Tire-Road Friction Coefficient Estimation based on Tire Sensors and Lateral Tire Deflection: Modeling, Simulations and Experiments

Tire-Road Friction Coefficient Estimation based on Tire Sensors and Lateral Tire Deflection: Modeling, Simulations and Experiments Sanghyun Hong, Gurk...
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Tire-Road Friction Coefficient Estimation based on Tire Sensors and Lateral Tire Deflection: Modeling, Simulations and Experiments Sanghyun Hong, Gurkan Erdogan, Karl Hedrick, Francesco Borrelli 

Abstract—Estimation of the tire-road friction coefficient is fundamental for vehicle control systems. Tire sensors enable the friction coefficient estimation based on signals extracted directly from tires. This paper presents a tire-road friction coefficient estimation algorithm based on tire lateral deflection obtained from lateral acceleration. The lateral acceleration is measured by wireless 3D accelerometers embedded inside the tires. The proposed algorithm first determines the contact patch using a radial acceleration profile. Then, the portion of the lateral acceleration profile, only inside the tire-road contact patch, is used to estimate the friction coefficient through a tire-brush model and a simple tire model. The proposed strategy accounts for orientation-variation of accelerometer body frame during tire rotation. The effectiveness and performance of the algorithm is demonstrated through finite element model simulations and experimental tests with small tire slip angles on different road surface conditions. Keywords: tire-road friction coefficient, tire-road contact patch, lateral deflection model, in-tire accelerometer, tire brush model

1.

T

INTRODUCTION

ire-road friction force plays a key role in maintaining the stability and the controllability of vehicle dynamic motion. The estimation of the tire-road friction coefficient is of fundamental importance in every active safety system and has been studied for more than four decades. An excellent review can be found in [2, 12, 13]. Many of the existing methods in the industry use only measurements from the vehicle sensors and cannot be applied to steady state driving conditions, e.g. traveling ahead at a constant vehicle speed. Unsteady state driving conditions, i.e. acceleration or deceleration, are necessary to identify the tire-road friction coefficient [1-3, 12-15]. This paper focuses on a paradigm-shift which uses a sensor embedded in the tire [7, 10,16-18] and can work in steady state driving conditions. This paper uses the idea of the friction coefficient estimation based on lateral deflection profiles proposed in [6]. Lateral direction here refers to the tire axis perpendicular to the wheel vertical plane on which the tire longitudinal axis lies. The tire has small stiffness in the lateral direction, and this is the basis of the proposed approach. In [5] we have demonstrated the potential of the friction coefficient estimation using the lateral acceleration with finite element model (FEM) simulations. This paper builds on the algorithm proposed in [5] and explains the changes to the models and the approach required for its “real world” application. “Real world” here refers to the use of actual measurements from wireless 3D accelerometers in the estimation algorithm. In this case, of particular importance is the issue of orientation-variation of accelerometer body frame which invalidates our approach presented in [5]. We propose a new estimation algorithm which first determines the contact patch and then uses the acceleration measurements only the inside of the contact patch. A corresponding simplified lateral deflection model is presented and discussed in Section 2.3. A simple optimization problem uses the simplified model and the measured data to estimate the curvature of the lateral deflection profile inside the contact patch. The lateral force and the aligning moment are calculated based on the estimated curvature and the pneumatic trail relationship. In the last section of the paper the effectiveness and performance of the proposed approach is demonstrated through experimental tests with small tire slip angles on asphalt and ice-covered roads. 2.

FRICTION COEFFICIENT ESTIMATION USING MEASUREMENT FROM ACCELEROMETER ON TEST VEHICLE

This section presents the new friction coefficient estimation algorithm. Before proceeding further we first briefly list the main steps of the algorithm proposed in [5]. ALGORITHM 1

Tire-road friction coefficient estimation

This work is supported by the National Science Foundation under Grant No. 1239323 and Tyre Systems & Vehicle Dynamics Division of Pirelli Tyres SpA. Sanghyun Hong is with University of California at Berkeley, CA 94720-1740 USA (phone: 510-642-6933; e-mail: [email protected]). Gurkan Erdogan is with University of California at Berkeley, CA 94720-1740 USA (phone: 612-751-1410; e-mail: [email protected]). Karl Hedrick is with University of California at Berkeley, CA 94720-1740 USA (phone: 510-642-2482; e-mail: [email protected]). Francesco Borrelli (corresponding author) is with University of California at Berkeley, CA 94720-1740 USA (phone: 510-316-6925; e-mail: [email protected]).

Input: Lateral acceleration during tire rotation Output: Tire-road friction coefficient µ Begin 1: De-trend the lateral acceleration profile by subtracting its mean value. 2: Double-integrate the lateral acceleration profile to obtain the lateral deflection profile. 3: Estimate 𝛼 (∶= −

) and 𝛽 (∶=

) of a

parabolic lateral deflection model 𝑦 = 𝛼𝑥 + 𝛽𝑥 + 𝛾 [4] by using curve-fitting technique, where y is the lateral deflection and x the coordinates in the tire longitudinal direction. 4: Estimate the lateral force Fy and the aligning moment Mz: 𝛼 𝐹 = −2𝑐 𝑀 =𝑐 𝛽 and 𝑐 are where the stiffness parameters 𝑐 assumed to be known. 5: Estimate the tire-road friction coefficient µ by using a tire brush model and the estimated Fy and Mz. End Although we keep the basic approach of Algorithm 1 (use of simple physics-based models and estimate the parameters from measured lateral acceleration), the new algorithm uses the different lateral deflection model, acceleration signals and estimation technique. We will show that the proposed changes are fundamental in order to succeed in the friction coefficient estimation during experiments. In particular, the new algorithm (i) first determines the tire-road contact patch, (ii) uses the contact patch to filter the lateral acceleration profile in order to be robust to orientation-variation of accelerometer body frame, and (iii) uses a new form of lateral deflection model which leads to a different estimation of Mz than Algorithm 1. All the steps are detailed next. 2.1. Determination of Tire-Road Contact Patch Interaction between the tire and the road occurs at the tire-road contact patch. The tire-road contact patch is defined as the region of the tire in contact with the road [8]. Next we propose the use of a radial acceleration profile to determine the tire-road contact patch. The proposed approach is different than ones proposed in [7] and [10] which utilize zero, maximum and minimum radial acceleration values. Later in this paper the contact patch estimation will be used to improve the robustness of the tire-road friction coefficient estimation algorithm. The contact patch on the tire circumferential center line is identified by determining the locations of leading and trailing edges. The leading and trailing edges are the foremost and rearmost edges of the contact patch, respectively. In order to determine the locations of the leading and trailing edges, the side view of a simple tire model in Figure 1 is investigated. The positions of the accelerometer at two different time instants are depicted as dots. An angular position 𝜃 starts from the top of the tire, while another angular position ∅ starts from the center of the contact patch; 𝜃 = 𝜋 − ∅.

