An optimized formula for the two-point resistance of a cobweb resistance network and its potential application

Abstract

In recent years, the exploration and application of resistance networks have expanded significantly, and solving the equivalent resistance between two points of a resistance network has been an important topic. In this paper, we focus on optimizing the formula for calculating the two-point resistance of an m × n cobweb resistance network with 2r boundary conditions. To improve the computational efficiency of the equivalent resistance between two points, the formula is optimized by using the optimal approximation property of Chebyshev polynomials in combination with hyperbolic functions, and the derivation process is simplified. We discuss the equivalent resistance formulas in several special cases and compare the computational efficiency of the equivalent resistance formulas before and after optimization. Finally, we make an innovative attempt at path planning through potential formulas and propose a heuristic algorithm based on cobweb potential function for robot path planning in a cobweb environment with obstacles.

1. Introduction

With the continuous progress of modern science, many fields are facing increasingly complex problems. Further exploration by the researchers found that building a resistance network model (Kirchhoff, 1847; Pennetta et al., 2004; Ferri and Antonini, 2007; Owaidat et al., 2012; Rhazaoui et al., 2013; Liu et al., 2016; Hadad et al., 2018; Xu et al., 2021; Zhang et al., 2021) can help solve some of these problems. Since Kirchhoff’s law (Kirchhoff, 1847) laid the foundation for the research of resistance networks, after more than 170 years of in-depth exploration, the research of resistance networks has made remarkable progress and researchers have proposed some new methods.

In the early days, the research of resistance networks focused on infinite network structures, and Cserti et al. (2002) opened a completely new path for the exploration of infinite resistance networks by introducing Green’s function technique. Subsequently, the equivalent resistance values between any two points in various infinite lattice resistance structures were accurately computed using this method (Cserti et al., 2011), and the method was extended to deal with perturbed lattice problems containing one missing bond (Giordano, 2007). Since then, Green’s function techniques have been gradually applied to the research of infinite networks, and some new theoretical results have been obtained (Giordano, 2007; Hijjawi et al., 2008; Guttmann, 2010; Kook, 2011; Asad, 2013a, 2013b; Asad et al., 2013). However, an infinite network is an ideal structure without considering boundary conditions, and Green’s function techniques are not applicable to finite networks. The initial exploration of finite networks can be traced back to the research of Klein and Randić (1993), which was subsequently explored in more depth by Klein (2010) and Yang and Klein (2013). Meanwhile, Wu FY (2004) proposed a Laplacian matrix method to calculate the resistance between arbitrary nodes in a finite resistance lattice, which has been widely used in subsequent studies (Tzeng and Wu, 2006; Essam et al., 2014, 2015; Izmailian and Kenna, 2014, 2015; Izmailian et al., 2014). Specifically, Essam and Wu (2009) used the method to successfully find the exact value of the site-to-site resistance and its asymptotic expansion under free boundary conditions, while Izmailian and Huang (2010) further extended it to compute the resistance values under many different boundary conditions. However, the resistance between arbitrary nodes with complex boundary conditions cannot be solved using the Laplacian method.

To overcome the shortcomings of previous theories, Tan et al. (2013) creatively introduced the recursive transformation (RT) method, which opened up a new path for the study of resistance networks. Compared with the traditional Laplacian method that relies on two direction matrices, this method requires only one direction matrix. This not only simplifies the solving process but also makes the solution more convenient. For instance, it can be used to solve the equivalent resistance between two points in a cobweb with complex boundary conditions (Tan and Fang, 2015). Subsequently, Tan utilized the RT method to explore the electrical characteristics of resistance networks and made significant contributions to the development of the field (Tan, 2017, 2022, 2023a, 2023b; Tan ZZ and Tan Z, 2020a, 202...).