Research on Active Collision Avoidance Control of Vehicles Based on Estimation of Road Surface Adhesion Coefficient

Abstract

In order to solve the problem that intelligent vehicle active collision avoidance systems have different decision-making results under different road conditions, the square-root cubature Kalman filtering algorithm is used to estimate the road adhesion coefficients, which are introduced into the safety distance model and combined with the fireworks algorithm for braking and steering weight coefficient allocation to ensure that the vehicle can safely avoid collision. The simulation results show that the square-root cubature Kalman filter has higher estimation accuracy and robustness compared with the cubature Kalman filter, and a more reasonable collision avoidance control can be adopted in the subsequent collision avoidance control. Therefore, the proposed new estimation method of road adhesion coefficients proves effective in mitigating vehicle collision risks.

1. Introduction

With the continuous advancement of intelligent vehicle technologies, particularly in the field of active safety systems, the deployment of Advanced Driver-Assistance Systems (ADASs) has become increasingly widespread [1]. Maintaining driving safety—especially in complex and dynamic traffic environments—remains a fundamental requirement of intelligent vehicles. Among various active safety components, active collision avoidance systems play an irreplaceable role in preventing accidents by enabling vehicles to anticipate and avoid potential hazards in real time [2].

One of the most critical parameters affecting the performance of collision avoidance systems is the road surface adhesion coefficient, which directly reflects the frictional interaction between the tires and the road surface. This parameter determines the vehicle’s ability to accelerate, brake, and steer safely. An accurate and timely estimation of road adhesion allows the collision avoidance system to select an optimal control strategy that is adapted to the prevailing road conditions, thereby enhancing the safety of the vehicle and its occupants.

Various approaches have been proposed to estimate or identify the road surface adhesion coefficient. Jin-Oh [3] introduced a method that estimates adhesion based on the vehicle’s lateral dynamics response using a real-time identification algorithm fed by Differential GPS (DGPS) and gyroscopic sensors. While accurate, such methods are highly dependent on sensor availability and require high-precision positioning.

Šabanovič et al. [4] applied deep convolutional neural networks to classify six typical road surfaces, achieving an 88% recognition rate. However, their method mainly focuses on road surface type classification, and the indirect mapping from texture to adhesion coefficient may lack robustness in mixed or transitional road conditions. Similarly, Zhuo Yu et al. [5] utilized lidar reflection intensity combined with probabilistic fusion across multiple databases. This laser-based approach shows potential but can be significantly affected by environmental factors such as fog or dirt.

Image-based estimation methods, such as those using HSV color features and gray-level co-occurrence matrices [6,7], have been integrated with Hidden Markov Models (HMMs) to track road states. However, these methods are sensitive to ambient lighting and typically lack generalizability in low-visibility conditions. Junhui Lu [8] introduced a BP neural network that correlates environmental factors like solar radiation and pavement temperature with adhesion estimation. While useful in specific scenarios, its performance relies heavily on meteorological data accuracy.

Machine learning-based approaches, such as the SVM method proposed by Hongqiang Wu [9], have improved feature extraction and partial occlusion handling but still suffer from challenges in real-time applicability and generalization across variable environments. Additionally, Xianyao Ping et al. [10] enhanced adhesion estimation by introducing a strong tracking theory into the Unscented Kalman Filter (UKF), improving accuracy but increasing algorithmic complexity and parameter sensitivity. In terms of control strategy, Yuho Song [11] and Mingyue Yan [12] proposed predictive controllers and multi-objective fuzzy decision-making algorithms for active collision avoidance, focusing on high-level motion planning. However, they did not comprehensively consider the dynamic influence of real-time adhesion variation in their decision-making processes.

In response to the aforementioned limitations, this paper proposes an integrated framework that combines road adhesion coefficient estimation with a coordinated control strategy for active collision avoidance, featuring the following key innovations:

• Accurate Adhesion Estimation Using SCKF: A real-time method for estimating the road surface adhesion coefficient is proposed based on the Square-Root Cubature Kalman Filter (SCKF), which demonstrates improved numerical stability and superior performance compared to traditional UKF-based approaches under nonlinear and non-Gaussian conditions.
• Safety Distance Modeling with Adhesion Awareness: A dynamic safety distance model is established that explicitly considers the estimated road adhesion coefficient and the relative motion state of the leading vehicle, enabling more adaptive and context-aware collision risk assessment.
• Novel Dual-Mode Control Strategy: A hybrid steering–braking cooperative collision avoidance controller is developed by integrating double PID control and fireworks algorithm-based optimization. This approach ensures real-time adaptability and effect control allocation under varying road conditions.
• Integrated Decision Logic: A decision-making logic module is designed to determine the appropriate avoidance maneuver (steering, braking, or combined) based on both the adhesion coefficient and the vehicle’s dynamic state, thereby enhancing system robustness in complex scenarios.

In conclusion, this paper introduces advanced state estimation algorithms, establishes a dynamic safety model, designs a cooperative controller and an integrated decision-making mechanism, and constructs a highly precise, adaptable, and generalizable active collision avoidance system framework.

2. Vehicle Dynamic Modeling

2.1. Three-Degree-of-Freedom Vehicle Dynamics Model

This paper employs a three-degree-of-freedom vehicle dynamics model. Based on this model, the Square-Root Cubature Kalman Filter algorithm is applied to estimate the road surface adhesion coefficient. The force analysis of the vehicle is illustrated in Figure 1, and the following assumptions are made: the origin of the vehicle coordinate system coincides with the center of mass of the vehicle; the air resistance and rolling resistance during vehicle motion are neglected; the mechanical characteristics of the four tires are identical; the effect of the suspension is ignored, and the vehicle body is rigidly connected to the chassis; roll and yaw motions of the vehicle are not considered.

In Figure 1, CoG is the center of mass of the vehicle; $F_{xfl},F_{xfr},F_{xrl},F_{xrr}$ are the longitudinal forces of the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively; $F_{yfl},F_{yfr},F_{yrl},F_{yrr}$ are the lateral forces of the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively; $L_a,L_b$ are the distances between the center of mass and the front and rear axes, respectively; and B_f, B_r are the front and rear wheelbases, respectively.

