A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors
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Article A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors Dongxin Xu , Yueqiang Han, Chang Ge, Longtao Qu, Rui Zhang and Guoye Wang * College of Engineering, China Agricultural University, Beijing 100083, China; xudongxin1996@163.com (D.X.); hanyueqiang123123@163.com (Y.H.); 18811738966@163.com (C.G.); 15839191315@163.com (L.Q.); zrabrr@126.com (R.Z.) * Correspondence: guoye@cau.edu.cn; Tel.: +86-010-6273-6856 Abstract: Vehicle drifting control has attracted wide attention, and the study methods are divided into expert-based and theory-based. In this paper, the vehicle drifting control was based on the vehicle drifting state characteristics. The vehicle drifting state parameters were obtained by the theory-based vehicle drifting motion mechanism analysis based on a nonlinear vehicle dynamics model, which was used to express the vehicle characteristics, together with the UniTire model, by considering the vehicle longitudinal, lateral, roll, and yaw motions. A vehicle drifting controller was designed by the model predictive control (MPC) theory and a linear dynamics model with the linearized expressions of nonlinear tire forces based on the consideration of measurable errors. The control targets were the vehicle drifting state parameters obtained by calculation, and the controller performance was proved by simulation in MATLAB/Simulink, demonstrating that the controller is good to realize vehicle drifting motions. The same target drifting motions were realized at different original states, which proved that the vehicle drifting control is possible with the designed controller. Keywords: vehicle drifting motion; model predictive control; vehicle drifting control Citation: Xu, D.; Han, Y.; Ge, C.; Qu, L.; Zhang, R.; Wang, G. A Model Predictive Control Method for 1. Introduction Vehicle Drifting Motions with With the rise of autonomous driving technology and the increasing demand for Measurable Errors. World Electr. Veh. vehicle motion performance, it is focused on the fact that the vehicle moves under drifting J. 2022, 13, 54. https://doi.org/ conditions which means slip ratios are large and tire forces reach the maximum. 10.3390/wevj13030054 The vehicle can accomplish the kind of drifting, fast turning that racing car drivers Academic Editor: Carlo Villante make in professional competitions, which provides a reference for the study. Professional Received: 10 February 2022 drivers accurately control rear-wheel slip ratios and front-wheel steering angles to realize Accepted: 15 March 2022 drift safely based on driving experiences, in which the vehicle has lost none of turning Published: 18 March 2022 control ability and rear-axle forces reach the maximum. According to the reference, it needs to adjust the steering angle when tire forces reach the maximum to accomplish the vehicle Publisher’s Note: MDPI stays neutral drifting motion control. However, it is extremely easy to case vehicle rollover accidents with regard to jurisdictional claims in during drifting motions. How to accomplish vehicle drifting motion control safely is the published maps and institutional affil- important content of the study. iations. The research methods are divided into two kinds, which are expert-based methods and the theory-based methods. Expert-based methods are applied gradually in the controller design, which teaches vehicle controllers to learn how to drive under limited conditions Copyright: © 2022 by the authors. based on the operating data of racing car drivers. High-speed driving data of drivers’ Licensee MDPI, Basel, Switzerland. behavior in circular cornering were collected by a measuring system and are available free This article is an open access article online in [1], which provides references for similar studies. Most expert-based method distributed under the terms and studies designed controllers by learning vehicle operational drifting motions as professional conditions of the Creative Commons drivers based on neural networks (NNs), such as references [2–4], which can achieve Attribution (CC BY) license (https:// autonomous drifting motions accurately and safely. Expert-based methods can be applied creativecommons.org/licenses/by/ to not only drifting control, such as papers [5,6], which can realize autonomous driving 4.0/). without operational data of drivers. However, expert-based methods in drifting control World Electr. Veh. J. 2022, 13, 54. https://doi.org/10.3390/wevj13030054 https://www.mdpi.com/journal/wevj
World Electr. Veh. J. 2022, 13, 54 2 of 17 need so much operational data of drivers to retrain the controller if replacing different vehicles that it is very difficult for most research institutes and universities. Theory-based methods are applied to analyze drifting motion mechanisms and to design controllers based on theoretical models, which can be independent of the operational data of drivers. To analyze the drifting motion mechanism, vehicle main status parameters need to be measured to maintain the vehicle equilibria in drifting motions. The vehicle can accomplish a circle drifting motion with constant state parameters including the velocity, sideslip angle, and yaw rate. Most studies obtained constant drifting parameters based on the equilibrium and stability analysis of a 3-DOF (degree of freedom) model with the neglect of rollovers, such as reference [7]. In addition, most controllers were designed to realize vehicle drifting motions in steady states based on the turning analysis of a 3-DOF model including longitudinal, lateral, and yaw motions, such as reference [8]. A mixed open-loop and closed-loop control strategy was presented based on the linear–quadratic regulator (LQR) theory to perform a transient drifting-corning trajectory in [9]. According to steady-state condition calculation, an LQR controller was designed to accomplish vehicle drifting motions in [10], and the LQR is the most commonly used in controller design. The problems of autonomous drifting and stabilization around an equilibrium state were studied and simulated in a 3D (three dimensions) car racing simulator in [11]. Based on the vehicle equilibrium analysis by phase portrait, a torque-vectoring-based control strategy was proposed to help the driver in high-sideslip maneuvers, namely in vehicle drifting conditions, in [12]. A model-free adaptive control algorithm was used to design a steady-state drifting controller based on the analysis of the vehicle drifting dynamics in [13]. Model predictive control (MPC) is widely used in the automotive industry in [14]. The MPC theory was applied to design an open-loop controller to achieve accurate vehicle drift to the parking position in [15], but the open-loop control system lacks the self-correction ability. A robust controller