## Graphical Visualization For Commutative Sequential Rotations

Conference

2002 Annual Conference

Location

Publication Date

June 16, 2002

Start Date

June 16, 2002

End Date

June 19, 2002

ISSN

2153-5965

Conference Session

Graphics Applications in ME

Page Count

8

Page Numbers

7.595.1 - 7.595.8

DOI

10.18260/1-2--10641

Permanent URL

https://peer.asee.org/10641

633

#### Abstract NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session 2238

Graphical Visualization for commutative Sequential Rotations

Alamgir A. Choudhury, Mitchel J. Keil, Jorge Rodriguez

Western Michigan University

Abstract

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.

1. Introduction

A mathematical model for design and analysis of interconnected mechanical bodies must deal with both translational and rotational motion. In kinematics, vectors are often used to analyze motion when commutativity is applicable to the motion sequence. Sequential finite rotations are encountered in the motion of many bodies such as robotic links. Finite rotations are not commutative [1]. Therefore, in treatment of such motion, one has to ensure that the method does not contradict this characteristic of rotational motion. In design and analysis involving complex rotational motion of interconnected mechanical links, students and practicing professionals in the field need to have a clear comprehension of this problem. Due to the lack of commutativity of finite rotations, in order to achieve a desired position and orientation as a mechanical link undergoes a sequence of rotations, the order of rotation [2] is apriori. The noncommutativity of finite rotations has long been proven

Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright ©2002, American Society for Engineering Education