identification of vehicle suspension parameters by design optimization
TRANSCRIPT
This article was downloaded by: [University of Colorado at Boulder Libraries]On: 21 December 2014, At: 00:39Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Click for updates
Engineering OptimizationPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/geno20
Identification of vehicle suspensionparameters by design optimizationJ.Y. Teya, R. Ramlia, C.W. Khengb, S.Y. Chongc & M.A.Z. Abidind
a Advanced Computational and Applied Mechanics Research Group,Department of Mechanical Engineering, Faculty of Engineering,University of Malaya, Kuala Lumpur, Malaysiab Department of Computer Science, Faculty of Information andCommunication Technology, Universiti Tunku Abdul Rahman, JalanUniversiti, Perak, Malaysiac School of Computer Science, The University of NottinghamMalaysia Campus, Malaysiad Proton Professor Office, Proton Holdings Bhd., MalaysiaPublished online: 19 Jun 2013.
To cite this article: J.Y. Tey, R. Ramli, C.W. Kheng, S.Y. Chong & M.A.Z. Abidin (2014) Identificationof vehicle suspension parameters by design optimization, Engineering Optimization, 46:5, 669-686,DOI: 10.1080/0305215X.2013.795558
To link to this article: http://dx.doi.org/10.1080/0305215X.2013.795558
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization, 2014Vol. 46, No. 5, 669–686, http://dx.doi.org/10.1080/0305215X.2013.795558
Identification of vehicle suspension parameters bydesign optimization
J.Y. Teya, R. Ramlia*, C.W. Khengb, S.Y. Chongc and M.A.Z. Abidind
aAdvanced Computational and Applied Mechanics Research Group, Department ofMechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia;
bDepartment of Computer Science, Faculty of Information and Communication Technology,Universiti Tunku Abdul Rahman, Jalan Universiti, Perak, Malaysia; cSchool of Computer Science,
The University of Nottingham Malaysia Campus, Malaysia; dProton Professor Office,Proton Holdings Bhd., Malaysia
(Received 7 June 2012; final version received 25 March 2013)
The design of a vehicle suspension system through simulation requires accurate representation of thedesign parameters. These parameters are usually difficult to measure or sometimes unavailable. This arti-cle proposes an efficient approach to identify the unknown parameters through optimization based onexperimental results, where the covariance matrix adaptation–evolutionary strategy (CMA-es) is utilizedto improve the simulation and experimental results against the kinematic and compliance tests. This speedsup the design and development cycle by recovering all the unknown data with respect to a set of kine-matic measurements through a single optimization process. A case study employing a McPherson strutsuspension system is modelled in a multi-body dynamic system. Three kinematic and compliance testsare examined, namely, vertical parallel wheel travel, opposite wheel travel and single wheel travel. Theproblem is formulated as a multi-objective optimization problem with 40 objectives and 49 design param-eters. A hierarchical clustering method based on global sensitivity analysis is used to reduce the numberof objectives to 30 by grouping correlated objectives together. Then, a dynamic summation of rank valueis used as pseudo-objective functions to reformulate the multi-objective optimization to a single-objectiveoptimization problem. The optimized results show a significant improvement in the correlation betweenthe simulated model and the experimental model. Once accurate representation of the vehicle suspensionmodel is achieved, further analysis, such as ride and handling performances, can be implemented for furtheroptimization.
Keywords: hierarchical clustering; global sensitivity analysis; design of experiments; kinematic andcompliance analysis
1. Introduction
In the vehicle development process, experimenting with physical prototype remains an importanttask (Rauh 2003). Often, these experiments are conducted by designers to obtain parameters forsimulation and validation. However, the process of achieving accurate correlations between sim-ulations and experiments is often difficult. According to Blundell (1997), this requires expensiveset-ups of instruments and testing facilities. It is a time-consuming task that can take up to 21person-days per axle to complete a vehicle characterization. Once the design parameters have beenobtained, a fine-tuning process is necessary to simplify the model. This involves improving thecorrelation between the simulated model and the experimental model repeatedly. Conventionally,
*Corresponding author. Email: [email protected]
© 2013 Taylor & Francis
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
670 J.Y. Tey et al.
final tuning is performed by experienced engineers through trial-and-error experimentation basedon human perception (Kuo et al. 2008). In addition, the nature of the vehicle dynamic problem israther complex and highly nonlinear, so the process is time consuming.
In the past, design sensitivity analysis was introduced to analyse a mechanical system forredesign and modification of the vehicle’s passive suspension setting (Nalecz and Wicher 1988).However, design sensitivity analysis has its limitations. It identifies the key parameters only butdoes not provide the optimal solution. Other researchers have implemented design sensitivity anal-ysis to identify the key parameters and combined them with gradient-based optimization to solvesuspension design problems (Lee, Won, and Kim 2009). However, this method has a drawback inthe context of kinematics and compliance. It often demands a set of design parameters to fulfil mul-tiple objectives such as camber change, caster change and toe change, among others. In order toperform a gradient-based optimization for the above setting, the multi-objective optimizationproblem is usually formulated to a single-objective optimization problem through a weightedsum model. However, the weighting required for each objective is not known beforehand andit is difficult to determine a proper value for each objective that can produce the best globalminimum solution.
