# Sensitivity and reliability analysis of articulated heavy vehicle

Hamid Badi, Frédéric Bernardin, Mohamed Bouteldja, Michel Fogli,
Claude-Henri Lamarque

<figure><img src="image_1.png" /></figure>

## To cite this version:

Hamid Badi, Frédéric Bernardin, Mohamed Bouteldja, Michel Fogli, Claude-Henri Lamarque. Sensitivity and reliability analysis of articulated heavy vehicle. SIA, Oct 2011, Mulhouse, France. 8p. hal-05289861

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Distributed under a Creative Commons Attribution 4.0 International License
# Sensitivity and reliability analysis of articulated heavy vehicle

H. Badi¹, F. Bernardin², M. Bouteldja³, M. Fogli⁴ & C.H. Lamarque⁵

¹LCPC, Public Works Research Laboratory, 58 Boulevard Lefebvre, 75015 Paris, FRANCE

²CETE de Lyon, Département Laboratoire de Clermont-Ferrand, 8 rue Bernard Palissy, 63014 Clermont-Ferrand

³CETE de Lyon, Département Laboratoire de Lyon, 25 avenue Franois Mitterand, 69500 Bron

⁴ University of Clermont-Ferrand, Campus Les Cézeaux, 63175 Aubière, FRANCE

⁵ ENTPE, France's State School of Civil Engineering, Rue Maurice Audin, 69518 Vaulx-en-Velin, FRANCE

email: ¹hamid.badi@ifma.fr, ²frederic.bernardin@developpement-durable.gouv.fr,

³mohamed.bouteldja@developpement-durable.gouv.fr, ⁴claude.lamarque@entpe.fr & ⁵ michel.fogli@polytech.univ-bpclermont.fr

**ABSTRACT:** The aim of this paper is to develop a stochastic based approach to prevent rollover and jackknifing risk of articulated heavy vehicle. A specific 6-DOF heavy vehicle model is developed and safety criteria are used. Parameters involved in the model are modeled by random variables or stochastic processes in order to take into account uncertainties. These parameters are deduced from two different sensitivity analysis methods. Then, structural reliability methods are employed to assess the probability of failure and compared to Monte-Carlo based method.

**KEY WORDS:** articulated heavy vehicle; rollover; jackknifing; stochastic dynamics; sensitivity analysis; reliability analysis

## 1 INTRODUCTION

At present time, the truck transport is one of the most important activity of the country's economy. According to the French road safety statistics (ONISR) for year 2008 [1], accidents involving heavy vehicles have serious consequences for road users and incidents induce major congestions or damage to the environment or the infrastructure at disproportionate economic costs. The risk of having dead people in accidents involving trucks is multiplied by 2.4 in comparison to the same risk computed for accidents involving only light vehicles, mainly because of the important gross mass difference between light vehicles and trucks. Many real-time active safety systems have been developed to control vehicle stability in dangerous conditions : rollover, jackknifing, roadway departure... :

ABS Anti-Lock Braking System,

EPS Electronic Stability Program,

RSP Roll Stability Program,

VEST Volvo Enhanced Stability Technology.

Nevertheless all these systems don't detect and warn drivers early enough for preventing risky situations. Consequently, the aim is to develop a driving support system, embedded in the heavy vehicle and communicate with on road equipments, to warn about risk's level early enough.

Little work is available in the literature concerning combined probabilistic analysis and vehicle dynamics studies. Is mostly deals with suspensions analysis under uncertain characteristics and loading [2]. One Icelandic study [3] covers the wind-related accident issue with reliability analysis, but no warning system has been envisaged. More recently, classical reliability methods are deployed in [4], [5] to prevent rollover risk of single heavy vehicle and roadway departure risk of lighty vehicle.

## 2 PRINCIPLE OF THE PROPOSED APPROACH

This paper propose of a probabilistic method for identifying dangerous situations of risk of articulated heavy vehicle. To this end, we develop a stochastic heavy vehicle model derived from a 6-DOF deterministic heavy vehicle model. Next, two safety criteria related to rollover and jackknifing risk are introduced. In practice, parameters involved in the model are not known with absolute precision. Thus, it's necessary to take into account the random nature of these parameters by modeling them as random variables or stochastic processes. A part of the work presented here aims to exhibit the revelant entry variables that must be known to be representative of the system uncertainties. So, we perform and compare two differents sensitivity analysis methods in order to identify influent parameters and to reduce the number of stochastic parameters involved in the stochastic model. Finally, probability of failure is assessed through standard structural reliability methods by using the well-known FORM/SORM methods.

