Estimation of tire-road friction coefficient based on frequency domain data fusion

Mechanical Systems and Signal Processing 85 (2017) 177–192

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Estimation of tire-road friction coefficient based on frequency domain data fusion Long Chen, Yugong Luo, Mingyuan Bian, Zhaobo Qin, Jian Luo, Keqiang Li n State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, PR China

a r t i c l e i n f o

abstract

Article history: Received 15 April 2016 Received in revised form 30 July 2016 Accepted 2 August 2016

Due to the noise of sensing equipment, the tire states, such as the sideslip angle and the slip ratio, cannot be accurately observed under the conditions with small acceleration, which results in the inapplicability of the time domain data based tire-road friction coefficient (TRFC) estimation method. In order to overcome this shortcoming, frequency domain data fusion is proposed to estimate the TRFC based on the natural frequencies of the steering system and the in-wheel motor driving system. Firstly, a relationship between TRFC and the steering system natural frequency is deduced by investigating its frequency response function (FRF). Then the lateral TRFC is determined by the steering natural frequency which is only identified using the information of the assist motor current and the steering speed of the column. With spectral comparison between the steering and driving systems, the data fusion is carried out to get a comprehensive TRFC result, using the different frequency information of the longitudinal and lateral value. Finally, simulations and experiments on different road surfaces validated the correctness of the steering system FRF and the effectiveness of the proposed approach. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Tire-road friction coefficient Data fusion Active front steering system Dynamic tire model Natural frequency

1. Introduction The active front steering (AFS), as a kind of steering-based vehicle stability control (VSC) systems, can generate the driver's desired lateral force and yaw moment more quickly with no intervention of braking, compared to the direct yaw moment (DYC)-based VSC [1]. In addition, the application of the motor has various merits in the control and functional aspects, such as high motor response, and precise torque generation [2]. Besides, it can also be treated as an accurate feedback sensor. In recent years, the motor-based vehicle state estimation has become a hot spot of research [3]. For longitudinal control, some researchers have investigated the advantages of the in-wheel motor on longitudinal velocity estimation [4,5]. For VSC, the most important vehicle state is the sideslip angle. By combining a linear vehicle model with the steering system model, a simple observer can be used to estimate sideslip when yaw rate and steering angle are measured, if the steering torque can easily be determined from the current applied to the steering assist motor. In addition to the sideslip angle, the information of the tire -road friction coefficient is indispensable for accurate stability control [6], as the tire forces primarily determine the states of the steering wheel motion. The TRFC μmax in this paper can be defined as Equation (1) [7].

n

Corresponding author. E-mail address: [email protected] (K. Li).

http://dx.doi.org/10.1016/j.ymssp.2016.08.006 0888-3270/& 2016 Elsevier Ltd. All rights reserved.

178

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

μ max =

Fx2 + Fy2 Fz max

(1)

where Fx , Fy and Fz are respectively the longitudinal, lateral and vertical tire forces. This definition shows that μmax can be estimated from longitudinal or lateral dynamics response. Different approaches have been proposed about the estimation of the tire-road friction coefficient [8–10]. The wellknown method is the slope based one. Hedrick [9] and Gustafssonp [10] deemed that the TRFC can be considered as a function of the slope between the slip ratio or sideslip angle and friction coefficient curve. The longitudinal dynamics-based TRFC estimation methods requires large longitudinal tire slip ratios, so the vehicle should accelerate or brake sufficiently in order to provide reliable TRFC estimates [6]. Such a requirement may constitute limitation as the longitudinal tire slip is typically small for normal driving conditions. Bartram [11] found that the tire/road contact interface can influence the driveline vibrations. Following this research and the study of [12], Chen [13] deduced a relation between the TRFC and the high in-wheel motor (IWM) drive system natural frequency using coupling analysis [14]. Based on this relationship, a method is proposed to detect the difference of road condition with the merit that the motor can capture the high frequency information of the driving torque [13]. Compared to the vehicle longitudinal dynamics-based TRFC estimation methods, the vehicle lateral dynamics-based methods do not require excessive longitudinal motion excitations but can be used only when the vehicle is turning [8] The driver needs to keep on turning the steering wheel so that the persistence of excitation condition, which ensures convergence of TRFC, can be satisfied. Hong [15] tried to estimate TRFC using the measured lateral force from sensors installed in the tire. Wang [16] and Wang [17] proposed a sequential tire cornering stiffness coefficient and tire–road friction coefficient estimation method for the four-wheel independently-actuated electric vehicles. This method can estimate TRFC without affecting the vehicle desired motion control and trajectory tracking objectives. But the sideslip angle is still relative large, which means that this approach can’t overcome the limitation of steering maneuver. As the tire self-aligning torque (SAT) exhibits high sensitivity to road friction at low slip angles and can be calculated with the electric power steering torque, Matsuda [18] applied a method to estimate front tire road friction accurately at low lateral acceleration. But, when the vehicle is moving straightforward, the sensor noise and error influence the estimation accuracy of the sideslip angle. Meanwhile, the active front steering system which consists of the assist motor, the steering column and the tire, is highly electromechanical. However, only simplified rigid tire models are used in most of the published papers regarding the steering control. The dynamic tire properties cannot be sufficiently captured, which are caused by different wheel and tire parameters. Briefly, these all result in the low accuracy of the friction estimation the slope-based methods if only the timedomain information is used. Modern advanced driver assistant system(ADAS) such as adaptive cruise control and lane keeping system, try to ensure the safety of the vehicle, under the case that the tire and the vehicle are operating within the stable and/or safe conditions. This means that there is a small slip ratio and sideslip angle. The TRFC estimation is indispensable to calculate the safety distance to surrounding objects. Chen's approach [13] is proposed for this case using longitudinal dynamics. If Chen's is also suitable for the lateral situation, a similar frequency relation should exist in the active front steering system, and can be used to estimate TRFC. With the development of data fusion theory, Li [19] and Chen [20] tried to implement it in TRFC estimation to reach a more accurate performance under complex maneuver. As AFS and IWM are two key technologies which will be implemented in the future electric vehicles, the assist steering motor and the driving motor can act as different road condition sensors. So the data fusion of these information resources should be also investigated to improve the robustness of the estimation. A TRFC estimation method based on frequency domain data fusion under the conditions with small acceleration is presented in this paper. For implementing the method, firstly, the impact of the lateral road friction on the natural frequency of the steering system should be found. A simplified motor model and a first-order delay-based dynamic tire model are considered to establish the column-assist-type steering system. As a result of analysis of the steering dynamics in the frequency domains using its frequency response function (FRF), the equation of the natural frequency is obtained. It infers that the squared natural frequency changes linearly along with the road friction. Then, with the identification method in [13], the lateral TRFC is determined using only the signal of the assist motor current and the steering column rotation speed. Spectral comparison hereafter is conducted between AFS and IWM, to propose the data fusion approach. It combines the lateral and longitudinal TRFC to get a comprehensive and accurate value. The correctness of the equation of the natural frequency and advantages of the total estimation scheme is validated by the simulation and the experimental results under small sideslip angle conditions.

