Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness

The mobility of a linkage is determined by the constraints imposed on its members. The constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The instantaneous motions are determined by the constraints, rather than by the c-space geometry. Shaky linkages are prominent examples that exhibit a higher instantaneous than finite DOF even in regular configurations. Inextricably connected to the mobility are kinematic singularities that are reflected in a change of the instantaneous DOF. The local analysis of a linkage, aiming at determining the instantaneous and finite mobility in a given configuration, hence needs to consider the c-space geometry as well as the constraint system. A method for the local analysis is presented based on a higher-order local approximation of the c-space adopting the concept of the tangent cone to a variety. The latter is the best local approximation of the c-space in a general configuration. It thus allows for investigating the mobility in regular as well as singular configurations. Therewith the c-space is locally represented as an algebraic variety whose degree is the necessary approximation order. In regular configurations the tangent cone is the tangent space. The method is generally applicable and computationally simple. It allows for a classification of linkages as overconstrained and underconstrained, and to identify singularities.

ON THE "GENERALIZED SLATER" APPROXIMATION

We work within the Density Functional Theory (DFT), in the Local Density Approximation (LDA) with Self Interaction Correction (SIC). We show that, thanks to a formulation which employs two different sets of orbitals, the equations can be written in the form of eigenvalues equations, leading to single electron interpretation. However, the resulting hamiltonian is non-local. We propose to find it's best local approximation within using the Optimized Effective Potential (OEP) method. The resulting approximate theory is denominated "Generalized Slater". We show that this new scheme cures the pathologies of the standard SIC-Slater or SIC-KLI approximations.