# A Survey on Low-Thrust Trajectory Optimization Approaches

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## Abstract

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## 1. Introduction

## 2. Concurrent Engineering Requirements

- Flexibility: high versatility to cope with a wide range of scenarios is demanded, as well as the ability to optimize discrete decision-making and mission planning.
- Robustness: the sensitivity to the input parameters has as low as possible.
- Speed: they have to be fast, since it is not possible to spend long computation times during concurrent design studies.
- Accuracy: they must provide meaningful results, yet high-fidelity is not required. An accurate trajectory will be required during the detailed design, once a mission candidate is selected.
- Automation: minimal user-interaction is desired to reduce man-power cost.
- Optimality: near-optimal solutions are deemed acceptable.

## 3. Multiobjective Hybrid Optimal Control

#### 3.1. Hybrid Dynamical System

- Autonomous switching: An autonomous switching occurs when the continuous state trajectory crosses the discontinuity surface in the continuous state-time space (see Figure 1a). In this case, the discontinuity surface depends only on the continuous state and on time, i.e., $s=s(\mathit{x},t)$. The switching causes the discrete state to change, whereas the continuous states before and after the switching are equal, i.e., $\mathit{x}\left({t}_{i}^{+}\right)=\mathit{x}\left({t}_{i}^{-}\right)$ and $\mathit{q}\left({t}_{i}^{+}\right)=\varphi (\mathit{x},\mathit{q},\mathit{v},{t}_{i}^{-})$. In the new discrete state, the continuous state trajectory follows different equation of motions than in the previous discrete state. In spacecraft systems, autonomous switching occurs, for example, when the electric engine is switched-off due to power availability constraints (e.g., the spacecraft crosses through the Earth-shadow or it is far from the Sun).
- Controlled switchings: Controlled switching differs from autonomous switching in that the discontinuity surface is not a function of the continuous state but it depends on the controls, i.e., $s=s(\mathit{v},t)$. Therefore, the discrete event occurs in the control-time space (see Figure 1b). Controlled switching models logical decisions that can be made at a desired point of time to change the system dynamics, e.g., switching-off the electric engine for propellant savings reasons.
- Autonomous impulses: An autonomous impulse resets the value of the continuous state, when the continuous state trajectory hits the discontinuity surface (see Figure 2a). In a similar fashion than autonomous switching, the discontinuity surface depends only on the continuous state and on time, i.e., $s=s(\mathit{x},t)$. However, after an autonomous impulse, the discrete state, and thus the differential equations, remains unchanged, whereas the continuous state jumps according to the transition maps function, i.e., $\mathit{x}\left({t}_{i}^{+}\right)=\varphi (\mathit{x},\mathit{q},\mathit{v},{t}_{i}^{-})$ and $\mathit{q}\left({t}_{i}^{+}\right)=\mathit{q}\left({t}_{i}^{-}\right)$. Examples for autonomous impulses in spacecraft dynamics are gravity assisted-maneuvers, since a discrete change is the heliocentric velocity is experienced when it encounters a planet in space and time.
- Controlled impulses: The difference of controlled impulses to autonomous ones is that the impulse is triggered by a discontinuity surface that depends on the controls, i.e., $s=s(\mathit{v},t)$. Similarly to controlled switchings, the event occurs in the control-time space (see Figure 2b). Incrementing the velocity of a spacecraft by an instantaneous firing of a chemical engine is an example of a controlled impulse.

