MB simulations for vehicle dynamics: reduction

through parameters estimation

Gubitosa Marco

The aim of this activity is to propose a methodology applicable for parameters estimation in

vehicle dynamics, aiming at generating reduced models to be adopted for functional analyses

and real time simulations with the focus on enabling a model conversion scheme, allowing

building a communication bridge between the 1D and the 3D simulation domains.

The benchmark case, i.e. the reference high fidelity model, is defined in the 3D multibody

environment of LMS Virtual.Lab Motion, while the simplified model is developed

symbolically with the help of Maple and implemented in the block scheme oriented interface

of Imagine.Lab AMESim. To estimate the physical and structural parameters of the detailed

benchmark model for use in the functional model, it is important to make clear how and to

what extent the response of the system depends on each parameter. Therefore sensitivity

analysis and optimization loops are programmed to firstly define the most effective

contribution to the behaviour of the simplified model and secondly asses a good correlation in

the dynamic performance.

Reference vehicle model The use of MBS software allows the modeling and simulation of a range of vehicle subsystem

representing the chassis, engine, driveline and body areas of the vehicle as shown in Figure 1,

where is intended that multibody system models for each of those areas are integrated to

provide a detailed representation of the complete real vehicle.

Figure 1: Integration of subsystems in a full vehicle model and detail of the vehicle dynamic areas of interest

Here also the modeling of road and driver are included as elements of what is considered a

full vehicle system model. Restricting the discussion of full vehicle system models to a level

appropriate for the vehicle dynamics, a detailed modeling of the suspension systems, anti roll

bars, steering system and tires is needed as evidenced with ovals in Figure 1. Of course to

complete the model and give the possibility to run simulations with an acceptable realistic

level of the vehicle’s response, models of driver inputs as well as engine and driveline final

effects should be comprised. Besides the inertia characteristics of the sprung (and un-sprung)

masses have to be included. A creation of a virtual environment is than an aspect that can be

considered.

Set up of the MB model

Figure 2: Front double wishbone and Rear 5-link suspension (example of the Acura RL and TSX 2008)

The double wishbone front suspension is constructed with short upper wishbones, lower

transverse control arms and longitudinal rods whose front mounts absorb the dynamic rolling

stiffness of the radial tires. The spring shock absorbers are supported via fork-shaped struts on

the transverse control arms in order to leave space to the crank shafts and are fixed within the

upper link mounts.

As its name indicates, the rear suspension employs five links. The hub-carrier/spring-shock-

absorber mount is located by five tubular links: a trailing link, lower link, lower control link,

upper link, and upper leading link. Car manufacturers claim that this system gives even better

road-holding properties, because all the various joints make the suspension almost infinitely

adjustable.

The linkages of the suspension parts are realized partially with bushings, representing the

compliance elements, and for some of them ideal joints have been included, realizing a

kinematically constrained mechanism as represented in Figures 3 and 4.

The bushing element available in Virtual Lab Motion defines a six degree-of-freedom element

between two bodies, producing forces along and torques about the three principal axes of the

element attachments. The bushing characteristics are defined as a combination of six values of

stiffness and six values of damping which are normally defined by non linear spline curves.

The equation below describes the formulation for forces in the bushing.

F1 = Kz + DŜ + FK(z) + FD(Ŝ)

F2=-F1

where

F2 and F1 are the force vectors applied to body 1 and 2

K and D are the stiffness and damping matrixes

z and Ŝ are the relative displacement and velocity vectors between the two bodies

FK(z) and FD(Ŝ) are the forces expressing stiffness and damping as functions of relative

displacement and velocity in a nonlinear sense.

A similar formulation is used for the calculation of torque reactions in function of relative

rotation and rotation velocity between the connected bodies.

Table 1: Connection types (Joints and Bushings) for the front suspension

Figure 3: Linkage of the Front Suspension system

Table 2: Connection types (Joints and Bushings) for the rear suspension

Figure 4: Linkage of the Rear Suspension system

Assumptions and force elements considered

Antiroll Bars

They're also known as sway-bars or anti-sway-bars. The function of the anti-roll bars is to

reduce the body roll inclination during cornering and to influence the cornering behavior in

terms of under- or over-steering. The anti-roll bar is usually connected to the front, lower

edge of the bottom suspension joint. It passes through two pivot points under the chassis,

usually on the subframe and is attached to the same point on the opposite suspension setup.

