Keywords

1 Introduction

In race cars, operating at the limit of tire performance, enhancing cornering performance is crucial to minimize lap times. The most important of these is the steady-state performance at the corner apex, which determines the minimum speed during the cornering process, and requires maximizing the grip of the four tires and balancing the yaw moment. Typical measures for improving cornering performance include various approaches aimed at uniformly improving overall performance. These may include initiatives such as mass reduction, tire performance improvement, and increased downforce. In scenarios where cornering performance potential is constrained by regulations, the steady-state cornering capability of a rear-wheel-drive vehicle primarily hinges on achieving equilibrium between lateral forces acting on all four tires and longitudinal forces exerted on the left and right rear tires. Ensuring that the lateral forces on all four tires reach their limits is crucial. The key lies in the distribution of torque applied to the left and right rear tires, known as torque-vectoring (TV). While TV is typically induced by a passive differential mechanism, achieving precise control for each vehicle in motion presents challenges. Active yaw moment control systems have been extensively researched to improve vehicle maneuverability and stability in recent years. Among these, active TV systems, including those investigated by our research group [1] have garnered significant attention. Different TV control laws that change the dynamic characteristics have been studied so far. Among them, a torque distribution strategy proportional to the vertical load was devised to suppress stability changes associated with load transfer during acceleration and deceleration [2].

This study presents an analytical investigation of the contribution of a TV control law proportional to the vertical load to maximizing the steady-state lateral acceleration of a rear-wheel-drive race car. Although this control law has been recognized and much investigated for some time, the purpose of this paper is to emphasize its importance compared to conventional passive systems for maximizing the steady-state lateral acceleration of race cars. The analysis method used is the moment method [3, 4], which can comprehensively analyze the behavior changes caused by nonlinear tire characteristics. This analysis method has the advantage of being able to quantify stability indices for whether or not a turn can be maintained. The analysis results were then compared with those obtained with conventional passive systems.

2 Vehicle Dynamics Model

2.1 Equations of Motion with 7 Degrees of Freedom

As shown in Fig. 1(a), the vehicle is constrained on a two-dimensional plane, exhibiting 7 degrees of freedom, which encompass the rotational motion of each wheel. To accurately the model’s behavior, roll/pitch load transfers were calculated based on the height of the center of gravity and the tire positions under static equilibrium conditions. In addition, anti-roll stiffness distribution was considered for rolling effects.

Fig. 1.
figure 1

(a) Free body diagram showing all external forces applied to a vehicle body. (b) Free body diagram showing all external forces on the tire and wheel assembly.

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The equations of translational motion for two degrees of freedom in the longitudinal and lateral directions at the vehicle’s center of gravity are shown in Eqs. (1) and (2). Equation (3) shows the equation of rotational motion in the yaw direction.

$$ m\left( {\dot{V}_{x} - rV_{y} } \right) = - (F_{yFL} + F_{yFR} )\sin \delta + F_{xRL} + F_{xRR} - F_{d} , $$
(1)
$$ m\left( {\dot{V}_{y} + rV_{x} } \right) = (F_{yFL} + F_{yFR} )\cos \delta + F_{yRL} + F_{yRR} , $$
(2)
$$ I_{zz} \dot{r} = l_{f} \left( {F_{yFL} + F_{yFR} } \right){\text{cos}}\delta - l_{r} \left( {F_{yRL} + F_{yRR} } \right) + t_{f} /{2}\left( { - F_{yFL} + F_{yFR} } \right){\text{sin}}\delta + t_{r} /{2}\left( {F_{xRL} - F_{xRR} } \right), $$
(3)

2.2 Application to Moment Method

The moment method is a quasi-static analysis method that uses a combination of the steering wheel angle δsw and the sideslip angle β at a given longitudinal velocity Vx as inputs. Developed in the 1990s, primarily as an analytical tool for race cars, this method offers a significant advantage in that it can comprehensively represent the relationship between the vehicle’s lateral acceleration ay and yaw moment N for each input condition. In this study, as shown in Eq. (4), the time derivatives of the longitudinal velocity Vx and lateral velocity Vy were constrained to zero in the moment method calculation.

$$ \left[ {a_{y} ,N} \right] = {\text{f}}\left( {\delta_{sw} ,\beta ,V_{x} } \right)\,{\text{with}}\,\dot{V}_{x} = \dot{V}_{y} = 0, $$
(4)

Specifically, a moment diagram was created as shown in Fig. 2. The horizontal axis represents the lateral acceleration, indicating the cornering performance, whereas the vertical axis represents the yaw moment. The inputs are represented by two lines: the red line represents the constant steering angle, and the black dashed line represents the constant sideslip angle. The endpoint of the constant steering angle line represents the limit of the front tire, whereas the endpoint of the constant sideslip angle line signifies the limit of the rear tire.

