Machine-Tool error observer design with application to thermal error tracking

A parameter identification procedure for identifying the parameters of a volumetric error model of a large and complex machine tool usually requires a large number of observations of volumetric error components in its workspace. This paper demonstrates the possibility of applying optimal observation/experimental design theories to volumetric error model parameter identification of a large 5-Axis machine with one redundant axis. Several designs such as A-, D-and K-optimal designs seek to maximize the amount of information carried in the observations made in an experiment. In this paper, we adapt these design approaches in the construction of machine-Tool error observers by determining locations in the workspace at which components of volumetric errors must be measured so that the underlying error model parameters can be identified. Many of optimal designs tend to localize observations at either the center or the boundary of the workspace. This can leave large volumes of the workspace inadequately represented, making the identified model parameters particularly susceptible to model inadequacy issues. Therefore, we develop constrained optimization algorithms that force the distribution of observation points in the machine's workspace. Optimal designs provide the possibility of efficiency (reduced number of observations and hence reduced measurement time) in the identification procedure. This opens up the possibility of tracking thermal variations of the volumetric error model with periodic quick measurements. We report on the design, implementation and performance of a constrained K-optimal in tracking the thermal variations of the volumetric error over a 5.5 hour period of operations with measurements being made each hour.