The longitudinal equation of motion of the vehicle is

$$m{a}_x=(F_{xfl}+F_{xfr})\mathrm{c}\mathrm{o}\mathrm{s}\delta -(F_{yfl}+F_{yfr})\mathrm{s}\mathrm{i}\mathrm{n}\delta +F_{xrl}+F_{xrr}$$

(1)

where m is the mass of the vehicle, a_x is the longitudinal acceleration of the vehicle, and δ is the front wheel’s angle of rotation

The lateral equation of motion of the vehicle is

$$m{a}_y=(F_{xfl}+F_{xfr})\mathrm{s}\mathrm{i}\mathrm{n}\delta -(F_{yfl}+F_{yfr})\mathrm{c}\mathrm{o}\mathrm{s}\delta +F_{yrl}+F_{yrr}$$

(2)

where a_y is the lateral acceleration of the vehicle, while F_yrl and F_yrr are the lateral forces of the left and right rear wheels, respectively.

The yaw motion equation of the vehicle is

$$\begin{array}{c}J\dot{\omega }=(F_{xfr}-F_{xfl}){\displaystyle \frac{B_f}{2}}\mathrm{c}\mathrm{o}\mathrm{s}\delta +(F_{yfl}-F_{yfr}){\displaystyle \frac{B_f}{2}}\mathrm{s}\mathrm{i}\mathrm{n}\delta +(F_{xrr}-F_{xrl}){\displaystyle \frac{B_r}{2}}+(F_{xfl}+F_{xfr})L_a\mathrm{s}\mathrm{i}\mathrm{n}\delta \\ +(F_{yfl}+F_{yfr})L_a\mathrm{c}\mathrm{o}\mathrm{s}\delta -(F_{yrl}+F_{yrr})L_b\end{array}$$

(3)

where J is the rotational inertia of the vehicle around the z axis, ω is the angular velocity of the transverse pendulum, and Mz is the return torque.

2.2. Dugoff Tire Model

To accurately describe the relationship between the road adhesion coefficient and tire force, it is essential to select an appropriate tire model. Currently, tire models are primarily categorized into theoretical models, empirical models, and semi-empirical models. Among these, the theoretical models are derived by simplifying the physical model into a mathematical form, offering high accuracy. However, their main drawback lies in the complexity of the equations, which makes it difficult to meet real-time computational requirements. Empirical and semi-empirical models, on the other hand, are primarily based on extensive experimental tire data to fit expressions for various tire parameters. These models are generally more concise and sufficiently accurate compared to theoretical models. Commonly used models include the Magic Formula [13], the Dugoff model [14], and the UniTire model [15]. Considering the advantages of the Dugoff tire model—such as fewer parameters, lower computational demand, and ease of road adhesion coefficient separation—and taking into account the algorithm’s real-time requirements, the Dugoff tire model [16] was selected for tire force calculation. The corresponding tire force analysis is illustrated in Figure 2.

Expressions for the longitudinal force of the tire:

$$F_{xy}={\mu }_y{F}_{xy}^0={\mu }_y{F}_{xy}{C}_{xy}{\displaystyle \frac{s_y}{1-s_y}}f(l)$$

(4)

Expressions for the lateral force of the tire:

$$F_{yy}={\mu }_y{F}_{yy}^0={\mu }_y{F}_{xy}{C}_{yy}{\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n} {\alpha }_y}{1-s_{\bar{y}}}}f(l)$$

(5)

where $f(l)=\{\begin{array}{c}1,l\ge 1\\ l(2-l),l<1\end{array}$, $l={\displaystyle \frac{(1-s_{ij})(1-\epsilon {\nu }_x\sqrt{C_{xij}^2{s}_{ij}^2+C_{yij}^2{\mathrm{t}\mathrm{a}\mathrm{n}}^2{a}_{ij}})}{2\sqrt{C_{xij}^2{s}_{ij}^2+C_{yij}^2{\mathrm{t}\mathrm{a}\mathrm{n}}^2{a}_{ij}}}}$, where i = f and r represent the front and rear wheels, respectively; j = l and r represent the left and right wheels. respectively; μ_ij is the road adhesion coefficient of each wheel; C_xij and C_yij are the longitudinal stiffness and lateral deviation stiffness of each wheel’s tire, respectively; s_ij is the tire slip rate of each wheel; α_ij is the side deviation angle of each wheel tire; ε is the velocity influencing factor; F_zij is the vertical load of the tires of each wheel; $F_{xij}^0,F_{yij}^0$ are the normalized tire force of each wheel, ignoring the influence of the road adhesion coefficient, which is conducive to algorithm programming; f(l) is the model correction factor; and l is the characteristic parameter of tire slip.

The slip rate is calculated as follows:

$$s_{ij}=\{\begin{array}{c}{\displaystyle \frac{{\nu }_{ij}-{\omega }_{ij}R}{{\nu }_{ij}}}, braking\\ {\displaystyle \frac{{\omega }_{ij}R-{\nu }_{ij}}{{\nu }_{ij}}}, drive\end{array}$$

(6)

where v_ij is the longitudinal velocity of each wheel, ω_ij is the angular velocity of each wheel, and R is the effective rolling radius of the wheel.

The formula for calculating the wheels’ center velocity is

$$\left\{\begin{array}{l}{v}_{cfl}=(v_x+{\displaystyle B_f{\dot{\omega }}_{fl}/2})\cos \delta +(v_y+L_a{\dot{\omega }}_{fl})\sin \delta \\ v_{cfl}=(v_x-{\displaystyle B_f{\dot{\omega }}_{fr}/2})\cos \delta +(v_y+L_a{\dot{\omega }}_{fr})\sin \delta \\ v_{crl}=v_x+{\displaystyle B_r{\dot{\omega }}_{rl}/2}\\ v_{crr}=v_x-{\displaystyle B_r{\dot{\omega }}_{rr}/2}\end{array}\right.$$

(7)

where, v_cfl, v_cfr, v_crl, and v_crr are the velocities at the center of the left front wheel, the right front wheel, the left rear wheel, and the right rear wheel, respectively; ω_fl, ω_fr, ω_rl, and ω_rr are the angular velocities of the left front wheel, the right front wheel, the left rear wheel, and the right rear wheel, respectively; and v_y is the lateral speed of the vehicle.

