"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, alongside faculty adviser Robyn Woollands, validated their algorithm by simulating swarms of two, three, or four CubeSats transporting modular components between a servicing vehicle and a telescope undergoing repairs in orbit.

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

One of the greatest challenges in this research is the vast scale of space distances. For instance, the James Webb Space Telescope orbits approximately 1.5 million kilometers away at the Sun-Earth Lagrange Point 2 (L2). This location provides gravitational equilibrium, making it an ideal position for deep-space observatories.

"Without getting too technical, we used indirect optimization methods to guarantee that the output solution is fuel optimal. Direct methods do not guarantee that."

Bommena explained that their approach incorporates anti-collision constraints directly into the optimal control formulation, ensuring safe maneuvering at all times.

"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 advancement from their research is 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."

The project, which took approximately 18 months, saw a breakthrough moment for Bommena during a long-haul 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."

While the primary goal of this research is to improve safety and efficiency in space servicing and assembly, Bommena emphasized that their methodology has broader applications in trajectory optimization under various constraints.

Research Report:Indirect Trajectory Optimization with Path Constraints for Multi-Agent Proximity Operations

Related Links
University of Illinois Grainger College of Engineering
Microsat News and Nanosat News at SpaceMart.com