# Design of a Laparoscopic Robot System Based on Spherical Magnetic Field
## 2 Control Principle of the Laparoscopic Robot System  
In our design, the control of the rotor consists of two parts: (1) the position control and (2) the orientation control. As shown in Figure 3, the position control involves the shift along the \(x\) and \(y\) axes, the orientation control involves the deflection perpendicular to the \(z\) axis (i.e., the nutation angle \(\alpha\)), and the rotation along the \(z\) axis (i.e., the precession angle \(\beta\)). Figure 1: The prototype of laparoscopic robot system: (**a**) is the driving device (the stator) which is consist of five electromagnets, and (**b**) is the laparoscopic robot (the rotor). Figure 2: (**a**) shows the distribution of electromagnets in the stator and the IPMs in the rotor; (**b**) shows the working principle diagram of laparoscopic robot. The position control of the rotor is achieved by repositioning of the stator. If the position of the stator changes, the position of the magnetic field created by the stator will be changed, and the rotor will follow the movement of the stator under the magnetic force. The orientation of the rotor is determined by both orientation and input electric currents of the stator. By changing these parameters, the intensity and orientation of the magnetic field will be changed, and so are the magnetic force and torque acting upon the IPMs of the rotor. Therefore, the orientation of the rotor can be controlled. To simplify the control model, the rotation motion of the rotor is achieved by rotating the stator, and the deflection motion of the rotor is controlled by changing the input electric currents. In order to achieve the deflection control of the rotor, the relationship between the combination of five input electric currents of the stator and the deflection angle of the rotor need to be established. For that purpose, the magnetic flux density of the rotor (\(\mathbf{B}_{i}(\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{S}_{m},\mathbf{R}_{c},\Phi_{c},\mathbf{I}_{i})\) (\(i\) from 1-5) created by every coil of the stator and the magnetic moment \(\mathbf{M}_{j}(\mathbf{k},\mathbf{S}_{m})\) (\(j\) from 1-5) of every IPM in the rotor should be determined at first, where \(\mathbf{B}_{i}\) is concerned with the location \(\mathbf{p}(\mathbf{x},\mathbf{y},\mathbf{z})\), the diameter of wires \(\mathbf{R}_{c}\), the dimension of magnetic iron core \(\mathbf{S}_{m}\), the numbers of coil turns \(\Phi_{c}\), and the electric current \(\mathbf{I}_{i}\) through every coil, whereas the magnetic moment \(\mathbf{M}_{j}\) is determined by the material \(\mathbf{k}\) and the dimension of IPMs \(\mathbf{S}_{m}\). By mechanical design and parameter selection, the expression of the magnetic flux density \(\mathbf{B}_{i}\) of every coil, as well as the magnetic moment \(\mathbf{M}_{j}\) of every IPM can be obtained. Subsequently, the magnetic force and torque of coil to IPM can be solved. By superimposing the force and torque of every electromagnet to all five IPMs, the relationship between input current and the force \(\mathbf{F}(\overrightarrow{\mathbf{I}},\mathbf{P})\) and torque \(\mathbf{T}(\overrightarrow{\mathbf{I}},\mathbf{P})\) from the stator to the rotor can be determined, and both of them are determined by the input current and the orientation of the rotor \(\mathbf{P}\). Then, by analysing the force and torque acting on the rotor, the relationship between the input current and the deflection angle of the rotor \(\mathbf{\alpha}(\overrightarrow{\mathbf{I}})\) in the balanced condition can be determined. In the following part, the process of mechanical design and parameter selection will be introduced in detail. Figure 3: (**a**) shows the connection between rotor and stator. (**b**) shows the structure and degrees of freedom (DOFs) of the rotor. In the conceptual design, the rotor consists of a head, a wireless unit, a power unit, an IMU, a control unit, and a camera. In this paper, only the head is fabricated. The coordinate described in (**b**) is fixed to the center of the hemispherical head of the robot.