** Identification of Two-bodies model**

**4.1 Multi-Objective Optimization**

**4.1.3 Simulation Results**

Criteria→*C** _{i}*(

*t*

*) =*

_{k}^{1}

_{2}.P

(*y*_{i}* _{m}*(

*t*

*)*

_{k}_{−}

*y*

*(*

_{i}*t*

*))*

_{k}^{2}Sensitivity→

*S*

*(*

_{i}*t*

*) =*

_{k}

_{dθ}

^{dy}

_{j}

^{i}Gradient→*G** _{i}*(

*t*

*) =*

_{k}^{P}(

*y*

_{i}*(*

_{m}*t*

*)*

_{k}_{−}

*y*

*(*

_{i}*t*

*)).S*

_{k}*i*

(4.9)

Where*y*_{i}* _{m}* are the measured outputs,

*y*

*are model simulated output and*

_{i}*t*

*is the current simple time.*

_{k}As an example, to identify the parameter*θ*_{1}, 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
*ma** _{y}* =

*F*

*+*

_{yf}*F*

*, we get*

_{yr}*y*

_{1}=

*ψ*

^{¨}such that:

*mv*˙* _{y}*+

*θ*

_{1}

*ψ*¨+

*mhφ*¨+

*mv*

_{x}*ψ*˙ =

*ma*

*(4.10) From the above equation, we can deduce the expression of*

_{y}*y*1 and compute then its sensitivity

*S*1 =

_{dθ}

^{d}

^{ψ}^{¨}

_{1}and its gradient

*G*1. Next, the gradient algorithm is applied to estimate the value of the ﬁrst parameter

*θ*1. Further, the algorithm is repeated for each parameter after selecting the corresponding scenario as described a follows:

1. From an initial starting point*θ** _{i}*(

*t*

_{0}), we calculate the criterion

*C*

*(*

_{i}*t*

_{0})and the gradient

*G*

*(*

_{i}*t*

_{0}). 2. Compute the new value:

*θ*

*(*

_{i}*t*

*) =*

_{k}*θ*

*(*

_{i}*t*

_{k}_{−}

_{1})

_{−}

*αG*

*(*

_{i}*t*

_{k}_{−}

_{1})

3. Compute a new value of*C** _{i}*(

*t*

*)and*

_{k}*G*

*(*

_{i}*t*

*)taking*

_{k}*θ*

*(*

_{i}*t*

*).*

_{k}4. If the second criterion is smaller keep the new parameter value of the corresponding*θ** _{i}*(

*t*

*). Increase*

_{k}*α*for eﬃciency and increment the counter

*k*+1.

5. Else, keep the old value of*θ** _{i}*(

*t*

_{k}_{−}

_{1})and reduce

*α*to 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 of*θ. 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

0 50 100 150

-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,
*θ** _{i}* = [

*e*34,

*e*36,

*e*44,

*e*45,

*e*46,

*a*45,

*a*46,

*e*55,

*a*66]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 *θ*0 *θ*_{r}*θ*_{i}

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 to*BikeSim* 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 from*BikeSim* in blue.

0 50 100 150

Figure 4.2: Comparison between the roll, steer angles and lateral speed of the identified
model and the*BikeSim*motorcycle 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.

0 50 100 150

Figure 4.3: Comparison between the identified model and the*BikeSim* 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.

0 50 100 150

Figure 4.4: Comparison between the identified model and the*BikeSim*motorcycle 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 *α*in 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 IC** _{i}*=

where*y*_{i}* _{m}*are the actual observations containing

*n*samples and

*y*

*are the corresponding predictions, resulting from estimated parameters.*

_{i}Table 4.2quantiﬁes the identiﬁcation errors between the*BikeSim* 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* *v*_{y}*ψ*˙

*T IC* 0.0974 0.1172 0.2924 0.0565
*F IT*_{%} 79.9940 77.7522 66.2456 79.9202

**#** *φ*˙ *δ*˙ *F*_{yf}*F*_{yr}

*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 with*BikeSim*data
demon-strates the potential of the identiﬁcation procedure.

88 Chapter 4. Identification of Two-bodies model