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Development of a parameter identification method for MF-Tyre/MF-Swift applied to parking and low speed manoeuvres.

ABSTRACT

A vehicle parking manoeuvre is characterized by low or zero speed, small turning radius and large yaw velocity of the steered wheels. To predict the forces and moments generated by a wheel under these conditions, the Pacejka Magic Formula model has been extended to incorporate the effect of spin (turn slip model) in the past years. The extensions have been further developed and incorporated in the MFTyre/MF-Swift 6.2 model. This paper describes the development of a method for the identification of the turn slip parameters. Based on the operating conditions of a typical parking manoeuvre, the dominant parameters of the turn slip model are firstly defined. At an indoor test facility, the response of a tyre under the identified operating conditions is measured. An algorithm is developed to identify the dominant turn slip parameters from the measured responses. Wherever possible, the algorithm is based on direct analytical relationships between the turn slip model parameters and the measured signals, otherwise an iterative method is applied. By means of comparing vehicle simulation results to instrumented vehicle measurements, the turn slip model, including the identified parameters, is validated. The results show the effectiveness of the developed method.

CITATION: Lugaro, C, Schmeitz, A., Ogawa, T, Murakami, T. et al., "Development of a Parameter Identification Method for MF-Tyre/MF-Swift Applied to Parking and Low Speed Manoeuvres," SAE Int. J. Passeng. Cars - Mech. Syst. 9(2):2016.

INTRODUCTION

The development and validation of advanced driver assistance systems (ADAS) is nowadays performed based on computer aided engineering (CAE). CAE relies on computer models that predict the response of the involved systems under certain operating conditions. For the development of electric power steering systems and parking assist systems, it is important to have an accurate prediction of the forces and moments generated by the tyres when the spin, the side slip angle and the longitudinal slip change with time and the longitudinal vehicle speed is low or zero. In this context, the Pacejka Magic Formula model has been extended to incorporate the effect of spin (turn slip model) [1]. The extensions have been further developed and incorporated in the MF-Tyre/MF-Swift-6.2 model [2]. The extensions of the model can be summarized as follows:

* Adaptations of the Magic Formula (MF) equations that describe the forces and moments generated by a rolling tyre, by introduction of the spin as an input quantity [1].

* Introduction of a model that can predict the aligning moment generated by a tyre in standstill conditions [1], the Van der Jagt model [3] is used as a basis.

* Introduction of a transition model that, as function of the forward speed, can smoothly switch between the rolling tyre model and the standstill model [1].

Because of the unavailability of suitable test facilities, only a reduced set of parameters can be directly identified. The parameters related to the standstill model can be identified based on measured data; the MF spin parameters related to the rolling tyre have until now been set to the default values as suggested in [1]. To validate the turn slip model with the reduced set of identified parameters, a test vehicle, equipped with sensors to measure the forces and moments on the wheels, has been used in [4]. Both standstill and low speed manoeuvres have been conducted; the results show that the turn slip model, with a reduced set of identified parameters, can qualitatively predict the aligning moment and lateral force generated by a tyre during a typical parking manoeuvre.

In this study, a method for a complete identification of the turn slip parameters for both the standstill and rolling tyre is developed. This is accomplished with the following steps:

* definition of the operating conditions of a typical parking manoeuvre;

* determination of the dominant turn slip model parameters for the given operating conditions;

* identification of the parameters based on experimental data acquired on a Flat-Trac test rig;

* validation of the parameterized model based on experimental data acquired on a test vehicle.

After this introduction, firstly the turn slip theory is reviewed. A discussion follows over the operating conditions and dominant parameters during a typical parking manoeuvre. Next, a description of the conducted experiments and the results of the identification of the turn slip parameters is given. Subsequently, validation results of the identified parameters are shown and discussed. Finally, the conclusions of this study are summarized and recommendations for future work are given.

