Aerospace and Electronic Systems Magazine August 2017 - 40
Control Theoretic Approach to Gyro-Free Inertial Navigation Systems
vector is directly measured by the gyro triad. However, the
GF-IMU measures angular acceleration, and then integration
is required to calculate the angular velocity vector. That integration reflects a drawback of the GF approach, because errors
in the accelerometer measurements permeate to the angular
acceleration calculation; therefore, their integration leads to a
time-varying error in the estimation of angular velocity vector.
Another GF approach enables a direct calculation of the angular velocity but requires more than six accelerometers.
2. Coarse alignment. Several INS grades enable the possibility of
self-coarse alignment, in which the initial roll and pitch angles
are determined by the accelerometers and the initial yaw angle
by the gyros while measuring Earth's rotation rate. Yet the GFIMU cannot measure Earth's rotation rate; therefore, the initial
yaw angle cannot be determined regardless of the accelerometer grade [27]. In that manner, the GF-IMU resembles a lowgrade INS not capable of coarse alignment and any solution
used there to determine the initial yaw angle will probably be
feasible for the GF-INS.
3. Angular acceleration vector. Several applications require information on the angular acceleration vector. In a classical INS,
this requirement issues the differentiation of the measured angular velocity vector, while in the GF-IMU, the angular acceleration vector is directly measured.
4. Multiple IMUS. GF-INS theory can also be applied in multiple
distributed IMUs to obtain improved angular rate measurements and specific force estimation.
5. Power consumption. As stated in [12] and [23], because of the
required mechanical excitation of MEMS gyros, the power
consumption of a gyro is about 20 times higher compared with
a mechanically passive accelerometer. A GF-INS architecture
can therefore be used to implement a system with a lower
power consumption, which is mostly valuable for handheld
devices.
6. Current MEMS technology. The emergence of low-cost
MEMS accelerometers with rapidly increasing performance
(gyro technology is still falling behind) and expectations for
new technologies to breakthrough and increase accelerometer
capabilities [3] give a boost for implementation of GF-INS architectures.
UNIFIED GF APPROACH
In this section, we review the unified GF approach as derived by
Klein [15], including GF-IMU output, GF-INS equations of motion, and corresponding error models. These models are used in the
remainder of the article for stability and observability analysis and
are given here for the completeness of the presentation.
b
k
A single axis accelerometer output, f , model is given by
40
T
{f
b
ib
+ ΩbibΩbib ρ kb − ρ kb × ω ibb
}
b b T eb T
eb T Ωb Ωb ρ b
f1b ρ1 × e1
1 b
1 1b 1b 1
ωib +
.
=
f ibb
b
T
T
f b
eb T Ω b Ωb ρ b
N
ρ N × eNb eNb
N Nb Nb N
Y
H N ×6
(1)
(2)
M N ×1
Taking into account that N ≥ 6, we use Moore-Penrose pseudoinverse and recover the GF-IMU output from (2). The output consisting of the specific force and angular acceleration vectors expressed
in the body frame is as follows:
ω ibb
−1
T
T
b = hY − hM , h = H H H
f ib
(3)
Equation (3) is the general expression for the GF-IMU output
valid for any GF configuration, that is, for any number of accelerometers and regardless of their positions and orientations. We
would like to connect the accelerometers errors and the IMU errors. To that end, linearizing of (3) yields
δω ibb
b
b = hδ Y − hAδωib
δ fib
(4)
where
eb T P
1 1
A=
T
eb P
N
N
N ×3
and
ω y ρ yk + ω z ρ zk
Pk = ω y ρ xk − 2ω x ρ yk
ω ρ − 2ω ρ
x zk
z xk
(5)
ω x ρ yk − 2ω y ρ xk ω x ρ zk − 2ω z ρ xk
ω x ρ xk + ω z ρ zk ω y ρ zk − 2ω z ρ yk .
ω z ρ yk − 2ω y ρ zk ω x ρ xk + ω y ρ yk
(6)
Rearranging the matrices h and hA in a convenient form leads
to the GF-IMU error-state model
δω ibb hω1 hω 2 δ Y
b =
b
δ f ib hy1 hy 2 δωib
GF-IMU
f kb = ekb
where k is the accelerometer index, ekb is its orientation (sensitivity)
axis, Ωbib is the skew symmetric form of the relative angular velocity between the body and the inertial frame, ρ kb is the location of
the accelerometer, and is the relative angular acceleration between
the body and the inertial frame, all expressed in the body's coordinates system. Given the GF acceleration measurements, the problem is to construct a standard linear system formulation to yield
the classical IMU output. It is assumed that all accelerometers are
mounted on a rigid body. Because rigid bodies have 6 degrees of
freedom (DOF), at least six accelerometers are required to solve
for the motion. Using (1), for n accelerometers,
(7)
where hω1 is the matrix formed by taking the upper three rows of h,
hω2 is the matrix formed by taking the upper three rows of hA, hy1
IEEE A&E SYSTEMS MAGAZINE
AUGUST 2017
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