on pedal kickback, and the mistake that misled the industry

Back when I first got interested in bicycle design, I purchased a license for Linkage. It’s a great tool for experimenting with geometry and quickly getting familiar with bike kinematics, or for analysing the characteristics of existing designs. While I never actually designed a single bike using the program, it is clear many people do, and rely on it heavily.

Since I design and analyse my bikes entirely within a single CAD-program, I had to develop my own method for plotting kinematics curves. Most of the time, the results lined up well with what Linkage produced, except for one curve: pedal kickback. That one was always noticeably different. As I never really had any problems with pedal kickback so far, I decided I must have made a mistake, decided to ignored the issue and never tried to solve it…

ENTER a high pivot design with an idler pulley wheel! The holy grail: a rearward axle path, high anti-squat with little to no pedal kickback. It sounded like I could have my cake and eat it too! Unsurprisingly, the bicycle industry jumped on this trend a while ago for all the reasons above.

The M0DI, a high pivot and idler equipped enduro bike.

You may have noticed, but the M0DI has a Pinion gearbox instead of the traditional derailleur setup. This means the torque and speed conversion happens inside the gearbox, rather than at the rear axle, where it would normally depend on which cog of the cassette is being used.
This means that any pedal kickback is actually multiplied by the gear you are in! For the Pinion gearbox, this ranges from 1.82 in the easiest gear to 0.3 for the hardest gear. Knowing this, and seeing the results from my own program, I really wondered how this was going to feel on the trail.
Linkage showed around a 10° of pedal kickback at full travel for a 1:1 transmission ratio, while my calculations gave me just under 21° , twice as much!

And finally the time came to stop postponing the inevitable: let’s see who’s wrong.

The formulas Linkage uses to calculate pedal kickback, are found here. My method involves graphically representing the chain throughout the suspension travel and directly measuring the resulting motion at the pedals. It preserves the same degrees of freedom for the rear wheel as Linkage: the frame remains stationary and horizontally, while the rear wheel is free to roll forward or backward to account for changes in chainstay length.

When I model a single pivot bike in both CAD program and Linkage, they both produce the same pedal kickback result. When an idler pulley is introduced, things start to differ.So I started with a simple model, a design similar to the M0DI in the low anti-squat setting.

Below are 3 mechanically equivalent layouts:
- All crank-to-wheel transmission ratios are 1:1.
- The chain orientation relative to the swingarm is identical in all layouts.

Layout 1: Idler(18T) mounted coaxially with the swingarm pivot.

This layout shows great promise, the pedal kickback is nearly eliminated and even turns negative at a certain point in the travel, theoretically slackening the chain between the rear wheel and the cranks, definitely ridding you of any noticeable feedback.

BUT what if we change the layout to a mechanically equivalent jackshaft version:

Layout 2: Jackshaft, coaxial with the swingarm pivot, with two 18T sprockets.

Or another mechanically equivalent version comprising of a gearbox with a 30/18 ratio:

Layout 3: Gearbox in frame (E.g. Effigear) coaxial with the swingarm pivot, with an 18T output sprocket and internal ratio of 30/18).

When comparing layouts 2 and 3, we see that they both maximally produce 10.9° of pedal kickback. Layout 1, however, appears to magically cancel it out. Interestingly, according to Linkage, all three layouts produce the same anti-squat curve.

Modelling any of the above layouts in my CAD program yields a maximum pedal kickback of 11.4°. We may attribute this slight discrepancy to rounding or numerical tolerances, as modelling the M0DI in both my CAD software and Linkage produces virtually identical results, differing by only 0.04°.

Based on this, I conclude that Linkage's calculation of pedal kickback requires revision and should not be relied upon for designs that include an idler attached to the swingarm and when using an idler coaxially to the swingarm pivot, be sure to select the “jack-shaft” option if you want real pedal kickback results. Furthermore, I’m confident that my own method yields accurate and consistent results across different layouts.

Perhaps the final takeaway is this:

Always check your sources and don’t take anything at face value.

* I’ve contacted the developers of Linkage regarding this issue, and they mentioned they would look into it. I also welcome any feedback or alternative interpretations.