REVIEW OF THE TURN SLIP THEORY

The MF-Tyre/MF-Swift model is based on the well-known Magic Formula model [1] that describes the steady-state force and moment response of a tyre subjected to pure and combined slip conditions, rolling at a given speed, inclination (camber) angle and vertical load (steady-state slip model). To take into account the transient behaviour of the tyre, the flexibility and damping of the tyre carcass (carcass model) and the relaxation behaviour of the contact patch (contact patch slip model) are accounted for to determine the input quantities (slips) for the steady-state slip model.

The turn slip functionality introduced in the MF-Tyre/MF-Swift model consists of extensions of the Magic Formula model, a contact patch slip model that describes the relaxation behaviour of the turn slip in the contact patch, a standstill model that predicts the aligning moment of the tyre subjected to a large steering angle in standstill conditions and finally a speed transition model that can gradually switch between the rolling and the standstill conditions. An overview of these extensions is given in this section; for simplicity, mainly the calculation of the aligning moment is considered. For the complete description of the theory, it is referred to the original work of Pacejka [1].

Steady-State Slip Model

The tyre spin is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [OHM] is the rotational velocity of the wheel around its spin axis, y is the inclination angle and [[epsilon].sub.y] is the camber reduction factor for the camber to become comparable with turn slip. The contribution of wheel yaw rate to spin ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is described by the component [[phi].sub.t] (turn slip), the contribution of camber to spin is described by the component [[phi].sub.c].

The general form of the Magic Formula (MF) is used to describe the aligning moment that is generated by a tyre under the effect of spin:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The product of the coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] describes the aligning moment stiffness against spin; this quantity can be calculated from the aligning moment stiffness against camber:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [R.sub.0] is the free tyre radius.

The same applies for the lateral force characterized by the following stiffness against spin:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Figure 1 depicts qualitatively the normalized aligning moment [M.sub.zn] as calculated by Equation (2) and the normalised lateral force [F.sub.yn], as a function of the normalized spin. The spin is normalized by the half contact patch length a, producing the unit-less quantity a[phi]. The aligning moment and the lateral force are respectively normalized by their peak quantities.

Interactions between turn slip, longitudinal slip and side slip are modelled by the weighting functions: [[zeta].sub.1],[[zeta].sub.2], [[zeta].sub.3], [[zeta].sub.5], and [[zeta].sub.6] [1]. These functions are all defined in the cosine MF form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [f.sub.i] is a monotonically increasing function. For example, [[zeta].sub.3] is the reduction factor of the cornering stiffness when turn slip occurs:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Carcass Model

The yaw velocity of the contact patch is calculated by equilibrating the yaw moments applied to the contact patch. The yaw stiffness [c.sub.[psi]], damping [k.sub.[psi]] and defection angle B of the tyre carcass determine this equilibrium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The carcass defection angle is the difference between the contact patch [[psi].sub.c] and wheel yaw angle [psi], i.e. [beta] = [[psi].sub.c] - [psi]. [M.sub.z] is the aligning moment of the contact patch due to slip. Note that in Equation (7) the yaw inertia moment of the contact patch is neglected.

Contact Patch Slip Model

The relaxation behaviour of the contact patch takes into account that a certain distance (relaxation length) must be travelled before the contact patch can respond to a variation of the spin. Note that, at a given speed, the turn slip is proportional to the contact patch yaw velocity: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

According to Pacejka [1], the relaxation behaviour of the contact patch can be decomposed into the concurrent effects of the contact patch length and of the tread width. A step response of the effect of the contact patch length is characterized by an initial slope that, after having reached a peak, asymptotically approaches zero; such behaviour can be modelled by subtracting two first-order responses ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), each asymptotically leading to the same level but starting with different slopes. The difference ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is proportional to the defection angle of the tread due to transient spin. In order to also account for the tread width in the transient response of the aligning moment due to spin, a first-order response characterized by the relaxation length [[sigma].sub.c] is further introduced ([[??]'.sub.c]).

