Piecewise rational rotation-minimizing motions via data stream interpolation

TL;DR

This paper introduces a new geometric method for constructing smooth, rotation-minimizing motions along space curves using rational quintic curves, enabling effective interpolation of position and orientation streams.

Contribution

It presents a novel geometric characterization and a local algorithm for interpolating position streams with rational rotation-minimizing frames, improving motion design techniques.

Findings

The method achieves globally continuous rational rotation-minimizing frames.

Numerical experiments demonstrate the effectiveness on synthetic and real data.

The approach provides smooth, interpolated motions with controlled orientation.

Abstract

When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric $G^1$ spline curve whose segments are quintic RRMFs, with a globally continuous piecewise rational rotation-minimizing frame. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams.