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Controlling Tendons to Modulate Stiffness of a Planar-to-Spatial Tendon-Driven Continuum Manipulator Under External Uncertain Forces

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Abstract

Continuum manipulators (CM) are soft and flexible manipulators with large numbers of degrees of freedom and can perform complex tasks in an unstructured environment. However, their deformability and compliance can deviate distal tip under uncertain external interactions. To address this challenge, a novel tension-based control scheme has been proposed to modulate the stiffness of a tendon-driven CM, reducing the tip position errors caused by uncertain external forces. To minimize the tip position error, a virtual spring is positioned between the deviated and the desired tip positions. The proposed algorithm corrects the manipulator deviated tip position, improving tension distribution and stiffness profile, resulting in higher stiffness and better performance. The corresponding task space stiffness and condition numbers are also computed under different cases, indicating the effectiveness of the tension control scheme in modulating the manipulator's stiffness. Experimental validation conducted on an in-house developed prototype confirms the practical feasibility of the proposed approach.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Vipin Pachouri & Pushparaj Mani Pathak

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  1. Vipin Pachouri
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Appendix

Appendix

The elements of the structure matrix for a two-section planar to spatial TDCM are shown below:

$${A}_{11}={\widehat{e}}_{{A}_{2}{A}_{1}}\frac{\partial {}_{ }{}^{{P}_{1}}{A}_{2}}{\partial {\theta }_{1}};{{A}_{12}=\widehat{e}}_{{B}_{2}{B}_{1}}\frac{\partial {}_{ }{}^{{P}_{1}}{B}_{2}}{\partial {\theta }_{1}}$$
$${A}_{21}={\widehat{e}}_{{A}_{3}{A}_{2}}\frac{\partial {}_{ }{}^{{P}_{1}}{A}_{3}}{\partial {\theta }_{2}};{A}_{22}={\widehat{e}}_{{B}_{3}{B}_{2}}\frac{\partial {}_{ }{}^{{P}_{1}}{B}_{3}}{\partial {\theta }_{2}}$$
$${A}_{31}={\widehat{e}}_{{A}_{4}{A}_{3}}\frac{\partial {}_{ }{}^{{P}_{1}}{A}_{4}}{\partial {\theta }_{3}};{A}_{32}={\widehat{e}}_{{B}_{4}{B}_{3}}\frac{\partial {}_{ }{}^{{P}_{1}}{B}_{4}}{\partial {\theta }_{3}}$$
$${A}_{41}={\widehat{e}}_{{A}_{5}{A}_{4}}\frac{\partial {}_{ }{}^{{P}_{1}}{A}_{5}}{\partial {\theta }_{4}};{A}_{42}={\widehat{e}}_{{B}_{5}B}.\frac{\partial {}_{ }{}^{{P}_{1}}{B}_{5}}{\partial {\theta }_{4}}$$
$${A}_{43}=\left[{\widehat{e}}_{{C}_{6}{C}_{5}}\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{6}}{\partial {\theta }_{4}}+\left({\widehat{e}}_{{C}_{5}{C}_{1}}-{\widehat{e}}_{{C}_{6}{C}_{5}}\right)\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{5}}{\partial {\theta }_{4}}\right];{A}_{44}=\left[{\widehat{e}}_{{D}_{6}{D}_{5}}\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{6}}{\partial {\theta }_{4}}+\left({\widehat{e}}_{{D}_{5}{D}_{1}}-{\widehat{e}}_{{D}_{6}{D}_{5}}\right)\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{5}}{\partial {\theta }_{4}}\right]$$
$${A}_{53}={\widehat{e}}_{{C}_{6}{C}_{5}}\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{6}}{\partial {\theta }_{5}};{A}_{54}={\widehat{e}}_{{D}_{6}{D}_{5}}\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{6}}{\partial {\theta }_{5}}$$
$${A}_{63}={\widehat{e}}_{{C}_{7}{C}_{6}}\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{7}}{\partial {\theta }_{6}};{A}_{64}={\widehat{e}}_{{D}_{7}{D}_{6}}\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{7}}{\partial {\theta }_{6}}$$
$${A}_{73}={\widehat{e}}_{{C}_{8}{C}_{7}}\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{8}}{\partial {\theta }_{7}};{A}_{74}={\widehat{e}}_{{D}_{8}{D}_{7}}\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{8}}{\partial {\theta }_{7}}$$
$${A}_{83=}{\widehat{e}}_{{C}_{9}{C}_{8}}\frac{\partial {}_{ }{}^{{P}_{1}}{C}_{9}}{\partial {\theta }_{8}};{A}_{84}={\widehat{e}}_{{D}_{9}{D}_{8}}\frac{\partial {}_{ }{}^{{P}_{1}}{D}_{8}}{\partial {\theta }_{8}}$$
$${A}_{13}={A}_{14}={A}_{23}={A}_{24}={A}_{33}={A}_{34}={A}_{51}={A}_{52}={A}_{61}={A}_{62}={A}_{71}={A}_{72}={A}_{81}={A}_{82}=0$$

The expression of elastic energy of ith the segment is devised as follows: The elastic energy models the effect of flexibility devised using the slope of the backbone \(\left(\frac{\partial {W}_{i}\left({x}_{i},t\right)}{\partial {x}_{i}}\right)\), modulus of elasticity \((E)\) and moment of inertia of the backbone, where \({D}_{{\text{Outer}}}\) and \({D}_{{\text{Inner}}}\) are the outer and inner diameters of the backbone.

\({U}_{{e}_{i}}=\frac{1}{2}{\left(EI\right)}_{i}\underset{0}{\overset{{l}_{i}}{\int }}{\left(\frac{{\partial }^{2}{W}_{i}\left({x}_{i},t\right)}{\partial {x}_{i}^{2}}\right)}^{2}{\text{d}}{x}_{i}; {\text{where}} {I}_{i}=\frac{\pi }{64}\left({D}_{{\text{Outer}}}^{4}-{D}_{{\text{Inner}}}^{4}\right)\)

Substituting the slope from the Euler–Bernoulli beam theory in the above expression produces the equation below:

$${U}_{{e}_{i}}=\frac{1}{2}{\left(EI\right)}_{i}\underset{0}{\overset{{l}_{i}}{\int }}\left(\frac{{\theta }_{i}}{{L}_{i}}\right){\text{d}}{L}_{i}=\frac{1}{2}{\left(EI\right)}_{i}\frac{{\theta }_{i}^{2}}{{l}_{i}}$$

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Pachouri, V., Pathak, P.M. Controlling Tendons to Modulate Stiffness of a Planar-to-Spatial Tendon-Driven Continuum Manipulator Under External Uncertain Forces. J Bionic Eng 21, 821–841 (2024). https://doi.org/10.1007/s42235-023-00473-4

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