The transient slip quantity [[??]'.sub.M] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Parameter [[epsilon]*.sub.[??]] accounts for the effect of the tread width, whereas [[epsilon].sub.[phi]12] accounts for the contact length. [[??]'.sub.M] is the transient spin that is used as input for the steady-state slip model to calculate the aligning moment in dynamic conditions.

The three turn slip states [[??]'.sub.c],[[??]'.sub.1] and [[??]'.sub.2] are calculated by the following three first-order differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Non-Rolling Tyre Model

The model described in the preceding sections can deal with both the situation of a rolling and a non-rolling tyre. In case of a non-rolling tyre, it reduces to a torsional spring with stiffness [C.sub.M[psi]] . This is valid as long as the contact patch is mainly in adhesion, i.e. for small yaw angles of the wheel. When a larger yaw angle is applied, sliding becomes important and the [M.sub.z] response becomes nonlinear up to the moment of full sliding, i.e. friction limit. In that case, a more adequate model must be used. Experimental evidence shows that the Van der Jagt model [3] is suitable for this application. The model consists of the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

From this model, the gradient of the aligning moment when p = 0 or [M.sub.z] = 0 reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

For large values of the yaw angle, the moment equals:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Equation (11) and Equation (12) state that, for small values of the yaw angle the response is that of a torsional spring, for large values of the yaw angle the aligning moment tends to the maximum value that can be achieved when full sliding occurs. Between these two situations, there is a transition controlled by the parameter [c.sub.0].

Figure 2 depicts the normalized aligning moment calculated from Equation (10). The aligning moment and yaw angle are respectively normalized by their peak values.

Speed Transition Model

To assure a gradual transition between the rolling and non-rolling tyre models, a weighting function is applied to the moment response:

[M.sub.z] = [W.sub.vlow][M.sub.z.non-roll] + (1 - [W.sub.vlow]) [M.sub.z,roll] (13)

Where [M.sub.z,roll] is the aligning moment in rolling conditions (Steady-state slip model), [M.sub.z,non-roll] is the aligning moment in non-rolling conditions (Non-rolling tyre model).

The coefficient [w.sub.vlow] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

OPERATING CONDITIONS OF A PARKING MANOEUVRE

Typically, a parking manoeuvre consists of braking a vehicle to a standstill, starting from a standstill and driving at a low speed (in both forward and backward directions), steering the wheels up to large steering angles when the vehicle is at a standstill as well as during driving. Because the longitudinal speed is low, the lateral acceleration is also low; in this study it is further assumed that also the longitudinal acceleration is low (a driver accelerates and brakes in a gentle manner during a parking manoeuvre).

Because the accelerations are low, the vertical load on the tyres is close to the value determined in static conditions, the longitudinal tyre slip is close to zero and the side slip angle of the tyres is close to the toe angle. The toe angle and camber angle are a function of the steering angle and depend on the suspension and steering system design. The test vehicle used for the experiments in this study shows approximately a vertical load in the range [3000, 4000] N, a camber angle in the range [-5, 5] deg and a toe angle in the range [-2.5, 2.5] deg.

The turn slip during a parking manoeuvre is a function of the vehicle motion. Because the longitudinal slip and side slip angle are small, the vehicle motion is calculated with a bicycle model based on kinematic relationships. The maximum steering angle that the inner steering wheel of the test vehicle can achieve (30 deg) is used as input signal. The model is depicted in Figure 3.

The vehicle model input quantities (V and [delta]) are sufficient to define the vehicle motion. The vehicle yaw rate reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where the value of l = 2.5 m is used for the wheel base; it corresponds to the value of a typical compact car. The yaw rate of the front (steered) wheel is the sum of the yaw rate of the vehicle and the steering rate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Considering that the side slip angle equals zero, the tyre contact patch velocity equals the velocity [V.sub.x], i.e. [V.sub.c] = [V.sub.x]. The turn slip corresponds therefore to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

To determine the operating conditions for the turn slip model, a reference parking manoeuvre is specified. The vehicle model input signals (V and S) and the turn slip model input signals ([??] and [[phi].sub.t]) of the reference manoeuvre are depicted in Figure 4.

