Analysis of Mechanisms by the Dual Inertia Operator

Abstract

The application of dual numbers to kinematics is based on the principle of transference that extends vector algebra to dual vector (motor) algebra. No such direct extension exists however, for dynamics. Inertia binor is used to obtain the dual momentum, from which the dual equations of motion are derived. This derivation raises the dual dynamic equations to six dimensions, and in fact, it does not act on the dual vector as a whole, but rather on its real and dual parts as two distinct real vectors. In this investigation, the dual inertia operator is introduced. This gives the mass a dual property which has the inverse sense of Clifford’s dual unit, namely, it reduces a motor to a rotor proportional to the vector part of the former, allowing direct relation of dual force to dual acceleration. As a result, the same equation of momentum which holds for linear motion, holds also for angular motion if dual force, dual acceleration, and dual inertia, replace their real counterparts. This approach was implemented in a symbolic computer program. By adding dual number algebra, the program is able to handle dual quantities. Furthermore, applying the dual inertia rules, the dual equations of motion are obtained by replacing real with dual quantities as it is illustrated in the example of a three-degrees-of-freedom robot.