How do you derive the Kinematic Differential Equation of the Euler Parameters? I just want to know how we get the final matrix.For e4dot, e4 = (1/2)sqrt(1 + C11 + C22 + C33), e4dot = (1/4)*(1 + C11 + C22 + C33)^(-1/2) * (C11dot + C22dot + C33dot). From the C11dot, C22dot, and C33 dot equations we get e4dot = -(1/2)*(w1e1 + w2e2 + w3e3). I get how to get e4. How do I get the other 3 Euler Parameters? Please give detailed steps.The final equations should look like the image. The image shows a mathematical expression involving matrices. The expression can be used for educational purposes related to linear algebra or physics, particularly in the context of rotational dynamics or quaternion mathematics.The expression is as follows:\[ \begin{bmatrix} \dot{\epsilon}_1 \\ \dot{\epsilon}_2 \\ \dot{\epsilon}_3 \\ \dot{\epsilon}_4 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} \epsilon_4 & -\epsilon_3 & -\epsilon_2 & \epsilon_1 \\ \epsilon_3 & \epsilon_4 & -\epsilon_1 & \epsilon_2 \\ -\epsilon_2 & \epsilon_1 & \epsilon_4 & \epsilon_3 \\ -\epsilon_1 & -\epsilon_2 & -\epsilon_3 & \epsilon_4 \end{bmatrix}\begin{bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ 0 \end{bmatrix} \]### Explanation:1. **Left Side**: Represents a column vector consisting of the derivatives of the variables \(\epsilon_1\), \(\epsilon_2\), \(\epsilon_3\), and \(\epsilon_4\).2. **Middle Matrix**: A 4x4 matrix that operates on the vector involving angular velocities \(\omega_1\), \(\omega_2\), and \(\omega_3\). The matrix elements are a combination of \(\epsilon_1\), \(\epsilon_2\), \(\epsilon_3\), and \(\epsilon_4\) with appropriate signs.3. **Right Side Vector**: Contains the angular velocities \(\omega_1\), \(\omega_2\), and \(\omega_3\), with a zero as the fourth element.This formulation is typical in quaternion algebra, which is often used to represent orientations and rotations in three-dimensional space.

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Answered: How do you derive the Kinematic…

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Author:Sadiku, Matthew N. O.

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How do you derive the Kinematic Differential Equation of the Euler Parameters? I just want to know how we get the final matrix.

For e4dot, e4 = (1/2)sqrt(1 + C11 + C22 + C33), e4dot = (1/4)*(1 + C11 + C22 + C33)^(-1/2) * (C11dot + C22dot + C33dot). From the C11dot, C22dot, and C33 dot equations we get e4dot = -(1/2)*(w1e1 + w2e2 + w3e3).

I get how to get e4. How do I get the other 3 Euler Parameters? Please give detailed steps.

The final equations should look like the image.

The image shows a mathematical expression involving matrices. The expression can be used for educational purposes related to linear algebra or physics, particularly in the context of rotational dynamics or quaternion mathematics.

The expression is as follows:

\[
\begin{bmatrix}
\dot{\epsilon}_1 \\
\dot{\epsilon}_2 \\
\dot{\epsilon}_3 \\
\dot{\epsilon}_4
\end{bmatrix}
= \frac{1}{2}
\begin{bmatrix}
\epsilon_4 & -\epsilon_3 & -\epsilon_2 & \epsilon_1 \\
\epsilon_3 & \epsilon_4 & -\epsilon_1 & \epsilon_2 \\
-\epsilon_2 & \epsilon_1 & \epsilon_4 & \epsilon_3 \\
-\epsilon_1 & -\epsilon_2 & -\epsilon_3 & \epsilon_4
\end{bmatrix}
\begin{bmatrix}
\omega_1 \\
\omega_2 \\
\omega_3 \\
0
\end{bmatrix}
\]

### Explanation:

1. **Left Side**: Represents a column vector consisting of the derivatives of the variables \(\epsilon_1\), \(\epsilon_2\), \(\epsilon_3\), and \(\epsilon_4\).

2. **Middle Matrix**: A 4x4 matrix that operates on the vector involving angular velocities \(\omega_1\), \(\omega_2\), and \(\omega_3\). The matrix elements are a combination of \(\epsilon_1\), \(\epsilon_2\), \(\epsilon_3\), and \(\epsilon_4\) with appropriate signs.

3. **Right Side Vector**: Contains the angular velocities \(\omega_1\), \(\omega_2\), and \(\omega_3\), with a zero as the fourth element.

This formulation is typical in quaternion algebra, which is often used to represent orientations and rotations in three-dimensional space.

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