## 2 Torque Sensor Design
### Vertical Model  
Torque is defined as the product of the force and the moment arm length. Utilizing this simple definition, it is possible to measure torque without commercial torque sensor. Figure 1 shows a schematic of the proposed torque sensor using a load cell with simple other components. The lever arm and the base arm have a number of assembly holes to fix the ends of the load cell on them. In order to measure high torque with the limited capacity of the load cell, long lever and base arms are essential. However, it is physically impossible to make them infinitely long for high inertia and large bending deformation. Especially, if the arms are not rigid enough, the assembled sensor can show considerable hysteresis. Therefore, it is required to predict the deformation of the two arms to minimize the sensor inaccuracy. If the lever arm and the base arm have same cross-sectional dimensions: width \(b\) and height \(h\), their area moment of inertia are expressed as \(I_{L}=b_{L}{h_{L}}^{3}/12\) and \(I_{B}=b_{B}{h_{B}}^{3}/12\), where the subscripts \(L\) and \(B\) represent _lever_ and _base_, respectively. Furthermore, if a counter-clockwise torque is applied to the motor shaft, the two arms will deform for the vertical reaction force \(F\) applied by the load cell as shown in Figure 2. It is assumed that the two arms are nearly parallel to each other and the load cell is assembled vertically for the simple vertical model. Then, the Euler-Bernoulli beam theory yields the deflections of the arms on the load cell position as follows:  
\[\delta_{L}=\frac{FL^{3}}{3E_{L}I_{L}} \tag{1}\]  
\[\delta_{B}=\frac{FL^{3}}{3E_{B}I_{B}}, \tag{2}\]where \(E_{L}\) and \(E_{B}\) denote Young's modulus of the lever and the base arms, and they have the same length of \(L\). In addition, the longitudinal deformation of the load cell \(\delta_{LC}\) by the applied normal force \(F\) can be expressed as follows by using the stiffness of the load cell \(k_{LC}\): \(\delta_{LC}\) = \(F\)/\(k_{LC}\) or \(\delta_{LC}\) = \(\tau\)/\(L\)\(k_{LC}\). If their deformation is small, the resultant angular deflection or the error of the measured angle will be  
\[\theta_{T}\approx\frac{\delta_{L}+\delta_{B}+\delta_{LC}}{L} \tag{3}\]  
Here, if the two arms have same cross-sectional dimension and Young's modulus: \(E_{B}\) = \(E_{L}\) and \(I_{B}\) = \(I_{L}\), the angle error or torsional deformation can be simplified as follows:  
\[\theta_{T}=\tau\left(\frac{2L}{3E_{L}I_{L}}+\frac{1}{L^{2}k_{LC}}\right) \tag{4}\]  
Figure 1: Schematics of the proposed torque sensors using load cells: (**a**) a circular torque sensor using four load cells and (**b**) a range-adjustable torque sensor using a load cell. Figure 2: Angular deflection resulting from bending of the lever and base arms for external load. The longitudinal deformation of the load cell is not depicted here. As a result, torsional stiffness of this torque sensor \(K_{T}=\tau/\theta_{T}\) can be expressed as follows:  
\[K_{T}=\frac{3E_{L}I_{L}L^{2}k_{LC}}{3E_{L}I_{L}+2L^{3}k_{LC}} \tag{5}\]  
For example, if the linear stiffness of the load cell is \(k_{LC}=0.5\times 10^{6}\) N/m with the 200 mm long two arms of \(b_{L}=5\), \(h_{L}=20\) (mm) and \(E_{L}=69\) GPa, the maximum payload capacity of the load cell \(F_{max}=50\) N or maximum applied torque \(\tau_{max}=10\) N-m yields: \(\delta_{L}+\delta_{B}+\delta_{LC}=1.26\) mm and \(\theta_{T}=0.36^{\circ}\). This angular deflection will be negligible for most applications, and moreover it can be reduced by design modification. It will also work to substitute the arms material with another of higher elasticity. The torsional stiffness of this sensor is determined by two terms, as can be seen in Equation (4): bending of the two arms and axial deformation of the load cell. Depending on the length \(L\), the governing term that affects the torsional deformation and stiffness of the sensor changes. Figure 3 shows the changes in torsional deformation and stiffness by changing the length \(L\) in the example just above.