**sensors**

**sensors**

**ISSN 1424-8220**
www.mdpi.com/journal/sensors

*Article*

**Vehicle Dynamic Prediction Systems with On-Line Identification**

**of Vehicle Parameters and Road Conditions**

**Ling-Yuan Hsu and Tsung-Lin Chen**⋆

Department of Mechanical Engineering, National Chiao Tung University, University Road 1001, Hsinchu, Taiwan; E-Mail: lance1214@gmail.com

⋆

Author to whom correspondence should be addressed; E-Mail: tsunglin@mail.nctu.edu.tw; Tel.: +886-3-571-2121 (ext. 55127); Fax: +886-3-572-0634.

*Received: 31 August 2012; in revised form: 17 October 2012 / Accepted: 29 October 2012 /*
*Published: 13 November 2012*

**Abstract: This paper presents a vehicle dynamics prediction system, which consists of**
a sensor fusion system and a vehicle parameter identification system. This sensor fusion
system can obtain the six degree-of-freedom vehicle dynamics and two road angles without
using a vehicle model. The vehicle parameter identification system uses the vehicle dynamics
from the sensor fusion system to identify ten vehicle parameters in real time, including
vehicle mass, moment of inertial, and road friction coefficients. With above two systems,
the future vehicle dynamics is predicted by using a vehicle dynamics model, obtained from
the parameter identification system, to propagate with time the current vehicle state values,
obtained from the sensor fusion system. Comparing with most existing literatures in this
field, the proposed approach improves the prediction accuracy both by incorporating more
vehicle dynamics to the prediction system and by on-line identification to minimize the
vehicle modeling errors. Simulation results show that the proposed method successfully
predicts the vehicle dynamics in a left-hand turn event and a rollover event. The prediction
inaccuracy is 0.51% in a left-hand turn event and 27.3% in a rollover event.

**Keywords: dynamics predictions; sensor fusion system; vehicle parameter identifications;**
road condition identifications

**1. Introduction**

In recent years, many vehicle control research propose using the future vehicle dynamics information
to assist drivers’ maneuvers. For example, the vehicle path predictions can provide the future position
error for the vehicle guidance controls. Compared with the conventional “look-down” sensing system
that provides current position error, the path prediction not only provides the information that are
easier perceived by human drivers, but also provides additional information of road conditions, weather
*conditions, etc. [*1,2]. As another example, many researchers propose using vehicle rollover predictions
as anti-rollover measures [3–5]. The benefit is that the rollover prediction can trigger the control input
earlier than the approaches without prediction. Consequently, the advance of the control action both
saves actuation power and improves the driving safety. These examples highlight the importance of
vehicle dynamics predictions.

In general, the future vehicle dynamics is predicted by using a vehicle mathematics model to numerically propagate current state values with time. Therefore, a vehicle dynamics prediction system needs a mathematic model and current vehicle state values [1–6]. There are two concerns regarding the mathematic model. First, many reports employ simplified vehicle models in the dynamics prediction, such as 2 DOF model [1], 4 DOF [2], and 2 DOF yaw-roll model [1–3,6]. The prediction results may only be acceptable for limited purposes and driving conditions. The inaccurate prediction result can be understood that, from the control viewpoint, the dynamics prediction is an open loop system. Thus, any model simplification would affect prediction accuracy to some extent. Our previous work [7] shows that even excluding the vehicle pitch dynamics from a 6 DOF vehicle model would result in an obvious error in the rollover prediction. One of the key components that are often neglected in the vehicle dynamics modeling is the road angle. Many research reports have shown that road angles have direct influences on the vehicle dynamics [8–10]. Our previous report [7] also show that, under the same driving maneuvers, a vehicle would roll over on a sloped road but would be under control on a flat road. Therefore, it is important to include the road angle in the vehicle dynamics prediction.

The second concern is the parameter uncertainty in the vehicle model. Most dynamics prediction
methods use presumed vehicle parameters in the vehicle mathematics model [1–3,7]. This approach
can simplify the prediction problem but may be inaccurate in practice. This is because the vehicle
property (mass, moment of inertia) and road conditions (road angles, friction coefficients) may change
from different driving situations, such as number of passengers, amount of fuel, road surface profiles,
*weather conditions, etc. Therefore, it is preferred that the parameters associated with the vehicle*
model can be identified in real time. From system identification viewpoint, the success of parameter
identification depends on the model accuracy, persistent excitation, and signal-to-noise ratio (SNR) of
the excitation signals [11]. Therefore, the vehicle parameter identification problem is closely related
to the accompanied sensor fusion system that provides the measurements of the vehicle dynamics.
In literatures, many vehicle parameter identification systems have been proposed to identify vehicle
*parameters such as vehicle masses, moments of inertia, road-tire frictions, etc. [*9,12–14]. So far, we have
not found a research report that identifies more than six vehicle parameters using their sensor systems.

As mentioned earlier, another key factor of the vehicle dynamics prediction is the sensor system for obtaining current vehicle state values. Since the vehicle system is highly nonlinear and many

of its dynamics cannot be measured directly, the vehicle dynamics are often obtained by two ways: one is the observer-based sensor fusion system; the other is the kinematics-based sensor fusion system. The observer-based method needs a vehicle model and less number of sensors. On the contrary, the kinematics-based method does not require a vehicle model but needs more sensors. Since the state estimation accuracy of the observer-based method greatly relies on the incorporated model accuracy, the observer-based method is less preferred as compared with the kinematics-based method. Lots of kinematics-based sensor fusion systems employ a GPS and an IMU (three-axis accelerometer and three-axis gyroscope) to measure 6 degree-of-freedom (DOF) motions of an object. This sensor fusion system has been widely used in many applications, such as aircraft systems [15], ships [16], and road vehicles [17–19]. However, it has some drawbacks when applied to road vehicles. First, the rotation angles obtained by integrating the angular rates may suffer from the initial value ambiguity problem and the error accumulation problem. Second, both GPS and IMU are inertial sensors. Thus, the vehicle attitude and road angles are mingled together in the GPS and IMU measurements. Third, the measurement accuracy of the GPS is inadequate to determine the vehicle displacement in the vertical direction.

From pervious discussion, it can be concluded that a precise vehicle model is important for the accuracy of the dynamics prediction. How precise this vehicle model can be is determined by the accompanied sensor fusion system that provides current vehicle state values both for the state propagations and for the real time identification of vehicle parameters. In our previous work [20], we propose a kinematic based senor fusion system that employs a three-antenna global positioning system (GPS), an inertial measurement unit (IMU), and four suspension displacement sensors. This sensor fusion system can obtain the 6 DOF vehicle dynamics and two road angles. Based on this sensor system, we develop a vehicle dynamics prediction system in this paper using a vehicle model that is more complicated than most of the existing approaches. Moreover, the parameters of this vehicle model that may change in different driving conditions are identified in real time. In this paper, the dynamics prediction procedures, the parameter identification algorithms, the parameter observability, the SNR influence, and the prediction accuracy are all discussed in detail.

