## 2 Torque Sensor Design
### Generalized Obique Model  
Generalizing the vertical model described previously, the load cell can be installed slantly on purpose as shown in Figure 4. In this oblique model, if the load cell can rotate freely about the joints connecting with the base and lever arms, only pure axial force is applied to the load cell. Here, \(L_{L}\) and \(L_{B}\) mean the lengths of the lever arm and the base arm, respectively, while \(d_{L}\) and \(d_{B}\) represent the vertical distances from the motor shaft to the arms. \(L_{LC}\) means the length of load cell between the upper and lower connecting points on the lever and base arms. With this information, the following  
Figure 3: Changes in (**a**) torsional deformation and (**b**) stiffness by changing the length \(L\): \(k_{LC}=0.5\times 10^{6}\) N/m for the load cell, \(b_{L}=b_{B}=5\) mm, \(h_{L}=h_{B}=20\) mm and \(E_{L}=E_{B}=69\) GPa for the two arms. moment arm length of the exerted force on the lever arm by the load cell, \(d_{LC}\), can be provided by applying [PERSON]'s formula to the area of the triangle \(OBC\):  
\[d_{LC}=\frac{\sqrt{4{h_{L}}^{2}{h_{B}}^{2}-{({h_{L}}^{2}+{h_{B}}^{2}-{L_{LC}}^{2} )}^{2}}}{2{L_{LC}}} \tag{6}\]  
Here, the length \(h_{L}\) and \(h_{B}\) are the hypotenuse of the two right angle triangles \(OCD\) and \(OAB\), respectively, and they can be substituted with the followings:  
\[h_{L}=\sqrt{{L_{L}}^{2}+{d_{L}}^{2}} \tag{7}\]  
\[h_{B}=\sqrt{{L_{B}}^{2}+{d_{B}}^{2}}. \tag{8}\]  
If the lever arm line \(CD\) is parallel with the base arm line \(AB\), Equation (6) can be expressed as  
\[d_{LC}=\frac{d_{B}{L_{L}}+d_{L}{L_{B}}}{{L_{LC}}} \tag{9}\]  
or  
\[d_{LC}=\frac{d_{B}{L_{L}}+d_{L}{L_{B}}}{\sqrt{{(L_{B}-{L_{L}})}^{2}+{d_{B}}^{2}}} \tag{10}\]  
without \(L_{LC}\). This expression must be more desirable, because \(L_{LC}\) can be adjusted to make the lever arm parallel to the base arm in practical use. Again, if the motor axis is on the lever arm line \(CD\) (\(d_{L}=0\)), the moment arm is simplified as follows. \[d_{LC}=\frac{d_{B}{L_{L}}}{{L_{LC}}} \tag{11}\]  
With this method utilizing the oblique load cell installation, an extremely short moment arm can be accomplished. Figure 5 shows an example to obtain the short moment arm using this oblique model. It is evident that a small \(d_{LC}\) can be achieved by small \(d_{B}\) and \(L_{L}\) with a fixed length of \(L_{B}\). As a result, it is possible to make \(d_{LC}=6.608\) mm when \(L_{B}=200\), \(d_{B}=20\) and \(L_{L}=50\) mm, even with the parallel \(AB\) and \(CD\): in this case, \(L_{LC}=151.327\) mm. They are very practical values for use, and \(d_{B}\) cannot  
Figure 4: Schematic of the oblique load cell setup for higher sensitivity. The length of the line segment \(OE\) is defined as the moment arm \(d_{LC}\) for the force applied by the load cell. be zero in a general way for physical limitation. Furthermore, in order to achieve shorter moment arm, it is needed to return to Equations (6)-(8). Utilizing Equation (6), the following condition can be obtained to make \(d_{LC}=0\):  
\[L_{LC}{}^{2}=\left(h_{L}\pm h_{B}\right)^{2}. \tag{12}\]  
Among the four possible solution sets for this relation, the most intuitive one with regard to Figure 4 is following: \(h_{B}=L_{LC}+h_{L}\). It means that the upper connecting point of the load cell on the lever arm \(C\) needs to be close to the straight line between the motor axis and the lower connecting point of the load cell on the base arm \(OB\). If the lever arm line \(CD\) is below the motor axis \(O\), or \(d_{L}\) has negative length, the moment arm \(d_{LC}\) can be much shorter, as can be seen from Equations (9) and (10). Although the line \(CD\) is above the axis \(O\), a negative value of \(L_{L}\) can similarly lead to a very short moment arm. Even when the lengths of the all components are fixed, it is still possible to reduce the moment arm \(d_{LC}\) by adjusting the load cell length \(L_{LC}\).