A Novel Parameter Estimation Scheme for Vehicle Suspension Systems Based on Response and Test Track Prioritization

Abstract

In this paper, a system identification methodology based on vehicle response and test track prioritization is presented. The proposed method can be used to perform time-domain parameter estimation by simply driving the vehicle on different road profiles, thus eliminating the necessity of having a special test setup or excitation input feed. The method is based on exploiting the dependence of the identifiability of vehicle parameters on vehicle responses and test tracks. To prioritize the test tracks, the responses and test tracks are weighed based on their capabilities to estimate the vehicle parameters, and these weights are obtained in a preliminary analysis of a seven-degree-of-freedom ride model developed in MATLAB^® (R2021b). An optimization framework based on a differential evolution scheme with an optional archive scheme (JADE) is used to estimate the twelve vehicle parameters. The proposed system identification scheme is verified on tests using a high-fidelity vehicle model developed in ADAMS^® (29.2)using several tracks with different severities. Experimental validation is performed on a test vehicle, and it is demonstrated that the vehicle suspension system parameters can be identified accurately with fast convergence performance.

1. Introduction

System identification is routinely used to estimate the parameters of the mathematical model of a system from the measured data [1]. Estimating vehicle model parameters has been an intriguing problem for the evaluation of ride comfort [2]. A detailed literature survey is presented in [3]. In general, the parameter estimation procedures can be divided into two-groups: time-domain approaches [4] and frequency-domain approaches [5]. Moreover, in terms of the data collection aspect, the studies can be divided into two-groups: system identification using excitation data [6] and system identification on a vehicle driven on different road profiles [7].

Time-domain system identification using excitation input is the major method followed in the literature. Best and Gordon proposed a robust technique to identify suspension parameters based on linear regression of impulse–momentum equations by using randomised time intervals for integration [8]. Kanchwala used a hardware-in-the-loop vehicle simulator, applied rectangular bump motion to the vehicle and used torque inputs for excitation in all-wheel-drive vehicles [9]. Kim and Ro used a linearised seven-degree-of-freedom model and a singular perturbation method to identify the parameters of an active suspension system [10]. Zhou et al. used a Jacobian approach to determine suspension and damping rates of a planar double-wishbone suspension mechanism and compared their results against ADAMS simulations [11]. Balike et al. also worked on a double-wishbone quarter car model to estimate suspension damping when subjected to harmonic and bump/pothole excitation [12].

Numerous frequency-domain system identification approaches using specific excitation signals are also available in the literature. Attia et al. utilized IMU signals only to estimate the transfer functions for wheels and active suspension actuators in the presence of noise and uncertainties [13]. Similarly, Garcia and Patino used accelerometer inputs for suspension parameter estimation and compared estimation results of a Kalman filter, particle filter and neural networks [14]. Thite et al. used a matrix inversion approach to incorporate vehicle dynamic constraints in a full-scale four-post rig for system identification with noisy experimental data [15]. Cossalter et al. worked on motorcycles and used assumptions for the distribution of vehicle forward speed to identify suspension characteristics with a frequency-domain approach [16]. Cui and Kurfess used a sine sweep to identify the parameters of a non-linear full vehicle model and the shock absorber hysteresis characteristics on an experimental test rig [6]. Demic used dynamic programming to identify the vehicle vibration parameters by applying experimental data collected from a motor vehicle with noisy instrumentation [17].

Similar to this paper, employing different road profiles for vehicle ride parameter estimation has been studied in the literature. In the time domain, Thaller et al. drove their vehicle over bumps and used the continuous time response data demonstrating the vehicle dynamic response to estimate the suspension parameters [4]. Similarly, Reiterer et al. used four accelerometer measurements on a vehicle traversing over a single bump to estimate vehicle parameters using subspace identification, model updating and direct continuous time system identification techniques [18]. Kanchwala benefited from sinusoidal test track profiles in the time domain, and he proposed a method to identify the suspension system parameters using a Laplace domain reduced order to handle the complexity between a full car and a quarter car model [7]. Kohlhuber et al. estimated vehicle and tire parameters for a vehicle with different passenger and luggage loads using a Kalman filtering-based approach in real-life driving profiles [19]. Ventura et al. modelled a vehicle as a combination of 42 robotic formalisms and identified dynamic parameters of each formalism under sinus steering motion and straight-line tests with sudden deceleration [20].

Driving a vehicle over small bumps and extracting vehicle parameters in the frequency domain has been widely studied in the literature. Zhang et al. used a vehicle frequency response function and the data indicating the displacements of the vehicle–road contact points to estimate this function [21]. Pan et al. trained and tested a deep neural network on the data collected during bumpy road travel and showed that their method can estimate the suspension system parameters in real-time [22]. Dong et al. used a subspace identification technique with measured vertical accelerations of the unsprung mass as input to the system identification procedure in addition to the other measured outputs [23].

The vehicle’s suspension and tires isolate the occupants from ground disturbances and suspension models are studied thoroughly in the literature [24,25]. Depending on the vehicle, suspension models can be broadly classified into: quarter car, half car and full car models [26]. The quarter car model is simple and does not account for the effects of three other wheels. The half car model is midway in complexity and allows for fore/aft or sideways interaction but not both. The full car model is computationally complex but accounts for all of the above-mentioned effects. We have considered a seven-degree-of-freedom (dof) full car suspension model for this study. The system identification methodology is outlined in the adjoining section.

2. Proposed Approach

In this section, we provide a brief overview of the proposed methodology. For system identification of the vertical dynamics of a vehicle, in addition to the vehicle suspension and tire parameters, the centre of gravity (C.G.) location, unsprung and sprung masses and their inertias have to be estimated, as they greatly influence the ride performance [18,27]. Among this set of parameters, some can be measured easily (e.g., the wheelbase), while others are difficult to estimate (e.g., the pitch and roll inertia). In addition, the suspension parameters are dependent on the operating condition (tire stiffness is a function of inflation pressure and load) and also vary with the life of the vehicle (damper performance deteriorates with time due to reduction in oil viscosity, resulting in a decrease in the damping force). The system parameters of interest are defined along with the difficulty level involved in their estimation in

Table 1. Vehicle ride parameters of interest (P) with the amount of effort required for their estimation.