Figure 1. Simple Tire Model (STM)

We use the following assumptions for the simple tire model above. Assumption 1: The deformation of a tire material element on the circumferential line does not affect a neighboring material element. As a result, both the leading (x1) and trailing (xt) edges have sharp corners and the outside of the contact patch has perfect circular shape, i.e. a constant radius r. Assumption 2: The accelerometer, attached on the center line of the tire inner liner, revolves around the wheel-center at a constant angular velocity 𝜃̇. The kinematics of the accelerometer is investigated based on the simple tire model with above assumptions. Outside the contact patch, the radial acceleration 𝑎 , is the centripetal acceleration. It is a negative constant value since the radial axis is ( ⃗ ) = 𝑟̈ − 𝑟𝜃̇ 𝑒⃗ + 𝑟𝜃̈ + 2𝑟̇ 𝜃̇ 𝑒⃗ , where oriented outwards. Its kinematics is the radial component, i.e. 𝑟̈ − 𝑟𝜃̇ , of 𝑟⃗̈ = 𝑒⃗ and 𝑒⃗ are the unit vectors in the radial and tangential directions, respectively. Due to the assumption of the constant radius r, the radial acceleration outside the contact patch 𝑎 , is: 𝑎

,

= −𝑟𝜃̇

(1)

Inside the contact patch, the radius r2 is a function of the angular position 𝜃. It is 𝑟 = −𝑟 𝑠𝑒𝑐𝜃, where rl is a constant “loaded radius” of the tire in Figure 1, i.e. distance between the wheel-center and the center of the contact patch. The radial acceleration inside the contact patch 𝑎 , is the radial component of the second derivative of 𝑟⃗ = 𝑟 𝑒⃗ , i.e. 𝑟⃗̈ = 𝑟̈ − 𝑟 𝜃̇ 𝑒⃗ + 𝑟 𝜃̈ + 2𝑟̇ 𝜃̇ 𝑒⃗ . Since 𝑟̈ = −𝑟 𝜃̇ 𝑠𝑒𝑐𝜃(𝑡𝑎𝑛 𝜃 + 𝑠𝑒𝑐 𝜃), we can derive the radial acceleration 𝑎 , : 𝑎

,

= −2𝑟 𝜃̇ 𝑠𝑒𝑐𝜃𝑡𝑎𝑛 𝜃

(2)

Plotting 𝑎 , and 𝑎 , against the angular position 𝜃 shows that the radial acceleration for one tire rotation has an impulse-like shape. Figure 2 plots a radial acceleration, i.e. 𝑎 , in Equation 1 and 𝑎 , in Equation 2, against the angular position 𝜃 ranging from 0o through 360o. The radial acceleration in Figure 2 corresponds to a simulation where the angular velocity 𝜃̇ is set to 33.9685 𝑟𝑎𝑑/𝑠, the angular position at the leading edge ∅ to 15o, the unloaded radius 𝑟 to 0.3322 𝑚, and the loaded radius 𝑟 to 0.3222 𝑚.

Figure 2. [STM] Radial Acceleration from Simple Tire Model

In Figure 2, one can notice that the radial acceleration inside the contact patch has a convex shape. The convex shape is also

observed in the radial acceleration profiles collected from the experiments. Figure 3 depicts a set of measured radial acceleration profiles (details on the experimental setup will be given later in this paper). Note that the experimental radial acceleration profiles are not identical to those in Figure 2 since the assumptions in the simple tire model do not hold in a real tire.

Figure 3. [Exp.] Radial Acceleration from an Experiment

However, similarly to our oversimplified model in Figure 2, an abrupt change in the radial acceleration occurs at the leading and trailing edges. This implies that the derivative of the radial acceleration, i.e. radial jerk, has two large absolute values at the leading and trailing edges. We propose to locate the two large absolute values in the derivative of the radial acceleration (and thus locate the leading and trailing edges) in order to estimate the contact patch. Figure 4 shows an example of a tire FEM simulation. The solid line in Figure 4-(a) is the radial acceleration profile, and the dash line in Figure 4-(b) the magnitude of the derivative. The leading and trailing edges are estimated by the positions of the two peaks in Figure 4-(b). In the tire finite element model (FEM), the tire rolls freely on a flat road surface with a given tire slip angle. All details on the tire FEM model can be found in [5] and for the sake of brevity we report here only the most important features. We model a rubber body, an inner-liner, two layers of belts and a rigid bead. The geometry of all the tire components is symmetric with respect to the wheel vertical plane. The rubber body is modeled as the rubber material that have both hyperelastic and viscoelastic properties. All the material types and element types used in each tire component are the same as those presented in [5]. The rolling simulation setting on a flat road surface is also identical to the one in [5]. In particular we use the same tire inflation, tire normal load and translational velocity. There is only one difference from [5]. In the simulation reported in this paper we use a mesh density uniformly distributed every 1o throughout the tire circumference. The tire FEM in [5] uses the reduced mesh density in the upper half of the tire. The uniform mesh density helps to build a more realistic setup for the accelerometer with a constant sampling frequency. Note that the radial jerk is used to estimate the contact patch rather than the radial acceleration itself. In fact, the two peaks of the simple tire model in Figure 2 might not be clearly observed in the real tire, and therefore the localization of them might not be easy (notice the actual shape of experiment in Figure 3 and FEM simulation in Figure 4-(a)). However, the two peaks in the magnitude of the radial jerk are clearly identifiable as shown in Figure 4-(b).

