Black Ice Detection Based on Tire Friction Coefficient Estimation of Vehicle Longitudinal Model

Abstract

Black ice is a deadly hazard on the road because it is visually transparent and difficult to identify by driver’s naked eye while driving. Because tire friction on a black icy road surface is obviously smaller than normal road, the braking distance significantly increases and leads to severe traffic accidents. Road hazard detection such as black ice has been actively attempted so far, usually focusing on methodology using intelligent vision systems (e. g., cameras). However, current image-based methods are prone to reduced low accuracy due to their susceptibility to vibrations transmitted from road surfaces to vehicles. In addition, incorporating cameras and light detection and ranging sensors increases the complexity and computational burden, especially when extending their functionality to include road surface classification. Therefore, we investigate the potential of new road surface classification based on vehicle longitudinal velocity and tire effective radius estimation from vehicle longitudinal model. This study explores a sensor-fusion type indirect road surface classification algorithm based on Kalman filtering.

1 Introduction

Black ice is a deadly hazard on the road because it is visually transparent and difficult to identify by driver’s naked eye while driving. The braking distance significantly increases and leads to severe traffic accidents because tire friction on a black icy road surface is obviously smaller than normal road [1]. Road hazard detection such as potholes and black ice has been actively attempted so far, usually focusing on method-ology using intelligent vision systems (e. g., cameras). However, current image-based methods are prone to reduced low accuracy due to their susceptibility to vibrations transmitted from road surfaces to vehicles. In addition, incorporating cameras and light detection and ranging (LiDAR) sensors increases the complexity and computational burden, especially when extending their functionality to include road surface classification. The information of road conditions detected from smart tire sensors can be shared through a cloud server when the first vehicle encounters the road hazards. This approach seems to be promising because subsequent vehicles can use this shared information to navigate and avoid road hazards such as potholes, potentially reducing computational demands and enhancing efficiency, as shown in Fig. 1. Considering that tires are the only components of the vehicle system in direct contact with the road surface, they inherently possess high potential for detecting road hazards such as black ice. The sensor fusion-based indirect method then can be effective supplement system to smart tire sensors or act as a fail-safe system where smart tire sensors may not function properly because sensor fusion-based indirect method through CAN bus is currently being effectively utilized in various real-time vehicle control systems with a relatively high sampling frequency (e. g., 1 kHz).

Fig. 1.

Schematic overview of the proposed road surface classification both smart tire sensor built in pneumatic tires (direct, low sampling frequency) and sensor fusion based tire friction coefficient estimation (indirect, high sampling frequency) [2]

In this study, we explore new approach to achieve new cost-effective means of classifying road surface (i.e., tire friction coefficient) by measuring (or estimating) slip ratio and tire tractive force in real time, as shown in Fig. 2 This indirect sensor fusion method can compensate the accuracy of road surface classification in situations where smart tire sensors face limitations.

Fig. 2.

Tire tractive force vs slip ratio showing road surface-dependent tire friction coefficient

2 Slip Ratio and Vehicle Longitudinal Velocity Estimator

2.1 Vehicle Longitudinal Model

In this study, the wheel rotation dynamic model and vehicle longitudinal model of electric vehicles is used to estimate vehicle longitudinal velocity and the tire effective radius [3]. Assuming that the vehicle is driving on flat road ($\theta = 0$) and the braking force is inactive, the vehicle driving model in longitudinal direction can be governed by

$$ I_\omega \dot{\omega } = T_m - RF_x - RC_{rr} mg $$

(1)

$$ \gamma_m m\dot{v}_x = F_x - 0.5\rho C_d A_f v_x^2 - C_{rr} mg $$

(2)

where $I_\omega$ denotes the roll inertia, $\omega$ denotes the wheel angular velocity, $T_m$ denotes the motor output torque, $R$ denotes the tire effective radius, $F_x$ denotes the tractive force, $C_{rr}$ denotes the rolling resistance coefficient, $m$ denotes the vehicle mass, $\gamma_m$ represents the mass factor, $v_x$ denotes the vehicle longitudinal velocity, $\rho$ denotes the air mass density, $C_d$ represents the aerodynamic drag coefficient, and $A_f$ represents the vehicle frontal area. Compared to conventional vehicles with internal combustion engines combined with multi-stepped automatic transmissions, the tractive force of electric vehicles can be easily determined by converting motor output torque as

$$ F_x = \frac{T_m G\eta }{R} $$

(3)

where $G$ denotes the reduction gear ratio, and $\eta$ denotes the mechanical efficiency.

