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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
and linear accelerations described by the vectors u^ and a^
respectively, then it holds that
£^ A. 0 ut
(EME) 0
where A, and A, are n x3 and m x3 matrices. The rows
of A, (resp. Aa) contain the direction cosine vectors of the
n angular rate sensor axes (resp. of the 1 accelerometers).
Note that in equation | the errors of the inertial sensors
have been neglected.
At this point, two avenues can be explored for the opti-
mal exploitation of the redundant inertial information con-
tained in the observation vector /^ where (^ — ((2 , (5^ )T:
either redundancy is dealt with in the observation space or
in the state space (also called the parameter space). The
next two sections explore the two possibilities.
3.1.1 Dealing with redundancy in the observation spa-
ce: synthetic 3D axis IMU generation. Note that the
A, and A, matrices transform data from the actual sensor
axes to the three orthogonal axes of the predefined body
frame b. Thus, one could think of an imaginary non-redun-
dant IMU aligned to the b frame axes and centered at the b
frame origin. This imaginary IMU will be called synthetic.
Equation 1 can be rewritten as an error equation for the
measured n+m amounts by introducing the corresponding
residual vectors v» and vp.
p^ nat 17 | A, (on
(ley We
Equation 2 is the basis for the transformation of the actual
redundant observations /^ into a standard set of 3 + 3 ob-
servations of the synthetic IMU. All what has to be done is
to solve for w^ and a^ in equation 2. Considering the re-
dundant nature of the problem, w^ and a^ can be estimated
by least-squares in the usual way. This would lead to two
orthogonal projectors, II, for the angular rates and II, for
the linear accelerations, that blend the raw redundant data
into the synthetic IMU through the equations
Wb = I and a* — ne 3)
where
Ts zb a fp
MES (4. Cu gb As) Aw Clo gn (4)
and 1
d esse T T ~—1
Hn. =— (Aa C £b Au) Aa Cp £^ ©)
where Cy go and Cy go are the covariance matrices of the
raw inertially sensed vectors /^. and (^ respectively.
The above simple [plain least-squares] procedure has some
advantages. It allows the use of off-the-shelf existing INS
and INS/GPS software as the non-standard redundant IMU
output is converted into the usual IMU output. It allows
for the epoch-by-epoch, realistic estimation of sensor noise
and of covariance matrices for the synthetic IMU angular
161
rate sensor and accelerometer triads respectively. It allows
for defective sensor detection, identification and isolation
depending on the number and distribution of sensors by
standard geomatic data-snooping and gross-error detection
techniques based on hypothesis testing in linear models.
And it eliminates the need for adaptive Kalman filtering,
and other dangerous mathematical acrobacies.
However, the procedure has its drawbacks. Calibration of
the synthetic IMU is certainly possible and is, to a large
extent, acceptable, as linear combinations of the system-
atic errors can be estimated by the Kalman filter for the
synthetic IMU. But they cannot be back projected into the
single actual sensors and the noise estimates may be in-
flated by these unknown errors. The situation can be dealt
with in a number of ways, but it is not the ideal one.
3.1.2 Dealing with redundancy in the state space: ex-
tended INS mechanization equations. In order to over-
come the limitations of combining the redundant raw ob-
servations in the observation space the well known inertial
mechanization equations (Jekeli, 2001) can be easily mod-
ified to accommodate the redundant data. The following
set of extended inertial mechanization equations include all
redundant sensors and calibration states that, for the sake
of simplicity, are limited to biases (o” and a”) obeying a
Gauß-Markov first order stochastic process model. They
are written in a cartesian geocentric coordinate system for
an Earth-Centered-Earth-Fixed (ECEF) type of reference
system.
Et ou uU un
$5. REIL 4-aP 4- we) — 207,7 Fr g^ (a7)
Hj - RP(Oho0hOLULc-o rw) ©
à =. A0" +w
dee A An
In equation 6 above, o, 9 0 and t, wy, w,, wo; and Wa
are white noise generalized processes.
The above modeling allows for the calibration of the ac-
tual sensors at the external aiding epochs provided that the
geometric and dynamic properties of the motion guarantee
the observability of the system. Moreover, it can be com-
bined with the procedure of section 3.1.1 as the predicted
systematic errors can be eliminated from the raw sensed
data from the outset.
The main drawback of the procedure is that it requires the
modification of the INS and INS/GPS software. Whether
this is a problem or not, depends on non technical factors
that are out of the scope of the paper.
32 TWO INERTIAL MEASUREMENT UNITS
One, not yet mentioned, problem of SRIMUS is that, from
a practical market standpoint, they do not exist. Or al-
most. Some laboratories have built their own SRIMU and
the company L3 manufactures and sells them. However,
if the use of redundant inertial observations proves to be
of practical interest, a straightforward way to go is to use
