## Introduction

Active safety systems including antilock braking system, yaw stability control system, and rollover prevention control system have been equipped widely in automotive industry [13]. With the development of self-driving car, many advanced driving assistance systems are being studied, such as adaptive cruise control [4], vehicle collision avoidance [5], and active front steering system [6,7]. Besides, further research still needs to be done to develop more advanced technologies. Knowledge of vehicle dynamics states is crucial for both active safety control systems but also self-driving technologies. It is widely known, chassis stability and performance depend strongly on the forces generated at the tire contact patch. As a consequence, tire-road information must be monitored.

How to obtain vehicle dynamics states has been widely discussed in the previous literatures. Three kinds of methods can be utilized to obtain tire forces: (1) estimation algorithms, (2) physical or empirical-tire-model calculation, and (3) direct measurement. However, tire-force measurement equipment is very expensive and the calibration of a tire model requires extensive tests, so the tire force estimation algorithm is practical. Some researchers focused on the estimation of tire forces. For example, Ray proposed an extended Kalman filter (EKF) observer to estimate the state and longitudinal and lateral tire force histories of a nine degrees-of-freedom (DOF) vehicle [8]. Then, the EKF was utilized as a robust tire force estimator, which was validated against results from strain-gaged wheel rims [9]. Afterward, many EKF-based tire force estimators were designed [1012]. Two observers based on the EKF and unscented Kalman filter (UKF) were proposed by Ref. [13], and experimental results demonstrated that UKF is a superior alternative, especially when the system presents strong nonlinearities. Some tire-force estimation algorithms have been applied to vehicle dynamics control. For example, Hsiao proposed an observer-based traction force control scheme and it could be robust with respect to variations of road conditions and tire model uncertainties [14]. Cho et al. designed a scheme for longitudinal/lateral tire-force estimation and it was integrated into a unified chassis control system, i.e., vehicle stability control [15].

It is noted that tire-road information including tire force and road friction coefficient is important for vehicle active safety control and self-driving control. Though many tire-force estimators so far have been designed, their performance still needs to be improved. When the vehicle is at nonlinear state, EKF-based estimators' accuracy decreases because of the linearization method utilized in EKF, which only ensures first-order accuracy [27]. UKF-based estimators' stability requires the prediction matrix to be positive definite [28]. Various road conditions cause uncertainty fluctuation of tire-road friction coefficient, so its observation still need to be further researched. Researches about the observation of tire-road information are still insufficient.

Considering easy availability and good economy, this paper proposes a novel adaptive unified monitoring system (AUMS) to simultaneously observe the tire-road friction coefficient and tire forces, i.e., vertical, longitudinal, and lateral tire forces of each wheel. The main contributions of this paper are as follows. The vertical, longitudinal, and lateral tire forces of all the four wheels are estimated, respectively, by an adaptive unified sliding mode observer (AUSMO). Second, the road-tire friction coefficient of each individual wheel is observed through the designed mode-switch observer (MSO). The designed MSO contains two modes: when the vehicle is under driving or brake, a slip slope method (SSM) is used to estimate the road-tire friction coefficient of each wheel, and a recursive least-squares (RLS) identification method is utilized in the SSM; when the vehicle is under steering, a comprehensive friction estimation method is adopted. Moreover, the proposed AUMS estimates the individual wheel tire forces and tire-road friction coefficients using only common signals measured by sensors on-board. The paper is organized as follows: The related models are presented in Sec. 2. Section 3 gives a detailed description about the design of the proposed AUMS. Its performance is validated by not only matlab/simulink and carsim co-simulation but also real vehicle tests. The conclusion is presented in Sec. 5.

