Keywords

1 Introduction

Electric vehicles are increasingly recognized as a viable alternative to traditional internal combustion engine vehicles due to their perceived sustainability [1]. The use of electric motors in these vehicles facilitates the implementation of multiple-motor powertrain layouts [2], enhancing vehicle dynamics through improved control capabilities. Notably, the adoption of In-Wheel Motors (IWMs) is widespread [3], as they allow independent control of driving torque at each wheel, enabling the implementation of Torque Vectoring Control (TVC). Indeed, Torque Vectoring (TV) controls vehicle lateral dynamics by generating a yaw moment through differential longitudinal forces on the wheels of the same axle.

Over the years, several TV controllers have been proposed [4, 5], with their performance typically evaluated using objective metrics that measure how closely they track reference quantities for lateral dynamics, such as yaw rate and sideslip angle. Recently, there has been a growing trend to incorporate driver preferences into TV control logics, allowing the selection of driving modes [6, 7] to slightly modify the behavior of the controlled vehicle. This innovation tailors the vehicle response to driver desires, potentially increasing vehicle effectiveness and acceptability through appropriate feedback to the driver. Indeed, drivers’ subjective evaluations of vehicle performance still are a fundamental process for car manufacturers [8]. However, there is a trend toward correlating subjective evaluations with objective indexes to reduce reliance on human testing in the future. For instance, ride comfort ratings are typically correlated with acceleration measurements at the seat level [9, 10]. On the other hand, vehicle handling ratings are typically correlated with system deadbands, delays, and gains in the vehicle's response to driver inputs [11,12,13].

This paper presents an in-depth analysis of the subjective and objective assessment of various torque vectoring control strategies through Driver-in-the-Loop simulations. The objective analysis provides a quantitative framework for evaluating the effectiveness of different torque vectoring control strategies. Conversely, the subjective assessments encompass factors such as perceived vehicle responsiveness, handling confidence, and overall driving satisfaction, offering crucial insights into user experience. Additionally, the study investigates the relationship between objective measurements and subjective evaluations, aiming to establish a correlation between quantitative metrics and human perception. This integrated approach enhances the understanding of torque vectoring control strategies’ effectiveness and their real-world impact on driver satisfaction.

2 Torque Vectoring Controllers Design

Three alternative torque vectoring (TV) control strategies have been developed to enhance the cornering performance of the DinamiΣ PRC DP14 Formula SAE competition vehicle of Politecnico di Milano, shown in Fig. 1. These control strategies differ in their architectures, utilizing various combinations of feedforward and feedback yaw moment contributions. The objective of the feedback control action, regardless of its specific implementation, is to track the desired yaw rate response of the vehicle, which is defined as a function of vehicle speed and steering wheel angle input.

Fig. 1.
figure 1

DP14 Formula SAE vehicle of Politecnico di Milano.

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Each TV controller features a unique formulation for defining the yaw moment applied to the vehicle, as detailed in the following:

  • PI Torque Vectoring Controller: The PI torque vectoring controller consists in a pure feedback control aiming at tracking the desired yaw rate through the minimization of the yaw rate error (\(\varepsilon_{\dot{\psi }}\))

    $$ \begin{array}{*{20}c} {M_{z,PI} = k_p \left( {v_x } \right) \cdot \varepsilon_{\dot{\psi }} + k_i \left( {v_x } \right) \cdot \int {\varepsilon_{\dot{\psi }} dt} } \\ \end{array} $$
    (1)

  • PI + DD Torque Vectoring Controller: The PI + DD toque vectoring controller consists of the PI feedback control action described above with the addition of a DD feedforward control action function of the input steering wheel angle (\(\delta\)) and steering wheel angle rate (\(\dot{\delta }\)) commanded by the driver

    $$ \begin{array}{*{20}c} {M_{z,PI + DD} = M_{z,PI} + M_{z,DD} = M_{z,PI} + k_\delta \left( {v_x } \right) \cdot \delta + c_\delta \left( {v_x } \right) \cdot \dot{\delta }} \\ \end{array} $$
    (2)

