New State Update Equation for the Unscented Kalman Filter

A FUNDAMENTAL tenet of a Kalman-Bucy filter, usually referred to as the Kalman filter [1-3] (KF), is that the system to which it is applied can be described through a linear model. The simplicity of its formulation is so powerful that its concept has been extended to nonlinear systems in almost every imaginable application [3,4]. The unscented Kalman filter [5] (UKF) is among the latest and most compelling extensions of Kalman-type filters, particularly for systems that are markedly nonlinear. Despite its allure, however, the UKF has also been shown to diverge for highly nonlinear systems [6]. In the following sections, we first review both the KF and the UKF and then present a new formulation for the state update equation of the latter. In the theoretical derivation of the new coefficients for the new update equation, we consider several approximations. The error that arises from these approximations is compared with the error that arises from truncating the UKF update equation to the first order. Numerical simulations of a simple nonlinear model relevant to formation flying configurations show that the improved UKF can be several orders of magnitude more accurate than the standard UKF.