The reference manoeuvre contains typical input conditions that occur during a parking manoeuvre: stay at a standstill (time interval [0, 2] s), steer standstill (time interval [2, 4] s), drive-off during steering (time interval [4, 6] s), steer at constant speed (time interval [6, 7] s) and drive in a circular turn (time interval [7, 10] s).

From Figure 4 it can be concluded that the turn slip is in general very large (i.e. infinite) when the vehicle is standstill and relatively low as soon as the vehicle is driving: assuming a half contact patch length of 0.05 m, a turn slip of 0.2 rad/m becomes after normalization equal to 0.01 which is very close to the origin in Figure 1. The large value of the steering angle permits the aligning moment to reach its maximum value during the standstill phase (time interval [2, 4] s). Table 1 and Table 2 summarize the operating conditions respectively for the standstill and the low speed situations.

In standstill conditions, the non-rolling tyre model is applied; this model is characterized by the parameters [C.sub.M[phi]] and [M.sub.z[phi][infinity]] (Equation (11) and Equation (12)). At low speed conditions, the steady-state slip model is applied; this model is characterized, for low values of the turn slip, by the parameters [C.sub.M[phi]] and [C.sub.F[phi]] (Equation (3) and Equation (4)). It is important to note that, during the transition from standstill to low speed and vice versa, the turn slip changes between an extremely high and a low value very quickly.

Consequently, [C.sub.M[phi]], [M.sub.z[phi][infinity]], [C.sub.M[phi]] and [C.sub.F[phi]] are the most important parameters to be identified. If turn slip acts in combined slip mode with side slip, the reduction factors [[zeta].sub.2], [[zeta].sub.3], [[zeta].sub.5] and [[zeta].sub.6] must also be taken into account.

EXPERIMENTS AND IDENTIFICATION OF THE TURN SLIP PARAMETERS

To produce data for the identification of the turn slip parameters, experiments are conducted on a MTS Flat-Trac test rig. The measured tire has dimensions 195/50R16 and an inflation pressure of 240 kPa.

The experiments are divided into two categories: standstill and low speed experiments. The first is used for the identification of the non-rolling tyre model parameters with the operating conditions described in Table 1; the second is used for the identification of steady-state turn slip model parameters with the operating conditions described in Table 2.

Standstill Experiments and Parameter Identification

For the identification of the turn slip parameters in standstill conditions, the wheel is positioned on a fixed surface, steered left and right up to a maximum angle under the effect of different vertical loads and camber angles. As summarized in Table 3, 3 vertical loads and 3 camber angles are used, producing 9 experiments.

Figure 5 depicts an example of the experiments conducted to identify the standstill turn slip parameters.

Typically the aligning moment is represented as a function of the steering angle. This is shown in Figure 6 for zero camber angle and the three vertical loads indicated in Table 3.

From the non-rolling tyre model theory (Equation (11) and Equation (12)), it can be concluded that:

* [C.sub.M[phi]] is equal to the derivative of the aligning moment with respect to the yaw angle changed by sign (note that the yaw angle [phi] corresponds in this case to the steering angle [delta]).

* [M.sub.z[phi][infinity]] is equal to the maximum value of the aligning moment.

These two quantities are directly derived from the measured data for each of the 9 experiments in Table 3. In the turn slip model theory, they are function of two sets of parameters ([P.sub.CM[phi]] and [P.sub.Mz[phi][infinity]]) and the input conditions:

[C.sub.M[phi]] = [f.sub.cM[phi]]([P.sub.CM[phi]],Fz,Y) [M.sub.z[phi][infinity]] = [f.sub.Mz[phi][infinity]]([P.sub.Mz[phi][infinity]], Fz,Y) (18)

As an example, in [1] the peak of spin moment is modelled as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where the parameter [q.sub.cr[phi]1] controls the level of the aligning moment that is further a function of the peak lateral friction coefficient [[mu].sub.y] , the vertical force F and the tyre free radius [R.sub.0]. [F.sub.z0] is the nominal load of the tyre.