  • DD + ESC Torque Vectoring Controller: The DD + ESC torque vectoring controller supplements the DD feedforward control action with an ESC-like discontinuous feedback contribution applying a discontinuous yaw moment contingent upon whether the absolute value of the yaw rate error falls below or exceeds a specified threshold (\(\varepsilon_{\dot{\psi },TH}\)).

    $$ \begin{array}{*{20}c} {M_{z,DD + ESC} = M_{z,DD} + M_{z,ESC} = M_{z,DD} + \left\{ {\begin{array}{*{20}c} 0 & {IF{ }\left| {\varepsilon_{\dot{\psi }} } \right| < \varepsilon_{\dot{\psi },TH} } \\ {k_p \left( {v_x } \right) \cdot \varepsilon_{\dot{\psi }} } & {IF{ }\left| {\varepsilon_{\dot{\psi }} } \right| \ge \varepsilon_{\dot{\psi },TH} } \\ \end{array} } \right.} \\ \end{array} $$
    (3)

    For an effective and reliable performance of the proposed torque vectoring controllers, all their gains are scheduled as function of vehicle longitudinal speed (\(v_x\)).

3 Driver-in-the-Loop Simulator Testing Campaign

The effects of different TV control logics for a Formula SAE vehicle are assessed in this study through the adoption of Driver-In-the-Loop simulations at a static driving simulator. This allows testing in a safe environment while also ensuring a proper repeatability of the boundary conditions among the performed tests. Therefore, the resulting subjective evaluations from drivers can be considered as feedback about the torque vectoring control logic. Indeed, the 14 DOFs vehicle model implemented in VI-CarRealTime software has not been modified throughout the testing campaign.

The static driving simulator experimental campaign has been conducted by involving six drivers possessing significant familiarity with the FSAE competition vehicle both on track and in its development utilizing the same static driving simulator employed in this study. A tailored test track has been designed for the present study, where a double lane change maneuver is proposed. To ensure comprehensive evaluation, drivers were tasked with completing a minimum of 10 laps of the test track with each TV controller, rating vehicle response on a scale of 1 to 10, focusing on four key aspects being control, stability, easiness and repeatability.

4 Results

The performance of the proposed torque vectoring controllers is evaluated at first on a subjective basis through the ratings gathered from drivers. Then, an objective evaluation is performed to understand the effectiveness of each TV control logic in improving vehicle lateral dynamics. In the end, a correlation between subjective ratings and objective indexes is performed to understand which are the most important objective quantities that drive the human perception of vehicle lateral dynamics control.

4.1 Subjective Evaluation

The subjective evaluation of the proposed torque vectoring controllers relies on drivers’ evaluations, which are reported in Fig. 2 for the double lane change maneuver. In there, the median rating is highlighted with a red bar, with the blue box that indicates the 25th and 75th percentiles of the ratings distribution. On average, the PI + DD TVC results the best control logic on a subjective basis for all the inspected rating categories, while the PI TVC results the control logic with the highest spread of ratings. This means that the addition of the feedforward component to the yaw moment generation is positively perceived by the driver. This could be due to a prompter vehicle response to the driver inputs, which also results in a more consistent rating in easiness and repeatability categories having a more direct vehicle behavior as function of the steering input. The DD + ESC TVC generally results the worst performing control logic, probably due to its discontinuous control action which is not positively perceived by humans.

Fig. 2.
figure 2

Subjective evaluation of the proposed TV controllers about the vehicle cornering response during the double lane change maneuver.

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4.2 Objective Evaluation

In this study, the objective analysis of vehicle handling performance is conducted using specifically defined Key Performance Indicators (KPIs) for transient maneuvers. To ensure consistency in the objective KPIs across different drivers, a normalization method has been implemented to minimize the influence of driving style on the evaluation process. The normalization is expressed as follows.

$$ KPI_{j,norm} = \frac{{KPI_j - KPI_{PI} }}{{\left| {KPI_{PI} } \right|}} $$
(4)

With this approach, the \(KPI_{j,norm}\) values represent the relative variation of \(KPI_j\) obtained using the \(j^{th}\) TV control logic, compared to \(KPI_{PI}\) obtained with the PI TV control logic, which serves as the reference.

