**4. Discussion**

The experimental data of straight and circular trajectories are rearranged in Tables 1 and 2, respectively. Higher magnification leads to a better resolution but a smaller working range. The COD, straightness, and roundness all indicate the precision of the controlled motion. All COD values of 45◦ linear motions are approaching 0.999, which means that the controller can compensate for the crosstalk between two DOFs effectively.


**Table 1.** Experiment data of straight trajectories.

\* The condition of smallest error; \*\* the condition of fastest speed.

The smallest straightness 0.291 μm in Table 1 and roundness 2.380 μm in Table 2 both occur at the setting of 10-time objective and one-pixel closeness. The fastest speeds, 4.274 μm/s in Table 1 and 3.607 μm/s in Table 2, both occur at the setting of four-time objective and five-pixel closeness. Referring to Figure 12, plotting straightness/roundness versus the average speed, an obvious correlation can be found between the error and the speed.


**Table 2.** Experiment data of circular trajectories.

\* The condition of smallest error; \*\* the condition of fastest speed.

**Figure 12.** (**a**) Relation between the average speed and the straightness of linear motions in Table 1. (**b**) Relation between the average speed and the roundness of circular motions in Table 2. Positive correlation can be found between the error and the speed.

After analyzing the experimental data, we found that "minimum walkable voltage" is position-relative and time-varying. Figure 13 shows the minimum walkable voltage of 700 μm linear translational motion for three repeated tests. Around 420 μm, the local friction force is greater than average; therefore, a higher voltage is needed to keep going forward. The results are similar, however, not identical over three tests because the contact condition had been changed over time. There does not exists a globally consistent minimum walkable voltage value, although the contact surfaces had been carefully polished. In summary, dynamically adjusting the driving voltage is a practical method to deal with the changing friction force.

As the voltage is increased, the rotor suddenly overcomes the maximum static friction and has the chance to step away from the desired trajectory. This phenomenon is illustrated in the zoomed screens of Figures 9b and 10b. At those places with discontinuous friction force, the tracking error becomes greater than average. Eventually, our visual servo system can pull it back and reduces the error effectively.

**Figure 13.** Minimum walkable voltage of 700 μm linear translational motion for three repeated tests.

Depending on the demands of a specific task, a balance between performance and budget is to be expected. Both the camera's pixel density and the objective's magnification affect precision. On the other hand, the camera's frame rate and the computer's image processing ability help to achieve higher speed. The direction of our future research is to build a coaxial multi-camera visual servo system. The large FOV image navigates the coarse motion quickly. The small FOV image controls the fine motion at a relatively low speed. The scheduling method between coarse and fine motions will release the power of the proposed 2-DOF nano-stepping motor. The third DOF, vertical to the image plane, will also be added. This will let the AFM probe fabricate microstructures.
