Kinematic calibration of a 3-P(Pa)S parallel-type spindle head considering the thermal error

Calibration is considered to be the most effective way to improve the accuracy of parallel kinematic machine tools (PKMTs). However, ordinary calibrations only considered the time-invariant errors (manufacturing error), neglecting some time-variant errors, a significant one of which is thermal error. Therefore, in this paper, the influence of thermal error was considered in the calibration of a 3-P(Pa)S parallel-type spindle head. First, a new kinematic model of the spindle head was proposed, which is closer to the real physical model, so the thermal error of the spindle head can be considered in the model. Second, the structural parameters of the spindle head were expressed as the sum of the ideal parameters, the manufacturing errors, and the thermal errors. Third, the pose (position and orientation) of the end effector and the temperature of the spindle head were measured. The positions of the temperature sensors were selected using the global temperature sensitivity index (GTSI), which is derived from the global sensitivity index (GSI). Thus, by setting a standard temperature, the thermal error of the structural parameter can be obtained. Fourth, the influence of the thermal error was inputted into the identification equation for calibration, so the results are the structural parameters at the standard temperature (20 °C). To solve the ill-conditioning problem, a Regularization method was used in the identification. Finally, the calibration was verified on a 3-P(Pa)S-XY machine tool. The RTCP test, performed immediately after the measurement, shows that the maximum position error after the calibration is 0.019 mm at the tilt angle of 30° and 0.037 mm at 20°. In addition, the RTCP test after a temperature change shows that the calibration considering the thermal error can improve the average position accuracy from 0.025 mm to 0.015 mm. The calibration method in this paper is expected to be applicable for other machine tools.