## 3 DETERMINISTIC ARTICULATED HEAVY VEHICLE MODEL

This section presents the deterministic model of an articulated heavy vehicle and its validation. Then safety criteria, respect to rollover and jackknifing risks, are chosen to define the riskiness of a situation. Finally, a validation is conducted uppon the safety criterion values by comparison with a complete simulator (Prosper).

### 3.1 Heavy vehicle dynamics model

The framework of this study is focused on an articulated heavy vehicle composed of a rigid tractor with 2 axles and a rigid semitrailer with 3 axles. These two components are linked at the
fifth wheel as shown in figure 1. The tractor is the superposition of an unsprung mass (axles and tires) and a sprung mass (cabine and chassis). The trailer is composed of an unsprung mass (axles and tires) and a sprung mass (chassis and trailer). Unsprung and sprung masses are connected together with suspensions. In this study, the following two basic assumptions are used :

1 the cornering manoeuvre realized at constant vehicle speed,
2 no breaking and acceleration actions by the driver.

Under these hypotheses, the considered dynamic component of
the motion are both the yaw tractor, the translation of the vehicle
in the horizontal plane, the tractor roll angle and the articulation
angle between the tractor and the trailer. To construct the model
of the heavy vehicle we complete the previous assumptions with
the following ones :

3 the pitch and bounce dynamics motions are neglected,

4 tractor and trailer frames are modelled as rigid bodies,

5 the total axles with axle suspension are reduced to roll
suspension only.

To obtain the dynamics equations of simplified heavy vehicle,
the classical Lagrangian formulation is used.

<figure><img src="image_3.png" /><figcaption>Figure 1. Articulated heavy vehicle</figcaption></figure>

The proposed model is based on 6 degrees of freedom (DOF) yaw-roll model derived from a 5-DOF [6], [7]. To get the dynamics equation of the model we consider the motion the two sprung masses in the coordinate system (see fig. 2). (X<sub>E</sub>, Y<sub>E</sub>, Z<sub>E</sub>) is the earth-fixed coordinate system, (X<sub>t</sub>, Y<sub>t</sub>, Z<sub>t</sub>) and (X<sub>st</sub>, Y<sub>st</sub>, Z<sub>st</sub>) are respectively the tractor and semi-trailers'sprung masses coordinate systems fixed at the center of gravity (C.G) of each body. (X<sub>u</sub>, Y<sub>u</sub>, Z<sub>u</sub>) is the tractor's unpring mass coordinate defined at the center plane of the front axle with Z<sub>u</sub> parallel to Z<sub>E</sub>. The relative motion of (X<sub>u</sub>, Y<sub>u</sub>, Z<sub>u</sub>) with respect to the earth-fixed coordinate system (X<sub>E</sub>, Y<sub>E</sub>, Z<sub>E</sub>) describes the translation motion of the tractor in the horizontal plane and its yaw motion along the Z<sub>E</sub>-axis. The roll motion is described by motion of the coordinate system (X<sub>t</sub>, Y<sub>t</sub>, Z<sub>t</sub>) associated with the coordinate system (X<sub>u</sub>, Y<sub>u</sub>, Z<sub>u</sub>). The articulation angle between the tractor and trailer can be described by relative motion of the coordinate (X<sub>t</sub>, Y<sub>t</sub>, Z<sub>t</sub>) with respect to the coordinate system (X<sub>st</sub>, Y<sub>st</sub>, Z<sub>st</sub>).

<figure><img src="image_4.png" /><figcaption>Figure 2. Systems of coordinate</figcaption></figure>

From these coordinate systems, we introduce the generalized
coordinate system (x<sub>E</sub>, y<sub>E</sub>, z<sub>E</sub>, ψ, φ, ψ<sub>f</sub>), defined as follows :

x_E is the position of the tractor C.G. in the direction of the X_E-
axis,

y_E is the position of the tractor C.G. in the direction of the Y_E-
axis,

z_E is the position of the tractor C.G. in the direction of the Z_E-
axis,

Ψ is the yaw angle of the tractor,

φ is the roll angle if the tractor,

Ψ<sub>f</sub> is the angle between tractor and trailer at fifth wheel.

