reduction of multibody dynamic models in automotive...

Ramin Masoudi1Department of Systems Design Engineering,

University of Waterloo,

Waterloo, ON N2L 3G1, Canada

e-mail: [email protected]

Thomas UchidaDepartment of Bioengineering,

Stanford University,

Stanford, CA 94305-5448


John McPheeProfessor

Department of Systems Design Engineering,

University of Waterloo,

Waterloo, ON N2L 3G1, Canada


Reduction of Multibody DynamicModels in Automotive SystemsUsing the Proper OrthogonalDecompositionThe proper orthogonal decomposition (POD) is employed to reduce the order of small-scale automotive multibody systems. The reduction procedure is demonstrated usingthree models of increasing complexity: a simplified dynamic vehicle model with a fullyindependent suspension, a kinematic model of a single double-wishbone suspension, anda high-fidelity dynamic vehicle model with double-wishbone and trailing-arm suspen-sions. These three models were chosen to evaluate the effectiveness of the POD given sys-tems of ordinary differential equations (ODEs), algebraic equations (AEs), anddifferential-algebraic equations (DAEs), respectively. These models are also componentsof more complicated full vehicle models used for design, control, and optimization pur-poses, which often involve real-time simulation. The governing kinematic and dynamicequations are generated symbolically and solved numerically. Snapshot data to constructthe reduced subspace are obtained from simulations of the original nonlinear systems.The performance of the reduction scheme is evaluated based on both accuracy and com-putational efficiency. Good agreement is observed between the simulation results fromthe original models and reduced-order models, but the latter simulate substantiallyfaster. Finally, a robustness study is conducted to explore the behavior of a reduced-order system as its input signal deviates from the reference input that was used to con-struct the reduced subspace. [DOI: 10.1115/1.4029390]

1 Model Reduction

The demand for high-fidelity simulation models in the automo-tive industry has been increasing in recent decades. A state-of-the-art design and analysis of automotive systems, from suspensionsto powertrains, benefit from more realistic simulation models inseveral respects. Most significantly, experimental costs can bereduced by providing virtual design environments. Furthermore,high-fidelity models can incorporate more complex physical phe-nomena in the simulation of mechanical systems, which can resultin a more reliable, identifiable, and controllable model. Moderncomputers with fast processors coupled with sophisticatedcommercial software packages built on advanced computationalalgorithms provide the essential tools to create complex dynamicmodels of automotive systems.

For a dynamic system to be proper for a particular application,it should perform accurately with a minimal degree of complexity[1]—that is, elementary principles in the dynamic behavior of thesystem must not be violated, but the computational effort must beminimized for a particular application of the simulation model(e.g., system identification or advanced control design purposes).Since computation time and storage requirements play key rolesin computational aspects of simulation systems analysis, the taskof model order reduction is quickly flourishing in the design ofdynamical systems. Reducing the complexity of a model whilepreserving the input–output behavior of the system is an essentialpart of the design process and can facilitate simulation when facedwith computational limitations.

Nearly all reduction schemes, regardless of the sophisticatedmathematical techniques or different physical perspectives onwhich they are based (such as energy and characteristic speeds),follow the basic strategy of determining the components whosecontributions dominate the dynamic response of the system andmust be retained in the reduced-order model. Model reductionapproaches can be divided into two broad classes: structure-preserving schemes and altered-structure methods. The formerfocuses on reducing the model while still employing physical ele-ments of the original system to govern the dynamic behavior ofthe reduced-order model (e.g., singular perturbation and energymethods); the latter exploits abstract math-based transformations(e.g., aggregation algorithm and projection methods), where thephysical significance of each state variable is not preserved.

A broad survey of model order reduction schemes was con-ducted by Ersal et al. [1], who performed a comparative study ofestablished approaches and classified those approaches intofrequency-, projection-, optimization-, and energy-based methods.Although most model reduction algorithms are inspired by a par-ticular physical interpretation, few preserve the real physicalstructure of the original model. The notion of tracking the energy/power flowing through the components of a physical system[2–4], performing singular perturbation analyses [5,6] or usingsensitivity information [7] are among the realization-preservingpractices in model simplification.