2. System modeling 2.1. Active front steering system The active front steering system in this article is a column-assist-type, whose assist motor steers the front wheel through the worm gear structure by rotation of the steering column, shown as Fig. 1 [21].

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

179

Fig. 1. Schematic diagram of AFS.

Considering the torque balance of the steering column, the dynamics of this system can be modeled as:

Th + Tm G m −

Tα = Bd θḋ + Jd θ¨d Gs

(2)

where, Th is the torque input of the driver, Tm is the motor torque, Gm is the worm gear ratio, Tα is the aligning torque of the front axle, Gs is the gear ratio of the steering system form the steering wheel angle to the steering angle of the front wheel. θd is the steering angle of the column, Bd is damping coefficient, Jd is the equivalent moment of inertia. 2.2. Tire model This paper only studies the normal driving case when the slip ratio and the sideslip angle α are both relatively small. To reasonably simplify this problem, only lateral motion is considered in this steering system modeling. The longitudinal and vertical vehicle dynamics, and road slope impact are neglected. Thus, the steady tire aligning torque is then described as Eq. (3) using algebraic sideslip/torque relationships.

TαS = k α α α= −

vy + l f ω r θd + Gs vx

(3)

where TαS is the steady aligning torque, k α and α are the aligning stiffness and the sideslip angle of the front axle, respectively. Due to the considered condition, the vertical load is constant and the slip ratio is around zero. The impacts of the vertical load and the slip ratio on the aligning stiffness are ignored. So the latter one can be assumed to change nearly linear with the lateral road friction. vx and vy is the longitudinal and lateral velocity of the vehicle at the point of the center of gravity(CoG), ωr is the yaw rate of the vehicle around the vertical axis. l f is the distance from the front axle to CoG. To describe the dynamic reaction of the tire force under disturbances, the first order systems can be used to derive the dynamic aligning torque TαD from the steady aligning torque force TαS [22]. D τTα̇ + Tα D = Tα S

(4)

The time constant τ is calculated using the relaxation length ry as follows.

τ=

ry vx

(5)

The non-linear characteristics of ry is ignored in order to minimize the complexity of the calculation process, so ry is set as a constant in this paper. 2.3. Permanent magnet synchronous motor (PMSM) model Under the following assumption that hysteresis loss and the saturation of the ferrite core in the electric motor are neglected, the torque of the PMSM is determined by a mathematical model of armature currents id and iq (d - and q -axis,

180

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

respectively).

Tm = Pn ψa iq + Pn (L d − L q ) id iq

(6)

where Pn is the number of pole pairs; L d and L q are inductances at d - and q-axis, respectively; ψa is the magnet flux-linkage, Tm is the output torque from the motor. Under the assumption that L d equals to L q , Eq. (6) can be re-arranged as follows:

Tm = Pn ψa iq = Kiq

(7)

3. Relationship between the TRFC and the steering system natural frequency 3.1. Deduction of the frequency response function In this paper, due to the convenience of measurement, the current of the motor and θḋ are set as the input and output of the AFS system. A frequency response function is needed to build a relationship between these two states. Assuming that the sum of Th and Tm Gm consists of a constant value and high-frequency one, it can be shown as follows.

Th + Tm G m = T0 + T1 sin (2πf ⋅t )

(8)

where T0 is a constant value denoting the low frequency information, and T1 sin (2πf ⋅t ) is the high frequency information, for which T1 is the amplitude, and f is the frequency. After substituting Eq. (8) into Eq. (2), the steering dynamic function can be described as Eq. (9).