#### 3.2. Problem Statement

## 4. Dynamical Modeling

#### 4.1. Continuous State Representation

- Cartesian State Vector (CSV): The most common model for describing a spacecraft trajectory refers to its position and velocity vectors. They are typically projected on an inertial Cartesian frame, such that ${\mathit{x}}_{CSV}=[{\mathrm{r}}_{\mathrm{x}},{\mathrm{r}}_{\mathrm{y}},{\mathrm{r}}_{\mathrm{z}},{\mathrm{v}}_{\mathrm{x}},{\mathrm{v}}_{\mathrm{y}},{\mathrm{v}}_{\mathrm{z}}]$. Here, $\left({\mathrm{r}}_{\mathrm{x}},{\mathrm{r}}_{\mathrm{y}},{\mathrm{r}}_{\mathrm{z}}\right)$ and $({\mathrm{v}}_{\mathrm{x}},{\mathrm{v}}_{\mathrm{y}},{\mathrm{v}}_{\mathrm{z}})$ are the projections of the position $\mathbf{r}\in {\mathbb{R}}^{3}$ vector, and of the velocity vector $\mathbf{v}\in {\mathbb{R}}^{3}$, respectively.
- Polar State Vector (PSV): They are mainly used for two-dimensional or planar representations of the problem dynamics. They consists on the following set: ${\mathit{x}}_{PSV}=(\mathrm{r},\theta ,\mathrm{v},\psi )$, where $\mathrm{r}$ is the distance to the central body, $\theta $ is the polar angle, $\mathrm{v}$ is the modulus of the velocity with respect to an inertial frame, and $\psi $ is the flight path angle.
- Classical Orbital Elements (COE): Another form of mathematical model to represent the spacecraft dynamics is in terms of classical orbit elements ${\mathit{x}}_{COE}=(a,e,i,\mathrm{\Omega},\omega ,M)$. They are named as the semimajor axis, eccentricity, inclination, right-ascension of the ascending node, argument of perigee, and mean anomaly, respectively. Instead of the true anomaly, the mean motion, the true anomaly or the eccentric anomaly can be used [18].
- Modified Equinoctial Elements (MEE): The other model for completely defining the state of the spacecraft is by the use of the set of modified equinoctial orbital elements ${\mathit{x}}_{MEE}=(p,f,g,h,k,L)$. Here, p is the semilatus rectum and L is named the true longitude. The elements $(f,g)$ are related to the projection of the eccentricity vector on the inertial frame, while $(h,k)$ are associated to the inclination of the orbit.

