# Variational Path Optimization Algorithm

## Description

This file is part of joint work between A. Rasoulzadeh and G. Nawratil at Center for Geometry and Computational Design (GCD), Vienna University of Technology (TU Wien). It is created on October 10th, 2019. ABSTRACT: The class of linear pentapods with a simple singularity variety is obtained by imposing architectural restrictions on the design of a linear pentapod in a way that the manipulator's singularity variety is linear in orientation/position variables. It turns out that such a simplification leads to crucial computational advantages while maintaining the machine's applications in some fundamental industrial tasks such as 5-axis milling and laser cutting. Assuming that a singularity-free path between a given start- and end-pose of the end-effector within the manipulator's workspace is known, an optimization process of this path is proposed in such a way that the robot increases its distance to the singularity loci while the motion is being smoothed. In this case the computation time of the optimization is improved as one deals with the pentapods having a simple singularity variety allowing symbolic solutions for the local extrema of the singularity-distance function. The whole process is called variational path optimization and takes place through defining a novel cost function. This optimization process takes the physical limits of prismatic joints and base spherical joints into account. NOTE: The variational path optimization algorithm is quite general and can be used in different cases related to optimizing a path with respect to the presence of obstacles in different dimensions. The files here contain a pure geometric demonstration of this algorithm in the case of path optimization in plane with respect to planar quadric curves as obstacles (Parabola and Ellipse) (cf. MATLAB + Maple files > Variational Path Optimization of the Planar Quadrics). HOW TO USE: Download the files from the folder "MATLAB + Maple Files". Then open the "Manual.pdf". This file demonstrates how to use the Graphical User Interface. NOTE: Videos (GIF files) of motions of a sample is provided for you in the folder "Sample Videos".