# A Fiber-Optic Six-Axis Force Sensor Based on a 3-UPU-Compliant Parallel Mechanism
## 4 Analysis of the Force Mapping Matrix for the Six-Axis Force Sensor  
### Static Analysis of the Sensor  
As analyzed above, the sensor can be equivalent to a 3-UPU configuration. The equivalent diagram is shown in Figure 7. The model consists of a fixed platform M, a moving platform m, and three UPU-structured branches. The two U-joints at the top and bottom of each force-measuring branch ensure that the branch is only subjected to tension or compression forces in the direction perpendicular to the axis of the flexible hinge. The realistic physical factors (such as friction, mechanical clearances, adhesive joint deformation and internal material damping) have a certain impact on the sensor model. However, these realistic physical factors are complex and some of them cannot be quantified in the sensor model. For the convenience of establishing the sensor model, we ignore these factors in static and stiffness analysis. The 6 \(\times\) 6 velocity Jacobian matrix \(J\) for the 3-UPU parallel structure can be expressed as:  
\[J=\frac{1}{2} \begin{pmatrix}0&0&2&-e&-\sqrt{3}e&0\\ 0&0&2&2e&0&0\\ 0&0&2&-e&\sqrt{3}e&0\\ 1&\sqrt{3}&0&0&0&2e\\ 2&0&0&0&0&-2e\\ 1&-\sqrt{3}&0&0&0&2e\\ \end{pmatrix}  \tag{4}\]  
where \(e\) (\(e\) = 40 mm) is the distance from the U-joint on the force-measuring branch to the center of the platform, as shown in Figure 7.  
Based on the principle of virtual work, the static model of the 3-UPU structure can be expressed as [20]:  
\[ \begin{bmatrix}F_{a1}\\ F_{a2}\\ F_{a3}\\ F_{c1}\\ F_{c2}\\ F_{c3}\end{bmatrix} =-\begin{pmatrix}J^{T}\\ \end{bmatrix}^{-1} \begin{bmatrix}F\\ T\end{bmatrix}  \tag{5}\]  
where \(F\) is the external force acting on the sensor; \(T\) is the external torque acting on the sensor; \(F_{a1}\), \(F_{a2}\) and \(F_{a3}\) are the forces along the direction of the three force-measuring branches; \(F_{c1}\), \(F_{c2}\) and \(F_{c3}\) are the forces perpendicular to the direction of three force-measuring branches. Figure 7: Equivalent 3-UPU parallel structure. For the force \(F_{a1}\) along the direction of the force-measuring branch, the equation is as follows:  
\[F_{a1}=(F_{a11}+F_{a12})\cos\alpha \tag{6}\]  
where \(F_{a11}\) and \(F_{a12}\) are the forces acting on the connecting rods under the action of force \(F_{a1}\) and \(\alpha\) is the angle between the connecting rod and the vertical direction, as shown in Figure 8a. Since the 3-UPU parallel mechanism has only three translational degrees of freedom, the force-measuring branch only experiences translational motion. For the force \(F_{c1}\) perpendicular to the direction of the force-measuring branch, the equation is as follows:  
\[F_{c1}=(F_{c11}+F_{c12})\cos\alpha \tag{7}\]  
where \(F_{c11}\) and \(F_{c12}\) are the forces acting on the connecting rods under the action of force \(F_{c1}\), as shown in Figure 8b. Based on the above analysis, Equation (5) can be rewritten as:  
Figure 8: Force analysis of the force-measuring branch. \[ \begin{bmatrix}F\\ T\end{bmatrix} =-J^{T}\cos\alpha \begin{bmatrix}F_{a11}+F_{a12}\\ F_{a21}+F_{a22}\\ F_{a31}+F_{a32}\\ F_{c11}+F_{c12}\\ F_{c21}+F_{c22}\\ F_{c31}+F_{c32}\end{bmatrix}  \tag{8}\]  
where \(F_{a21}\) and \(F_{a22}\) are the forces acting on the connecting rods under the action of force \(F_{a2}\), \(F_{c21}\) and \(F_{c22}\) are the forces acting on the connecting rods under the action of force \(F_{c2}\), \(F_{a31}\) and \(F_{a32}\) are the forces acting on the connecting rods under the action of force \(F_{a3}\), and \(F_{c31}\) and \(F_{c32}\) are the forces acting on the connecting rods under the action of force \(F_{c3}\). Due to the presence of two connecting rods on each force-measuring branch, there are, in total, six connecting rods. Through the analysis of Figure 8, the forces (\(F_{11}\), \(F_{12}\), \(F_{i3}\), \(F_{14}\), \(F_{i5}\), \(F_{i6}\)) of the six connecting rods can be derived as:  
\[ \begin{bmatrix}F_{11}\\ F_{22}\\ F_{13}\\ F_{14}\\ F_{15}\\ F_{6}\end{bmatrix} =