As more satellites, telescopes, and other spacecraft are built to be repairable, it will take reliable trajectories for service spacecraft to reach them safely. Researchers in the Department of Aerospace Engineering in The Grainger College of Engineering, University of Illinois Urbana-Champaign are developing a methodology that will allow multiple CubeSats to act as servicing agents to assemble or repair a space telescope. Their method minimizes fuel consumption, guarantees that servicing agents never come closer to each other than 5 meters, and can be used to solve pathway guidance problems that aren’t space related.

“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 adviser Robyn Woollands demonstrated the performance of the algorithm by simulating two, three or four vehicle swarms simultaneously transporting modular components between a service vehicle and a space telescope undergoing in-space servicing.

“These are difficult trajectories to compute and calculate, but we came up with a novel technique that guarantees its optimality,” Bommena said.

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.”

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.”

Another major outcome from 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.”

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.”

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.

This work was partially supported by Ten One Aerospace through a NASA STTR Phase I research grant.

The study, “Indirect Trajectory Optimization with Path Constraints for Multi-Agent Proximity Operations,” was written by Ruthvik Bommena and Robyn Woollands. It is published in The Journal of the Astronautical Sciences. DOI: 10.1007/s40295-024-00470-7