Energy Optimal Control Design for Steer-by-Wire ... - Semantic Scholar

M. Selçuk Arslan1 Department of Mechatronics Engineering, Yildiz Technical University, Barbaros Boulevard, Besiktas, Istanbul 34349, Turkey e-mail: [email protected]

Naoto Fukushima Fukushima Research Institute, Ltd., Tokyo 104-0032, Japan

1

Energy Optimal Control Design for Steer-by-Wire Systems and Hardware-in-the-Loop Simulation Evaluation A Steer-By-Wire (SBW) control scheme is proposed for enhancing the lateral stability and handling capability of a super lightweight vehicle by using the energy optimal control method. Tire dissipation power and virtual power, which is the product of yaw moment and the deviation of actual yaw rate from the target yaw rate, were selected as performance measures to be minimized. The SBW control scheme was tested using Hardware-In-the-Loop (HIL) simulation on an SBW test rig. The case studies performed were high-speed rapid lane change, crosswind, and braking-in-a-turn. HIL simulation results showed that the SBW control scheme was able to maintain vehicle stability. The proposed SBW control design taking advantage of the full range steering of front wheel, significantly improves the vehicle handling capability. The results also demonstrate the importance of SBW control for super lightweight vehicles. [DOI: 10.1115/1.4029719]

Introduction

Studies on SBW mainly focus on active steering control, which we use in this work to describe the control improving driver’s steering command to enhance vehicle stability and handling, and steering feel. In the active steering control area, SBW has the potential to increase vehicle performance by improving its handling and stability [1–6]. As indicated in the work of Ackermann et al. [3], active steering is an efficient technique, since the safety margin of a vehicle can be enhanced by making the most of physical limits in terms of the tire forces. In his previous work [7], Ackermann showed that making the yaw rate unobservable from the lateral acceleration enables the vehicle to be robust to yaw disturbances. Application of that approach on a test vehicle for a case of l-split braking demonstrated that active steering of front wheels enhances the active safety significantly. In Ref. [8], an active control steering approach combining the feedforward and feedback H1 controllers was reported. The control objectives given as the enhancement of vehicle stability region, rejection of disturbances, enhancement of the damping of vehicle responses, and exhibiting a desired steady-state behavior were accomplished by active steering. Although not intended for SBW systems, there are some researches in the literature proposed yaw rate control methods for lightweight vehicles [9–13]. Particularly, studies in Refs. [11–13] showed the effectiveness of yaw moment control, as we also emphasized the impact of yaw moment in this work, by applying designed optimal control methods. Despite the fact that research efforts on SBW systems started in the early 1990s, application of the technology to passenger cars has not yet been practical, because of the reliability issues. In Ref. [5], a new side-slip estimation scheme based on global positioning system and inertial navigation system sensor measurements was proposed. Handling characteristics from understeering to neutral steering were tuned by the SBW control system which employed the developed side-slip estimation approach. Zheng and Anwar [4] proposed a gain scheduled control system which seeks a compromise between robust decoupling of yaw and lateral motions and yaw damping. The control system was implemented in an 1 Corresponding author. Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 24, 2014; final manuscript received January 21, 2015; published online March 19, 2015. Assoc. Editor: Junmin Wang.

