0 2 4 6 8 10 12 14 16
−7
−6
−5
−4
−3
−2
−1 0 1 2 3
θ v’ (rad/s)
time (s) Steering wheel dissipates its energy
without high−frequency oscillations
Figure 5.10 – Angular rate of the steering wheel in closed loop. The system dissipates its energy without oscillations at low frequency.
5 6 7 8 9 10 11 12 13 14 15
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Frequency (Hz)
|F t(f)|
HIL Setup in Open loop HIL Setup in Closed loop
Open Loop Peak at 8.4 Hz
Figure 5.11 – FFT of the signals of the torsion of the steering wheel in open (blue) and in closed loop (red).
5.5 Validation of performances of the torque obser-
Chapitre 5. Experimental validation NeCS-Car benchmark
is given in input to the booster stage to provide the correct steering amplification. In other words, good results of the estimation contribute to make the entire system work properly.
The observer is designed under the hypothesis that it is possible to measure the angular rate ˙θs and the torsion Ft = k(θv − θs) of the steering column. On the experimental setup, ˙θs and Ft are simulated signal. The driver’s exerted torque τv, the steering-wheel positionθv and the tire-road friction torque τa, are measured on the experimental setup.
5.5.1 Performances of the observer with the simulated tire-road friction torque
To validate the performances of the observer, let us consider the case of a trajectory exerted by a driver, who can manoeuvre with two hands at 10 :10 in clock-code. The oscillation annealing control is active, but the steering assistance is inactive.
During these preliminary validation tests, NeCS-Car is not connected to the control station and the friction torque is simulated for a speed of the vehicle of 0 km h−1. The applied trajectory on the steering wheel is shown in Fig. 5.12. The driver turns the steering wheel many times over a test period of about 40 s. The comparison between the
0 5 10 15 20 25 30 35
−80
−60
−40
−20 0 20 40 60 80
θ v (deg)
time (s)
Figure 5.12 – Trajectory applied from the operator on the steering wheel to validate performances of the observer.
estimated torque ˆτv and the measured oney2 =τv2 is shown in Fig. 5.13. The numerical noise, that affects the measure of ˙θs, and the measure noise, that acts on Ft do not modify the global performances observer. It is possible to calculate the estimation error
0 5 10 15 20 25 30 35
−5
−4
−3
−2
−1 0 1 2 3 4 5
Driver’s torque (Nm)
time (s)
Measured signal y2 = τv2 Estimated τv
Figure 5.13 – Comparison between the measured driver’s exerted torque and the estimated one.
E1, that is defined as the following difference
E1 =τv2−τˆv (5.5.1)
This error is shown in Fig. 5.14. Even if the observation error increases when the angular speed of the steering column augment, it keeps in acceptable range. The angular speed of the steering wheel is shown in Fig. 5.15. This phenomenon is due to the influence of the mechanical noise of the steering wheel. As this noise depends from the angular rate, it is difficult to filter properly.
0 5 10 15 20 25 30 35
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Estimation error (Nm)
time (s)
Figure 5.14 – Error between the measured driver’s torque and the estimated one.
Chapitre 5. Experimental validation NeCS-Car benchmark
0 5 10 15 20 25 30 35
−3
−2
−1 0 1 2 3
θ v’ (rad/s)
time (s)
Figure 5.15 – Angular rate of the steering wheel.
In Fig. 5.16, the comparison between the estimated tire-road friction torque ˆτa and the output of the simulated model τa is shown. This comparison allows to analyse perfor- mances of the observer in this case.
0 5 10 15 20 25 30 35
−40
−30
−20
−10 0 10 20 30 40
Tire−road friction torque (Nm)
time (s)
Simulated signal τ
a
Estimated τ
a
Figure 5.16 – Comparison between the simulated tire-road friction torque and the estimated one.
In this case, the estimation error is defined as follows
E2 =τa−τˆa (5.5.2)
In Fig. 5.17, it is possible to see that this error is negligible and the observer performance are satisfying.
Note also that, by dividing the value assumed by the friction torque at low frequency (about 30 N m) with the gear ratio N1 = 13.67, it is possible to obtain the value of the driver’s torque during at the same time instant (about 2.2 N m), meaning that the LQ control preserves the impedance relation between two torques. This property cannot be verified at high frequency, because of the influence of the angular speed and acceleration of the steering wheel.
0 5 10 15 20 25 30 35
−30
−20
−10 0 10 20 30
Estimation error (Nm)
time (s)
Figure 5.17 – Error between the simulated tire-road friction torque and the estimated one.
5.5.2 Performances of the observer with the NeCS-Car
A further validation of the observer can be done using the torque input obtained with the NeCS-Car. These measures contribute also to validate experimentally the mathematical model used to simulate the tire-road friction torque.
First, let us consider performances of the observer w.r.t the real tire-road friction torque.
In Fig. 5.18, the estimated torque is compared with the measured one. In spite of the measurement error, the observer performances are globally satisfying. It is possible to diminish the transient error at the cost of a high gain of the observer.
Chapitre 5. Experimental validation NeCS-Car benchmark
0 5 10 15 20 25 30
−40
−30
−20
−10 0 10 20 30 40 50
Tire−road friction torque (Nm)
time (s) Measured signal y3 = τa3
Estimated τa
Figure 5.18 – The friction torque obtained from measures on the NeCS-Car is compared with the estimated tire-road friction torque.
In Fig. 5.19, the comparison between the simulated friction torque and the real measures obtained from the NeCS-Car is proposed for the same input trajectory. It is possible to observe that the simulated torque is realistic, if compared to the measured one.
14 16 18 20 22 24 26 28 30
−40
−30
−20
−10 0 10 20 30 40 50
time (s)
Tire−road friction torque (Nm)
Measured friction torque y3 = τa3 LuGre tire−road friction torque τa
Figure 5.19 – The tire-road friction torque, obtained from measures on the NeCS-Car, is compared with the tire-road friction torque obtained with the mathematical model of LuGre.