Soft-Constrained Estimation for Tethered Satellite Formations Under Probabilistic Sensor Failures

Motivated by the practical challenges of tethered satellite formations (TSFs), this article focuses on the state estimation problem for systems with limited payload capacity, subject to probabilistic sensor failures and complex soft constraints. As prior knowledge, soft constraints reveal the interdependence of internal states, providing insights into observability preservation under probabilistic sensor failures. Unlike existing approaches, we propose a soft-constrained estimation scheme that fully leverages prior constraint knowledge to preserve observability and enhance performance via a constrained particle filter (CPF). Within the Bayesian framework, the CPF fully leverages the soft constraints to truncate both the prior and posterior distributions. The convergence analysis is also presented. Based on this scheme, we investigate the maximum tolerable sensor failures for TSF. Surprisingly, it is proven that n-body TSF (n ≥ 3) with typical configurations can tolerate up to n − 1 positioning sensor failures. This proof enables mission designers to sustain system observability even with up to n−1 sensor failures, thereby obviating redundant configurations while ensuring orbital mission reliability. Extensive simulations validate the effectiveness of the proposed scheme and its filter performance.