SBW car via active front wheel steering. To deal with the vehicle yaw stabilization problem, G€uvenç et al. [6] developed a robust model regulator for an SBW system. In HIL simulation tests, the controlled vehicle succeeded in following the desired dynamics. In the automotive industry, HIL simulation has been widely used to test newly designed components or systems without the need for a vehicle. Zheng et al. [14] used an HIL simulation system to evaluate their proposed control system, which includes a yaw moment controller and wheel slip controller, to improve dynamic stability under critical lateral motions. In this study, we propose an energy optimal control scheme to improve the handling and stability of road vehicles. This control approach is mainly based on the minimization of both tire dissipation power and a virtual power. We use the tire dissipation power as a performance measure since the utilization of the tire lateral forces by the control system has improved the vehicle handling performance [15]. In the analysis of experiment results, it was understood that the terms appearing in the control law, which are related to the tire dissipation power, mainly contribute to the stability of the vehicle. As a second important performance measure, the virtual power defined as the product of yaw moment and the deviation of actual yaw rate from the target yaw rate is introduced. The optimal control method we used enables the utilization of such kind of novel performance criteria. Although, the virtual power does not have a physical meaning, the combination of yaw rate following function and yaw moment describes the performance of vehicle response to the steering input. The experimental results showed that the responsiveness of the vehicle was improved by the use of virtual power as a performance measure. The generated steering angle in the case of energy optimal control exhibited a distinguishing characteristic: the sudden changes in steering angle during the rapid lane change task affected the handling of the vehicle. Those sudden changes appear as a result of the preventive action of the control system to the increase in sideslip. Furthermore, the lateral displacement of the vehicle during the rapid lane change verified that the energy optimal control provides an increased responsiveness. To test the performance of the SBW control system, we have used the HIL simulation technique in our experiments. The hardware-under-test emulates the steering system of a conventional passenger car. The software part consists of an 11 degrees-of-freedom (11DOF) full vehicle model and the target control system. We chose three case studies, which are sufficient to evaluate the

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SBW control approach, and used them to perform simulations of a super lightweight vehicle. In super lightweight vehicles, the vehicle has relatively lower yaw moment of inertia compared to normal weight vehicles. Therefore, the vehicle’s response to steering input is more sensitive and grows worse with disturbances such as side-wind. By applying an effective control method, such problems can be eliminated. In this study, energy optimal control method was tested for challenging cases. Although the control of a super lightweight vehicle without SBW is a quite challenging task in certain conditions like in our case studies, the SBW control significantly improves the handling of such vehicles. The chosen super lightweight vehicle has the same volume as a minivan class automobile and is almost half its weight. The motivation for such a choice is that the development of environment friendly vehicles has received much attention from academia and car manufacturers. The need to increase fuel efficiency and decrease pollution has stimulated the search for new materials and structural designs to reduce the weight of vehicles [16]. Given that emerging developments in the automobile industry provide indications of the coming availability of super lightweight vehicles in the market, the use of SBW technology in lightweight vehicles seems inevitable. In order to validate the vehicle model used in HIL simulations, a set of data collected from an FR sedan in field tests was compared with data obtained from the simulation of same tests in a computer. The validation study showed that the nonlinear vehicle model is appropriate for conducting HIL simulations accurately. The details of validation study are given in the Appendix. The outline is as follows: In Sec. 2, first, the energy optimal control method is explained briefly. Second, the vehicle model used in the design of controller and the energy optimal control design are introduced. The last part of Sec. 2 is dedicated to the calculation of the desired yaw rate. Section 3 presents the description of HIL environment, which includes the full vehicle model, steering system model, and SBW test rig. In Sec. 4, the case studies are given and the HIL simulation results are discussed.

where q is the generalized coordinates vector, g is a scalar function denoting the control-performance, uT q_ is the power delivered to actuators by the controller, and R is a positive-value weighting factor. The combination of Eq. (1) with the integral of the powerbalance equation yields the criteria function. The criteria function is employed just to seek the ideal system. By formulating the criteria function as follows: ð J ¼ Ldt (2)

2

the scalar function, L, is defined to seek the necessary conditions of optimal control

