r_0$, $v_0$, and $m_0$ are taken from Table 4.

$$\begin{array}{cc}\hfill r& =r_0{r}_s\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill u& =v_0{u}_s\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill v& =v_0{v}_s\hfill \end{array}$$

(14c)

$$\begin{array}{cc}\hfill m& =m_0{m}_s\hfill \end{array}$$

(14d)

$$\begin{array}{cc}\hfill t& =t_f{t}_s\hfill \end{array}$$

(14e)

Using these new scaled states, the new equations of motion are described as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{r}_s}{d{t}_s}}& ={\displaystyle \frac{u_s{v}_0{t}_f}{r_0}}\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\theta }{d{t}_s}}& ={\displaystyle \frac{v_s{v}_0{t}_f}{r_s{r}_0}}\hfill \end{array}$$

(15b)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{u}_s}{d{t}_s}}& ={\displaystyle \frac{t_f}{v_0}}\left({\displaystyle \frac{Tsin\beta }{m_s{m}_0}}+{\displaystyle \frac{{\left(v_s{v}_0\right)}^2}{r_s{r}_0}}-{\displaystyle \frac{\mu }{{\left(r_s{r}_0\right)}^2}}\right)\hfill \end{array}$$

(15c)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_s}{d{t}_s}}& ={\displaystyle \frac{t_f}{v_0}}\left({\displaystyle \frac{Tcos\beta }{m_s{m}_0}}-{\displaystyle \frac{u_s{v}_s{v}_0^2}{r_s{r}_0}}\right)\hfill \end{array}$$

(15d)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{t}_s}{d{t}_s}}& =1\hfill \end{array}$$

(15e)

The new objective function, compared to ref. [29], is shown in Equation (16). The scaled boundary conditions are shown in the

Table 5. Scaled boundary and bounding values for time and states of the LEO-to-GEO problem.

Parameter	Unit	Initial Value	Final Value	Minimum Value	Maximum Value
$t_s$	nd	0	31.5021	0	31.5021
$r_s$	nd	0.1584	1	0.1584	1
$\theta$	deg	0	Free	0	100$\pi$
$u_s$	nd	0	0	0	1
$v_s$	nd	2.5127	1	1	3

.

$$J=t_f$$

(16)

3.2.2. Hamiltonian Construction

The Hamiltonian H combines the system dynamics with the costates and the control input. It is expressed as Equation (17).

$$H={\lambda }_{r_s}{\displaystyle \frac{d{r}_s}{d{t}_s}}+{\lambda }_{\theta }{\displaystyle \frac{d\theta }{d{t}_s}}+{\lambda }_{u_s}{\displaystyle \frac{d{u}_s}{d{t}_s}}+{\lambda }_{v_s}{\displaystyle \frac{d{v}_s}{d{t}_s}}+{\lambda }_{t_s}$$

(17)

Using PMP, we derive the optimal control laws for $\beta$ by minimizing H with respect to $\beta$. The partial derivative of H with respect to $\beta$ must be zero for optimality, and the thrust direction can be defined in Equation (18). In order for the Hamiltonian to be minimized at all times, Equation (18) is derived.

$${\beta }^{*}=\left\{\begin{array}{cc}arctan\left({\displaystyle \frac{{\lambda }_{u_s}}{{\lambda }_{v_s}}}\right)\hfill & \mathrm{if}{H}_1=\min \{H_1,H_2\}\hfill \\ arctan\left({\displaystyle \frac{{\lambda }_{u_s}}{{\lambda }_{v_s}}}\right)+\pi ,\hfill & \mathrm{if}{H}_2=\min \{H_1,H_2\}\hfill \end{array}\right.$$

(18)

3.2.3. Costate Dynamics

The costate dynamics ${\dot{\lambda }}_i$ are derived from the Hamiltonian by differentiating with respect to each state variable $x_i$, shown in Equation (19).

$${\dot{\lambda }}_i=-{\displaystyle \frac{\partial H}{\partial x_i}}$$

(19)

The costate dynamics are formulated in Equations (20a)–(20e).

