k Mechanisms, Transmissions, ^. ' a n d Automation in §ffs?Design • •:.;/" Instantaneous Properties of Multi-Degrees- of-Freedom Motions- Line Trajectories

J. M. McCarthy1 A line fixed in a body that has one degree of freedom will move along a ruled surface in the fixed reference frame. If the body has two degrees of freedom the set of available locations for the line is a two dimensional ex-tent of lines. Similarly three and four degree of freedom mo-tion provide three and four dimensional extents for the loca-tion of a line. The kinematics of multi-degree-of-freedom mo-tion seeks to characterize these extents of lines in terms of the prescribed motion. Previous work in this area has focussed on determining preferred coordinate systems and special configurations of lines. See, for example, Nayak and Roth (1981), Pennock and Yang (1985), and Stanisic and Pennock (1986). In this paper the authors introduce the geometric parameters, coL, p, and p*, to describe the /^-dimensional line extents. They examine the first order instantaneous kinematic properties in each case, n = 1, 2, 3, and 4, and the second order properties for n = 1, 2. This work is an important contribution to the kinematic theory of multi-degree-of-freedom motion, and I highly recommend its careful and critical study to those interested in this area. In McCarthy (1986), I discuss the geometric parameters which characterize a ruled surface, and how they appear in the scalar and dual vector formulations. In what follows, I would like to establish the connection between the geometric parameters formulated in this paper and those traditionally used in the differential geometry of ruled surfaces. In the equations (4)-(9), the authors derive the fundamental properties of a ruled surface without reference to its directrix