Furthermore, the gradient-based approach has drawbacks in optimizing nonlinear problems(Datoussaid,Verlinden, and Conti 2002).Although the approach has gained a reputation in solvingvarious types of passive suspension system, it often requires auxiliary equations (derivation of thekinematic equations of the suspension motion), which are usually difficult to derive and implementowing to the complexity of the vehicle dynamic system. Rocca and Russo (2002) employed specialcoding to formulate the suspension kinematic equations as the method of identification process sothat the gradient-based algorithm was able to optimize the limited number of design variables. Ingeneral, gradient-based optimization approaches are more efficient than evolutionary algorithms,but they could get trapped in local optima instead of the global optimum design. From thisperspective, the gradient-based approaches are less robust and lack explorative features comparedto evolutionary algorithms when employed to search for the optimal design in a large-scaleoptimization problem. Furthermore, evolutionary algorithms such as evolution strategies haveadvantages over classical gradient approaches in that analysis of the problem characteristic isnot required prior to the optimization process (Datoussaid, Verlinden, and Conti 2002). Theyare capable of avoiding derivatives, which allows simpler implementation of the optimizationprocess for high-complexity vehicle models and efficiency in handling discontinuities in the designspace. This allows the optimization algorithm to be directly coupled with the existing multi-bodydynamic system in a form of generate-and-test framework, where the evolutionary algorithmgenerates a new design and the objective value is evaluated through a software simulation such asADAMS. This process repeats until the termination condition is met, e.g. the correlation betweenthe simulated model and the experimental model is above a certain threshold.
In this article, a methodology is proposed that uses an evolutionary algorithm, the covariancematrix adaptation–evolutionary strategy (CMA-es), as the optimization tool. This tool was shownto be robust in solving 25 black-box benchmark problems, as demonstrated at the Congress on Evo-lutionary Computation (CEC) in 2005 (Auger and Hansen 2005; Hansen 2006a, 2010; García et al.2009). In addition, it does not require any tedious parameter tuning (parameters of the algorithmare derived statistically), except for population size, which has to be set by the user. This makesthe algorithm more efficient and suitable for engineering applications compared with the gradient-based approach. The optimization algorithm incorporates global sensitivity analysis (GSA) andhierarchical clustering to reduce the number of objective functions and dimensionality of theproblem by grouping similar affecting design parameters. The proposed methodology not onlyallows for the recovery of the unknown design parameters but also rebuilds the physical equivalentmodel in the multi-body dynamic simulation software (MSC.ADAMS) through correlation withthe experimental results.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 671
2. Background
2.1. Global sensitivity analysis
GSA is a statistical method of evaluating the correlation between the design parameters on eachobjective function (Mastinu, Gobbi, and Miano 2006). It is different from conventional sensitivityanalysis, in which the analysis of the objective function centres on small variations in the systemparameters. GSA describes the behaviour of the system when parameters are varied across broadranges of the entire feasible design domain. It provides dimensionless measurements on howstrongly each design parameter correlates with the corresponding objective function. GSA can beemployed for a generic objective function with a generic design parameter, as well as between twogeneric objectives in order to discover the relationship between the objective functions (Mastinu,Gobbi, and Miano 2006). In this article, GSA (Spearman’s rank correlation) is employed in theearly stage to identify the relationship between design parameters and each measured objective togive a clear picture of how each design parameter contributes to the respective measured objective.This is performed through evaluation of a set of different design parameter combinations generatedby the Latin hypercube sampling method.
Latin hypercube is a sampling method that subdivides the design parameter domain along eachdimension into n subintervals and ensures that one sample lies in each subinterval. This methodis a more efficient approach than pseudo-random sampling and is capable of generating disperseand uniform samples across each dimension space.
Spearman’s rank correlation is a quantitative GSA method based on rank regression analysis.It provides robust estimation of global sensitivity as it can measure correlations of nonlinearrelationships between two sets of variables. For a sample of size n of two variables x and y, theSpearman’s rank ρ is calculated as follows:
ρ =∑n
i=1[Rix − Rx][Ri
y − Ry]√∑ni=1[Ri
x − Rx]2[Riy − Ry]2
(1)
where Rix and Ri
y are ranks and Rx and Ry are the mean values of the ranks of x and y.The value of ρ can vary between +1 and −1. Values close to 1 indicates a strong correlation
between x and y, values close to −1 indicate strong inverse correlation, and values close to 0indicate the absence of correlations or the presence of a non-monotonic correlation.
2.2. Hierarchical clustering
Cluster analysis is an important statistical method used in a variety of fields. It helps to ‘groupeither the data unit or the variables into clusters such that elements within a cluster have a highdegree of “natural association” among themselves while the clusters are “relatively distinct” fromone another’ (Anderberg 1973). There are several clustering methods, such as self-organizingmap (SOM), k-means, fuzzy c-mean and hierarchical clustering. However, there is no single bestclustering method that suits various types of problem (Milligan 1980; Mangiameli, Chen, andWest 1996). In this article, the idea of employing a clustering method is to group the objectivesto provide similar affecting design parameters. It aims to reduce the objective redundancy orredundant objectives and simulation tests. This will also simplify the problem as the number ofobjectives used in the optimization can be reduced significantly. Hierarchical clustering is chosenbecause prior information found from GSA can be employed to form a cluster tree. This providesa clear interpretation on how each objective is related to each other, which is important in the finalstage of fine-tuning the vehicle suspension.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
672 J.Y. Tey et al.
The hierarchical clustering method groups the objectives with similar affecting design param-eters (calculated based on Spearman’s coefficient) into a cluster by creating a cluster tree ordendrogram. In this article, an agglomerative hierarchical cluster strategy is employed. Thealgorithm first calculates the distance matrix between each objective. Then, the cluster tree isformed based on the linkage criteria (average distance, centroid distance, weighted distance,median distance, etc.). Visualization of the cluster tree formation can be plotted using a den-drogram. Finally, the number of clusters can be decided by the user or by using the maximumthreshold limit of linkage distance criteria to form a cluster.
