Two kinds of experiments have been designed to investigate the performance of the micro-probing system. First, the behaviour of the probing system in the stiff mode, where the system can work as a normal micro-scale probing system, is considered. This initial investigation determines long-term stability, sensitivity of the probe in 3D and repeatability. Second, the probing system is investigated in the flexible mode, and the ability of the probe to switch between stiff and flexible modes is examined. For example, is the probing system able to repeatedly switch from stiff to flexible mode with no resultant off-set errors, and will the sensitivity of the probing system be changed as a result of switching between modes?
4.2. Performance in Stiff Mode
In order to accurately measure the position of a probed surface, a set of coefficients are required that can relate the sensor measurements to the Cartesian position of the stylus tip. For the probing system presented, it is not convenient to directly observe Cartesian displacements,
i.e., displacement along the x, y and z axes, as shown in
Figure 9. Instead, it is necessary to first relate sensor response to motion along the more convenient δ
1, δ
2, δ
3 and δ
4 displacement vectors described below, and then relate these to motion in a Cartesian frame (this is hereafter referred to as calibration). To calibrate the probing system in the stiff mode, the stylus tip was displaced over an incremental range using the nano-positioning stage. To minimise the effect of humidity-related drift, the calibration was conducted over a one minute period for each direction. To correctly align the nano-positioning stage for the three lateral displacements, the triangular body of the probe fixture was used to provide datum faces (see
Figure 9b). To approach the probe tip, 100 nm incremental steps were carried out until contact was registered by the micro-probing system. Once contact had been made with the stylus tip, the output from the three capacitive sensors was recorded over a displacement range of 14 µm, with incremental steps of 2 µm.
The output from the three capacitive sensors, for δ
1, δ
2, δ
3 and δ
4 displacement vectors, can be seen in
Figure 10a–d, respectively. It is clear that all capacitive sensors read almost the same output voltage for displacements in the z direction, suggesting that the structure does not twist when displaced vertically. However, the displacement of the stylus tip in the y direction leads to rotation of the intermediate body about the x axis, where sensor B and C respond in opposite sign, but similar magnitude, and sensor A remains constant due to its positioned at the centre of rotation.
The data in
Figure 10 was used to form a 3 × 4 matrix of sensitivity coefficients based on a least-squares linear fit. A sensitivity matrix can be approximated using the first-order Taylor series when the working range is small [
18], see Equation (1)
The coefficients in Equation (1) relate the displacement of the stylus tip in the four directions described (see
Figure 9) to the three capacitive sensor voltage responses. Equation (2) [
19] was used to give the inverse to the coefficients in Equation (1), thus
In Equations (1) and (2), the first subscript denotes the Cartesian axis they relate to, and the second subscript denotes the sensor. Finally, Equation (3) relates the sensor voltage directly to the displacement of the stylus tip with respect to x, y and z axes
4.4. Performance in Flexible Mode
As described in
Section 2.2, stiffness control in the flexible mode is achieved using a frequency-based closed-loop control system, which requires the probing system to be vibrating. To avoid any potential errors due to probe tip vibration, the control system must be stopped prior to contact. This has the unwanted effect of allowing the piezo-electric actuator to drift without control. The amount of drift depends on the recent drive history of the actuators and creep can, therefore, be reduced by initialising the piezo-electric actuator using a set of load cycles prior to use [
20]. The number of cycles needed to reduce the influence of the creep was investigated. The closed loop control was run for one minute to maintain the frequency at 600 Hz, and then the controller was switched to open loop to allow the drift to be measured.
Figure 12 shows the result of an experiment where the vertical frequency was controlled to (599.997 ± 0.013) Hz and after one minute, the closed-loop control was switched off. The probe tip amplitude of the vertical mode of vibration was estimated from the AC voltage of the capacitive sensors and calculated to be approximately 2.4 µm. The sweeping chirp signal remains on to monitor the resonant frequency. It is clear that the piezoelectric actuators drift, which results in the first steep downward slope shown in
Figure 12. Once the control system is restarted, the frequency rapidly returns to the 600 Hz set point. The test was repeated after a further minute, and it can be seen that the drift-related frequency change was greatly reduced. This trend was found to continue, and after a three minute period, drift was around 7 nm, as shown in
Figure 13. Considering these results, ongoing tests of the probe in the flexible mode were conducted following an initial period of five minutes to reduce actuator creep.
Using the transfer function described in Equation (4) (see below), the impact of actuator creep on the stylus tip displacement was calculated. From
Figure 13, it can be seen that after one and three minute periods, the actuator creep is around 35 nm and 7 nm at the stylus tip respectively. It is clear that the creep in three piezo-electric actuators was reduced as a result of repeating and recycling after a further minute.
In order to investigate the sensitivity of the probing system in the flexible mode, it was calibrated using the same procedure as for the stiff mode. The position of the nano-positioning stage was maintained between the calibration of the stiff and flexible mode in each direction to minimise error. The calibration was completed within one minute, reasonably limiting drift to less than 6 nm, as described above.
