Comprehensive Chassis Control Strategy Based on Coordination of Traction/Braking Distribution and Active Roll Control

Abstract

A coordinated control of front/rear traction, four-wheel independent braking, and an active roll control system is proposed in this paper to improve overall vehicle cornering performance. The control algorithm is structured in a hierarchical form: the supervisory controller is used to calculate the desired yaw rate and roll angle based on the driver input and vehicle states; the upper-level controller is utilized to decide the target longitudinal tire forces, yaw moment, and anti-roll moment based on the sliding mode control method; and the control allocator is used to optimally map the virtual control input to the control commands of specific actuators. Co-simulations of MATLAB/Simulink and CarSim were conducted to verify the performance enhancement of the proposed controller. The results suggest that the proposed controller can effectively enhance the driving performance in limit cornering while maintaining lateral stability, and it reduces tire usage, as indicated by the index of tire dissipation energy when compared to control methods such as AWD, AWD-ESC, and ICC.

1. Introduction

1.1. Research Background

Coordinated chassis control—an advanced integration of various chassis control modules such as Four-Wheel Drive (4WD) systems, Active Front Steering (AFS), and Electronic Stability Control (ESC)—has emerged as a highly effective approach aimed at fully leveraging the control degrees of freedom inherent in the vehicle chassis. Alongside those modules, the integration of Rear-Wheel Steering (RWS) and Active Suspension Systems (ASS) [1] has garnered attention in research endeavors focusing on coordinated or integrated chassis control systems. As the proliferation of chassis control modules mounted on vehicles continues, research into the coordination of these systems has garnered significant interest, particularly in tandem with the development of intelligent vehicles. However, engine-based vehicles still stand as the majority in the automotive market, which necessitates research on the coordinated control of engine vehicles.

1.2. Related Works

The most intuitive integration is that of active steering and differential braking as the advantage is readily apparent in improving lateral stability. Saikia et al. [2], for instance, integrated AFS and ESC within a hierarchical framework to enhance vehicle stability. Their approach featured an upper-level controller, which was tasked with generating a modified front steering of AFS and the yaw moment of ESC through utilizing an adaptive sliding mode control algorithm, while a lower-level controller is instead focused on distributing the yaw moment and front steering to the actuators. Yim et al. contributed substantially to the integration of AFS and ESC by proposing unified chassis control strategies [3,4], subsequently synthesizing a new control strategy aimed at mitigating the under-steer induced by AFS employment during lateral motion [5]. Moreover, they conducted comparative analyses of four frequently employed algorithms in resolving unified chassis control problems, as well as in assessing aspects such as control accuracy, stability, and enhanced vehicle performance [6].

In normal driving scenarios, AFS can manage mild external disturbances and optimize the vehicle’s lateral performance by adjusting the driver’s steering input [7,8,9]. In high-speed turning scenarios, ESC intervenes to decelerate the vehicle and, in conjunction with AFS, generates the additional yaw moment. However, when negotiating sharp corners at relatively high speeds, the uneven distribution of tire forces may lead to single-tire saturation, especially in engine vehicles without rich sources of sensors, thus limiting the capacity of ESC or AFS to control vehicle dynamics. To address this challenge, research efforts have explored the inclusion of ASS, which is characterized by its ability to adjust the vertical forces on each tire.

Zhang et al. [10] proposed integrating the control of AFS and ASS utilizing a two-layer controller. The upper-level controller aimed to coordinate these two control modules, while the lower-level controller was responsible for distributing the actual control inputs to the actuators. Similarly, Zhao et al. [11] advocated for integrating three control modules—AFS, ESC, and ASS—employing a three-level hierarchical structure. While this hierarchical structure enhances vehicle lateral and vertical performance, its efficacy depends heavily on a precise sensing of tire forces, which is not applicable for engine vehicles.

Active suspension typically features high costs with four independent spring-damper mechanisms, making it challenging to control [1]. As a cost-effective alternative, the active roll control system (ARC) can exert anti-roll moments during emergency cornering, this enhancing ride comfort and vehicle stability [12]. By adjusting roll moments in real time, ARC influences tire roll and cornering stiffness nonlinearly [13], thereby affecting vehicle handling performance through the redistribution of roll moments across the front and rear axles during cornering. Despite its potential, there has been limited research exploring the integration of ARC with other control modules.

An integration of ESC, ARC, and AFS employing three fuzzy logic controllers to regulate each module was proposed in [14], albeit via representing a simplistic amalgamation of individual controllers. Li et al. [15] integrated active steering, direct yaw moment, and ARC utilizing a two-loop control algorithm. While structurally clear, their control allocation algorithm may not suffice for generating optimal tire control inputs, which potentially lead to control conflicts between ESC and ARC. Her et al. [16] proposed integrating front/rear 4WD, ESC, and ARC. While their approach facilitates desired longitudinal acceleration and yaw motion, the absence of an index regarding total friction force or tire slips may limit the vehicle performance in harsh scenarios.

To deal with these limitations, in this paper, we propose a novel coordinated control scheme that leverages mechanical AWD, four-wheel independent braking, and active stabilizer-based ASS (i.e., ARC) to enhance the vehicle motions in both longitudinal and lateral directions, thereby improving straight-line acceleration and cornering stability. The major difference of this research from most of the latest studies is the engine-based chassis configuration, as well as factors such as the limit on tire force distribution and lack of abundant sensors resulting in evidently different and more rigorous constraints in developing the controller. The primary contribution of this research lies in the development of a control allocation algorithm incorporating a slip-monitoring and vertical/lateral motion coupling mechanism, thus enabling the controller to redistribute actuations adaptively and expand the friction circle of specific tires, thereby enhancing lateral handling.

The remainder of this paper is organized as follows: First, we present the system dynamics based on a planar dynamics model and elaborate on the input/output systems of individual control modules. Subsequently, we design a coordinated chassis control algorithm integrating the three control modules based on an optimal control allocation method. Finally, the performance of the proposed controller is verified through simulations with the MATLAB/Simulink 2022a and CarSim 2019 co-simulation platform, and then some conclusions are drawn based on our findings.

