# On Analytical Modeling for Fast Multi-Objective
Torque Allocation in Over-Actuated IWM Vehicles

Fadel Tarhini, Reine Talj, and Moustapha Doumiati

**Abstract**—Efficient torque allocation in over-actuated vehicles poses a central challenge in the domain of advanced vehicle control. These vehicles, featuring redundant actuators, provide an exceptional avenue for enhancing performance, stability, and efficiency. This paper presents a pioneering tendency for torque allocation in the context of over-actuated vehicles, particularly in-wheel motor (IWM) driven electric vehicles. We introduce a systematic methodology grounded in analytical modeling, allowing for the efficient reconciliation of multiple, often conflicting objectives. The explicit functions are analytically modeled to enhance stability and energy economy. Additionally, a fuzzy logic-based torque allocation strategy is developed and compared, along with other literature methods, with the analytical models. Simulations are conducted in a joint simulation between Simulink/Matlab and SCANeR Studio vehicle dynamics simulator, followed by validation on a real-world dataset. Our findings elucidate the proficiency of the analytical models on vehicle performance, stability, computational efficiency, and energy consumption.

**Index Terms**—Torque Allocation, Over-actuated Vehicles, Analytical Modeling, Fuzzy Logic, Energy Economy, Stability

## NOMENCLATURE

<table><tr><td>p, p<sub>1</sub>, k, k<sub>1</sub>, k<sub>2</sub>, q, n</td><td>Torque allocation parameters [-]</td></tr><tr><td>g<sub>1</sub>, g<sub>2</sub>, g<sub>3</sub>, g<sub>4</sub></td><td>Sub-functions of the parameter q [-]</td></tr><tr><td>h<sub>1</sub>, h<sub>2</sub></td><td>Sub-functions of the parameter p<sub>1</sub> [-]</td></tr><tr><td>f<sub>1</sub>, f<sub>2</sub></td><td>Sub-functions of the parameter k<sub>1</sub> [-]</td></tr><tr><td>T<sub>m</sub></td><td>Total motor torque [N.m]</td></tr><tr><td>M<sub>z</sub></td><td>Corrective yaw moment [N.m]</td></tr><tr><td>α<sub>1</sub>, α<sub>2</sub>, α<sub>3</sub>, α<sub>4</sub>, α<sub>5</sub></td><td>Energy allocation parameters</td></tr><tr><td>ν<sub>1</sub>, ν<sub>2</sub>, ν<sub>3</sub>, ν<sub>4</sub></td><td>Reflection of M<sub>z</sub> in specific ranges</td></tr><tr><td>τ<sub>1</sub>, τ<sub>2</sub>, τ<sub>3</sub>, τ<sub>4</sub></td><td>Reflection of T<sub>m</sub> in specific ranges</td></tr><tr><td>ν<sub>i</sub>, ν̄<sub>i</sub></td><td>Lower and upper bounds of ν<sub>i</sub> [N.m]</td></tr><tr><td>τ̄<sub>i</sub>, τ̄<sub>i</sub></td><td>Lower and upper bounds of τ<sub>i</sub> [N.m]</td></tr><tr><td>ξ<sub>1</sub>, ξ<sub>2</sub>, λ<sub>1</sub>, λ<sub>2</sub></td><td>Shaping parameters for p<sub>1</sub> [-]</td></tr><tr><td>ζ<sub>1</sub>, ζ<sub>2</sub>, σ<sub>1</sub>, σ<sub>2</sub></td><td>Shaping parameters for k<sub>1</sub> [-]</td></tr><tr><td>γ<sub>1</sub>, γ<sub>2</sub>, γ<sub>3</sub>, γ<sub>4</sub>, γ<sub>5</sub></td><td>Transients-state intensity of q [-]</td></tr><tr><td>χ<sub>1</sub>, χ<sub>2</sub>, χ<sub>3</sub>, χ<sub>4</sub></td><td>Transients-state intensity of p<sub>1</sub>, k<sub>1</sub> [-]</td></tr><tr><td>u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub></td><td>Desired values at the extremities of p<sub>1</sub></td></tr><tr><td>v<sub>1</sub>, v<sub>2</sub>, v<sub>3</sub></td><td>Desired values at the extremities of k<sub>1</sub></td></tr><tr><td>ij</td><td>[rl, rr, fr, fl]</td></tr><tr><td>T<sub>b,ij</sub></td><td>Motor ij braking torques [N.m]</td></tr><tr><td>T<sub>d,ij</sub></td><td>Motor ij driving torques [N.m]</td></tr></table>

This work is carried out within the framework of the V3EA project "Electric, Energy Efficient, and Autonomous Vehicle" (2021-2025), funded by the Research National Agency (ANR) of the French government.

Fadel Tarhini and Reine Talj are with Université de Technologie de Compiègne, CNRS, Heudiasyc UMR 7253, Compiègne, France (email: fadel.tarhini@hds.utc.fr, reine.talj@hds.utc.fr).

Moustapha Doumiati is with École Supérieure d'Électronique de l'Ouest (ESEO)-IREENA UR 4642, Angers, France (email: moustapha.doumiati@eseo.fr).

<table><tr><td>l<sub>r</sub>, l<sub>f</sub></td><td>Wheelbase to the rear and front [m]</td></tr><tr><td>t<sub>r</sub>, t<sub>f</sub></td><td>Rear and front tracks [m]</td></tr><tr><td>F<sub>z</sub></td><td>Load vertical forces [N]</td></tr><tr><td>F<sub>x</sub></td><td>Tire longitudinal forces [N]</td></tr><tr><td>F<sub>y</sub></td><td>Tire lateral forces [N]</td></tr><tr><td>a<sub>x</sub></td><td>Longitudinal acceleration [m/s<sup>2</sup>]</td></tr><tr><td>a<sub>y</sub></td><td>Lateral acceleration [m/s<sup>2</sup>]</td></tr><tr><td>μ</td><td>Road adherence coefficient [-]</td></tr><tr><td>ω<sub>ij</sub></td><td>Motor ij rotational velocity [rad/s]</td></tr><tr><td>η<sub>ij</sub></td><td>Motor ij efficiency [-]</td></tr><tr><td>κ<sub>1</sub></td><td>Front/rear load distribution ratio [-]</td></tr><tr><td>κ<sub>2</sub></td><td>Right/left load distribution ratio [-]</td></tr><tr><td>κ</td><td>Function of κ<sub>2</sub> in terms of M<sub>z</sub> [-]</td></tr><tr><td>δ</td><td>Vehicle steering angle [rad]</td></tr><tr><td>ψ</td><td>Vehicle yaw angle [rad]</td></tr><tr><td>β</td><td>Vehicle side-slip angle [rad]</td></tr><tr><td>φ<sub>r</sub></td><td>Road inclination angle [rad]</td></tr><tr><td>m<sub>e</sub>, Υ</td><td>Mean and standard deviation [-]</td></tr><tr><td>P<sub>bo</sub></td><td>Battery output power [W]</td></tr></table>

## I. INTRODUCTION

In the contemporary landscape of transportation, Electric Vehicles (EVs) have emerged as a pivotal solution to mitigate environmental concerns and contribute to a sustainable future. These vehicles rely on intricate vehicle control systems, which serve as the linchpin for coordinating vehicle dynamics control and energy management. Advanced Driver Assistance Systems (ADAS) [1], driven by sophisticated control mechanisms including Active Front Steering (AFS) and Direct Yaw Control (DYC), represent a fundamental development in the realm of vehicle control. These mechanisms can be coordinated within a control architecture to achieve various control objectives. The objectives can solicit the overall motion of the vehicle, as in path-tracking [2], or focus on targeting chassis control, as in enhancing stability and maneuverability [3], [4]. For instance, a non-linear integrated control strategy based on Model Predictive Control (MPC) is proposed in [5] to maintain vehicle stability using AFS and differential braking. [6] focused on the finite frequency fuzzy H∞ control problem for uncertain active vehicle suspension systems to enhance driving comfort. Meanwhile, in the context of autonomous vehicles, [7] proposed a path-tracking control approach based on an explicit model formulation for the look-ahead distance, while [8] presented a safety and comfort-guaranteed path-tracking under several road geometry constraints.

The integration of in-wheel motors has emerged as a revolutionary force, redefining the traditional paradigms of vehicle
design and operation, and heralding a new era in the evolution
of electric mobility. The resulting over-actuated system from
the redundant actuators offers a multitude of advantages,
including precise and fast torque control, enhanced efficiency,
simplified vehicle design, and optimized regenerative braking
[9], [10], [11]. However, incorporating in-wheel motors results
in an increase in the unsprung mass which in turn aggravates
the road holding and ride comfort performance. To solve this
issue, [12] presented a decentralized dynamic event-triggered
communication for active suspension control of the in-wheel
motor. Prior studies have explored diverse objectives when
dealing with the control of over-actuated vehicles. For in-
stance, [13] presented a holistic adaptive multiple-model MPC
approach with the aim of path-following. [14] considered path
preview to develop an efficient MPC for high-speed lateral
motion control, while [15] employed MPC to coordinate path-
tracking and yaw moment control. It is worth noting that, in
the context of lateral dynamics control, the primary sources
of uncertainty stem from the variations in tire cornering
stiffness which is affected by road adherence and tire side-
slip angle [16]. These uncertainties can deteriorate the control
performance and lead to critical safety incidents. Hence, this
necessitates the adoption of more resilient and robust control
methodologies for the vehicle [17], [18], [19] such as Sliding
Mode Control (SMC), which is adopted in this work.

The adoption of hierarchical control architectures has
proven instrumental and highly pragmatic in the context of
autonomous over-actuated vehicles [20], [21]. The hierarchi-
cal strategy involves organizing control systems into distinct
levels, typically an upper level responsible for generating
control inputs aligned with several objectives, and a lower
level tasked with executing the control commands through
the designated actuators. This approach facilitates modular
development and scalability, as it permits independent design,
optimization, and modification of each control level. In partic-
ular, [22] presented a hierarchical control scheme designed to
keep the vehicle handling and improve yaw stability control
using the coordination of AFS and DYC. Similar common
works suggested the implementation of an upper-level con-
troller responsible for generating a corrective yaw moment
to ensure vehicle stability [23], [24], [25]. However, under
extreme conditions, the realization of the yaw moment control
command may encounter impediments, particularly when tire
slippage remains unaddressed. Therefore, it is of paramount
importance to integrate tire slippage considerations into the
torque allocation strategy at the lower-level control.

A. Related Works

Torque allocation entails the strategic distribution of torque
among a surplus of actuators beyond the fundamental require-
ments for basic vehicle control. Frequently, these problems
are approached by employing mathematical optimization tech-
niques [26], to solve constrained control allocation problems
that minimize various cost functions [27]. The targeted cost
functions might be designed to solely minimize energy con-
sumption [28], [29] or enhance vehicle stability [30], [31].
Nevertheless, vehicle safety stands as an indispensable objec-
tive within the global control architecture. Simultaneously, the

utilization of in-wheel motors presents a distinct advantage in
pursuit of energy economy. Effectively harmonizing these two
pivotal objectives, safety and energy economy, represents a
central challenge, as they frequently conflict with each other.
Hence, certain studies tended towards the pursuit of multi-
objective optimization. In particular, MPC-based torque allo-
cation is widely employed for multi-objective torque allocation
[32], [33]. MPC approaches are valuable for their predictive
capabilities and the flexibility they offer in handling con-
straints. However, in addition to high sensitivity to modeling
errors and poor adaptation to varying road conditions, this
strategy faces high computational complexity making its real-
time execution very challenging, especially when considering
multi-objectives. Other works considered online optimization
algorithms for multi-objective optimization. For instance, [34]
presented a two-step solution using the second-generation
nondominated sorting genetic algorithm and the hybrid genetic
tabu search algorithm. These methods are particularly useful
for finding optimal solutions across conflicting objectives,
but they still entail significant computational complexity and
are susceptible to premature convergence. However, these
algorithms entailed significant computational complexity and
exhibited susceptibility to premature convergence, as well
as challenges in reaching a global Pareto front during opti-
mization. More recently, deep reinforcement learning-based
techniques have emerged as promising alternatives for torque
allocation [35], [36]. These model-free approaches excel at
managing complex decision-making tasks and offer the poten-
tial for adaptive learning in dynamic environments. However,
their success heavily depends on the availability of com-
prehensive and representative datasets. Moreover, they face
limitations in terms of interpretability, safety, and robustness,
which are critical for practical applications.

Other recent studies have addressed the computational bur-
den resulting from optimization algorithms in torque alloca-
tion. For instance, [37] proposed an offline energy-efficient
torque allocation method based on an algorithm and compared
it with an online optimization method. While the results
showed significant energy savings, the allocated torques ex-
hibited oscillations, and vehicle stability was not an objective.
The optimization problem in [38] is divided into two sub-
problems to simplify the solution through the modification
and discretization of the feasible torque space. This method
enhanced computational efficiency but focused solely on en-
ergy economy, disregarding the stability objective, and focused
mainly on longitudinal dynamics for torque allocation. In
[39], the approach anticipates vehicle instability and adjusts
motor torques preemptively, aiming to maintain stability before
traditional stability control measures are activated. Although
this method enhanced stability, it did not consider energy
economy. [40] proposed a real-time heuristic approach consid-
ering solely energy management. Other works have proposed
torque allocation strategies that consider energy economy and
stability while aiming for real-time feasibility. For instance,
[41] introduced an optimal torque vectoring control strategy
based on a two-level distribution formula to reduce energy
consumption while ensuring handling stability. Although ef-
fective, the manual setting of the optimization algorithm's
maximum iterations to achieve real-time performance results in suboptimal solutions. Similarly, [42] presented a hierarchi-cal control scheme that balances handling stability and energy conservation, but the use of MPC in the lower layer incurs significant computational effort, which poses challenges for real-time implementation, especially in dynamic environments. [43] employed nonlinear MPC to address vehicle stability and energy conservation under extreme conditions, but this approach also faces inherent computational challenges. Au-thors of [44] presented an MPC strategy for energy-efficient yaw-stabilizing torque vectoring in four in-wheel motor elec-tric vehicles. However, the approach is over-dependent on weight tuning in different situations and is computationally heavy. There remains a notable gap in the literature for multi-objective torque allocation strategies that simultaneously address stability and energy economy while remaining compu-tationally efficient. Most existing methods either compromise one objective for another or face computational challenges, highlighting the need for a comprehensive approach that bal-ances these objectives effectively and efficiently. Furthermore, for autonomous vehicles operating at moderate to high speeds, merely improving real-time applicability for torque allocation may not suffice. There is a crucial need for more rapid approaches that can meet the system demands effectively.

