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Radius case study: optical bench measurement and uncertainty including stage error motions

This paper provides a case study for identifying radius measurement uncertainty on a commercially-available optical bench using a homogeneous transformation matrix (or HTM)-based formalism. In this approach, radius is defined using a vector equation, rather than relying solely on the recorded displacement between the confocal and cat's eye null positions (i.e., the projection of the true displacement between these positions on the transducer axis). The vector-based approach enables the stage error motions, as well as other well-known error sources, to be considered through the use of HTMs. An important aspect of this mathematical radius definition is the intrinsic correction for measurement biases, such as cosine error (i.e., misalignment between the stage motion and displacement transducer axis) which would lead to an artificially small radius value if the traditional projection-based radius measurand were employed. Experimental results and measurement techniques are provided for the stage error motions, which are then combined with the setup geometry to determine the radius of curvature for a spherical artifact. Comparisons are shown between the vector-based radius calculation, traditional radius computation, and independent measurements using a coordinate measuring machine. The measurement uncertainty for the vector-based approach is determined using Monte Carlo simulation and is compared to experimental results.