The formula for calculating the deviation angle of a tire is

$$\left\{\begin{array}{l}{\alpha }_{fl}=-\delta +\arctan {\displaystyle \frac{v_y+L_a{\dot{\omega }}_{fl}}{v_x+B_f{\dot{\omega }}_{fl}/2}}\\ {\alpha }_{fr}=-\delta +\arctan {\displaystyle \frac{v_y+L_a{\dot{\omega }}_{fr}}{v_x-B_f{\dot{\omega }}_{fr}/2}}\\ {\alpha }_{rl}=\arctan {\displaystyle \frac{v_y-L_a{\dot{\omega }}_{rl}}{v_x+B_r{\dot{\omega }}_{rl}/2}}\\ {\alpha }_{rr}=\arctan {\displaystyle \frac{v_y-L_a{\dot{\omega }}_{rr}}{v_x-B_r{\dot{\omega }}_{rr}/2}}\end{array}\right.$$

(8)

The formula for calculating the vertical load of the wheel is

$$\left\{\begin{array}{c}{F}_{zfl}={\displaystyle \frac{mg{L}_b}{2L}}-{\displaystyle \frac{m{a}_x{h}_g}{2L}}+{\displaystyle \frac{m{a}_y{h}_g{L}_b}{L{B}_f}}\\ F_{zfr}={\displaystyle \frac{{mgL}_b}{2L}}-{\displaystyle \frac{{ma}_x{h}_g}{2L}}-{\displaystyle \frac{{ma}_y{h}_g{L}_b}{{LB}_f}}\\ F_{zrl}={\displaystyle \frac{{mgL}_b}{2L}}+{\displaystyle \frac{{ma}_x{h}_g}{2L}}-{\displaystyle \frac{{ma}_y{h}_g{L}_a}{{LB}_r}}\\ F_{zrr}={\displaystyle \frac{{mgL}_b}{2L}}+{\displaystyle \frac{{ma}_x{h}_g}{2L}}+{\displaystyle \frac{{ma}_y{h}_g{L}_a}{{LB}_r}}\end{array}\right.$$

(9)

where L = L_a + L_b is the wheelbase, g is the acceleration due to gravity, and h_g is the height of the center of mass.

3. Design of Pavement Adhesion Coefficient Estimator

3.1. Square-Root Volume Kalman Filter Algorithm

The Cubature Kalman Filter (CKF) algorithm is a novel nonlinear Gaussian filtering method introduced by Canadian researchers in 2009 [17]. It has been widely applied in fields such as aerospace [18], target tracking [19,20], and battery state-of-charge (SOC) estimation [21]. The CKF selects cubature points based on the third-order spherical–radial cubature rule and achieves high computational accuracy. Compared with the Unscented Kalman Filter (UKF), the cubature points in the CKF are symmetrically distributed in a lower-dimensional space, and all points share equal weights, eliminating the need for prior parameter tuning. However, due to numerical rounding errors and other computational factors, the CKF covariance matrix may lose positive definiteness, leading to reduced observer stability.

To address the limitations of the Cubature Kalman Filter algorithm in state parameter estimation, this paper proposes the square-root cubature Kalman filtering algorithm, which avoids the non-negative definiteness of the matrix and the instability of the observer caused by the square of the covariance matrix of the CKF by employing orthogonal triangular decomposition and enhances both estimation accuracy and algorithmic robustness. The estimation flowchart is illustrated in Figure 3.

The parameter variables of the system are determined as x = [μ_fl,μ_fr,μ_rl,μ_rr]^T, according to Equations (1)–(3) of the vehicle dynamics. Define the observed quantity of the system as y = [a_x,a_y,$\dot{\omega }$]^T and the input quantity of the system as u = [δ,$F_{xij}^0$,$F_{yij}^0$]^T. The specific estimation process is as follows:

Step 1: For a general nonlinear system, the state-space equation can be written as follows:

$$\{\begin{array}{c}\dot{\mathit{x}}\left(t\right)=f\left(\mathit{x}\right(t),\mathit{u}(t\left)\right)+\mathit{w}\left(t\right)\\ \mathit{y}\left(t\right)=h\left(\mathit{x}\right(t),\mathit{u}(t\left)\right)+\mathit{v}\left(t\right)\end{array}$$

(10)

where f(x(t), u(t)) and h(x(t), u(t)) are the equation of state and measurement equation of the estimated system, respectively; w(t) is the process noise; and v(t) is the measurement noise.

Step 2: The state and observation equations of the estimator can be expressed as follows:

$$\dot{\mathit{x}}(t)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mu }_{fl}\\ {\mu }_{fr}\\ {\mu }_{rl}\\ {\mu }_{rr}\end{array}\right]+\mathit{w}(t)$$

(11)

$$\mathit{y}(t)=\left[\begin{array}{c}h\left(\mathrm{1,1}\right) h(1,2) h\left(\mathrm{1,3}\right) h\left(\mathrm{1,4}\right)\\ h\left(\mathrm{2,1}\right) h\left(\mathrm{2,2}\right) h\left(\mathrm{2,3}\right) h\left(\mathrm{2,4}\right)\\ h\left(\mathrm{3,1}\right) h\left(\mathrm{3,2}\right) h\left(\mathrm{3,3}\right) h\left(\mathrm{3,4}\right)\end{array}\right]\left[\begin{array}{c}{\mu }_{\mathrm{f}\mathrm{l}}\\ {\mu }_{\mathrm{f}\mathrm{r}}\\ {\mu }_{\mathrm{r}\mathrm{l}}\\ {\mu }_{\mathrm{r}\mathrm{r}}\end{array}\right]+\mathit{v}(t)$$