## B. Motivations and Contributions

In the realm of torque allocation for over-actuated vehicles, literature shows that prevailing approaches predominantly re-volve around computationally demanding approaches without adequately considering the trade-off between optimality and computational efficiency. Research studies frequently overlook the comprehensive examination of the global system architec-ture and the inherent delays among its constituent elements, including sensor acquisition, control, and actuators, in addition to perception, decision-making, and planning in the case of autonomous vehicles. Existing torque allocation methods, while valuable in certain contexts, exhibit notable limitations that necessitate the exploration of alternative methodologies. Especially for multi-objective optimization in torque alloca-tion, focusing on both energy economy and stability, existing methods often face significant challenges, including compu-tational complexity, managing trade-offs between conflicting objectives, ensuring real-time operation, and achieving scalab-ility across different vehicle types and driving conditions.

In the context of autonomous vehicles, optimization tech-niques are frequently required at other levels especially at the motion planning level. Learning-based methods are re-quired at the decision-making level and MPC is widely used at the control level. Introducing another optimization-based method at the actuator level could significantly increase the computational complexity of the entire system, potentially rendering it costly and its real-time operation infeasible. Therefore, it demonstrates the necessity for a rapid, robust, and multi-objective-oriented torque allocation methodology. In this paper, we introduce a novel methodology for multi-objective torque allocation founded on analytical modeling. Unlike existing torque allocation methodologies which of-ten rely on optimization-based or learning-based approaches,

our proposed methodology represents a fundamental shift by developing analytical models. This innovative approach opens new avenues for developing torque allocation strategies grounded in analytical modeling. The multi-objectives include energy economy, driving comfort, and stability. Our proposed methodology carries the attributes of scalability, expeditiousness, efficiency, and robustness while maintaining accessibility, ease of implementation, and interpretability: 1) The use of explicit mathematical functions ensures rapid computation, making the method highly efficient for real-time applications; 2) The analytical models are generalized for ground vehicles and their explicit and interpretable nature minimizes the need for extensive calibration; 3) The approach provides a clear, analytical framework that is inherently interpretable and scalable, easily implemented on various ground vehicles with minimal adjustment; 4) Unlike other methods that require comprehensive datasets or sophisticated optimization algorithms, our model's explicit functions are accessible and easily replicable; 5) The methodology is designed to be robust against variations in vehicle dynamics and driving conditions, enhancing its practical utility across vehicle models and operational scenarios.

Particularly, it demonstrates superior performance compared to the four strategies introduced in [37], including online and offline optimization techniques. In comparison to these methods, it exhibits a significant improvement in handling instantaneous allocation by mitigating induced oscillations through rapid allocation, all while attaining higher energy efficiency and improved stability. The analytical models are validated in a joint simulation between Simulink/MatLab and SCANeR™ Studio vehicle dynamics simulator, where the simulation findings demonstrate the effectiveness of the approach on expeditiousness, vehicle stability and comfort, and energy economy.

The main contributions of this paper can be elaborated as

* Inferring a set of rules for torque allocation based on multiple objectives including stability, comfort, and energy economy. These rules are designed to coordinate the actuators effectively, aiming to minimize tire slippage and enhance energy efficiency, while maintaining the control architecture objectives.

* Development of an original methodology based on analytical modeling for multi-objective torque allocation in over-actuated vehicles. The innovative methodology represents a fundamental shift from literature approaches by leveraging analytical models. The inherent characteristics of our proposed methodology include scalability, expeditiousness, efficiency, robustness, and interpretability.

* Establishment of a novel fuzzy logic-based torque allocation and comparing it with the analytical models to assess relative performance.

* Validation of our proposed methodology on the SCANeR Studio vehicle dynamics simulator and on a real-world dataset to demonstrate its effectiveness on energy economy, stability, comfort, and computational efficiency.

The rest of the paper is structured as follows: Section II introduces the torque allocation configuration and establishes the rules for coordinating the multiple objectives. Section
<figure><img src="image_1.png" /><figcaption>Fig. 1: Schematic diagram of the global control architecture</figcaption></figure>

III thoroughly presents the analytical modeling, and Section IV presents the fuzzy logic-based torque allocation strategy. The validation of the approaches is presented in Section V, followed by a general conclusion in Section VI.

## II. MULTI-OBJECTIVE PROBLEM FORMULATION

### A. Global Control Architecture

The global control architecture is depicted in Fig. 1. The architecture is developed in a cascaded structure, encompassing upper and lower levels of control. The objectives pursued within this architecture comprise lateral control (path-tracking), longitudinal velocity control, as well as stability and maneuverability control. In the upper level, control inputs are generated based on the Super-Twisting Sliding Mode control. Stability and maneuverability are adaptively switched based on the prevailing driving situation. In cases where a risk of instability is detected by a decision-layered stability monitor, the system directly prioritizes the stability objective while temporarily relaxing maneuverability, and vice versa. Stability is reinstated through the side-slip angle $\beta$ control, while maneuverability is achieved by regulating the vehicle's yaw rate $\dot{\psi}$. $\dot{\psi}$ and $\beta$ are controlled to track the yaw rate and the side-slip angle of the reference linear bicycle model [45]. The aforementioned objectives are realized by generating a corrective yaw moment, $M_z$, utilizing Direct Yaw Control (DYC). Longitudinal control is realized by controlling the vehicle's longitudinal velocity $V_x$ to track a generated reference speed profile $V_{x,des}$. This profile is constructed by accounting for road curvature while strictly adhering to the constraint of maintaining lateral acceleration below a threshold of $4 m/s^2$. This process is effectively achieved through the realization of the generated total torque $T_m$. Lastly, lateral control is attained by minimizing the lateral error between the vehicle and the reference path, specifically at a look-ahead distance situated ahead of the vehicle (see [7]). Hence, path-tracking is achieved by regulating the steering angle $\delta_c$ using Active Front Steering (AFS). The decentralized controllers for the four objectives are adopted from [46], applying second order robust Sliding Mode control.

At the lower level, the total torque $T_m$ and the yaw moment $M_z$ are received by a torque allocation unit to generate four traction and four braking torques among the motors. Torques are distributed based on a torque allocation configuration

posterior to the computation of the four parameters $p, k, q, n$ defined later. Two distinct strategies have been devised for parameter calculation: the first relies on analytical modeling, while the second employs fuzzy logic control. Ultimately, the controlled steering and the generated torques are fed into the fully dynamic model “Callas” of the vehicle within the professional SCANeR Studio simulator [47]. The SCANeR Studio simulator is widely recognized and utilized in the automotive industry, ensuring high fidelity and reliability in our validation process [47].

### B. Torque Allocation Configuration

The generated control inputs $T_m$ and $M_z$ are realized by the in-wheel motors actuators using a torque allocation strategy. The configuration for the torque distribution is adopted from [37] and shown in Fig. 2. The constraints for torque allocation are imposed into four parameters $p, k, q, n$ and modulated as linear constraints $p, k, q, n \in [0, 1]$ (see [37]). $T_m$ is weighted by $p$ on the front axle and $(1-p)$ on the rear one. $M_z$ is generated on the two axes of rotation on the rear and front axles. Hence, it is weighted by $k$ on the rear axle and $(1-k)$ on the front one. Finally, the distribution of the yaw moments weighted on the rear and front axles, $kM_z$ and $(1-k)M_z$, involves a combination of braking and traction torques. Hence, braking is weighted by $q$ while traction by $(1-q)$ for the rear axle; similarly by $n$ for the front axle. Therefore, according to the direction of $M_z$, torques are distributed as in (1) if $M_z \le 0$ (counter-clockwise) and according to (2) otherwise,

$$ 
\left\{ 
\begin{array}{l}
T_{b,rl} = q \frac{-r}{t_r} k M_z \\
T_{b,fl} = n \frac{-r}{t_r} (1-k) M_z \\
T_{b,rr} = T_{b,fr} = 0 \\
T_{d,rl} = (1-p) \frac{T_m}{2} \\
T_{d,rr} = (1-p) \frac{T_m}{2} + (1-q) \frac{-r}{t_r} k M_z \\
T_{d,fl} = p \frac{T_m}{2} \\
T_{d,fr} = p \frac{T_m}{2} + (1-n) \frac{-r}{t_r} (1-k) M_z
\end{array}
\right. 
; \text{ for } M_z \le 0 \qquad (1) $$

where $p, k, q, n \in [0, 1]$, $T_{b,ij}$ and $T_{d,ij}$ are the braking and traction torques of the motor $ij$ respectively, $r$ is the wheel radius, $t_r$ is the half rear track, and $ij = [rl, rr, fr, fl] = [\text{rear left (1)}, \text{rear right (2)}, \text{front right (3)}, \text{front left (4)}]$. This configuration is pivotal as it guarantees the attainment of both $T_m$ and $M_z$ for any combination of values within the
<figure><img src="image_2.png" /><figcaption>Fig. 2: Torque allocation configuration for a ccw $M_z$. $T_r$ and $T_f$ are intermediate torque variables that weight the total rear and front torques.</figcaption></figure>

range [0, 1] for $p, k, q, n$. Consequently, $p, k, q, n \in [0, 1]$ can be assigned according to several objectives while adhering to the overarching objectives of the architecture.

$$ 
\left\{
\begin{array}{l}
T_{b,rr} = q \frac{r}{t_r} k M_z \\
T_{b,fr} = n \frac{r}{t_r} (1-k) M_z \\
T_{b,rl} = T_{b,fl} = 0 \\
T_{d,rr} = (1-p) \frac{T_m}{2} \\
T_{d,rl} = (1-p) \frac{T_m}{2} + (1-q) \frac{r}{t_r} k M_z \\
T_{d,fr} = p \frac{T_m}{2} \\
T_{d,fl} = p \frac{T_m}{2} + (1-n) \frac{r}{t_r} (1-k) M_z
\end{array}
\right.
\quad \text{; for } M_z > 0 \quad (2)
$$

It is worth noting that it is possible to generate $M_z$ by applying a difference of traction torques only via $q \le 0$ and/or $n \le 0$, or realize $M_z$ using a difference of braking torques only via $q \ge 2$ and/or $n \ge 2$. However, excessive application of traction torques to generate $M_z$ significantly increases the vehicle's energy consumption. Conversely, excessive application of braking torques to generate $M_z$ significantly decreases the velocity of the vehicle. In order to compensate for the velocity loss and to track the desired speed profile, the vehicle must employ substantial traction torques, thus increasing energy consumption. Therefore, as energy economy is a primary objective in this study and to avoid excessive traction/braking, the parameters $q, n$ as well as $p, k$ are limited in [0, 1].

### C. Energy Consumption

The output power consumption of the battery, $P_{bo}$, can be expressed in (3) as the total power consumption of the four in-wheel motors, presuming that the power losses of the battery are negligible [37].

$$ P_{bo} = \sum_{i=1}^{4} P_i = \sum_{i=1}^{4} \frac{T_i \omega_i}{\eta_{s,i}^{\text{sign}(T_i)}} \quad (3) $$

where $T_i$ [$N.m$] and $\omega_i$ [$rad/s$] correspond respectively to the motor $i$ total traction/braking torque and its rotational velocity. The efficiency $\eta_s = \eta_d$ [-] corresponds to the motoring/driving efficiency if $T_i \ge 0$, while it aligns with regenerative braking efficiency ($\eta_s = \eta_b$) otherwise. The efficiency maps for the PD18 DC electric in-wheel motor [48] have been constructed as a function of motor torque and rotational velocity, with dedicated representations for motoring

<figure><img src="image_3.png" /><figcaption>Fig. 3: Motor efficiencies: motoring ($\eta_d$) and braking ($\eta_b$)</figcaption></figure>

(as depicted in Fig. 3 (left)) and regenerative braking (Fig. 3 (right)). Consequently, the accumulated energy consumption at instant $t_i$ is the entire energy consumption from the initial instant $t_0$ until $t_i$.

### D. Multi-Objective Torque Allocation Rules

The main objectives desired to be achieved through torque allocation are stability and energy economy. Indeed, the control input $M_z$ guarantees retrieving the vehicle's stability in the event of a stability loss. However, $M_z$ is generated to address the overall vehicle chassis as a unified system, without specifically accounting for the stability of each tire, including tire slippage concerns. Under some extreme situations where a tire is slipping, the generation of $M_z$ is hindered, regardless of the magnitude of the applied torque. Hence, to evaluate the performance of each wheel, the Tire Working-load Usage (TWU) (4) as defined in [49] is examined.

$$ \text{TWU} = \sum_{i=1}^{4} \text{TWU}_i = \sum_{i=1}^{4} \frac{F_{x,i}^2 + F_{y,i}^2}{\mu F_{z,i}^2} \quad (4) $$

where $F_{x,i}$, $F_{y,i}$, and $F_{z,i}$ are respectively the longitudinal, lateral, and vertical load forces acting on the tire $i$, and $\mu$ represents the adherence coefficient. TWU delineates the periphery of the friction circle. A diminished TWU$_i$, indicative of an expanded friction circle for the wheel $i$, signifies an augmented adhesion to the ground, thereby enabling the wheel to handle higher torque levels. It is important to stress that, while TWU plays a pivotal role in the collective wheel slip of all four tires, the individual tire-specific TWU$_i$ assumes critical significance, as it serves as a factor in maintaining each tire's slip below the critical threshold of 1.

#### 1) Rules to adapt $p$ and $k$

The parameters $p$ and $k$ are determined in a cascaded two-step approach. The parameter $p$, intended for allocating $T_m$, is adapted with the variation of $M_z$ along with $T_m$. Similarly, $k$ corresponding to distributing $M_z$, is adjusted to the changes in $T_m$ along with $M_z$. In the initial step, new parameters $p_1$ and $k_1$ are derived within the interval [0.5, 1] through a multi-objective approach that considers the variation of $T_m$ and $M_z$. Subsequently, in the second stage, $p \in [0, 1]$ and $k \in [0, 1]$ are determined based on $p_1$ and $k_1$ with a dedicated emphasis on energy economy. The parameters $p$ and $k$ are a seamless extension of $p_1$ and $k_1$ across the entire interval of [0, 1] with the values of $p$ and $k$ within [0, 0.5] symmetrically aligned with their counterparts $p_1$ and $k_1$. In other terms, $p$ and $k$ are assigned either as $p_1$ and $k_1$ in the range [0.5, 1] or as $(1-p_1)$
<caption>**Algorithm 1** Total torque ($T_m$) energy economic distribution</caption>

**procedure** GET-P($T_m$, $\omega_i$, $\eta_{s,i}$)
    if $T_m \ge 0$ then
        **if** $\frac{\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} + \frac{\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} \ge \frac{\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} + \frac{\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}}$ **then**
            $p \to 1$ &>& Activate front axle drive
        **else**
            $p \to 0$ &>& Activate rear axle drive
    end if
    **else**
        **if** $\frac{\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} + \frac{\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} \ge \frac{\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} + \frac{\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}}$ **then**
            $p \to 0$ &>& Activate rear axle drive
        **else**
            $p \to 1$ &>& Activate front axle drive
    end if
    **end if**
    **end if**
    **end procedure**

and $(1-k_1)$ in the range $[0, 0.5]$. Hence, if $p_1$ increases towards 1, $T_m$ will be distributed on the front axle (corresponding to $p=1$) or on the rear axle (corresponding to $p=0$). Similarly, if $k_1$ increases towards 1, $M_z$ will be directed toward the rear axle (corresponding to $k=1$) or shifted to the front axle (corresponding to $k=0$). On the contrary, when $p_1$ and $k_1$ decrease to 0.5, $T_m$ (respectively $M_z$) is distributed uniformly on the four wheels. Hence, $p_1$ and $k_1$ are defined to weight the distribution of $T_m$ and $M_z$, respectively, between two-axle-drive and one-axle-drive configurations. Subsequently, $p$ and $k$ are assigned by identifying whether the one-axle-drive is the rear or front axle.

Assuming a counter-clockwise $M_z$ and expanding the motor torques into longitudinal forces, we have

$$F_{x,1} = qk \frac{M_z}{2} \frac{1}{t_r} + \frac{1-p}{2r} T_m \quad (5a)$$

$$F_{x,2} = (1-q)k \frac{M_z}{2} \frac{1}{t_r} + \frac{1-p}{2r} T_m \quad (5b)$$

$$F_{x,3} = (1-q)(1-k) \frac{M_z}{2} \frac{1}{t_r} + \frac{p}{2r} T_m \quad (5c)$$

$$F_{x,4} = q(1-k) \frac{M_z}{2} \frac{1}{t_r} + \frac{p}{2r} T_m \quad (5d)$$

Let $a = \frac{1}{2r}$, $c_1 = q \frac{1}{2t_r}$ and $c_2 = (1-q) \frac{1}{2t_r}$, then

$$F_{x,1} = (1-p)T_m \times a + kM_z \times c_1 \quad (6a)$$

$$F_{x,2} = (1-p)T_m \times a + kM_z \times c_2 \quad (6b)$$

$$F_{x,3} = pT_m \times a + (1-k)M_z \times c_2 \quad (6c)$$

$$F_{x,4} = pT_m \times a + (1-k)M_z \times c_1 \quad (6d)$$

Considering a small change in the range of $\omega$, the in-wheel motor, in its nominal operating range, works in higher efficiency zones as the demanded torque increases (see Fig. 3). Subsequently, as $T_m$ increases, it becomes imperative to raise $p_1$ to ensure that the one-axle driving motors operate with enhanced efficiency. In the same context, when $T_m$ elevates, $k_1$ is increased despite its role in distributing $M_z$. This adjustment ensures that the in-wheel motor operates within higher efficiency zones, regardless of the value assigned

<caption>**Algorithm 2** Yaw moment ($M_z$) energy economic distribution</caption>

**procedure** GET-K($M_z$, $q$, $n$, $\omega_i$, $\eta_{s,i}$)
    if $M_z \le 0$ then
        **if** $\frac{-q\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} \ge \frac{-n\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}} + \frac{(1-n)\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}}$ **then**
            $k \to 1$ &>& Generate $M_z$ on the rear axle
        **else**
            $k \to 0$ &>& Generate $M_z$ on the front axle
    end if
    **else**
        **if** $\frac{-q\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} \ge \frac{-n\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} + \frac{(1-n)\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}}$ **then**
            $k \to 1$ &>& Generate $M_z$ on the rear axle
        **else**
            $k \to 0$ &>& Generate $M_z$ on the front axle
    end if
    **end if**
    **end if**
    **end procedure**

to $q$. Based on (6), for higher $T_m$ where $p_1$ increases, raising the value of $k_1$ allows the axle not tasked with distributing $T_m$ to allocate $M_z$. This adjustment ensures that TWU$_i$, which is proportional to $F_{x,i}$, is not excessive on a single pair of wheels.

Conversely, high magnitudes of $M_z$ indicate critical situations encountered by the vehicle. In such circumstances, the primary concern is ensuring vehicle stability. For $M_z$ to be generated accurately, individual tire slippage must be averted, i.e. each wheel must receive the optimal torque within the bounds of its friction circle. In such a scenario, it is advisable to reduce both $p_1$ and $k_1$ towards 0.5 to distribute $T_m$ and $M_z$ among the four wheels. However, $p_1$ and $k_1$ will be attenuated toward achieving a balanced dynamic load distribution, rather than a fixed value of 0.5. Hence, define the ratio of load distribution between front and rear axles as,

$$\kappa_1 = \frac{F_{z,fi}}{F_{z,ri}} = \frac{F_{z,fr} + F_{z,fl}}{F_{z,rr} + F_{z,rl}} \quad (7)$$

The strategy involves maintaining a proportionality between the ratio of longitudinal forces and the associated loads. Hence according to (1) and (2) and after reformulation, the lower bound of $p_1$ is $\frac{\kappa_1}{1+\kappa_1}$ and that for $k_1$ is $\frac{1}{1+\kappa_1}$.

Following the determination of $p_1$ and $k_1$ in the first step, $p$ and $k$ are calculated subsequently with a focus on minimizing energy consumption. The minimization of the vehicle's energy consumption is equivalent to minimizing the instantaneous total power consumption of the four motors, with the assumption that battery power losses are negligible [50]. Hence, considering (3) as a cost function depending on $\{T_i(p, k, q, n), \omega_i, \eta_{s,i}, T_m, M_z\}$, substitute (1) and (2) in (3) according to the sense of $M_z$. The analysis of the cost function enables us to assign $p$ and $k$ according to the sign of $T_m$, $M_z$, and a combination of $\omega_i$ and $\eta_i$ between the front and rear axles. Algorithm 1 depicts the strategy for energy economic distribution of $T_m$ and Algorithm 2 presents the same for $M_z$. According to Algorithm 1 and following the computation of $p_1$, a front axle drive implies that $p = p_1$ whilst $p = 1 - p_1$ for a rear axle drive. While recalling Algorithm 2, depending
<figure><img src="image_4.png" /><figcaption>Fig. 4: Load distribution on the vehicle's left and right sides</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 5: Schematic diagram of the torque allocation process</figcaption></figure>

on the sign of the combination involving $q$ and $n$, $k = k_1$ corresponds to the generation of $M_z$ on the rear axle whereas $1 - k_1$ otherwise.

## 2) Rules to adapt $q$ and $n$

The parameters $q$ and $n$ are assigned to coordinate between traction and braking torques to allocate $kM_z$ and $(1 - k)M_z$ respectively. Hence, the primal factor for adapting these parameters is the magnitude of $|M_z|$. Three distinct ranges are established for $|M_z|$ drawing an analogy with the errors in yaw rate and side-slip angles. Low $|M_z|$ ($< \bar{M}_z$) characterizes normal driving conditions in the presence of a minor error on the yaw rate. Mid $|M_z|$ ($\bar{M}_z < |M_z| < \bar{M}_z$) represents situations with elevated yaw rate error or low side-slip angle error. Finally, high $|M_z|$ ($> \bar{M}_z$) corresponds to critical scenarios featuring a significant error in the side-slip angle. The bounds $\{\bar{M}_z, \bar{M}_z\}$ will be discussed later.

Given the similarity in the objectives desired to be achieved on the vehicle's opposite sides, between the front and rear axles, $q$ and $n$ are set to be the same. The DYC controller operates continuously to ensure both maneuverability and stability. Consequently, if $M_z$ is consistently generated through braking, it may lead to driving discomfort and accelerated wheel wear. Therefore, for situations with low $|M_z|$, the assignment of $q, n$ as 0 exclusively for traction torques serves the purpose of reducing wheel wear, enhancing driving comfort, and facilitating higher longitudinal velocity. Conversely, for high $|M_z|$, which frequently occur during intense cornering or abrupt steering maneuvers, prioritizing safety necessitates deceleration and the avoidance of traction torques. As a result, $q$ and $n$ are set to 1, thereby earning high energy through the regenerative braking system. Consider (5) and let $c_3 = k \frac{1}{2t_r}$ and $c_4 = (1 - k) \frac{1}{2t_r}$, then we have (8).

$$F_{x,1} = (1 - p)T_m \times a + qM_z \times c_3 \quad (8a)$$

$$F_{x,2} = (1 - p)T_m \times a + (1 - q)M_z \times c_3 \quad (8b)$$

$$F_{x,3} = pT_m \times a + (1 - q)M_z \times c_4 \quad (8c)$$

$$F_{x,4} = pT_m \times a + qM_z \times c_4 \quad (8d)$$

In the presence of high $|M_z|$, the assignment of the parameter $p$ depends on the dynamic load distribution between the front and rear axles, consequently ensuring the balanced distribution for the traction part of $F_{x,i}$. Furthermore, it is worth noting that during cornering, the friction circle for braking torques typically exceeds that of traction torques (see Fig. 4). As a result, in such scenarios, setting $q$ to 1 allows for the attainment of the aforementioned objectives without incurring

substantial increases in TWU$_i$.

Lastly, for situations of mid $|M_z|$, $p$ and $k$ are biased toward opposite axles. So according to (8), precise consideration must be given to the assignment of $q$ to prevent an undue escalation of individual TWU$_i$. Additionally, according to [37], achieving a front/rear balanced load distribution is deemed energy-efficient in critical and less-critical situations. Consequently, an approach is formulated to assign $q$ according to a right/left dynamic load distribution, thereby balancing TWU$_i$ and promoting energy savings.

Define the load distribution ratio between the right and left sides of the vehicle as

$$\kappa_2 = \frac{F_{z,ir}}{F_{z,il}} = \frac{F_{z,fr} + F_{z,rr}}{F_{z,fl} + F_{z,rl}} \quad (9)$$

To maintain proportionality between the ratio of the longitudinal forces and the corresponding loads (setting $\kappa_2 = \frac{F_{z,ir}}{F_{z,il}}$), substituting the torques for generating $M_z$ (1) and (2) in (9), and according to the sense of $M_z$, $\kappa_2$ can be reformulated as follows.

$$v = \frac{q(r/t_r)kM_z + q(r/t_f)(1 - k)M_z}{(1 - q)(r/t_r)kM_z + (1 - q)(r/t_f)(1 - k)M_z} \quad (10a)$$

$$\kappa_2 = \begin{cases} v & \text{if } M_z \ge 0 \\ 1/v & \text{if } M_z < 0 \end{cases} \quad (10b)$$

Hence according to this distribution, $q$ is reformulated as

$$q_{dyn} = \begin{cases} \frac{\kappa_2}{1+\kappa_2} & \text{if } M_z \ge 0 \\ \frac{1}{1+\kappa_2} & \text{if } M_z < 0 \end{cases} \quad (11)$$

Therefore, the strategy entails assigning the higher torque magnitude as the braking torques, and the lower magnitude as the traction torques. Fig. 4 shows the right/left load distribution of the vehicle during cornering. In the situation of left cornering, where greater loads are imposed on the vehicle's right side, resulting in larger friction circles ($\kappa_2 > 1$), the generated yaw moment is clockwise ($M_z > 0$). In this context, the coefficient for the braking torques is the higher one, given by $\frac{\kappa_2}{1+\kappa_2}$ while $\frac{1}{1+\kappa_2}$ ($1 - \frac{\kappa_2}{1+\kappa_2}$) for the traction torques. Conversely, in the case of right cornering, where $\kappa_2 < 1$ and the greater magnitude pertains to $\frac{1}{1+\kappa_2}$ for the braking torques.

Therefore, in the cases of moderate $|M_z|$, $q$ is adapted towards $q_{dyn}$. Similarly, when confronted with high $|T_m|$, the allocation of $p$ and $k$ is directed toward a one-sided axle configuration. Hence, in order to achieve the balance of $F_{x,i}$ (8), $q$ is adjusted to align with $q_{dyn}$ as well.

