A generalized solution to smoothing and Out-of-Sequence processing

The increasing trend towards multi-sensor systems is driving a requirement for distributed tracking algorithms. In such systems, communication links often suffer from varying delays, which leads to timely disordered data at the fusion center. The challenge of online processing delayed data is generally known as the Out-of-Sequence (OoS) problem. An exact solution to this problem is given by the Accumulated State Density (ASD) filter. However, the Rauch-Tung-Striebel retrodiction¹ is inherently integrated to the ASD approach. This implies that a certain number of states has to be updated at each filtering step. In this paper, we present the generalized solution to the OoS problem. In particular, we derive the exact update formula for single states, which refer to older and newer instants of time than a given measurement. As a consequence, OoS data can be processed without applying retrodiction. This also includes a novel smoothing scheme, which uses the exact cross-covariances of a measurement and a past state without any recursion. A numerical evaluation shows that this smoothing scheme is about 25% faster than the Rauch-Tung-Striebel equations while still being exact.