Optimal Guidance and Collision Risk Evaluation for Circular Relative Orbit Injections

This work addresses the problem of safe and efficient injection into a Circular Relative Orbit (CRO) in proximity operations. A trajectory optimization framework is developed using Sequential Quadratic Programming (SQP) to account for the nonlinear terminal constraint imposed by the CRO geometry. The objective is to minimize the control effort while ensuring precise orbit acquisition and collision avoidance. Two injection scenarios are considered: (i) static release from the CRO center and (ii) dynamic insertion via radial boost during a rendezvous approach. The maneuver performance is evaluated in terms of total ΔV, peak acceleration, and collision risk, using instantaneous and time-propagated probabilistic metrics. Simulation results demonstrate the trade-offs between maneuver duration, orbit radius, and safety margins, highlighting the increased risk associated with prolonged proximity to the chief and the advantages of dynamic separation in reducing collision probability.