SBW Control Design

2.1 Energy Optimal Control Method. Energy optimal control is a new method for control of both linear and nonlinear mechanical systems [17]. Although this method has similarities with the classical optimal control, it is not an extension of the existing optimal control theory. In the classical optimal control, generally, calculus of variations produces nonlinear two-point boundary-value problems, which require numerical solutions since the solution cannot be obtained analytically. Therefore, in general, control signals cannot be generated in real-time. However, in the energy optimal control method, the optimization problem is solved by the closed-loop itself in real-time, as is explained below. While closed-loops in classical control theory provide necessary feedback to the system to attain the optimal state, in the energy optimal control, closed-loop produces the solutions of simultaneous differential equations in real-time. One of the simultaneous differential equations represents the controlled system, and the other is the control law which describes the ideal motion of the same system. In the design of energy optimal control, the aim is to make the behavior of controlled system same with the behavior of the control law, which holds the ideal characteristics of the controlled system. In the closed-loop, the control law part generates the output of the ideal system, then, this output is applied to the controlled system. In this way, the behavior of the controlled system becomes similar to the behavior of ideal system. The ideal system is obtained by minimizing a criteria function as in the classical optimal control theory, but the criteria function is redefined in the energy optimal control method. The criteria function consists of a: (1) power-balance equation of the controlled system, (2) control-performance, and (3) performance measure describing the input power delivered to the controlled 071005-2 / Vol. 137, JULY 2015

system. It is important to note that the performance measure called control-performance can be of any form. This feature of the energy optimal control enables to define a performance measure without any restriction, whereas the classical optimal control theory works well with performance measures of quadratic form. The control method is basically the calculation of control law from the output of the controlled system, and input of the control law back to the plant. To define the criteria function, first, the power-balance equation, P, for a controlled mechanical system is calculated. Each equation of motion corresponding to each DOF is multiplied by the corresponding velocity so that the sum of all resulting equations yields the power-balance equation. In designing the criteria function, how a power-balance equation can be used is determined by the control engineer. As well as using the power-balance equation of whole system, a power equation considered to be the most effective in control can be used. As an example for the latter, instead of using the whole power-balance equation of the vehicle, only the total power equation describing the dissipation in tires is selected in this work. The system’s performance measure function, J 0 , is then determined ð _ _ q€Þ þ RuT qgdt (1) J 0 ¼ fgðq; q;

_ q€Þ _ q€Þ þ Ra uT q_ þ Rb Pðq; q; L ¼ gðq; q;

(3)

Minimizing the integral of L means that the control-performance and the input energy weighted by Ra, and stored and dissipated energy weighted by Rb are to be minimized. The necessary condition to minimize the integral of L is the Euler equation. By applying the Euler equation to L and rearranging the result, the following expression which is analogous to a control law is derived:

uT ¼

ð ð 1 @P @P d @P @g @g d @g Rb dt þ þ dt þ Ra @q @ q_ dt @ q€ @q @ q_ dt @ q€ (4)

This control law represents the ideal system with inverse input–output property. The partial differential terms of P in the control law (4) are very important, since the dissipation terms coming from P play an important role in the stability. On the other hand, the effect of the partial differential terms of g on the controlled system depends on the selected control-performance. As for advantages of this method, in existing optimal control approaches, the solution of nonlinear differential equations in real-time is required. However, in the energy optimal control method, the real-time optimal control law is derived without solving the nonlinear differential equation. Hence, the nonlinear characteristics of systems can be reflected to the optimal control law. The method is easy to apply, not much mathematically involved and allows the use of performance indices in various forms. Transactions of the ASME


2.2 Energy Optimal Control Design. In this section, an SBW control scheme is designed using the control method explained in Sec. 2.1. Note that in the energy optimal control method, by definition, the total power equation of the system is used to obtain the criteria function. Here, instead of the total power, we use only the tire dissipation power in the criteria function, since it sufficiently represents the system dynamics required for the control purpose. The vehicle model used here is illustrated in Fig. 1. In the design of the controller, the following simple 3DOF model describing the lateral motion of a vehicle is used mðu_ vrÞ ¼ Fx mðv_ þ urÞ ¼ Fy Iz r_ ¼ Mz

