in the
ig the
ncipal
ane of
. The
(3)
agers,
| CCD
n and
>, the
m, are
dr”, as
at the
ational
ed by
t the
sensor
ations :
/e not
1 and
n be
se are
ays be
2) are
| time,
ir” are
1g the
um of
quired,
on the
ude of
1er be
titude,
lel, or
stereo
laser
hnique
use it
same
in be
ration
us
r" and
1sually
inertial
eivers
of the Global Positioning System (GPS). Although other
observables can be used, they are less suited for the task
at hand and will not be treated here. To obtain position
and orientation as functions of time, they are modelled as
functions of the time-dependent IMU and GPS
observables. The resulting model is a system of first-
order differential equations in which Ar" and R", are
variables. In engineering applications, such a system is
often called a state vector model. It is the model
underlying Kalman filtering and is therefore well-suited
for both system integration and optimal real-time
estimation. The advantage of using a state vector model
lies in the possibility of imposing smoothness conditions
on the solution by the definition of covariances for the
state vector elements and of the spectral densities.
The models for determining georeferencing parameters
from either IMU or GPS observables will not be
presented in detail. A brief discussion is given in Schwarz
et al. (1993), while Wei and Schwarz (1990) should be
consulted for details on the IMU model and Schwarz et
al. (1989) for details on the GPS model. What follows is
a brief descriptive account of the salient features of each
system and the reasons for integrating them.
A strapdown IMU outputs three components of the
specific force vector and three components of the angular
velocity vector in the body frame system. To use these
observables to derive position, velocity, and attitude in an
earth-fixed coordinate system, the attitude between the
measurement frame and the earth-fixed frame must be
determined as a function of time. This is accomplished
by determining the initial attitude in the so-called
alignment procedure, by correcting the measured angular
velocities for earth rotation and by then integrating them
in the earth-fixed frame. Since attitude is now known as a
function of time, the specific force measurements can be
transformed into the earth-fixed frame. By subtracting
gravity from the transformed measurements, vehicle
acceleration is obtained. By integrating acceleration once
with respect to time, velocity is obtained, by integrating
twice, position is obtained. The earth-fixed frame is in
principle arbitrary but is usually chosen either as a local
geodetic frame or a geocentric Cartesian frame. Because
of the integrations involved in the process, initial errors,
caused for instance by sensor biases, grow quickly with
time. Thus, a free inertial system will show systematic
errors in position, velocity, and attitude which oscillate
with a period of 84 minutes, the so-called Schuler period.
The presence of these errors is the major reason why
integration with GPS is advantageous and why it results
in a far superior determination of the georeferencing
parameters.
GPS observables are either of the pseudorange type or
of the carrier phase type. Models to transform the
resulting range equations into positions and velocities are
well-known, see, for instance, Wells et al. (1986). In the
process, orbital models as well as atmospheric models
are needed and the Earth rotation rate is again assumed
to be known. By locating one receiver at a known master
station and referencing the moving receiver to it, major
errors in the GPS measurement can be eliminated.
These differential procedures can be applied to
pseudorange measurements as well as to carrier phase
measurements and will be assumed as the modus
operandi in the following. In the typical case of one
69
ground receiver and one moving receiver, only the
translational vector r"(t) can be determined because one
antenna does not fix a vector within the rigid body and
thus the determination of rotational parameters is not
possible. Three body-mounted antennas, preferably
orthogonal to one another, are the minimum requirement
for the determination of R™p(t). The attitude matrix in the
body frame is obtained by using double differentiated
carrier phase measurements, see El-Mowafy (1994) for
details. The distance between antennas must be
considered as constant and accurately known and a
proper initialization of the R™,-matrix is required while the
system is stationary.
Thus, the georeferencing parameters, position and
attitude, can be obtained from either an IMU or a multiple
receiver GPS system. The stand-alone accuracy of GPS
and INS is given in Tables 1 and 2. Table 1 shows the
positioning and attitude determination capability of GPS
for different observables and processing methods. In
general, these results are achievable in post-mission
mode. The table shows that all required positioning
accuracies can be met but that operational constraints
may be necessary to satisfy the requirements of high
accuracy applications. For details under which
circumstances these results are achievable, see Cannon
and Lachapelle (1992), Lachapelle et al. (1994), Shi and
Cannon (19949. In general, the relative accuracy over
short time spans, say one minute, are better than the
numbers quoted.
Model Separation Accuracy Mode
Pseudorange 100 m horizontal | real time
point positioning 150 m in height
(single rev) using
precise orbits & 1-2m horizontal post
clock corrections 2-4min height mission
Smoothed 10 km 0.5 - 3 m horizontal
pseudorange 0.8 - 4 m in height post
mission
Pseudorange 500 km 3 - 7 m horizontal (or real
differential 4-8 min height time)
positioning
10 km 3 - 20 cm horizontal
5 - 30 cm in height
Carrier phase 50 km 15 - 30 cm horizon. post
differential 20 - 40 cm in height | mission
positioning
200 km 15 - 30 cm horizon.
(with precise | 20 - 40 cm in height
orbits)
Attitude 1m 18 - 30 arcminutes post
determination 5m 4 - 6 arcminutes mission
10m 2 - 3 arcminutes (or real
time)
Table 1: GPS Positioning and Attitude Accuracies.
Table 2 shows position, velocity, and attitude
performance of intertial navigation systems (INS) for two
different accuracy classes. Because INS errors are a
function of time, they are quoted for different time
intervals. Most of the time-dependent errors follow a
systematic pattern and can therefore be greatly reduced
by appropriate update measurements. The residual noise
level for a navigation-grade INS after GPS-updating will
usually be close to the value given for the one second
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B6. Vienna 1996
