6 Fundamentals of Filtering
213
claimed that overall the unscented Kalman filter is a more direct generalisation of the
Kalman filter rather than the extended Taylor series one. Indeed, the same reasoning
addressed at the beginning of the section on the definition of linear filtering arises.
There is no theoretical need of linearity on the dynamical and measurement models
to approximate the posterior as a normal distribution, and the unscented transform is
a powerful tool to achieve such approximation by only evaluating a set of Gaussian
moment equations for the selected propagated samples [26].
As for the other filters, plenty of variants, heuristics and generalisations were
developed for the unscented Kalman filter. Among them, some involve the use of
different numbers of sigma points resulting in higher-order techniques, as already
outlined in Sect. 6.2.2.2. An alternative filter can also be formulated to account
for a non-additive noise contribution, by augmenting the state vector with process
and observation noises before using the unscented transformation [34, 51]. For the
difference in the derivations by state augmentation, see Wu et al. [64]. Extensive
references for the original unscented filter and its numerous extensions can be found
in the literature [59].
Although historically developed and extensively employed with Gaussian priors
and posteriors, which provide a clear and simple result, it is worth to recall that
there is no need to rely on the Gaussianity assumption at all in the unscented
transformation.
6.3.4 Gaussian Filter Framework
The current section dealt entirely with filtering techniques which, directly or indirectly, approximate the posterior distribution p(x k |y 1:k ) as Gaussian. The extended
Kalman filter constructs indirectly a Gaussian posterior by linearising the nonlinear
transformations, therefore ensuring the conservation of the distribution’s Gaussian
nature (see Sec. 6.2.1). The unscented Kalman filter directly fits a Gaussian distribution to the propagated samples by matching the first two moments of the resulting
discrete density.
The latter idea was shown to be a particular case of a general framework for
Gaussian assumed density filter [28, 42, 51, 65]. The goal is again to approximate the
posterior density p(x k |y 1:k ) as Gaussian, regardless of the nonlinearity properties of
the dynamical and measurement models.
For a general nonlinear transformation z = g(x), with x ∼ N (ˆ x, P x ), the
moment matching approximation is constructed as in Eq. (6.78). Now, the moments
are computed by the expectation operator. Explicitly:
p(x, z) = N x,z
ˆ
x
ˆ
z
,
P C
C T S
,
where now:
213
claimed that overall the unscented Kalman filter is a more direct generalisation of the
Kalman filter rather than the extended Taylor series one. Indeed, the same reasoning
addressed at the beginning of the section on the definition of linear filtering arises.
There is no theoretical need of linearity on the dynamical and measurement models
to approximate the posterior as a normal distribution, and the unscented transform is
a powerful tool to achieve such approximation by only evaluating a set of Gaussian
moment equations for the selected propagated samples [26].
As for the other filters, plenty of variants, heuristics and generalisations were
developed for the unscented Kalman filter. Among them, some involve the use of
different numbers of sigma points resulting in higher-order techniques, as already
outlined in Sect. 6.2.2.2. An alternative filter can also be formulated to account
for a non-additive noise contribution, by augmenting the state vector with process
and observation noises before using the unscented transformation [34, 51]. For the
difference in the derivations by state augmentation, see Wu et al. [64]. Extensive
references for the original unscented filter and its numerous extensions can be found
in the literature [59].
Although historically developed and extensively employed with Gaussian priors
and posteriors, which provide a clear and simple result, it is worth to recall that
there is no need to rely on the Gaussianity assumption at all in the unscented
transformation.
6.3.4 Gaussian Filter Framework
The current section dealt entirely with filtering techniques which, directly or indirectly, approximate the posterior distribution p(x k |y 1:k ) as Gaussian. The extended
Kalman filter constructs indirectly a Gaussian posterior by linearising the nonlinear
transformations, therefore ensuring the conservation of the distribution’s Gaussian
nature (see Sec. 6.2.1). The unscented Kalman filter directly fits a Gaussian distribution to the propagated samples by matching the first two moments of the resulting
discrete density.
The latter idea was shown to be a particular case of a general framework for
Gaussian assumed density filter [28, 42, 51, 65]. The goal is again to approximate the
posterior density p(x k |y 1:k ) as Gaussian, regardless of the nonlinearity properties of
the dynamical and measurement models.
For a general nonlinear transformation z = g(x), with x ∼ N (ˆ x, P x ), the
moment matching approximation is constructed as in Eq. (6.78). Now, the moments
are computed by the expectation operator. Explicitly:
p(x, z) = N x,z
ˆ
x
ˆ
z
,
P C
C T S
,
where now: