In conventional steering system, a feedback torque is produced from the contact between tire and road surface and its flows through mechanical column shaft directly to driver. This allows the driver to sense the steering feel during driving. However, in steer by wire (SBW) system, the elimination of the mechanical column shaft requires the system to generate the feedback torque which should produce similar performance with conventional steering system. Therefore, this paper proposes a control algorithm to create the force feedback torque for SBW system. The direct current measurement approach is used to estimate torque at the steering wheel and front axle motor as elements to the feedback torque, while, adding the compensation torque for a realistic feedback torque. The gain scheduling with a linear quadratic regulator controller is used to control the feedback torque and to vary a steering feel gain. To investigate the effectiveness of the proposed algorithm, a real-time hardware in the loop (HIL) methodology is developed using Matlab XPC target toolbox. The results show that the proposed algorithm is able to generate the feedback torque similar to EPS steering system. Furthermore, the compensation torque is able to improve the steering feel and stabilize the system.
The latest generation of integrated steering system is steer by wire (SBW) system which is also known as independent steering system. It eliminates the need of mechanical column shaft between the steering wheel and the front axle system and was replaced with sensor, actuators, and electronic controller unit (ECU) as shown in Figure
Difference between SBW system and conventional steering system.
There are several characteristics of SBW system [
The main objective of this paper is to proposes a control algorithm to generate the force feedback torque for driver steering feel in the vehicle SBW system. The subsystem of SBW system which is steering wheel and front axle system is modelled and the control algorithm is explained. To verify the effectiveness of the proposed algorithm, the hardware in the loop (HIL) is presented and interfaced using a Matlab XPC target toolbox. The result was then is compared with electric power steering (EPS) system [
The SBW system is divided into two subsystems that consist of steering wheel and front axle system. The system diagram of subsystem is shown in Figure
System diagram: (a) steering wheel and (b) front axle system [
The main purpose of steering wheel motor is to generate the steering feedback torque for driver steering feel. Figure
Steering wheel system parameter.
Parameter | Description | Value | Unit |
---|---|---|---|
|
Motor resistance | 5.64 | ohm |
|
Motor inductance | 0.017 | henry |
|
Motor constant | 0.024 | Nm |
|
Motor inertia | 0.0036 | kgm2 |
|
Motor damping | 0.0068 | Nm/(rad/s) |
|
Lumped torque stiffness | 0.025 | Nm |
|
Voltage source | — | Volt |
The mathematical equations of the steering wheel system are written as follows.
The steering motor angular displacement is
The function of the front axle system is to ensure the front tire angle follows the steering wheel angle command according to the steering ratio [
Front axle system parameter [
Parameter | Description | Value | Unit |
---|---|---|---|
|
Motor resistance | 4.64 | ohm |
|
Motor inductance | 0.015 | henry |
|
Motor constant | 0.032 | Nm |
|
Motor inertia | 0.0062 | kgm2 |
|
Motor damping | 0.0036 | Nm/(rad/s) |
|
Lumped torque stiffness | 0.025 | Nm |
|
Rack damping coefficient | 0.015 | henry |
|
Rack lumped mass | 0.032 | Nm |
|
Rack linkage stiffness | 0.00062 | kgm2 |
|
Offset of king pin axis | 0.00036 | m |
|
Pinion gear radius | 0.025 | m |
|
King pin damping coefficient | 0.00062 | kgm2 |
|
Lumped front wheel inertia | 0.00036 | kgm2 |
|
Lumped torque stiffness | 0.025 | Nm |
Block diagram of front axle subsystem.
The mathematical equation of the front axle DC motor is written as follows.
The front axle motor angular displacement is
The parameter states are written as follows:
The stability characteristics of a steer-by-wire vehicle are affected by the influence of front tires via the feedback of the vehicle dynamic response. Thus, it is necessary to develop a single track-linear vehicle model as mentioned by author [
Single track-linear vehicle model.
The following assumptions are considered for normal driving maneuverer. These assumptions could simplify the vehicle model [ friction force in the vehicle speed is constant, angle for right and left tire is approximately the same.
With the aforementioned assumptions, the derived equations of the motion of yaw rate (
The parameter of a linear vehicle model directly can be measured, but some parameters need to be estimated. There are several methods to estimate the model parameter [
Linear vehicle model [
Parameter | Description | Value | Unit |
---|---|---|---|
|
Front cornering stiffness | 40000 | N/rad |
|
Rear cornering stiffness | 35000 | N/rad |
|
Length from center front wheel to (C.O.G) vehicle | 1.4 | m |
|
Length from center rear wheel to C.O.G vehicle | 1 | m |
|
Mass of the vehicle | 1535 | kg |
|
Inertia of vehicle | 2149 | kgm2 |
|
Speed of vehicle | — | km/h |
|
Front tire lateral force | — | Nm |
|
Rear tire lateral force | — | Nm |
|
Front tire slip angle | — | Nm |
|
Rear tire slip angle | — | Nm |
It has been known that the force feedback torque for driver steering feel should be created artificially, due to the elimination of mechanical column shaft in SBW system.
Moreover, the proportion of the steering feel obtained should be similar to conventional steering system. For this reason, the steering wheel in SBW system that is equipped with the motor actuator is used to generate the steering feel by controlling the total feedback torque (
Overview of force feedback torque control algorithm for SBW system.
Based on Figure
On the other hand, any change of the road surface could change the feedback torque magnitude of the front axle motor. Furthermore, the LQR controller of the front axle system is used to synchronize the front tire angle (
Thus, based on the proposed control algorithm, the total force feedback torque to the steering wheel system is written as follows:
Author [
The torque based on the function of vehicle speed is given as follows [
Steering torque based on torque map at different speed.
Thus, for this reason the front axle motor torque and steering wheel motor torque are added to the torque map. These ensure that the driver senses the feedback torque from road contact. The front axle motor torque is chosen due to its proportion to the road surface. The reaction torque of the road surface gives the effect to the rack and pinion system and it connects to the front axle DC motor. Thus, the front axle motor torque would always be proportional to the road surface. Any changes in road surface will influence the front axle motor torque. In order to measure the torque at the front axle motor, the direct current measurement technique is used to estimate the torque.
The direct relationship between torque and current of a motor can be defined [
Figure
Steering wheel and front axle motor torque at 80 km/h.
The phase compensation torque consists of the inertia torque and damping torque potential to recreate the artificial steering feel of the conventional steering system. Moreover, it is intended to adjust the steering feel, reduce the vibration, and stabilize the system. Due to this fact, the element of compensation torque is taken in order to improve the feedback torque for a driver steering feel and to stabilize the system.
The damping torque (
The inertia and damping torque responses are shown in Figure
Inertia and damping torque.
Proposed force feedback torque response at different vehicle speed.
The LQR controller provides the best possible performance to control and stabilize the system by changing the location of the poles of the system to the optimal location for a time response, overshoot, and steady state. This is done by designing the state feedback control
The wheel synchronization or directional control is a basic characteristic for steering function, where a front tire angle should follow driver input steering wheel angle according to the steering ratio. This is done by controlling the angle of the front axle motor. In the LQR control schemes, a feedback gain matrix is designed to achieve some compromise between the use of control effort, the magnitude, and the speed of response that will guarantee a stable system. Figure
Basic control block diagram of LQR controller.
The optimal LQR problem is often defined more generally and consists of finding controller transfer matrix that minimizes the cost function
Assume that the steering ratio is 15 : 1 and the gains of the LQR controller are
Front axle system response for wheel synchronization.
Steering wheel angle
Front axle motor angle
Rack displacement
Front tire angle
The results show that the angle of the front axle motor has a fast response compared to the EPS steering system. Moreover, the LQR controller is able to reduce the delay caused by the self aligning torque created from the contact between the tire and road surfaces. This not only improved the vehicle stability, but also ensured that the driver has a more confident level during manoeuvre. Compared to the EPS steering system, the delay caused by mechanical column shaft connected to the steering wheel system. Furthermore, it can be seen that the front tire angle is at almost fairly close to
In certain situations, the dynamic behaviour of the process changed in accordance with the conditions of the process. This situation could affect the stability of the system if not to control properly. Thus, it is possible to change the gains of the controller by monitoring the operating conditions of the process to have a stable system. The gain scheduling (GS) technique is used to change the gain of the LQR controller in order to control the feedback torque and the steering feel gain (
The relationship between a gain scheduling with LQR controller is shown in Table
The gains of LQR controller and steering feel.
|
|
|
|
|
|
---|---|---|---|---|---|
(0–30) | (±180° < |
0.03 | 0.65 | 0.001 | 0.14 |
(31–100) | (±46° < |
0.03 | 0.84 | 0.001 | 0.60 |
(101–120) |
|
0.05 | 0.95 | 0.001 | 0.95 |
The gains of the LQR controller are defined using Bryson’s rule method and the steering feel gain is defined using intuitive and has a range from 0.1 to 1. This range has been defined for various speeds: 10 km/h, 80 km/h, and 120 km/h after comparing with EPS steering system [
Comparing torque response between LQR and LQR + GS controller for different coefficient (
Comparison of RMS torque values of LQR and LQR + GS controller.
Reference | LQR | LQR + GS | |
---|---|---|---|
5.6361 | 5.0795 | 5.5450 | |
6.1218 | 6.8339 | 5.7243 | |
6.6184 | 7.2888 | 6.2080 |
Comparison of torque response of LQR and LQR + GS controller at different
Input steering wheel angle
Figure
Steering feel gains response at various vehicle speed.
Different steering feel gain at 80 km/h
Different steering feel gain at various speed
For hardware in the loop (HIL) setup, the front axle system and linear vehicle model are simulated on the host computer. The steering wheel system is replaced by HIL mechanism. The position of the steering wheel motor is measured using a rotary encoder with 500 pulses per revolution. To measure the steering wheel torque, the current sensor is connected in series with the DC motor driver. The steering wheel DC motor has a torque 12.9 Nm with 100 rpm. The single track linear vehicle model is used to generate the estimation self aligning torque as element to the feedback torque and disturbance to the front axle system. The HIL system is executed under 1 ms sampling time and interfaced using Matlab-XPC target toolbox configuration as shown in Figures
Steering wheel system, hardware overview.
