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A recursive least square (RLS) algorithm for estimation of vehicle sideslip angle and road friction coefficient is proposed. The algorithm uses the information from sensors onboard vehicle and control inputs from the control logic and is intended to provide the essential information for active safety systems such as active steering, direct yaw moment control, or their combination. Based on a simple two-degree-of-freedom (DOF) vehicle model, the algorithm minimizes the squared errors between estimated lateral acceleration and yaw acceleration of the vehicle and their measured values. The algorithm also utilizes available control inputs such as active steering angle and wheel brake torques. The proposed algorithm is evaluated using an 8-DOF full vehicle simulation model including all essential nonlinearities and an integrated active front steering and direct yaw moment control on dry and slippery roads.

The performance of a vehicle active safety system depends on not only the control algorithm, but also on the estimation of some key states if they can not directly be measured. Among these states to be estimated online, vehicle sideslip angle and tire-road friction coefficient have been extensively studied in the literature. It is noted that road friction is also used for determination of target response to driver’s steering inputs in the entire range of operation.

There are several strategies [

A common feature of most state observer and KF/RLS based algorithms for estimation of sideslip angle is that they rely heavily on an accurate tire model, which may vary during vehicle operation. To overcome the limitation, Hac and Simpson [

In [

The main idea of most slip-based friction estimation approaches is to predict the maximum friction based on the collected low-slip and low-friction data at normal driving, where normally acceleration/deceleration is less than 0.2 g [

Estimation of vehicle sideslip angle relies heavily on an accurate tire model, but the computing power required in such detailed models easily exceeds the control cycle time. For example, the famous Magic Formula tire model can very accurately represent the force and moment properties. However, due to trigonometric and exponential functions associated with such formulation with several associated coefficients, the time required for online calculation, far exceeds the control cycle time. On the other hand, if tire and vehicle dynamic models do not include the required details, estimation accuracy will be lost. Consequently, several efficient and accurate tire models with simpler expressions of tire forces have been investigated in the literature. In other words, a compromise between accuracy and complexity of tire model is required for online implementation.

In this paper, an RLS algorithm is proposed to estimate sideslip angle and road friction for online application during activation of active front steering and direct yaw moment control. Two main means are adopted to reduce the computational time of the algorithm. The first one is use of a modified Dugoff tire model, for which simple expressions of tire forces are used and parametric differences with respect to tire normal forces can be easily functionalized using polynomials. The second one is online linearization of the model and iterative computation for the proposed algorithm which are distributed to different control cycles without sacrificing estimation accuracy. A significant merit of the algorithm lies in the fact that it can provide estimates with reasonable accuracy without additional sensors. It makes adequate use of the data available through the control logic for correction steering angle and wheel brake torques. Comparison between estimated results and simulation data using Matlab/Simulink and an 8-DOF full vehicle model shows that the proposed algorithm is promising for practical use in active safety systems.

The algorithm considered in this paper is intended for real-time online estimation of sideslip angle and road friction for the vehicle stability control systems using active front steering and direct yaw moment control. Due to the limited computational authority of microprocessors used in vehicle control systems, the algorithm should not be too complex. A simple two-degrees-of-freedom (DOF) vehicle model is used to develop the estimation algorithm. Shown in Figure

Scheme of the proposed algorithm for sideslip angle and road friction estimation.

The estimator in Figure

In the vehicle model, tire forces are computed according to the estimated vehicle states

Shown in Figure

A two DOF vehicle model with four wheels.

Note that the vehicle velocity at the center of gravity,

Wheel load transfer is included in calculation of tire normal force as follows:

During a typical intervention of AFS/DYC, the tires often operate at or near the friction limit and combined-slip conditions may arise. Therefore, a nonlinear tire model capable of simulating the friction ellipse phenomena is required. A modified Dugoff tire model is used here for the estimator. First, lateral tire forces at pure-slip conditions are calculated using the modified Dugoff model and longitudinal forces are determined from the brake torques. Then, the lateral forces are further amended according to the magnitude of the longitudinal force. Variation of tire-road friction with respect to slip is included in the calculation of the lateral forces.

The pure-slip lateral force is first calculated for dry asphalt road with a nominal tire-road friction coefficient

The coefficient

Tire forces at pure-slip conditions based on a Magic Formula model and a modified Dugoff model.

When DYC is activated, brake torque is applied on some of the wheels. For simplification, tire longitudinal forces are calculated using the following equation

The target brake torque can always be realized without delay;

The longitudinal slip rate

Further,

When slip angle is less than 10° (this is almost always true as mentioned above), or equivalently

Comparison between two functions.

The recursive least square algorithm introduced here was developed based on a simple two-degrees-of-freedom vehicle model. Due to the nonlinearities involved in the equations, online linearization becomes a dynamic part of the algorithm.

As shown in Figure

At this point, functions

Equation (

Now the unknown parameter vector and the state vector can be defined using the following equations:

The state vector error can be expressed as

Define an index

By adopting the formulations given above and using the procedure in [

On certain situations such that when estimated road friction coefficients or sideslip angle are far from current operating point, a new operating point is needed and linearization of the functions

The following equations are used to compute the partial derivatives at

Computation of the partial derivatives using (

When any of the following conditions holds, substitute

If the value of estimated parameters changes too quickly, restrictions for their increment are applied. In this paper, maximum increment for

To facilitate understanding of parameter estimation using the proposed RLS algorithm, the main steps of the procedure are outlined as follows.

