Estimation of Longitudinal Tire Force Using Nonlinearity Observer

Tire forces are the major forces propelling the road vehicles. They significantly affect the dynamic behavior of the vehicles. Estimation of the tire forces is essential in vehicle dynamics and control. This paper presents an observer-based scheme for estimation of the longitudinal tire force of electric vehicles in real time. The observer is based on a nonlinearity observer method. The pole-placement technique is used for determination of the observer gains. Simulation results demonstrate that the observer is able to estimate the tire force successfully. The experiments are implemented on a single-wheel electric vehicle test rig. The test rig comprises an electric motor driven wheel and a free-rolling drum simulating vehicle-on-road situations. Experimental results confirm the effectiveness of the present scheme.


Introduction
Electric vehicles (EVs) have become very attractive for replacing conventional internal combustion engine vehicles because of environmental and energy problems [1].The research and development of EVs and hybrid EVs have been investigated on various topics such as, for example, propulsion systems [2], power converters [3], and motion control [4].
Traction control plays an important role in vehicle motion control because it increases drive efficiency, safety, and stability.Tire forces are essential in traction control.They are the vehicular propulsive forces produced by friction between the rolling wheel and the road surface.The characteristic of the friction between the wheel and the road surface is very nonlinear.It mainly depends on the wheel slip and the tire/road surface condition.In [5], an approach to estimate the tire-road friction during normal drive is presented.The approach is based on a Kalman filter to give estimates of the slip-slope.In [6], an on-line least-squares method is used to estimate the parameters concerned with a friction force margin.The effect of the estimation is evaluated by applying the method to the breaking control.A slip-based method to estimate the maximum available tire-road friction during breaking is developed in [7].The method is based on the hypothesis that the slope of the slip curve at the low-slip region during normal driving can indicate the maximum friction coefficient.In [8], vehicle-dynamics-based methods for tire-road friction coefficients estimation are reviewed.The methods include slip-slope-based, lateralehicle-dynamics-based, and an EKF-based estimation methods.In [9], three different observers are developed for the estimation of slip ratios and longitudinal tire forces.The observers include one that utilizes engine torque, break torque, and GPS measurements, one that utilizes torque measurements and an accelerometer, and one that utilizes GPS measurements and an accelerometer.
This paper presents an observer-based scheme for estimation of the longitudinal tire force of EVs using a nonlinearity observer.Simulation and experimental studies are used to illustrate the effectiveness of the present scheme.A single-wheel test rig is used as an experimental test bench.

Nonlinearity Observer
The nonlinearity observer developed in [10,11] nonlinearity matric.is an unknown nonlinear function, which is estimated by the observer.
The fundamental idea of the observer is to approximate by a fictitious system ( ) By substituting Equation (2) into Equation ( 1), the observer can be chosen as where x and v are the observer gain matrices that must be chosen such that the observer is asymptotically stable.In this paper, the pole-placement technique is used to obtain the observer gains.When L L  H I and are chosen, the observer is reduced to a proportional-integral (PI) observer [11] and the estimated nonlinearity is given by

Single-Wheel Test Rig
The single-wheel test rig used in the experiments is shown in Figure 1.It consists of a drum set, a wheel set and a measurement/control unit.The role of the drum set is to let the wheel to behave like rolling on the road.The diameter and width of the drum are approximately 1.0 and 0.3 meters, respective.The wheel set consists of a tire, a 4-inch-rim wheel, a brushed DC motor, and loading masses.The wheel is directly driven by the motor thought a rigid shaft.The measurement/control unit consists of a signal condition circuit and a PC computer.The computer is installed with a 12-bit analog/digital interface board.The reader is referred to [12] for more details of the test rig.
The mathematical model of the wheel set can be simply written as where 1  is the rotational speed of the wheel, in V is the input voltage applied to the motor, F  is the tire force, 1 is the outer radius of the tire, 1 r J is the total inertia of the rotating part including the wheel set, 1 is the equivalent rotational damping constant, T C K is the torque constant of the motor, E K is the back-EMF constant of the motor, is the resistance of the motor windings, and R 0 K is the effective gain of the motor drive.
The mathematical model of the rotating drum can be expressed as where 2  is the rotational speed of the drum, 2 is the outer radius of the drum, 2 r J is the total inertia of the drum, and is the equivalent rotational damping constant.

C
The tire force F  can be expressed as where N is the normal force between the wheel and the drum, ( )   is the friction (or adhesion) coefficient,  is the driving slip ratio.The slip ratio is defined as where is the vehicle velocity.Here, since the motion of the vehicle velocity is simulated by the rotation of the drum, is substituted by the circumferential velocity of the drum, i.e., The parameters of the model are determined by direct measurements and simulation tuning by comparing with real experimental data.The first set of the parameters are the mechanical parameters whose values are determined directly from measurements.The values are summarized in Table 1.24.95 0.5 where of , , and 2 are experimentally determined by measuring the step responses of the speeds of the wheel and the drum.Figure 2 shows an example of the step response.The initial velocities of the wheel and the drum are zero.There is also a step change of in at time = 500 sec.After conducting several simulation tuning, the approximated values of the parameters are found to be , , and .In simulations, are, respectively, the slip ratio and the friction coefficient when the maximum friction occurred.
Simulation results of the same operating condition done in the experiment are shown in Figure 3.The results confirm the validness of the simulation model.Note that there is high-slip occurred in the first 50 sec.The simulation model is also able to detect this behavior, with some small errors.

Observer Design
To design an observer, it is assumed that the output is   .Note that, in actual vehicles, 1  can be easily obtained from an ABS speed sensor and 2  , which represents the vehicle velocity, can be computed from the velocity of a non-driven wheel or GPS measurements.Equations ( 11) and ( 12) can be combined and written in the form of Equation (1) as 4 0 12.4 48 where Here, F  is the nonlinearity estimated by the observer.From Equation (3) and let and , the observer can be expressed as where . The pole-placement technique is used here to obtain the observer gain L .The desired poles in continuous-time domain are simply chosen to be .1, 2, 3    In implementations, the observer is implemented in a digital computer.The sampling period is set to be 0.5 second.Thus, the corresponding discrete-time desired poles are 0.60 . Using the pole placement technique, it results the following observer gain 65, 0.3679, 0.2231 0.1612 0.1309 0.0021 0.7766 0.0859 0.0232 To verify the effectiveness of the observer, simulations are conducted in MATLAB/Simulink.The simulation results are shown in

Experimental Results
The sampling rate of 0.5 sec is used in all experiments.The same operation condition used in the simulations is repeated in the experiments.The experimental results are shown in Figure 5.They indicate that the observer is successfully implemented.Note that, in experiments, the true value of the tire force is unknown because it is not measurable.Since the experimental responses are very similar to those in the simulations, this implies that the estimates of the tire force is convincing.

Conclusions
Observer-based estimation of tire forces for road vehicles is studied in this paper.The observer is based on the nonlinearity observer developed in [10,11].The poleplacement technique is used for determination of the observer gains.Both simulation studies and experiments are conducted in the paper.In the experiments, a single-wheel test rig is used as a testing platform.Both simulation and experimental results show the effectiveness of the present observer.

Figure 1 .
Figure 1.Photograph of the single-wheel test rig.