2. Chassis System Dynamics

2.1. Overall Vehicle Dynamics Modeling

As shown in Figure 1, the vehicle driveline consists of an electromagnetic center coupler and a limited slip differential (LSD). In this configuration, the driving torque originating from the engine can be transferred from the front axle to the rear axle, and from the left wheel to the right wheel in the rear axle. The cornering vehicle dynamics can be represented by the planar model illustrated in Figure 2, and the vehicle roll dynamics model is shown in Figure 3.

As shown in Figure 1, the mechanical type of 4WD configuration is suitable for an engine-driven vehicle, and it can improve the potential limit of lateral stability when the vehicle accelerates when compared to the kind of 4WD system that only has a center coupler [17]. The overall vehicle dynamics can be expressed as follows:

$$\left\{\begin{array}{l}m{\dot{v}}_x=(\overline{F_{xfl}}+\overline{F_{xfr}})\cos {\delta }_f-(\overline{F_{yfl}}+\overline{F_{yfr}})\sin {\delta }_f+(\overline{F_{xrl}}+\overline{F_{xrr}})+m{v}_y\gamma \\ m{\dot{v}}_y=(\overline{F_{yfl}}+\overline{F_{yfr}})\cos {\delta }_f+(\overline{F_{xfl}}+\overline{F_{xfr}})\sin {\delta }_f+(F_{yrl}+F_{yrr})-m{v}_x\gamma \\ I_z\cdot \dot{\gamma }=[(\overline{F_{yfl}}-\overline{F_{yfr}})\sin {\delta }_f-(\overline{F_{xfl}}-\overline{F_{xfr}})\cos {\delta }_f]t_w/2-(F_{yrl}+F_{yrr})l_r-(\overline{F_{xrl}}-\overline{F_{xrr}})t_w/2\\ I_x\ddot{\varphi }=m_s{a}_y{h}_s-C_{\varphi }\dot{\varphi }-K_{\varphi }\varphi +m_{s}g{h}_s\varphi -M_{\varphi ,ARC}\\ \overline{F_{xii}}=F_{xii,4WD}-F_{xii,ESC},\overline{F_{yii}}=F_{yii}-\Delta F_{yii},ii=fl,fr,rl,rr\end{array}\right.,$$

(1)

where $v_x$ denotes the longitudinal velocity; $v_y$ the lateral velocity; $\gamma$ the yaw rate; $a_y$ the lateral acceleration; ${\delta }_f$ the front wheel steering angle; $m$ the mass of the vehicle; $m_s$ the sprung mass; $l_f,l_r$ the distance from the center of gravity to the front/rear axle; $l$ the wheel base; $t_w$ the tread (track width); ${\overline{F}}_{xii}$, $ii=fl,fr,rl,rr$ the longitudinal forces of the front/rear wheels; ${\overline{F}}_{yii}$ the lateral forces on the individual wheels; $I_z$ the yaw moment of inertia about the z-axis; $I_x$ the roll moment of the inertia about the x-axis; $C_{\varphi }$ and $K_{\varphi }$ the damping ratio and stiffness of the suspension; $\varphi$ the roll angle of the sprung mass; and $M_{\varphi ,ARC}$ the anti-roll moment exerted by ARC. The longitudinal forces and the steering angle are used as the control input in the controller design in the following sections.

2.2. Individual System Modeling

The 4WD actuator consists of a center coupling and a rear limited-slip differential, the detailed modeling methods of which can be found in [18,19]. The four-wheel independent braking actuations are represented by the individual braking force, which are then converted to corresponding master cylinder pressure as the actual control input. The ARC actuator consists of a motor supplying the driving force that generates the anti-roll moments, a reduction gear that amplifies the motor torque, a housing, and a pair of stabilizers [20]. The control commands of the ARC system are subject to the specification of the ARC actuator, which is a brushless DC motor.

In this paper, the systems are all represented by incorporating a first-order delay mechanism with specific lag constants to accurately reflect real-world actuator responses, the detailed representation method of which can be found in [12,16]. The proposed comprehensive controller capitalizes on the constrained front/rear axles and left/right wheels that are driving the torques generated by the 4WD system, the four-wheel-independent braking capabilities facilitated by ESC, and the auxiliary front/rear roll moment provided by ARC. These inputs are strategically utilized to optimize vehicle control.

3. Coordinated Control Algorithm

To exploit the over-actuated characteristics of the chassis to push the tire forces to the limit in emergency turning, the effects of different control modules on the vehicle dynamics, particularly on yaw motion, should be taken into consideration, as shown in Figure 4.

The direct and indirect performance improvement can be induced simultaneously by both 4WD and ESC, as shown in Figure 4, because the vehicle longitudinal/lateral dynamics are coupled. The effect of ARC on the yaw motion, however, is quite different from the two torque-vectoring control modules. ARC can alter the front/rear axle roll stiffness and the characteristics of the tire forces by distributing the anti-roll moments of the front/rear stabilizer. The net result of the weight transfer across an axle in turning is that the total lateral forces on an axle decrease, and it requires the tires on that axle to generate larger slip angles to achieve a yaw moment balance [21]. The anti-roll moment on the front axle increasing leads to more of a load transfer at the rear axle and less at the front axle; thus, the cornering stiffness of the front wheels increases, and the vehicle exhibits more understeering characteristics. Then, the lateral acceleration performance of the vehicles with active stabilizers in cornering can be improved.

With the analyses given above, the coordinated chassis control algorithm can be designed, as shown in Figure 5.