(12)

where $h(\mathrm{1,1})={\displaystyle \frac{F_{xfl}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta -F_{yfl}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta }{m}}$, $h(1,2)={\displaystyle \frac{F_{xfr}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta -F_{yfr}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta }{m}}$, $h(\mathrm{1,3})={\displaystyle \frac{F_{ x\mathrm{r}\mathrm{l}}^0}{m}}, h(\mathrm{1,4})={\displaystyle \frac{F_{ x\mathrm{r}\mathrm{r}}^0}{m}}$, $h(\mathrm{2,1})={\displaystyle \frac{F_{x\mathrm{f}\mathrm{l}}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta +F_{y\mathrm{f}\mathrm{l}}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta }{m}}$, $h(\mathrm{2,2})={\displaystyle \frac{F_{xfl}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta +F_{yfl}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta }{m}}$, $h(\mathrm{2,3})={\displaystyle \frac{F_{yrl}^0}{m}},h(\mathrm{2,4})={\displaystyle \frac{F_{yrr}^0}{m}}$, $h(\mathrm{3,1})={\displaystyle \frac{(F_{yfl}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta -F_{xfl}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta )B_f/2+(F_{xfl}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta +F_{yfl}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta )L_f}{J}}$, $h(\mathrm{3,2})={\displaystyle \frac{(F_{xfr}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta -F_{yfr}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta )B_f/2+(F_{xfr}^0\mathrm{s}\mathrm{i}\mathrm{n}\delta +F_{yfr}^0\mathrm{c}\mathrm{o}\mathrm{s}\delta )L_f}{J}}$, $h(\mathrm{3,3})=-{\displaystyle \frac{F_{xrl}^0{B}_r/2+F_{yrl}^0{L}_r}{J}},h(\mathrm{3,4})={\displaystyle \frac{F_{xrr}^0{B}_r/2-F_{yrr}^0{L}_r}{J}}$.

Step 3: The Singular Value Decomposition (SVD) method is used to decompose the state volume error covariance matrix:

$${\mathit{G}}_{k-1|k-1}=svd\left({\mathit{P}}_{k-1|k-1}\right)$$

(13)

where G_{k − 1|k − 1} is the state error covariance matrix, P_{k − 1|k − 1} is the state quantity error covariance matrix at the (k − 1) moment, and svd( ) is the matrix decomposition function.

Step 4: Calculate the volume point:

$${\mathit{x}}_{i,k-1}={\mathit{G}}_{k-1|k-1}{\xi }_i+{\hat{\mathit{x}}}_{k-1} i=1,2\dots ,2n$$

(14)

where $\widehat{\mathit{x}}$_{k − 1} is the predicted value of the state quantity at the moment (k − 1), n is the dimension of the state quantity, ξ_i = $\mathit{B}$ is the cubature point weight matrix, and B = [In × n − In × n].

Step 5: Calculate the volume point after iteration of the equation of state:

$${\mathit{x}}_{i,k|k-1}^{*}=f({\mathit{x}}_{i,k-1},\mathit{u})$$

(15)

Step 6: State One Step Prediction:

$${\hat{\mathit{x}}}_{k|k-1}={\displaystyle \frac{1}{2n}}{\displaystyle \sum _{i=1}^{2n}}{\mathit{x}}_{i,k|k-1}^{*}$$

(16)

where $\widehat{\mathit{x}}$_{k|k − 1} is the state prediction at moment k.

Step 7: Calculate the square root of the error covariance matrix of the predicted values of the state quantities:

$${\mathit{G}}_{k|k-1}=qr\left(\right[{\mathit{x}}_{k|k-1}^{*},{\mathit{J}}_{Q,k|k-1}\left]\right)$$

(17)

where ${\mathit{x}}_{k|k-l}^{*}={\displaystyle \frac{1}{2n}}\left[\right({\mathit{x}}_{i,k|k-1}^{*}-{\hat{\mathit{x}}}_{k|k-1}\left)\right]$ is the state-volume-weighted center matrix, J_{Q,k|k − 1} = chol(Qk) is the square-root factorization of Qk, Qk is the observer process noise covariance matrix, chol( ) is the matrix Cholesky decomposition operation, and qr( ) is the orthogonal triangular decomposition operation of the matrix.

Step 8: Calculate the updated volume points:

$${\mathit{x}}_{i,k|k-1}={\mathit{G}}_{k|k-1}{\xi }_i+{\hat{\mathit{x}}}_{k|k-1}$$

(18)

Step 9: Propagate the volume points by substituting them into the volume equation:

$${\mathit{y}}_{i,k|k-1}=h({\mathit{x}}_{i,k|k-1},\mathit{u})$$

(19)

$${\hat{\mathit{y}}}_{k|k-1}={\displaystyle \frac{1}{2n}}{\displaystyle \sum _{i=1}^{2n}}{\mathit{y}}_{i,k|k-1}$$

(20)

where $\widehat{\mathit{y}}$_{k|k− 1} is the estimated value of the observation at moment k. The following is an example of the estimation of the observation at moment k:

Step 10: Calculate the square root of the error covariance matrix of the predicted values of the observations:

$${\mathit{G}}_{zz,k|k-1}=qr\left(\right[{\mathit{y}}_{k|k-1},{\mathit{J}}_{R,k}\left]\right)$$

(21)

where $y_{k|k-1}={\displaystyle \frac{1}{2n}}\left[\right(y_{i,k|k-1}-{\hat{y}}_{k|k-1}\left)\right]$ is the observation-weighted center matrix, J_{R,k|k − 1} = chol(Rk) is the square-root factorization of Rk, and Rk is the observer measurement noise covariance matrix.

Step 11: Calculate the mutual covariance matrix:

$${\mathit{P}}_{\mathrm{x}\mathrm{y},k|k-1}={\mathit{x}}_{k|k-1}{\mathit{y}}_{k|k-1}^{\mathrm{T}}$$

(22)

where ${\mathit{x}}_{k|k-1}={\displaystyle \frac{1}{\sqrt{2n}}}\left[\right({\mathit{x}}_{i,k|k-1}-{\hat{\mathit{x}}}_{k|k-1}\left)\right]$.

