Construction of periodic adapted orthonormal frames on closed space curves

Abstract

The construction of continuous adapted orthonormal frames along C-1 closed-loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid-body motions along smooth closed paths. The construction is illustrated through the simplest non-trivial context - namely, C-1 closed loops defined by a single Pythagorean-hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two-parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of pi. The desired frame is constructed through a rotation applied to the normal-plane vectors of the Euler-Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C-1 closed-loop PH curves is possible, although this incurs transcendental terms. However, the C-1 closed-loop PH quintics admit particularly simple rational periodic adapted frames. (C) 2019 Elsevier B.V. All rights reserved.