Design of slip-based traction control system for EV and validation using co-simulation between Adams and Matlab/Simulink

Abstract

This paper focuses on controlling traction for a four-wheel electric vehicle by using the longitudinal slip ratio control technique. By keeping the slip ratio value inside an optimal limit, it can be ensured that the maximum driving force is obtainable by increasing the friction force between tire and road. The usefulness of the sliding mode control method is to provide robust performance from the parameter uncertainties at different road conditions. A control law is formulated based upon the Lyapunov stability approach to assure the sliding action. To satisfy the robustness, a vehicle model is made in Matlab, and it is simulated based on various parameter values. The slip ratios at different parameter values are plotted for open loop and closed loop. Then considering the vehicle kinematics and dynamics, a 3D CAD model using Catia is developed. Then exporting the model to Adams to use it as a plant model for the vehicle, co-simulation has been achieved by keeping the slip-based traction controller in Matlab/Simulink. Matlab/Simulink and Adams/View simulation validate the proposed method.

Keywords

Co-simulation electric vehicle slip ratio sliding mode control traction control tire–road friction force

1. Introduction

In the road transport field, internal combustion (IC) engine-based vehicles have become unmitigated leaders. In the last few years, due to the expansion of powerful and durable batteries and higher oil prices, rapid growth of electric vehicles (EV) has been initiated to decrease air pollution and gain an encouraging achievement in comparison to IC engined vehicles. There are different structures of EVs according to the propulsion system. The most significant fact in EVs is that no mechanical differential is needed and the developed torque from each motor is controlled autonomously. Therefore, this permits selection of positive torque and negative torque in two independent wheels to simplify protection algorithms such as an electronic stability program (ESP).^1,2 The traction control is developed for these types of the car based on the individual wheel so that in crucial stages individual motors can achieve torque independently and the friction between the wheel and the road reaches its highest value.³

Traction control systems have existed for a long time, but largely for vehicles with an IC engine. In recent times, various works linked to the traction control for EVs have been done. Traction control design built upon maximum transmissible torque estimation without the knowledge of the tire–road condition is described by Hori et al.^4,5 In the study by Johansen et al.,⁶ traction control depends on the evaluation of the road surface condition. An adaptive slip reversal controller for EVs based on backstepping is reported by Ting and Lin.⁷ Gain scheduling-based wheel slip ratio control for a brake system with electro-mechanical brakes has been presented.⁸ Traction control considering vehicle stability is mentioned in studies by Sakai et al. and Shino et al.^9,10 An algorithm for traction control of EVs based on four independent wheel drives has been reported.¹¹ An estimate for slip ratio without knowledge of vehicle speed is proposed,¹² and single wheel-based longitudinal traction control is also described.¹³ An antilock braking system (ABS) incorporating active suspension control has been proposed,¹⁴ and an optimal slip ratio controller based on the LQ method has also been revealed.¹⁵ Slip ratio control based upon state feedback for ABS is described by Yi et al.¹⁶

In the last decade, rigorous research on the assessment of the road surface condition has been carried out. With the use of slip measurements, the road surface conditions can be determined.¹⁷ Ono et al. described a road surface condition evaluation technique featuring use of the wheel’s rotating speed.¹⁸ An estimate of road surface condition in real time is reported.¹⁹ Road condition estimation by designing a driving force observer has also been mentioned.²⁰ In the study by Sui and Johansen,²¹ a horizon computation of the friction force between tire and road is proposed. Model following control (MFC) has been described,²² as has a model predictive PID method (MP-PID).²³ In the steady road surface condition when parameters like vehicle mass, road coefficient, etc. are known, both methods show excellent performance. Due to the constant changing road condition, formation of a robust controller is important. Sliding mode control (SMC) performs better in terms of robustness with nonlinearities and uncertainties in parameters.

