# Image-Based Tactile Deformation Simulation and Pose Estimation for Robot Skill Learning
## 2 Materials and Methods
### Model Proposal
#### 2.2.1 Chain Point in 2D Cross-Section
The NSGA-II algorithm, on the other hand, is a random, evolution-based optimization technique that can handle complex optimization problems with multiple objectives and intricate constraints. For example, in [22], the algorithm is used to identify the optimal oil platform location based on constraints such as well location, platform capacity, cost, and more. When applied to solving deformed chain shapes, the SLSQP algorithm is expected to provide reasonably accurate solutions in a short amount of time, while the NSGA-II algorithm is expected to yield more accurate solutions over a longer time period. (2) Contact Surface Chain Point Calculation  
In this section, the contact condition between TacTip and a pillow shaped part with a cylindrical surface is analyzed. The process of calculating the points on the contact surface chain involves two key steps: first, determining the location of the contact surface boundary, and second, accurately identifying the positions of the points along the contact surface itself. To calculate the contact surface boundary, as illustrated in Figure 7, the transparent Tactip skin exhibits properties similar to frosted glass. In the absence of contact, the view through this skin appears blurry when observed by the camera. However, upon making contact with the object, the contact surface becomes distinctly clear, contrasting sharply with the blurred background. By applying a binarization technique to the captured image, the boundary of the contact surface can be clearly delineated. From this perspective, the boundary of the contact surface presents as a rectangle, with its length and width denoted as \(l\) and \(d\), respectively. By utilizing these parameters, the height of contact surface \(h_{c}\) and the z-coordinate values for the boundary points along the length side \(Z_{l}\) and the width side \(Z_{w}\) can be calculated. Figure 6: Composition of each chain. \[h_{c}=R_{p}-\sqrt{{R_{p}}^{2}-(\frac{d}{2})^{2}} \tag{2}\]  
\[Z_{l}=-R_{T}+h-h_{c} \tag{3}\]  
\[Z_{w}=-R_{T}+h-(R_{p}-\sqrt{{R_{p}}^{2}-{Y_{w}}^{2}}) \tag{4}\]  
Given that the x and y coordinates of the contact boundary are identifiable from the transparent Tactip's imagery, and that the z-coordinates are calculated using the aforementioned formulas, the complete x-y-z coordinates of the contact boundary can now be accurately determined. These data are critical for precisely shaping the chains from the Tactip base to the contact boundary through the optimization algorithm. To accurately determine the exact position of the on contact surface points, a method of unwrapping and fitting of Tactip and object surface is employed. First, the center of the object's contact surface is identified, serving as the focal point for unwrapping the chains. Key chain points closed to this center are anchored, while the remaining points are arranged on the same x-y plane, maintaining their initial orientation and uniform spacing, equivalent to the chain's sectional length. Simultaneously, the object's contact surface is unwrapped relative to this central point. This technique efficiently locates the contact boundary points on the deformed chain, as demonstrated in Figure 8.  
The method for collision detection between the object and the TacTip simplifies the process by designating a contact point if its distance from the origin is less than the TacTip's radius. This approach provides greater computational efficiency compared to more complex algorithms, such as Triangle Mesh Collision. Once the contact points are identified, they projected onto the x/y plane, and the Graham Scan algorithm is used to generate a convex hull around them, efficiently determining the boundary sequence. However, the actual contact surface on the TacTip is slightly smaller than the initial boundary. By identifying the center of the contact surface, the boundary is scaled down to  
Figure 8: Unwrapping of TacTip skin and object surface for accurate boundary location. Figure 7: Locating contact surface boundary. provide a more accurate representation.