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130 Autonomous Mobile Robots

where b is the skew matrix form of ω b and
nb bn
b b b n n
ω bn = ω − C (ω + ω ) (3.65)
ie
en
n
ib
n
n
The symbols ω and ω represent the rotation rate of the ECEF frame relative
ie en
to an inertial frame and the rotation rate of the navigation frame relative to
the earth frame (i.e., transport rate), respectively. Both vectors are resolved
in the navigation frame. The second term on the right of (3.65) compensates
the gyro measurements for the rotation rate of the navigation frame relative to
an inertial frame.
The dynamic equations for an INS system have different forms for different
navigation frames. Detailed derivations of the navigation equations with respect
to different navigation frames can be found in various references, for example,
in References 27, 34, and 41–43. The navigation equations for a terrestrial
navigation system operating in the local geodetic frame are:

n
n
n
n
n
˙ v = f − (2ω + ω ) × v + g l n (3.66)
ie
e
en
e
n
T
where v =[v n , v e , v d ] is the velocity with respect to the Earth expressed
e
T
in the local geodetic frame (i.e., navigation frame), f n =[f n , f e , f d ] is the
n
specific force resolved to this navigation frame, and g is the local gravity
l
vector expressed in the navigation frame. The local gravity vector is
i i i
g l = g − ω ×[ω × R ] (3.67)
ie
ie
which accounts for the mass attraction of the earth g and the centripetal accel-
eration caused by the Earth’s rotation. Note that g l is the acceleration sensed
by a stationary accelerometer located on the surface of the earth. Note also that
g l is a function of position.
Given an initial velocity, Equation (3.66) integrates acceleration to estimate
the velocity as a function of time. Prior to integration, the measured specific
force vector is corrected for Coriolis effects (second term) and gravity (third
term). Given an initial position, integration of velocity provides an INS estimate
of position. Given high rate IMU measurements (and a sufficiently fast com-
puter), the INS can integrate the above equations to provide high rate estimates
of position, velocity, attitude, angular rate, and acceleration. Since the INS is an
integrative process, the INS attenuates the high frequency measurement noise
from the IMU, but amplifies low frequency measurement errors such as biases.
Calibration and removal of the INS state and IMU instrument errors can be
accomplished through EKF data fusion, once the designer obtains a dynamic
model for the INS and IMU error processes.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 130 — #32