A Novel Torque Vectoring Approach to Enhance Driving Experience

Abstract

The current state of Direct Yaw Control literature has reached an advanced level, yielding compelling results in stabilizing vehicle behavior, and enhancing overall performance. However, conventional approaches exhibit limitations in addressing highly transient lateral dynamics, particularly during conditions of rapidly increasing vehicle sideslip angles, leading to poor controllability and reduced intuition for the driver. This paper introduces a novel extension to the established yaw rate tracking technique, integrating the trivial yaw rate feedback control with a less conventional model-based feedforward term, complemented by an innovative sideslip rate tracking loop. The primary objective of this term is to smoothen vehicle cornering response, effectively dampening oscillations in the vehicle’s behavior without compromising time delays to driver inputs. The intended outcome is not only enhancing safety but also delivering a more intuitive and enjoyable driving experience. The effectiveness of the approach is demonstrated by the results obtained through detailed Driver-in-Motion (DiM) sessions.

1 Introduction

In recent years, the research field of vehicle control algorithms has witnessed a significant surge in research endeavors, particularly directed towards enhancing handling, performance, and stability. Among the various performance-oriented algorithms, the Torque Vectoring (TV) stands out as the most widely adopted, whose concept is to impose a desired vehicle yaw rate ($\omega_z$) or sideslip ($\beta$), derived from an ideal reference model, by generating an additional yaw moment ($M_z$) through proper control of actuators. A comprehensive coverage on the state-of-the-art feedback (FB) and feedforward (FF) algorithms is depicted in [1], therefore the authors will refrain from discussing further references at this stage. This paper proposes a novel approach to TV algorithms, building upon the conventional yaw rate tracking method. The proposed method combines commonly used yaw rate feedback with a model-based feedforward control module and a novel damping term aimed at minimizing overshoots of vehicle sideslip rate ($\dot{\beta }$) (Fig. 1). A comparison between the passive vehicle and the controlled one through the proposed approach highlights how the latter results in a notably better damped and smooth vehicle response during transient cornering, with significantly reduced overshoot and oscillations in the closed loop tracking, without inducing any additional delay in the vehicle response. While discussing such topics, the focus is placed on how such novel algorithm is specifically designed for production road cars: emphasis is set on debugging efficiency and calibration simplicity, while ensuring high performance, which are crucial aspects for the cost-effective development of luxury hypercars.

2 Control Scheme

Fig. 1.

Torque vectoring control scheme.

2.1 Reference Model

Aligned with the application efficiency, a simplified nonlinear bicycle model combined with a 4-parameter Pacejka tire formulation is selected to generate the reference vehicle response, guaranteeing both modeling precision and simplicity. The mathematical representation of this modeling approach can be described :

$$ \dot{\omega }_z = \frac{1}{J_z }\left( {F_{y,f} l_f \cos \delta - F_{y,r} l_r } \right) $$

(1)

where $J_z$ is the vehicle rotational inertia around its z-axis, $F_{y,i}$ the axles lateral force, δ the wheel steer angle, and $l_i$ the distance between the axle and the center of gravity.

It is important to note that due to the presence of the lateral force, estimations of longitudinal speed ($v_x$), vertical load on the tires ($F_{z,i}$), and vehicle sideslip are essential. While the first two can be estimated with reasonable precision using standard algorithms, accurately estimating sideslip in all driving conditions remains a challenge.

To address this and enhance robustness while minimizing the calibration effort, the sideslip angle is computed by integrating the vehicle model using measurements of steering wheel angle and longitudinal speed. The dynamics are then evolved using Eq. 1 combined with the sideslip rate dynamics, derived by:

$$ a_y = \frac{1}{M}\left( {F_{y,f} \cos \delta + F_{y,r} } \right) $$

(2)

$$ \dot{\beta } = \frac{1}{v_x }\left( {a_y - \omega_z v_x } \right) $$

(3)

Here, $a_y$ represents the lateral acceleration and $M$ the total mass of the vehicle.

Since the model is based on a non-linear tire formulation, saturation limits are inherently considered, particularly with respect to the peak lateral force. Additionally, to capture the combined tire behavior, a scaling factor is applied to $F_{y,i}$ such that when high slip ratios $k_i$ are experienced, the maximum lateral force is significantly reduced.

