How to derive Non-normalized quaternion with respect to time?

• I
• Roni BM
In summary, the time derivative of a normalized quaternion, $$\hat{q}$$, is given by $$\frac{d\hat{q}}{dt}=\frac{1}{2}\hat{q}\cdot \omega$$ where $$\cdot$$ denotes quaternion multiplication. However, for a non-normalized quaternion q, the derivative can be calculated using the chain rule as $$\dot{q}=\left|q\right|\dot{\hat{q}}+\hat{q}\frac{d\left|q\right|}{dt}$$ which can result in a complicated term due to the arbitrary path in a four dimensional real space.
Roni BM
I know that for normalized quaternion, $$\hat{q}$$, the derivative is given by $$\frac{d\hat{q}}{dt}=\frac{1}{2}\hat{q}\cdot \omega$$ where $$\cdot$$ denotes the quaternion multiplication.

I want to calculate the time derivative of a non-normalized quaternion q.

I tried to calculate the derivative by using the chain rule, $$\dot{q}=\left|q\right|\dot{\hat{q}}+\hat{q}\frac{d\left|q\right|}{dt}$$ and I got a very complicated term. I wonder if I am having a wrong approach and if there is a known formula?

I assume you have a time dependent radius ##|q|##, which means you have an arbitrary path in a four dimensional real space. So without any further information, the expression is necessarily general and arbitrary.

1. What is a non-normalized quaternion?

A non-normalized quaternion is a mathematical object that represents a rotation in 3-dimensional space. It is composed of four components: a scalar component and three vector components. Unlike a normalized quaternion, the length of a non-normalized quaternion is not equal to 1.

2. How is a non-normalized quaternion derived?

A non-normalized quaternion can be derived by taking the derivative of a normalized quaternion with respect to time. This involves calculating the derivative of each component of the quaternion separately.

3. Why is it important to derive a non-normalized quaternion with respect to time?

Deriving a non-normalized quaternion with respect to time is important because it allows us to track changes in the rotation of an object over time. This is useful in many applications, such as computer graphics, robotics, and aerospace engineering.

4. What are the applications of a non-normalized quaternion?

A non-normalized quaternion has many applications in mathematics, physics, and engineering. It is commonly used to represent rotations in 3-dimensional space, and it is also used in computer graphics, animation, and robotics.

5. How do you use a non-normalized quaternion in calculations?

To use a non-normalized quaternion in calculations, you can first convert it to a matrix representation. This allows you to perform operations such as multiplication, addition, and inversion. You can then convert the resulting matrix back to a quaternion if needed.

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