The vector equation describing the dynamics of the heavy
vehicle is obtained using the Lagrangian formulation. It's of the
form :

$$
\begin{align}
& M(q(t), p)\ddot{q}(t) + C(q(t), \dot{q}(t), p)\dot{q} + G(q(t), p, \delta(t)) \nonumber \\
& \qquad = F_g(p, \delta(t)), \quad t \ge 0 \tag{1}
\end{align}
$$

where $q = (x, y, z, \psi, \phi, \psi_f)^T$ is the generalized coordinates vector in which we set to simplify : $x = x_E, y = y_E$ and $z = z_E$ ; $F_g$ is the vector of generalized forces, $M$ is the inertial matrix that is symmetric positive definite, $C(q, \dot{q})\dot{q}$ is the combined Coriolis and centrifugal forces and $G$ is the gravity vector. The generalized forces $F_g$ represent the effect of external forces acting on the vehicle body. These later result from the tire-road interface and suspensions defined in terms of longitudinal and lateral tire forces and vertical forces. Generalized forces depend on steering angle $\delta$. We use a specific steering angle profile, shown in fig. 3, which allows to observe rollover and jackknifing risks. All heavy vehicle's parameters, defined in Table 3.1, are gathered in the vector $p \in \mathbb{R}^{n_p}$, where $n_p = 25$. Then, we rewrite the vehicle model as a first-order differential equations (ODE) system as :

<figure><img src="image_5.png" /><figcaption>Figure 3. Steering angle profile</figcaption></figure>

$$
\dot{\mathbf{u}}(t) = f(t, \mathbf{u}(t), p, \delta(t)), \quad t \ge 0 \quad (2)
$$

where **u** = [q<sup>T</sup>, ḋ<sup>T</sup>]<sup>T</sup> and *f* is a function from **R**<sub>+</sub> × **R**<sup>12</sup> × **R**<sup>25</sup> ×
**R** into **R**<sup>12</sup>.

This equation is solved by using classical Runge-Kutta method
of order 4. Such a model easily brings forth useful insights
of dynamic phenomena (yaw-roll) with fast computation time,
compared to mulibody approach. The system is simulated
and validated in the Matlab environment with a specific ODE
Fortran solver. Parameters are obtained from Prosper truck
simulator [8]. In practice, heavy vehicle simulation requires only 50 ms compared to a quasi-realtime simulator (Simulink, Prosper, TruckSim...). These improvements allow to embedde our heavy vehicle simulator into sensitivity and reliability algorithms.

## 3.2 Safety criteria

The framework of this work is focused on two risk situations : 1 - the rollover, which is a lateral unstability due to a lateral load transfer.

2 - the jackknifing, which is a loss of control that causes the rotation of the tractor with respect to the semitrailer.

For each risk we propose and choose a safety criterion to assess and detect dangerous situations. Rollover is a well-known phenomena and its analysis is spread enough in the litterature [9]. Unfortunately, jackknining risk is a more complicated phenomena and there is few safety criteria.

### 3.2.1 Rollover criteria

Rollover is one of the most frequent accidents (20%) and causes significant damages to the vehicles and injuries to its driver and passengers. Several anti-rollover systems and rollover warnings systems were developed to assist and warn the driver [10], [11]. Several rollover indicators can be found in the literature : SRT (Static Rollover Threshold), RPER (Rollover Prevention Energy Reserve) , CSV (Critical Sliding Velocity)...

<figure><img src="image_6.png" /><figcaption>Figure 4. Rollover on curve with a dry surface.</figcaption></figure>

In our study, rollover risk evaluation is based on the maximum of a rollover risk indicator, namely the load transfer ratio (LTR), which corresponds to the load transfer between the left and the right sides of the vehicle. The resulting expression of this indicator is defined as :

$$LTR := \frac{F_{z,l} - F_{z,r}}{F_{z,l} + F_{z,r}} \qquad (3)$$

where $F_{z,l}$ and $F_{z,r}$ are respectively the left and the right normal forces (see fig. 5). In pratice, when LTR is equal to 0, the heavy vehicle has stable roll dynamics. The risk becomes high as this indicator goes towards ±1. Ackermann [11] simplified the LTR criterion, when overlooking unsprung mass, as follows :

$$LTR = \frac{2m_2^s}{m_2 T_{w,3g}} \left[ (h_r + h_2 \cos \phi)(\ddot{y} + v_0 r - h_2 \ddot{\phi}) + gh_2 \sin \phi \right] \qquad (4)$$

where $T_{w,3}$, $h_r$, $v_0$, $h_2$, $m_2^s$, $r$, $g$ are defined in Table 3.1 and $m_2 = m_2^s + m_2^l$. Kamnik [12] proposed an improved nonlinear rollover criterion for articulated heavy vehicle called LLT. This later is accurate and coincide to the Ackermann criterion when neglecting unsprung mass contribution.

<figure><img src="image_7.png" /><figcaption>Figure 5. Lateral dynamics of heavy vehicle.</figcaption></figure>

### 3.2.2 Jackknifing criterion

Jackknifing is characterized by a loss of stability in the yaw motion of the articulated system. It occurs for several reasons :

1. when the rear wheels of the tractor are blocked,

2. when the vehicle applies in turn the brakes abruptly (see fig. 6),

3. when the road is slipping (low adherence).