(5)

where u is the longitudinal velocity, v is the lateral velocity, and r is the yaw rate. The mass of the vehicle is m and the moment of inertia about the z axis is Iz. In Fig. 1, d is the steering command and d* is the controller input. The total longitudinal force, Fx, and lateral tire force, Fy, are Fx ¼ Fx1 þ Fx2 þ Fx3 þ Fx4 and Fy ¼ Fy1 þ Fy2 þ Fy3 þ Fy4. Here, Fxi and Fyi are longitudinal and lateral tire forces as shown in Fig. 1. Since the contribution of longitudinal tire force to yaw moment, Mz, is negligibly small compared to that of the lateral force, the vehicle yaw moment can be simplified to Mz ¼ lf(Fy1 þ Fy2) – lr(Fy3 þ Fy4). Here, lf and lr denote the distances from CG to front axle and to rear axle, respectively. Then, as a first step in describing the criteria function, the sum of each tire dissipation power, P, based on Eq. (5), can be given by P ¼ Fy v þ Mz r

(6)

Another important energy that we require is the input power delivered to front wheels. The control effort Pu ¼ Uðv þ lf rÞ

(7)

simply describes the control input power to the vehicle. Here, U denotes the input force on front wheels. Besides the power equations (6) and (7), a control-performance measure representing the control target needs to be found. For this purpose, we introduce the virtual power g ¼ R1 ðrd rÞMz

(8)

by taking advantage of the flexibility in the energy optimal control method. In Eq. (8), R1 is the weighting factor, and the desired yaw rate, rd is defined in Sec. 2.3. As it is well known, in a terminal control problem, deviation of the final state of a system from its desired value is tried to be minimized. Similarly, we define the virtual power to represent the deviation of actual yaw power from the desired yaw power. Here, the yaw power is the product of yaw rate, r, and the yaw moment, Mz. Although Eq. (8) simply implies the response of vehicle to the input, it does not have a direct physical meaning here. Now combining the power equations (6)–(8), the function L can be described as follows: L ¼ g þ Ra Pu þ Rb P

(9)

with the weighting factors Ra and Rb. In Eq. (9), the first term on the right-hand side represents the ideal characteristics of the vehicle, which is determined by the control engineer. The second term is the power to be minimized, which is delivered to actuators by the control system. A negative value of this term means the energy regeneration and the controller receives energy from the vehicle. The third term represents the tire dissipation power to be minimized. The minimization of this term implies the control effort making the motion of vehicle stable. Employing the Euler equation to minimize the criteria function, (9), the control law is obtained @Fy R1 @Mz Rb @Mz Mz þ ðr rd Þ Mz þ vþ r U¼ R a lf @r Ra lf @r @r (10) Since the control law (10) describes the side force, it should be divided by the front wheel cornering stiffness, Kf, to find the control input, d*. The lateral force acting on the vehicle, Fy is generated as a result of the steering angle, d. In the same way, U is the lateral input force generated as a result of the correcting steering angle, d*, calculated by the controller. The lateral tire force at one of the front tires can be defined as Fy1 ¼ Kf(d hv), where hv is the angle between the longitudinal axis of the vehicle and the velocity vector. By neglecting the effect of hv, the lateral input force can also be written as U ¼ Kfd*, or, d* ¼ U/Kf In the design of energy optimal control, the designed control system has been kept as simple as possible. Since the objective is lateral control of the vehicle, the tire dissipation power, which is mainly related with the lateral motion, is selected. Some dynamics of the vehicle, such as roll and tire dynamics, are not included in the control model. In doing so, it is desired that the vehicle has an ideal motion. For instance, by excluding the roll motion, it is aimed that the vertical force variation at each wheel, so the area of contact patch, would not have an effect on the cornering motion. Since the roll motion has an adverse effect on the stability of vehicle during cornering, and so its presence means a nonideal motion, involvement of the roll motion into the ideal system is contrary to the energy optimal control approach. On the other hand, the vehicle model includes the motion of suspension system, and therefore, the effect of roll motion also appears in the control law through the states of vehicle.