3.1. Supervisory Controller Design

3.1.1. Desired Yaw Rate

As shown in Figure 6, the 3-DOF vehicle dynamics model consists of an additional control yaw moment. As such, based on (1), the dynamics can be represented by

$$\left[\begin{array}{c}\dot{\beta }\\ \dot{\gamma }\end{array}\right]=\left[\begin{array}{cc}-\frac{C_{\alpha f}+C_{\alpha r}}{m{v}_x}& \frac{l_r{C}_{\alpha r}-l_f{C}_{\alpha f}}{m{v}_x^2}-1\\ \frac{l_r{C}_{\alpha r}-l_f{C}_{\alpha f}}{I_z}& -\frac{l_r^2{C}_{\alpha r}+l_f^2{C}_{\alpha f}}{I_z{v}_x}\end{array}\right]\left[\begin{array}{c}\beta \\ \gamma \end{array}\right]+\left[\begin{array}{c}\frac{C_{\alpha f}}{m{v}_x}\\ \frac{l_f{C}_{\alpha f}}{I_z}\end{array}\right]{\delta }_f+\left[\begin{array}{c}0\\ \frac{1}{I_z}\end{array}\right]M_z,$$

(2)

$$a_y=v_x(\dot{\beta }+\gamma ).$$

(3)

Based on the assumption of a small steering angle, the reference yaw rate can be calculated with the following derivative equation [22]:

$${\dot{\gamma }}_{ref}=-\frac{l{C}_{\alpha f}{C}_{\alpha r}}{I_z\left(C_{\alpha f}+C_{\alpha r}\right)}\cdot \left[\frac{l}{v_x}+\left(\frac{m{l}_r}{l{C}_{\alpha f}}-\frac{m{l}_f}{l{C}_{\alpha r}}\right)\cdot v_x\right]\cdot {\gamma }_{ref}+\frac{l{C}_{\alpha f}{C}_{\alpha r}}{I_z\left(C_{\alpha f}+C_{\alpha r}\right)}\cdot {\delta }_f.$$

(4)

As there is the limitation of tire–road friction, then the target yaw rate can be obtained as

$${\gamma }_{target}=\min \left(\frac{{\gamma }_{ref}}{1+{\tau }_{e}s},0.85\cdot \frac{\mu g}{v_x}\right),$$

(5)

where ${\gamma }_{Target}$ denotes the target yaw rate, and ${\tau }_e$ is a time constant.

3.1.2. Desired Roll Angle

When the vehicle negotiates a corner, a rolling moment is generated by the centrifugal force at the center of mass, and then the passive reaction forces of the suspension start to resist the rolling motion. The desired roll angle of the ARC system can be acquired based on the lateral acceleration [23], and the relation between the desired roll angle of the ARC system and the lateral acceleration is shown in Figure 7.

Then, the desired roll angle can be generated through a lookup table as follows:

$${\varphi }_{des}=f\left(a_y\right).$$

(6)

3.2. Upper-Level Controller Design

3.2.1. Target Longitudinal Tire Force

Different from vehicle of a drive-by-wire configuration, the vehicle prototype applied in this paper is driven by an engine, and the power is transmitted through mechanical devices in the powertrain; thus, as the driver controls the longitudinal dynamics vehicle through the accelerating/braking pedal, the target longitudinal tire forces can be calculated by the indicated engine torque acquired from the CAN (Control Area Network) bus for accelerating scenarios. Similarly, the target braking forces can be generated through the multiplication of the braking gain and the master cylinder braking pressure obtained from CAN. The calculation can be expressed as follows:

$$F_{x,\mathrm{target}}=\left\{\begin{array}{c}g(T_{eng}),\mathrm{if}\mathrm{APS}>0\\ h(p_{mc}),\mathrm{if}\mathrm{BPS}>0\end{array}\right.,$$

(7)

where APS denotes the accelerating pedal signal, BPS is the braking pedal signal, and $g(\cdot )$ and $h(\cdot )$ can be two complex functions developed with data-driven method. However, with the emphasis on controller design in this paper, we set the two functions as linear ones with the gains tuned in advance.

3.2.2. Target Anti-Roll Moment

The purpose of the target anti-roll moment is to yield the desired roll angle of the sprung mass, the calculation of which is based on Equation (1):

$$I_x\ddot{\varphi }=m_s{a}_y{h}_s-C_{\varphi }\dot{\varphi }-K_{\varphi }\varphi +m_{s}g{h}_s\varphi -M_{\varphi ,ARC}.$$

(8)

Sliding mode control is employed here to eliminate possible disturbances and shivering during the control process. The sliding surface $s_{\varphi }$ and the sliding condition for the ARC to drive the roll angle and roll rate to the desired value can be defined as

$$s_{\varphi }=\left({\dot{\varphi }}_{des}-\dot{\varphi }\right)+{\eta }_{\varphi }\left({\varphi }_{des}-\varphi \right),\frac{1}{2}\frac{d}{dt}{s}_{\varphi }^2=s_{\varphi }{\dot{s}}_{\varphi }\le -{\lambda }_{\varphi }\left|s_{\varphi }\right|\left({\lambda }_{\varphi }>0\right),$$

(9)

where ${\eta }_{\varphi }$ is the weighting factor. Then, the target total anti-roll moment for the ARC system can be obtained by

$$M_{\varphi ,\mathrm{target}}=m_s{a}_y{h}_s+\left(m_{s}g{h}_s-K_{\varphi }\right)\varphi +\left({\eta }_{\varphi }{I}_x-C_{\varphi }\right)\dot{\varphi }-I_x\left({\ddot{\varphi }}_{des}+{\eta }_{\varphi }{\dot{\varphi }}_{des}\right)+k_{\varphi }\mathrm{sat}\left(\frac{\dot{\varphi }+{\eta }_{\varphi }\varphi }{{\mathrm{\Phi }}_{\varphi }}\right)$$

