Finite energy reachable set and its optimal ellipsoid approximations in relative orbital dynamics

To prevent distortions in estimation caused by abnormal thrust magnitudes, finite energy reachable sets are essential for estimating the maximum state range of a spacecraft during engine failures. This paper examines finite energy reachable sets defined by a combination of initial state sets and weighted energy constraints. Through rigorous mathematical proofs, it is demonstrated that the reachable boundaries form the union of several convex sets. Due to the computational complexity involved in determining the envelope of this union, which poses challenges for real-time engineering applications, an equivalent transformation of the finite energy reachable set into the Minkowski sum of the initial state gain set and the control gain set is proposed. Recognizing the lower computational power of space-grade processors compared to personal computers, this paper develops optimal trace inner and outer ellipsoid approximations of the ellipsoidal Minkowski sum, avoiding reliance on matrix eigenvalue computations through matrix analysis. The result is a significant improvement in the accuracy of the approximate solution with comparable efficiency.