G m ( T0 + T1 sin (2πf ⋅t ) ) −

TαD = Bd θḋ + Jd θ¨d Gs

(9)

Taking the derivative on both sides of Eq. (9), it becomes: D

G m ( 2πf ) T1 cos (2πf ⋅t ) −

Tα̇ = Bd θ¨d + Jd θd⃛ Gs

(10)

Then, in order to express the steering dynamics using the steady aligning torque, Eq. (11) can be obtained by multiplying Eq. (10) by τ and adding to Eq. (9) results, shown as follows:

G m T0 + G m T1 ⎡⎣ ( 2πfτ ) cos (2πf ⋅t ) + sin (2π f ⋅t ) ⎤⎦ D

−τ

Tα̇ TD − α = Bd θḋ + ( τBd + Jd ) θ¨d + τ Jd θd⃛ Gs Gs

(11)

Considering to express the tire self-aligning torque by the sideslip angle (Eqs. (3), (4) and (5)), the equation above can be simplified as:

G m T0 + aG m T1sin (2πf ⋅t + ϕ) −

kα α Gs

= Bd θḋ + ( τBd + Jd ) θ¨d + τ Jd θd⃛

(12)

2

where a = 1 + ( 2πfτ ) , ϕ = arctan ( 2πfτ ). To eliminate the constant value, Eq. (12) is differentiated with respect to time, shown as Eq. (13):

kα α̇ Gs ¨ ⃛ ⃜ = Bd θd + ( τBd + Jd ) θd + τ Jd θd 2πfaG m T1 cos (2πf ⋅t + ϕ) −

(13)

Only the straightforward case is studied in this paper which means that the steady values of the steering wheel angle, the lateral velocity vy and the yaw rate ωr are nearly zero. As the moment inertia and the mass of vehicle are rather larger than the one of steering system, so the derivative of the yaw rate ωr and the lateral velocity vy can be treated as zero, compared to the one of the steering angle. vẏ + l f ω̇ r ≈ 0. The longitudinal velocity is constant, so vẋ ≈ 0. Under these assumptions, α̇ can be expressed as:

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

⎛ vy + l f ω r ⎞′ θḋ +⎜ ⎟ G vx ⎝ ⎠ ̇ ̇ + v l vy + l f ω r ω θ y f ̇r = − d + − vẋ G vx vx2 θ̇ ≈ − d G

181

α̇ = −

(14)

Then Eq. (13) can be rewritten as:

2πfaG m T1 cos (2πf ⋅t + ϕ) k θ̇ = α 2d + Bd θ¨d + ( τBd + Jd ) θd⃛ + τ Jd θd⃜ Gs

(15)

Assuming:T2 = T1 cos (2πf ⋅t + ϕ)

(16)

Eq. (15) can be rearranged as:

k α θḋ + Bd θ¨d + ( τBd + Jd ) θd⃛ + τ Jd θd⃜ Gd2

2π faG m T2 =

(17)

Assuming that ωd = θḋ . Both sides of Eq. (17) are re-arranged by the Laplace transform, resulting in the following equation:

2πfaG m T2 ( λ ) ⎡k ⎤ = ⎢ α2 + Bd λ + ( τBd + Jd ) λ2 + τ Jd λ 3⎥ ωd ( λ ) ⎣ Gs ⎦

(18)

where λ is the Laplacian operator. Then, ignoring the highest-order term τ Jd λ3, the transfer function from the control signal of the motor to the steering column dynamics can be shown as:

T2 ( λ ) ωd ( λ )

kα

=

G s2

+ Bd λ + ( τBd + Jd ) λ2 2πfaG m

(19)

In the end, the frequency response that expresses the relationship between the steering torque and θḋ is deduced as Eq. (19). To easily verify the above deduction result using an experiment, the steering torque input by the driver can be treated as zero in the case that the driver does not hold the steering wheel. So the steering torque can be measured using the value of the motor current. Thus, Eq. (20) is obtained by rearranging Eq. (19) with Eq. (7).

iq (λ ) = ω (λ )

kα G s2

+ Bd λ + ( τBd + Jd ) λ2 2

2πf 1 + ( 2πfτ ) KG m

(20)

3.2. Deduction of the steering system natural frequency As discussed in the section Introduction, this paper will investigate whether a relationship exists between the tire-road friction coefficient and the steering system natural frequency, similar to the one found in the reference [13]. The steering system natural frequency cannot be obtained directly from Eq. (19). In order to reasonably simplify Eq. (19), the empirical value range of the natural frequency should be known by checking the different parameters’ impact on the amplitude ratio between the steering column rotation speed and the motor current. The amplitude ratio is shown as:

182

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

4000

4000

Gm=4

kα=2000Nm/rad

3000

A(ωd)/A(i) (ras/s/A)

A(ωd)/A(i) (rad/s/A)

kα=1000Nm/rad kα=4000Nm/rad 2000

1000

0

0

0.5

1

1.5 2 Frequency (Hz)

2.5

Gm=12

2000

1000

0

3

Gm=8

3000

0

0.5

1 Frequency (Hz)

1.5

2

gear ratio 1200 vx=5m/s

Gs =10

vx=15m/s

Gs =15

A(ωd)/A(i) (rad/s/A)

