Derivation of the Rotation Matrix for an Axis-Angle Rotation Based on an Intuitive Interpretation of the Rotation Matrix

Abstract

In this paper, we present the derivation of the rotation matrix for an axisangle representation of rotation. The problem is of finding out the rotation matrix corresponding to the rotation of a reference frame, by a certain angle, about an arbitrary axis passing through its origin. The axis-angle representation is particularly useful in computer graphics and rigid body motion. We have used an intuitive interpretation of the rotation matrix for this derivation. The intuitive interpretation is as follows: the columns of the rotation matrix are the coordinate axes unit vectors of the rotated frame as seen from the fixed frame. This interpretation of the rotation matrix helps in quick computation of the rotation matrices required to obtain the required rotation matrix. The required rotation matrix can be computed from just two rotation matrices which are easy to find and intuitive to understand. The derivation presented in simpler to understand than presented in most books and can be grasped with minimum geometric visualization.

Floating wave energy harvester: a new perspective

“…However, when the float experienced ripples, CS sensor orientation rotated concerning CS world , causing the accelerometer's Z-axis acceleration to deviate from the vertical Frontiers in Energy Research frontiersin.org acceleration. To address this, the equivalent rotation angle-axis pair (Hota and Kumar, 2022) was employed to describe the chosen oriented reference system. Here, the new position {1} of the bodybound CS results from a rotation of the CS {0} by an angle θ around an axis k [k x k y k z ] T .…”