Continuous Approximate Synthesis of Planar and Spatial Mechanisms

机构综合是近千年来发展起来的机构学的一个重要分支。一个机制是一个相互连接的组件的集合，这些组件被单独称为链路。两个相邻链环之间的连接是一个接头。四杆机构是一类重要的机构，它由一个相对固定的刚性连杆组成，输入和输出连杆通常通过旋转或线性关节连接。输入和输出连杆另外连接到第四连杆，该第四连杆将输入的运动耦合到输出的运动，从而形成闭环。四杆机构在函数生成（输入和输出连杆的运动根据代数函数进行协调）、刚体导引和轨迹生成方面有重要的应用。当前的方法是规定联接器、连杆4的位置和取向的有限集合，或者输入和输出连杆角度对的值的有限集合，然后使用数值程序来合成连杆的几何形状，该连杆的几何形状可以以最小的误差近似期望的运动，但是仅相对于规定的值。这被称为离散近似合成，因为离散指定的配置的有限数量。每个配置之间发生的情况没有考虑在内，这往往会使设计过程效率低下。该研究计划包括开发一个全新的机构科学分支，我们称之为连续近似综合（CAS）。而不是离散集，我们建议使用微积分和几何扩展到连续的无限集的有限集。因为我们考虑了整个运动范围，所以产生的连杆代表了可行的最佳连杆，在整个运动范围内以最小的误差执行。这项工作的长期目标是为任何类型的刚性机械联动装置创建设计工具，串联连接（类似于单个人类手臂）或并联连接（类似于两个人类手臂彼此连接，在胸部和手部，形成闭环），相对于特定误差产生最佳联动装置。本研究的近期目标包括四个方面：1）证明对于平面函数生成的四杆机构，最小化临界非线性性能指标的机构可以通过求解一组线性方程组来确定; 2）将结果应用于任意平面和空间闭环机构; 3）将结果推广到刚体导引的CAS; 4）将结果进一步推广到轨迹生成。CAS的概念已经得到了加拿大机制设计界的大力支持，因此，预计拟议的工作将对加拿大和国际上机制设计界的研究和工业部门产生重大的积极影响。**

Mechanism synthesis is an important branch of mechanism science that has developed over the last millennium. A mechanism is a collection of interconnected components individually called links. The connection between two adjacent links is a joint. Four-bar linkages are an important class of mechanism that consist of a relatively fixed rigid link to which an input and an output link are attached, typically by rotary, or linear joints. The input and output links are additionally connected to a fourth link which couples the motion of the input to that of the output, forming a closed loop. Four-bar linkages have important applications to function generation (the motions of the input and output links are coordinated according to an algebraic function), rigid body guidance, and trajectory generation. The current approach is to prescribe a finite set of positions and orientations of the coupler, link 4, or a finite set of values for pairs of input and output link angles, and then to use numerical procedures to synthesise the geometry of a linkage that can approximate the desired motions with the least error, but relative to only the prescribed values. This is termed discrete approximate synthesis because of the finite number of discrete specified configurations. What occurs between each configuration is not accounted for and tends to make the design process inefficient. This research program consists of developing a fundamentally new branch of mechanism science, which we call continuous approximate synthesis (CAS). Instead of discrete sets, we propose to use calculus and geometry to expand the finite sets into continuous infinite sets. Because we take into account the entire range of motion, the resulting linkage represents the very best that is viable, performing with the smallest possible error throughout the entire range of motion. The long term goal of this work is to create design tools for any type of rigid mechanical linkage, connected both serially (analogous to a single human arm) or in parallel (analogous to two human arms connected to each other, both at the chest and hands, forming a closed loop), that yields the very best linkage relative to a particular error. There are four short term goals, comprising the focus of this grant application: 1) prove that for planar function generating four-bar linkages, the one which minimises a critical non-linear performance index can be identified by solving a set of linear equations; 2) adapt the results to arbitrary planar and spatial closed loop linkages; 3) extend the results to CAS for rigid body guidance; 4) further extend the results to trajectory generation. The concept of CAS has already received strong support from the Canadian mechanism design community, it is therefore expected that the proposed work will make a significant positive impact on the research and industrial sectors of the mechanism design community both within Canada and internationally. **