Go to Step

Go to Step

Similarly,

Go to Step

Go to Step

Go to Step

Matrix

If any of the conditions in (

From the viewpoint of online application, each of the steps is intended to be executed within one control cycle.

The algorithm is evaluated using the data from simulation of an AFS/DYC-based integrated control system. Simulation of a double lane change maneuver is conducted using Matlab/Simulink. A nonlinear 8-DOF vehicle model along with a combined-slip tire model and a single-point preview driver model is used. Control commands are executed through correction steering angle on front wheels and brake torque applied on one of the four wheels.

The data for the steering angles at front wheels, brake torques on the four wheels, yaw rates, lateral acceleration, and vehicle speeds are used as inputs to the RLS based estimator. Estimated results of vehicle sideslip angles and road friction coefficients are compared with those from the simulation of double lane change maneuver using Matlab/Simulink. This enables the reader to evaluate whether the results are sufficiently precise to be used in control.

Two scenarios of double lane change maneuvers are involved: one is on high friction road surface and the other is on low friction road surface, and the target vehicle speeds for the two scenarios are 110 km/h and 40 km/h, respectively.

The initial sideslip angle and nominal tire-road friction coefficients on both sides are assumed to be 0, 0.8, and 0.8, respectively. The forgetting factor

Double lane change on high-

Inputs to estimator

Estimated results and comparison with actual data

Double lane change on low-

Inputs to estimator (Part one)

Inputs to estimator (Part two)

Estimated results and comparison with actual data

In each figure, the first one or two diagrams illustrate the inputs to the estimator. Estimated results are plotted in the second diagram, together with the actual data for comparison. When integrated control quits from intervention, the estimated sideslip angle and nominal tire-road friction coefficients are reset to their initial values. This is due to the fact that, with the current sensors onboard vehicles equipped with active safety systems, it is not possible to determine the surface coefficient of adhesion as long as vehicle remains within the linear range of operation [

For the double lane change maneuver performed on dry road with

Figure

For evaluating the accuracy of the above estimated sideslip angle using the RLS algorithm, some results cited from [

Estimated results and comparison with experimental data for vehicle performing a double lane change maneuver, cited from [

On dry asphalt: 90 km/h, maximum ^{2}

On snow: 75 km/h, maximum ^{2}

To evaluate robustness of the proposed RLS algorithm with repect to certain variantons that may occur during vehicle operation (mass, moment of inertia, tire cornering stiffness etc.), more simulation was performed. As an example, Figure ^{2}, ^{2} for the estimator, while those for the double lane change maneuver are ^{2}, ^{2}. Though partly deviated from the actual states, the estimates are generally acceptable. In Figure

Double lane change with variation of vehicle inertia properties

Double lane change on high-

Double lane change on low-

A model-based recursive least square algorithm for estimation of sideslip angle and road friction using data from the active front steering and dynamic yaw control logic is proposed. The estimates are evaluated through simulation of double lane change maneuvers using Matlab/Simulink. The results indicate that the strategy of estimation is valid and successful without using additional sensors, on both high and low friction road surfaces. Robustness of the algorithm is evaluated through more simulation with variation of vehicle inertia properties, and results show that the estimates are generally acceptable but the parameters for the algorithm need to be further tuned.

Though not yet included in our investigation, we propose that the RLS algorithm developed in this research be combined with kinematic formulation to enhance estimation accuracy during abrupt change of sideslip angle.

Future work of the research may include evaluation of the methodology through hardware-in-the-loop and road tests and implementation of the estimation algorithm on a vehicle stability enhancement system for online applications.

Front left

Front right

Rear left

Rear right

Front

Rear

Center of gravity

Indicator for estimated value

Indicator for error between measured and estimated values.

Horizontal distance between vehicle COG and front axle

Longitudinal acceleration of vehicle

Lateral acceleration of vehicle

Matrix for RLS algorithm

Horizontal distance between vehicle COG and rear axle

Tire longitudinal slip stiffness

Tire cornering stiffness

Peak value of lateral force of tire

Longitudinal tire force in tire

Lateral tire force in tire

Lateral tire force in tire

Peak value of lateral tire force in tire

Lateral tire force at pure lateral sliding in tire

Vertical force on tire

Vertical static force on tire

COG height of total vehicle mass with respect to ground

Roll moment of inertia (about vehicle

Yaw moment of inertia (about vehicle

Wheel base

Total vehicle mass

Sprung mass of vehicle

Yaw rate

Tire static loaded radius

Tire lateral slip rate

Brake torque vector for all the four wheels, defined as

Brake torque on a single wheel

Time

Wheel track

Longitudinal velocity

Velocity vector at vehicle COG

Lateral velocity

State vector

Tire sideslip angle

Vehicle sideslip angle at COG

Total applied steer angle at wheels

Applied steer angle at wheels, result of driver’s input

Correction steer angle at wheels supplied by AFS

Longitudinal slip rate

Angle between velocity vector and vehicle

Forgetting factor

Parameter vector to be estimated

Index

Tire-road nominal friction coefficient.

This work is supported by National Science Fund of China, with an approval number of 50475003.