Identification of Two-bodies model
4.1 Multi-Objective Optimization
4.1.3 Simulation Results
Criteria→Ci(tk) = 12.P
(yim(tk)−yi(tk))2 Sensitivity→Si(tk) = dθdyji
Gradient→Gi(tk) =P(yim(tk)−yi(tk)).Si
(4.9)
Whereyim are the measured outputs,yi are model simulated output andtk is the current simple time.
As an example, to identify the parameterθ1, one select a short manoeuvrer for the lateral motion in which, the second derivative of the steering angle is neglected, i.e. δ¨≈0. From equations (4.2) and knowing that may =Fyf+Fyr, we gety1=ψ¨ such that:
mv˙y+θ1ψ¨+mhφ¨+mvxψ˙ =may (4.10) From the above equation, we can deduce the expression of y1 and compute then its sensitivity S1 = dθdψ¨1 and its gradientG1. Next, the gradient algorithm is applied to estimate the value of the first parameterθ1. Further, the algorithm is repeated for each parameter after selecting the corresponding scenario as described a follows:
1. From an initial starting pointθi(t0), we calculate the criterionCi(t0)and the gradientGi(t0). 2. Compute the new value: θi(tk) =θi(tk−1)−αGi(tk−1)
3. Compute a new value ofCi(tk)andGi(tk)taking θi(tk).
4. If the second criterion is smaller keep the new parameter value of the correspondingθi(tk). Increaseα for efficiency and increment the counterk+1.
5. Else, keep the old value ofθi(tk−1)and reduceαto seek a nearest local minimum.
6. Evaluate the stopping criteria for exit loop: accuracy on the criteria, the gradient, maximum number of effective iteration, tolerance between the last two values ofθ. For more detail see Algorithm (1).
4.1.3 Simulation Results
The test described in this section is carried-out on the BikeSim simulator. The accessible parameters are captured from the motorcycle’s datasheet. The algorithm is tested during a track test on handling road course. Figures 4.1 show the rider steering torque and the trajectory of the conducted manoeuvrer. The
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-40 -20 0 20
-200 -100 0 100 200 300 400
-200 -150 -100 -50 0 50 100 150
Figure 4.1: Riders torque and motorcycle path.
identification results of the multi-objective optimisation are summarized in the following Table 4.1. Where, θi = [e34,e36,e44,e45,e46,a45,a46,e55,a66]is the identified vector of interest which depends on the physical parameters of motorcycle. The other parameters are concluded from equation (4.5).
86 Chapter 4. Identification of Two-bodies model Table 4.1: Multi-Objective-Optimisation method.
Initial P rior values Identif ication
Parameters θ0 θr θi
The objective is to validate the result of the identification method on the two-bodies motorcycle model.
After the identification process, the estimated parameters are used to simulate the PTWV lateral dynamics and hence to validate the dynamics behaviour with respect toBikeSim data. To assess the performance of the algorithm, we report the track test generated by the input in figure4.1. With the kind of this trajectory, the motorcycle lateral dynamics is largely excited. The following simulation results introduce the states of the PTWV lateral two-bodies dynamics in dashed red and the actual ones fromBikeSim in blue.
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Figure 4.2: Comparison between the roll, steer angles and lateral speed of the identified model and theBikeSimmotorcycle data.
Figures4.2illustrate the time variation of the roll angle, steering angle and the lateral speed that characterize the lateral dynamics recorded from BikeSim. These variables are compared to their corresponding states computed from the identified parameters. We can note that the errors between the identified states (dashed red) and the actuals data are practically acceptable except the lateral speed. Even if the considered scenario seems to be aggressive, the lateral velocity is a dynamic state that is not very excited in the PTWV dynamics.
Indeed, figures4.3depict the comparison of the actual roll, yaw and steer rates with the estimated ones.
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Figure 4.3: Comparison between the identified model and theBikeSim motorcycle data.
4.1. Multi-Objective Optimization 87 In the same way, figures4.4show the cornering front and rear forces as well as the lateral acceleration, these identified states are closely similar to the actual data. The results show that the model perfectly reconstructs the most of the dynamic states, hence the cornering behaviour of the motorcycle is well identified.
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Figure 4.4: Comparison between the identified model and theBikeSimmotorcycle data.
One can notice some differences between the identified model and the actual data. These identification errors come from the PTWV two-bodies modelling assumptions whereas BikeSim uses a non-linear eight-body model. Also, the accuracy of the identified parameters of the model depends on the initial parameter value and the step rate αin each iteration of the algorithm. Further, we can notice also a large amplitude of the roll angle (about 35 deg). However, it will be recalled that the two-bodies model is theoretically valid only for small variations around the equilibrium position in a straight line (φ=0). This extreme scenario shows that, we are reaching the limits of the model.
This section ends with a study of the identification errors. The errors between the identified model and the BikeSim states are quantified by mean of the Theil Inequality Coefficient (TIC), is the standardized root mean-squared error, used in the sensitivity analysis to measure the model predictive accuracy and to facilitate comparison between the actual and identified model (Woschnagg and Cipan, 2004). Note that, the identification errors are quantified also in terms of errors variation percentage, this coefficient is denoted
"FIT", to compare the performance of the models that we have estimated.
T ICi=
whereyimare the actual observations containingnsamples andyiare the corresponding predictions, resulting from estimated parameters.
Table 4.2quantifies the identification errors between theBikeSim variables and the identified states of the two-bodies model, confirms the efficiency with acceptable errors. The TIC is bounded by 0 and 1, the lower boundary is the ideal case of perfect identification. The small values of TIC and the percentage of the fit values, show a good forecast accuracy and prove the reliability of the model.
# States
φ δ vy ψ˙
T IC 0.0974 0.1172 0.2924 0.0565 F IT% 79.9940 77.7522 66.2456 79.9202
# φ˙ δ˙ Fyf Fyr
T IC 0.1199 0.1962 0.1372 0.1113 F IT% 77.3422 62.7593 74.4299 78.6061
Table 4.2: Analysis of the estimated states for the tests conducted.
The results, from the identification process, are very promising. The comparison withBikeSimdata demon-strates the potential of the identification procedure.
88 Chapter 4. Identification of Two-bodies model