**2. Euler Angles and Coordinate Systems**

Three coordinate systems are introduced to describe a vehicle moving on a sloped road (see, Figure1). These three coordinate systems are: global frame{g}, road frame {r}, and vehicle frame {v}. Similar

to conventional research, the global frame is fixed to a point on Earth, while the vehicle frame is fixed to the center of gravity (CG) of the vehicle and rotates with the vehicle. The additional road frame is introduced to describe the vehicle dynamics on a sloped road, which is fixed to a road and rotated with the road terrain.

Three sets of Euler angles are used to describe the relationships between any two out of three coordinate systems. The first set of Euler angles (ψg, θg, φg), which are referred to in this paper as

*absolute attitude of the vehicle (global frame vs. vehicle frame). The rotation order of this set of Euler*
angles is yaw-pitch-roll. Its direction cosine matrix (Cv_{g}) can be written as follows:

Cv_{g} = R(x, φg) R(y, θg) R(z, ψg) (1)
R(x, φg) =
1 0 0
0 cos(φg) sin(φg)
0 − sin(φg) cos(φg)
R(y, θg) =
cos(θg) 0 sin(θg)
0 1 0
− sin(θg) 0 cos(θg)
R(z, ψg) =
cos(ψg) sin(ψg) 0
− sin(ψg) cos(ψg) 0
0 0 1

**Figure 1. A schematic plot a vehicle and four coordinate systems (global frame, road frame,**
vehicle frame and auxiliary frame).

{g}
*y*
*z*
*x*
*x*
*z* *y*
{r}
4
1
2
3
4
4
1
1
2
2
3
3
global frame
road frame
{v}
CG
*y*
*z*
*x*
*f*
*t*
2
*r*
*t*
2
*r*
*l*
*f*
*l*
vehicle frame
*g*
*m _{tot}*

*x*

*z*

*y*{a} aux-frame

The second set of Euler angles (θr, φr, ψr), which are referred to in this paper as the “road grade

angle”, “road bank angle” and “road curve angle”, are used to describe the road grade profiles (global
*frame vs. road frame). The rotation order of this set of Euler angles is pitch-roll-yaw. Since a vehicle*
may move on a terrain irrelevant to the human-defined road path, it is impossible to determine the road
curve angle from vehicle dynamics. Thus, it is assumed to be zero (ψr = 0) for simplicity. Its direction

cosine matrix (Cr

g) can be written as:

Cr_{g} = R(z, ψr) R(x, φr) R(y, θr) (2)

The third set of Euler angles (ψv, θv, φv), which are referred to in this paper as the “vehicle yaw

*road plane (road frame vs. vehicle frame). The rotation order of this set of Euler angles is yaw-pitch-roll.*
Its direction cosine matrix (Cv_{r}) can be written as:

Cv_{r} = R(x, φv) R(y, θv) R(z, ψv) (3)

Since two sets of Euler angles are enough to describe the relationships between three coordinate systems, complying with the above angle definitions, the following relationship can be established for these angles:

Cv_{g}= Cv

rC r

g (4)

An additional auxiliary frame (aux-frame) is obtained by rotating the z-axis of the road frame until the x-axis of the road frame is aligned with the x-axis of the body frame. The aux-frame is used because it describes vehicle translational motions in an intuitive manner while preserving the information of other vehicle dynamics relative to the road level. In the following vehicle modeling, vehicle translational motions are described in the aux-frame, and the rotational motions are described by anglesψv, θv, φv.

**3. A Sensor Fusion System for Road Vehicles**

Since lots of the vehicle dynamics cannot be directly measured by individual sensors, a sensor fusion system is constructed to obtain the vehicle dynamics on a sloped road. The proposed sensor fusion system consists of a group of sensors, a kinematic model related to those sensor outputs, and a state estimation algorithm. They are discussed in the following.

*3.1. Sensor Selections*

3.1.1. Three-Antenna GPS System

Different from a conventional GPS system, a three-antenna GPS is used here because it not only provides absolute position measurements (xgps

g , ygpsg , zgpsg ) but also absolute angle measurements

(φgps

g , θggps, ψgpsg ). Both information are relative to the global frame, and the reported angle measuring

error is approximately 0.1◦

from a test vehicle [21,22]. 3.1.2. Suspension Displacement Sensors

Four suspension displacement sensors are installed at four corners of a vehicle. The suspension deflection can be related to the vehicle attitude and vertical displacement of the vehicle CG, both relative to the road frame:

zsusr =

−lr(H1sus+ H2sus) − lf(H3sus+ H4sus)

2l_{f} + 2l_{r} (5)

θvsus = sin

−1 H

sus

1 + H2sus− H3sus− H4sus

2l_{f}+ 2l_{r}

φsusv = sin

−1 −H

sus

1 + H2sus+ H3sus− H4sus

(2t_{f}+ 2t_{r}) cos θv

where the superscript (sus) denotes the physical quantities measured by the suspension displacement

the displacement of suspension at the corner i; the subscript (i) refers to four suspension corners in a

way: 1→ front-left, 2 to 4 in a clockwise direction; lf and lr are the distances from CG to the front and

rear axis, respectively; t_{f} and t_{r} are one half of the distances of the front and rear track, respectively.
3.1.3. Inertial Measurement Unit

An IMU sensor is installed at the center of gravity of the vehicle to measure the 6 DOF movements. They are used here to improve the estimation accuracy of the vehicle dynamics.

*3.2. A Kinematic Model*

As discussed in this paper, three sets of Euler angles (nine angles in total) parameterize this vehicle attitude determination system. The relationships stated in Equation (4) provide three constrained equations; the 3-antenna GPS system provides the measurements of three absolute vehicle angles (φg, θg, ψg); the measurements from suspension displacement sensors provide the values of two vehicle

attitude (φsus

v , θvsus); the road curve angle is assumed to be zero (ψr = 0). Therefore, even without a

kinematic model, those nine angles can be solved.

In addition to the angle measurements above, the vehicle rotational dynamics are also present in the IMU measurements, GPS position measurements, and suspension displacement measurements. In order to improve the robustness and accuracy of the angle determination, all the sensor measurements should be used. Thus, the estimation of vehicle dynamics is done for the rotational dynamics and translational dynamics simultaneously. In that case, since Equation (4) provide three constrained equations, six angles must be employed as system states to describe the vehicle attitude, and those six angle states are chosen as (φg, θg, ψg, φr, θr, ψv) for the ease of subsequent fusion algorithm derivation. Furthermore, in

order to apply existing state estimation techniques to this problem, the “governing equations” of these unknown angles must be obtained beforehand and added to the conventional kinematic model. Since it is impractical to either use additional sensors to measure the change rate of the last three angles or obtain this information for a specific case, the change rates of the last three angles are assumed to be zero.

