A Convex Formulation of the Soft-Capture Problem

TL;DR

This paper addresses the soft-capture problem for a chaser approaching a tumbling, uncooperative target in orbit. It introduces a convex formulation by relaxing the field-of-view constraint into a second-order cone form and handles collision-avoidance via a sequential convex programming loop that leverages a differentiable collision metric (DCOL). The proposed method yields safe, minimum-fuel trajectories by solving a convex SOCP for the initial guess and then applying iterative linearized corrections with an L1 control cost, making it suitable for flight software while maintaining rigorous safety guarantees. Experimental results show convergence and robustness up to target tumble rates of $10^\circ/\mathrm{s}$ across hundreds of scenarios, with high success rates and modest runtimes per iteration.

Abstract

We present a fast trajectory optimization algorithm for the soft capture of uncooperative tumbling space objects. Our algorithm generates safe, dynamically feasible, and minimum-fuel trajectories for a six-degree-of-freedom servicing spacecraft to achieve soft capture (near-zero relative velocity at contact) between predefined locations on the servicer spacecraft and target body. We solve a convex problem by enforcing a convex relaxation of the field-of-view constraint, followed by a sequential convex program correcting the trajectory for collision avoidance. The optimization problems can be solved with a standard second-order cone programming solver, making the algorithm both fast and practical for implementation in flight software. We demonstrate the performance and robustness of our algorithm in simulation over a range of object tumble rates up to 10°/s.

Figures (9)

• Figure 1: The soft-capture maneuver exhibits four phases: 1) The chaser (beige) approaches the target (pink) while maintaining its line of sight. 2) Impulsive burns help avoid collision with the client (thrust vector in red). 3) Corrective burns align the capture end-effector with the target capture frame. 4) the chaser is within the capture ball and turns on its capture end-effector.
• Figure 2: Left: Collision detection methods find a non-differentiable minimum distance between the 3D meshes. Right: DCOL tracy_differentiable_2023 finds the minimum inflating factor $\alpha$ between the convex hulls leading to an intersection with fast Jacobian computations.
• Figure 3: $\{O\}$ is the reference frame. The target body frame $\{T\}$, rotating with an angular velocity vector $\omega_t$, coincides with the center of $\{O\}$. $D_t$ is the capture point on the target vehicle. The chaser body frame $\{C\}$, with relative position $r$ and velocity $v$ expressed in $\{O\}$, is centered at the center of mass of the chaser. $u$ is the thrust vector. $D_c$ is the chaser capture point.
• Figure 4: Illustration of the final algorithm. The future trajectory of the target $q_t(t), \omega_t(t)$ is predicted using the current estimate of the state of the target $q_t^0, \omega_t^0$. The prediction, along the current state of the chaser $r^0, v^0$, is input to an initial convex optimization problem without collision-avoidance constraints to obtain the first iterate of the trajectory $x^1, u^1$ for the sequential convex programming (SCP) pipeline. The SCP linearizes the collision constraints and iteratively solves convex subproblems until the safety criteria $\forall k: \alpha_k > 1$ is met to yield $x^{sol}, u^{sol}$. If the SCP reaches a set maximum number of iterations, the algorithm reports infeasibility.
• Figure 5: The initial trajectory (left) resulting from solving Problem 2 is infeasible. The SCP procedure (Problem 3) adjusts the "tail" of the trajectory to avoid collision with the target (right).