Figure 14a,b show the calibration time against the output voltage from the three capacitive sensors for displacement in the first and second directions respectively. The step height was 1 µm, and the datum point was taken as 100 nm beyond the first detected contact, to ensure a reliable contact between stylus and surface.
The transfer function resulting from the calibration of the four directions is given in Equation (4).
In order to illustrate the linearity of the probe’s response to displacement, the residual is plotted in
Figure 15a,b for the z and y directions respectively. It is clear that the maximum residuals in the z and y directions are approximately 23 nm and 4 nm, respectively.
To demonstrate that changing between modes does not significantly affect the calibration, a calibration was first performed in flexible mode, and then the probing system was set to stiff mode before returning to flexible mode for a repeat calibration. The coefficients generated from each calibration cycle are presented in in
Table 2. There is a small change in the coefficients, approximately 0.4% and 0.9% in the z and y directions, respectively. To illustrate the effect of the repeated calibrations, the second set of calibration coefficients for direction 1 was applied to the first set of data, and a new residual error was calculated. For the z direction, the maximum residual errors were found to be 23 nm, and changed to 17 nm. In the y direction, the maximum residual errors were initially 4 nm and changed to 5 nm, which is an acceptably low change in residual error.
As well as determining the effect that switching from stiff to flexible mode has on sensitivity, it was also necessary to determine any unwanted stylus tip position offset error. This was determined with the knowledge that the load applied to the beams will result in a deflection of the stylus tip, which is most dominant in the z direction [
12]. If the probing system was not able to return to the same initial zero offset when switched to flexible mode, it would directly impact the probing system error. The only option to reduce this error would be to recalibrate the probe each time the flexible mode was switched on, which would not be practical. As both natural frequency and displacement are functions of applied compressive force, it is possible to define displacement as a function of frequency; thus, for a given frequency there is an unambiguous corresponding displacement. The ability of the probing system to maintain a consistent zero position when set to flexible mode will therefore be limited by two factors. These factors are the ability of the control system to maintain a stable frequency, and the sensitivity of frequency to displacement change; for example, if the control system is able to maintain a perfect frequency with no error, then the displacement must also be perfectly controlled. However, if there is some error, then the degree to which this causes the stylus tip to be deflected will depend of the sensitivity of stylus tip deflection to frequency change; a low sensitivity will result in a low positional error and
vice versa.
To overcome this issue, the frequency-based probe stiffness control system (
Section 2), was used. To determine the capability of the frequency-based probe stiffness control system to maintain stylus tip position, the relationship between stylus tip position (displacement in the z direction) and the natural frequency of the structure was determined; the results can be seen in
Figure 16. This data was collected by setting the frequency control system to target 600 Hz, and then switching off the control system so only a steady state voltage is applied to the actuators. Under these conditions the actuators are free to creep under the influence of environmental conditions. This results in a steady increase in the applied force which results in the observed reduction in frequency from 600 to 598.4 Hz. While the frequency was reducing, the output from the capacitive sensors was monitored to allow the resulting displacement of the stylus tip to be measured.
The relationship is effectively linear over the small range of frequency deviations from the nominal 600 Hz control position. From a linear least-squares fit, the sensitivity of the deflection of the probe tip in the z direction as a function of tuned frequency is 26 nmHz
−1. The control system is able to tune the frequency within a range of 0.075 Hz (see inset on
Figure 12), so based on this the deflection, the probe tip can be controlled to within 2 nm in the z direction. Using frequency-based control, therefore, provides an effective method of controlling the flexible mode stiffness.
The displacement that occurs between the “at rest” unloaded positions when making repeated switches between stiff to flexible control modes, henceforth referred to as switching repeatability, was determined. To measure the switching repeatability, the control mode was switched every ten seconds for a total of nine cycles, and the tip displacement monitored throughout. For this experiment the stylus tip was not constrained, ensuring true “at rest” position were measured. The results are shown in
Figure 17. To obtain the data plotted in
Figure 17, the DC voltage from the three capacitive sensors was recorded, in stiff and flexible mode, and then flexible and stiff calibrations were used to calculate the stylus tip positions. It is clear that the probe was deflected by a total magnitude of approximately 18 µm when transitioning to flexible mode.
The frequency according to
Figure 17 can be controlled within 0.12 Hz (standard deviation). Therefore, the stiffness variability in the flexible mode can be calculated based on Equation (5); where
,
and
denote the standard deviation of the vertical stiffness, the standard deviation of the tuned frequency and the effective mass. The effective mass in the vertical direction is estimated based on the FEA (65 mg), this gives a stiffness variability of 0.04 N/mm, which is 4.5% of the nominal value at 600 Hz.
Table 3 shows the repeatability for the performance of the control system switching between stiff and flexible mode. The maximum standard deviation is 12 nm in the z direction. This standard deviation is relatively high compared with the estimated capability to control the deflection within 2 nm. It is expected that, if longer periods than ten seconds were used to allow increased settling time for the control loop, the standard deviation would be reduced.