Identifying the bicycle model

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.

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.

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.

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).