P	Units	Description	Estimation
$K_f$	N/mm	Front wheel rate	Moderate
$K_r$	N/mm	Rear wheel rate	Moderate
$C_f$	N-s/mm	Front damping	Difficult
$C_r$	N-s/mm	Rear damping	Difficult
$K_t$	N/mm	Tire stiffness	Difficult
$C_t$	N-s/mm	Tire damping	Difficult
l	m	Wheel base	Easy
$l_f$	m	C.G. from front axle	Moderate
b	m	Half track width	Easy
$M_s$	kg	Sprung mass	Moderate
$M_{\mathrm{uf}}$	kg	Front unsprung mass	Easy
$M_{\mathrm{ur}}$	kg	Rear unsprung mass	Easy
$I_{xx}$	kg-m2	Roll inertia	Difficult
$I_{yy}$	kg-m2	Pitch inertia	Difficult

. The parameters are divided into three categories based on the amount of effort required for their estimation: easy, moderate and difficult.

We propose a novel system identification procedure to estimate these twelve parameters that influence vehicle vertical dynamics. The parameter estimation on time-domain data is based on a least-square optimization scheme. There are a wide variety of optimization methods; however, value-based techniques like evolutionary optimization or genetic algorithms are more suitable due to the complex dependency of the optimization parameters on the cost definition. As an example, differential evolution (DE) is a heuristic global optimization approach that has faster convergence and necessitates fewer control variables as compared to genetic algorithms [28]. Among various modifications of DE, Zhang et al. proposed JADE: adaptive differential evolution with an external archive, in which the historical data provide the information regarding the direction of the progress and improve diversification with adaptation [29]. In this paper, we employed the JADE algorithm as the optimization procedure.

The purpose of this paper is to investigate the effect of different test tracks and vehicle parameters on responses to develop a system identification framework. The vehicle responses are acquired using accelerometers mounted on the wheel axles and suspension-to-body attachment points and a gyro mounted on the vehicle chassis. The time-domain data are collected while driving the vehicle at different speeds on selected test tracks. The sensors—namely, the accelerometers and IMU—can be easily installed, thereby eliminating the need for a specialized and expensive test rig.

Next, mapping between responses and the strength of various test tracks to excite these responses is developed. Three test tracks used for this study are the chassis-twist, herringbone and washboard, which all have different bump amplitudes and wavelengths. Finally, the vehicle is driven on these test tracks, and response data are measured. The proposed optimization procedure based on JADE evaluates these data measures and systematically estimates the vehicle ride parameters. The contribution of this paper is threefold: (1) the dependency level between estimation parameters and vehicle responses is extracted, (2) evaluation of the test tracks based on their performance to excite vehicle responses is performed, and (3) in the proposed systematic approach using data collected on different test tracks, the vehicle parameters are estimated. The proposed method is verified on the data acquired in the simulation and co-simulation environments, and its accuracy is validated on experimental data.

During system identification, the spread of the estimated parameters in the population throughout the optimization iterations—namely, offspring—is evaluated. It is known that after a certain number of offspring, the parameters should converge to the original values. However, the decline in the spread of the estimates in the population is considered as the dependency of the system identification procedure on the test tracks. It is observed that some test tracks outperform in terms of faster decline in the spread of the estimates of some parameters. Selection of the test tracks is proposed at this stage. On the selected test tracks, the tests are repeated on a higher-fidelity vehicle model in an ADAMS–MATLAB co-simulation environment to verify the proposed estimation algorithm. Following this verification test, the system identification procedure is applied to the actual vehicle. The proposed methodology adopted in the paper is explained using a flowchart in Figure 1.

We start with the test vehicle for which the parameters need to be estimated in of Figure 1. Next, we model the vehicle ride characteristics using a seven-dof model. Next, we validate the model against ADAMS simulation results in and then investigate the field-test-based approach in . In the simulation approach, full car multi-body dynamics models of the vehicle and test tracks are developed in the ADAMS environment [30]. In the field-test-based method, the vehicle is driven on a variety of deterministic test tracks, and the responses are measured using a set of accelerometers and a gyro. The simulation results and test track measurements are used for model fitting using an improved JADE algorithm, as shown in . For this purpose, the tracks and responses are weighed against the parameters to be estimated.

The structure of the paper is as follows: In Section 3, mathematical modelling of the vehicle—including the parameters to be estimated, the experimental setup and the instrumentation used in the validation tests and the ADAMS model used in the verification tests—are presented. In Section 4, the proposed method, response prioritization and track prioritization scheme are presented. In Section 5, the results are presented and the accuracy of the method is validated. The paper concludes with Section 6.

3. Vehicle Model and Experimental Setup

3.1. Mathematical Modelling

To study vehicle ride comfort, a 7-dof vehicle model including suspension is considered to exhibit adequate characterisation. Figure 2 shows the full car ride model. $A_i, B_i$ and $C_i$ correspond to the axle, suspension-to-body attachments and ground–wheel contacts, respectively.

The vehicle ride model has 4 dof corresponding to wheel jounces $z_1$ through $z_4$ and an additional 3 dof for bounce z, roll $\varphi$ and pitch $\theta$, respectively. The ground displacements at four wheel-contact points are given by $u_1$ through $u_4$, respectively, and vertical dynamics are represented by (1)–(7).

