Design and Analysis of Kinematic Couplings for Modular Machine ...

3.2.3 The Interface TransformationIf some of the dimensional perturbations in the error chain are known or can be measured, introductionof an interface transformation can reduce the error at the tool point. Normally, when a kinematic couplinginterface is mounted, its interface transformation is an identity matrix; hence by the reciprocal path ofcoordinate transformations:T Groove – Work = T Ball – TPHowever, the existence of any error component in the chain means this equality is untrue, hence:(3.1)T Groove – Work ≠ T Ball – TP. (3.2)The mismatch is the error from the work point to the tool point, expressed in terms of the transformationsas:T Error – full = ( T Ball – TP ) – 1 T Groove – Work. (3.3)If the interface transformation is known exactly, the forward loop to the tool point including the interfacetransformation exactly matches the reverse loop. However, because error in the measurement systemis a part of the error estimate, a residual error transformation exists:T Error – resid = ( T interface T Ball – TP) – 1 T Groove – Work. (3.4)However, the residual error is far less in magnitude (expressed as a Cartesian distance by the arg() notation)than the full error:arg( T Error – resid ) < T Error – fullarg( ). (3.5)This chapter contends that the interface error transformation is in large part due to positional and rotationalmisalignments of the kinematic coupling’s mating curved and flat contact surfaces. For a canoe ballinterface the mating surfaces are the spherical canoe ball units and the vee grooves, and for a three-pininterface are the shouldered pins in the top interface plate and the mating vertical contact planes in the bottominterface plate. Once the positions and orientations of the contacting pieces are known, deducing theinterface transformation from the kinematic constraints of the coupling is a simple deterministic calculation.Absent error of the measurement system, the interchangeability of calibrated kinematic coupling46