based on LMIs (linear matrix inequality) was designed to realize the vehicle drifting motions by considering uncertain disturbances in [16], which requires a greater computer because of the computational complexity. Although there are the T-S fuzzy model and the LPV method, which can reduce the computational complexity of robust controllers such as reference [17], there are so many variable parameters in the controller model in this paper and [16] that these methods are not applicable. Considering the computational complexity and the self-correction ability, the controller is a close-loop controller based on the MPC theory with measurable errors. From the above analysis, the paper studies how to realize the vehicle drifting motions by analyzing the vehicle drifting motion mechanism and designing a suitable controller based on the MPC theory. The vehicle dynamics model was established by considering longitudinal, lateral, roll, and yaw motions and rolling safety with the nonlinear tire model UniTire. The vehicle drifting motion mechanism was analyzed to obtain the vehicle drifting state parameters by the theory-based method, as shown in a previous study [16]. The paper designs a drifting controller based on the MPC theory with measurable errors to accomplish the optimal tracking problem of target drifting motions, and the controller was proved by MATLAB/Simulink simulations. 2. Vehicle Dynamics Model The vehicle dynamics model can express the vehicle characteristics, which is the basis of the drifting mechanism analysis and the drifting controller design. This section describes the vehicle drifting dynamic characteristics by a vehicle nonlinear dynamics model with the UniTire model, which is a nonlinear tire model to express the tire dynamics characteristics during complex wheel motions. 2.1. Vehicle Dynamics Equations The vehicle drifting analysis was based entirely on an applicable vehicle dynamics model, which also took the place of the real car in simulation and played an important role in the controller design. The vehicle dynamics model was used to describe the dynamics
role in the controller design. The vehicle dynamics model was used to describe the dy- namics characteristics of the real car with simplification. The longitudinal and lateral ve- hicle velocities were variational, and the vehicle was rotational around the vertical di- rection and was dangerous because of the rotation around the longitudinal direction. To World Electr. Veh. J. 2022, 13, 54 3 of 17 ensure rolling safety, there was no one side tire off the ground during the vehicle drifting motions in this paper. The 4-DOF vehiclecharacteristics nonlinearofdynamics model the real car with was established simplification. to describe The longitudinal and lateralthe vehi- vehicle velocities were variational, and the vehicle was rotational around the vertical direction cle longitudinal, lateral,and yaw, and rolling was dangerous motions because witharound of the rotation the nonlinear tiredirection. the longitudinal modelTo inensure Figure 1. The chassis coordinate system rolling is expressed safety, there was no one sidebytirethe off x-y-z coordinate the ground system, during the vehicle themotions drifting vehicle in this paper. longitudinal direction is expressed by the x axis, and the vehicle lateral direction is ex- The 4-DOF vehicle nonlinear dynamics model was established to describe the vehicle pressed by the y axis, as shown in longitudinal, [16].yaw, lateral, Theandcoordinate rolling motions systems of the front with the nonlinear andin rear tire model Figuretires 1. The chassis coordinate system is expressed by are expressed by the xtf-ytf and the xtr-ytr coordinate systems, respectively, and the tire the x-y-z coordinate system, the vehicle longitudinal direction is expressed by the x axis, and the vehicle lateral direction is ex- revolution directions are expressed pressed by asthe by the y axis, xtf-axis shown in [16].and the xtr-axis. The coordinate systems of the front and rear tires are expressed by the xtf -ytf and the xtr -ytr coordinate systems, respectively, and the tire revolution directions are expressed by the xtf -axis and the xtr -axis. (a) (b) Figure 1. Vehicle dynamics model: Figure (a)dynamics 1. Vehicle perspective view; model: (a) (b) plan perspective view;view. (b) plan view. The vehicle dynamics equations are as shown in Equations (1)–(4) to describe vehicle The vehicle dynamics equations dynamics areinas characteristics shown Figure 1: in Equations (1)–(4) to describe vehi- cle dynamics characteristics in. Figure 1: . . mv cos β − mv sin β γ + β + mb hb ψγ = Fx f cos δ f − Fy f sin δ f + Fxr − Fd , (1) ̇ cos − sin ( + . ̇ ) + ℎ ̇ = . cos .. − sin + − , (1) mv sin β + mv cos β γ + β − mb hb ψ = Fx f sin δ f + Fy f cos δ f + Fyr , (2) . .. Iż γ − Ixz ψ = l f ̈ Fx f sin δ f + Fy f cos δ f − lr Fyr , (3) ̇ sin + cos ( + ) − ℎ = sin + cos + , (2) .. . . . . Ix ψ − Ixz γ − mb hb v sin β + v cos β γ + β = mb ghb sin ψ − Kφ ψ − Cφ ψ, (4) ̇ −m is the where ̈ vehicle = ( sin mass, mb + vehicle is the cos )mass body − (sprung , mass), β is the vehicle (3) sideslip angle, γ is the vehicle yaw rate, ψ is the vehicle roll angle, δ f is the front-wheel steering angle, v is the vehicle velocity, l f and lr are the distances from gravity center to ̈ − ̇ − the ℎ front and rear axles, respectively, ḣ is the height of the gravity center from the ̇ roll (4) ( ̇ sin + cos ( + b)) = ℎ sin − − , axis, Ix , Iz , and Ixz are moments of inertia with respect to the roll and yaw axes, Fx f , Fy f , Fz f , Fxr , Fyr , and Fzr are the longitudinal, lateral, and vertical forces at front and rear tire, where is the vehiclerespectively, mass, K φ is the and Cφ vehicle body mass are the suspension (sprung roll stiffness mass), and roll damping is the vehi- coefficients, cle sideslip angle, is the vehicle respectively, Fd = 0.5ρyaw v cos β) is the a Cd A f (rate, 2 is aerodynamic the vehicle dragroll and ρ a , Cd , andisA fthe force,angle, front-wheel steering angle,are the air density, is thethevehicle aerodynamic drag coefficient, velocity, andand the frontal are thearea of the vehicle, distances from respectively. gravity center to the frontConsideringand reartheaxles, respectively, load transfer caused by the ℎ lateral is the height force, ofand the front- therear-axle gravity normal loads are expressed as: center from the roll axis, , , and are moments of inertia with respect to the roll and yaw axes, , , , , , and are Fz f +the Fzr −longitudinal, mg = 0 lateral, and vertical , (5) forces at front and rear tire, respectively, f and l F − l F r zr + ∑ F h = 0 are the suspension roll stiffness zf x g and roll damping coefficients, respectively, = 0.5 ( cos )2 is the aerodynam- ic drag force, and , , and are the air density, the aerodynamic drag coefficient, and the frontal area of the vehicle, respectively.
