As robotic systems flourish, reliability has become a topic of paramount importance in the human–robot relationship. The Jacobian matrix in screw theory underpins the design and optimization of robotic manipulators. Kernel properties of robotic manipulators, including dexterity and singularity, are characterized with the Jacobian matrix. The accurate specification and the rigorous analysis of the Jacobian matrix are indispensable in guaranteeing correct evaluation of the kinematics performance of manipulators. In this paper, a formal method for analyzing the Jacobian matrix in screw theory is presented using the higher-order logic theorem prover HOL4. Formalizations of twists and the forward kinematics are performed using the product of exponentials formula and the theory of functional matrices. To the best of our knowledge, this work is the first to formally analyze the kinematic Jacobian using theorem proving. The formal modeling and analysis of the Stanford manipulator demonstrate the effectiveness and applicability of the proposed approach to the formal verification of the kinematic properties of robotic manipulators.

中文翻译：

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螺旋理论中运动学雅可比行列式的形式分析

随着机器人系统的蓬勃发展，可靠性已成为人机关系中最重要的话题。螺杆理论中的雅可比矩阵支持机器人机械手的设计和优化。机器人机械手的核心属性，包括灵巧性和奇异性，用雅可比矩阵来表征。雅可比矩阵的准确规范和严格分析对于保证正确评估机械手的运动学性能是必不可少的。在本文中，使用高阶逻辑定理证明器 HOL4 提出了一种分析螺旋理论中雅可比矩阵的形式化方法。使用指数公式的乘积和泛函矩阵理论对扭曲和正向运动学进行形式化。据我们所知，这项工作是第一个使用定理证明正式分析运动雅可比行列式的工作。斯坦福机械手的形式建模和分析证明了所提出的方法对机器人机械手运动学特性的形式验证的有效性和适用性。