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Identifying the bicycle model

by Jason Moore — last modified Feb 29, 2012 03:55 PM

I've been working on a small application to visualize and interact with the data resulting from identifying the bicycle model from all of my data. This is a preliminary look at how it is coming along.

I'm using the basic grey box identification technique as described in my previous posts [1], [2]. The basic idea is to fit a 4th order model to the experimental data with steer torque as the input and the four states (roll angle, steer angle, roll rate and steer rate) as the outputs. This has proved to give really good output matching when comparing the identified model to the experimentally measured output data and the models seem to show some repeatability. So far I've been exploring the runs that have no perturbations. I've run two identification routines on the data. The first assumes no process noise (output error model) and the second attempts to compute the Kalman gain matrix that captures that describes the process noise. I'm still blurry on what is going on behind the scene in the Matlab system id toolbox and am continually reading Ljung1999 to try to wrap my head around it all. Here are a few things that are of note with the ID process:

  1. If I identify the process noise Kalman gain matrix, the resulting model output to measured output fits are all poorer. This makes little sense to me. The output error version assumes that the Kalman gain matrix is zero, so the output error model is in a sub space of the full system description. With that in mind, it seems the process noise identification would always result in a model with better output fit that one without. And if K=0 gives a better output fit, then the minimization process when K is a free parameters should find K=0 for the minima.
  2. I'm identifying an unstable model for almost all of my runs (the instrumented bicycle with rider has a weave critical speed at about 7 m/s). This presents some issues and the identified model doesn't always give a stable output for the measured steer torque input. I don't necessarily get a good model result from both the output error form and the process noise form. Sometimes they both converge to a good solution, sometimes one or the other does, and sometimes they both don't.
I know I have some holes in the identification that need to be tended to but I've started exploring the resulting data regardless.
Bicycle ID GUI Coefficient Plot
A screen shot of the bicycle ID application showing the coefficient plot. The blue dots are the identified entries of dynamics equations (as opposed to kinematic) in the A and B matrices of the linear Whipple model. This is for the output error model. The buttons on the right allow you to select subsets of data depending on the rider, environment, speed bin, maneuver, and model output fit quality. The lower buttons allow you to adjust the first principles models (the linear Whipple model is currently shown) based on the physical parameters.
Bicycle ID GUI Bode Plot
This is the Bode plot view for the steer torque to roll angle and steer angle transfer functions for the 2 m/s speed bin. The blue lines give the mean +/- sigma of the experimentally identified transfer functions. The red line is the Whipple model for that speed. Notice that the spread in the magnitudes are pretty large, but the mean is very similar to the Whipple model, except for a 5-8 dB difference in magnitude.
Bicycle ID GUI Coefficient Plot Process Noise
This is the same view as above, except with the identified data when the process noise is accounted for. Notice that the spread decreases and the Whipple model does a pretty good job fitting the data for many of the coefficients. The steer equation (lower 5 graphs) is a power fit than the roll equation. The spread in the data for the roll angle coefficient is large and I'm not sure what kind of model woudl "fit" the data well. But the steer angle coefficient just needs a larger multiplier to velocity^2 and the steer torque coefficient needs to drop down a bit (which the inclusion of the riders arms in the model, will take care of).
These are my ideas on what to do next:
  • I plan to add in a root loci plot so that the identified eigenvalues can be compared to the model.
  • Fit some curves through the coefficient data. I think I can generate the bicycle model from the data exclusively, assuming that each coefficient is either constant, linear or quadratic with respect to speed. This would basically remove most of the first principles based modeling of the bicycle from the picture and I can move on to identifying the rider. I think that I need to use linear mixed effects (multi-level) modeling to properly do these fits, otherwise my repeated experiments and riders won't be properly accounted for.
  • I could free up some particular parameters in my bicycle model including some new parameters that would allow me to a grey box identification on the data with respect to a small set of parameters that I most uncertain in. This would retain the use of the first principles model, methodology. 
  • The model parameter adjustment needs some speed up. I think my data subset algorithms are slowing things down because of the way I programmed it. I have noticed that changing the trail and other parameters associated with the front frame of the bicycle have little effect on the resulting curves unless you exaggerate them unrealistically. Since there are 20+ parameters to fiddle with, it is hard to get a feel of how you could push the model curves around to better match the data.
  • I'm going to add in the model with the rider's arms, as I believe it will fit the data better than the Whipple model. For sure on the steer torque term in the steer acceleration equation.
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