shown as Equation (1).
The tire velocities along the wheel’s longitudinal and lateral axe
rear wheels are given as follows:
World Electr. Veh. J. 2022, 13, 54
= cos cos + ( sin + )4sin
of 17
{ ,
= − cos sin + ( sin + ) cos
where h g is the height of the gravity center, and ∑ Fx is the vehicle longitudinal force shown
as Equation (1).
= cos
The tire velocities along the wheel’s longitudinal{ and lateral axes of the front
, and rear
wheels are given as follows: = sin −
where , , v v cos β cos
x f ,=and δare
f + v the
sin βlongitudinal
+ l f γ sin δ f and lateral velocitie
, (6)
rear wheel centers, vy frespectively.
= −v cos β sin δ f + v sin β + l f γ cos δ f
v xr = v cos β
2.2. Tire Force vyr = v sin β − lr γ
, (7)
whereUniTire,
v x f , vy f , v xra unified
, and nonlinear
vyr are the longitudinaltire model,
and lateral can ofcalculate
velocities longitudin
the front and rear
wheel centers,
forces respectively.
and overturning, rolling resistance, and aligning tire moments
vehicle dynamics simulation and control under complex wheel motion
2.2. Tire Force
this paper,
UniTire, the tire nonlinear
a unified model was applied
tire model, to calculate
can calculate longitudinal
longitudinal and late
and lateral tire
forces and overturning, rolling resistance, and aligning tire moments and be applied to
the neglect of tire moments. With the consideration of longitudinal an
vehicle dynamics simulation and control under complex wheel motion inputs in [18]. In
tions, thethe
this paper, tire
tirecoordinate system
model was applied is described
to calculate longitudinalin
andFigure 2,forces
lateral tire andwith
the tire m
the neglect of tire moments. With the consideration of longitudinal and lateral tire motions,
asthefollows. The index = , was adopted to denote the front and r
tire coordinate system is described in Figure 2, and the tire model is described as
tively,
follows.toThesimplify
index i = fexpressions.
, r was adopted to denote the front and rear axles, respectively, to
simplify expressions.
Figure 2. Tire coordinate system.
Figure 2. Tire coordinate system.
The longitudinal and lateral slip ratios, Sxi and Syi and the normalized longitudinal,
lateral, and combined slip ratios, φxi , φyi , and φi , at each tire are defined as [19]:
The longitudinal and lateral slip ratios, and and the no
xi ) / ( ωi rei ) , , and , at each t
dinal, lateral, and combined i rei − vratios,
Sxi = (ωslip
, (8)
Syi = −vyi /(ωi rei )
[19]:
φxi = (Kxi Sxi)/(µ xi Fzi )
= − )⁄( )
φyi = Kq / µyi(Fzi ,
yi Syi (9)
{
φ = φ 2+φ 2
i xi yi = − ⁄( )
,
= ( )⁄( )
= ( )⁄( ),
̅ ⁄ , =
= ̅
⁄ , (10)
where ̅ is the normalized resultant force at the tire.
Satisfactory with the boundary condition of the physical tire model in [20], the
World Electr. Veh. J. 2022, 13, 54 5 of 17
semi-physical expression of the UniTire tire model is described as:
2 2 1 3
where ωi is ̅the
= − −
1 −rotation
wheel
−( +12)
angular velocity, rei is the wheel effective rolling radius,
Kxi and Kyi are the longitudinal slip and cornering
stiffness of the tire, respectively, and
̅ and lateral
the=longitudinal
µ xi and µyi are
friction coefficients between tire and road
2
surface, respectively. √( )2 + , (11)
In the simplified physical tire model as [19], the longitudinal and lateral forces are
described as:
= ̅
Fxi = µ xi Fzi Fi φxi /φi , F2yi = µyi2Fzi Fi φyi /φi , (10)
√( ) +
where Fi
{
is the normalized resultant force at the tire.
Satisfactory with the boundary condition of the physical tire model in [20], the semi-
where is the curvature factor of the combined slip resultant force, and is the
physical expression of the UniTire tire model is described as:
modification factor to express the variationtrend of the resultant force.
