A Novel Collision-Free Homotopy Path Planning for Planar Robotic Arms.

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.

GHM method for obtaining rationalsolutions of nonlinear differential equations.

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

Direct application of Padé approximant for solving nonlinear differential equations.

UNLABELLED: This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. AMS SUBJECT CLASSIFICATION: 34L30.

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

ABSTRACT: In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. AMS SUBJECT CLASSIFICATION: 34L30.