*2.2. Control System*

As investigated, the proposed nano-stepping motor works up to 300 Hz but its velocity is not linear versus the driving frequency [1]. This phenomenon implies a suitable frequency range. After observing its stepping behavior under a microscope, we find that the device works smoothly between 70–90 Hz. Therefore, the driving frequency 80 Hz (i.e., 12.5 ms per sawtooth waveform) is fixed in the following experiments. For every task, the desired trajectory is separated into a series of checkpoints from start to finish. The amount of checkpoints depends on the intended precision and total length of the desired trajectory.

Referring to Figure 4, we have developed a motion control program to let the rotor's position to sequentially trace the checkpoints. Whenever the rotor's position gets close enough to the current checkpoint, the control program commands the motor to trace the next checkpoint until the whole task is finished. Combining forward/backward with translation/rotation, there are four decisions that can be made. After Cartesian–polar coordinate transformation, the control program calculates its decision related to the current checkpoint, and then moves the motor step by step. If the current position is identical to the previous one, the driving voltage is increased gradually to overcome the local friction force. The driving voltage range is 15 Vpp to 30 Vpp [1]. The control program dynamically adjusts the "minimum walkable voltage", which achieves precise steps without being stuck. The "closeness" value determines how closely the desired trajectory should be followed. Lower closeness values lead to less errors but take a longer time. The control system has to sacrifice accuracy for speed. The effect will be discussed in the following sections.

**Figure 4.** The control flow diagram of the control program. The goal is to trace a series of checkpoints. The driving voltage is increased gradually if the rotor is stuck.

## **3. Experimental Result**

The four-time and 10-time microscope objectives create fields of view (FOV) of 1 × 1 mm<sup>2</sup> and 0.38 × 38 mm2, respectively. Three kinds of motion trajectories—straight lines, circles, and pentagrams—are demonstrated in this section.

#### *3.1. Straight Line Trajectory*

#### 3.1.1. Objective of Four-Time Magnification

As illustrated in Figure 5, the task trajectory is from the third quadrant to the first quadrant with a distance of 300 μm. Once the imaged position deviates from the desired path, the visual servo system corrects it back. Figure 5a,b shows the closeness values of one pixel (i.e., 1 μm) and five pixels (i.e., 5 μm), respectively. In Figure 5a, the coefficient of determination, the straightness, and the consumed time are 0.9993, 1.184 μm, 165 s, respectively. In Figure 5b, the coefficient of determination (COD), the straightness, and the consumed time are 0.9987, 1.506 μm, 108 s. The results reflect the trade-off between accuracy and speed.

Figure 6 shows the experimental results when the walking distance is extended to 500 μm, while other conditions are maintained. In Figure 6a, the coefficient of determination, the straightness, and the consumed time are 0.9995, 1.579 μm, 278 s, respectively. In Figure 6b, COD, the straightness, and the consumed time are 0.9993, 1.812 μm, 117 s. The results present a similar trend to Figure 5.

**Figure 5.** The experiment results of 300 μm straight line trajectories under the FOV of four-time objective. The closeness values are 1 μm and 5 μm in (**a**) and (**b**), respectively. The red line represents the desired path. The black dots are the imaged position of the marker driven by the 2-DOF nano-stepping motor.

**Figure 6.** The experiment results of 500 μm straight line trajectories under the FOV of four-time objective. The closeness values are 1 μm and 5 μm in (**a**) and (**b**), respectively. The red line represents the desired path. The black dots are the imaged position of the marker driven by the 2-DOF nano-stepping motor.

#### 3.1.2. Objective of 10-Time Magnification

Replacing the microscope objective with a higher magnification enhances the image resolution and improves the motion accuracy. As illustrated in Figures 7 and 8, the distances from their start points to end points are 100 μm and 200 μm, respectively. Figures 7a and 8a show the results of closeness values of one pixel (i.e., 0.38 μm). The closeness values are five pixels (i.e., 1.9 μm) in Figures 7b and 8b. The detailed experimental results are listed in Table 1.

**Figure 7.** The experiment results of 100 μm straight line trajectories under the FOV of 10-time objective. The closeness values are 0.38 μm and 1.9 μm in (**a**) and (**b**), respectively. The red line represents the desired path. The black dots are the imaged position of the marker.

**Figure 8.** The experiment results of 200 μm straight line trajectories under the FOV of 10-time objective. The closeness values are 0.38 μm and 1.9 μm in (**a**) and (**b**), respectively. The red line represents the desired path. The black dots are the imaged position of the marker.
