The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

Rolling Manifolds of Different Dimensions

If $(M,g)$ and (\hM,\hg) are two smooth connected complete oriented Riemannian manifolds of dimensions $n$ and \hn respectively, we model the rolling of $(M,g)$ onto (\hM,\hg) as a driftless control affine systems describing two possible constraints of motion: the first rolling motion $\Sigma_{NS}$ captures the no-spinning condition only and the second rolling motion $\Sigma_{R}$ corresponds to rolling without spinning nor slipping. Two distributions of dimensions (n + \hn) and $n$, respectively, are then associated to the rolling motions $\Sigma_{NS}$ and $\Sigma_{R}$ respectively. This generalizes the rolling problems considered in \cite{ChitourKokkonen1} where both manifolds had the same dimension. The controllability issue is then addressed for both $\Sigma_{NS}$ and $\Sigma_{R}$ and completely solved for $\Sigma_{NS}$. As regards to $\Sigma_{R}$, basic properties for the reachable sets are provided as well as the complete study of the case (n,\hn)=(3,2) and some sufficient conditions for non-controllability

Proposal For a Study of Commonsense Physical Reasoning

This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-80-C-0505.Our common sense views of physics are the first coin in our intellectual capital; understanding precisely what they contain could be very important both for understanding ourselves and for making machines more like us. This proposal describes a domain that has been designed for studying reasoning about constrained motion and describes my theories about performing such reasoning. The issues examined include qualitative reasoning about shape and physical processes, as well as ways of using knowledge about motion other than "envisioning". Being a proposal, the treatment of these issues is necessarily cursory and incomplete.MIT Artificial Intelligence Laboratory Department of Defense Advanced Research Projects Agenc

Trajectory planning using reachable-state density functions

Presents a trajectory planning algorithm for mobile robots which may be subject to kinodynamic constraints. Using computational methods from noncommutative harmonic analysis, the algorithm efficiently constructs an approximation to the robot's reachable-state density function. Based on a multiscale approach, the density function is then used to plan a path. One variation of the algorithm exhibits time complexity that is logarithmic in the number of steps. Simulations illustrate the method

A novel haptic model and environment for maxillofacial surgical operation planning and manipulation

This paper presents a practical method and a new haptic model to support manipulations of bones and their segments during the planning of a surgical operation in a virtual environment using a haptic interface. To perform an effective dental surgery it is important to have all the operation related information of the patient available beforehand in order to plan the operation and avoid any complications. A haptic interface with a virtual and accurate patient model to support the planning of bone cuts is therefore critical, useful and necessary for the surgeons. The system proposed uses DICOM images taken from a digital tomography scanner and creates a mesh model of the filtered skull, from which the jaw bone can be isolated for further use. A novel solution for cutting the bones has been developed and it uses the haptic tool to determine and define the bone-cutting plane in the bone, and this new approach creates three new meshes of the original model. Using this approach the computational power is optimized and a real time feedback can be achieved during all bone manipulations. During the movement of the mesh cutting, a novel friction profile is predefined in the haptical system to simulate the force feedback feel of different densities in the bone