## Vehicle Dynamics Models

The vehicle longitudinal dynamics model as shown in Fig. 1, is adopted to observe the road friction coefficient. It includes the longitudinal motion of vehicle body and rotational motion of four wheels as shown in below equations:
$mu˙=Fxf+Fxr−Faero$
(1)
$Iwω˙ij=Tdij−Tbij−RFxij$
(2)
Fig. 1
Fig. 1
Close modal
A simplified four-wheel vehicle dynamics model is utilized in the design of AUMS to estimate the lateral tire forces (see Fig. 2). The longitudinal, lateral and yaw motion equations which use the tire forces of four wheels are as presented as follows:
$m(u˙−Vyφ)=Fxrl+Fxrr−(Fyfl+Fyfr)sin δ+(Fxfl+Fxfr)cos δ$
(3)
$mu(β˙+φ)=Fyrl+Fyrr+(Fyrl+Fyrr)cos δ+(Fxfl+Fxfr)sin δ$
(4)
$Izφ˙=(Fyfl+Fyfr)a cos δ+(Fyfl−Fyfr)w2sin δ−(Fyrl+Fyrr)b+(Fxfl+Fxfr)a sin δ−(Fxfl−Fxfr)w2cos δ−(Fxrl−Fxrr)w2$
(5)
Fig. 2
Fig. 2
Close modal
The relationship between the longitudinal tire force and longitudinal slip ratio of each tire is shown in Fig. 3. The longitudinal slip ratio is determined by the difference between the vehicle velocity $u$ and wheel speed $Rωij$ as presented in the below equation:
$λij={Rωij−uRωij,(u˙>0)Rωij−uu,(u˙≤0)$
(6)
Fig. 3
Fig. 3
Close modal

When a tire works in its linear region, the longitudinal tire force generated at each wheel is proportional to its slip ratio for any road surface and normal force. As we all know, there is no clear definition about when the tire begins to work in its nonlinear region, so this paper only considers the linear tire region.

## Design of the Adaptive Unified Monitoring System

Precise measurement of tire-road friction coefficient and tire forces is difficult to implement without expensive equipment, for example, wheel force transducers of Kistler (RoaDyn® Wheel Force Sensor S635, Kistler, Winterthur, Switzerland). They are too expensive to be widely used in mass-produced cars. Therefore, the monitoring of tire-road information utilizing either accurate models or improved estimation algorithms is essential. Considering easy availability and good economy, this paper proposes a novel AUMS to simultaneously observe the tire-road friction coefficient and tire forces. The diagram of AUMS is shown in Fig. 4, it consists of a vertical tire force calculation block, a unified observer of lateral and longitudinal tire force, and a road-tire friction coefficient estimation block. First, the vertical tire forces can be calculated considering vehicle body roll and load transfer. The longitudinal and lateral tire forces are estimated by the AUSMO. Then, the road-tire friction coefficient is observed through the designed MSO which utilizes RLS identification method and comprehensive friction estimation method.

Fig. 4
Fig. 4
Close modal

### The Calculation of Vertical Tire Force.

Vehicle attitude angles cannot be measured in automobiles which commonly do not equip the inertial measurement unit. Therefore, this paper uses a practical method of calculating vertical tire force considering the effects of vehicle body roll and load transfer [3]. The calculation does not need vehicle pitch angle and roll angle which cannot be measured by common sensors on board. The vertical tire force $Fzij$ can be calculated as follows:
$F̂zfl=mg2(ba+b−hgu˙g(a+b)−bhgVyφ˙gw(a+b))$
(7)
$F̂zfr=mg2(ba+b−hgu˙g(a+b)+bhgVyφ˙gw(a+b))$
(8)
$F̂zrl=mg2(aa+b+hgu˙g(a+b)−ahgVyφ˙gw(a+b))$
(9)
$F̂zrr=mg2(aa+b+hgu˙g(a+b)+ahgVyφ˙gw(a+b))$
(10)

### The Adaptive Unified Sliding Mode Observer of Individual Wheel Tire Forces.

The individual wheel vertical tire forces can be obtained by the above subsection considering the effects of vehicle body roll and load transfer. The longitudinal and lateral tire forces can be observed by the designed unified tire force observer which is based on the sliding mode observer. The unified observer estimates both longitudinal and lateral tire forces of four wheels simultaneously, and it only uses the common signals measured by the sensors on-board.

