Human Control of a Bicycle
|
Jason K. Moore
May 10, 2012
|
$$
q = [\phi \quad \delta]^T
$$
\(\phi\) : 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
Lateral Knee, Knee Bounce
Physical Parameter Measurement
Eaton 1973
$$
Y_p(s)Y_c(s) = \frac{\omega_c e^{-\tau s}}{s}
$$
$$
G_{nm}(s) = \frac{\omega_{nm}^2}{s^2 + 2 \zeta_{nm} \omega_{nm}s + \omega_{nm}}
$$
$$
\hat{y}(t) = a_1 y(t-1) + \ldots + a_n y(t-n) + b_0 u(t) + \ldots + b_m u(t-m)
$$
$$
\theta = [a_1, \ldots, a_n, b_0, \ldots, b_m]^T
$$
$$
\phi(t) = [y(t-1), \ldots, y(t-n), u(t), \ldots, u(t-m)]^T
$$
$$
\hat{y}(t|\theta) = \phi(t)^T\theta
$$
$$
Z^N = \{u(1), y(1), \ldots, u(N), y(N)\}
$$
$$
V_N(\theta, Z^N) = \frac{1}{N} \sum^N_{t=1}(y(t)-\hat{y}(t|\theta))^2
$$
$$
\hat{\theta} = min_\theta V_N(\theta, Z^N)
$$
$$
\dot{x}(t) =
\mathbf{F}x(t) + \mathbf{G}u(t)\\
\begin{bmatrix}
\dot{\phi} \\
\dot{\delta} \\
\ddot{\phi} \\
\ddot{\delta}
\end{bmatrix}
=
\begin{bmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
a_{\ddot{\phi}\phi} & a_{\ddot{\phi}\delta} &
a_{\ddot{\phi}\dot{\phi}} & a_{\ddot{\phi}\dot{\delta}}\\
a_{\ddot{\delta}\phi} & a_{\ddot{\delta}\delta} &
a_{\ddot{\delta}\dot{\phi}} & a_{\ddot{\delta}\dot{\delta}}
\end{bmatrix}
\begin{bmatrix}
\phi \\
\delta \\
\dot{\phi} \\
\dot{\delta}
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 \\
0 & 0\\
b_{\ddot{\phi}T_\delta} & b_{\ddot{\phi}F_{c_l}}\\
b_{\ddot{\delta}T_\delta} & b_{\ddot{\delta}F_{c_l}}
\end{bmatrix}
\begin{bmatrix}
T_\delta\\
F_{c_l}
\end{bmatrix}
$$
`
$$
\mathbf{M} \ddot{q} + v \mathbf{C}_1 \dot{q} + [g \mathbf{K}_0 + v^2 \mathbf{K}_2] q = T + H F
$$
$$
\Gamma \Theta = Y
$$
$$
\hat{\Theta} = [\Gamma^T\Gamma]^{-1}\Gamma^T Y
$$
$$
\frac{\delta}{\delta_c}(s)
$$
$$
\frac{\dot{\phi}}{\dot{\phi}_c}(s)
$$
$$
\frac{\phi}{\phi_e}(s)
$$
$$
\frac{\psi}{\psi_e}(s)
$$
$$
\frac{y_Q}{y_{Qe}}(s)
$$
$$
\frac{y_Q}{y_{Qc}}(s)
$$
$$
\frac{y_Q}{F}(s)
$$
-
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.
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