Keywords

1 Introduction

Vehicle control systems that enable cornering at high sideslip angle (drifting) have recently been the focus of significant research effort. These approaches have been developed mainly in the context of Autonomous Vehicles (AVs) to demonstrate that the AV controllers are able to operate in regimes that are challenging and non-intuitive for average human drivers [1,2,3,4,5].

Drift-assist control, where the driver still maintains authority over the control of the vehicle in the context of Advanced Driver Assist Systems (ADAS), has been introduced by several high-performance vehicle manufacturers (“Variable Drift Control” of McLaren [6], “Drift Mode” of BMW [7]). These systems allow the vehicle to operate outside the envelope of stability control systems, or even provoke instability to achieve high sideslip angle [8, 9]. However, maintaining a drift condition, even with drift-assist control, currently still requires driving skills beyond average and understanding of the non-intuitive vehicle behavior in such operating regimes. One of the reasons is the steering input provided by the driver can prevent the control system from achieving its task, if it is not properly coordinated with the wheel torque control of the drift assist system [10].

In this paper, we introduce a concept of drift-assist system to stabilize the vehicle in high sideslip cornering using torque vectoring interventions against uncoordinated steering inputs (steering noise) by the driver. In particular, we calculate drifting equilibria for vehicles equipped with four-wheel torque vectoring capability, and explore the effects of steering angle in the required wheel-slip for four wheels to achieve the same equilibrium states. This study reveals a conflict in maintaining the same yaw rate and vehicle sideslip angle when the steering angle changes. Therefore, we formulate a trajectory tracking task where the vehicle speed and path angle rate (instead of both yaw rate and sideslip angle) are sought to be stabilized by means of torque vectoring when the steering angle is being disturbed. The control task is formulated as a nonlinear programming problem to demonstrate the feasibility of the concept, and a control structure is designed and further validated in CarMaker.

2 Drift Equilibria for Four-Wheel Drive Vehicle

Based on typical parameters of a high-performance vehicle, a four-wheel drive vehicle model (Fig. 1) is considered for longitudinal/lateral dynamics (speed \(v\), yaw rate \(r\) and sideslip angle \(\beta \)), with steering \(\delta \) and four-wheel slip ratio \({s}_{ri} (i=fl,fr,rl,rr)\) as inputs. The model considers nonlinear tire characteristics and load transfer effects.

Fig. 1.
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The four-wheel drive vehicle model for drift equilibrium analysis.

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Unlike rear-wheel drive drift analysis where there are only two controlled variables (steering and throttle), now with four-wheel drive that allows different slip ratio for each wheel, there are more unknown variables involved. But with three equilibrium equations (\(\dot{v}, \dot{r}, \dot{\beta }=0\)) we can only solve for three variables. Therefore, we set the steering \(\delta \) and the slip ratio \({s}_{ri}\) as known before solving the equilibria [11]. With equal slip ratio for wheels of the same axle (\({s}_{fl,fr}={s}_{rf},{s}_{rl,rr}={s}_{rr}\)), the solved equilibria at different \(\delta \) are shown in Fig. 2 (counterclockwise drifting with counter steering) as surfaces marked in different color. One of the most important findings of this steady-state analysis is the steering effects on the drift equilibria: i.e., when the counter steering increases from −10° to −20° (green surface to orange surface in Fig. 2), to maintain the previous \({r}_{eq}\), the rear wheel slip ratio \({s}_{rr}\) has to increase and \({s}_{rf}\) to decrease, however for keeping the sideslip \({\beta }_{eq}\), the trend is opposite that \({s}_{rr}\) has to decrease and \({s}_{rf}\) to increase. This leads to a preliminary conclusion that \({r}_{eq},{\beta }_{eq}\) are unlikely to be both stabilized when the steering has changed (the effects of different slip ratio for wheels on the same axle will be discussed in detail in our future work).

Fig. 2.
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The four-wheel drive drift equilibria at different steering angle \(\updelta \).

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3 Nonlinear Programming Based Steering Noise Cancelling

As discussed in previous section, when the drift equilibrium is disturbed by additional steering inputs, \({r}_{eq},{\beta }_{eq}\) are not likely to be stabilized separately. Therefore, we discuss the steering noise cancelling problem in the aspect of path angle rate \(\dot{\varphi }\), which is the sum of yaw rate \(r\) and sideslip rate \(\dot{\beta }\) (Fig. 1), and directly decides the trajectory. To explore whether stabilizing \(\dot{\varphi }\) is possible, nonlinear programming method [12] is applied to the previous four-wheel vehicle dynamics model in a counterclockwise drift, and the cost function \(J\) is written as:

$$J={\varepsilon }_{1}{\left(\dot{\varphi }-\dot{{\varphi }_{eq}}\right)}^{2}+{\varepsilon }_{2}{(v-{v}_{eq})}^{2}$$
(1)

where \({\varphi }_{eq}\), \({v}_{eq}\) are the equilibria path angle rate and speed, \({\varepsilon }_{1},{\varepsilon }_{2}\) are the weighting coefficients. The result is shown in Fig. 3, a sinusoidal noise (180°) is added to the equilibrium steering wheel angle (−380°). With torque vectoring that independently controls the slip ratio for each wheel, the desired speed, path angle rate and the trajectory could be kept against the steering noise from unskilled driver.