#### 4.2. Continuous Controls

- Blended Control (BC): The optimal thrust steering that maximize the variation (i.e., increase or decrease) of a set of orbital elemenst element independently or each other, ${\mathit{u}}_{x}\left(\mathit{x}\right)\in {\mathbb{R}}^{{n}_{x}}$ are computed as a function of the position in the orbit. They are commonly obtained analytically. Then, the complete control law to simultaneously modify all the elements of the state vector results from the following weighted sum:$${\mathit{u}}^{*}({W}_{x},t)=\sum {G}_{x}\left(t\right){\mathit{u}}_{x}\left(\mathit{x}\right)$$
- Calculus of Variations based (COV) The Pontryagin Minimum Principle (PMP) [20] is used to obtain the optimal control history. For a minimum-time continuous optimal control problem, the optimal thrust direction will have the following form:$${\mathit{u}}^{*}(\lambda ,t)=-{\displaystyle \frac{M\left(\mathit{x}\right)\lambda \left(t\right)}{\left|\right|M\left(\mathit{x}\right)\lambda \left(t\right)\left|\right|}}$$
- Lyapunov Control (LC): It defines an energy-like (i.e., a positive-definite) scalar Lyapunov function of the state $V(\Delta x\left(t\right),{W}_{x})\in \mathbb{R}$. Here, $\Delta x\left(t\right)=\mathit{x}\left(t\right)-{\mathit{x}}_{f}$, and ${\mathit{x}}_{f}$ is the target state. The set of constant parameters or static controls ${W}_{x}\in {\mathbb{R}}^{{n}_{x}}$ are to be determined as part of the solution. The Lyapunov function has to fulfill the following condition:$$\dot{V}\left({W}_{x}\right)={\nabla}_{x}V(\Delta x,{W}_{x})\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{f}(\mathit{x},\mathit{u})\le 0$$The thrust steering law is then obtained by minimizing the variation of $\dot{V}$ with respect to the control law (i.e., making it as negative as possible) as follows:$${\mathit{u}}^{*}(\mathit{z},t)=arg\underset{u}{min}{\nabla}_{x}V(\Delta x\left(t\right),{W}_{x})\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{f}(\mathit{x},\mathit{u})$$Notably, this control law naturally drives the spacecraft to the desired final state, avoiding the need to include the final boundary conditions in the problem.
- Shape-based Approaches (SB): In this approach, the state vector $\mathit{x}\left(t\right)$, usually the trajectory, is assumed to have a predefined form, e.g., $\mathit{x}=\mathit{x}(\mathit{z},t)$, where $\mathit{z}\in {\mathbb{R}}^{{n}_{z}}$ are the set of parameters to be determined. The control law is obtained by forcing the EOM to be satisfied:$${\mathit{u}}^{*}(\mathit{z},t):\dot{\mathit{x}}(\mathit{z},t)-\mathrm{f}(\mathit{x}(\mathit{z},t),{\mathit{u}}^{*},t)=0$$An analytical solution for the control is derived therefrom. Note that the obtained control may not satisfy the constrained related to the maximum thrust available. Thus it may lead to unfeasible trajectories. The solution may not fulfill the boundary constraints, thus they must be included as part of the problem.
- Neurocontroller (NC): The problem of finding an optimal strategy that leads to an optimal trajectory is thus transformed into the determination of the optimal network transfer function $N:\mathcal{X}\times {\mathbb{R}}^{{n}_{z}}\times \mathbb{R}\u27f6\mathcal{U}$. This function acts as a map from the current spacecraft state $\mathit{x}$, the desired final state ${\mathit{x}}_{f}$, and the network’s internal parameters $\mathit{z}\in {\mathbb{R}}^{{n}_{z}}$ to the instantaneous steering. Thus, it holds that:$${\mathit{u}}^{*}(\mathit{z},t)=N(\mathit{z},{\mathit{x}}_{f},\mathit{x},t)$$The controller parameters $\mathit{z}\in {\mathbb{R}}^{{n}_{z}}$ are to be determined as part of the solution.
- Finite Fourier Series (FFS). The low-thrust steering history is assumed to be represented by a Finite Fourier series expansion, such that:$${\mathit{u}}^{*}({a}_{k},{b}_{k},t)=\sum _{k=0}{a}_{k}\left(t\right)cos\left({\displaystyle \frac{2\pi k\theta}{\Delta \theta}}\right)+{b}_{k}\left(t\right)cos\left({\displaystyle \frac{2\pi k\theta}{\Delta \theta}}\right)$$

#### 4.3. Discrete States

#### 4.4. Discrete Controls

#### 4.5. Continuous Dynamics

- Analytical solutions: Analytical techniques were at the origin of spacecraft trajectory optimization. They seek to obtain closed-forms solutions for the dynamical systems, such that the EOM do not need to be integrated.$$\dot{\mathit{x}}=\mathrm{f}(\mathit{x},\mathit{u},t)\u27f6\mathit{x}=\mathit{x}(\mathit{x},\mathit{u},t)$$These techniques are only available for special cases. Two well-known and widely used analytical solutions are the Kepler and Stark models. A graphical representation of these techniques along with the continuous model is represented in Figure 4.
- -
- Kepler Model (KM): It is a reduced model that uses pure Keplerian arcs connected at nodes with impulsive velocity vector discontinuities that approximate the effect of performing a low-thrust maneuver during the Keplerian arc.
- -
- Stark Model (SM): The Stark model yields exact closed-form solutions for a spacecraft in a two-body gravitational field subject to a thrust acceleration that is inertially constant in both magnitude and direction.

