q = [ϕ δ] T

$q = [\phi \quad \delta]^T$

ϕ $\phi$ : roll angle

δ $\delta$ : steer angle

T = [T ϕ T δ] T

$T = [T_\phi \quad T_\delta]^T$

Tϕ $T_\phi$ : roll torque

Tδ $T_\delta$ : steer torque

M q ¨ + v C 1 q ˙ + [g K 0 + v 2 K 2] q = T

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

Bicycle Control Observations

Lateral Knee, Knee Bounce

Physical Parameter Measurement

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

Instrumented Bicycle

Perturbation Experiments

Eaton 1973

Y p (s) Y c (s) = ω c e - τ s s

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

G n m (s) = ω 2 n m s 2 + 2 ζ n m ω n m s + ω n m

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

System Identification

y ˆ (t) = a 1 y (t - 1) + \dots + a n y (t - n) + b 0 u (t) + \dots + b m u (t - m)

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

θ = [a 1, \dots, a n, b 0, \dots, b m] T

$\theta = [a_1, \ldots, a_n, b_0, \ldots, b_m]^T$

ϕ (t) = [y (t - 1), \dots, y (t - n), u (t), \dots, u (t - m)] T

$\phi(t) = [y(t-1), \ldots, y(t-n), u(t), \ldots, u(t-m)]^T$

y ˆ (t | θ) = ϕ (t) T θ

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

Z N = {u (1), y (1), \dots, u (N), y (N)}

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

V N (θ, Z N) = 1 N \sum t = 1 N (y (t) - y ˆ (t | θ)) 2

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

θ ˆ = m i n θ V N (θ, Z N)

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

x ˙ (t) = F x (t) + G u (t) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ϕ ˙ δ ˙ ϕ ¨ δ ¨ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ 00 a ϕ ¨ ϕ a δ ¨ ϕ 00 a ϕ ¨ δ a δ ¨ δ 10 a ϕ ¨ ϕ ˙ a δ ¨ ϕ ˙ 01 a ϕ ¨ δ ˙ a δ ¨ δ ˙ ⎤ ⎦ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ϕ δ ϕ ˙ δ ˙ ⎤ ⎦ ⎥ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ 00 b ϕ ¨ T δ b δ ¨ T δ 00 b ϕ ¨ F c l b δ ¨ F c l ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ [T δ F c l]

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

M q ¨ + v C 1 q ˙ + [g K 0 + v 2 K 2] q = T + H F

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

Γ Θ = Y

$\Gamma \Theta = Y$

Θ ˆ = [Γ T Γ] - 1 Γ T Y

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

δ δ c (s)

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

ϕ ˙ ϕ ˙ c (s)

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

ϕ ϕ e (s)

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

ψ ψ e (s)

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

y Q y Q e (s)

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

y Q y Q c (s)

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

y Q F (s)

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

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.

Human Control of a Bicycle

Bicycle Dynamics

Bicycle Control Observations

Rider Biomechanics

Physical Parameter Measurement

Instrumented Bicycle

Perturbation Experiments

Human Control

System Identification

Wrap Up