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{rs}}{d{t}_s}}& =-{\displaystyle \frac{\partial H}{\partial r_s}}={{\displaystyle \frac{{\lambda }_{rs}{t}_f{u}_s{v}_0}{r_s}}}^2-{\displaystyle \frac{{\lambda }_{us}{t}_f}{v_0}}\left({\displaystyle \frac{2\mu }{r_0^2{r}_s^3}}-{\displaystyle \frac{v_0^2{v}_s^2}{r_0{r}_s^2}}\right)+{\displaystyle \frac{t_f{v}_0{v}_s({\lambda }_{\theta }-{\lambda }_{vs}{u}_s)}{r_0{r}_s^2}}\hfill \end{array}$$

(20a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{\theta }}{d{t}_s}}& =-{\displaystyle \frac{\partial H}{\partial \theta }}=0\hfill \end{array}$$

(20b)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{u_s}}{d{t}_s}}& =-{\displaystyle \frac{\partial H}{\partial u_s}}={\displaystyle \frac{v_0{t}_f}{r_s}}\left({\displaystyle \frac{{\lambda }_{vs}{v}_s}{r_0}}-{\lambda }_{rs}\right)\hfill \end{array}$$

(20c)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{v_s}}{d{t}_s}}& =-{\displaystyle \frac{\partial H}{\partial v_s}}={\displaystyle \frac{v_0{t}_f({\lambda }_{vs}{u}_s-{\lambda }_{\theta }-2{\lambda }_{u_s}{v}_s)}{r_0{r}_s}}\hfill \end{array}$$

(20d)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{t_s}}{d{t}_s}}& =-{\displaystyle \frac{\partial H}{\partial t_s}}=-{\displaystyle \frac{{\left(T{t}_f\right)}^2{I}_{sp}{g}_0({\lambda }_{us}sin\beta +{\lambda }_{vs}cos\beta )}{v_0{(T{t}_f{t}_s-I_{sp}{g}_0{m}_0)}^2}}\hfill \end{array}$$

(20e)

The derivation of the costate dynamics and optimal control laws allows for successful implementation of the indirect method to solve the LEO-to-GEO transfer problem.

3.2.4. Numerical Continuation Strategy

Note that the following continuation strategy is a result of engineering judgment and experience with several OCPs. Propagate the trajectory for a very short duration of 0.01 days in a scaled form using the initial conditions shown in Table 5. The guess value for each costate is chosen as 0.1. Quick convergence is obtained for this seed trajectory, as it is less complicated than the entire trajectory.

In the first continuation set comprising five steps, the final values of the states $r_{s_f}$, $u_{s_f}$, and $v_{s_f}$, are varied until the desired values. The value of thrust is decreased from 2000 N to 20 N in the next continuation set of 100 steps. The tolerance values are then changed from $1\times {10}^{-3}$ to $1\times {10}^{-6}$ in five steps in the final continuation set. The continuation process for this problem is summarized in

Table 6. Continuation strategy for Case 1 of the LEO-to-GEO transfer problem.

Continuation	Seed	#1	#2	#3
Steps		5	100	5
$r_{s_f}$ [nd]	$r_0$/$r_f$	1
${\theta }_f$ [rad]	0
$u_{s_f}$ [nd]	0	0
$v_{s_f}$ [nd]	$v_0$/$v_f$	1
$t_{s_f}$ [nd]	0
T [N]	2000		20
Solver Tolerance	$1\times {10}^{-3}$			$1\times {10}^{-6}$

. It should be noted that, due to this continuation process, the UTM needs more computation time to obtain the final optimal solution than GPOPS-II.

3.3. Results

Figure 1 presents the states r, $\theta$, u, and v versus time plots. On each of the plots there are three different methods of solving this case. It is evident from Figure 1 that each of these methods yields identical solutions. The solution structure of u is oscillatory, while the other three states have a simpler solution structure.

Figure 2 shows the flight path of the entire transfer, showing the expected spiral path of a low-thrust transfer. The thrust steering control oscillates to higher magnitudes with time.

Figure 3 shows the costates and Hamiltonian for this case. There is a slight blip in the Hamiltonian time-history plot. The Hamiltonian is an explicit function of time and therefore does not necessarily need to be constant according to optimal control theory [29]. The magnitude scales of the costates vary for this case.

Case 1 is used to demonstrate that the UTM obtains the same results as the traditional indirect method used in ref. [29] and GPOPS-II. The UTM framework simplifies the application of indirect methods, making it superior to traditional indirect methods from the formulation and problem-solving standpoint.

4. Case 2: Minimum-Time Very-Low-Thrust Low Earth Orbit-to- Geostationary Earth Orbit Transfer

4.1. Problem Statement

Case 2 is more complex than Case 1, as it resembles a realistic low-thrust electric propulsion rocket engine. In Case 1, the thrust of the spacecraft is higher than what is feasible with current technology. For this case, we vary the thrust values to much lower values compared to ref. [29], which used a constant thrust value of 20 N. These lower thrust values were considered extremely difficult to generate an optimal solution using traditional indirect methods.