$$\begin{array}{cc}\hfill M_s\ddot{z}=& -K_f{z}_{\mathrm{fl}}-K_f{z}_{\mathrm{fr}}-K_r{z}_{\mathrm{rl}}-K_r{z}_{\mathrm{rr}}\hfill \\ \hfill & -C_f{\dot{z}}_{\mathrm{fl}}-C_f{\dot{z}}_{\mathrm{fr}}-C_r{\dot{z}}_{\mathrm{rl}}-C_r{\dot{z}}_{\mathrm{rr}}-M_{s}g\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill I_{xx}\ddot{\theta }=& K_f{l}_f{z}_{\mathrm{fl}}+K_f{l}_f{z}_{\mathrm{fr}}+C_f{l}_f{\dot{z}}_{\mathrm{fl}}+C_f{l}_f{\dot{z}}_{\mathrm{fr}}\hfill \\ \hfill & -K_r{l}_r{z}_{\mathrm{rl}}-K_r{l}_r{z}_{\mathrm{rr}}-C_r{l}_r{\dot{z}}_{\mathrm{rl}}-C_r{l}_r{\dot{z}}_{\mathrm{rr}}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill I_{yy}\ddot{\varphi }=& -K_{f}b{z}_{\mathrm{fl}}-K_{r}b{z}_{\mathrm{rl}}-C_{f}b{\dot{z}}_{\mathrm{fl}}-C_{r}b{\dot{z}}_{\mathrm{rl}}\hfill \\ \hfill & +K_{f}b{z}_{\mathrm{fr}}+K_{r}b{z}_{\mathrm{rr}}+C_{f}b{\dot{z}}_{\mathrm{fr}}+C_{r}b{\dot{z}}_{\mathrm{rr}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill M_{\mathrm{uf}}{\ddot{z}}_1=& K_f{z}_{\mathrm{fl}}+C_f{\dot{z}}_{\mathrm{fl}}-K_t{z}_{\mathrm{t},\mathrm{fl}}-C_t{\dot{z}}_{\mathrm{t},\mathrm{fl}}-M_{\mathrm{uf}}g\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill M_{\mathrm{uf}}{\ddot{z}}_2=& K_f{z}_{\mathrm{fr}}+C_f{\dot{z}}_{\mathrm{fr}}-K_t{z}_{\mathrm{t},\mathrm{fr}}-C_t{\dot{z}}_{\mathrm{t},\mathrm{fr}}-M_{\mathrm{uf}}g\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill M_{\mathrm{ur}}{\ddot{z}}_3=& K_r{z}_{\mathrm{rl}}+C_f{\dot{z}}_{\mathrm{rl}}-K_t{z}_{\mathrm{t},\mathrm{rl}}-C_t{\dot{z}}_{\mathrm{t},\mathrm{rl}}-M_{\mathrm{ur}}g\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill M_{\mathrm{ur}}{\ddot{z}}_4=& K_r{z}_{\mathrm{rr}}+C_f{\dot{z}}_{\mathrm{rr}}-K_t{z}_{\mathrm{t},\mathrm{rr}}-C_t{\dot{z}}_{\mathrm{t},\mathrm{rr}}-M_{\mathrm{ur}}g\hfill \end{array}$$

(7)

(1)–(3) represent the ride, pitch and roll dynamics, respectively, while (4)–(7) represent the four wheel jounce motions. The effective suspension travel (difference between the body motion and unsprung mass motion at individual wheel corners) and wheel travel (effective unsprung mass motion obtained by subtracting the ground excitation) of the front left/right and the rear left/right are given by (8):

$$\begin{array}{cc}\hfill z_{\mathrm{fl}}=z_1-y_1,& z_{\mathrm{fr}}=z_2-y_2,\hfill \\ \hfill z_{\mathrm{rl}}=z_3-y_3,& z_{\mathrm{rr}}=z_4-y_4,\hfill \\ \hfill z_{\mathrm{t},\mathrm{fl}}=y_1-u_1,& z_{\mathrm{t},\mathrm{fr}}=y_2-u_2,\hfill \\ \hfill z_{\mathrm{t},\mathrm{rl}}=y_3-u_3,& z_{\mathrm{t},\mathrm{rr}}=y_4-u_4.\hfill \end{array}$$

(8)

In (8), $z_1$ through $z_4$ are the vertical displacements of the suspension-to-body attachment points that are given by (9):

$$\begin{array}{cc}\hfill z_1=z-l_f\theta +b\varphi ,& z_2=z-l_f\theta -b\varphi ,\hfill \\ \hfill z_3=z+l_r\theta +b\varphi ,& z_4=z+l_r\theta -b\varphi .\hfill \end{array}$$

(9)

In (9), it has been assumed that the vehicle pitch and roll motion happens about its centre of gravity, which is at a distance of $l_f$ from the front and $l_r$ from the rear axle; it is assumed that there is left–right symmetry, and the half track width of the vehicle is given by b, as shown in Figure 2.

3.2. Experimental Setup

The experimental validation of the proposed method is done using an all-terrain vehicle (ATV) built for the BAJA SAE-India student competition, as shown in Figure 3. The vehicle is equipped with a double-wishbone front suspension and a semi-trailing-arm rear suspension (for details, see [31]). The front suspensions have Float 3 and the rear suspensions have Float 3 EVOL R shocks [32]. The suspensions and tires are characterized on an MTS 850.25 damper test rig and MTS 370.10 elastomer test rig at the National Automotive Test Tracks (NATRAX) facility of the Government of India at Indore [33]. The suspension and tire mounted on the test rig is also shown in Figure 3a.

The vehicle chassis parameters (sprung mass, C.G. location and roll and pitch inertias) are obtained from the detailed CAD model of the vehicle, as shown in Figure 3b. The vehicle parameters P obtained from rig testing (suspension and tire) and the CAD model are reported in

Table 2. Baseline vehicle parameters P known beforehand.

Parameters	${\mathit{k}}_{\mathit{f}}$	${\mathit{K}}_{\mathit{f}}$	${\mathit{k}}_{\mathit{r}}$	${\mathit{K}}_{\mathit{r}}$	${\mathit{c}}_{\mathit{f}}$	${\mathit{C}}_{\mathit{f}}$	${\mathit{c}}_{\mathit{r}}$	${\mathit{C}}_{\mathit{r}}$	l
Values	25	10.5	15	6.5	1.5	0.63	1.36	0.59	1.6
Parameters	$K_t$	$C_t$	$M_{\mathrm{uf}}$	$M_{\mathrm{ur}}$	$I_{xx}$	$I_{yy}$	$M_s$	$l_f$	b
Values	40	0.2	15	12.5	257	246	240	0.85	1.2

.

