Kinematic analysis of universal split heads for milling machines

Most milling machines are equipped with split heads aimed at orienting the tool at any angle with respect to the workpiece. Typical examples are given in Fig. 1. The general geometry of these heads can be reduced to the linkage shown in Fig. 2. Each particular architecture is achieved by choosing angle θ between the member axis and by selecting the orientation of the tool with respect to the final member. The head features two degrees of freedom, embodied by the joint rotations α and β, through which the configuration can be changed to adjust the orientation ofthe tool. The kinematic analysis of the head can be posed in two ways: a) direct analysis if the values of the joint rotations, α and β, are given and the orientation of the tool axis with respect to the machine frame is sought; b) inverse analysis if the desired orientation of the tool axis is given and the joint rotations, α and β, are sought that enforce that orientation. This paper provides the analytical solution of both direct and inverse problems. The work falls within a research program carried out with the partnership of a renowned machine tool manufacturer of Reggio Emilia.