Step 12: Calculate the filter gain matrix:

$${\mathit{K}}_k={\mathit{P}}_{\mathrm{x}\mathrm{y},k|k-1}({\mathit{G}}_{\mathrm{z}\mathrm{z},k|k-1}{\mathit{G}}_{\mathrm{z}\mathrm{z},k|k-1}^{\mathrm{T}}{)}^{-1}$$

(23)

Step 13: Update the square-root factor of the error covariance matrix:

$${\hat{\mathit{x}}}_k={\hat{\mathit{x}}}_{k|k-1}+{\mathit{K}}_k(\mathit{y}-{\hat{\mathit{y}}}_{k|k-1})$$

(24)

Step 14: Update the square-root factor of the error covariance matrix:

$${\mathit{G}}_k=qr\left(\right[{\mathit{x}}_{k|k-1}-{\mathit{K}}_k{\mathit{y}}_{k|k-1},{\mathit{K}}_k{\mathit{J}}_{R,k|k-1}\left]\right)$$

(25)

According to the settings of the SCKF algorithm used in this paper,

Table 1. Summary of key parameters of SCKF.

Name of Parameter	Value/Setting	Notes
State vector x	[μfl,μfr,μrl,μrr]	Four-wheel road adhesion coefficient
Observation variable y	[ax,ay,ω]	Longitudinal acceleration, lateral acceleration, yaw angular velocity
State dimension n	4	Determine the number of volume points
Process noise covariance Qk	Model calibration	Indicating modeling errors and disturbances
Measurement noise covariance Rk	Sensor calibration	Indicates the measurement accuracy of the sensor
Decomposition method	SVD + QR + Cholesky	Ensuring the positive definiteness of the covariance and numerical stability indicates the measurement accuracy of the sensor
Update step size	0.01 s	Synchronized with vehicle dynamics simulation

summarizes the key parameters and their meanings. These parameters were selected based on the vehicle dynamics characteristics and sensor accuracy, which not only ensures the estimation accuracy but also improves the numerical stability.

3.2. Model for Estimating Pavement Adhesion Coefficient

In order to verify the effectiveness of the proposed algorithm in this section, the CarSim/Simulink co-simulation platform was employed. The CKF algorithm and SCKF algorithm were implemented to estimate the pavement adhesion coefficient, and the estimated values of the algorithms were compared with the reference values provided by CarSim, so as to verify the accuracy and validity of the SCKF algorithm. The estimation flowchart is illustrated in Figure 4.

In CarSim, the initial speed of the vehicle was set to 80 km/h, the throttle opening was 0, the transmission was in neutral, the brake wheel cylinder pressure was 0.3 MPa, the road surface adhesion coefficient setting was set to 0.8 and 0.4, and the simulation time was 10 s. Simulation experiments were carried out, and the simulation results are presented in Figure 5 and Figure 6.

To quantitatively assess the accuracy of the estimation algorithms, the root-mean-square error (RMSE) was employed to measure the deviation between the estimated values and the actual values, as presented in

Table 2. RMSE values for different road adhesion coefficients.

Road Adhesion Coefficients	Tire	RMSE (CKF)	RMSE (SCKF)	Improvement Rate (%)
μ = 0.4	Left front wheel	0.1162	0.0138	88.1
	Right front wheel	0.1183	0.0170	85.6
	Left rear wheel	0.0790	0.0086	89.1
	Right rear wheel	0.0845	0.0062	92.7
μ = 0.8	Left front wheel	0.1064	0.0201	81.1
	Right front wheel	0.0999	0.0239	76.1
	Left rear wheel	0.0712	0.0115	83.9
	Right rear wheel	0.0704	0.0096	86.4

. By comparing the simulation curves, it can be observed that both the Cubature Kalman Filter (CKF) algorithm and the Square-Root Cubature Kalman Filter (SCKF) algorithm exhibit a tendency to converge towards the standard values. In comparison with the CKF algorithm, the SCKF algorithm demonstrates higher convergence accuracy. This is because the SCKF algorithm utilizes the orthogonal triangular decomposition method to ensure the positive definiteness of the covariance matrix, whereas the CKF algorithm’s use of square-root decomposition results in a negative-definite covariance matrix.

4. Establishment of Safe Distance Model

4.1. Vertical Safe Distance Model

As shown in Figure 7, the complete braking process consists of the driver’s reaction time τ₁, the braking mechanism’s action time τ₂ (where τ₂′ and τ₂′′ represent the time required to eliminate the braking clearance and the time for the braking pressure to increase, respectively), the continuous braking time τ₃, and the time for the braking to be canceled τ₄.

Let the intelligent vehicle and the front vehicle be traveling in the same lane, in the safe braking distance model shown in Figure 8. In the figure, v₀ and v_xfront are the initial speeds of the intelligent car and the front car, respectively; d_br is the distance between the two cars when they are not braked; S is the distance traveled by the intelligent car when it is braked; S_front is the distance traveled by the front car when it is braked; and d_safe is the minimum safe distance.

Depending on the different states of motion of the front vehicle, the safe distances between the intelligent vehicle and the front vehicle are as follows:

Critical safe distance for stationary working conditions of the vehicle in front:

$$d_{br}=v_0({\tau }_2^{\prime }+{\displaystyle \frac{{{\tau }_2}^{″}}{2}})+{\displaystyle \frac{v_0^2}{2a_{bmax}}}+d_{safe}$$

(26)

Critical safe distance at uniform speed of the vehicle in front:

$$d_{br}=(v_0-v_{xfront})\times ({\tau }_2^{\prime }+{\displaystyle \frac{{\tau }_2^{″}}{2}})+{\displaystyle \frac{(v_0-{\nu }_{xfront}{)}^2}{2a_{bmax}}}+d_{safe}$$

(27)

Critical safe distance under emergency braking conditions of the vehicle in front:

$$d_{br}=({\nu }_0-{\nu }_{xfront})\times ({\tau }_2^{\prime }+{\displaystyle \frac{{\tau }_2^{″}}{2}})+{\displaystyle \frac{{\nu }_0^2}{2a_{bmax}}}+{\displaystyle \frac{{\nu }_{xfront}^2}{2a_{xfront}}}+d_{sufe}$$

(28)

where a_xfront is the braking deceleration of the front vehicle; a_bmax is the maximum braking deceleration of the intelligent vehicle, which is related to the road surface adhesion coefficient, which is effectively estimated as detailed in the previous section; τ₁ is the reaction time of the driver of the intelligent vehicle; and τ₂′ and τ₂″ are the time required for the brake to start acting and the time for the brake force to continue acting, respectively.

