A geometric method for computation of datum reference frames

A datum reference frame (DRF) is a coordinate system used to locate and orient part features. Constructing a DRF from a set of datum features is a complicated process involving: a) specifying a valid combination and the precedence of the datum features which define the DRF; b) developing datum from datum features of the part; and c) determining the position and orientation of the DRF from the datums. We develop a geometric theory for establishing DRFs. The theory is based on the observation that a datum feature such as a plane, a cylinder or a sphere has a symmetry subgroup G₀ under the action by the group SE(3) of rigid motions in R³. Thus, the configuration space of a datum feature can be identified with the homogeneous space SE(3)/G₀, and the problem of datum development can be posed as a minimization problem in SE(J)/G₀. We give conditions under which a datum feature qualifies to be a secondary or a tertiary datum. We present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem in the homogeneous spaces of SE(3). We develop simple algorithms to solve these problems, and give simulation results illustrating efficiency and simplicity of the approach