2.3. Optimization and covariance matrix adaptation–evolutionary strategy
Optimization is a process of locating the design parameter x, such that the objective functionf produces minimum output. Optimization that requires maximum output can be formulatedby changing the sign of the function output (Fletcher 1987). The optimization space is usuallybounded by a hypercube:
minx f (x), f : �d → �,subject to xlowerbound ≤ x ≤ xupperbound
(2)
Modern optimization assumes the underlying problem characteristics to be a black box, wherethe problem is hard to solve with mathematical analysis or has no analytical solution, for exampleoptimization involving simulation. In evolutionary computation, this can be resolved by adap-tive sampling of the space according to its functional response. This process is outlined as agenerate-and-test framework, where the algorithm keeps a set of current best solutions (referredto as population) and generates a new set of solutions randomly (commonly known as variationor mutation) according to certain probability density functions based on the current best solu-tions. This process is repeated for k generations (loops) until a termination condition is satisfied(e.g. resource allocation is utilized). Out of the many evolutionary algorithms, e.g. evolutionaryprogramming (Xin, Yong, and Guangming 1999), genetic algorithm (Goldberg 1989) and evo-lution strategies (Rechenberg 1971), CMA-es is selected because it uses statistical informationto define the search path towards the global minimum/maximum. CMA-es has been shown tooutperform other algorithms, for example in the CEC 2005 Session on Real-Parameter Optimiza-tion. This method generated a good average score for solving 25 black-box benchmark problems(Hansen 2006b). In addition, CMA-es has been successfully implemented to solve actual prob-lems in various engineering applications (Salehi, Young, and Mousavi 2008; Colutto et al. 2010;Gregory, Bayraktar, and Werner 2011). CMA-es is a nonlinear optimization method that usesstatistical techniques to optimize the objective function through an iterative process. Initially, itgenerates a set of candidate samples from a predefined multivariate normal distribution. In thefollowing iteration, the mean vector and the covariance matrix of the multivariate normal distribu-tion are updated using the best candidate solution found from the previous iteration. The updatedmultivariate normal distribution is then used to sample a new population of candidates and theiteration process continues until it converges (Hansen 2006a) (Figure 1).
3. Identification process
The proposed identification process begins with preliminary vehicle data to construct the suspen-sion subsystem in MSC.ADAMS/CAR. Design parameters and objectives measurements requiredto be optimized are identified. This is followed by Latin hypercube sampling and GSA to anal-yse the relationship between the design parameters against the measured kinematic objectives.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 673
Set (population size)
Initialize state variables (m, , C = I, p = 0, pc = 0)
While terminate not meet
For i=1: //Sample new and evaluate them
xi = sample multivariate normal distribution
(mean = m, covariance matrix 2C )
fi = fitness (xi)
end
Sort fitness of new samples, and calculate the new mean.
m //Update new mean points and move mean to better solution
p //Update isotropic evolution path
pc
σ
// Update anisotropic evolution path
C // Update covariance matrix
s // Update sigma, step-size using isotropic path length
return, m
l
l
s s
l
s
l
Figure 1. Pseudo-code of CMA-es.
Through this analysis, the information on the relationship between the design parameters and theobjectives can be further examined through a clustering method to cluster those objectives thathave similar factors to the design parameters. This helps to reduce the number of objectives to beoptimized while enhancing the efficiency of the algorithms since a smaller number of objectivesis required during optimization. Finally, CMA-es is employed to optimize the design parametersso that the measured objectives correlate with the experimental measurements. The workflow inimproving the correlation between simulation and experimental results is shown in Figure 2. Theinteraction between each process will be further discussed and explained in the following sections.
3.1. Modelling and simulation
A McPherson strut suspension model is developed in the MSC.ADAMS/CAR environment. Themodel is constructed using rigid body components connected with bushes and joints. The hardpoint locations are based on an initial computer-aided design (CAD) drawing or are determinedthrough estimation. In addition, bushing profiles in all six degrees of freedom, i.e. x, y and z—translational and rotational about x, y and z—are modelled as linear stiffness functions to simplifythe model.
In this study, the model consists of 21 degrees of freedom. The model topology is shown inFigure 3 and general information about the vehicle set-up is given in Appendix 1 (Table A1).An identification process is required to recover a total of 49 design parameters consisting of hardpoints, bushing stiffness and anti-roll bar stiffness. The wheel is modelled as a simple linear springelement. The wheel centre is fixed throughout the optimization process to maintain the track widthof the vehicle. The range of each selected parameter to be used during the optimization processmust be well defined. A large parameter range will cause the optimization search process to takelonger to converge; however, a small design range may not cover the location of the best solutionthat compromises all measured objectives. In the case study, the design space of each designvariable is shown in Appendix 1 (Table A3) and Figure 3.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
674 J.Y. Tey et al.
Figure 2. Workflow of design optimization.
