Continuous Approximate Synthesis of Planar and Spatial Mechanisms

机构综合是近千年来发展起来的机构科学的一个重要分支。机制是相互连接的组件的集合，这些组件分别称为链路。两个相邻链节之间的连接是一个关节。四杆机构是一类重要的机构，它由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在其上。输入和输出链节另外连接到第四链节，该第四链节将输入的运动耦合到输出的运动，形成闭合回路。四杆机构在函数生成(根据代数函数协调输入和输出连杆的运动)、刚体导引和轨迹生成方面有着重要的应用。目前的方法是规定连杆4的位置和方向的有限集合，或输入和输出连杆角度对的有限数值集合，然后使用数值程序来综合能够以最小误差近似期望运动的连杆几何形状，但仅相对于规定值。这被称为离散近似合成，因为离散指定构型的数目有限。每种配置之间发生的事情不会被考虑在内，并且往往会使设计过程效率低下。这一研究计划包括发展一个全新的机械科学分支，我们称之为连续近似综合(CAS)。我们建议用微积分和几何将有限集扩展成连续的无限集，而不是离散集。因为我们考虑了整个运动范围，所以产生的连杆代表了最好的可行的，在整个运动范围内以最小的可能误差执行。这项工作的长期目标是为任何类型的刚性机械连杆创建设计工具，这些连杆既可以串联(类似于单只人类手臂)，也可以并行连接(类似于两只人类手臂在胸部和手部相互连接，形成一个闭合环)，从而产生相对于特定误差的最好连杆。有四个短期目标，构成了这项拨款申请的重点：1)证明对于平面函数生成的四杆机构，可以通过求解一组线性方程组来识别使临界非线性性能指标最小的机构；2)使结果适用于任意平面和空间闭合环机构；3)将结果推广到CAS中用于刚体制导；4)进一步将结果扩展到轨迹生成。CAS的概念已经得到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国内和国际机制设计界的研究和工业部门产生重大积极影响。**

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. **