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Gait transition and orbital stability analysis for a biped robot based on the V-DSLIP model with torso and swing leg dynamics

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Abstract

In this paper, the variable legs stiffness damper-spring-loaded inverted pendulum with a rigid torso and swing leg dynamics is proposed. First, the dynamic equations are derived using the Euler–Lagrange method, and the impact model is derived based on momentum theorem. Second, the feedback linearization controller is designed to track the desired trajectory and regulate the swing leg orientation and the attitude of the torso. Third, the gait switching strategy is presented to realize waking gait transition by controlling the legs stiffness and the hip torques; thus, the average walking speed can be changed. Fourth, the Poincaré mapping method is used to analyze the orbital stability of the biped robot system. Finally, computer simulations are carried out to verify the effectiveness of the presented method. The simulation results show that the controller is effective for the transition between two natural gaits, it can track the desired walking gait and keep the torso upright simultaneously. Since all the eigenvalues of the Jacobian matrix are located inside the unit circle, it is also concluded that the proposed controller is robust against external disturbances and the system is orbital stable.

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Acknowledgements

This work was supported by the National Natural Science Foundation (NNSF) of China under Grant 12172059.

Funding

Liao has received research support from the National Natural Science Foundation(NNSF) of China under Grant 12172059.

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Authors and Affiliations

  1. School of Automation, Beijing Information Science and Technology University, Beijing, 100192, China

    Fakang Liao, Yali Zhou & Qizhi Zhang

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  1. Fakang Liao
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  2. Yali Zhou
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  3. Qizhi Zhang
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Correspondence to Fakang Liao.

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Appendices

Appendix A: Single support model

First, calculate the kinetic energy K and the potential energy P of the biped robot system, then the Lagrangian function L can be written as:

$$ \begin{aligned} L = & \frac{{M + m + m_{f} }}{2}\left( {\dot{x}^{2} + \dot{y}^{2} } \right) + \frac{1}{2}I\dot{\alpha }^{2} \\ & + \frac{m}{2}\left( {l^{2} \dot{\alpha }^{2} + 2l\dot{x}\dot{\alpha }\cos \alpha - 2l\dot{y}\dot{\alpha }\sin \alpha } \right) \\ & + \frac{{m_{f} }}{2}\left( {L_{0}^{2} \dot{\beta }^{2} + 2L_{0} \dot{x}\dot{\beta }\cos \beta + 2L_{0} \dot{y}\dot{\beta }\sin \beta } \right) \\ & - Mgy - mg\left( {y + l\cos \alpha } \right) - m_{f} g\left( {y - L_{0} \cos \beta } \right) \\ & - \frac{1}{2}\left( {k_{0} + u_{1} } \right)\left( {L_{0} - \sqrt {x^{2} + y^{2} } } \right)^{2} \\ \end{aligned} $$
(A1)

where g is the gravitational acceleration.

Secondary, substitute (A1) into (2), then the dynamic equations for the SS phase can be obtained as follows:

$$ \begin{gathered} (M + m + m_{f} )\ddot{x} + ml\ddot{\alpha }\cos \alpha - ml\dot{\alpha }^{2} \sin \alpha \hfill \\ \quad + m_{f} L_{0} \ddot{\beta }\cos \beta - m_{f} L_{0} \dot{\beta }^{2} \sin \beta \hfill \\ \quad - (k_{0} + u_{1} )\left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right)x - \frac{{c\dot{L}_{1} }}{{L_{1} }}x = F_{1} \hfill \\ \end{gathered} $$
(A2)
$$ \begin{gathered} (M + m + m_{f} )\ddot{y} - ml\ddot{\alpha }\sin \alpha - ml\dot{\alpha }^{2} \cos \alpha \hfill \\ \quad \quad \quad + m_{f} L_{0} \ddot{\beta }\sin \beta + m_{f} L_{0} \dot{\beta }^{2} \cos \beta + (M + m + m_{f} )g \hfill \\ \quad \quad - (k_{0} + u_{1} )\left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right)y - \frac{{c\dot{L}_{1} }}{{L_{1} }}y = F_{2} \hfill \\ \end{gathered} $$
(A3)
$$ \begin{gathered} (I + ml^{2} )\ddot{\alpha } + ml(\ddot{x}\cos \alpha - \ddot{y}\sin \alpha ) \hfill \\ \quad \quad - mgl\sin \alpha = F_{3} \hfill \\ \end{gathered} $$
(A4)
$$ \begin{gathered} m_{f} L_{0}^{2} \ddot{\beta } + m_{f} L_{0} \left( {\ddot{x}\cos \beta + \ddot{y}\sin \beta } \right) \hfill \\ \quad \quad + m_{f} gL_{0} \sin \beta = F_{4} \hfill \\ \end{gathered} $$
(A5)