Figure 3. Front suspension in MSC.ADAMS/CAR.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 675
Figure 4. Gradient information at static position measured for kinematic profiles curve.
In this model, three test cases are chosen based on the experimental kinematic and compliancetests, namely, vertical parallel wheel travel, opposite wheel travel, and single wheel vertical travel(Appendix 1, TableA2). Each of the test cases has its own measured objectives, as shown in Table 2.All the measured objectives are in the form of response curves. It is necessary to translate eachcurve into a single value representing a unique characteristic before conducting GSA. Therefore,the objectives have to be measured in the form of gradients at the static position (Figure 4).Gradient information has its own advantages over other measurement approaches such as theabsolute maximum, absolute minimum and root mean square values, as it inherits the shape andcharacteristic of the curve in a single value. However, gradient information is not able to inheritthe offset information of the curve. Therefore, the values that measure the offset of the curve withrespect to the changes in design variables in the static position are also employed in optimization.
3.2. Latin hypercube sampling and global sensitivity analysis
The Latin hypercube sampling method is utilized to sample 500 samples of the input parameterswithin a selected range given in Appendix 1 (Table A3). In order to maintain the feasibility ofvarious combinations or the shape of the suspension locations, the drop links to the strut, springupper mount point and lower mount point are varied with respect to the strut top vector and strutslider axis point rather than having their own design spaces. This is to prevent illogical positioningof those components where the spring position may yield large offset from the damper locationand form an independent component floating in space.
The relative movement of the location is calculated as follows:
−→Psup = −→
ST + a∗dsu (3)−→Pslp = −→
ST + a∗dsl + Vsl (4)−→Pdts = −→
ST + a∗ddts + Vdts (5)
where a is a unit vector between the strut top and strut slider axis point, dsu, dsl and ddts are thedistances between the strut top and the location point (Figure 4), Vsl and Vdts are the perpendiculardistances between the coordinate and the vector between the strut top and strut slider axis point,−→Psup is the vector of the spring upper mounting point,
−→ST is the vector of the strut top point,−→
Pslp is the vector of the spring lower mounting point, and−→Pdts is the vector of the drop link to the
mounting strut point (Figure 5).
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
676 J.Y. Tey et al.
Figure 5. Schematic diagram of the strut linkage.
The entire sample is evaluated with this model and developed in MSC.ADAMS/CAR. Spear-man’s rank correlation is applied to examine the correlation between the design variables andthe objectives. A hypothesis test of no correlation against the alternative that there is a nonzerocorrelation is conducted. A p value of more than 0.05 shows that it is significantly different fromzero and will be rejected as it indicates no correlation. This helps to filter out the non-correlateddesign variables and measured objectives, hence simplifying the process of clustering.
3.3. Clustering
In hierarchical clustering, each observation is the objective function of design parameters. Itconsists of the Spearman correlation coefficients with respect to the objective function. TheEuclidean distance calculates the distance matrix between the observed pairs. For two vectors(x1, x2, . . . , xn), (y1, y2, . . . , yn) the Euclidean distance is calculated as follows:
deuc =√√√√
n∑i=1
(xi − yi)2 (6)
A linkage function uses linkage criteria to define the linkages between observed pairs to formthe hierarchical cluster tree. Many methods of linkage criteria can be utilized, such as averagedistance, centroid distance and median distance. The selection of a proper linkage criterion isimportant to form an accurate tree. Therefore, the linkage function is selected based on thecophenetic correlation coefficient (CPCC) (Sokal and Rohlf 1962). The CPCC is a measureof how faithfully the tree represents the dissimilarities among observations. In this article, thehierarchical tree is formed using the average distance linkage function as the linkage criterion.It is found that the function gives the highest score of 0.957 in the CPCC. The closer the valueof the CPCC is to 1, the more accurately the tree will be represented compared to other linkagecriteria (complete linkage = 0.925, single linkage = 0.949, weighted linkage = 0.956).
In the dendrogram plot (Figure 6), linkage distance on the x-axis represents the similaritybetween two objectives. Objectives with the closest affecting factor will have a shorter linkagedistance, as shown in Figure 6. By visualization, it is hard to determine the number of clusters asthey grow larger. Therefore, a new method of selecting the number of clusters is proposed using
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 677
Figure 6. Dendrogram of the hierarchical cluster tree and cluster formation. VP = vertical parallel; VO = verticalopposite; SW = single wheel; Pos = position; unless otherwise mentioned, all objectives are measured in gradients.
the silhouette index and rank method. The silhouette index measures the similarity of data withinits cluster by comparing it to data of other clusters. The silhouette index is defined by:
s(i) =min
k(bi,k) − ai
maxk
(ai, mink
(bi,k))(7)
where ai is the mean distance between the data i and the individual data in its cluster, and bi,k isthe minimum average distance between i and the individual data of another cluster, k (Lamrousand Taileb 2006).
This measurement is in a continuous range of [−1, 1]. Values closer to +1 indicate that theobject is distinctive from other clusters, showing that a good cluster is formed. In contrast, if thevalue closer to −1, the object is badly grouped in the cluster. For values closer to zero, the objectis not distinctive from one cluster to another.
The number of clusters formed increases with increasing silhouette index. However, the interestin this study is to reduce the number of objectives or clusters while maintaining the highest possiblesilhouette index. This can be treated as a conflicting objective to be optimized. Therefore, a rangeof maximum number of clusters is employed. The respective formation of the cluster group ismeasured with its mean values of silhouette index. The set of the number of clusters formed andsilhouette index is ranked into a range of [0, 1]. The number of clusters formed is normalized withthe minimum set of 0 (Equation 8), whereas for the silhouette index, the maximum value is setas 0 (Equation 9). The corresponding minimum summation of both values is the optimal result of
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
678 J.Y. Tey et al.
Figure 7. Plot of mean silhouette index versus number of clusters.
the number of clusters formed.
RSi = max(s) − s(i)
max(s) − min(s)(8)
RCi = C(i) − min(C)
max(C) − min(C)(9)
where RSi is a rank value of the silhouette index, RCi are rank values for the number of clusters,and i refers to the set of combination between the silhouette index and number of clusters formed.
A plot of mean silhouette index versus the number of clusters formed is shown in Figure 7. Theplot shows the reliability of this silhouette index with the rank method. The plot indicates thatin general, increasing the number of clusters will result in the mean silhouette index increasing.However, the increasing function is not linearly dependent. Increasing the number of clustersformed will not necessarily result in an increase in better cluster formation as the mean silhouetteindex may drop owing to overclustering of data. Through the method of calculation using thesilhouette index with the rank values, the result shows an optimal number of 30 clusters, whichcorresponds to a mean silhouette index of 0.97. The silhouette plot of the 30 clusters is shownin Figure 8. This suggests that each cluster is significantly distinctive from one another as thesilhouette index is close to 1. Figure 7 also shows that there is a significant increment in meansilhouette below 30 clusters. After the optimal cluster formation, the increment of the silhouetteindex becomes marginal, resulting in insignificant improvement as the number of clusters formedincreases.
Through this method, the number of objectives required to be optimized is reduced to 30(Figure 6: one objective chosen per cluster) from the 40 objectives.
3.4. Optimization (CMA-es)
The exploitation and exploration of optimization are limited to the design variable bounds speci-fied in Appendix 1 (Table A3). The goal of this optimization process is to minimize the difference
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 679
Figure 8. Silhouette plot of 30 cluster forms.
between simulated and experimental results. Based on previous hierarchical clustering, the objec-tives required to be minimized are grouped, and one objective is selected from each cluster toform a vector of objective functions. Through this simplification process, the required objectivesto be optimized are reduced. This enhances the efficiency of the optimization algorithm as smallernumbers of objectives are required to be evaluated during the test. However, the objectives canbe further simplified by introducing the dynamic summation of rank method. This reduces thecomplexity of the multi-objective function into a single objective, which then can be coupled withCMA-es. Through this method, a pseudo-objective replaces the vector of objective functions. Thismethod ranks each dimension objective based on current generation population size with the rankvalues falling between 0 and 10.
Rij =Objij − min
j(Objj)
maxj
objj − minj
(Objj)× 10 (10)
Fitnessi =n∑j
Rij (11)
where n is the total number of samples in the population size, i represents individual samples, andj represents the number of objectives. The summation of the rank values represents the fitnessvalues of each sample.
This method is introduced to prevent the optimization algorithm from creating a wrong selectionpressure towards a region of dominated solutions. This is because the scale of each objectivevalue differs from one to another. For the same percentage difference between each objective, thesummation of objectives with large-scale values will dominate the contribution in fitness valuesover the objectives with smaller scale values. For example, if Euclidean distance measurementwere employed, it would mislead the search direction of the optimization algorithm towards themost dominant solution that only fulfils the dominant objectives (Figures 9 and 10).
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
680 J.Y. Tey et al.
Figure 9. Comparison between Euclidean distance measurement and rank function measurement for a population sizeof 100 samples.
Figure 10. Histogram of absolute values of percentage difference against objective score.
An empirical study was conducted between the dynamic summation of rank method and theEuclidean distance method. A random generation consisting of 100 sets of design parameters wasevaluated using both methods. In Figure 9, the results show that both measurement methods givea different set of design parameters that produces the best minimum results (highlighted withcircles in Figure 9). The absolute value of the percentage difference between the target values andsimulated values was calculated for minimum points of both methods. Figure 10 indicates that thedynamic summation of rank method obtained a significantly smaller percentage difference than
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 681
the Euclidean distance measurement. Therefore, by employing the dynamic summation of rankmethod, the optimization search direction will move towards the most non-dominant solution,which is a compromise of all objectives targeted. In addition, a penalty method is introducedto penalize those infeasible design variables that lead to kinematic motion lock-up. Suspensionlinkage motion lock-up may occur owing to the poor design of suspension linkage connectionpoints, violating the four-bar linkage motion constraint. In this case, infeasible design variableswill be eliminated before the sorting process and thus avoid the search path moving towardsinfeasible design parameters.
There is no single robust termination criterion that can be used to determine the conver-gence of the optimization problems. The default termination condition employed by CMA-esis limited by the number of function calls of 1000N2, where N is the number of design vari-ables. N = 49 requires 2.401 × 106 seconds, making the solving time to terminate around 79.4weeks (calculated based on 20 seconds measured per function call to complete the three tests inMSC.ADAMS/CAR). Owing to the long termination time, the termination condition is reduced
Figure 11. Optimization process between CMA-es and MSC.ADAMS/CAR.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
682 J.Y. Tey et al.
to 2250 function calls, which reduces the solving time to approximately 5 days. The populationsize used in the optimization process is 42. An archive stores each generation solution. Afterthe termination condition has been reached, the archive data will be used to calculate the bestsolution throughout the generation of the pseudo-objective fitness values using Equations (10)and (11). Since the dynamic summation rank method for the optimization process is ranked basedon the current generation, the best solution in each generation is not comparable to the previousgeneration’s best solution. Based on the fitness values, the best 20 solutions are selected and thecorresponding standard deviation and mean are calculated. Through this method, the best opti-mized design variables and objectives values can be defined statistically and the robustness of thesolution can be examined.
