Optical EngineeringNonlinear estimation comparison for ballistic missile tracking
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Nonlinear filtering for state estimation has been recently advanced to include more accurate and stable alternatives. The extended Kalman filter (EKF), the first and most widely used approach (applied as early as the late 1960s and developed into the early 1980s), uses potentially unstable derivative-based linearization of nonlinear process and/or measurement dynamics. The unscented Kalman filter (UKF), developed after around 1994, approximates a distribution about the mean using a set of calculated sigma points. The central difference filter (CDF), or divided difference filter (DDF), developed after around 1997, uses divided difference approximations of derivatives based on Stirling's interpolation formula and results in a similar mean, but a different covariance from the EKF and using techniques based on similar principles to those of the UKF. We compare the performance of these three approaches to the coordinate conversion problem within the larger problem of ballistic missile tracking under various sensor configurations, target dynamics, measurement update/sensor communication rates, and measurement noise. The coordinate conversion problem here specifically deals with conversion of spherical measurements to Cartesian estimates and vice versa. The importance of filter stability in some cases is emphasized as the EKF shows possible divergence due to linearization errors and overconfident state covariance, while the UKF shows possibly slow convergence due to overly large state covariance approximations. The CDF demonstrates relatively consistent stability, despite its similarities to the UKF. The requirement that the CDF and UKF expected state covariances be positive definite is demonstrated to be unrealistic in a case involving multisensor fusion, indicating the necessity for the reportedly more robust and efficient square-root implementation. Strategies for taking advantage of the strengths (and avoiding the weaknesses) of each filter are proposed.