Model reduction schemes have generally been designed forlarge-scale systems, where there are many candidate componentsfor reduction. The application of model reduction strategies tosmall-scale systems, such as automotive systems, where computa-tionally efficient models are essential for design and control pur-poses, is the primary motivation for the present work. We seekstrategies for applying existing reduction schemes to approximatesmall-scale systems. In particular, we explore the use of the POD,a projection scheme, because it is a versatile approach for

1 Corresponding author.Contributed by the Design Engineering Division of ASME for publication in the

JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 2,2014; final manuscript received December 12, 2014; published online February 11,2015. Assoc. Editor: Rudranarayan Mukherjee.

Journal of Computational and Nonlinear Dynamics MAY 2015, Vol. 10 / 031007-1Copyright VC 2015 by ASME

Downloaded From: http://computationalnonlinear.asmedigitalcollection.asme.org/ on 05/14/2015 Terms of Use: http://asme.org/terms

reducing nonlinear dynamic systems. The POD does not preservethe physical structure of the original higher-dimensional system;rather, it distils information obtained from a nonlinear dynamicsystem as it responds to a particular set of inputs.

POD, also called the Karhunen–Loeve decomposition, has beenwidely used for reducing nonlinear systems. A broad range ofapplications, from vibration and turbulence analysis to heat diffu-sion system analysis and control [8,9], have established the PODas a useful and versatile tool for reducing various engineeringproblems. The POD projection-based model reduction frameworkaims to form a lower-dimensional approximation from a generalnonlinear dynamic system using singular value decomposition(SVD), which facilitates retaining as many “interesting” phenom-ena as possible. Data from experiments or high-fidelity simulationmodels are used to construct the basis onto which the originalstates of the system are projected.

The ability of the POD to reduce nonlinear systems distin-guishes this reduction scheme from Krylov-space procedures andbalanced truncation methods, which are designed for linear time-invariant systems. Marquez et al. [8] used the POD and Galerkinprojections to reduce a distributed reactor model, then developeda model-predictive control scheme based on the reduced-ordermodel. A holistic reduction approach involving both model andcontroller (in contrast to “design-then-reduce” and “reduce-then-design” paradigms) was proposed by Atwell [10] to increase theperformance and robustness of a reduced-order, control-orientedproblem. The application of POD to characterizing and reducingboth linear and nonlinear mechanical systems, such as vibro-impacting continuous systems and coherent spatial structures, wasstudied by Kerschen et al. [11]. Masoudi et al. [12] applied PODto reduce small-scale systems in automotive applications, includ-ing vehicle dynamics, battery, and suspension systems. The per-formance and robustness of the reduction scheme were examinedusing complex physics-based models governed by ODEs andDAEs. They used the �-embedding approach to obtain ODEs fromthe original DAEs and AEs governing the dynamic behavior ofthe battery and suspension models, respectively.

In this work, the equations governing the kinematics anddynamics of automotive systems are generated symbolically, thenoptimized for efficient simulation. In particular, we examine threemodels of increasing complexity: a simplified dynamic vehiclemodel with a fully independent suspension, a kinematic model ofa single double-wishbone suspension, and a high-fidelity dynamicvehicle model with double-wishbone and trailing-arm suspen-sions. The POD, a popular reduction scheme for large-scalenonlinear systems, is applied to explore its efficacy in reducingsmall-scale nonlinear multibody systems, which is essential fordesign, control, and optimization tasks—particularly in automo-tive systems applications. Capability of POD in reducing thekinematic model of the double-wishbone suspension system withAEs as governing equations is also examined, which shows agreat improvement compared to the results presented earlier [12].Furthermore, we study the robustness of the reduced-ordersuspension model to inputs different from those used to performthe reduction. Suitable parameters for the reduction scheme aredetermined based on the desired precision (i.e., the acceptableamount of deviation from the original state trajectory) andthe computational efficiency demanded by the particularapplication.

