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

The construction of space curves with rational rotation- minimizing frames (RRMF curves) by the interpolation of $G^1$ Hermite data, i.e., initial/final points $\mathbf{p}_i$ and $\mathbf{p}_{f}$ and frames $(\mathbf{t}_i, \mathbf{u}_i,\mathbf{v}_i)$ and $(\mathbf{t}_{f},\mathbf{u}_{f},\mathbf{v}_{f})$, 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 $\mathbf{p}_{f}-\mathbf{p}_i$, 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.