Abstract
Target tracking from non-invertible measurement sets, for example, incomplete spherical coordinates measured by asynchronous sensors in a sensor network, is a task of data fusion present in a lot of applications. Difficulties in tracking using extended Kalman filters lead to unstable behavior, mainly caused by poor initialization. Instead of using high complexity numerical batch-estimators, we offer an analytical approach to initialize the filter from a minimum number of observations. This directly pertains to multihypothesis tracking (MHT), where in the presence of clutter and/or multiple targets (i) low complexity algorithms are desirable and (ii) using a small set of measurements avoids the combinatorial explosion. Our approach uses no numerical optimization, simply evaluating several equations to find the state estimates. This is possible since we avoid an over-determined setup by initializing only from the minimum necessary subset of measurements. Loss in accuracy is minimized by choosing the best subset using an optimality criterion and incorporating the leftover measurements afterwards. Additionally, we provide the possibility to estimate only sub-sets of parameters, and to reliably model the resulting added uncertainties by the covariance matrix. We compare two different implementations, differing in the approximation of the posterior: linearizing the measurement equation as in the extended Kalman filter (EKF) or employing the unscented transform (UT). The approach will be studied in two practical examples: 3D track initialization using bearingsonly measurements or using slant-range and azimuth only.
Highlights
Target tracking from incomplete polar or spherical measurements, like bearings only or range only, is a topic that has received close investigation, for example, target motion analysis and related questions of observability [1, 2]
We can use the unscented transform (UT) for which we need to find a function g which maps from v and z − w to p
To give an example of which subset of measurements has the best initialization geometry, we plot the choices in Figure 5 for the extended Kalman filter (EKF) implementation
Summary
Target tracking from incomplete polar or spherical measurements, like bearings only or range only, is a topic that has received close investigation, for example, target motion analysis and related questions of observability [1, 2]. In the case of incomplete polar or spherical measurements, the full target state vector is not directly observable, since we cannot invert the measurement function. This makes the initialization of the extended Kalman filter with an initial state estimate and corresponding covariance crucial for its performance, otherwise the filter can become unstable [6]. In our case, this cannot be accomplished by direct inversion of the measurement function. The concatenation of two functions f and g, where f : Rm→Rn and g : Rn→Rp, is expressed as g ◦ f : Rm→Rp
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