It is worth noting that our proposed strategy accounts for
<figure><img src="image_6.png" /><figcaption>Fig. 6: Variation of g₂ and g₄ as a function of γ</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 7: Parameter q for κ = 0.5 (left) and κ = 0.7 (right)</figcaption></figure>

the influence of front wheel steering on the allocation of output torques, particularly during cornering maneuvers. By integrating the right/left load distribution ratio $κ_2$, our strategy dynamically adjusts the torque output of the motors to optimize vehicle dynamics and enhance cornering performance. The load forces $F_{z,ij}$ are defined as (12) derived from [51].

$$F_{z,fl} = m \left[ \frac{l_r}{l_r+l_f} g \cos(\phi_r) - \frac{h}{l_r+l_f} g \sin(\phi_r) - \frac{h}{l_r+l_f} a_x \right] \times \left[ \frac{1}{2} - \frac{h}{l_f g} a_y \right] \quad (12a)$$

$$F_{z,fr} = m \left[ \frac{l_r}{l_r+l_f} g \cos(\phi_r) - \frac{h}{l_r+l_f} g \sin(\phi_r) - \frac{h}{l_r+l_f} a_x \right] \times \left[ \frac{1}{2} + \frac{h}{l_f g} a_y \right] \quad (12b)$$

$$F_{z,rl} = m \left[ \frac{l_f}{l_r+l_f} g \cos(\phi_r) + \frac{h}{l_r+l_f} g \sin(\phi_r) + \frac{h}{l_r+l_f} a_x \right] \times \left[ \frac{1}{2} - \frac{h}{l_f g} a_y \right] \quad (12c)$$

$$F_{z,rr} = m \left[ \frac{l_f}{l_r+l_f} g \cos(\phi_r) + \frac{h}{l_r+l_f} g \sin(\phi_r) + \frac{h}{l_r+l_f} a_x \right] \times \left[ \frac{1}{2} + \frac{h}{l_f g} a_y \right] \quad (12d)$$

where $a_x, a_y$ are the longitudinal and lateral accelerations, $l_r, l_f$ are the distances between the center of gravity to the rear and front axles respectively, $t_r, t_f$ are the rear and front tracks, $h$ is the height of the center of gravity, $g$ is the gravitational acceleration, $m$ is the vehicle's mass, and $\phi_r$ is the inclination angle of the road. Therefore, the load distribution ratios $κ_1$ and $κ_2$ takes into consideration the effect of the vehicle mass distribution, longitudinal and lateral accelerations, and road gradient. An overall flow scheme for the torque allocation process is depicted in Fig. 5.

### III. FAST CONTINUOUS TORQUE ALLOCATION

In this section, analytical models are developed for the parameters $p_1$ (respectively $p$), $k_1$ (respectively $k$), and $q$ as a function of $T_m$, $M_z$, $κ_1$, $κ_2$, $ω_i$, and $η_i$. The parameters and the shaping of the constructed functions adhere to the principles established in the preceding section. The formulation of the functions is based on the following concepts: 1) imposing constraints on boundedness and saturation to confine the outputs within their limits [0, 1], 2) employing a non-linear relationship to capture intricate dependencies, 3) ensuring continuity and differentiability, and 4) integrating a modifiable, smooth transition state to facilitate adaptability in adjusting the output behavior.

#### A. Analytical Modeling of q

Let $κ$ be a function of $κ_2$ depending on the sense of $M_z$ such that

$$κ = \operatorname{sign}(M_z) \times \frac{κ_2}{1 + κ_2} + \frac{1 - \operatorname{sign}(M_z)}{2} \quad (13)$$

where $κ = κ_2/(1 + κ_2)$ if $\operatorname{sign}(M_z) = +1$, $κ = 1/(1 + κ_2)$ if $\operatorname{sign}(M_z) = -1$, and $κ = 0.5$ if $\operatorname{sign}(M_z) = 0$. Due to the irrelevance of the value of $q$ when $M_z = 0$, $κ$ is a representation of $q_{dyn}$ (11).

To range the subsequent functions transition according to the situations governed by {$M_z, \overline{M}_z$}, let's introduce {$ν_i, \overline{ν}_i$} ($i$ as an index to each sub-function) as the lower and upper bounds in specific ranges of $M_z$. Similarly, {$τ_i, \overline{τ}_i$} to be the lower and upper bounds of $T_m$. The function governing the parameter $q$ is a composite of two distinct sub-functions, each operating within its respective domain. This distinction arises due to the fact that the behavior of $q$ as a function of $T_m$ for low $M_z$ exhibits dissimilar patterns from its variation for high $M_z$. Hence, define the first sub-function $q_1$ as

$$q_1 = g_1(M_z) \cdot g_2(T_m) \quad (14)$$

where,

$$g_1(M_z) = \frac{1}{1 + e^{-\frac{2\gamma_1}{\nu_1-\nu_1}(|M_z| - \frac{\nu_1+\nu_1}{2})}} - \frac{1}{1 + e^{-\frac{2\gamma_2}{\nu_2-\nu_2}(|M_z| - \frac{\nu_2+\nu_2}{2})}} \quad (15a)$$

$$g_2(T_m) = \frac{\kappa}{1 + e^{-\frac{2\gamma_3}{\tau_1-\tau_1}(|T_m| - \frac{\tau_1+\tau_1}{2})}} \quad (15b)$$

where $ν_1, \overline{ν}_1, ν_2$, and $\overline{ν}_2$ respectively delineate the extremities characterizing the transitional phases associated with the ascending and descending components of $g_1$, and $τ_1$ and $\overline{τ}_1$ denote the respective endpoints of the transient state within $g_2$. $γ_1$ and $γ_2$ signify the gradient intensity of the transient-state centroid within the ascending and descending components of $g_1$, respectively, while $γ_3$ denotes the gradient intensity of the transient-state centroid in $g_2$. Additionally, $κ$ (13) is integrated into $g_2$ as an upper bound for $q_1$.

Similarly, define the second sub-function $q_2$ as:

$$q_2 = g_3(M_z) \cdot g_4(T_m) \quad (16)$$

where,

$$g_3(M_z) = \frac{1}{1 + e^{-\frac{2\gamma_4}{\nu_2-\nu_2}(|M_z| - \frac{\nu_2+\nu_2}{2})}} \quad (17a)$$

$$g_4(T_m) = 1 - \frac{1 - \kappa}{1 + e^{-\frac{2\gamma_5}{\tau_2-\tau_2}(|T_m| - \frac{\tau_2+\tau_2}{2})}} \quad (17b)$$

where $ν_2$ and $\overline{ν}_2$ for $g_3$ are the same extremities as for the descending component of $g_1$, and $τ_2$ and $\overline{τ}_2$ denote the boundaries of the transient state of $g_4$. $γ_4$, respectively $γ_5$, correspond to the gradient intensity of the transient-state centroid in $g_3$ and $g_4$. Finally, $(1 - κ)$ is integrated into $g_4$.
<figure><img src="image_8.png" /><figcaption>Fig. 8: Reshaping points for $p_1$ and $k_1$</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 9: Parameter $p_1$ for $\kappa_1 > 1$ (left) and $\kappa_1 < 1$ (right)</figcaption></figure>

<figure><img src="image_10.png" /></figure>

to impose a lower bound of value $\kappa$ to $q_2$.

Therefore, the analytical model of the parameter $q$ is given by (18) as a combination of $q_1$ and $q_2$. The change in the shape of the functions $g_2$ and $g_4$, bounded by $\kappa$, as a variation of $\gamma_i$ ($i = \{1, 2, 3, 4, 5\}$) is given in Fig. 6. According to this figure and to accurately bound the transient state between $[\tau_i, \bar{\tau}_i]$ ($i = 1, 2$), we chose $\gamma_3 = \gamma_5 = 4$. Similarly for $g_1$ and $g_3$ in terms of $[\nu_i, \bar{\nu}_i]$ ($i = 1, 2$), choose $\gamma_1 = \gamma_2 = \gamma_4 = 4$.

$$q = q_1(T_m, M_z) + q_2(T_m, M_z) = g_1.g_2 + g_3.g_4 \quad (18)$$

Let $\underline{\nu}_1 = 100, \bar{\nu}_1 = 300, \underline{\nu}_2 = 400, \bar{\nu}_2 = 700$ and $\underline{\tau}_1 = 100, \bar{\tau}_1 = \underline{\tau}_2 = 300$ and $\bar{\tau}_2 = 750$. Fig. 7 depicts the shape of the parameter $q$ as $\kappa$ varies. For normal driving where the loads on the vehicle's opposite sides are the same, $\kappa_2$ is 1 and so $\kappa = 0.5$. The shape of $q$ in this situation is presented in Fig. 7-(left), while Fig. 7-(right) reveals $q$ for $\kappa = 0.7$. Therefore, based on the estimated tire loads $F_{z,i}$, $\kappa$ is calculated at each step, and the shape of the function of $q$ is adapted accordingly.

## B. Analytical Modeling of p

The parameter $p_1$ is defined as a combination of two subfunctions, $h_1(T_m)$ and $h_2(M_z)$. As $p_1$ ranges in $[0.5, 1]$, certain shaping parameters are integrated to adapt and conform the functions to the intended range, while upholding the established rules. To this end, we define $h_1$ and $h_2$ as follows,

$$h_1(T_m) = \xi_1 \frac{1}{1 + e^{-\frac{2\chi_1}{\bar{\tau}_3 - \bar{\tau}_3}(|T_m| - \frac{\bar{\tau}_3 + \bar{\tau}_3}{2})}} + \lambda_1 \quad (19a)$$

$$h_2(M_z) = \xi_2 \frac{1}{1 + e^{-\frac{2\chi_2}{\bar{\nu}_3 - \bar{\nu}_3}(|M_z| - \frac{\bar{\nu}_3 + \bar{\nu}_3}{2})}} + \lambda_2 \quad (19b)$$

where $\xi_1, \lambda_1$ and $\xi_2, \lambda_2$ are shape parameters for $h_1$ and $h_2$ respectively. $\chi_i$ correspond, as $\gamma_i$, to the intensity of the slope of the transient-state centroid of $h_i$ where $i = \{1, 2\}$. $\underline{\tau}_3$ and $\bar{\nu}_3$ represent the extremities of the transient state of $h_1$ and $h_2$ respectively. As $p_1$ must increase with the increase of $|T_m|$, while conversely decreasing with the increase of $|M_z|$, it can be formulated as (20).

$$p_1 = h_1(T_m) - h_2(M_z) \quad (20)$$

In order to reshape $p_1$, $\xi_i$ and $\lambda_i$ are calculated and updated at each step while maintaining $\chi_i$, $\tau_3$, $\bar{\tau}_3$, $\underline{\nu}_3$, and $\bar{\nu}_3$ ($i = \{1, 2\}$) at constant, predefined values. Note that $\xi_i$ and $\lambda_i$ indicate the location of the four boundaries of $p_1$ while the remnant parameters define the characteristics of the function.

Let $A(\bar{\tau}_3, \underline{\nu}_3)$, $B(\tau_3, \bar{\nu}_3)$, $C(\bar{\tau}_3, \underline{\nu}_3)$, and $D(\bar{\tau}_3, \underline{\nu}_3)$ be the four extremity points of $\bar{h}_1$ and $\bar{h}_2$ (see Fig. 8). The value of the function $p_1$ at these points can be represented by (21).

$$p_{1,A} = \left[ \frac{1}{1 + e^{-\chi_1}} \right] \xi_1 - \left[ \frac{1}{1 + e^{\chi_2}} \right] \xi_2 + \lambda \quad (21a)$$

$$p_{1,B} = \left[ \frac{1}{1 + e^{\chi_1}} \right] \xi_1 - \left[ \frac{1}{1 + e^{\chi_2}} \right] \xi_2 + \lambda \quad (21b)$$

$$p_{1,C} = \left[ \frac{1}{1 + e^{\chi_1}} \right] \xi_1 - \left[ \frac{1}{1 + e^{-\chi_2}} \right] \xi_2 + \lambda \quad (21c)$$

$$p_{1,D} = \left[ \frac{1}{1 + e^{-\chi_1}} \right] \xi_1 - \left[ \frac{1}{1 + e^{-\chi_2}} \right] \xi_2 + \lambda \quad (21d)$$

Let $h_1$ and $h_2$ have the same gradient intensity ($\chi_1 = \chi_2 = \chi$). As $\chi$ is a constant chosen in accordance with the desired transient state behavior, one has to determine the three variables $\xi_1, \xi_2$ and $\lambda = \lambda_1 - \lambda_2$. Consequently, it suffices to modify the restructuring of points A, B, and C.

$$\xi_2^* = (u_3 - u_2) \frac{(1 + e^{-\chi})(1 + e^{\chi})}{e^{-\chi} - e^{\chi}} \quad (22a)$$

$$\lambda^* = \frac{1}{e^{\chi} - e^{-\chi}} \left[ -(1 + e^{-\chi})u_1 + (1 + e^{\chi})u_2 - \frac{e^{-\chi} - e^{\chi}}{1 + e^{\chi}} \xi_2^* \right] \quad (22b)$$

$$\xi_1^* = (1 + e^{-\chi})(u_1 + \frac{1}{1 + e^{\chi}} \xi_2^* - \lambda^*) \quad (22c)$$

