Aerospace and Electronic Systems Magazine August 2017 - 14
Flight Data Assessment of Tightly Coupled PPP/INS
vki | k −1 = vki −1| k −1 + Δv i + γ iτ
(6)
Here, the updated INS velocity estimate is the summation of the
previous velocity estimate (k − 1 | k − 1) and the IMU measured
change in velocity, Δvi. In addition, the component of acceleration
that is due to Earth's gravity in ECI, γi, must be modeled and integrated over the time step τ. For this application, Earth's gravity
was modeled using the EGM-2008 gravity field [45] out to degree
and order 200.
Position Update
Finally, the position is updated from the previous position estimate
via trapezoidal integration:
(
rki k −1 = rki −1 k −1 + vki k −1 + vki −1 k −1
) τ2
(7)
Although the estimation filter used in the study is realized in an
ECI frame, for georeferencing applications, position must be
known in an Earth-centered, Earth-fixed (ECEF) frame. Therefore,
in practice, before the estimated position and velocity are output
to the end user, a position and velocity transformation from ECI to
ECEF is conducted that is based on Earth's rotation rate, the polar
motion model, and time past which the ECI and ECEF frames were
coincident (i.e., typically the J2000 reference epoch). The specific
ECI-to-ECEF transformation realized within RTGx is the transformation adopted by the International Earth Rotation Service [46].
ERROR-STATE MODEL
As discussed earlier, a portion of the estimated state vector consists
of the INS error states. As such, the dynamics of the INS error
states must also be modeled. From the error-state dynamic equations, a state transition matrix (STM) is formed via a power-series
expansion of the Jacobian of the system of equations to propagate
the error states from one time step to another. This section reviews
the elements of the INS error-state STM. For more details, comprehensive discussions of INS error-state models are offered by
Groves [39] and Jekeli [43], upon which the discussion herein is
based.
Attitude Error Propagation
A small-angle assumption is made when modeling the dynamics
of the attitude errors. This allows the attitude error to be expressed
as a vector of perturbations in the body frame with respect to the
i
inertial frame, δΨ ib. If the attitude error is expressed as a DCM,
then the error associated with the attitude is
i
δΨ
ib ∧ ≈ I 3 − δ Cb ,
(8)
∧
δΨ
where ib represents the skew-symmetric matrix composed
of small-angle perturbations. The attitude error partial derivatives
are calculated by differentiating the skew-symmetric matrix populated with the small-angle perturbations:
14
i
δΨ
ib ∧ ≈ δ Cb
(9)
where the time derivative attitude error is
δΨ
i b b i b i
ib ∧ ≈ Cb ΩibCi + CbCi Ωib ,
(10)
which reduces to
ˆ ≈ Cˆ i b gyro .
δΨ
ib
b b
(11)
Velocity Error Propagation
In the inertial frame, a body's acceleration is the summation of the
specific force, measured by the accelerometer, and the gravitation
acceleration, which must be modeled:
(12)
a i = vi = f i + γ i
Thus, taking the time derivative of (12) yields
δ vi = f i − f i + γ i − γ i = δ f i − δγ i ,
(13)
which shows that the errors associated with the derivative of the
velocity in the inertial frame are composed of two parts: the error associated with the specific force (i.e., components 1 and 3 in
the later summation) and the error associated with modeling the
gravitation acceleration (i.e., component 2 in the summation):
(
)
2 g rˆi i
δ vˆi = − Cˆ bi fˆ i δΨ ib + e
rˆ δ r i + Cˆ bi bbaccel
T
res rˆ
i2
(14)
In particular, because of the transformation of the IMU-specific
force measurements from the body frame to ECI, the errors associated with the specific force in ECI are a combination of accelerometer measurement errors (i.e., bbaccel) in the body frame and
attitude errors (i.e., δΨib). Furthermore, to model Earth's gravity,
the position in ECI must be known (i.e., δri).
Position Error Propagation
Finally, because the mechanization frame is ECI, the time derivative of the position error is the velocity error:
δ ri = δ vi
(15)
Sensor Bias Dynamics
For IMU accelerometer and rate gyroscope biases, the dynamics
are modeled as first-order Gauss-Markov processes.
System Dynamics and Transition Matrices
Using the previously defined error-state dynamics, the system matrix is defined as follows:
IEEE A&E SYSTEMS MAGAZINE
AUGUST 2017
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