Fig. 3.
figure 3

The steering noise cancelling results of nonlinear programming.

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4 Steering Noise Cancelling Controller and Simulation Results

In the preceding section, we discussed the possibility to cancel the steering effects of an inexperienced driver and maintain the desired trajectory for drifting with torque vectoring. In this section, we design a steering noise cancelling controller and evaluate it in CarMaker. As shown in Fig. 4, the proposed structure could be divided into three parts, which will be discussed respectively.

Fig. 4.
figure 4

The structure of the steering noise cancelling and trajectory tracking controller.

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4.1 Steering Noise Cancelling and Trajectory Tracking

Known the desired trajectory radius \({R}_{des}\) and the disturbed steering angle \(\delta \) (which deviates from the equilibrium \({\delta }_{eq}\) with a noise of \({\delta }_{noise}\)), we firstly calculate a new equilibrium sideslip angle \({\beta }_{eq}\) as feedforward based on the previous equilibrium map (Fig. 2). Next, to account for the lateral deviation \({e}_{d}\) from the desired trajectory, a feedback sideslip angle \({\beta }_{c}\) based on a Proportional-Derivative (PD) controller is adopted. Then, the desired sideslip angle \({\beta }_{des}\) could be written as Eq. (2), in which \({k}_{pd},{k}_{dd}\) are coefficients. The desired trajectory could be tracked as higher sideslip generates more centripetal force which drives the vehicle drift deeper into the corner thus reduce the trajectory radius, and vice versa.

$${\beta }_{des}= {\beta }_{eq}+{\beta }_{c}= {\beta }_{eq}-{k}_{pd}{e}_{d}-{k}_{dd}d{e}_{d}/dt$$
(2)

4.2 Sideslip Stabilization

To stabilize the sideslip angle at the desired \({\beta }_{des}\), the equilibrium yaw rate \({r}_{eq}\) is considered together with the sideslip error \({\beta }_{err} ({\beta }_{err}={\beta }_{des}- \beta )\). And the desired yaw rate \({r}_{des}\) could be written as Eq. (3), in which \(\tau \) is a time constant.

$${r}_{des}= {r}_{eq}-{\dot{\beta }}_{err}= {r}_{eq}-{\beta }_{err}/\tau $$
(3)

4.3 Yaw Rate Stabilization and Torque Vectoring

With the desired yaw rate \({r}_{des}\) and the yaw rate error \({r}_{err}={r}_{des}-r\), a Proportional-Integral (PI) controller is implemented to calculate the desired yaw moment \({\Delta M}_{z}\) for torque vectoring, as shown in Eq. (4), in which \({k}_{pr},{k}_{ir}\) are constant coefficients. Then, together with the total drive force \({F}_{xdes}\) in order to maintain the equilibrium speed \({v}_{eq}\), also assuming all wheel torque is delivered as drive force [13], the drive torque for each wheel \({T}_{fl,fr,rl,rr}\) could be written as Eq. (5). Here \(w\) is the wheel track width, \({r}_{w}\) is the wheel rolling radius, and \({c}_{tf}\) is a coefficient to decide the torque allocation between front and rear wheels.

$${\Delta M}_{z}={k}_{pr}{r}_{err}+{k}_{ir}\int {r}_{err}dt$$
(4)
$$\begin{aligned}& \quad {T}_{fl,fr}={c}_{tf}(1/2{F}_{xdes}\mp \Delta {M}_{z}/w){r}_{w} \\ & {T}_{rl,rr}=(1-{c}_{tf})(1/2{F}_{xdes}\mp {\Delta M}_{z}/w){r}_{w}\\ \end{aligned}$$
(5)

4.4 Simulation Results

To evaluate the above steering noise cancelling controller, a circular track with 20 m radius is built in CarMaker. The vehicle is drifting in counterclockwise direction until 2 s a steering noise with increasing frequency and amplitude is added to imitate the steering behavior of an inexperienced human driver. As shown in Fig. 5, the steering effects are effectively reduced by varying the sideslip angle and yaw rate, that the lateral deviation is lower than 1 m and the desired trajectory is basically maintained as shown in Fig. 6.

Fig. 5.
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Vehicle motion with steering noise cancelling controller.

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Fig. 6.
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Trajectory with steering noise cancelling controller.

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5 Conclusion

In this paper we propose a driver assist system concept for the stabilization of the vehicle during drifting, as undesired steering inputs (steering noise) from inexperienced drivers can disturb the drift equilibria and deviate the vehicle from the desired track. Therefore, this paper presents a steering noise cancelling and trajectory tracking concept to cancel the steering effects from the driver by torque vectoring. In detail, first by equilibrium analysis specifically for four-wheel drive vehicle, the steering effects on drift equilibria are analyzed, which indicates a contradiction between stabilizing yaw rate and sideslip angle. Then, by nonlinear programming, it is proven that the path angle rate together with the speed which determines the trajectory could be regulated with torque vectoring against the steering noise, by independently controlling the slip ratio for all four wheels. Finally, a control structure is proposed and further evaluated in CarMaker, where the vehicle successfully maintains drifting on the desired circular track under a sinusoidal steering disturbance.