###### Abstract  
The freeform surface quality is limited by the measurement, and form error cannot be convergent via compensation machining. This paper proposed a surface matching measurement strategy based on the least squares principle and iterative precision adjustment to precisely obtain surface form error after manufacturing by single point diamond turning. Through the coordinate transformation of translation and rotation, the measured surface was aligned with theoretical surface at the same coordinate system. The corresponding simulation was carried out to verify the performance of the proposed method, and the simulation results indicated that this method can achieve accurate alignment in a sub-nanometer range. Finally, with XY polynomial freeform surface as the ideal surface, compensation experiments were undertaken. The form error of freeform converged continuously after compensation machining for three times, during which the form accuracy of PV and RMS were down to 335 nm and 34 nm respectively from 1.4 um and 173 nm. The results showed that capability of the proposed compensation method was verified and the form accuracy could be improved effectively. freeform surface; compensation machining; surface matching +
Footnote †: journal: Applied Sciences  
**Citation:** [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Matching Measurement Strategy and Form  
**Citation:** [PERSON]; [PERSON]; [PERSON] [PERSON]. machining [13; 14]. Due to the geometric complexity of freeform surface, the characterization of surface quality lacks definitive standards and techniques. There are many types of ultra-sophisticated measuring equipment, and notwithstanding freeform surface shape can be measured, the shape error as the compensation feedback cannot be attained directly [15]. Effective compensation feedback can be obtained only via precise re-mounting of freeform surface. Initially, confirm that the consistency of measurement coordinates with design coordinates. Then the projection of the measured point on the theoretical coordinates after coordinate transfer is calculated. Ultimately by computing the distance of measured surface from the theoretical design surface in theoretical coordinates, the form error as compensation feedback is obtained. Related studies have explored the matching method of the freeform surface. [PERSON] et al. [16] presented an integrated form characterization method based on the five-point pre-fixture, dual cubic B-spline and an iterative precision adjustment algorithm with coordinate transfer to measure the form accuracy of ultra-precision freeform surface. [PERSON] et al. [17] proposed a fiducial-aided on-machine positioning method to precisely position optical freeform surfaces during the precision manufacturing cycle, by using the developed on-machine measuring devices to get the accurate positions of the fiducials after remounting in the machining coordinate system. [PERSON] et al. [18] developed a closed-loop FTS system to improve freeform machining accuracy, surface sampling and reconstruction strategies and robust surface filtration algorithms that are adapted to regulate the data flow for both freeform surface characterisation and the optimisation of compensation toolpath from machined error maps. [PERSON] et al. [19] proposed a new compensation strategy based on the combination of on-machine and off-machine measurement. These methods intend to realize optimal alignment by moving and rotating coordinates so that the feature points, lines, and surfaces of the measured surface and the designed surface overlap completely. Because the accuracy of these method for characterizing freeform surfaces with form error is not every high, it is difficult for the final surface machined by compensation to reach sub-micron precision [20; 21]. In this paper, a new method based on the least squares principle and iterative precision adjustment is put forward to obtain surface form error, in the meantime, the compensation method of STS freeform processing was developed to improve the surface quality.