Basic tracking using nonlinear continuous-time dynamic models [Tutorial]

Basic tracking using nonlinear 3D monostatic and bistatic measurements

“…The ability to handle general state transformations opens the door to applications in which complicated nonlinear transformations, such as refraction effects or nonlinear target dynamics, can be taken into account. A Basic Tracking Using Nonlinear 3D Measurements companion tutorial in [71] discusses multistatic tracking with refractive errors, and a separate companion tutorial in [72] discusses tracking using nonlinear continuous-time dynamic models.…”

“…This tutorial considers only discrete-time models arising from continuous-time, linear time-invariant systems of the form (42) where A and B are termed the drift and diffusion matrices, respectively, and is the derivative of a Wiener process. The companion tutorial in [72] addresses the use of more general nonlinear continuous-time dynamic models. Let be the ith element in .…”

“…In the simulation sections, the initial state of the target is fixed, meaning that (68) where δ represents the Dirac delta function and μ 0 is the chosen initial location. Using the linear dynamic model of Section V, assuming that a uniform sampling interval is used, and omitting the subscript of k on F and Q, which are constant when the sampling interval is uniform, the PDF of the state at time k is (superscripts denote matrix powers) (69) The sum in (69) can be written as a block-diagonal matrix with block elements 39 (70) As derived in [294], the Bayes information matrix at step k + 1 may be calculated recursively as (71) where the initial condition is and the parameters of interest simplify to (72) 38 In the literature, both are typically represented by J. Here, a superscript of B is used to differentiate the two matrices.…”

“…State prediction can be performed for discrete-time dynamic models as in Figure 9, for additive noise, or as in Figure 10, for general noise, or using (103) and (104) for linear dynamics. Nonlinear, continuous-time dynamics models can be handled as described in companion tutorial [72].…”

Stochastic Feedback Based Continuous-Discrete Cubature Kalman Filtering for Bearings-Only Tracking

“…While state models for most target tracking algorithms are described as stochastic differential equations (SDEs), CD filtering methods are quite different from traditional methods. Thus, mathematical models for CD filtering methods are more complicated than traditional ones, while the CD methods are potentially more accurate [ 9 ]. The common form of the stochastic system can be expressed in the form of stochastic differential equation (SDE): where is the n -dimensional target state vector, is known as the drift function, is the diffusion matrix, and is a Brownian motion, which is known as a Wiener process.…”