The evaluation of the relative objective performance of the proposed TV controllers during a double lane change maneuver is presented in Fig. 3. The results indicate that both the PI + DD and DD + ESC TVCs achieve lower section times compared to the PI TVC, with enhanced cornering performance attributed to an increased yaw rate gain in response to steering wheel angle inputs. Additionally, the PI + DD TVC demonstrates a significantly reduced hysteresis cycle area for the yaw rate response as a function of the steering wheel angle input, indicating improved repeatability of cornering response. However, this improvement is accompanied by several instances where the steering wheel torque opposes the steering wheel angle input, making vehicle control more challenging for the driver.

Fig. 3.
figure 3

Effects of the proposed TV controllers on the double lane change vehicle cornering response.

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4.3 Subjective-Objective Evaluation Correlation

The objective evaluation of vehicle cornering performance is conducted in this study on a relative basis, using the PI TV control logic as a reference, thus also the subjective assessment is reverted to a relative basis for correlation purposes. This method eliminates driver bias in the evaluation of the TV control logics by considering only the variation in subjective evaluations (\(Rating_i\)) of the \(i^{th}\) driver relative to those obtained from the same driver using the reference TV control logic (\(Rating_{PI}\)).

$$ Rating_{i,rel} = Rating_i - Rating_{PI} $$
(5)

The primary tool used for analyzing the level of correlation between subjective ratings and objective indexes is the Spearman rank correlation coefficient. This statistical measure evaluates the strength and direction of a monotonic relationship between two variables. The results regarding the correlation of subjective ratings with various objective KPIs is reported in Fig. 4 for the double lane change maneuver.

Fig. 4.
figure 4

Spearman correlation between subjective ratings and objective KPIs during double lane change maneuvers.

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The analysis of control ratings reveals an inverse correlation with the steering wheel angle rate and sideslip angle rate KPIs. This is logical, as lower values for these rates likely correspond to fewer driver corrections needed to complete the maneuver. Additionally, control ratings are strongly correlated with changes in the yaw rate/steering hysteresis loop area. Stability ratings exhibit a strong correlation with section time and a strong inverse correlation with the yaw rate gain relative to steering input. This indicates that higher stability is perceived by the driver when the double lane change maneuver is completed at a lower average speed and the vehicle is slightly less responsive to steering input. Easiness ratings show a significant inverse relationship with the RMS value of the steering wheel angle rate and the standard deviation of the tires’ utilization factor. This suggests that the vehicle is perceived as easier to drive when it requires fewer corrections. Repeatability ratings exhibit a very strong correlation with the hysteresis loop area of yaw rate relative to steering wheel angle input, indicating that drivers perceive increased repeatability when the vehicle's yaw rate response is more predictable based on the steering wheel input.

5 Conclusions

The impact of three different torque vectoring control strategies on vehicle dynamics has been evaluated in this study. The standard approach, which relies on objective KPIs, has been enhanced by incorporating subjective evaluations from six drivers through a testing campaign on a static driving simulator. The subjective evaluations identify the PI + DD controller as the best perceived by humans. Additionally, the objective evaluation of torque vectoring controllers indicates that the PI + DD controller is the one most significantly enhancing vehicle cornering response, though it increases driver workload. The correlation between subjective and objective evaluations then is used to explain the reasons behind the improved or worsened ratings. Specifically, the PI + DD torque vectoring controller demonstrates the smallest hysteresis loop area between vehicle yaw rate response and steering wheel angle input, which correlates with higher ratings of controller capabilities. Additionally, the PI + DD controller also causes lower sideslip angle speeds, making the vehicle less challenging to drive and resulting in higher subjective ratings for vehicle easiness. This research not only contributes to optimizing torque vectoring control strategies but also provides a methodology for bridging the gap between objective performance metrics and subjective driver experience. These findings can guide the development of future torque vectoring systems, leading to vehicles that excel in objective performance measures while aligning with drivers’ subjective preferences.