This phenomena is more frequent when then trailer is empty or when the load is badly distributed in the trailer. Theoretically, jackknifing is detected the relative angle $\psi_f$ is greater than $\pi/2$ (see fig. 7)

<figure><img src="image_8.png" /><figcaption>Figure 6. Jackknifing with an over-steer angle around a curve on a wet/snowy/icy road surface.</figcaption></figure>

Jackknifing is characterized by the friction indicator [13] :

$$\mu_{min} = \frac{F_y}{\cos(\psi_f)F_z}$$

where $F_y$ and $F_z$ are respectively the lateral and normal road forces applied to the heavy vehicle. When $\mu_{min} \le 0.1$ the heavy vehicle remains stable.

In our work, we use the criterion proposed in [14] and based on the estimation of the articulation yaw angle $\psi_f$. In this paper the jackknifing criterion is defined as $C_m \le 0$, where :

$$C_m = \dot{r}_1 \cdot (r_2 - r_1) = \dot{r}_1 \cdot r$$

where $r_1$ and $r_2$ are defined on figure 7.

# 4 SENSITIVITY ANALYSIS

Among all the parameters of the mechanical model, some present a marked random variability. Therefore it is crucial for the credibility of the application to take into account this reality via a suited stochastic modeling. The object of this section is to identify the minimal family of variables that must be considered
<table><caption>Table 1. Main heavy vehicle characteristics.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Tractor's sprung mass</td><td>m<sub>1</sub><sup>s</sup></td><td>kg</td><td>4533</td></tr><tr><td>Tractor's unsprung mass</td><td>m<sub>1</sub><sup>u</sup></td><td>kg</td><td>1156</td></tr><tr><td>Semitrailer's sprung mass</td><td>m<sub>2</sub><sup>s</sup></td><td>kg</td><td>5833</td></tr><tr><td>Semitrailer's unsprung mass</td><td>m<sub>2</sub><sup>u</sup></td><td>kg</td><td>1469</td></tr><tr><td>Tractor's moment of inertia relative to axis x</td><td>I<sub>x1</sub></td><td>Kg.m<sup>2</sup></td><td>3115</td></tr><tr><td>Tractor's moment of inertia relative to axis y</td><td>I<sub>y1</sub></td><td>Kg.m<sup>2</sup></td><td>20000</td></tr><tr><td>Tractor's moment of inertia relative to axis z</td><td>I<sub>z1</sub></td><td>Kg.m<sup>2</sup></td><td>45000</td></tr><tr><td>Semitrailer's moment of inertia relative to axis x</td><td>I<sub>x2</sub></td><td>Kg.m<sup>2</sup></td><td>60000</td></tr><tr><td>Semitrailer's moment of inertia relative to axis y</td><td>I<sub>y2</sub></td><td>Kg.m<sup>2</sup></td><td>71318</td></tr><tr><td>Semitrailer's moment of inertia relative to axis z</td><td>I<sub>z2</sub></td><td>Kg.m<sup>2</sup></td><td>100000</td></tr><tr><td>Distance between tractor C.G. and front wheel axle</td><td>l<sub>1</sub></td><td>m</td><td>1.46</td></tr><tr><td>Distance between tractor C.G. and rear wheel axle</td><td>l<sub>2</sub></td><td>m</td><td>2.66</td></tr><tr><td>Distance between joint (fifth wheel) and trailer rear wheel axle</td><td>l<sub>3</sub></td><td>m</td><td>7.5</td></tr><tr><td>Relative x position between tractor's C.G. to fifth wheel</td><td>d<sub>1</sub></td><td>m</td><td>2.24</td></tr><tr><td>Relative y position between tractor's C.G. to fifth wheel</td><td>d<sub>2</sub></td><td>m</td><td>0.16</td></tr><tr><td>Relative x position between semitrailer's C.G. to fifth wheel</td><td>d<sub>3</sub></td><td>m</td><td>5.85</td></tr><tr><td>Relative y position between semitrailer's C.G. to fifth wheel</td><td>d<sub>4</sub></td><td>m</td><td>0.155</td></tr><tr><td>Tractor front axle track width</td><td>T<sub>w1</sub></td><td>m</td><td>2.04</td></tr><tr><td>Tractor rear axle track width</td><td>T<sub>w2</sub></td><td>m</td><td>1.78</td></tr><tr><td>Trailer rear axle track width</td><td>T<sub>w3</sub></td><td>m</td><td>2.15</td></tr><tr><td>Distance from tractor roll center to C.G.</td><td>h<sub>r</sub></td><td>m</td><td>0.3</td></tr><tr><td>Tractor's C.G. height</td><td>h<sub>1</sub></td><td>m</td><td>0.348</td></tr><tr><td>Semitrailer's C.G. height</td><td>h<sub>2</sub></td><td>m</td><td>0.256</td></tr><tr><td>Wheel radius</td><td>r</td><td>m</td><td>0.6</td></tr><tr><td>Adherence coefficient</td><td>μ</td><td>—</td><td>0.95</td></tr><tr><td>Initial velocity</td><td>ν<sub>0</sub></td><td>m.s<sup>-1</sup></td><td>15</td></tr><tr><td>Gravity acceleration</td><td>g</td><td>m.s<sup>-2</sup></td><td>0.81</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Figure 7. Heavy vehicle configuration during jackknifing.</figcaption></figure>