4.2. Steering Safe Distance Model

The potential collision types that may occur during lane changing are front corner collisions, mid-vehicle collisions, and rear corner collisions, as illustrated in Figure 9.

Assuming that the vehicle performs a lane change for collision avoidance in an adjacent lane without interference from other vehicles, the rear corner point is considered to be the last possible collision point based on a fifth-degree polynomial trajectory planning method. To ensure collision avoidance, it is sufficient to guarantee that the intelligent vehicle’s right rear corner point, relative to the vehicle’s center of mass, maintains a distance greater than half of the front vehicle’s lateral width. This ensures that all parts of the following vehicle will not collide with the front vehicle. The collision avoidance schematic is illustrated in Figure 10.

In Figure 10, O is the center of mass of the smart vehicle; α and γ are the angles between the vehicle axis and the line connecting the vehicle’s center of mass to the vehicle’s right rear and right front corner points, respectively; tc is the moment of critical collision; and θ is the vehicle’s heading angle.

The distance L_OE between the center of mass of the intelligent vehicle and the outer edge of the front vehicle is calculated as follows:

$$L_{OE}=\sqrt{(L_{rr}^2+(W_r/2{)}^2}·\cos (\pi /2-\theta -a)$$

(29)

where L_rr is the distance from the center of mass of the intelligent vehicle to the rear edge of the front vehicle, and W_r is the width of the intelligent vehicle.

When te/2, v_y reaches its maximum value, θ also reaches its maximum value. The calculation formula is

$${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\arctan {\displaystyle \frac{15y_d}{8v_x{t}_e}}$$

(30)

In order to prevent the rear vehicle from colliding with the leading vehicle, the intelligent vehicle should meet the following requirements in the lateral direction:

$${\displaystyle \frac{W_f}{2}}=y_e\left[10\right({\displaystyle \frac{t_c}{t_e}}{)}^3-15({\displaystyle \frac{t_c}{t_e}}{)}^4+6({\displaystyle \frac{t_c}{t_e}}{)}^5]-\sqrt{(L_{rr}^2+({\displaystyle \frac{W_r}{2}}{)}^2}·\cos ({\displaystyle \frac{\pi }{2}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}-a)$$

(31)

where W_f is the width of the front vehicle.

The critical collision time t_c can be obtained by solving Equation (31), so the safe distance under the conditions of stationary front vehicle, uniform speed of front vehicle, and braking condition of front vehicle can be obtained, which is calculated as follows:

Corresponding critical safety distance for steering collision avoidance under stationary conditions of the front vehicle d_{st_steer}:

$$d_{st\_steer}=v_x{t}_c+L_{oc}\cos (\gamma -\theta (t_c\left)\right)-L_{ff}+d_{\mathit{safe}}$$

(32)

The corresponding steering collision avoidance critical safety distance d_{con_steer} when the front vehicle is traveling at a constant speed with speed v_front:

$$d_{con\_steer}=(v_x-v_{front})t_c+L_{oc}\cos (\gamma -\theta (t_c\left)\right)-L_{ff}+d_{\mathit{safe}}$$

(33)

When the front vehicle is decelerating with the maximum braking deceleration a_fmax, the corresponding critical safe distance for steering to avoid collision is d_{br_steer}:

$$d_{br,steer}=({\nu }_x-{\nu }_{front}+{\displaystyle \frac{a_{fmax}{t}_c}{2}})t_c+L_{oc}\cos (\gamma -\theta (t_c))-L_{ff}+d_{\mathit{safe}}$$

(34)

5. Steering Brake Joint Collision Avoidance Controller Design

5.1. Determination of the Domain of the Steering and Braking Distribution Coefficients

Brake steering cooperative control is different from the single control mode. During the collision avoidance process, braking and steering are coupled with each other, and the longitudinal force and lateral force of the tire should meet the adhesion condition:

$$\sqrt{F_x^2+F_y^2}\le \mu F_z$$

(35)

The friction circle is shown in Figure 11. The right half of the axis in the figure represents a right turn, the left half represents a left turn, the upper half represents drive control, and the lower half represents brake control. As the steering wheel angle and braking intensity continuously increase, the longitudinal force and lateral force acting on the tire also increase [22].

From the formula, the relationship between the longitudinal acceleration and the lateral acceleration is obtained:

$$\sqrt{(\lambda a_{xmax}{)}^2+(\gamma a_{ymax}{)}^2}\le \mu g$$

(36)

where λ represents the braking distribution coefficient, and $\gamma$ represents the steering distribution coefficient.

The selection of braking and steering distribution coefficients plays a critical role in the collision avoidance effect. The selection principle for the domain of braking and steering distribution coefficients is as follows:

(1)

Domain of the Braking Distribution Coefficient:

The braking system applies braking based on the maximum frictional force that the ground can provide, and its domain is [0, 1].

(2)

Definition Domain of the Distribution Coefficient:

This paper draws on the research results of Nathaniel [23], as shown in

Table 3. Classification of lateral acceleration intensity levels.

Level	Lateral Acceleration	Intensity
Normal level	0 ≤ ay ≤ (0.1−0.0013u)g	Low
Strong level	(0.1−0.0013u)g ≤ ay ≤ (0.22−0.002u)g	Medium
Restricted level	(0.22−0.002u)g ≤ ay ≤ 0.67aymax	High
Maximum level	0.67aymax ≤ ay ≤ 0.85aymax	Very high

, to classify the lateral acceleration and thereby determine the domain of the definition of the steering distribution coefficient.