It is essential to emphasize that these equations describe the vehicle response to a given input steering wheel angle at a specific longitudinal speed. If the model is provided with ideal vehicle parameters, used as reference model calibration, it will yield an idealized transient behavior. Hence, for a specific set of calibration parameters, a unique model capable of producing a desired yaw rate and sideslip angle response can be defined, thus covering feedforward modeling and closed-loop reference definition.

At this stage, it is crucial to not introduce any control variables into the formulation, as this would alter the fundamental behavior, as the goal is to derive an ideal response that can then be deliver by the vehicle through torque vectoring. Control variables will be introduced in the next section for computing the feedforward action.

2.2 Feedforward Term

The feedforward action entails estimating the additional yaw torque needed to achieve the reference behavior in an open-loop manner, relying solely on knowledge of both the physical vehicle and the reference vehicle. The rationale behind this approach is to compare the yaw torque produced by the passive vehicle with the yaw torque that the target vehicle would generate under identical input conditions (such as steering wheel angle and longitudinal speed). The difference between these values indicates the adjustment that torque vectoring should apply to achieve the target response.

Enhancing Eq. 1 with the vehicle parameters and incorporating the torque vectoring contribution, the yaw moment generated by the real vehicle is described by equation:

$$ \dot{\omega }_z = \frac{1}{{\hat{J}_z }}\left( {\hat{F}_{y,f} \hat{l}_f \cos \delta - \hat{F}_{y,r} \hat{l}_r + M_{z,FF} } \right) $$

(4)

Alternatively, the ideal vehicle response can be calculated by exclusively evaluating Eq. 1 with the parameters of the target vehicle:

$$ \dot{\omega }_z = \frac{1}{{\overline{J}_z }}\left( {\overline{F}_{y,f} \overline{l}_f \cos \delta - \overline{F}_{y,r} \overline{l}_r } \right) $$

(5)

The required feedforward action can be computed by equating Eq. 7 and Eq. 8, as:

$$ M_{z,FF} = \frac{{\hat{J}_z }}{{\overline{J}_z }}\left( {\overline{F}_{y,f} \overline{l}_f \cos \delta - \overline{F}_{y,r} \overline{l}_r } \right) - \left( {\hat{F}_{y,f} \hat{l}_f \cos \delta - \hat{F}_{y,r} \hat{l}_r } \right) $$

(6)

Most of common state-of-the-art open-loop control methods are derived by mathematically inverting the vehicle model based on a single specific reference variable, such as yaw rate or sideslip angle. However, direct inversion is not always feasible due to the transfer function realization matter, typically requiring a simplified tire model. Instead, the proposed approach does not require model inversion and establishes a law that represents the response of the target vehicle whilst including multiple reference quantities, thus allowing for enhanced yet straightforward and intuitive reference calibration.

2.3 Feedback Term

The innovative approach proposed here incorporates a feedback tracking system for the sideslip rate in conjunction with the feedforward control. The primary advantage of this method lies in the introduction of a damping element within the loop. This ensures an improved driving experience for the driver by delivering a smoother transient response, significantly reducing instances of over- or undershooting of the sideslip angle, as well as avoiding snappy vehicle responses whilst offering an intuitive driving feeling.

For the sake of illustration, a proportional controller $R_1 = - \left| {K_1 } \right|$ is used. Such a gain must be negative to ensure asymptotic stability, as it can be proven that the transfer function between the yaw torque $M_z$ and sideslip rate $\dot{\beta }$ has a negative gain, thus:

$$ M_z = - \left| {K_1 } \right|\left( {\dot{\beta }_{ref} - \dot{\beta }} \right) $$

(7)

Equation 3 and Eq. 7 can be combined to derive a more explicit control law:

$$ M_z = \left| {K_1 } \right|\left( {\omega_{z,ref} - \omega_z } \right) - \frac{{\left| {K_1 } \right|}}{v_x }\left( {a_{y,ref} - a_y } \right) $$

(8)

Equation 8 shows how the proposed control approach can be interpreted as a multivariable control system. The sideslip rate feedback inherently integrates a yaw rate tracking loop with a lateral acceleration feedback, whose gain dynamically varies with vehicle speed. However, this term has a negative gain opposing the yaw rate tracking action, therefore necessitating a tradeoff between yaw rate tracking and the damped transient responses.

