_{1}

The dependency of the steady-state yaw rate model on vehicle weight and its distribution is studied in this paper. A speed-dependent adjustment of the yaw rate model is proposed to reduce the yaw rate estimation error. This new methodology allows the calibration engineer to minimize the yaw rate estimation error caused by the different weight conditions without going through the calibration process multiple times. It is expected that this modified yaw rate model will improve the performance of Electronic Stability Control (ESC) systems such as response time and robustness.

Yaw rate [

In this paper, the impact of vehicle loading and its distribution on the estimation of yaw rate is analyzed. An improved methodology for yaw rate estimation is proposed without adding much complexity to the calibration process. The steady-state model [

To illustrate the idea of the new methodology, we use the bicycle model for the vehicle [

where L is the wheel base (m), V is the vehicle speed (m/s), K is the understeer gradient, and g is the gravitational acceleration (m/s^{2}). Solving for R, we get

But the yaw rate is given by

where r is the yaw rate (deg/s) and 57.3 is the conversion factor from radian to degree [

The understeer gradient is given by

where the front and rear corning coefficients

the range of 25% - 200% of rated load is plotted. Equations (4), (5), and

In order to study the impact of weight and its distribution on the yaw rate, two vehicle loading conditions are analyzed: 1) The weight is increased from LLVW to GVW while maintaining the same center of gravity (CG) location; 2) GVW with CG at different locations. These two cases are more general than the typical GVW loading condition specified by the vehicle OEM since the CG location can be shifted to various positions.

If the weight is added at the CG of LLVW, the CG will not change. However, there are two additional constraints: the loads to the front and real axels must be within the corresponding axel loading ratings. As a result, the extra weight added to the vehicle may be less than the difference between GVW and LLVW. The exact maximum weight that can be added at the CG location for LLVW can be easily calculated.

The following set of vehicle parameters are used to calculate the yaw rate estimation:

§ L = 9.4 ft;

§ GVW = 5700 lb;

§ LLVW = 4600 lb;

§ LLVW distribution: front 55%, rear 45%;

§ Rated load for tires = 2050 lb;

§ Front axle rating = 3000 lb;

§ Rear axle rating = 3200 lb;

§ Steering ratio = 17 degree;

§ Cornering coefficient:

For simplicity, the steering wheel angle is assumed to be 20 degrees. The impact of steering angle will be discussed later. Using the formulas given in section 2 one can calculate the yaw rate for any given vehicle speed. The resulting yaw rate of the following two extreme cases can be plotted as a function of the vehicle speed: one is the LLVW, which is close to the calibration condition for the controller; the other is with the heaviest loading without violating the GVW and axle rating limitation (Labeled as GVW in

Intuitively, all other weights should have the yaw rate between the two extreme cases. This was verified with MATLAB simulation. Note that the difference between LLVW and GVW in ^{2} in the denominator increases as the vehicle speed increases.

In general, when the load is added to the vehicle, the CG location can shift. To analyze this condition, the weight of the vehicle is assumed to be GVW. The position of the CG for GVW is varied such that the axel load ratings are not exceeded. A one dimensional search in MATLAB showed that there were two extreme positions to adding weights to the GVW condition without violating the

axle rating, one is close to the front axle and the other is close to the rear axle. Weight can be added up to the GVW condition to any points between these two extreme positions without violating the axle rating.

The two extreme cases are plotted in comparison to the LLVW condition in

Comparing to

Based on the analysis in Sections 3 and 4, it is easy to see that one should not use the yaw rate estimation model calibrated at the LLVW condition, since the LLVW condition is very close to one of the extreme cases. If the vehicle loading condition can be estimated using the data from brake control action such ABS, traction control, or ESC, then a simple interpolation of the LLVW and GVW curve would provide an improved yaw rate estimation. Of course, the estimation error will depend on how good the loading condition estimation is. The loading condition can be estimated based on front and rear wheel slips, vehicle longitudinal/lateral accelerations, and other information that is available to the ESC system.

If the loading condition cannot be estimated, one can take the average of the two extreme cases, i.e., the GVW_rear curve in

Assuming that the ESC system is calibrated under the LLVW loading condition, one can simply add a yaw rate compensation that is equal to the difference between the optimal loading curve and the LLVW loading curve in

Recall that a simplifying assumption was made in order to plot the curves in Figures 2-4, that is the steering wheel angle is constant. In order to come up with the actual compensation amount, the results plotted in Figures 2-4 are repeated for different values of steering wheel angle. The yaw rate estimation correction amount is plotted in

Next, the robustness of this new methodology will be analyzed. The cornering coefficient curve will have a tolerance. It is desirable to understand what impact any change in the cornering coefficient would have on the overall result.

In

cornering coefficient is increased.

The impact of the weight and its distribution on the yaw rate estimation during steady-state corning is illustrated using a simple bicycle model. Through simulation in MATLAB, the loading condition is identified as an important source of yaw rate estimation error. With the proposed simple change to the calibration process, the yaw rate estimation error can be greatly reduced. One can apply this methodology using more complicated vehicle models, such as a Carsim model. By plotting the yaw rate vs speed function with different steering wheel angles and different loading conditions, one can identify the two extreme loading

conditions. The average of the yaw rate for the two extreme cases can be used to find the optimal load position. The compensation amount for yaw rate estimation can be calculated. All these steps can be carried out in a simulation environment. The result can be easily implemented in the existing ESC software. The estimation error is reduced as a result, without any loading information or need for recalibration.

The preliminary result in this paper can be further improved by adding a load condition estimation algorithm, which is still under investigation. The idea is to first estimate the loading condition (how much weight and the CG location) before ESC activation using the longitudinal/lateral acceleration signals and wheel speed signals; for a given loading condition, the yaw rate compensation amount can be calculated off line and store in the software. This would further reduce the impact of loading condition on the yaw rate estimation. Even if complete loading condition cannot be identified, partial information such as the total weight can allow one to further reduce the yaw rate estimation error.

Zhan, W. (2017) A New Methodology for Reducing Yaw Rate Estimation Error. World Journal of Engineering and Technology, 5, 12-20. http://dx.doi.org/10.4236/wjet.2017.51002