Bicycle Dynamics

Delft University of Technology 

Intro: This work started in the Fall of 2002 when I came to Cornell to have my sabbatical leave with Andy Ruina. His line of research began in 1985 when Jim Papadopoulos came to Cornell to work with him (or visa versa). Take a look at this TEDxDelft talk for an intro on our work:

TEDxDelft talk

First take a look at the Cornell Yellow Bicycle Demo website to see what is so special about uncontrolled bicycles. Highly unstable at rest but very stable at moderate forward speed!  Or look at this VIDEO: Bicycle Stability Demonstration (MP4) (3 MB).

We would like to think that this is the definitive review paper on the linearized equations of the motion for a bicycle:

J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007 ``Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,'' Proceedings of the Royal Society A 463:1955-1982. doi:10.1098/rspa.2007.1857,  or  preprint+ESM pdf(578k).

Please take a look at the Benchmark Bicycle website.

Media Coverage: This paper has drawn some attention in the media. Here you will find some of the media coverage.

A Matlab Graphical User Interface (GUI) to the calculation of the stability of a bicycle, the implementation of the linearized equations of motion from the 2007 PRS-A paper, is JBike6, free download here.

And in this paper we show that neither front-wheel spin angular momentum nor trail are necessary for self-stability of an uncontrolled bicycle:

J. D. G. Kooijman, J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, and A. L. Schwab, "A bicycle can be self-stable without gyroscopic or caster effects", Science 15 April 2011: 332(6027), 339-342. [DOI:10.1126/science.1201959]

Preprint, supplementary text, videos and photos are available at the TMS Bicycle website.

An international three-day symposium on Bicycle and Motorcycle Dynamics, BMD2010, has been held at Delft University of Technology, Delft, The Netherlands, October 20-22, 2010. The aim of this symposium is to bring together leading scientists and researchers in the field of bicycle and motorcycle dynamics and control, in a broad sense.

History of bicycle steer and dynamics equations.

Even now, after we’ve been building them for 100 years, it’s very difficult to understand just why a bicycle works - it’s even difficult to formulate it as a mathematical problem.” — Freeman Dyson interviewed by Stewart Brand in Wired News, February 1998.

Over the past 140 years, scores of other people have studied bicycle dynamics, either for a dissertation, a hobby or sometimes as part of a life’s work on vehicles. This sparse and varied research on the dynamics of bicycles modelled as linked rigid bodies was initially reviewed in Hand (1988). A detailed history of bicycle dynamics studies with a large bibliography, which we update as we go along, can be found here.

    Hand, R. S. 1988 ''Comparisons and stability analysis of linearized equations of motion for a basic bicycle model.'' MSc thesis, Cornell University, Ithaca, NY.

Some hard to find papers:
- B. D. Herfkens. De stabiliteit van het rijwiel (”The Stability of the bicycle”). Technical Report S-98-247-50-10-’49, Instituut voor rijwielontwikkeling, ’sGravenhage, September 1949. (Translated by J.D.G. Kooijman 2006).
- A. J. R. Doyle, The skill of bicycle riding, PhD thesis, University of Sheffield, UK, 1987.
- A. J. R. Doyle, The essential contribution to bicycle riding, in J. Patrick and K. Duncan eds., Training, human decision making, and control, Elsevier Science Publisher BV North Holland,  pp.351-370, 1988.


Nov 5, 2012: TED talk by Arend Schwab at TEDxDelft2012 on "Why bicycles do not fall":


Feb 8, 2012: Andrew Dressel is finishing his bicycle tire testrig, see:

Nov 30, 2011: Peter de Lange successfully defended his Engineering Dynamics MSc thesis and graduate Cum Laude:

P.D.L. de Lange, Rider Control Identification in Bicycling, a preliminary analysis, MSc thesis, Delft University of Technology, Nov 2011. (pdf 2.3Mb)

Feb 9, 2011: Eric-Jan van den Ouden successfully defended his Engineering Dynamics  MSc thesis:

J.H. van den Ouden, Inventory of bicycle motion for the design of a bicycle simulator, MSc thesis, Delft University of Technology, Jan 2011. (pdf 2.2Mb)

Sep 22, 2010: This month the bicycle lab moved to the new lab space (room 4A-1-04) which is on the first floor, close to our offices. We are very happy with the new lab; more than 80 m2 of light and clean space with lots of room for table-top experimental setups and high visibility from the corridor because of the large glass windows.

