Victor Manuel Jimenez-Fernandez

This work presents the modified Lagrange interpolating polynomial (MLIP) method, which aims to provide a straightforward procedure for deriving accurate analytical approximations of a given function. The method introduces an exponential function with several parameters which multiplies one of the terms of a Lagrange interpolating polynomial. These parameters will adjust their values to ensure that the proposed approximation passes through several points of the target function, while also adopting the correct values of its derivative at several points, showing versatility. Lagrange interpolating polynomials (LIPs) present the problem of introducing oscillatory terms and are, therefore, expected to provide poor approximations for the derivative of a given function. We will see that one of the relevant contributions of MLIPs is that their approximations contain fewer oscillatory terms compared to those obtained by LIPs when both approximations pass through the same points of the function to be represented; consequently, better MLIP approximations are expected. A comparison of the results obtained by MLIPs with those from other methods reported in the literature highlights the method’s potential as a useful tool for obtaining accurate analytical approximations when interpolating a set of points. It is expected that this work contributes to break the paradigm that an effective modification of a known method has to be lengthy and complex.

En este artículo se presenta una metodología didáctica basada en el algoritmo numérico de Newton-Raphson, diseñada para enseñar a estudiantes de educación media superior cómo calcular raíces cuadradas y cúbicas. El enfoque de esta metodología está basado en el criterio de paro de convergencia cuadrática de Newton-Raphson, permitiendo en dos iteraciones una precisión de hasta cuatro dígitos significativos o más en las raíces obtenidas. Asimismo, esta metodología puede apoyarse en la descomposición de factores para su aplicación. Además, se incluyen ejemplos numéricos frecuentemente propuestos a los alumnos de educación media, con el fin de brindarles práctica en la aplicación del algoritmo enseñado en escuelas secundarias y preparatorias. La aplicación de este algoritmo trae como beneficio estimular el cálculo mental en los estudiantes.

This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.

Achieving the smart motion of any autonomous or semi-autonomous robot requires an efficient algorithm to determine a feasible collision-free path. In this paper, a novel collision-free path homotopy-based path-planning algorithm applied to planar robotic arms is presented. The algorithm utilizes homotopy continuation methods (HCMs) to solve the non-linear algebraic equations system (NAES) that models the robot’s workspace. The method was validated with three case studies with robotic arms in different configurations. For the first case, a robot arm with three links must enter a narrow corridor with two obstacles. For the second case, a six-link robot arm with a gripper is required to take an object inside a narrow corridor with two obstacles. For the third case, a twenty-link arm must take an object inside a maze-like environment. These case studies validated, by simulation, the versatility and capacity of the proposed path-planning algorithm. The results show that the CPU time is dozens of milliseconds with a memory consumption less than 4.5 kB for the first two cases. For the third case, the CPU time is around 2.7 s and the memory consumption around 18 kB. Finally, the method’s performance was further validated using the industrial robot arm CRS CataLyst-5 by Thermo Electron.

This article presents the Modified Integral Homotopy Expansive Method (MIHEM) which is utilized to find power series solutions for linear ordinary differential equations about ordinary points. This method is a modification of the integral homotopy expansive method. The proposal consists in providing a versatile, easy to employ and systematic method. Thus, we will see that MIHEM requires only of elementary integrations and that the initial function will be always the same for the linear ordinary differential equations of the same order which contributes to ease the procedure. Therefore, it is expected that this article contributes to change the idea that an effective method has to be long and difficult, such as it is the case of Power Series Method (PSM). This method expresses a differential equation as an integral equation, and the integrand of the equation in terms of a homotopy. We will see along this work the convenience of this procedure.

This work presents the design of a biosensor for detection of Escherichia coli bacteria. The biosensor consists of an interdigitated electrodes array to detect the impedance changes, as a function of frequency, when bacteria is captured in different areas. The design was first simulated with COMSOL Multiphysics software to obtain the impedance magnitude and phase curves, after that the transfer function for every case was established with an algorithm in MATLAB software to propose an equivalent circuit in order to determine the electrical components that dominate the impedance response in every frequency range of the curve. Results show that impedance changes in the biosensor are mainly due to the bacteria captured between interdigitated electrodes, whereas the bacteria captured only over interdigitated electrodes produce a very small change in the impedance. In order to confine the bacteria between electrodes, a microchannel is added, increasing the biosensor sensitivity.

The ability to plan a multiple-target path that goes through places considered important is desirable for autonomous mobile robots that perform tasks in industrial environments. This characteristic is necessary for inspection robots that monitor the critical conditions of sectors in thermal, nuclear, and hydropower plants. This ability is also useful for applications such as service at home, victim rescue, museum guidance, land mine detection, and so forth. Multiple-target collision-free path planning is a topic that has not been very studied because of the complexity that it implies. Usually, this issue is left in second place because, commonly, it is solved by segmentation using the point-to-point strategy. Nevertheless, this approach exhibits a poor performance, in terms of path length, due to unnecessary turnings and redundant segments present in the found path. In this paper, a multiple-target method based on homotopy continuation capable to calculate a collision-free path in a single execution for complex environments is presented. This method exhibits a better performance, both in speed and efficiency, and robustness compared to the original Homotopic Path Planning Method (HPPM). Among the new schemes that improve their performance are the Double Spherical Tracking (DST), the dummy obstacle scheme, and a systematic criterion to a selection of repulsion parameter. The case studies show its effectiveness to find a solution path for office-like environments in just a few milliseconds, even if they have narrow corridors and hundreds of obstacles. Additionally, a comparison between the proposed method and sampling-based planning algorithms (SBP) with the best performance is presented. Furthermore, the results of case studies show that the proposed method exhibits a better performance than SBP algorithms for execution time, memory, and in some cases path length metrics. Finally, to validate the feasibility of the paths calculated by the proposed planner; two simulations using the pure-pursuit controlled and differential drive robot model contained in the Robotics System Toolbox of MATLAB are presented.