Fig. 2.
figure 2

Schematic diagram describing the moment method.

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The trim line, where the yaw moment was balanced, and the trimmed maximum lateral acceleration were used as indices of cornering performance. In addition, as a stable system must maintain turning for a steady-state, we focused on the stability index. The slope of the constant steering angle in the Fig. 2 indicates stability; specifically, the stability index SI shown in Eq. (5) was used.

$$ SI = \partial N/\partial \{ a_{y} m(l_{f} + l_{r} )\} , $$
(5)

3 Study of Control Law for Active TV Systems

In this study, our focus lies in examining whether the yaw moment, resulting from the disparity in longitudinal forces, can alter the trimming point of the cornering performance depicted in Fig. 2, consequently improving overall cornering performance. Another important consideration is the interdependence between the lateral and longitudinal forces generated by the tires. Figure 3 shows the lateral force values against the slip ratio along the line of constant slip angle, as determined using the tire model. The convex shape of the graph, with its apex at zero slip ratio, indicates the undesirability of employing extremely high slip ratios when aiming to maximize lateral force on the tire for enhanced lateral acceleration. This issue remains unresolved with existing passive differentials. In passive systems, TV is governed by the grip state of the left and right tires, affected by factors such as the frictional torque of the clutch or the yaw angular velocity. Consequently, one wheel may experience strain, resulting in a high slip ratio and a loss of the lateral force necessary for improving cornering performance.

Fig. 3.
figure 3

Lateral force against slip ratio calculated from tire model.

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The control law of the torque vectoring system proportional to the vertical load in this study based on the relationship between slip ratio and lateral force for improving the problem and cornering perfor-mance of the passive system is shown in Eq. (6).

$$ T_{RL,RR} = T_{in} \left[ {p\left\{ {\left| {F_{zRL,RR} /F_{zRL} + F_{zRR} } \right| - 0.{5}} \right\} + 0.{5}} \right], $$
(6)

Here, Tin represents the input torque. This control law allocates torque to the rear wheels according to the vertical load distribution on the left and right sides. The parameter p is set, where p = 1 signifies distribution in accordance with the vertical load distribution on the left and right sides. For p = 0, TV was performed, equivalent to an open differential.

4 Analysis to Validate the Proposed Control Law

The proposed active TV control law was compared to the existing open differential, limited-slip differential (LSD), and locked differential during steady-state turning to determine its effects on lateral acceleration and stability. The analysis was conducted on a vehicle used in Formula SAE events [5], where competition vehicles were built by students from universities worldwide. Test data specific to FSAE vehicles were used for tires [6] and differentials. Table 1 lists the fundamental vehicle parameters. Figure 4 shows the moment diagram of each system. The longitudinal velocity was 15 m/s, and the steering wheel angle was calculated for combinations in the range of 0°–100°, with the sideslip angle ranging from −5°–5°.

Table 1. FSAE vehicle parameters
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The lateral acceleration of the proposed system improved by approximately 7.8% compared to the maximum value of the passive system. Among the passive systems, the locked differential system exhibits the lowest lateral acceleration.

Fig. 4.
figure 4

Moment diagram for each system.

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Figure 5 shows the stability index at the trim line for each system. The larger the stability index in the negative direction, the higher the stability; if the value is in the positive direction, the vehicle cannot continue turning. In the passive system, the directly connected locked differential exhibited the highest stability, up to approximately 10 m/s2 of lateral acceleration. In the region where the lateral acceleration exceeded 10 m/s2, the other systems were more stable, and the situation was reversed. The proposed control law for TV was found to be unstable, indicating a trade-off between stability and steady-state cornering performance. If stability becomes an issue at speeds higher than those used in this study, the TV ratio in Eq. (6) should be adjusted closer to zero. Two factors contributed to the increase in the lateral acceleration of the proposed system. First, the increase in the slip ratio of the inner rear wheel was suppressed to increase the lateral force, as the control law intended. In fact, the slip ratio was reduced by approximately 66% compared to the direct-coupling case. The second factor is thought to be that the addition of yaw moment increased the grip in the negative yaw moment region of the moment diagram to zero.

Fig. 5.
figure 5

Trimmed lateral acceleration relative to the stability index.

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5 Conclusion

Moment method analysis compares the proposed torque vectoring control law with the passive system and shows that significant performance improvement can be expected without weight reduction or additional downforce. However, since the reduction in stability, which is considered to be a trade-off with lateral acceleration, could be a serious problem, the same analysis will be performed at higher vehicle speeds where the yaw damping is reduced. In addition, since the estimation of vertical loads is necessary to fully apply the control law to reality, the estimation and calculation of load changes due to load transfers and aerodynamics will also be investigated.