as random using a sensitivity analysis. We investigate two different sensitivity analysis methods. On the one hand, a local sensitivity method based on a perturbation method is applied. On the other hand, a global sensitivity method is performed by computing the Sobol indices.

## 4.1 Local sensitivity

Let $r$ be the control variable associated with the chosen safety criterion, defined as follows:

$$r(t) := g(\mathbf{u}(t), p, \delta(t)) \quad (5)$$

where $u$ is the solution of the equation (2).

Here, the local sensitivity analysis consists in calculating, at the design point [15], the partial derivatives of $r$ with respect to

the component $p_i$ of $p$. Let $s_i$ be the sensitivity of $r$ with respect to $p_i$. It's given by:

$$s_i(t) := \partial_{p_i} r(t) = \partial_{\mathbf{u}} g \cdot \partial_{p_i} \mathbf{u} + \partial_{p_i} g \quad (6)$$

According to [16], the quantity $d_i := \partial_{p_i} \mathbf{u}$ is the solution of the following differential equation:

$$\dot{d}_i(t) = \partial_{\mathbf{u}} f \cdot d_i + \partial_{p_i} f \quad (7)$$

We get as much equations (7) as the number $n_p$ of parameters. Equations (2) and (7) are solved together using a coupled Euler scheme (time integration) and Newton algorithm to solve point fixed problem at each integration time. We used an appropriate software called CVODES [17], developed by the Lawrence Livermore National Laboratory, to compute sensitivity indices and solve ODE/DAE. Results for rollover safety criteria are presented on figure 8. It shows that among the initial $n_p = 25$ parameters, only 5 parameters have a significant influence on the rollover response: $h_r, l_2, l_3, h_1, v_0$.

## 4.2 Global sensitivity by Sobol' indices

The parameters are gathered in the vector $p \in \mathbb{R}^{n_p}$ modeled as a continuous $\mathbb{R}^{n_p}$-valued random variable denoted $\mathbf{P} = (P_1, ..., P_{n_p})$ for which the following hypotheses are made: (H1) its components are mutually independent,
<figure><img src="image_10.png" /><figcaption>Figure 8. Evolution of sensitivity indicator applied to LTR rollover criteria.</figcaption></figure>

(H2) two distributions are alternatively considered for each of its components : a truncated Gaussian distribution and an uniform distribution,

(H3) all the components of **P** follow simultaneously the same type of distribution,

(H4) all the components of **P** have the same coefficient of variation.

Each random variable $P_i$ verifies :

$$ \mathbb{E}(P_i) = p_i $$

$$ \sigma(P_i) = \kappa \mathbb{E}(P_i) = \kappa p_i $$

where $\sigma(P_i)$ and $\kappa$ are respectively the standard deviation and the coefficient of variation of $P_i$. In the following, this later is taken equal to 0.01.

In order to take into account pertubations on steering angle $\delta$, this one is modeled as bounded stochastic process of the form [5] :

$$ \Delta(t) = \delta(t) + \Lambda(t) $$

with :

$$ \Lambda(t) = \varepsilon \sin(vt + \sigma W(t) + 2\pi\Theta), \quad t \in \mathbb{R}_+ $$

where $\varepsilon$, $v$, $\sigma$ are given real positive constants, $W$ is a standard real Wiener process and $\Theta$ is a random variable uniformly distributed on $[0, 1]$ and independent of $W$.

With these stochastic modelings, the response system **u** is now a vector random process **U** which depends on $(\mathbf{P}, \Delta)$ and the control variable $r$ becomes a random process **R** depending on **P** and $\Delta$ :

$$ R(t) = \mathcal{S}(\mathbf{P}, \Delta(t)) = \mathcal{S}(P_1, \dots, P_{n_p}, \Delta(t)). \quad (8) $$

The Sobol index $s_i^R$ [18], [19] associated with the random variable $P_i$ is a deterministic function of $t$ defined from the conditional expectation $\mathbb{E}[R|P_i]$, according to the formula :

$$ s_i^R = \frac{\mathrm{Var}(\mathbb{E}[R|P_i])}{\mathrm{Var}(R)} = \frac{V_i}{V} \quad (9) $$

where $\mathrm{Var}(\cdot)$ denotes the variance.