2 Proper Orthogonal Decomposition

Model order reduction is the process of reducing the size of asystem by either projecting the original (linear or nonlinear) spaceinto a space of lower dimension or removing components whosecontributions to the phenomena of interest are negligible. Energy/power [2–4], singular perturbation [5,6], and sensitivity analysis[7] strategies can be used to identify the most meaningful compo-nents. Mathematically, the original equations can be expressed asfollows:

X:

_xðtÞ ¼ f ðt; xðtÞ;uðtÞÞyðtÞ ¼ gðt; xðtÞ; uðtÞÞ

�(1)

where xðtÞ 2 Rn is the state vector with initial conditionsxð0Þ ¼ x0; yðtÞ 2 Rm is the output vector, and uðtÞ 2 Rp are theinputs to the system. The reduced-order form can then beexpressed as follows:

X:

_xðtÞ ¼ f ðt; xðtÞ; uðtÞÞyðtÞ ¼ gðt; xðtÞ;uðtÞÞ

�(2)

where xðtÞ 2 Rr and yðtÞ 2 Rm are, respectively, the state andoutput vectors in the reduced-order space, uðtÞ are the same inputsapplied to the original system (Eq. (1)), and r � n. The reducedsystem R is assumed to be generated so as to satisfy the followingcondition:

jjyðtÞ � yðtÞjj � njjuðtÞjj 8uðtÞ 2 U (3)

where n is the tolerance and U is the domain of the inputs to themodel. The ultimate objective is to build a proper reduced-ordermodel that is both accurate and simple. The highly nonlinearnature of the dynamic models used in this work restricts the prob-lem to a few particular model order reduction methods.

The philosophy behind the POD is to provide an orthogonalbasis to represent “snapshots” from the response of a dynamicalsystem, collected from theoretical or experimental data, in a least-squares optimal sense. Mathematically, the task is to project theoriginal space of the dynamic system into a lower-dimensionalsubspace for the trajectory x using a projection operator S of rankr � n. The optimal projection is obtained by minimizing thefollowing residual [13]:

jjx� Sxjj22 ¼ðtf

0

jjxðtÞ � SxðtÞjj22dt (4)

A symmetric, positive, semidefinite correlation matrix (theGramian) is constructed to solve the optimization problem

C ¼ðtf

0

xðtÞxðtÞTdt (5)

which is then spectrally decomposed using an orthogonal set ofeigenvectors

C ¼ S Q½ � K1

K2

� �ST

QT

� �(6)

where K1 ¼ diagðk1;…; krÞ;K2 ¼ diagðkrþ1;…; knÞ, and k1 � k2

� � � � � kn. The decomposition expressed in Eq. (6) is simply aSVD of the correlation matrix. The projection operator S used toapproximate the reduced subspace (of dimension r) can beevaluated as follows [13]:

S ¼ SST (7)

which minimizes the residual

jjx� Sxjj22 ¼Xn

j¼rþ1

kj ¼ trace K2ð Þ (8)

In fact, orthogonal columns of S span the optimal subspace ofdimension r. In applying the POD technique, the mapping x ¼ Sxrestricts the ODEs represented by Eq. (1) to the obtained optimalsubspace; the reduced-order system will be in the following form:

_xðtÞ ¼ STf ðt;SxðtÞ;uðtÞÞ (9)

031007-2 / Vol. 10, MAY 2015 Transactions of the ASME


3 Model Order Reduction of Automotive Systems

To examine the efficacy of the proposed reduction scheme insmall-scale dynamic systems, three well-established multibodymodels of automotive systems (whose governing dynamic equa-tions are of different types) have been considered in this paper. Acomparative study in terms of computational cost and relativeerror between the original and reduced-order models is conductedto explore the performance of the proposed approach in variousapplications in the automotive industry. All models are developedsymbolically in this work using linear graph theory introduced inMapleSim [14], an advanced modeling and simulation tool, tofacilitate analysis of these models using different numericalparameters.

The process of applying the proposed model reduction schemecan be summarized as follows:

(1) Construct the response matrix using m observations (snap-shots) from the system response

X ¼ x1 x2 … xm½ � 2 Rn�m

(2) Form a zero-mean ensemble from the discretized snapshotdata

Xij ¼ Xij �1

m

Xm

j¼1

Xij ) X ¼ X � x

(3) Perform a SVD on the statistically modified snapshots

X ¼ UDVT

where U 2 Rn�n is a left-singular matrix, V 2 Rm�m is aright-singular matrix, and D 2 Rn�m is a rectangular diago-nal matrix, whose entries are the singular values of themodified snapshot matrix X.