where \(L_{1} = \sqrt {x^{2} + y^{2} }\) is the length of the left leg.

Third, according to the principle of virtual work, the generalized force is computed as:

$$ \begin{aligned} F_{1} = & \frac{y}{{x^{2} + y^{2} }}\tau_{1} \\ F_{2} = & - \frac{x}{{x^{2} + y^{2} }}\tau_{1} \\ F_{3} = & - \tau_{1} - \tau_{2} \\ F_{4} = & - \tau_{2} \\ \end{aligned} $$
(A6)

Then, the dynamic equations in compact matrix form can be written as:

$$ \begin{gathered} {\varvec{M}}_{{{\varvec{ss}}}} {\ddot{\mathbf q}}_{{{\varvec{ss}}}} + {\varvec{C}}_{{{\varvec{ss}}}} \dot{\mathbf{q}}_{{{\varvec{ss}}}} - {\varvec{F}}_{{{\varvec{ss}}}} + {\varvec{G}}_{{{\varvec{ss}}}} \hfill \\ \quad \quad = {\varvec{B}}_{{{\varvec{ss}}1}} \tau_{1} + {\varvec{B}}_{{{\varvec{ss}}2}} \tau_{2} + {\varvec{B}}_{{{\varvec{ss}}3}} u_{1} \hfill \\ \end{gathered} $$
(A7)

where

$$ {\varvec{M}}_{{{\varvec{ss}}}} = \left[ {\begin{array}{*{20}c} {M + m + m_{f} } & 0 & {ml\cos \alpha } & {m_{f} L_{0} \cos \beta } \\ 0 & {M + m + m_{f} } & { - ml\sin \alpha } & {m_{f} L_{0} \sin \beta } \\ {ml\cos \alpha } & { - ml\sin \alpha } & {ml^{2} + I} & 0 \\ {m_{f} L_{0} \cos \beta } & {m_{f} L_{0} \sin \beta } & 0 & {m_{f} L_{0}^{2} } \\ \end{array} } \right] $$
$$ {\varvec{C}}_{{{\varvec{ss}}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\begin{array}{*{20}c} { - ml\dot{\alpha }\sin \alpha } & { - m_{f} L_{0} \dot{\beta }\sin \beta } \\ { - ml\dot{\alpha }\cos \alpha } & {m_{f} L_{0} \dot{\beta }\cos \beta } \\ 0 & 0 \\ 0 & 0 \\ \end{array} } \\ \end{array} } \right],\quad {\varvec{G}}_{{{\varvec{ss}}}} = \left[ {\begin{array}{*{20}c} 0 \\ {\left( {M + m + m_{f} } \right)g} \\ { - mgl\sin \alpha } \\ {m_{f} gL_{0} \sin \beta } \\ \end{array} } \right] $$
$$ {\varvec{F}}_{{{\varvec{ss}}}} = \left[ {k_{0} \left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right) - \frac{{c\dot{L}_{1} }}{{L_{1} }}} \right]\left[ {\begin{array}{*{20}c} x \\ y \\ 0 \\ 0 \\ \end{array} } \right] $$
$$ {\varvec{B}}_{{{\varvec{ss}}1}} = \left[ {\begin{array}{*{20}c} {\frac{y}{{x^{2} + y^{2} }}} \\ { - \frac{x}{{x^{2} + y^{2} }}} \\ { - 1} \\ 0 \\ \end{array} } \right],\quad {\varvec{B}}_{{{\varvec{ss}}2}} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ { - 1} \\ \end{array} } \right],\quad {\varvec{B}}_{{{\varvec{ss}}3}} = \left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right)\left[ {\begin{array}{*{20}c} x \\ y \\ 0 \\ 0 \\ \end{array} } \right] $$