(10)

with

$$\mathrm{sat}\left(\frac{\dot{\varphi }+{\eta }_{\varphi }\varphi }{{\mathrm{\Phi }}_{\varphi }}\right)=\left\{\begin{array}{c}\mathrm{sgn}\left(\dot{\varphi }+{\eta }_{\varphi }\varphi \right),\left|\frac{\dot{\varphi }+{\eta }_{\varphi }\varphi }{{\mathrm{\Phi }}_{\varphi }}\right|>1\\ \left|\frac{\dot{\varphi }+{\eta }_{\varphi }\varphi }{{\mathrm{\Phi }}_{\varphi }}\right|,\left|\frac{\dot{\varphi }+{\eta }_{\varphi }\varphi }{{\mathrm{\Phi }}_{\varphi }}\right|\le 1\end{array}\right.,$$

(11)

where ${\mathrm{\Phi }}_{\varphi }$ is the thickness of the boundary layer. In Equation (10), $k_{\varphi }$ is a constant satisfying

$$k_{\varphi }\ge {\lambda }_{\varphi }{I}_x.$$

(12)

3.2.3. Target Yaw Moment

The primary target for the handling performance control is to precisely track the target yaw rate and the vehicle sideslip angle. The target yaw rate is calculated in the supervisory controller. Here, we set the desired sideslip angle as ${\beta }_{des}=0$, as a large sideslip motion is undesirable. Then, via adopting sliding mode theory, we defined the sliding surface and the reaching law as

$$s_{yaw}=\gamma -{\gamma }_{des}+{\xi }_{Mz}\beta ,{\dot{s}}_{yaw}=-{\lambda }_{yaw}{s}_{yaw}({\lambda }_{yaw}>0).$$

(13)

Then, through combining Equations (2) and (3), we can obtain the following equation:

$$\dot{\gamma }=-\frac{2C_{\alpha f}{C}_{\alpha r}}{C_{\alpha f}+C_{\alpha r}}\cdot \frac{{(l_f+l_r)}^2}{I_z{v}_x}\cdot \gamma +\frac{m(l_f{C}_{\alpha f}-C_{\alpha r}{l}_r)}{(C_{\alpha f}+C_{\alpha r})I_z}{a}_y+\frac{2C_{\alpha f}{C}_{\alpha r}}{C_{\alpha f}+C_{\alpha r}}\cdot \frac{(l_f+l_r)}{I_z}{\delta }_f+\frac{1}{I_z}{M}_z$$

(14)

The additional yaw moment for optimizing the vehicle stability can then be obtained by applying (13) as

$$M_z=\left(\frac{2C_{\alpha f}{C}_{\alpha r}}{C_{\alpha f}+C_{\alpha r}}\cdot \frac{{(l_f+l_r)}^2}{v_x}-I_z{\lambda }_{yaw}\right)\cdot \gamma -\frac{2C_{\alpha f}{C}_{\alpha r}(l_f+l_r)}{C_{\alpha f}+C_{\alpha r}}{\delta }_f-\left(\frac{m(l_f{C}_{\alpha f}-C_{\alpha r}{l}_r)}{C_{\alpha f}+C_{\alpha r}}+\frac{I_z{\lambda }_{yaw}}{v_x}\right)\cdot a_y+I_z\cdot {\dot{\gamma }}_{target}-K_{\gamma }\cdot \mathrm{sat}(\frac{S_{yaw}}{{\mathrm{\Phi }}_{yaw}}),$$

(15)

where ${\mathrm{\Phi }}_{yaw}$ is the thickness of the boundary layer.

3.3. Control Allocator

The control allocation was designed to allocate the over-actuated control inputs to the specific actuators. In this paper, the actuation distribution issue was converted into an optimization problem, in which the goal and constraints are fivefold: 1. minimization of the mapping error; 2. minimization of the tire slip motion; 3. minimization of ESC exertion; 4. minimization of the error between optimal allocation and the guideline; and 5. alignment of the control inputs with specific constraints.

3.3.1. Minimization of the Mapping Error

As the control allocation problem will be formulated as an optimization problem, the virtual control that was calculated in previous sections will not be exactly the same in the actual control input, as shown in Equation (16):

$$\left\{\begin{array}{l}{F}_{x,target}=\underset{F_{x,4WD}(\mathrm{acceleration}\mathrm{pedal})}{\underbrace{F_{xf,4WD}+F_{xrl,4WD}+F_{xrr,4WD}}}-\underset{F_{x,ESC}(\mathrm{brake}\mathrm{pedal})}{\underbrace{(F_{x,fl,ESC}+F_{x,rl,ESC}+F_{x,rl,ESC}+F_{x,rr,ESC})}}\\ M_{z,target}=\underset{\mathrm{driving}\mathrm{force}\mathrm{distribution}}{\underbrace{(-F_{x.rl,4WD}+F_{x.rr,4WD})\cdot l_f}}+\underset{\mathrm{braking}\mathrm{force}\mathrm{distribution}}{\underbrace{\frac{t_w}{2}\cdot (F_{x.fl,ESC}-F_{x.fr,ESC}+F_{x.rl,ESC}-F_{x.rr,ESC})}}\\ M_{\varphi ,\mathrm{target}}=M_{f,ARC}+M_{r,ARC}\end{array}\right..$$

(16)

Note that the lateral dynamics of tires is affected by vertical motion; to make full use of these characteristics, we related the front/rear driving force and roll moment distribution with the front/rear tire slip difference as

$${\alpha }_f-{\alpha }_r={\varpi }_{c,4WD}\left(F_{xf,4WD}-F_{xrl,4WD}-F_{xrr,4WD}\right)+{\varpi }_{c,4WD}\left(M_{f,ARC}-M_{r,ARC}\right).$$

(17)