An optimization routine is used to determine the values of the parameters included in [P.sub.CM[phi]] and [P.sub.Mz[phi][infinity]] that minimize the error between the measured and calculated values of [C.sub.M[phi]] and [M.sub.z[phi][infinity]].

The measured and simulated aligning moment are shown in Figure 7. Figure 8 depicts the same results with the aligning moment versus the steering angle, for zero camber angle and the three vertical loads indicated in Table 3. It is observed that the model can well predict the aligning moment at different levels of vertical load in conditions of adhesion and full sliding. The nonlinear transition from full adhesion to full sliding conditions is however less accurate; this is due to a hardcoded and constant value of parameter [c.sub.0], see Equation (10), in the current implementation of the MF-Tyre/MF-Swift model. Figures of all other combinations of operating conditions show comparable results; for simplicity these figures are omitted.

Low Speed Rolling Experiments and Parameter Identification

For the identification of the turn slip parameters at low speed, the wheel is positioned on the surface of the Flat-Trac (moving at 2 km/h), while steered left and right up to a maximum angle under the effect of different vertical loads and camber angles. As summarized in Table 4, 3 vertical loads, 3 camber angles and 3 maximum steering angles are used, producing 27 experiments.

The experiments with a maximum steering angle of 5 deg and 10 deg are conducted such that one full steering cycle (i.e. from left to right and back to left) has a period of 1 s; the experiments with a maximum steering angle of 15 deg are conducted such that one full steering cycle has a period of 1.41 s. Figure 9 depicts an example of the executed experiments.

Typically the aligning moment and the lateral force are represented as a function of the steering angle. This is shown respectively in Figure 10 and Figure 11 for zero camber angle, nominal load and the three maximum steering angles indicated in Table 4.

It is important to consider that the experiments described above have one implication: by steering a rolling wheel, both turn slip and side slip are produced at the same time. The side slip angle is proportional to the yaw angle (i.e. steering angle), the turn slip is proportional to the yaw velocity (i.e. steering velocity): the two signals are therefore phase-shifted by [[pi]/2] rad. Furthermore, the excitation frequencies of 1 Hz and 0.707 Hz do not allow the transient behaviour of the tyre to be neglected. The consequences are that the effect of interactions between turn slip and side slip and the effect of transient behaviour cannot be separated: the full MF-Tyre/MF-Swift model with combined slip and transient behaviour has to be used to identify the turn slip parameters. In this situation, it is not possible to derive direct relationships between model parameters and measured signals; therefore, for the parameters identification, a different approach is used.

In the turn slip model theory, the aligning moment and the lateral force are functions of two sets of parameters ([P.sub.Mz[phi]]. and [P.sub.Fy[phi]]) and the input signals that vary with time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

An optimization routine is used to determine the parameter values that minimize the error between the measured and calculated values of [M.sub.z] and [F.sub.y]. The calculated [M.sub.z] and [F.sub.y] are determined, for every optimization iteration (i.e. a configuration of the parameters sets), by running a simulation with the MF-Tyre/MF-Swift model.

The measured and simulated aligning moment and lateral force are shown in Figure 12. Figure 13 and Figure 14 depict the same results with respectively the aligning moment and the lateral force versus the steering angle, for zero camber angle, nominal load and the maximum steering angles indicated in Table 4. It is observed that the model can predict the aligning moment and lateral force in different conditions relatively well; the lateral force results are in general larger in magnitude than the measurement. Figures of all other combinations of operating conditions show comparable results; for simplicity these figures are omitted.

VALIDATION OF THE TURN SLIP MODEL WITH EXPERIMENTS CONDUCTED ON A TEST VEHICLE

The validation of the turn slip model with the identified parameters is conducted by comparing signals measured on a test vehicle with signals produced by simulation of a multibody vehicle model.