Block diagram of the steering wheel-XPC target interfaced.
The assumptions through the experiment are listed as follows: the vehicle cruises at constant speed, the condition of dry asphalt road (
The experiments are done under high, medium, and low speed manoeuvre [
Although the model of steering wheel system has been derived, the model validation is necessary to ensure the system is fairly able to adequate for real time application. This section describes the motor model validation of steering wheel system. The block diagram is shown in Figure
Block diagram motor model validation for steering wheel system.
The input signal is
Parameter estimation for DC motor steering wheel system.
Parameter | Value | Estimation value | Unit |
---|---|---|---|
|
4.405 | 5.64 | ohm |
|
0.010 | 0.017 | henry |
|
1.459 | 0.024 | Nm |
|
0.002 | 0.0036 | (kgm2) |
|
0.004 | 0.0068 | Nm/(rad/s) |
|
1.65 | 0.025 | Nm |
Motor model validation: comparison between simulation and measurement.
Rate change of motor position
Position of the motor
Current of the motor
Both analog and digital devices have a trait of noise or unwanted features that have effect on the system performance such as electromagnetic interference. The noise could be either random or white noise. Attenuation of this noise is often as a primary goal in control system application and in particular when controlling the process system. The average value of noise typically is zero which will give misreading information to the process and it would not be possible to control the process. In order, to reduce the noise inference, filtering technique is used. A filter is used to remove unwanted noise. It removes some noise frequencies to suppress interfacing signal and reduce background noise. There are many types of filtering techniques such as low pass filter that has been used in this study. The low pass filter is used to filter out the interference noise from the output of the current sensor.
Figure
Noise filtering using low pass filter.
A steering torque comparison without and with compensation torque is shown in Figure
Steering torque influence on compensation torque.
Without compensation torque
With compensation torque
It is observed that, by adding the additional compensation torque, the steering torque response curve trends fairly close to EPS steering system. Thus, it helps to improve the driver steering feel during manoeuvre and stabilize the system. However, with the wide range adjustment gains of the compensation torque, it could improve the steering torque response.
The experimental steering torque results at medium speed manoeuvre are shown in Figure
Steering torque estimation and steering wheel angle response at 80 km/h.
Steering wheel angle at 80 km/h
Steering torque estimation at 80 km/h
At high speed manoeuvre, the input steering wheel angle that is more than
Steering torque estimation response at 120 km/h.
Steering wheel angle at 120 km/h
Steering torque estimation at 120 km/h
Steering torque estimation against steering wheel angle at 120 km/h
In general, the centering of the steering wheel performance or steering wheel returnability is better when the steering torque gradient is large and the hysteresis is small [
The steering torque results at low speed manoeuvre are shown in Figures
Steering torque estimation response at 10 km/h.
Steering wheel angle
Steering torque estimation at 10 km/h
Steering torque estimation against steering wheel angle at 10 km/h
The experimental steering torque is almost comparable to EPS torque, but the amplitude is decreases. The torque is decreased due to the fact that the vehicle speed is reduced which is effect on reducing the demand to the total feedback torque. Moreover, the large turning of the steering wheel increases the backlash of the motor which then contributes to the decrease of the motor torque. Thus, driver will sense softer during steering the wheel during maneuver. However, it has small different torque against EPS steering. Wide range adjustment of gains in proposed control algorithm may help improve the steering torque response.
The proposed force feedback torque estimation and control algorithm have been described. The proposed control algorithm is considered as an effort to generate the feedback torque for driver steering feel in the vehicle SBW system. The mathematical model of SBW subsystem consists of steering wheel and front axle systems that are modelled. The effectiveness of the control algorithm is verified through the experimental hardware in the loop (HIL) interfaced with Matlab XPC target toolbox software. According to the experimental results, the proposed control algorithm is able to generate the feedback torque and the driver is able to sense steering feel similar to that obtained in EPS steering system. The LQR controller with gain scheduling based on steering wheel angle and vehicle speed function provides better torque control that allows rejection of the uncertainty torque from a road condition. Furthermore, the steering feel gain is able to increase and lower the steering torque with improved steering stability at high speed and it improved the vehicle manoeuvre at low speed. It is also found that compensation torque is able to improve the feedback torque and to stabilize the system. However, it has small different torque magnitude that occurs due to the backlash of steering wheel motor and uncertainty disturbance of mechanical properties. However, with a wide range adjustment of the gains on the proposed control algorithm, it may improve and provide realistic steering torque for better driver steering feel.
In future work, the proposed control algorithm will be tested in real vehicle in order to have more realistic steering torque for further investigation and analysis.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is fully supported by the Ministry of Higher Education Malaysia and University Technology Malaysia under FRGS (vote: 4F370) and Research University Grant (vote: 00G64). This work is also supported by Proton Holding Berhad.