# High-Speed Cell Assembly with Piezo-Driven Two-Finger Microhand
## 2 System Setup  
### Two-Finger Microhand Structure  
The dexterous two-finger microhand is used to grasp and release cells by adjusting the distance between the two needle tips. One of the two fingers is an active finger, which is a 3-PRR (3-prismatic-revolute-revolute) parallel mechanism driven by three piezoelectric actuators, as shown in Figure 1a. The end-effector is installed on the parallel mechanism, which is driven by piezoelectric ceramics (NEC TOKIN AE0203D16, Shiroishi, Japan), as indicated by the green strip in Figure 1a, with a maximum deformation length of 17 \(\upmu\)m. The schematic diagram of its joint design is also shown in Figure 1a, including three prismatic joints formed by piezoelectric actuators and eight revolute joints formed by flexible hinges. The motion of every joint is achieved through the designed flexure hinge. The positioning accuracy of the end-effector can reach sub-millimeter levels and, in practical micromanipulations, we have chosen a minimum displacement resolution of 1 \(\upmu\)m.  
During operation, the mechanism is installed in a way that the plane formed by the X and Y axes coincides with the horizontal operating plane, while the Z-axis aligns with the vertical direction perpendicular to the focal plane of the microscope. Due to the asymmetry of the 3-PRR structure in this paper, algebraic methods would result in a large computational burden. Therefore, a geometric approach was employed for the inverse kinematic analysis. As shown in Figure 1a, motion chain 1 is composed of \(P1\), \(R1\), \(R6\), and \(R7\). Motion chain 2 is composed of \(P2\), \(R2\), and \(R4\). Motion chain 3 is composed  
Figure 1: (**a**) 3D parallel mechanism model and its joint design. (**b**) Kinetic chain 1 and kinetic chain 2 or 3.  
of \(P3\), \(R3\), and \(R5\). As the revolute joint \(R1\) cannot achieve motion when the prismatic joint \(P1\) moves independently, \(R1\) is considered a rigid body when analyzing motion chain 1. As illustrated in Figure 1b, for motion chain 1, with the displacement of the prismatic joint \(P1\), the links FA and AB reach the positions of \(F^{\prime}A^{\prime}\) and \(A^{\prime}\)B. The red and blue circles represent the trajectories of points \(A^{\prime}\) and \(F^{\prime}\), respectively. The length FF\({}^{\prime}\) can be calculated based on the geometric relationships in this figure, denoted as \(\Delta p_{1}\). Motion chains 2 and 3 are symmetric, and the inverse kinetic analysis can be conducted for one, with the joint displacement of the other being opposite. Additionally, the geometric kinematics analysis for motion chain 2 is shown in Figure 1b. With the displacement of the prismatic joint \(P2\), the links AB and BC reach the positions of \(A^{\prime}B^{\prime}\) and \(B^{\prime}\)C. The red and blue circles represent the trajectories of points \(B^{\prime}\) and \(A^{\prime}\), respectively. The length of AA\({}^{\prime}\) can be calculated based on the geometric relationships in this figure, denoted as \(\Delta p_{2}\). The \(\Delta p_{3}\) in motion chain 3 is of the same magnitude as \(\Delta p_{2}\) but in the opposite direction. The inverse kinematics analysis for motion chain 1 involves the rotational angle \(\alpha\) around the Z-axis, while the inverse kinematics analysis for motion chains 2 and 3 involves the rotational angle \(\beta\) around the Y-axis, as shown in Figure 1a. Translational motion in the X-axis direction is achieved by simultaneous equal displacements of the three prismatic joints. Therefore, when the target pose of the end-effector relative to its original position is represented as [\(x\)\(\alpha\)\(\beta\)], the displacement changes of the three prismatic joints can be deduced by synthesizing the kinematic relationships of each motion chain.