# 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  
This mathematical model is designed to accurately represent the cross-section of TacTip's skin using an assembly of chains. Each chain in the model is characterized by a fixed endpoint, a constant sectional length, and a limited angular variation between consecutive sections. This chain-based approach is advantageous for its precise representation of the TacTip's skin shape, allowing for detailed tracking of specific skin positions and aligning well with our hypotheses. In the model, each chain represents a cross-section of the TacTip's skin. The chains consisting of multiple sections, each representing a distinct segment of the skin. Increasing the number of sections within each chain enhances the resolution of the simulation, providing a more detailed and accurate representation of the TacTip's skin. The structure of the chains is such that every section is connected to its immediate neighbors. This interconnectedness is maintained by indexing each section, a crucial feature for the tracking individual skin segment movements post-deformation. This indexing is essential for analyzing how specific segments of the skin shift and deform. Figure 3: Contact feedback with pillow part. **Initial Chain Calculation**  
The process of determining each section's position within a chain is systematic. Initially, each chain is configured to approximate the lower half of a circle, a radius with a radius corresponding to the cross-section of the TacTip it represents. The total number of sections in a chain is obtained by dividing the chain's total length by the length of each section. For a chain with \(n\) sections, there will be \(n+1\) ends, which connect to the TacTip's base or to adjacent sections, as shown in Figure 4. The angle between two adjacent sections is calculated by dividing \(\pi\) by the number of sections. The orientation angle for each section is determined by cumulatively summing the angles between sections. The coordinates of each section's end are plotted on the Tactip's 2D cross-sectional plane, with \(x\) representing the horizontal and \(y\) representing the vertical directions. These coordinates are computed using the horizontal distance \(l\cos(\theta)\) and the vertical distance \(l\sin(\theta)\), where \(l\) is the sectional length and \(\theta\) is the angle between sections. \[\left\{ \begin{array}{l}x_{n}=\sum_{n=1}^{n}l\cdot\cos\theta_{n}\\ y_{n}=\sum_{n=1}^{n}l\cdot\text{sin}\theta_{n}\end{array} \right. \tag{1}\]  
**Displaced Chain Calculation**  
This model intricately segments the displaced chain into three distinct parts, enabling a detailed analysis of the Tactip's deformation. The two side sections of the chain represent its shape extending from the base to the boundary of the contact surface, while the central portion vividly illustrates the Tactip surface in direct contact with the object, as shown in Figures 5 and 6.  
established hypotheses, there is a predetermined maximum angle change between sections, and the overall shape change needed to be minimized. The deformation modeling is thus formulated as a constrained optimization problem. In this context, the angle changes between sections are considered as the variables to be optimized. The constraints include the coordinates of the endpoints and the upper limit on the angle change. Hence, the primary objective of the optimization is to minimize the cumulative displacement of each section from its original position. To effectively tackle this optimization challenge, we utilized two algorithms: Sequential Least Squares Programming (SLSQP) [19] from SciPy and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [20] from Platypus. The SLSQP algorithm is a gradient-based optimization method that is typically highly efficient for solving smooth, single-objective optimization problems. For instance, in [21], SLSQP, along with other algorithms, is applied to optimization problems involving inviscid and viscous transonic flows.