2 −( E2 + 1 ) φ 3
To obtain a more accurate tire description,
1 − e−φi − Eφi (11)
Fi =Equation was
12 i applied to the vehicle
F = µ F F q λd φxi
dynamics model in simulations. In addition, toxi simplify xi zi i the drifting
(λd φxi )2 +φyi 2 ,
motion analysis, the
(11)
simplified longitudinal and lateral tire forces
were employed.
Fyi = µyi Fzi Fi
q
φyi The utilization of the fric-
2 2
(λd φxi ) +φyi
tion coefficient at each tire stays the same at the ultimate value in drifting, so that the
force on each tire reaches
where its
E is maximum. Combining
the curvature factor Equations
of the combined (8)–(10),
slip resultant force, the
and λlongitudinal,
d is the modifica-
tion factor to express the variation
lateral, and resultant forces on each tire are derived as: trend of the resultant force.
To obtain a more accurate tire description, Equation (11) was applied to the vehicle
dynamics model in simulations. In addition, to simplify the drifting motion analysis, the
2
=simplified
( )⁄( and
longitudinal lateral
), =
tire √ were
forces + 2 =
employed. The utilization
,
(12)
of the friction
coefficient at each tire stays the same at the ultimate value in drifting, so that the force on
where is the friction
eachcoefficient
tire reaches itsbetween
maximum.the Combining
tire and Equations (8)–(10),
the road the longitudinal, lateral, and
surface.
resultant forces on each tire are derived as:
q
2.3. Roll Safety Analysis
Fxi = K S F
xi xi yi / K S
yi yi , Fi = Fxi 2 + Fxi 2 = µi Fzi , (12)
Vehicle wheels lift
whereoffµifrom the ground
is the friction more
coefficient often
between the than not
tire and the as a surface.
road result of large roll-
ing motion under drift conditions. The vehicle has a risk of rollover, when the vertical
2.3. Roll Safety Analysis
force on one side wheel equals zero, otherwise, there is no risk [21–23]. Therefore, the
Vehicle wheels lift off from the ground more often than not as a result of large rolling
vehicle roll model is motion
classified
underinto
drifttwo conditions:
conditions. before
The vehicle has a and
risk ofafter thewhen
rollover, wheelthe lift-off, as
vertical force
shown in Figure 3a,b.on one side wheel equals zero, otherwise, there is no risk [21–23]. Therefore, the vehicle
roll model is classified into two conditions: before and after the wheel lift-off, as shown in
Figure 3a,b.
(a) (b)
Figure 3. Roll dynamics model: (a) before the wheel lift-off; (b) after the wheel lift-off. The vehicle
postures with gray broken lines express the vehicle motions without rolling.
̈ − ̇ − ℎ ( ̇ sin + cos ( + ̇ )) = ℎ sin −
cos , (13)
2
where and are moments of inertia with respect to the roll and yaw axes after
wheel lift-off, respectively.
World Electr. Veh. J. 2022, 13, 54 6 of 17
Combined with Equations (13) and (4), the relational expression between the safe
roll angle, the safe roll rate, and the safe roll angular acceleration in critical states can be
obtained. The roll dynamics expression after the wheel lift-off in Figure 3b is shown as the follow:
.. .
. . tb
3. Vehicle Drifting Based on
Ixψ ψModel Predictive
− Ixzψ γ − mh g v sin β Control
+ v cos β γ + β = mgh g sin ψ − mg
2
cos ψ, (13)
This section designs
whereaIxψ controller
and Ixzψ arebased onofthe
moments model
inertia predictive
with respect control
to the roll and yawtheory towheel
axes after re-
alize drifting motions. The respectively.
lift-off, vehicle model is nonlinear, and the control model is linear to
Combined
simplify the controller design. with Equations (13) and (4), the relational expression between the safe
roll angle, the safe roll rate, and the safe roll angular acceleration in critical states can
The control system was described as a state-space representation with a linearized
be obtained.
tire model to optimize the controller design. The linearized tire model was obtained
based on the UniTire3.model,
Vehicle Drifting Based on Model Predictive Control
which is similar to the linearized lateral tire force as shown
This section designs a controller based on the model predictive control theory to realize
in [24]. Based on Section 2.2, the longitudinal tire force mainly relates to the longitudinal
drifting motions. The vehicle model is nonlinear, and the control model is linear to simplify
slip ratio, while the lateral tire design.
the controller force mainly relates to the slip angle, when the vertical
The control
force, the tire velocity, and the friction system was described are
coefficient as a state-space
invariant.representation
The linearizedwith a expres-
linearized
tire model to optimize the controller design. The linearized tire model was obtained based
sions of the longitudinal and lateral tire forces are derived as Equation (14)
on the UniTire model, which is similar to the linearized lateral tire force as shown in [24].
Based on Section 2.2, the longitudinal tire force mainly relates to the longitudinal slip ratio,
while the lateral tire force mainly relates to the slip angle, when the vertical force, the
the
tire velocity, and ≈ friction ̂ ( − ̂are
̃ = coefficient + ̂ The linearized expressions of the
) invariant.