Introducing the additional assumption that the tire effective radius is in a quasi-static state ($\dot{R} \approx 0$), the state space equation can then be expressed as Eq. (4).

$$ \begin{gathered} \dot{x} = f(x) = \left[ {\begin{array}{*{20}c} {\dot{x}_1 } \\ {\dot{x}_2 } \\ {\dot{x}_3 } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{T_m }{{I_\omega }} - \frac{T_m G\eta }{{I_\omega }} - \frac{{RC_{rr} mg}}{I_\omega }} \\ {\frac{T_m G\eta }{{\gamma_m mR}} - \frac{\rho C_d A_f }{{2\gamma_m m}}v_x^2 - \frac{{gC_{rr} }}{\gamma_m }} \\ 0 \\ \end{array} } \right] \hfill \\ x = \left[ {\begin{array}{*{20}c} {x_1 } & {x_2 } & {x_3 } \\ \end{array} } \right]^T = \left[ {\begin{array}{*{20}c} \omega & {v_x } & R \\ \end{array} } \right]^T ,\,\,y = \left[ {\begin{array}{*{20}c} \omega & {a_x } \\ \end{array} } \right]^T \hfill \\ \end{gathered} $$

(4)

Consequently, slip ratio $\lambda$ is derived from the estimated $v_x$ and $R$ using Eq. (5) [4]

$$ \lambda = \frac{v_w - v_x }{{v_x }} = \frac{R\omega }{{v_x }} - 1 $$

(5)

3 Design of Extended Kalman Filter

The nonlinear system model Eq. (4) is linearized using the Jacobian respect to the defined state variables and is discretized using Euler’s method for being applicable to the discrete-time extended Kalman filter (EKF). To design EKF, we discretized the continuous-time state equation as follows [5]:

$$ \begin{gathered} x_{k + 1} = f_k \left( {x_k } \right) + w_k \hfill \\ y_k = h_k \left( {x_k } \right) + v_k \hfill \\ \end{gathered} $$

(6)

where $w_k$ is a system-noise vector and $v_k$ is a measurement noise vector. The extended Kalman filter assumes the differentiability of the state-change function instead of the linearity of the model. The nonlinear system model was linearized using a Jacobian matrix as follows:

$$ A_k = \left. {\frac{\partial f_k }{{\partial x}}} \right|_{x_{k - 1} } ,\,\,H_k = \left. {\frac{\partial h_k }{{\partial x}}} \right|_{\hat{x}_k } $$

(7)

Matrices A and H of the system model were linearized using Eq. (7) are expressed as follows:

$$ F_k = \frac{\partial f}{{\partial x}} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - \frac{{C_{rr} mg}}{I_\omega }} \\ 0 & { - \frac{\rho C_d A_f x_2 }{{\gamma_m m}}} & { - \frac{T_m G\eta }{{\gamma_m mx_3^2 }}} \\ 0 & 0 & 0 \\ \end{array} } \right],\,\,H_k = \frac{\partial h}{{\partial x}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & { - \frac{\rho C_d A_f x_2 }{{\gamma_m m}}} & { - \frac{T_m G\eta }{{\gamma_m mx_3^2 }}} \\ \end{array} } \right] $$

(8)

The overall estimation process using the EKF algorithm represents as follow [5].