Denote $\mathcal{U} = [u_1 \ u_2 \ u_3]$ the desired values for A, B, and C respectively. In order to integrate the lower bound of $p_1$, set $u_3$ to $\frac{\kappa_1}{1+\kappa_1}$ (7). Consequently, substituting $[p_{1,A} \ p_{1,B} \ p_{1,C}]^T = [u_1 \ u_2 \frac{\kappa_1}{1+\kappa_1}]^T$ in (21) and using the Gaussian elimination method, the solution $\{\xi_1^*, \xi_2^*, \lambda^*\}$ can be determined iteratively using (22). Beyond points A, B, C, and D, $p_1$ saturates, and its values in the vicinity of its extremities closely approximate, with small errors, those of these points. In order to account for these errors and to stay in the range $[0, 1]$, let $u_1 = 0.99$ and $u_2 = 0.9$. Similar to $\gamma$, set $\chi = 4$ for the same reasoning, and let $\underline{\nu}_3 = 300, \bar{\nu}_3 = 700$ and $\bar{\tau}_3 = 300, \underline{\tau}_3 = 800(N.m)$. Therefore, the shape of the analytical model of $p_1$ is given in Fig. 9 as a function of $\kappa_1$. Fig. 9-(left) shows when the front axle loads exceed those of the rear axle, then $\kappa_1 > 1$ and consequently $\frac{\kappa_1}{1+\kappa_1} > 0.5$, while Fig. 9-(right) depicts the shape of $p_1$ when $\kappa_1 < 1$.

$$\alpha_1 = \left[ \frac{\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} + \frac{\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} \right] - \left[ \frac{\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} + \frac{\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}} \right] \quad (23a)$$

$$\alpha_2 = |M_z| - \frac{\bar{\nu}_3 + \underline{\nu}_3}{2} + \varepsilon \quad (23b)$$

Following the estimation of the tire loads, $\kappa_1$ is calculated and
<figure><img src="image_11.png" /><figcaption>Fig. 10: Projection of $k_1$ on $M_z$ axis: $f(\chi)$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11: Parameter $k_1$ for $\kappa_1 > 1$ (left) and $\kappa_1 < 1$ (right)</figcaption></figure>

$p_1$ is reshaped accordingly. Subsequently, $p$ is determined with the objective of minimizing energy consumption. Referring to Algorithm 1, we define $\alpha_1$ as (23), where $\alpha_2$ is defined to ascertain whether $|M_z|$ falls after the centroid of the transient state of $h_2$. According to Algorithm 1, $p$ is assigned either as $p_1$ or $1-p_1$ based on the sign of $\alpha_1$ and $T_m$. However, for high $|M_z|$ ($\alpha_2 > 0$) and to maintain stability (towards $\frac{\kappa_1}{1+\kappa_1}$), $p$ is confined to the value of $p_1$, regardless of the sign of $T_m$ and $\alpha_1$.

$$p = \left[ \text{sign}(T_m \cdot \alpha_1) \times p_1 + \frac{1 - \text{sign}(T_m \cdot \alpha_1)}{2} \right] \times \frac{1 - \text{sign}(\alpha_2)}{2} + p_1 \times \frac{1 + \text{sign}(\alpha_2)}{2} \quad (24)$$

Based on Algorithm 1, $p$ can be determined based on the sign of $T_m \cdot \alpha_1$. For $T_m \cdot \alpha_1 > 0$, $p = p_1$, while $p = 1-p_1$ otherwise. Therefore, $p$ is formulated as (24), where $p$ is determined according to Algorithm 1 if $\text{sign}(\alpha_2) = -1$, and assigned as $p_1$ if $\text{sign}(\alpha_2) = +1$. To avoid unnecessary values for $p$ when $\text{sign}(\alpha_2) = 0$, an infinitesimal $\epsilon \in \mathbb{R}$ is integrated into $\alpha_2$ to relax this issue.

### C. Analytical Modeling of $k$

In a context analogous to $p_1$ where the variation of $k_1$ is similar to that of $p_1$, define $k_1$ as a composition of two subfunctions $f_1(T_m)$ and $f_2(M_z)$ such that,

$$f_1(T_m) = \zeta_1 \frac{1}{1 + e^{-\frac{2\chi_3}{\tau_4 - \tau_4}(|M_t| - \frac{\tau_4 + \tau_4}{2})}} + \sigma_1 \quad (25a)$$

$$f_2(M_z) = \zeta_2 \frac{1}{1 + e^{-\frac{2\chi_4}{\nu_4 - \nu_4}(|M_z| - \frac{\nu_4 + \nu_4}{2})}} + \sigma_2 \quad (25b)$$

where $\zeta_1$, $\sigma_1$ and $\zeta_2$, $\sigma_2$ are shape parameters for $f_1$ and $f_2$ respectively. $\chi_3$ and $\chi_4$ are similar to $\chi$ for $f_1$ and $f_2$ respectively. $\tau_4$, $\tau_4$ and $\nu_4$, $\nu_4$ signify the endpoints of the transient state of $f_1$ and $f_2$ respectively. Correspondingly, $k_1$ is defined as,

$$k_1 = f_1(T_m) - f_2(M_z) \quad (26)$$

The shape of $k_1$ is adapted by simultaneously updating the shaping parameters $\zeta_i$ and $\sigma_i$ with $i = \{1, 2\}$. Following Fig. 8, define the boundary points for $k_1$ as: $A(\tau_4, \nu_4)$, $B(\tau_4, \nu_4)$, $C(\tau_4, \nu_4)$, and $D(\tau_4, \nu_4)$. Let $\sigma = \sigma_1 - \sigma_2$ and $\bar{\chi}_3 = \bar{\chi}_4 = \chi$. Denote $\mathcal{V}$ be the vector of the desired values of $k_1$ at A, B, and C, and $X = [\zeta_1 \ \sigma \ \zeta_2]^T$.

$$\mathcal{A} = \begin{bmatrix} \frac{1}{1+\frac{1}{1}-\chi} & 1 & -\frac{1}{1+e^{\chi}} \\ \frac{1+e^{\gamma}}{1} & 1 & -\frac{1+e^{\chi}}{1} \\ \frac{1}{1+e^{\chi}} & 1 & -\frac{1}{1+e^{-\chi}} \end{bmatrix}; \quad \mathcal{V} = \begin{bmatrix} k_{1,A} \\ k_{1,B} \\ k_{1,C} \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} \quad (27)$$

Referring to (21), formulate the problem as $\mathcal{AX} = \mathcal{V}$ in (27). Set $v_3 = \frac{1}{1+\kappa_1}$ (7) to integrate the lower bound for $k_1$. Subsequently, $\bar{X}^*$ can be determined at each step by solving (28) iteratively.

$$\zeta_2^* = (v_3 - v_2) \frac{(1 + e^{-\chi})(1 + e^{\chi})}{e^{-\chi} - e^{\chi}} \quad (28a)$$

$$\sigma^* = \frac{1}{e^{\chi} - e^{-\chi}} \left[ -(1 + e^{-\chi})v_1 + (1 + e^{\chi})v_2 - \frac{e^{-\chi} - e^{\chi}}{1 + e^{\chi}} \zeta_2^* \right] \quad (28b)$$

$$\zeta_1^* = (1 + e^{-\chi})(v_1 + \frac{1}{1 + e^{\chi}} \zeta_2^* - \sigma^*) \quad (28c)$$

Fig. 10 depicts the projection of $k_1$ on the yaw moment axis for $T_m = 0$. It shows the effect of increasing the gradient intensity of the transient state in $[\nu_4, \bar{\nu}_4]$ between the boundaries $v_2$ and $\frac{1}{1+\kappa_1}$. As indicated in $p_1$, $\chi$ is chosen equal to 4.

$$\alpha_3 = \left[ \frac{-q\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} \right] - \left[ \frac{-q\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} \right] \quad (29a)$$

$$\alpha_4 = \left[ \frac{-q\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} \right] - \left[ \frac{-q\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} + \frac{(1-q)\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}} \right] \quad (29b)$$

In order to ensure that $k$ is confined to $[0, 1]$, let $v_1 = 0.99$ along with $v_2 = 0.8$, and $\nu_4 = 200, \bar{\nu}_4 = 600, \tau_4 = 300$ and $\bar{\tau}_4 = 700(N.m)$. Therefore, following the computation of $\kappa_1$, the shape of $k_1$ is adjusted accordingly. Fig. 11-(left) presents the shape of the analytical model of $k_1$ when the front axle loads transcend that of the rear axle ($\frac{1}{1+\kappa_1} < 0.5$), while Fig. 11-(right) reveals $k_1$ for $\kappa_1 < 1$.

The assignment of $k$ is handled according to Algorithm 2 either as $k_1$ or $1-k_1$ with the aim of minimizing energy consumption. To this end, we define $\alpha_3$ and $\alpha_4$ as (29). Referring to Algorithm 2, $k_1$ is determined based on the sign of $\alpha_3$ if $M_z \le 0$, and on the sign of $\alpha_4$ if $M_z > 0$. With reformulation, $\alpha_3$ and $\alpha_4$ are constructed based on $\alpha_1$ and the difference in the rotational velocities of the wheels receiving traction torques (refer to Fig. 4).

$$\alpha_3 = -q \cdot \alpha_1 + \left[ \frac{\omega_2}{\eta_{s,2}^{\text{sign}(T_m)}} - \frac{\omega_3}{\eta_{s,3}^{\text{sign}(T_m)}} \right] \quad (30a)$$

$$\alpha_4 = -q \cdot \alpha_1 + \left[ \frac{\omega_1}{\eta_{s,1}^{\text{sign}(T_m)}} - \frac{\omega_4}{\eta_{s,4}^{\text{sign}(T_m)}} \right] \quad (30b)$$

$$\alpha_5 = |M_z| - \frac{\bar{\nu}_4 + \nu_4}{2} + \varepsilon \quad (30c)$$

where $\alpha_5$ is defined for the same reasoning as $\alpha_2$.

$$k_2 = \frac{1 + \text{sign}(M_z)}{2} \left[ \text{sign}(\alpha_4) \times k_1 + \frac{1 - \text{sign}(\alpha_4)}{2} \right] + \frac{1 - \text{sign}(M_z)}{2} \left[ \text{sign}(\alpha_3) \times k_1 + \frac{1 - \text{sign}(\alpha_3)}{2} \right] \quad (31)$$
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12: Fuzzy logic inputs: yaw moment $M_z$ and total torque $T_m$</figcaption></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 13: Fuzzy logic outputs: parameters $p_1$, $k_1$, and $q$</figcaption></figure>

Fig. 10 reveals the ranges depending on the sign of $\alpha_5$. Based on Algorithm 2, parameter $k_2$ is determined as (31). Finally, $k$ is determined based on Algorithm 2 where $k = k_2$ when $\text{sign}(\alpha_5) = -1$ and assigned directly as $k_1$ if $\text{sign}(\alpha_5) = +1$ to maintain stability.

$$k = k_2 \times \frac{1 - \text{sign}(\alpha_5)}{2} + k_1 \times \frac{1 + \text{sign}(\alpha_5)}{2} \quad (32)$$

It is imperative to stress that the functions' parameters of $p_1$, $k_1$, and $q$ are tunable. 1) The shape parameters $\xi_i$, $\zeta_i$, $\sigma$, and $\lambda$ ($i = \{1, 2\}$) depending on $\kappa_1$ and $\kappa_2$ designate the bounds on $p_1$, $k_1$ and $q$. These bounds are specified based on the desired objectives. To maintain tire stability and avert slipping, it is advisable to keep $u_3$ and $v_3$ as defined. $u_1$, $u_2$ and $v_1$, $v_2$ can be adjusted in a manner that boosts energy efficiency when increased, while enhancing vehicle stability when decreased, contingent upon the condition that $u_2 > u_3$ and $v_2 > v_3$. Furthermore, it is essential to bear in mind that depending on the values of $\gamma$ and $\chi$, the functions saturate after exceeding $u_1$ and $v_1$. This consideration is vital to ensure that $p_1$ and $k_1$ remain within the specified range. 2) The transient state parameters $\gamma$, $\chi$, $\tau_i$, $\bar{\tau}_i$, $\nu_i$, $\bar{\nu}_i$ where $i = \{1, 2, 3, 4\}$ for $p_1$, $k_1$, $q$ may require adjustment according to the controller parameters. The controller gain directly influences the transient-state extremities while the response time affects the choice of the gradient intensity of the transient-state centroid. The controller gain directly influence the range of variation of $|T_m|$ and $|M_z|$. For instance, larger vehicle mass might reflect a higher range, thereby requiring an increase in $\tau_i$, $\bar{\tau}_i$, $\nu_i$, $\bar{\nu}_i$ and decrease in $\chi$ and $\gamma$, and vice versa.

Therefore, the analytical models of $p_1$, $k_1$, and $q$ are established in a generalized manner under a rule-based formulation. The functions' parameters are subjected to tuning depending on the controller characteristics and the desired objectives.

Hence, this novel approach for multi-objective torque allocation carries the attributes of scalability, expeditiousness, simplicity of tuning, and ease of implementation.