˙

φr = 0 (6)

˙θr = 0

˙

ψv = 0

Thus, a kinematic model that can coordinate the outputs of IMU, GPS and suspension displacement sensors is: ˙x = f (x, u) (7) = Fx + Gacc Aacc x Aacc y Aacc z + Ggyro ωgyro x ωgyro y ωgyro z x = " xg, yg, zg, ˙xg, ˙yg, ˙zg, φg, θg, ψg, . . . φr, θr, ψv #T

u=hAacc

x , Aaccy , Aaccz , ωgyrox , ωygyro, ωzgyro

iT
F=
"
03×3 I3×3 03×6
09×12
#
Gacc=
h
03×3 Cvg
−1
06×3
iT
Ggyro =
h
0_{3×6} Cω
−1
0_{3×3}i
T
Cω =
1 0 − sin θg

0 cos φg cos θgsin φg

0 − sin φg cos θgcos φg

where (Aacc

x , Aaccy , Aaccz ) represents the measurements from a three-axis accelerometer;

(ωgyro

x , ωgyroy , ωzgyro) represents the measurements from a three-axis gyroscope; (xg, yg, zg) and

( ˙xg, ˙yg, ˙zg) represent position and velocity observed in the global frame; Cω describes the relation

between angular rate and rate of change of Euler angles, which can be found in [23].
*3.3. State Estimation Algorithm*

For a dynamics model shown in Equation (7), a state observer that can estimate each state value is constructed as follows:

˙ˆx = f (ˆx, u) + L (h(x) − h(ˆx)) (8)

where the(ˆ) denotes the estimated state value; h(x) is the system output equation; L is the matrix of

observer gains. In this paper, the extended Kalman filter is chosen as the state estimation algorithm to
calculate the observer gain. The standard procedures of the extended Kalman filter can be found in [11].
The system output equation h(x) in Equation (8) is carefully chosen as follows to ensure the system
observability.
h(x) = hygps_{1} , y_{2}gps, ysus
1 , ysus2
iT
(9)
y_{1}gps = hxgps
g , yggps, zgpsg , φgpsg , θgpsg , ψggps
i
y_{2}gps = hCv
g(1, 1), Cvg(1, 2)
i
= hCv
rCrg(1, 1), CvrCrg(1, 2)
i
ysus_{1} = hCv_{r}(1, 3), Cv
r(2, 3), Cvr(3, 3)
i
= hCv
gCrg
−1
(1, 3), Cv
gCrg
−1
(2, 3), Cv
gCrg
−1
(3, 3)i
ysus2 = zrsus
= Cr
g(3, 1) xg+ Crg(3, 2) yg+ Crg(3, 3) zg

where C(m, n) denotes the element in the mth row and the nth column of the matrix C. The output

equation y_{1}gps provides the locations and attitude of the vehicle in the global frame and its values is
obtained from the measurements of a three-antenna GPS. The output equation y_{2}gps is a function of
(φr, θr, ψv) and its values are calculated from the measurements of a three-antenna GPS. The output

equation y_{1}sus is a function of (φr, θr) and its values are calculated from the measurements of the

suspension displacement sensors. The output equation ysus

2 is related to a function of (φr, θr) and

(xg, yg, zg) and its values are then calculated from the measurements of the suspension displacement

sensors in Equation (5) and three-antenna GPS.

It should be emphasized that the (ygps_{2} , ysus_{1} ) are two sets of output equations, and each consists
of 2 to 3 nonlinear algebraic equations. Each equation consists of multiplication terms of two or more
trigonometric functions from the corresponding direction cosine matrix. In most cases, only one equation
in each set of output equations is enough to ensure the observability of state estimations. However,
since it involves multiplications of trigonometric functions, the estimation would fail at certain angles.
Therefore, redundant equations are used to ensure the success at all angles.

*3.4. Multi-Rate Kalman Filter*

Since the outputs of the GPS, IMU and suspension displacement sensors are unsynchronized and contaminated by different noise characteristics (see Table 1), instead of using a conventional extended Kalman filter, a multi-rate extended Kalman filter [24] is chosen to coordinate these sensor outputs. The algorithm of a multi-rate Kalman filter is similar to that of a conventional extended Kalman filter with the only difference in correcting the estimated state values. When the GPS measurement is available, the estimated state value is updated by the measurements of the GPS and the suspension displacement sensors. When the GPS measurement is unavailable, the estimated state value is updated only by the measurements of the suspension displacement sensors. It is done by the following:

**when GPS measurements are available,**

h(x) = hygps_{1} , y_{2}gps, ysus
1 , ysus2
iT
(10)
h(ˆx) = hyˆgps_{1} , ˆy_{2}gps, ˆysus
1 , ˆysus2
iT

**when GPS measurements are unavailable,**

h(x) = h0_{1×6}, 01×2, ysus1 , ysus2
iT
(11)
h(ˆx) = h01×6, 01×2, ˆysus1 , ˆysus2
iT
*3.5. Alpha-Beta Filter*

The above state estimation process can provide noiseless information for the displacement, velocity, and attitude of the vehicle, but not for the angular velocity and accelerations. Without this information, the subsequent vehicle parameter identification can be much complicated. Potentially, the angular velocity and accelerations can also be obtained from Kalman filtering by including those two states as system states in Equation (7). However, this approach may need a fictitious noise and increase the computation complexity. Hence, the alpha-beta filter is used to obtain the values for the angular velocity and acceleration.

**Table 1. Sensor output rates and noise characteristics.**

Output Noise

frequency Standard deviation

GPS 5 Hz 0.4◦

(attitude measurement)

GPS 5 Hz horizontal: 1m

(position measurement) vertical: 3m

Suspension 1 Hz 0.001m displacement sensor Accelerometer 1 kHz 0.02m/s2 Gyroscope 1 kHz 0.08◦ /s Tachometer 1 kHz 2◦ /s

The measurement bias is not considered.

