This paper introduces the design and the optimization of a probe holder robot for tele-echography applications. To define its kinematic architecture, an approach based on motion capture of an expert's gestures during ultrasound examinations was proposed. The medical gestures analyzed consisted of ultrasound probe movements and were used to characterize the kinematic specifications of the proposed manipulator. The selected architecture was a Spherical Parallel Mechanism (SPM) with 3 degrees of freedom (DoF) and its optimal synthesis was performed using real-coded Genetic Algorithms (GA). The optimization criteria and constraints were established thanks to the collaboration of medical experts and were successively formulated and solved using mono-objective and multi-objective functions.

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.

We study Hamiltonian systems which generate extremal flows of regularvariational problems on smooth manifolds and demonstrate thatnegativity of the generalized curvature of such a system impliesthe existence of a global smooth optimal synthesis for the infinitehorizon problem.We also show that in the Euclidean case negativity of the generalized curvature is a consequence ofthe convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.