## IV. FUZZY TORQUE ALLOCATION

In the same context of rule-based formulation for the parameters $p$, $k$, and $q$, a second approach is developed based on fuzzy logic control. The fuzzy logic controller receives $|T_m|$ and $|M_z|$ as inputs and generate $p_1$, $k_1$, and $q$ as outputs. Hence, $p_1$, $k_1$, and $q$ are determined in the first step based on the fuzzy logic controller, and subsequently, $p$ and $k$ are determined in the second stage. The framework of fuzzy logic control consists of three stages: fuzzification, inference engine, and defuzzification [52]. In the fuzzification step, the membership functions of the inputs and outputs are defined as a combination of Gaussian functions (33) to provide smooth transitions.

$$f_g(x) = e^{-\frac{(x-m_e)^2}{2\Upsilon^2}} \quad (33)$$

where $m_e$ is the Gaussian mean and $\Upsilon$ is the standard deviation. The membership functions of the fuzzy inputs are shown in Fig. 12, while Fig. 13 depicts that of the fuzzy outputs. Five sets are defined for the inputs: {Small (S), Medium-Small (MS), Medium-Medium (MM), Medium-Big (MB), Big (B)}, yet with distinct ranges for $|M_z|$ and $|T_m|$. As for the fuzzy outputs, nine sets are defined as: {Small-Small (SS), Small-Medium (SM), Small-Big (SB), Medium-Small (MS), Medium-Medium (MM), Medium-Big (MB), Big-Small (BS), Big-Medium (BM), Big-Big (BB)}. However, the shape and the range of the fuzzy sets vary between $p_1$, $k_1$, and $q$. In accordance with the prescribed rules for parameter adaptation, it has been ascertained that $p_1$ is subject to a lower bound $\frac{\kappa_1}{1+\kappa_1}$, $k_1$ is constrained by $\frac{1}{1+\kappa_1}$, and $q$ tends to a mid-range
<table><caption>TABLE I: Fuzzy inference rules for $p_1$</caption><thead><tr><th rowspan="2">p<sub>1</sub></th><th colspan="5">T<sub>m</sub></th></tr><tr><th>S</th><th>MS</th><th>MM</th><th>MB</th><th>B</th></tr></thead><tbody><tr><th>S</th><td>MM</td><td>MB</td><td>BS</td><td>BM</td><td>BB</td></tr><tr><th>MS</th><td>MS</td><td>MM</td><td>MB</td><td>BS</td><td>BM</td></tr><tr><th>MM</th><td>SB</td><td>MS</td><td>MM</td><td>MB</td><td>BS</td></tr><tr><th>MB</th><td>SM</td><td>SB</td><td>MS</td><td>MM</td><td>MB</td></tr><tr><th>B</th><td>SS</td><td>SM</td><td>SB</td><td>MS</td><td>MM</td></tr></tbody></table>

<table><caption>TABLE II: Fuzzy inference rules for $k_1$</caption><thead><tr><th rowspan="2">k<sub>1</sub></th><th colspan="5">T<sub>m</sub></th></tr><tr><th>S</th><th>MS</th><th>MM</th><th>MB</th><th>B</th></tr></thead><tbody><tr><th>S</th><td>MM</td><td>MB</td><td>BS</td><td>BM</td><td>BB</td></tr><tr><th>MS</th><td>MS</td><td>MM</td><td>MB</td><td>BS</td><td>BM</td></tr><tr><th>MM</th><td>SB</td><td>MS</td><td>MM</td><td>MB</td><td>BS</td></tr><tr><th>MB</th><td>SM</td><td>SB</td><td>MS</td><td>MM</td><td>MB</td></tr><tr><th>B</th><td>SS</td><td>SM</td><td>SB</td><td>MS</td><td>MM</td></tr></tbody></table>

<table><caption>TABLE III: Fuzzy inference rules for $q$</caption><thead><tr><th rowspan="2">q</th><th colspan="5">T<sub>m</sub></th></tr><tr><th>S</th><th>MS</th><th>MM</th><th>MB</th><th>B</th></tr></thead><tbody><tr><th>S</th><td>SS</td><td>SM</td><td>SB</td><td>MS</td><td>MM</td></tr><tr><th>MS</th><td>SM</td><td>SB</td><td>MS</td><td>MM</td><td>MM</td></tr><tr><th>MM</th><td>MM</td><td>MM</td><td>MM</td><td>MM</td><td>MM</td></tr><tr><th>MB</th><td>BM</td><td>BS</td><td>MB</td><td>MM</td><td>MM</td></tr><tr><th>B</th><td>BB</td><td>BM</td><td>BS</td><td>MB</td><td>MM</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>Fig. 14: Fuzzy outputs control space: parameters $p_1$, $k_1$, and $q$ respectively</figcaption></figure>

tendency value denoted as $\kappa$ (13). Nevertheless, for the fuzzy structure, the bounds for $p_1$, $k_1$, and $q$ are restricted to 0.5 instead of a combination of progressively updated load ratios. In other words, the starting point of the parameters $p_1$, $k_1$ and the mid point of $q$ can't be modified at each iteration as performed in the analytical modeling case; hence they are set to 0.5. Hence, the fuzzy sets are defined for $p_1$ and $k_1$ in the range [0.5, 1], while in the interval [0, 1] for $q$.

The rule base of the inference engine is defined based on the chosen values of $\tau_i$ and $\nu_i$ where $i = \{1, 2, 3, 4\}$ while upholding the established rules. Specifically, the extremities of each parameter are determined in line with the constructed rules. The flow of the fuzzy rules for each parameter, that is, the transition between two adjacent rules along the rows and columns (refer to Tables I, II, III) aligns with the values of $\tau_i$ and $\nu_i$. This alignment is evident from the fact that the shape of the fuzzy control space in Fig. 14 closely resembles that of the analytical models shown in Figures 7, 9, 11, albeit in a discretized form. Fuzzy rules are defined in terms of the values of $|T_m|$ and $|M_z|$ to infer $p_1$, $k_1$, and $q$. Tables I, II, and III reveal the rules for $p_1$, $k_1$, and $q$ respectively. The fuzzy implication is solved using the Mamdani inference method (see [52]), and the defuzzification process is carried out using the centroid method. Based on the membership functions and the rule base, the fuzzy control space is rendered for the fuzzy outputs in Fig. 14.

Following the computation of $p_1$, $k_1$, and $q$ in the fuzzy logic controller, $p$ and $k$ are determined based on $\alpha_1$, $\alpha_3$, and $\alpha_4$. Whilst $\alpha_2$ and $\alpha_5$ are excluded from consideration due to the constancy of the fuzzy membership functions, with the bounds confined to 0.5. Therefore, within a context akin to the analytical modeling approach, the values of $p$ and $k$ are established based on Algorithm 1 and Algorithm 2, subsequent to the inference of $p_1$ and $k_1$.

## V. SIMULATION RESULTS

The control architecture is implemented and validated in a co-simulation between Simulink/MatLab and the professional SCANeR™ Studio vehicle dynamics simulator. The vehicle parameters are given in Table IV. The developed approaches for torque allocation are integrated into the architecture and contrasted under identical testing conditions. In addition to our approach, we assess the performance of four alternative methods proposed by [37], including online optimization, offline optimization, and two strategies predicated on static and dynamic load distribution. The online and offline optimization strategies are executed based on instantaneous minimization of (3). These strategies are aptly labeled as follows: “continuous” for the continuous analytical models, “fuzzy” for the fuzzy logic strategy, “online” and “offline” for online and offline optimization respectively, and “constant” and “dynamic” for the static and dynamic load distribution respectively. For benchmarking energy savings, we use the reference strategy denoted as “classical,” in which the parameters $p, k, q, n$ are uniformly set to 0.5. This implies that $T_m$ and $M_z$ are uniformly distributed across the four wheels, with an equal split between braking and traction torques.

In order to have a better insight into the behavior of each approach, the following cost variables are introduced: 1) accumulated energy consumption $E_m$, 2) vehicle longitudinal acceleration $a_x$, 3) Tire Working-load Usage TWU (4), and 4) Stability Index SI (34) as defined in [46] where the coefficients are specified in [45].

$$SI = |2.49\dot{\beta} + 9.55\beta| \quad (34)$$

where the vehicle enters the instability region for SI $\ge$ 1. Hence the quantitative indicators are: 1) $max(E_m)$ for energy evaluation; $E_g(\%)$ (35) for energy savings, which represents the energy saving of the torque allocation strategies with
<table><caption>TABLE IV: Vehicle Parameters</caption><thead><tr><th>Symbol</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td><i>m</i></td><td>Vehicle's Total Mass</td><td>1286.4 [kg]</td></tr><tr><td><i>K</i><sub>t</sub></td><td>Tire stiffness coefficient</td><td>467000 [N/m]</td></tr><tr><td><i>C</i><sub>t</sub></td><td>Tire damping coefficient</td><td>500 [N.s/m]</td></tr><tr><td><i>t</i><sub>f</sub></td><td>Half front axle</td><td>0.773 [m]</td></tr><tr><td><i>t</i><sub>r</sub></td><td>Half rear axle</td><td>0.773 [m]</td></tr><tr><td><i>l</i><sub>f</sub></td><td>Distance between CG and the front axle</td><td>1.0385 [m]</td></tr><tr><td><i>l</i><sub>r</sub></td><td>Distance between CG and the rear axle</td><td>1.6015 [m]</td></tr><tr><td><i>I</i><sub>z</sub></td><td>Yaw moment of inertia</td><td>1970 [kg.m<sup>2</sup>]</td></tr><tr><td><i>g</i></td><td>Gravitational constant</td><td>9.81 [m/s<sup>2</sup>]</td></tr><tr><td><i>μ</i></td><td>Road adherence coefficient</td><td>for dry surface = 1</td></tr><tr><td><i>r</i></td><td>Wheel effective radius</td><td>0.30759 [m]</td></tr><tr><td><i>C</i><sub>f</sub></td><td>Front tire cornering stiffness</td><td>76776 [N/rad]</td></tr><tr><td><i>C</i><sub>r</sub></td><td>Rear tire cornering stiffness</td><td>76776 [N/rad]</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>Fig. 15: Vehicle trajectory and longitudinal velocity - sc1</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16: Energy consumption comparison - sc1</figcaption></figure>

respect to the reference classical one,

$$E_g = \frac{E_{m,\text{classical}} - E_m}{E_{m,\text{classical}}} \times 100 \quad (35)$$

where $E_m$ and $E_{m,\text{classical}}$ represent the energy consumption of the vehicle using the tested torque allocation strategy and the classical approach, respectively; 3) $\max(SI)$ for stability assessment; 4) a comfort index (CI) representing the smoothness of $a_x$ (image of jerk) which is defined as (36).

$$CI = \sum_{i=0}^{n-1} (a_x^{i+1} - a_x^i)^2 \quad (36)$$

The index CI is designed based on the effect of jerk profile on driving comfort, studied in [53]. The lateral acceleration $a_y$ is not included in CI since the speed profile ensures driving comfort based on $a_y$. Hence, CI directly captures the longitudinal force applied by the in-wheel motor, to ensure a focused analysis of longitudinal dynamics and their impact on driving comfort.

A case study is carried out to assess the vehicle performance across three different scenarios. The first scenario involves navigating double high-curvature turns, representing a challenging road geometry with sharp bends. This scenario was selected to examine how our methodology manages extreme curvature and speed transitions, focusing on energy efficiency, vehicle stability, and comfort under less-critical driving situations. The second scenario simulates a typical

urban environment with a mix of straight and curved roads, representing normal driving conditions. The third scenario features driving on roads with varying levels of adherence, such as slippery surfaces, specifically designed to test the method's stability and comfort under critical conditions where traction is compromised. Finally, the fourth scenario uses a real-world dataset to demonstrate the robustness and practical advantages of our method, showcasing its effectiveness and adaptability in real-life driving conditions.

### A. Scenario 1: Hard road geometry

The first scenario is executed on the track given in Fig. 15 with a reference velocity profile set to 60 km/h. The stability index of the vehicle for the distinct torque allocation approaches is shown in Fig. 17, and the TWU of each approach is presented in Fig. 17. SI serves as an indicator for assessing the overall stability of the vehicle, whereas $TWU_i$ ($\{i = 1, 2, 3, 4\}$) inspects the stability of each individual tire $i$, accounting for its slippage. It can be noticed that, at the instant where TWU of the online and offline strategies are excessive, their corresponding SI surpasses the threshold of 1, indicating an entry into an unstable region. Indeed, for this region, the stability controller generates the required yaw moment and restores vehicle stability. Nevertheless, the continuous and fuzzy strategies, as the dynamic strategy, effectively keep SI below 0.8, demonstrating their ability to maintain superior stability without the need for external intervention. The longitudinal acceleration of each strategy is depicted in Fig. 17. It is evident that the online strategy exhibits significant oscillations, a phenomenon that not only induces discomfort but also has the potential for adverse consequences on the in-wheel motors. The offline strategy reveals higher overshoot in some instances, whereas the continuous and fuzzy strategies consistently deliver a smoother and more comfortable drive. The quantitative indicators CI and $\max(SI)$ presented in Table V demonstrate that both the continuous and fuzzy strategies outperform the online and offline optimization methods.

Note that the values for $\max E_m$ in Table V are not provided for the online and offline strategies in some scenarios because the vehicle veered off the road in this scenario. Additionally, the absence of values for $E_g$ indicates that there were no energy savings in this scenario for the respective torque allocation strategy compared to the classical approach. The accumulated energy consumption of all the strategies is revealed
<figure><img src="image_21.png" /><figcaption>Fig. 17: Simulation results representing: Stability Index SI, Tire Working-load Usage, and longitudinal acceleration - sc1</figcaption></figure>

