# A Flexible Surgical Instrument for Robot-Assisted Minimally Invasive Surgery
## 3 Kinematics Modeling and Analysis
The flexible part of the instrument is composed of \(n\) segments connected by pin rolls in series, the length of each segment is denoted by a, the length of the end effector is denoted by b, and the position on (\(o_{nx}\), \(o_{ny}\), and \(o_{nz}\)) of the pin roll, \(A_{n}\), and \(P\) (\(P_{x}\), \(P_{y}\), and \(P_{z}\)) of the end effector in the base system \(o_{0}\)-\(x_{0}y_{0}z_{0}\) can be obtained as follows:  
\[o_{nx}=\sum_{i=1}^{n}a\sin i\theta\times\cos\delta P_{x}=\left[b\sin(n+1)\theta+\sum_{i=1}^{n}a\sin i \theta\right]\cos\delta \tag{3}\]  
\[o_{ny}=\sum_{i=1}^{n}a\sin i\theta\times\sin\delta P_{y}=\left[b\sin(n+1)\theta+\sum_{i=1}^{n}a\sin i \theta\right]\sin\delta \tag{4}\]  
\[o_{nz}=\sum_{i=1}^{n}a\cos i\theta P_{z}=b\cos(n+1)\theta+\sum_{i=1}^{n}a\cos i\theta \tag{5}\]  
RMIS generally uses a master-slave operation mode, and the motion information of the master manipulator controlled by the surgeon is mapped onto the surgical instrument to follow the operation performed by the surgeon's hands. To ensure that the surgical instrument can reach the mapped position, it is necessary to solve the inverse kinematics of the instrument to obtain the motion of each segment. The motion of the instrument can be decomposed into bending angle \(\theta\) and rotation angle \(\delta\), which can be calculated by formulas (3) and (4). Figure 2.0  
The bending angle \(\theta\) of the designed instrument was in the range of 0-15\({}^{\circ}\), the rotation angle \(\delta\) was in the range of 0-360\({}^{\circ}\), the length \(a\) of each segment was 20 mm, the length \(b\) of the end effector was 50 mm, and the number of the segment was seven. According to the above forward kinematics analysis, the working space of the flexible instrument was calculated in MATLAB software ([[https://www.mathworks.com/products/matlab.html](https://www.mathworks.com/products/matlab.html)]([https://www.mathworks.com/products/matlab.html](https://www.mathworks.com/products/matlab.html))). As shown in Figure 4, the working space of the flexible instrument was approximately half a sphere with a radius of 150 mm. A flexible instrument can reach any position in the surgical space with several DOFs of motion. Figure 2.1 Workspace of the flexible instrument: (**a**) instrument coordinate system; (**b**) instrument plane motion trajectory; (**c**) instrument workspace trajectory; (**d**) instrument workspace. In this study, the bending angle \(\theta\) was solved using Newton-Raphson method following the basic equations:  
Figure 2.2  
Figure 2.3  
where _f(\(\theta\))_ is the equation containing the variable \(\theta\), \(\theta^{n}\) represents the value of the variable \(\theta\) at the \(n\)th iteration, and \(\Delta\theta^{n}\) is the difference of the variable \(\theta\) at two adjacent iterations. When solving the equations, \(\theta^{0}\) is used as the initial iteration point to start the iterative calculation, \(\Delta\theta^{n}\) can be obtained by Equation (8), \(\theta^{n+1}\) can be obtained by Equation (7), and so forth until \(\Delta\theta^{n}\) reaches the required error range, where \(\theta^{n}\) is the solution of the system of equations. Thus, the bending angle \(\theta\) can be obtained. The bending motion of the surgical instrument was achieved by controlling the lengths of the two cables, C\({}_{1}\) and C\({}_{3}\), and the relationship between the segment bending angle \(\theta\) and the lengths of cables C\({}_{1}\) and C\({}_{3}\) was calculated through the following steps. As shown in Figure 5a, S\({}_{i}\) is the center of the lower end face of hole H\({}_{i}\)/ before the segment R\({}_{i}\) is bent. S\({}_{i}\)/ is the center of the lower end face of hole H\({}_{i}\)/ after the instrument is bent, and F\({}_{i}\)/ is the center of the upper-end face of hole H\({}_{i}\)/. For each segment, there is a hollow cavity and a solid part with holes H\({}_{i}\)/. The diameter of the holes H\({}_{i}\)/ is the same as the diameter of the cables.