## 5 Conclusions  
In summary, an accurate inner profile measurement and geometric parameters evaluation method for a high aspect ratio aspheric workpiece using two measuring probes is proposed. The geometric parameters such as the radius of the bottom spherical shape as well as the steepness and straightness of the steep sidewall can be successfully evaluated. A lateral displacement gauge is used to measure the steep sidewall of the inner profile, and an axial displacement gauge is used to measure the deep underside of the inner profile of the workpiece. To qualify the measurement accuracy, the systematic errors associated with the measurement procedure, including the miscalibration, misalignment and the roundness error of the gauge probes, as well as the slide motion error of the four-axis motion platform, are all evaluated and separated from the measurement results. The measured profile is evaluated after making an accurate alignment of the probes and workpiece as well as compensation of the slide motion error and probing sphere roundness error. To obtain the entire inner profile of the workpiece, a point cloud registration algorithm was applied to stitch the inner sidewall profile measured by the lateral gauge and the inner underside profile measured by the axial gauge. To verify the performance of the proposed measurement system and the methods, the inner profile error of a high aspect ratio aspheric workpiece with a maximum inner radius of 40 mm, a maximum inner depth of 140 mm and a steep sidewall angle approaching 85\({}^{\circ}\)(conicity of 5.11\({}^{\circ}\)) was evaluated in experiments. The radius of the bottom sphere was evaluated as 29.5923 mm, and the roundness was evaluated as 0.4808 mm. The steepness of the sidewall was evaluated as 84.9671\({}^{\circ}\), and the straightness was evaluated as 0.2794 mm. Some of the experimental results are compared with that of a coordinate measuring machine (CMM). The comparison result, which has a difference of about \(\pm\)1.350 \(\upmu\)m, has verified the feasibility of the proposed measurement system. In addition, it is also noted that the measurement can be performed by integrating the two-probe measuring system to a commonly used multi-axis motion stage instead of using the state-of-art CMM, which is thus potential for on-machine measurement of the high aspect ratio aspheric part. The advantage of the proposed method is that the inner and outer surfaces of the high aspect ratio aspheric workpiece can be assessed accurately with high compatibility for on-machine measurement. The integration of the on-machine measurement and the generalization of the measurement for other parts with different forms will be considered for potential improvements. Future works include comprehensive uncertainty analysis of the results such as the repeatability error and the misalignment error of the two probes, compensation of the machined surface using the evaluation results, measuring other high aspect ratio  
Figure 20: Measurement results of the inner profile of the high aspect ratio aspheric workpiece. (**a**) Comparison between theory profile and measurement profile; (**b**) measurement error of the high aspect ratio aspheric workpiece compared with the CMM measurement. parts by using the developed two-probe system, as well as automation of the measurement system to reduce the uncertainties in alignment and measurement procedure. Conceptualization, Y.-L.C., B.-F.J. and X.X. ; methodology, Y.-L.C. and P.H. ; software, P.H. ; validation, P.H. ; formal analysis, P.H. ; investigation, P.H. and W.-H.Z. ; resources, P.H. and W.-H.Z. ; data curation, P.H. ; writing--original draft preparation, Y.-L.C. and P.H. ; writing--review and editing, Y.-L.C. and X.X. ; visualization, Y.-L.C. and X.X. ; supervision, Y.-L.C. ; project administration, Y.-L.C. and B.-F.J.; funding acquisition, Y.-L.C. and B.-F.J. All authors have read and agreed to the published version of the manuscript. This research is supported by the National Key R&D Program of China (Grant No.