(4) Collect the columns of U corresponding to the r largest sin-gular values to construct the projection matrix Ur .

(5) Project the original space into the reduced-order space

xðtÞ ¼ Ur xðtÞ þ x

(6) Represent the dynamic system in the reduced-order space

_xðtÞ ¼ UTr f Ur xðtÞ þ x;uð Þ; x0 ¼ UT

r x0 � xð Þ

3.1 Dynamic Analysis of a Vehicle Model With a FullyIndependent Suspension. The handling and braking behavior ofautomobiles are of fundamental importance in vehicle dynamics.Several models with different levels of fidelity have been devel-oped by automotive researchers for various dynamic analyses andcontrol design purposes. Sayers and Han [15] introduced a simpli-fied, yet sufficiently reliable, multibody dynamic model of ageneric four-wheeled vehicle with a fully independent suspension.As illustrated in Fig. 1, four lumped masses (ms), each represent-ing one-quarter of all suspension components, are connected tothe vehicle chassis by prismatic joints; springs (Ks) and dampers(Cs) placed in parallel represent the suspension compliance. Eachwheel is connected to its corresponding lumped mass with arevolute joint (h(t)), which allows the wheel to spin. Verticallyoriented revolute joints on the front wheels (d(t)) are used to steerthe vehicle through a trajectory that is either predetermined,calculated by a driver model, or prescribed by a human driver.The tire–road interaction is modeled using the Pacejka MagicFormula [16]. A multibody dynamic model of this 14 degrees offreedom (DOF) vehicle system is constructed, resulting in a set ofODEs of the following form:

_xðtÞ ¼ fGVðt; xðtÞ;uðtÞÞ (10)

The symbolic equations are then optimized and exported toMATLAB for further analysis and model order reduction.

The vehicle performs a single-lane-change maneuver (as shownin Fig. 2) on a flat surface, and the data from the full-ordersimulated model are used to construct the response matrix ofsnapshots. The initial speed of the vehicle is 20 m/s and all wheeltorques are assumed to be zero throughout the 25 s simulation. Toincrease the accuracy of the estimated subspace in the POD proce-dure, we divide the simulation into control periods of duration 0.5s and estimate the optimal subspace in each control period. Thisapproach resembles the operation of onboard model-predictivecontrollers, which update at regular intervals. Simulation resultsusing the original and reduced-order models are shown in Fig. 2(vehicle trajectory) and Fig. 3 (pitch and yaw angles); errors forpitch and yaw angles are shown in Fig. 4, with root-mean-squareerrors of 5:18� 10�4 rad and 7:59� 10�5 rad, respectively.Clearly, the simulation results obtained using the reduced-ordermodel closely match those obtained using the original model;

Fig. 1 Vehicle model with a fully independent suspension. Thedynamics of this 14DOF system are governed by a set of pureODEs.

Fig. 2 Simulated trajectory of the independent-suspensionvehicle using original and reduced-order models

Journal of Computational and Nonlinear Dynamics MAY 2015, Vol. 10 / 031007-3


however, the simulation time was reduced by 20%, demonstratingthe utility of this reduction scheme. In the reduced-order model,the number of states has been reduced from 14 to 3—that is, onlythree states are required to retain most of the information contentin the response. Note that the simulation time can be reduced sub-stantially by expanding the control period—provided some devia-tion is permissible between the trajectory of the reduced-ordersystem and that of the original model (e.g., if the reduced-ordermodel is used within a model-based controller).

3.2 Kinematic Analysis of a Single Double-Wishbone Sus-pension System. A schematic of a double-wishbone suspension,which was used by Uchida and McPhee [17] in a kinematic analy-sis study, is shown in Fig. 5(a). The ability to adjust many aspectsof its kinematics makes the double-wishbone suspension popularon high-performance vehicles [18]; its load-handling capabilitiesalso make it suitable for use on the front axle of medium- andheavy-duty vehicles [19]. The upper and lower control arms areconnected to the wheel carrier with spherical joints (S) and to thechassis with revolute joints (R). One end of the tie rod is con-nected to the wheel carrier with a spherical joint; a universal joint(U) at the other end connects it to either the rack (on the frontaxle) or the chassis (on the rear axle). All geometric suspensionparameters are provided in Ref. [14]. The system is modeled usingjoint coordinates q ¼ ff; g; n;u h;l h; a;b; sg, as labeled inFig. 5(b). The configuration of the universal joint is specified byits angles of rotation about the global Z-axis (a) and the rotatedX-axis (b); ff; g; ng represent the 3-2-1 Euler angles associated

with the spherical joint between the upper control arm and thewheel carrier [17].

