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Say we have an object (e.g., end-effector of a robot) that is defined in a general coordinate system by a Cartesian position and Quaternion.

since the Quaternion is with respect to a general and fixed coordinate system, it is rather not intuitive for me how to calculate this Quaternion if the object is rotated locally.

That is, assuming a local coordinate system on this object, while keeping the position of it also fixed, I want to calculate a new Quaternion vector if the object is rotated by small angles (α, β, γ, or pitch, yaw, and roll) with respect to this local frame/coordinate-system.

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It is often useful to create a 2D example and use rigid-body transforms (i.e., $T\in\text{SE}(2)$, where ${\small T=\begin{bmatrix}R&t\\0&1\end{bmatrix}}$, $R\in\text{SO}(2),t\in\mathbb{R}^2$). In this way, the position and rotation are compactly encoded in a single element.

When working through these types of problems, it is important to clearly define notation and coordinate frames. For example, here we will write $T^a_b$ to describe the coordinate frame $\mathcal{F}^b$ with respect to $\mathcal{F}^a$. In other words, $T^a_b$ transforms data in $b$ and expresses it in $a$ (e.g., see more here).

In the following diagram, I have drawn the world frame axis $\mathcal{F}^W$ and an object that has a position and orientation with respect to that world frame (i.e., $T^W_b$). This object has its own coordinate frame $\mathcal{F}^b$. Now, a further transformation $T^b_{b'}$ occurs, where it just so happens that there was no translation. This final coordinate body position and orientation can be expressed in terms of the world frame by composition as

$$ T^W_{b'} = T^W_b T^b_{b'}. $$

Now, to answer your question more concretely, we can consider just the unit quaternion encoding the orientation of $\mathcal{F}^{b'}$ w.r.t $\mathcal{F}^W$, that is, for $q\in S^3$,

$$ q^W_{b'} = q^W_b q^b_{b'}. $$

In your setup, it sounds like you have $q^W_b$ and some Euler angles describing the rotation $q^b_{b'}$. Given everything above, your question seems to become "how do I convert Euler angles to a quaternion," for which many answers and software packages exist (Wikipedia). Please take care to know if your Euler angles are expressed as intrinsic or extrinsic and the correct rotation order. I recommend using something like MATLAB's eul2quat or, in Python, transformations.py (embedded in ROS as tf.transformations) to verify.

coordinate transformations from world to body to body'

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