Order-free rotations and transformations

Abstract

A serial multibody with a common point and n controllable revolute joints, or a serial multibody, such as a manipulator, with n controllable joints can be moved with any arbitrary order of joint actuations. The final orientation and configuration of the bodies are independent of the order of activation of the joints. We introduce two theorems and develop order-free transformation matrices to show that under specific conditions, the order of transformations of such a multibody is immaterial. The theorems will be applied to a detector camera, and an Eulerian wrist. To apply the order-free transformations to a spherical wrist, we present a classification for the spherical wrists and show that there are only three types of spherical wrists. The Eulerian spherical wrist is mathematically and physically the most applied one.

Keywords:

• Order-free transformations
• Order-free rotations
• Multibody kinematics
• Spherical wrist classification

Additional information

Notes on contributors

R N Jazar

Reza N. Jazar is a professor of mechanical engineering, receiving his master’s degree from Tehran Polytechnic in 1990, specialising in robotics. In 1997, he acquired his PhD from Sharif Institute of Technology in nonlinear dynamics and applied mathematics. Prof Jazar is a specialist in classical and nonlinear dynamics, and has extensive experience in the field of dynamics and mathematical modelling. He has worked in numerous universities worldwide, and through his years of work experience, he has formulated many theorems, innovative ideas, and discoveries in classical dynamics, robotics, control, and nonlinear vibrations. “Razi Acceleration”, Theory of Time Derivative, OrderFree Transformations, Caster Theory, Autodriver Algorithm, Floating-Time Method, Energy-Rate Method, and RMS Optimisation Method are some of his discoveries and innovative ideas. Some of his recent discoveries in kinematics and dynamics were introduced in Advanced Dynamics for the first time. Prof Jazar has made complicated theories understandable by providing some real and applied examples. He has written over 200 scientific papers and technical reports and has authored several more books including Advanced Dynamics; Theory of Applied Robotics; and Vehicle Dynamics.