The alpha-beta filter is a steady-state filter for noisy signals. Its algorithm is shown as follows:

ˆ xα(k + 1) = Aαxˆα(k) + Kα(k + 1) [z(k + 1) − ˆz(k + 1)] Aα = " 1 Ts 0 1 # (12) Kα(k + 1) = h α, β/Ts iT

where xˆα is a set of state vector for a two-dimensional model; z is the measured first dimensional

state; z is the estimated value of z; Aˆ α is the system matrix of the two-dimensional model; Ts is

the sampling period; the feedback gains α and β are chosen empirically. The detailed information of

alpha-beta filters can be found in [11]. For example,xˆα can be chosen as[ωx, ˙ωx]T; the corresponding

z is the measured rotational velocity from the gyroscope (ωgyro

x ). Thus, through the alpha-beta filter,

the rotational acceleration can be obtained without the direct differentiation.
**4. A Vehicle Model for the Dynamics Predictions**

As mentioned earlier, a dynamics prediction system needs a precise vehicle model. Furthermore,
some of the parameters in that vehicle model should be identified in real time. To meet both requirements,
we propose the following vehicle model for the dynamics prediction, which consists of 6 DOF vehicle
*dynamics, road angles, tire-road friction, nonlinear suspension, etc.*

mtot(¨xa− ˙yaψ˙v) = Fx,tire+ mtotGx (13)

mtot(¨ya+ ˙xaψ˙v) = Fy,tire+ mtotGy

Ix˙ωx = (Iy− Iz)ωyωz+ Mx Iy˙ωy = (Iz− Ix)ωzωx+ My Iz˙ωz = (Ix− Iy)ωxωy + Mz Iω˙ωi = −riFa,tire,i+ Ti (i = 1 ∼ 4) Gx Gy Gz = R(z, ψv) R(x, φr) R(y, θr) 0 0 −g

whereg is the Earth gravity; (xa, ya, za) represents the three-axis displacement of the vehicle CG

observed in the aux-frame; (Fx,tire, Fy,tire, Fz,spring) are the translational forces generated by tires and

suspension systems; (Mx, My, Mz) are the external torques acting on the vehicle CG along three

axes of the vehicle frame, which are the functions of forces (Fx,tire, Fy,tire, Fz,spring), vehicle attitude

(ψv, θv, φv), and vehicle geometry [25]; (Ix, Iy, Iz) are the moment of inertia of the vehicle body along

three axes of the vehicle frame; (ωx, ωy, ωz) are the rotational velocities of the vehicle body along three

axes of the vehicle frame;ωirepresents the angular rate of each tirei; Fa,tire,iis the longitudinal adhesive

force generated by tirei; Ti are the wheel torque transmitted to the tirei; riis the effective rolling radius

of a tirei; Iωis the moment of inertia of a tire.

The suspension system is modeled as a nonlinear spring-mass-damper system. Thus, the translational force generated by the suspension can be described as follows [7]:

Fz,spring,i = Ks,iHi+ Ds,iH˙i+ mu,ig (14)

Ks,i = C1eC2(Hi −C3) (i = 1 ∼ 4) Hi = ( Hi, f or Hi > −mu,ig/Ks,i

−mu,ig/Ks,i, f or Hi ≤ −mu,ig/Ks,i

whereKs,i is the spring stiffness of the suspension i and C1, C2, C3 parameterize the stiffness;Ds,i is

the damper coefficient of the suspensioni; mu,iis the unsprung mass of the suspension corneri.

The adhesive force generated by tire is a highly nonlinear function of variables including slip ratios,
*slip angles, vertical loads, etc. [*26,27]. However, under normal vehicle maneuvering, the adhesive force
is almost linearly proportional to those variables. Thus, a linear tire model [13,14] is used to describe
the longitudinal and lateral tire forces (Fa,tire, Fb,tire) for simplicity.

Fa,tire,i = Cλ,iλi (15)

Fb,tire,i = Cα,iαi

whereCλ,iandCα,iare the tracking stiffness and the cornering stiffness of the tirei, respectively; λi and αi are the slip ratio and slip angle, respectively. The translational forces represented in the x-axis and the

y-axis of the aux-frame (Fx,tire, Fy,tire) can be obtained as follows: Fx,tire =

X

(Fa,tire,icos δi− Fb,tire,isin δi) (16)

Fy,tire = X

(Fa,tire,isin δi+ Fb,tire,icos δi).

Noted that two rear wheel angles (δ3, δ4) are zeros for a front-steer vehicle, and two front wheel

Ackerman principle [28]. The vehicle model for the dynamics prediction is thus constructed and shown in Equations (13)–(16).

**5. Vehicle Parameter Identification Systems**

In this approach, the parameters shown in the vehicle model in Equations (13)–(16) needs to be identified using the vehicle dynamics obtained from the sensor fusion system in Equation (7). Note that the vehicle dynamics from the sensor fusion system is presented in the global frame, while the above vehicle model is presented in the aux-frame. Thus, the estimated vehicle dynamics are transformed into the aux-frame using the matrices shown in Equations (1)–(3), prior to the vehicle parameter identification.

After feeding the vehicle dynamics, Equation (13) becomes a set of 10 linear equations with 12 unknown vehicle parameters. Hence, the number of the cornering stiffness is reduced from four to two because the cornering stiffness is similar at two sides of the vehicle [13,14]. In that case, the model of the lateral tire force is simplified as follows:

Fb,tire,f = Fb,tire,1+ Fb,tire,2 (17)

≃ Cα,f(α1+ α2)/2

Fb,tire,r = Fb,tire,1+ Fb,tire,2

≃ Cα,r(α3+ α4)/2

where

Cα,f = Cα,1+ Cα,2

Cα,r = Cα,3+ Cα,4.

*5.1. Recursive Least-Square Algorithms*

In order to apply the recursive-least-square (RLS) algorithm to identify vehicle parameters, the vehicle parameters and the corresponding measured dynamics are rearranged into the following format:

QArlsWW

−1

x= Qb (18)

where, x represent the vehicle parameters; Arlsand b are the vehicle dynamics both from direct sensor

measurements and the output of the sensor fusion system; Q is the weighting matrix; W is the scaling matrix. By choosing matrices Q, W, and an initial guess of the vector x, the vector x is solved recursively by the following steps [29]:

P(k + 1) = P(k) − P(k)AQ(k + 1)T[I + AQ(k + 1)· P(k)AQ(k + 1)T −1 AQ(k + 1)P(k) xQ(k + 1) = xQ(k) − P(k + 1)AQ(k + 1)T· [Q(k + 1)b(k + 1) − AQ(k + 1)xQ(k)] AQ(k + 1) = Q(k + 1)Arls(k + 1)W(k + 1) xQ(k + 1) = W(k + 1) −1 x(k + 1) (19)

*5.2. Vehicle Parameter Identifications*

The following vehicle parameters are identified using the above RLS algorithms: vehicle mass (mtot),

moment of inertia of the vehicle body (Ix, Iy, Iz), tracking stiffness (Cλ,1∼4), and the cornering stiffness

(Cα,f, Cα,r). These parameters are identified in real time because their values can be changing in each

driving condition. The other vehicle parameters such as the spring stiffnessKs, the damper coefficient Ds, the unsprung mass mu, the rolling radius of tire r, and the moment of inertia of the tire Iω are

assumed to be known values.

In this case, it is possible to manipulate the signal processing steps and formulate four independent RLS problems for identifying the above ten parameters, which can greatly reduce the computation loads and efforts of searching the optimal Q and W matrices. Those four independent RLS problems are “mass identification”, “tracking stiffness identification”, “cornering stiffness identification”, and “moment of inertia identification”.

