Mathematics
A rotational testbed may be comprised of any number of rotating axes. We will designate the total number of axes as \(A\) and an individual axis as \(a\) where \(a = 1 \dots A\). The most typical rotational testbed include:
A basic centrifuge, or a single axis rate table \(\rightarrow A = 1\)
A centrifuge with a counter-rotating platform, or a two axis rate table \(\rightarrow A = 2\)
A centrifuge with a two-axis platform, or a three axis rate table \(\rightarrow A = 3\)
The axis designated as \(a=1\) is the axis to which a system under test (SUT) is rigidly attached. Axes are then number outwardly such that a rotation of the axis moves all lower axes. For example, axis \(a=2\) only moves axis \(a=1\) and axis \(a=3\) moves both axes \(a=2\) and \(a=1\).
Angular Rate
For a single axis testbed \((A=1)\), the rate of rotation of the body \((\mathrm{b})\) frame relative to the inertial \((\mathrm{i})\) frame resolved in the inertial frame is the sum of the rates of rotation of all the intermediate frames.
\[\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) = \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t) + \boldsymbol{\omega}^\mathrm{i}_{\mathrm{e}\mathrm{n}}(t) + \boldsymbol{\omega}^\mathrm{i}_{\mathrm{n}{\zeta_1}}(t) + \boldsymbol{\omega}^\mathrm{i}_{{\zeta_1}{\mu_1}}(t) + \boldsymbol{\omega}^\mathrm{i}_{{\mu_1}{\rho_1}}(t) + \boldsymbol{\omega}^\mathrm{i}_{{\rho_1}\mathrm{m}}(t) + \boldsymbol{\omega}^\mathrm{i}_{\mathrm{m}\mathrm{b}}(t)\]
Note
A detailed discussion of the pertinent coordinate frames and their relational parameters may be found in Coordinate Frames.
All of the right hand terms other than \(\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\) are difficult to measure in the inertial frame. Therefore, we convert them to the frames in which each parameter may be easily measured to get:
\[\begin{split}\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{\omega}^\mathrm{e}_{\mathrm{e}\mathrm{n}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \mathbf{C}^\mathrm{e}_\mathrm{n}\; \boldsymbol{\omega}^\mathrm{n}_{\mathrm{n}{\zeta_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \mathbf{C}^\mathrm{e}_\mathrm{n}\; \mathbf{C}^\mathrm{n}_{\zeta_1}\; \boldsymbol{\omega}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \mathbf{C}^\mathrm{e}_\mathrm{n}\; \mathbf{C}^\mathrm{n}_{\zeta_1}\; \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \boldsymbol{\omega}^{\mu_1}_{{\mu_1}{\rho_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \mathbf{C}^\mathrm{e}_\mathrm{n}\; \mathbf{C}^\mathrm{n}_{\zeta_1}\; \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \mathbf{C}^{\mu_1}_{\rho_1}\; \boldsymbol{\omega}^{\rho_1}_{{\rho_1}\mathrm{m}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \mathbf{C}^\mathrm{e}_\mathrm{n}\; \mathbf{C}^\mathrm{n}_{\zeta_1}\; \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \mathbf{C}^{\mu_1}_{\rho_1}\; \mathbf{C}^{\rho_1}_\mathrm{m} \; \boldsymbol{\omega}^\mathrm{m}_{\mathrm{m}\mathrm{b}}(t)\end{split}\]
Using the relational parameters discussed in Coordinate Frames, the angular rate reduces to:
\[\begin{split}\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\boldsymbol{\omega}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\boldsymbol{\omega}^{\mu_1}_{{\mu_1}{\rho_1}}(t)\end{split}\]
Performing the same procedure for a two axis testbed \((A=2)\) and a three axis testbed \((A=3)\) resolves to:
\[\begin{split}\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\boldsymbol{\omega}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\boldsymbol{\omega}^{\mu_2}_{{\mu_2}{\rho_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\boldsymbol{\omega}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\boldsymbol{\omega}^{\mu_1}_{{\mu_1}{\rho_1}}(t)\end{split}\]
