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

of the balls and grooves exist, the frames are subject to translational and rotational error offsets. The translationalerrors are reflected exactly at the tool point, yet the rotational errors are magnified by the distanceto the tool point. For example, when the tool point is 2 m from the coupling centroid, a 0.0001 radian errorabout one axis results in a sine error of 0.2 mm translation at the tool point. Hence, while this error may berepeatable to the micron level, its existence makes “out of the box” use of kinematic couplings without calibrationimpractical when reasonable errors of securing the balls and grooves to the modules are present.This presents a problem to the concept of complete modularity, in which interfaces should be “plug andplay”, requiring no on-line measurement to eliminate the interface error after installation.In establishing a deterministic error model, this work presents a method through which error of kinematicinterface interchangeability can be minimized by a simple computerized calibration routine. Furthermore,the deterministic behavior can be parametrized in terms of the error contributions, enabling designdecisions that minimize cost and maintain the desired error budget by optimally specifying manufacturingtolerances on the couplings and interface plates.3.2.1 Components of Interface ErrorThe error at the interface can be broken into several causes and components, and then transformed intoan error at the tool point by deriving the serial chain of HTM’s relating the position of the cell measurementsystem to the position of the tool. The sources of TCP error from the kinematic interface can bedivided into random errors and systematic (hysteretic) errors. The components of error within these categoriesare:Random Errors:1. Positional tolerances of the mounting holes in the interface plate holding the balls and in the interfaceplate holding the grooves.2. Flatness of the ball holding plate and groove holding plate.3. Feature and form errors in the ball and groove mounts.4. Error in the measurement system. For a standard laser tracker, this is a worst-case value of 0.01mm per meter between the tracker head and the measurement point.5. Errors in the process of fitting the ball and groove mounts to the pallets, predominately manifested42