(a) Radial Acceleration

(b) Magnitude of Derivative of Radial Acceleration Figure 4. [FEM] Radial Acceleration and Magnitude of its Derivative

The effectiveness of the proposed methodology is demonstrated through the tire FEM simulations. We compare the proposed approach with the approach using contact pressures on elements of the tire FEM to estimate the leading and trailing edges. The contact pressure values are reported from the FEM simulation software. Angular positions corresponding to the leading and trailing edges estimated by both methodologies are summarized in Table 1. Table 1 Angular positions at leading and trailing edges Normal Force

Leading Edge (Proposed / Pressure)

Trailing Edge (Proposed / Pressure)

3800 N 4500 N 6000 N

173o / 172o 172o / 171o 170o / 169 o

187 o / 188o 188 o / 189o 191 o / 192o

2.2. Orientation-Variation of Accelerometer Body Frame The lateral deflection profile is obtained through the double-integration of the lateral acceleration profile. However, the lateral axis of the accelerometer body frame is not always perpendicular to the wheel vertical plane during one full tire rotation. Therefore, the measured lateral acceleration is different from the actual lateral acceleration unlike the assumption in [5]. Orientation of the accelerometer body frame varies with the tire slip angle. The accelerometer follows the tire circumferential inner center line as the tire rotates. If a certain tire slip angle is applied on the tire, a friction force is generated inside the contact patch. Then, the tire deforms in the direction of the friction force vector. The tire inner center line also deforms in the same direction. Thus, the orientation of the accelerometer body frame continues to vary as the accelerometer follows the deformed circumferential inner center line. Figure 5-(b) shows a lateral deflection profile from bottom view of the tire FEM in Figure 5-(a). The tire FEM moves in the direction of the velocity V at a tire slip angle α = 2.0o. The angular position in Figure 5-(b) starts from and ends at the top position of the tire. As shown in Figure 5-(b), the lateral direction (y) of the accelerometer body frame is not parallel to the actual lateral direction (η) which is perpendicular to the wheel vertical plane. Note that the orientation-variation of the accelerometer body frame occurs in the entire 3-dimensional space, while Figure 5-(b) only presents the bottom view.

(a) [FEM] Tire Finite Element Model

(b) [FEM] Lateral Deflection Figure 5. Orientation-Variation of Accelerometer Body Frame

Due to the orientation-variation, the accelerometer does not measure the lateral acceleration desired by Algorithm 1. Figure 6-(a) shows two lateral acceleration profiles of the tire FEM. Dash line is an acceleration profile measured by the lateral axis of the wheel-center frame, i.e. axis η in Figure 5-(a), and solid line is that measured by the lateral axis of the accelerometer body frame, i.e. axis y in Figure 5-(a). The wheel-center frame rotates as its lateral axis (η) keeps pointing at the direction perpendicular to the wheel vertical plane, i.e. actual lateral direction. As shown in Figure 6-(a), the lateral acceleration measured by the accelerometer, i.e. solid line, is different from the actual one, i.e. dash line.

(a) [FEM] Lateral Acceleration with respect to Wheel-Center and Accelerometer Body Frames

(b) [Exp.] Lateral Acceleration Inside Contact Patch Figure 6. Lateral Acceleration

Inside the contact patch, however, the orientation-variation of the accelerometer body frame is negligible. As shown in Figure 6-(a), there is little difference between both acceleration profiles inside the contact patch. In Figure 6-(a), L indicates the leading edge and T the trailing edge. Therefore, this paper uses the lateral acceleration profile only inside the contact patch to estimate the friction coefficient. Before the double-integration to obtain the lateral deflection, the lateral accelerations are set to zero outside the contact patch as shown in Figure 6-(b). Note that, differently than the friction coefficient estimation algorithm proposed in [5], the mean value of the lateral acceleration is not subtracted. 2.3. New Form of Lateral Deflection Model In [4] the lateral deflection profile inside the contact patch is modeled as a parabolic polynomial: 𝑦 = 𝛼𝑥 + 𝛽𝑥 + 𝛾 𝛼 ∶= −

𝐹 2𝑐

, 𝛽 ∶=

𝐹 𝑀 , 𝛾 ∶= 𝑐 𝑐

(3)

However, since we want to use the lateral accelertion only inside the contact patch, this model has to be modified. Next we explain how. 2.3.1. Lateral Deflection Produced by Lateral Acceleration Only Inside Contact Patch Figure 7 depicts an actual lateral deflection, i.e. dash-dot line, which can be obtained from the lateral acceleration measured with “no” the orientation-variation of the accelerometer body frame. The lateral deflection produced by the lateral acceleration only inside the contact patch is a solid line in Figure 7. This results from the zero lateral acceleration outside the contact patch and zero boundary conditions for double-integration at the leading edge (time 𝑡 = 𝑡 ), i.e. zero lateral velocity 𝑣 (𝑡 ) = 0 and zero lateral deflection 𝑦 = 0.

Figure 7. [Exp.] Produced Lateral Deflection Inside Contact Patch

In order to describe the lateral acceleration, we will use the differential form of the parabolic lateral deflection model, i.e. the double-derivative of Equation 3:

𝑑𝑥 𝑑 𝑦 = 2𝛼 𝑑𝑡 𝑑𝑡

+ (2𝛼𝑥 + 𝛽)

𝑑 𝑥 𝑑𝑡

(4)

2.3.2. Classification of Lateral Acceleration Profiles The lateral acceleration measured on a high friction road, e.g. asphalt road, will be handled differently for the double-integration than that on a low friction road, e.g. ice road. A simple criteria will be established to judge roughly whether or not the lateral acceleration is measured on a road with high friction coefficient. The criteria stems from the following observation: a small lateral deflection is generated inside the contact patch even at zero tire slip angle. Figure 8-(a) presents the lateral deflections from bottom view of the tire FEM in different friction coefficient μ and tire slip angle α conditions. This changes of the lateral deflections at different friction coefficient and tire slip angle conditions has been also experimentally validated in [6]. The contact patches of the lateral deflections in Figure 8-(a) are magnified in Figure 8-(b), 8-(c) and 8-(d), respectively. In Figure 8-(b), the lateral deflection profile at “zero” tire slip angle has a certain type of curve rather than a straight line parallel to the longitudinal axis. The tire deforms in the vertical direction due to a normal load, and it results in “compressive” force inside the contact patch [11]. This creates the small lateral deflection inside the contact patch even at the zero tire slip angle. Note that this kind of lateral deflection is different from the lateral deflection by “friction” force at a “non-zero” tire slip angle. The lateral deflection induced by the compressive force inside the contact patch depends on many tire properties. For instance tire manufacturing elements, such as a radial body ply and a bias ply, affects the tire stiffness in the vertical direction, and can result in the anisotropic characteristic of the stiffness. Different tire tread design can also lead to different lateral deflection at the zero tire slip angle. Furthermore, the compressive lateral deflection is affected by the hyperelasticity of the individual tire rubber body. This is also captured by our tire FEM model. The hyperelasticity prevents the tire circumferential line from compressing. As a result, the center line deforms under the effect of the compressive force. The compressive component of the lateral deflection is a clue to the rough classification of the lateral accelerations. On a low friction road, e.g. ice road, the lateral deflection by the compressive force still remains dominantly even at a “non-zero” tire slip angle as dash-dot line in Figure 8-(c). The dash-dot line is a lateral deflection profile of the tire FEM at a tire slip angle α = 2.0o on a road with a friction coefficient μ = 0.1. However, on a high friction road, e.g. asphalt, the compressive lateral deflection component vanishes and the frictional lateral deflection component is dominant as solid line in Figure 8-(d). The solid line is a lateral deflection profile at a tire slip angle α = 2.0o on a road with a friction coefficient μ = 0.9.

