Aerospace and Electronic Systems Magazine August 2017 - 39

vector (which requires at least six accelerometers), which in turn
requires integration to obtain the required angular velocity vector?
An example for a six-accelerometer configuration was designed
by Chen et al. [5], presenting a cube-shaped optimal configuration
enabling estimation of the linear and angular acceleration vectors,
thus making them the output of the GF-INS. Tan and Park [25]
examined the feasibility of other configurations of six accelerometers. Later, Genin et al. [10] used nine accelerometers arranged
as three triplets of parallel axes. With nine accelerometers at hand,
the quadratic terms of the angular velocity were also estimated;
however, the direction of the angular velocity was not determined.
Chen and Park [6] used also a nine-accelerometer configuration
in a two-stage approach. A six-accelerometer configuration at the
first stage was used to deal with the unbounded drift error, and then
they add a separate triaxial accelerometer, which they refer to as
observer-accelerometer, for the second stage. A GF-INS configuration with 12 accelerometers was examined by Zappa et al. [30],
which show that 12 accelerometers are required for a unique and
nonsingular solution. To that end, they formulated geometric rules
for the placement of the accelerometers, and most publications on
12-accelerometer configurations follow these rules [8].
Schopp et al. [23] argue that nine accelerometers are enough
to directly capture arbitrary spatial motion and in particular the
angular velocity. For the indirect approaches, an arbitrary accelerometer configuration model expressed in the Earth-centered,
Earth-fixed (ECEF) reference frame was derived and embedded in
the INS mechanization equations and corresponding error model
[20]. In that model, linear acceleration, instead of specific force,
was estimated using the GF configuration while taking into account that the gravity acting on each accelerometer location is different and differs from the gravity acting on the center of the body
frame. Inspired by that research, practical GF kinematic equations
of motion and corresponding error-state models, fitting a configuration with any set of accelerometers, were derived [15]. In addition, an analytic error assessment of the North Channel dynamics
was provided, enabling insight into the parameters affecting the
GF-INS position error, and a comparison between INS and GFINS is performed, based on the derived analytical expressions. In
both papers [15] and [20], the models were derived for any number
of accelerometers and for any arbitrary location and orientation of
those accelerometers.
Once the number of accelerometers is given, the question of
where to locate them poses a performance tradeoff related to the
AUGUST 2017

distances (arm lengths) between accelerometers in the GF-INS
configuration. As discussed in [11] and [12], longer arm lengths
are desired because they reduce the effect of noise, but arm lengths
that are too large may be impractical. The geometry of the accelerometer configuration has a similar but separate effect. In terms
of geometry, good geometry of the configuration increases the observability of linear and angular accelerations, while bad geometry
of the configuration can make the variables poorly observable or
even unobservable. Thus, the focus of their work was enhancing
the observability of the measurements by using geometry and several criteria, such as condition number or geometric dilution of
precision.
Besides addressing the number of accelerometers in the configurations and the GF-INS as a standalone system, several publications address the issue of fusion of GF-INS with other sensors,
mainly global positioning system (GPS). The six-accelerometer
cube was fused with GPS in the loosely coupled approach by Park
and Tan [21] and later by Marques Filho et al. [17] to estimate the
performance of such fusion. Williams et al. [28] used a single gyro
to aid the GF-IMU consisting of five triaxial accelerometers and
then used the output in a GPS/INS tightly coupled approach.
Many implementations of GF-INS were reported in the literature. The angular rate and accelerations of a rocket were estimated
using a GF configuration [7]. The GF configuration was used for
sensing tremor in handheld microsurgical instruments [1], [16] because of its ability to measure directly the angular acceleration.
Other applications include motion analysis during sports activities
[4], pedestrian navigation [28], and automotive applications [19].
Angular acceleration measured by a GF-INS was suggested as
means to calibrate the accelerometers in situations of limited or no
translational observations, which is a technical challenge for many
applications, such as Mars exploration missions [29].
The theory of classical INS is well established in the literature,
many excellent textbooks are available [9], [13], [26], and INSs are
regularly used worldwide in various applications. However, GFINS theory is still evolving as an exciting new field, and products
of GF-INS are usually found in academia for research. Keeping
that in mind, when considering the GF-INS architecture relative to
the classical INS architecture, several differences arise:
1. Angular velocity vector calculation. Strapdown INS equations
of motion require angular velocity information to propagate
the attitude states. In the classical INS, the angular velocity

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine August 2017