The kinematics of the double-wishbone suspension system aredesigned to control and optimize the motion of the wheel effi-ciently. Furthermore, the large number of design and functionalparameters involved in this suspension system make model reduc-tion an essential tool for design purposes. A multibody model ofthe double-wishbone suspension is developed and the governingdynamic equations are obtained in symbolic form using a Maplepostprocessor. We apply the POD reduction scheme to the kine-matic analysis of this system, which demonstrates the versatilityof the reduction technique in treating various nonlinear problems.A sinusoidal input of amplitude 0.1 m and frequency 0.1 Hz isapplied to the wheel axle (in the Z-direction) for 10 s—that is,zðtÞ ¼ 0:1 sin ð2p0:1Þtð Þ. We apply the POD approach to reducethe number of states from 8 to 3, use a control period of 5 s, andcollect ten snapshots in each control period. Simulation results forthe spindle (wheel carrier) translational displacement in theY-direction and the second Euler angle are shown in Fig. 6 for theoriginal and reduced-order models. Once again, the original andreduced-order models produce nearly identical results (as shownin Fig. 7); the root-mean-square errors are 7:57� 10�5m and2:4� 10�5rad for the spindle translation and second Euler angle,respectively. In this case, the reduced-order model simulates 65%faster than the original model.

3.3 Dynamic Analysis of a High-Fidelity Vehicle Model.The simulation of complex mechanical systems can benefit fromthe use of high-fidelity components in several applications (e.g.,control design and optimization); however, low-order simulationmodels are preferred in real-time applications and virtual designenvironments. In this section, we consider a high-fidelity model ofa vehicle with double-wishbone front suspensions and trailing-arm rear suspensions, as shown in Fig. 8.

Fig. 3 Simulated pitch and yaw angles for the independent-suspension vehicle using original and reduced-order models

Fig. 4 Errors between the original and reduced-order modelsfor pitch and yaw angles in the independent-suspension vehicle

Fig. 5 Schematic of the single double-wishbone suspensionsystem (a) and its corresponding topological graph (b). Thekinematics of this single-degree-of-freedom suspensionsystem are described by a set of nonlinear AEs.



The model is developed using the graph-theoretic approach,from which we obtain fully symbolic DAEs. We then apply thePOD technique to generate a more computationally efficientmodel for simulation. The vehicle is modeled using 26 general-ized coordinates coupled by 12 algebraic constraints, thereby

resulting in 14DOF. The dynamic behavior of the multibodysystem is governed by DAEs of the following form:

E _xðtÞ ¼ fHFVðt; xðtÞ;uðtÞÞ (11)

where E is singular due to the algebraic constraints. The DAEsdescribed by Eq. (11) are converted into a system of pure ODEsusing Baumgarte constraint stabilization [20], which combines theconstraint equations at the position, velocity, and acceleration lev-els to prevent the accumulation of constraint violations. The Fialatire model [21] is used to simulate the interaction between the tirecontact patch and the road.

A single-lane-change maneuver is simulated with an initialvehicle speed of 20 m/s. All numeric parameters associated withthe vehicle body, suspension systems, and tires are obtained fromRef. [14]. Simulation results obtained from the original model anda reduced-order model with only one state, along with the corre-sponding errors, are shown in Figs. 9 and 10, once again demon-strating excellent agreement. Root-mean-square errors for thelateral speed and vertical oscillation speed are 0.0198 m/s and0.0143 m/s, respectively. In this case, the reduced-order modelsimulates 23% faster than the original model when using a controlperiod of 0.1 s and five state snapshots in each control period.

