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Astrodynamics & Spacecraft Trajectory Design

Astrodynamics (orbital mechanics) is a field that studies the motion of objects (including satellites, space probes, rockets, comets and asteroids). While characterizing the motion of objects is an important task, our efforts are focused on space mission design and control. Specifically, we are interested in determining "efficient" orbital maneuvers to transfer spacecraft from an initial orbit to a desired orbit. In general, there are multiple goals in space missions, which may include establishing an orbit around a target body, performing an orbit raising maneuver or achieve a rendezvous status with an object (e.g., a spacecraft in another orbit). Designing transfer trajectories is a common aspect of all these problems and our research is focused on developing dependable algorithms for designing efficient maneuvers. 

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Over the past six decades, significant technological developments have been made that drastically improve, and arguably, revolutionize the capacity of space probes to accomplish missions beyond what had been possible using earlier technologies. Specifically, the recent breakthroughs that have taken place in advanced Solar Electric Propulsion (SEP) systems are noticeable in reducing launch mass and enabling inexpensive missions. Electric engines have desirable efficiencies for space mission design. They have high specific impulse values, but they generate significantly small thrust values. This makes the task of trajectory design quite challenging. The above figure is representative low-thrust trajectory from earth to asteroid Dionysus. The orbit of the asteroid has a high eccentricity and its orbit is highly inclined with respect to the ecliptic. When an electric propulsion system (electric thruster) is used to design a fuel optimal trajectory, the resulting trajectory will consist of multiple revolutions around the sun. In addition, for fuel-optimal trajectories (namely, those trajectory that consume the least amount of propellant), it turns out that the optimal operation logic for a thruster is not to always operate. Instead, the trajectory consists of ballistic and maximum thrust arcs.

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Our research uses both optimal control theory and enhanced smoothing methods to regularize space trajectory optimization problems and make them more amenable to numerical treatment. The resulting trajectories may consist of planet-centric and interplanetary phases (or even gravity-assisst maneuvers). In addition, the overall structure of the resulting trajectories is predominantly governed by the type of propulsion system. There are several types of propulsion systems including chemical rocket propulsion, electric propulsion and solar-sail systems. We are developing enhanced methods (based on optimal control theory) to design low-thrust and impulsive trajectories. For instance, above figure depicts a trajectory that makes five revolutions around the sun. The red arrows show the regions in space where the engine is operating at its full capacity. The magnitude of the vector is constant, but its orientation changes in time to guide the spacecraft to its target point (which is an asteroid). This particular trajectory takes nearly 9.7 years. Designing these trajectories is not trivial. Therefore, we adopt a "break and conquer" paradigm by formulating and solving a simpler problem. Then, we increase the complexity of the problem in a successive manner such that we obtain the solution to the actual problem. For instance, figure below shows the time history of the thrust profile for different values of a smoothing parameter that allows us to solve the original fuel-optimal trajectory design problem from the Earth to asteroid Dionysus in a significantly simpler manner. In this case, we start by setting a large value (say 1 in this case) to a smoothing parameter. Then, we reduce the value of the smoothing parameter down to 0 and we recover the original bang-off-bang thrust profile, which is the optimal solution for this problem.

ED_Thrust_Continuation.jpg

Smooth Incorporation of Shadow Constraints

The methods that we have proposed in our research allows us to incorporate shadow constraints within the indirect formulation of optimal control theory using a recently introduced hyperbolic tangent smoothing (HTS) method. Using the HTS method, boundary-value problems are converted into two-point boundary-value problems. In addition, the domain of convergence of the resulting two-point boundary-value problems enlarges significantly such that we are able to solve these challenging problems more efficiently. Below is a demonstration for the application of the method for an orbit raising transfer trajectory design from a geostationary transfer orbit (GTO) to a geostationary earth orbit (GEO). In this problem, a cylindrical Earth shadow model is considered and the Sun is assumed to be located at a far distance on the positive side of the x axis. The spacecraft makes eight revolutions around the Earth to increase its energy and to eventually merge into the GEO orbit. As the spacecraft enters the shadow regions, its solar arrays are no longer able to generate power and its thrust value drops to zero. As the spacecraft exits the shadow region, the solar powers are able to generate power and the spacecraft uses the full capacity of its engine to raise the orbit. There are additional constraints that have to be considered during the trajectory design and our research is focused on developing algorithms that allow us to solve these challenging problems.

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Selected Publications

  1. Taheri, Ehsan, Ilya Kolmanovsky, and Ella Atkins. "Enhanced smoothing technique for indirect optimization of minimum-fuel low-thrust trajectories." Journal of Guidance, Control, and Dynamics (2016): 2500-2511. https://arc.aiaa.org/doi/full/10.2514/1.G000379

  2. Taheri, Ehsan, and John L. Junkins. "Generic Smoothing for Optimal Bang-Off-Bang Spacecraft Maneuvers." Journal of Guidance, Control, and Dynamics 41.11 (2018): 2470-2475. https://arc.aiaa.org/doi/full/10.2514/1.G003604

  3. Junkins, John L. and Ehsan Taheri, "Exploration of Alternative State Vector Choices for Low-Thrust Trajectory Optimization." Journal of Guidance, Control, and Dynamics 42.1 (2018): 47-64.​ https://arc.aiaa.org/doi/full/10.2514/1.G003686

  4. Taheri, Ehsan, Ella M. Atkins, and Ilya Kolmanovsky. "Performance Comparison of Smoothing Functions for Indirect Optimization of Minimum-Fuel Low-Thrust Trajectories." 2018 Space Flight Mechanics Meeting. 2018. https://arc.aiaa.org/doi/pdf/10.2514/6.2018-0214

  5. Woollands, Robyn, Ehsan Taheri, and John L. Junkins. "Efficient Computation of Optimal Low Thrust Gravity Perturbed Orbit Transfers." The Journal of the Astronautical Sciences (2019): 1-27. https://link.springer.com/article/10.1007/s40295-019-00152-9

  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://www.sciencedirect.com/science/article/pii/S0273117717301072

  7. Shimoun, Jared, Ehsan Taheri, et al. "A Study on GPU-Enabled Lambert's Problem Solution for Space Targeting Missions." 2018 Annual American Control Conference (ACC). IEEE, 2018.​ https://ieeexplore.ieee.org/abstract/document/8431478

  8. Taheri, Ehsan, et al. "On Satellite Orbit Decay Compensation in Low Earth Orbits." 2018 Space Flight Mechanics Meeting. 2018. https://arc.aiaa.org/doi/pdf/10.2514/6.2018-2226

  9. Wu, Di and Ehsan Taheri. "Complex Interplanetary Trajectories Design with Low-Thrust Based Motion Primitives." 2018 Space Flight Mechanics Meeting. 2018. https://arc.aiaa.org/doi/pdf/10.2514/6.2018-0215

  10. Robyn Woollands and Ehsan Taheri, "Efficient Computation of Optimal Low Thrust Perturbed Orbit Transfers with Shadow Constraints," 2019 AAS/AIAA Astrodynamics Specialist Conference, Portland, ME, August 11-15, 2019, AAS 19-688.

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