From this definition, $s_i^R$ ranges from 0 to 1. A small value means that uncertainty on $P_i$ has few influence on the variability

of $R$ and consequently, in this case, $P_i$ can be considered as a deterministic parameter. On the contrary if $s_i^R$ is closed to 1, $P_i$ must keep its status of random variable. Second order indices reflect the interaction between two parameters. They are defined by :

$$ s_{i,j}^R = \frac{\mathrm{Var}(\mathbb{E}[R|(P_i, P_j)]) - V_i - V_j}{\mathrm{Var}(R)} = \frac{V_{i,j} - V_i - V_j}{V}, \quad i \neq j \quad (10) $$

Higher orders, until the $n_p$-th order, are defined in the same way. These sensitivity indices satisfy the fundamental property :

$$ \sum_{1 \le i \le n_p} s_i^R + \sum_{1 \le i < j \le n_p} s_{i,j}^R + \dots + \sum_{1 \le i_1 < \dots < i_{n_p}} s_{i_1, \dots, i_{n_p}}^R = 1 \quad (11) $$

Hence, the analysis starts with the first order indices $s_i^R$ calculation. Then, if $\sum_i s_i^R \approx 1$ then all the higher orders are negligible. Otherwise, interactions exists in the safety criterion model, and further calculations are needed.

The sensitivity of $R$ with respect to $\Delta$ is estimated from the Iooss's work [20].

Sobol's indices are computed with a huge sample $10^6$ to obtain a good first order indices. In fact, sums of Sobol indices are plotted on figure (9), (10), (11) and (12) and are closed to 1.

Figures 9) and 10) show the obtained results for the rollover and jackknifing criteria, using the Sobol sensitivity analysis without perturbations on the steering angle. Results exhibit that only 6 parameters $d_3, l_2, l_3, T_{w,3}, h_2, v_0$ are really influent on rollover and only 5 parameters $d_3, l_2, l_3, h_1, h_2, v_0$ on jackknifing. Figures 11) and 12) show the same results in the case where the random perturbation $\Lambda$ of the steering angle is taken into account. Results enhance that only 7 parameters $d_3, l_2, l_3, T_{w,3}, h_2, v_0, \delta$ are really influent on rollover and only 6 parameters $d_3, l_2, l_3, h_1, h_2, v_0, \delta$ on jackknifing.

<figure><img src="image_11.png" /><figcaption>Figure 9. Evolution of first order Sobol indices for rollover criteria without steering perturbation.</figcaption></figure>

## 4.3 Conclusions

With both methods of sensitivity analysis, we get quite the same influent parameters. Global sensitivity gives finer results and allows to take into account the perturbation on steering angle but it's slower than local sensitivity analysis. However here, for simplicity and to apply standard reliability analysis, the steering
<figure><img src="image_12.png" /><figcaption>Figure 10. Evolution of first order Sobol indices for jackknifing criteria without steering perturbation.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 11. Evolution of first order Sobol indices for rollover criteria with steering perturbation</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12. Evolution of first order Sobol indices for jackknifing criteria with steering perturbation</figcaption></figure>

angle perturbation is supposed is not taken into considerations. As a result, the only random parameters considered are at most the six random variables $P_1 = v_0, P_2 = h_r, P_3 = l_2, P_4 = l_3, P_5 = h_2, P_6 = T_{w,3}$.

# 5 RELIABILITY ANALYSIS

## 5.1 General principle and statement of the problem

In the following the vector random parameter $\mathbf{P} = (P_1, ..., P_6)^T$ is assumed to be defined on the probability space $(\Omega, \mathcal{F}, \mathcal{P})$, where $\Omega$ is a sample space, $\mathcal{F}$ is $\sigma$-algebra on $\Omega$ and $\mathcal{P}$ is a probability

measure on $\mathcal{F}$. According to (H1)-(H4) hypotheses (cf. section 4.2), the distribution is known and admits a probability density denoted in that follows $f_{\mathbf{P}}$.

