Piecewise rational rotation-minimizing motions via data stream interpolation

TL;DR

The paper tackles the problem of interpolating 3D position streams with orientation while maintaining rotation-minimizing frames (RMFs). It develops a local $G^1$ Hermite interpolation framework using quintic Pythagorean-hodograph (PH) curves whose RMFs are rational (RRMFs), focusing on quintics of class I. A novel geometric characterization on the unit sphere enables a local algorithm that computes the segment, its RMF, and a compatible tangent while preserving a globally continuous RMF; a bisection strategy solves the angular alignment required to match endpoint displacements. The method extends to a global spline extension, with a simple point-insertion mechanism ensuring solvability for arbitrary input streams and real-time feasibility, demonstrated through numerical experiments on synthetic and synthetic-like data.

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.

Figures (15)

• Figure 1: Example of a degree 4 tangent indicatrix (solid black line) of a regular PH quintic with its spherical control polygon (dashed black line) and spherical control points ${\bf s}_i, i=0,\ldots,4$ (black dots).
• Figure 2: Two unit vectors ${\bf v}$, ${\bf w}$ and their unit bisector ${\bf b}({\bf v},{\bf w})$ all depicted as spherical points (black dots) on the unit sphere, together with the great circle ${\mathcal{C}}({\bf v},{\bf w})$ (red line) and the negatively oriented normalized cross product ${\bf n}({\bf v}, {\bf w})$, represented as a vector (black arrow) applied at the point ${\bf b}({\bf v},{\bf w})$.
• Figure 3: The spherical points ${\bf s}_0, \ldots, {\bf s}_4$ (black dots) of an RRMF5-I curve and the corresponding spherical control polygon (dashed black line). Each great circle (red lines) where ${\bf s}_1$ (left), ${\bf s}_2$ (center), and ${\bf s}_3$ (right) are located is also shown.
• Figure 4: Three fixed spherical points ${\bf s}_0$, ${\bf s}_4$ and ${\bf s}_2\in{\mathcal{C}}({\bf s}_0,{\bf s}_4)$ (black dots), together with three possible positions (left, center, right) of the spherical point ${\bf s}_1$ (red dots) $\in {\mathcal{C}}({\bf s}_0,{\bf s}_2)$ (red lines) and corresponding positions of ${\bf s}_3$ (blue dots) $\in {\mathcal{C}}({\bf s}_0,{\bf s}_2)$ (blue lines) characterizing an RRMF5-I. The corresponding spherical control polygon (dashed black line) is also shown.
• Figure 5: Left: the spherical control points ${\bf s}_i, i=0,\ldots,4,$ (black dots) of Example \ref{['ex1']}, the related spherical control polygon (dashed black line), and the coincident tangent indicatrices $\mathbf{t}(t)$ and $\tilde{\mathbf{t}}(\tilde{t}\,)$ (black line). The points $\mathbf{t}\left(1/2\right)$ (red dot) and $\tilde{\mathbf{t}} \left(1/2\right)$ (blue dot) are also shown. Right: the corresponding RRMF5-I curves ${\bf r}(t)$ and $\tilde{{\bf r}}(\tilde{t}\,)$ (black lines), sharing the same initial point, with the associated ${\bf f}_2$ (black line) and ${\bf f}_3$ (red and blue lines for ${\bf r}$ and $\tilde{{\bf r}}$, resepectively) RMF vectors.

Theorems & Definitions (49)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Corollary 2.4
• Remark 2.5
• Remark 2.6
• Proposition 2.7
• Lemma 2.8
• Proposition 2.9
• proof