A(ωd)/A(i) (rad/s/A)

1800

vx=45m/s

1200

600

0

0

0.5

1 Frequency (Hz)

1.5

2

Gs =20

800

400

0

0

0.5

1

1.5 2 Frequency (Hz)

2.5

3

(d) Fig. 2. Amplitude ratio. (Parameters for Fig. 2(a): ka ¼2000Nm/rad; y r ¼0.05m; m G ¼ 4; x v ¼ 5m/s; K ¼ 6Nm/A; d B ¼0.03Nm-s/rad; d J ¼0.1 kg-m2. Parameters for Fig. 2(b): ka ¼ 2000Nm/rad; y r ¼0.05m; s G ¼20; m G ¼ 4; K ¼6Nm/A; d B ¼0.03Nm-s/rad; d J ¼0.1 kg-m2. Parameters for Fig. 2(c): ka ¼ 2000Nm/rad; y r ¼0.05m; s G ¼ 20; x v ¼5m/s; K ¼6Nm/A; d B ¼ 0.03Nm-s/rad; d J ¼ 0.1 kg-m2. Parameters for Fig. 2(d): ka ¼ 2000Nm/rad; y r ¼ 0.05m; m G ¼ 4; x v ¼5m/s; K ¼6Nm/A; d B ¼0.03Nm-s/rad; d J ¼0.1 kg-m2).

A (iq ) i q ( j 2π f ) = ω ( j 2π f ) A (ω) kα

=

G s2

2

− ( τBd + Jd )( 2πf ) + jBd ( 2πf ) 2

2πf 1 + ( 2πfτ ) KG m kα

=

G s2

−

(

B d ry vx

2π f

)( 2πf ) + jB ( 2πf ) 1+( ) KG 2

+ Jd

2πfry 2 vx

d

m

(21)

Using Eq. (21), the curves of amplitude ratio- frequency for different system parameters and states are shown as Fig. 2. It can be clearly found in Fig. 2 that the resonance phenomenon exists in the active front steering system. The frequency at the peak point is called the steering system natural frequency in this paper. It is shown in Fig. 2(a) that the aligning stiffness strongly impacts he natural frequency. As the aligning stiffness has a linear relationship with the road friction, the impact above can be used to estimate the TRFC. Fig. 2(b) and (c) imply that even though the longitudinal velocity and the worm gear ratio can influence the curve shape, the natural frequency keeps fixed. Similar to the aligning stiffness, the natural frequency varies linearly along with the steering ratio. No matter how these four kinds of parameters change, the empirical value of natural frequency is smaller than 3 Hz. 2

With this empirical value illuminated above, ωd = θḋ 1 + ( 2πfry/vx ) conversion can be performed on Eq. (21).

can be simplified as 1. Then, the approximate

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

183

⎡ A (iq ) ⎤ min ⎢ ⎥ ⎣ A (ω) ⎦ ⎡ ⎢ = min ⎢ ⎢ ⎢⎣

kα G s2

−

(

B d ry vx

⎡ ⎢ ≈ min ⎢ ⎢ ⎢⎣

kα G s2

vx

kα G s2

2

− Jd ( 2πf ) 2πf KG m

As the imaginary item real item. If

Gs2

d

m

⎤ 2 − Jd ( 2πf ) + jBd ( 2πf ) ⎥ ⎥ 2πf KG m ⎥ ⎥⎦

⎡ ⎢ jBd = min ⎢ + ⎢ Gm ⎣

kα

2

2πfry 2

2π f

⎤

)( 2πf ) + jB ( 2πf ) ⎥⎥ ⎥ 1+( ⎥⎦ ) KG

+ Jd

jBd Gm

⎤ ⎥ ⎥ ⎥ ⎦

(22)

is constant, the minimum value of the amplitude ratio depends on the absolute value of the 2

equals to Jd ( 2πf ) , the real item is zero, the amplitude ratio reaches the minimum, which means the re-

sonance happens at this frequency. So, the natural frequency (NF) can be expressed as:

f0 =

1 2π G s

kα Jd

(23)

Obviously, those regular patterns shown in Fig. 2 can be reasonably explained by Eq. (23). Considering from another point of view, Fig. 2 verifies the correctness of Eq. (23). As the straightforward case is studied in this paper, the sideslip angle is relatively small. As described in tire modeling, the aligning stiffness has a linear relationship with the road friction [22], which can be shown as the following equation:

μ max = a μmax k α + b μmax

(24)

where, a μmax and bμmax are two constants, which can be obtained by a linear regression using the measured aligning stiffness data on different road surfaces. Combining Eq. (23) with (24), the NF can be expressed as a function of the TRFC.

f0 =

1 2π G s

μ max − b μmax Jd a μmax

(25)

AsGs , a μmax , bμmax and Jd are considered as constants for a vehicle, a linear relationship is expressed between the road friction and the squared natural frequency as Eq. (25), which makes it possible to estimate the former one if the latter one can be estimated by a method using the dynamic response of the active front steering system. The characteristic that the natural frequency is unrelated to the longitudinal velocity, which is ideal for estimating the friction coefficient, can solve the problem of the traditional method for small sideslip angle conditions that its accuracy is sensitive to the error and noise of velocity estimation.

