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Steering control and stability analysis for an autonomous bicycle: part I—theoretical framework and simulations

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Abstract

This paper presents a theoretical and experimental study on the dynamics and stability of an autonomous bicycle. The first part of the paper proposes a theoretical framework for dynamics modeling and stability analysis of a controlled bicycle. The autonomous bicycle is driven by maintaining a constant angular speed of the rear wheel and is controlled by a linear control law between the steer and lean angles of the bicycle. Noting that the control laws can serve as two servo-constraints imposed on the bicycle system, we use the tools of geometric mechanics to reduce the dynamics of the autonomous bicycle into a two-dimensional dynamic system. Theoretical analysis shows that the two-dimensional dynamic system exhibits important mathematical properties, which can be used to determine the control parameters guaranteeing the motion stability of the bicycle. Based on the stability analysis, we demonstrate that the control law not only enhances the stability of the bicycle in uniform straight motion, but also allows it to be stabilized in uniform circular motion. Additionally, this paper provides an explanation for interesting phenomena observed in rider-controlled bicycles, such as the driving rules of steering toward a fall and counter-steering behavior. Furthermore, we conducted numerical simulations using the physical parameters of a self-designed autonomous bicycle to validate the theoretical development. The second part of the paper investigates how uncertainties in the bicycle system affect the stability of relative equilibria, and presents experimental results observed from the autonomous bicycle.

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Data availability

The datasets generated and analyzed during the current study are not publicly available since the authors need to do further research on them, but are available from the corresponding author on reasonable request.

Notes

  1. The bicycle system is a nonholonomic Voronets’ system [49].

  2. In the following paragraph, a single term containing repeated indices (no more than twice) always implies summation of that term over all the values of the index.

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Acknowledgements

This work has been supported by the National Natural Science Foundation of China (NSFC:11932001, U2241264) and Key Research Project of Zhejiang Lab (No. G2021NB0AL03).

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Authors and Affiliations

  1. Research Center for Intelligent Robotics, Research Institute of Interdisciplinary Innovation, Zhejiang Lab, Hangzhou, 311100, China

    Jiaming Xiong

  2. Zhejiang Engineering Research Center for Intelligent Robotics, Hangzhou, 311100, China

    Jiaming Xiong

  3. State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, 100871, China

    Ruihan Yu & Caishan Liu

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  1. Jiaming Xiong
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Appendix A: Coefficients in Eq. (51)

Appendix A: Coefficients in Eq. (51)

The concrete expressions of the coefficients in Eq. (51) are given as follows:

$$\begin{aligned} P_{1,\theta }({\varvec{0}})&=(m_rR_r+m_fR_f+m_sz_s+m_hz_h)g, \\ P_{1,\delta }({\varvec{0}})&=-\frac{gc\cos \lambda }{w}\left( m_sx_s+m_hx_h\right) -m_fgR_f\sin \lambda \\&\quad +m_hg\left( (w+c-x_h)\cos \lambda -z_h\sin \lambda \right) , \\ c_{113}({\varvec{0}})&=0, \;\;\;\;c_{133,\theta }({\varvec{0}})=0, \\ c_{133,\delta }({\varvec{0}})&=-\frac{R_r\cos \lambda }{R_fw}\left( I_{f,yy}R_r + I_{r,yy}R_f \right. \\&\quad \left. + m_fR_rR_f^2 +m_rR_r^2R_f + m_sz_sR_rR_f \right. \\ {}&\quad \left. + m_hz_hR_rR_f\right) , \\ c_{123}({\varvec{0}})&=\frac{c_\lambda }{2wR_f}\left( (I_{h,xx}-I_{h,zz})R_rR_fs_\lambda c_\lambda \right. \\&\quad \left. +2I_{h,xz}R_rR_fc_\lambda ^2 +I_{s,xz}R_rR_f-I_{f,yy}wR_r\right. \\&\quad \left. -m_sR_rR_fz_s(c+x_s)-m_hR_rR_fz_h(c+x_h) \right. \\&\quad \left. -I_{h,xz}R_rR_f-I_{r,yy}cR_f-I_{f,yy}cR_r\right. \\&\quad \left. -m_rcR_r^2R_f-m_fR_rR_f^2(c+w)\right) , \\ m_{11}({\varvec{0}})&=(I_{h,xx}-I_{h,zz})c_\lambda ^2-I_{h,xz}s_{(2\lambda )}+m_fR_f^2\\&\quad +m_rR_r^2+m_sz_s^2+m_hz_h^2+I_{f,xx}+I_{h,zz}\\&\quad +I_{r,xx}+I_{s,xx}, \\ m_{12}({\varvec{0}})&=\frac{1}{w}\left( 2I_{h,xz}cc_\lambda ^3+(I_{h,xx}-I_{h,zz})cs_\lambda c_\lambda ^2\right. \\&\quad +(c(m_hwz_h{-}m_sx_sz_s{-}m_hx_hz_h){+}m_hw^2z_h\\&\quad \left. +c(I_{s,xz}-I_{h,xz})+w(I_{h,xz}-m_hx_hz_h))c_\lambda \right. \\&\quad \left. -w(I_{f,xx}+I_{h,zz}+m_fR_f^2+m_hz_h^2)s_\lambda \right) , \end{aligned}$$

where \(s_{(\cdot )}=\sin (\cdot )\), \(c_{(\cdot )}=\cos (\cdot )\), and g is gravity acceleration.

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Xiong, J., Yu, R. & Liu, C. Steering control and stability analysis for an autonomous bicycle: part I—theoretical framework and simulations. Nonlinear Dyn 111, 16705–16728 (2023). https://doi.org/10.1007/s11071-023-08729-8

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