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A study on the stiffness of the end of the suspended master–slave remote control manipulator

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Abstract

In the hot chamber of the nuclear industry, operators can only use master–slave remote control manipulators to perform various actions when fuel reprocessing is required to avoid danger. Compared with the master manipulator, the coordinate transformation matrix and the Jacobian matrix are established. The slave manipulator in the thermal chamber requires higher load and higher precision operation, but it has not been widely explored. In this paper, through the kinematic analysis of the manipulator, the coordinate transformation matrix and the Jacobian matrix of the suspended master–slave remote control manipulator are established, and the total stiffness model of the end of the manipulator is established according to the mapping of the two deflections of the end of the manipulator. Besides, the joint stiffness and arm link stiffness of the manipulator were solved separately. To demonstrate the applicability of the model, the corresponding simulation calculation is carried out. First, we use ANSYS to carry out finite element static analysis on different load conditions of the driven mechanism, and the end deflection values are solved by simulation and compared with the theoretically calculated values, the results are in good agreement, so the validity of the model is proved. Then, the experimental verification has been carried out through the suspended master–slave remote control manipulator test platform, and the results show the accuracy of the stiffness model and the finite element analysis results. This model provides a reference advice for the calculation of the stiffness of the end of the suspended master–slave remote control manipulator.

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Funding

Funding is work was supported by the Autonomous Research Project of Zhejiang University Robotics Research Institute and also supported by Zhejiang science and technology planning project [2022C01054].

Author information

Authors and Affiliations

  1. State Key Laboratory of Fluid Power and Mechatronic System, College of Mechanical Engineering, Zhejiang University, Hangzhou, 310027, China

    Junxia Jiang, Qun Dong, Yan Li & Xiaoou Zhong

  2. Key Laboratory of Advanced Manufacturing Technology of Zhejiang Province, College of Mechanical Engineering, Zhejiang University, Hangzhou, 310027, China

    Junxia Jiang, Qun Dong, Yan Li & Xiaoou Zhong

Authors
  1. Junxia Jiang
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  2. Qun Dong
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  3. Yan Li
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  4. Xiaoou Zhong
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Corresponding author

Correspondence to Yan Li.

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Technical Editor: Rogério Sales Gonçalves.