The test vehicle is equipped with:

* wheel force transducers that measure the forces and moments exchanged at the interface between the tyre and the wheel rim (e.g. lateral force, aligning moment);

* wheel trackers that measure the relative position of the wheel with respect to the vehicle body (e.g. steering angle, camber angle, vertical displacement);

* a steering wheel sensor that measures the steering wheel angular position and the applied torque;

* an optical velocity sensor that measures the vehicle longitudinal speed and the body side slip angle.

The input quantities to the multibody vehicle model are the steering wheel angle ([[delta].sub.steer]) and the driving/braking torque for each of the front wheels ([M.sub.yFL],[M.sub.yFR]). Figure 15 shows the test vehicle that is used for the validation, Figure 16 depicts a graphical representation of the multibody vehicle model; the red arrows indicate the model inputs.

The experiments are divided into two categories: standstill and low speed experiments. The first is used for the validation of the non-rolling tyre model of MF-Tyre/MF-Swift with the operating conditions described in Table 1; the second is used for the validation of the rolling steady-state behaviour of MF-Tyre/MF-Swift with the operating conditions described in Table 2.

Vehicle in Standstill

For the validation of the turn slip parameterization in standstill conditions, the vehicle is placed with the front wheels on a flat surface. The surface is made with a safety walk antiskid plate. Then, with the brakes engaged, the steering wheel is completely steered to the left and right directions. In this case, unlike the standstill experiments conducted on the Flat-Trac, the camber angle, the lateral position and longitudinal position of the wheel change as function of the steering angle. The maximum steering angle and the vertical load on the wheel are functions of the vehicle specifications. The typical operating conditions applied during this test are indicated in Table 1.

The forces and moments generated by one wheel are measured by means of a wheel force transducer. Figure 17 depicts the measured and simulated aligning moment. The simulation results match qualitatively well with the measured signals, but they appear to be larger in magnitude. This difference is likely caused by a different coefficient of friction between the Flat-Trac surface used for the parameters identification and the safety walk surface used for the vehicle validation tests. In the figure, also the aligning moment, calculated from the MF-Tyre/MF-Swift model without turn slip extension, is depicted. It is observed that the model with turn slip extension can much better predict the aligning moment during steering at a standstill.

After correction of the coefficient of friction in the model, to better represent the safety walk surface, the simulated aligning moment better matches the measured one, see Figure 18.

Low Speed Turn

For the validation of the turn slip parameters in low speed conditions, an open loop cornering manoeuvre is conducted. The vehicle is firstly driven straight-on at a speed of approximately 1 m/s, then the steering angle is rapidly increased up to the maximum value and after that it is kept constant. Figure 19 depicts the vehicle speed, steering angle and turn slip signals during the manoeuvre. Note that the turn slip is larger when the steering angle increases (Equation (16) and Equation (17)).

Figure 20 depicts the measured and simulated aligning moment for one wheel. The turn slip model results match well with the measurements. In the figure, also the aligning moment, calculated from the MF-Tyre/MF-Swift model without turn slip extension, is depicted. It is seen that the model with turn slip extension can better predict the aligning moment during driving at low speed, especially during the steering phase in the time frame [8, 10] s where the turn slip is large. In the figure, oscillations of the measured aligning moment can be noticed; their frequency is around 0.55 Hz which corresponds to the wheel rotational frequency. The oscillations are likely caused by tire non-uniformity

Figure 21 depicts the measured and simulated lateral force for one wheel. Both simulation model results, with and without the turn slip extension, match qualitatively well with the measurements, but they show larger values of the force after about 10 s. The reason of this discrepancy might be explained by the dependency of the steady-state slip characteristics (e.g. cornering stiffness) on the forward speed; the MF-Tyre/MF-Swift MF model parameters were previously identified at a larger speed (typically 60 km/h) while the manoeuvre shown in the figure corresponds to 4 km/h. This inconsistency in parameter identification affects the aligning moment response (e.g. Figure 20) in a much lesser extent, as this response is dominated by turn slip, whose model parameters are also identified at low speed. Note also that, during a low speed turn, the turn slip effect on the lateral force (difference between models with and without turn slip) is very little compared to the effect of the suspension toe angle that causes a side slip angle. Regarding the speed effect, also Van der Jagt [3] found a decrease of about 12 % in cornering stiffness between 60 km/h and 10 km/h. This leads to similar differences in the lateral force as are observed in Figure 21.