<table><caption>TABLE V: Cost variables (quantitative indicators) of the torque allocation strategies</caption><thead><tr><th rowspan="2">Scenario cases</th><th rowspan="2">Cost variables</th><th colspan="6">Cost variables of the torque allocation strategies in different scenarios</th></tr><tr><th>Constant</th><th>Dynamic</th><th>Offline</th><th>Fuzzy</th><th>Online</th><th>Continuous</th></tr></thead><tbody><tr><td rowspan="4">Scenario 1</td><td>Max E<sub>m</sub> (kJ)</td><td>342.575</td><td>340.077</td><td>298.5</td><td>294.216</td><td>275.306</td><td>260.655</td></tr><tr><td>E<sub>g</sub> (%)</td><td>2.094</td><td>2.808</td><td>14.691</td><td>15.915</td><td>21.319</td><td>25.506</td></tr><tr><td>CI (m<sup>2</sup>/s<sup>4</sup>)</td><td>1</td><td>0.661</td><td>1.893</td><td>0.895</td><td>2.247</td><td>1.211</td></tr><tr><td>Stability (Max SI)</td><td>0.776</td><td>0.744</td><td>1.291</td><td>0.737</td><td>1.083</td><td>0.746</td></tr><tr><td rowspan="4">Scenario 2</td><td>Max E<sub>m</sub> (kJ)</td><td>1033.56</td><td>1033.26</td><td>925.444</td><td>987.147</td><td>904.641</td><td>876.409</td></tr><tr><td>E<sub>g</sub> (%)</td><td>0.451</td><td>0.479</td><td>10.864</td><td>4.921</td><td>12.886</td><td>15.588</td></tr><tr><td>CI (m<sup>2</sup>/s<sup>4</sup>)</td><td>0.816</td><td>0.781</td><td>3.632</td><td>1.354</td><td>5.492</td><td>1.506</td></tr><tr><td>Stability (Max SI)</td><td>0.776</td><td>0.727</td><td>1.2</td><td>0.753</td><td>1.115</td><td>0.72</td></tr><tr><td rowspan="4">Scenario 3</td><td>Max E<sub>m</sub> (kJ)</td><td>931.573</td><td>896.077</td><td>-</td><td>910.115</td><td>-</td><td>822.384</td></tr><tr><td>E<sub>g</sub> (%)</td><td>10.24</td><td>13.65</td><td>-</td><td>12.306</td><td>-</td><td>20.76</td></tr><tr><td>CI (m<sup>2</sup>/s<sup>4</sup>)</td><td>1.067</td><td>1.016</td><td>> 100</td><td>2.252</td><td>5.278</td><td>2.045</td></tr><tr><td>Stability (Max SI)</td><td>0.776</td><td>0.757</td><td>19.468</td><td>0.806</td><td>3.978</td><td>0.803</td></tr><tr><td rowspan="4">Scenario 4</td><td>Max E<sub>m</sub> (kJ)</td><td>2308.69</td><td>2293.93</td><td>1832.98</td><td>1904.91</td><td>1829.9</td><td>1838.17</td></tr><tr><td>E<sub>g</sub> (%)</td><td>-</td><td>-</td><td>18.08</td><td>14.866</td><td>18.218</td><td>17.848</td></tr><tr><td>CI (m<sup>2</sup>/s<sup>4</sup>)</td><td>10.47</td><td>10.934</td><td>11.192</td><td>10.124</td><td>11.216</td><td>10.029</td></tr><tr><td>Stability (Max SI)</td><td>1.298</td><td>1.167</td><td>6.704</td><td>1.49</td><td>4.469</td><td>1.15</td></tr></tbody></table>

in Fig. 16. It is apparent that the highest energy consumption is marked by the classical strategy, while the continuous strategy exhibits the lowest energy consumption. Referring to Table V, the percentage of energy saved is determined by the relative energy change between the strategy in interest and the classical strategy. The fuzzy strategy presented a compromise between offline and online optimization, while the continuous strategy marked the highest energy saving with roughly 26%. It is important to note that, the optimization strategies are carried out to minimize the instantaneous power consumption of the motors. However, when employing a combination of traction and braking torques in line with velocity profile tracking, it is important to note that minimizing instantaneous power consumption does not necessarily equate to minimizing energy consumption over the entire time horizon. This elucidates the rationale behind the continuous strategy of exhibiting a superior energy economy.

## B. Scenario 2: Urban map environment

Fig. 19 depicts the urban map used for the second scenario. The vehicle velocity is observed to track the generated reference velocity profile $V_{x,des}$, as defined previously. Fig. 18 represent the Stability Index, Tire Working-load Usage, and vehicle longitudinal acceleration, while Fig. 20 shows the accumulated energy consumption of each strategy. The offline strategy, along with the online, revealed a higher magnitude of SI as a result of wheel slip reflected by TWU. In contrast,

the fuzzy strategy manifests its effectiveness in maintaining vehicle stability. Hence, although it revealed a lower energy economy than the online and offline strategies, the fuzzy strategy displays an overall superior level of stability. The continuous strategy, on the other hand, demonstrates a significant energy economy—with regard to the scenario representing daily normal driving—while upholding vehicle stability as the dynamic strategy. Fig. 18 depicts that, in contrast to the continuous and fuzzy strategies, the offline strategy exhibited notable overshoots with fluctuations. The magnitudes of these oscillations are more pronounced in online optimization, which subsequently deprives the vehicle of comfort. It is important to take into consideration the wheel slippage not only global stability. From Table V, our continuous strategy has revealed the lowest max(SI) and significantly enhanced CI compared to the online and offline strategies, while also maintaining the highest energy economy.

## C. Scenario 3: Road adherence variation

The third scenario is performed on the track in Fig. 19, similar to Scenario 2, with an identical velocity profile. However, unlike the initial two scenarios where road adherence is kept 1, scenario 3 comprises a continuous variation in road adherence $\mu$. The scenario is delineated as follows: adherence undergoes continuous variation, starting with mid-adherence at the outset, transitioning to a portion with low adherence, and ultimately escalating to a high-adherence portion (see Fig. 21).
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 18: Simulation results representing: Stability Index SI, Tire Working-load Usage, and longitudinal acceleration - sc2</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 19: Vehicle trajectory and velocity - sc2</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 20: Energy consumption comparison - sc2</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 21: Parameters $p_1, k_1, q$ function of $T_m, M_z, \mu$ of the continuous strategy - sc3</figcaption></figure>

The reduction in $\mu$ is directly proportional to the increment in TWU (4), subsequently leading to higher tire slippage and a reconfiguration of the stability envelope, denoted as $SI \ge SI_n$, where $SI_n < 1$. The exact value of $SI_n < 1$ is of minor relevance, as our primary aim is to illustrate the comparative ratio of values in SI between the continuous and fuzzy strategies in contrast to the online and offline approaches. The resulting Stability Index, Tire Working-load Usage, and the longitudinal acceleration of each strategy are depicted in Fig. 22. Whilst Fig. 21 reveals the output parameters $p_1, k_1, q$ of the analytical models as a variation of $T_m, M_z$, and $\mu$. The offline optimization exhibited dramatic deterioration behavior across all levels. Its longitudinal acceleration exceeds

$6 \text{ m/s}^2$ with significant oscillations. Notably, a stability index of SI = 19 coupled with TWU = 23 corresponds to an exceedingly critical scenario, posing a substantial risk of severe road accident. The stability index of the online strategy is noticed to exceed 1 multiple times, reaching a peak of $\approx 4$, an outcome that is highly unfavorable. Furthermore, the longitudinal acceleration associated with this strategy displays more major oscillations compared to the initial two scenarios, consequently exerting significant stress on the motors, not to mention driving discomfort. On the contrary, the continuous and fuzzy strategies revealed a superior performance in terms of both individual tire stability and global vehicle stability, in addition to highly comfortable driving (refer to Table V).

## D. Scenario 4: Real-world dataset

The final scenario unfolds on a track constructed from a real-world dataset. The dataset is a well-established dataset that has been employed in [53] and is commonly used for motion planning of autonomous vehicles. It includes real road information alongside a desired velocity profile as well as recorded data about surrounding vehicles. In our study, we focus exclusively on the real road information, as our objective
<figure><img src="image_28.png" /><figcaption>Fig. 22: Simulation results representing: Stability Index SI, Tire Working-load Usage, and longitudinal acceleration - sc3</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 23: Satellite image and map construction of the real trajectory dataset</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Fig. 24: Vehicle trajectory and velocity - sc4</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 25: Energy consumption comparison - sc4</figcaption></figure>

is not related to motion planning involving other vehicles. The simulations are conducted using a full nonlinear vehicle model adopted from [4]. The real track is shown in Fig. 23, and the vehicle trajectory is depicted in Fig. 24, along with the longitudinal velocity. The accumulated energy consumption of all the strategies is presented in Fig. 25, and the quantitative indicators for this scenario are given in Table V. It should be noted that the dataset waypoints are sparse, and thus the constructed form of the lane center-line could lead to a jerky motion. This justifies the elevated values of comfort index CI and stability index SI. As observed in Table V, although the energy savings of the continuous strategy are slightly lower than those of the online and offline methods, the continuous strategy significantly improves stability and enhances comfort. Specifically, the max(SI) values of 6.7 for the offline strategy and 4.4 for the online method indicate high instability in the vehicle, in contrast to the value of 1.15 for the continuous strategy, which can be effectively mitigated by the stability controller.

As we delve into more realistic real-world scenarios, it becomes crucial for the torque allocation methodologies to dynamically adjust to road-changing conditions. This includes dynamic load distribution during cornering and maintaining vehicle stability based on tire working-load usage. Additionally, the online construction of road maps (in the case of considering autonomous vehicles) can occasionally result in inaccuracies and sparse data points, as observed in this dataset, leading to jerky vehicle motion. This underscores the robustness of our approach and its practical applicability for on-road operations.

## E. Recap

The conducted scenarios have revealed the importance of the opted strategy for torque allocation. Within the framework of a multi-level control architecture, it has been demonstrated that the yaw moment, generated by the upper level, might not be effective in restoring vehicle stability in cases where tire slippage remains unaddressed at the lower level. Scenario 1 manifested that the fuzzy strategy revealed a compromise in energy consumption between offline and online optimization with better overall stability, in less-critical situations. It also exposed the major effectiveness of the continuous strategy in terms of stability and energy economy. Scenario 2 highlighted the remarkable energy economy of the continuous strategy
<table><caption>TABLE VI: Computational time required for calculating the allocation parameters *p*, *k*, *q*, *n* (s)</caption><thead><tr><th rowspan="2">Scenario cases</th><th colspan="4">Execution time for the allocation parameters computation (s)</th><th rowspan="2">Simulation time (s)</th></tr><tr><th>Offline</th><th>Continuous</th><th>Fuzzy</th><th>Online</th></tr></thead><tbody><tr><td>Scenario 1</td><td>0.208</td><td>0.243</td><td>42.15</td><td>105.83</td><td>43</td></tr><tr><td>Scenario 2</td><td>0.714</td><td>0.748</td><td>118.73</td><td>342.95</td><td>135</td></tr><tr><td>Scenario 3</td><td>0.697</td><td>0.755</td><td>121.284</td><td>330.64</td><td>135</td></tr><tr><td>Scenario 4</td><td>0.765</td><td>0.855</td><td>133.91</td><td>334.551</td><td>134</td></tr></tbody></table>

<figure><img src="image_32.png" /><figcaption>Fig. 26: Computational time required for the torque allocation methods (ms).</figcaption></figure>

in daily urban driving. The fuzzy strategy exhibited a lower energy economy than the offline and online strategies in exchange for vehicle stability. The third scenario disclosed the superior performance for the continuous and fuzzy strategies on the individual tire and overall vehicle stability in critical situations. The final scenario demonstrates significant improvements in the primary objectives of energy economy and stability, as well as the practical applicability and robustness of our approach using real-world datasets.

The overall execution time of the computation of the allocation parameters *p*, *k*, *q* has been recorded in all strategies and averaged by 10 runs. The simulations are conducted under an Intel Core i9-12950HX CPU 2.3-GHz laptop. Table VI depicts the recorded data in each scenario, where the simulation time refers to the total time of the test. One can notice that the online strategy does not operate in real-time on this hardware and demands specialized and costly equipment, and the fuzzy strategy may exhibit challenges in real-time operation on this hardware. On the other hand, the continuous strategy revealed an extremely fast computation of the parameters which is close to that of the offline strategy.

To further address the computational complexity in real-time, we have validated the proposed approaches using Simulink Real-Time (Fig. 26). The system operates at 500 Hz to accurately capture the intricate dependencies in the full nonlinear vehicle model. The average computational time at each time step (in ms) is:

* 2.8ms for the online strategy, with frequent oscillations exceeding 10ms at certain instants.

* 0.8ms for the fuzzy strategy, but occasionally spikes above 20ms.

* 0.0003ms and 0.00038ms for the offline and continuous approaches respectively.

This indicates that the proposed continuous torque allocation strategy is well-suited for real-time application, showing substantial improvement in computational efficiency over both the fuzzy and online strategies. In light of this, the continuous strategy exhibited a superior performance in terms of performance, stability, comfort, energy economy, and computational efficiency.

The developed rules can be applied to *p*, *k*, *q*, *n* using conventional control method such as Fuzzy Logic Control, similar to the application of our analytical modeling approach. The fuzzy torque allocation strategy serves as a benchmark to highlight the strengths of our analytical models. By comparing the fuzzy approach with our analytical models, we demonstrate that the analytical models are more robust, scalable, and computationally efficient. Additionally, the analytical models offer greater parametric adjustability and ease of tuning compared to the fuzzy approach. For instance, the load distribution ratios $\kappa_1, \kappa_2, \kappa$ are incorporated into the analytical models but cannot be directly applied to the fuzzy logic approach without incorporating adaptable membership functions and rules. Overall, the inclusion of the fuzzy logic approach serves to showcase its performance and demonstrate its potential as a method for torque allocation. It provides a valuable comparison, illustrating the enhanced trade-off between energy economy and stability that our analytical models offer.

Note that the proposed torque allocation method offers several advantages, including reduced computational complexity through explicit mathematical function formulation, balanced multi-objectives including energy economy, stability, and driving comfort, enhanced interpretability over learning-based methods, ease of implementation, scalability, and accessibility. This method can be generalized for over-actuated vehicles where torque allocation is needed for a generated total torque $T_m$ and a corrective yaw moment $M_z$. However, there are also potential disadvantages to consider. While the methodology can be generalized, it requires a delicate tuning phase of several parameters and thorough testing. Additionally, the current torque allocation methodology is formulated specifically for ground vehicles. To extend this approach to other types of vehicles, the analytical models should be adapted to the new problem formulation.

## VI. CONCLUSION

In this paper, we introduced a novel methodology for multi-objective torque allocation in over-actuated vehicles. By formulating analytical models based on established rules for coordinating multiple objectives, we achieved torque allocation that minimizes energy consumption and tire slippage while maintaining the overarching control architecture objectives. Additionally, we developed and compared a fuzzy logic-based torque allocation strategy with the analytical models. Our validation results underscore the advantages of analytical models in terms of stability, scalability, comfort, energy economy, and computational efficiency. As a perspective, future research will concentrate on the experimental validation of the proposed torque allocation method on real-world platforms, with the
goal of refining the models based on empirical data. Rapid control prototyping (RCP) and hardware-in-the-loop (HIL) experiments are planned to further validate the real-time performance of the proposed strategy. Key areas for future work include exploring the adaptability of the proposed method to various vehicle types and expanding the scope to incorporate fault-tolerant control strategies. Additionally, efforts will be directed towards generalizing our models for application in diverse vehicle types, such as aerial and agricultural vehicles, thereby broadening the applicability of our torque allocation methodology.

