Super-Clothoids

Piecewise clothoids are 2D curves with continuous, piecewise linear curvature. Due to their smoothness properties, they have been extensively used in road design and robot path planning, as well as for the compact representation of hand-drawn curves. In this paper we present the Super-Clothoid model, a new mechanical model that for the first time allows for the computing of the dynamics of an elastic, inextensible piecewise clothoid. We first show that the kinematics of this model can be computed analytically depending on the Fresnel integrals, and precisely evaluated when required. Secondly, the discrete dynamics, naturally emerging from the Lagrange equations of motion, can be robustly and efficiently computed by performing and storing formal computations as far as possible, recoursing to numerical evaluation only when assembling the linear system to be solved at each time step. As a result, simulations turn out to be both interactive and stable, even for large displacements of the rod. Finally, we demonstrate the versatility of our model by handling various boundary conditions for the rod as well as complex external constraints such as frictional contact, and show that our model is perfectly adapted to inverse statics. Compared to lower-order models, the super-clothoid appears as a more natural and aesthetic primitive for bridging the gap between 2D geometric design and physics-based deformation.

• Paper: [PDF]
• Accompanying video: [MP4]
• More ressources to come.

We were happy to use the source code provided by Ilya Baran and his co-authors for fitting curves with piecewise clothoids, allowing us to couple our model to a user-friendly interface and then to build a full pipeline from hand-drawn design to physics-based animation. Thanks to Alexandre Derouet-Jourdan for his help in this process. Thanks also to Romain Casati for his code on articulated rigid bodies. More ressources from us to come...