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Robotic arms are extensively employed across various disciplines, within both structured and unstructured workspaces, as well as across numerous applications. These components serve to address a multitude of challenges, thus enabling their integration into collaborative tasks with humans. In order to address these emerging challenges, there is a growing demand for more efficient path-planning algorithms. Hence, this study introduces a technique based on homotopy continuation methods, offering a distinctive and innovative approach to tackling these issues. The methodology proposed in this study enables the modeling of robotic arms with multiple degrees of freedom. It is proficient in operating within narrow corridors and achieves a collision-free path, provided the workspace allows for it. Due to its deterministic nature, the solution path is consistently reproducible. Furthermore, the computation times and RAM memory consumption achieved for hyper-redundant robotic arms fall within the range of seconds and kilobytes (KB). For anthropomorphic or classical robotic arms, computation times are on the order of milliseconds, as evidenced by the case studies presented in this paper. The method was validated using a Thermo Electron robotic arm, specifically the CRS CataLyst-5 model. This validation demonstrated its potential for application in the field of industrial robotics.

In this paper, we propose a new sliding window edge-oriented filter that computes the output pixels using a cross-correlation-like equation derived from the principles of fractional calculus (FC); thus, we call it the “fractional cross-correlation filter” (FCCF). We assessed the performance of this filter utilizing exclusively edge-preservation-oriented metrics such as the gradient conduction mean square error (GCMSE), the edge-based structural similarity (EBSSIM), and the multi-scale structural similarity (MS-SSIM); we conducted a statistical assessment of the performance of the filter based on those metrics by using the Berkeley segmentation dataset benchmark as a test corpus. Experimental data reveal that our approach achieves higher performance compared to conventional edge filters for all the metrics considered in this study. This is supported by the statistical analysis we carried out; specifically, the FCCF demonstrates a consistent enhancement in edge detection. We also conducted additional experiments for determining the main filter parameters, which we found to be optimal for a broad spectrum of images. The results underscore the FCCF’s potential to make significant contributions to the advancement of image processing techniques since many practical applications such as medical imaging, image enhancement, and computer vision rely heavily on edge detection filters.

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 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.

This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.

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.

During the phenomena modelling process in the different areas of science and engineering is common to face nonlinear equations without exact solutions; thus, the need of employing numerical methods to obtain such solutions. Therefore, in order to provide new possibilities for the isolation of variables, we propose a novel family of transcendental functions with new algebraic properties including their integration and differentiation rules. Likewise, in order facilitate the numerical evaluation for every new family set of functions, a highly accurate series of approximations is proposed by employing analytical expressions in terms of known transcendental functions and polynomials combinations. By the use of known functions for the proposed approximations, makes possible the use of any programming language for their respective implementation. In this article, three interesting case studies are presented with applications on: coastal engineering, transmission lines span on electrical engineering, and the planar one-dimensional Bratu equation. Finally, based on the results from study cases, it can be concluded that Leal-functions will have relevant impact in all areas of physics and mathematics, by providing new tools to scientists and engineers for the proposal of new mathematical models and numerical/analytical analysis, design and implementation of new theories and technological innovations.

This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials. Such characteristic makes of LPs a handy and powerful tool to approximate different kind of differential equations including: singular problems, initial condition and boundary-valued problems, equations with discontinuities, coupled differential equations, high-order equations, among others. Additionally, we show how the process to obtain the polynomials is straightforward and simple to implement; generating a compact, and easy to compute, expression. Even more, we present the process to approximate Gelfand's equation, an equation of an isothermal reaction, a model for chronic myelogenous leukemia, Thomas-Fermi equation, and a high order nonlinear differential equations with discontinuities getting, as result, accurate, fast and compact approximate solutions. In addition, we present the computational convergence and error studies for LPs resulting convergent polynomials and error tendency to zero as the order of LPs increases for all study cases. Finally, a study of CPU time shows that LPs require a few nano-seconds to be evaluated, which makes them suitable for intensive computing applications.