4 Discussion

Due to the nonlinear nature of the dynamic systems consideredin this work, the reduced-order models may deviate from the orig-inal models in some situations. We investigated the robustness ofthe POD approach by applying a perturbed input to the lower-dimensional single double-wishbone system. In particular, weincreased the frequency of the input function by 20%, but contin-ued to use the unperturbed signal in the reduction procedure. Asshown in Fig. 11, increasing the frequency of the input functiondoes not affect the translational displacement of the spindle, butits second Euler angle diverges from that of the original model atthe end of the simulation. Note, however, that the response of areduced-order model need not exactly match that of the originalsystem. In general, we define an acceptable threshold of deviationbased on the required performance of the reduced-order model,which depends on the application. Decreasing the control periodused in the POD algorithm would reduce the deviation observedin Fig. 11, but would increase the computational expense.

Now consider Fig. 12, where we performed a similar experi-ment, but increased the amplitude of the input function by 20%.

Fig. 6 Simulated time histories of the spindle (wheel carrier)translational displacement (a) and second Euler angle (b) forthe double-wishbone suspension system using original andreduced-order models

Fig. 7 Errors between the original and reduced-order modelsfor spindle translational displacement and second Euler anglein the double-wishbone suspension system

Fig. 8 High-fidelity vehicle model with double-wishbone andtrailing-arm suspension systems. The dynamics of this 14DOFfreedom system are governed by a set of DAEs.



In this case, the results do not diverge substantially, indicatingthat the projection operator is insensitive to the amplitude of theinput signal (i.e., a similar projection would have been obtained ifthe high-amplitude input signal was used in the reduction proce-dure). Examining the sensitivity of a reduced-order model to per-turbations in the inputs is a useful tool for assessing the suitabilityof a model in a particular application, and for determiningwhether the model is proper. Figure 13 illustrates root-mean-square errors for the spindle translational displacement and secondEuler angle of the double-wishbone suspension for different per-turbations in the frequency and amplitude of the input signal,using the same parameters for the reduction scheme as before.The obtained results indicate that increasing the amplitude or thefrequency of the input signal does not necessarily generate more

deviations from the base reduced-order system, which can beattributed to the nonlinear behavior of the dynamic system. Fur-thermore, the spindle second Euler angle is nearly more sensitive

Fig. 9 Simulated trajectory of the high-fidelity vehicle modelusing original and reduced-order models

Fig. 10 Simulated time histories of lateral speed (top) and ver-tical oscillation speed (bottom) of the chassis, along with thecorresponding errors, in the high-fidelity vehicle model usingoriginal and reduced-order models

Fig. 11 Simulated time histories of spindle translational dis-placement (top) and second Euler angle (bottom) for thedouble-wishbone suspension system using original andreduced-order models, where a perturbed input (increased fre-quency) is applied to the lower-dimensional model

Fig. 12 Simulated time histories of spindle translationaldisplacement (top) and second Euler angle (bottom) for thedouble-wishbone suspension system using original andreduced-order models, where a perturbed input (increasedamplitude) is applied to the lower-dimensional model



to perturbations in frequency and amplitude of the input signalcompared to the spindle translational displacement.

To further establish the performance of the reduction schemeand the proposed numerical algorithm, we consider the case inwhich the input signal frequency is increased by 100%, for whichthere is a significant error between the original model and thereduced-order model for the spindle second Euler angle (Fig. 14).It is important to note that, by increasing the input signal fre-quency, some of the components ignored in the reduced-order

model will begin contributing to the dynamic response of thesystem—that is, the energy content in the system will no longerbe predominantly contained in the three modes considered for thereduced-order model. By increasing the number of states in thereduced-order model to 6, the results are once again close to thoseof the original model, as expected (see Fig. 14). Although, thesimulation time for the model with six states was longer comparedto the model with three states, it is still faster than the originalmodel by 50%, indicating the great performance and robustness ofthe POD reduction scheme.