The longitudinal tire forces of four wheels can be estimated utilizing only the wheel angular speed. In order to design the longitudinal tire force estimation part of the proposed AUSMO, the Eq. (2) is transformed into the state-space model as follows:
${x˙1=ω˙ij=1Iw(Tdij−Tbij)−RIwFxijy=ωij$
(11)
Then, the system state observer can be designed as shown in the following equation:
$ω̂˙ij=1Iw(Tdij−Tbij)−RIwF̂xij+Lij(ωij−ω̂ij)$
(12)
where Lij represents the observer's gain. The error of system state variable is defined as the sliding mode surface, i.e., $s=ωij−ω̂ij$. y denotes the system measurement output. Then, we design the Lyapunov function as follows:
$V=s22$
(13)
The derivation of Eq. (13) can be presented as follows:
$V˙=ss˙=(ωij−ω̂ij)(ω˙ij−ω̂˙ij)$
(14)
Combined the Eqs. (12) and (14), we can recalculate the derivation of Lyapunov function as follows:
$V˙=sRIwF̂xij+s(−RIwFxij−Lijs)$
(15)
The longitudinal tire force $Fxij$ is unknown and bounded input variable, so there always exists positive constants $ε$, if it is big enough, which can satisfy the following equation:
$|−RIwFxij−Lijs|<ε$
(16)
Then, combined with Eqs. (15) and (16), the derivation of Lyapunov function can be presented as the following equation:
$V˙=sRIwF̂xij+s(−RIwFxij−Lijs)≤sRIwF̂xij+sε≤sRIwF̂xij+|s|ε$
(17)
We define that sgn(s) and $F̂xij$ as follows:
$sgn(s)={1,s>00,s=0−1,s<0$
(18)
$F̂xij=−IwRεsgn(s)$
(19)
Then, Eq. (17) can be transformed as follows:
$V˙≤sRIwF̂xij+|s|ε=−sεsgn(s)+|s|ε=0$
(20)

As shown in Eq. (20), the derivation of Lyapunov function satisfies $V˙≤0$, the system is Lyapunov stable. Therefore, the longitudinal tire force can converge to the sliding mode surface.

The derivation of the error of system variable can be calculated as follows:
$ω̃˙ij=−εsgn(s)−RIwFxij−Lijs$
(21)
When the system variable converges to the sliding mode surface, $ω̃˙ij=0$. Therefore, the longitudinal tire force can be observed as follows:
$F̂xij=−IwRεsgn(s)−IwRLijs$
(22)

where $ε$ is the observer's sliding mode gain, and $Lij$ is the observer's feedback gain.

Considering big tremble due to the time delay and system inertia, we use the concept of boundary layer to decrease the estimation error. The saturation function $sat(s)$ is used to replace the $sgn(s)$, and the tremble can be weakened greatly
$sat(s)={sφ,|s|≤φsgn(s),|s|>φ$
(23)

where $φ>0$ is the thickness of the boundary layer.

Then, the longitudinal tire force estimation part of the proposed AUSMO can be presented as follows:
$F̂xij=−IwRε⋅sat(s)−IwRLijs$
(24)

Similarly, the lateral tire force estimation part of the proposed AUSMO can estimated the lateral tire forces of four wheels, utilizing only the signals the wheel steering angle, and yaw rate.

Based on the 2DOF single-track vehicle model, $Fyf$ and $Fyr$ are decoupled as follows:
${Fyf=b⋅m⋅ay+Izφ˙−Fxf(a+b)sin δ(a+b)cos δFyr=a⋅m⋅ay−Izφ˙a+b$
(25)
We can transform Eq. (25) into the following formula:
${x˙2=φ˙=−bmIway+(a+b)cos δIwFyf+(a+b)sin δIwx˙2=φ˙=amIway−a+bIwFyr$
(26)
Thus, based on the Eq. (26), we can design the lateral tire force estimation part of the proposed AUSMO which is similar to Eq. (24) as follows:
${F̂yf=Iw(a+b)cos δ[εyfsat(s)+Lyf(φ−φ̂)]F̂yr=−Iwa+b[εyrsat(s)+Lyr(φ−φ̂)]$
(27)
Then, the lateral tire force of each wheel can be calculated as follows:
${F̂yfl=F̂zflF̂zfl+F̂zfrF̂yfF̂yfr=F̂zfrF̂zfl+F̂zfrF̂yfF̂yrl=F̂zrlF̂zrl+F̂zrrF̂yrF̂yrr=F̂zrrF̂zrl+F̂zrrF̂yr$
(28)
Therefore, the proposed AUSMO can be presented finally as shown in the following equation:
${F̂xij=−IwRεsgn(s)−IwRLijsF̂yfj=F̂zfjF̂zfl+F̂zfr{Iw(a+b)cos δ[εyfsat(s)+Lyf(φ−φ̂)]}F̂yrj=F̂zrjF̂zrl+F̂zrr{−Iwa+b[εyrsat(s)+Lyr(φ−φ̂)]}$
(29)

### Estimation of Individual Wheel Tire-Road Friction Coefficients.

The designed MSO contains two modes: when the vehicle is under driving or brake, an SSM is used to estimate the road-tire friction coefficient, and a RLS is utilized in the SSM; when the vehicle is under steering, a comprehensive friction estimation method is adopted.