The aim of this case is to showcase the ability of the UTM framework to solve very complex low-thrust orbit transfer problems involving several revolutions, which have traditionally been considered prohibitive to solve. The constants, bounds, parameters, and Hamiltonian expression for this case remain the same as in Case 1. The only major change in this case occurs in the thrust values.

4.2. Solution Process Using the Uniform Trigonometrization Method

The Hamiltonian and the costate equations remain the same as in Case 1. The only major difference from a formulation standpoint is the continuation process. The continuation set on thrust values is changed from 20 N to 1 N in 180 steps.

4.3. Results

The state results obtained using the UTM for a constant thrust of 1 N are shown in Figure 4. The duration of transfer increases significantly from 3.97 days obtained in Case 1 to 77.65 days in this case. The number of oscillations for u increases significantly. The other states are relatively simpler in structure. The results for state $\theta$ are specified in terms of the revolutions needed to complete the LEO-to-GEO transfer.

The trajectory plot, as shown in Figure 5, has a very high number of revolutions, making this problem very complex and extremely difficult to solve. GPOPS-II was unable to converge for the thrust value of 1 N, demonstrating the complexity of this case. The steering control also has a significant increase in the number of oscillations, as seen in Figure 5.

The costates and Hamiltonian time-history plots are captured in Figure 6. The costates corresponding to the velocities u and v have a very high amount of oscillations compared to Case 1. The Hamiltonian stays at 0 value except for a short duration where it has significant jumps. For this case too, the Hamiltonian is an explicit function of time, and hence, we see such a variation instead of a constant value throughout the trajectory.

Table 7. Results summary using the UTM for LEO-to-GEO orbit transfer problem with Isp as 6000 s and different thrust values.

Thrust	Time of Flight (Days)		Revolutions
(N)	UTM	PSM	(#)
20	3.9716	3.9716	25.5565
10	7.8226	7.8229	50.8917
5	15.5621	15.5621	101.5922
1	77.6476	Did not converge	507.5287

compares the results obtained using the UTM for Cases 1 and 2. Intermediate results are also specified to show an exponential increase in complexity as the thrust values are lowered from 20 N to 1 N. This also highlights the advantage of the UTM to generate such intermediate trade study results, which give more insight into the nature of the LEO-to-GEO transfer problem. Note that to calculate the number of revolutions shown in Table 7, we divided $\theta$ by $2\pi$.

Thus, the results of Case 2 demonstrate the robustness of UTM for realistic low-thrust transfers. As the thrust was reduced from 20 N to 1 N, the flight time increased from 3.97 to 77.6 days, which is an increase of around 19 times. The number of revolutions increased from around 25 in Case 1 to over 500 in this case, an increase of approximately 20 times. Despite this dramatic change in mission duration and numerical complexity, the UTM maintained smooth and stable control. The next case is simpler than this case, but showcases the strength of UTM to perform trade studies for useful problem parameters.

5. Case 3: Minimum-Time, Low-Thrust, Very-High-Isp, Low Earth Orbit-to-Geostationary Earth Orbit Transfer

5.1. Problem Statement

This case remains similar to Case 1, with the only change being higher Isp values. Very-high-Isp microwave ion engines have been studied for future space missions [36,37]. Such higher values of Isp correspond to cleaner and more efficient rocket fuel, which is desired for future space exploration [36]. This case serves to evaluate the numerical behavior of the UTM under high-efficiency propulsion rather than geometrical complexity. This case showcases the ability of the UTM to quickly perform trade studies on a parameter of interest.

5.2. Solution Process Using the Uniform Trigonometrization Method

The Hamiltonian and the costate equations remain the same as in Case 1. The only major difference from a formulation standpoint is the continuation process. One more continuation set is added to Case 1, through which the Isp value is changed from 6000 s to 10,000 s in five steps.

5.3. Results

The results obtained in Case 3, shown in Figure 7, Figure 8 and Figure 9, are very similar to those obtained in Case 1. The time duration and the number of revolutions around Earth for LEO-to-GEO transfers are slightly higher for Case 3 compared to Case 1. As the Isp increases from 6000 to 10,000 s, there is only a minor change in flight duration. This is because the mass flow rate is inversely proportional to Isp. Thus, the effective thrust acceleration decreases as Isp increases.

The results for Case 3 are summarized and compared with the corresponding results for Case 1 in

Table 8. Result summary using the UTM for LEO-to-GEO orbit transfer problem with thrust of 20 N and different Isp values.

