# Stochastic Models and Control of Anchoring Mechanisms for Grasping in Microgravity
## 3 Stochastic Models for Grasping  
### Surface Properties  
The surface properties are among the factors affecting the maximum adhesion that the anchoring mechanism can provide. Obviously, the microspine array can more easily engage with a rough surface. On smooth surfaces, such as glass, the microspine array has difficulty providing stable adhesion. The influence of surface properties on the microspine is discussed below. There are two surface properties that determine whether the microspines can grasp the surface: the spatial distribution of asperities and the angle of asperities. The spatial distribution of asperities affects the possibility of microspine-asperity engagement. More asperities per unit length makes it more likely for the microspine to engage when it contacts the surface, and the microspine travels a shorter distance along the surface to engage. [PERSON] has studied the relationship between the tangential force and angle [33]. In this section, we discuss the spatial distribution of asperities in more detail. Assuming that asperities are randomly distributed on the surface, we can model the appearance of asperities as a random process. Therefore, the distance between adjacent asperities is described by random distance, and the number of asperities per unit distance is certain. It also shows that the microspine starts to travel in the gap of two asperities on the surface, and the longer the distance the microspine travels before catching the next asperity, the greater the probability of engagement. For the shape-modeling of asperities, a more intuitive triangle model is applied instead of the semicircle model used in the Hertz contact model, because the relationship between the probability of engagement and the travel distance is more worthy of attention, rather than the magnitude or direction of the contact force. The two-dimensional model of asperities is shown in Figure 2.  
The red triangles in Figure 2 are asperities, the length of the asperity is \(l_{1}\), the length of adjacent asperities is \(L\), and the distance between them is \(\Delta\). If the loading angle is appropriate, the microspine will not disengage under the tension after catching an asperity, rather than travel on the surface. That is to say, when the microspine contacts the surface, it cannot engage immediately when it is located in the area with a blue background (non-ideal region). It needs to travel a certain distance on the surface under the tension to engage. Therefore, there are only two cases when the microspine contacts with the surface: when  
Figure 2: Two-dimensional model of asperities. it engages with an asperity, and when it is located in a non-ideal region. It is a binomial distribution, and the probability of engagement can be expressed as:  
\[P_{initial}=\alpha=1-\frac{\Delta+\frac{\mathbb{I}_{2}}{2}}{L} \tag{1}\]  
If a microspine is located in a non-ideal region, the probability that it will be located at any point in the region is the same. The probability is related to the number of discrete points in the non-ideal region. It is obvious that if the microspine located in the non-ideal region is closer to the previous asperity, it needs to travel a longer distance to catch the next asperity. The distance that the microspine travels in the non-ideal region is related to its position. The discrete model of the non-ideal region is shown in Figure 3.  
The non-ideal region is divided into four parts equally, and the length of each part is called the step size. If the microspine travels one step on the surface, all the points in the D region can catch the next asperity, while the points in other regions still stay in the non-ideal region. If the microspine travels three steps, only points in the A region cannot engage, but points in the C and D regions only need to travel two and one step, respectively, to engage.