When the vehicle is traveling at a low speed and the road adhesion coefficient is high, a smaller distribution coefficient should be selected; otherwise, a larger steering distribution coefficient should be chosen. The range of maximum lateral acceleration for different road surfaces is presented in

Table 4. Range of maximum lateral acceleration values based on steering allocation coefficient.

Road Adhesion Coefficient μ	Maximum Lateral Acceleration γaymax
0.2	[(0.05–0.0013u)g, 0.45g]
0.4	[(0.05–0.0013u)g, 0.32g]
0.8	[(0.05–0.0013u)g, 0.11g]

.

5.2. Brake–Steering Coefficient Allocation Based on the Fireworks Algorithm

In this study, the fireworks algorithm (FWA) is not used as a generic optimization tool but is tailored for brake–steering control allocation to satisfy the constraints of the friction circle and vehicle stability.

(1)

Search Space Definition:

The search variables are a two-dimensional vector (λ, γ), where λ is the braking distribution coefficient and γ is the steering distribution coefficient. Their ranges are determined according to the friction circle conditions, Equations (35) and (36), and stability boundaries, ensuring that the allocation does not cause the tires’ longitudinal and lateral forces to exceed limits.

(2)

Fitness Function Construction:

To balance collision avoidance performance and vehicle stability, the fitness function Equation (44) is constructed as a weighted sum of the normalized braking deceleration Equation (41), safety distance Equation (42), and lane-change time Equation (43). The weights are selected according to a safety-first principle.

(3)

Search Strategy:

Local Refinement: High-fitness solutions generate more sparks within a small search radius, enabling fine-tuning of the allocation coefficients;

Global Exploration: Low-fitness solutions search over a larger radius, increasing the probability of escaping local optima;

Constraint Repair: Each newly generated solution is projected back to the feasible domain to satisfy physical and safety constraints.

(4)

Application Results:

Through iterative optimization, the FWA can quickly converge to the optimal (λ, γ), achieving dynamic optimal allocation of braking and steering adhesion utilization. Under different adhesion conditions, this significantly shortens the collision avoidance distance and improves the vehicle’s attitude stability.

The framework of the fireworks algorithm is mainly composed of the following four parts: the explosion operator, variation operation, mapping rule, and selection strategy [24]; the flowchart composing the framework of the fireworks algorithm is shown in Figure 12.

The longitudinal acceleration based on the brake distribution factor is

$$a_{xm}=\lambda a_{xmax}$$

(37)

where λ denotes the brake distribution coefficient.

The lateral acceleration based on the steering distribution coefficient is

$$a_y={\displaystyle \frac{2\pi y_e}{t_{e\_\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}}}}sin\left({\displaystyle \frac{2\pi t}{t_{e\_\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}}}}\right)$$

(38)

where the lane-change time based on the steering allocation factor is

$$t_{e\_\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{e}}=\sqrt{{\displaystyle \frac{2\pi y_e}{\gamma \mu g}}}$$

(39)

where γ denotes the steering distribution factor.

5.3. Fireworks Algorithm Evaluation Function Establishment

Normalization is carried out by polarization:

$${x_n}^{\prime }={\displaystyle \frac{x_n-x_{n\min }}{x_{n\max }-x_{n\min }}}$$

(40)

where x_nmax denotes the maximum value of the variable and x_nmin denotes the minimum value of the variable.

The normalized sum of braking deceleration speed, safe braking distance, and lane-change time is selected as the value of the fitness function, which is calculated as follows:

Brake deceleration a_xm′ normalization formula:

$${a_{xm}}^{\prime }={\displaystyle \frac{\lambda -{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$$

(41)

Safety distance d₀′ normalization formula:

$${d_0}^{\prime }={\displaystyle \frac{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}-\lambda {\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\lambda {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}-\lambda {\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$$

(42)

Transit time t_e′ normalization formula:

$${t_e}^{\prime }={\displaystyle \frac{\sqrt{{\gamma }_{\min }{\gamma }_{\max }}-\sqrt{\gamma {\gamma }_{\min }}}{\sqrt{\gamma {\gamma }_{\max }}-\sqrt{\gamma {\gamma }_{\min }}}}$$

(43)

In summary, the fitness function is

$$F=a_x^{\prime }+d_0^{\prime }+t_e^{\prime }$$

(44)

5.4. PID-Based Steering and Braking Controller

After obtaining the optimal allocation coefficients (λ, γ), PID controllers are used to track the desired longitudinal and lateral accelerations.

(1)

Control Objectives:

Brake Controller Input: The difference between the desired longitudinal acceleration and the current actual longitudinal acceleration (Equation (45));

Steering Controller Input: The difference between the desired lateral acceleration and the current actual lateral acceleration (Equation (46)).

(2)

Tuning Procedure:

Initial Estimation: The feasible range of PID gains is estimated based on a linearized vehicle dynamics model;

Ziegler–Nichols Coarse Tuning: The ultimate gain Ku and ultimate oscillation period Tu are measured to calculate the initial parameters;

Performance Optimization: Parameters are fine-tuned according to the ITAE criterion, maximum overshoot, and actuator smoothness;

Global Optimization with FWA: Within the stable domain of (λ, γ), the fireworks algorithm further optimizes the PID parameters to improve tracking accuracy and control smoothness.

(3)

Execution:

The PID outputs for braking and steering are converted into brake pressure and steering angle inputs via the inverse braking system model and inverse steering system model (Equations (47) and (48), respectively), thereby implementing coordinated brake–steering collision avoidance control.

The input to the brake controller is the difference between the desired longitudinal acceleration $\lambda a_x$ and the actual longitudinal acceleration $a_x$ of the vehicle at the current moment; that is,

$$e_x\left(t\right)=\lambda a_x\left(t\right)-a_x\left(t\right)$$

(45)

The input to the steering controller is the difference between the desired lateral acceleration $\gamma a_y$ and the actual lateral acceleration $a_y$ of the vehicle at the current moment; that is,

$$e_y\left(t\right)=\gamma a_y\left(t\right)-a_y\left(t\right)$$

(46)

The expression of the PID algorithm in the continuous time domain is

$$u(t)=K_p[e(t)+{\displaystyle \frac{1}{T_i}}{\int }_0^{t}e(t)dt+T_d{\displaystyle \frac{de\left(t\right)}{dt}}]$$

(47)

$$k_p=K_p, k_i={\displaystyle \frac{K_p}{T_i}}, k_d=K_p{T}_D$$

(48)

where K_p is the proportional gain, T_i is the differential time constant, T_d is the integral time constant, k_p is the proportional coefficient, k_i is the integral coefficient, and k_d is the differential coefficient.