• 1) Initial estimation

  $$ \hat{x}_0 = E\left[ {x_0 } \right] $$

  (9)
  $$ P_0 = E\left[ {\left( {x_0 - \hat{x}_0 } \right)\left( {x_0 - \hat{x}_0 } \right)^T } \right] $$

  (10)

• 2) Prediction

  $$ \hat{x}_k^- = f_{k - 1} \left( {\hat{x}_{k - 1} } \right) $$

  (11)
  $$ P_k^- = AP_{k - 1} A^T + Q $$

  (12)

• 3) Kalman-gain calculation

  $$ K_k = P_k^- H^T \left( {HP_k^- H^T + R} \right)^{ - 1} $$

  (13)

• 4) State correction

  $$ \hat{x}_k = \hat{x}_k^- + K_k \left( {y_k - h\left( {\hat{x}_k^- } \right)} \right) $$

  (14)
  $$ P_k = P_k^- - K_k HP_k^- $$

  (15)

4 Estimation of Tire Friction Coefficient

Based on two estimated variables (slip ratio and vehicle longitudinal velocity), tire friction coefficient was estimated. The simulation of the integrated vehicle model is performed assuming a scenario of a vehicle driving on four types of roads: icy, snowy, wet, dry. We assumed the tire friction coefficients for each type of road (icy, snowy, wet, dry) to be 0.1, 0.2, 0.4, and 0.8 respectively. The proposed estimation algorithm was implemented in MATLAB®. A classification algorithm is additionally required to classify road surface conditions based on the estimated vehicle longitudinal velocity and the tire effective radius. Deep learning model (e. g., support vector machine) is one of the methods to design the algorithm.

To evaluate the estimation performance, the simulation of the vehicle model was performed in a scenario that the vehicle accelerates from a standstill. The vehicle longitudinal velocity and the tire effective radius estimation results of performing the simulation are shown in Fig. 3 (a) and (b). It can be observed that the vehicle longitudinal velocity was correctly estimated compared to the measured values. A peak point in the tire effective radius is observed during the initial acceleration phase, and as the vehicle longitudinal stabilizes, the tire effective radius also remains constant. The same simulation was performed for each tire friction coefficient ($\mu = 0.1,\,\,0.2,\,\,0.4,\,\,0.8$). With the assumption of motor output torque as the known input, tractive force is calculated by substituting the tire effective radius into Eq. (3). The estimated vehicle longitudinal velocity and the tire effective radius were substituted into Eq. (5) to calculate the slip ratio. The variations in tractive force concerning the variations in slip ratio for each tire friction coefficient are shown in Fig. 3 (c). It is observed that the shape of the graph varies for each tire friction coefficient. This variation can be used to classify the road surface conditions. However, in a range with a small slip ratio, it is challenging to classify wet and dry roads, as shown in Fig. 3 (d). Also, there are technical limitations to utilize such smart tire sensors capable of measuring the tire friction coefficients. Smart tire sensors typically have a low sampling frequency (e. g., 1 Hz) to save the life time. Given the vehicle's movement per second (e. g., at a speed of 50 km/h, driving 14 m per second), this low sampling frequency is not suitable for road surface classification. Moreover, high-frequency sensors based on smart tires, developed to address this issue, are facing challenges in commercialization due to battery life and durability concerns.

5 Conclusions

In this study, we proposed new method of estimating tire friction coefficient based on rigorous slip ration estimation enabled by tire effective radius and longitudinal vehicle velocity estimation using extended Kalman filtering. The estimated slip ratio and tractive force are used to estimate tire friction coefficient. The proposed method offers advantages in terms of cost and computational aspects compared to image-based methods. However, at a certain level of tire friction coefficient (e. g., $\mu = 0.4,\,\,0.8$) the variations of the slip ratio versus tractive force are not clearly distinct. Additionally, this study did not reach the stage of designing classification algorithms. For future direction, we will design classification algorithm such as support vector machine (SVM).

Fig. 3.

Simulation result of estimation; (a) vehicle longitudinal velocity at $\mu = 0.1$, (b) Tire effective radius at $\mu = 0.1$, (c) Calculated tractive force vs estimated wheel slip ratio