Continuous Approximate Synthesis of Planar and Spatial Mechanisms

机构综合是近一千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接。两个相邻的连杆之间的连接是一个接头。四杆连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，输入连杆和输出连杆通常由旋转或线性关节连接。输入和输出环节另外连接到第四环节，第四环节将输入的运动耦合到输出的运动，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协调）、刚体制导和轨迹生成方面具有重要的应用。目前的方法是规定耦合器，连杆4的有限位置和方向，或输入和输出连杆角度对的有限值集，然后使用数值程序合成连杆的几何形状，该连杆可以以最小的误差近似所需的运动，但相对于仅规定的值。这被称为离散近似综合，因为离散的指定构型的数量有限。每个配置之间发生的情况没有考虑在内，而且往往会使设计过程效率低下。这一研究计划包括发展机制科学的一个全新分支，我们称之为连续近似综合（CAS）。我们建议用微积分和几何来代替离散集，将有限集展开为连续无限集。因为我们考虑了整个运动范围，所以最终的连杆代表了可行的最佳连杆，在整个运动范围内以尽可能小的误差执行。这项工作的长期目标是为任何类型的刚性机械连杆创建设计工具，这些连杆可以串行连接（类似于单臂）或并联连接（类似于两条人的手臂在胸前和手上相连，形成一个闭环），从而产生相对于特定误差的最佳连杆。有四个短期目标，包括本资助申请的重点：1)证明平面函数生成四杆机构，可以通过求解一组线性方程来确定最小的关键非线性性能指标；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CAS中用于刚体指导；4)进一步将结果扩展到轨迹生成。CAS的概念已经得到了加拿大机构设计界的大力支持，因此预计拟议的工作将对加拿大和国际机构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.