{ , (14)
longitudinal and lateral ̂ ( are
≈ ̃ tire=forces ̂ ) +as ̂Equation
derived
−
(14)
(
where the linearized expressions are based Fon xi ≈approximation
Fexi = Ĉxi (κi − κ̂i ) +points,
F̂xi is the TYDEX
F ≈ Feyi = Ĉyi (αi − α̂i ) + F̂yi , (14)
̃ ̃
longitudinal slip ratio, and are the approximate values of the longitudinal tire
yi
force and the lateral tire
whereforce,
the linearized ̂ , ̂ , are
and ̂ ,expressions ̂ , based
̂ , and ̂ are thepoints,
on approximation known κi isslip ratio,
the TYDEX
longitudinal slip ratio, F
exi and Feyi are the approximate values of the longitudinal tire force
slip angle, slip stiffness, cornering stiffness, longitudinal force, and lateral force of the
and the lateral tire force, and κ̂i , α̂i , Ĉxi , Ĉyi , F̂xi , and F̂yi are the known slip ratio, slip angle,
approximate point, respectively. The relationship between the slip angle and the ap-
slip stiffness, cornering stiffness, longitudinal force, and lateral force of the approximate
proximate value of the lateral
point, tire force
respectively. and the between
The relationship relationship the slip between
angle and thethe slip ratiovalue
approximate andof
the approximate value theof the tire
lateral longitudinal
force and the tire force are
relationship betweenshown in ratio
the slip Figure 4. approximate value
and the
of the longitudinal tire force are shown in Figure 4.
(a) (b)
Figure 4. The UniTire tire model
Figure 4. Thewith
UniTireantire
affine
modelapproximation: (a) the longitudinal
with an affine approximation: tire force
(a) the longitudinal ̂ α̂;i ;
at at
tire force
(b) the lateral tire force (b) ̂ .lateral tire force at κ̂i .
at the
Combing Equations (1)–(4) and (14), the vehicle continuous-time state space represen-
Combing Equations
tation(1)–(4)
is shownand
as: (14),
the
.
vehicle continuous-time state space repre-
x(t) = Ac x(t) + Bcu u(t) + Bcd d(t)
sentation is shown as: y t =C x t
, (15)
() c ()
World Electr. Veh. J. 2022, 13, 54 7 of 17 h . iT h iT where x(t) = v, β, γ, ψ, ψ , u(t) = δ f , κ f , κr , and y(t) are the state vector, the input h iT vector, and the output vector, respectively, d(t) = dδ , d f , dr is the measurable error vector, and the state-space vehicle model is suitable for this study though the nonlinear transformation with the Jacobian matrices Ac , Bcu , and Bcd . The discrete-time model of the above continuous-time model is shown as: x(k + 1) = Ad x(k) + Bdu u(k) + Bdd d(k ) , (16) y ( k ) = Cc x ( k ) RT RT where Ad = eAc Ts , Bdu = 0 s eAc τ dτ ·Bcu , and Bdd = 0 s eAc τ dτ ·Bdd with sampling time Ts . Based on the fundamental of MPC, the latest measured value is the initial condition and the predictive horizon and control horizon are Nu and Np while predictive horizon Nu is less than or equal to control horizon Np . The cost function to be minimized is designed as: N N −1 J( k ) = ∑i=p1 Qy (i)(y(k + i) − ye (k + i))2 + ∑i=u0 Qu (i )(u(k + i ) − ue (k + i ))2 , (17) where ye (k + i ) and ue (k + i ) are the target values of y(k + i ) and u(k + i ), respectively. The predictive output of the vehicle dynamic system is derived as: Cc Bdd y( k + 1) Cc Ad y( k + 2) Cc Ad 2 Cc Ad Bdd + Cc Bdd .. .. .. . . . ∑iN=u0−1 Cc Ad i Bdd C A Nu y(k + Nu ) c d = .. x(k)+ .. d(k)+ .. . . . ∑ Nu + j−1 C A i B y(k + Nu + j) Cc Ad Nu + j i =0 c d dd .. .. .. . . . Cc Ad Np N p −1 y k + Np ∑i=0 Cc Ad i Bdd } } } Y( k ) CY BYd Cc Bdu 0 ··· 0 0 (18) Cc Ad Bdu Cc Bdu ··· 0 0 .. .. .. .. .. . . . . . Cc Ad Nu −1 Bdu Cc Ad Nu −2 Bdu ··· Cc Ad Bdu Cc Bdu u( k ) .. .. .. .. .. u( k + 1) . . . . . , j .. . Cc Ad Nu + j−1 Bdu Cc Ad Nu + j−2 Bdu Cc Ad j+1 Bdu ∑ Cc Ad i Bdu ··· u(k + Nu − 1) i =0 .. .. .. .. .. } . . . . . U( k ) Np − Nu Cc Ad Np −1 Bdu Cc Ad Np −2 Bdu · · · Cc Ad Np − Nu +1 Bdu ∑ Cc Ad i Bdu i =0 } BYu where 0 ≤ j ≤ Np − Nu . In order to calculate the minimum value of the cost function, the following is proposed: SQy (Y(k) − Ye (k)) z( k ) = , (19) SQu (U(k) − Ue (k)) T and Ue (k) = [ue (k), · · · , ue (k + Nu − 1)] T . where Ye (k) = ye (k + 1), · · · , ye k + Np Integrating Equations (18) and (19), the following equation can be obtained as:
World Electr. Veh. J. 2022, 13, 54 8 of 17
SQy (CY x(k ) + BYu U(k ) + BYd d(k) − Ye (k )) SQy BYu SQy (Ye (k) − CY x(k) − BYd d(k))
z( k ) = = U( k ) − , (20)
SQu (U(k) − Ue (k)) SQu SQu Ue (k )
} }
Az Bz
In addition, the cost function in Equation (17) becomes J(k) = z(k) T z(k ). The optimal
solution minJ(k) is derived as:
minz(k) T z(k ), z(k) = Az U(k ) − Bz , (21)
The extremum condition of Equation (21) is shown as:
dz(k) T z(k)
= 2Az T (Az U(k) − Bz ) = 0 (22)
dU(k)
The matrix Az is the non-zero, and thus, the second derivative is as follows:
d2 z(k ) T z(k)
= 2Az T Az > 0. (23)
dU(k)2
Therefore, the solution of Equation (21) is the minimum solution of Equation (17), and
the sequence of the optimal input vectors can be obtained:
−1
U ( k ) = Az T Az Az T Bz . (24)
The weight coefficients in the matrices SQy and SQu take an important part in the
solution of Equation (24), and the real-time computational burden depends on the values of
predictive horizon Nu and control horizon Np . Through enough computation, the solution
can be obtained.
