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Page 2 of 15 Chen et al. Complex Eng Syst 2023;3:8 I http://dx.doi.org/10.20517/ces.2022.50

1. INTRODUCTION
Worldwide,energycrisesandenvironmentalpollutionarethefundamentalreasonsdrivingthedevelopmentof
electric vehicles (EVs) [1,2] . For any type of vehicle, vehicle handling stability, which determines driving safety,
is a significant performance measure. Among various types of EVs, four-wheel independent drive (4WID)
EVs come with four in-wheel motors that can simultaneously reduce energy consumption and increase vehicle
stability [3,4] . Given that the use of independent in-wheel motors facilitates independent installation of drive
systems, this approach allows each wheel to regulate its driving force, which provides more possibilities to
enhance vehicle performance in terms of maneuverability and stability [5,6] . However, because of the time-
varying nonlinear characteristics of vehicles, 4WID EV stability and effective torque distribution algorithms
remain suboptimal.

The greatest advantage of 4WID vehicles is that the four hub motors can be controlled independently, meaning
that the motors can work in their respective high efficient range and optimal attachment range to the extent
possible. Given that vehicle stability is essential for traffic safety, many scholars have focused on the key issues
related to vehicle stability. In this context, the understeer coefficient in quasi-steady-state maneuvers has been
studied extensively, with a focus on typical lateral dynamics controls, such as active front steering and yaw
moment control [7–9] . Lenzo et al. derived a relationship between the understeer coefficient and yaw moment,
and they obtained an apparently surprising result at low speeds: the rear-wheel-drive (RWD) architecture
provided the highest level of understeer, and the yaw moment due to the longitudinal forces of the front tires
was significant under high lateral accelerations and steering angles [10] . Analogously, the concept of relaxed
static stability (RSS) was proposed and utilized to guide the configuration of the 4WID configuration and
to design the overall 4WID vehicle structure with the aim of improving vehicle stability” without affecting
the intended meaning [11] . In Ref. [12] , the influences of the electric motor’s output power limit, road friction
coefficient, and torque response of each wheel on stability control were elucidated. Chen et al. used a double-
layer control algorithm to determine the desired yaw moment and longitudinal forces of four tires with the
aim of improving vehicle stability [13] . The authors of [14] added a layer to the aforementioned algorithm [13]
to judge whether a vehicle is in a stable state by implementing the phase plane method before the two layers.
For stability control of 4WID vehicles, sliding mode control and its improved version are the most commonly
used methods [15,16] . An integral sliding mode control (ISMC) approach was proposed for 4WID vehicles to
generate differential drive force to assist the steering process in the absence of adequate lateral tire force [17] .
However, sliding mode control tends to oscillate near the sliding surface. Peng et al. proposed a 7-degree-of-
freedom (DoF) model-predictive control (MPC) method to improve vehicle stability [18] . However, in their
case, discrete MPC linearization was slightly rough, which may lead to inaccurate results.

Although a few researchers have drawn attention toward this knowledge, the problems of ensuring vehicle
stability and torque allocation still cannot be solved quickly and accurately for the following reasons: (1) 4WID
EVs are highly nonlinear and time-varying system, and the use of simple processes will reduce the system
accuracy; (2) The four in-wheel motors are not decoupled and need to be coordinated simultaneously; and (3)
Unpredictability of the iteration steps in the traditional optimization algorithm may lead to a scenario where
the torques applied to the four tires do not reach the respective optimal values in real time. In Ref. [16] , the
minimum total adhesion rate algorithm was used to allocate torque to each wheel. However, this method may
leadtolocaloptimizationorlargedifferencesintheadhesionratesofdifferenttires. Forthisreason,wepropose
a hierarchical control algorithm that includes a nonlinear-MPC-based upper algorithm for obtaining the total
longitudinal force and direct yaw moment, and an equal-adhesion-rate-rule-based lower torque allocation
algorithm. The main contributions of this study are as follows: (1) an extended 3-DOF reference vehicle model
is built that can be integrated with the traditional 2-DOF reference vehicle model; (2) Exact expressions are
derived for the first-order derivatives of TV-MPC; and (3) A torque allocation algorithm based on the equal
adhesion rate rule of the bottom-level controller is proposed to ensure full utilization of the adhesion rate. The
structure of the hierarchical control algorithm proposed herein is illustrated in Figure 1.