Hence, the two suspensions are not any more connected only due to the subframe and the

chassis, but effectively they are joined together through the anti-roll bar. This connection

clearly affects each one-sided bouncing.

Figure 5: Anti-roll bar loaded by vertical forces

In the model here implemented it has been considered a lumped torsional stiffness granted

to a bushing element located at the mid-point connection of the two bodies representing the

left and right portions of the antiroll-bar.

Damper, springs and end stops

Figure 6: Example of a spring dumper structure with notable elements listed

The force elements included in the strut here shown are re reproduced in the MB model as

non-linear splines for the damping characteristic and linear stiffnesses for the main

spring and end stops.

Figure 7: Setting the damping curves

Front Suspension

Stiffness 48000 N/m Coil Spring

Preload 5800 N

AntiRoll Bar Torsional Stiffness 2000 Nm/rad

Stiffness 350000 N/m Bump Stop

Clearance 40 mm

Stiffness 680000 N/m Rebound Stop

Clearance 50 mm

Rear Suspension

Stiffness 36000 N/m Coil Spring

Preload 3500 N

AntiRoll Bar Torsional Stiffness 500 Nm/rad

Stiffness 350000 N/m Bump Stop

Clearance 26 mm

Stiffness 700000 N/m Rebound Stop

Clearance 75 mm

Table 3: Characteristics of the Force Elements

Steering system

The simplest and also most common steering system to be created is the rack and pinion

steering system. Firstly the rotations of the steering wheel are transformed by the steering box

to the rack travel which is travels along a straight rail activated by the rotations of a pinion. At

the extremities of the rack two tie rods permit the transformation of this translational

movement to the rotation around the steering axis of the suspension. Hence the overall

steering ratio depends on the ratio of the steering box and the kinematics of the steering

linkage.

Table 4: Connection types for the steering mechanism

Figure 7: Representation of the joints and configuration of the mechanism of the steer

In Figure 7 the hierarchical organization of the joints is shown. Here is possible to see (in the

block scheme) two green arrows indicating the revolute joint of the steer on the chassis and

the translational joint of the rack. This means that there is a correlation between the two (set

by a relative driver) which is programmed by the steering ratio.

Tires modelling

An accurate modelling of the tire force elements is achieved by including the so called TNO-

MF tire (version 6.0), which is based on the renowned ‘Magic Formula’ tire model of

Pacejka. The model takes as input a series of parameters (i.e. a vector with more than 100

elements) for each calculation to be performed, which are empirically determined coefficients

that address the complexity of the model.

Equations of motion

Virtual.Lab Motion is based on a Cartesian coordinates approach for the assembly of the

equations of motion. The solver uses Euler parameters to represent the rotational degrees of

freedom (avoiding therefore the intrinsic singularity of the angular notation) and Lagrangian

formulation for the assembly and generation of equations of motion. The joints between

bodies are expressed in a set of algebraic equations, subsequently assembled in a second

derivative structure, obtaining finally a set of Differential Algebraic Equations (DAEs)

packable in the following form:

( ) ( )( )

( )

=

γ

qqQ

λ

q

0qΦ

qΦqM tT ,, &&&

Here

M is the mass matrix

q is the vector of the generalized coordinates

Q is the vector of the generalized forces applied to the rigid bodies

λλλλ is the vector of the Lagrange multipliers

ΦΦΦΦ is the Jacobian of the constraint forces

γ the right-hand-side of the second derivative the constraint equations

This model includes 52 bodies; therefore a total of 52 x 7 = 364 configuration parameters

are used by the pre-processor to build the set of equations of motion.

For the settling configuration, in which the vehicle is just let rest on the ground with null

initial conditions, joints and drivers are for a total of 234, therefore leaving the system with

130 degrees of freedom.

While setting up a manoeuvre, instead, additional constraint is added to the system in terms of

position driver on the steering wheel, commanded in open loop, and forces are acting on the

wheel’s revolute joints to represent the driving torque. Moreover, non-zero initial conditions

at velocity and position level are added to every body.

Between the different solvers solution proposed in the Virtual.Lab Motion (here below

reported), the BDF has been selected, granting a good stable behaviour for such a stiff system.

