Using Mathematical Models and Extensions to Solve a Robot’s Equidistant Returning Path

The main purpose of this project is to study the Rusty the Robot problem where a robot travels equidistantly from the starting point back to the starting point. We mainly use mathematical models such as trigonometric functions, recursive sequences, and mathematical induction to explore, deduce and demonstrate findings, while using mathematical computer software for calculation and verification. In the beginning, Geogebra was used to explore the original question and allow us to obtain some properties and results. Then, the topic was extended and the relationship between the number of steps and angles was explored, as well as the recursive relationship between the landing points during the travel process. Next, we altered the starting positions of Rusty to see how this affects our findings. Finally, the two intersecting straight lines are extended into three straight lines intersecting at the same point. Based on the angles between them, we discuss the number of returning paths from the starting point to the starting point in an equidistant traveling method. Based on the mathematical model constructed, this study constructs a return path diagram for the robot to return to the starting point in an equidistant manner according to different starting point positions and obtains the recursive relationship of the relative positions on this return path. During the research process of this study, some interesting mathematical theories were obtained. It is expected that these results can be applied to related fields of AI robot movement patterns in the future.