IEEE - Aerospace and Electronic Systems - July 2022 - 7

Kassas et al.
radio environment (i.e., estimating the unknown
states of the SOPs xs;i
M
i¼1.
Radio SLAM mode: GNSS observables are
unavailable. Here, SOP observables aid the INS (if
available) to simultaneously estimate the navigatormounted
states xr
while continuing to refine estimates
of the SOPs' states.
NAVIGATOR DYNAMICS MODEL
xins ¼
In a tightly coupled radio SLAM framework, either an INS
kinematics model or a dynamics model for the navigator is
utilized to perform the EKF time update. In what follows,
a description of each is discussed.
INSKINEMATICSFORMULATION
Let fbg denote a body frame fixed at the navigator,
and let fgg denote a global frame, e.g., the Earthcentered
inertial frame [36]. Moreover, let ub 2 R3
represent
the three-dimensional (3-D) orientation vector of
the body frame with respect to the global frame and
grb 2 R3 the 3-D position vector of the navigator
expressed in fgg. Given the INS's true 3-D rotational
rate vector
bv 2 R3 in the body frame and its 3-D
acceleration gab 2 R3 in the global frame, the standard
strapdown kinematics equations can be expressed in
continuous time as
u_ bðtÞ¼ bvðtÞ
g€rbðtÞ¼ gabðtÞ:
(1)
(2)
The 3-D orientation vector of the body frame with
respect to the global frame can be represented by the 4-D
quaternion vector b
gq 2 R4.
The navigator's IMU is assumed to contain a triadT
imu;aaT
imu
of
the angular rate
gyroscope and a triad-accelerometer, which produce
measurements zimu , vT
and specific force, which are modeled as
vimuðkÞ¼ bvðkÞþ bgyrðkÞþ ngyrðkÞ
aimuðkÞ¼ R b
hi
gq
hi
where R b
(3)
gqðkÞðÞgabðkÞggðkÞ þbaccðkÞþnaccðkÞ (4)
is the equivalent rotation matrix of b
gq; gg
is the acceleration due to gravity acting on the navigator
in the global frame; bgyr 2 R3 and bacc 2 R3
are the
gyroscope and accelerometer biases, respectively; and
ngyr and nacc are measurement noise vectors, which are
modeled as zero-mean white noise sequences with
covariances Qngyr
and Qnacc , respectively. Integrating
IMU specific forcedatatoperform atimeupdateof the
position and velocity in a rotating coordinate frame
introduces a centrifugal and Coriolis term due to the
JULY 2022
IEEE A&E SYSTEMS MAGAZINE
rotation rate of the Earth [37]. However, a short integration
interval is considered in this article, where the variation
of the Coriolis force was considered negligible for
simplicity. Further details about neglecting the Coriolis
force over short integration intervals can be found in
Morales and Kassas [27].
The gyroscope and accelerometer biases in (3) and (4) are
dynamic and stochastic; hence, they must be estimated in the
EKFaswell. As such,the INS16-state vector is given by
hiT
T
b
gq
;
grT
b ;
g _rT
b ;bbT
gyr;bbT
acc
where _rb 2 R3 is the 3-D velocity of the navigator.
The INS states evolve in time according to
xins k þ 1ðÞ¼ fins xinsðkÞ; bvðkÞ; gabðkÞ
where fins is a vector-valued function of standard strapdown
kinematic equation [38], which discretizes (1) and
(2) by integrating bv and gab to produce b
gqðk þ 1Þ, rbðk þ
1Þ, and _rbðk þ 1Þ, and uses a velocity random walk model
for the biases, which is given by
bgyrðk þ 1Þ¼ bgyrðkÞþ wgyrðkÞ
baccðk þ 1Þ¼ baccðkÞþ waccðkÞ
where wgyr and wacc are process noise vectors that drive
the in-run bias variation (or bias instability) and are modeled
as white noise sequences with covariance Qwgyr and
Qwacc , respectively.
DYNAMICSMODELFORMULATION
Generally, the navigator's dynamics can be described as
x_ ðtÞ¼ ffxxðtÞ;uuðtÞ;t
½þ wðtÞ
where x is the navigator's state vector, u represents known
exogenous inputs, and w is the process noise. Depending
on the navigator's platform (pedestrian or ground, aerial, or
maritime) and motion, different dynamic models can be
used to describe the navigator's dynamics, such as polynomial
(e.g., white noise acceleration and Wiener process
acceleration), singer acceleration, mean-adaptive acceleration,
semi-Markovjump process, circular motion, curvilinear
motion, coordinated turn, among others [39].
A simple, yet effective dynamical model that has been
successfully employed for navigators with " low dynamics, "
which sufficiently captures the dynamics between
EKF measurement updates is the white noise acceleration,
given by
x_ pvðtÞ¼ Apv xpvðtÞþDpv ~wpvðtÞ
Apv ¼
; Dpv ¼
033 I33
033 033
033
I33
7
(5)
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