This paper presents a joint trajectory smoothing and tracking framework for a specific class of targets with smooth motion. We model the target trajectory by a continuous function of time (FoT), which leads to a curve fitting approach that finds a trajectory FoT fitting the sensor data in a sliding time-window. A simulation study is conducted to demonstrate the effectiveness of our approach in tracking a maneuvering target, in comparison with the conventional filters and smoothers. Note to Practitioners —Estimation, such as automatically tracking and predicting the movement of an aircraft, a train, or a bus, plays a key role in our daily life. In this paper, we provide a new approach for the online estimation of the target trajectory function by means of fitting the time-series observation, which accommodates the lack of quantifiable knowledge about the target motion and of the statistical property of the sensor observation noise. The resulting trajectory function can be used to infer either the past or the present state of the target. Engineering-friendly strategies are provided for computationally efficient implementation. The proposed approach is particularly appealing to a broad range of real-world targets that move in smooth courses, such as passenger aircraft and ships.
More relevantly to our approach, efforts have been devoted to continuous time trajectory estimation via data fitting in different disciplines. De facto, signal processing stems from the interpolation and extrapolation of a sequence of observations @cite_82 . Data fitting is a self-contained mathematical problem and a prosperous research theme by its own, which has proven to be a powerful and universal method for pattern learning and time series data prediction, especially when adequate analytical solutions may not exist. Moreover, the recursive LS algorithm reformulated in state-space form was recognized a special case of the Kalman filter (KF) @cite_48 @cite_21 .
@cite_48 @cite_21 @cite_82
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This discussion is directed to least-squares estimation theory, from its inception by Gauss1 to its modern form, as developed by Kalman.2 To aid in furnishing the desired perspective, the contributions and insights provided by Gauss are described and related to developments that have appeared more recently (that is, in the 20th century). In the author's opinion, it is enlightening to consider just how far (or how little) we have advanced since the initial developments and to recognize the truth in the saying that we stand on the shoulders of giants.'' Adaptive filtering algorithms fall into four main groups: recursive least squares (RLS) algorithms and the corresponding fast versions; QR- and inverse QR-least squares algorithms; least squares lattice (LSL) and QR decomposition-based least squares lattice (QRD-LSL) algorithms; and gradient-based algorithms such as the least-mean square (LMS) algorithm. Our purpose in this article is to present yet another approach, for the sake of achieving two important goals. The first one is to show how several different variants of the recursive least-squares algorithm can be directly related to the widely studied Kalman filtering problem of estimation and control. Our second important goal is to present all the different versions of the RLS algorithm in computationally convenient square-root forms: a prearray of numbers has to be triangularized by a rotation, or a sequence of elementary rotations, in order to yield a postarray of numbers. The quantities needed to form the next prearray can then be read off from the entries of the postarray, and the procedure can be repeated; the explicit forms of the rotation matrices are not needed in most cases. >