To make space servicing safer and more efficient, researchers created a fuel-efficient, collision-free trajectory optimization method for CubeSats.
Their algorithm allows small spacecraft to work together in assembling or repairing space telescopes, overcoming deep-space distance challenges through advanced mathematical modeling. A critical breakthrough in their research occurred mid-flight when a persistent numerical issue was finally resolved. Beyond space, this methodology has broad applications for other trajectory planning problems.
Paving the Way for Spacecraft Servicing
As more satellites, telescopes, and spacecraft are designed for in-orbit repairs, ensuring service spacecraft can reach them safely is crucial. Researchers at the University of Illinois Urbana-Champaign’s Department of Aerospace Engineering are developing a method that enables multiple CubeSats to work together to assemble or repair space telescopes. Their approach minimizes fuel use, ensures the CubeSats stay at least five meters apart to prevent collisions, and can even be applied to non-space-related navigation problems.
Ensuring Safe and Efficient CubeSat Operations
“We developed a scheme that allows the CubeSats to operate efficiently without colliding,” said aerospace Ph.D. student Ruthvik Bommena. “These small spacecrafts have limited onboard computation capabilities, so these trajectories are precomputed by mission design engineers.”
Bommena and his faculty advisor, Robyn Woollands, tested the algorithm by simulating swarms of two, three, or four CubeSats transporting modular components between a servicing vehicle and a space telescope undergoing repairs.
“These are difficult trajectories to compute and calculate, but we came up with a novel technique that guarantees its optimality,” Bommena said.
Overcoming the Challenges of Deep-Space Distances
Bommena said the most difficult aspect is the scale of the distances. The James Webb Space Telescope’s orbit is about 1.5 million kilometers away, at the Sun-Earth Lagrange Point 2. It’s where the gravitational force of the Sun and Earth balance each other, making it the perfect place in space for deep-space observation satellites to maintain orbit while facing away from the Sun.
“Without getting too technical, we used indirect optimization methods to guarantee that the output solution is fuel optimal. Direct methods do not guarantee that.”
“We also incorporated the anti-collision path inequality constraints into the optimal control formulation as a hard constraint, so the spacecraft do not violate the constraint at any point during the trajectory.”
Breaking the Complexity Barrier with Single-Arc Trajectories
Bommena explained that traditional direct or indirect methods with constraints, such as collision avoidance, break the trajectory into multiple arcs, increasing the complexity exponentially.
“Our methodology allows the trajectories to be solved as single arcs. We are just going from the starting point directly to the destination point. It’s more fuel optimal and more computationally efficient.”
Developing a New Model for Deep-Space Navigation
Another major outcome of the research is the development of a novel target-relative circular restricted three-body problem dynamical model.
“We needed to mitigate the numerical challenges that come from the large distance between the Sun and the Earth,” Bommena said. “To do that, we first shifted the center of the frame along the x-axis from the Sun-Earth barycenter to the location of Lagrange point L2 and then derived the equations of motion relative to the target spacecraft. We also introduced a new distance unit by applying a scaling factor that proportionally adjusts in relation to the original distance measurement.”
A Breakthrough at 30,000 Feet
Bommena said he and Woollands worked on this project for about a year and a half. His breakthrough came on a long-distance flight.
“The math was working on paper. The major problem we had was wrestling with numerics. I was coding during a long flight. I tried a couple of things and suddenly the solution converged. At first, I didn’t believe it. That was a very exciting moment and the next few days felt awesome.”
Beyond Space: Versatile Applications of the Methodology
Bommena said although the application for this work is to make in-space servicing and assembly safer and more efficient, the methodology they developed is very versatile and can be used in other trajectory optimization scenarios with different constraints.
Reference: “Indirect Trajectory Optimization with Path Constraints for Multi-Agent Proximity Operations” by Ruthvik Bommena, and Robyn Woollands, 4 December 2024, The Journal of the Astronautical Sciences.
DOI: 10.1007/s40295-024-00470-7
This work was partially supported by Ten One Aerospace through a NASA STTR Phase I research grant.
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