## 5 Framework
### Stage 1: Device Calibration and Estimation of Human Thumb Joint Angles  
The developed device needs to be personalized to the user, and as such, the first step in the calibration procedure is to obtain a set of user\(-\)specific measurements. The user\(-\)specific limits for the MCP \((\theta_{MCP\ min/max})\) and IP \((\theta_{IP\ min/max})\) joints are measured using a goniometer, and their ranges of motion are determined. The location of the CMC joint is identified by extending the thumb and palpating the prominent bony structure at its base, where the first metacarpal meets the trapezium bone. Then, the dimensions for the metacarpal length \((L_{mp})\) and MCP knuckle thickness \((t)\) are measured using a vernier caliper/ruler. The user must wear the device using the bracelet such that the AA pot joint axis is parallel and coincident or as close to the CMC AA joint visually. Based on the personalization procedure discussed in Section 4.2, the user must adjust the device to ensure an unrestricted range of motion for the thumb while performing a set of prescribed  
Figure 4: Graphical layout of the four\(-\)bar linkage of the device (links \(a\), \(g\), \(b\), \(f\) in solid lines). The base frame axes are \(X_{B}\) and \(Z_{B}\), and CMC FE joint frame axes are \(X_{FE}\) and \(Y_{FE}\). pure FE or AA motions. The user must perform all the prescribed motions and record the FE and AA pot--joint potentiometer readings at their respective limits for each thumb motion. Then, the user must measure the height of the AA pot joint center from the CMC joint (\(g\)) along the direction of pot joint height adjustment (see Figure 3). Next, the user must flex/extend the MCP and IP joints and record the flex sensor readings at their motion limits. After calibration, the pot joint FE and AA potentiometer readings must be correlated to the human CMC joint motion. When a user wears the tracking device, the FE motion of the thumb is transferred to the FE axis pot joint through an equivalent four--bar kinematic linkage shown in Figure 4 with links \(a,f,b,g\). The CMC FE joint angle, \(\theta_{FE}\), is the angular displacement of link \(a\) relative to \(X_{B}\) (\(X\) axis of the base frame in Figure 4) and is estimated using Equation (1):  
\[\theta_{FE}=\alpha+\mu \tag{1}\]  
where \(\alpha\) is the angular displacement of link \(a\) relative to link \(g\), and \(\mu=\arctan\left(t_{0}/L_{mp}\right)\) is the angular offset of the metacarpal phalange from link \(a\). Note that \(t_{0}(=t/2+t_{1})\) is the position offset of the thumb ring pivot along the negative \(Y_{FE}\) direction, where \(t\) is the thickness of the MCP knuckle, and \(t_{1}\) is the thumb ring pivot joint offset from the base of the ring to the joint center along the \(X_{FE}\) direction (see Figure 4), which is obtained from the 3D model of the thumb ring. The angular displacement (\(\alpha\)) of link \(a\) is estimated using Equation (2):  
\[\alpha=\arccos\left(\frac{g^{2}+d^{2}-b^{2}}{2 gd}\right)+\arccos\left(\frac{a^ {2}+d^{2}-f^{2}}{2 ad}\right) \tag{2}\]  
where \(d^{2}=(g^{2}+b^{2}-2 gb\cos(\pi-\theta_{pot,FE}))\), \(\theta_{pot,FE}\) is the angular displacement of the FE pot joint relative to \(X_{B}\) (the \(X\) axis of the kinematic base frame in Figure 4), and \(a^{2}=(t_{0}^{2}+L_{mp}^{2})\). The link lengths \(f\) and \(b\) are obtained from the 3D model of the tracking device. The FE axis pot joint angular displacement, \(\theta_{p,FE}\), is estimated using Equation (3):  
\[\theta_{p,FE}=\left(\frac{v_{FE}-v_{FE,min}}{\Delta v_{FE,max}}\right)\Delta \theta_{p,FE}+\theta_{p,FE,min} \tag{3}\]  
where \(\Delta\theta_{p,FE}=(\theta_{p,FE,max}-\theta_{p,FE,min})\approx 90^{\circ}\) is the range of FE pot joint motion, \(\theta_{p,FE,min}=-45^{\circ}\) is the minimum angular displacement, and \(\Delta v_{FE}=(v_{FE,max}-v_{FE,min})\approx 1.6\) V and \(v_{FE,min}\approx 1.7\) V are the voltage range and minimum voltage reading, respectively. Figure 5: Framework for mapping human thumb to an anatomically different robot thumb motion.