SUMMARY/CONCLUSIONS

In the past years, the turn slip model has been incorporated in MFTyre/MF-Swift 6.2. A method for parameter identification of the turn slip model had however not yet been established: until now only a few parameters could be directly identified. In this paper, the development of a method for the identification of the turn slip parameters is described. Simulation results of a vehicle model with the MF-Tyre/MF-Swift tyre model, including the identified turn slip parameters, are compared with experimental results. It is concluded that, provided the road surface friction and the steady-state characterisation of the lateral force in pure slip conditions are correctly modelled, the turn slip model with the identified parameters is able to reproduce with a good correlation the measured signals. In addition, simulation results of the MF-Tyre/MF-Swift tyre model without the turn slip model active are also compared. It is shown that the turn slip model is essential for accurate results during a parking manoeuvre.

RECOMMENDATIONS

The Flat-Trac test rig that was at disposal to conduct the low speed experiments does not permit a tyre in steady-state pure turn slip conditions to be excited. The consequences are that the parameter identification algorithm has to deal with a combined transient slip situation. New experiments shall be designed to overcome this issue.

The Magic Formula extensions to include turn slip are based on a previously identified steady-state characterization that is typically done using experiments at a speed of around 60 km/h, while parking manoeuvres are conducted at speeds close to zero. As suggested from Figure 21, the dependency of the lateral force on the forward speed cannot be disregarded. In the effort of developing a model valid for both parking manoeuvres and driving at high speed, an extension of the Magic Formula model that includes the forward speed as an input signal shall be developed.

As suggested from Figure 17 and Figure 18, the road surface friction plays a major role on the forces and moments generated by a tyre during a parking manoeuvre. To compensate for friction differences, a method should be developed to easily scale the friction to the road surface type.

REFERENCES

[1.] Pacejka, H.B., "Tire and Vehicle Dynamics, third Edition", (Butterworth-Heinemann, 2012), ISBN 978-0-08-097016-5.

[2]. MF-Tyre/MF-Swift (Version 6.2), TASS International, https://www.tassinternational.com/delft-tyre.

[3.] Jagt, P. van der, "The Road to Virtual Vehicle Prototyping", Ph.D. thesis, Mechanical Engineering Department, Eindhoven University of Technology, Eindhoven, The Netherlands, 2000.

[4.] Schmeitz, A., Hofstad, R. van der, Versteden, W., Niedermeier, F, et al., "Application and Validation of the MF-Swift Model for Parking Manoeuvres," Proceedings of the FISITA 2014 World Automotive Congress, F2014-IVC-069, 2014.

CONTACT INFORMATION

Carlo Lugaro MSc.

