## Monday, October 5, 2009

### The G-G Diagram Applied To Bicycle Racing

In the midst of our world of possibilities for data analysis in cycling performance, I was wondering if we could consider extending our scope a bit to other things.

As we speak, more models of GPS systems, power meters and heart rate monitors are being designed and released with are basically just different answers to the same old questions : "what's my speed", "what's my power", "what's my heart rate".

But what about other fundamental questions and problems imposed by bike racing? How could we explore and quantify those parameters?

As an illustration, one topic for discussion rarely brought to the table in our daily techno babble is that of acceleration and its impact on safety. Acceleration is an elementary concept in physics and we all know how important it can be towards team race strategies we see today in cycling. Is it getting the importance it may deserve from an analysis standpoint? I'm not sure.

The question is : Could an acceleration analysis be helpful to the racing cyclist and how?

Consider a human-bike system in an Individual Time Trial, a bike race against the clock on a circuit of predetermined length and design. Often in a race of professional caliber such as this, the standard deviation of the data set of results from the top 5 placers is mere seconds.

While time trials are usually steady power output races, if we considered a very technical race course with lots of curves and S-bends, we could say that a majority of the racer's effort is concentrated towards finding the optimum line between the bends while controlling the bike's speed through braking and acceleration. Miscalculations here could cost seconds and a possible podium spot.

Consider two racers, A and B, who chose different paths around a section of this hypothetical race course. Let's also say that they were riding bicycles of the same design, with the same handling qualities.
Fig 1 : Paths of two cyclists in a section of an ITT course. COG = center of gravity.

Who emerged faster at the right end of this section? Racer A or B?

Of course, we wouldn't know because we don't have enough information, you would say. The information missing here is that of the cyclists' acceleration.

To maintain a curved path on the ground, any vehicle, be it a bicycle or a Formula 1 race car, must be moving sideways as well as forward. Hence, there are two components to the bicycle's acceleration. They are :

1. Lateral or centripetal acceleration (LA), whose vector points towards the center of curvature of the road. A bicycle turns because of applied lateral tire forces. LA is affected by tire-road friction characteristics, angle of lean, square of the speed of the bike and radius of the curve, R. If lean is too large (i.e. rider tilts too much into the circle), centripetal force will be too much and the bike will start turning into a circle with radius smaller than R. If lean is not large enough, there won't be sufficient force to keep the bike on the circle and the bike will veer off, turning in a circle with radius larger than R.

2. Tangential or longitudinal acceleration (TA), whose vector points in the fore-aft direction of the bike rider. It is decided by pedal torque, aerodynamic drag forces, and traction limit of the tires.

If m is mass of bicycle-rider system, v its speed along the course, R the radius of curvature of the curve, t is time and g is the acceleration due to earth's gravity, then the above two are defined mathematically in terms of g-force as :

The vectors of these components and their resultant roughly look like this :

Fig 2 : The tangential and lateral vectors of acceleration represented graphically. A reversal of lateral acceleration vector (purple) signifies reversal of direction while a reversal in tangential acceleration vector (red) signified reversal of speed with time.

If we attached perpendicularly oriented accelerometers at the center of mass of the rider-bike system for the 2 riders in Fig 1 , and if we captured lateral acceleration (LA) and tangential acceleration (TA) through a data recording system, the data points when plotted on a graph could look like this (shown just for illustration, not to be taken for granted).

Fig 3 : This plot shows an example g-g diagram (a composite of friction circles for both wheels) for two bicycle riders on the same section of the race course on the same bicycle. It consists of a forward acceleration, turning and braking regions. Data points for each cyclist is shown in red and blue. A rough boundary envelopes these points for both riders. It must be noted that the g-g envelope/boundary for each of the cyclists is not fixed and depends on the bicycle, maximum tire friction force, human skill level and environmental conditions imposed on tire-road contact. This is the performance envelope for the bicycle-rider system. Outside this safe envelope, racing a bicycle could be dangerous.