Additionally, analytical solutions can be derived under constant radial or tangential thrust without space perturbations, even including some environmental effects, such as the Earth oblateness. - Asymptotic solutions: The propulsive acceleration is considered as a perturbation effect acting on a well-known or unperturbed trajectory (e.g., a Keplerian orbit). Thus, the perturbed trajectory can be approximated as a series expansion:$$\dot{\mathit{x}}=\mathrm{f}(\mathit{x},\mathit{u},t)\u27f6\mathit{x}(\u03f5,t)\approx {\mathit{x}}_{0}\left(t\right)+\u03f5{\mathit{x}}_{1}\left(t\right)+\mathcal{O}\left({\u03f5}^{2}\right)$$
- Averaging techniques: The method of averaging consists in the elimination of high-frequency components from the EOM by averaging over a short time scale (typically the orbital period). The averaged equations contains only secular and long-periodic terms.$$\dot{\mathit{x}}=\mathrm{f}(\mathit{x},\mathit{u},t)\u27f6\dot{\overline{\mathit{x}}}={\displaystyle \frac{1}{\mathrm{T}}}{\int}_{t}^{t+\mathrm{T}}\mathrm{f}(\mathit{x}\left(t\right),\mathit{u}\left(t\right),t)dt$$

#### 4.6. Discrete Dynamics

#### 4.6.1. Flybys

#### 4.6.2. Engine on-off Switchings

## 5. Objective Functions

- Single-objective: The goal is to search for a solution in the feasible set that provides the minimum value of a scalar-valued function, i.e., ${n}_{j}=1$. In this case, a single-point solution, under mild regularity assumptions, is obtained. From a mathematical point of view, a feasible solution $({\mathit{u}}^{*},{\mathit{v}}^{*})$ is optimal if it satisfies the following condition:$$J({\mathit{u}}^{*},{\mathit{v}}^{*})\le J(\mathit{u},\mathit{v}),\phantom{\rule{1.em}{0ex}}\forall \mathit{u}\in \mathcal{U}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\forall \mathit{v}\in \mathcal{V}\phantom{\rule{1.em}{0ex}}$$
- Multiobjective: The aim is to minimize a vector-valued function formed by ${n}_{j}>1$ conflicting criteria, i.e., $J=[{J}_{1},{J}_{2},\dots ,{J}_{{n}_{j}}]$. The solution in the objective space typically consists of a $({n}_{j}-1)$-dimensional hypersurface [23] known as the Pareto-optimal set (Pareto-optimal set is also known as Pareto front, Pareto frontier, Pareto-efficient set or nondominated front.) [24]. A feasible solution $({\mathit{u}}^{*},{\mathit{v}}^{*})$ is weak Pareto-optimal if there does not exit another feasible solution $(\mathit{u},\mathit{v})$ that could improve all the objectives simultaneously such that:$${J}_{i}(\mathit{u},\mathit{v})\le {J}_{i}({\mathit{u}}^{*},{\mathit{v}}^{*}),\phantom{\rule{1.em}{0ex}}\forall i\in \{1,\dots ,{n}_{j}\}\phantom{\rule{1.em}{0ex}}\forall \mathit{u}\in \mathcal{U}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\forall \mathit{v}\in \mathcal{V}$$Otherwise, the point $({\mathit{u}}^{*},{\mathit{v}}^{*})$ is said to be dominated.

## 6. Approaches and Solutions for COCPs

#### 6.1. Indirect, Direct, and Dynamic Programming Approaches

- Indirect Approach: In the indirect approach, the goal is to solve the multipoint boundary value problem (MPBVP) that results from applying the PMP [20]. The PMP characterizes the first-order necessary conditions that an optimal solution must satisfy. its derivation involves the determination of the states and costates, which must obey the Euler-Lagrange equation. Notably, the minimum principle allow to obtain the continuous control as a function of the state and costate at each instant, explicitly or numerically. Furthermore, a set of additional constraints, namely transversality, and complementary conditions, must be satisfied [26].
- Direct Approach: The basic idea of direct methods is to transcribe the COCP into a nonlinear programming problem (NLP), where the objective function (Equation (3)) is “directly” optimized. The transcription process requires the discretization of the control variables in a time-grid. The goal of a NLP problem is to determine a vector of unknown decision variables that comply with a set of nonlinear constraints, including equality and inequality restrictions. An optimal solution to the NLP problem has to fulfill first-order necessary optimality conditions. These conditions are known as the Karush-Kuhn-Tucker conditions (KKT) [27,28]. The NLP is then numerically solved using well-known optimization techniques [13].
- Dynamic Programming Approach: The method of Dynamic Programming is based on the Bellman’s principle of optimality [29]: “An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” Even though Dynamic Programming was originally developed for discrete-time systems, it was extended to continuous-time problems. The continuous-time equivalent of the Bellman’s principle resulted in the Hamilton-Jacobi-Bellman (HJB) theorem [30]. In this case, a set of partial differential equations must be solved first.