In the case of SMC, the system is forced to attain a predesigned surface and then slides along in it. The main aim of SMC is to achieve a sliding surface in finite time and then to stay on it. Chattering incidents occur due to the rapid change of the control inputs in the variable structure. The chattering phenomenon is avoided by using the boundary layer technique. During the design, the integral part is incorporated in the control law to reduce the steady-state error and enhance the transient response. Li and Kawabe described a slip elimination technique using SMC theory,²⁴ while in Lyapunov stability has also been described.²⁵ Li and Kawabe represented controller robustness performance with changing mass of the vehicle as a major parameter and the idea of constructing series road using different parameters.²⁴ Control algorithms designed for a nonlinear system using the Matlab/Simulink environment and their validation techniques have beeb reported.²⁶ The basic knowledge of multi-body design software Catia V5 and part design using it is reported by Zamani.²⁷ Assembling the design parts like wheel and chassis to make a complete model has also been demonstrated.²⁸ A model export technique from Catia to Adams is descrived by Cheraghpour et al.²⁹ Adams/View interfacing for beginners and creating parts using it is reported by Junwen and Liaozhong.³⁰ Model export from Adams to Simulink, co-simulation procedure and controlling of damper forces have also been documented.³¹ Development and experimental validation technique for an in-wheeled EV has also been described.³²

The rest of this paper is organized as follows. Sections 2 and 3 define EV modeling and formulation of the control law. In Section 4, simulation results in Matlab are obtained. In Section 5, the vehicle CAD model is designed in Catia and modified in Adams. Co-simulation is setup between Adams and Simulink in Section 6. In the next section, co-simulation results are explained. Finally, a conclusion is drawn in the last section.

2. Traction control system

A traction control system helps to maximize the grasp and balance of the vehicle by estimating wheel speed upon the roadway during the acceleration phase. In different road conditions, when skidding occurs, it helps the car to overcome that phase and accelerate smoothly by applying brakes or reducing engine power. To design the traction control system, the EV has to be modeled.

2.1. EV Modeling

Three coordinate frames are specified to define the EV, i.e., body-fixed frame (BFF), wheel fixed frame (WFF), and inertial frame (IF). The origin of BFF is the mass center of the vehicle. In Figure 1, $x^{B}$ , $y^{B}$ , and $z^{B}$ denote longitudinal, lateral, and vertically upward direction and roll, pitch, and yaw are symbolized as $ϕ$ , $θ$ , and $ψ$ , respectively. The wheel center is taken as the origin of the WFF.

Figure 1.

BFF, IF, and WFF shown for a vehicle.

For four WFF, the wheel is denoted by superscripts $w$ . The frontal or tail, left or right side of the vehicle are symbolized as $a = {f, r}$ , $b = {l, r}$ . The side slip angle is defined as the angle between a spinning wheel’s actual direction of travel and the direction of the velocity of the car. The side slip angle $μ$ is represented as follows³³:

ξ_{ab} = \tan^{- 1} (\frac{V_{ytab}}{V_{xtab}})

(1)

μ_{ab} = ϵ_{b} - ξ_{ab}

(2)

where $ξ_{ab}$ and $ϵ_{b}$ are described as the angle between velocity vectors and angle given by the driver. $V_{ytab}$ and $V_{xtab}$ are $y$ and $x$ components of the velocity of the car.

The geometrical summation of slip ratio and side slip angle is the resulting slip³³:

τ_{r, ab} = \sqrt{τ_{xab}^{2} + τ_{yab}^{2}}

(3)

where $τ_{xab}$ and $τ_{yab}$ are the slip ratio and side slip. In Table 1, $V_{tab}$ is the velocity of car, $r$ is effective tire radius, and $ν_{ab}$ is the wheel angular velocity.

Table 1.

Explanation of side slip and slip ratio.

	Braking	Driving
	$(ν_{ab} r \cos (μ_{ab}) \leq V_{tab})$	$(ν_{ab} r \cos (μ_{ab}) > V_{tab})$
Slip ratio $τ_{xab}$	$\frac{ν_{ab} r \cos (μ_{ab}) - V_{tab}}{V_{tab}}$	$\frac{ν_{ab} r \cos (μ_{ab}) - V_{tab}}{ν_{ab} r \cos (μ_{ab})}$
Side slip $τ_{yab}$	$\frac{ν_{ab} r \cos (μ_{ab})}{V_{tab}}$	$\tan (μ_{ab})$

Torque balance around the $z^{B}$ -axis and force balancing around the $x^{B}$ and $y^{B}$ directions can be represented as follows³⁴:

\begin{matrix} m (\overset{\cdot\cdot}{x} - \overset{\cdot}{ψ} \overset{\cdot}{y}) = {F_{xfr} + F_{xfl} + F_{xrr} + F_{xrl}} \\ - {F_{ro, fr} \cos (δ_{r}) + F_{ro, fl} \cos (δ_{l}) \\ + F_{ro, rr} + F_{ro, rl}} - F_{d} \end{matrix}