5.2.1. Mass Identification

The translational dynamics in z direction in the vehicle model in Equations (13) and (14) can be rearranged as: q11Arls,1w11w −1 11x1 = q11b1 (20) where x1 , mtot Arls,1 , ¨ˆza− ˆGz b1 , 4 X i=1

KsHisus+ DsH˙isus+ mu,ig.

whereq11andw11are the elements in weighting and scaling matrices, respectively. ˙Hisusis the derivative

of the suspension displacement and is obtained by the alpha-beta filter via the input from the suspension displacement measurements.

5.2.2. Tracking Stiffness Identification

The wheel dynamics in the vehicle model in Equations (13) and (15)–(16) can be rearranged as:

Q2Arls,2W2W
−1
2 x2 = Q2b2 (21)
where
x2 , Cλ,1, Cλ,2, Cλ,3, Cλ,4
T
A_{rls,2} _{, diag{−r}_{1}λˆ1, −r2λˆ2, −r3λˆ3, −r4λˆ4}
b_{2} _{,} hIω˙ˆω1− T1, Iω˙ˆω2− T2, Iω˙ˆω3− T3, Iω˙ˆω4− T4
iT
Q2 , diag{q21, q22, q23, q24}
W2 , diag{w21, w22, w23, w24}.

Noted that the matrix Arls,2 is singular when one of the slip ratios is zero. It can be understood that

the tracking stiffness cannot be identified when there is no traction force. Moreover, the angular rates of four tires are directly measured by tachometers (see Table1) and conditioned by the alpha-beta filter. The slip ratios are calculated using the measurements from the sensor fusion system and the tachometers. The applying wheel torque is assumed to be obtained from torque sensors.

5.2.3. Cornering Stiffness Identification

The longitudinal and lateral dynamics in the vehicle model Equations (13) and (15)–(17) can be
rearranged as:
Q3Arls,3W3W
−1
3 x3 = Q3b3 (22)
where
x_{3} , Cα,f, Cα,r
T
Arls,3 ,
"
−αˆ1+ ˆα2
2 sin
δ1+δ2
2 0
ˆ
α1+ ˆα2
2 cos
δ1+δ2
2
ˆ
α3+ ˆα4
2
#
b3 ,
"

mtot(¨ˆxa− ˙ˆyaψ˙ˆv− ˆGx) −P Cλ,iˆλicos δi mtot(¨ˆya+ ˙ˆxaψ˙ˆv− ˆGy) −P Cλ,iλˆisin δi

#

Q3 , diag{q31, q32}

W3 , diag{w31, w32}.

Noted that the matrix Arls,3 is singular when the summation of the front (or rear) two slip angles

is zero or the summation of the front steering wheel angles is zero. Again, it can be understood that the cornering stiffness cannot be identified when there is no lateral force. Moreover, the slip angles are calculated using the measurements from the sensor fusion system and the steering wheel angle.

5.2.4. Moment of Inertia Identification

The rotational dynamics in the vehicle model in Equation (13) can be rearranged as:

Q_{4}A_{rls,4}W_{4}W−1
4 x4 = Q4b4 (23)
where
x4 , Ix, Iy, Iz
T
Arls,4 ,
˙ˆωx −ˆωyωˆz ωˆyωˆz
ˆ
ωzωˆx ˙ˆωy −ˆωzωˆx
−ˆωxωˆy ωˆxωˆy ˙ˆωz
b4 , [Mx, My, Mz]T
Q4 , diag{q41, q42, q43}
W4 , diag{w41, w42, w43}

In Equation (23), the angular velocities and accelerations are provided by the alpha-beta filter via the input from the gyroscope measurements.

**6. Vehicle Dynamics Prediction**

From a system observability viewpoint [30], if both the governing equations of a dynamic system and state values at a time instant are given, the state values at any time instant can be calculated accordingly. Stemming from this concept, one can predict the vehicle dynamics using a vehicle model and current state values. In this case, the current vehicle dynamics is obtained from the sensor fusion system shown in Equations (7)–(9); the vehicle model for propagating the current vehicle states is shown in Equations (13)–(16); the parameters in that vehicle model is estimated using four RLS algorithms in Equations (20)–(23). The block diagram of this signal processing is shown in Figure2.

**Figure 2. Block diagram of the vehicle dynamics prediction system.**

α-β filter

α-β filter α-β filter

Vehicle Parameter Identification (RLS)

Sensor Fusion Algorithm

(multi-rate EKF) Sensors IMU 3-antenna GPS Suspension disp. sensors Tachometers l l l l Vehicle Dynamics Prediction Driver Maneuvers Steerging angle gas /brake pedal

Vehicle
parameters
l
l
Vehicle dynamics
Road angles
**7. Simulation Results**

Numerical simulations are used to demonstrate the feasibility of the proposed dynamics prediction method. In these simulations, a vehicle moves at a longitudinal speed of 90 km/h. The steering wheel angles and the generated tire torques are both varying with time at the frequency of 1 Hz (see, Figure3). The road bank angle is 2◦

and the road grade angle is −2◦

. A full-state vehicle model, which is a
nonlinear 6 DOF vehicle model and consists of 20 states and road angles [7], is used to mimic the real
vehicle dynamics on this slope road. This full-state vehicle model differs from the vehicle model shown
in Equation (13) only in the tire model, wherein the nonlinear tire model “magic formula” [26] is used.
The selected sensors and their output characteristics are listed in Table1. The sampling frequency of the
simulations is 100 Hz. No other disturbance is applied to the vehicle system unless otherwise specified.
*7.1. Vehicle Dynamics Estimations*

The simulation results of the proposed sensor fusion system are shown in Figure4, where the state values from the full-state vehicle model are drawn in dashed-blue line, the sensor outputs are drawn in dashed-dotted-green line, and the state values from the output of the sensor fusion system are drawn in solid-red line. Simulation results indicate that the proposed sensor fusion system can accurately obtain the 6 DOF vehicle dynamics and two road angles. The estimation error of each vehicle state, which

is defined as the standard deviation of the difference between the simulated vehicle dynamics and the sensor fusion outputs, is also shown in Figure4.

