# The Development of a Soft Robot Hand with Pin-Array Structure
## 4 Analyze
### Analysis of the Grasping Force  
This section mainly analyzes the grasping force that one pin can provide. For the convenience of research, assume that the CTSA-II hand is grabbing a sphere with a radius \(R_{o}\) placed at (\(x_{o}\), \(y_{o}\)), ignoring the elasticity of the membrane. The force analysis can be simplified as shown in Figure 9. This section will mainly analyze how the characteristics of the sphere, the size of the pin, and the size of the base affect the grasping force of one pin. The distribution force of a pin by membrane satisfies:  
\[(p_{0}-p_{i})\cdot 2\pi rl=N\cdot ql \tag{1}\]  
In the gripping process, the pin is equivalent to a cantilever beam of which one end is fixed at the base. Through static analysis, the force and moment that base acts on one pin follow:  
\[F_{b}=F_{o}-ql \tag{2}\]  
\[M_{b}+\int_{0}^{l}q\mathrm{ydy}-F_{o}y_{0}=0 \tag{3}\]  
Figure 8: The overall force situation: 1, base: 2, spring; 3, balloon membrane; 4, bendable sliding pins. Figure 9: Force analysis of single pin. To calculate \(F_{o}\), this paper uses the method of material mechanics. Divide the pin into two parts according to the contact point between the pin and the object, and the bending moments \(M_{i}(y)(i=1,2)\) of the two parts satisfy:  
\[\left\{ \begin{array}{ll}M_{1}(y)+\int yydy+M_{b}=0&(0<y\leq y_{0})\\ M_{2}(y)+\int yydy+M_{b}-F_{o}y_{0}=0&(y_{0}\leq y<l)\end{array} \right. \tag{4}\]  
We solve \(F_{o}\) by the approximately differential equation of the deflection curve [27]:  
\[\frac{d^{2}\omega_{i}}{dy^{2}}=\frac{M_{i}(y)}{EI}(i=1,2) \tag{5}\]  
In this differential equation, the boundary conditions are:  
\[\omega_{i}(0)=0,\left.\frac{d\omega_{1}}{dy}\right|_{y=0}=0,\omega_{1}(y_{0}) =\omega_{2}(y_{0}),\left.\frac{d\omega_{1}}{dy}\right|_{y=y_{0}}=\left.\frac{d \omega_{2}}{dy}\right|_{y=y_{0}} \tag{6}\]  
By jointly solving Equations (1)-(6), the deflection distribution is:  
\[\left\{ \begin{array}{ll}\omega_{1}(y)=\frac{-\frac{\omega^{4}}{2}+\frac{1}{ 2}(\frac{1}{2}ql^{2}-F_{o}y_{0})\cdot y^{2}}{EI}\\ \omega_{2}(y)=\frac{-\frac{\omega^{4}}{2}+\frac{1}{2}ql^{2}y^{2}-F_{o}y^{2}+ \frac{1}{2}F_{o}y^{3}}{EI}\end{array} \right. \tag{7}\]  
At the point where the pin is in contact with the object, the deflection is determined by:  
\[\omega_{1}(y_{0})=r-x_{o}-R_{o} \tag{8}\]  
So,  
\[F_{o}=\frac{-24 EI(r-x_{o}-R_{o})+\frac{(p_{0}-p_{1})\cdot 2\pi r}{N}(6l^{2}{y_{0} }^{2}-{y_{0}}^{4})}{12{y_{0}}^{3}} \tag{9}\]  
Equation (9) shows almost all the factors that affect \(F_{o}\), including the position and size of the target object, the elastic modulus, the size of the pin, the size of the base, etc. Equation (9) has important guiding significance for the design of the CTSA-II hand. From Equation (9), \(F_{o}\) is linear with \(p_{i}\). If \(p_{i}\) is smaller, \(F_{o}\) is larger. However, relatively speaking, the smaller the \(p_{i}\), the greater the power consumption required. Therefore, the pressure selected during the crawling process needs to be weighed. However, in order to ensure the gripping force, the internal pressure of the device should satisfy:  
\[p_{i}<p_{0}-\frac{12 EIN(r-x_{o}-R_{o})}{(6l^{2}{y_{0}}^{2}-{y_{0}}^{4})\pi r} \tag{10}\]  
For the parameters \(r\), it can be seen from Equation (9) that the gripping force is linear with \(r\), but there are many factors affecting the partial conductance of \(r\), as shown in Equation (11):  
\[\frac{\partial F_{o}}{\partial r}=\frac{(p_{0}-p_{i})\pi l^{2}}{N{y_{0}}}- \frac{2 EI}{{y_{0}}^{3}} \tag{11}\]  
It can be seen that it is not possible to determine whether it is positive or negative. For the parameter \(N\), from the perspective of a single unit, it is an inverse relationship between \(F_{o}\) and \(N\). However, as a whole, the increase of \(N\) will increase the number of pins that can provide the gripping force, which may bring better performance. The appropriate \(N\) is not determined through this discussion, and it may be better determined by experiments.