May 11, 2010: Marc Evertse successfully defended his MSc thesis:

M.V.C. Evertse, Rider Analysis using a fully instrumented motorcycle, MSc thesis, Delft University of Technology, May 2010. (pdf 1.9Mb)

Aug 19, 2010: After many years of searching and negotiating we are now able to post a selection of the Calspan reports on Bicycle and Motorcycle Dynamics research. Thanks to the effort of Jim Papadopoulos and Doug Milliken!

May 2010 Joep Mutsaerts builds a LEGO Mindstorms NXT based bicycle robot in an assignment for the TUDelft 3mE/PME course Mechatronic System Design II. The lateral stability control is done by the concept of "steer into the undesired fall" and comes down to only one line of code in the controller: SteerMotorVoltage=8*RearFrameLeanRate.

Here you can download the design (NXTbike-GS Design.pdf) and the building instructions (NXTbike_GS_Building_Instructions.pdf).

Feb, 2010 Recently Jason Moore from UCDavis started a google discussion group on single track vehicle dynamics.

July 16, 2009 VPRO Noorderlicht Nieuws made a 5 minute item out of our Bicycle Treadmill experiments. The interview is in Dutch:

Jan 9-11, 2009: For three days we have been doing full motion capture of the rider and the bicycle while riding on the large(3X5 m) VU Amsterdam treadmil. This to investigate rider motions on a bicycle. The Optotrack Certus Motion Capture System with active markers was used (thanks to the VU!) with 31 markers (20 on the rider, 11 on the bicycle) at a sample freq of 100 Hz. The test comprised: normal pedalling, towing, hands-free pedalling, pedalling with line tracking. The belt speed (=forward bicycle speed) was varied from 5 to 30 km/h in steps of 5 km/h and then down in the same manner. At every speed we measured for 60 seconds. Going down we also tried 5,4,3,2,.. km/h until one could not stabilize any more. Every test was done with three riders and two bicycles form Batavus, the Browser and a Stratos.

Here you see Jodi geared-up with 20 markers and a safety harness on the Batavus Stratos riding on the VU treadmill. Below you see a video of Jason on the same bicycle in a pedalling run at 10 km/h and to the right of that a visualisation of the marker motions for that same run. We analyze the large data sets by Principal Component Analysis.
Jason, pedalling, Batavus Stratos, 10 km/h Measured marker motions same run

Oct 28, 2008: Today we did a number of exploratory tests on the big VU treadmill (3X5m) to have a first look at human rider control. The bicycle was a Batavus Browser town bicycle which was prepared by Jodi and Jason and is equipped with sensors for: steer angel, steer rate, rear frame lean and yaw rate, forward speed (2), and pedaling cadence. A video camera was added mounted on the rear frame looking at the rider to identify rider motions relative to the rear frame.

We looked at normal pedaling, towing and pedaling with lateral perturbations in the forward speed range of 5-30 km/h. Here you can find three compilations of the experiments:


Normal Cycling



Cycling and Perturbing


April 16, 2007: Today we put the instrumented bicycle from Jodi' MSc work on the big VU treadmill. We feel more confident about the usage of this treadmill for our bicycle handling experiments if the instrumented bicycle on the treadmill shows the same lateral dynamic behaviour as on the road. It turned out to be rather tricky to launch the bicycle and catch it in the  fall. Jodi got really handy by first launching it from standing aside and then stepping to the rear where he could steer, perturb and catch it. The weave speed  of the bicycle on the road is 4.0 m/s (14.4 km/h) [The bicycle is laterally unstable below the weave s