(a) Lateral Deflections

(b) Friction Coefficient μ: 0.1, Tire Slip Angle α: 0.0o

(c) Friction Coefficient μ: 0.1, Tire Slip Angle α: 2.0o

(d) Friction Coefficient μ: 0.9, Tire Slip Angle α: 2.0o Figure 8. [FEM] Lateral Deflections by Compressive Force and Friction Force

Therefore, the lateral accelerations will be classifed into two categories: Category 1 - lateral accelerations “without compressive lateral deflection component” and Category 2 - lateral accelerations “with compressive lateral deflection component”. The reason for classifying the lateral accelerations (rather than deflection) is that the lateral acceleration in category 1 might include a random and persistent acceleration. Figure 9 presents experimental examples of lateral accelerations only inside the contact patch on the asphalt road. Although all lateral acceleration profiles in Figure 9 have similar dominant curves, those are shifted by random and constant accelerations (the shifts are exaggerated for visibility). It is presumed that the random and constant shifting is a consequence of the tire deformation; this phenomenon is under investigation. The random shift at each tire rotation does not influence the fundamental profile of the lateral acceleration as observed in Figure 9. However, when double-integrating, as required by the next steps of the friction coefficient estimation algorithm, the effect will be amplified. This will be illustrated in the Experiment Results section.

Figure 9. [Exp.] Shifted Lateral Accelerations Inside Contact Patch

According to the simple tire model in Figure 1, the longitudinal acceleration, i.e. 𝑎 =

within Equation 4, can be

approximated as the negated tangential acceleration inside the contact patch, i.e −𝑎 . The tangential acceleration 𝑎 is the coefficient of 𝑒⃗ in 𝑟⃗̈ = 𝑟̈ − 𝑟 𝜃̇ 𝑒⃗ + 𝑟 𝜃̈ + 2𝑟̇ 𝜃̇ 𝑒⃗ , and 𝑟̇ = −𝑟 𝜃̇𝑠𝑒𝑐𝜃𝑡𝑎𝑛𝜃. Therefore, the longitudinal acceleration 𝑎 under the assumption of 𝜃̈ = 0 is:

𝑎 ≈ −𝑎 = 2𝑟 𝜃̇ 𝑠𝑒𝑐𝜃𝑡𝑎𝑛𝜃

(5)

Note that the longitudinal acceleration is positive and decreases to zero as shown in Figure 10 over the leading region, i.e. from the leading edge to the center of the contact patch: 𝑎 ≥ 𝑥 ≥ 0. The longitudinal velocity 𝑣 is approximated as the negated tangential velocity 𝑣 . The tangential velocity 𝑣 is the coefficient of 𝑒⃗ in 𝑟⃗̇ = 𝑟̇ 𝑒⃗ + 𝑟 𝜃̇ 𝑒⃗ , where 𝑟 = −𝑟 𝑠𝑒𝑐𝜃. The longitudinal velocity is approximated as: 𝑣 ≈ −𝑣 = 𝑟 𝜃̇𝑠𝑒𝑐𝜃

(6)

Figure 10. [STM] Longitudinal Acceleration (𝑎 ≈ −𝑎 )

After substituting Equation 5 and 6 into Equation 4 we obtain the lateral acceleration as a fucntion of the angular position and the angular velocity. The lateral acceleration in Figure 11 depicts the case where its corresponding lateral deflection is on the negative side of the lateral axis as the solid line in Figure 8-(d) and has no compressive deflection component. Note that both α and β of the parabolic lateral deflection model in Equation 3 represent the curvature and the slope, respectively. Those have positive signs for the lateral deflection on the negative side of the lateral axis. In Figure 11, the angular velocity 𝜃̇ is set to 33.9685 𝑟𝑎𝑑/𝑠, the angular position at the leading edge ∅ to 15° , the unloaded radius 𝑟 to 0.3322 𝑚, the loaded radius 𝑟 to 0.3222 𝑚, 𝛼 to 3, and 𝛽 to 0.05. As shown in Figure 11, the lateral acceleration decreases over the leading region and has its maximum value at the leading edge.

Figure 11. Lateral Acceleration corresponding to Lateral Deflection without Component from Compressive Force

We use Figure 11 to establish the criteria to classify the lateral accelerations. The criteria is given in Table 2, where Acc(k) indicates the lateral acceleration at the kth point in Figure 11. If the lateral deflection is on the positive side of the lateral axis, the same criteria in Table 2 can be applied after inverting the lateral acceleration and deflection profiles. Table 2 Classification of Lateral Acceleration If Acc(1) > Acc(2) > Acc(3) AND Mean(Acc(2) : Acc(3)) > 0 Then Lateral Acceleration corresponding to Lateral Deflection without Deflection Component from Compressive Force

Else Lateral Acceleration corresponding to Lateral Deflection with Deflection Component from Compressive Force

2.3.3. Derivation of New Form of Lateral Deflection Model The classified lateral accelerations will be integrated twice to obtain the lateral deflections. As for the lateral acceleration in category 1, i.e. lateral acceleration corresponding to the lateral deflection without compressive component, the lateral acceleration model is reported in Equation 7. Note that it includes the shift term B(t) due to the random shift in the lateral acceleration. The shift term B(t) is assumed to be a constant value B. = 2𝛼

+ (2𝛼𝑥 + 𝛽)

+ 𝐵(𝑡)

(7)

The term B(t) is a random constant number during each tire rotation. We suggest to remove the term by subtracting the lateral acceleration at the center of the contact patch. Since the longitudinal acceleration at the center of the contact patch, i.e. (time t=tc) is zero as shown in Figure 10, the lateral acceleration at the center of the contact patch is Then, the term B is removed from the lateral acceleration in Equation 7 by subtracting

= 2𝛼

+ 𝐵.

from Equation 7 as reported in

Equation 8. −

= 2𝛼



+ (2𝛼𝑥 + 𝛽)

(8)

Equation 8 is integrated twice to obtain Equation 9. ∬



𝑑𝑡𝑑𝑡 = 𝛼 1 −

( )

(𝑥 − 𝑥 )

(9)

