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Passive walking biped robot model with flexible viscoelastic legs

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Abstract

To simulate the complex human walking motion accurately, a suitable biped model has to be proposed that can significantly translate the compliance of biological structures. In this way, the simplest passive walking model is often used as a standard benchmark for making the bipedal locomotion so natural and energy efficient. This paper is devoted to a presentation of the application of internal damping mechanism to the mathematical description of the simplest passive walking model with flexible legs. This feature can be taken into account by using the viscoelastic legs, which are constituted by the Kelvin–Voigt rheological model. Then, the update of the impulsive hybrid nonlinear dynamics of the simplest passive walker is obtained based on the Euler–Bernoulli’s beam theory and using a combination of Lagrange mechanics and the assumed mode method, along with the precise boundary conditions. The main goal of this study is to develop a numerical procedure based on the new definition of the step function for enforcing the biped start walking from stable condition and walking continuously. In our previous work, it was shown that the period-one gait cycle can be produced by adding the proportional damping model with high damping ratio to the elastic legs. The present paper overcomes the practical limitations of this damping model and similarly demonstrates the stable period-one gait cycles for the viscoelastic legs. The study of the influence of various system parameters is carried out through bifurcation diagrams, highlighting the region of stable period-one gait cycles. Numerical simulations clearly prove that the overall effect of viscoelastic leg on the passive walking is efficient enough from the viewpoint of stability and energy dissipation.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Babol Noshirvani University of Technology, Shariati Street, P.O. Box 484, Babol, 47148-71167, Mazandaran, Iran

    Masoumeh Safartoobi, Morteza Dardel & Hamidreza Mohammadi Daniali

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Appendix A: Definition of the normalized matrices of the flexible model with viscoelastic legs

Appendix A: Definition of the normalized matrices of the flexible model with viscoelastic legs

In this appendix, we derive the upper triangular elements of the symmetric inertia matrix and the right-hand side vector in Eq. (20) for the first mode shape (\(n=1\)) as following:

$$ \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {M_{11} } \\ \ldots \\ \end{array} } & {\begin{array}{*{20}c} {M_{12} } \\ {M_{22} } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {M_{13} } \\ {M_{23} } \\ \end{array} } & {\begin{array}{*{20}c} {M_{14} } \\ {M_{24} } \\ \end{array} } \\ \end{array} } \\ {\begin{array}{*{20}c} \ldots & \ldots \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {M_{33} } \\ \ldots \\ \end{array} } & {\begin{array}{*{20}c} {M_{34} } \\ {M_{44} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\ddot{\theta }} \\ {\ddot{v}_{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\ddot{\phi }} \\ {\ddot{p}_{1} } \\ \end{array} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {F_{1} \left( {{\varvec{q}},\dot{\user2{q}}} \right)} \\ {F_{2} \left( {{\varvec{q}},\dot{\user2{q}}} \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} {F_{3} \left( {{\varvec{q}},\dot{\user2{q}}} \right)} \\ {F_{4} \left( {{\varvec{q}},\dot{\user2{q}}} \right)} \\ \end{array} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {F_{{{\varvec{v}}1}} \left( {\dot{\user2{q}}} \right)} \\ {F_{{{\varvec{v}}2}} \left( {\dot{\user2{q}}} \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} {F_{{{\varvec{v}}3}} \left( {\dot{\user2{q}}} \right)} \\ {F_{{{\varvec{v}}4}} \left( {\dot{\user2{q}}} \right)} \\ \end{array} } \\ \end{array} } \right\} $$
$$ \begin{aligned} M_{11} = & \frac{5}{3} + \overline{v}_{1}^{2} \left( \tau \right)\left( {\int\limits_{0}^{1} {\varphi_{1}^{2} \left( {\overline{x}_{1} } \right)} d\overline{x}_{1} + \varphi_{1}^{2} \left( 1 \right)} \right) \\ & + \overline{p}_{1}^{2} \left( \tau \right)\int\limits_{0}^{1} {\psi_{1}^{2} \left( {\overline{x}_{2} } \right)} d\overline{x}_{2} + m_{H} \left( {1 + \overline{v}_{1}^{2} \left( \tau \right)\varphi_{1}^{2} \left( 1 \right)} \right) \\ \end{aligned} $$
$$ M_{12} = - \left( {\mathop \int \limits_{0}^{1} \overline{x}_{1} \varphi_{1} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\varphi_{1} \left( 1 \right)} \right) $$
$$ M_{13} = - \left( {\frac{1}{3} + \overline{p}_{1}^{2} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} } \right) ,\quad M_{14} = - \mathop \int \limits_{0}^{1} \overline{x}_{2} \psi_{1} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} $$
$$ \begin{gathered} M_{22} = \left( {\mathop \int \limits_{0}^{1} \varphi_{1}^{2} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\varphi_{1}^{2} \left( 1 \right)} \right) \hfill \\ M_{23} = 0,\quad M_{24} = 0 \hfill \\ \end{gathered} $$
$$ \begin{gathered} M_{33} = \left( {\frac{1}{3} + \overline{p}_{1}^{2} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} } \right),\quad M_{34} = \mathop \int \limits_{0}^{1} \overline{x}_{2} \psi_{1} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} \hfill \\ M_{44} = \mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} \hfill \\ \end{gathered} $$
$$ \begin{aligned} F_{1} \left( {{\varvec{q}},\dot{\user2{q}}} \right) = & - 2\dot{\overline{\theta }}_{st} \left[ {\overline{v}_{1} \left( \tau \right)\dot{\overline{v}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \varphi_{1}^{2} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\overline{v}_{1} \left( \tau \right)\dot{\overline{v}}_{1} \left( \tau \right)\varphi_{1}^{2} \left( 1 \right)} \right. \\ & \left. { + \overline{p}_{1} \left( \tau \right)\dot{\overline{p}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} } \right] + 2\dot{\overline{\phi }}\left[ {\overline{p}_{1} \left( \tau \right)\dot{\overline{p}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} } \right] \\ & - \cos \left( {\overline{\theta }_{st} \left( \tau \right) - \gamma } \right)\overline{v}_{1} \left( \tau \right)\left[ {\mathop \int \limits_{0}^{1} \varphi_{1} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\varphi_{1} \left( 1 \right)} \right] \\ & + \cos \left( {\overline{\theta }_{st} \left( \tau \right) - \overline{\phi }\left( \tau \right) - \gamma } \right)\overline{p}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} \\ & + \left( {\frac{3}{2} + m_{H} } \right)\sin \left( {\overline{\theta }_{st} \left( \tau \right) - \gamma } \right) - \frac{1}{2}\sin \left( {\overline{\theta }_{st} \left( \tau \right) - \overline{\phi }\left( \tau \right) - \gamma } \right) \\ \end{aligned} $$
$$ \begin{aligned} F_{2} \left( {{\varvec{q}},\dot{\user2{q}}} \right) = & \overline{v}_{1} \left( \tau \right)\left( {\mathop \int \limits_{0}^{1} \varphi_{1}^{2} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\varphi_{1}^{2} \left( 1 \right)} \right)\dot{\overline{\theta }}_{st}^{2} \\ & - \sin \left( {\overline{\theta }_{st} \left( \tau \right) - \gamma } \right)\left( {\mathop \int \limits_{0}^{1} \varphi_{1} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} + \left( {1 + m_{H} } \right)\varphi_{1} \left( 1 \right)} \right) - D\overline{v}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \varphi_{1}^{\prime \prime 2} \left( {\overline{x}_{1} } \right)d\overline{x}_{1} \\ \end{aligned} $$
$$ \begin{aligned} F_{3} \left( {{\varvec{q}},\dot{\user2{q}}} \right) = & 2\left( {\dot{\overline{\theta }}_{st} - \dot{\overline{\phi }}} \right)\overline{p}_{1} \left( \tau \right)\dot{\overline{p}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{2} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} \\ & - \cos \left( {\overline{\theta }_{st} \left( \tau \right) - \overline{\phi }\left( \tau \right) - \gamma } \right)\overline{p}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1} \left( {\overline{x}_{2} } \right)d\overline{x}_{2} \\ & + \frac{1}{2}\sin \left( {\overline{\theta }_{st} \left( \tau \right) - \overline{\phi }\left( \tau \right) - \gamma } \right) \\ \end{aligned} $$
$$ \begin{aligned} F_{4} ({\varvec{q}},\dot{\user2{q}}) = & \left( {\dot{\overline{\theta }}_{st} - \dot{\overline{\phi }}} \right)^{2} \overline{p}_{1} (\tau )\int\limits_{0}^{1} {\psi_{1}^{2} \left( {\overline{x}_{2} } \right)} d\overline{x}_{2} \\ & + \sin \left( {\overline{\theta }_{st} \left( \tau \right) - \overline{\phi }\left( \tau \right) - \gamma } \right)\int\limits_{0}^{1} {\overline{\psi }_{1} (\overline{x}_{2} )} d\overline{x}_{2} - D\overline{p}_{1} (\tau )\int\limits_{0}^{1} {\psi_{1}^{\prime \prime 2} \left( {\overline{x}_{2} } \right)} d\overline{x}_{2} \\ \end{aligned} $$
$$ F_{v1} \left( {\dot{\user2{q}}} \right) = 0 ,\quad F_{v2} \left( {\dot{q}} \right) = - \overline{\eta }D\dot{\overline{v}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \varphi_{1}^{\prime \prime \prime \prime } \left( {\overline{x}_{1} } \right)d\overline{x}_{1} $$
$$ F_{{{\varvec{v}}3}} \left( {\dot{\user2{q}}} \right) = 0,\quad F_{{{\varvec{v}}4}} \left( {\dot{\user2{q}}} \right) = - \overline{\eta }D\dot{\overline{p}}_{1} \left( \tau \right)\mathop \int \limits_{0}^{1} \psi_{1}^{\prime \prime \prime \prime } \left( {\overline{x}_{2} } \right)d\overline{x}_{2} . $$

To see the matrices \(Q^{ + }\) and \(Q^{ - }\) in the jump equations, the readers are referred to our former study in [41].

$$ Q^{ + } \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\theta_{st} } \\ {v_{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} \phi \\ {p_{1} } \\ \end{array} } \\ \end{array} } \right\}^{ + } = Q^{ - } \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\theta_{st} } \\ {v_{1} } \\ \end{array} } \\ {\begin{array}{*{20}c} \phi \\ {p_{1} } \\ \end{array} } \\ \end{array} } \right\}^{ - } . $$

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Safartoobi, M., Dardel, M. & Mohammadi Daniali, H. Passive walking biped robot model with flexible viscoelastic legs. Nonlinear Dyn 109, 2615–2636 (2022). https://doi.org/10.1007/s11071-022-07600-6

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