## DYNAMIC ANALYSIS OF A SELF-BALANCING ONE-WHEEL SCOOTER

LIU Yanzhu,1)

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240

Abstract

The dynamics of a self-balancing one-wheel scooter is analyzed in this paper. The one-wheel scooter is simplified as a two-body system composed of a frame with rider and a wheel. Under the nonholonomic constraint conditions of pure rolling of the wheel, the dynamic equations of the system are derived by use of the theorem of angular momentum. The acceleration and deceleration, the turning motion of the scooter, and the standing stability of the rider are examined theoretically. The stability criteria of self-balancing procedure by control system are derived.

Keywords： self-balancing scooter; one-wheel scooter; gyroscopic torque; stability theory of motion

LIU Yanzhu. DYNAMIC ANALYSIS OF A SELF-BALANCING ONE-WHEEL SCOOTER. Mechanics in Engineering, 2021, 43(6): 1002-1005 DOI:10.6052/1000-0879-21-226

### 图4

$\begin{eqnarray} &&\mathbf v_{P} =\left[ {v_{cx} -R\varOmega -\left( {R+l} \right)\dot{{\theta }}} \right]\mathbf i+\\&&\qquad\left[ {v_{cy} +\left( {R+l} \right)\dot{{\psi }}} \right]\mathbf j+v_{cz} \mathbf k \end{eqnarray}$

$\begin{eqnarray} \left.\begin{array}{l} v_{cx} -R\varOmega -\left( {R+l} \right)\dot{{\theta }}=0\\ v_{cy} +\left( {R+l} \right)\dot{{\psi }}=0\\ v_{cz} =0 \end{array}\right\} \end{eqnarray}$

$\begin{eqnarray} \mathbf v_{c} \mathbf{=}\left[ {R\varOmega +\left( {R+l} \right)\dot{{\theta }}} \right]\mathbf i-\left( {R+l} \right)\dot{{\psi }}\mathbf j \end{eqnarray}$

$\begin{eqnarray} \frac{{\tilde{{\rm d}}}\mathbf v_{c} }{{\rm d}t}+\mathbf \omega\times \mathbf v_{c} ={m\mathbf g}+\mathbf{F} \end{eqnarray}$

$\begin{eqnarray} \left.\begin{array}{l} F_{x} =m\left[ {R\dot{{\varOmega }}+\left( {R+l} \right)\ddot{{\theta }}-g\theta } \right] \\ F_{y} =m\left[ {R\varOmega \dot{{\varphi }}-\left( {R+l} \right)\ddot{{\psi }}+g\psi } \right] \\ F_{z} =m\left( {g-R\varOmega \dot{{\theta }}} \right) \\ \end{array}\right\} \end{eqnarray}$

$\begin{eqnarray} \frac{\tilde{\rm d}\mathbf{L}_{1} }{{\rm d}t}+\mathbf \omega\times \mathbf{L}_{1} =\mathbf{M}_{c} +\mathbf{F}\times \mathbf R \end{eqnarray}$

$\begin{eqnarray} \left( {J+mRl} \right)\ddot{{\theta }}-mgR\theta +J\dot{{\varOmega }}=M_{{c}} \end{eqnarray}$

$\begin{eqnarray} \frac{\tilde{\rm d}\mathbf{L}}{{\rm d}t}+\mathbf \omega\times\mathbf{L}=\mathbf{F}\times \left( {\mathbf{R+l}} \right) \end{eqnarray}$

$\left[ {A+m\left( {R+l} \right)^{2}} \right]\ddot{{\psi }}-\left({J+mRl} \right)\varOmega \dot{{\varphi }}-\left( {R+l} \right)mg\psi =0$

$\left[ {B+m\left( {R+l} \right)^{2}} \right]\ddot{{\theta }}+\left({J+mRl} \right)\dot{{\varOmega }}-\left( {R+l} \right)mg\theta =0$

$C\ddot{{\varphi }}+J_{0} \varOmega \dot{{\psi }}=0$

$\begin{eqnarray} M_{c} =k\theta \end{eqnarray}$

$\begin{eqnarray} J\dot{{\varOmega }}=\left( {k+mgR} \right)\theta -\left( {J+mRl}\right)\ddot{{\theta }} \end{eqnarray}$

$\begin{eqnarray} \left[ {B+m\left( {R+l} \right)^{2}-\left( {1+\alpha } \right)\left( {J+mRl} \right)} \right]\ddot{{\theta }}+\left[ {k\left( {1+\alpha } \right)-mg\left( {l-\alpha R} \right)} \right]\theta =0 \end{eqnarray}$

$\begin{eqnarray} \left( {1+\alpha } \right)\left( {J+mRl} \right)=\frac{1}{m_{\ast }}\left( {m_{\ast }R+ml} \right)^{2} \end{eqnarray}$

$\begin{eqnarray} B\lambda^{2}+\left[ {k\left( {1+\alpha } \right)-mg\left( {l-\alpha R} \right)} \right]\theta =0 \end{eqnarray}$

$\begin{eqnarray} k>mg\left( {\frac{l-\alpha R}{1+\alpha }} \right) \end{eqnarray}$

$\begin{eqnarray} M_{c} =k\theta +k_{1} \dot{{\theta }} \end{eqnarray}$

$\begin{eqnarray} &&B\lambda^{2}+k_{1} \left( {1+\alpha } \right)\lambda +\big[ k\left( {1+\alpha } \right)-\\&&\qquad mg\left( {l-\alpha R} \right) \big]\theta =0 \end{eqnarray}$

$\begin{eqnarray} \lambda^{2}\left( {a\lambda^{2}+b} \right)=0 \end{eqnarray}$

$\begin{eqnarray} \left.\begin{array}{l} a=C\left[ {A+m\left( {R+l} \right)^{2}} \right]\\ b=J_{0} \left[{J_{0} +mR\left( {R+l} \right)} \right]\varOmega_{0}^{2} -Cmg\left( {R+l} \right) \end{array}\right\} \end{eqnarray}$

$\begin{eqnarray} \varOmega_{{0,\min}} =\sqrt {\frac{Cmg\left( {R+l} \right)}{J_{0} \left[ {J_{0} +mR\left( {R+l} \right)} \right]}} \end{eqnarray}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Liu Yanzhu.

Talk on segway human transporter

Mechanics in Engineering, 2009, 31(6): 95-96 (in Chinese)

Liu Yanzhu.

A miniature segway-talk on self-balancing scooter

Mechanics in Engineering, 2017, 39(2): 208-210 (in Chinese)

Onewheel: 独轮电动滑板让你纵横驰骋, https://www.ifanr.com/354598

Liu Yanzhu.

On the stability of bicycle

Mechanics in Engineering, 2012, 34(2): 90-93 (in Chinese)

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