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Whipple model validity

by Jason Moore — last modified Jan 13, 2012 05:30 PM

A look at the validity of the Whipple model with respect to the data we've collected.

Our current human controller identification hinges on the assumption that the Whipple model is a valid model of the bicycle/rider system. I've noticed on even the best full system (including controller) identifications, that the steer torque is under predicted (i.e. we measure a higher steer torque needed to force the Whipple model's outputs to our measured data). This isn't necessarily a surprise, as the Whipple model includes at least two assumptions with respect to our system that could certainly account for the mis-identification: 1. no tire to floor interaction model and 2. the rider is completely rigid.

The essential linear Whipple model with steer torque as the only input takes this form:

$$ \begin{bmatrix} \dot{\phi} \\ \dot{\delta} \\ \ddot{\phi} \\ \ddot{\delta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \begin{bmatrix} \phi \\ \delta \\ \dot{\phi} \\ \dot{\delta} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ b_{31} \\ b_{41} \end{bmatrix} T_\delta $$

One of the most basic identification tasks would be to find the \(a\) and \(b\) entries. This assumes that the system is 4th order, that the simple derivative kinematic differential equations hold, and the input only acts on the acceleration equations. The second way to arrive at those matrix entries is to compute them using the Whipple model (or some other fourth order model) and a realistic set of parameters for the model.

If we do a good job at measuring the states and the inputs during the experiment and do a good job at measuring the Whipple model parameters (mass, inertia, geometry) and that the Whipple model is a good model of our bicycle-rider system, a given measured steer torque input to the model should give the same state trajectories as we measured and the identified matrix entries should match the ones calculated from the measured Whipple model parameters. Both of these can give an estimate of the validity of the Whipple model for our system.

The following graph shows the state trajectories for an unperturbed run (no lateral perturbation). The black line (data) is the filter (30 hz low pass) experimental data. The blue line (whippleModel) is the trajectory of the Whipple model states forced by the experimentally measured steer torque. The A and B matrix entries were computed by measuring the Whipple model parameters. The green line is the state trajectories of the identified 4th order model, which was free to adjust the variable A and B entries as shown above.

Whipple Validation

Two things are immediately clear: 1. A fourth order model can be found that relates the steer torque to the state trajectories and 2. The amplitudes of the state trajectories for the Whipple model are much larger in amplitude than the experimental data. The first is a good sign, as higher order models are probably not needed but this also means that either the measurements are very poor or that the Whipple model is a poor model for our system. I'm leaning to the later. Steer torque is the only measurement that I have less confidence in, but that confidence is still high. This is a comparison of the state and input matrices calculated from measured parameters and from the identification process:

>> whippleModel.A

ans =

         0         0    1.0000         0
         0         0         0    1.0000
    8.4510  -18.4667   -0.1481   -1.2529
   18.5136  -10.8271    8.4775  -17.4473

>> identifiedModel.A

ans =

         0         0    1.0000         0
         0         0         0    1.0000
    8.5007  -18.5512    0.2315   -2.2580
   40.3916  -50.5260   -2.5057  -11.2178

>> whippleModel.B   

ans =

         0
         0
   -0.0971
    5.5584

>> identifiedModel.B

ans =

         0
         0
    0.1652
    1.4777

This is of course only for one run, but the other's I've examined have similar issues. This seems to say that for a given measured state trajectory, the steer torque measured is much larger than required to force the Whipple model to that trajectory. We rigidified the rider, except for his arms. The first thing I will do to address the torque discrepancies is to use a Whipple model with additional passive arms (these can be added without increasing the order of the system). The second, would be to explore a tire model.

Update - January 22, 2012

I checked the 4th order model I derived with the arm's effectivlely adding inertia to the fork/handlebar assembly. My intution was that if I added passively moving arms then more steer torque would be required for a given trajectory in steer and roll. But this initial check showed the opposite.

Whipple Arm Model Validation
A comparison of the time trajectories of three 4th order bicycle models given the same steering torque input.

Looking at the A and B matrix entries is also interesting. Note that the roll equation phi and delta terms match the Whipple model and the identified model, but the steer equation phi and delta terms are very different than the identified model.

armsA = [0         0    1.0000         0
         0         0         0    1.0000                           
         8.7171  -18.6499   -0.0368   -1.4557                      
         4.3115   -1.3594    2.4701   -7.0037];                    
         
armsB = [0
         0
         -0.1019                                                   
         5.5687];

We've rationalized to ourselves somewhat that the roll equation is more likely to be correct from first principles modeling because it is primarily a function of the states, steer torque doesn't affect roll acceleration that much directly. But steer torque plays a large roll in the steer equation, affecting steer acceleration which in turn affects the roll. So getting better matches in the roll equation than the steer equation may make more sense.

The fact that the model with the arm inertial effects didnt' seem to improve the fits, points strongly to the fact that we don't model the tire/road interaction beyond a knife edge, pure rolling, no slip model. There is little doubt in my mind that an accurate tire model would account for the steer torque to roll/steer discrepancies seen in these graphs. The motorcycle dynamics researchers already knew this and have even been warning the knife edge bicycle model folks for while now [Limebeer2006] is an example. In a way, the riderless bicycle experimental validation of the Whipple model by [Kooijman2006] was a lead in the wrong direction. It seems clear that a bicycle with a rider must be modeled with the ground contact torques due to tires properly modeled. We are going to try to move forward with the rider control identification even though the Whipple model is likely poor for our experiments. One way to do this would be to use the identified models instead of models derived from first principles. This would give us a 4th order model with proper steer torque to steer/roll relationships so that the controller identification can be performed. I may also be able to augment the Whipple model with some stiffness terms that show up in the roll equation and work to identify them, giving a pseudo tire model (at least the extra torques would be accounted for). But, this seems to be very interesting finding, as many bicycle researchers make use of the Whipple model for studies relying on some belief that it is valid, when it is probably not.

Update - January 23, 2012

Another way to check the validity on the models is to take the measured coordinates, rates and accelerations then plug them into the equations of motion, solving for the inputs as a function of time. The following graph shows the inputs generated from the models compared to what I measured.
Input Comparisons
The roll and steer torques as a function of time for run number 638. The model inputs were generated by solving the equations of motion for the inputs and plugging in the measured coordinates, rates and accelerations.
I notice several things about this one run:
  1. The Whipple model under predicts the steer torque by a factor of about 1.75.
  2. The Whipple model with the arm inertial effects under predicts the steer torque by a factor of about 1.2, which is better than the Whipple model alone. This is opposite what I concluded from the previous simulations.
  3. The roll torque is not zero. It is basically bouncing around in +/- 10 Nm. I can only attribute this to the uncertainties in the measured coordinates, rates and accelerations. I would have thought it would be lower. I'm not really sure how to explain that.
  4. Both models give practically the same roll torque solution.
Check this thread for some more thoughts on this problem from other researchers: https://groups.google.com/forum/?hl=en#!topic/stvdy/pMcj4zaWnmA
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