and
\[\begin{split}\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\boldsymbol{\omega}^{\zeta_3}_{{\zeta_3}{\mu_3}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\boldsymbol{\omega}^{\mu_3}_{{\mu_3}{\rho_3}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\boldsymbol{\omega}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\boldsymbol{\omega}^{\mu_2}_{{\mu_2}{\rho_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\boldsymbol{\omega}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\boldsymbol{\omega}^{\mu_1}_{{\mu_1}{\rho_1}}(t)\end{split}\]
This progression continues for any positive number of testbed axes. However, these three equations are enough to notice a pattern. As a result, we may define the following variables:
\[\begin{split}\boldsymbol{\omega}_0(t) &= 0\\[1em]
\boldsymbol{\omega}_1(t) &= \mathbf{C}^\mathrm{h}_{\zeta_1}\; \left[\boldsymbol{\omega}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t) + \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \boldsymbol{\omega}^{\mu_1}_{{\mu_1}{\rho_1}}(t) + \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \mathbf{C}^{\mu_1}_{\rho_1}(t)\; \boldsymbol{\omega}_0(t)\right]\\[1em]
\boldsymbol{\omega}_2(t) &= \mathbf{C}^\mathrm{h}_{\zeta_2}\; \left[\boldsymbol{\omega}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t) + \mathbf{C}^{\zeta_2}_{\mu_2}(t)\; \boldsymbol{\omega}^{\mu_2}_{{\mu_2}{\rho_2}}(t) + \mathbf{C}^{\zeta_2}_{\mu_2}(t)\; \mathbf{C}^{\mu_2}_{\rho_2}(t)\; \boldsymbol{\omega}_1(t)\right]\\[1em]
\boldsymbol{\omega}_3(t) &= \mathbf{C}^\mathrm{h}_{\zeta_3}\; \left[\boldsymbol{\omega}^{\zeta_3}_{{\zeta_3}{\mu_3}}(t) + \mathbf{C}^{\zeta_3}_{\mu_3}(t)\; \boldsymbol{\omega}^{\mu_3}_{{\mu_3}{\rho_3}}(t) + \mathbf{C}^{\zeta_3}_{\mu_3}(t)\; \mathbf{C}^{\mu_3}_{\rho_3}(t)\; \boldsymbol{\omega}_2(t)\right]\end{split}\]
Using these definitions, the three angular rate equations simplify to:
\[\begin{split}\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) &= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\omega}_1(t)\\[1em]
\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) &= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\omega}_2(t)\\[1em]
\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) &= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\omega}_3(t)\end{split}\]
Now, using the definitions and final angular rate equation forms we may define modular equations for the angular rate of a general testbed \((A>0)\) as:
\[\begin{split}\boldsymbol{\omega}_0(t) &= 0\\[1em]
\boldsymbol{\omega}_a(t) &= \mathbf{C}^\mathrm{h}_{\zeta_a}\; \left[\boldsymbol{\omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \boldsymbol{\omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t) + \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\; \boldsymbol{\omega}_{a-1}(t)\right]\\[1em]
\boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) &= \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\omega}_A(t)\end{split}\]
Angular Acceleration
The angular acceleration of a general testbed is produced by applying the sum and product rules to the angular rate equations to provide:
\[\begin{split}\begin{align}
\dot{\boldsymbol{\omega}}_0(t) &= 0\\[1em]
\dot{\boldsymbol{\omega}}_a(t)
&= \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\dot{\boldsymbol{\omega}}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\\
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\dot{\mathbf{C}}^{\zeta_a}_{\mu_a}(t)\; \boldsymbol{\omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\\
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\dot{\boldsymbol{\omega}}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\\
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\dot{\mathbf{C}}^{\zeta_a}_{\mu_a}(t)\;