When the vehicle is under steering, the nonlinear extent of the yaw rate $φ$ and ay can be used to obtain the error of the friction which may compensate the detected lateral acceleration to estimate the road friction under steering conditions. The nonlinear extent of ay reflects the understeer condition and the nonlinear extent of $φ$ reflects the oversteer conditions [29]. Therefore, the comprehensive friction estimation method in Ref. [24] is adopted to estimate the road-tire friction under steering conditions.

When the vehicle is under driving and brake, the MSO works in another mode. The SSM is designed which utilizes RLS to observe the road-tire friction coefficient of each individual wheel under driving or brake conditions. The road-tire friction coefficient estimation part of the proposed AUMS uses the vertical and longitudinal tire forces of individual wheel estimated by the above AUSMO, and the individual wheel longitudinal slip ratio which can be calculated by Eq. (6). The longitudinal tire force of individual wheel is proportional to its own slip ratio for any given road surface and normal vertical tire force, which can be presented as the following equation [17]:
$Fxij=κij⋅λij⋅Fzij$
(30)
The slip-slop can change with the road surface, i.e., the tire-road friction coefficient. Therefore, it can be used to observe the value of tire-road friction coefficient of individual wheel. The estimation relationship between tire-road friction coefficient and slip-slop can be presented as linear equation as shown in the following equation:
$μij=ξκ⋅κij+ξ0$
(31)

where $ξκ$ and $ξ0$ are constants, and they can be calibrated by the simulation.

Then, the value of slip-slop $κij$ is essential to observe $μij$. We use RLS-based parameter identification method to obtain $κij$ in real time. The standard parameter identification format is shown in the following equation:
$y(t)=φT(t)⋅θ(t)+e(t)$
(32)

In this paper, the longitudinal tire force of each individual wheel is the system output $y(t)$. $φ(t)=λij⋅Fzij$ denotes the system input. $θ(t)=κij$ is the unknown parameter. $e(t)$ is the identification error between $y(t)$ and the estimated value $φT(t)⋅θ(t)$.

The iterative steps of the RLS-based parameter identification method are as follows.

First, we use the vertical, longitudinal tire forces, and slip ratio of each individual wheel estimated by the proposed AUSMO to obtain the system output $yk=F̂xij,k$ and system input $φk=λij,k⋅F̂zij,k$.

Second, the identification error $ek$, that is the difference between the system's actual output at this step and the output predicted in the previous step.
$ek=yk−φkTθk−1=F̂xij,k−λij,k⋅F̂zij,k⋅κij,k−1$
(33)
Then, the proposed method calculates the update gain $Kk$ and the covariance matrix $Pk$
$Kk=Pk−1⋅φk1+φkT⋅Pk−1⋅φk=Pk−1⋅λij,k⋅F̂zij,k1+F̂zij,k⋅λij,k⋅Pk−1⋅λij,k⋅F̂zij,k$
(34)
$Pk=Pk−1−Pk−1⋅φk⋅φkT⋅Pk−11+φkT⋅Pk−1⋅φk=Pk−1−Pk−1⋅λij,k⋅F̂zij,k⋅F̂zij,k⋅λij,k⋅Pk−11+F̂zij,k⋅λij,k⋅Pk−1⋅λij,k⋅F̂zij,k$
(35)
Finally, we update the estimated parameter as follows:
$κij,k=κij,k−1+Kk⋅ek$
(36)

The slip slop $κij$ can be estimated by the Eq. (36), and then based on the Eq. (31), the tire-road friction coefficient at each individual wheel can be observed.

## Simulation and Experiment Results

To verify the effectiveness of the proposed AUMS, a group of simulation tests and real car experiments are carried out. The results of both simulations and real car experiments show that the proposed AUMS has great performance of tire-road information monitoring.

