Tyre curve fitting and validation

Precision and Accuracy
In the fields of engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to its actual (true) value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. Although the two words can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.
A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. Eliminating the systematic error improves accuracy but does not change precision. A measurement system is called valid if it is both accurate and precise.

Precision verses Accuracy; The target analogy
Precision is the degree of reproducibility while accuracy is the degree of veracity. The analogy used here to explain the difference between precision and accuracy is the target comparison. In this analogy, repeated measurements are compared to arrows that are shot at a target.

Tyre curve fitting and validation - White-Smoke

Precision describes the size of the arrow cluster. (When only one arrow is shot, precision is the size of the cluster one would expect if this were repeated many times under the same conditions.) When all arrows are grouped tightly together, the cluster is considered precise since they all struck close to the same spot, even if not necessarily near the bullseye. The measurements are precise, though not necessarily accurate.
Accuracy describes the closeness of arrows to the bullseye at the target center. Arrows that strike closer to the bullseye are considered more accurate. The closer a system's measurements to the accepted value, the more accurate the system is considered to be.
However, it is not possible to reliably achieve accuracy in individual measurements without precision. If the arrows are not grouped close to one another, they cannot all be close to the bullseye. (Their average position might be an accurate estimation of the bullseye, but the individual arrows are inaccurate.)

With all types of tyre testing we are limited to a small sample set. The repeatability between samples is not always good. The methodology for collecting the data is often not optimum. There is often a lot of noise (thermal, wear and hysteresis related) within the data.
Remember, with MF-Tyre curve fitting we are trying to fit mathematically generated curves to raw data. It is an empirical model, the curves themselves are not based on laws of nature related to tyre physics.
Historically tyre data has been collected in a two dimensional fashion. For example, a set of nominal vertical loads are chosen, then slip angle or slip ratio sweeps are made. This is then repeated at different camber angles. If we are lucky they may be repeated at different pressures. As we have seen in part 1, it is not a two dimensional problem! A surface is produced by the collection of data, which is three dimensional. With the addition of camber and pressure the problem becomes multidimensional.

To illustrate this conundrum in more detail, what follows is a comparison of conventional methodology of tyre data collection and fitting against alternative "design of experiment" (DOE) methods. At the end of the day we require precision and accuracy.
The chart below shows a common method of sampling tyre data from a stability rig. Note that four normal test loads were chosen then sample sweeps in slip ratio and slip angle were made. The first observation is that the samples are clustered together giving high precision at each test load but not covering the whole surface, which may have given more accuracy to the results. You may ask; Why is the data collected in this way? Well, they produce nice two dimensional curves to which the MF-Tyre curves are then fitted. That makes complete sense, or does it!

Tyre curve fitting and validation - White-Smoke

A three dimensional representation of the raw data is shown in the next chart. Note that with this amount of noise, the fitting of MF-Tyre curves directly to the raw data has a low degree of confidence.

Tyre curve fitting and validation - White-Smoke

It is however possible to resample these surfaces before fitting the curves. Several different methods could be applied. The surface shown below uses a similar method to Kriging. The aim is to maintain all the characteristics and trends within the surface but remove most of the noise. The MF-Tyre curves can then be fitted to this resampled surface with a higher degree of confidence.

Tyre curve fitting and validation - White-Smoke

The charts below show both approaches to curve fitting. The one on the left was fitted to the raw data directly. These are the MF-Tyre coefficients as issued by the tyre manufacturer. Note that the deviation from the curves is shown in the lower left box in this chart. The chart on the right is fitted using the resampled data. The deviation from the curves is a lot less with the resampled surface.

Tyre curve fitting and validation - White-SmokeTyre curve fitting and validation - White-Smoke

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