# A Flexible Surgical Instrument for Robot-Assisted Minimally Invasive Surgery
## 3 Kinematics Modeling and Analysis  
When the flexible part of a surgical instrument is bent, the bending curvature of each segment is constantly based on the motion characteristics of the instrument [31]; thus, the bending angle of each segment is equal. The flexible instrument in Reference [31] was designed with multiple bending DOFs, and the forward kinematics of the instrument were established using the spatial geometry method, which increases the difficulty of kinematic modeling, and the multiple bending DOFs also increase the difficulty of precisely controlling flexible instruments. The flexible instrument designed in this paper had a bending DOF, which avoided redundant DOFs. When the flexible instrument does not rotate in motion, the instrument is only bent in a plane, and it is easy to establish the forward kinematics of the instrument using the plane geometry method, and it is convenient for precise control of the instrument. The following describes the kinematic model of a flexible instrument. To describe the motion state of the segment R\({}_{i}\), the coordinate system \(o_{i-1}-x_{i-1}y_{i-1}z_{i-1}\) was built at the geometric center of the pin roll, A\({}_{k-1}\). The positive direction of the \(x_{i-1}\) axis was pointed from \(o_{i-1}\) to the hole H\({}_{i}\)1, the positive direction of the \(y_{i-1}\) axis was pointed from \(o_{i-1}\) to the hole H\({}_{i}\)4, and the positive direction of the \(z_{i-1}\) axis was determined by the right-hand rule. The method of establishing the coordinate system \(o_{i-1}-x_{i-1}y_{i-1}z_{i-1}\), which is shown in Figure 2.  
Footnote 1: The \(x_{i-1}\) axis was pointed from \(o_{i-1}\) to the hole H\({}_{i}\)1, and the positive direction of the \(z_{i-1}\) axis was determined by the right-hand rule. The method of establishing the coordinate system \(o_{i-1}-x_{i-1}y_{i-1}z_{i-1}\) was similar to the coordinate system \(o_{i-1}-x_{i-1}y_{i-1}z_{i-1}\), which is shown in Figure 2.  
The motion of segment R\({}_{i}\) can be decomposed into two angles: the bending angle \(\theta\) of the segment R\({}_{i}\) around the pin roll and the rotation angle \(\delta\) around the z\({}_{i-1}\) axis. The length of segment R\({}_{i}\) is denoted by \(a\), and the position \(o_{i}\) (\(o_{ix}\), \(o_{iy}\), and \(o_{iz}\)) in the system \(o_{i-1}\)-x\({}_{i-1}y_{i-1}z_{i-1}\) can be obtained as follows:  
\[o_{ix}=a\text{sin}\theta\times\text{cos}\delta\quad o_{iy}=a\text{sin}\theta \times\text{sin}\delta\quad o_{iz}=a\text{cos}\theta \tag{1}\]  
The rotation angle \(\delta\) can be obtained by the coordinates of \(o_{i}\):  
\[\delta=\arctan\frac{o_{iy}}{o_{ix}} \tag{2}\]  
To analyze the kinematics of the flexible instrument, a multisegment coordinate system was established through the same method used for a single segment. The base coordinate system \(o_{0}\)-x\({}_{0}\)y\({}_{0}\)z\({}_{0}\) was established at the geometric center of the pin roll, A\({}_{0}\), and the coordinate system \(o_{1}\)-x\({}_{1}\)y\({}_{1}\)z\({}_{1}\) was established at the geometric center of the pin roll, A\({}_{1}\), etc. \(P\) (\(P_{x}\), \(P_{y}\), and \(P_{z}\)) represents the position of the end effector, which is shown in Figure 3.  
Figure 3: Kinematics model of the flexible surgical instrument. Figure 2: Kinematics model of a single segment.