Angle-Set Conventions

Ways to rotate around an axis with rotational matrices.

$$R_{\text{euler}}(x', y', z') = R_{\text{fixed}}(z, y, x) \Rightarrow R_{x'} \cdot R_{y'} \cdot R_{z'} = R_{x} \cdot R_{y} \cdot R_{z}$$

Fixed Angles $$ R_Z(45\deg) \cdot \ ^\text{Base}_\text{TCP}R \Rightarrow \text{Rotate toolhead } 45 \deg \text{round the \textit{base} z-axis} $$

Euler Angles $$ ^\text{Base}_\text{TCP}R \cdot R_Z(45\deg) \Rightarrow \text{Rotate toolhead } 45 \deg \text{round the \textit{toolhead} z-axis} $$

Fixed Angles

The coordinate system stays fixed.

Here we left-multiply (from right to left) the rotational matrices.

“Each rotation is performed about an axis of the coordinate system from the last step” - Iñigo Iturrate

The coordinate system changes whenever you do a rotation.