Appendix B: Double support model

Using the derivation method used in the SS phase, the dynamic equations of the DS phase can be obtained as:

$$ \begin{gathered} {\varvec{M}}_{{{\varvec{ds}}}} {\ddot{\mathbf q}}_{{{\varvec{ds}}}} + {\varvec{C}}_{{{\varvec{ds}}}} \dot{\mathbf {q}}_{{{\varvec{ds}}}} - {\varvec{F}}_{{{\varvec{ds}}1}} - {\varvec{F}}_{{{\varvec{ds}}2}} + {\varvec{G}}_{{{\varvec{ds}}}} \hfill \\ \quad = {\varvec{B}}_{{{\varvec{ds}}1}} \tau_{1} + {\varvec{B}}_{{{\varvec{ds}}2}} u_{1} + {\varvec{B}}_{{{\varvec{ds}}3}} u_{2} \hfill \\ \end{gathered} $$
(B1)

where

$$ {\varvec{M}}_{{{\varvec{ds}}}} = \left[ {\begin{array}{*{20}c} {M + m} & 0 & {ml\cos \alpha } \\ 0 & {M + m} & { - ml\sin \alpha } \\ {ml\cos \alpha } & { - ml\sin \alpha } & {ml^{2} + I} \\ \end{array} } \right],\quad {\varvec{C}}_{{{\varvec{ds}}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\begin{array}{*{20}c} { - ml\dot{\alpha }\sin \alpha } \\ { - ml\dot{\alpha }\cos \alpha } \\ 0 \\ \end{array} } \\ \end{array} } \right] $$
$$ {\varvec{F}}_{{{\varvec{ds}}{1}}} = \left[ {k_{0} \left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right) - \frac{{c\dot{L}_{1} }}{{L_{1} }}} \right]\left[ {\begin{array}{*{20}c} x \\ y \\ 0 \\ \end{array} } \right],\quad {\varvec{F}}_{{{\varvec{ds}}{2}}} = \left[ {k_{0} \left( {\frac{{L_{0} }}{{L_{2} }} - 1} \right) - \frac{{c\dot{L}_{2} }}{{L_{2} }}} \right]\left[ {\begin{array}{*{20}c} {x - a} \\ y \\ 0 \\ \end{array} } \right] $$
$$ {\varvec{G}}_{{{\varvec{ds}}}} = \left[ {\begin{array}{*{20}c} 0 \\ {\left( {M + m} \right)g} \\ { - mgl\sin \alpha } \\ \end{array} } \right],\quad {\varvec{B}}_{{{\varvec{ds}}1}} = \left[ {\begin{array}{*{20}c} {\frac{y}{{x^{2} + y^{2} }}} \\ {\begin{array}{*{20}c} { - \frac{x}{{x^{2} + y^{2} }}} \\ { - 1} \\ \end{array} } \\ \end{array} } \right],\quad {\varvec{B}}_{{{\varvec{ds}}2}} = \left( {\frac{{L_{0} }}{{L_{1} }} - 1} \right)\left[ {\begin{array}{*{20}c} x \\ y \\ 0 \\ \end{array} } \right] $$
$$ {\varvec{B}}_{{{\varvec{ds}}3}} = \left( {\frac{{L_{0} }}{{L_{2} }} - 1} \right)\left[ {\begin{array}{*{20}c} {x - a} \\ y \\ 0 \\ \end{array} } \right],\quad L_{2} = \sqrt {(x - a)^{2} + y^{2} } $$

L2 is the length of the right leg and a is the step size, a = x + L0cosθ.