- Single shooting: The trajectory is integrated using time-marching methods from ${t}_{0}$ upon reaching the final time ${t}_{f}$. In this case, the initial state (and costates) are unknowns to be determined, and boundary constraints are imposed at the end of the integration.
- Multiple shooting: The time interval $[{t}_{0},{t}_{f}]$ is broken up into $N+1$ subintervals. The trajectory is integrated over each subinterval $[{t}_{i},{t}_{i+1}]$ with the initial values of the state (and adjoints) at each subinterval being unknowns that need to be determined. Additionally, continuity conditions have to be enforced at the interface of each subinterval.
- Collocation: The states (and costates) are discretized over a predefined time-grid, such that they are known only at discrete points. The system-governing equations are transformed into discrete defect constraints, which relate the values at the beginning of the subinterval to the values at the end. Different methods are characterized by the choice of quadrature rule to approximate the differential equations between each two subintervals: local and global collocation methods.

#### 6.2. Gradient-Based, Heuristic, and Hybrid Solutions

- Gradient-based: In a gradient-based method, an initial guess is made of the unknown decision vector $\mathit{z}$. At the ${k}^{th}$ iteration, a search direction ${\mathit{p}}_{k}$, and a step length ${\alpha}_{k}$, are determined. The search direction provides a direction in ${\mathbb{R}}^{{n}_{z}}$ along which to change the current value ${\mathit{z}}_{k}$, while the step length provides the magnitude of the change. The update from ${\mathit{z}}_{k}$ to ${\mathit{z}}_{k+1}$ has the form: ${\mathit{z}}_{k+1}={\mathit{z}}_{k}+{\alpha}_{k}{\mathit{p}}_{k}$. The iterations proceed until the KKT conditions are met. To compute the search direction, these methods require the user provide information for the gradient of the constraint and the objective function (if necessary). The most widely used methods are classified as sequential quadratic problems (e.g., SNOPT, NPSOL) or interior point methods (e.g., IPOPT, KNITRO). Extensive information about their implementations can be found in Refs. [34,35], respectively.
- Heuristic: The search is performed in a stochastic/metaheuristic manner without requiring gradient information. The most known class of heuristics are evolutionary algorithms. They start by generating a set of candidate solutions or individuals ${\mathit{z}}_{i,0}$ for $i=1,\dots ,n$, termed population. Thereafter, the population is iteratively modified by applying a set of stochastic rules $\Pi :\mathcal{Z}\u27f6\mathcal{Z}$, which may incorporate random processes, such that the population at ${(k+1)}^{th}$ iteration is computed as ${\mathit{z}}_{i,k+1}=\Pi \left({\mathit{z}}_{i,k}\right)$, and the iterations proceed until a stopping criteria is met (e.g., max number of iterations). The candidate with the lowest cost is deemed as the solution to the problem. Well known stochastic rules are genetic algorithms (GA) [36], which emulate evolutionary processes in genetics, and particle swarm optimization (PSO) [37], which is based on the idea of swarms of animals.
- Hybrid: Hybrid approaches combine a set of rules exploiting gradient-information and a set of rules based on heuristics searches to iteratively operate over a solution or a set of candidate solutions. Gradient-information is exploited to drive the constraints to zero, while heuristic rules are applied to efficiently explore large design domains or to manage integer variables. They are typically combined on a two-loop approach. The heuristic solver operates over a subset of decision variables in the outer loop. In the inner loop, the remaining subset of design parameters are optimized with the gradient-based method.

