Aerospace and Electronic Systems Magazine September 2017 - 7

Layh and Gebre-Egziabher
 = ( I − B 2 ) M1 B 2M 2 

(11)

where P1,+k and P2,+ k are the a posteriori covariances computed by the
individual parallel filters (i.e., the KF or EKF measurement update

(

)

+
+
equations). The variable P12,
is the postblending crossk = P21, k
covariance between the estimation errors xˆ 1,+ k and xˆ +2, k, computed
as follows:

−
 P12,
+
k
P12,

=

1
k
 0

0 T
 2
0

(12)

where 1 and 2 are matrices that depend on the Kalman gains and
−
measurement model of the parallel filters. The variable P12,
k is the a
+
priori cross-covariance between the estimation errors xˆ 1, k and xˆ +2, k .
It is propagated using the dynamic models of the parallel filters.
For clarity and continuity of this discussion, the expressions used
−
for calculating 1 and 2, as well as propagating P12,
k forward in
time, are not given here. Instead, the equations and associated derivation details can be found in Appendix B of this article.

Figure 3.

Goldy FCS shown with an expansion port for incorporating additional
sensors.

CASE STUDY
As a case study of the filtering approach described in this article, the integrated navigation system found in the FCS shown in
Figure 3 was decentralized. This is the Goldy FCS developed by
the UAV Laboratory at the University of Minnesota [24]-[26]. It
is an open-source FCS whose algorithms and construction details
can be found at [27]. Its basic sensor suite includes an IMU with
an integrated a magnetometer triad (Analog Devices AD16405), a
GNSS receiver (Hemisphere Crescent original equipment manufacturer [OEM] board) and a pair of pressure transducers (AMSYS
AMS5812). The pressure transducers are used for barometric altimeter and airspeed measurement. It also has an expansion port
(shown in Figure 3) that can accommodate additional sensors such
as cameras or other radiofrequency receivers. For example, this
port has been used to integrate a cellphone modem as a signal of
opportunity navigation sensor [8]. In its nominal configuration, it
uses a loosely integrated GNSS/INS as the primary state estimator.
For the case study presented here, the sensor fusion algorithms
were restructured as shown in Figure 4. This figure shows one
particular implementation of the generic decentralization scheme
depicted in Figure 2. The outputs from three parallel filters are
fused using two blending filters. The first blending filter fuses the
attitude estimates from a GNSS/INS filter and an AHRS filter. This
filter is called the attitude blender. The second blending filter fuses
the position estimates from a GNSS/INS filter and an airspeedbased DR filter. This filter is called the position blender. Each of
these filters is described in more detail next.

GNSS/INS FILTER
The first of the three parallel filters is an integrated GNSS/INS that
features a loose integration architecture, as described in [13]. It
uses GPS position and velocity measurements to aid an INS. The
SEPTEMBER 2017

Figure 4.

The decentralized filtering architecture implemented on the Goldy FCS
shown in Figure 3.

IMU outputs are used to generate a position, velocity, and attitude
solution at a rate of 50 Hz. A 1-Hz measurement update from GPS
is used to arrest drift errors inherent in INSs. The GPS updates also
allow estimation of the inertial sensor biases.
The state vector for the GNSS/INS filter is denoted x1 ∈ R15×1
and consists of the following states: latitude (Λ), longitude (λ), altitude (h), north velocity (Vn), east velocity (Ve), down velocity (Vd),
roll angle (ϕ), pitch angle (θ), yaw angle (ψ), three gyro biases
(bp, bq, and br), and three accelerometer biases (bax, bay, and baz).
In the validation experiments described later in this article, the attitude states remained well within the range in which the Euler
π
angle singularity is not an issue (i.e., θ << radians). For cases in
2
which this might an issue, a switch to either a quaternion or a direction cosine matrix parameterization of attitude can be made. While

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