Given a [Position Vector] , we can describe it’s orientation as a coordinate system attached to that point. This new coordinate system \(\{B\}\) is relative to the reference coordinate system \(\{A\}\). Describing \(\{B\}\) with respect to \(\{A\}\) will tell us the orientation of the point with respect to \(\{A\}\). We can use a [Rotation Matrix] to describe the orientation of \(\{B\}\) with respect to \(\{A\}\).
There are multiple representations of orientation. Some of them are:
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X-Y-Z fixed angles: This describes the rotation along the x-axis, y-axis and z-axis simultaneously with angles \(\gamma\), \(\beta\) and \(\alpha\) respectively. The rotations are performed along the axis of the fixed coordinate system \(\{A\}\). This is also referred to as roll, pitch and yaw. The rotation matrix for X-Y-Z representation of an orientation is given by \(R_{xyz} = R_z(\alpha) R_y(\beta) R_x(\gamma)\).
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Z-Y-X Euler angles: Rotate \(B\) along \(^BZ\), then along \(^BY\) and finally around \(^BX\). Here the rotations are performed along the axis of the moving coordinate system.
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Z-Y-Z Euler angles: We first rotate along \(^BZ\), then along \(^BY\) and finally along \(^BZ\). The rotations happen along the axes of the moving coordinate system.