#### 6.3. Discussion

## 7. Approaches and Solutions for HOCPs

## 8. Existing Low-Thrust Optimization Tools

#### 8.1. Analytical Solutions

#### 8.2. Indirect Methods

#### 8.3. Direct Methods

#### 8.4. Predefined Control Laws

#### 8.5. Dynamic Programming Methods

## 9. Conclusions

- Optimize alternative objectives: it has been seen that typically, either propellant mass or time-of-flight are optimized. However, mission designer may be interested into minimizing the radiation absorbed during the passage through the Van-Allen radiation belts to reduce the damage into the solar panel, or into minimizing the time-spent in eclipse. Additionally, when including spacecraft design along with the trajectory optimization, other performance indexes, such as spacecraft total mass or target on-station mass may have to be included.
- Reduce computational time: among the presented tools, GA-EMTG is able to automatically find the sequence of gravity assists for an interplanetary mission with respect to multiple-objectives, requiring minimal user-interaction, and providing medium fidelity solutions. However, computational times range from several hours to days. Therefore, faster assessments at the cost of fidelity and optimality are desirable.
- Extend the capability of preliminary design tools to include mission constraints: low-thrust trajectory optimization tools used for the preliminary design due to their speed, such as implementing predefined control laws, do not have the ability to impose important mission constraints, which may imply that the obtained trajectory is not feasible. Thus, advancing into the incorporation of constraints into such tools, either by a penalty function or by a different predefined control law, will significantly enhance the success during the preliminary design.
- Increase the efficiency of searching over wider design spaces: presented hybrid and heuristic tools are able to work for a limited combinatorial complexity of the problem. However, they are not well-suited for solving problems such as asteroid tours, debris-removal missions, or asteroid mining mission, where the are thousands of available options. Improving the capability of searching over this broad spaces will enable the of more ambitious low-thrust missions. A potential approach would be to develop dedicated heuristic algorithms able to efficiently optimize over large sequences os visited bodies (e.g., asteroids, debris), possibly incorporating artificial intelligence into the approach.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Name | Ref | Company/Org./Author | Approach | Solution | Obj. | Dynamics | States | Transfers |
---|---|---|---|---|---|---|---|---|

VARITOP | [42] | JPL | Single Shooting | GB | SO | PR-TBP | CSV | IT |

SEPTOP | [43] | JPL | Single Shooting | GB | SO | PR-TBP | CSV | IT |

NEWSEP | [44] | JPL | Single Shooting | GB | SO | PR-TBP | CSV | IT |

SAIL | [45] | JPL | Single Shooting | GB | SO | PR-TBP | CSV | IT |

HILTOP | [46] | Space Flight Sol. | Single Shooting | GB | SO | PR-TBP | CSV | IT |

