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.
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.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fa7c7246e-cc8d-4ae6-b3ae-d0578dd32a72%2Fxmbtjg_processed.png&w=3840&q=75)
Transcribed Image Text: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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 6 images

Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you

Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press

Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON

Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education

Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press

Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON

Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education

Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY

Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning

Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY