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Figure 7 - from "Sigma-point Kalman filters for integrated navigation"

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arrow_backarrow_forwardFigure 3 Instrumented X-Cell-90 helicopter in flight.

Figure 3 Instrumented X-Cell-90 helicopter in flight.

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where the optimal prediction x, corresponds to the expec- tation of a nonlinear function of the random variables &;,_ 1 and vz—1 (see Eq. (5)). Similar interpretation holds for the optimal prediction of the observation y, in Eq. (6). The optimal gain term K;,, is expressed as a function of posterior covariance matrices in Eq. (7). Note these terms also require taking expectations of a nonlinear function of the prior state estimate RVs. This recursion provides the optimal minimum mean-squared error (MMSE) linear es- timator of x; assuming all relevant random variables in the system can be efficiently and consistently modeled by maintaining their first and second order moments, i.e., they can be accurately modeled as Gaussian random variables GRVs). We need not assume linearity of the system model f(-) and h(-). The Kalman filter calculates the optimal quantities in Eqs. 5), (6) and (7) exactly in the linear case, and can be viewed as an efficient method for analytically propagating a GRV through linear system dynamics. For nonlinear models, however the extended Kalman filter (FK F) annroximates where x, is the optimal prediction of the state at time k conditioned on all of the observed information up to and including time & — 1, and y, is the optimal prediction of the observation at time k. P,, is the covariance of x;,, and Ps, is the covariance of y; = yz — y,, termed the innovation. The optimal terms in this recursion are given hy
Figure 1 Weighted sigma-points for a 2 dimensional Gaus- sian random variable (RV). These sigma-points lie along the major eigen-axes of the RV’s covariance matrix and com- plete captures the first and second order statistics of the RV. The height of each sigma-point indicates its relative weight.
where w}” and wf, are scalar weights. Note, all weights need not be positive valued. In fact, depending on the spe- cific sigma-point approach at hand, certain weights on the cross-terms are set equal to zero, i.e., w;; = 0 for certain {i, 7;7 4 7}. The specific values of the weights (w) and the scaling factors (¢) depend on the type of sigma-point ap- proach used: These include the unscented transformation and the mean and covariance of y are approximated using a weighted sample mean and covariance of the posterior sigma-points,
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Figure 2 2D example of sigma-point approach.
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mented state RV to calculate the corresponding sigma-point set, {AE ;;¢=0,....2L} where WZ, € R’xtlv+ln, The pseudo-code for the SPKF is given below: The specific type of resulting SPKF is determined by the choice of sigma-point selection scheme (weights & scal- ing factors) as well as the specific method by which the propagated sigma-points are combined in order to calcu- late the posterior covariance matrices. In Appendix A we
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Eric Wan
Rudolph van der Merwe, Eric Wan
Sigma-point Kalman filters for integrated navigation

Abstract: in Electrical Engineering from Stanford University. His research is in algorithms and architectures for adaptive signal processing and machine learning. He has ongoing projects in autonomous unmanned aerial vehicles, estimation and probabilistic inference, integrated navigation systems, time series prediction and modeling, and speech enhancement. He holds several patents in adaptive signal processing and has authored over 50 technical papers in the field.

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