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# Identification of the bicycle model for a set of runs

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The preliminary results for identification of the model for a series of runs.

I'm working on the bicycle model identification mentioned that I started on in a previous post. My plan is to work on identifying a simple fourth order model as was shown but for a series of runs (I only looked at one of my best fitting runs previously). I started with all of the runs with no lateral disturbances (Balance and Line Tracking) from both the treadmill and the pavilion floor. This gave me about 110 usable runs. I then ran basic identification to find the $$a$$ and $$b$$ coefficients in this following model:

$$\begin{bmatrix} \dot{\phi} \\ \dot{\delta} \\ \ddot{\phi} \\ \ddot{\delta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ a_{\ddot{\phi}\phi} & a_{\ddot{\phi}\delta} & a_{\ddot{\phi}\dot{\phi}} & a_{\ddot{\phi}\dot{\delta}}\\ a_{\ddot{\delta}\phi} & a_{\ddot{\delta}\delta} & a_{\ddot{\delta}\dot{\phi}} & a_{\ddot{\delta}\dot{\delta}} \end{bmatrix} \begin{bmatrix} \phi \\ \delta \\ \dot{\phi} \\ \dot{\delta} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ b_{\ddot{\phi}T_\delta} \\ b_{\ddot{\delta}T_\delta} \end{bmatrix} T_\delta$$

No process noise model was used (I'll try that next). I was able to identified 98 of the ~110 runs with a mean fit for all for outputs (roll angle, steer angle, roll rate and steer rate) of at least 0%. The following graphs plot the coefficients versus speed. The dots are the identified coefficient and lines are the ones predicted from the Whipple model. The color of the dots represent the riders: Jason - red, Luke - blue, Charlie - magenta. Each plot shows only data with fit threshold of at least a given percentage. The 20% and higher plots are generally "good" fitting data.

An example fit from steer torque to the four outputs for a treadmill run.
An example fit from steer torque to the for outputs for a pavilion floor run.

Most of the runs are able to be identified by a fourth order model and fit pretty well. This is a good sign, as a higher order model will not be needed and I can restrict the bicycle identification to variations in a fourth order model. The data in the coefficient plots may look scattered but I think it may reveal some interesting things that need to be examined further:

• The rider inertia and seating position do not seem to significantly change the Whipple model coefficients (i.e. the differences are small). It would seem reasonable to use one model, instead of one for each rider.
• The Whipple model does a better job at predicting the roll equation than the steer equation. In particular the $$a_{\ddot{\phi}\delta}$$ is predicted really well, even at the low fit thresholds. The $$b_{\ddot{\phi}T_\delta}$$ has opposite sign as the model.
• The $$a_{\ddot{\delta}\delta}$$ term shows proportionality to $$v^2$$ but the multiplication factor needs to be a bit larger for a better fit.
• The $$a_{\ddot{\delta}\phi}$$ term seems to be the most variable and the bias is off.
• The $$b_{\ddot{\delta}T_\delta}$$ term seems fairly constant, but has a slight bias. This bias may be important for good prediction of the steer torque. This term only has mass and inertia as the coefficients and I feel like those were probably measured pretty well.

There few ideas to proceed:

1. Set up a statistical model for these coefficients and find the best fit model for comparison to my various first principle models.
2. Add the arms model to the plots here. I think that the arm's model will generally show a better fit. I've still got some coding to do to get it to work with all riders.
3. Run this for the dual input data (steer torque and lateral perturbation). The inputs here may have power at a broader bandwidth and may excite all the modes of the bicycle better.
4. Set up a grey box identification for the Whipple model and minor extensions which do not increase the degrees of freedom. Then pick the few parameters with the most uncertainty in their values (i.e. mass is ery certain, trail isn't). Let the identification find these parameters instead of the coefficients as done here. This may allow for better model fits.
5. Explore the analytical nature of these coefficients with respect to the Whipple model. It can reveal which physical parameters would be good to let be free in the identification.