Five-Bar Linkage Model of the Bicycle-Rider System

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This Demonstration models the motion of the bicycle-rider system by a five-bar planar linkage with the crank axis and the hip as fixed joints; the pedal, ankle, and knee are moving joints.


If we take the foot angle as a function of the crank angle, we end up with the crank angle as the only degree of freedom.


Contributed by: Erik Mahieu (July 2012)
Open content licensed under CC BY-NC-SA



This Demonstration shows only the left leg of the rider and the left crank on the bicycle chainring. All angles are measured in the counterclockwise direction relative to the positive axis.

An extensive study of the kinematics of the bicycle-rider system can be found in [1].

An analytical solution for the geometry of the five-bar linkage system can be obtained using the Freudenstein equation [2] .

The relation between the foot angle and the crank angle was developed experimentally by Bolourchi and Hull [3]: .


[1] L. Malfait, G. Storme, and M. Derdeyn. "Comparative Biomechanical Study of Circular and Non-circular Chainrings for Endurance Cycling at Constant Speed." (May 2, 2010)

[2] A. Ghosal, "The Freudenstein Equation: Design of Four-Link Mechanisms," Resonance, 15(8), 2010 pp. 699–710.

[3] F. Bolourchi and M. L. Hull, "Measurement of Rider Induced Loads During Simulated Bicycling," in Conference Proceedings of the 3rd International Symposium on Biomechanics in Sports (ISBS85) Greeley, Colorado (J. Terauds and J. N. Barham, eds.), Konstanz, Germany: International Society of Biomechanics in Sports, 1985.

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