## 2 Measurement System and Basic Measurement Principle  
As shown in Figure 1, the measurement system is established based on a four-axis motion platform with \(XYZ\) slides and a spindle. A schematic of the error of the inner profile of the high aspect ratio aspheric workpiece is shown in Figure 2. The right side of the single cross-section can be measured by rotating the spindle and the workpiece by 180\({}^{\circ}\) after measuring the left side of the single cross-section with a precision rotation stage. The workpiece has a tapered cone shape with a steep sidewall and a depth much larger than its opening diameter. The measurement target is its entire inner profile, including the steep side part and the shallow bottom part at a deep depth. The high aspect ratio aspheric workpiece is mounted on the spindle, which can be driven by the \(XY\)-slide. A lateral displacement gauge (Probe A) for measuring the inner sidewall profile and an axial displacement gauge (Probe B) for measuring the inner underside profile of the high aspect ratio aspheric workpiece are both mounted on the \(Z\)-slide of the motion platform. The side part of the inner profile can be measured by Probe A while driving the \(Z\)-slide and the \(Y\)-slide to keep the probe within its measurement range. The inner underside profile can be measured by Probe B while only driving the \(Y\)-slide. Figure 3 shows the measurement procedure for evaluating the entire inner profile of the high aspect ratio aspheric workpiece. The inner sidewall profile measurement and the inner underside profile measurement are measured by Probe A and Probe B, respectively, and then the measurement results are stitched together by applying a point cloud registration algorithm [24]. Figure 1: Schematic of the constructed two-probe measurement system. Figure 2: Schematic of error of inner profile of the high aspect ratio aspheric workpiece. Firstly, the inner sidewall profile of the high aspect ratio aspheric workpiece is measured using Probe A. During the measurement procedure, Probe A is employed to scan by starting from the top edge of the high aspect ratio aspheric workpiece and ending at the limited position of its measuring range, which has a horizontal distance \(d_{A}\) from the central axis of the high aspect ratio aspheric workpiece. The trajectory of the center of the probing sphere of Probe A can be obtained by:  
\[y_{ball\_A}=y_{slide}+\Delta y_{ball\_A} \tag{1}\]  
\[z_{ball\_A}=z_{slide}+\Delta z_{ball\_A} \tag{2}\]  
where \(y_{ball\_A}\) and \(z_{ball\_A}\) are the \(Y\) coordinate and the \(Z\) coordinate of the center of the probing sphere of Probe A, respectively, \(y_{slide}\) and \(z_{slide}\) are the \(Y\)-slide position and \(Z\)-slide position obtained from the four-axis motion platform, \(\Delta y_{ball\_A}\) is the \(Y\)-axis displacement of the probing sphere obtained as the output of Probe A, and \(\Delta z_{ball\_A}\) is the undesired \(Z\)-axis displacement of the probing sphere of Probe A associated with its \(Y\)-axis displacement, which shall be calibrated. This calibration of the undesired \(Z\)-axis displacement is explained later. Since the inner sidewall profile of the workpiece has a normal distance, which is the radius \(r_{A}\) of the probing sphere of Probe A, from the trajectory of the measuring Probe A, the inner sidewall profile of the workpiece can be evaluated based on the trajectory and the information of the probing sphere radius. Then, the inner underside profile of the workpiece is measured using Probe B. During the measurement procedure, Probe B is employed to scan starting from the central axis of the workpiece and ending at its limited measuring range position, having a horizontal distance from the central axis of the high aspect ratio aspheric workpiece of \(d_{B}\). The trajectory of the probing sphere of Probe B can be obtained by:  
\[y_{ball\_B}=y_{slide} \tag{3}\]  
Figure 3: Measurement procedure with the two measuring probes and profile stitching.