4. Frequency domain data fusion based tire-road friction coefficient estimation As discussed in the section Introduction, the tire road friction coefficient estimation method proposed in this paper is based on frequency domain data fusion, shown as Fig. 3. In Fig. 3, μmax −x and μmax −y mean respectively the estimated TRFC using longitudinal or lateral dynamics response. This method consists of two steps: Firstly, via system identification theory and recursive least squares filter, the TRFCs for each direction are estimated. Then, the final value is obtained by combining the TRFCs for each direction through two first order filters. 4.1. Natural frequency based TRFC estimation There is a detailed introduction about this natural frequency based TRFC estimation method for longitudinal dynamics and the process to reduce the effects of sensor noise in [13]. So, in this paper, only the differences are discussed when it is applied for lateral dynamics. From the perspective of equipment, the steering system natural frequency is obtained using the auto-regressive exogenous model, with the information of assist motor and steering torque sensor.

184

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

Fig. 3. Process of the estimation method.

The other difference is in the RLS process. Eq. (25) should be used to estimate the lateral TRFC, according to the identified natural frequency. 4.2. Frequency domain data fusion As shown in Fig. 2(a), the empirical value of the steering system natural frequency is 1–3 Hz. Comparatively, the empirical value of the in-wheel motor drive system is 10 times of the former one, about 10–30 Hz, shown as Fig. 4. If the estimated TRFCs for each direction can be regarded as two measurements for the same signal subjected to different disturbances, they are significantly distinct from the viewpoint of frequency response and the model error. The estimated longitudinal TRFC can quickly response to the changing of road conditions, but the accuracy can be easily influenced by the model(the relation function between the IWM natural frequency and the longitudinal TRFC) error and the “sensing error”. In the research, “sensing error” refers to the error in the frequency domain, which actually means the noise or unstable and instantaneous vibration of the signal in the time domain. So, comparatively, even though the estimated lateral TRFC cannot

0.4 ks=60000N ks=80000N ks=100000N

A(ω)/A(iq)

0.3

C

0.2 A

0.1 B

0

0

5

10 15 20 Frequency(Hz)

25

Fig. 4. Natural frequency of the in-wheel motor drive system.

30

20

0.05

10

0.025

Sideslip angle (rad)

Motor current (A)

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

0

-10

-20

185

0

-0.025

0

3

6

9

12

15

-0.05

18

0

3

6

Time (s)

9

12

15

18

12

15

18

Time (s)

60

3

Aligning Torque (Nm)

Rotation speed (rad/s)

2 1 0 -1

30

0

-30

-2 -3

0

3

6

9

12

15

18

-60

0

3

6

9 Time (s)

Time (s)

Fig. 5. Simulation data.

quickly response to the changing of road conditions, it is relatively robust against the sensing instantaneous vibration and model errors. So, at low frequencies, the value from the lateral TRFC estimation is mainly used against model errors. In order to accurately capture the dynamic changing of road condition, the higher-frequency information of the longitudinal TRFC estimation should be extracted. To implement this idea, first order filters [23] are found to be adequate for the combination of the two directions’ TRFC. The combination format is proposed as Eq. (26).

μ max =

τ0 s 1 μ + μ τ 0 s + 1 max − y τ 0 s + 1 max − x

(26)

where τ0 is a time constant, which is chosen by simulation experience to get a relatively good result.

5. Simulation and analysis To verify the deduced FRF and frequency domain data fusion proposed in this paper, a bicycle vehicle model which integrated the dynamic tire model, AFS discussed in System modeling and the in-wheel motor drive system introduced in [13] was built in Matlab/Simulink (Table 1). In this simulation, the longitudinal tire driving force was set as noise around zero in order to keep a constant velocity. The steering torque input by the driver can be assumed as zero in the case that the driver does not hold the steering wheel. The motor torque demand was zero added a kind of band-limited white noise, applied at the beginning, to fulfill the precondition of this proposed method. But it is not said that this method only can be used in the case that the motor torque is zero. The natural frequency of the steering system is inherent, not influenced be the value of motor torque. The motor current feedback values were shown as Fig. 5(a). The TRFC was 0.8 in the time range 0–9 s, and 0.2 in 9–18 s. The simulation results were shown as Fig. 5(b)–(d). Fig. 5(b)–(d) shows that there was noise in different signals. Even though the motor current was a same noise in the whole time range, it can be seen clearly in Fig. 5(b)–(d) that the oscillation cycle was different before and after the time point 9 s, which illustrates the TRFC has a strong impact on the steering system dynamic response and the deduction procedure is reasonable. The sideslip angle and the aligning toque were rather small, which fulfilled the assumption of natural frequency function. In this situation, the states such as the sideslip angle are very difficult to estimate. In order to clearly show the inapplicability of existing methods, the simulated “true” values of the tire sideslip angle were used in slope-Kalman based estimation

186

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

Table 1 Main parameters of the simulation model. Parameters

Value

Vehicle mass (kg) Wheel base (mm) Load ratio (front axle: rear axle) Yaw inertia(kg m2) ry (m)

