Stability analysis for the Whipple bicycle dynamics

Abstract

It has been known that bicycle stability is closely linked to a pair of ordinary differential equations (ODEs). The linearization technique used to derive these ODEs, nevertheless, has yet to be thoroughly examined. For this purpose, we conduct an analysis of the dynamics of the Whipple bicycle, starting with the contact kinematics, using the Gibbs–Appell method. The effort results in a complete nonlinear model with minimal dimensions, from which equilibrium points during the bicycle’s straight and circular motions can be determined. The model can be linearized around these points via a perturbation analysis under no additional assumptions. Given the non-hyperbolic nature of the equilibria, we apply the center manifold theorem to analyze their stability, offering a rigorous derivation of the (well-know) exponential stability of the bicycle in its leaning and steering motions. Finally, a dimensionless index is defined to characterize the influence of physical parameters on the bicycle stability.

Appendices

Appendix A: Analytic solutions of the geometric constraint equations at the wheel–ground contacts

Using the second equation in (11) and the first equation in (12) to eliminate $z$, we obtain

$$ A\cdot c_{\vartheta _{f}} - B\cdot s_{\vartheta _{f}}=C, $$

where

$$ C=-\frac{1}{R_{f}} \bigl((\xi _{r} s_{\varphi }- \zeta _{r} c_{\varphi }+R_{r})c_{\theta }+A\xi _{f} \bigr). $$

Combining the above equation with the second equation in (12), we easily obtain

$$ A^{2}+B^{2}=C^{2}, $$

which, together with $s_{\varphi }^{2}+c_{\varphi }^{2}=1$, results in the identity

$$ \kappa _{1}s_{\varphi }^{2}+\kappa _{2}s_{\varphi }+\kappa _{3}=-(\kappa _{4}s_{\varphi }+\kappa _{5})c_{\varphi }, $$

where $\kappa _{i},\;i=1,\ldots ,5$ are functions with respect to $\theta$ and $\delta$. This equation can be written as a quartic polynomial equation about $s_{\varphi }$ with coefficients that depend on $\theta$ and $\delta$ only:

$$ \bigl(\kappa _{1}s_{\varphi }^{2}+\kappa _{2}s_{\varphi }+\kappa _{3} \bigr) ^{2}=(\kappa _{4}s_{\varphi }+\kappa _{5})^{2} \bigl(1-s_{\varphi }^{2} \bigr). $$

Therefore, an analytic solution for the geometric constraint $\varphi =\varphi (\theta ,\delta )$ can be obtained. Accordingly, the explicit expression for another geometric constraint, $z=z(\theta ,\delta )$, can also be found.

Appendix B: Coefficients of the velocity constraints

The expression of the coefficient matrix of the velocity constraints, designated by $\mathbf{W}$, is

$$\begin{aligned} \mathbf{W}= \left [ \textstyle\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 1 & 0 & 0 & \mathit{as}_{\theta }c_{\psi }-\mathit{bs}_{\psi }& \mathit{ac}_{\theta }s_{\psi }& \mathit{ac} _{\psi }-\mathit{bs}_{\theta }s_{\psi }& 0 & -R_{r}c_{\psi }& 0 \\ 0 & 1 & 0 & \mathit{as}_{\theta }s_{\psi }+\mathit{bc}_{\psi }& -\mathit{ac}_{\theta }c_{\psi }& \mathit{as}_{\psi }+\mathit{bs}_{\theta }c_{\psi }& 0 & -R_{r}s_{\psi }& 0 \\ 0 & 0 & 1 & 0 & -\mathit{as}_{\theta }& -\mathit{bc}_{\theta }& 0 & 0 & 0 \\ 1 & 0 & 0 & W_{44} & W_{45} & W_{46} & W_{47} & 0 & W_{49} \\ 0 & 1 & 0 & W_{54} & W_{55} & W_{56} & W_{57} & 0 & W_{59} \\ 0 & 0 & 1 & 0 & W_{65} & W_{66} & W_{67} & 0 & 0 \\ \end{array}\displaystyle \right ], \end{aligned}$$