Let $Z$ be the safety margin associated with the control variable $\mathcal{R}$, such that:

$$Z = r_0 - \max_{t \in \mathcal{T}} |\mathcal{R}(t)|$$

where $r_0$ is a given limit value and $\mathcal{T}$ is the observation interval. This real random variable is a function of $\mathbf{P}$:

$$Z = G(\mathbf{P})$$

where $G$ is the limit state function associated with the safety criterion chosen for the study. It is a measurable mapping from $\mathbb{R}^6$ into $\mathbb{R}$ which defines two complementary subsets of $\mathbb{R}^6$, $D_s$ and $D_f$, such that:

$$D_s = \{\mathbf{p} \in \mathbb{R}^6 : G(\mathbf{p}) > 0\}$$

$$D_s = \{\mathbf{p} \in \mathbb{R}^6 : G(\mathbf{p}) \le 0\}$$

called respectively the safety domain and the failure domain. Two events are associated with these domains, the safety event $E_s \in \mathcal{F}$ and the failure event $E_f \in \mathcal{F}$, such that:

$$E_s = \{\omega \in \Omega : G(\mathbf{P}(\omega)) > 0\}$$

$$E_f = \{\omega \in \Omega : G(\mathbf{P}(\omega)) \le 0\}$$

and which verify

$$E_s \cup E_f = \Omega, E_s \cap E_f = \emptyset$$

Once known the pdf $f_{\mathbf{P}}$ and defined the events $E_s$ and $E_f$, an important objective of the reliability analysis is then to evaluate the probabilities $P(E_s)$ and $P(E_f)$ (respectively called safety probability and failure probability) given by:

$$P_s = P(E_s) = \int_{\mathbb{R}^6} f_{\mathbf{P}}(p) \mathbb{1}_{D_s} dp$$

$$P_f = P(E_f) = \int_{\mathbb{R}^6} f_{\mathbf{P}}(p) \mathbb{1}_{D_f} dp$$

and such that:

$$P_s = 1 - P_f$$

In practice, an exact calculation of $P_f$ is not possible and a Monte-Carlo procedure must be used. In classical reliability approach, it is customary to transform the initial formulation into a standard formulation in which the vector of the random parameters follows a standard Gaussian distribution. This leads to construct a regular transformation $\mathbf{T}$, with inverse $\mathbf{T}^{-1}$, such the random variable $\mathbf{X}$ can be written:

$$\mathbf{P} = \mathbf{T}^{-1}(\mathbf{Y})$$

where $\mathbf{Y}$ is a $\mathbb{R}^6$-valued standard Gaussian random variable. Carrying out the change of variable $y = \mathbf{T}(p)$, probability of failure becomes:

$$P_f = \int_{\mathbb{R}^6} \mathbb{1}_{\Delta_s}(y) \phi_6(y) dy$$

where $\phi_6$ if the standard Gaussian pdf on $\mathbb{R}^6$ and $\Delta_f = \mathbf{T}(D_f)$.
<figure><img src="image_15.png" /><figcaption>Figure 13. Approximation of $\Delta_f$ and Hasofer-Lind index representation, in two dimensions</figcaption></figure>

FORM approximation consists in replacing the failure domain $\Delta_f$ by a half-space $\Delta_f^L$ tangent to $\Delta_f$ at the design point $M^*$ (see figure 13). Using this approximation, the failure probability $P_f$ can be approximated by :

$$P_f \approx P_f^L = \int_{\mathbb{R}^6} 1_{\Delta_f^L}(y) \varphi_6(y) dy = \Phi(-\beta_{HL})$$

where $\Phi$ is the one-dimensional standard Gaussian distribution function and $\beta_{HL}$ is the Hasofer-Lind index [15] defined by :

$$\beta_{HL} := \min_{M \in \Delta_f} \|OM\| = \|OM^*\|$$

In practice, $\beta_{HL}$ is computed by the Rackwitz-Fiessler algorithm [15]. It is possible to approximate the limit-state function by a quadratic surface at the design point, that leads to the SORM method. In this case, the probability of failure can be approximated by the following formula :

$$P_f \approx P_f^S = \Phi(-\beta_{HL}) \prod_{1 \le i \le 5} \left( 1 + \frac{\varphi(\beta_{HL})}{\Phi(-\beta_{HL})} \kappa_i \right)^{-1/2} \quad (12)$$

where $\kappa_i$ denote the main curvatures of the limit-state function at $M^*$.

Monte-Carlo method is employed to estimate the "exact" probability of failure. A set of $10^6$ simulated realizations of $P$ is used for an initial velocity ranges from $20 \text{ m.s}^{-1}$ to $24 \text{ m.s}^{-1}$. Each calculation requires about one hour of computations on a standard PC but we have improved the numerical procedure by distributing the computations on a 27-nodes cluster which allows to provide the set of all failure probabilities in one hour. The MatLAB toolbox FERUM [21] has been used to evaluate $P_f^L$ and $P_f^S$. Figure 14 compares the probabilities of failure given by Monte-Carlo method and the FORM/SORM methods. We can see that $P_f^L$ and $P_f^S$ are correct approximations of $P_f$.

