多自由度机构真实运动线汇理论与误差补偿的不变量方法研究

A new systematic line congruence theory of error-inclusive kinematic geometry for multi-DOF mechanisms is presented and studied for the first time in this project; meanwhile, a new invariant method for kinematic accuracy analysis and error compensation of multi-DOF mechanisms is set up based on the line congruence theory. Firstly, the line congruence studied in differential geometry is naturally introduced to represent the error-inclusive motion of an actual multi-DOF mechanism, because the trajectory traced by an arbitrary line of the executive link in a multi-DOF mechanism happens to be a line congruence. The error-inclusive motion of the link is decomposed into the multi-DOF rotational motion along the spherical image curves of the line congruence and the multi-DOF translational motion along the focal surface of the line congruence. The invariants of the family of spherical image curves and the focal surface for the error-inclusive motion are deduced, whose identical equations are used to expound the kinematic and geometric properties of the error-inclusive multi-DOF motion. Then, the multidimensional complete mappings between the invariants of the line congruence and those of the ruled surface are discussed, which reveal the relationships among the error-inclusive motion of the mechanism, the error-inclusive motion of the kinematic pairs and the structure errors of the links. The invariants based kinematic models of the actual multi-DOF mechanisms are constructed. Based on the kinematic models, the kinematic accuracy analysis and error compensation of the actual multi-DOF mechanisms are realized by combining the measurement, identification and inverse mapping of the invariants. Finally, the experimental researches and applications of the line congruence theory and the invariants method are carried out using the precision equipment, including the precision machine tool, the multi-axis rotary table and the multi-DOF industrial manipulator. All of the proposed researches will provide a new theoretical basis and some technical means to improve the kinematic accuracy of the actual multi-DOF mechanisms, especially for the accuracy of multi-DOF motion. These will also take some new contents and applications for kinematic geometry.

首次将微分几何学的线汇理论引入真实多自由度机构的运动分析与误差补偿。以执行构件上直线的轨迹线汇描述多自由度机构的真实运动规律，将构件的多自由度误差运动分解为沿线汇球面像曲线族的多自由度转动与沿线汇焦曲面的多自由度平动，导出线汇球面像曲线族与线汇焦曲面的误差运动不变量阐明构件误差运动的整体性质，建立多自由度机构真实运动线汇理论；以多参数线汇不变量与单参数直纹面不变量之间的多维完全映射阐明执行构件误差运动、运动副误差运动、构件误差三者之间的关系，构造真实多自由度机构的运动学模型，结合多维不变量的测量、辨识与逆映射，形成多自由度机构全工作空间运动位置及姿态误差整体补偿的不变量方法。以精密机床、多轴转台、多关节机器人等为对象，开展多自由度机构真实运动线汇理论与误差补偿方法的应用研究。本项目研究将为提高精密装备的多自由度运动精度提供新的理论基础与技术手段，丰富机构运动几何学的内涵并拓展新的应用领域。

本项目以精密机床、多轴转台、多关节机器人等真实多自由度机构为应用对象，开展其运动精度测评与误差补偿的理论方法及应用技术研究，大幅提高了应用对象的运动精度。项目聚焦真实多自由度机构的多参数运动，引入微分几何学的线汇理论与方法，围绕多自由度机构的真实运动线汇理论、多自由度误差运动的整体性质与变化规律、机构运动误差的整体补偿方法、真实多自由度机构的误差运动试验等四个方面进行了深入研究。项目形成了多自由度机构真实运动的线汇表示方法，将多自由度机构的误差运动性质转换为直线轨迹线汇的几何性质，并以线汇的不变量描述；揭示了多自由度误差运动的整体性质及其与局部性质的关系，找出了运动构件上误差运动的特征几何要素，区分了机构的可控主运动与误差运动，替代了运动测量的阿贝原理与布莱恩准则；建立了真实多自由度开式机构的运动学模型，阐明了构件误差与运动副误差对机构执行构件误差运动的影响规律；提出了多自由度开式机构真实运动参数完备性测量原理，实现了由回转副与移动副组成机构的运动参数的完整测量，从理论上消除了测量仪器安装位置及其误差的对测量结果的影响；形成了多自由度开式机构参数辨识与误差补偿的不变量方法，实现了机构构件误差与运动副误差的辨识以及机构满工作空间的误差补偿，将应用对象工作空间内的总体运动误差减少50%以上。本项目将以理想机构单参数运动为内涵的机构运动几何学理论发展到包含误差运动的真实机构多参数运动几何学理论，丰富了机构运动几何学的内涵；同时，项目研究为提高精密机床、光电雷达、多关节机器人等装备的多自由度运动精度提供了经济有效的方法，拓展了机构运动几何学的应用领域。项目成果还可结合真实机构的运动精度测量数据形成多自由度机构运动精度的测评方法，也可深入探讨多自由度误差运动的形成机理，为精密装备的精度设计与调控、装配工艺参数设计以及误差溯源等提供理论依据，具有重要的理论意义与广泛的应用价值。

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