# Quaternion derivative ambiguity

• softec17
In summary, there is a discrepancy in the definition of the quaternion derivative in the book Quaternions and Rotation Sequences by Jack B. Kuipers and other published papers. While the book defines it as q(t)\overline{\omega}(t), the papers define it as \frac{1}{2}q(t)\omega(t). This could be due to different nomenclature or a factor of 2 difference in the definition of \omega. The quaternion is commonly used to transform a vector from the fixed frame to the body frame, while \omega(t) represents the angular velocity of the body axis with respect to the fixed frame.
softec17
In Quaternions and Rotation Sequences by Jack B. Kuipers (pg. 264-265)
the quaternion derivative is defined as:

$$\frac{dq}{dt}=q(t)\overline{\omega}(t)$$

But in many published papers, I have seen the derivative defined instead as

$$\frac{dq}{dt}=\frac{1}{2}q(t)\omega(t)$$

Why is there a discrepancy? Is there some nomenclature that is different. I am assuming \omega(t) to be the angular velocity of the body axis wrt the fixed frame and the quaternion is used to transform a vector from the fixed frame to the body frame.

Thanks for any help.

Probably the two definitions of $$\omega$$ differ by the factor of 2.

## 1. What is quaternion derivative ambiguity?

Quaternion derivative ambiguity refers to the issue of multiple possible ways to define the derivative of a quaternion function. This is due to the non-commutative nature of quaternions, meaning that the order in which quaternion multiplication is performed affects the result. Therefore, there is no single, universally accepted definition of quaternion derivative.

## 2. Why is quaternion derivative ambiguity important?

Quaternion derivative ambiguity is important because it affects the accuracy and validity of calculations involving quaternion derivatives. Depending on the chosen definition, the resulting derivatives may differ significantly, leading to different interpretations and conclusions. It is necessary to carefully consider and specify the definition of quaternion derivative in order to accurately represent and analyze quaternion functions.

## 3. How is quaternion derivative ambiguity addressed?

One approach to addressing quaternion derivative ambiguity is to use the right- and left-hand derivatives, which account for the non-commutativity of quaternion multiplication. Another method is to use a symmetric definition, which takes the average of the right- and left-hand derivatives. It is also important to clearly define and specify the order in which quaternion multiplication is performed to avoid ambiguity.

## 4. What are the potential consequences of ignoring quaternion derivative ambiguity?

Ignoring quaternion derivative ambiguity can lead to incorrect results and interpretations in calculations. This can have significant consequences in applications where accuracy is crucial, such as robotics, computer graphics, and physics simulations. It is important to carefully consider and address quaternion derivative ambiguity in order to ensure the validity of results.

## 5. Are there any benefits to quaternion derivative ambiguity?

While quaternion derivative ambiguity can pose challenges in calculations, it also allows for more flexibility and versatility in representing and analyzing quaternion functions. Different definitions of quaternion derivative may be more suitable for different applications, and the ability to choose between them can be beneficial. However, it is important to carefully consider and specify the chosen definition to avoid ambiguity and ensure accurate results.

• Calculus
Replies
1
Views
920
• Aerospace Engineering
Replies
2
Views
2K
• Topology and Analysis
Replies
4
Views
509
• Quantum Physics
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
5
Views
695
• Classical Physics
Replies
21
Views
1K
• Classical Physics
Replies
1
Views
560
• Classical Physics
Replies
10
Views
7K
• Mechanics
Replies
2
Views
715
• Classical Physics
Replies
1
Views
645