Time derivatives of screws with applications to dynamics and stiffness

Abstract

International audienceScrew quantities provide geometric insight into three-dimensional mechanics modeled by rigid bodies and lumped parameters. Four distinct cases of time differentiation are examined by combining fixed and moving body derivatives (fundamental to rigid body mechanics) with material and local derivatives (fundamental to continuum mechanics). Three combinations always yield another screw quantity while the most common, the material derivative with respect to the fixed body, does not. Two fundamental formulations are examined with this last derivative, Euler's Laws and the gravitational loading of an elastic system. By coincidence, the formulations appear screw-like when they are expressed at the center-of-mass but, in contrast to actual screw formulations, they do not retain invariant forms when expressed at arbitrary points

A family of spherical parallel manipulators with two legs

“…The reduced acceleration state must be considered as the time derivative of the velocity state, via a helicoidal field, for details the reader is referred to Gallardo and Rico [32] and Lipkin [33].…”

Stationary acceleration of Frenet curves

“…Now the intrinsic Serret-Frenet formulas are where κ and τ are the curvature and torsion functions of the curve, respectively. The Darboux vector has the properties see [ 6 ], Section 10.2. This means that can be written for the anti-symmetric matrix , which is corresponding to ω .…”

“…Finding the curve with given curvature and torsion functions involves solving a system of differential equations given by the Serret-Frenet relations. This is not straightforward, and solutions are only known in a very few cases as studied by Lipkin [ 6 ] in 2005 and Selig [ 4 ] in 2007. In this work, the ideas of Zefran and Kumar [ 3 ] and Selig [ 4 ] are revisited.…”

Kinematics and dynamics Hessian matrices of manipulators based on screw theory

“…HUNT [31] pointed out that "the acceleration of a rigid body is a very different thing as compared with first-order kinematic quantities, because it cannot be associated with an axis that has a pitch, there being, ordinarily, only one point of the body that, at any instant, has zero acceleration". LIPKIN [32] indicated that the derivative of a twist is not a screw quantity since it does not satisfy the shifting property, and suggested that it is called the spatial acceleration. This is also the reason of the "pseudodual vector" of YANG.…”