Note that the boundary conditions at the leading edge for the double-integration are 𝑣 (𝑡 ) = 0 and 𝑦 = 0 because of zero lateral acceleration outside the contact patch. In Equation 9 we introduce the average longitudinal velocity over the contact patch 𝑣̅ = . In Equation 9, 𝑣 (𝑡 ) is the longitudinal velocity at the center of the contact patch. As discussed previously, the longitudinal velocity 𝑣 is approximated as the negated tangential velocity, i.e. 𝑣 ≈ −𝑣 = 𝑟 𝜃̇𝑠𝑒𝑐𝜃. At the center of the contact patch (𝜃 = 𝜋) the longitudinal velocity 𝑣 (𝑡 ) is 𝑣 (𝑡 ) = −𝑟 𝜃̇

(10)

Note that the longitudinal speed at the center of the contact patch |𝑣 (𝑡 )| is the smallest inside the contact patch, but not close to zero. Also, the difference is not large between the longitudinal velocity at the leading edge 𝑣 (𝑡 ) and at the center of the contact patch 𝑣 (𝑡 ). This is because the coordinate frame in Figure 8 is fixed at the “center of contact patch” and under a motion of “translation” in the direction of the velocity V. If the coordinate frame is fixed at the “ground”, 𝑣 (𝑡 ) would be almost zero since the tire material element on the inner center line is close to the instantaneous center of rotation. The average longitudinal velocity 𝑣̅ is approximated to be the longitudinal velocity at the leading edge, i.e. 𝑣̅ ≈ 𝑣 (𝑡 ). The reason is that the difference in 𝑣 (𝑡) over the contact patch is not large. Moreover, in our setup, it is not possible to measure 𝑣 (𝑡) throughout the contact patch for calculating the average longitudinal velocity 𝑣̅ . Therefore, the average longitudinal velocity 𝑣̅ is computed as: 𝑣̅ = 𝑟 𝜃̇ 𝑠𝑒𝑐𝜃 = −𝑟 𝜃̇sec ∅

(11)

In Equation 11, the angular position at the leading edge in the simple tire model is 𝜃 = 𝜋 − ∅ , where ∅ is the angular position . starting from the center of the contact patch and ∅ = 𝑠𝑖𝑛 Finally, the model for the lateral deflection with “no” compressive component is obtained by substituting Equation 10 and Equation 11 into Equation 9:

𝑦=∬



In Equation 12, 𝛼 is defined as 𝛼 ∶= −

(𝑥 − 𝑥 )

𝑑𝑡𝑑𝑡 = 𝛼

(12)

. Note that the lateral acceleration at the center of the contact patch should be

subtracted before the double-integration to use the lateral deflection model in Equation 12. Furthermore, Equation 12 contains longitudinal coordinate values inside the contact patch, i.e. x, and unloaded radius runloaded. Those are parameters that can be estimated more accurately and easily than the longitudinal velocity at the center of the contact patch 𝑣 (𝑡 ) and the average longitudinal velocity 𝑣̅ in Equation 9. For the lateral deflection with a compressive component, the original lateral acceleration model in Equation 4 is integrated twice with the same boundary conditions. The corresponding lateral deflection model is: 𝑦=∬ 2.4.

𝑑𝑡𝑑𝑡 = 𝛼(𝑥 − 𝑥 )

(13)

Estimation of Lateral Force and Aligning Moment

In order to estimate the lateral force Fy, 𝛼 defined as 𝛼 ∶= −

in Equation 12 and 13 should be estimated first. A least

squares curve-fitting technique is applied to the lateral deflections y for estimating 𝛼. The lateral deflection y is produced from the lateral acceleration “only inside” the contact patch. The curve-fitting technique is based on the lateral deflection models in Equation 12 and 13 depending on the criterion defined in Table 2. Note that the lateral acceleration at the center of the contact patch is subtracted from the original one before the double-integration if the lateral acceleration belongs to the category 1, i.e. the lateral accelerations corresponding to the lateral deflections without compressive component. The lateral deflections for a certain number of tire rotations, n, are used when applying the curve-fitting technique. This reduces a statistical deviation in the lateral force Fy estimation. If n=5, for example, the lateral deflection profiles yk-4, … , yk for (k-4)th, … ,kth tire rotations are used in the curve-fitting to estimate 𝛼 for the kth tire rotation as solid lines in Figure 12. In the case of the lateral deflection without compressive component, 𝛼 is estimated by solving an optimization problem in Equation 14 based on the model in Equation 12. In Equation 14, 𝑥 = [𝑥 , ; … ; 𝑥 , ] represents a column vector whose elements are the longitudinal coordinate values from the leading edge 𝑥 , through the trailing edge 𝑥 , for the kth tire rotation. Also 1 𝑥

represents a column vector whose elements are one and whose size is the same as 𝑥 . The square symbol in

⋮ 𝑥

,

−1∙𝑥

,

,

refers to the element-wise square. 𝑥

𝛼 =



,

𝛼

𝑥

𝑎𝑟𝑔 𝑚𝑖𝑛

𝑥 ⋮

,

𝛼

𝑥

,

−1∙𝑥 ,



,



𝑦 , ⋮ 𝑦 ,

(14)

𝑦

,

−1∙𝑥 ,

,



⋮ 𝑦

, ,

The estimated lateral deflection yf,k for the kth tire rotation is: 𝑥 𝑦

,

=𝛼

,

𝑥



,

−1∙𝑥

,

(15)

,

Figure 12 shows an example of resulting estimation, i.e. dash line, obtained from Equation 15 and an estimate of 𝛼 . The new lateral deflection profile yf,k for the kth tire rotation is stored in the place of the original lateral deflection profile yk so that it can be used for the future curve-fitting at (k+1)th, (k+2)th, … tire rotations. is a given bending stiffness. The lateral force Fy is calculated with the estimated 𝛼 as presented in Equation 16, where 𝑐 𝐹 = −2𝛼𝑐

(16)

Figure 12. [Exp.] Curve-Fitting for 𝛼

The aligning moment Mz is estimated through a pneumatic trail. The lateral deflection models in Equation 12 and 13 do not include 𝛽 defined as 𝛽 ∶= . As reported in Equation 18, the aligning moment Mz can be estimated by the pneumatic trail t in Equation 17, i.e. the relationship between Fy and Mz. The pneumatic trail is the distance between the center of the contact patch and the position where the (resultant) lateral force Fy is applied to generate the aligning moment Mz. Therefore, η in Equation 17 can be any value between 0 and 1 as long as the nonlinear equations to be discussed in the next step can converge to a solution. In other words, η is a tuning parameter in this paper. 𝑡=−