ETOPH | [47] | CNES | Single Shooting | GB | SO | PR-TBP | CSV | IT |

ITOP | [48] | Aerospace Corp. | Single Shooting | GB | SO | PR-TBP | MEE | PC |

LT20 | [49] | Milano Univ. | Single Shooting | GB | SO | PR-TBP | MEE | PC |

Tfmin | [50] | CNES | Single Shooting | GB | SO | PR-TBP | COE | PC |

- | [51] | Kéchichian | Single Shooting | GB | SO | PR-TBP | MEE | PC |

T-3D | [52] | Thales | Single Shooting | GB | SO | PR-TBP + AVG | MEE | G |

SOFTT | [53] | Thales | Single Shooting | GB | SO | PR-TBP + AVG | - | PC |

ELECTRO | [54] | OHB | Single Shooting | GB | SO | PR-TBP + AVG | MEE | PC |

MIPELEC | [55] | CNES | Single Shooting | GB | SO | PR-TBP + AVG | MEE | PC |

SEPSPOT | [56] | NASA | Single Shooting | GB | SO | PR-TBP + AVG | MEE | PC |

GA-SEPTOP | [57] | JPL | Single Shooting | HY | MO | PR-TBP | CSV | IT |

LOTTO | [58] | SES Engineering | Single Shooting | GB | SO | PR-TBP | MEE | PC |

- | [59] | Torino Univ. | Single Shooting | HS | SO | PR-TBP | CSV | IT |

- | [60] | Pontani et al. | Single Shooting | HS | SO | PR-TBP | PSV | IT |

- | [61] | Lee et al. | Single Shooting | HS | MO | PR-TBP | CSV | IT |

BNDSCO | [62] | Hamburg. Univ | Multiple Shooting | HS | SO | - | - | G |

LOTNAV | [63] | Deimos Space | Multiple-shooting | GB | SO | CSV | PR-NBP | IT |

- | [64] | Meng et al. | Multiple-Shooting | GB | SO | PR-TBP | MEE | PC |

- | [65] | Olympio | Gradient method | - | SO | PR-NBP | PSV | G |

Name | Ref | Company/Org./Author | Approach | Solution | Obj. | Dynamics | States | Transfers |
---|---|---|---|---|---|---|---|---|

ASTOP | [66] | Space Flight Solutions | Single Shooting | GB | SO | PR-NBP | CSV | IT |

COPERNICUS | [67] | Texas Univ., JSC | Multiple Shooting | GB | SO | PR-NBP | CSV | G |

jTOP | [68] | Tokio Univ., JAXA | Multiple Shooting | GB | SO | PR-NBP | CSV | G |

DITAN | [69] | ESA, Milano Univ. | Collocation | GB | SO | PR-NBP | CSV | G |

MODHOC | [70] | Strathclyde Univ. | Collocation | HY | MO | PR-NBP | CSV | G |

DIRETTO | [71] | Milano Univ. | Collocation | GB | SO | PR-NBP | CSV | G |

MAVERICK | [72] | Colorado Boulder Univ. | Collocation | GB | SO | PR-NBP | CSV | G |

MColl | [73] | NASA. | Collocation | GB | SO | PR-NBP | CSV | G |

COLT | [74] | Purdue Univ. | Collocation | GB | SO | PR-NBP | CSV | G |

GMAT | [75] | NASA | Collocation | GB | SO | - | - | G |

STK | [76] | AGI | Collocation | GB | SO | - | - | G |

OTIS | [77] | GCR, Boeing | Collocation | GB | SO | - | - | G |

POST | [78] | NASA | Single Shooting | GB | SO | - | - | G |

SOCS | [79] | Boeing | Collocation | GB | SO | - | - | G |

DIDO | [80] | TOMLAB | Collocation | GB | SO | - | - | G |

GPOPS | [81] | Univ. of Florida | Collocation | GB | SO | - | - | G |

OPTELEC | [82] | Airbus | Multiple Shooting | GB | SO | PR-TBP | MEE | PC |

MANTRA | [83] | ESA | Multiple-shooting | GB | SO | PR-NBP | CSV | G |

LOTOS | [84] | ASTOS Solutions | Collocation | GB | SO | PR-TBP | MEE | PC |

XIPSTOP | [85] | Boeing | Collocation | GB | SO | PR-TBP | MEE | PC |

GALLOP | [86] | JPL, Purdue Univ. | Multiple-Shooting | GB | SO | KM | CSV | IT |

COLTT | [87] | Colorado Boulder | Multiple-Shooting | GB | SO | KM | CSV | IT |

LInX | [88] | J.H. Univ., Nabla Zero | Multiple-Shooting | GB | SO | KM | CSV | IT |

BOLTT | [89] | Colorado Boulder | Multiple-Shooting | GB | SO | KM | CSV | IT |

MALTO | [90] | JPL | Multiple-Shooting | GB | SO | KM | CSV | IT |

EMTG | [91] | GSFC, Illinois Univ. | Multiple-Shooting | HY | MO | KM | CSV | IT |

PaGMO | [92] | ESA | Multiple-Shooting | HY | SO | KM | CSV | IT |

GA-GALLOP | [93] | Purdue Univ. | Multiple-Shooting | HY | MO | KM | CSV | IT |

- | [94] | Zuiani et al. | Multiple-Shooting | GB | SO | SM | CSV | IT |

DIFINC | [95] | Coverstone et al. | Differential Inclusion | GB | SO | PR-TBP | CSV | IT |

- | [96] | Gerald et al. | Single Shooting | HS | SO | PR-TBP | PSV | IT |

- | [97] | Pontani et al. | Single Shooting | HS | SO | PR-TBP | PSV | IT |

**Table 3.**Representative Tools Implementing Direct Methods with Predefined Control laws for Low-Thrust Trajectory Optimization.