**Figure 3. The driving maneuvers for the illustrative simulation. The upper plot is the steering**
wheel angle and the lower plot is the wheel torques applying on four tires. The frequency
is 1 Hz.
0 1 2 3 4 5 6 7 8 9 10.25
360
180
0
180
360 Steering Maneuvers
deg
0 1 2 3 4 5 6 7 8 9 10.25
300
200
100
0
100
200
300

Wheel Torques Applying on Tires

N

m

Time (sec)

**Figure 4. Comparisons of the vehicle dynamics from the simulated vehicle dynamics, the**
sensor outputs, and the sensor fusion system outputs. The vehicle dynamics are presented
in the global frame. The error standard deviations are calculated from the 5th second to the
10th second.
0 5 10.25
0
50
100
150
200
m
Logitudinal Displacement xg

Real vehicle dynamics Sensor output Sensor fusion system

0 5 10.25 0 10 20 30 m Lateral Displacement yg 0 5 10.25 0 5 10 15 m Vertical Displacement zg Time (sec) 0 5 10.25 10 15 20 25 m/s Logitudinal Velocity x⋅g 0 5 10.25 0 2 4 6 m/s Lateral Velocity y⋅g 0 5 10.25 0 0.5 1 m/ s Vertical Velocity z⋅g Time (sec) 0 5 10.25 -5 0 5 10 deg

Absolute Roll Angle φg

0 5 10.25 -4 -2 0 2 deg

Absolute Pitch Angle θg

0 5 10.25 0 10 20 30 deg

Absolute Yaw Angle ψg

Time (sec) 0 5 10.25 1 1.5 2 2.5 deg

Road Bank Angle φr

0 5 10.25 -2.5 -2 -1.5 -1 deg

Road Grade Angle θr

0 5 10.25 0 10 20 30 deg

Vehicle Yaw Angle ψv

Time (sec)
**Error ST D: 0.05 deg**
**Error ST D: 0.034 deg**
**Error ST D: 0.048 deg**
**Error ST D: 0.028 deg**
**Error ST D: 0.027 deg**
**Error ST D: 0.023 deg**
**Error ST D: 0.13 m/s**
**Error ST D: 0.05 m/s**
**Error ST D: 0.016 m/s**
**Error ST D: 0.31 m**
**Error ST D: 0.14 m**
**Error ST D: 0.27 m**

*7.2. Vehicle Parameter Identifications*

The vehicle dynamics and sensor fusion outputs presented in the global frame (show in Figure 4) are converted into the aux-frame and shown in Figure 5. These state values, along with direct sensor measurements conditioned by the alpha-beta filter, are used for the vehicle parameter identification. The identification results are shown in Figures 6 and 7, where the identified vehicle parameters are drawn in the solid-red line, and the real vehicle parameters are drawn in the dashed-blue line. The “relative inaccuracy” of estimation, which is defined as (real value − identified

value)/(real value) [31], of the mass and moment of inertial are calculated to be (mtot, Ix, Iy, Iz)

= (5.17 × 10−3

%, 0.12%, 5.05%, 4.37%). The estimation accuracy is good mainly because the

incorporated suspension displacement sensors are relatively accurate. On the other hand, as shown in Figure 7, the estimation of tracking stiffness and cornering stiffness do not converge well and because there is no corresponding tracking stiffness and cornering stiffness in the full-state vehicle model. The feasibility of the tire stiffness estimation is discussed in the next section.

**Figure 5. Comparisons of the vehicle dynamics from the simulated vehicle dynamics and**
the sensor fusion system outputs. The vehicle dynamics are presented in the aux-frame.

0 5 10.25 0 100 200 m Longitudinal Displacement xa

Real vehicle dynamics Sensor fusion system

0 5 10.25 10 15 20 25 m/ s Longitudinal Velocity x⋅a 0 5 10.25 4 2 0 2 m Lateral Displacement ya 0 5 10.25 4 2 0 2 4 m/s Lateral Velocity y⋅a 0 5 10.25 0.13 0.12 0.11 0.1 m Vertical Displacement za Time (sec) 0 5 10.25 0.1 0 0.1 m/s Vertical Velocity z⋅a Time (sec) 0 5 10.25 40 20 0 20 40 deg/s Rotational Velocity ωx 0 5 10.25 10 0 10 20 deg/s Rotational Velocity ωy 0 5 10.25 50 0 50 deg/s Rotational Velocity ωz Time (sec) 0 5 10.25 40 60 80 rad/s

Wheel Angular Rate ω1

0 5 10.25 40

60 80

rad/s

Wheel Angular Rate ω2

0 5 10.25 40

60 80

rad/s

Wheel Angular Rate ω3

0 5 10.25 40

60 80

rad/s

Wheel Angular Rate ω4

Time (sec)
**Error STD: 0.32 m **
**Error STD: 0.41 m**
**Error STD: 0.00025 m**
**Error STD: 0.12 m/s**
**Error STD: 0.032 m/s**
**Error STD: 0.01 m/s**
**Error STD:0.81 deg/s**
**Error STD: 1.74 deg/s**
**Error STD:2.82 deg/s**
**Error STD: 0.023 rad/s**
**Error STD: 0.022 rad/s**
**Error STD: 0.023 rad/s**
**Error STD: 0.021 rad/s**

*7.3. Vehicle Dynamic Predictions*

Continuing from previous simulations, the driver is assumed to hold still the steering wheel and the gas/brake pedal at the time instant 10.25 s to do a left-hand turn on the road. The prediction system is turned on at the 10.25 s to predict the vehicle dynamics for the next 4.75 s, using the vehicle dynamics from the sensor fusion system at the 10.25 s and the vehicle model with the parameters identified from Figures 6 and 7. Since the steering wheel and the gas/brake pedal are the same during 10.25 to 15 s, both the vehicle dynamics and the prediction results during this period can be shown in the same plot for comparison. In Figure8, the real vehicle dynamics are drawn in the dashed-blue line, and the predicted vehicle dynamics are drawn in the solid-green line. According to the simulation results, the proposed method can predict the vehicle dynamics accurately. The prediction error of each state can be found in the plot. The relative-inaccuracy of this prediction, averaged from vehicle displacements (xa, ya, za)

**Figure 6. The identification of the vehicle mass and moment of inertia. The mean values are**
calculated from the 15th second to the 10th second.

0 2 4 6 8 10.25 1720 1725 1730 1735 1740 1745 1750 kg Total Mass mto t

Real vehicle parameters Vehicle parameter identification system

0 2 4 6 8 10.25
350
400
450
500
550
600
Nm/s
2
Moment of Inertia Ix
0 2 4 6 8 10.25
2000
2200
2400
2600
2800
N
m/s
2
Moment of Inertia Iy
Time (sec)
0 2 4 6 8 10.25
3000
3200
3400
3600
3800
4000
N
m/s
2
Moment of Inertia Iz
Time (sec)
**Mean V alue: 1739.91 kg**
**Mean V alue: 2462.95 Nm/s2**
**Mean V alue: 419.50 N m/s2**
**Mean V alue: 3354.30 Nm/s2**

**Figure 7. The identification of the tire tracking stiffness and cornering stiffness. The mean**
values are calculated from the 15th second to the 10th second.