= 𝜂𝑥 (0 ≤ 𝜂 ≤ 1)

(17)

𝑀 = −𝑡𝐹 = −𝜂𝑥 𝐹

(18)

2.5. Estimation of Friction Coefficient The tire-road friction coefficient is estimated through a tire brush model, the lateral force Fy and the aligning moment Mz [4]. The tire brush model in Equation 19, 20 and 21 correlates the tire-road friction coefficient µ, the tire slip angle α, and the sliding tire slip angle αsl with the lateral force Fy and the aligning moment Mz. The sliding tire slip angle αsl represents the limit tire slip angle before the tire starts sliding, cpy in Equation 21 a given lateral stiffness of a tire tread element, Fz the normal force and a the half length of the tire-road contact patch assumed to be 𝑎 = 𝑥 . The friction coefficient µ is estimated by solving the nonlinear equations in Equation 19, 20 and 21. ( )

𝐹 = 3𝜇𝐹

(

𝑀 = −𝜇𝐹 𝑎 𝜇=

)

( ) (

( )

1− )

(

1−3

( )

+

)

(

( ) (

)

+3

(19)

) ( ) (

)

𝑡𝑎𝑛(𝛼 )



( ) (

)

(20) (21)

3.

EXPERIMENTAL RESULTS

Experiments presented in this section aims at validating the effectiveness of the proposed approach in differentiating two road surface conditions, i.e. asphalt and ice, at small tire slip angles and at different vehicle speeds. In this section, the new friction coefficient estimation algorithm is applied to two classes of experimental maneuvers. The maneuvers in both classes are performed at different vehicle speeds and at small tire slip angles on different road surface conditions. One class is a “slow lane change”, in which the steering-wheel is turned by around ±5o, and the test vehicle travels at vehicle speeds of 32, 48, 64, 80, 96 kph (= 20, 30, 40, 50, 60 mph). The other class is a “steady-state surface transition”, where the vehicle travels straight on a road with successive different road surface conditions (asphalt-ice). Among all four tires the proposed friction coefficient estimation algorithm is applied to “rear-left (RL)” tire. The rear-left tire has a toe angle, i.e. built-in tire slip angle, of 0.15o. Front tires can be steered unintentionally during driving, whereas rear tires are fixed. Therefore, the front tires have more noise than the rear ones. 3.1. Experiment Setup The experiments were conducted at the Smithers Winter Test Center with the support by the Tyre Systems & Vehicle Dynamics Division of Pirelli Tyres SpA and the Research and Innovation Center of Ford Motor Company.

The test vehicle, i.e. Volvo XC90 equipped with Pirelli Scorpion Ice & Snow tires (235/60R18) with 2.3 bar air pressure, has two data acquisition (DAQ) systems. One is for the acceleration measurement from the tire sensors, i.e. 3D wireless accelerometers attached on the center line of the tire inner liner. The other is for the measurements, e.g. steering angle and wheel angular velocity etc., from the vehicle sensors and those, e.g. yaw-rate, forward vehicle velocity, traveled-distance etc., from GPS/IMU. Time-synchronization is required in storing the measurement data. The acceleration data collected from four tires are sent to a computer. However, the data from GPS/IMU are collected along with the data from other vehicle sensors transmitted via CAN bus before being sent to the computer as shown in Figure 13. In other words, the acceleration data use a route different from that for GPS/IMU and other vehicle sensors. An impulse signal is used for the time-synchronization. The accelerations from the tires and vehicle velocity data are used in the friction coefficient estimation algorithm. Measurements from other vehicle sensors and GPS/IMU are used to check and understand what is happening on the test vehicle.

Figure 13. Experiment Setup

3.2. Slow Lane Change In most experiments of the “slow” lane change maneuver, the steering-wheel is turned by around ±5o. This means the front wheels are turned by around ±0.31o since the steering ratio of the test vehicle is known to be 15.9. Figure 14 presents absolute values of the front-wheel steering angle for all experiments in the slow lane change maneuver. Circle point at the center is the mean value. Dark and light colored bands indicate σ (standard deviation) and 2σ which covers 68.2% and 95.4% of samples, respectively, on the assumption of a normal distribution. The front wheel steering angle is less than 1o in Figure 14, and shows that tires of the test vehicle have the small tire slip angles in the slow lane change maneuver.

Figure 14. Front Wheel Steering Angle

Ice Speed Mean σ

32 0.41 0.27

32 0.29 0.16

48 0.14 0.12

48 0.13 0.10 Asphalt

Speed Mean σ

48 0.18 0.12

48 0.18 0.12

64 0.15 0.10

64 0.15 0.10

80 0.12 0.10

80 0.13 0.09

96 0.13 0.09

96 0.09 0.08

Table 3. Mean and Standard Deviation of Front Wheel Steering Angle

3.2.1. Existing Friction Coefficient Estimation Algorithm The existing algorithm [5] could not differentiate the road surface conditions between the asphalt and the ice with the test vehicle. Figure 15 presents the friction coefficient estimations in an experiment on the asphalt and ice roads, where the vehicle speed is 48 kph (≈ 30 mph). Horizontal lines indicate the mean values of the estimated friction coefficients. As shown in Figure 15, for the most time the estimated friction coefficients on the ice road are greater than those on the asphalt road. Furthermore, some friction coefficient estimations coincide.

Figure 15. Existing Friction Coefficient Estimation Algorithm

3.2.2. New Friction Coefficient Estimation Algorithm By applying the new friction coefficient estimation algorithm proposed in this paper, the road surface conditions, i.e. asphalt and ice, could be differentiated successfully. Figure 16-(a) shows the estimated friction coefficients at the vehicle speed 48 kph (≈ 30 mph). The mean value of the estimated friction coefficients are depicted as a horizontal line. Dark and light colored bands indicate σ (standard deviation) and 2σ which covers 68.2% and 95.4% of the estimated friction coefficients, respectively, on the assumption of a normal distribution. As shown in Figure 16-(a), the estimated friction coefficients on the asphalt road are greater than those on the ice road, and the gap between two road surface conditions is large enough to clearly differentiate two road surface conditions.