From measured suspension properties using the test rig ($k_{f,r}$ and $c_{f,r}$), the equivalent suspension properties at the wheels ($K_{f,r}$ and $C_{f,r}$) are obtained after multiplying by the square of the motion ratios. These motion ratios are functions of the suspension geometry and can be determined either from the ADAMS half car suspension models or can be measured experimentally [34]. In this paper, the motion ratios of the BAJA ATV are measured experimentally by raising and lowering the wheel of interest and measuring the spring compression and extension; for details of these experiments, see [35].

The vehicle ride performance is characterised by recording the acceleration responses of the suspension-to-body attachment points $B_i$ and the wheel accelerations measured at axle locations $A_i$. For this purpose, the vehicle is instrumented with MEMS-based ADXL326 accelerometers (a total of eight). These accelerometers are mounted on wheel axle locations and suspension-to-body pivot points using magnetic mounting strips (see Figure 4a). The data acquisition system consists of a FAT32 microSD card module and an Arduino ATMega328P microcontroller board. These systems are compatible with a breadboard, making it easy to connect them together and power the circuit using a standard 9V battery. The data acquisition system is mounted on the vehicle dashboard (see Figure 4b).

In addition to this, a gyro sensor and a vertical accelerometer is also mounted at the C.G. location of the vehicle in order to record the pitch and roll rates and the vertical acceleration of the sprung mass, which are used for vehicle characterisation in addition to the axle and body point acceleration responses.

3.3. ADAMS Model

Before validating the proposed method on the experimental setup, we develope an ADAMS model of the vehicle with fidelity higher than that of the 7-dof mathematical model to be used in an intermediate verification step. The finite-element model of the chassis is developed in Nastran and imported to ADAMS. The four mounting locations $B_i$ are defined as interface nodes. In addition, the model has three more interface nodes denoted as D, E and F corresponding to the centre-of-mass locations of the driver, engine and battery, respectively (see Figure 5 right). Three rigid bodies with mass and inertia properties representing the driver, engine–driveline assembly and battery are attached to these interface nodes. The inertia properties of the driver are obtained from [36] and those of the engine and battery are obtained from the vehicle CAD model.

Four spring–dashpot pairs with the equivalent properties from Table 2 are attached between points $B_i$ on the car body and the unsprung masses at wheel axle $A_i$. The unsprung masses are connected to ground contact points $C_i$ by springs and dashpots representing the tyre properties (see Table 2).

For performing ADAMS simulation, a flexible multi-body dynamics model is developed. The vehicle chassis is made of a roll-cage type structure. The effect of frame flexibility is captured by means of the FE model (Figure 5 right) of the roll cage, as shown in Figure 5 left.

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 ($\widehat{R}$) 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).

$$e_{i,P_{est}}=\sqrt{\sum _k{({\widehat{R}}_{i,k}\left(P_{est}\right)-R_{i,k}\left(P_{real}\right))}^2}$$

(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).

$$\underset{P_{est}}{min}{w}_i{e}_{i,P_{est}}$$

(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 $P_{real}$ 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 $w_i{e}_{i,P_{est}}$.

$$\begin{array}{cc}\hfill \overline{{\sigma }_l}& =\frac{{\sigma }_l}{{\mu }_l}\hfill \\ \hfill s_l& =\frac{\overline{{\sigma }_l}}{\sum \overline{{\sigma }_l}}\hfill \end{array}$$

(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.

$$w_i=P2T·s$$

(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. To obtain a mapping between tracks and parameters before running the optimization procedure, using the mathematical model, first a parameters-to-response mapping is generated and then a responses-to-tracks mapping is created. These mappings are merged to obtain a global parameters-to-tracks mapping.

4.1. Optimization Framework: JADE

In parameter estimation problems, evolutionary algorithms are often utilized. In this paper, adaptive differential evolution with an external archive (JADE) [29] is employed as the optimization framework. JADE has advantages in its usage of crossover and mutation functions similar to other evolutionary algorithms [28] and adaptation for the parameters of mutation and crossover factors in an ’evolution of the evolution’ concept similar to other differential evolutionary algorithms [37]. Furthermore, since the best candidate to be used in mutated offspring generation is selected from the percentage of the best population members instead of the best one itself, premature convergence is avoided. Moreover, it benefits from an external archive whose members are used in differential mutation, which is a further precaution against impoverishment.

Mutation: In JADE with an external archive, in addition to the population set, denoted by $\mathbf{P}$, an archive of solutions, denoted by $\mathbf{A}$, is preserved. A mutation vector is generated:

$${\mathbf{v}}_{i,g}={\mathbf{x}}_{i,g}+F_i\left({\mathbf{x}}_{\mathrm{best},g}^p-{\mathbf{x}}_{i,g}\right)+F_i\left({\mathbf{x}}_{r1,g}-{\tilde{\mathbf{x}}}_{r2,g}\right)$$

(14)

where g indicates the generation number, i indicates the ID in the population, ${\mathbf{x}}_{i,g}$ is the vector in $\mathbf{P}$ going through the mutation operation, $F_i$ is the mutation factor associated with the vector ${\mathbf{x}}_{i,g}$, ${\mathbf{x}}_{\mathrm{best},g}^p$ is the vector randomly chosen as one of the top 100$p\%$ individuals in $\mathbf{P}$ with $p\in 0,1$, ${\mathbf{x}}_{r1,g}$ is a vector randomly chosen from $\mathbf{P}$ satisfying ${\mathbf{x}}_{r1,g}\ne {\mathbf{x}}_{i,g}$, and finally, ${\tilde{\mathbf{x}}}_{r2,g}$ is a vector randomly chosen from $\mathbf{P}\cup \mathbf{A}$ satisfying ${\tilde{\mathbf{x}}}_{r2,g}\ne {\mathbf{x}}_{i,g}$ and ${\tilde{\mathbf{x}}}_{r2,g}\ne {\mathbf{x}}_{r1,g}$.