5 Conclusion

This work focuses on the reduction of small-scale, nonlinearmultibody models in automotive systems using the POD, a well-established reduction scheme for large-scale models. Wedemonstrated the versatility of the POD approach by generatingreduced-order systems governed by pure ODEs, nonlinear AEs,and DAEs. Specifically, we analyzed the dynamics of a simplifiedvehicle model with a fully independent suspension, the kinematicsof a single double-wishbone suspension system, and the dynamicsof a high-fidelity vehicle model with double-wishbone andtrailing-arm suspensions. The reduced-order models, we obtainedproduced simulation results that were in close agreement with theresults obtained using the original models while demanding sub-stantially less computational effort. A sensitivity study indicatedthat the reduced-order kinematic model of the double-wishbonesuspension is more robust to perturbations in amplitude than fre-quency of the input signal. Ultimately, we conclude that the PODis a reliable model reduction tool for small-scale, nonlinearmultibody systems.

To summarize, we verified the performance of the POD insmall-scale systems, focusing on multibody models used in auto-motive systems; established applicability of the reduction schemeto kinematic analysis of multibody systems; proposed an efficientcomputer algorithm capable of retaining the accuracy of reduced-order systems; and investigated the robustness of reduced-ordermodels, when applying a different input from the one used toextract the projection bases. It is important to note that thereduced-order models are primarily intended for use in high-fidelity yet computationally low-cost full vehicle models incontrol-oriented and design optimization problems (e.g., powermanagement systems design in hybrid and plug-in hybrid electricvehicles). In these applications, reducing the size of each compo-nent can play an important role when simulating the system as awhole, especially if real-time performance is required (e.g., hard-ware-in-the-loop and software-in-the-loop applications). The errorplots we have shown demonstrate that a reduced-order model canfollow the trajectory of the original nonlinear model precisely.Depending on the requirements of the application, however, thecomputational cost and time may be further improved by reducingthe accuracy of the reduced-order model.

Acknowledgment

The authors gratefully acknowledge the financial support of theNSERC/Toyota/Maplesoft Industrial Research Chair program.

References[1] Ersal, T., Rideout, D. G., Louca, L. S., Stein, J. L., and Fathy, H. K., 2008, “A

Review of Proper Modeling Techniques,” ASME J. Dyn. Syst., Meas. Control,130(6), p. 061008.

[2] Louca, L. S., Stein, J. L., and Hulbert, G. M., 1998, “A Physical-BasedModel Reduction Metric With an Application to Vehicle Dynamics,” Pro-ceedings of the 4th IFAC Nonlinear Control Systems Design Symposium,Enschede, The Netherlands.

[3] Rideout, D. G., Stein, J. L., and Louca, L. S., 2005, “System Partitioning andImproved Bond Graph Model Reduction Using Junction Structure PowerFlow,” Proceedings of the International Conference on Bond Graph Modeling,Vol. 5, pp. 43–50.

Fig. 13 Root-mean-square errors for spindle translational dis-placement and second Euler angle of the double-wishbone sus-pension in terms of various perturbations in frequency (top)and amplitude (bottom) of the input

Fig. 14 Simulated time histories of the second Euler angle forthe double-wishbone suspension system using original andreduced-order models, where a perturbed input is applied tothe lower-dimensional model with three states (top) and sixstates (bottom)



http://dx.doi.org/10.1115/1.2977484

[4] Louca, L. S., and Yildir, B. U., 2006, “Modelling and Reduction Techniquesfor Studies of Integrated Hybrid Vehicle Systems,” Math. Comput. Modell.Dyn. Syst., 12(2–3), pp. 208–218.

[5] Kokotovic, P. V., Malley, R. E. O., Jr., and Sannuti, P., 1976, “Singular Pertur-bations and Order Reduction in Control Theory—An Overview,” Automatica,12(2), pp. 123–132.

[6] Liu, Y., and Anderson, B. D. O., 1989, “Singular Perturbation Approximationof Balanced Systems,” Int. J. Control, 50(4), pp. 1379–1405.

[7] Azad, N. L., Sanketi, P. R., and Hedrick, J. K., 2012, “Determining Model Ac-curacy Requirements for Automotive Engine Coldstart Hydrocarbon EmissionsControl,” ASME J. Dyn. Syst. Meas. Control, 134(5), p. 051002.