Acronym Name Code based on Type Strengths

PECE

Predict -Evaluate-Correct-Evaluate Adams-Bashforth-Moulton Method

Shampine-Gordon’s DE

Explicit, Multistep Discontinuous Systems and

Non-stiff systems

BDF Backwards Difference

Formulation DASSL Implicit, Multistep

Smooth, stiff, Systems

RK Runge-Kutta DOPRI5 Explicit, Singlestep Extremely

discontinuous systems

Table 5: Solvers available in Virtual.Lab Motion

Simplified modelling approach

ψψψ &&& ,, Yaw angle, yaw velocity and yaw acceleration

ϕϕϕ &&&,, Roll angle, roll velocity and roll acceleration

ββ &, Car-body sideslip angle, velocity at center of gravity

v Absolute car-body velocity at center of gravity

δ1 δ2 Steering angle of the front wheels and rear

λ1, λ2 Coefficient for camber angle induced by roll (front and rear)

a1, a2 Front and rear wheelbases

b1, b2 Front and rear half tracks

h relative position of roll center with respect to car-body CG

M Total mass of the vehicle

Ixx Roll inertia of the vehicle

Izz Yaw inertia of the vehicle

Kφ Total anti-roll stiffness (Kr1 + Kr2)

bφ Total roll damper rate

C1, C2 Camber stiffness, resp. for front and rear axle

g Constant of gravity (defaulted to 9.80665 m/s

δaxle elas + δtire Total front and rear sideslip (axle + tire)

δ1 axle kin Steering angle of the wheel due to axle kinematics - Front axle

δ1 axle elas Toe angle of the wheel due to axle elasto-kinematics - Front axle

δ2 axle kin Steering angle of the wheel due to axle kinematics - Rear axle

δ2 axle elas Toe angle of the wheel due to axle elasto-kinematics - Rear axle

axleaxleaxle bkM ,, Mass, stiffness and damping of the axle in lateral direction

vy1 Lateral deformation velocity of the axle (front and rear)

Table 6: List of symbols for the simplified model

The domain of lateral vehicle dynamics is here investigated. As mentioned before a range of

possible approaches has been reported to model the dynamics of a vehicle. Depending on the

field of study and the accuracy required, the details to be included vary considerably. The

solution for this dilemma, and a trustworthy help to the vehicle dynamics engineer comes

from the adoption of a modular simplified modeling approach.

The simplified model for lateral dynamics studies proposed in the following is a four wheels

chassis model with medium wheel approach for front and rear axles. The structure of this

model, well known in the literature, is meant for handling modeling and lateral dynamics

studies and has 3 DOF: yaw velocity (ψ& ), carbody sideslip angle at center of gravity (β ) and

roll angle (φ). The equation of motion are obtained cascading the overall dynamics to the

following set of differential equations, expressed in what are normally called quasi-

coordinates and generated by forces and moments balance of the Newton-Euler approach.

Moreover, linearization in the McLaurin series (assuming to be in steady state conditions and

close enough to the equilibrium position) brings to the condensed formulation:

( )[ ]

( ) ( )[ ]

=++−+−+

−=

=−+

∑

∑

tiresyxx

tireytireyzz

tiresy

FhkbhvMhMhI

FaFaI

FhvM

0

2

2211

ϕϕϕβψϕ

ψ

ϕβψ

ϕϕ&&&&&&&

&&

&&&&

Hence a global motion is allowed with respect to the ground, including also the car-body roll

effect on the generalized sideslip and yaw velocity (due to roll center heights and axle

kinematics). In the state equations, the relative position of roll center with respect to the car-

body center of gravity can be computed with relative height of roll center above front and rear

axle:

++

+−=−= 2

21

11

21

20 rrGG h

aa

ah

aa

ahhhh

In addition the load transfer between left and right is included (effect while negotiating a turn)

which brings in the variation of lateral force available at the tires, computed via:

tirez

tireya

FaF δ⋅

⋅−=

4

3 arctan2sin

The total axle sideslip angle gives an extra contribution to the lateral force acting on the axle

due to the synthesis parameters of camber stiffness (C1λ1, C2λ2) and lateral stiffness and

damping. Therefore the combination of axle kinematic steering angle (axle kin) and axle

compliance contribution (axle elas) is considered; the tire slip angle can be hence written as:

( ) elasaxlekinaxle

y

fronttirev

va11

11 δδψ

βδ −−−

+=&

( )tireaxleaxleaxletireyelasaxle bkMFD

δδ ,,,2

1

1

1 −=

In addition the relaxation length is considered as a simple first order filter with fixed time

constant.

