Bicycle Dynamics

$$q = [\phi \quad \delta]^T$$ $\phi$ : roll angle
$\delta$ : steer angle
$$T = [T_\phi \quad T_\delta]^T$$ $T_\phi$ : roll torque
$T_\delta$ : steer torque
$$\mathbf{M} \ddot{q} + v \mathbf{C}_1 \dot{q} + [g \mathbf{K}_0 + v^2 \mathbf{K}_2] q = T$$

Bicycle Control Observations

Rider Biomechanics

Steer-Roll-Yaw

Pedaling

Lateral Knee, Knee Bounce

Physical Parameter Measurement

pypi.python.org/pypi/BicycleParameters
pypi.python.org/pypi/yeadon

Instrumented Bicycle

Perturbation Experiments

Human Control

Eaton 1973

$$Y_p(s)Y_c(s) = \frac{\omega_c e^{-\tau s}}{s}$$

$$G_{nm}(s) = \frac{\omega_{nm}^2}{s^2 + 2 \zeta_{nm} \omega_{nm}s + \omega_{nm}}$$

System Identification

$$\hat{y}(t) = a_1 y(t-1) + \ldots + a_n y(t-n) + b_0 u(t) + \ldots + b_m u(t-m)$$

$$\theta = [a_1, \ldots, a_n, b_0, \ldots, b_m]^T$$ $$\phi(t) = [y(t-1), \ldots, y(t-n), u(t), \ldots, u(t-m)]^T$$

$$\hat{y}(t|\theta) = \phi(t)^T\theta$$

$$Z^N = \{u(1), y(1), \ldots, u(N), y(N)\}$$

$$V_N(\theta, Z^N) = \frac{1}{N} \sum^N_{t=1}(y(t)-\hat{y}(t|\theta))^2$$

$$\hat{\theta} = min_\theta V_N(\theta, Z^N)$$

$$\dot{x}(t) = \mathbf{F}x(t) + \mathbf{G}u(t)\\ \begin{bmatrix} \dot{\phi} \\ \dot{\delta} \\ \ddot{\phi} \\ \ddot{\delta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ a_{\ddot{\phi}\phi} & a_{\ddot{\phi}\delta} & a_{\ddot{\phi}\dot{\phi}} & a_{\ddot{\phi}\dot{\delta}}\\ a_{\ddot{\delta}\phi} & a_{\ddot{\delta}\delta} & a_{\ddot{\delta}\dot{\phi}} & a_{\ddot{\delta}\dot{\delta}} \end{bmatrix} \begin{bmatrix} \phi \\ \delta \\ \dot{\phi} \\ \dot{\delta} \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0\\ b_{\ddot{\phi}T_\delta} & b_{\ddot{\phi}F_{c_l}}\\ b_{\ddot{\delta}T_\delta} & b_{\ddot{\delta}F_{c_l}} \end{bmatrix} \begin{bmatrix} T_\delta\\ F_{c_l} \end{bmatrix}$$

$$\mathbf{M} \ddot{q} + v \mathbf{C}_1 \dot{q} + [g \mathbf{K}_0 + v^2 \mathbf{K}_2] q = T + H F$$

$$\Gamma \Theta = Y$$

$$\hat{\Theta} = [\Gamma^T\Gamma]^{-1}\Gamma^T Y$$

$$\frac{\delta}{\delta_c}(s)$$

$$\frac{\dot{\phi}}{\dot{\phi}_c}(s)$$

$$\frac{\phi}{\phi_e}(s)$$

$$\frac{\psi}{\psi_e}(s)$$

$$\frac{y_Q}{y_{Qe}}(s)$$

$$\frac{y_Q}{y_{Qc}}(s)$$

$$\frac{y_Q}{F}(s)$$

Wrap Up

• The bicycle is seemingly simple but is a very non-trivial dynamic system with both speed dependent stability and non-minimum phase behavior.
• The rider can control the bicycle using a multitude of actuators, but generally chooses to control the bicycle through the steer due to it having the most authority.

• The bicycle/rider can be modeled by a 4th order first principles model and is generally good at basic prediction, but the classic models have been shown to be deficient in the correct prediction of the steer dynamics.
• The human control can be modeled by a simple five gain feedback structure with a simple neuromuscular model that follows the dictates of the crossover model.

Jason K. Moore (moorepants@gmail.com)

moorepants on Github, G+, Twitter, Linkedn

Project Website: biosport.ucdavis.edu

dissertation: moorepants.github.com/dissertation

source code: github.com/moorepants

This work was supported by the National Science Foundation under Grant No 0928339.