## 2 Materials and Methods
### Analytical Modelling of the Proposed IAC Actuator
The temperature (\(T_{w}\)) should be between the final martensite temperature (\(M_{F}\)) and the final austenite temperature (\(A_{F}\)) in Table 3.  
\[ \begin{split}\rho_{s}V_{s}c_{p}\frac{dT_{w}}{dt}&=I^{2 }R-h_{s}\frac{\pi D_{s}^{2}}{4}(T_{w}-T_{e})\ \text{(Heat transfer in SMA spring)}\\ T_{hc}&=T_{e}+\frac{I^{2}R}{h_{s}\frac{\pi D_{s}^{2} }{4}}\Big{(}1-e^{-(\frac{\phi_{h}}{\phi_{h}\varphi})t}\Big{)}\ \ \ \text{(Heating cycle)}\\ T_{cc}&=T_{e}+(T_{w}-T_{e})e^{-(\frac{\phi_{h}}{\phi _{h}\varphi})t}\ \ \ \text{(Cooling cycle)}\end{split}  \tag{5}\]  
where \(\rho_{s}\), \(V_{s}\), \(c_{p}\), \(I\), \(R\), \(h_{s}\), \(t\), \(T_{e}\) stand for the SMA spring's density, total volume, specific heat, electric current, electric resistance, convection coefficient, operational time, and ambient temperature, respectively. The volume fraction (\(\theta\)) defines the process of the transformation model. From the volume fraction, this model is divided into three classes such as the martensite (M) to the austenite (A) phase, clarifying the austenite fraction volume (\(\theta_{A}\)), austenite (A) to martensite (M) phase to define the martensite fraction volume (\(\theta_{M}\)), and twinned (\(M_{tw}\)) to detwinned (\(M_{dt}\)) martensite phase to derive the detwinned martensite fraction volume (\(\theta_{dt}\)). During the phase transformation process, we set the transformation boundaries of martensite (\(\theta_{M}\)) and austenite fraction volume (\(\theta_{A}\)), with a maximum (\(\theta_{M}\)) of 1 and 0 for (\(\theta_{A}\)), as shown in Figure 6. Equations (6) and (7) express the phase transformation processes of volume fraction (\(\theta\)). This equation is established based on the phase transformation equation model of the SMA-driven compliant rotary actuator demonstrated by [PERSON] et al. in 2018 [42] and the operating condition of the proposed actuator. The induced stress during the austenite fraction should be less than the stress of the martensite fraction. However, both stresses could cause the operational strain below the transformation strain (\(\epsilon_{tr}\)). \[ \begin{split}\theta_{M}&=\theta_{Ai}\Big{(}\frac{1 }{2}\Big{(}cos\frac{\pi(\delta T_{M}-\delta\sigma_{M})}{M_{S}-M_{F}}+1\Big{)} \Big{)}^{z_{a}}\ \text{(}A\ to\ M\ \text{transformation process)}\\ \theta_{A}&=(\theta_{Mi}-1)\Big{(}\frac{1}{2} \Big{(}cos\frac{\pi(\delta T_{A}-\delta\sigma_{A})}{A_{S}-A_{F}}+1\Big{)} \Big{)}^{z_{b}}+1\ \text{(}M\ \text{to}\ A\ \text{transformation process)}\\ \theta_{dt}&=(1-\theta_{A})(\theta_{dti}-1)\Big{(} \frac{1}{2}\Big{(}cos\frac{\pi\delta\sigma}{\
abla\sigma}+1\Big{)}\Big{)}^{ z_{c}}+(1-\theta_{A})\ \text{(}M_{tw}\ \text{to}\ M_{dt}\ \text{transformation process)}\\ \text{where,}\end{split}  \tag{6}\]  
\[ \begin{split} z_{a}&=\Big{(}1+\Big{(}\frac{\delta T _{M}-\delta\sigma_{M}}{M_{S}-M_{F}}\Big{)}\Big{(}\frac{\alpha_{1}-1}{\alpha_{ 1}}\Big{)}\Big{)}\alpha_{1},\\ z_{b}&=\Big{(}1-\Big{(}\frac{\delta T_{A}-\delta \sigma_{A}}{A_{F}-A_{S}}\Big{)}\Big{(}\frac{\alpha_{2}-1}{\alpha_{2}}\Big{)} \Big{)}\alpha_{2},\\ z_{c}&=\Big{(}1-\frac{\delta\sigma(\alpha_{3}-1)}{ \
abla\sigma_{3}}\Big{)}\alpha_{3},\\ \
abla\sigma&=\sigma_{0}-\sigma_{1}\,\delta\sigma= \sigma_{w}-\sigma_{i}\,\delta\sigma_{M}=\frac{\sigma_{w}}{\beta_{M}},\ \delta\sigma_{A}=\frac{\sigma_{w}}{\beta_{A}},\\ \delta T_{M}&=T_{w}-M_{s},\ \delta T_{A}=T_{w}-A_{S} \end{split}  \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Reference** \\ \hline Martensite starting temperature (\(M_{S}\)) & 38 \({}^{\circ}\)C & \\ Martensite final temperature (\(M_{F}\)) & 26.5 \({}^{\circ}\)C & \\ Austenite starting temperature (\(A_{S}\)) & 42 \({}^{\circ}\)C & [40] \\ Austenite final temperature (\(A_{F}\)) & 50.5 \({}^{\circ}\)C & \\ Slope of Austenite limit curve (\(\beta_{A}\)) & 7.2 Nmm\({}^{-2}\)K\({}^{-1}\) & \\ Slope of martensite limit curve (\(\beta_{M}\)) & 6.92 Nmm\({}^{-2}\)K\({}^{-1}\) & \\ Maximum transformation strain (\(\epsilon_{tr}\)) & 2.3\% & [41] \\ \hline \hline \end{tabular}
\end{table}