# Variational Path Optimization of Linear Pentapods with a Simple Singularity Variety

## 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. HOW TO USE: In order to use the algorithm please type "variational_path_optimization" in the MATLAB "Command Window". The code then asks the user a range of questions from "architectural aspects of your manipulator" to plot options. For most of these questions the user can simply ignore (if he/she does not wish a very specific optimization or plot) by pressing enter. However some of these questions are obligatory to answer. Therefore we ask the user to type "help variational_path_optimization" in in the MATLAB "Command Window" for a thorough description of his/her available options. Please note that a DEFAULT SETTING is made available for the code by which the user can observe how the algorithm works on a predefined simple pentapod and a predefined singularity-free initial motion (as a sample we recommend using the special curve "twisted"). Finally, if the user has access to MAPLE, then he/she can visualize the full results alongside the shape of the manipulator. In order to do so please run "variational_path_optimization.m" on a case, then execute the files "pentapod.mw" and "plot.mw" consecutively. NOTE: 9 videos (GIF files) of motions of a sample is provided for you in the folder "Sample Videos". WARNING: Note that the rest of the MATLAB functions in the folder are just nested functions in the file "variational_path_optimization.m". The help option is also available for all these nested functions which demonstrates their specific role within the main code. The MATLAB files will be subject to minor updates including a GUI