## 2 Materials and Methods
### Cell Transportation Strategy
#### 2.2.2 Control Strategy  
Figure 4 illustrates the control sequences of the indirect transportation of biological cells with our HOT system. First, an operator would determine a target cell, which has the phenotypes of interest, then set the transport requirements through the GUI. Afterward, visual servo control was performed to transport the cell by manipulating the microtool with optical traps. In our study, the microtool must move toward a consistent direction while pushing the cell, because the cells were manipulated in a non-prehensile manner. Otherwise, the cell may slip away from the microtool. Therefore, this study focuses only on translational and rotational movements of the microtool. Specifically, a translational movement is used to push the cell to move along a straight path, and a rotational movement is used to adjust the microtool's direction when the actual direction does not coincide with the desired direction.  
Figure 3: Robot-aided holographic optical tweezer (HOT) cell manipulation system. (**a**) Experimental setup. (**b**) Schematic illustration.  
A geometry-based calibration method was introduced to set up the mapping relationship between the world and microtool coordinate systems. Figure 5 shows the kinematic diagram of the microtool, where \([X_{w},Y_{w}]\) denotes the world coordinate system; \([X_{R},Y_{R}]\) is the microtool coordinate system, which is located at the microtool center \((O_{R})\); \(D\) pertains to the central point of the rod that connects the two optical handling spheres A and B of the microtool; \(\overset{\rightarrow}{O_{R}D}\) is the robot head direction, which is coincident with that of the \(X_{R}\) axis \(\theta\) is the rotational angle between the world and the microtool coordinate systems; and \((a,\,b)\) is the coordinate value of \(O_{R}\) in the world coordinate system. The mapping relationship between the world and microtool coordinate systems can be derived from the geometry of the microtool and expressed as follows:  
\[\left[\begin{array}{c}x_{R}\\ y_{R}\end{array}\right]=\left[\begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}\right]\left[\begin{array}{c}x_{W}\\ y_{W}\end{array}\right]+\left[\begin{array}{cc}-b&-a\\ a&-b\end{array}\right]\left[\begin{array}{c}\sin\theta\\ \cos\theta\end{array}\right] \tag{1}\]  
where \((x_{R},y_{R})\) is the coordinate value of a certain point in the microtool coordinate system, and \((x_{W},y_{W})\) is its corresponding value in the world coordinate system.  
The kinematic diagram of the microtool shows that the microtool can act similarly to an omnidirectional microrobot [44], whose translational and rotational movements can be controlled independently and simultaneously. Specifically, the microtool pose can be controlled by manipulating the three handling spheres (see A, B, and C in Figure 5) with three optical traps. In this paper, we assumed that the movement velocities of the handling spheres were the same as those of the optical traps. This observation can be justified by experimentally setting the appropriate movement velocities of the optical traps. The effect of the inertia force of the cells is negligible due to the low Reynolds  
Figure 4: Control sequences of indirect transport of biological cells with the robot-aided HOT system by manipulating the microtool.  
Figure 5: Kinematic diagram of the microtool.  
number in the liquid environments [45], such that the cells were supposed to be stably pushed with the microtool.