(4a)

\begin{matrix} m (\overset{\cdot\cdot}{y} - \overset{\cdot}{ψ} \overset{\cdot}{x}) = {F_{yfr} + F_{yfl} + F_{yrr} + F_{yrl}} \\ - {F_{ro, fr} \sin (δ_{r}) + F_{ro, fl} \sin (δ_{l})} \end{matrix}

(4b)

\begin{matrix} J_{z} \overset{\cdot\cdot}{ψ} = \frac{l_{w}}{2} {F_{xfr} + F_{xfl} + F_{xrr} + F_{xrl}} \\ + l_{f} (F_{yfr} + F_{yfl}) + l_{r} (F_{yrr} + F_{yrl}) \end{matrix}

(4c)

where $m$ is the mass of the vehicle, $J_{z}$ is the mass moment of inertia around the $z^{B}$ axis, $F_{xab}$ is the acting friction on wheel $ab$ in the vehicle longitudinal axis, $F_{yab}$ is the acting friction on wheel $ab$ in the lateral vehicle axis, $F_{ro, ab}$ is rolling resistance, and $F_{d}$ is the aerodynamic drag force. From Figure 2 wheel dynamics of one wheel model can be obtained as follows³⁵:

J_{ta} . {\overset{\cdot\cdot}{ν}}_{ab} = T_{mb} - T_{bab} sign (ν_{ab}) - c . ν - r . F_{xtab}

(5)

where $J_{ta}$ is wheel inertia of front or rear wheel, $T_{mb}$ is driving torque, $T_{bab}$ is the braking torque, $c$ is the viscous friction coefficient, and $F_{xtab}$ is the acting friction on wheel $ab$ in the $x^{tab}$ direction. From the Burckhardt tire model, the friction coefficient is modeled as³⁶:

η_{ab} = d_{1} (1 - e^{- d_{2} . τ_{r, ab}}) - d_{3} . τ_{r, ab}

(6)

where $d_{1}$ , $d_{2}$ , and $d_{3}$ are dependent coefficients for the road upper surface. The aerodynamic drag and rolling resistance can be taken as³⁷:

F_{d} = \frac{ϱ}{2} A e_{D} {\overset{\cdot\cdot}{x}}^{2} sign (\overset{\cdot}{x})

(7)

F_{ro, ab} = e_{ro} F_{zab}

(8)

where $A$ is the frontal vehicle area, $ϱ$ is the density of air, $e_{D}$ is drag coefficient, $F_{zab}$ is normal force and $e_{ro}$ is the rolling coefficient. The value of $F_{xab}$ , $F_{yab}$ and $F_{xtab}$ derived as³⁸:

F_{xab} = \frac{F_{zab} η_{ab}}{τ_{r, ab}} {τ_{xab} \cos (δ_{b} - μ_{ab}) - τ_{yab} \sin (δ_{b} - μ_{ab})}

(9)

F_{yab} = \frac{F_{zab} η_{ab}}{τ_{r, ab}} {τ_{xab} \cos (δ_{b} - μ_{ab}) + τ_{yab} \sin (δ_{b} - μ_{ab})}

(10)

F_{xtab} = \frac{F_{zab} η_{ab}}{τ_{r, ab}} {τ_{xab} \cos (μ_{ab}) + τ_{yab} \sin (μ_{ab})} .

(11)

Figure 2.

Free body diagram of one wheel.

2.2. Model simplification

To control the longitudinal slip ratio of the vehicle, it is required to limit the motion in the longitudinal direction only. This is stated as $\overset{\cdot}{y} = \overset{\cdot\cdot}{y} = \overset{\cdot}{ψ} = \overset{\cdot\cdot}{ψ} = δ_{b} = μ_{ab} = 0$ . Hence the equations of dynamic model turn into the following:

m {\overset{\cdot}{V}}_{c} = 4 F_{x} - F_{ro} - F_{d} .