0 2 4 6 8 10.25 0 2 4x 10 5 N Tracking stiffiness Cλ,1 0 2 4 6 8 10.25 0 2 4x 10 5 N Tracking stiffiness Cλ,2 0 2 4 6 8 10.25 -1 0 1 2x 10 5 N Tracking stiffiness Cλ,3 0 2 4 6 8 10.25 -1 0 1 2x 10 5 N Tracking stiffiness Cλ,4 Time (sec) 0 2 4 6 8 10.25 3 3.5 4 4.5 5x 10 4 N/rad Cornering stiffiness Cα,f 0 2 4 6 8 10.25 3 3.5 4 4.5 5x 10 4 N/rad Cornering stiffiness Cα,r Time (sec)

**Mean value: 40029.38 N/rad**

**Mean value: 39013.83 N/rad**
**Mean value: 74550.63 N**

**Mean value: 79404.84 N**

**Mean value: 31618.82 N**

**Mean value: 50426.31 N**

In another example, the driver holds the steering wheel still but generate 1,000N · m torques on two

tires on the right hand side (T2 = T3 = 1, 000 N · m). In that case, the vehicle is likely to rollover

due to the excess yaw moment applying to the vehicle. The simulation results (see Figure 9) show that the vehicle roll angle is larger than 90◦

at the 11.5 s, which indicates a rollover incident. The dynamic prediction system can predict the rollover event. The relative-inaccuracy of this prediction is 27.3%, which is calculated from 10.25 s to 11.5 s and averaged from vehicle states including vehicle displacements (xa, ya, za) and vehicle attitude (ψv, θv, φv). The prediction accuracy is worse than that

**Figure 8. Predictions of the vehicle dynamics in a left-hand turn event. The prediction**
inaccuracy is 0.51% on average, calculated from the 10.25th second to the 15th second.

10 11 12 13 14 15 180 200 220 m Longitudinal Displacement xa 10 11 12 13 14 15 5 10 m/ s Longitudinal Velocity x⋅a 10 11 12 13 14 15 4 3 2 1 m Lateral Displacement ya 10 11 12 13 14 15 4 2 0 2 m/ s Lateral Velocity y⋅a 10 11 12 13 14 15 0.14 0.12 0.1 0.08 m Vertical Displacement za 10 11 12 13 14 15 0.2 0 0.2 m/s Vertical Velocity za⋅ 10 11 12 13 14 15 0 100 200 250 deg

Vehicle Yaw Angle ψv

10 11 12 13 14 15

0 50 100

deg/s

Vehicle Yaw Rate ψ⋅v

10 11 12 13 14 15 0.5 1 1.5 2 deg

Vehicle Pitch Angle θv

10 11 12 13 14 15

5 0 5

deg/s

Vehicle Pitch Rate θv⋅

10 11 12 13 14 15 0 5 10 15 deg

Vehicle Roll Angle φv

10 11 12 13 14 15 40 20 0 20 40 deg/s

Vehicle Roll Rate φ⋅v

10 11 12 13 14 15 10 20 30 40 rad/s

Wheel Angular Rate ω_{1}

Time (sec) 10 11 12 13 14 15 20 30 40 rad/s

Wheel Angular Rate ω2

Time (sec) 10 11 12 13 14 15 10 20 30 40 rad/s

Wheel Angular Rate ω3

Time (sec) 10 11 12 13 14 15 10 20 30 40 rad/s

Wheel Angular Rate ω4

Time (sec)

Real vehicle dynamics Vehicle dynamic prediction system

Error STD: 0.575 m

Error STD: 0.006 m

Error STD: 0.0001 m

Error STD: 0.091 m/s

Error STD: 0.002 m/s Error STD: 0.798 deg

Error STD: 0.0003 deg Error STD: 0.012 deg

Error STD: 0.021 m/s

Error STD: 0.777 deg/s

Error STD: 0.027 deg/s Error STD: 0.300 deg/s

Error STD: 1.607 rad/s Error STD: 0.824 rad/s Error STD: 0.897 rad/s

Error STD: 3.511 rad/s

**Figure 9. Predictions of the vehicle dynamics in a rollover event. The prediction system**
successfully predicts the rollover incident. The prediction inaccuracy is 27.3% on average,
calculated from the 10.25th second to the 11.5th second.

10 10.5 11 11.5 12 180 190 200 m Longitudinal Displacement xa 10 10.5 11 11.5 12 0 5 10 15 m/ s Longitudinal Velocity x⋅a 10 10.5 11 11.5 12 5 0 m Lateral Displacement ya 10 10.5 11 11.5 12 6 4 2 0 2 m/ s Lateral Velocity y⋅a 10 10.5 11 11.5 12 0.2 0 0.2 0.4 0.6 m Vertical Displacement za 10 10.5 11 11.5 12 1 0 1 2 3 m/s Vertical Velocity za⋅ 10 10.5 11 11.5 12 0 50 100 150 deg

Vehicle Yaw Angle ψv

10 10.5 11 11.5 12 0 50 100 150 deg/s

Vehicle Yaw Rate ψ⋅v

10 10.5 11 11.5 12 1 0 1 2 deg

Vehicle Pitch Angle θv

10 10.5 11 11.5 12 20 10 0 10 deg/s

Vehicle Pitch Rate θ⋅v

10 10.5 11 11.5 12

0 50 100

deg

Vehicle Roll Angle φv

10 10.5 11 11.5 12 200 0 200 400 600 deg/s

Vehicle Roll Rate φ⋅v

10 10.5 11 11.5 12 25 30 35 40 rad/s

Wheel Angular Rate ω1

Time (sec)

10 10.5 11 11.5 12

20 40

rad/s

Wheel Angular Rate ω2

Time (sec) 10 10.5 11 11.5 12 20 40 60 rad/s

Wheel Angular Rate ω3

Time (sec) 10 10.5 11 11.5 12 36 37 38 39 rad/s

Wheel Angular Rate ω

4

Time (sec)

Real vehicle dynamics Vehicle dynamic prediction system

Error STD: 0.003 m

Error STD: 0.021 m

Error STD: 0.525 m/s

Error STD: 0.224 m/s

Error STD: 0.681 m/s Error STD: 0.843 deg

Error STD: 0.0008 deg Error STD: 3.672 deg

Error STD: 3.401 deg/s

Error STD: 0.153 deg/s

Error STD: 232.035 deg/s

Error STD: 3.666 rad/s

Error STD: 2.074 rad/s Error STD: 11.187 rad/s

Error STD: 0.773 rad/s Error STD: 0.077 m

**8. Discussion**

Although the vehicle rollover accident can be foreseen in this case, the prediction is a bit inaccurate. According to the parameter identification results shown in Figures6and7, this error is likely due to the model difference used in the prediction process and in the simulated vehicle dynamics. In turns, it leads to two possibilities: (1) the vehicle dynamics estimated from the sensor fusion system are not accurate enough for the traction/cornering stiffness identification; (2) the adhesive tire forces in the rollover event

are located in the nonlinear regime and the linear tire model is inadequate to describe them. To clarify this, Figure10presents the relation between longitudinal tire force and slip ratio, and Figure11presents the relation between lateral tire force and slip angle. As shown in Figure10, the longitudinal tire force estimated from the sensor fusion system (drawn in blue dots) and from the simulated vehicle dynamics (drawn in green stars) are not the same. Thus, the sensor noise associated with the selected sensors does affect the estimation of longitudinal dynamics to certain extent. In turns, the identified traction stiffness (drawn in solid-black line) cannot be accurate. On the other hand, as shown in Figure11, the lateral tire force estimated from the sensor fusion system are close to the simulated vehicle dynamics. Thus, the identified cornering stiffness is more accurate than the identified traction stiffness.