1274 2578 7:13 1523 0.05

Longitudinal relaxation length rx (m) vx (m/s) The wheel radius(m) The wheel rotation inertia(kg m2) Gs (dimensionless) Gm (dimensionless) K (Nm/A) a μmax − y (rad/Nm)

0.5 15 0.3 1 15.9 4 4 0.0002

b μmax − y (dimensionless)

0

Jd (kg m2) Bd (Nm s/deg) τ0 (s)

0.03 0.5

0.1

Reference value Slope-based TRFC NF-based TFRC without signal noise NF-based TFRC with signal noise

1

TRFC (-)

0.8 0.6 0.4 0.2 0

0

3

6

9 Time (s)

12

15

18

Fig. 6. Estimated lateral friction coefficient of different approaches.

process, instead of the estimated sideslip angle. Due to the robustness of this method to the signal noise already discussed in [13], the signals(steering wheel rotation speed and motor current) shown in Fig. 5 can be directly used in the NF finder. The friction coefficient estimation results of the well-known slope-based method discussed in [10] and the proposed steering system natural-frequency-based approach are compared under different signals, shown as Fig. 6. Fig. 6 shows the comparison results. The solid line is the reference value. The dotted line is the slope-based one. Even though there was no error in the sideslip angle estimation process, the result of the traditional method cannot coincide with the reference value well because the dynamic property of tire is neglected. The dashed line is the estimation result of the proposed NF-based method without signal noise and the dot-dashed one is with signal noise. For most of the time interval, these two lines coincided together with the reference value, which illustrates the NF-based method's accuracy and robustness against the signal noise. The convergence time of NF-based approach was 2 s under high-friction road condition, and 3 s under low-friction road condition. This is related to the difference of the oscillation cycle. A longer oscillation cycle will result in the increment of the convergence time for frequency domain estimation. It is the reason why the data fusion should be introduced to improve the performance of the NF-based lateral TRFC method. The same current demand was applied to the in-wheel motor as Fig. 5(a). The wheel rotation speed was obtained as Fig. 7. Fig. 7(a) shows the wheel rotation speed with noise, which was used in the following NF-based longitudinal TRFC estimation. To clearly reflect the impact of TRFC on the wheel rotation speed, the true value without noise was shown as Fig. 7(b). It can be seen clearly that the signal before 9 s was more compressed than the one after 9 s. The oscillation cycle of the in-wheel motor system in Fig. 7(b) was rather shorter than the one of the AFS in Fig. 5(b), which verified the discussion in the sub-section Frequency domain data fusion. The advantage of the NF-based longitudinal TRFC estimation method had been introduced in [13]. Thus in this paper, only the comparison results of single direction TRFC and the combined one which was obtained by frequency domain data fusion are discussed, shown as Fig. 8.

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

187

Wheel rotation speed (rad/s)

52

51

50

49

48

0

3

6

9 Time (s)

12

15

18

0

3

6

9 Time (s)

12

15

18

Wheel rotation speed (rad/s)

52

51

50

49

48

Fig. 7. Wheel rotation speed.

1

Reference value NF-based lateral TFRC NF-based longitudinal TFRC Combined TFRC

TRFC (-)

0.8 0.6 0.4 0.2 0

0

3

6

9 Time (s)

12

15

18

Fig. 8. Estimated friction coefficient of different approaches.

The dashed line is the result of the NF-based longitudinal TRFC. It can quickly track the reference value change, but the overshoot and the oscillation due to the signal noise are relatively severe. Among these three estimation curves, the dotted one combined the respective advantages of others. It can stably follow the true value like the lateral TRFC during the interval 3–9 s and 12–18 s, and exactly reflect the change like the longitudinal one during the interval 0–3 s and 9–12 s. Although the convergence time of the combined one is 1.5 s, still longer than the best convergence time of the literature that studies the large lateral acceleration case, the result of the proposed method can be beneficial to the feedforword vehicle stability control. In this simulation section, the results about the frequency domain data fusion and the natural-frequency-based lateral friction estimation were presented. The correctness of the FRF deduction was further verified by these results. The NF-based advantage about the robustness to the signal noise was shown through the comparison to the slope-Kalman-based method. Using frequency domain data fusion, the estimated combined TRFC can achieve quick response to the changing of road conditions in high frequency domain and stability to the signal noise in lower frequency domain.

188

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

6. Experiment and analysis The experiment of the NF-based longitudinal TRFC estimation method had been discussed in [13]. Because the in-wheel motor controllers of the experimental vehicle which was equipped an AFS system cannot fulfill the quick response requirement of the frequency domain data fusion, this section only validates the frequency response of the steering system and the NF-based lateral TRFC estimation method. The experimental vehicle is shown in Fig. 9, whose main parameters are listed in Table 2. The measurements including the lateral and longitudinal speeds of the vehicle were obtained from the RT3100 system. Information on the steering wheel rotation speed was obtained by a steering-wheel angle sensor which was mounted on the steering column. Due to the limited test site, the track could only consist of one kind of surface with a same friction level. Snow surface and cement surface were separately paved on the same track. Trials were carried out under the following conditions, for the repeatability of the experiment and safety reasons.