Isp	Time of Flight	Revolutions
(N)	(days)	(#)
6000	3.9716	25.5565
7000	3.9921	25.6449
8000	4.0077	25.7106
9000	4.0199	25.7609
10,000	4.0297	25.8009

. GPOPS-II results were similar to those obtained using the UTM for this case. The GPOPS-II results are not included in this case for brevity.

6. Case 4: Minimum-Fuel Low-Thrust Low Earth Orbit-to-Geostationary Earth Orbit Transfer

6.1. Problem Statement

In the previous cases, the performance index was formulated as a time minimization problem, but for many real-world missions, minimizing the fuel cost is a more relevant objective. In this case, we reformulate the problem to minimize the fuel used during the LEO-to-GEO transfer. The objective function is shown in Equation (21a), where $ϵ$ is an error parameter and $cos{T}_{\mathrm{TRIG}}$ is an error control. The scaled mass, $m_s$, has its own equation specified in Equation (21b). Note that in this case the final time is set to 4 days.

$$\begin{array}{cc}\hfill J& =-m_{s_f}+{\int }_{t_{s_0}}^{t_{s_f}}ϵcos{T}_{\mathrm{TRIG}}dts\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{m}_s}{d{t}_s}}& =-{\displaystyle \frac{t_f}{m_0}}\left({\displaystyle \frac{T}{Isp{g}_0}}\right)\hfill \end{array}$$

(21b)

where $m_f$ is the final mass of the spacecraft. The dynamics remain the same as in Case 1. Unlike the earlier fixed-thrust problems, in this case, thrust is a bounded control variable, with thrust varying between 0 N and 20 N. Formulating a problem like this usually introduces a bang–bang structure to the control, which has traditionally been very hard to solve using indirect methods due to the discontinuities in the control structure.

6.2. Solution Process Using the Uniform Trigonometrization Method

The UTM naturally handles bounded and bang–bang controls, making it well suited to solve this problem. The control substitution for thrust was expressed using the process described in Section 2. This ensures that thrust remains continuous while still respecting its bounds. The trigonometric transformation also prevents unrealistic rapid on–off cycling.

Since the thrust control varies between 0 N and 20 N, the following trigonometric form is used to specify the thrust using the UTM.

$$T=10\left(\sin T_{TRIG}+1\right)$$

(22)

The Hamiltonian for this case is now given as follows.

$$H={\lambda }_{r_s}{\displaystyle \frac{d{r}_s}{d{t}_s}}+{\lambda }_{\theta }{\displaystyle \frac{d\theta }{d{t}_s}}+{\lambda }_{u_s}{\displaystyle \frac{d{u}_s}{d{t}_s}}+{\lambda }_{v_s}{\displaystyle \frac{d{v}_s}{d{t}_s}}+{\lambda }_{m_s}{\displaystyle \frac{d{m}_s}{d{t}_s}}+{\lambda }_{t_s}$$

(23)

The optimal control law can then be evaluated for the control $T_{\mathrm{TRIG}}$ as shown in Equation (24). Note that the optimal control law for $\beta$ stays the same as Equation (18).

$$T_{\mathrm{TRIG}}^{*}=\left\{\begin{array}{cc}arctan\left({\displaystyle \frac{1}{2ϵ}}\left({\displaystyle \frac{{\lambda }_{us}sin\beta +{\lambda }_{vs}cos\beta }{m_s}}-{\displaystyle \frac{{\lambda }_{ms}}{I_{sp}{g}_0}}\right)\right),\hfill & \mathrm{if}{H}_1=\min \{H_1,H_2\}\hfill \\ arctan\left({\displaystyle \frac{1}{2ϵ}}\left({\displaystyle \frac{{\lambda }_{us}sin\beta +{\lambda }_{vs}cos\beta }{m_s}}-{\displaystyle \frac{{\lambda }_{ms}}{I_{sp}{g}_0}}\right)\right)+\pi ,\hfill & \mathrm{if}{H}_2=\min \{H_1,H_2\}\hfill \end{array}\right.$$

(24)

Since Case 4 involves a new equation of motion for m, there is a corresponding costate equation given as follows. Note that all these equations involve a new expression for the thrust control.

$${\displaystyle \frac{d{\lambda }_{m_s}}{d{t}_s}}=-{\displaystyle \frac{\partial H}{\partial m_s}}={\displaystyle \frac{10(sin\left(T_{\mathrm{TRIG}}\right)+1)t_f({\lambda }_{us}sin\beta +{\lambda }_{vs}cos\beta )}{m_0{m}_s^2{v}_0}}$$

