基于轴不变量的机器人动力学建模与力位控制

Because of the need of lightweight robot development and axle-variants which precisely express the construction of multi-axle system (MAS) and own elegant math operations, it is necessary to establish MAS modelling and force-position control theory based on axle-variants..Firstly, present directed Span tree of MAS, establish motion-chain Symbolic system with the chain of partial ordering corner marks and tree chain operators, and prove the system’s compatibility to classical theories. The chain of partial ordering corner marks and projection operator help precisely express the partial ordering relations and reference frame of fixed and free tensors; the tree-chain operators allow us take the topological relation of MAS into account when dynamic equations are established..Second, the properties and calculation of axle-variants are proven axiomatically, and establish Jacobi, configuration, velocity and acceleration equations in term of axle-variants. Furthermore, for tree and parallel chain system with or without axial frictions, based on Kane method, respectively establish recursive dynamic equations in term of axle-variants, and solve the problem which classical analytic dynamics are not applicable for MAS with high degrees of freedom..Last, develop robot joints with linear relationship between driven electric current and output torque and a lightweight six-axle manipulator. And achieve the flexibility and agility of the manipulator based on the above recursive dynamic equations and force control of fuzzy sliding mode to prove the valid of the MAS theory.

首先，提出多轴系统（MAS）有向Span树，通过轴不变量表征关节运动，由固定轴矢量表征结构参数及DH参数；结合链及张量理论，建立有偏序指标及链操作符的运动链符号演算系统，证明其对传统理论的兼容性；通过链指标及投影符表征张量的作用及参考关系；通过链操作符表达系统的拓扑结构。.其次，公理化地证明轴不变量的属性及运算关系；建立基于轴不变量的迭代式雅克比、位形、速度及加速度方程；进而，基于凯恩法，针对树链、并链及是否存在轴摩擦，分别建立统一的基于轴不变量的迭代式MAS动力学方程。解决传统分析方法难以应用于高自由度MAS动力学显式建模的难题。.最后，研制驱动电流与输出力矩呈线性的空轴关节、轻量六轴机械臂；基于上述动力学方程及模糊变结构的力控制，实现机器人控制的敏捷性与柔顺性。实验证明该理论的有效性。

项目背景：以研制高精度机械臂为目标，建立《基于轴不变量的多轴系统建模与控制》理论，开发演示验证系统。.本项目以高精度机械臂为应用对象，建立了基于轴不变量的多轴系统正逆运动、正逆动力学与控制原理，开发了相应的软件系统，通过工程应用证明了有效性。.以现代集合论的“链”理论及“张量”理论为基础，建立了具有运动链指标系统的三维空间操作代数；以之为基础，创立了“基于轴不变量的多轴系统建模与控制”理论。.首先，提出了以“不变性、对偶性、参数化、程式化及公理化”为特征的同构方法论，指导开展融合数学、力学、计算机及控制的多轴系统跨学科理论及工程技术研究。.接着，基于同构方法论，提出树链、拓扑及度量三大公理；创建了以自然坐标系及轴不变量为基础的三维矢量空间操作代数，从而建立了基于轴不变量的多轴系统正运动学原理。.继之，提出“居-吉布斯”分析四元数，证明其相关性质，进而提出多元矢量多项式系统，并证明其线性计算复杂度及求解原理；建立了基于轴不变量的RBR、BBR、半通用及通用机械臂逆解原理，它们具有精确性及实时性，不存在计算奇异性。从而建立了基于轴不变量的多轴系统逆运动学原理。.最后，基于三维空间操作代数及拉格朗日分析力学原理，证明了树链及闭链刚体系统的“居-凯恩”动力学定理，能够简洁地表征任一轴的显式动力学方程，实现了完全参数化的动力学建模与控制。从而建立了基于轴不变量的多轴系统动力学与控制原理。.上述正逆运动学与动力学，既是以空间操作为核心的运动链语言系统，又是包含算法结构的迭代式伪代码系统，保证了多轴系统建模与控制软件的准确性、可靠性及实时性。