The mutation factor $F_i$ is selected for each individual ${\mathbf{x}}_{i,g}$ of generation g according to a Cauchy distribution with location parameter ${\mu }_F$ and scale parameter 0.1. If $F_i>1$, then it is truncated to $F_i=1$, and if $F_i<0$, then it is regenerated. The value ${\mu }_F$ is an adaptive parameter (see below for parameter adaptation).

$$F_i={randc}_i\left({\mu }_F,0.1\right)$$

(15)

Crossover: This is performed after mutation and a binomial selection are done:

$$u_{j,i,g}=\left\{\begin{array}{cc}{v}_{j,i,g},\hfill & \mathrm{if}rand(0,1)\le C{R}_i\mathrm{or}j=j_{rand}\hfill \\ x_{j,i,g},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(16)

where $rand(0,1)$ is a uniform random number generator outputting within the interval of $[0,1]$ independently for each generation g, individual i and component j. The parameter $j_{\mathrm{rand}}$ is a random integer generator with the same size as the individual length, and  $C{R}_i$ is the crossover probability generated independently for each individual ${\mathbf{x}}_i$ according to a normal distribution of mean ${\mu }_{CR}$ and a standard deviation of 0.1. It is truncated to $[0,1]$. The variable ${\mu }_{CR}$ is an adaptive parameter

$$C{R}_i={randn}_i({\mu }_{CR},0.1)$$

(17)

Selection and Parameter Updates: The next spring ${\mathbf{x}}_{i,g+1}$ is set equal to the one with the minimum cost function between ${\mathbf{x}}_{i,g}$ and ${\mathbf{u}}_{i,g}$. If ${\mathbf{u}}_{i,g}$ is selected, then ${\mathbf{x}}_{i,g}$ is moved to the archive set $\mathbf{A}$, and  ${\mu }_{CR}$ and ${\mu }_F$ are labelled as successful parameters. At each generation g, randomly selected solutions in the archive are removed to keep the size of $\mathbf{A}$ less than or equal to the population size, if necessary.

For each generation g, denote $S_{CR}$ as the set of all successful crossover probabilities and denote $S_F$ as the set of all successful mutation factors. Then for the next generation, ${\mu }_{CR}$ and ${\mu }_F$ are calculated according to:

$$\begin{array}{cc}\hfill {\mu }_{CR}& =(1-c)·{\mu }_{CR}+c·{mean}_A\left(S_{CR}\right)\hfill \\ \hfill {\mu }_F& =(1-c)·{\mu }_F+c·{mean}_L\left(S_F\right)\hfill \end{array}$$

(18)

where c is a positive constant between 0 and 1, ${mean}_A(·)$ is the arithmetic mean, and ${mean}_L(·)$ is the Lehmer mean:

$${mean}_L\left(S_F\right)=\frac{{\sum }_{F\in S_F}{F}^2}{{\sum }_{F\in S_F}F}$$

(19)

Initially, ${\mu }_{CR}$ and ${\mu }_F$ can be chosen as $0.5$. For details about JADE, please refer to [29].

4.2. Test Tracks

Accelerated fatigue tracks are widely used to test vehicle suspensions for ride comfort and durability. These tracks consist of a variety of surfaces from low-to-high severity and are designed to produce accelerated ageing of the vehicle’s structure and components. Fatigue tracks contain a variety of surfaces like Belgium pave, pot holes, twist track, washboard, resonance track, herringbone, rough road, sine sweep, etc. There are a total of 28 different profiles that are available at the National Automotive Test Tracks (NATRAX) where the experimental study was performed. These tracks enable manufacturers to decide their own course of durability with various options.

The test tracks and tests performed for the proposed parameter identification procedure at the NATRAX facility are given in

Table 3. Details of the vehicle running on continuous displacement tracks with low (L) and high (H) severity levels S at low (L) and high (H) speeds V.

Track	S	Mag	Pitch	V	Value
		(mm)	(mm)		(kmph)
1. Chassis-twist	L	150	2000	L	7.5
		25	600	H	15.5
	H	200	4000	L	5.5
		25	600	H	14.5
2. Herringbone	L	10	500	L	9
		25	600	H	14
	H	20	800	L	14
		25	600	H	19
3. Washboard	L	15	400	L	7.5
				H	9.5
	H	25	600	L	11
				H	13

. These tracks have a simple displacement profile.

• Chassis-Twist Track: This is a 4 m-wide track. The left and right track ends are sinusoids separated in phase by $\pi$ radians (see Figure 7b). Between the two extreme profiles on the right and the left, the variation is linear. During testing, a narrow vehicle can be driven on one side of the track to increase the intensity of excitation at one wheel at the cost of reducing the intensity at the other.
• Herringbone Track: In this track, the bumps are inclined at an angle of 20 degrees across the width of the track (Figure 7a). The road inputs for the left and right wheels are no longer identical; the phase difference between them depends on the vehicle width.
• Washboard Track: This is the simplest sinusoidal track. Inputs to the left and right wheels are identical.

The vehicle used for field testing has an unloaded tire radius of 300 mm. Since the wavelength of the Herringbone sinusoidal track is comparable to the tire radius, although the road profile is sinusoidal, the actual displacement input to the wheel is not sinusoidal. The road surface geometry has to be converted into an equivalent vertical input to be applied at the wheel–ground contact points $C_i$ of the vehicle ride model. A simple road contact kinematic model that calculates the effective wheel displacement is employed.

