Graphical Visualization For Commutative Sequential Rotations

The analysis of rotational motion in articulated mechanisms, and the subsequent design of a system involving sequential rotations is often a tedious task. Thus, simply comprehending the rotational motion required to orient a rigid link or associated reference frame can be quite challenging for students as well as practicing designers. The introduction of a special chain rule allows users to treat finite rotations as vectors i.e. the option to apply the commutative law to the sequence of a specified set of rotations. This chain rule consists of a lemma for forward and backward propagation of the prescribed rotations. The proposed methodology has been demonstrated by using a commercial graphics package to visualize the initial and final orientations of a rigid body following three identical finite rotations composed in three different sequences. The unique final orientation of the link in each of the cases confirms the commutative nature of the constrained rotations. Using this method, the consequence of sequential finite rotations becomes easy to comprehend. Therefore, it can be used as a tool for learning the concepts associated with finite rotations as well as in kinematic analysis involving sequential rotations.