This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

In this paper, we explored the application of the fractional calculus to digital watermarking. In order to reach this goal, we chose a set of watermarking systems, and then, we altered the watermark detection step of each system. The novelty of our proposal is that we deviated from the trend of proposing new embedding domains computed by using fractional calculus principles. Instead, we proposed to reformulate the detection equation set. Then, this paper contributes to a new equation set for detecting watermarks based on fractional calculus principles. In addition, we provided the proper order for the derivative; this equation set replaces the previous one, given as a result that the detection rate increased up to 88% for the best case compared to the original equation set. The main modification was to replace the system's equations to another set that we derived by using the Riemann-Liouville fractional operator. To verify that if this approach is effective, we performed tests on grayscale images of size 512 × 512; the test covered several attacks including noise addition, JPEG compression, and scaling. The results showed that the modified systems detected more watermarks than their counterparts. The proposed scheme presented better results for 15 out of 21 different tests and the other 6 ended in a draw. It is important to remark that the behavior holds for other embedding domains, for example, the discrete cosine transform coefficients and the singular values decomposition values, which suggests the regularity of the approach.

Hydrogenated microcrystalline silicon (µc-Si:H) and epitaxial silicon (epi-Si) films have been produced from SiF4, H2 and Ar mixtures by plasma enhanced chemical vapor deposition (PECVD) at 200 °C. Here, both films were produced using identical deposition conditions, to determine if the conditions for producing µc-Si with the largest crystalline fraction (XC), will also result in epi-Si films that encompass the best quality and largest crystalline silicon (c-Si) fraction. Both characteristics are of importance for the development of thin film transistors (TFTs), thin film solar cells and novel 3D devices since epi-Si films can be grown or etched in a selective manner. Therefore, we have distinguished that the H2/SiF4 ratio affects the XC of µc-Si, the c-Si fraction in epi-Si films, and the structure of the epi-Si/c-Si interface. Raman and UV-Vis ellipsometry were used to evaluate the crystalline volume fraction (Xc) and composition of the deposited layers, while the structure of the films were inspected by high resolution transmission electron microscopy (HRTEM). Notably, the conditions for producing µc-Si with the largest XC are different in comparison to the fabrication conditions of epi-Si films with the best quality and largest c-Si fraction.

In the design of a controller for grasping objects through a robotic manipulator, there are two key problems: to find the position of the object to be grasped accurately, and to apply the appropriate force to each finger to handle the object properly without causing undesirable movement of it during its manipulation. A proportional-integral-derivative (PID) controller is widely used to grasp objects in robotics; however, its main shortcomings are its sensitivity to controller gains, sluggish response, and high starting overshooting. This research presents three coupled (position/force) controllers for object manipulation using an assembled robotic manipulator (i.e., a gripper attached to a robotic arm mounted on a mobile robot). Specifically, an angular gripper was employed in this study, which was composed of two independent fingers with a piezoelectric force sensor attached to each fingertip. The main contributions of this study are the designs and implementations of three controllers: a classic PID controller, a type-I controller, and a type-II fuzzy controller. These three controllers were used to find an object to be grasped properly (position) and apply an equivalent force to each finger (force).

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

The applicability of the path planning strategy to robotic manipulators has been an exciting topic for researchers in the last few decades due to the large demand in the industrial sector and its enormous potential development for space, surgical, and pharmaceutical applications. The automation of high-degree-of-freedom (DOF) manipulator robots is a challenging task due to the high redundancy in the end-effector position. Additionally, in the presence of obstacles in the workspace, the task becomes even more complicated. Therefore, for decades, the most common method of integrating a manipulator in an industrial automated process has been the demonstration technique through human operator intervention. Although it is a simple strategy, some drawbacks must be considered: first, the path’s success, length, and execution time depend on operator experience; second, for a structured environment with few objects, the planning task is easy. However, for most typical industrial applications, the environments contain many obstacles, which poses challenges for planning a collision-free trajectory. In this paper, a multiple-query method capable of obtaining collision-free paths for high DOF manipulators with multiple surrounding obstacles is presented. The proposed method is inspired by the resistive grid-based planner method (RGBPM). Furthermore, several improvements are implemented to solve complex planning problems that cannot be handled by the original formulation. The most important features of the proposed planner are as follows: (1) the easy implementation of robotic manipulators with multiple degrees of freedom, (2) the ability to handle dozens of obstacles in the environment, (3) compatibility with various obstacle representations using mathematical models, (4) a new recycling of a previous simulation strategy to convert the RGBPM into a multiple-query planner, and (5) the capacity to handle large sparse matrices representing the configuration space. A numerical simulation was carried out to validate the proposed planning method’s effectiveness for manipulators with three, five, and six DOFs on environments with dozens of surrounding obstacles. The case study results show the applicability of the proposed novel strategy in quickly computing new collision-free paths using the first execution data. Each new query requires less than 0.2 s for a 3 DOF manipulator in a configuration space free-modeled by a 7291 × 7291 sparse matrix and less than 30 s for five and six DOF manipulators in a configuration space free-modeled by 313,958 × 313,958 and 204,087 × 204,087 sparse matrices, respectively. Finally, a simulation was conducted to validate the proposed multiple-query RGBPM planner’s efficacy in finding feasible paths without collision using a six-DOF manipulator (KUKA LBR iiwa 14R820) in a complex environment with dozens of surrounding obstacles.

