If however, the parametrized field is nonlinear, such as in the case of the
sum of Gaussians (SOG) with unknown centers or variance, the recursive
equations need to be set up using the Taylor series expansion of H.
A set of simulations was performed for the case of simultaneous vehicle
localization and linear parameter field estimation. In a 1D case (a linear field
with two unknowns ??’ intercept and slope), we considered data sets generated
under Gaussian noise assumptions, using nominal coe?±cient values a0 = 2,
a1 = 0.5 (intercept and slope), and measurement noise covariance of 0.5, state
measurement noise covariance of 0.1, and state transition noise covariance of
0.1. The convergence of the coe?±cient estimates to their nominal values is
shown in Figures 6 (a)??’(d) for two di?®erent sampling sequences. For Figures
6(a) and (c), the sampling sequence is the default one (1,2,), while Figures 6(b)
and (d) were obtained by minimizing the error covariance, e.g., the adaptive
sampling algorithm moves the vehicle from location Xn to Xn+1, such that
m(X) = ||Pn||???, m(Xn+1) ?‰? m(X), (???)X ??? ??–.
Note that the slope estimate of the second sampling sequence converges faster
to its nominal value. The optimal sampling sequence is similar to the one in
Figure 2(a) because the localization error is still smaller than the measurement
error. The optimal sampling sequences are identical to the ones reported in
[40] using the closed-form parameter variance.
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