Identification of the open loop dynamics of a manually controlled
bicycle-rider system
Jason K. Moore and Mont Hubbard
University of California, Davis
November 11, 2013
No rider or rigid rider
Stable?
Sometimes
Primary Inputs
Torque/forces between the vehicle rigid bodies and/or ground, e.g. steer
torque, ground reaction forces
External torques and forces, e.g. gust of wind
Primary Outputs
Functions of the vehicle states and inputs, e.g. steer angle
Non-rigid rider
Stable?
Probably never
Primary Inputs
Torque/forces between the vehicle and rider rigid bodies and/or ground,
e.g. Muscle forces (or joint torques), ground reaction forces
External torques and forces, e.g gust of wind
Primary Outputs
Functions of the vehicle and rider states and inputs, e.g. elbow flexion
Do our models predict reality?
Validation/Identification of open loop rigid/no rider systems
CALSPAN
Rice, Roland, Manning, Lunch, Kunkel, Milliken, Davis, Cassidy; 1970-176
Man-Machine Dynamics in the Stabilization of Single-Track Vehicles
Eaton, D. J.; 1973
Experimental Validation of a Model for the Motion of an Uncontrolled
Bicycle
Kooijman, J. D. G; 2007/8/9
Limitations
- Tests require open loop stability
- Limited speed range
- Instability leads to short data collection
- Limited bandwidth inputs
- Only identify one mode at a time
Validation/Identification vehicle + non-rigid rider
CALSPAN
Rice, Roland, Manning, Lunch, Kunkel, Milliken, Davis, Cassidy; 1970-76
Lateral dynamics of an offroad motorcycle by system identification
James, S. R..; 2002
Experimental Study of Motorcycle Transfer Functions for Evaluating
Handling
Biral, F., Bortoluzzi, D., Cossalter, V., and Lio, M.; 2003
Comparison of experimental data to a model for bicycle steady-state
turning
Cain, S. M. and Perkins, N. C.; 2012
Limitations
- Plant includes passive rider dynamics
- Plant may include neuromuscular dynamics
- Manual excitation: human operating bandwidth
- Closed loop identification
There are very few validations of bicycle/motorcycle models!
Some are questionable, some have decent results.
Bicycle validation is very weak.
Comprehensive data at many speeds and input frequencies is missing.
What is our plant?
Models
From first principles
Whipple model
Whipple model + arm inertial effects
Both linearized about the nominal configuration
Identified
4th order grey box model based on the Whipple model
|
|
Instrumented bicycle
Data
~1.7 million time samples from each of about 30 sensors sampled at 200
hertz (about 2.4 hours of real time)
Grey Box Model Structure
Canonical linear form of the Whipple bicycle model:
$$
\mathbf{M} \ddot{q}(t) + v \mathbf{C}_1 \dot{q}(t) + [g \mathbf{K}_0 + v^2
\mathbf{K}_2] q(t) = T(t) + H F(t)
$$
Coordinates:
$$q(t) = [\phi(t) \quad \delta(t)]^T$$
Torques:
$$T(t) = [0 \quad T_\delta(t)]^T$$
Lateral Force:
$$F(t)$$
Known and Unknown
Measured time varying values:
$$q(t), \dot{q}(t), v(t), T(t), F(t)$$
Estimated (numerical differentiation and filtering):
$$\ddot{q}(t)$$
Measured constants:
|
$$
\mathbf{M} =
\begin{bmatrix}
\underline{M_{\phi\phi}} & M_{\phi\delta} \\
M_{\delta\phi} & M_{\delta\delta}
\end{bmatrix}
$$
|
$$
\mathbf{C}_1 =
\begin{bmatrix}
0 & C_{1\phi\delta} \\
C_{1\delta\phi} & C_{1\delta\delta} \\
\end{bmatrix}
$$
|
$$
\mathbf{K}_0 =
\begin{bmatrix}
\underline{K_{0\phi\phi}} & K_{0\phi\delta} \\
K_{0\delta\phi} & K_{0\delta\delta}
\end{bmatrix}
$$
|
|
$$
\mathbf{K}_2 =
\begin{bmatrix}
0 & K_{2\phi\delta} \\
0 & K_{2\delta\delta}
\end{bmatrix}
$$
|
$$
H =
\begin{bmatrix}
\underline{H_{\phi F}} \\
H_{\delta F}
\end{bmatrix}
$$
|
$$\underline{g}$$
|
Linear Least Squares
Two equations
$$
\mathbf{\Gamma}_{\phi} \Theta_{\phi} = Y_{\phi}, \quad
\mathbf{\Gamma}_{\delta} \Theta_{\delta} = Y_{\delta}
$$
Solution
$$
\hat{\Theta}_{\phi,\delta} = [\mathbf{\Gamma}_{\phi,\delta}^T
\mathbf{\Gamma}_{\phi,\delta}]^{-1} \mathbf{\Gamma}_{\phi,\delta}^T
Y_{\phi,\delta}
$$
Variance Accounted For
$$
\textrm{VAF}_{\phi,\delta} = 1 - \frac{\vert \vert
\mathbf{\Gamma}_{\phi,\delta}\hat{\Theta}_{\phi,\delta} - Y_{\phi,\delta} \vert \vert}
{\vert \vert Y_{\phi,\delta} - \bar{Y}_{\phi,\delta} \vert \vert}
$$
Linear Least Squares
$$
\small{
\begin{align}
&\begin{bmatrix}
\ddot{\delta}(1) &
v(1) \dot{\delta}(1) &
g \delta(1) \\
\vdots & \vdots & \vdots\\
\ddot{\delta}(N) &
v(N) \dot{\delta}(N) &
g \delta(N) \\
\end{bmatrix}
\begin{bmatrix}
M_{\phi\delta} \\
C_{1\phi\delta} \\
K_{0\phi\delta}
\end{bmatrix}\\
&=
\begin{bmatrix}
H_{\phi F} F(1)
- M_{\phi\phi} \ddot{\phi}(1)
- C_{1\phi\phi} v(1) \dot{\phi}(1)
- K_{0\phi\phi} g \phi(1)
- K_{2\phi\phi} v(1)^2 \phi(1)
- K_{2\phi\delta} v(1)^2 \delta(1) \\
\vdots\\
H_{\phi F} F(N)
- M_{\phi\phi} \ddot{\phi}(N)
- C_{1\phi\phi} v(N) \dot{\phi}(N)
- K_{0\phi\phi} g \phi(N)
- K_{2\phi\phi} v(N)^2 \phi(N)
- K_{2\phi\delta} v(N)^2 \delta(N) \\
\end{bmatrix} \nonumber
\end{align}
}
$$
LSS Variance Accounted For
Treadmill Simulation
Flat Ground Simulation
Simulation Variance Accounted For
Linear Model Comparison
Linear Model Comparison
Linear Model Comparison
Conclusions
Good stuff
- A fourth order model can predict the measured motion.
- A single model predicts well for all riders and both environments
- The Whipple model captures the change in motion, but not the torque to
output gain
- A simple gain could improve the Whipple model, could be represented by
a static scrub torque
- The addition of arm inertia improves outputs, except in the roll
angle
Questionable Stuff
- Data driven, not clear connections to first principles, e.g. trail
prediction
- The VAF from the linear fit doesn't give same results as simulation
VAF
- All models predict significant roll torques even if there were no
applied lateral forces.
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