Simulation of Riding a Bicycle Using Simulink

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Training to ride a bicycle in a race requires a rider to maintain different cadences for the give situation. One situation where most riders try to maintain a cadence is hill climbing. In order to train for hill climbing a rider needs to have hills to climb. If the rider lives in an area without hills then training for hills becomes more difficult.

To help a rider train for hill climbing in areas without hills a device that is built into a bike to simulate hills is proposed.  To aid in the design of such a device a simulation of a bicycle was built in Simulink. A fuzzy logic controller was designed to control the cadence through the manipulation of the applied force. Using another fuzzy logic controller a gear shifting was attempted. The simulation data about how a bicycle performs was generated with good accuracy.

The cadence controller was able to control the cadence with little to no overshoot and a small about of steady state error. The gear shifting system caused Simulink to fail due to the singularity created when the gear changed. Overall the bicycle simulation could be used to develop a hill simulation device. The addition of a gear selection system would be beneficial to the testing of a hill simulation device because it would allow the device to be tested during gear changes.


Bicycle Force Transmission:

For a bicycle the force applied by the rider is transmitted to the rear wheel through the chain, gears and crank.  A basic representation of a bicycle power transmission system . Where,  is the length of the crank, is the radius of the front gear,  is the radius of the rear gear, and  is the radius of the rear wheel.  A list of all variables and notation .  A force is applied by the rider on a petal connected to a crank.

Force Transmission System of a Bicycle.

Force Transmission System of a Bicycle.

Crank Force Approximation:

The way the rider applies force to the pedal was measured by (Okajima, 1983).  The directions and magnitudes of the forces are very complicated to model.  It shows the forces on the pedal as well as foot and leg position as the crank goes around one revolution. The forces appear to be cyclical, maximum when the crank is pointing forward and minimum when the crank points backward.  The force perpendicular to the crank appears to be at a maximum when the crank is horizontal.

Pedal Forces.

Pedal Forces.


Bike Simulation Without Control:

Entire Model

The equations given in Chapter 2 were used to build a model in Simulink of a bicycle.  It  shows the resulting model.  The various blocks in the model are color coded . The model consists of inputs, subsystems, and blocks which together perform the calculations needed to model the bike.  The inputs route to blocks or subsystems that make calculations and then send the results to the next set of blocks or subsystems and then finally fed back for the next iteration.  The subsystems are sets of blocks grouped together because they perform a particular calculation and are used to remove clutter in the main window.

The Inputs in the Model Relating to the bike and Ride.

The Inputs in the Model Relating to the bike and Ride.

Bike Simulation With Cadence Control:

Entire Model with Cadence Controller

The bicycle simulation with cadence (crank RPM) control .The addition of the cadence controller is on the left side of the figure.  The right side of the figure is the bike simulation described in the previous sections.

Bike Simulation with Gear Shifting:

The bicycle simulation with gear shifting . The gear shifting controller is on the left side of the figure.  The right side of the figure is the bike simulation.  In this version of the model the power calculation subsystem has not been added due to problems with the model and gear shifting.


Bike Simulation without Control:

To verify that the bike simulation is correctly representing a bicycle a recreation of a chart provided by Wilson,  was made using data generated by the bicycle simulation.  The chart was made using the data. and plugging it into equation 2‑21.  The ground was level so there was no slope resistance and there was no wind so the aerodynamic force was only due to velocity.

Power Curve From P.140

Power Curve From P.140

Bike Simulation with Cadence Controlled Through Force:

A fuzzy logic controller was used to adjust the force applied by the rider to maintain a preferred cadence.  It shows the cadence as the simulation runs for a gear ratio of 0.310 (front gear = 42 teeth and rear gear = 13 teeth).  The cadence levels off as time progresses due to the aerodynamic force matching the propulsion force when the velocity becomes high enough.  In the cadence stays at a reasonable value, something a rider could possibly sustain for a short period of time.  However, setting the gear ratio to 1.45 (front gear = 22 teeth and rear gear = 32 teeth) the cadence climbs to a level not achievable by a human and only after a few seconds and reaches over a 1000 rpm.


To complete the verification of the bicycle model physical testing need to be done to verify if the applied force translates to the velocity calculated by the simulation.  The setup for the test would be to outfit a bike with a speedometer and a force sensor on the pedal or the crank.  Recording the speed and the force as the rider rides in a particular gear so this information can be plotted.  The same setup for the bike and rider can be inputted into the model and the results plotted.  Comparing the two plots would reveal if the model correctly makes this calculation.


A simulation of a bicycle was created using the Simulink simulation environment. Using the simulation data of how a bicycle performs can be generated with good accuracy.  The rate the crank turns can be controlled with a fuzzy logic style controller that adjusts the force applied to the crank.  Using a fuzzy logic controller a gear shifting was attempted, but did not work due to errors it caused in Simulink.

The bicycle simulation in its current level of development can be used as a platform to test design concepts of a hill climbing simulation device. Controlling the cadence through the rider force will allow the testing to be done at the revolutions per minute that the device will be used.  Valuable design and control data can be learned by running the device with the bicycle simulation.  However, without the gear shifting how the hill climbing simulation device responds to a change in gear cannot be tested.

Source: California State University
Author: Jason Thomas Parks

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