Fig. 1

Kinematic model of an automobile

Journal of Dynamic Systems, Measurement, and Control

2.3 Desired Yaw Rate. The desired yaw rate used in the control approach described above is calculated in this section. In most yaw stability control approaches, the desired yaw rate is JULY 2015, Vol. 137 / 071005-3


formulated for the steady-state cornering with a constant steering angle for a single-track vehicle model. It is expressed as a function of the steering angle and longitudinal velocity [18]. In our approach, we use the following formula which has the same form as the simplified magic formula: rd ¼ DsinðCtan1 ðBdÞÞ

(11)

with 1 D ¼ c1 þ c2 ; u B¼

C ¼ c0

1 1 u CD 1 þ Au2 lf þ lr

A ¼ m

(12)

lf Kf lr Kr 2Kf Kr ðlf þ lr Þ2

Fig. 2

and 8 > DsinðCtan1 ðBdÞÞ; > < rd ¼ D; > > : D;

Ctan1 ðBdÞ p=2 Ctan1 ðBdÞ > p=2

(13)

3

the steering actuator is transmitted to the road wheels through the gearbox, rack, and tie-rod. The relation between the angular position of the motor shaft and the steering angle of the road wheel is given by

Ctan1 ðBdÞ < p=2

In Eq. (12), c0 is the shape factor parameter, c1 and c2 are the peak factor parameters, Kf is front wheel cornering stiffness, and Kr is rear wheel cornering stiffness. In order to maintain the tire cornering capability within certain limits to prevent the vehicle from experiencing excessive lateral acceleration, rd is constrained as given in Eq. (13). In Eq. (12), product of B and CD corresponds to the aforementioned desired yaw rate formulation.

HIL Simulation Environment

In this study, we are concerned with the use of HIL simulation in active steering control to evaluate the performance of designed control methods for an SBW system of a super lightweight vehicle. For this purpose, an SBW test rig that represents the steering system of an automobile was designed. The control method described in Sec. 2.2 was developed for the SBW test rig. The mathematical model of the vehicle, controller, and the physical system were configured and interfaced so as to constitute a closed-loop. In Secs. 3.1–3.3, this HIL simulation environment is introduced without entering into details. 3.1 Vehicle Model. To present realistic results, a detailed nonlinear model is used in the simulations. The mathematical model is an 11DOF full vehicle model, which consists of the chassis, suspension, and wheel dynamics. The sprung mass has 6DOF, and each of four wheels, as an unsprung mass, has 1DOF. The vehicle is a front wheel steered vehicle, thus, the steering angle adds 1DOF more. The mathematical model of vehicle is not given here, but, the reader is referred to Refs. [18] and [19] for the mathematical models used in this study. The model used for tires is the well-known brush tire model [20,21]. The self-aligning torque generated in a tire is also modeled based on the brush tire model for both braking and acceleration. In the tire model, the friction coefficient was calculated as a function of side-slip velocity for a dry road condition. 3.2 Steering System Model. The steering system considered here consists of a steering actuator, rack, gearbox including pinion and worm gears, tie-rod, and electromechanical system emulating a pair of road wheels, Fig. 2. Since the steering-wheel command is generated automatically in the experiments, the steering wheel system is not included into the model. The motion generated by 071005-4 / Vol. 137, JULY 2015

SBW test rig

hm ¼ nNd

(14)

where n is the worm gear ratio and N is the ratio between the pinion and road wheel. The dynamic equation of the steering system is written as Im h€m þ

1 Tw þ Tf þ TSAT ¼ Tm nN

(15)

where Im is the moment of inertia of rotor (steering motor). The steering motor torque and the self-aligning torque are denoted by Tm and TSAT, respectively. The torque occurring around the kingpin axis, Tw, is assumed as equivalent to the moment generated by the rotational motion of a mass about that axis. The details are given in Sec. 3.3. By defining this torque as follows: Tw ¼ 2Iw d€ ¼

