The Dual Inertia Operator and Its Application to Robot Dynamics

Abstract

The principle of transference states that when dual numbers replace real ones all laws of vector algebra, which describe the kinematics of rigid body with one point fixed, are also valid for motor algebra, which describes a free rigid body. No such direct extension exists, however, for dynamics. Rather, the inertia binor is used to obtain the dual momentum, from which the dual equations of motion are derived. This raises the dual dynamic equations to six dimensions, and in fact, does not act on the dual vector as a whole, but on its real and dual parts as two distinct real vectors. Moreover, in order to obtain the dual force as a derivative of the dual momentum in a correct order, real and dual parts have to be artificially interchanged. In this investigation the dual inertia operator, which allows direct relation of dual momentum to dual velocity, is introduced. It 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. In a way analogous to the principle of transference, the same equation of momentum and its time derivative, which holds for a linear motion, holds for both linear and angular motion of a rigid body if dual force, dual velocity, and dual inertia replace their real counterparts. It is shown that by systematic application of the dual inertia for derivation of the dual momentum and the dual energy, both Newton-Euler and Lagrange formulations of equations of motion are obtained in a complete three-dimensional dual form. As an example, these formulations are used to derive the inverse dual dynamic equations of a robot manipulator.

Dual numbers representation of rigid body dynamics

“…This paper expands, in its first part, the results reported in Brodsky and Shoham's paper [7] and gives a more general expression for the dynamic equations which enables the calculation of the dual momentum and the dual force with respect to an arbitrary point. Then, the moment-product is applied to obtain the energy of a rigid body.…”

“…Only recently, a new dual number formulation of a rigid body dynamics was suggested that casts the equations throughout the derivation into a three-dimensional form [7,32]. This approach is detailed in the next section.…”

“…The recently introduced dual inertia operator [7] suggests a solution to this problem. By giving the mass a dual property, results have shown that it is possible to remain in a three-dimensional dual space throughout the entire derivation of the dynamic equations.…”

Derivation of dual forces in robot manipulators

“…1 The derivation of the inverse dynamic equations of this robot requires the calculation of the Jacobian matrices to each one of the robot links center-of-mass. As mentioned above, (Section 2), the Jacobian matrix is obtained explicitly from the product of the dual transformation matrices, the third column of dual transformation matrix 3 A Ã 0 (rotation about z 0 ), the ®rst column of 3 A Ã 1 (rotation about x 1 ) and the second column of 3 A Ã 2 (translation along y 2 ):…”

“…(6) in a form that includes also the dynamic loads. Using the virtual-work and D'Alembert principles [1] and applying the dual inertia operator [3] the following equation is obtained:…”