An Optimal Handling Bicycle

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

BMD 2016

Milwaukee, Wisconsin, September 21-23

Bicycle Designers' Holy Grail

How can a vehicle designer decide whether changes to the vehicle's geometry, layout, components, or control system are better or worse for the human's intended use of the vehicle?

Our Goal

Develop an analytical method to simultaneously predict the human's perception of handling and the human's ability to perform adequately.

What is handling?

General definition:

The ease and precision with which the human can complete a given control task.

ease is a measure of both human perception of ease and physiological measures of ease

precision is the objective measurements to perform specific tasks

Our Method

1) The Plant: Construct a linear open loop vehicle/rider model

2) The Controller: Design a human-like compensatory feedback control structure

3) Choose the Gains: Given the plant's parameter values, choose controller parameter values for human-like control.

4) Handling Quality Metric: Estimate the activity in the inner roll tracking loop for handling predictions.

Hess, Ronald, Jason K. Moore, and Mont Hubbard. “Modeling the Manually Controlled Bicycle.” IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 42, no. 3 (May 2012): 545–57. doi:10.1109/TSMCA.2011.2164244.

1) Plant

The plant describes the open loop dynamics of the system

Our plant is defined as:

open loop vehicle dynamics of the bicycle combined with open loop biomechanical dynamics of the rider

Inputs: Rider Compensation Error

Outputs: Vehicle/Rider States

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

  1. Control is compensatory in nature with no feed-forward or preview elements.
  2. 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.
  3. The input to the bicycle model is steer torque, $T_\delta$ (as opposed to steer angle, $\delta$).


  1. The closed roll rate loop should exhibit similarities to the rate feedback loops found in more general laboratory tracking tasks of a human operator.
  2. The system should be simple. In our case, only simple gains are needed to stabilize the system and close all the loops.
  3. We should see evidence of the crossover model in the roll, heading, and lateral deviation loops.

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.

Inner Loops

The three primary sequential feedback loops required to command a roll angle.

Outer Loops

The outer two sequential feedback loops required to command heading and front wheel lateral deviation.

3) Gain Selection

  • Inner two gains are selected such that steer and roll loops each have pole pairs with a 0.15 damping ratio.
  • Outer gains are selected such that the crossover model holds starting at 2 rad/s.

Closed Loop Simulations

The controller is robust for a wide variety of plant values and ensures that task performance is a constant.

Controller Validation Experiments

Controller Validation Results

The controller captures the linear portion of control

Moore, Jason K. “Human Control of a Bicycle.” Doctor of Philosophy, University of California, 2012.

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.

Previous Aircraft Findings

Hess, R. A. “Simplified Approach for Modelling Pilot Pursuit Control Behavior in Multi-Loop Flight Control Tasks.” Proceedings of the Institute of Mechanical Engineers, Part G., Journal of Aerospace Engineering 220, no. 2 (April 2006): 85–102.

Comparison of common bicycles

Hess, Ronald, Jason K. Moore, and Mont Hubbard. “Modeling the Manually Controlled Bicycle.” IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 42, no. 3 (May 2012): 545–57. doi:10.1109/TSMCA.2011.2164244.


  • The handling quality metric decreases with speed all of the investigated bicycle designs.
  • Variations in physical parameters affect handling in different relative amounts depending on speed.

Summary of HQM Method

  1. The human selects their "gains" such that they can perform tasks as good as possible.
  2. The activity in the roll rate feedback loop indicates how well the vehicle handles.

We now have a single value, max(HQM), that predicts handling.

What is bad handling?

So, I built this bicycle this past year...

How does Falkor Handle?

Falkor's Eigenvalues

Optimal Handling?

Can we discover the set of physical parameters that minimize the handling quality metric?

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 Bicycle

Handling and Stability

Does this bicycle exist?

Optimal Bicycle Simulation

Optimal Bicycles at Various Speeds


  • There is evidence that our handling quality metric correlates to actual handling.
  • In general, handling is a strong function of speed regardless of the particular physical design.
  • The handling of typical safety-style bicycles is a strong function of physical parameters but only at low speeds.
  • Unitutive geometry results from the optimal handling based design process.
  • We saw little evidence that open loop stability of the plant correlates with handling.

More info