2 € Iw hm nN

we write the following equation for the steering torque: ! 2Iw € Im þ hm þ Tf þ TSAT ¼ Tm ðnNÞ2

(16)

(17)

where Iw is calculated by summing the moments of inertia of rotor and extra weight, and hm denotes the steering motor angle. The friction of the steering system is modeled by Coulomb friction Tf ¼

Fs sgnðh_m Þ N

(18)

where Fs is the Coulomb friction constant. To evaluate the performance of the model given in Eq. (17), we analyzed its frequency response. The frequency response of the real system for a given sinusoidal input is compared with that of simulation model. Based on those analyses, a proportional þ derivative (PD) motion controller was designed for the torque control of steering actuator and was verified by the tests carried out in the physical system. 3.3 SBW Test Rig. The SBW test rig was designed to test the control algorithms is an emulator system that replicates the steering system of a conventional automobile. The present system and general architecture of the test rig are shown in Figs. 2 and 3, Transactions of the ASME


respectively. The embedded software part mainly consists of the vehicle model and controllers. The generated control commands are sent to the actuators and the sensor information from the hardware is fed to both vehicle model and control system. The vehicle model and controllers were designed in Simulink and the I/O connections with the controller board were made in Simulink by using Real-Time Interface software. The built code of simulation model was embedded to the controller board. The controller board is a dSPACE DS1103-1GHzPX4CLP to which the hardware and software parts are interfaced. The steering actuator is a brushless alternating current (AC) servo motor. The worm gear attached to the rotor drives the pinion gear shaft. The rack connected to the tie-rod is driven by the pinion gear, where the gear ratio, N, is equal to steering ratio. In this test rig, the road wheels are mainly emulated by a pair of AC servo direct drive motors. When a self-aligning torque is calculated by the tire model during the maneuver of the vehicle, it is used as a torque command to drive the motors. Thus, the generation of selfaligning torque in a road wheel is emulated by a direct drive motor. On the other hand, the torque about the kingpin axis is also emulated in this test rig. A knuckle arm fixed to the rotor is connected to the tie-rod at one end. At the other end of the knuckle arm, a weight is attached to it. The rotation of this equipment and rotor emulates the rotation of road wheel about kingpin axis.

4

Evaluation of Control Schemes on SBW Test Rig

The designed control method has been tested by HIL simulation for three cases, which are the rapid lane change, crosswind, and braking-in-a-turn. The initial vehicle speed, vx, was 100 km/h for all cases. The authors of this study performed a computer simulation only investigation almost the same vehicle model in a previous study [22]. Very similar results were obtained for these three cases. In the experiments given in the Secs. 4.1–4.3, the steering wheel and braking force inputs were generated by the computer. 4.1 High-Speed Rapid Lane Change. A rapid lane change maneuver is often needed to avoid obstacles in real-life situations, and it is a very useful case to test the stability and handling of a vehicle in experiments. In the open-loop lane change maneuver, the vehicle displaces laterally while maintaining lane-keeping at a longitudinal speed and aligns itself with the adjacent lane [23]. In a high-speed rapid lane change case, the lateral displacement of the vehicle generally becomes larger than the desired displacement. The corrective maneuvers made by drivers are usually late and excessive when driving at high speed. In this case study, the sine wave of amplitude 120 deg and of frequency 0.5 Hz was applied as a steering command. Neither acceleration nor brake was applied during the lane change task. In the application of SBW control, it is expected that the yaw rate and lateral velocity converges to zero at the end of the maneuver.

