# Design of a Laparoscopic Robot System Based on Spherical Magnetic Field
## 3 Structural Design of Laparoscopic Robot System
### Design of the Stator  
To ensure the accuracy of the following linear modeling of magnetic flux density \(\mathbf{B_{i}}\), it is necessary to magnetize the iron cores along the radial direction as much as possible when determining the dimension \(\mathbf{S_{m}}\) of the inserted iron cores. According to [14], the ferromagnet can be magnetize linearly when the external magnetic field is controlled within a certain range, which can be expressed as:  
\[M=\chi H_{i} \tag{2}\]  
where \(M\) represents the magnetization intensity of the ferromagnet. \(\chi\) is the magnetic susceptibility. \(H_{i}\) is the internal magnetic field of the ferromagnet. In Equation (2), \(H_{i}\) is composed of external magnetic field \(H\) and internal demagnetizing field \(H_{d}\):  
\[H_{i}=H+H_{d} \tag{3}\]  
The intensity of demagnetizing field \(H_{d}\) is concerned with the demagnetization tensor \(N\), which can be expressed as:  
\[H_{d}=-NM \tag{4}\]  
\(N\) is related to the shape of the ferromagnet. For ferromagnets with symmetric axes, when the magnetic coordinate is aligned with the principal axis of the ferromagnet, \(N\) is a diagonal matrix, and the diagonal line of which consists of demagnetization factor \(n_{x}\), \(n_{y}\) and \(n_{z}\) (\(n_{x}\), \(n_{y}\), and \(n_{z}\), respectively, indicate the demagnetization factor along \(x\), \(y\), and \(z\) axis). By combining Equations (3) and (4), one can obtain:  
\[M=\chi_{a}H \tag{5}\]  
where:  
\[\chi_{a}=diag(\frac{\chi}{1+n_{x}\chi},\frac{\chi}{1+n_{y}\chi},\frac{\chi}{1+ n_{z}\chi}) \tag{6}\]  
According to [14], we have:  
\[n_{x}+n_{y}+n_{y}=1 \tag{7}\]  
The demagnetization factors \(n_{x}\), \(n_{y}\), and \(n_{z}\) only concern with the shape of the ferromagnet. For a cylindrical ferromagnet, the greater the ratio of the length to the diameter, the closer these parameters approach the following conditions:  
\[n_{x}=n_{y}=\frac{1}{2},\ n_{z}=0 \tag{8}\]  
Since the magnetic susceptibility \(\chi\) of a ferromagnet is very large, according to Equations (5)-(8), we can tell that the greater the ratio of the length to the diameter, the greater the axial component and the smaller the radial component of magnetization intensity \(M\). Accordingly, the ferrite cores are more inclined to be magnetized along the radial direction. According to [15], the relationship between the ratio of the length to the diameter and the demagnetization coefficient \(n_{z}\) is shown in Table 1.  
According to Table 1, when the ratio of length to diameter exceeds 10, \(n_{z}\) is less than 0.02, and the axial component of \(\chi_{a}\) is more than 20 times larger than its radial component. In this condition, the cylindrical ferromagnet is considered to be axially magnetized. Therefore, the dimension of these ferrite cores is preliminarily selected as diameter = 6 mm and length = 100 mm. After the size of the ferrite core is determined, the electric current \(\mathbf{I}_{t}\) through every coil, the turns of electromagnets \(\Phi_{c}\), as well as the distance between the rotor and the stator (\(\mathbf{L}_{b}\) in Figure 5) can be subsequently determined. To obtain a greater magnetic flux intensity, more coil turns should be adopted, however, since the electromagnets of the stator are tilting to a same point (the point \(\mathbf{O}\) shown in Figure 5), there is a relationship between the distance of the stator and the rotor and the greatest amount of coil turns without interference between the electromagnets of the stator. In Figure 5, \(L_{d}\) is the distance between the IPMs and the coil. \(L_{r}\) represent the distance between the upper surface of IPMs and the sphere center, \(r_{c}\) is the thickness of the coil \(\delta_{r}\) is the thickness of shell and \(r_{f}\) is the radius of the magnetic iron core. To avoid the interference among coils, we have:  
\[tan\frac{\pi}{8}\geq\frac{r_{c}+r_{f}+\delta_{r}}{L_{r}+L_{d}} \tag{9}\]  
where \(L_{r}\) is 10.58 mm, \(r_{f}\) is 3 mm and \(\delta_{r}\) is 1 mm. Thus, one can obtain:  
\[r_{c}\leq 0.38+0.414L_{d} \tag{10}\]