where $a=\zeta _{r}\cos {\varphi }-\xi _{r}\sin {\varphi }-R_{r}$, $b=\zeta _{r}\sin \varphi +\xi _{r}\cos \varphi$, and the other symbols are defined as follows:²

$$\begin{aligned} W_{44}= &-R_{f}s_{\vartheta _{f}}(c_{(\lambda -\varphi )}c_{\psi }s_{ \theta }+s_{(\lambda -\varphi )}s_{\psi }) \\ &+(\xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\lambda -\varphi )}c_{\psi }c _{\delta }s_{\theta }-c_{(\lambda -\varphi )}s_{\psi }c_{\delta }-c _{\psi }s_{\delta }c_{\theta }), \\ W_{45}= &-R_{f}s_{\vartheta _{f}}c_{(\lambda -\varphi )}s_{\psi }c_{ \theta }+( \xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\lambda -\varphi )}c _{\delta }c_{\theta }+s_{\delta }s_{\theta })s_{\psi }, \\ W_{46}= &-R_{f}s_{\vartheta _{f}}(s_{(\lambda -\varphi )}s_{\psi }s _{\theta }+c_{(\lambda -\varphi )}c_{\psi }) \\ &+(\xi _{f}+R_{f}c_{\vartheta _{f}}) (-c_{(\lambda -\varphi )}s_{\psi }c _{\delta }s_{\theta }+s_{(\lambda -\varphi )}c_{\psi }c_{\delta }), \\ W_{47}= &(\xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\varphi -\lambda )}s _{\psi }s_{\delta }s_{\theta }-c_{(\lambda -\varphi )}c_{\psi }s_{ \delta }-s_{\psi }c_{\delta }c_{\theta }), \\ W_{49}= &R_{f}c_{\vartheta _{f}}(-c_{(\lambda -\varphi )}s_{\psi }s _{\theta }+s_{(\lambda -\varphi )}c_{\psi }) \\ &+R_{f}s_{\vartheta _{f}}(-c_{(\lambda -\varphi )}c_{\psi }c_{\delta }+s_{\psi }s_{\delta }c_{\theta }-s_{(\lambda -\varphi )}s_{\psi }s _{\theta }c_{\delta }), \\ W_{54}= &-R_{f}s_{\vartheta _{f}}(c_{(\lambda -\varphi )}s_{\psi }s _{\theta }-s_{(\lambda -\varphi )}c_{\psi }) \\ &+(\xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\lambda -\varphi )}s_{\psi }c _{\delta }s_{\theta }+c_{(\lambda -\varphi )}c_{\psi }c_{\delta }-s _{\psi }s_{\delta }c_{\theta }), \\ W_{55}= &R_{f}s_{\vartheta _{f}}c_{(\lambda -\varphi )}c_{\psi }c_{ \theta }-( \xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\lambda -\varphi )}c _{\delta }c_{\theta }+s_{\delta }s_{\theta })c_{\psi }, \\ W_{56}= &R_{f}s_{\vartheta _{f}}(s_{(\lambda -\varphi )}c_{\psi }s_{ \theta }-c_{(\lambda -\varphi )}s_{\psi }) \\ &+(\xi _{f}+R_{f}c_{\vartheta _{f}}) (c_{(\lambda -\varphi )}c_{\psi }c _{\delta }s_{\theta }+s_{(\lambda -\varphi )}s_{\psi }c_{\delta }), \\ W_{57}= &(\xi _{f}+R_{f}c_{\vartheta _{f}}) (s_{(\lambda -\varphi )}c _{\psi }s_{\delta }s_{\theta }-c_{(\lambda -\varphi )}s_{\psi }s_{ \delta }+c_{\psi }c_{\delta }c_{\theta }), \\ W_{59}= &{R_{f}c_{\vartheta _{f}}(c_{(\lambda -\varphi )}c_{\psi }s _{\theta }+s_{(\lambda -\varphi )}s_{\psi })} \\ &+R_{f}s_{\vartheta _{f}}(-c_{(\lambda -\varphi )}s_{\psi }c_{\delta }-c_{\psi }s_{\delta }c_{\theta }+s_{(\lambda -\varphi )}c_{\psi }s _{\theta }c_{\delta }), \\ W_{65}= &(\xi _{f}+R_{f}c_{\vartheta _{f}}) (-s_{(\lambda -\varphi )}s _{\theta }c_{\delta }+c_{\theta }s_{\delta })+R_{f}s_{\vartheta _{f}}s _{\theta }c_{(\lambda -\varphi )}, \\ W_{66}= &-(\xi _{f}+R_{f}c_{\vartheta _{f}})c_{(\lambda -\varphi )}c _{\theta }c_{\delta }-R_{f}s_{\vartheta _{f}}c_{\theta }s_{(\lambda - \varphi )}, \\ W_{67}= &{ (\xi _{f}+R_{f}c_{\vartheta _{f}}) (-s_{(\lambda -\varphi )}c _{\theta }s_{\delta }+s_{\theta }c_{\delta })}. \end{aligned}$$

