Keywords

1 Background

Many researchers have been working on vehicle motion control by distributing driving force to each wheel for a long time. However, realizing the control requires a complex control device to distribute the driving force, so it has not yet become widely used for mass-produced vehicles. A movement that may change this situation is “electrification”, which is currently occurring. Unlike internal combustion engines, multiple electric motors can be installed in a vehicle, making it easier to implement driving force distribution control.

The driving force distribution controls that have been proposed so far have various aims, such as stabilizing steering characteristics, improving steering response, equalizing tire workloads, and improving disturbance stability [1,2,3,4,5,6]. All of those are important target characteristics, however there are still questions in control development, such as which combination of them is optimal, whether there are any that act redundantly, and whether the control gain has no choice but to rely on tuning. This paper discusses a method to uniquely derive the ideal driving force control logic.

2 Effect of Tire Lateral Force Restrained by Steering Angle

Focusing on vehicle motion control methods that use tire longitudinal forces, in most cases control commands are based on the yaw moment generated by the difference in driving force between the left and right wheels. In other words, control using tire longitudinal forces is generally performed by force (moment) control. On the other hand, tire lateral force is generally controlled not by force but by steering angle. In this chapter, in order to clarify the dynamic characteristics of the tire lateral motion, the function of lateral force is analyzed and discussed using equations of motion.

The lateral and yaw motion equations of a general bicycle model are shown in Eqs. (1) and (2), and the front and rear tire slip angles αf, αr are shown in Eqs. (3) and (4).

$$mV\left(r+\dot{\beta }\right)=-2{K}_{f}{\alpha }_{f}-2{K}_{r}{\alpha }_{r}+{F}_{yd}$$
(1)
$${I}_{z}\dot{r}=-2{K}_{f}{\alpha }_{f}{l}_{f}+2{K}_{r}{\alpha }_{r}{l}_{r}$$
(2)
$${\alpha }_{f}=\beta +\frac{{l}_{f}r}{V}-{\delta }_{f}$$
(3)
$${\alpha }_{r}=\beta -\frac{{l}_{r}r}{V}-{\delta }_{r}$$
(4)

where, m is the vehicle mass, V is the vehicle speed, r is the raw rate, β is the slip angle at the CoG (center of gravity), Kf, Kr are the cornering stiffness of the front and rear tires, Iz is the yaw inertial moment, lf, lr are the distance from the CoG to front and rear axles, \(\delta_{f} ,\delta_{r}\) are the front and rear steer angles, and Fyd is the lateral disturbance force.

Focusing on the lateral motion of the vehicle, the slip angle β for the steering input and lateral disturbance force is derived from the above equations. And it is expressed as Eq. (5), where, in order to roughly understand the principle, terms with high orders of s are deleted. This is also represented by the block diagram shown in Fig. 1.

$$\beta \left(s\right)\approx \frac{\frac{1}{2}-\frac{{V}^{2}}{l{C}_{r}g}}{\left(\frac{1+{I}_{zn}}{2}\right)\frac{V}{{C}_{a}g}s+1+A{V}^{2}}{\delta }_{f}+\frac{\frac{1}{2{C}_{a}}}{\left(\frac{1+{I}_{zn}}{2}\right)\frac{V}{{C}_{a}g}s+1+A{V}^{2}}\frac{{F}_{yd}}{mg}$$
(5)

where, l is the wheelbase, Cf, Cr are the normalized cornering stiffnesses, Ca is the reciprocal of the sum of reciprocals of Cf and Cr, Izn is the normalized yaw inertial moment, A is the stability factor, g is the gravitational acceleration, and s is the Laplace operator.

Fig. 1.
figure 1

Block diagram representing vehicle lateral motion.

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This system uses tire cornering stiffness Ca as a control gain to reduce the error between the target steady slip angle and the actual slip angle due to steering angle input, while also compensating for external lateral disturbance force. It can be understood that it has the function of a skyhook damper installed on the lateral side of the vehicle. This is why it is effective in stabilizing the vehicle against disturbances without applying any feedback control.