(12)

For longitudinal motion $F_{xab} = F_{xtab} = F_{zab} . η_{ab}$ . Friction force resides in the orientation of $τ_{xab}$ , $τ_{yab} = 0$ . Then the model is reduced by presuming that the viscous wheel torques are inconsiderable in comparison to the tractive torques. Thus $c . ν_{ab} = 0$ . As the tractive forces generated at each wheel are equal, single wheel dynamics is enough to derive the tractive forces:

ν_{ab} = \frac{T_{mb}}{J_{ta}} - \frac{r . F_{xab}}{J_{ta}}

(13)

Also, $F_{zab} = m . g$ , where $g$ denotes the gravitational constant. The ratio between the force during driving and normalized tire force gives friction coefficient $η (f (d), τ)$ , which is a function of road coefficient $d$ and wheel slip $τ$ . The slip ratio can be given as:

τ = \frac{V_{t} - V_{c}}{V_{t}}

(14)

where $V_{t} = r . ν$ is the wheel velocity and $V_{c}$ is the vehicle velocity. At $τ = 0$ , there is no skidding in between the touching point of tire and road and $τ = 1$ shows complete skidding of the wheel.

Figure 3 demonstrates the relation between friction coefficient $η_{x}$ and slip ratio $τ_{x}$ under divergent road conditions. It can be observed that $η_{x}$ reaches a maximum value when $τ_{x}$ is between $0.1$ and $0.2$ for all road surface conditions, except for the cobblestone dry road condition as it achieves its maximum at $0.4$ . At those points, the driving force also obtains the highest value. The friction coefficient reduces after the crossing of the peak point in the slip ratio curve. In order to obtain the maximum driving force, the slip ratio needs to be controlled in that range, where the maximum friction coefficient is available.

Figure 3.

Slip ratio versus friction coefficient.

2.3. Calculation of reference slip ratio

From Equation (6) we can say that the friction coefficient depends on the predefined function $f (d)$ and $τ$ . In the case of asphalt dry road surface conditions, the values of $d_{1}$ , $d_{2}$ , and $d_{3}$ are evaluated from Figure 3 as follows:

d_{1} = 1.2799

d_{2} = 22.999

d_{3} = 0.525 .

After inserting these values, the equation becomes the following:

η (f (d), τ) = 1.2799 (1 - \exp^{- 22.999 τ}) - 0.525 τ .

(15)

The function $η (f (d), τ)$ attains the maximum value at a fixed value of $τ$ , which can be procured as below. Suppose the following;

\frac{d η (f (d), τ)}{d τ} = 0,

(16)

After putting Equation (15) into this, we obtain the following:

τ = \frac{\ln (0.01783)}{- 22.999}

(17)

τ = 0.175 .

(18)

For the asphalt dry road condition, when $τ = 0.175$ is met the maximum value of driving force is reached. Hence, for slip ratio, the reference value $τ_{r}$ is set to $0.175$ in case of the asphalt dry road condition. The value of the reference slip ratio for other road surface conditions are evaluated in the same procedure:

For Asphalt Wet τ_{r} = 0.1326

Concrete Dry τ_{r} = 0.1705

Cobblestone Dry τ_{r} = 0.4002

Cobblestone Wet τ_{r} = 0.1415

Snow τ_{r} = 0.0595

Ice τ_{r} = 0.0398 .

To obtain the maximum driving force, the slip ratio needs to be retained at the reference value at all times. But the road condition is always changing, so when the weight of loads such as passengers and luggage changes, the mass of the vehicle is altered. Therefore slip ratio needs to be controlled under the conditions of changing vehicle mass and road conditions.

3. Design of SMC

As the parameters are changing, the controller needs to be robust enough to deal with this kind of situation. Therefore a sliding mode controller is designed in this case.

3.1. SMC with integral effort

To control the slip ratio, a sliding mode controller with addition to an integral part was proposed by Li and Kawabe.²⁴ The dynamics of the system can be taken as:

\overset{\cdot}{τ} = p + q . T_{m}

(19)

where $τ \in R$ represents the system state of slip ratio described in Equation (14). Here $T_{m}$ denotes the control input. After the differentiation of Equation (14) with time:

\overset{\cdot}{τ} = \frac{- {\overset{\cdot}{V}}_{c} + (1 - τ) {\overset{\cdot}{V}}_{t}}{V_{t}},

(20)

Then putting Equations (12) and (13) and $F_{x}$ in Equation (20):

\begin{matrix} p = - \frac{1}{V_{t}} {[4 g η (f (d), τ) - g e_{ro} - \frac{ϱ}{2 . m} A e_{D} {\overset{\cdot}{x}}^{2}] \\ + \frac{r^{2}}{J_{t}} (1 - τ) mg η (f (d), τ)} \end{matrix}

(21)

q = \frac{(1 - τ) r}{J_{t} V_{t}} .

