Variational Path Optimization Algorithm

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".

本文件是维也纳工业大学（TU Wien）几何与计算设计中心（Center for Geometry and Computational Design，GCD）A. Rasoulzadeh与G. Nawratil合作研究成果的一部分，创建于2019年10月10日。

ABSTRACT：
具有简洁奇异位形簇的线性五自由度并联机构（linear pentapod）类，通过对线性五自由度并联机构的设计施加架构约束得到，使得该操作器（manipulator）的奇异位形簇（singularity variety）在姿态/位置变量中呈线性形式。实践表明，这种简化不仅能带来显著的计算优势，同时可保留该机构在5轴铣削、激光切割等核心工业任务中的应用场景。

假设已知操作器工作空间内，末端执行器（end-effector）给定初始位姿与目标位姿之间的无奇异位形路径，本文提出一种路径优化方法，在平滑运动的同时，使机器人与奇异位形轨迹（singularity loci）之间的距离最大化。由于此类机构的奇异位形簇较为简洁，可通过符号求解法获得奇异距离函数的局部极值，因此该优化过程的计算效率得以提升。整个流程通过定义全新的代价函数（cost function）实现，被称为变分路径优化（variational path optimization）。该优化过程同时考虑了移动副（prismatic joints）与基座球铰（spherical joints）的物理运动极限。

NOTE：变分路径优化算法具有较强的通用性，可应用于不同维度下针对障碍物的路径优化场景。本数据集包含该算法在平面路径优化场景中的纯几何演示，其中障碍物为平面二次曲线（抛物线与椭圆），详见MATLAB + Maple文件中的“Variational Path Optimization of the Planar Quadrics”。

HOW TO USE：
从“MATLAB + Maple Files”文件夹中下载所有文件，随后打开“Manual.pdf”文档，该文档将演示如何使用图形用户界面（Graphical User Interface）。

NOTE：示例运动的GIF视频文件已存放于“Sample Videos”文件夹中。