The equations above can be written as follows:

$$\underset{v_1(t)[4\times 1]}{\underbrace{\left[\begin{array}{c}{F}_{x,\mathrm{target}}\\ M_{z,\mathrm{target}}\\ M_{\varphi ,\mathrm{target}}\\ {\alpha }_f-{\alpha }_r\end{array}\right]}}=\underset{B_1[4\times 9]}{\underbrace{\left[\begin{array}{ccccccccc}1& 1& 1& -1& -1& -1& -1& 0& 0\\ 0& -l_f& l_f& 0.5t_w& -0.5t_w& 0.5t_w& -0.5t_w& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 1\\ {\varpi }_{c,4WD}& -{\varpi }_{c,4WD}& -{\varpi }_{c,4WD}& 0& 0& 0& 0& {\varpi }_{c,4WD}& -{\varpi }_{c,4WD}\end{array}\right]}}\cdot \underset{u\left[9\times 1\right]}{\underbrace{u(t)}},$$

(18)

where $u(t)={[F_{xf,4WD},F_{xrl,4WD},F_{xrr,4WD},F_{x,fl,ESC},F_{x,rl,ESC},F_{x,rl,ESC},F_{x,rr,ESC},M_{f,ARC},M_{r,ARC}]}^{\mathrm{T}}$ denotes the actuator control input, and $v_1(t)$ and $B_1$ are defined as in the equation. Then, the cost function for the first goal in formulating the optimization problem is constructed as follows:

$$J_1={‖W_{v1}(B_{1}u(t)-v_1(t))‖}^2,$$

(19)

where $W_{v1}$ is a diagonal matrix to add weight to the corresponding terms.

3.3.2. Minimizing the Tire Slip Motion

Excessively large tire slip ratio/slip angles can lead to compromised driving, braking forces, or even a lateral instability of the vehicle; thus, to improve the overall performance of the vehicle, even in extreme conditions, monitoring and regulation should be conducted on the slip motion of the tires. Then, the tire overall slip penalty method in [21] can be applied here.

Next, the hyperbolic type of tire slip motion penalty function is formulated as follows:

$$S_i=0.1\cdot \overline{\mu }{\widehat{F}}_z/\left(-a_{hyper}\cdot \frac{(\left|{\lambda }_i\right|-0.5\cdot {\lambda }_{peak,i})}{{\lambda }_{peak,i}}+\sqrt{{\left\{a_{hyper}\cdot \frac{(\left|{\lambda }_i\right|-0.5\cdot {\lambda }_{peak,i})}{{\lambda }_{peak,i}}\right\}}^2+b_{hyper}^2}\right),$$

(20)

where ${\lambda }_i$ is the overall slip of the wheel, the calculation of which can be found in [8]; ${\lambda }_{peak,i}$ is the peak value of ${\lambda }_i$; $a_{hyper},b_{hyper}$ are constants to be set by the designer; and $\overline{\mu }$ is the road friction coefficient, which was estimated by an observer in [24]. The varying trend of the penalty function with a tire overall slip is shown in Figure 8.

The hyperbolic penalty function increases monotonically with individual tire slips, and this is based on which tire saturation cost function can be formulated as follows:

$$\left[\begin{array}{cccc}{p}_{fl}& 0& 0& 0\\ 0& p_{fl}& 0& 0\\ 0& 0& p_{fl}& 0\\ 0& 0& 0& p_{fl}\end{array}\right]\cdot \underset{{[F_{xfl},F_{xfr},F_{xrl},F_{xrr}]}^T}{\underbrace{B_{lf}\cdot u(t)}}=\underset{[4\times 9]}{\underbrace{B_2}}\cdot \underset{[9\times 1]}{\underbrace{u(t)}},$$

(21)

$$J_2={‖W_{u2}(B_2\cdot u(t))‖}^2,$$

(22)

where $B_{lf}$ is the control conversion matrix with a suitable size; $B_2$ is a matrix for exerting penalties; and $W_{u2}$ is the diagonal weight matrix.

3.3.3. Minimization of the Error between Optimal Allocation and the Guidelines

The challenge with optimal allocation lies in the selection, which is often dependent on the expertise and intuition of engineers, of weights. Previous research has attempted to address this by designing gains or weights as functions of velocity or lateral acceleration [25,26], albeit with effectiveness limited to specific cases. In this study, we employed the feed-forward allocation method [27] and the minimum lap time solution [28] for race cars on pre-defined race tracks to guide the longitudinal tire force distribution. This approach generates guidelines that steer command inputs of the allocation toward sub-optimal solutions. As proposed in [28], the sub-optimal longitudinal torque distribution is designed as

$$\left\{\begin{array}{l}{\rho }_0=1/\left(\frac{F_{zrl}+F_{zrr}}{F_{zfl}+F_{zfr}}\cdot \frac{a_x}{a_x\cos {\delta }_f+a_y\sin {\delta }_f}+1\right)\\ {\rho }_1=F_{zfr}/\left(F_{zfr}+F_{zfl}\right),{\rho }_2=F_{zrr}/\left(F_{zrr}+F_{zrl}\right)\end{array}\right.,$$

(23)

where ${\rho }_0$ is the front–rear torque distribution ratio; and ${\rho }_1$ and ${\rho }_2$ are the left–right wheel torque distribution ration on front and rear axles, respectively. It can be seen from Equation (26) that approximating the ratios of the tire workload to vehicle weight corresponds with the limit handling that maximizes the use of tire friction circle.

As for the guideline regarding the distribution of front–rear anti-roll moments, through considering the indirect effect of the roll moment on the yaw motion of the vehicle, a proportional–integral control method based on yaw rate error was adopted in this paper to generate the distribution ratio as follows:

$${\rho }_{ARC}=k_p\left({\gamma }_{des}-\gamma \right)+k_i{\displaystyle \int \left({\gamma }_{des}-\gamma \right)}dt+{\rho }_{ARC,0},$$

(24)

where $k_p$ and $k_i$ are the proportional and integral gains, respectively; and ${\rho }_{ARC,0}$ is the initial distribution ratio of the front–rear roll stiffness.

