# Stochastic Models and Control of Anchoring Mechanisms for Grasping in Microgravity
## 3 Stochastic Models for Grasping
### Stiffness Model of the Microspine  
The microspine can be regarded as a variant of a planar serpentine spring. A microspine consists of multiple identical units. They are arranged in order through a pattern connected end to end. Generally speaking, the microspine has a certain stiffness in the X and Y directions of its plane. The movement and rotation in Z direction are limited by adjacent microspines. The microspine with square structure is adopted in this paper. The microspine is divided into 10 parts. Since the base is located below the 10 th, the deformation of this part is not considered. The unit number, load position and direction of the microspine are shown in Figure 7a. It is assumed that under the action of external force, the microspine produces only elastic deformation but no plastic deformation. According to [PERSON]'s second theorem, the deformation of 1 can be expressed as:  
\[\delta_{1x}=\int\frac{M}{EI}\frac{\partial M}{\partial F_{i}}dy=\int_{0}^{l} \frac{F_{x}y}{EI}ydy=\frac{F_{x}l^{3}}{3 EI} \tag{6}\]  
Figure 6: Relationship between the pull force and the adhesion probability. \[\delta_{1y}=\int\frac{M}{EA}\cdot\frac{\partial F_{y}}{\partial F_{i}}dx=\int_{0}^{d }\frac{F_{x}y}{EA}dx=\frac{F_{y}l}{EA} \tag{7}\]  
where \(E\) is the elastic model of the microspine, \(I\) is the section moment of inertia of the microspine, \(l\) is the length of 1, and \(A\) is the cross-sectional area of the microspine. The deformation of 2 can be obtained:  
\[\delta_{21x}=\int\frac{F_{x}}{EA}\cdot\frac{\partial F_{y}}{\partial F_{i}}dy= \int_{0}^{d}\frac{F_{x}}{EA}dy=\frac{F_{x}d}{EA} \tag{8}\]  
\[\delta_{22x}=\int\frac{M}{EI}\cdot\frac{\partial M}{\partial F_{i}}dy=\int_{0} ^{d}\frac{F_{x}l}{EI}ldy=\frac{F_{x}l^{2}d}{EI} \tag{9}\]  
\[\delta_{2y}=\int\frac{M}{EI}\cdot\frac{\partial M}{\partial F_{i}}dy=\int_{0} ^{d}\frac{F_{y}x}{EI}xdx=\frac{F_{y}d^{3}}{3 EI} \tag{10}\]  
Considering that the configuration and size of each unit of the microspine are the same, the deformation of the microspine in the \(X\) direction can be expressed as:  
\[\delta_{x}=5\delta_{1}+3\delta_{2}\]  
If the cross-sectional area of the microspine is expressed as \(A=ab\) and the cross-sectional moment of inertia is expressed as \(I=\frac{\delta_{0}^{3}b}{12}\), then the stiffness of the microspine in the \(X\) direction can be expressed as:  
\[k_{x}=\frac{F_{x}}{\delta_{x}}=\frac{12 EI}{20l^{3}+36l^{2}d+3a^{2}d} \tag{11}\]  
The stiffness of the rest of the microspine in the \(Y\) direction can be expressed as:  
\[\delta_{31y}=\int\frac{F_{y}}{EA}\cdot\frac{\partial F_{y}}{\partial F_{i}}dx =\int_{0}^{l}\frac{F_{y}}{EA}dx=\frac{F_{y}l}{EA} \tag{12}\]  
\[\delta_{32y}=\int\frac{M}{EI}\cdot\frac{\partial M}{\partial F_{i}}dy=\int_{0 }^{l}\frac{F_{y}d}{EI}ddy=\frac{F_{y}d^{2}l}{EI} \tag{13}\]  
Figure 6:0  
Figure 6:1  
Figure 6:2  
Figure 6:3  
Similarly to calculating the deformation in the \(X\) direction, the deformation of the microspine in the \(Y\) direction can be expressed as the superposition of multiple deformation of the same part:  
Figure 6:4  
The stiffness in the \(Y\) direction can be written as:  
Figure 6:5Based on the stiffness model and the stochastic model for grasping, the driving-force model of the anchoring mechanism can be expressed as:  
Figure 6:6  
where \(n\) is the number if microspines, \(K_{M}\) is the stiffness of the microspine, \(P(x)\) is the probability of a microspine engage with asperities, and \(x\) is the lifting distance of drive disc. For verifying the deduced stiffness model, the finite element software was used to simulate the deformation of the microspine caused by the force. In Figure 6(b), the model is fixed on the left. Then the deformation of the microspine in the direction of the applied force was recorded. The parameters of the microspine are as follows: \(l=14\) mm, \(E=7000\) MPa, \(a=2\) mm, \(b=3\) mm, \(d=4.5\) mm. The simulation results are shown in the figure below.