(25)

Numerical Continuation

Propagate the trajectory for 4 days (approximate result obtained for Case 1) in a scaled form using the initial conditions shown in Table 5. The guess value for each costate is chosen as 0.1. Quick convergence is obtained for this seed trajectory, since it is easy to solve and generates a nice starting trajectory to build upon. In the first continuation set comprising just two steps, the final values of the states $r_{s_f}$, $u_{s_f}$, and $v_{s_f}$ are varied until the desired values. The value of $ϵ$ is reduced from 0.1 units to $1\times {10}^{-9}$ units in the next continuation set comprising 60 steps. The solver tolerance value is changed from $1\times {10}^{-3}$ to $1\times {10}^{-6}$ in two steps in the final continuation set. The continuation process for this problem is summarized in

Table 9. Continuation strategy for Case 4 of the LEO-to-GEO transfer problem.

Continuation	Seed	#1	#2	#3
Steps		2	60	2
$r_s$ [nd]	$r_0$/$r_f$	1
$\theta$ [rad]	0
$u_s$ [nd]	0	0
$v_s$ [nd]	$v_0$/$v_f$	1
$m_s$ [nd]	1
$t_s$ [nd]	0
$ϵ$	0.1		$1\times {10}^{-9}$
Solver Tolerance	$1\times {10}^{-3}$			$1\times {10}^{-6}$

.

6.3. Results

The solution exhibits a bang–bang thrust profile, which is consistent with theoretical expectations for fuel minimization problems. The results for this case are summarized in

Table 10. Results summary for Case 4 of the LEO-to-GEO transfer problem.

Parameter	Case 4	Case 1
Final mass $m_f$ (kg)	1385.3	1383.4
Propellant used (kg)	114.7	116.6
Time of flight (days)	4	3.97
Revolutions (#)	25.5398	25.5565

and shown in Figure 10, Figure 11, Figure 12 and Figure 13. The computation times for the four cases of LEO-to-GEO transfer problems are presented in

Table 11. Computation time results for the four cases of LEO-to-GEO transfer problem solved using the UTM and PSM.

Case	UTM	PSM	Total Trajectories Using UTM
(#)	Computation Time (s)	Computation Time (s)	(#)
1	28.00	14.71	110
2	2862.44	Did not converge	190
3	29.12	14.34	115
4	448.67	61.13	64

.

Figure 12 shows the thrust time history. There is a significant period in the flight where thrust is zero, which is different from all previous cases considered. The major difference is with the lower fuel consumption, which results in a slightly longer flight time due to the zero-thrust coasting seen during the transfer.

7. Conclusions and Future Work

This study establishes that the Uniform Trigonometrization Method (UTM) can overcome the numerical sensitivity that has traditionally limited the use of indirect methods for low-thrust low Earth orbit (LEO)-to-geostationary Earth orbit (GEO) transfers. Using trigonometric substitutions in the controls, the UTM guarantees bounded, continuous, and smooth control solutions in a simple and effective manner.

In this study, four different cases of the LEO-to-GEO problem were solved. The results of Case 1 indicate that the UTM is superior to the traditional indirect methods that were used in ref. [29]. A very complex version of the LEO-to-GEO problem was solved in Case 2. For this case, both indirect and direct methods were originally failing to converge but were successfully solved using the UTM, demonstrating its robustness for complex trajectory design. Going below thrust values of 1 N for Case 2 led to extremely long computation times, which, hence, were not solved for.

In Case 3, solutions were obtained simply and efficiently using the UTM for very high Isp values in the LEO-to-GEO transfer problem. In Case 4, the problem becomes more difficult as the thrust becomes an additional control compared to the other three cases. Despite this added challenge, the UTM was able to solve this problem quite easily. We found that the numerical continuation strategy, employed in the UTM, is essential for convergence in very low-thrust, long-duration orbit transfer problems.

The results not only confirm that the UTM produces stable results and satisfies the conditions of optimality in the Hamiltonian but also show that the UTM is a useful tool for real-world mission planning. This shows that UTM is a very useful advancement in the world of low-thrust optimization that allows for more efficient and reliable trajectory design for future space missions. Although the UTM framework itself is established, this study demonstrates its first application to realistic long-duration, low-thrust transfers, achieving convergence where other methods fail. This highlights the practical value of UTM rather than the theoretical algorithmic novelty.

The present results are based on unperturbed planar dynamics; however, the UTM framework can be easily extended to include perturbations and higher-fidelity equations of motion. Exploring perturbed, higher-fidelity, and out-of-plane transfers is part of future work.