In this paper, we explore the advantages of a fractional calculus based watermarking system for detecting Gaussian watermarks. To reach this goal, we selected a typical watermarking scheme and replaced the detection equation set by another set of equations derived from fractional calculus principles; then, we carried out a statistical assessment of the performance of both schemes by analyzing the Receiver Operating Characteristic (ROC) curve and the False Positive Percentage (FPP) when they are used to detect Gaussian watermarks. The results show that the ROC of a fractional equation based scheme has 48.3% more Area Under the Curve (AUC) and a False Positives Percentage median of 0.2% whilst the selected typical watermarking scheme has 3%. In addition, the experimental results suggest that the target applications of fractional schemes for detecting Gaussian watermarks are as a semi-fragile image watermarking systems robust to Gaussian noise.

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.

In recent times, autonomous robots have become more relevant, aiming not only to be an extension of mobility and human performance but also allowing them to independently solve specific problems such as finding free-collision paths within some defined environments. In order to achieve this, several techniques have been developed, like action-reaction algorithms, sampling-based algorithms, and deterministic algorithms such as the Homotopy Path Planning Method (HPPM). This work presents, for the first time, a complete deterministic collision-free path planning scheme implemented in FPGA, which is mounted on a Scribbler 2 robot from Parallax. Then, an automatic algorithm of the repulsion parameter for the HPPM method is presented, using as a reference the minimum distance between the center of each obstacle with respect to the homotopic ideal path; furthermore, an algorithm is proposed for discriminating dead-end routes and collision risk trajectories, which allows us to obtain a feasible free-collision path that takes into account the robot dimensions. Besides, comparative performance tests have been carried out against other path-finding methods from the low degrees of freedom (low DoF) and sampling-based planners. Our proposal exhibits path calculation times which are 5 to 10 times faster on FPGA implementation, compared to the other methods and 10 to 100 times faster on PC implementation also compared to the rest. Similar results are obtained with regards to memory consumption, namely 20 to 200 times lower on FPGA implementation and 10 to 100 times lower on PC implementation.

The slider-crank mechanism (SCM) is one of the most important mechanisms in modern technology. It appears in most combustion engines including those of automobiles, trucks, and other small engines. The SCM model considered here is an index-three nonlinear system of differential-algebraic equations (DAEs), and therefore difficult to integrate numerically. In this work, we present the application of the differential transform method (DTM) to obtain an approximate analytical solution of the SCM model in convergent series form. In addition, we propose a posttreatment of the power series solution with the Padé resummation method to extend the domain of convergence of the approximate series solution. The main advantage of the proposed technique is that it does not require an index reduction and does not generate secular terms or depend on a perturbation parameter.

Galvanometer mirror systems are a fundamental tool used in many research fields to deploy curves over virtually any surface. Even though the point-based methodology is the current standard used to achieve this task, it has the shortcoming of using a format of coordinates (expressed as a massive list of points) to represent the curve to be displayed, requiring large memory arrays. An alternative methodology is the so-called Piecewise-Linear which representation format is based on the use of a mathematical Piecewise-Linear formulation where the curves to be drawn are treated as a parametric system composed of two positional equations, X and Y, related to each other by an artificial parameter \mu. In comparison against the point-based method, Piecewise-Linear exhibits attractive advantages such as: memory saving and improved sharpness for projected curves.

This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter.

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature.

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.