Based on the allocation guideline, the cost function can be arranged as follows:

$$\underset{v_3\left[6\times 1\right]}{\underbrace{\left[\begin{array}{c}{F}_{xfl}^b\\ F_{xrl}^b\\ F_{xrl}^b\\ F_{xrr}^b\\ M_{f,ARC}^b\\ M_{r,ARC}^b\end{array}\right]}}=\left[\begin{array}{c}\left(1-{\rho }_0\right){\rho }_1{F}_{x,\mathrm{target}}\\ \left(1-{\rho }_0\right)\left(1-{\rho }_1\right)F_{x,\mathrm{target}}\\ {\rho }_0{\rho }_2{F}_{x,\mathrm{target}}\\ {\rho }_0\left(1-{\rho }_2\right)F_{x,\mathrm{target}}\\ \left(1-{\rho }_{ARC}\right)M_{\varphi ,\mathrm{target}}\\ {\rho }_{ARC}{M}_{\varphi ,\mathrm{target}}\end{array}\right]=\underset{B_3\left[6\times 9\right]}{\underbrace{\left[\begin{array}{ccccccccc}0.5& 0& 0& -1& 0& 0& 0& 0& 0\\ 0.5& 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\cdot }}u(t),$$

(25)

$$J_3={‖W_{v3}(B_3\cdot u(t)-v_3(t))‖}^2,$$

(26)

where $F_{xii}^b(ii=fl,fr,rl,rr)$ and $M_{j,ARC}^b(j=f,r)$ denote the pre-defined guideline for longitudinal tire forces and the anti-roll moment on front/rear axle, respectively; and $W_{v3}$ is the diagonal weight matrix.

3.3.4. Minimization of ESC Exertion

Braking is almost always undesirable in normal driving, and it can cause unnecessary tire slip. As such, the cost function for minimizing ESC exertion can be generated as follows:

$$J_4={‖W_{u4}u(t)‖}^2,$$

(27)

where $W_{u4}$ is the diagonal weight matrix for corresponding actuations.

3.3.5. Actuator Control Limit Constraints

To effectively implement the control algorithm in practical applications, it is imperative to consider the constraints on the actuators’ operation, including physical limits and rate limits. Notably, the control of each actuator is intricately linked to vehicle dynamics. Consequently, the operational range of actuators at any given moment is constrained by real-time vehicle dynamics.

In contrast to vehicles equipped with in-wheel motors, the vehicle prototype utilized in this study features limited front/rear and left/right driving torque distribution ratios. The driveline of the 4WD configuration described in this paper comprises a center coupler (as depicted in Figure 4) and a rear axle slip-limit clutch. Through experimental testing and theoretical analyses of the prototype vehicle employed in this research, it was determined that the front/rear traction distribution ratio was capped at 0.6, and the rear axle left/right wheel traction distribution ratio was set at 0.5. Additionally, longitudinal traction force is bounded by the corresponding vertical load according to the following relationship:

$$F_{xii,\max }=\sqrt{{(\mu F_{zii})}^2-F_{yii}^2},$$

(28)

where $F_{xii,\max }$ is the maximum of the individual traction force. The latter portion of Equation (32) was omitted due to the practical challenges associated with measuring lateral tire forces. Nonetheless, the previously mentioned tire saturation penalty can effectively mitigate the overestimation of the upper limit of the longitudinal force. Consequently, based on the aforementioned analyses, the constraints for the 4WD actuator can be formulated as follows:

$$\left\{\begin{array}{c}0\le F_{xf,4WD}\le \min (\sqrt{{(\mu F_{zf})}^2},T_{trans}/r)\\ 0<F_{xi,4WD}\le \min (0.5\cdot 0.6\cdot T_{trans}/r,\sqrt{{(\mu F_{zi})}^2}),i=rl,rr\end{array}\right.,$$

(29)

where $r$ denotes the wheel effective radius, $F_{xf,4WD}$ is the front wheels’ traction force, and $F_{zf}$ is the front wheels’ vertical load.

The control command of an ARC system should be in accordance with the specification of the actuator, which is a brushless DC motor. Then, the maximum anti-roll moment and the limit torque rate can be defined as follows:

$$0\le \left|M_{\varphi ,ARC}\right|\le M_{\varphi ,\max },0\le r_{\varphi ,ARC}\le r_{\varphi ,\max }.$$

(30)

The brake actuation is mainly limited by the physical limitations of the compression actuator. As such, the constraints of ESC can be expressed as follows:

$$0\le F_{xii,ESC}\le p_{\max }\cdot {\sigma }_{ESC},ii=fl,fr,rl,rr,$$

(31)

where ${\sigma }_{ESC}$ is the gain of braking system, and $p_{\max }$ is the upper limit of the M/C pressure.

Then, the actuators limit constraints can be rewritten in a compact form as

$$\left\{\begin{array}{c}{u}_{\min }=[0{,0,0,0,0,0,0,0,0]}^{\mathrm{T}}\\ u_{\max }={\left[\begin{array}{l}{F}_{f,4WD,\max },F_{rl,4WD,\max },F_{rr,4WD,\max },F_{fl,ESC,\max },\\ F_{fr,ESC,\max },F_{rl,ESC,\max },F_{rr,ESC,\max },M_{f,\max },M_{r,\max }\end{array}\right]}^{\mathrm{T}}\end{array}\right..$$

(32)