(22)

The values of $m$ , $r$ , $e_{ro}$ , and $e_{D}$ are taken as:

m_{min} \leq m \leq m_{max}

r_{min} \leq r \leq r_{max}

e_{ro, min} \leq e_{ro} \leq e_{ro, max}

e_{D, min} \leq e_{D} \leq e_{D, max} .

As the value of function $p$ is not known in Equation (19) so we have to estimate it by $\hat{p}$ . The estimation error is bounded by a function $H = H (τ)$ , which is known:

| p - \hat{p} | \leq H

(23)

The estimation of $\hat{p}$ can be given as:

\begin{matrix} \hat{p} = - \frac{1}{V_{t}} {[4 g η (f (\hat{d}), τ) - g {\hat{e}}_{ro} - \frac{ϱ}{2 \hat{m}} A {\hat{e}}_{D} {\overset{\cdot}{x}}^{2}] \\ + \frac{{\hat{r}}^{2}}{J_{t}} (1 - τ) \hat{m} g η (f (\hat{d}), τ)} \end{matrix}

(24)

where $\hat{m}$ , ${\hat{e}}_{ro}$ , ${\hat{e}}_{D}$ , ad $\hat{r}$ are the estimated mass, rolling resistance, drag coefficient, and radius of the tire, respectively. The estimated values are chosen as:

\hat{m} = \frac{m_{min} + m_{max}}{2}

(25a)

{\hat{e}}_{ro} = \frac{e_{ro, min} + e_{ro, max}}{2}

(25b)

{\hat{e}}_{D} = \frac{e_{D, min} + e_{D, max}}{2}

(25c)

\hat{r} = \frac{r_{min} + r_{max}}{2} .

(25d)

From these evaluations, the error of the estimation can be presented by:

\begin{matrix} | p - \hat{p} | \leq \frac{g}{V_{t}} {4 | η (f (d), τ) - η (f (\hat{d}), τ) | - | e_{ro} - {\hat{e}}_{ro} | \\ - \frac{ϱ}{2} A {\overset{\cdot}{x}}^{2} | e_{D} - {\hat{e}}_{D} | + \frac{(1 - τ) | r^{2} - {\hat{r}}^{2} |}{J_{t}} | m η (f (d), τ) \\ - \hat{m} η (f (\hat{d}), τ) |} . \end{matrix}

(26)

Let’s presume the following:

\begin{matrix} H = \frac{g}{V_{t}} {4 | η (f (d), τ) - η (f (\hat{d}), τ) | - | e_{ro} - {\hat{e}}_{ro} | \\ - \frac{ϱ}{2} A {\overset{\cdot}{x}}^{2} | e_{D} - {\hat{e}}_{D} | + \frac{(1 - τ) | r^{2} - {\hat{r}}^{2} |}{J_{t}} | m η (f (d), τ) \\ - \hat{m} η (f (\hat{d}), τ) |} . \end{matrix}

(27)

3.1.1. Sliding surface

When we presume that $\bar{τ}$ is the variable of interest, the order of the system reduced to one. In normal case, the sliding function can be taken as:

f_{n} (τ, t) = \bar{τ} .

(28)

The error between the generated slip value and the reference slip value represented as $\bar{τ} = τ - \hat{τ}$ . When an integral part is added with the normal sliding function $f_{n} (τ, t)$ , the new sliding function can be presented as:

f (τ, t) = \bar{τ} + G_{i} \int_{0}^{t} \bar{τ} (x) . dx

(29)

where $G_{i}$ is the integral gain, $G_{i} > 0$ .

3.1.2. Foundation of control law

The main concept of SMC is to establish a function $f$ of tracking error and then design a control law that makes the function zero despite the disturbance and parameter uncertainties. The sliding phase dynamics is taken as:

\overset{\cdot}{f} = 0 .

(30)

After differentiation of Equation (29):

\overset{\cdot}{f} (τ, t) = (\overset{\cdot}{τ} - {\overset{\cdot}{τ}}_{r}) + G_{i} (τ - τ_{r}),

(31)

and putting this in Equation (30) gives the following:

(\overset{\cdot}{τ} - {\overset{\cdot}{τ}}_{r}) + G_{i} (τ - τ_{r}) = 0 .

