## 4 Methodology  
The purpose of this research framework is to achieve accurate and fast-converging parameter estimation for a vehicle suspension system. The method uses time-series data, either in simulation or from the experimental setup, that contain the acceleration of the wheels and body and the orientation of the vehicle body. While driving the vehicle at different speeds on the three selected test tracks--namely, chassis-twist, herringbone and washboard with different bump amplitudes and wavelengths--the accelerometer and gyroscope sensor data are recorded or generated synthetically. These data are used for parameter identification, and the benefit of the proposed method is that it does not necessitate an expensive test rig. The _twelve_ parameters (\(P\)) presented in Table 1 are estimated using the square of the error between the ideal responses (\(\hat{R}\)) obtained from the mathematical model in (1)-(7) and the measured responses (\(R\)) recorded during the driving experiment. There are ten different tracks, as will be explained in Section 4.2. For track \(i\), the errors in the responses can be written as in (10). \[e_{i,P_{est}}=\sqrt{\sum_{k}(\hat{R}_{i,k}(P_{est})-R_{i,k}(P_{real}))^{2}} \tag{10}\]  
where \(k\) indicates the instant measurement. For the same length of measurements, for each track \(i\), response errors \(e_{i,P_{est}}\) can be calculated. The optimization objective is to minimize the weighted sum of these errors. The optimization problem can be defined as in (11). \[\min_{P_{est}}w_{i}e_{i,P_{est}} \tag{11}\]  
The novelty of the proposed method is the adjustment of these track weights \(w_{i}\) adaptively during the optimization procedure to exploit the test track data and achieve faster-converging and more accurate estimation. To solve the optimization problem in (11), adaptive differential evolution with an external archive (JADE) is utilized. The details of this optimization framework are presented in Section 4.1. In this optimization procedure, a population of estimates of the parameters \(P_{est}\) is evolved towards the real parameters  
Figure 5: CAD model of chassis (on the **left**) and FE model of simplified chassis with pipe curvatures removed and displacement inputs given at wheel contacts (on the **right**) \(C_{i}\). \(P_{real}\) throughout the iterations. The track weights \(w_{i}\) sum to 1 and are recalculated in the proposed method at each optimization iteration. To determine the adaptation in \(w_{i}\), at the end of each optimization iteration, the variance of the parameter estimates in the population, \(\sigma_{l}\) are calculated for each parameter, and normalized with the mean of that parameter. Let us indicate the parameter index is with subscript \(l\). The calculated self mean-normalized parameter variances are later normalized with all 12 parameters in the set, a normalized spread of the parameters \(s_{l}\) is calculated in (12). Note that \(s_{l}\) is obtained for the output population of optimization iteration with current weights \(w_{i}e_{i},p_{est}\). \[ \begin{split}\sigma_{l}&=\frac{\sigma_{l}}{\mu_{l} }\\ s_{l}&=\frac{\sigma_{l}}{\sum\sigma_{l}}\end{split}  \tag{12}\]  
A parameter-to-track mapping is generated, as will be explained in Section 4.3. This procedure produces a matrix \(P2T\) that maps the importance of the parameters to the importance of the tracks; the weights of the tracks \(w_{i}\) are updated as given in (13). To preserve continuous and consistent optimization, before continuing with the next iteration, the total cost is updated based on the new track weights. \[w_{i}=P2T\cdot s \tag{13}\]  
The summary of the proposed method is presented in Figure 6. The novelty of the proposed method is in the adaptation of the track weights: those tracks that demonstrate the effect of the parameters with a bigger spread in the population at the output of the optimization iteration will have bigger weights.