\mathbf{C}^{\mu_a}_{\rho_a}(t)\;
\boldsymbol{\omega}_{a-1}(t)\\
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\dot{\mathbf{C}}^{\mu_a}_{\rho_a}(t)\;
\boldsymbol{\omega}_{a-1}(t)\\
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\mathbf{C}^{\mu_a}_{\rho_a}(t)\;
\dot{\boldsymbol{\omega}}_{a-1}(t)\\[1em]
\dot{\boldsymbol{\omega}}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \dot{\mathbf{C}}^\mathrm{i}_\mathrm{n}(t)\;
\boldsymbol{\omega}_A(t)
+ \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\;
\dot{\boldsymbol{\omega}}_A(t)
\end{align}\end{split}\]
Applying the first derivative of the DCMs (see DCM Derivatives) to remove the need for calculating the derivatives numerically, produces the angular acceleration equations:
\[\begin{split}\begin{align}
\dot{\boldsymbol{\omega}}_0(t) &= 0\\[1em]
\dot{\boldsymbol{\omega}}_a(t)
&= \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\dot{\boldsymbol{\omega}}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\\\notag
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\boldsymbol{\omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\\\notag
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\dot{\boldsymbol{\omega}}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\\\notag
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\mathbf{C}^{\mu_a}_{\rho_a}(t)\;
\boldsymbol{\omega}_{a-1}(t)\\\notag
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\boldsymbol{\Omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\;
\boldsymbol{\omega}_{a-1}(t)\\\notag
&+ \mathbf{C}^\mathrm{h}_{\zeta_a}\;
\mathbf{C}^{\zeta_a}_{\mu_a}(t)\;
\mathbf{C}^{\mu_a}_{\rho_a}(t)\;
\dot{\boldsymbol{\omega}}_{a-1}(t)\\[1em]
\dot{\boldsymbol{\omega}}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\;
\mathbf{C}^\mathrm{i}_\mathrm{n}(t)\;
\boldsymbol{\omega}_A(t) + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \dot{\boldsymbol{\omega}}_A(t)
\end{align}\end{split}\]
Linear Position
As with the angular rate, the linear position of the \(\mathrm{b}\) frame relative to the \(\mathrm{i}\) frame resolved in the \(\mathrm{i}\) frame for a single axis testbed \((A=1)\) is the sum of the positions of all the intermediate frames.
\[\boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
= \boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)
+ \boldsymbol{r}^\mathrm{i}_{\mathrm{e}\mathrm{n}}(t)
+ \boldsymbol{r}^\mathrm{i}_{\mathrm{n}{\zeta_1}}(t)
+ \boldsymbol{r}^\mathrm{i}_{{\zeta_1}{\mu_1}}(t)
+ \boldsymbol{r}^\mathrm{i}_{{\mu_1}{\rho_1}}(t)
+ \boldsymbol{r}^\mathrm{i}_{{\rho_1}\mathrm{m}}(t)
+ \boldsymbol{r}^\mathrm{i}_{\mathrm{m}\mathrm{b}}(t)\]
All of the right hand terms except \(\boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\) are difficult to measure in the \(\mathrm{i}\) frame. Therefore, we convert them to frames in which the parameters may be easily measured:
\[\begin{split}\boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\boldsymbol{r}^\mathrm{n}_{\mathrm{n}{\zeta_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\boldsymbol{r}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\boldsymbol{r}^{\mu_1}_{{\mu_1}{\rho_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\mathbf{C}^{\rho_1}_\mathrm{m}\;
\boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}(t)\end{split}\]
Using the relational parameters discussed in Coordinate Frames, the linear position reduces to:
\[\begin{split}\boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)
&= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\boldsymbol{r}^\mathrm{n}_{\mathrm{n}{\zeta_1}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\boldsymbol{r}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\mathbf{C}^{\rho_1}_\mathrm{m}\;
\boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\end{split}\]
Performing the same procedure for a two axis testbed \((A=2)\) and a three axis testbed \((A=3)\) gives:
\[\begin{split}\boldsymbol{r}^\mathrm{i}_{\mathrm{i}b}(t)
&= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\boldsymbol{r}^\mathrm{n}_{\mathrm{n}{\zeta_2}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\boldsymbol{r}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\boldsymbol{r}^{\rho_2}_{{\rho_2}{\zeta_1}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\boldsymbol{r}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\mathbf{C}^{\rho_1}_\mathrm{m}\;
\boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\end{split}\]
and
\[\begin{split}\boldsymbol{r}^\mathrm{i}_{\mathrm{i}b}(t)
&= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\boldsymbol{r}^\mathrm{n}_{\mathrm{n}{\zeta_3}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\boldsymbol{r}^{\zeta_3}_{{\zeta_3}{\mu_3}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\boldsymbol{r}^{\rho_3}_{{\rho_3}{\zeta_2}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\boldsymbol{r}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\boldsymbol{r}^{\rho_2}_{{\rho_2}{\zeta_1}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\boldsymbol{r}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t)\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}}\\\notag
&+ \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\;
\mathbf{C}^\mathrm{e}_\mathrm{n}\;
\mathbf{C}^\mathrm{n}_{\zeta_3}\;
\mathbf{C}^{\zeta_3}_{\mu_3}(t)\;
\mathbf{C}^{\mu_3}_{\rho_3}(t)\;
\mathbf{C}^{\rho_3}_{\zeta_2}\;
\mathbf{C}^{\zeta_2}_{\mu_2}(t)\;
\mathbf{C}^{\mu_2}_{\rho_2}(t)\;
\mathbf{C}^{\rho_2}_{\zeta_1}\;
\mathbf{C}^{\zeta_1}_{\mu_1}(t)\;
\mathbf{C}^{\mu_1}_{\rho_1}(t)\;
\mathbf{C}^{\rho_1}_\mathrm{m}\;
\boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\end{split}\]
Like with the angular rate, the progression continues for any positive number of testbed axes and a pattern is identified. If we define the following:
\[\begin{split}\mathbf{C}^{\zeta_1}_{\rho_1}(t) &= \mathbf{C}^{\zeta_1}_{\mu_1}(t)\; \mathbf{C}^{\mu_1}_{\rho_1}(t)\\[1em]
\mathbf{C}^{\zeta_2}_{\rho_2}(t) &= \mathbf{C}^{\zeta_2}_{\mu_2}(t)\; \mathbf{C}^{\mu_2}_{\rho_2}(t)\\[1em]
\mathbf{C}^{\zeta_3}_{\rho_3}(t) &= \mathbf{C}^{\zeta_3}_{\mu_3}(t)\; \mathbf{C}^{\mu_3}_{\rho_3}(t)\\[1em]
\boldsymbol{\alpha}_0(t) &= \boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}} + \mathbf{C}^{\rho_1}_\mathrm{m}\; \boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\\[1em]
\boldsymbol{\alpha}_1(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_1}} + \mathbf{C}^\mathrm{h}_{\zeta_1} \left[\boldsymbol{r}^{\zeta_1}_{{\zeta_1}{\mu_1}}(t) + \mathbf{C}^{\zeta_1}_{\rho_1}(t)\; \boldsymbol{\alpha}_0(t)\right]\\[1em]
\boldsymbol{\alpha}_2(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_2}} + \mathbf{C}^\mathrm{h}_{\zeta_2} \left[\boldsymbol{r}^{\zeta_2}_{{\zeta_2}{\mu_2}}(t) + \mathbf{C}^{\zeta_2}_{\rho_2}(t)\; \boldsymbol{\alpha}_1(t)\right]\\[1em]
\boldsymbol{\alpha}_3(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_3}} + \mathbf{C}^\mathrm{h}_{\zeta_3} \left[\boldsymbol{r}^{\zeta_3}_{{\zeta_3}{\mu_3}}(t) + \mathbf{C}^{\zeta_3}_{\rho_3}(t)\; \boldsymbol{\alpha}_2(t)\right]\end{split}\]
the three angular rate equations simplify to:
\[\begin{split}\boldsymbol{r}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) &= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_1(t)\\[1em]
\boldsymbol{r}^\mathrm{i}_{\mathrm{i}b}(t) &= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_2(t)\\[1em]
\boldsymbol{r}^\mathrm{i}_{\mathrm{i}b}(t) &= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_3(t)\end{split}\]