Figure 8: Representation of the vehicle model in Imagine.Lab AMESim for studying lateral dynamics

The set of equations therefore obtained is presented in the form of an ODE system, but with

an implicit loop for the computation of the lateral force on the tire: infact it depends on the

lateral slip that is computed from the lateral force again.

In this condition AMESim automatically selects a solver based on DASSL (Differential

Algebraic System Solver), therefore a sort of BDF formula is used.

Whether a full ODE system was provided the solver would have been selected between the

ADAMS or Gear’s method, actually both included in the same algorithm (LSODA) which

switches between the two based on an index to identify the stiffness of the problem.

Approach for the Estimation scheme

The systematic approach proposed is distinguished then in two steps:

1. in the first step, the Assured and Calculated parameters are provided as input to the

simplified models and considered as fixed

2. in the second step optimization algorithms run to determine the Estimated parameters in a

loop that aims to minimize multiple objectives.

Assured Obtained from the high fidelity model (the real vehicle) without

experiments

Calculated Can be computed a priori from known parameters or by basic

measurements

Estimated These include synthesis parameters and reduced model

topological definition classified based on LH-DOE

Table 7: Phenomenological classification of system’s parameters

Case study of Lateral Dynamics

In general the motion equations governing a mechanical system are in the form of second

order differential equations. Three types of solutions can be computed from this mathematical

formulation, corresponding to three types of driving circumstances: the steady-state, the

stability solution and the frequency response. To be able to explore those three domains,

different maneuvers have been selected as comparison between the high fidelity model and

the simplified model: slalom, step steer and a form of open loop double lane change. Since

it is commonly accepted that yaw rate relates mainly to what a driver sees and lateral

acceleration relates to the human feelings, both aspects will be considered as output

parameters. The identification method illustrated here is then based on the error minimization

calculated in three different time windows, as shown here below, considering basically the

equilibrium starting condition in error 1, transient behavior in error 2 and steady state after

distortion (steering input) in error 3.

0 1 2 3 4 5 6 7 8 9 10-10

0

10

20

30

Time [s]

ya

w r

ate

[d

eg

ree

/s]

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

Time [s]

late

ral accele

ratio

n [G

]

1 2 3

Figure 9: Distinction of the errors domain

The following table summarizes the parameter classification adopted for the lateral dynamics

model.

Sub model Parameters Definition

mass of vehicle Assured

yaw inertia Calculated

roll inertia Calculated

front wheelbase Assured

rear wheelbase Assured

front half track Assured

rear half track Assured

height of centre of gravity Calculated

height of front roll centre (absolute) Estimated

height of rear roll centre (absolute) Estimated

front anti-roll stiffness Estimated

rear anti-roll stiffness Estimated

roll damper rating Estimated

steer angle ratio Estimated

toe coefficient induced by roll - front & rear Estimated

camber coefficient induced by roll - front & rear Estimated

3 DOF Vehicle Model

front castor offset Estimated

compliance gain - front & rear Estimated

lumped mass axis - front & rear Estimated

compliance spring - front & rear Estimated Axis compliances

compliance damper - front & rear Estimated

slope at the origin - maximum cornering stiffness

Calculated

slope at the origin - vertical load for maximum cornering stiffness

Calculated Tire Models

camber stiffness Calculated

Steering Mechanism radius of the pinion Calculated

Table 8: Parameters classification for the lateral dynamics case study

Before the optimization, an accurate sensitivity analysis is run. The number of sampling

points in the Latin Hypercube-DOE is set to 3000. Since the number of parameters is high, the

importance of selection the appropriate excitation for the target parameters is a crucial process

since one must avoid ending up with an ill-posed inverse problem. The objective functions of

each stage are hence defined by results of the sensitivity analysis.

The subsequent optimization is divided into three stages, cascading from the highest (most

contributing) to lower sensitive parameters with respect to the selected cost functions. The

combinations of Design Variables and Object functions are summarized in the below Table,

where part 1, 2 and 3 refer to the error classification scheme previously defined.