 The initial vehicle speed was zero.  The driver tried to keep the vehicle moving longitudinally, and did not hold the steering wheel.  The vehicle was accelerated slowly and steadily. 6.1. TRFC's impact on the natural frequency The values of the equivalent rotation inertia and the gear ratio of the steering system, due to the limitations of the apparatus, cannot be measured. Because only the case that the vehicle moves straight at small acceleration is considered, these system parameters can be treated as constants. So, by simplifying Eq. (25), the relationship between the natural frequency and the lateral TRFC will be shown as Eq. (27).

f 02 = c μmax μ max + d μmax

(27)

where c μmax and d μmax are constants. The NFs for different surfaces were estimated to determine the values of c μmax and d μmax , by the NF finder using the data collected in the experiment, shown in Figs. 10 and 11.

Fig. 9. Experimental vehicle.

Table 2 Main parameters of the experimental vehicle. Parameters

Symbol

Values

Vehicle mass Height of c.g. above ground Distance from front axle to CoG.

m (kg) h (mm) l f (mm)

520 40 1254

Distance from rear axle to CoG. Moment of inertia around z axis Nominal torque Wheelbase Radius of the tire

lr (mm) Iz (kg/m2) Te (N m) B (mm) R (mm)

836 1110 350 1500 280

8

1

6

0.5 Yawrate (rad/s)

Longitudinal velocity (m/s)

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

4

0

-0.5

2

0

189

0

2

4

6

8

-1

10

0

Time (s)

2

4

6

8

10

6

8

10

Time (s)

10

2

6

Motor current (A)

Lateral velocity (m/s)

8 1

0

4 2 0

-1

-2 -4

-2 0

2

4

6

8

0

10

2

4 Time (s)

Time (s)

Steering rotation speed (rad/s)

1.5 1 0.5 0 -0.5 -1 -1.5 0

2

4

6

8

10

Time (s)

Fig. 10. Data collected on the snow surface.

The vehicle kinematic variables were shown in Fig. 10(a)–(c) and Fig. 11(a)–(c). The vehicle was accelerated slowly and steadily. The lateral motion was quite small, which could ensure that the sideslip angle at CoG was around zero. Due to the slight steering operation, the yaw rate fluctuated around zero, indicated that the steering angle of the front axle was also small. All these variables further guaranteed that the sideslip angle of the front axle was around zero, which met the precondition of the NF-based TRFC estimation method. The natural frequency of the steering system was estimated by Natural frequency finder using the steering rotation speed and the assist motor current data, shown as Fig. 12. Regarding empirical knowledge, the road friction coefficients of the compacted snow and cement are respectively 0.2 and 0.8. It was shown in Fig. 12 that the natural frequencies of a certain steering system were not the same on different kinds of road surface, which verified the effect of TRFC on the natural frequency. The value of the snow surface was 1.6 Hz, whereas, the one of the cement road was 2.0 Hz. Then, c μmax and d μmax can be determined by solving Eq. (27) with this frequency information as following.

cμ

max

= 2.4; dμ

max

= 2.08

190

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

6

0.06

0.02

4

Yawrate (rad/s)

Longitudinal velocity (m/s)

0.04

2

0 -0.02 -0.04 -0.06

0

0

2

4

6

8

-0.08

10

0

2

4

Time (s)

6

8

10

6

8

10

Time (s)

6

2

Motor current (A)

Lateral velocity (m/s)

4 1

0

2 0 -2

-1 -4 -2

0

2

4

6

8

-6

10

0

2

4 Time (s)

Time (s)

Steering rotation speed (rad/s)

0.5

0

-0.5

-1 0

2

4

6

8

10

Time (s)

Fig. 11. Data collected on the cement surface.

6.2. Estimation of the lateral TRFC based on the natural frequency Based on the estimated NF, and the deduced c μmax , d μmax , the TRFC was obtained by the RLS-based road friction estimator, as shown in Fig. 13. Fig. 13(a) shows that the estimated TRFC of the snow road, after 2 s′ delay, fluctuated between 0.17 and 0.25, around the corresponding reference value. Meanwhile, the estimated value of the cement road fluctuated approximately 0.8. So this proposed method can accurately distinguish different kinds of road that the wheel is rolling on. Figs. 10(a) and 11(a) show that, for the whole interval between 0 and 10 s, the vehicle was accelerated slowly. Therefore, the estimated NF in Fig. 12 verified that the correctness of the NF function from the point that NF were not affected by the longitudinal velocity. Furthermore, the robustness of the proposed approach was validated for the different velocities by the results in Fig. 13.

7. Conclusions This paper proposes a TRFC estimation method based on frequency domain data fusion under the conditions with small acceleration. It fuses the NF-based estimated tire-road friction coefficient in each direction through first order filters. The

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

Fig. 12. Estimated natural frequency on different surfaces. 1 Estimated TRFC Reference value

TRFC (-)

0.8

0.6

0.4

0.2

0

0

2

4

6

8

10

Time (s)

1 Estimated TRFC Reference value

TRFC (-)

0.8

0.6

0.4

0.2

0

0

2

4

6 Time (s)

Fig. 13. Estimated TRFC.

8

10

191

192

L. Chen et al. / Mechanical Systems and Signal Processing 85 (2017) 177–192

equation of the steering system natural frequency, as the lateral theoretical foundation, is deduced. The correctness of this equation and advantages of the total estimation scheme is validated by the simulation and the experimental results on different road conditions. The contributions of this paper can be drawn as follows:

 A frequency response function of the steering system is obtained, which can provide insight into the steering dynamics in the frequency domain for straight driving cases with a small sideslip angle.