## 6 CONCLUSION

Two sensitivity analysis (a local one and a global one) have been presented for a mechanical model describing the dynamics of the articulated heavy vehicle. They have shown that only 6

<figure><img src="image_16.png" /><figcaption>Figure 14. Evolution of probability of failure $P_f^L$ and $P_f$ with respect to the initial velocity.</figcaption></figure>

parameters among the 25 initial parameters are really influent towards rollover and jackknifing safety criteria. An application of structural reliability methods to the proposed mechanical model has shown that the failure probability can be correctly approximated using the classical FORM and SORM methods.

## ACKNOWLEDGMENTS

This work is supported by the french Public Works Research Laboratory.

## REFERENCES

[1] ONISR, "La sécurité routière en france. bilan de l'année 2008," Tech. Rep., Observatoire national interministériel de sécurité routière, 2008.

[2] W. Gao, N. Zhang, and J. Dai, "A stochastic quarter-car model for dynamic analysis of vehicles with uncertain parameters," *Vehicle System Dynamics*, vol. 46, no. 12, pp. 1159-1169, 2008.

[3] R. Sigbjørnsson and J.T. Snaebjørnsson, "Probabilistic assessment of wind related accidents of road vehicles: A reliability approach," *Journal of Wind Engineering and Industrial Aerodynamics*, vol. 74, pp. 1079-1090, 1998.

[4] Y. Sellami, *Approche Fiabiliste et Mécanique pour la Prédiction de Risques d'Accident de Poids Lourds.*, Ph.D. thesis, Université de Nantes, 2008.

[5] G. Rey, *Approche probabiliste de la sécurité des véhicules légers en zones accidentogènes*, Ph.D. thesis, Université Blaise Pascal, 2010.

[6] T. Tomisuka and C. Chieh, "Modeling and control of articulated vehicles," Tech. Rep., University of California, Berkeley, 1997.

[7] M. Bouteldja, *Modélisation des interactions dynamiques Poids-Lourds/Infrastructures pour la sécurité et les altertes*, Ph.D. thesis, UVSQ, LCPC, Novembre 2005.

[8] Sera-Cd, "www.sera-cd.com,".

[9] R.W. Goldman, M. El-Gindy, and B.T. Kulakowski, "Rollover dynamics of road vehicles: Literature survey," *Heavy Vehicle Systems, Int. J. of Vehicle Design*, vol. 8, no. 2, 2001.

[10] P. Gaspar, I. Szaszi, and J. Bokor, "Rollover stability control in steer-by-wire vehicles based on an lpv method," *International Journal of Heavy Vehicle Systems*, vol. 13, no. 1, pp. 125-143, 2006.

[11] J. Ackermann and D. Odenthal, "Damping of vehicle roll dynamics by gain scheduled active steering," in *Proc. European Control Conf. Karlsruhe.*, 1999.

[12] R. Kamnik, F. Boettiger, and K. Hunt, "Roll dynamics and lateral load transfer estimation in articulated heavy freight vehicles," *Journal of Automobile Engineering*, vol. 217, no. 11, pp. 985-997, 2003.

[13] M. Buissières and B. Falah, "Outil permettant d'analyser la stabilité des véhicules lourds," Tech. Rep., Rapport technique du service de la normalisation technique, Ministère des transports du Québec, 2001.

[14] M. Bouteldja, A. Koita, V. Dolcemascolo, and J.C. Cadiou, "Prediction and detection of jackknifing problems for tractor semi-trailer," in *IEEE Vehicle Power and Propulsion Conference*, Windsor, England, 2006, pp. 1-6.

[15] O. Ditlevsen and H.O. Madsen, *Structural reliability methods*, Wiley, 2007.
[16] L. Petzold and T. Maly, "Numerical methods and software for sensitivity analysis of differential-algebraic systems," *Appl. Numer. Math*, vol. 20, pp. 57–79, 1997.

[17] R. Serban and C. Hindmarsh, "Cvodes: the sensitivity-enabled ode solver in sundials," in *Proceedings of IDETC/CIE 2005*, Sept. 2005.

[18] I. M. Sobol, "Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates," *Math. Comput. Simul.*, vol. 55, no. 1-3, pp. 271–280, 2001.

[19] A. Saltelli, S. Tarantola, and F. Campolongo, *Sensitivity analysis in practice: a guide to assessing scientific models*, John Wiley & Sons Inc, 2004.

[20] B. Iooss and M. Ribatet, "Global sensitivity analysis of computer models with functional inputs," *Reliability Engineering & System Safety*, vol. 94, pp. 1194–1204, 2009.

[21] J.M. Bourinet, "Ferum 4.1 user's guide," Tech. Rep., IFMA, 2010.