An application of potential function in robot path planning and three optimized formulas for equivalent resistance

The study proposed an innovative path planning algorithm based on the potential function of a special case of the cobweb resistor network, addressing the path planning problem in globe environments with obstacles. For the non-regular $m\times n$ cobweb resistor network with arbitrary longitude, we found that by introducing Chebyshev polynomial of the second class, the precise equivalent resistance formulas could be optimized effectively. Compared with the original formula, optimized equivalent resistance formulas significantly reduced the time cost in large-scale data calculations. Furthermore, we have plotted 3D views of the equivalent resistance formulas for several special cases and conducted simulation experiments on the computational efficiency of the original and optimized formulas at different data scales, verifying the superiority of the optimized formulas. These findings provided new perspectives and tools for the computation of resistor networks and the design of path planning algorithms.