Kinematic Modeling, Linearization and First-Order Error Analysis

The kinematic analysis of parallel kinematic machines (PKM) is a challenging field, since PKM are complex multi-body systems involving a couple of closed kinematic loops. It is well-known that the forward kinematic function has in general no closed-form solution, and that up to 40 different real solutions may exist for general geometry (Husty, 1996; Dietmaier, 1998). Therefore, an efficient and handy method is needed in practise, e.g. for design, simulation, control, and calibration. The analysis of manufacturing and assembly errors of manipulators is a topic that is highly relevant for practical applications because the magnitude of these errors is directly coupled to the total cost of production of the manipulator. In this setting, there exist intensive studies on how to estimate the error of certain moving points, e.g. the tool center point, in terms of the errors in the components of the mechanism (Brisan et al., 2002; Jelenkovic & Budin, 2002; Kim & Choi, 2000; Song et al., 1999; Zhao et al., 2002), as well as how to allocate cost-optimal tolerances to a mechanism (Chase et al., 1990; Ji et al., 2000). In this paper, an approach to estimate the first-order influence of geometric errors on target quantities is suggested in which linearization is performed by considering the force transmission of the manipulator. This enables one to obtain a comprehensive model of linearized geometric sensitivities at a low computational cost. Error analysis for serial manipulators is relatively easy because one can establish an analytical expression for the forward kinematics which maps the generalized joint and link coordinates to the spatial displacements of the end-effector. There are numerous methods to parameterize the forward kinematics, where the approach of Denavit and Hartenberg (1955) is the most popular one. Once one has a closed-form expression for the forward kinematics, one can take derivatives of it (with respect to the geometric parameters one is interested in) and use these as sensitivity coefficients. In general, one introduces the sensitivity parameters in such a way that they vanish at the nominal configuration. This is always possible by introducing corresponding constant offsets where necessary. For example, consider a robot involving a universal joint, and assume that the sensitivity to errors in the fulfilment of the intersection property of the axes is to be analyzed. This can be done by adding a parameter for the normal distance between the joint axes which is zero in the nominal design, and with respect to which the partial derivative will yield the sought sensitivity. However, such a method for sensitivity analysis results in a model with a