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Appendix

Appendix

$$ \begin{aligned} R_{11} = & - s\theta_{6} \left( {c\theta_{4} s\theta_{1} + s\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{6} \left( {c\theta_{5} \left( {s\theta_{1} s\theta_{4} - c\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & + \left. {s\theta_{5} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{21} =\, & c\theta_{6} \left( {c\theta_{5} \left( {c\theta_{1} s\theta_{4} + c\theta_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)s\theta_{1} - s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & - \left. {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)s\theta_{1} + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)s\theta_{1} } \right)} \right) \\ & + s\theta_{6} \left( {c\theta_{1} c\theta_{4} - s\theta_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)s\theta_{1} - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)s\theta_{1} } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{31} =\, & c\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - s\theta_{4} s\theta_{6} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{12} =\, & s\theta_{6} \left( {c\theta_{5} \left( {s\theta_{1} s\theta_{4} - c\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & + \left. {s\theta_{5} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{6} \left( {s\theta_{1} c\theta_{4} + s\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{22} =\, & c\theta_{6} \left( {c\theta_{1} c\theta_{4} - s\theta_{4} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - s\theta_{6} \left( {c\theta_{5} \left( {c\theta_{1} s\theta_{4} + c\theta_{4} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & - \left. {s\theta_{5} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{32} =\, & - s\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{6} s\theta_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{3} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{13} =\, & s\theta_{5} \left( {s\theta_{1} s\theta_{4} - c\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{5} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{23} =\, & - s\theta_{5} \left( {c\theta_{1} s\theta_{4} + c\theta_{4} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{5} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{33} =\, & c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right) \\ & - c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{14} =\, & d_{6} \left( {s\theta_{5} \left( {s\theta_{1} s\theta_{4} - c\theta_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) - c\theta_{5} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - d_{4} \left( {c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right) + a_{2} c\theta_{1} c\left( {\theta_{2} + 90^\circ } \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{24} =\, & a_{2} c\left( {\theta_{2} + 90^\circ } \right)s\theta_{1} - d_{6} \left( {s\theta_{5} \left( {c\theta_{1} s\theta_{4} + c\theta_{4} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & + \left. {c\theta_{5} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - d_{4} \left( {s\theta_{1} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\theta_{1} s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} R_{34} =\, & d_{1} + d_{6} \left( {c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right. \\ & - \left. {c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & + d_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)) + a_{2} s\left( {\theta_{2} + 90^\circ } \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1 \times 1} =\, & - \left( {c\theta_{4} s\theta_{6} + c\theta_{5} c\theta_{6} s\theta_{4} } \right)\left( {d_{6} \left( {c\theta_{5} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & + d_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right) - a_{2} c\left( {\theta_{2} + 90^\circ } \right) \\ & - d_{6} s\theta_{4} s\theta_{5} \left( {c\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + s\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right)} \right) - c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & + \left. {s\theta_{4} s\theta_{6} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1y1} =\, & d_{6} s\theta_{4} s\theta_{5} (s\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right. \\ & - \left. {c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & - c\theta_{6} s\theta_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)) \\ & - \left( {c\theta_{4} c\theta_{6} - c\theta_{5} c\theta_{6} s\theta_{4} } \right)\left( {d_{6} \left( {c\theta_{5} c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right) \\ & + d_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right) - a_{2} c\left( {\theta_{2} + 90^\circ } \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1z1} =\, & s\theta_{4} s\theta_{5} \left( {d_{6} \left( {c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right) + c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right)} \right. \\ & + \left. {d_{4} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right) - a_{2} c\left( {\theta_{2} + 90^\circ } \right)} \right) \\ & - d_{6} s\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right) + c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right)} \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{ax1} =\, & c\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right) \\ & - s\theta_{4} s\theta_{6} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{ay1} =\, & - s\theta_{6} \left( {s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right. \\ & + \left. {c\theta_{4} c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right)} \right) \\ & - c\theta_{4} s\theta_{6} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{az} = &\, c\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)c\left( {\theta_{3} - 90^\circ } \right) - s\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right) \\ & - c\theta_{4} s\theta_{5} \left( {c\left( {\theta_{2} + 90^\circ } \right)s\left( {\theta_{3} - 90^\circ } \right) + c\left( {\theta_{3} - 90^\circ } \right)s\left( {\theta_{2} + 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1x2} = &\, \left( {c\theta_{6} \left( {s\theta_{5} s\left( {\theta_{3} - 90^\circ } \right) - c\theta_{4} c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) + c\left( {\theta_{3} - 90^\circ } \right)s\theta_{4} s\theta_{6} } \right)\left( {d_{6} \left( {c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{4} s\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right) + d_{4} c\left( {\theta_{3} - 90^\circ } \right)} \right) \\ & - \left( {c\theta_{6} \left( {c\left( {\theta_{3} - 90^\circ } \right)s\theta_{5} + c\theta_{4} c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right) - s\theta_{4} s\theta_{6} s\left( {\theta_{3} - 90^\circ } \right)} \right)\left( {d_{6} \left( {c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{4} s\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) - a_{2} + d_{4} s\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1y2} = &\, \left( {s\theta_{6} \left( {s\theta_{5} c\left( {\theta_{3} - 90^\circ } \right) + c\theta_{4} c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right) + s\left( {\theta_{3} - 90^\circ } \right)s\theta_{4} c\theta_{6} } \right)\left( {d_{6} \left( {c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right) - c\theta_{4} s\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) - a_{2} + d_{4} s\left( {\theta_{3} - 90^\circ } \right)} \right) \\ & - \left( {s\theta_{6} \left( {s\left( {\theta_{3} - 90^\circ } \right)s\theta_{5} - c\theta_{4} c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) - s\theta_{4} c\theta_{6} c\left( {\theta_{3} - 90^\circ } \right)} \right)\left( {d_{6} \left( {c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right) + c\theta_{4} s\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right) + d_{4} c\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ \begin{aligned} J_{1z2} = &\, \left( {d_{6} \left( {c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{4} s\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right) + d_{4} c\left( {\theta_{3} - 90^\circ } \right)} \right)\left( {c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{4} s\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) \\ & - \left( {c\theta_{5} c\left( {\theta_{3} - 90^\circ } \right) - c\theta_{4} s\theta_{5} s\left( {\theta_{3} - 90^\circ } \right)} \right)\left( {d_{6} \left( {c\theta_{5} s\left( {\theta_{3} - 90^\circ } \right) + c\theta_{4} s\theta_{5} c\left( {\theta_{3} - 90^\circ } \right)} \right) - a_{2} + d_{4} s\left( {\theta_{3} - 90^\circ } \right)} \right); \\ \end{aligned} $$
$$ J_{1 \times 3} =\, \left( {d_{4} + d_{6} c\theta_{5} } \right)\left( {s\theta_{4} s\theta_{6} - c\theta_{4} c\theta_{5} c\theta_{6} } \right) - d_{6} c\theta_{4} c\theta_{6} s\theta_{5}^{2} ; $$
$$ J_{1y3} =\, d_{6} c\theta_{4} c\theta_{6} s\theta_{5}^{2} + \left( {d_{4} + d_{6} c\theta_{5} } \right)\left( {c\theta_{6} s\theta_{4} + c\theta_{4} c\theta_{5} s\theta_{6} } \right); $$
$$ J_{1 \times 3} =\, c\theta_{4} s\theta_{5} \left( {d_{4} + d_{6} c\theta_{5} } \right) - d_{6} c\theta_{4} c\theta_{5} s\theta_{5} $$

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Jiang, J., Dong, Q., Li, Y. et al. A study on the stiffness of the end of the suspended master–slave remote control manipulator. J Braz. Soc. Mech. Sci. Eng. 44, 483 (2022). https://doi.org/10.1007/s40430-022-03774-7

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