Stages Design Variables Objective Functions

Stage1 steer angle ratio part3_error_yaw_rate

roll damper rating part2_error_lateral_acceleration

front anti-roll stiffness

rear anti-roll stiffness

Slalom

part3_error_lateral_acceleration

front anti-roll stiffness

rear anti-roll stiffness part1_error_yaw_rate

compliance gain - front

Stage2

compliance gain - rear

Double Lane Change

part3_error_yaw_rate

toe coefficient induced by roll - front & rear

part3_error_yaw_rate

lumped mass axis - front & rear

Slalom

part1_error_lateral_acceleration

height of rear roll centre part1_error_yaw_rate

height of front roll centre

camber coefficient induced by roll - front & rear

compliance spring - front

compliance spring - rear

part3_error_yaw_rate

front castor offset

Double Lane Change

part3_error_lateral_acceleration

compliance damper - front part2_error_yaw_rate

Stage3

compliance damper - rear

Step steer part2_error_yaw_rate

Table 9: Combination of design variables and object functions for each stage

As the multi-objective optimization of the third stage reaches the stopping criteria, the

differential evolution algorithm provides the Pareto set (collection of optimal solutions) that

minimizes the concurrent objective functions. As clarified by the objective contribution plot

(Figure 10) a parameter set that happens to minimize one cost function, is instead acting

negatively for another objective. The optimal point is selected afterwards based on a trade-off

between the single contributions. The results of the final comparison between the high fidelity

model and simplified lateral dynamic model are shown in Figure 11 a, b, c, d. In particular

Figure 11 d proposes a cross validation of the model by applying to it a manoeuvre for which

parameters estimation has not been performed (i.e. random steering action in time).

Figure 10: Objective contribution plot of optimization of stage 3 with respect to the 7 objective functions of the

different maneuvers of slalom (SLALOM), double lane change (LANE), step steer (CSA)

0 2 4 6 8 10-20

-10

0

10

20

Time [s]

ya

w r

ate

[d

eg

ree/s

]

simplified

reference

0 2 4 6 8 10-0.4

-0.2

0

0.2

0.4

Time [s]

late

ral a

ccele

ratio

n [G

]

simplified

reference

0 2 4 6 8 10-20

-10

0

10

20

Time [s]

ya

w r

ate

[d

eg

ree

/s]

simplified

reference

0 2 4 6 8 10-1

-0.5

0

0.5

1

Time [s]

late

ral a

cce

lera

tio

n [G

]

simplified

reference

(a) (b)

0 2 4 6 8 10-5

0

5

10

15

Time [s]

ya

w r

ate

[d

eg

ree

/s]

simplified

reference

0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

Time [s]

late

ral a

cce

lera

tion

[G

]

simplified

reference

0 2 4 6 8 10-5

0

5

Time [s]

ya

w r

ate

[d

eg

ree

/s]

simplified

reference

0 2 4 6 8 10-0.2

-0.1

0

0.1

0.2

Time [s]

late

ral a

cce

lera

tio

n [G

]

simplified

reference

(c) (d)

Figure 11: Comparison of the behavior of simplified and reference model for a) slalom maneuver, b) double

lane change, c) step steer, d) polynomial steering angle

Conclusion

An observation can be made regarding the simplified model validity. Simplified vehicle

models are often used by control engineers for control design and online implementation of

on-board safety systems. Typically, these models are intensively used for efficient cycle

computation within a limited validity range: several physical phenomena and dynamics

effects are not included. This means that from the analysis of these models, the controls

engineer does not learn about possible interactions of the neglected dynamics with the control

law and furthermore, the effect of uncertainty in the input parameters on the controlled system

performance is not assessed. An example is indeed in the notable gap between the prediction

of the simplified model and the high fidelity multibody simulation when large lateral

acceleration is required.

0 2 4 6 8 10-20

0

20

40

60

Time [s]

ya

w r

ate

[d

eg

ree

/s]

simplified

reference

0 2 4 6 8 10-0.5

0

0.5

1

1.5

Time [s]

late

ral a

cce

lera

tio

n [G

]

simplified

reference

Figure 10: Step steer manoeuvre with large steering input (i.e. high lateral acceleration)

For a proper insight in the physical performance of the end product (i.e. the active vehicle on

the road), an improved engineering process is needed to guarantee the vehicle and the

controller performance even in the presence of unmodelled physical effects and uncertainty in

the input parameters. A work in progress is indeed in this direction, aiming at achieving

higher model accuracy while keeping them as simple as possible.