International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
two independent, standard inertial units. But this is eas-
ier said than done. Time synchronization becomes criti-
cal, the relative orientation (position and attitude) of the
units must be measured and/or calibrated precisely. And
these geometrical constrains have to be transferred to ei-
ther the system dynamic observation model —the equa-
tions of motion— or to the static observation model —the
measurement equations.
As with the SRIMUs, various modeling approaches are
possible. One approach, working in the observational space,
combines the two units into a single one. A second ap-
proach, in the state space, is to define a single inertial unit
containing all the sensors of the two units. A third ap-
proach, in the state space, is to navigate the two units and
to impose geometrical constrains to the navigation results.
In the next sections the first and, particularly, the third ap-
proach are discussed. For this purpose, assume: that a dual
configuration with two inertial units, named one and two
rwspectively, is given; that the inertial units body frames
are bl and b2 respectively; that unit ? measurements are
wP and fP', for i = 1,2; that their relative orientation
(u?2, RP?) does not change with time and that it is known
through direct measurements or through calibration.
3.2.1 Dealing with redundancy in the observation spa-
ce: synthetic 3D axis IMU generation. In principle, if
the relative orientation between the two inertial units is
known, it is possible to reduce all inertial measurements to
a single synthetic IMU. That is, if the inertial unit 1 is cho-
sen as a reference for the synthetic unit, then the original
measurements of unit 2, #3? and f3? must be transformed
—“corrected”— to w2! and f3!. While the transforma-
tion for angular rates is straightforward w5! = R}Lw5? the
transformation for linear accelerations is a bit more com-
plex because of the combined effect of the uj? lever-arm
and vehicle motion. The correction to be applied is simi-
lar to the “size-effect” correction (Savage, 2000) and can
be obtained after some algebraic manipulations and some
simplification assumptions. Once the correction is applied
the situation is similar to that of section 3.1.1 for the syn-
thetic generation of a standard non-redundant IMU and the
discussion is not repeated here.
3.2.2 Dealing with redundancy in the state space: geo-
metrically constrained dual navigation. Geometrically
constrained dual navigation is the navigation of the two
units, subject to the geometric constrains of their constant
relative orientation. The dynamic observation model for
dual navigation is composed of the inertial mechanization
equations of the two inertial units plus the dynamic ob-
servation model for the relative orientation parameters or
states between the inertial units. The static observation
model for dual navigation is, essentially, a set of three vec-
tor equations describing the relative orientation between
the inertial units. The two models are given in the next two
paragraphs.
Dynamic Observation Model. The following equations
are the standard inertial mechanization equations for the
two inertial units extended with two sets of differential
162
equations for uw?! and R}2. The relative orientation param-
eters between the IMUs are modeled as random constants
in case their direct measurement is not accurate enough.
ah
dé = MAUR 4 ait +wp) — 206.01 + 9° (af)
Ha = Ru (Qi + bi + 2° 07) + Wurm)
ou — gol v uw,
a zs oj git d
RN 0 (7)
e. oz 9
go = V5 Fa
TA EE EE dro)
Riz = Rip (QF +00, +008) + Wire)
o = —[» op? Wo
és = —Q9 af Wes
In equation 7, 81,01, 95, 02 7» 0 and the terms w,,,, wy, ,
Wiss sim yin Way aas Wins Woon uy» Won 204 Wig, SC While
noise generalized processes.
Static Observation Model. In addition to the measure-
ments provided by the external navigation aids, dual nav-
igation exploits the following relative orientation relation-
ship.
Ot, = AR IRR.
Oi, = Tp RENE UE as (8)
Ow, - 424 RE, (OS +00 (07) 0 as
In equation 8, v,., v4, v, are zero-mean normally distributed
random variables.
The processing strategy for the above dynamic and static
observation models is the usual one. The state of the sys-
tem is predicted by solving the stochastic differential equa-
tion 7 and the static observation model is used to feed the
system with the relative orientation constrains. Note that,
in principle, the filter corresponding to equation 7 can be
applied after each prediction step. Note, as well, that, al-
though not discussed here, any other external navigation
aids can be integrated in this model, at their own frequency
and in the usual way.
3.3 Comparative analysis
In the preceding sections, two algorithmic approaches have
been proposed for each one ofthe redundant configurations
considered. One approach combines the redundant data
in the observation space (synthetic IMU generation) and
and the other in the state space (extended INS navigation
equations and geometrically constrained navigation of dual
IMU configurations).
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