and the linear position of of a general testbed \((A>0)\) is calculated as:
\[\begin{split}\mathbf{C}^{\zeta_a}_{\rho_a}(t) &= \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\[1em]
\boldsymbol{\alpha}_0(t) &= \boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}} + \mathbf{C}^{\rho_1}_\mathrm{m}\; \boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\\[1em]
\boldsymbol{\alpha}_a(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_a}} + \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{r}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t)\right]\\[1em]
\boldsymbol{r}^\mathrm{i}_{\mathrm{i}b}(t) &= \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}} + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_A(t)\end{split}\]
Linear Acceleration
The linear acceleration of a general testbed is produced by applying the sum and product rules to the linear position equations twice to provide:
\[\begin{split}\mathbf{C}^{\zeta_a}_{\rho_a}(t) &= \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\[1em]
\dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t) &= \dot{\mathbf{C}}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t) + \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \dot{\mathbf{C}}^{\mu_a}_{\rho_a}(t)\\[1em]
\ddot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t) &= \ddot{\mathbf{C}}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t) + 2\; \dot{\mathbf{C}}^{\zeta_a}_{\mu_a}(t)\; \dot{\mathbf{C}}^{\mu_a}_{\rho_a}(t) + \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \ddot{\mathbf{C}}^{\mu_a}_{\rho_a}(t)\\[1em]
\boldsymbol{\alpha}_0(t) &= \boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}} + \mathbf{C}^{\rho_1}_\mathrm{m}\; \boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\\[1em]
\dot{\boldsymbol{\alpha}}_0(t) &= 0\\[1em]
\ddot{\boldsymbol{\alpha}}_0(t) &= 0\\[1em]
\boldsymbol{\alpha}_a(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_a}} + \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{r}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t)\right]\\[1em]
\dot{\boldsymbol{\alpha}}_a(t) &= \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{v}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \dot{\boldsymbol{\alpha}}_{a-1}(t)\right]\\[1em]
\ddot{\boldsymbol{\alpha}}_a(t) &= \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{a}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \ddot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t) + 2\; \dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \dot{\boldsymbol{\alpha}}_{a-1}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \ddot{\boldsymbol{\alpha}}_{a-1}(t)\right]\\[1em]
\boldsymbol{a}^\mathrm{i}_{\mathrm{i}b}(t)
&= \ddot{\mathbf{C}}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}
+ \ddot{\mathbf{C}}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_A(t)
+ 2\; \dot{\mathbf{C}}^\mathrm{i}_\mathrm{n}(t)\; \dot{\boldsymbol{\alpha}}_A(t)
+ \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \ddot{\boldsymbol{\alpha}}_A(t)\end{split}\]
We can then apply the DCM Derivatives to produces the final linear acceleration equations:
\[\begin{split}\mathbf{C}^{\zeta_a}_{\rho_a}(t) &= \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\[1em]
\dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t) &= \boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)
+ \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \boldsymbol{\Omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\[1em]
\ddot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t) &= \dot{\boldsymbol{\Omega}}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\
&+ \boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\
