# Human-Robot Cooperative Control Based on Virtual Fixture in Robot-Assisted Endoscopic Sinus Surgery
## 2 Robot Motion Constraint Model Based on Virtual Fixture
### Virtual Fixtures for Endoscopic Simus Surgery
#### 2.2.2 Forbidden Virtual Fixture Based on Hyperboloid  
The FRVF can limit the operating range of the robot and prevent the robot from entering a specific area, thereby avoiding collision between the instrument and tissue during operation. For the FRVF,  
Figure 2: Schematic diagram of GVF based on the spatial curve (**a**) The robot end is on the guiding curve, (**b**) the robot end is not on the guiding curve. how to construct appropriate geometries to constrain the workspace of the robot is an important issue to be considered in the application of the surgical robot. Considering that the endoscope and instrument swing around the nostril to form a workspace similar to the \"funnel\" shape [36], a single-leaf hyperboloid is used as the geometry of the FRVF. The schematic diagram of the FRVF based on a single-leaf hyperboloid is shown in Figure 3. Assuming that the position of the robot end is \(P(x,y,z)\), and the velocity of the robot end is \(V_{in}\) (direction as \(\overline{\text{PD}}\)), the equation of the single-leaf hyperboloid, S, is shown below:  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=d \tag{7}\]  
The action process of the FRVF based on a single-leaf hyperboloid is as follows:  
1) According to the position of the robot end and Equation (7), the spatial relationship between \(P(x,y,z)\) and the single-leaf hyperboloid is judged. 2) If \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\leq d+\frac{z^{2}}{c^{2}}\), the robot end, \(P(x,y,z)\), is inside the single-leaf hyperboloid, the FRVF does not work, and the robot end is in the free motion state, its velocity, \(V_{in}\), remains unchanged. 3) If \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}>d+\frac{z^{2}}{c^{2}}\), the robot end is outside the single-leaf hyperboloid, the velocity of the robot end will be changed by the FRVF. If \(V_{in}\)\(\overline{\text{P}^{\text{S}}\text{B}}>0\), the velocity of the robot end is in the same direction as the normal vector, \(\overline{\text{P}^{\text{S}}\text{B}}\), of the tangent plane, \(\overline{\text{P}^{\text{S}}\text{AC}}\), indicating that the robot end is leaving the prohibited area of the VF; if \(V_{in}\)\(\overline{\text{P}^{\text{S}}\text{B}}<0\), the velocity of the robot end is opposite to the normal vector, \(\overline{\text{P}^{\text{S}}\text{B}}\), of the tangent plane, \(\overline{\text{P}^{\text{S}}\text{AC}}\), indicating that the robot is entering the prohibited area of the VF. In the above two cases, the velocity of the robot needs to be adjusted. By projecting the velocity vector, \(V_{in}\), onto the normal, \(V_{in}\)\({}^{\perp}\), and tangent planes, \(V_{in}\)\({}^{\parallel}\) (the \(V_{in}\)\({}^{\perp}\) direction is along the direction of \(\overline{\text{P}^{\text{S}}\text{B}}\), the \(V_{in}\)\({}^{\parallel}\) direction is along the direction of \(\overline{\text{P}^{\text{S}}\text{D}}\)), the following can be obtained:  
\[\left\{\begin{array}{l}V_{in}=V_{in}\)\({}^{\parallel}+V_{in}\)\({}^{\perp}\)\\ V_{in}\)\({}^{\perp}=V_{in}\)-\(normal(\overline{\text{P}^{\text{S}}\text{B}})\\ V_{in}\)\({}^{\parallel}=V_{in}-V_{in}\)\({}^{\perp}\)  
Figure 3: Schematic diagram of the FRVF based on a single-leaf hyperboloid. (**a**) The funnel-shaped endoscope workspace, (**b**) the FRVF based on a single-leaf hyperboloid.  
Introducing the forbidden stiffness coefficient, \(k\in[-1,1]\), the input velocity can be expressed as follows:  
\[V_{in}=V_{in}{}^{\parallel}+k\cdot V_{in}{}^{\perp} \tag{9}\]  
When \(V_{in}{}^{\perp}\) and \(\overline{\text{P}^{\text{s}}\text{B}}\) is in the same direction, if \(k<0\), the robot end will continue to enter the prohibited area; if \(k\geq 0\), the robot end will leave the prohibited area. When \(V_{in}{}^{\perp}\) and \(\overline{\text{P}^{\text{s}}\text{B}}\) is in the reverse direction, if \(k<0\), the robot end will leave the prohibited area; if \(k\geq 0\), the robot end will continue to enter the prohibited area.