Variational Path Optimization Algorithm
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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）内A. Rasoulzadeh与G. Nawratil合作研究成果的一部分，创建于2019年10月10日。 摘要：通过对线性五杆并联机器人（linear pentapod）的设计施加架构约束，使得该操作机的奇异位形簇（singularity variety）在姿态与位置变量中呈线性形式，由此得到一类具有简单奇异位形簇的线性五杆并联机器人。研究表明，这类简化设计在保留该类机器人在5轴铣削、激光切割等核心工业任务中的应用场景的同时，还能带来显著的计算优势。 假设已知该操作机工作空间内，末端执行器给定初始位姿与终止位姿之间的无奇异位形路径，本文提出一种该路径的优化方法：可在平滑运动的同时，使机器人远离奇异位形轨迹。由于这类五杆并联机器人具有简单的奇异位形簇，可对奇异距离函数的局部极值进行符号求解，因此该优化过程的计算效率得到大幅提升。整个优化过程通过定义一种新型代价函数实现，被称为变分路径优化（variational path optimization），且该过程同时考虑了移动副（prismatic joints）与基座球铰（base spherical joints）的物理极限。 注：该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存在障碍物的路径优化相关场景。本文附带的文件针对平面内以平面二次曲线（抛物线与椭圆）为障碍物的路径优化场景，给出了该算法的纯几何演示（详见MATLAB + Maple文件 > 平面二次曲线的变分路径优化）。 使用方法：从"MATLAB + Maple Files"文件夹中下载相关文件，随后打开"Manual.pdf"，该文档将演示如何使用配套的图形用户界面。 补充说明："Sample Videos"文件夹中提供了某示例运动的GIF格式视频文件供参考。