The overall flow of the optimization process is shown in Figure 11. The software-in-the-loopoptimization approach is adopted in this optimization process. New samples generated from themultivariate normal distribution will be employed to generate suspension assembly files andbushing component files required by MSC.ADAMS/CAR to model the suspension subsystems.Kinematic tests are performed in MSC.ADAMS/CAR to evaluate the new design points. Theresulting files are read by the algorithm to compute the gradient information of the kinematicprofiles; then, the dynamic rank method is used to compute the fitness values. The optimizationprocess will continue to iterate until the termination condition is met.
4. Results and discussion
4.1. Empirical results
The best solution is defined by the mean and standard deviation of the 20 optimized solutions. Itscorresponding objective values are shown in Tables 1 and 2. The standard deviations for the 20
Table 1. Optimized design variables.
Hard point Optimized mean coordinates (mm) Standard deviation
Lower ball joint (−15.4, −688.6, −39.1) (0.0433, 0.0434, 0.1664)Lower control arm (FR) (−2.1, −383.2, −47.1) (0.3081, 0.0264, 0.1585)Lower control arm (RR) (309.3, −346.7, −29.1) (0.1991, 0.1176, 0.1499)Strut axis point (1.6, −581.8, 217.2) (0.2519, 0.0087, 0.0669)Strut top (39.5, −536.9, 558.9) (0.0707, 0.0176, 0.0793)Trackrod inner (163.0, −317.6, 85.9) (1.0177, 0.1718, 0.3296)Trackrod outer (136.7, −642.7, 105.3) (0.423, 0.1429, 0.3747)Stabilizer bar to droplink (65.1, −533.7, 70.6) (0.1064, 0.1126, 1.6639)
Bushing component Optimized mean bushingstiffness (K-N/mmR-N/deg)
Standard deviation
Bushing FR (Kx,Ky,Rx,Ry,Rz) (9483.4, 8811.48,32268.7, 760.3, 6019.3)
(0.7932, 2.3855, 0.3812,0.4409, 0.464)
Bushing RR (Kx,Ky,Rx,Ry,Rz) (3721.1, 21425.1, 5496.9,18353, 4399.7)
(1.9401, 1.9732, 3.991,1.1673, 0.481)
Bushing ARB (Kx,KyKz,Rx,Ry,Rz) (36581.3, 2625.1,39097.8, 24386, 10380,1160.5)
(1.4675, 3.5607, 0.3798,1.7948, 0.8275, 0.2876)
Bushing at strut top (Kx,Ky,Kz,Rx,Ry,Rz) (5401.2, 33553.1, 2463.9,16494, 23121, 3396.7)
(0.3919, 0.7944, 0.411,0.3721, 0.3027, 1.1289)
Torsional stiffness (N/deg) 930.6 0.0345Camber angle (deg) −0.52 0.0248Toe angle (deg) 0.0032 0.0243
Note: K = translational stiffness; R = rotational stiffness.
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 683
Table 2. Optimized objective values.
Vertical Improvement Vertical Improvement Single wheel Improvementparallel test gains (%) opposite test gains (%) travel gains (%)
Suspension ride rate −23.15 Toe change 124.08 Suspension ride rate 23.81Suspension ride rate
(pos)−2.70 Toe change (pos) 198.48 Suspension ride rate
(pos)−3.08
Toe change 137.59 Camber change −10.06 Toe change 1.24Toe change (pos) 377.06 Camber change (pos) 64.40 Toe change (pos) 409.66Camber change 13.91 Roll caster 43.93 Camber change −11.71Camber change (pos) 68.04 Roll toe change 122.60 Camber change (pos) 69.53Lateral displacement
hub81.83 Roll toe change (pos) 184.83 Lateral displacement
hub82.47
Suspension lateraldisplacement TCP
249.00 Roll camber 0.08 Suspension lateraldisplacement TCP
451.89
Suspension foreaftdisplacement hub
96.36 Roll camber (pos) −8.45 Suspension foreaftdisplacement hub
−64.28
Caster change 33.17 Roll force 20.33 Caster Change 44.12Ride rate −20.51 Roll force (pos) −2.91 Ride rate 21.84Lateral displacement
TCP244.31 Roll rate 19.51 Ride rate (pos) −3.08
Foreaft displacementTCP
42.36 Roll rate (pos) −2.91 Lateral displacementTCP
383.14
Foreaft displacementTCP
117.93
Note: pos = values at static position; unless otherwise mentioned, all objectives are measured in gradients; TCP = tyre contact patch.
optimized solutions are small, indicating that the optimization converges near the best solution. InTable 2, the corresponding optimal design parameters are able to fulfil most of the objectives, i.e.by comparing the gradient of the simulated results with those from the experiments. Most of theobjectives show significant improvement. Only average experimental results at static positionsare taken as targeted values for optimization. It should be noted that the simulated results areunable to fit into the experimental curve with hysteresis behaviour. The simulation environmenthas perfect symmetry of the suspension system and linear bushing characteristics compared to theexperimental vehicle suspension, which has manufacturing tolerances and hysteresis of rubberbushings. As such, it is hard to satisfy all objectives. Compromises of the results are found tofit all objectives at their respective mean values. The phenomenon previously described can bevisualized through a comparison plot of experimental versus an optimized model and a non-optimized model in Figure 12 (four objectives are selected out of 40 objectives to demonstratethe phenomena described).