3.4. Optimal Distribution of the Control Inputs

Upon generating the target virtual inputs and the constraints of the actuators, the comprehensive distribution of the control inputs can be formulated as a weighted least-squares (WLS) problem:

$$\begin{array}{rr}u(t)& =\arg \underset{U_{\min }\le u\le U_{\max }}{\min }\left[{\xi }_{v1}\cdot J_1+{\xi }_{u2}\cdot J_2+{\xi }_{v3}\cdot J_3+{\xi }_{u4}\cdot J_4\right]\hfill \\ & =\arg \underset{U_{\min }\le u\le U_{\max }}{\min }\left\{{‖\underset{A}{\underbrace{\left(\begin{array}{c}{\xi }_{v1}^{1/2}\cdot W_{v1}\\ {\xi }_{u2}^{1/2}\cdot W_{u2}\cdot B_2\\ {\xi }_{v3}^{1/2}\cdot W_{v3}\cdot B_3\\ {\xi }_{u4}^{1/2}\cdot W_{u4}\end{array}\right)}}\cdot u-\underset{b}{\underbrace{\left(\begin{array}{c}{\xi }_{v1}^{1/2}\cdot W_{v1}\cdot v_1\\ 0\\ {\xi }_{v3}^{1/2}\cdot W_{v3}\cdot v_3\\ 0\end{array}\right)}}‖}^2\right\}\\ & =\arg \underset{\underset{\_}{U}\le u\le \overline{U}}{\min }\left({‖A\cdot u-b‖}^2\right)\hfill \end{array},$$

(33)

where ${\xi }_{v1}$, ${\xi }_{u2}$, ${\xi }_{v3}$ and ${\xi }_{u4}$ are the tuning weights for individual cost. The WLS problem presented above can be solved with the active-set algorithm detailed in [29]. Then, the optimally distributed control commands will be calculated and converted to the actual control inputs for 4WD, ESC, and ARC, i.e., the control current input of 4WD, the brake pressure of ESC, and the control current input of ARC [22].

4. Simulation and Comparison

To verify the effectiveness and superiority of the proposed controller, two scenarios were simulated with the MATLAB/Simulink and CarSim co-simulation platform, as shown in Figure 9. The first scenario is a circular turning with acceleration to evaluate the handling stability of the controller in a driving scenario. The second experiment is a double-lane change together with a given braking M/C pressure deceleration, which is comparatively common in collision avoidance occasion.

4.1. Simulation Settings

To put the control algorithm into practice, some essential vehicle states such as the longitudinal vehicle velocity, vehicle sideslip angle, tire slip angle, and tire cornering stiffness needed to be estimated. A typical vehicle is already equipped with sensors such as four wheel speed sensors, a steering wheel angle sensor, a yaw rate sensor, and a longitudinal/lateral acceleration sensor. With more emphasis put on the control algorithm, the longitudinal velocity estimation method proposed in [30] was adopted here, and the vehicle and tire sideslip angles were observed utilizing the estimation method detailed in [31,32]. The cornering stiffness, which is associated with the tire slip angle, can be obtained along with the estimation of tire slip angles.

The proposed controller was compared to three other control systems with respect to the driving and handling performance: (1) the front/rear traction distribution system equipped with an anti-lock braking system (ABS), which w marked as ‘AWD’; (2) a simple integration of AWD with a differential brake controller (marked as AWD-ESC); and (3) a previously proposed chassis integrated controller coordinating front/rear traction distribution, four-wheel independent brake, and front/rear ARC system (marked as ICC) [16]. The vehicles all had front wheel drive. The proposed controller differs from the ICC system mainly in the calculation of the desired motion and formulation of the control allocation algorithm.

The prototype vehicle built in CarSim was a mid-size SUV, and the parameters of the vehicle are listed in Table 1.

4.2. Circular Turning with Acceleration

In the simulation of circular turning with acceleration, the road was a 500-foot circular track with one lane. The tire–road friction coefficient was set to 0.8. The driver model used in the simulation was a built-in model of CarSim that is responsible for generating the steering angle and accelerating/decelerating pedal input in a closed-loop maneuver to track the pre-defined path. The driver preview interval was set to 0.7 s. In this simulation, the driver started to accelerate the vehicle at 1s with the throttle opening spiking and holding to 0.5. The simulation results are shown in Figure 10, Figure 11 and Figure 12.

As can be seen in Figure 10A, the AWD vehicle showed some instability with regard to lane keeping compared with the other three systems since the effect of handling the improvement of the AWD system on the vehicle was limited with the front/rear traction distribution configuration. The AWD-ESC system, however, was able to maintain the vehicle stability well due to the independent wheel braking configuration.

As shown in Figure 10B, the proposed algorithm can reduce the steering effort of the driver effectively compared with the AWD and AWD-ESC systems. In addition, due to the tire force distribution, the driver models of those two systems have to exert more steering input to keep the vehicle in lane. As shown in Figure 10C,D, the proposed algorithm exhibits the best overall performance in terms of longitudinal vehicle speed and lateral acceleration. Figure 10E,F show the anti-roll performance of the four systems, where ICC and the proposed algorithm showed similarly superior performances than the other two systems regarding the roll rate and the roll angle error of the sprung mass, which will give the driver and passengers a better driving and riding experience.

There are two points worth noting in Figure 10: (1) the lateral acceleration of ICC is larger than the proposed algorithm at some points because ICC put more emphasis on the maximization of lateral tire force, which consequently leads to some loss of longitudinal speed compared with the proposed algorithm, as shown in Figure 10C; (2) the roll angle of the proposed algorithm is relatively smaller and reaches the ‘steady state’ faster than ICC, and because it takes the effect of front/rear anti-roll moment distribution on handling stability into consideration, the relatively steady state of vehicle at the velocity limit on the given track will in turn make the rolling dynamics of the sprung mass stable.

To quantize the usage of each tire, the tire dissipation energy (TDE) of the four algorithms was calculated based on the tire dissipation power (TDP) with the following equation:

$$\mathrm{TDP}={\displaystyle \sum _{ii=\left\{fl,fr,rl,rr\right\}}\left(\left|{\overline{v}}_{xii}\cdot F_{xii}\right|+\left|{\overline{v}}_{yii}\cdot F_{yii}\right|\right)},$$

(34)

where ${\overline{v}}_{xii}$ and ${\overline{v}}_{yii}$ are the longitudinal and lateral slip velocity, which can be obtained by multiplication of tire slip ratio and slip angle with the longitudinal vehicle speed, respectively; and $F_{xii}$ and $F_{yii}$ are the longitudinal and lateral tire forces, respectively.

Figure 10. Simulation results when accelerating circular turning.