**Figure 10. The relations between the slip ratio and the longitudinal tire adhesive force.**

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 -1000 -500 0 500 1000 Slip ratio N

Front - left longitudinal force

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 -1000 -500 0 500 1000 1500 2000 2500 3000 3500 Slip ratio N

Front - right longitudinal force

-0.04 -0.02 0 0.02 0.04 -1000 -500 0 500 1000 1500 2000 2500 3000 3500 Slip ratio N

Rear - right longitudinal force

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 -1000 -500 0 500 1000 Slip ratio N

Rear - left longitudinal force

Sensor fusion system Real vehicle dynamics

Vehicle dynamics in the left-hand turn event

Vehicle dynamics in the rollover event Identified tracking stiffness coefficient

**Figure 11. The relations between the slip angle and the lateral tire adhesive force.**

-0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 1.5x 10 4

Slip angle (rad)

N

Front lateral force

-0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 1.5x 10 4

Slip angle (rad)

N

Rear lateral force

Sensor fusion system Real vehicle dynamics

Vehicle dynamics in the left-hand turn event

Vehicle dynamics in the rollover event Identified cornering stiffness coefficient

The tire forces of two prediction cases are also shown in Figures10and11. The tire forces are drawn in red circles for the left-turn event and in cyan squares for the rollover event. The tire forces of the left-hand turn event are very close to the forces calculated from the identified traction/cornering stiffness, except for the force at the rear-left tire. Thus, the prediction of the left-hand turn dynamics is quite accurate. On the other hand, the tire forces of the rollover event are far from the forces calculated from the identified traction/cornering stiffness. Thus, the prediction of the rollover dynamics is a bit inaccurate.

From above discussion, one may propose using nonlinear tire models, such as Pacejka’s magic formula [26] or Dugoff tire model [27], for the friction coefficient identifications and dynamics predictions. Our experiences show that it is possible to identify more parameters of a nonlinear tire model as long as the nonlinear tire adhesive forces are present in the measured vehicle dynamics (persistent excitation condition). However, the tire adhesive forces are within in the linear regime in most driving conditions. In that case, including a nonlinear tire model in the identification process would not gain any advantage but cause convergence problems.

This vehicle parameter identification is challenging mainly because the system has a low degree-of-observability [32,33], and it gets worse when large amount of noise is present in the sensor measurements. This effect can be investigated by the signal-to-noise ratio (SNR), where the signal is referred to as the estimated vehicle dynamics from the sensor fusion system, while the noise is referred to as the estimation error. To show how the SNR affects this parameter identification, we use the identification of the moment of inertia as an example. As shown in Table 2, the estimation error can be minimized when the SNR is large, and the relative error approaches (1.94%, 2.68%, 0.33%) for (Ix, Iy, Iz). The best estimation accuracy is limited by the numerical errors and the model errors from the

linear tire model assumption. To enlarge the SNR, either the range of vehicle dynamics needs to be enlarged or the noise from the sensor measurement needs to be minimized. The vehicle dynamics cannot have a large span due to its strong stability. On the other hand, using high-end sensors would reduce the noise but incur higher cost.

**Table 2. The relations between the estimation error of sensor fusion system and the relative**
inaccuracy of the parameter identification.

relative inaccuracy Ix Iy Iz

infinite 1.94% 2.68% 0.33%

SNR 30 dB 2.50% 14.62% 1.52%

20 dB 14.91% 60.28% 10.41%

10 dB 69.05% 93.63% 54.11%

One alternative to improve this parameter identification is to increase the degree-of-observability by choosing proper weighting and scaling matrices shown in Equation (19). To show the effectiveness of this approach, we use the identification of the moment of inertia as an example and assume no noise in the sensor measurement. The choice of weighting matrix (Q4) changes the minimum eigenvalue of the

estimation matrix (AT_{rls,4}QT_{4}Q4Arls,4) and results in different degree-of-observability [32,33]. Table3

shows the larger minimum eigenvalues of the resulting matrix, the better parameter observability and the faster convergence rate of the parameter identification.

**Table 3. Different weighting matrices result in different convergence rate.**
convergence ratea _{I}
x Iy Iz
diag{0.625, 0.625, 0.625} 0.08 s 1.63 s 0.24 s
Q_{4} b _{diag}_{{0.125, 0.125, 0.125} 6.19 s 18.2 s} _{3.32 s}
diag{0.025, 0.025, 0.025} 36 s Nanc _{25 s}

diag{0.005, 0.005, 0.005} Nanc _{Nan}c _{Nan}c

a_{The convergent rate is defined at the time when estimated value reaches 90% of the real value.}

b_{Q}

4and W4are both designed as a diagonal matrix. Q4varies in each case,
while W_{4}is kept the same as diag{1, 1, 1}.

c_{The value “Nan” means that the convergence time is too long to calculate.}

**9. Conclusions**

A vehicle dynamics prediction system, consisting of a kinematics-based sensor fusion system and a vehicle parameter identification system, is proposed and verified by simulation results. The sensor fusion system can obtain the 6 DOF vehicle dynamics and the two road angles accurately. The estimation error for each vehicle dynamics is shown in Figure 4. The vehicle parameter identification system uses the dynamics information from the sensor fusion system to identify ten vehicle parameters in real time. The identification inaccuracy of the vehicle mass and moment of inertia is less than 5.05%. Using the vehicle dynamics from the sensor fusion system and the vehicle model from the parameter identification system, the prediction system successful predicts the vehicle dynamics in a left-hand turn event and a rollover event. The prediction inaccuracy is 0.51% in the left-hand turn event and 27.3% in the rollover event.

The prediction accuracy of the rollover event is worse than that of the left-hand turn event. It is mainly because the identified linear tire model cannot accurately describe the nonlinear tire adhesive force in the rollover event. Using a nonlinear tire model for the dynamics prediction is possible but not practical in this case, because the nonlinear tire behaviors are not excited in normal vehicle maneuvers.

The prediction accuracy of two scenarios suggests that modeling error of the unsprung mass system may greatly affect the accuracy of the parameter identification and thus the dynamics predictions. Therefore, a detail modeling and/or real-time system identification of the unsprung mass system may be needed to improve the feasibility of this approach. Besides, this research also shows that the vehicle parameter identification is challenging because the system has a low degree-of-observability. Therefore, increasing the SNR of the sensor systems and careful designs of the weighting matrix of the identification algorithm are recommended.

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