Road Contact Kinematic Model: Consider a rigid wheel going over a sinusoidal track as shown in Figure 8a. The track amplitude is A, the wavelength is L, the wheel radius is R, the coordinates of the wheel centre O are $(\overline{x},\overline{u})$, and the wheel–ground contact point P is $(x,u)$. From geometrical considerations,

$$\begin{array}{c}\hfill x=\overline{x}+Rsin\theta ,\end{array}$$

(20)

$$\begin{array}{c}\hfill u=Asin\left(\frac{2\pi x}{L}\right),\end{array}$$

(21)

$$\begin{array}{c}\hfill \frac{du}{dx}=tan\theta .\end{array}$$

(22)

Differentiating (20) gives

$$\dot{x}=\dot{\overline{x}}+R\dot{\theta }cos\theta ,$$

(23)

where $\dot{\overline{x}}$ is the longitudinal velocity V (taken as a constant). Substituting (21) into (22) and differentiating, we get

$$\dot{\theta }=-A{\left(\frac{2\pi }{L}\right)}^{2}sin\left(\frac{2\pi x}{L}\right)\dot{x}{cos}^2\theta .$$

(24)

(23) and (24) yield

$$\begin{array}{c}\hfill \dot{x}=\frac{V}{1+AR{cos}^3\theta {\left({\displaystyle \frac{2\pi }{L}}\right)}^{2}sin\left({\displaystyle \frac{2\pi x}{L}}\right)},\dot{\theta }=\frac{-Av{\left({\displaystyle \frac{2\pi }{L}}\right)}^{2}sin\left({\displaystyle \frac{2\pi x}{L}}\right){cos}^2\theta }{1+R{cos}^3\theta A{\left({\displaystyle \frac{2\pi }{L}}\right)}^{2}sin\left({\displaystyle \frac{2\pi x}{L}}\right)}\end{array}$$

(25)

Equation set (25) is solved numerically in MATLAB using ode45 with initial conditions $x=\overline{x}={\displaystyle \frac{L}{4}},\theta =0$ at $t=0$.

A typical solution is shown in Figure 8b.

This ground excitation is used as input for solving the differential equations of motion of the ride model to compute vehicle responses.

4.3. Response and Track Prioritization

Response Prioritization: The vehicle ride behaviour is objectively evaluated by means of pitch and roll rates ($\dot{\theta },\dot{\varphi }$) and body and axle acceleration responses ${\ddot{z}}_i,{\ddot{y}}_i\forall i\in 1-4$. These vehicle responses (physics) on a particular test track are a function of the vehicle parameters. To understand the effects of parameters on the responses of interest, each parameter is varied one at a time. Each of these vehicle parameters is increased by 25% of its baseline value for the vehicle simulation performed at low and high speeds on the low-severity herringbone track. The effects of the parametric sweep on the vehicle responses ($\dot{\varphi },\dot{\theta },\ddot{z},A_b\left({\ddot{z}}_i\right),A_w\left({\ddot{y}}_i\right)$) are quantitatively estimated in terms of percentage variation of the peak value of the responses obtained using the baseline and the new sets of parameters. As an example, the results of the simulation responses of a vehicle traversing on a low-severity herringbone track are tabulated in

Table 4. Parameter P response mapping for low- and high-speed tests for low-severity herringbone track. The effects of parameters on the amplitudes of roll and pitch rates and bounce, body and axle point acceleration responses is reported.

P	V	$\dot{\mathit{\varphi }}$	$\dot{\mathit{\theta }}$	$\ddot{\mathit{z}}$	${\ddot{\mathit{z}}}_1$	${\ddot{\mathit{z}}}_3$	${\ddot{\mathit{y}}}_1$	${\ddot{\mathit{y}}}_3$
$I_{xx}$	L	⌀	−22	1	−12	3	−3	2
	H	−4	−19	⌀	−9	⌀	⌀	⌀
$I_{yy}$	L	−18	⌀	⌀	⌀	−2	⌀	−3
	H	−25	⌀	⌀	1	⌀	⌀	⌀
$M_s$	L	⌀	−3	−13	−6	−15	⌀	⌀
	H	9	−2	−13	−6	−12	⌀	⌀
$l_f$	L	−6	18	2	22	15	7	5
	H	⌀	20	⌀	19	12	−3	3
$K_f$	L	3	3	6	5	6	−3	⌀
	H	1	1	2	⌀	⌀	5	⌀
$K_r$	L	−3	⌀	−2	⌀	⌀	3	−3
	H	⌀	−4	1	⌀	⌀	⌀	3
$C_f$	L	12	12	8	14	7	−4	−3
	H	7	4	2	6	1	10	⌀
$C_r$	L	6	2	8	3	9	⌀	−6
	H	⌀	2	⌀	⌀	4	⌀	−10
$M_{uf}$	L	7	2	6	8	8	−7	⌀
	H	9	−5	5	3	8	−11	⌀
$M_{ur}$	L	⌀	⌀	−2	⌀	⌀	⌀	−5
	H	−4	⌀	−5	−3	−5	⌀	⌀
$K_t$	L	3	10	3	6	8	11	10
	H	17	19	15	15	12	20	19
$C_t$	L	2	4	2	3	4	6	2
	H	−10	−7	−6	−3	−4	⌀	⌀

for low- and high-speed tests ($V\in$ L and H, respectively). The responses that are unaffected by variation of a certain parameter are denoted with ⌀.

For developing this parameter-to-response mapping, the dependence of a parameter to a particular response as presented in Table 4 is translated into a dependency-star rating as shown in

Table 5. Parameter P-to-response R mapping.

P	$\dot{\mathit{\varphi }}$	$\dot{\mathit{\theta }}$	$\ddot{\mathit{z}}$	${\mathit{A}}_{\mathit{b}}\left({\ddot{\mathit{z}}}_{\mathit{i}}\right)$	${\mathit{A}}_{\mathit{w}}\left({\ddot{\mathit{y}}}_{\mathit{i}}\right)$
$I_{xx}$	★	★ ★ ★	★	★ ★	★
$I_{yy}$	★ ★ ★	★	★	★	★
$M_s$	★	★	★ ★ ★	★ ★ ★	★
$l_f$	★	★ ★ ★	★	★ ★ ★	★ ★
$K_f$	★	★	★	★	★
$K_r$	★	★	★	★	★
$C_f$	★ ★	★ ★	★	★ ★	★
$C_r$	★	★	★	★	★ ★
$M_{uf}$	★ ★	★	★	★ ★	★
$M_{ur}$	★	★	★	★	★
$K_t$	★ ★	★ ★	★ ★	★ ★	★ ★ ★
$C_t$	★ ★	★	★	★	★

. The ★ rating indicates the level of the dependence, i.e., whether the particular response is loosely or strongly affected by the underlying vehicle parameter of interest. Table 5 is the average of the dependencies of each parameter on every response for all the tracks considered in Table 3.

