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C. Greco and M. Vasile
dynamics is used to propagate the covariance information. In the same fashion,
the nonlinear observation model is used to compute the difference between the
received and predicted observations, while the linearised measurement model is
used to update the state estimate and its corresponding covariance matrix. All the
Jacobian matrices are evaluated at the reference trajectory {·}| ¯
x propagated as in step
3 of Algorithm 3. This reference state is updated after each update step to the best
estimate.
As in the classic Kalman filter, the extended Kalman filter can be derived in the
state transition notation. However, the procedure and resulting algorithm are similar
to the classic case, and it will not be presented here. Another possible derivation
is achieved by expanding directly the involved nonlinear transformations in Taylor
series [51]. The same result can be generated by a statistical reasoning with least
squares approach [58].
Several variants and heuristics exist to improve the basic extended Kalman filter.
A solid improvement to this filter is realised with local iterations of the update steps
5–8 when the measurements of nonlinearity are critical [17, 29]. The iterations are
called local as they are realised at a fixed time. Iterating is a regular tool for solving
nonlinear problems with linear sub-steps. In the same fashion, the original update
routine would involve a nonlinear measurement model, but there is no closed-form
solution unless a linearisation is performed. Therefore, after the updated estimate
and covariance are computed, the reference values can be updated to their values
¯
x
−
k = ˆ
x
+
k and P
−
k = P
+
k and steps 5–8 repeated linearising with respect to these
quantities until the changes in the optimal estimate are under a certain threshold.
In Sect. 6.2.2.1, it was shown that if the quadratic term is retained, additional
terms should be included in the mean and covariance propagation (see Eqs. 6.42
and 6.43) through the nonlinear transformations, resulting in the so-called secondorder extended Kalman filter. The higher order in the Taylor expansion helps
to cope with the neglected system nonlinearities, at the expense of increased
preliminary analytical derivations and increased computational burden. For the
complete derivation, see Särkkä [51].
Plenty of variants and heuristics have been developed for generic or specific
problems. In the vast literature, extensive references with a particular focus on the
practical approaches are available [26, 66]. The extended Kalman filter is widely
applied in navigation and orbit determination problems for space applications due
to its simplicity and effectiveness [51, 58]. However, the filter may fail if the initial
guess is far from the real state, a situation in which the linearised dynamics is not
properly representative of the true trajectory evolution. Another drawback is that
usually this method relies on the explicit derivative computation of the dynamical
and measurement models, requiring rather lengthy and error-prone derivations.
Therefore, generally the extended Kalman filter is not suitable for black box
systems. Numerical finite-difference schemes can be implemented for the derivative
computation, however resulting in worse computational performance.
C. Greco and M. Vasile
dynamics is used to propagate the covariance information. In the same fashion,
the nonlinear observation model is used to compute the difference between the
received and predicted observations, while the linearised measurement model is
used to update the state estimate and its corresponding covariance matrix. All the
Jacobian matrices are evaluated at the reference trajectory {·}| ¯
x propagated as in step
3 of Algorithm 3. This reference state is updated after each update step to the best
estimate.
As in the classic Kalman filter, the extended Kalman filter can be derived in the
state transition notation. However, the procedure and resulting algorithm are similar
to the classic case, and it will not be presented here. Another possible derivation
is achieved by expanding directly the involved nonlinear transformations in Taylor
series [51]. The same result can be generated by a statistical reasoning with least
squares approach [58].
Several variants and heuristics exist to improve the basic extended Kalman filter.
A solid improvement to this filter is realised with local iterations of the update steps
5–8 when the measurements of nonlinearity are critical [17, 29]. The iterations are
called local as they are realised at a fixed time. Iterating is a regular tool for solving
nonlinear problems with linear sub-steps. In the same fashion, the original update
routine would involve a nonlinear measurement model, but there is no closed-form
solution unless a linearisation is performed. Therefore, after the updated estimate
and covariance are computed, the reference values can be updated to their values
¯
x
−
k = ˆ
x
+
k and P
−
k = P
+
k and steps 5–8 repeated linearising with respect to these
quantities until the changes in the optimal estimate are under a certain threshold.
In Sect. 6.2.2.1, it was shown that if the quadratic term is retained, additional
terms should be included in the mean and covariance propagation (see Eqs. 6.42
and 6.43) through the nonlinear transformations, resulting in the so-called secondorder extended Kalman filter. The higher order in the Taylor expansion helps
to cope with the neglected system nonlinearities, at the expense of increased
preliminary analytical derivations and increased computational burden. For the
complete derivation, see Särkkä [51].
Plenty of variants and heuristics have been developed for generic or specific
problems. In the vast literature, extensive references with a particular focus on the
practical approaches are available [26, 66]. The extended Kalman filter is widely
applied in navigation and orbit determination problems for space applications due
to its simplicity and effectiveness [51, 58]. However, the filter may fail if the initial
guess is far from the real state, a situation in which the linearised dynamics is not
properly representative of the true trajectory evolution. Another drawback is that
usually this method relies on the explicit derivative computation of the dynamical
and measurement models, requiring rather lengthy and error-prone derivations.
Therefore, generally the extended Kalman filter is not suitable for black box
systems. Numerical finite-difference schemes can be implemented for the derivative
computation, however resulting in worse computational performance.