TASS International

Automotive Campus 15, 5708 JZ Helmond, The Netherlands

Work phone: +31 88 8277 124

carlo.lugaro@tassinternational.com

NOMENCLATURE

a - Half of contact patch length

[c.sub.[phi]] - Carcass yaw stiffness

[k.sub.[phi]] - Carcass yaw damping

[C.sub.F[gamma]] - Lateral force stiffness against camber

[C.sub.F[phi]] - Lateral force stiffness against spin

[C.sub.My].. - Moment stiffness against camber

[C.sub.M[phi]] - Moment stiffness against spin

[C.sub.M[psi]] - Moment stiffness against yaw angle

[F.sub.y] - Lateral tyre force

[F.sub.yn] - Normalised lateral tyre force

[F.sub.z] - Vertical tyre force

[F.sub.z0] - Nominal vertical load

[K.sub.M[psi]].. - Moment damping against yaw velocity

l - Wheel base

[M.sub.z] - Aligning moment generated by the tyre

[M.sub.zn] - Normalised aligning moment generated by the tyre

[M.sub.z[phi][infinity]] - Peak of spin moment

[M.sub.yFL] - Input moment on the front left wheel of the testing vehicle

[M.sub.yFR] - Input moment on the front right wheel of the testing vehicle

[P.sub.CM[PHI]] - Set of parameters that model [C.sub.M[phi]].

[P.sub.Mz[phi][infinity]] - Set of parameters that model [M.sub.z[phi][infinity]]

[P.sub.Fy[phi]] - Set of parameters that model [F.sub.y] in turn slip conditions

[P.sub.Mz[phi]] - Set of parameters that model M in turn slip conditions

R - Radius of curvature

[R.sub.0] - Free/unloaded tyre radius

[V.sub.c] - Magnitude of the speed of wheel centre

[V.sub.x] - Longitudinal speed component of wheel centre

[V.sub.low] - Speed threshold for transition between rolling and standstill

[W.sub.vlow] - Weighting factor for transition between rolling and standstill

[alpha]- Side slip angle of the wheel

[beta] - Tyre yaw torsion angle

[gamma] - Camber angle of the wheel

[delta] - Steer angle of front wheels

[[delta].sub.steer] - Steering wheel angle of the testing vehicle

[[epsilon].sub.[gamma]] - Camber stiffness reduction factor

[[zeta].sub.1] - Peak longitudinal force reduction factor

[[zeta].sub.2] - Peak side force reduction factor

[[zeta].sub.3] - Cornering stiffness reduction factor

[[zeta].sub.5] - Pneumatic trail reduction factor

[[zeta].sub.6] - Residual moment sharpness reduction factor

K - Longitudinal slip of the wheel

[[mu].sub.y] - Peak lateral friction

[phi] - Spin slip

[[phi].sub.t] - Turn slip

[[phi].sub.ct] - Turn slip of the contact patch

[[phi]'.sub.M]- Transient spin that contributes to the aligning moment

[psi]- Yaw angle of the wheel

[PSI] - Yaw velocity of the wheel

[[psi].sub.c] - Yaw angle of the contact patch

[[psi].sub.v] - Yaw velocity of the vehicle body

[OHM] - Wheel spin velocity

Carlo Lugaro

TASS International

Antoine Schmeitz

TNO

Toshiya Ogawa and Tetsuya Murakami

Toyota Motor Corporation

Sonny Huisman

TASS International

Table 1. Operating conditions in standstill

Vertical load    [F.sub.Z][N]            [3000,4000]

Camber angle     [gamma] [deg]            [- 5,5]
Wheel yaw angle  [psi] [deg]              [-30,30]
Turn slip        [[??].sub.t][rad/m]  [+ or -][infinity]

Table 2. Operating conditions at low speed

Vertical load      [F.sub.z][N]         [3000,4000]

Camber angle       [gamma] [deg]        [-  5,5]
Longitudinal slip  [kappa][-]             0
Side slip angle    [alpha] [deg]        [-2.5,2.5]
Turn slip          [[??].sub.t][rad/m]  [-0.2,0.2]

Table 3. Input conditions for the standstill experiments

Vertical load [N]  Maximum steering angle  Camber angle [deg]
                   [deg]

1960               20                      -4
3430               20                       0
4900               20                       4

Table 4. Input conditions for the low speed rolling experiments

Vertical load [N]  Maximum steering angle  Camber angle [deg]
                   [deg]

1960                5                       -4
3430               10                        0
4900               15                        4
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Author:Lugaro, Carlo; Schmeitz, Antoine; Ogawa, Toshiya; Murakami, Tetsuya; Huisman, Sonny
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Jun 1, 2016
Words:5875
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