Appendix C: Coefficients of the linearized equations

The coefficients of the linearized equations (32) are given as follows:

$$\begin{aligned} M_{11}= &(I_{h,\mathit{xx}}-I_{h,\mathit{zz}})c_{\lambda }^{2}-I_{h,\mathit{xz}}s_{(2\lambda )}+m _{f}R_{f}^{2}+m_{r}R_{r}^{2} \\ &+m_{b}z_{b}^{2}+m_{h}z_{h}^{2}+I_{f,\mathit{xx}}+I_{h,\mathit{zz}}+I_{r,\mathit{xx}}+I_{b,\mathit{xx}}, \\ M_{12}= &\frac{1}{w}\big(2I_{h,\mathit{xz}} \mathit{cc}_{\lambda }^{3}+(I_{h,\mathit{xx}}-I_{h,\mathit{zz}}) \mathit{cs} _{\lambda }c_{\lambda }^{2} \\ &+\bigl(c(m_{h}\mathit{wz}_{h}-m_{b}x_{b}z_{b}-m_{h}x_{h}z_{h}+I_{b,\mathit{xz}}-I_{h,\mathit{xz}}) \\ &+m_{h}w^{2}z_{h}+w(I_{h,\mathit{xz}}-m_{h}x_{h}z_{h}) \bigr)c_{\lambda }-w\bigl(I_{f,\mathit{xx}}+I _{h,\mathit{zz}} \\ &+m_{f}R_{f}^{2}+m_{h}z_{h}^{2} \bigr)s_{\lambda }\big), \\ M_{21}= &M_{12}, \\ M_{22}= &\frac{1}{w^{2}}\big((I_{h,\mathit{zz}}-I_{h,\mathit{xx}})c^{2}c_{\lambda } ^{4}+2I_{h,\mathit{xz}}c^{2}s_{\lambda }c_{\lambda }^{3} \\ &+\bigl(m_{h}w^{4}+2m_{h}(c-x_{h})w^{3}+ \bigl(m_{h}\bigl(c^{2}-4\mathit{cx}_{h}+x_{h}^{2} \\ &-z_{h}^{2}\bigr)-m_{f}R_{f}^{2} \bigr)w^{2}+2c\bigl(m_{h}x_{h}^{2}-m_{h} \mathit{cx}_{h}+I _{h,\mathit{zz}} \\ &+I_{f,\mathit{xx}}\bigr)w+c^{2}\bigl(m_{b}x_{b}^{2}+m_{h}x_{h}^{2}+I_{b,\mathit{zz}}+I_{f,\mathit{xx}}+I _{h,\mathit{xx}} \\ &+I_{r,\mathit{xx}}\bigr)\bigr)c_{\lambda }^{2}+2 \bigl(-m_{h}z_{h}w^{3}+m_{h}z_{h}(x_{h}-c)w ^{2} \\ &+c(I_{h,\mathit{xz}}+m_{h}x_{h}z_{h})w \bigr)s_{\lambda }c_{\lambda } \bigr)+m_{f}R _{f}^{2}+m_{h}z_{h}^{2}+I_{f,\mathit{xx}}+I_{h,\mathit{zz}}, \\ C_{11}= &0,\\ C_{12}= &-\frac{R_{r}c_{\lambda }}{\mathit{wR}_{f}}\bigl(m_{f}R_{r}R_{f}^{2}+m_{r}R _{r}^{2}R_{f}+I_{r,\mathit{yy}}R_{f}+I_{f,\mathit{yy}}R_{r} \\ &+m_{b}z_{b}R_{r}R_{f}+m_{h}z_{h}R_{r}R_{f} \bigr), \\ C_{21}= &0, \\ C_{22}= &-\frac{R_{r}^{2}c_{\lambda }}{w^{2}R_{f}}\big(\bigl(m_{h}w^{2}+m _{h}(c-x_{h})w-m_{b}x_{b}c-m_{h}x_{h}c \bigr)R_{f}c_{\lambda } \\ &-\bigl(m_{h}R_{f}z_{h}+m_{f}R_{f}^{2}+I_{f,\mathit{yy}} \bigr)\mathit{ws}_{\lambda }\big), \\ D_{11}= &0, \\ D_{12}= &\frac{c_{\lambda }}{\mathit{wR}_{f}}\big((I_{h,\mathit{xx}}-I_{h,\mathit{zz}})R_{r}R _{f}s_{\lambda }c_{\lambda }+2I_{h,\mathit{xz}}R_{r}R_{f}c_{\lambda }^{2} \\ &-m_{r}\mathit{cR}_{r}^{2}R_{f}+I_{b,\mathit{xz}}R_{r}R_{f}-I_{f,\mathit{yy}} \mathit{wR}_{r}-m_{b}R_{r}R _{f}z_{b}(c+x_{b}) \\ &-m_{h}R_{r}R_{f}z_{h}(c+x_{h})-I_{h,\mathit{xz}}R_{r}R_{f}-I_{r,\mathit{yy}} \mathit{cR}_{f}-I _{f,\mathit{yy}}\mathit{cR}_{r} \\ &-m_{f}R_{r}R_{f}^{2}(c+w) \big), \\ D_{21}= &\frac{R_{r}}{R_{f}}I_{f,\mathit{yy}}c_{\lambda }+ \biggl(\frac{R_{r}}{R _{f}}I_{f,\mathit{yy}}+I_{r,\mathit{yy}} \biggr)c_{\lambda }\frac{c}{w}, \\ D_{22}= &\frac{R_{r}c_{\lambda }}{w^{2}}\big(I_{h,\mathit{xz}} \mathit{ws}_{\lambda }+2I _{h,\mathit{xz}}\mathit{cs}_{\lambda }c_{\lambda }^{2}+m_{f}R_{f}w^{2}s_{\lambda } \\ &+m_{h}\mathit{wcz}_{h}s_{\lambda }+m_{h} \mathit{wx}_{h}z_{h}s_{\lambda }+m_{f} \mathit{wR}_{f}\mathit{cs} _{\lambda } \\ &+\bigl(m_{b}x_{b}^{2}c-m_{h}x_{h}w^{2}+m_{h} \mathit{wx}_{h}^{2}+I_{h,\mathit{xx}}c+I_{f,\mathit{xx}}c+I _{f,\mathit{xx}}w \\ &+I_{h,\mathit{zz}}w-m_{h}w^{2}c-m_{h} \mathit{wc}^{2}+m_{h}c^{2}x_{h}+m_{h} \mathit{cx}_{h}^{2}+I _{b,\mathit{zz}}c \\ &+I_{r,\mathit{xx}}c+m_{b}x_{b}c^{2} \bigr)c_{\lambda }+(I_{h,\mathit{zz}}-I_{h,\mathit{xx}})\mathit{cc} _{\lambda }^{3}\big), \\ K_{011}= &-(m_{f}R_{f}+m_{r}R_{r}+m_{b}z_{b}+m_{h}z_{h}), \\ K_{012}= &-\frac{1}{w}\bigl(\bigl(m_{h}w^{2}+m_{h}(c-x_{h})w-m_{b}x_{b}c-m_{h}x _{h}c\bigr)c_{\lambda } \\ &-(m_{h}z_{h}+m_{f}R_{f}) \mathit{ws}_{\lambda }\bigr), \\ K_{021}= &K_{012}, \\ K_{022}= &-s_{\lambda }K_{012}. \end{aligned}$$