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The sensitivity of the vehicle responses, driver control parameters and performance indices to variations in the understeer coefficient of the vehicle are evaluated considering an understeer, oversteer and a neutral steer vehicle. The variations in the understeer coefficient (Table 2.3) are attained by varying the cornering stiffness of the front and rear tires. The understeer coefficients (Kus) for the oversteer, neutral and nominal vehicle are

taken as -0.004, 0 and +0.004, respectively. The driver model parameters are identified through minimization of the reported performance index subject to limit constraints, defined in Eq. (2.17), while the forward speed is considered as 20 m/s. The identified model parameters and performance indices are summarized in Table 2.10 together with

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the success or failure of each model considering the three vehicle handling scenarios. The variations in the lead time and the compensatory gains are also presented in Figure 2.14.

The simulation results again illustrate considerable variations in control parameters of the selected driver models with variations in the handling characteristics of the vehicle. Both the compensatory and single-point preview models exhibit higher position and orientation compensatory gains (Ky and KΨ) with increase in understeer coefficient of the

vehicle, as shown in Figures 2.14(b) and 2.14(c), and Table 2.10. This suggests greater compensatory action demand when driving an understeer vehicle. The driver model based on the compensatory/anticipatory control strategy (model 7), however, yields substantially lower lateral position compensatory gain Ky with increasing values of Kus

(Figure 2.14b). The orientation gain also increases for the understeer vehicle but is nearly constant for the over and neutral steer vehicles. Further, this driver model exhibits greatest sensitivity to variations in the understeer coefficient that is evident from substantial variations in its compensatory and anticipatory gains (Ky, Kψ, KC and Ka). The

models results, however, do not show a definite trend in TL with variations in understeer

coefficient, as seen in Figure 2.14(a). The results obtained for the compensatory models in the absence of preview (models 1 and 2) show that the lead time constant TL increase

with the understeer coefficient, while the single- and two point preview strategies (models 3, 4 and 7) suggest an opposite trend. The models based on multi-point preview (models 5 and 6), however, assume a constant TL. The compensatory and two-point

preview models (models 1 and 7) exhibit greater lag time constant TI with increasing in

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Table 2.10: Influences of variations in understeer coefficient of the vehicle on the identified driver model parameters and corresponding performance measures

Driver

Model Understeer coefficient

Test Result Ky† (rad/m) KΨ (rad/rad) KC (rad.m) Ka (rad.m) TL ‡ (s) TI * (s) ̇ Co m pe ns at ory m od el s Model 1 Oversteer Pass 0.360 - - - 0.149 0.033 0.361 0.051 0.033 0.069 0.188 Neutral Pass 0.530 - - - 0.198 0.040 0.374 0.063 0.030 0.081 0.206 Understeer Pass 0.669 - - - 0.291 0.045 0.613 0.093 0.028 0.145 0.347 Model 2 Oversteer Pass 0.563 0.174 - - 0.274 0.200 1.274 0.744 0.121 0.109 0.300 Neutral Pass 0.823 0.284 - - 0.282 0.200 1.555 0.972 0.099 0.134 0.351 Understeer Pass 0.840 0.360 - - 0.292 0.200 1.933 0.960 0.103 0.240 0.631 Prev ie w c om pe ns at or y mo de ls Model 3 Oversteer Pass 0.013 - - - 1.144 0.300 1.442 1.370 0.055 0.014 0.003 Neutral Pass 0.016 - - - 1.122 0.300 1.456 1.375 0.058 0.020 0.004 Understeer Pass 0.021 - - - 1.112 0.300 1.503 1.404 0.058 0.034 0.007 Model 4 Oversteer Pass 0.038 - - - 0.601 0.200 0.235 0.148 0.043 0.026 0.017 Neutral Pass 0.052 - - - 0.545 0.200 0.265 0.149 0.039 0.047 0.028 Understeer Pass 0.071 - - - 0.540 0.200 0.301 0.153 0.031 0.087 0.029 Model 5 Oversteer Pass 0.028† _{0.704 - - 2.000 0.000 0.311 0.286 0.009 0.014 0.001} Neutral Pass 0.028† _{0.705 - - 2.000 0.000 0.404 0.360 0.017 0.025 0.001} Understeer Pass 0.028† _{0.745 - - 2.000 0.000 0.935 0.861 0.033 0.038 0.002} Model 6 Oversteer Pass 0.028† _{0.763 - - 2.000 0.000 0.330 0.295 0.008 0.025 0.002} Neutral Pass 0.028† 0.704 - - 2.000 0.000 0.448 0.387 0.017 0.041 0.002 Understeer Pass 0.028† _{0.705 - - 2.000 0.000 1.100 0.978 0.047 0.073 0.003} Co m pe ns at ory A nt ic ip at or y Model 7

Oversteer Pass 1.3E-03 0.737 0.465 1.025 0.230 0.101 0.141 0.030 0.013 0.025 0.074

Neutral Pass 8.6E-04 0.807 0.475 0.966 0.229 0.113 0.162 0.033 0.011 0.041 0.078

Understeer Pass 5.0E-05 0.930 0.260 1.488 0.224 0.121 0.222 0.041 0.009 0.070 0.101

† _K

y, the compensatory gain, for the multi-point preview models (models 5 and 6) is reported corresponding to the first preview point.

‡ _T

L represents the lead time constant (model 1) and summation of individual lead times (model 2), preview time in models employing only

preview (models 4, 5, 6 and 7), and lead and preview time (model 3)

* _{The lag time, T}

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Figure 2.14: Comparison of (a) the lead time constant; (b) lateral position gain constant; and (c) orientation gain constant of selected driver models subject to variations of

understeer coefficient (speed= 20 m/s)

The selected driver models are further evaluated considering the identical performance index, defined in Eq. (2.10). Table 2.10 also presents the values of total performance

(c)

(a)

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index and its constituents. The results suggest that a vehicle with oversteer tendency yields lower total performance index , path tracking performance , and steering effort indices ( and _̇), irrespective of the modeling strategy. The compensatory model 1 [17], without the driver preview and time delays can effectively track the desired path by converging to lower time lag at the selected forward speed (20 m/s). The compensatory models (models 1 and 2) in the absence of preview process, however, yield the greatest steering effort ( and _̇). The use of multi-point preview strategy (models 5 and 6), on the other hand, yields the lowest steering effort demands.

The simulation results suggest that employing the human driver preview process would help enhance the path tracking performance and reduce the steering effort demand, as observed for variations in the vehicle mass and vehicle speed. The path tracking performance and steering effort of the single-point preview control models (models 3 and 4) can be further enhanced through two- or multi-point preview strategies. The multi- point preview strategy coupled with internal vehicle model path prediction strategy yields least demand on the driver’s steering effort. Consideration of a constant preview interval, however, yields a higher level of path deviation. The compensatory/anticipatory driver model [45] (Model 7) results in superior path tracking performance with over 50% lesser steering demand compared to the compensatory models. From the results, it is evident that two-point preview and internal vehicle model path prediction strategies yield most effective vehicle control with variations in its handling characteristics.

The sensitivity of the peak directional responses of the vehicle and driver’s steering effort, and performance indices to variations in the Kus are evaluated using Eq. (2.16) and

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represents an increase in a response quantity with respect to the nominal understeer vehicle (Kus = 0.004). As it would be expected, the results show that decreasing the

understeer coefficient (approaching neutral and oversteer tendency) invariably lowers the peak steering angle ( ) and peak steering rate ( ̇ ), and the corresponding performance indices ( and _̇).

Table 2.11: Variation in the peak directional responses of the selected driver models to changes in the understeer coefficient (speed=20 m/s)

Driver Model Under-steer _coefficient Sensitivity of the peak values of directional responses (%) (deg) ̇ (deg/s) (m) (m/s2)

Model 1 Oversteer _Neutral -22.4 _-19.6 -18.5 _-22.0 -6.3 _-8.2 7.9 _6.3 Model 2 Oversteer -33.6 -32.0 -15.6 -6.5 Neutral -25.3 -26.1 0.1 -0.6 Model 3 Oversteer _Neutral -34.5 _-24.4 -34.4 _-24.3 -1.3 _-1.1 0.8 _0.2 Model 4 Oversteer _Neutral -47.3 _-27.1 -29.4 _-8.2 -2.0 _-1.6 -7.9 _-3.6 Model 5 Oversteer _Neutral -33.4 _-16.3 -18.2 _-6.6 -55.3 _-53.7 19.1 _12.7 Model 6 Oversteer _Neutral -33.1 _-17.4 -15.1 _-5.7 -29.4 _-35.4 20.7 _14.2 Model 7 Oversteer _Neutral -39.9 _-24.1 -32.5 _-31.7 -15.3 _{-14.5 -0.8} 0.9

The driver model employing single-point preview strategy (models 3 and 4) [30,40] yield substantially lower sensitivity of peak , and , while the peak steering angle and steering rate and corresponding performance indices ( and _̇) vary substantially to satisfy the path tracking requirements. The simulation results further show that multi- point preview models [67,68] exhibit relatively greater sensitivity of peak path deviation

, and the position and orientation performance indices ( and ) to variations in the understeer coefficient. This can be mostly attributed to the constant preview time assumption of the multi-point preview models.

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Table 2.12: Variation in the total performance index and its constituents of the selected driver models to changes in the understeer coefficient (speed=20 m/s)

Driver Model Understeer coefficient Sensitivity of the performance indices (%)

̇

Model 1 Oversteer _Neutral -41.1 _-38.9 -45.5 _-32.5 19.8 _7.6 -52.6 _-43.9 -45.7 _-40.5 Model 2 Oversteer _Neutral -34.1 _-19.6 -22.5 _1.3 17.0 _-4.8 -54.5 _-44.2 -52.5 _-44.4 Model 3 Oversteer _Neutral -4.0 _-3.1 -2.4 _-2.1 -5.0 _0.4 -58.2 _-42.7 -55.7 _-42.7 Model 4 Oversteer _Neutral -21.9 _-11.9 -3.4 _-2.6 39.1 _26.8 -70.0 _-46.0 -42.0 _-4.1 Model 5 Oversteer _Neutral -66.7 _-56.8 -66.8 _-58.2 -72.4 _-49.0 -63.0 _-34.6 -35.0 _-15.3 Model 6 Oversteer _Neutral -70.0 _-59.3 -69.8 _-60.4 -81.9 _-63.4 -66.2 _-43.0 -31.9 _-14.4 Model 7 Oversteer _Neutral -36.2 _-26.8 -27.5 _-21.3 49.1 _24.5 -64.4 _-41.4 -27.3 _-23.1