[8] Marquez, A., Oviedo, J. J. E., and Odloak, D., 2013, “Model Reduction UsingProper Orthogonal Decomposition and Predictive Control of Distributed Reac-tor System,” J. Control Sci. Eng., 2013(3), pp. 1–19.

[9] Prabhat, P., Balakrishnan, S. N., Look, D. C. L., Jr., and Padhi, R., 2003,“Proper Orthogonal Decomposition Based Modeling and Experimental Imple-mentation of a Neurocontroller for a Heat Diffusion System,” Proceedings ofthe American Control Conference, Vol. 3, pp. 2652–2657.

[10] Atwell, J. A., 2000, “Proper Orthogonal Decomposition for Reduced OrderControl of Partial Differential Equations,” Ph.D. thesis, Virginia PolytechnicInstitute and State University, Blacksburg, VA.

[11] Kerschen, G., Golinval, J.-C., Vakakis, A. F., and Bergman, L. A., 2005, “TheMethod of Proper Orthogonal Decomposition for Dynamical Characterizationand Order Reduction of Mechanical Systems: An Overview,” Nonlinear Dyn.,41(1–3), pp. 147–169.

[12] Masoudi, R., Uchida, T., and McPhee, J., 2013, “Comparison of Model OrderReduction Methods for Automotive Systems,” Proceedings of the ECCOMASThematic Conference on Multibody Dynamics, pp. 153–154.

[13] Ebert, F., 2010, “A Note on POD Model Reduction Methods for DAEs,” Math.Comput. Modell. Dyn. Syst., 16(2), pp. 115–131.

[14] Maplesoft, 2014, “Vehicle Model With Double-Wishbone Front and Trailing-Arm Rear Suspensions,” MapleSim Model Gallery: Application ExamplesCreated in MapleSim, Maplesoft, Inc. Available at: http://www.maplesoft.com/products/maplesim/ModelGallery/

[15] Sayers, M. W., and Han, D., 1996, “A Generic Multibody Vehicle Model forSimulating Handling and Braking,” Veh. Syst. Dyn., 25(S1), pp. 599–613.

[16] Pacejka, H., 2012, Tire and Vehicle Dynamics, 3rd ed., Elsevier, Oxford, UK.[17] Uchida, T., and McPhee, J., 2012, “Driving Simulator With Double-Wishbone

Suspension Using Efficient Block-Triangularized Kinematic Equations,” Multi-body Syst. Dyn., 28(4), pp. 331–347.

[18] Reimpell, J., and Stoll, H., 2001, Automotive Chassis: Engineering Principles,2nd ed., Butterworth-Heinemann, Oxford, UK.

[19] Cherian, V., Jalili, N., and Ayglon, V., 2009, “Modelling, Simulation, andExperimental Verification of the Kinematics and Dynamics of a DoubleWishbone Suspension Configuration,” Proc. Inst. Mech. Eng. Part D, 223(10),pp. 1239–1262.

[20] Baumgarte, J., 1972, “Stabilization of Constraints and Integrals of Motion inDynamical Systems,” Comput. Methods Appl. Mech. Eng., 1(1), pp. 1–16.

[21] Fiala, E., 1954, “Seitenkr€afte am Rollenden Luftreifen,” VDI Z., 96(2),pp. 973–979.



http://dx.doi.org/10.1080/13873950500068690

http://dx.doi.org/10.1080/13873950500068690

http://dx.doi.org/10.1016/0005-1098(76)90076-5

http://dx.doi.org/10.1080/00207178908953437

http://dx.doi.org/10.1115/1.4006217

http://dx.doi.org/10.1155/2013/763165

http://dx.doi.org/10.1007/s11071-005-2803-2

http://dx.doi.org/10.1080/13873951003740041

http://dx.doi.org/10.1080/13873951003740041

http://www.maplesoft.com/products/maplesim/ModelGallery/

http://www.maplesoft.com/products/maplesim/ModelGallery/

http://dx.doi.org/10.1080/00423119608969223

http://dx.doi.org/10.1007/s11044-011-9299-y

http://dx.doi.org/10.1007/s11044-011-9299-y

http://dx.doi.org/10.1243/09544070JAUTO1153

http://dx.doi.org/10.1016/0045-7825(72)90018-7

reduction of multibody dynamic models in automotive...

Documents