Fig. 3

HIL environment


The HIL simulation results are shown in Figs. 4–7. As shown in Fig. 6, in the case of no control, the vehicle does not return to the state of zero yaw rate at the end of the maneuver, since the vehicle side-slip angle increases excessively. The energy optimal control shows deviations with respect to the target yaw rate. However, considering that difference in following the desired yaw rate, the performance of energy optimal control can be assessed by the lateral displacement in Fig. 4. The yaw moment, M, appearing in Eq. (10), gives rise to a yaw rate larger than the desired yaw rate. Lateral displacement of the vehicle at 1 s in Fig. 4), which is the standard for avoidance performance [24], is greater in the controlled case. This indicates the increase in the responsiveness of the vehicle system. The lateral velocity was prevented from being excessively generated by the effect of yaw moment (Fig. 5). This effect can be seen clearly in Fig. 7, when the energy optimal control and no control cases are compared. Before the steering angle has a large influence on the lateral velocity, two sudden reverse steering actions of the front wheels appear around 0.5–1 s and 1.5–2 s time ranges as shown in Fig. 7. The reason for the emergence of such a steering action in the energy optimal control is the introduction of the virtual power, which is given in Eq. (8). The power needed to reduce the error between the actual yaw rate and desired yaw rate to zero can be defined by assuming the actual yaw moment is equal to target yaw moment. By using this virtual power in the function L, a control action resisting to the generation of actual yaw moment is created. 4.2 Crosswind Stability. A crosswind is generally an unexpected and dangerous disturbance to a running vehicle. The initial moment generated in a vehicle under the effect of crosswind plays an essential role on the stability of a vehicle. The first reaction of a driver to a high crosswind generally causes the vehicle become out of control. Even if the driver manages to control the vehicle, the deviation from the intended course may result in an accident. On the other hand, since the effect of crosswind on a super lightweight vehicle is larger compared to a normal weight vehicle, the crosswind performance of a super lightweight vehicle is critical to its stability. Therefore, control of yaw rate and lateral displacement through active steering control by SBW is extremely important. In the experiment, the test vehicle passes thorough the crosswind area with a fixed steering-wheel angle. The speed of the wind is chosen as 20 m/s. It has been shown that the speed of gust winds causing accidents is generally above 20 m/s [25]. The length of the crosswind area is about 30 m. By applying the SBW control, the steering control input is expected to response quickly

Fig. 4 Lateral displacements

JULY 2015, Vol. 137 / 071005-5


Fig. 8 Yaw rates

Fig. 5 Lateral velocities

to a crosswind disturbance. Also, it is important that the yaw rate response of vehicle to a crosswind remains close to its initial value. Figures 8–9 show that the control method was very effective in maintaining the stability of the vehicle. In Fig. 8, the yaw rate response in the action of control indicates that the yaw angle of the vehicle under the effect of the crosswind is so small that the corrective input of the driver would be hardly needed. As shown in Fig. 9, the quick change of steering angle just after the vehicle entering the crosswind area indicates that a driver would be almost unaffected by the effect of crosswind.

Fig. 6 Yaw rates

Fig. 7

Front wheel steering angles

071005-6 / Vol. 137, JULY 2015

4.3 Braking-In-A-Turn. The brake oversteer phenomenon occurs when the driver applies the brakes suddenly during a cornering [26]. In any driving condition, the deceleration causes the vertical tire loads to decrease in the rear wheels and to increase in the front wheels. Transfer of the vertical tire load toward to the front wheels leads to an increase of the tire side force at the front wheels. Therefore, application of brakes during turning generally causes the vehicle to experience oversteer. As the weight of a vehicle increases, the conditions causing the brake oversteer become ineffective. Brake oversteer can result in more critical situations as the weight of a vehicle decreases, as in super lightweight vehicles. In order to further confirm the performance of the SBW control, a braking-in-a-turn simulation was carried out as the third case

Fig. 9

Front wheel steering angles

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Fig. 10 Braking forces and steering input Fig. 13 Lateral velocities

study. The steering angle command was applied from 0 s to 4 s as a ramp input. As shown in Fig. 11, the steering angle was kept constant from 4 s, the time when the lateral acceleration reaches at 8 m/s2, which was chosen as the test condition. After the yaw rate becomes steady, the braking forces are constantly applied to the front and rear wheels by quickly increasing and then fixing them at 300 N (front) and 200 N (rear). The time histories of applied braking forces and steering input are shown in Fig. 11. The yaw rate and lateral velocity responses are given in Figs. 12 and 13. From these results, it is clear that brake oversteer occurs and the vehicle has a risk of spin in the case of no control. By applying the designed SBW controls, the brake oversteer was suppressed. In the control approach, the effect of stabilizing yaw moment is seen within the first second after the start of braking. Since drivers mostly react after the first second of braking to compensate for the course deviation [27], the interference of the control system is desired within this first second. In conclusion, the energy optimal control provides stability.