Equation (24) is the solution of the open-loop system. This study designs a close-loop
controller, and the first sample u(k) of Equation (24) can be used to compute the optimal
steering angle and slip ratios to control vehicle drifting motions, which is equivalent to
predictive horizon Nu equaling to one with the real-time computing.
4. Results
The driving performance of the vehicle drifting is shown as follows by analyzing the
motion mechanism, and the satisfying performance of the MPC controller was expressed
as the simulation results in MATLAB/Simulink.
The vehicle main parameters are shown in Table 1. The vehicle drifting motion
mechanism was analyzed in [16]. Considering the absolute maximum steering angle was
0.7 rad in practice, the velocity limitations are shown in Figure 5, which was obtained by the
calculation to describe the maximum and minimum velocities of the vehicle in steady-state
drifting motions in different road conditions and suggested the vehicle drifting motion can
improve the vehicle dynamics performance.
World Electr. Veh. J. 2022, 13, 54 9 of 17 Table 1. The main parameters of the vehicle in simulations. Parameter Symbol Unit Value m kg 1126.7 mb kg 1111 Iz kg·m2 2038 lf m 1.265 lr m 1.335 hb m 0.136 hg m 0.518 Af m2 1.6 World Electr. Veh. J. 2022, 13, x FOR PEER REVIEW g N/kg 9.8 µ = µ f = µr 0.65 Figure 5.5.The Figure Thevelocity limitations velocity in steady-state limitations drifting motions in steady-state in motions drifting different road conditions.road in different The conditi vehicle can drift at greater velocities with better road conditions. vehicle can drift at greater velocities with better road conditions. The main vehicle state parameters in drifting motions were obtained by analyzing Table 1. Themechanism. the drifting main parameters of thefour There were vehicle in of groups simulations. vehicle drifting state parameters in Table 2 to reveal the characteristics of the vehicle drifting state parameters. Group (a) was Parameter one group of FigureSymbol Unit when the circle equaled 12 m 5 to suggest the maximum velocity Value in the steady-state drifting motion, and group (b) waskg one group in Figure 5 to suggest1126.7 the minimum velocity. Groups (c) and (d) were randomly kgchosen among all analysis results1111 as target groups in simulations to verify the controller performance. All TYDEX longitudinal tires in groups (a), (b), (c), and slip ratios of the rear kg·m 2 (d) were obtained by matching the 2038 and the UniTire model. theoretical tire forces m 1.265 m 1.335 Table 2. The main state parameters of the vehicle drifting. ℎ m 0.136 ℎR (m) v (m/s) β (rad) mγ (rad/s) δf (rad) κr 0.518 (a) 12 8.73 −0.06 m2 0.73 0.06 0.03 1.6 (b) 12 6.82 −0.90 0.91 −0.7 0.95 (c) 12 8.05 −0.51 N/kg0.77 −0.26 0.61 9.8 (d) = 20= 10.5 −0.49 0.6 −0.26 0.51 0.65 (e) 42 15 −0.51 0.41 −0.26 0.53 The main vehicle state parameters in drifting motions were obtained by an the drifting mechanism. There were four groups of vehicle drifting state param Table 2 to reveal the characteristics of the vehicle drifting state parameters. Group one group of Figure 5 to suggest the maximum velocity when the circle equaled
World Electr. Veh. J. 2022, 13, 54 10 of 17 All original states were uniform linear motions with different velocities to verify the controller performance. The 6 m/s original velocity was lower than the target in group (c), the second original velocity equaled 10 m/s and was higher than the target in group (c), and the original velocity equaling 8.1 m/s was very close to the target in group (c), of which the main intention was to demonstrate that the designed controller can be used to realize the vehicle drifting motions with different no-limited original states. The last two simulations with 9 m/s and 13.7 m/s original velocities for the target groups (d) and (e), respectively, were to verify that the controller can realize more than just a target drifting motion. The simulation results, shown in Figures 6–8, suggested that the same target vehicle drifting motion (group (c) in Table 2) can be realized with different no-limited original states and all final drifting motions were in the vicinity of the target. Combined with the simulation result in Figure 9, it suggested that the controller can realize different target drifting motions with no-limited original states. Figure 10 suggests that the controller can realize a larger target drifting motion with no-limited original states. The steering angles and the TYDEX longitudinal slip ratios of the front and rear tires were the controlled variables in this study. Considering vehicle practical working conditions, there were amplitude and gain limitations of the control parameters including the steering angle and tire slip ratios in the controller design. The final steering angles in Figure 6d, Figure 7d, Figure 8d, Figure 9d, and Figure 10d and the final TYDEX longitudinal slip ratios of the rear tires in Figure 6e, Figure 7e, Figure 8e, Figure 9e, and Figure 10e were close to the targets in small error ranges. The TYDEX longitudinal slip ratios of the front tires, in order to better realize the vehicle drifting motions, were arranged in the range from 0.15 to −0.15, where the longitudinal tire force was remarkably correlated linearly with the TYDEX longitudinal slip ratio, and the corresponding simulation results are shown in Figures 6e, 7e, 8e, 9e and 10e. Based on the controlled variables, the vehicle main states parameters, including the velocities in Figures 6a, 7a, 8a, 9a and 10a, the sideslip angles in Figures 6b, 7b, 8b, 9d and 10d, and the yaw rates in Figures 6c, 7c, 8c, 9c and 10c, were close to the targets within the little bit larger margins of errors than the controlled variables, because the UniTire semi-empirical model was not identical to the theoretical analysis. The radiuses of the vehicle motion curves approach to the targets in Figures 6f, 7f, 8f, 9f and 10f, and the vehicle motion curves are shown in Figures 6g, 7g, 8g, 9g and 10g where the coordinated origins are the starting points. Because all vehicle original states parameters had big differences with the target states in values including the sideslip angle and yaw rate and the target control parameters including the steering angle, the controller took some time to realize drifting motions, and there were some fluctuations in the results, which all finally converged nearby the targets. According to the simulation results, the vehicle drifting characteristics were revealed, and the designed controller was tested by realizing vehicle drifting motions which started with uniform linear motions. The vehicle drifting motion can take full advantage of the vehicle dynamics performance, and the vehicle drifting control study will be combined with automatic driving to improve driving safety in further studies.
