Abstract A geometrically exact analytical model of cable-driven parallel manipulators operating in a three-dimensional space is formulated and discussed in this work. The model aims at investigating the exact positioning of a rigid body in a Cartesian space by a m-cables-driven parallel manipulator taking into account the cables mass, their elasticity and, consequently, the sag effect. The direct kinematic problem is formulated by means of 3 ( m + 1 ) nonlinear equations expressing compatibility and equilibrium in the same number of unknowns, namely, 3m components of the constraint reactions and 3 finite rotations. A novel approach to the solution of the inverse kinematic problem is formulated by adding a set of m target equations to obtain 4 m + 3 nonlinear algebraic equations solved in terms of an equal number of unknowns. The latter are defined by the 3 ( m + 1 ) increments of the variables of the direct problem and m additional unknowns, i.e., the increments of the cable unstretched lengths. The elastostatic problem of point mass end-effector is then derived. Consequently, the direct and the inverse kinematic problems are formulated by means of 3m and 4m nonlinear equations, respectively, in the equal number of unknowns. Examples with an increasing number of cables are considered, showing that, although solving the inverse kinematic problem may imply very slack cables configurations, the proposed approach allows to find a solution.

中文翻译：

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用于末端执行器定位的电缆驱动并联机械手的几何精确三维建模

摘要 本文提出并讨论了在三维空间中运行的缆索驱动并联机械手的几何精确解析模型。该模型旨在研究由 m 电缆驱动的平行机械手在笛卡尔空间中精确定位刚体，同时考虑到电缆质量、它们的弹性以及下垂效应。直接运动学问题是通过 3 ( m + 1 ) 个非线性方程来表达的，这些方程表示在相同数量的未知数下的相容性和平衡，即约束反应的 3m 分量和 3 个有限旋转。求解逆运动学问题的新方法是通过添加一组 m 个目标方程来获得 4 m + 3 个以等量未知数求解的非线性代数方程。后者由直接问题的变量的 3 ( m + 1 ) 个增量和 m 个附加未知量（即，电缆未拉伸长度的增量）定义。然后导出点质量末端执行器的弹性静力学问题。因此，直接和逆运动学问题分别通过 3m 和 4m 非线性方程在相等数量的未知数下制定。考虑了电缆数量不断增加的示例，表明尽管解决逆运动学问题可能意味着非常松弛的电缆配置，但所提出的方法允许找到解决方案。直接和逆运动学问题分别通过 3m 和 4m 非线性方程在相等数量的未知数下制定。考虑了电缆数量不断增加的示例，表明尽管解决逆运动学问题可能意味着非常松弛的电缆配置，但所提出的方法允许找到解决方案。直接和逆运动学问题分别通过 3m 和 4m 非线性方程在相等数量的未知数下制定。考虑了电缆数量不断增加的示例，表明尽管解决逆运动学问题可能意味着非常松弛的电缆配置，但所提出的方法允许找到解决方案。