Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C¹ 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, C¹ 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 π. 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 C¹ closed–loop PH curves is possible, although this incurs transcendental terms. However, the C¹ closed–loop PH quintics admit particularly simple rational periodic adapted frames.