In the plot, only a single curve is plotted to represent the simulated result. Owing to thesymmetrical geometry topology of the suspension model, identical results will be generated forboth left and right wheels. From the plot, it is also demonstrated that gradient information as anobjective measurement can be used to represent the characteristic of the curve, i.e. the simulatedcurve closely emulates the experimental results. In addition, the result of the optimized solutionsuggests that the pseudo-objective (Equations 10 and 11) is able to generate the correct selectionpressure for CMA-es to search for the best solution that fits all objectives.
Using this methodology, the optimized design parameters of the vehicle suspension modelare reasonably well represented, i.e. they closely reproduce the physical suspension system. Itis also shown that the optimized model can recover the unknown bushing stiffness and designhard points. It is crucial to have a well-correlated suspension model at the subsystem level inthe early design stage. For instance, the optimized solution can be further examined by robustanalysis as the solutions are statically determined. By analysing the standard deviation of thedesign parameters, it is possible to identify the robustness of the suspension design against its
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
684 J.Y. Tey et al.
Figure 12. Comparison plot of measured objectives.
kinematic performance. The model can be further assembled into a full vehicle model by couplingit with other subsystems in MSC.ADAMS/CAR for complete systems analysis of the vehicle.
5. Conclusion
The proposed methodology has been demonstrated to correlate the simulation model with thephysical suspension system through an evolutionary optimization method based on experimentalresults. This method is capable of correlating large design parameters and objectives in a single-run optimization process. A systematic workflow of the methodology is presented with the initialmodel built in MSC.ADAMS/CAR. This is followed by executing the design of experimentsto explore the design space and analysing the correlation between the design parameters andobjectives measured. Then, using the results from GSA, hierarchical clustering is employed toform a hierarchical cluster tree. This reduces the number of objectives from 40 to 30 objectives.CMA-es is coupled with a pseudo-objective to optimize the design parameters. The optimizedresults indicate that the optimized design parameters are able to accurately predict the kinematiccharacteristics of the physical suspension system measured in the experiments. This shows thatthe simulated model is accurate and can be coupled with full vehicle assembly for further analysison characteristics of the vehicle model at the systems level (i.e. based on ride and handlingperformance). This will help to improve the results of full vehicle simulation as the subsystem
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
Engineering Optimization 685
suspension model has been validated with the experimental results. This methodology will reducethe development process in vehicle suspension design.
Acknowledgements
The authors wish to express their gratitude to the Ministry of Science and Technology (MOSTI) of Malaysia for thefinancial support extended to this research project (TF0608C073), entitled ‘Computationally Optimized Fuel-EfficientConcept (COFEC) Car’. The authors would also like to convey their appreciation to PROTON BHD, Universiti Malaya(UM), Universiti Kebangsaan Malaysia (UKM), Universiti Teknologi Malaysia (UTM), Universiti Putra Malaysia (UPM),UniKL and MIMOS BHD for collaborative work.
References
Anderberg, M. R. 1973. Cluster Analysis for Applications. New York: Academic Press.Auger, A., and N. Hansen. 2005. “A Restart CMA Evolution Strategy with Increasing Population Size.” In IEEE Congress
on CEC 2005, Edinburgh, UK, September 2–5, 2: 1769–1776. Zurich, Switzerland: IEEE.Blundell, M. V. 1997. “Influence of Rubber Bush Compliance on Vehicle Suspension Movement.” Materials & Design
19 (1): 29–37.Colutto, S., F. Fruhauf, M. Fuchs, and O. Scherzer. 2010. “The CMA-ES on Riemannian Manifolds to Reconstruct Shapes
in 3-D Voxel Images.” IEEE Transactions on Evolutionary Computation 14 (2): 227–245.Datoussaid, S., O. Verlinden, and C. Conti. 2002. “Application of Evolutionary Strategies to Optimal Design of Multibody
Systems.” Multibody System Dynamics 8 (4): 393–408.Fletcher, R. 1987. Practical Methods of Optimization: Vol. 2. Constrained Optimization. Chichester: John Wiley & Sons.García, S., D. Molina, M. Lozano, and F. Herrera. 2009. “A Study on the Use of Non-parametric Tests for Analyz-
ing the Evolutionary Algorithms’ Behaviour: A Case Study on the CEC’2005 Special Session on Real ParameterOptimization.” Journal of Heuristics 15 (6): 617–644.
Goldberg, D. E. 1989. Genetic Algorithms in Search, Optimization & Machine Learning. Reading, MA: Addison-Wesley.Gregory, M. D., Z. Bayraktar, and D. H. Werner. 2011. “Fast Optimization of Electromagnetic Design Problems Using
the Covariance Matrix Adaptation Evolutionary Strategy.” IEEE Transactions on Antennas and Propagation 59 (4):1275–1285.