Track Prioritization: In the next step of the procedure, a response-to-track mapping matrix is constructed to identify the dominant effect of a particular response on a particular test track of interest. As an illustration of the rationale behind this, it is observed in the vehicle simulation performed on a low-severity herringbone track at low and high speeds that as the vehicle speed V is increased from low to high (↑), the change in the response of interest is shown as a percentage increase or decrease as compared to its value at low speed. The values are reported in

Table 6. Response to speed mapping for low-severity herringbone track.

V	$\dot{\mathit{\varphi }}$	$\dot{\mathit{\theta }}$	$\ddot{\mathit{z}}$	${\ddot{\mathit{z}}}_1$	${\ddot{\mathit{z}}}_3$	${\ddot{\mathit{y}}}_1$	${\ddot{\mathit{y}}}_3$
↑	−45	−53	24	21	55	60	68

. A similar analysis is performed for each test track in Table 3 to obtain a dependency-star rating mapping between responses and tracks.

The final step in the procedure is to determine the overall relation between the parameters and the tracks. This mapping is obtained by combining the parameter-to-response and response-to-track mappings. In

Table 7. Parameter P-to-track mapping: Rating of test tracks in terms of their convergence performance.

Track	${\mathit{I}}_{\mathbf{xx}}$	${\mathit{I}}_{\mathbf{yy}}$	${\mathit{M}}_{\mathit{s}}$	${\mathit{l}}_{\mathit{f}}$	${\mathit{K}}_{\mathit{f}}$	${\mathit{K}}_{\mathit{r}}$	${\mathit{C}}_{\mathit{f}}$	${\mathit{C}}_{\mathit{r}}$	${\mathit{M}}_{\mathbf{uf}}$	${\mathit{M}}_{\mathbf{ur}}$	${\mathit{K}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{t}}$
$T_{LL}$	★★★	★★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★ ★
$T_{LH}$	★	★ ★ ★	★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★	★ ★ ★
$T_{HL}$	★	★ ★	★	★	★ ★	★	★ ★	★	★	★	★ ★	★
$T_{HH}$	★ ★	★ ★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★	★ ★	★ ★	★ ★	★ ★ ★
$H_{LL}$	★ ★	★ ★	★ ★	★ ★ ★	★ ★	★ ★	★ ★	★ ★ ★	★	★ ★ ★	★ ★	★ ★
$H_{LH}$	★	★	★ ★	★ ★ ★	★	★	★	★	★ ★ ★	★	★	★
$H_{HL}$	★ ★ ★	★ ★ ★	★	★	★ ★	★ ★	★ ★ ★	★ ★ ★	★	★ ★ ★	★ ★ ★	★ ★
$H_{HH}$	★ ★	★	★ ★	★ ★	★	★ ★	★ ★	★ ★	★ ★ ★	★	★	★ ★
$W_{LL}$	★	★	★	★	★	★	★	★	★ ★	★ ★	★	★ ★
$W_{LH}$	★ ★ ★	★ ★	★ ★ ★	★ ★	★	★ ★ ★	★	★ ★	★ ★ ★	★ ★	★ ★ ★	★
$W_{HL}$	★ ★	★ ★ ★	★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★ ★	★ ★	★ ★	★ ★ ★	★ ★ ★	★ ★ ★
$W_{HH}$	★ ★ ★	★	★ ★ ★	★ ★	★	★	★ ★	★	★	★	★	★

, T, H and W stand for chassis-twist, herringbone and washboard tracks, respectively. The first subscript stands for the track severity, and the second subscript stands for the vehicle speed: for instance, $T_{LL}$ is the low-speed test on the low-severity chassis-twist track.

5. Results

The proposed system identification method based on track and response prioritization for the suspension system parameter estimation problem is tested using a hierarchical approach:

• Proof-of-Concept Tests: These tests are performed in a MATLAB environment. The wheel excitations are generated by the road kinematic model. The true vehicle responses are generated by the 7-dof model with known parameters. A system identification procedure based on the track and response prioritization estimates the parameters, reducing the errors between the estimated responses and true responses. In Section 5.1, (1) the evolution of spread of the parameter estimations throughout generations is presented, and (2) the normalized estimation errors for different test tracks are demonstrated and compared against our track prioritization scheme. These two results demonstrate the feasibility of the proposed method.
• Verification Tests: These are performed in a MATLAB–ADAMS co-simulation environment where the high-fidelity vehicle model with the known parameters generates the true responses. A system identification procedure based on track and response prioritization estimates the parameters by reducing the errors between the responses estimated with the 7-dof model and the true responses. In Section 5.2, body and axle accelerations and the orientation rates for a selected track are demonstrated for the ADAMS model and the 7-dof vehicle model with the estimated parameters. The results verify the capability of the proposed method to execute with the data acquired from a sophisticated model or with real data.
• Validation Tests: These tests are performed on the real test vehicle. The vehicle is driven on a real road, and the true parameters are measured and logged on the system. Similar to the above two models, the system identification procedure based on track and response prioritization estimates the parameters by reducing the errors between the response estimated with the 7-dof model and the true responses. In Section 5.3, (1) the error between the true suspension system parameters and the final parameter estimates are tabulated, and (2) the normalized errors between real test data and the 7-dof vehicle model with estimated parameters in terms of body and axle accelerations and the orientation rates for the test tracks are demonstrated. These two results validate the proposed method on real test vehicle experiments.

5.1. MATLAB Simulation Results

In Figure 9, the evolution of spread of parameter estimation throughout generations is presented. It is important to note that for different tracks, the evolution of spread of the estimations in the population changes; in other words, some knowledge related to the track and the spread of the parameters is available, and for different tracks the behaviour changes. The second point to note is that the solid green line attained with the proposed method has a greater spread in all the parameters, indicating the reduced sampling impoverishment achieved with the proposed method. The results are obtained after 100 Monte Carlo tests.

