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.

Multiple-Target Homotopic Quasi-Complete Path Planning Method for Mobile Robot Using a Piecewise Linear Approach.

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.

Transforming the canonical piecewise-linear model into a smooth-piecewise representation.

A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine their parameters, the capability of controlling not only the smoothness grade, but also the approximation accuracy at specific breakpoint locations, a lower or equal overshooting for high order derivatives in comparison with other approaches, and the additional advantage of being expressed in a reduced mathematical form with only two types of inverse functions (logarithmic and exponential). By numerical simulation examples, this proposal is verified and well-illustrated.

Speed-up hyperspheres homotopic path tracking algorithm for PWL circuits simulations.

In the present work, we introduce an improved version of the hyperspheres path tracking method adapted for piecewise linear (PWL) circuits. This enhanced version takes advantage of the PWL characteristics from the homotopic curve, achieving faster path tracking and improving the performance of the homotopy continuation method (HCM). Faster computing time allows the study of complex circuits with higher complexity; the proposed method also decrease, significantly, the probability of having a diverging problem when using the Newton-Raphson method because it is applied just twice per linear region on the homotopic path.

Laplace transform homotopy perturbation method for the approximation of variational problems.

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.