Figure 10. Simulation results when accelerating circular turning.

The TDEs of each tire and the overall values were calculated, as shown in Figure 11 and Table 2. In Figure 11 and Table 2, the TDE of the proposed algorithm was the smallest of the four systems, and the dissipation energy distributed to the four tires was roughly even, which is exactly the aim of the algorithm, i.e., reducing excessive tear and wear on one tire. The TDE of AWD-ESC was smaller than that of ICC mostly due to the relatively low vehicle speed, while the AWD system was remarkably larger than the other three systems owing to the instability of the vehicle.

Figure 11. The total TDE for the circular turning scenario.

Figure 11. The total TDE for the circular turning scenario.

4.3. Double-Lane Change with Deceleration

A double-lane change in the deceleration simulation was conducted to replicate a common collision avoidance scenario. The initial speed was set to 85 km/h, and the tire–road friction coefficient was set to 0.8. The built-in driver model in CarSim was utilized for the simulation, with a driver preview interval of 0.75 s. It was assumed that, at 2 s, the driver initiated braking by applying a force of 70 N on the brake pedal, thus resulting in a master cylinder pressure of approximately 2.56 MPa. The average longitudinal deceleration was 0.4g, and the peak lateral acceleration reached 0.6g, thus indicating a relatively aggressive braking maneuver.

The simulation results are depicted in Figure 12, Figure 13 and Figure 14. Due to the optimal coordination of the Active Roll Control (ARC) and Electronic Stability Control (ESC), the steering effort required by the proposed algorithm was slightly less than that required by the integrated chassis control (ICC), as shown in Figure 12A. The handling performance in terms of yaw rate error and vehicle sideslip angle is illustrated in Figure 12C,D. The root mean square values of the yaw rate error ${\mathrm{RM}}_{\gamma }$ and the vehicle sideslip angle ${\mathrm{RM}}_{\beta }$ under different systems are listed in Table 3. Without an appropriate braking force distribution algorithm, the yaw rate error and sideslip angle of the All-Wheel Drive (AWD) system were significantly larger than those of the other three systems. The proposed controller evidently yielded the best outcomes out of the four systems in terms of yaw and sideslip motion.

Figure 12. Simulation results of the second scenario.

Figure 12. Simulation results of the second scenario.

Figure 13 and Figure 14 present the tire dissipation power of each tire and the total dissipation energy, and the exact values of TDE in each system are listed in Table 4. The front wheel total dissipation energy (TDE) of the AWD vehicle was relatively higher than that of the rear wheels, primarily due to the brake proportioning configuration. However, by optimally distributing the braking forces among the four wheels, the other three systems effectively reduced the dissipation energy.

Despite the similar handling performance of the proposed algorithm and the integrated chassis control (ICC) system shown in Figure 12, the proposed algorithm demonstrated reduced tire usage compared to ICC. This was evidenced by the indices of the tire slip ratio and slip angle, which are presented in Figure 15. While the ICC system employed a tire force management method to maximize the lateral tire force, it lacked a tire slip monitoring algorithm. Consequently, the friction limit constraint used in the ICC system’s tire force calculations did not ensure that the tire slip remained within the normalized slip circle, as illustrated in Figure 15.

4.4. Comprehensive Handling Course

The controller was further tested to verify the comprehensive performance. A handling course was chosen as the target path profile, as shown in Figure 16a, where the black arrow points to the running direction. In this test, the major target of the controllers was to finish the course in the minimum time without losing stability. To do so, an advanced driver model, which was developed in previous research [33], was adopted here to generate the race-car driver performance. The simulation was conducted on the entire handling course while we selected a representative part (shown surrounded by a black square in Figure 16a) to analyze the performance of each controller more closely. The main results regarding trajectory, steering wheel angle, longitudinal/lateral acceleration, and the longitudinal velocity are illustrated in Figure 16. As the comprehensive test was performed to measure the overall performance, the shortest lap time served as the main performance index. The results of the best lap time of the four controllers together with the corresponding TDE values are listed in Table 5.

As shown in the figure, the trajectories of the four controllers were basically identical except from the slight deviation of the ‘Base’ controller, which indicated the excessive slip motion of the tires in ‘Base’ controller, as can be seen in Table 5. On the other hand, Figure 16d shows the longitudinal acceleration of the four controllers, in which the one of the proposed controller was evidently larger than that of the other controllers, and similar observations could also be made in the result of the lateral acceleration, as shown in Figure 16e. Larger longitudinal and lateral accelerations allow for a larger overall longitudinal velocity, which implies a capability of decelerating later in cornering and accelerating faster in straight-line driving. This implication can be demonstrated by the overall lap time listed in Table 5, which was as near as 3.8% of reduced lap time when using the proposed controller, and it can be easily observed compared with that of the ‘AWD’ system.

In conclusion, the proposed algorithm can effectively maintain vehicle stability and enhance handling performance. Furthermore, it significantly reduces the total dissipation energy (TDE) during both accelerating circular turning and aggressive braking in double-lane change scenarios. In a comprehensive driving situation, the proposed controller can effectively improve the vehicle mobility by enlarging the envelope of maximum longitudinal/lateral accelerations, thus resulting in a shorter reaching time for the same destination.

5. Conclusions

A coordinated control algorithm integrating front/rear traction, four-wheel independent braking, and an anti-roll control system for a vehicle chassis is proposed in this paper. The aim was to enhance driving performance while maintaining handling stability during cornering scenarios. Through the coordination of the three control modules, the vehicle demonstrated improved handling capabilities with reduced energy dissipation, as validated by the co-simulations. The physical constraints were considered in the controller design, and the variable structure control method enabled an easy tuning process in engineering applications, which can help reduce the time and energy of engineers in tuning the controller.

As the effectiveness of the proposed coordinated control algorithm was verified to some extent in these co-simulations, future work should focus on real-vehicle tests with real-time implementations of the control algorithm.