6. Simulation Verification

6.1. Basic Software Parameter Settings

In this paper, the Audi A8, which comes with CarSim, is used for co-simulation, and the specific simulation parameters are shown in

Table 5. Table of model simulation parameters.

Parameter Symbol	Parameter Meaning	Numerical Value	Unit
m	Vehicle mass	1903	kg
Bf	Front wheelbase	1.6	m
Br	Rear wheelbase	1.6	m
La	Distance from center of mass to front axle	1.232	m
Lb	Distance from center of mass to rear axle	1.468	m
J	Inertia	4175	kg m2
R	Wheel effective rolling radius	325	mm
Ccf	Lateral stiffness of the front wheels	66,900	N/rad
Ccr	Lateral stiffness of the rear wheels	62,700	N/rad
hg	Center of gravity	590	mm

.

6.2. Combined Braking and Steering Obstacle Avoidance Conditions Under High-Adhesion Road Surfaces

According to the friction circle and vehicle stability constraints to determine the road surface adhesion coefficient of 0.8, the conditions under the brake distribution coefficient stability domain are [0.12, 0.87], while those under the steering distribution coefficient stability domain are [0.23, 0.85]. Through the fireworks algorithm for the brake distribution coefficient and steering distribution coefficient optimization, after 35 generations, the algorithm adaptability value is maintained at 0.759. The adaptability of the change is shown in Figure 13. At this time, the brake allocation coefficient and steering allocation coefficient are 0.16 and 0.75, respectively.

We set the speed of this vehicle to 72 km/h, the initial distance between the two vehicles to 60 m, the speed of the front vehicle to 72 km/h, the two vehicles traveling in the same direction, the front vehicle starting braking after 1 s, and the braking deceleration to 5 m/s²; the simulation results are shown in Figure 14. From Figure 14a–d, it can be seen that the vehicle starts collision avoidance from 89.2 m; at this time, the distance between the two vehicles is much smaller than the same conditions of steering collision avoidance distance between the two vehicles, lateral acceleration is controlled in the range of (−6, 6) m/s², the maximum brake deceleration is 6.82 m/s², and the vehicle’s traverse angle is controlled in the range of (−2°, 13.7°). In summary, the vehicle is able to track the desired lateral acceleration to complete the collision avoidance maneuver.

6.3. Combined Braking and Steering Obstacle Avoidance Conditions Under Medium-Adhesion Road Surfaces

According to the friction circle and vehicle stability constraints to determine the road surface adhesion coefficient of 0.4, the conditions under the brake distribution coefficient stability domain are [0.12, 0.87], while those under the steering distribution coefficient stability domain are [0.23, 0.85]. Through the fireworks algorithm for the brake distribution coefficient and steering distribution coefficient optimization, after 95 generations, the algorithm adaptability value is maintained at 0.503. The adaptability of the change is shown in Figure 15. At this time, the brake allocation coefficient and steering allocation coefficient are 0.16 and 0.81, respectively.

We set the speed of this vehicle to 72 km/h, the initial distance between the two vehicles to 40 m, the speed of the front vehicle to 72 km/h, the two vehicles traveling in the same direction, the front vehicle starting braking after 1 s, and the braking deceleration to 5 m/s²; the simulation results are shown in Figure 16. From Figure 16a–d, it can be seen that the vehicle starts collision avoidance from 80.1 m; at this time, the distance between the two vehicles is much smaller than the same conditions of steering collision avoidance distance between the two vehicles, lateral acceleration is controlled in the range of (−4, 4) m/s², the maximum brake deceleration is 3.31 m/s², the vehicle’s traverse angle is controlled in the range of (−2°, 8°). In summary, the vehicle is able to track the desired lateral acceleration to complete the collision avoidance maneuver.

7. Conclusions

This paper addresses the filter divergence problem caused by the asymmetry of the covariance matrix in existing pavement adhesion coefficient estimators, and it proposes an estimator based on the square-root cubature Kalman filtering algorithm (SCKF). The simulation results demonstrate that the proposed SCKF method significantly outperforms the traditional Cubature Kalman Filter (CKF) across different tires and road adhesion coefficients. Specifically, under low-adhesion conditions (μ = 0.4), the SCKF algorithm reduces the RMSE of each wheel by up to 92.7%, while under higher-adhesion conditions (μ = 0.8) the improvement rate ranges from 76.1% to 86.4%. These results confirm the superior estimation accuracy and filtering stability of the SCKF algorithm, effectively enhancing the safety and reliability of dynamic vehicle control.

Regarding the safe distance model and lane-change path planning, this study developed a safe distance model that accounts for various lead vehicle motion states and designs lane-change path functions based on fifth-degree polynomial trajectory planning. Furthermore, a decision logic for an active collision avoidance system was established, enabling the vehicle to make reasonable and timely collision avoidance maneuvers according to real-time traffic scenarios, thereby improving the proactivity and intelligence of collision avoidance.

To overcome the limitations of steering-only collision avoidance strategies under extreme conditions, a joint steering–braking collision avoidance strategy is proposed. Steering and braking allocation coefficients are optimized using the fireworks algorithm, and desired longitudinal and lateral accelerations are tracked via PID control to achieve closed-loop control. The simulation results verify that this approach initiates collision avoidance earlier and effectively regulates lateral acceleration and yaw angle, thereby enhancing the safety and stability of collision avoidance.

Future work will focus on applying the proposed algorithms to more complex road scenarios and multi-vehicle cooperative collision avoidance. Integration with deep learning methods will be explored to improve real-time performance and robustness. Additionally, implementation and testing on actual vehicle hardware platforms will be pursued to facilitate the industrialization of active safety technologies.