Design of rational rotation–minimizing rigid body motions by Hermite interpolation

Abstract

Abstract. The construction of space curves with rational rotationminimizing frames (RRMF curves) by the interpolation of G 1 Hermite data, i.e., initial/final points p i and p f and frames (t i , u i , v i ) and (t f , u f , v f ), is addressed. Noting that the RRMF quintics form a proper subset of the spatial Pythagorean-hodograph (PH) quintics, characterized by a vector constraint on their quaternion coefficients, and that C 1 spatial PH quintic Hermite interpolants possess two free scalar parameters, sufficient degrees of freedom for satisfying the RRMF condition and interpolating the end points and frames can be obtained by relaxing the Hermite data from C 1 to G 1 . It is shown that, after satisfaction of the RRMF condition, interpolation of the end frames can always be achieved by solving a quadratic equation with a positive discriminant. Three scalar freedoms then remain for interpolation of the end-point displacement p f − p i , and this can be reduced to computing the real roots of a degree 6 univariate polynomial. The nonlinear dependence of the polynomial coefficients on the prescribed data precludes simple a priori guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely-sampled data from a smooth curve. Modulation of the hodograph by a scalar polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end-point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid-body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.

Rational rotation-minimizing frames—Recent advances and open problems

“…Since RRMF quintics cannot match arbitrary first-order data in the rotation-minimizing rigid-body motion Hermite interpolation problem [19], the extra degrees of freedom afforded by degree 7 RRMF curves are important, and the obvious next step is to attempt a complete characterization of them.…”

“…This problem has been studied in [19] for the case of Class 1 RRMF quintics, which nominally have sufficient freedoms to satisfy the specified boundary conditions. It was shown in [19] that interpolation of the end frames is always possible, while interpolation of the displacement p f − p i reduces to finding the positive real roots of a degree 6 polynomial.…”

“…This problem has been studied in [19] for the case of Class 1 RRMF quintics, which nominally have sufficient freedoms to satisfy the specified boundary conditions. It was shown in [19] that interpolation of the end frames is always possible, while interpolation of the displacement p f − p i reduces to finding the positive real roots of a degree 6 polynomial. However, it is possible to construct examples in which this polynomial has no real roots, and the non-linear dependence of its coefficients on the input data precludes simple a priori conditions for the existence of solutions (although it can be verified in the asymptotic case of sufficiently-dense data, sampled from a smooth curve).…”

Rational swept surface constructions based on differential and integral sweep curve properties

“…The simplest non-trivial instances are the RRMF quintics, and are identified by a simple algebraic constraint on the coefficients of the quadratic quaternion polynomials that generate them [11]. The RRMF quintics have been used in the design of rational rotation-minimizing spatial motions, interpolating prescribed initial/final locations and orientations of a rigid body [13]. Figure 8 shows how the normal-plane orientation of a profile curve that is swept along a space curve can strongly influence the resulting surface shape.…”

Identification and “reverse engineering” of Pythagorean-hodograph curves

“…Once the control points have been identified as defining a PH curve, the reverse engineering procedures allow very accurate reconstruction of the complex or quaternion pre-image polynomials (for planar and spatial curves, respectively) in double-precision arithmetic, so the advantageous properties of PH curves can be fully exploited. These include closed-form solutions for arc lengths, offset curves, and elastic bending energy [4,17]; real-time CNC interpolators for constant or variable feedrates along curved paths [13,19]; a diverse family of sweep operations yielding rational surfaces [15]; and rational rotation-minimizing frames along space curves [7,9].…”