top of page

Shape-Based Methods for Low-Thrust Trajectory Design

This research thrust investigates the utility of shape-based methods for trajectory optimization of dynamical systems. In particular, they have been used to generate low-thrust spacecraft trajectories rapidly. Shape-based methods are among the popular techniques for rapid generation of feasible trajectories. These methods fall under the category of "inverse dynamics" methods.

EM-3DFFS300600_Traj3D.jpg
EM-3DFFS300600Th.jpg

Above figures depict a representative trajectory for a low-thrust transfer from earth to mars. This particular trajectory corresponds to 600 days of flight from the departure instant until the spacecraft reaches within the sphere of influence of mars. Note that "AU" stands for astronomical unit which is the distance from the Earth to the Sun. The thrust profile of the shape-based trajectory is also compared against the thrust profile of a fuel-optimal solution for the same boundary conditions and mission parameters. The optimal thrust profile exhibits the well-known bang-off-bang structure (three thrust arcs and three coast arcs). The difference between the propellant consumed along these two trajectories is negligible. 

​

Below figures depict two representative trajectories for a low-thrust transfer from low-Earth orbits (LEOs) to Low Moon orbits (LMOs) generated using the Finite Fourier Series (FFS) shape-based method. Generation of these trajectories is a challenging task since at each instant of time the spacecraft is subject to the simultaneous gravitational attraction of earth and moon, which leads to a more non-linear dynamical model. Depending on the energy of the initial and final orbits and characteristics of the propulsion systems, the resulting trajectories may consist of many revolutions around the primaries (namely, earth and moon). Typically, a trajectory is made up of three phases: a escape phase, an intermediate phase and a capture phase all of which are denoted in the following figures. Note that these trajectories are plotted in a synodica rotating frame. In the left plot, "DU" is the distance unit that is the distance between earth and moon (384400 km). In the right plot, the distances are expressed in terms of the radius of the earth (Re). 

ECRF_traj_C1_optimal.jpg
Traj_C4_Color.jpg

The standard approach to formulate and solve any optimization problem is to determine the time history of the control input(s) to achieve the best performance with respect to a prescribed goal. This is usually achieved via two broad classes of optimization methods: direct and indirect methods. However, there are many scenarios for mission design (e.g., the number of planets with which a gravity-assist maneuvers has to be performed). In the preliminary phase of mission design, many scenarios have to be assessed. Solving actual optimal control problems is a computationally demanding task. This need has provided the motivation for developing algorithms that not only can be used for rapid analysis purposes, but, their solutions can be used as "high-quality" initial guesses for high-fidelity design optimization tools. Development of the shape-based methods is a response to the above need. 

​

The core idea, in shape-based methods, is to first approximate (or assume a particular shape for) the time history of the states of the system (using some basis functions). Usually, position coordinates are approximated. However, it is possible to approximate velocity too. Then, appropriate derivatives of these approximate expressions are taken (with respect to time or any other prescribed independent variable). The resulting expressions are enforced to the equations of motions. It is possible to evaluate the acceleration (or force) needed to realize the resulting trajectories. This means that the considered approximations are true at any point in time, but optimality has been traded-off for speed. There are additional items that have to be taken into account. For instance, 1) What are the best approximating functions? 2) How should boundary conditions be handled? Researchers are looking for innovative ideas to address these and many similar problems, and as is always the case in life, there is no universal all-enabling method!

​

This research subject is focused on using Fourier series as the basis functions for approximating the trajectories. Unlike all of the other shape-based methods, we do not assume a fixed functional form for the states of the system, and instead, a flexible shape is considered, which can be altered based on the nature of the problem. For the majority of the considered problems, a truncated series (of Fourier terms) is shown to be sufficient for most of the preliminary analyses. Formulation of the Finite Fourier Series (FFS) shape-based method has evolved over time and its utility has been demonstrated on a number of problems, in particular, spacecraft low-thrust trajectory design. 

​

In all of the existing shape-based methods, constraints due to the propulsion system are treated in terms of acceleration level and not at the thrust level; the two types of constraints are not equivalent since the spacecraft mass varies dynamically. It is shown that FFS shape-based method is capable of handling thrust-level constraints. This feature can greatly facilitate low-thrust mission designs where various trajectory alternatives have to be assessed rapidly. 

Selected Publications

  1. Ehsan Taheri and Abdelkhalik, Ossama. "Shape based approximation of constrained low-thrust space trajectories using Fourier series." Journal of Spacecraft and Rockets 49.3 (2012): 535-546. https://doi.org/10.2514/1.58789

  2. Elsan Taheri and Abdelkhalik, Ossama. "Approximate on-off low-thrust space trajectories using Fourier series." Journal of Spacecraft and Rockets 49.5 (2012): 962-965. https://doi.org/10.2514/1.A32307

  3. Taheri, Ehsan, and Ossama Abdelkhalik. "Fast initial trajectory design for low-thrust restricted-three-body problems." Journal of Guidance, Control, and Dynamics 38.11 (2015): 2146-2160. https://doi.org/10.2514/1.G000878

  4. Taheri, Ehsan, and Ossama Abdelkhalik. "Initial three-dimensional low-thrust trajectory design." Advances in Space Research 57.3 (2016): 889-903. https://doi.org/10.1016/j.asr.2015.11.034

  5. Taheri, Ehsan, Ilya Kolmanovsky, and Ella Atkins. "Shaping low-thrust trajectories with thrust-handling feature." Advances in Space Research 61.3 (2018): 879-890. https://doi.org/10.1016/j.asr.2017.11.006

  6. Taheri, Ehsan, Nan I. Li, and Ilya Kolmanovsky. "Co-state initialization for the minimum-time low-thrust trajectory optimization." Advances in Space Research 59.9 (2017): 2360-2373. https://doi.org/10.1016/j.asr.2017.02.010

bottom of page