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0.3.0
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0.3.0
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where
is the skew symmetric form of 
, the body rate with respect to the navigation framesee
The body rate with respect to the navigation frame can be expressed as
![\begin{equation} \label{eq:eq-ImuIntegrator-Mechanization-n-Attitude-omega-nb}
\boldsymbol{\omega}_{nb}^b = \boldsymbol{\omega}_{ib}^b - \mathbf{C}_n^b [\boldsymbol{\omega}_{ie}^n + \boldsymbol{\omega}_{en}^n]
\end{equation}](form_201.svg)
this together with the skew-symmetric matrix transformation (see [17] Groves, ch. 2.3.1, eq. 2.51, p. 45)
leads to the skew symmetric form of 
The gimbal angles 
(roll), 
(pitch), 
(yaw) are related to the body rates as follows:
where
is the rotation about the x-axis (see [47] Titterton, ch. 3.6.3.2, eq. 3.46, p. 41)
is the rotation about the y-axis (see [47] Titterton, ch. 3.6.3.2, eq. 3.45, p. 41)
is the body rate with respect to the navigation frame, expressed in the body frame
see
instead of 
)The quaternion 
propagates as (see [47] Titterton, ch. 3.6.4.3, eq. 3.60-3.62, p. 44)
where 
This can be written in matrix form as
and this can be written in matrix form again as
see
The derivative of the velocity in local-navigation coordinates can be expressed as
![\begin{equation} \label{eq:eq-ImuIntegrator-Mechanization-n-Velocity}
\boldsymbol{\dotup{v}}^n
= \overbrace{\boldsymbol{f}^n}^{\hidewidth\text{measured}\hidewidth}
-\ \underbrace{(2 \boldsymbol{\omega}_{ie}^n + \boldsymbol{\omega}_{en}^n) \times \boldsymbol{v}^n}_{\text{coriolis acceleration}}
+\ \overbrace{\boldsymbol{\gamma}^n}^{\hidewidth\text{gravitation}\hidewidth}
-\ \mathbf{C}_e^n \cdot \underbrace{\left(\boldsymbol{\omega}_{ie}^e \times [ \boldsymbol{\omega}_{ie}^e \times \mathbf{x}^e ] \right)}_{\text{centrifugal acceleration}}
\end{equation}](form_219.svg)
where
is the velocity with respect to the Earth in local-navigation frame coordinates,
is the specific force vector as measured by a triad of accelerometers and resolved into local-navigation frame coordinates
is the turn rate of the Earth expressed in local-navigation frame coordinates
is the turn rate of the local frame with respect to the Earth-fixed frame, called the transport rate, expressed in local-navigation frame coordinates
the local gravitation vector (caused by effects of mass attraction)see
where
is the latitude, longitude and altitude above the ellipsoid,
is the North/South (meridian) earth radius,
is the East/West (prime vertical) earth radiussee
Different books have different forms of the quaternion propagation, therefore this shall be discussed here.
The matrix form for the propagation is (see [47] Titterton, ch. 11.2.5, eq. 11.33-11.35, p. 319)
which can be converted into a direction cosine matrix with
the quaternion is defined as (see [47] Titterton, ch. 3.6.4.2, eq. 3.57, p. 43)
Jekeli defines the equations from the body frame to an arbitrary frame, which in our case is the n-frame. Therefore his equations become ([24] Jekeli, ch. 4.2.3.1, eq. 4.20, p. 114)
The quaternion is defined by Jekeli over the direction cosine matrix which is ([24] Jekeli, ch. 4.2.3.1, eq. 4.22, p. 114)
Comparing Jekeli's DCM 
with Titterton's 
, it can easily be shown, that the quaternion is the complex conjugate (see [24] Jekeli, ch. 4.2.3.1, eq. 4.21,4.23, p. 114f)
Replacing the complex conjugated quaternion leads to the same equation as Titterton's 
Groves defines his Quaternion update in two arbitrary frames ([17] Groves, Appendix E, ch. E.6.2, eq. E.32, p. E-12)
This can be expresed for the body rate with respect to the navigation frame in body frame coordinates
which is the same as Titterton and Jekeli
1.13.2