3 Proposed Driving-Force-Control Method Acting as a Lateral Force

When a tire is moved laterally due to disturbance, lateral force is generated to resist it. However, such a reaction force does not occur in the tire longitudinal direction unless some kind of control is applied. This is because tires roll in the longitudinal direction. This is the reason why the right side of the equation of motion in Eqs. (1) and (2) consists only of the lateral-force elements of the tires.

Here, it is assumed that such geometric anisotropy in the longitudinal and lateral directions of the tire does not exist, and that force is generated in the longitudinal direction of the tire by the same mechanism as in the lateral direction. Then, the modified equations of motion are expressed as shown in Eqs. (6) and (7). The third and fourth terms on the right side of Eq. (7) are due to longitudinal force, and are defined to have the same structure as the first and second terms.

$$mV\left(r+\dot{\beta }\right)=-2{K}_{f}{\alpha }_{f}-2{K}_{r}{\alpha }_{r}$$
(6)
$${I}_{z}\dot{r}=-2{K}_{f}{\alpha }_{f}{l}_{f}+2{K}_{r}{\alpha }_{r}{l}_{r}+2{K}_{xl}{s}_{xl}\frac{t}{2}-2{K}_{xr}{s}_{xr}\frac{t}{2}$$
(7)

where, Kxl, Kxr are the driving stiffnesses, sxl, sxr are the slip ratios at the left and right wheels, and they are defined by Eqs. (8) and (9) so that they have the same configuration as the slip angles in Eqs. (3) and (4).

$$ s_{xl} = s_{d} - \frac{{\frac{{v_{wr} - v_{wl} }}{2}}}{V} + \frac{\frac{t}{2}}{V}\frac{V}{l}\frac{1}{{1 + AV^{2} }}\left( {\delta_{f} - \delta_{r} } \right) $$
(8)
$$ s_{xr} = s_{d} + \frac{{\frac{{v_{wr} - v_{wl} }}{2}}}{V} - \frac{\frac{t}{2}}{V}\frac{V}{l}\frac{1}{{1 + AV^{2} }}\left( {\delta_{f} - \delta_{r} } \right) $$
(9)

The first term sd is the slip ratio due to the driver's acceleration request, the second term is the slip ratio for the yaw rotation detected by the difference between the left and right wheel speeds vwl and vwr, and the third term is the slip ratio for the target yaw rate due to the driver's steering input. Here, if the slip ratio difference sxz between the left and right wheels caused by yaw motion is defined by Eq. (10), Eq. (7), which is the equation of motion for yaw, can be transformed into Eq. (11), and the third and fourth terms on the right side explains that the tire longitudinal force can be expressed in terms of translational slip and rotational slip.

$$ s_{xz} \mathop = \limits^{{{\text{def}}}} \frac{{s_{xr} - s_{xl} }}{2} = \frac{{\frac{{v_{wr} - v_{wl} }}{2}}}{V} - \frac{\frac{t}{2}}{V}\frac{V}{l}\frac{1}{{1 + AV^{2} }}\left( {\delta_{f} - \delta_{r} } \right) $$
(10)
$${I}_{z}\dot{r}=-2{K}_{f}{\alpha }_{f}{l}_{f}+2{K}_{r}{\alpha }_{r}{l}_{r}+2\left({K}_{xl}-{K}_{xr}\right){s}_{d}\frac{t}{2}-4{K}_{x}{s}_{xz}\frac{t}{2}$$
(11)

where, the Kx is the average value of left and right wheel driving stiffness Kxl and Kxr.