This plot is called a g-g diagram. The concept was described extensively by aerospace engineer and race car driver Will Milliken Jr.

The difference in riding techniques of the two riders resulted in two different maneuverability boundaries, which can also be looked at as the maximum potential of the rider-bike system in any technical section of the race course for the race conditions. A given individual can only generate limited g's of acceleration to get up to speed. Theoretical limits of deceleration are on the order of 0.5g for a crouched rider on level ground before a person flies over the handlebars. If they are riding two different bicycles with different handling qualities and tires, the g-g boundaries will be different in this case too.

The ultimate limit of the g-g boundaries is imposed by the acceleration capability of the bicycle, which is primarily determined by the grip between the tire and road surface. This is represented in the g-g diagram by the outer oval shaped boundary.

When a racer sits down and studies his g-g performance, he could get a graphical picture of how he utilized the components of acceleration at specific sites on the course and how his choice of equipment may have cost him.

With appropriate software, answers could be generated to questions such as : What's the range of my accelerations? Which direction did I spend more time cornering to? Did I decelerate too much before the sharp curves? Did I accelerate optimally after the apex? How much emphasis did I place on acceleration and deceleration? Could I have changed the ratios of these accelerations by riding differently and emphasizing various body movements? How would these have affected or improved my course times at the end? What really caused my wipe out at that sharp bend and was it related to the lean angle and speed with which I faced that bend? How does all this change with a change in my bicycle tires? Or bicycle design.

How may this be specifically applied to improve performance? I have some thoughts :

1) The tire and the bicycle could be engineered to widen the g-g boundary as much as possible throughout an expected range of operating conditions (load, surface, temperature) without bringing about potentially dangerous modes of oscillatory motion.

2) The rider could train and improve his skill level to ride and exploit these maximal g-g limits of his machine.

Perhaps knowing the performance envelope of the bike being ridden for specific operating conditions may also empower the cyclist with a feeling for when he can safely take risks to win a race.

Take a look at the following two videos. One shows the capturing of the acceleration vectors on a g-g circle for a remotely controlled toy car. The one below it shows what looks to be a real time friction circle generation from a computer simulation.

Its inspiring to watch these data recording and software applications. Perhaps we could have a neat cyclocomputer in the future that could show the bicycle's g-g diagram in real time, if its practicalities have been established. Maybe cycling commentators will start talking about g-g diagrams and other cool things in future race telecasts. Who knows. What do you think?

The G-G Diagram
Rate Of Cycling Uphill Explained
Wild Ideas For New Cycling Products Part 1
Wild Ideas For New Cycling Products Part 2

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Darryl (Cape Town SA) said...

What I enjoy most is catching up with how you take cycling to a whole new level.

Anonymous said...

"Maybe cycling commentators will start talking about g-g diagrams and other cool things in future race telecasts. Who knows. What do you think?"

Even as it stands now, popular race presenters do such a shabby job of discussing technical details about races. I don't imagine them doing a better job, leave alone expanding on the topics for future broadcasts.

Todd said...

This begs me to ask the question why we don't have handlebar mounted computers that can measure acceleration. Is it difficult to produce? Two to three decades of design still hasn't seen the arrival of this feature.

John B. said...

Interesting ideas! Thank you.

crispy said...

I don't know if there's a whole lot to be gained from applying these analytical techniques to cycling. Most of our riding is done under very low accelerations in all directions.

I think you could get most of the same benefits from just thinking to yourself, "What can I do differently to get away with turning my bicycle less?"

You could argue that knowing what your cornering force is would allow you to adjust how tightly you take corners, but a) max. cornering force is highly dependent on tire traction, which can change in an instant, and b) maximum cornering force is really limited by the bicycle's lean angle, so an inclinometer would actually be more useful here, because you know your max. lean angle is roughly 45 degrees.