Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France
Institut de Recherches en Communications et en Cybernetique de Nantes, 1 rue de la Noe, 44321 Nantes, France
(Submmited to "IEEE Transactions on Robotics", 2011)
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For robotic manipulators with passive joints, the stiffness matrices of separate kinematic chains are singular. So, most of existing stiffness analysis methods can not be applied directly and this problem is usually overcome by elimination the passive joint coordinates via geometrical constraints describing the manipulator assembly. However, such techniques degenerate if the number of passive joints is redundant and/or the resulting matrix is inherently singular.
To deal with such architectures in more efficient way, this paper proposes an analytical approach that allows obtaining both singular and non-singular stiffness matrices and which is appropriate for a general case, independent of the type and spatial location of the passive joints. The developed approach is based on the extension of the virtual-joint modelling technique and includes two basic steps which sequentially produce stiffness matrices of separate chains and then aggregate them in a common matrix.
In contrast to previous works, the desired stiffness matrix is presented in an explicit analytical form, as a sum of two terms. The first of them has traditional structure and describes manipulator elasticity due to the link/joint flexibility, while the second one directly takes into account influence of the passive joints. It is proved that, for each chain, the rank-deficiency of the resulting matrix is equal to the number of independent passive joints. To simplify analytical computations, it is proposed a recursive procedure that sequentially modifies the original matrix in accordance with the geometry of each passive joint. For the trivial cases, for which the passive joint axes are collinear to the axes of the base coordinate system, this modification is presented in the form of simple analytical rules.
Advantages of the developed technique are illustrated by application examples that deal with stiffness modelling of two Stewart-Gough platforms. They demonstrate its ability to produce both singular and non-singular stiffness matrices, and also show its feasibility for analytical computations. These examples give also some prospective for future work that include development of the dedicated techniques for the stiffness matrix aggregation in the case of non-rigid platform and an extension of these results for the case of manipulators with external loading.
The work presented in this paper was partially funded by the Region “Pays de la Loire”, France (project RoboComposite) and by the ANR, France (project COROUSSO).