In order to further transform the equations, 2Kxl and 2Kxr are expressed as Eqs. (12) and (13).

$$2{K}_{xl}={C}_{x}\left(\frac{mg}{2}-m{a}_{y}\frac{h}{t}\right)$$
(12)
$$2{K}_{xr}={C}_{x}\left(\frac{mg}{2}+m{a}_{y}\frac{h}{t}\right)$$
(13)

where, the Cx is the normalized driving stiffness, ay is the lateral acceleration, h is the CoG height, and t is the track at the front and rear. Here, we focus on the third term on the right side of Eq. (11). By substituting Eqs. (12) and (13) into the third term, it can be transformed into Eq. (14) by ignoring the driving resistance and assuming that Cxsd is equal to longitudinal acceleration ax/g.

$$ \begin{aligned} 2\left( {K_{xl} - K_{xr} } \right)s_{d} \frac{t}{2} = & - C_{x} ma_{y} hs_{d} \\ = &\,\, mgh\frac{{a_{x} }}{g}\frac{{a_{y} }}{g} \\ \end{aligned} $$
(14)

The above equation is the yaw moment that occurs when the driving force is distributed in proportion to each wheel load. This is the so-called ideal driving force distribution. It is known that in the linear region of tires, the yaw moment theoretically cancels out changes in steering characteristics due to longitudinal load transfer during turns [7]. Here, Eq. (11) can be rearranged as Eq. (15).

$${I}_{z}\dot{r}=-2{K}_{f}{\alpha }_{f}{l}_{f}+2{K}_{r}{\alpha }_{r}{l}_{r}+mgh\frac{{a}_{x}}{g}\frac{{a}_{y}}{g}-4{K}_{x}{s}_{xz}\frac{t}{2}$$
(15)

Considering the meaning of the derived longitudinal force term in the above equation, first, the third term on the right side means that the driver's required driving force is distributed in proportion to each wheel's load. Translating this into lateral motion, it corresponds to the generation of tire lateral force proportional to the load on each wheel during turns. As can be seen from Eq. (10), the fourth term plays the role of feedback compensation when an error occurs in the yaw rate estimated from the left and right wheel speed difference with respect to the target steady yaw rate determined from the driver’s steering input. A schematic diagram of this is shown in Fig. 2, which shows that for yaw motion, skyhook dampers are applied in the direction of rotation.

This function attempts to maintain the left and right wheel speed difference at a certain target value in response to the driver's steering input. Translating this into lateral motion, it corresponds to maintaining the steering angle difference between the front and rear wheels at a certain value in response to steering input. Since this control theory was derived by isotropicizing the longitudinal and lateral characteristics of tire, it is called as “Tire Isotropic Control (TIC)”.

Fig. 2.
figure 2

Schematic diagram of Tire Isotropic Control (TIC).

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4 Verification Using a Full Vehicle Simulation Model

The vehicle model has six degrees of freedom on its sprung mass, as well as degrees of freedom for each wheel to move up and down, rotate, and steer, and is equipped with suspension and tires with nonlinear characteristics. The yaw moment due to the tire longitudinal force shown in Eq. (15) is commanded by each wheel driving force.

Figure 3 shows the yaw rate, lateral acceleration, tire lateral and longitudinal forces, and tire workloads when a stepped steering is input at a speed of 80 km/h, and then a yaw moment disturbance is input. It can be confirmed that by applying longitudinal forces to each wheel appropriately, steering response and disturbance stability are improved while suppressing the tire workloads. This is the most efficient state in which the forces of each wheel do not cancel each other out, and it is possible to use the tire friction circles efficiently to the limit.

Fig. 3.
figure 3

Effect of TIC on steering response and yaw disturbance.

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5 Conclusions

Conventionally, designing driving force distribution control has involved adjusting appropriate control gains through trial and error. In this paper, we proposed an original control method that generates longitudinal force that imitates tire lateral force, inspired by the mechanism of tire lateral force generation. As a result, it was shown that it is possible to use the tire friction circle without wasting it with a simple control law, and because the control gains are derived from the equation, the work of trial and error is freed. It was also clarified that this control method enables steering response and disturbance stability to be achieved as if the tire cornering stiffness has been increased.

In this paper, we focused on vehicles that can independently control the driving force of the four wheels, but in the actual market, few vehicles equipped with four motors are produced. In the future, we plan to study how to apply this control method to more realistic two-motor or three-motor vehicles, or vehicles equipped with brake actuators for each wheel.