Aerospace and Electronic Systems Magazine September 2017 - 6

Graceful Degradation and Recovery from GNSS Interruptions
from the measurements available on the data bus. For clarity and
simplicity, Figure 2 shows only four parallel filters, but in practice
there can be more or fewer filters. The ith filter generates an estimate of the state vector xi. The vectors xi may or may not be replicas of each other or the navigation state vector x. However, they
do share some common elements with the navigation state vector
such that xi ∩ x ≠ ∅. Because the same sensor information may be
used in more than one of the parallel filters, some state vectors xi
may be correlated.
The states estimated by the bank of parallel filters are fused in
a set of blending filters. We refer to these filters as blenders. The
input to the blenders are the estimates xi and the associated state
error covariances Pi. The output of the blenders are the estimates
of attitude, position, and velocity that will be used by the FCS. The
only information the blenders need to know consists of the outputs
of the parallel filters. As long as each ith filter state is observable
from the measurements used by the respective filters, the blenders
require minimal knowledge of the structure of any of the upstream
filters. This has an important practical implication: a bank of parallel, standalone navigation systems with potentially correlated
outputs can be integrated with only minimal knowledge about the
internal workings of each system. In the section that follows, the
blender equations are derived.

BLENDER EQUATIONS

Variable

Definition

The blender equations are a weighted sum of the reshaped x1,k and
x2,k or the following:

(

)

(

)

(4)

B1 = I n× n − B 2

B 2 = Γ3 ( Γ1 − Γ 2 )

−1

(5)

where

)

Pk+

The a posteriori covariance for xˆ +k

xi,k

The true (unknown) ith filter state vector

xˆ

The a posteriori estimate of xi,k
The a posteriori covariance for xˆ i , k

(

+
+
T
T
Γ 2 = M 2 P21,
k M1 + M1P12, k M 2

(6)

)

(7)

)

(8)

The state estimation error covariance of the blender output xˆ +k is
denoted Pk+ and calculated as follows:

The a posteriori estimate of xk

(i ∈ {1, 2} and ∈  ni ×1)

(3)

The matrices B1 ∈  n× n and B 2 ∈  n× n are blender weights that ensure xˆ +k is an unbiased, minimum variance estimate (see Appendix
A) calculated as follows:

(

(x k ∈  )

Pi+, k

(2)

+
T
Γ3 = M1P1,+k M1T − M1P12,
kM2

The true (unknown) state vector
n×1

+
i,k

x k = M1x 2, k

(

Blending Filter Variable Definitions at Time Step t k

xˆ

(1)

Γ1 = M1 P1,+k M1T + M 2 P2,+ k M T2

Table 1.

+
k

x k = M1x1, k

xˆ +k = B1 M1xˆ 1,+ k + B 2 M 2 xˆ 2,+ k

The blender equations are described in this section, but a more
detailed derivation and discussion of them can be found in the appendices. For the work reported here, each of the blending filters
receives no more than two xˆ i. While a blender that can accommodate more inputs would provide added flexibility, for the system
discussed in this article, this was not found to be necessary. Extending the approach to deal with more blender inputs is a subject
of future work discussed in the article's conclusion. Therefore, in
the discussions that follow, it is assumed that the parallel filter index i ∈ {1, 2}.

Table 1 defines the variables that describe the inputs and outputs
of the blender at time step tk. The hat above variables indicates an
estimate of a quantity and distinguishes it from the variable's true
but unknown value. The blender inputs are the current best state
estimate and covariance from each parallel filter. It is assumed that
these are generated by a Kalman filter (KF) or an extended Kalman
filter (EKF). Furthermore, for the moment, it is assumed that these
are the a posteriori (postmeasurement update) state and covariance
estimate pairs (xˆ 1,+ k , P1,+k) and (xˆ +2, k , P1,+k). How asynchronous measurement update schedules can be handled is discussed in Appendix B
in the paragraph after (65).
The output of the blender is xˆ +k , which is an estimate of the desired state vector xk. Because the dimensions of x1,k, x2,k, and xk are
not the same, a pair of mapping matrices whose entries are 0 or 1
are used to reshape the blender inputs. These matrices are denoted
M1 ∈ R n× n1 and M 2 ∈  n× n2 and relate xk to x1,k and x2,k as follows:



Pk  E  xˆ k  x k  xˆ k  x k 




(9)

where

 P
   1, k
 P21,k
IEEE A&E SYSTEMS MAGAZINE



P12,
k
 
P2,k 

(10)

SEPTEMBER 2017

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine September 2017