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by Dale Lukas Peterson — last modified Apr 22, 2011 11:50 AM

I'm working on some new plots closer to the origin, I'll put them up when I'm done.  In the mean time, here is a description of the plot.

• x - axis is rear frame lean (phi), positive is a rightward lean
• y - axis is steer (delta), positive is rightward steer
• lean=steer=0 is the upright zero steer equilibrium
• the black curve which passes through  (0,0)  and (0,pi) represents static equilibrium
• the black curves which pass through the (pi/2, 0) and (-pi/2, pi) are the infinite speed steady turning configuration and the zero level curve of rear wheel normal force.  The normal force curve is the one in the interior of the two triangular regions and represents a physical limit if we assume wheels that can be pull down onto the ground.
• the two outer most curves that pass through the four corners of the plot represent configuration limits -- for a given steer, there is a minimum and maximum lean.
• steady turns can occur inside the two triangular regions.
• within these regions, green lines represent level curves of steer torque.  Notice that there are two zero-steer torque level curves
• The solid red and green regions represent stable steady turns.  Green indicates the eigenvalue closest to the j-omega axis is complex, red indicates it is real.

Thoughts
• This plot is generated with the nonlinear equations of motion of the Whipple bicycle model, using the numerical values of the physical parameters presented in the Meijaard et al (2007).
• The bottom right triangular region represents turns in which the bicycle is leaned and steer to the right
• The upper left triangular region represents turns in which the bicycle is leaned to the left and the handle bars are steered to the right but steer past approximately pi/2.
• Notice the following:
• There are turns very close to the origin (i.e., near the region of typical bicycle riding) which are on either side of the zero steer torque level curve.  The ones on the negative side of the curve are stable, and I think this is what Jim is getting at.  I think however his message is muddled by using stability in a loose sense.  There are turns very close to the origin, on the positive side of the zero-steer torque curve that are unstable.
• Stability of a set of ODE's is fundamentally a property of an equilibrium point.  The nonlinear bicycle equations of motion for the Whipple model have an infinite number of equilibrium points.  To say that the model is "stable" leaves out the critical piece of information, namely, what equilibrium point is being referred to.
• To use the stability of one equilibrium point to make conclusions about the stability of other equilibrium points is unfounded mathematically.  So to use the stability properties of the upright zero steer equilibria to make conclusions about the stability of turning equilibrium is also unfounded mathematically.  An example of where this would lead to trouble would be to use the stability properties of the upright zero-steer configuration at one speed to make conclusions about the stability properties about the same upright zero-steer configuration at another speed.  This would be especially problematic near a bifurcation.
• In the upright zero-steer configuration, speed is the only parameter to consider, but in a turn, there are effectively two: speed and configuration.  The set of all feasible steady turns is a 2-d surface in the lean-pitch-steer space.  At any point on that 2-d surface, there are only two local directions you can move in and remain within the set of feasible steady turns, hence two parameters. Needless to say, a lot more can happen when you have two parameters to vary instead of just one.
• There are right handed turns with larger lean and steer values, although not unrealistically large, that are stable and have a rightward steer torque.

### Using linear models to say something about nonlinear behavior

Posted by Jason Moore at Apr 22, 2011 02:49 PM
You say: "To use the stability of one equilibrium point to make conclusions about the stability of other equilibrium points is unfounded mathematically."

I'm sure this is true but often we are trying to make conclusions about the behavior of a real system (i.e. a bicycle). We use various types of models to do this. One model is the one you are showing (non-linear Whipple model) and another is the Whipple model linearized about the upright. I think that it is probably ok to make this statement:

"To use the stability of a model at one equilibrium point to make conclusions about the stability of a physical system at another equilibrium point is founded mathematically."

I think this is certainly true for using linear models to make statements about real systems. It seems to boil down to whether a model really represents a physical system and at what point does the model no longer capture the physical systems motion.

The linear Whipple model is a different model than the non-linear Whipple model, but they both say something about the motion of a bicycle and the non-linear model is surely more valid outside of the linear regime (this regime is subjective to some degree).

### not unrealistically large

Posted by jim papadopoulos at Apr 22, 2011 03:08 PM
Your final comment about larger lean and steer values leaves me wondering. I don't know if pi/4 is ever exceeded with bicycles, and then you are suggesting about ten degrees of steer, which I think is unlikely unless you are going pretty slowly. I really wonder if anyone can achieve it. And then, if they can, isn't the steer torque gradient in the red region just as supposed? (Edge of red region could be a minimum or maximum in torque?) Some zooming in would be good, here! Finer torque lines, and maybe show the friction lines too! Jim

### I disagree

Posted by Luke at Apr 22, 2011 03:08 PM
I disagree with this statement:

"To use the stability of a model at one equilibrium point to make conclusions about the stability of a physical system at another equilibrium point is founded mathematically."

For example, why would you use the stability properties of the linearized upright bicycle dynamic equations at 1 m/s to make inferences about the stability a physical bicycle at 10 m/s? I think it would only be appropriate to make conclusions about the stability of a real bicycle at the same speed, i.e. at 1 m/s.

### ok

Posted by Jason Moore at Apr 22, 2011 03:11 PM
I wouldn't use a model linearized about 1 m/s to make claims about 10 m/s. But I would feel comfortable using a model linearized about 1 m/s to make claims about a real bicycle moving at 1.1 m/s. Linear implies small deviations, but I don't know how to judge what is small and what isn't. I think it depends on the system.

I'll update my statement too:

"To use the stability of a model at one equilibrium point to make conclusions about the stability of a physical system at another equilibrium point can be founded mathematically."

### More plot detail needed

Posted by Mont Hubbard at Apr 22, 2011 04:17 PM
I can't wait to see the expanded plot about the origin in lean-steer plane.