## REFERENCES

[1] T.-C. Lin, S. Ji, C. E. Dickerson, and D. Battersby, "Coordinated control architecture for motion management in adas systems," *IEEE/CAA Journal of Automatica Sinica*, vol. 5, no. 2, pp. 432-444, 2018.

[2] X. Cao, T. Xu, Y. Tian, and X. Ji, "Gain-scheduling lpv synthesis $h_\infty$ robust lateral motion control for path following of autonomous vehicle via coordination of steering and braking," *Vehicle System Dynamics*, vol. 61, no. 4, pp. 1-24, 2022.

[3] X. Li, N. Xu, K. Guo, and Y. Huang, "An adaptive smc controller for evs with four iwms handling and stability enhancement based on a stability index," *Vehicle System Dynamics*, vol. 59, no. 10, pp. 1509-1532, 2021.

[4] A. Chokor, R. Talj, M. Doumiati, A. Hamdan, and A. Charara, "A comparison between a centralized multilayer lpv/$h_\infty$ and a decentralized multilayer sliding mode control architectures for vehicle's global chassis control," *International Journal of Control*, vol. 95, pp. 1-32, 07 2020.

[5] W. Botes, T. R. Botha, and P. S. Els, "Real-time lateral stability and steering characteristic control using non-linear model predictive control," *Vehicle System Dynamics*, vol. 61, no. 4, pp. 1063-1085, 2022.

[6] Z. Zhang, H. Li, C. Wu, and Q. Zhou, "Finite frequency fuzzy $h_\infty$ control for uncertain active suspension systems with sensor failure," *IEEE/CAA Journal of Automatica Sinica*, vol. 5, no. 4, pp. 777-786, 2018.

[7] F. Tarhini, R. Talj, and M. Doumiati, "Dynamic and real-time continuous look-ahead distance for autonomous vehicles: an explicit formulation," *Vehicle System Dynamics*, pp. 1-27, 2023.

[8] M. Waqas and P. Ioannou, "Automatic vehicle following under safety, comfort, and road geometry constraints," *IEEE Transactions on Intelligent Vehicles*, vol. 8, no. 1, pp. 531-546, 2023.

[9] Y. M. Mok, L. Zhai, C. Wang, X. Zhang, and Y. Hou, "A post impact stability control for four hub-motor independent-drive electric vehicles," *IEEE Transactions on Vehicular Technology*, vol. 71, no. 2, pp. 1384-1396, 2022.

[10] G. Park and S. B. Choi, "A model predictive control for path tracking of electronic-four-wheel drive vehicles," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 11, pp. 11 352-11 364, 2021.

[11] W. Wang, Y. Zhang, C. Yang, T. Qie, and M. Ma, "Adaptive model predictive control-based path following control for four-wheel independent drive automated vehicles," *IEEE Transactions on Intelligent Transportation Systems*, vol. 23, no. 9, pp. 14 399-14 412, 2022.

[12] I. Ahmad, X. Ge, and Q.-L. Han, "Decentralized dynamic event-triggered communication and active suspension control of in-wheel motor driven electric vehicles with dynamic damping," *IEEE/CAA Journal of Automatica Sinica*, vol. 8, no. 5, pp. 971-986, 2021.

[13] Y. Liang, Y. Li, A. Khajepour, and L. Zheng, "Holistic adaptive multi-model predictive control for the path following of 4wid autonomous vehicles," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 1, pp. 69-81, 2021.

[14] G. Chen, J. Yao, H. Hu, Z. Gao, L. He, and X. Zheng, "Design and experimental evaluation of an efficient mpc-based lateral motion controller considering path preview for autonomous vehicles," *Control Engineering Practice*, vol. 123, p. 105164, 2022.

[15] H. Peng, W. Wang, Q. An, C. Xiang, and L. Li, "Path tracking and direct yaw moment coordinated control based on robust mpc with the finite time horizon for autonomous independent-drive vehicles," *IEEE Transactions on Vehicular Technology*, vol. 69, no. 6, pp. 6053-6066, 2020.

[16] J. Guo, J. Wang, Y. Luo, and K. Li, "Robust lateral control of autonomous four-wheel independent drive electric vehicles considering the roll effects and actuator faults," *Mechanical Systems and Signal Processing*, vol. 143, p. 106773, 2020.

[17] G. Wang, Z. Zuo, and C. Wang, "Robust consensus control of second-order uncertain multiagent systems with velocity and input constraints," *Automatica*, vol. 157, p. 111226, 2023.

[18] X. Zhou, Z. Wang, H. Shen, and J. Wang, "Robust adaptive path-tracking control of autonomous ground vehicles with considerations of steering system backlash," *IEEE Transactions on Intelligent Vehicles*, vol. 7, no. 2, pp. 315-325, 2022.

[19] W. Wang, T. Ma, C. Yang, Y. Zhang, Y. Li, and T. Qie, "A path following lateral control scheme for four-wheel independent drive autonomous vehicle using sliding mode prediction control," *IEEE Transactions on Transportation Electrification*, vol. 8, no. 3, pp. 3192-3207, 2022.

[20] Z. Liu, S. Cheng, X. Ji, L. Li, and L. Wei, "A hierarchical anti-disturbance path tracking control scheme for autonomous vehicles under complex driving conditions," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 11, pp. 11 244-11 254, 2021.

[21] X. Pan, B. Chen, S. Timothou, and S. A. Evangelou, "A convex optimal control framework for autonomous vehicle intersection crossing," *IEEE Transactions on Intelligent Transportation Systems*, vol. 24, no. 1, pp. 163-177, 2023.

[22] N. Ahmadian, A. Khosravi, and P. Sarhadi, "Driver assistant yaw stability control via integration of afs and dyc," *Vehicle System Dynamics*, vol. 60, no. 5, pp. 1742-1762, 2022.

[23] C. Liu, H. Liu, L. Han, W. Wang, and C. Guo, "Multi-level coordinated yaw stability control based on sliding mode predictive control for distributed drive electric vehicles under extreme conditions," *IEEE Transactions on Vehicular Technology*, vol. 72, no. 1, pp. 280-296, 2023.

[24] P. Wang, H. Liu, L. Guo, L. Zhang, H. Ding, and H. Chen, "Design and experimental verification of real-time nonlinear predictive controller for improving the stability of production vehicles," *IEEE Transactions on Control Systems Technology*, vol. 29, no. 5, pp. 2206-2213, 2021.

[25] A. K. Mansour Ataei and S. Jeon, "Model predictive control for integrated lateral stability, traction/braking control, and rollover prevention of electric vehicles," *Vehicle System Dynamics*, vol. 58, no. 1, pp. 49-73, 2020.

[26] W. Cui, N. Cui, T. Li, Z. Cui, Y. Du, and C. Zhang, "An efficient multi-objective hierarchical energy management strategy for plug-in hybrid electric vehicle in connected scenario," *Energy*, vol. 257, p. 124690, 2022.

[27] C. Jing, H. Shu, R. Shu, and Y. Song, "Integrated control of electric vehicles based on active front steering and model predictive control," *Control Engineering Practice*, vol. 121, p. 105066, 2022.

[28] J. Wang, S. Gao, K. Wang, Y. Wang, and Q. Wang, "Wheel torque distribution optimization of four-wheel independent-drive electric vehicle for energy efficient driving," *Control Engineering Practice*, vol. 110, p. 104779, 2021.

[29] A. Parra, D. Tavernini, P. Gruber, A. Sorniotti, A. Zubizarreta, and J. Pérez, "On nonlinear model predictive control for energy-efficient torque-vectoring," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 1, pp. 173-188, 2021.

[30] Q. Li, J. Zhang, L. Li, X. Wang, B. Zhang, and X. Ping, "Coordination control of maneuverability and stability for four-wheel-independent-drive ev considering tire sideslip," *IEEE Transactions on Transportation Electrification*, vol. 8, no. 3, pp. 3111-3126, 2022.

[31] L. Zhai, T. Sun, and J. Wang, "Electronic stability control based on motor driving and braking torque distribution for a four in-wheel motor drive electric vehicle," *IEEE Transactions on Vehicular Technology*, vol. 65, no. 6, pp. 4726-4739, 2016.

[32] H. Wei, Q. Ai, W. Zhao, and Y. Zhang, "Modelling and experimental validation of an ev torque distribution strategy towards active safety and energy efficiency," *Energy*, vol. 239, p. 121953, 2022.

[33] M. Dalboni, D. Tavernini, U. Montanaro, A. Soldati, C. Concari, M. Dhaens, and A. Sorniotti, "Nonlinear model predictive control for integrated energy-efficient torque-vectoring and anti-roll moment distribution," *IEEE/ASME Transactions on Mechatronics*, vol. 26, no. 3, pp. 1212-1224, 2021.

[34] J. Huang, Y. Liu, M. Liu, M. Cao, and Q. Yan, "Multi-objective optimization control of distributed electric drive vehicles based on optimal torque distribution," *IEEE Access*, vol. 7, pp. 16 377-16 394, 2019.

[35] H. Wei, N. Zhang, J. Liang, Q. Ai, W. Zhao, T. Huang, and Y. Zhang, "Deep reinforcement learning based direct torque control strategy for distributed drive electric vehicles considering active safety and energy saving performance," *Energy*, vol. 238, p. 121725, 2022.

[36] H. Deng, Y. Zhao, A.-T. Nguyen, and C. Huang, "Fault-tolerant predictive control with deep-reinforcement-learning-based torque distribution for four in-wheel motor drive electric vehicles," *IEEE/ASME Transactions on Mechatronics*, vol. 28, no. 2, pp. 668-680, 2023.
[37] F. Tarhini, R. Talj, and M. Doumiati, "Dual-level control architectures for over-actuated autonomous vehicle's stability, path-tracking, and energy economy," *IEEE Transactions on Intelligent Vehicles*, vol. 9, no. 1, pp. 287-303, 2024.

[38] H. Chen, P. Du, Y. Wang, D. Jin, and X. Lian, "Dynamic energy-efficient torque allocation algorithm for in-wheel motor-driven vehicle," *Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering*, vol. 234, no. 7, pp. 1815-1825, 2020.

[39] M. Tian, Q. Zhang, D. Tian, L. Jin, J. Li, and F. Xiao, "Pre-stability control for in-wheel-motor-driven electric vehicles with dynamic state prediction," *IEEE Transactions on Intelligent Vehicles*, vol. 9, no. 3, pp. 4541-4554, 2024.

[40] S. Barcellona, M. Barresi, and S. Colnago, "Torque allocation and energy management strategy for a multi-motor electric vehicle," in *2023 IEEE Vehicle Power and Propulsion Conference (VPPC)*, 2023, pp. 1-6.

[41] X. Hu, H. Chen, Z. Li, and P. Wang, "An energy-saving torque vectoring control strategy for electric vehicles considering handling stability under extreme conditions," *IEEE Transactions on Vehicular Technology*, vol. 69, no. 10, pp. 10787-10796, 2020.

[42] J. Liang, F. Wang, J. Feng, M. Zhao, R. Fang, D. Pi, and G. Yin, "A hierarchical control of independently driven electric vehicles considering handling stability and energy conservation," *IEEE Transactions on Intelligent Vehicles*, vol. 9, no. 1, pp. 738-751, 2024.

[43] X. Hu, P. Wang, Y. Hu, and H. Chen, "A stability-guaranteed and energy-conserving torque distribution strategy for electric vehicles under extreme conditions," *Applied Energy*, vol. 259, p. 114162, 2020.

[44] S. H. Kim and K.-K. Kim, "Model predictive control for energy-efficient yaw-stabilizing torque vectoring in electric vehicles with four in-wheel motors," *IEEE Access*, vol. 11, pp. 37665-37680, 2023.

[45] R. Rajamani, *Vehicle Dynamics and Control*, 2nd ed. Springer, 2012.

[46] F. Tarhini, R. Talj, and M. Doumiati, "Multi-objective control architecture for an autonomous in-wheel driven electric vehicle," *IFAC-PapersOnLine*, vol. 56, no. 2, pp. 11470-11476, 2023.

[47] SCANeR - AVSimulation, SCANeR Studio Simulator. [Online]. Available: https://www.avsimulation.com/scaner/

[48] Protean Electric PD18 In-wheel Electric Motor Datasheet, Protean Electric. [Online]. Available: https://www.proteanelectric.com/f/2018/05/Pd18-Datasheet-Master.pdf

[49] Y. Tian, X. Cao, X. Wang, and Y. Zhao, "Four wheel independent drive electric vehicle lateral stability control strategy," *IEEE/CAA Journal of Automatica Sinica*, vol. 7, no. 6, pp. 1542-1554, 2020.

[50] N. Guo, X. Zhang, Y. Zou, B. Lenzo, G. Du, and T. Zhang, "A supervisory control strategy of distributed drive electric vehicles for coordinating handling, lateral stability, and energy efficiency," *IEEE Transactions on Transportation Electrification*, vol. 7, no. 4, pp. 2488-2504, 2021.

[51] R. Jazar, *Vehicle dynamics: Theory and application*, second edition, 01 2014.

[52] F. Tarhini, R. Talj, and M. Doumiati, "Adaptive look-ahead distance based on an intelligent fuzzy decision for an autonomous vehicle," in *2023 IEEE Intelligent Vehicles Symposium (IV)*, 2023, pp. 1-8.

[53] F. Tarhini, R. Talj, and M. Doumiati, "Safe and energy-efficient jerk-controlled speed profiling for on-road autonomous vehicles," *IEEE Transactions on Intelligent Vehicles*, pp. 1-16, 2024.