# Design of Shape Memory Alloy-Based Soft Wearable Robot for Assisting Wrist Motion
## 2 Design of Muscle-Like Actuator Based on Shape Memory Alloy (SMA) Coil Spring
Adjustable parameters for designing conventional spring are described in Figure 2. The relationship between force and displacement of the spring is Equation (1):  
\[\mathrm{F}=\frac{\mathrm{Gd}^{4}}{8\mathrm{D}^{3}\mathrm{n}}\delta \tag{1}\]F is the force, G is the shear modulus of the SMA, d is the wire diameter of the SMA, D is the coil diameter of the SMA, n is the number of coil turns, and \(\delta\) is the displacement of the SMA coil spring. Shear modulus is the property of the SMA material itself, whereas D, d, and n can be changed in the fabrication process. In the case of SMA, the shear modulus G changes depending on the temperature change. In addition, the transition temperature of the SMA, which is the temperature that the phase transformation from martensite to austenite starts, can be considered as a parameter to be selected depending on the environment in which the actuator is used. In order to determine the spring parameters that optimize the performances, five samples of coil springs that have different transition temperatures, wire diameter, and spring diameter were fabricated, and the force and displacement relationships of those samples were observed. The selected values of spring parameters and the transition temperature of these five samples are displayed in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample Number** & **SMA Materials** & \begin{tabular}{c} **Spring Diameter** \\ **(D)** \\ \end{tabular} & \
\begin{tabular}{c} **Number of Turns** \\ **(n)** \\ \end{tabular} &
\begin{tabular}{c} **Initial Length** \\ **(mm)** \\ \end{tabular} \\ \hline
1 & d = 0.5 mm & 3 mm & 55 & 50 \\
2 & Transition Temperature 40 °C & 2.8 mm & 55 & 50 \\
3 & & 2.5 mm & 55 & 50 \\
4 & d = 0.42 mm & 3 mm & 55 & 45 \\
5 & Transition Temperature 70 °C & 2.5 mm & 55 & 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of five shape memory alloy (SMA) coil spring samples. Figure 1: Schematic of the shape memory alloy (SMA)-based actuator. Figure 2: Parameters of coil spring. Several target performances of the proposed muscle-like actuator are determined for the wearable robot application. The actuator should be designed to produce high force in the desired deformation range. As preloading the SMA is required to be stretched, the target range of deformation is determined under load conditions. The target force required is 10 N per a single coil spring. The target maximum contraction ratio is selected to be greater than 40% of its initial stretched length, which exhibits a similar contraction ratio as that of a human muscle [60]. The target displacement range is selected as 50 mm from its initial stretched length when a 10 N load is applied to the spring samples. Target maximum length of the spring is selected up to 150 mm, which is a suitable size for a compact wearable robot for the wrist. In addition, a main consideration in actuator design is that the actuator should be as thin as possible to make the wearable robot portable and easy to wear. Finally, the maximum surface temperature has to be as low as possible to prevent burning of human skin since the SMA is activated by the increase of temperature. The experimental setup for the force-displacement relationship measurement of the spring samples is shown in Figure 3. The temperature of the SMA was measured by the thermocouple and was kept constant by feedback control. For the samples that have transition temperatures of 40 \({}^{\circ}\)C, the temperature was controlled at 30, 40, 50, 60 and 70 \({}^{\circ}\)C. For the samples that have transition temperature of 70 \({}^{\circ}\)C, the temperature was controlled at 30, 50, 70 and 90 \({}^{\circ}\)C.  
The graphs of the relationship between force and displacement of the sample springs at various temperatures are shown in Figure 4. Several characteristics of SMA can be observed in all the experimental results. As the temperature increases, the spring force in the same displacement increased as well.