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

While at the small scale, repeatability of kinematic couplings has been proven to be more than sufficientfor most industrial applications, interchangeability has not yet been studied in detail. The fundamentalcontribution of this analysis is a statistical model of mechanical interchangeability of the canoe ball andthree-pin coupling interfaces. Parametric relationships result in terms of the manufacturing tolerances onthe couplings and interface plates, the error of the coupling mounting and measurement processes, and thedetail of pre-calibration of the contact point locations. Simplification of this model to traditional ballgroovecouplings, or extension to inexact constraint quasi-kinematic couplings for which a minimumenergyconfiguration must be found, is straightforward future work.3.2 Global Error Model of a Kinematic Coupling InterfaceRecalling that the six-points of contact from a kinematic coupling exactly constrain a solid body, kinematiccouplings can be exploited as minimum geometric error interfaces for two main reasons. First, theforced point-line contact of the ball triangle on the groove lines reduces the positional error of the couplingcentroid to one-third the error of the coupling positions. Second, the deterministic point-line contactenables calculation of an error transformation between the nominal interface mating position and the truemating position, when the position and orientation offsets of the balls and grooves are known. Furthermore,at a very fine scale, the mechanical error motion of the interface from Hertzian contact between theballs and grooves can be determined [2]. However, at even extreme interface preloads and disturbancesthis error motion is very small compared to the magnitude of the kinematic transformation due to error inthe coupling placements from typical CNC machining processes. Because of this determinism, the kinematicerror of the coupling interface can be expressed in closed form.To start the error model, consider a general machine design application in which two modules matethrough an interface of canoe ball kinematic couplings, shown in Figure 3.1 for the base interface of anindustrial robot. The grooves sit on a fixed floor-mounted lower module, and the mating balls are attachedto the upper module. Reference coordinate frames are placed centroidally on the groove set (F groove ) andthe ball set (F ball ), and the couplings are secured using a sufficient (bolted or magnetic) preload. For themachining task, the tool center point (TCP) and co-located coordinate frame (F TCP ) are offset from the ball40