(a) η = 0.25

(b) η = 0.1 Figure 16. Estimated Friction Coefficient for Slow Lane Change (Asphalt and Ice)

Mean σ

Ice 0.013 0.006

Asphalt 0.994 0.04

(a) η = 0.25 Mean σ

Ice 0.005 0.003

Asphalt 0.442 0.017

(b) η = 0.1 Table 4. Mean and Standard Deviation of Estimated Friction Coefficient for Slow Lane Change (Asphalt and Ice)

The different choice of η in the estimation of the aligning moment in Equation 18 is also presented in Figure 16-(b). As discussed previously, any value between 0 and 1 can be chosen for η as long as the nonlinear equations from the tire brush model converge to a solution for the friction coefficient estimation. Although different values of η effect the order of the magnitude of the estimated friction coefficients, the differentiation of the road surface conditions is still achieved as shown in Figure 16-(a) and 16-(b). Figure 17-(a) shows the friction coefficient estimation results for all experiments at different vehicle speeds on each road surface condition. Each block represents an experiment at a certain vehicle speed. The circle point at the center is the mean value, while dark and light colored bands for each block indicate σ and 2σ. As shown in Figure 17-(a), the differentiation between the asphalt and the ice is achieved for all the experiments. The robustness of the proposed estimation algorithm is demonstrated through Figure 17-(b). The friction coefficients in Figure 17-(b) were estimated without the shift-removal in the lateral acceleration corresponding to the lateral deflection without compressive component. The mean value and the deviation are random for every experiment on the asphalt road as shown in Figure 17-(b), whereas those are consistent in Figure 17-(a).

(a) Removing Shift

(b) No Removing Shift Figure 17. Estimated Friction Coefficients for Slow Lane Change (All Experiments) : η = 0.25 Ice Speed Mean σ

32 0.004 0.012

32 0.018 0.008

Speed Mean σ

48 0.99 0.04

48 1.04 0.05

48 0.013 0.006 64 1.02 0.05

48 0.013 0.017 Asphalt 64 80 1.04 1.00 0.05 0.05

80 0.99 0.07

96 0.95 0.06

96 0.93 0.06

80 1.10 0.05

96 1.23 0.16

96 1.16 0.20

(a) Removing Shift Ice Speed Mean σ

32 0.004 0.006

32 0.018 0.008

Speed Mean σ

48 2.67 0.13

48 1.22 0.07

48 0.013 0.006 64 1.46 0.12

48 0.014 0.020 Asphalt 64 80 1.36 2.01 0.10 0.20

(b) No Removing Shift Table 5. Mean and Standard Deviation of Estimated Friction Coefficients for Slow Lane Change (All Experiments) : η = 0.25

3.3. Steady-State Surface Transition Figure 18 presents the friction coefficient estimations for an experiment traveling straight with “no steering” on successive road surface conditions, i.e. asphalt to ice, at a constant vehicle speed 48 kph (≈30 mph). The mean value of the measured front wheel steering angle is ±0.041o for this maneuver, which is smaller than those for the slow lane change maneuver in Table 3. Thus, the tire slip angle is mainly the toe angle 0.15o. As expected, at very small tire slip angle, the proposed estimation algorithm cannot differentiate the road surface conditions (since the proposed estimation algorithm is based on the lateral deflection profile). In Figure 18 two horizontal solid lines indicate the mean values of the estimated friction coefficients for each road surface condition, respectively. The mean value on the asphalt road, 0.544, is smaller than that on the ice road, 0.843. However, the “slow lane change” maneuver is close to the normal driving in real life rather than traveling ahead with “no steering”. Therefore, the friction coefficient estimation algorithm based on the lateral deflection is still a promising way to identify the road surface conditions.

Figure 18. Estimated Friction Coefficient for Steady-State Surface Transition

4.

CONCLUSION

In this paper the tire-road friction coefficient is estimated using acceleration data directly measured by a 3D wireless accelerometer attached on the inner center line of the tire. Compared with a previously developed estimation algorithm, the new estimation algorithm determines the contact patch, uses only information inside the contact patch and relies on the different lateral deflection model. The developed approach has been motivated by the issue of the orientation-variation of the accelerometer body frame. The new algorithm is applied to two classes of maneuvers in a series of experimental tests, i.e. steady-state surface transition and slow lane change. In the steady-state surface transition maneuvers, the built-in toe angle (0.15o) of the test vehicle is not enough to achieve successful differentiation between the asphalt and ice roads. In the slow lane change maneuver, the asphalt and ice road surface conditions are successfully and robustly differentiated by applying the new estimation algorithm. ACKNOWLEDGMENT The authors would like to thank Tyre Systems & Vehicle Dynamics Division of Pirelli Tyres SpA and Dr. E. Tseng, M. Fodor and M. McConnel at the Research and Innovation Center of Ford Motor Company for providing access to the experimental facility used in this research. REFERENCES [1] Lee, C.; Hedrick, J. K.; Yi, K., “Real-time slip-based estimation of maximum tire-road friction coefficient”, Mechatronics, IEEE/ASME Transactions on, vol.9, no.2, pp.454-458, June 2004 [2] Müller, S.; Uchanski, M.; Hedrick, J. K., “Estimation of the Maximum Tire-Road Friction Coefficient”, J. Dyn. Sys., Meas., Control, vol.125, pp.607-617, December 2003 [3] Hahn, J.O.; Rajamani, R.; Alexander, L., “GPS-based real-time identification of tire-road friction coefficient”, Control Systems Technology, IEEE Transactions on, vol.10, no.3, pp.331-343, May 2002 [4] Pacejka, H. B., “Tire and Vehicle Dynamics”, SAE International, Warrendale, PA, ISBN 978-0-7680-1702-1, 2002 [5] Erdogan, G.; Hong, S.; Borrelli, F.; Hedrick, K., “Tire Sensors for the Measurement of Slip Angle and Friction Coefficient and Their Use in Traction Control Systems”, SAE International Journal of Passenger Cars- Mechanical Systems, vol.4, no.1, pp.44-58, June 2011 [6] Erdogan, G.; Alexander, L.; Rajamani, R., “Estimation of Tire-Road Friction Coefficient Using a Novel Wireless Piezoelectric Tire Sensor”, Sensors Journal, IEEE, vol.11, no.2, pp.267-279, February 2011 [7] Savaresi, S.M.; Tanelli, M.; Langthaler, P.; Del Re, L., “New Regressors for the Direct Identification of Tire Deformation in Road Vehicles Via In-Tire Accelerometers”, Control Systems Technology, IEEE Transactions on, vol.16, no.4, pp.769-780, July 2008 [8] Rajamani, R., “Vehicle Dynamics and Control”, Springer verlag, New York, 2006 [9] Zheng, et al., “Real-time signal processing for vehicle tire load monitoring”, US Patent No. 6980925, 2005 (http://www.patentgenius.com/patent/6980925.html) [10] Braghin, F.; Brusarosco, M.; Cheli, F.; Cigada, A.; Manzoni, S.; Mancosu, F., “Measurement of contact forces and patch features by means of accelerometers fixed inside the tire to improve future car active control”, Vehicle System Dynamics, vol.44, supplement 1, pp.3-13, 2006 [11] Jazar, R. N., “Vehicle Dynamics: Theory and Application”, Springer verlag, New York, 2008 [12] Gustafsson, F., “Slip-based tire-road friction estimation”, Automatica, vol.33, no.6, pp.1087-1099, 1997 [13] Li, K.; Misener, J.A.; Hedrick, K., “On-board road condition monitoring system using slip-based tyre-road friction estimation and wheel speed signal analysis”, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, vol.221 no.1, pp.129-146, March 2007 [14] Ray, L., “Nonlinear tire force estimation and road friction identification: Simulation and experiments”, Automatica, vol.33, no.10, pp.1819-1833, 1997 [15] Yi, K.; Hedrick, K.; Lee, S.-C., “Estimation of Tire-Road Friction Using Observer Based Identifiers”, Vehicle System Dynamics, vol.31, issue 4, pp.233-261 [16] Cheli, F.; Audisio, G.; Brusarosco, M.; Mancosu, F. et al, “Cyber Tyre: A Novel Sensor to Improve Vehicle's Safety”, SAE Technical Paper 2011-01-0990, 2011, doi:10.4271/2011-01-0990 [17] Cheli, F.; Sabbioni, E.; Sbrosi, M.; Brusarosco, M. et al., “Enhancement of ABS Performance through On-Board Estimation of the Tires' Response by Means of Smart Tires”, SAE Technical Paper 2011-01-0991, 2011, doi:10.4271/2011-01-0991 [18] Singh, K.; Arat, M.; Taheri, S., “Enhancement of Collision Mitigation Braking System Performance Through Real-Time Estimation of Tire-road Friction Coefficient by Means of Smart Tires”, SAE Int. J. Passeng. Cars -Electron. Electr. Syst. 5(2):2012, doi:10.4271/2012-01-2014