Hansen, N. 2006a. The CMA Evolution Strategy: A Comparing Review. Dordrecht: Springer.Hansen, N. 2006b. Compilation of Results on the 2005 CEC Benchmark Function Set. ETH Zurich, Switzerland: Institute
of Computational Science.Hansen, N. 2010. The CMA Evolution Strategy: A Tutorial, 2005[online]. Available from http://www.bionik.tu-berlin.de/
user/niko/cmatutorial.pdfKuo, Y. P., N. S. Pai, J. S. Lin, and C. Y. Yang. 2008. “Passive Vehicle Suspension System Design Using Evolutionary
Algorithm.” In IEEE International Symposium on Knowledge Acquisition and Modeling Workshop Proceedings,Wuhan, China, December 21–22, 292–295. Piscataway, NJ: IEEE Press.
Lamrous, S., and M. Taileb. 2006. “Divisive Hierarchical K-Means.” Article presented at Computational Intelligence forModelling, Control and Automation, 2006 and International Conference on Intelligent Agents, Web Technologiesand Internet Commerce, International Conference, Sydney, Australia, November 28–December 1, 18. Piscataway,NJ: IEEE Press.
Lee, H. G., C. J. Won, and J. W. Kim. 2009. “Design Sensitivity Analysis and Optimization of McPherson SuspensionSystems.” In Proceedings of the World Congress on Engineering, 2, London, UK, July 1–3.
Mangiameli, P., S. K. Chen, and D. West. 1996. “A Comparison of SOM Neural Network and Hierarchical ClusteringMethods.” European Journal of Operational Research 93 (2): 402–417.
Mastinu, G., M. Gobbi, and C. Miano. 2006. Optimal Design of Complex Mechanical Systems with Applications to VehicleEngineering. Berlin: Springer.
Milligan, G. W. 1980. “An Examination of the Effect of Six Types of Error Perturbation on Fifteen Clustering Algorithms.”Psychometrika 43 (5): 325–342.
Nalecz, A. G., and J. Wicher. 1988. “Design Sensitivity Analysis of Mechanical Systems in Frequency Domain.” Journalof Sound and Vibration 120 (3): 517–526.
Rauh, J. 2003. “Virtual Development of Ride and Handling Characteristics for Advanced Passenger Cars.” Vehicle SystemDynamic 40: 135–155.
Rechenberg, I. 1971. Evolutionsstrategie—Optimierung technischer Systeme nach Prinzipien der biologischen Evolution.Stuttgart-Bad Cannstatt, Germany: Frommann-Holzboog.
Rocca, E., and R. Russo. 2002. “A Feasibility Study on Elastokinematic Parameter Identification for a Multilink Suspen-sion.” Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 216 (2):153–160.
Salehi, M., P. G. Young, and P. Mousavi. 2008. “Reverse Engineering of the Transcriptional Subnetwork in the Yeast CellCycle Pathway Using Dynamic Bayesian Networks and Evolutionary Search.” In IEEE Symposium on Computational
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014
686 J.Y. Tey et al.
Intelligence in Bioinformatics and Computational Biology, 2008 (CIBCB ’08), Sun Valley, Idaho, USA, September15–17, 106–111. Piscataway, NJ: IEEE Press.
Sokal, R. R., and F. J. Rohlf. 1962. “The Comparison of Dendrograms by Objective Methods.” Taxon 11 (2): 33–40.Xin,Y., L.Yong, and L. Guangming. 1999. “Evolutionary Programming Made Faster.” IEEE Transactions on Evolutionary
Computation 3 (2): 82–102.
Appendix 1
Table A1. Vehicle set-up.
Vehicle data Value
Tyre stiffness (N/mm) 252.18Wheel radius (mm) 277Total sprung mass (kg) 1175.6Wheel base (mm) 2605Wheel mass (kg) 27.5
Table A2. Test case set-up.
Test case Test case condition
Vertical parallel/opposite/single wheel test Bump distance 40 mmRebound distance 40 mmNumber of steps 30
Table A3. Variables of input parameters and the design range employed during optimization.
Design range (mm)
Point x y z
Lower ball joint [−15, 5] [−710, −690] [−50, −30]Lower control arm (FR) [0, 20] [−380, −360] [−50, −30]Lower control arm (RR) [310, 330] [−370, −350] [−40, −20]Strut axis point [−5, 15] [−610, −590] [125, 145]Strut top [20, 40] [−560, −540] [560, 580]Trackrod inner [150, 170] [−320, −300] [85, 105]Trackrod outer [120, 140] [−660, −640] [100, 120]Stabilizer bar to droplink [45, 65] [−550, −530] [50, 70]Wheel centre FixSpring lower mount Varies relative to the line between strut top and strutSpring upper mount axis pointDroplink to strut
Brushing component Estimated values Design range(K-N/mm R-N/deg) (K-N/mm R-N/deg)
Bushing FR (Kx,Ky,Rx,Ry,Rz) 500 [0, 50000]Bushing RR (Kx,Ky,Rx,Ry,Rz)Bushing ARB (Kx,KyKz,Rx,Ry,Rz)Bushing at strut top (Kx,Ky,Kz,Rx,Ry,Rz)Torsional stiffness 300 [30, 3000]
Characteristics Estimated values (deg) Design range (deg)Camber angle 0 [−1, 1]Toe angle 0
Dow
nloa
ded
by [
Uni
vers
ity o
f C
olor
ado
at B
ould
er L
ibra
ries
] at
00:
39 2
1 D
ecem
ber
2014