### Simulation Results.

We use carsim and matlab/simulink to develop the simulation platform. carsim provides a full-vehicle model with the actual vehicle parameters. Then, a complex scenario is designed and implemented to evaluate the performance of the proposed monitoring system. The simulation scenario consists of the driving maneuver, brake maneuver, and sine wave steering maneuver as shown in Fig. 5. The vehicle utilized in carsim first brakes with 2 MPa brake control pressure, and the velocity decreases from 33.3 m/s to about 11 m/s. From about 6 s to 24 s, the sine wave steering maneuver is conducted and the steering-wheel amplitude is 90 deg, and the frequency is 4 Hz. Then, the vehicle model accelerates with small throttle opening. Figures 5(b) and 5(c) give the longitudinal and lateral accelerations of the vehicle model.

Fig. 5
Fig. 5
Close modal

Figure 6 shows the individual longitudinal tire force of each wheel in the designed complex simulation scenario. The estimates longitudinal tire forces can converge to the corresponding actual values quickly. During the brake maneuver, the front longitudinal tire forces are about −1900 N, and the rear longitudinal tire forces are about −1000 N. when the vehicle model is under steering or driving, the rear tire forces are about zero. The front longitudinal tire forces are about 1000 N when the vehicle model accelerates.

Fig. 6
Fig. 6
Close modal

As shown in Fig. 7, during steering maneuver, the lateral tire forces of four wheels fluctuate and the estimated lateral tire forces are of high accuracy. Moreover, the lateral tire forces of rear axle are about zero as shown in Figs. 7(c) and 7(d) when the vehicle model is not under steering, but the actual lateral tire forces of front axle are about N, respectively, due to the accurate modeling of carsim. Wheel alignment parameters are considered in the design of carsim vehicle models, and it can cause two lateral forces in equilibrium of two front wheels, which is not necessary to consider in the lateral tire force estimation. Figure 8 gives the calculation results of individual vertical tire forces. The estimated values can meet the requirement of application precision since the estimated values can converge quickly to the actual values with high estimation accuracy. While, the vertical tire forces are calculated only considering the effects of vehicle body roll and load transfer. Future works may focus on more accurate observation of vertical tire forces which may rely on more specific sensors unequipped on board.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

Figure 9 shows the road-tire friction coefficient in the designed complex simulation scenario. It is set to be 0.4 before 2.5 s, and after 2.5 s, it is a constant of 0.9. The estimated values are of enough accuracy to be applied in the automotive industry. At about 24.5 s, the estimated values have a little large fluctuation due to the mode switching when the vehicle stop sine wave steering maneuver to accelerate. When the vehicle model is under severe brake on a low adhesion road, the load transfer is large and rear axle may occur slippage, so the estimated friction coefficient values of rear two wheels fluctuate a little violently before about 2.5 s, and similar phenomenon exists in the vertical tire forces as shown in Figs. 9(c) and 9(d).

Fig. 9
Fig. 9
Close modal

### Experiment Results.

The real car experiment is carried out utilize an A-class sedan, and it is equipped with multiple sensors, such as wheel force transducers of Kistler, steering angle sensor, wheel angular velocity sensor, and yaw rate and lateral acceleration sensors. Figure 10 demonstrates the test equipment of the real car experiment. The proposed AUMS can estimate tire-road information utilizing the signals measured by sensors on-board, i.e., steering angle sensor, and wheel angular velocity sensor and so on. The estimated values can be compared with the measured results by the wheel force transducers of Kistler.

Fig. 10
Fig. 10
Close modal

The real car test is carried out on an asphalt road. The test scenario is designed as follows. First, the car speeds up to about 15 m/s; then, the sine wave maneuver is implemented at about 15 m/s, finally, the car brakes suddenly. Figure 11(a) shows the longitudinal velocity and steering wheel angle in real car tests. The longitudinal acceleration is shown in Fig. 11(b), the maximum acceleration is about 5 m/s2, while the maximum deceleration is about −6 m/s2. The lateral acceleration is shown in Fig. 11(c), and the maximum lateral acceleration is about 6 m/s2.