Appendix C: State space representation

The SS phase:

$$ \dot{\mathbf{z}}_{{{\varvec{ss}}}} = {\varvec{f}}\left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right) + {\varvec{g}}_{{\tau_{1} }} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right)\tau_{1} + {\varvec{g}}_{{\tau_{2} }} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right)\tau_{2} + {\varvec{g}}_{1} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right)u_{1} $$
(C1)

where

$$ {\varvec{f}}\left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right) = \left( {\begin{array}{*{20}c} {\dot{\mathbf{q}}_{{{\varvec{ss}}}} } \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} \left( { - {\varvec{C}}_{{{\varvec{ss}}}} \dot{\mathbf{q}} + {\varvec{F}}_{{{\varvec{ss}}}} - {\varvec{G}}_{{{\varvec{ss}}}} } \right)} \\ \end{array} } \right),\quad {\varvec{g}}_{{\tau_{1} }} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}1}} } \\ \end{array} } \right) $$
$$ {\varvec{g}}_{{\tau_{2} }} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}2}} } \\ \end{array} } \right),\quad {\varvec{g}}_{1} \left( {{\varvec{z}}_{{{\varvec{ss}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}3}} } \\ \end{array} } \right) $$

The DS phase:

$$ \dot{\mathbf{z}}_{{{\varvec{ds}}}} = {\varvec{f}}\left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right) + {\varvec{g}}_{{\tau_{1} }} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right)\tau_{1} + {\varvec{g}}_{1} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right)u_{1} + {\varvec{g}}_{2} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right)u_{2} $$
(C2)

where

$$ {\varvec{f}}\left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right) = \left( {\begin{array}{*{20}c} {\dot{\mathbf{q}}_{{{\varvec{ds}}}} } \\ {{\varvec{M}}_{{{\varvec{ds}}}}^{ - 1} \left( {{\varvec{F}}_{{{\varvec{ds}}1}} + {\varvec{F}}_{{{\varvec{ds}}2}} - {\varvec{G}}_{{{\varvec{ds}}}} - C_{{{\varvec{ds}}}} \dot{\mathbf{q}}} \right)} \\ \end{array} } \right),\quad {\varvec{g}}_{{\tau_{1} }} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ds}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ds}}1}} } \\ \end{array} } \right) $$
$$ {\varvec{g}}_{1} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ds}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ds}}2}} } \\ \end{array} } \right),\quad {\varvec{g}}_{2} \left( {{\varvec{z}}_{{{\varvec{ds}}}} } \right) = \left( {\begin{array}{*{20}c} 0 \\ {{\varvec{M}}_{{{\varvec{ds}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ds}}3}} } \\ \end{array} } \right) $$

Appendix D: Relative degree derivation

In order to convert this nonlinear system into a standard linear form, first the relative degree γss = {γp, γα, γβ} of the error system needs to be determined.

For the position error (ep) system:

$$ \begin{aligned} \dot{e}_{p} = & \frac{{\partial e_{p} }}{{\partial {\varvec{z}}}}\dot{\mathbf{z}}_{ss} = \left( {\begin{array}{*{20}c} {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} & {\frac{{\partial e_{p} }}{{\partial \dot{\mathbf{q}}}}} \\ \end{array} } \right)\left( {{\varvec{f}} + {\varvec{g}}_{{\tau_{1} }} \tau_{1} + {\varvec{g}}_{{\tau_{2} }} \tau_{2} + {\varvec{g}}_{1} u_{1} } \right) \\ = & \underbrace {{\left( {\begin{array}{*{20}c} {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\dot{\mathbf{q}}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} \left( {{\varvec{F}}_{{{\varvec{ss}}}} - {\varvec{G}}_{{{\varvec{ss}}}} - C_{{{\varvec{ss}}}} \dot{\mathbf{q}}} \right)} \\ \end{array} } \right)}}_{{L_{f} e_{p} }} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}1}} } \\ \end{array} } \right)}}_{{L_{{g\tau_{1} }} e_{p} = 0}}\tau_{1} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}2}} } \\ \end{array} } \right)}}_{{L_{{g\tau_{2} }} e_{p} = 0}}\tau_{2} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}3}} } \\ \end{array} } \right)}}_{{L_{{g_{1} }} e_{p} = 0}}u_{1} \\ = & L_{f} e_{p} ({\varvec{q}},\dot{\mathbf{q}}) \\ \end{aligned} $$
(D1)