According to the simulation results, the vehicle drifting characteristics were re- vealed, and the designed controller was tested by realizing vehicle drifting motions which started with uniform linear motions. The vehicle drifting motion can take full ad- vantage of the vehicle dynamics performance, and the vehicle drifting control study will beJ.combined World Electr. Veh. 2022, 13, 54 with automatic driving to improve driving safety in further studies. 11 of 17 Veh. J. 2022, 13, x FOR PEER REVIEW 11 of 17 (a) (b) (c) (d) (e) (f) (g) Figure 6. The simulation Figure 6.result with the 6 result The simulation m/s original with thevelocity and thevelocity 6 m/s original target and groupthe(c): (a) the target ve-(c): (a) the group locity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip ratios of the front ratios and rear tires; of the front(f) andthe radius rear of the tires; (f) theradius motion curve; of the (g) curve; motion the motion (g) thecurve motionofcurve the ve- of the vehicle. hicle. All continuous All continuous lines suggest the simulation results, while the dashed lines suggest thetar- lines suggest the simulation results, while the dashed lines suggest the targets. gets.
Figure 6. The simulation result with the 6 m/s original velocity and the target group (c): (a) the ve- locity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion curve of the ve- hicle. All continuous lines suggest the simulation results, while the dashed lines suggest the tar- World Electr. Veh. J. 2022, 13, 54 12 of 17 gets. eh. J. 2022, 13, x FOR PEER REVIEW 12 of 17 (a) (b) (c) (d) (e) (f) (g) Figure 7. The simulation Figure 7.result with the 10 The simulation m/swith result original the 10velocity and velocity m/s original the targetandgroup (c): group the target (a) the(c): (a) the velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip slip ratios of the front ratiosand rear of the tires; front and(f) the rear radius tires; ofradius (f) the the motion curve; curve; of the motion (g) the(g)motion curve the motion of the curve of the vehicle. vehicle. All continuous All continuous lines suggest the simulation results, while the dashed lines suggest thethe lines suggest the simulation results, while the dashed lines suggest targets. targets.
Figure 7. The simulation result with the 10 m/s original velocity and the target group (c): (a) the velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the World Electr. Veh. J. 2022, 13, 54 targets. 13 of 17 eh. J. 2022, 13, x FOR PEER REVIEW 13 of 17 (a) (b) (c) (d) (e) (f) (g) Figure 8. The simulation Figure 8.result with the 8.1 The simulation m/s result original with the 8.1velocity and the m/s original target velocity andgroup (c): (a) the target the(c): (a) the group velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip slip ratios of the front ratiosand rear of the tires; front and(f) thetires; rear radius ofradius (f) the the motion curve; curve; of the motion (g) the(g)motion curve the motion of the curve of the vehicle. vehicle. All continuous lines suggest All continuous the simulation lines suggest results, the simulation while results, whilethe thedashed linessuggest dashed lines suggestthethe targets. targets.
Figure 8. The simulation result with the 8.1 m/s original velocity and the target group (c): (a) the velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the World Electr. Veh. J. 2022, 13, 54 14 of 17 targets. eh. J. 2022, 13, x FOR PEER REVIEW 14 of 17 (a) (b) (c) (d) (e) (f) (g) Figure 9. The simulation Figure 9.result with the 9result The simulation m/s with original the 9velocity and velocity m/s original the targetandgroup (d): group the target (a) the(d): (a) the velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip slip ratios of the front ratiosand rear of the tires; front and(f) thetires; rear radius ofradius (f) the the motion curve; curve; of the motion (g) the(g)motion curve the motion of the curve of the vehicle. vehicle. All continuous All continuous lines suggest the simulation results, while the dashed lines suggest thethe lines suggest the simulation results, while the dashed lines suggest targets. targets.