The proposed method ends the system identification routine with convergence to the final estimates faster compared to classical time-domain estimation using each track individually. In addition to the reduced sampling impoverishment achieved with the proposed method, the estimates obtained of these parameters are closer to the true values. In Figure 9, one can see that even though the performance on some individual tracks (i.e., low-speed test on high-severity chassis-twist track takes 294 offspring) in terms of the number of offspring required for convergence is better than our method, which takes 397, but the combined prediction accuracy is inferior. (Some parameters are estimated better than others when we consider individual tracks, but when the track and response prioritization scheme is applied, then both the parameter estimation and convergence improve.) To compare our track and response prioritization scheme responses against individual test track estimation schemes, we report the average sum estimation error of our method against individual test track results in

Table 8. Average estimation errors for different tracks against the proposed track prioritization scheme.

Track	Severity	Speed	Average Error (%)
Chassis-twist	Low	Low	19
		High	23
	High	Low	16
		High	14
Herringbone	Low	Low	15
		High	13
	High	Low	14
		High	15
Washboard	Low	Low	23
		High	19
	High	Low	25
		High	13
Our method			9

. This gives an idea of the overall performance in estimating the twelve vehicle parameters, as some are estimated better than others on different tracks. The average estimation error with the track prioritization scheme is 9%, which is much lower compared to those of individual tracks.

5.2. MATLAB–ADAMS Co-Simulation Results

In MATLAB–ADAMS co-simulation environments, the tests are repeated, and a similar trend to that in Figure 9 was obtained after 100 Monte Carlo tests. For the sake of brevity, we show the verification results in Figure 10. The results obtained from the ADAMS model are compared against the 7-dof model for the low-speed–low-severity herringbone test. It is observed that even when excluding the high-order dynamics not available in the MATLAB model, the responses match fairly well.

5.3. Experimental Results

In the validation tests, the responses from the 7-dof model are compared against the measured responses in Figure 11. It has been observed that the experimental measurements are in accordance with the simulation results with the parameter estimates obtained via the co-simulation study in the previous section. The axle point acceleration recorded field test responses show narrow peaks and wider troughs due to the fact that the tire radius is comparable to the track wavelength, and this has been captured well in the simulation as well owing to the road kinematic tire model as described in Section 4.3.

In

Table 9. Parameter estimation errors obtained by comparing the time-series responses with experimental measurements.

P	True Value (P)	Estimate ($\tilde{\mathit{P}}$)	Error (%)
$I_{xx}$	257	215	16
$I_{yy}$	246	239	3
$M_s$	240	225.6	6
$l_f$	0.85	0.91	7
$K_f$	10.5	11.2	7
$K_r$	6.5	5.7	12
$C_f$	0.63	0.67	6
$C_r$	0.59	0.65	10
$M_{uf}$	15	13.1	13
$M_{ur}$	12.5	10.5	16
$K_t$	40	37	7
$C_t$	0.2	0.19	5

, the final estimated parameters using the proposed method and the true parameters are presented. It is observed that the estimation error is less than 16% for all the parameters and as low as 3% for some of the parameters. Some of the parameters are estimated better than others. For instance, the error in the estimation of the sprung mass is only 6% as opposed to the unsprung mass estimation errors of 13 and 16% for the front and rear, respectively. In our modelling approach, we made some modelling assumptions: for example, the vehicle is represented by a 7-dof ride model that pitches and rolls about its centre of gravity, while in actual practice, these out-of-plane motions are encountered about the pitch and roll axes, respectively. It was assumed that the tire contact is maintained with the road surface at all times, but we observed that during field testing the wheel jumps off the road surface, leading to inaccuracies in the estimation of the unsprung mass. Moreover, it was assumed that the vehicle is a symmetric body with no product-of-inertia terms, which is certainly not true for an actual vehicle, leading to large deviations in the estimation of inertia properties; for instance, the error in the estimation of the roll inertia $I_{xx}$ is 16%.

Finally, in the present study, we assumed the suspension to be represented by a linear Kelvin–Voight model. The vehicle is equipped with double-wishbone front and semi-trailing-arm rear suspensions, for which the equivalent suspension properties are considered at the wheels by using the motion ratio, which is the usual practice followed by suspension designers. These are assumed constant in the present study, while in actual practice, the suspensions are usually non-linear and the motion ratio is a function of suspension geometry, which varies with wheel travel. This is a modelling assumption, and the variation with wheel travel can be considered in order to increase the model fidelity. A model of intermediate complexity can be developed to reduce the estimation errors; this can be a topic of research for future work.

6. Conclusions and Future Work

In this paper, a system identification procedure is introduced to determine the parameters of a vehicle that are essential for determining its ride characteristics. A 7-dof mathematical model for the vehicle is developed, and after careful investigation, 12 vehicle parameters are observed to require identification to achieve comprehensive understanding of the vehicle behaviour. A novel system identification procedure is proposed in the paper, which does not necessitate expensive test rigs and uses only accelerometer and gyroscope measurements obtained from the vehicle running different road test profiles at different speed profiles. The proposed system identification method is based on track and response prioritization. The effects of each parameter of interest on each vehicle response and the capabilities of each track to excite these responses are determined. An optimization routine based on JADE is used in time-domain parameter estimation. The weights of the tracks are adaptively changed according to the spread of the parameters in the population generated in each optimization iteration. The method achieves faster convergence by reducing the sampling impoverishment. A hierarchical test procedure is followed, including a proof-of-concept test run in a MATLAB environment, verification tests run in a MATLAB–ADAMS co-simulation environment, and validation tests run on data collected from the test vehicle. All the results, including the experimental tests, support the superiority of the proposed method.

In the present approach, we simulated and tested the vehicle traversing on deterministic tracks and used time-series response data for parameter estimation. In the future, the system identification results will be further validated with new experimental data for the vehicle traversing on non-deterministic tracks. The track-to-parameter weight mapping can be analytically acquired based on characteristics of the non-deterministic tracks, including the road excitation frequency characteristics. Furthermore, the proposed method can be extended to include a frequency-domain parameter estimation approach, as in [38].