Fig. 11 Lateral accelerations

Fig. 12 Yaw rates


5

Conclusion

An energy optimal control scheme for SBW systems was designed, and its performance evaluation from the stability and handling points of view was presented. To evaluate the performance of the control scheme, we developed an HIL environment. The SBW test rig as the hardware-under-test was interfaced with the software part by a controller board, in which the control scheme and vehicle simulation model were embedded. The control laws were tested using the HIL system for three case studies; high-speed rapid lane change, crosswind, and braking-in-a-turn. The vehicle model used in the experiment was designed as an 11DOF full vehicle model. We chose a super lightweight vehicle as the target vehicle in the experiments after considering the current trends in the automotive industry. This choice actually allowed the capabilities of SBW control to be clearly demonstrated. The uncontrolled super lightweight vehicle failed to achieve the desired performance levels in all three of the case studies. A vehicle with an SBW system has a front steering angle range of about 630 deg. Considering the complex dynamics of road vehicles, the difficulty lies in finding an effective control technique theoretically as well as independent from the steering of driver for such a large region of control variable. In this context, we can talk about four important elements of our control approach: tire dissipation power, desired yaw rate tracking, yaw moment, and expenditure of control effort. The control law JULY 2015, Vol. 137 / 071005-7


obtained by minimizing these measures includes important handling characteristics of a vehicle. As a conclusion, the experimental results from the implemented control system on the HIL test setup showed that the energy optimal control design achieved improvements in both the stability and handling of an SBW vehicle. Especially in the high-speed rapid lane change experiment, emergence of remarkable changes in the steering angle command contributed to also responsiveness of the vehicle.

Acknowledgment This work was principally funded by Japan Railway Construction, Transport and Technology Agency (JRTT) [Project ID No. 2007-04].

Appendix: Validation of the Vehicle Model In order to use the 11DOF nonlinear vehicle model in HIL simulations, we have checked the validity of the model by using the data collected from field tests. Since the focus of this study is on lateral dynamics, the transient and steady-state responses of system variables, which are yaw rate, roll rate, and lateral acceleration, were evaluated. As given below, the results of two standard tests [28,29] obtained from both field tests and computer simulations were compared.

Fig. 15 Lateral acceleration response of the vehicle

Constant Steering-Wheel Angle Test. In this single continuous test run, a constant steering-wheel angle is applied to the vehicle while speed is increased continuously at a slow rate. The steering-wheel angle was set to follow a low-speed path radius of 15 m. The results from field data and computer simulation are shown in Fig. 14. The close match between data can be seen in the graphic, where turning behaviors are compared by plotting the vehicle motion radius versus lateral acceleration. Sinusoidal Sweep Steering Test. This test was performed by driving the vehicle at a constant speed of u ¼ 120 km/h along a straight line path while steering in a sinusoidal manner. The steering amplitude was selected as 730 deg. The steering input frequencies swept from 0.6 to 16 rad/s. In this test, resonant frequencies in the vehicle’s response are excited and any possible issue in responses is investigated. Frequency responses of the yaw rate, roll rate, and lateral acceleration obtained from field tests and computer simulations are compared in Figs. 15–17.

Fig. 14 Vehicle motion radius versus lateral acceleration

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Fig. 16 Yaw rate response of the vehicle

Fig. 17 Roll rate response of the vehicle

Transactions of the ASME


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