### C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Hwan Pyo Moon, Rida T. Farouki
2018 Computer Aided Geometric Design
The problem of constructing a rational adapted frame (f1(ξ), f2(ξ), f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler-Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the
more » ... plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C 1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C 2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points. 1 The spatial motion of a rigid body is defined by specifying its position and orientation at each instant. The position may be specified as a parametric curve r(ξ) describing the path of a distinguished point (such as the center of mass), and the orientation may be specified by an orthonormal frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) defined along r(ξ). In many applications, an adapted frame that satisfies f 1 (ξ) = r (ξ)/|r (ξ)| -i.e., the first frame vector coincides with the tangent to the path r(ξ) -is desired, so that f 2 (ξ) and f 3 (ξ) span the curve normal plane. Clearly, there are infinitely many adapted frames, corresponding to different choices for the variation of the orientation of f 2 (ξ) and f 3 (ξ) along r(ξ). The rotation minimizing frame (RMF) or Bishop frame  is an important type of adapted frame (t, u, v) on a space curve, consisting of the curve tangent t and unit normal-plane vectors u and v that exhibit no instantaneous rotation about t -i.e., the tangent component of the frame angular velocity vanishes. Since polynomial/rational curves do not ordinarily admit rational RMFs, many schemes to approximate them have been proposed [8, 15, 16, 18, 19, 20, 23, 24, 25, 26] . On the other hand, the identification of space curves that admit exact rational RMFs has recently become a topic of active investigation [4, 6, 11, 12] . Such curves are necessarily a subset of the Pythagorean-hodograph (PH) curves , since only PH curves possess rational unit tangents. The Frenet frames of PH curves have also been studied for cubic helical spline curves  . The focus of the present study is the construction of rational adapted spline frames along a pre-defined spatial PH curve, that interpolate prescribed frame orientations at a sequence of specified curve points with C 1 or C 2 continuitywhere C 1 implies continuity of angular velocity, and C 2 implies continuity of angular velocity and angular acceleration. Smoothness of orientational motion is as important as that of translational motion in applications, since discontinuity of angular velocity and acceleration is physically impossible in the steering of devices such robot end effectors, unmanned aerial vehicles, or spacecraft. Algorithms to construct rigid body motions specified by rational adapted RMFs with given initial/final positions and orientations have recently been developed [7, 9]. However, since the computation of an RMF is an initial value problem, the path r(ξ) is an outcome of these algorithms, rather than being specified a priori. Interpolation algorithms to construct rigid body motions for general orientations, without imposing the adaptedness condition, have been proposed [14, 17, 21] . These algorithms also compute the rotations and the trajectories simultaneously. In the present study, we consider adapted motions along a prescribed spatial PH curve r(ξ) that matches a sequence of specified orientations at nodal points along it, with continuity of angular velocity and acceleration at those points. In this context, the rotation-minimizing condition is relaxed, since an RMF cannot (in general) match given orientations at distinct points along pre-defined curve. The Euler-Rodrigues frame (ERF) is a rational adapted frame defined on any spatial Pythagorean-hodograph curve  . The ERF serves as a starting point for the construction of other rational adapted frames, by applying a rationally-2