It is crucial to underline that while tracking a reference yaw rate is pivotal for guaranteeing performance, driving feel and safety are equally important. The proposed approach introduces an additional feedback loop that tracks the yaw rate with a positive feedback gain $R_2 = \left| {K_2 } \right|$:

$$ M_{z,FB} = \left( {\left| {K_1 } \right| + \left| {K_2 } \right|} \right)\left( {\omega_{z,ref} - \omega_z } \right) - \frac{{\left| {K_1 } \right|}}{v_x }\left( {a_{y,ref} - a_y } \right) $$

(9)

This additional term serves two purposes: firstly, to introduce feedback robustness by coping with modeling uncertainties, delays, and noise in the plant, and secondly, as a means of calibration tradeoff. Indeed, by appropriately scheduling of the feedback controllers, more emphasis can be placed on yaw rate tracking during steady-state conditions, while ensuring a more damped response during highly dynamic maneuvers.

While the first term of Eq. 9 represents a conventional yaw rate tracking loop, it is the inclusion of this additional term that distinguishes this proposed approach as novel in the literature. For the sake of illustration, proportional feedback gains and the linear tire formulation are adopted here to derive the proof of asymptotic stability. However, through extension, it is possible to prove stability for a non-linear model and more elaborate control algorithms. By linearizing and combining Eqs. 1–3, an extended state-space system can be derived, whereby the steering effect is considered a disturbance:

$$ \left[ {\begin{array}{*{20}c} {\dot{\omega }_z } \\ {\dot{\beta }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\omega_z } \\ {\beta } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {b_1 } \\ 0 \\ \end{array} } \right]M_z $$

(10)

By deriving the sideslip dynamics with respect to time, the state-space system can be extended to include the sideslip rate as a state variable:

$$ \left[ {\begin{array}{*{20}c} {\dot{\omega }_z } \\ {\dot{\beta }} \\ {\ddot{\beta }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & 0 \\ {a_{21} } & {a_{22} } & 0 \\ {a_{21} a_{11} } & {a_{21} a_{12} } & {a_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\omega_z } \\ \beta \\ \end{array} } \\ {\dot{\beta }} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {b_1 } \\ 0 \\ {a_{21} {b_1} } \\ \end{array} } \right]M_z $$

(11)

The stability condition for the state feedback gains can be derived using the Routh-Hurwitz criterion applied to the feedback matrix described by Eq. 12, obtained by substituting the control law $M_z = \left| {K_2 } \right|\left( {\omega_{z,ref} - \omega_z } \right) - \left| {K_1 } \right|\left( {\dot{\beta }_{ref} - \dot{\beta }} \right)$ into Eq. 11.

$$ A = \left[ {\begin{array}{*{20}c} {a_{11} - \left| {K_2 } \right|b_1 } & a_{12} & {\left| {K_1 } \right|b_1 } \\ {a_{21} } & {a_{22} } & 0 \\ {a_{21} a_{11} - \left| {K_2 } \right|a_{21} {b_1} }& a_{21} a_{12} & {a_{22}+ \left| {K_1 } \right|a_{21} {b_1} } \\ \end{array} } \right] $$

(12)

2.4 Tires Saturation

Although tires saturation has already been addressed in the feedforward modeling, the feedback term also influences the total yaw torque required. While it is relatively straightforward to develop a model-based saturation approach for the first component, deriving one solely for the feedback loop is not. In the interest of calibration efficiency and robustness, the total yaw torque is saturated with a calibratable function of tire slip ratios, steering wheel angle, and combined lateral acceleration vector.

3 Results and Conclusions

The proposed scheme has been very convincingly tested in numerous DiM and Dil-HiL sessions. Due to the limited space available, the results provided here will not focus on the comparison between the proposed and conventional approaches in terms of objective manuevres that can only offer a snapshot of the vehicle envelope. Likewise, elaborating on the brand’s DNA and its reflection on objective metrics mapped against our experts’ subjective attributes targets would not be feasible in this limited space. Instead the results presented (Fig. 2) focus on the benefits of the active vehicle on outright track performance, an attribute consiously balanced against intuitive dynamc responses, and benchmarked against the same vehicle optimally setup for track usage. Given the prerequisites of such a setup, it is evident that the benefits in dynamics and laptime (–3.1s) is the outcome of improved vehicle control and higher driver confidence.

The reference venue for the results presented is the Handling Track of the Nardo Technical Centre [2], a fast and highly technical track with a diverse combination of very high and low curvature corners as well as challenging elevation variance. The log extracts highlighting the delta in dynamics responses and driver effort through two particularly demanding track segments, whereby the dynamic composure achieved as well as the reference tracking ability of the proposed control scheme are evident.

Fig. 2.

NTC handling lap comparison, passive vs active; normalized states and control inputs.