Name | Ref | Company/Org./Author | Approach | Solution | Obj. | Dynamics | States | Transfers |
---|---|---|---|---|---|---|---|---|

HYTOP | [109] | Aerospace Corp. | Blended Control | GB | SO | PR-TBP | MEE | PC |

- | [110] | Yang Gao | Blended Control | GB | SO | PR-TBP + AN + AVG | COE | PC |

- | [111] | Yang Gao | COV-Based | GB | SO | PR-TBP + AVG | MEE | PC |

- | [112] | Strathclyde Univ | Blended Control | HY | MO | SM + AVG | COE | PC |

SEPDOC | [113] | Kluever et al. | Blended Control | GB | SO | PR-TBP + AVG | COE | PC |

- | [114] | Hudson et al. | Fourier-Expansion | GB | SO | PR-TBP + AN + AVG | COE | PC |

- | [115] | Chang et al. | Lyapunov Control | GB | SO | PR-TBP | CSV | PC |

LATOP | [116] | ESA | Lyapunov Control | HS | MO | PR-TBP | MEE | PC |

GA-Q-Law | [117] | JPL | Lyapunov Control | HS | MO | PR-TBP | MEE | PC |

STOUR-LTGA | [118] | JPL, Purdue Univ. | Shape-based | HS | SO | PR-TBP + AN | PSV | IT |

IMAGO | [119] | Pascale et al. | Shape-based | HS | SO | PR-TBP + AN | MEE | IT |

- | [120] | Wall et al. | Shape-based | HS | SO | PR-TBP + AN | PSV | IT |

- | [121] | Taheri et al. | Shape-based | HS | SO | PR-TBP + AN | PSV3 | IT |

- | [122] | Gondelach et al. | Shape-based | HS | SO | PR-TBP + AN | PSV3 | IT |

- | [123] | Roa et al. | Shape-based | HS | SO | PR-TBP + AN | PSV | IT |

MOLTO-IT | [22] | Morante et al. | Shape-based | HY | MO | PR-TBP + AN | PSV | IT |

MOLTO-OR | [124] | Morante et al. | Lyapunov Control | HS | MO | PR-TBP | MEE | PC |

InTrance-GA | [125] | DLR | Neural control | HY | SO | PR-TBP | CSV | IT |

**Table 4.**Representative Tools Implementing Dynamic Programming for Low-Thrust Trajectory Optimization.

Name | Ref | Company/Org./Author | Approach | Solution | Obj. | Dynamics | States | Transfers |
---|---|---|---|---|---|---|---|---|

MYSTIC | [126] | NASA | DDP | - | SO | PR-NBP | CSV | G |

- | [127] | Colorado Boulder Univ. | DDP | - | SO | PR-TBP | MEE | PC |

HDDP | [128] | Lantoine et al. | HDDP | - | SO | SM/KM | CSV | G |

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## Share and Cite

**MDPI and ACS Style**

Morante, D.; Sanjurjo Rivo, M.; Soler, M.
A Survey on Low-Thrust Trajectory Optimization Approaches. *Aerospace* **2021**, *8*, 88.
https://doi.org/10.3390/aerospace8030088

**AMA Style**

Morante D, Sanjurjo Rivo M, Soler M.
A Survey on Low-Thrust Trajectory Optimization Approaches. *Aerospace*. 2021; 8(3):88.
https://doi.org/10.3390/aerospace8030088

**Chicago/Turabian Style**

Morante, David, Manuel Sanjurjo Rivo, and Manuel Soler.
2021. "A Survey on Low-Thrust Trajectory Optimization Approaches" *Aerospace* 8, no. 3: 88.
https://doi.org/10.3390/aerospace8030088