Fig. 11
Fig. 11
Close modal

The actual tire forces can be measured by the wheel force transducers of Kistler and the measured signals contain some noise signals as shown in Figs. 1214, but the measured values of tire forces are of high precision. The estimated tire forces are compared with corresponding measured values.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

Figure 12 shows the individual longitudinal tire force of each wheel in real car test. During the brake maneuver, the maximum value of front longitudinal tire forces is about −4000 N, while the maximum value of rear longitudinal tire forces is about −1500 N. when the vehicle model is under steering or driving, the rear tire forces are about zero. The front longitudinal tire forces are larger than zero when the car accelerates before about 10 s. Similar to the simulation results, the proposed AUMS can estimate longitudinal tire forces of four wheels accurately.

Figure 13 shows the individual lateral tire force of each wheel in real car test. The errors between the estimated values and the actual values of lateral tire forces in the real car test are larger than those in the simulation. Sensors on-board exist inevitable measurement noises. The actual vehicle test condition is more complicated than that of simulation. Therefore, the observation accuracy is reduced. Anyway, the proposed AUMS is accurate enough to be applied in the automotive industry. Figure 14 gives the calculation results of individual vertical tire forces in our real car experiment, the calculated values coincides with the corresponding measured vertical tire forces.

Figure 15 shows the road-tire friction coefficient in real car test. The test is carried out on an asphalt road. The estimated road-tire friction coefficients of four wheels are all about 0.7. The actual road tire friction coefficient cannot be measured, but the estimated values seem reasonable. The estimated values have minor change before about 20 s. Afterward, a little large fluctuation appears, and the car is under severe steering. The proposed MSO works in steering estimation mode instead of driving or brake estimation mode.

Fig. 15
Fig. 15
Close modal

## Conclusion

This paper proposes a novel AUMS to simultaneously observe the tire-road friction coefficient and tire forces, i.e., vertical, longitudinal, and lateral tire forces. First, the vertical tire forces can be calculated considering vehicle body roll and load transfer. The longitudinal and lateral tire forces are estimated by the AUSMO. Then, the road-tire friction coefficient is observed through the designed MSO which utilizes RLS identification method and comprehensive friction estimation method. When the vehicle is under driving or brake, a SSM with the RLS identification method is used to estimate the road-tire friction coefficient of each wheel; when the vehicle is under steering, a comprehensive friction estimation method is adopted. The performance of the proposed AUMS is verified by both the matlab/simulinkcarsim cosimulation and the real car experiment. Results demonstrate the effectiveness of our proposed AUMS, and it provides accurate monitoring of tire-road information. However, the road-tire information is of great complexity, it not only includes tire forces and road-tire friction, but also contains tire side-slip angle and tire pressure, etc. Therefore, its observation still needs to be studied to be more exhaustive and robust in the future research.

## Funding Data

• National Key Research and Development Program of China (Grant No. 2017YFB0103902; Funder ID: 10.13039/501100002855).

• National Science Fund of the Peoples Republic of China (Grant No. 51675293; Funder ID: 10.13039/501100001809).

## Nomenclature

• $ay$ =

lateral acceleration

•
• $a,b$ =

distance from vehicle gravity center to front and rear axles, respectively

•
• $Faero$ =

air resistance force

•
• $Fxf,Fxr$ =

front and rear longitudinal tire forces in total, respectively

•
• $Fxij,Fyij,Fzij$ =

longitudinal, lateral, and vertical tire forces of four wheels, respectively

•
• $Fyf,Fyr$ =

front and rear lateral tire forces in total, respectively

•
• $g$ =

acceleration of gravity

•
• $hg$ =

height of sprung mass center

•
• $Iw$ =

inertia moment about the vehicle vertical axis

•
• $m$ =

vehicle mass

•
• $R$ =

•
• $Ttij,Tbij$ =

driving and brake torques of four wheels, respectively

•
• $u,Vy$ =

vehicle longitudinal and lateral velocity, respectively

•
• $w$ =

distance between the left and right wheel

•
• $αij$ =

side-slip angle of four wheels, respectively

•
• $β$ =

vehicle side-slip angle

•
• $δ$ =

steering angle of front wheel

•
• $κij$ =

slip-slop of individual wheel, respectively

•
• $λij$ =

slip ratio of four wheels, respectively

•
• $φ$ =

yaw rate

•
• $ωij$ =

wheel angular speed of four wheels, respectively

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