An Optimal Handling Bicycle¶

Jason K. Moore, Mont Hubbard, Ronald A. Hess
University of California, Davis

BMD 2016

Milwaukee, Wisconsin, September 21-23

https://www.popchartlab.com/products/the-evolution-of-bicycles

Whipple Bicycle Model¶

Simplest bicycle model that exhibits both non-minimum phase behavior and open loop stability.

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

Meijaard, J. P., Jim M. Papadopoulos, Andy Ruina, and A. L. Schwab. “Linearized Dynamics Equations for the Balance and Steer of a Bicycle: A Benchmark and Review.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (August 2007): 1955–82.

Simple Neuromuscular Model¶

Simple representation of limb dynamics as a second order underdamped system with sufficient bandwidth.

Input: Rider Compensation Error
Output: Steer Torque

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

where $\omega_{nm}=30 \textrm{ rad/s} \\ \zeta_{nm}=0.707$

2) Controller Design Requirements¶

Control is compensatory in nature with no feed-forward or preview elements.
Roll stabilization is the primary task, with path following addressed in the outer loops. The system should be stable in roll before closing the path following loops.
The input to the bicycle model is steer torque, $T_\delta$ (as opposed to steer angle, $\delta$ ).

Crossover Model¶

The human adopts a control behavior such that the open loop system behaves as an integrator at a desired crossover frequency with a neuromuscular delay.

$G_{human}G_{plant}(s) = \frac{\omega_c e^{-\tau_e s}}{s}$

The crossover frequency is correlated to the human's agressiveness.

McRuer, D. T., and E. Krendel. “Mathematical Models of Human Pilot Behavior.” Advisory Group on Aerospace Research and Development, January 1974.

4) Handling Quality Metric¶

rate control of any "primary vehicle response variable" is of fundamental importance from the standpoint of perceived vehicle handling qualities

$HQM = \left|\frac{\dot{\phi}(s)}{\phi_c(s)}\right| \frac{1}{|k_\phi|}$

The HQM defined above is a simple task independent measure that reflects the power in the human's roll rate feedback loop.

Optimal Problem Formulation¶

$\begin{aligned} & \underset{p}{\text{minimize}} & & max(\textrm{HQM}(p)) \\ & \text{subject to} & & \\ & & & -\infty < c < \infty \\ & & & \frac{m_H x_H + m_B x_B}{m_H + m_B} \leq w < \infty \\ & & & -\pi / 2 \leq \lambda \leq \pi/2 \\ & & & 0 < r_F < \infty \end{aligned}$ $p = [c, w, r_F, \lambda]$

An Optimal Handling Bicycle¶

Bicycle Designers' Holy Grail¶

Our Goal¶

What is handling?¶

Our Method¶

1) Plant¶

Our plant is defined as:¶

Whipple Bicycle Model¶

Simple Neuromuscular Model¶

2) Controller Design Requirements¶

Continued...¶

Crossover Model¶

Inner Loops¶

Outer Loops¶

3) Gain Selection¶

Closed Loop Simulations¶

Controller Validation Experiments¶

Controller Validation Results¶

4) Handling Quality Metric¶

Previous Aircraft Findings¶

Comparison of common bicycles¶

Realizations¶

Summary of HQM Method¶

What is bad handling?¶

How does Falkor Handle?¶

Falkor's Eigenvalues¶

Optimal Handling?¶

Optimal Problem Formulation¶

An Optimal Bicycle¶

Handling and Stability¶

Does this bicycle exist?¶

Optimal Bicycle Simulation¶

Optimal Bicycles at Various Speeds¶

Conclusions¶

More info¶