TABLES

Table 1 Angular positions at leading and trailing edges Normal Force

Leading Edge (Proposed / Pressure)

Trailing Edge (Proposed / Pressure)

3800 N 4500 N 6000 N

173o / 172o 172o / 171o 170o / 169 o

187 o / 188o 188 o / 189o 191 o / 192o

Table 2 Classification of Lateral Acceleration If Acc(1) > Acc(2) > Acc(3) AND Mean(Acc(2) : Acc(3)) > 0 Then Lateral Acceleration corresponding to Lateral Deflection without Deflection Component from Compressive Force Else Lateral Acceleration corresponding to Lateral Deflection with Deflection Component from Compressive Force

Ice Speed Mean σ

32 0.41 0.27

32 0.29 0.16

48 0.14 0.12

Speed Mean σ

48 0.18 0.12

48 0.18 0.12

64 0.15 0.10

48 0.13 0.10 Asphalt 64 80 0.15 0.12 0.10 0.10

80 0.13 0.09

96 0.13 0.09

96 0.09 0.08

Table 3. Mean and Standard Deviation of Front Wheel Steering Angle

Mean σ

Ice 0.013 0.006

Asphalt 0.994 0.04

(a) η = 0.25 Mean σ

Ice 0.005 0.003

Asphalt 0.442 0.017

(b) η = 0.1 Table 4. Mean and Standard Deviation of Estimated Friction Coefficient for Slow Lane Change (Asphalt and Ice)

Ice Speed Mean σ

32 0.004 0.012

32 0.018 0.008

Speed Mean σ

48 0.99 0.04

48 1.04 0.05

48 0.013 0.006 64 1.02 0.05

48 0.013 0.017 Asphalt 64 80 1.04 1.00 0.05 0.05

80 0.99 0.07

96 0.95 0.06

96 0.93 0.06

80 1.10 0.05

96 1.23 0.16

96 1.16 0.20

(a) Removing Shift Ice Speed Mean σ

32 0.004 0.006

32 0.018 0.008

Speed Mean σ

48 2.67 0.13

48 1.22 0.07

48 0.013 0.006 64 1.46 0.12

48 0.014 0.020 Asphalt 64 80 1.36 2.01 0.10 0.20

(b) No Removing Shift Table 5. Mean and Standard Deviation of Estimated Friction Coefficients for Slow Lane Change (All Experiments) : η = 0.25

FIGURE CAPTIONS                                

Figure 1. Simple Tire Model (STM) Figure 2. [STM] Radial Acceleration from Simple Tire Model Figure 3. [Exp.] Radial Acceleration from an Experiment Figure 4. [FEM] Radial Acceleration and Magnitude of its Derivative Figure 4-(a). Radial Acceleration Figure 4-(b). Magnitude of Derivative of Radial Acceleration Figure 5. Orientation-Variation of Accelerometer Body Frame Figure5-(a). [FEM] Tire Finite Element Model Figure5-(b). [FEM] Lateral Deflection Figure 6. Lateral Acceleration Figure 6-(a). [FEM] Lateral Acceleration with respect to Wheel-Center and Accelerometer Body Frames Figure 6-(b). [Exp.] Lateral Acceleration Inside Contact Patch Figure 7. [Exp.] Produced Lateral Deflection Inside Contact Patch Figure 8. [FEM] Lateral Deflections by Compressive Force and Friction Force Figure 8-(a). Lateral Deflections Figure 8-(b). Friction Coefficient μ: 0.1, Tire Slip Angle α: 0.0o Figure 8-(c). Friction Coefficient μ: 0.1, Tire Slip Angle α: 2.0o Figure 8-(d). Friction Coefficient μ: 0.9, Tire Slip Angle α: 2.0o Figure 9. [Exp.] Shifted Lateral Accelerations Inside Contact Patch Figure 10. [STM] Longitudinal Acceleration (𝑎 ≈ −𝑎 ) Figure 11. Lateral Acceleration corresponding to Lateral Deflection without Component from Compressive Force Figure 12. [Exp.] Curve-Fitting for 𝛼 Figure 13. Experiment Setup Figure 14. Front Wheel Steering Angle Figure 15. Existing Friction Coefficient Estimation Algorithm Figure 16. Estimated Friction Coefficient for Slow Lane Change (Asphalt and Ice) Figure 16-(a). η = 0.25 Figure 16-(b). η = 0.1 Figure 17. Estimated Friction Coefficients for Slow Lane Change (All Experiments) : η = 0.25 Figure17-(a). Removing Shift Figure17-(b). No Removing Shift Figure 18. Estimated Friction Coefficient for Steady-State Surface Transition