where \(L_{{g\tau_{1} }} e_{p} = L_{{g\tau_{2} }} e_{p} = \, L_{{g_{1} }} e_{p} = 0\). We proceed by calculating the second-order derivative:

$$ \begin{aligned} \ddot{e}_{p} = & \left( {\begin{array}{*{20}c} {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}\dot{\mathbf{q}}} \right)} & {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} \\ \end{array} } \right)\left( {{\varvec{f}} + {\varvec{g}}_{{\tau_{1} }} \tau_{1} + {\varvec{g}}_{{\tau_{2} }} \tau_{2} + {\varvec{g}}_{1} u_{1} } \right) \\ = & \underbrace {{\left( {\begin{array}{*{20}c} {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}\dot{\mathbf{q}}} \right)} & {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\dot{\mathbf{q}}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} \left( {{\varvec{F}}_{{{\varvec{ss}}}} - {\varvec{G}}_{{{\varvec{ss}}}} - C_{{{\varvec{ss}}}} \dot{\mathbf{q}}} \right)} \\ \end{array} } \right)}}_{{L_{f}^{2} e_{p} }} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}\dot{\mathbf{q}}} \right)} & {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}1}} } \\ \end{array} } \right)}}_{{L_{{g\tau_{1} }} L_{f} e_{p} }}\tau_{1} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}\dot{\mathbf{q}}} \right)} & {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{2} } \\ \end{array} } \right)}}_{{L_{{g\tau_{2} }} L_{f} e_{p} }}\tau_{2} \\ & + \underbrace {{\left( {\begin{array}{*{20}c} {\frac{\partial }{{\partial {\varvec{q}}}}\left( {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}\dot{\mathbf{q}}} \right)} & {\frac{{\partial e_{p} }}{{\partial {\varvec{q}}}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\varvec{M}}_{{{\varvec{ss}}}}^{ - 1} {\varvec{B}}_{{{\varvec{ss}}3}} } \\ \end{array} } \right)}}_{{L_{g1} L_{f} e_{p} }}u_{1} \\ =\, & L^{2}_{f} e_{p} ({\varvec{q}},\dot{\mathbf{q}}) + L_{{g\tau_{1} }} L{}_{f}e_{p} ({\varvec{q}})\tau_{1} \\ & + L_{{g\tau_{2} }} L{}_{f}e_{p} ({\varvec{q}})\tau_{2} + L_{g1} L{}_{f}e_{p} ({\varvec{q}})u_{1} \\ \end{aligned} $$
(D2)

where \(L_{{g\tau_{1} }} L_{f} e_{p} \ne 0,\quad L_{{g\tau_{2} }} L_{f} e_{p} \ne 0,\), and \(L_{{g_{1} }} L_{f} e_{p} \ne 0\) denote the (repeated) Lie-derivatives of the error functions along the vector fields of dynamics system. So the relative degree γp = 2. In the same way, we can get the relative degrees γα = γβ = 2. So, in the SS phase, the relative degree of the error system γss = {γp, γα, γβ} = {2, 2, 2}.

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Liao, F., Zhou, Y. & Zhang, Q. Gait transition and orbital stability analysis for a biped robot based on the V-DSLIP model with torso and swing leg dynamics. Nonlinear Dyn 108, 3053–3075 (2022). https://doi.org/10.1007/s11071-022-07364-z

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