Section: New Results

Mechanical rods

Reduced-coordinates models for rods such as the articulated rigid body model or the super-helix model  [39] are able to capture the bending and twisting deformations of thin elastic rods while strictly and robustly avoiding stretching deformations. In this work we are exploring new reduced-coordinates models based on a higher-order geometry. Typically, elements are defined by a polynomial curvature function of the arc length, of degree $d\ge 1$. The main difficulty compared to the super-helix model (where $d=0$) is that the kinematics has no longer a closed form. Last year we investigated the clothoidal case ($d=1$) in the 2d case [19] , relying on Romberg numerical integration. This year, in R. Casati's PhD's thesis, we extended this result to the full 3D case. The key idea was to integrate the rod's kinematics using power series expansion, and to design an accurate and efficient computational algorithm adapted to floating point arithmetics. Our method nicely propagates to the computation of the full dynamic of a linked chain of 3d clothoid. All these results will we submitted for publication early 2013.

Controlling the input shape of slender structures such as rods is desirable in many design applications (such as hairstyling, reverse engineering, etc.), but solving the corresponding inverse problem is not straightforward. In  [43] we noted that reduced-coordinates models such as the super-helix are well-suited for static inversion in presence of gravity. The main difficulty then amounts to fitting a piecewise helix to an arbitrary input curve. This year in A. Derouet-Jourdan's PhD's thesis, we solved this problem by extending to 3d the floating tangents algorithm introduced in 2d in  [43] . In this new method, only tangents are strictly interpolated while points are displaced in an optimal way so as to lie in a feasible configuration, i.e., a configuration that is compatible with the interpolation by a helix. Our method proves to be efficient and robust as it can successfully handle large and complex datasets from real curve aquisitions, such as the capture of hair fibers or the magnetic field of a star. This result was submitted for publication to Computer-Aided Geometric Design in Spring, and is currently under minor revision.