Figure 9. The simulation result with the 9 m/s original velocity and the target group (d): (a) the velocity; (b) the sideslip angle; (c) the yaw rate; (d) the steering angle; (e) the TYDEX longitudinal slip ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the World Electr. Veh. J. 2022, 13, 54 targets. 15 of 17 eh. J. 2022, 13, x FOR PEER REVIEW 15 of 17 (a) (b) (c) (d) (e) (f) (g) Figure 10. The simulation Figure 10.result with the 13.7 The simulation resultm/s original with the 13.7velocity and the m/s original target velocity andgroup (e): (a) the target groupthe(e): (a) the velocity; (b) the sideslip velocity;angle; (b) the(c) the yaw sideslip rate; angle; (d) yaw (c) the the rate; steering angle; (d) the (e) angle; steering the TYDEX longitudinal (e) the TYDEX longitudinal slip slip ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion ratios of the front and rear tires; (f) the radius of the motion curve; (g) the motion curve of the curve of the vehicle. vehicle. All continuous lines suggest All continuous the simulation lines suggest results, the simulation while results, whilethe the dashed linessuggest dashed lines suggest thethe targets. targets. 5. Conclusions How to control the vehicle moves under drifting conditions, which means slip ratios are large and tire forces reach the maximum, is a topical issue in the vehicle dynamics
World Electr. Veh. J. 2022, 13, 54 16 of 17 5. Conclusions How to control the vehicle moves under drifting conditions, which means slip ratios are large and tire forces reach the maximum, is a topical issue in the vehicle dynamics control. The vehicle drifting driving is too hard to be realized for most people, so there are expert-based methods and theory-based methods to study the vehicle drifting control. This paper designed a controller based on the MPC theory with the linearized expressions of nonlinear longitudinal and lateral tire forces to realize the vehicle drifting motion analyzed by the vehicle drifting motion mechanism based on the nonlinear dynamics model, together with the UniTire model, by considering vehicle longitudinal, lateral, roll, and yaw motions. The vehicle main state parameters were calculated by MATLAB/Simulink simulations. In addition, the designed controller performance was proved by realizing target drifting motions with different original states, which suggested the vehicle drifting motions can be accomplished by the designed controller to make vehicle drifting easier for most people. Author Contributions: Methodology, D.X.; investigation, Y.H., C.G., L.Q. and R.Z.; writing—original draft preparation, D.X.; writing—review and editing, G.W. and D.X.; project administration, G.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by “The National Natural Science Foundation of China (grant number: 51775548”) and “The National Natural Science Foundation of the Nei Monggol Autonomous Region (grant number: 2020MS05059”). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The study did not report any data. All data in this study can be obtained by calculation. Acknowledgments: The author(s) would like to thank all the researchers taking part in the experi- ment from College of Engineering, China Agricultural University. Conflicts of Interest: The authors declare no conflict of interest. References 1. Katzourakis, D.I.; Velenis, E.; Abbink, D.A.; Happee, R.; Holweg, E. Race-Car Instrumentation for Driving Behavior Studies. IEEE Trans. Instrum. Meas. 2012, 61, 462–474. [CrossRef] 2. Spielberg, N.A.; Brown, M.; Kapania, N.R.; Kegelman, J.C.; Gerdes, J.C. Neural network vehicle models for high-performance automated driving. Sci. Robot. 2019, 4. [CrossRef] [PubMed] 3. Acosta, M.; Kanarachos, S. Teaching a vehicle to autonomously drift: A data-based approach using Neural Networks. Knowl.-Based Syst. 2018, 153, 12–28. [CrossRef] 4. Cai, P.; Mei, X.; Tai, L.; Sun, Y.; Liu, M. High-Speed Autonomous Drifting with Deep Reinforcement Learning. IEEE Robot. Au-Tom. Lett. 2020, 5, 1247–1254. [CrossRef] 5. Jin, X.; Yin, G.; Wang, J. Robust fuzzy control for vehicle lateral dynamic stability via Takagi-Sugeno fuzzy approach. In Proceedings of the American Control Conference (ACC), Seattle, WA, USA, 24–26 May 2017. 6. Nguyen, A.T.; Rath, J.; Guerra, T.M.; Palhares, R.; Zhang, H. Robust Set-Invariance Based Fuzzy Output Tracking Control for Vehicle Autonomous Driving Under Uncertain Lateral Forces and Steering Constraints. IEEE Trans. Intell. Transp. Syst. 2021, 22, 5849–5860. [CrossRef] 7. Hindiyeh, R.Y.; Gerdes, J.C. Equilibrium analysis of drifting vehicles for control design. In Proceedings of the ASME 2009 Dynamic Systems and Control Conference, Los Angeles, CA, USA, 12–14 October 2009. 8. Velenis, E.; Katzourakis, D.; Frazzoli, E.; Tsiotras, P.; Happee, R. Steady-state drifting stabilization of RWD vehicles. Control Eng. Pract. 2011, 19, 1363–1376. [CrossRef] 9. Zhang, F.; Gonzales, J.; Li, K.; Borrelli, F. Autonomous Drift Cornering Mixed Open-loop and Closed-loop Control. IFAC Pap. 2017, 50, 1916–1922. [CrossRef] 10. Khan, M.A.; Youn, E.; Youn, I.; Wu, L. Steady State Drifting Controller for Vehicles Travelling in Reverse Direction. In Proceedings of the 15th International Bhurban Conference on Applied Sciences and Technology (IBCAST), Islamabad, Pakistan, 9–13 January 2018. 11. Zubov, I.; Afanasyev, I.; Gabdullin, A.; Mustafin, R.; Shimchik, I. Autonomous Drifting Control in 3D Car Racing Simulator. In Proceedings of the 9th International Conference on Intelligent Systems (IS), Funchal, Portugal, 25–27 September 2018.
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