Convex Maneuver Planning for Spacecraft Collision Avoidance

TL;DR

This work tackles autonomous collision avoidance for short-term conjunctions in dense LEO by recasting the nonconvex PoC-constrained maneuver design as a convex semidefinite program (SDP) using Shor's relaxation. By linearizing dynamics around a reference trajectory and introducing moment matrices, the PoC and thrust constraints become linear in the SDP, yielding globally optimal, minimum-energy low-thrust maneuvers when the relaxation is tight. The authors empirically demonstrate tightness (rank-1 moment matrices) across multiple examples and compare against nonlinear programming and SCP-based methods, showing robust performance and improved reliability without reliance on good initial guesses. When thrust or time limitations prevent meeting the target PoC, a penalty-based contingency minimizes collision risk while controlling control effort. The approach is validated with high-fidelity LEO simulations and a CDM, signaling a practical step toward fully autonomous, scalable space traffic management, with future work to incorporate true nonlinear dynamics and extend to other mission-design settings.

Abstract

Conjunction analysis and maneuver planning for spacecraft collision avoidance remains a manual and time-consuming process, typically involving repeated forward simulations of hand-designed maneuvers. With the growing density of satellites in low-Earth orbit (LEO), autonomy is becoming essential for efficiently evaluating and mitigating collisions. In this work, we present an algorithm to design low-thrust collision-avoidance maneuvers for short-term conjunction events. We first formulate the problem as a nonconvex quadratically-constrained quadratic program (QCQP), which we then relax into a convex semidefinite program (SDP) using Shor's relaxation. We demonstrate empirically that the relaxation is tight, which enables the recovery of globally optimal solutions to the original nonconvex problem. Our formulation produces a minimum-energy solution while ensuring a desired probability of collision at the time of closest approach. Finally, if the desired probability of collision cannot be satisfied, we relax this constraint into a penalty, yielding a minimum-risk solution. We validate our algorithm with a high-fidelity simulation of a satellite conjunction in low-Earth orbit with a simulated conjunction data message (CDM), demonstrating its effectiveness in reducing collision risk.

Figures (6)

• Figure 1: The probability-of-collision constraint accounts for the position uncertainties of both spacecraft, $C_{r1}$ and $C_{r2}$, combined and projected onto the b-plane --- the plane perpendicular to the relative velocity vector. The b-plane is then centered on the secondary spacecraft, yielding an ellipse centered on the secondary spacecraft. Points inside this ellipse have high $P_c$, so the feasible region is the blue-shaded, nonconvex set outside the ellipse. The red point marks the relative position $\Delta r_b$ of the primary spacecraft with respect to the secondary spacecraft in the b-plane.
• Figure 2: (Example 1) The boundary of the ellipse in (a) represents all the relative positions that have a $P_c =1 \times 10^{-6}$. With no maneuvers, the relative position of the primary spacecraft is represented by the brown marker. After the maneuvers in (b) are applied, the relative position is represented by the red marker, showing the success of the maneuvers to achieve the desired $P_c$ value.
• Figure 3: The solution obtained from successive linearizations, shown by the red marker, closely approximates the global optimum after 100 iterations around the ellipse. The heat map illustrates the variation in delta-v as a function of the linearization point on the ellipse. In contrast, the nonlinear trajectory optimization solution, shown by the brown marker, converges to a local minimum, which is suboptimal.
• Figure 4: The ratio between the largest and second-largest eigenvalues in each moment matrix is above $1 \times 10^4$, ensuring a rank 1 moment matrix and tight relaxation. For Example 3, only the case where delta-v limit is set to 0.004 $m/s$ is plotted. The other cases had a similar tightness result.
• Figure 5: (Example 2) The boundary of the ellipse in (a) represents all relative positions corresponding to a collision probability of $P_c =8 \times 10^{-6}$. The computed maneuvers in (b) satisfy the specified upper and lower bounds and, when applied, achieve the target $P_c$ value.