# Low-Cost Data-Driven Robot Collision Localization Using a Sparse Modular Point Matrix
## 2 Materials and Methods
### SMPM Method  
In our previous work, we observed that when using the collision point matrix template, the estimated positions of collision points are prone to confusion with their neighboring collision points. This implies that it is possible to estimate the occurrence of collisions at a particular collision point by leveraging the collision vibration information from its neighboring points. Building upon this idea, we propose a local modularization and sparsification approach for the collision point matrix template, as illustrated in Figure 1. The collision point matrix template mentioned here is identical to the one in ref. [4]. Collision points are defined as intersection points between the central axis of the charger and the plane in which the charging port is located. The template comprises collision points with 1 mm spacing, arranged in 17 rows and 17 columns, with its center located at the intersection of the central axis of the charging port and the plane in which the charging port is situated. In practical applications, the template can be scaled without altering the spacing between collision points. Modularization is achieved by exploiting the similarity in collision vibration signals between the estimated collision points and their neighboring points, eliminating the need to collect data for the estimated collision points during the data collection process. We refer to the estimated collision points that do not appear in the dataset as \"zero-shot points\", while the collision points used to estimate zero-shot points, requiring collection in the dataset, are defined as \"fully observable points\". In the process of implementing local modularization, we consider the information from fully observable points nearest to the zero-shot points to estimate collisions occurring at the zero-shot points. This process leads to the formation of a modular point matrix (MPM), as illustrated in the figure, comprising a central zero-shot point and its surrounding eight fully observable points. The MPM utilized results in a 1/9 reduction in the quantity of data collected, compared to the original collision point matrix template.  
Furthermore, it is crucial to consider whether utilizing all eight fully observable points is necessary for accurately estimating a collision occurrence at a zero-shot point. In theory, the collision vibration information obtained from these eight fully observable points may contain redundancy when estimating a zero-shot point. If this hypothesis holds true, eliminating the redundant fully observable points could further reduce the scale of the collision dataset. Taking this into consideration, we propose three sparsification methods for the MPM, as illustrated in Figure 2. The first sparsification method involves removing one of the fully observable points from the MPM. This approach results in eight sparse modular point matrices (SMPMs), denoted as Cell 1-1, Cell 1-2, \(\ldots\), Cell 1-8, obtained by sequentially removing one fully observable point in clockwise direction starting from the top left corner. The second sparsification method involves removing two fully observable points from the MPM. During this removal process, we consider two extreme cases: one that maximally preserves the zero-shot point's farthest adjacent points (resulting in Cell 2-1 and  
Figure 1: Illustration of the automatic charging equipment.  
Cell 2-2), and another that maximally preserves the zero-shot point's nearest adjacent points (resulting in Cell 2-3 and Cell 2-4). The third sparsification method involves removing four fully observable points from the MPM, specifically resulting in Cell 3-1, which excludes all of the nearest adjacent points, and Cell 3-2, which excludes all of the farthest adjacent points. The collision localization effects arising from the different sparsification methods are explained in detail in the experimental section.