&+ 2\; \boldsymbol{\Omega}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t)\; \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \boldsymbol{\Omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\
&+ \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \dot{\boldsymbol{\Omega}}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\
&+ \mathbf{C}^{\zeta_a}_{\mu_a}(t)\; \boldsymbol{\Omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \boldsymbol{\Omega}^{\mu_a}_{{\mu_a}{\rho_a}}(t)\; \mathbf{C}^{\mu_a}_{\rho_a}(t)\\[1em]
\boldsymbol{\alpha}_0(t) &= \boldsymbol{r}^{\rho_1}_{{\rho_1}\mathrm{m}} + \mathbf{C}^{\rho_1}_\mathrm{m}\; \boldsymbol{r}^\mathrm{m}_{\mathrm{m}\mathrm{b}}\\[1em]
\dot{\boldsymbol{\alpha}}_0(t) &= 0\\[1em]
\ddot{\boldsymbol{\alpha}}_0(t) &= 0\\[1em]
\boldsymbol{\alpha}_a(t) &= \boldsymbol{r}^\mathrm{h}_{\mathrm{h}{\zeta_a}} + \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{r}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t)\right]\\[1em]
\dot{\boldsymbol{\alpha}}_a(t) &= \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{v}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \dot{\boldsymbol{\alpha}}_{a-1}(t)\right]\\[1em]
\ddot{\boldsymbol{\alpha}}_a(t) &= \mathbf{C}^\mathrm{h}_{\zeta_a} \left[\boldsymbol{a}^{\zeta_a}_{{\zeta_a}{\mu_a}}(t) + \ddot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \boldsymbol{\alpha}_{a-1}(t) + 2\; \dot{\mathbf{C}}^{\zeta_a}_{\rho_a}(t)\; \dot{\boldsymbol{\alpha}}_{a-1}(t) + \mathbf{C}^{\zeta_a}_{\rho_a}(t)\; \ddot{\boldsymbol{\alpha}}_{a-1}(t)\right]\\[1em]
\boldsymbol{a}^\mathrm{i}_{\mathrm{i}b}(t)
&= \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\; \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\; \mathbf{C}^\mathrm{i}_\mathrm{e}(t)\; \boldsymbol{r}^\mathrm{e}_{\mathrm{e}\mathrm{n}}\\
&+ \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\; \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\; \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \boldsymbol{\alpha}_A(t)\\
&+ 2\; \boldsymbol{\Omega}^\mathrm{i}_{\mathrm{i}\mathrm{e}}(t)\; \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \dot{\boldsymbol{\alpha}}_A(t)\\
&+ \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \ddot{\boldsymbol{\alpha}}_A(t)\end{split}\]
Specific Force
The specific force \((f)\) sensed by the system under test resolved in the inertial frame is the sum of the linear acceleration of the body frame and gravitational acceleration.
\[\boldsymbol{f}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) = \boldsymbol{a}^\mathrm{i}_{\mathrm{i}b}(t) + \mathbf{g}^\mathrm{i}_{\mathrm{i}\mathrm{n}}\]
Gravity is difficult to measure in the inertial frame. However, it is easy to measure in the local navigation frame.
\[\begin{split}\mathbf{g}^\mathrm{n}_{\mathrm{i}\mathrm{n}} = \begin{bmatrix}0\\0\\-\mathrm{g}_n\end{bmatrix}\end{split}\]
Therefore, the specific force in the inertial frame is:
\[\boldsymbol{f}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t) = \boldsymbol{a}^\mathrm{i}_{\mathrm{i}b}(t) + \mathbf{C}^\mathrm{i}_\mathrm{n}(t)\; \mathbf{g}^\mathrm{n}_{\mathrm{i}\mathrm{n}}\]
SUT Inputs
\[\mathbf{x}^\mathrm{b}_{\mathrm{i}\mathrm{b}}(t) = \mathbf{C}^\mathrm{b}_\mathrm{i}(t)\; \mathbf{x}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)\]
So
\[\begin{split}\boldsymbol{\omega}^\mathrm{b}_{\mathrm{i}\mathrm{b}}(t) &= \mathbf{C}^\mathrm{b}_\mathrm{i}(t)\; \boldsymbol{\omega}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)\\[1em]
\dot{\boldsymbol{\omega}}^\mathrm{b}_{\mathrm{i}\mathrm{b}}(t) &= \mathbf{C}^\mathrm{b}_\mathrm{i}(t)\; \dot{\boldsymbol{\omega}}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)\\[1em]
\boldsymbol{f}^\mathrm{b}_{\mathrm{i}\mathrm{b}}(t) &= \mathbf{C}^\mathrm{b}_\mathrm{i}(t)\; \boldsymbol{f}^\mathrm{i}_{\mathrm{i}\mathrm{b}}(t)\\[1em]\end{split}\]