 Without estimating the sideslip angle and the self-aligning torque, the NF-based TRFC estimation approach only uses the 

signals of the assist motor current and the steering wheel rotation speed, which is not sensitive to the sensor noise and robust for different longitudinal velocities. Using frequency domain data fusion, the estimated combined TRFC can achieve quick response to the changing of road conditions in high frequency domain and stability to the signal noise in lower frequency domain.

Acknowledgments The authors are grateful for the sponsorship from the National Natural Science Foundation of China (Grant No. 51575295) and the National Key Research and Development Program of China (2016YFB0100905).

References [1] Y. Dai, Y. Luo, W. Chu, et al., Optimum tyre force distribution for four-wheel-independent drive electric vehicle with active front steering, Int. J. Veh. Des. 65 (4) (2014) 336–359. [2] J. Ni, J. Hu, Handling performance control for hybrid 8-wheel-drive vehicle and simulation verification, Veh. Syst. Dyn. 54 (8) (2016) 1098–1119. [3] M. Satoshi, Innovation by in-wheel-motor drive unit, Veh. Syst. Dyn. 50 (2012) 807–830. [4] W. Chu, State Estimation and coordinated Control for Distributed Electric Vehicles, Berlin, 2015. [5] S.Y. Ko, J.W. Ko, J.S. Lee SM Cheon, et al., Vehicle velocity estimation using effective inertia for an in-wheel electric vehicle, Int. J. Auto. Technol. 15 (5) (2014) 815–821. [6] R. Rajamani, D. Piyabongkarn, J.Y. Lew, J.A. Grogg, Algorithms for real-time estimation of individual wheel tire-road friction coefficients. In: Proceedings of the 2006 American Control Conference, 2006. [7] R. Rajamani, Vehicle Dynamics and Control, New York, 2005. [8] G. Baffeta, A. Chararaa, D. Lechner, Estimation of vehicle sideslip, tire force and wheel cornering stiffness, Control Eng. Pract. 17 (11) (2009) 1255–1264. [9] K. Yi, K. Hedrick, S.C. Lee, Estimation of tire-road friction using observer based identifiers, Veh. Syst. Dyn. 31 (1999) 233–261. [10] F. Gustafsson, Slip-based tire-road friction estimation, Automatica 33 (6) (1997) 1087–1099. [11] M. Bartram, G. Mavros, S. Biggs, A study on the effect of road friction on driveline vibrations, Proc. IMechE Part K: J. Multi-Body Dyn. 224 (4) (2010) 321–340. [12] T. Umeno, et al., Observer based estimation of parameter variations and its application to tyre pressure diagnosis, Control Eng. Pr. 9 (6) (2001) 639–645. [13] L. Chen, et al., Tire–road friction coefficient estimation based on the resonance frequency of in-wheel motor drive system, Veh. Syst. Dyn. 54 (1) (2016) 1–19. [14] F. Meng, Zhang Hui, Cao Dongpu, Chen Huiyan, System modeling, coupling analysis, and experimental validation of a proportional pressure valve with pulsewidth modulation control, IEEE/ASME Trans. Mechatron. 21 (3) (2016) 1742–1753. [15] S. Hong, G. Erdogan, K. Hedrick, F. Borrelli, Tyre–road friction coefficient estimation based on tyre sensors and lateral tyre deflection: modeling, simulations and experiments, Veh. Syst. Dyn. 51 (5) (2013) 627–647. [16] R. Wang, J. Wang, Tire–road friction coefficient and tire cornering stiffness estimation based on longitudinal tire force difference generation, Control Eng. Pr. 21 (2013) 65–75. [17] R. Wang, C. Hu, Z. Wang, et al., Integrated optimal dynamics control of 4WD4WS electric ground vehicle with tire-road frictional coefficient estimation, Mech. Syst. Signal Process. 60 (2015) 727–741. [18] Matsuda T, Jo S, Nishira H, Deguchi Y. Instantaneous estimation of road friction based on front tire SAT using Kalman filter. Sae Paper 2013-01-0680, 2013. [19] L. Li, K. Yang, G. Jia, et al., Comprehensive tire–road friction coefficient estimation based on signal fusion method under complex maneuvering operations, Mech. Syst. Signal Process. 56 (2015) 259–276. [20] L. Chen, M. Bian, Y. Luo, K. Li, Real-time identification of the tyre-road friction coefficient using an unscented Kalman filter and mean-square-errorweighted fusion, Proc. Inst. Mech. Eng. Part D: J. Automob. Eng. 230 (6) (2015) 788–802. [21] Y. Dai, Integrated Longitudinal and Lateral Motion Control of Distributed Electric Vehicles (PhD Thesis), Tsinghua University, China, 2013. [22] HB. Pacejka, Tire and Vehicle Dynamics, SAE, Warrendale, PA, USA, 2002. [23] D. Piyabongkarn, R. Rajamani, J.A. Grogg, et al., Development and experimental evaluation of a slip angle estimator for vehicle stability control, IEEE Trans. Control Syst. Technol. 17 (1) (2009) 78–88.