(32)

The reference slip ratio value is a constant hence, ${\overset{\cdot}{τ}}_{r} = 0$ . Then putting Equation (19) in Equation (32) we obtain the following:

p + q . T_{m} + G_{i} (τ - τ_{r}) = 0 .

(33)

The control input we obtain after resolving Equation (33) as:

T_{me} = \frac{1}{q} [- p - G_{i} (τ - τ_{r})] .

(34)

The estimated equivalent control input is obtained as:

{\hat{T}}_{me} = \frac{1}{q} [- \hat{p} - G_{i} (τ - τ_{r})] .

(35)

In spite of uncertainties on the dynamics of $p$ , the hitting control input of the sliding mode condition is defined as:

T_{mh} = \frac{1}{q} [- G sgn (f)],

(36)

where the following holds:

sgn (f) = {\begin{matrix} - 1, & f < 0 \\ 0, & f = 0 \\ 1, & f > 0 \end{matrix}

(37)

and $G$ represents the sliding gain. Then the control law is given by:

T_{m} = T_{me} + T_{mh}

= \frac{1}{q} [- \hat{p} - G_{i} (τ - τ_{r}) - G sgn (f)] .

(38)

The control law defined in the above equation is made robust for the bounded parameter uncertainties by an exact choice of $G$ . Now the sliding gain $G$ is selected by:

G = H + σ

(39)

where $H$ can be given by Equation (32) and $σ$ is strictly positive design parameter.

3.1.3. Stability analysis

To verify the stability of the control law, the Lyapunov function is chosen as:

V = \frac{1}{2} f^{2},

(40)

which meets the condition $V \to \infty$ for $| f | \to \infty$ . After differentiation of Equation (45):

\overset{\cdot}{V} = \frac{1}{2} \cdot \frac{d}{dt} f^{2} = f \cdot \overset{\cdot}{f} .

(41)

Putting Equations (19), (30), (35), (36), and (38) into Equation (40) it becomes the following:

\begin{matrix} \overset{\cdot}{V} & = f \cdot \overset{\cdot}{f} \\ = f [p - \hat{p} - G sgn (f)] \\ = f (p - \hat{p}) - G | f | \\ \leq | p - \hat{p} | | f | - G | f | . \end{matrix}

(42)

Substituting the value of $| p - \hat{p} |$ from Equation (27) in Equation (42) yields the following:

\begin{matrix} \overset{\cdot}{V} \leq H | f | - G | f | \\ \leq - (G - H) | f | \\ \leq - σ | f | \end{matrix}

(43)

where $G - H = σ > 0$ . Thus, $\overset{\cdot}{V}$ is negative definite. Therefore, the exact Lyapunov function is $V$ which makes $f = 0$ globally asymptotically stable equilibrium. This means that the trajectory of $\bar{τ}$ will converge asymptotically to $f = 0$ and then slide along it.

3.2. Chattering

Due to the variable structure of the SMC when it is applied to a digital controller chattering happens. As the control law switches from one structure to the other at an infinite frequency, the trajectories chatter around $f = 0$ and produce oscillations. Hence, the need for control effort is significantly increased. These oscillations are dependent on the dynamics of the system, discontinuity gain, sampling time and the time delay of the actuator. In order to diminish chattering, the hitting control $T_{mh}$ is upgraded by incorporating saturation function as:

T_{mh} = \frac{1}{q} [- G sat (\frac{f}{β})]

(44)

where the width of the boundary layer around the sliding surface $f = 0$ is $β$ and saturation function expressed as:

sat (\frac{f}{β}) = {\begin{matrix} - 1, & f < β \\ \frac{f}{β}, & - β \leq f \leq β \\ 1, & f > β \end{matrix} .

(45)

Now we achieved revised control law using Equations (38), (39), and (44) as:

T_{m} = \frac{1}{q} [- \hat{p} - G_{i} (τ - τ_{r}) - (H + σ) sat (\frac{f}{β})] .

(46)

4. Matlab simulation

Matlab/Simulink is used in the simulation. The simulation requires both constant and varying parameters. Constant and bounded parameters are given by Tables 2 and 3, respectively.

Table 2.

Fixed parameters.

Parameters	Interpretation
$g = 9.81$ m/s²	Gravity
$ϱ = 1.2041$ kg/m³	Air density
$L = 2.4$ m	Distance between wheels
$A = 2.24$ m²	EV area in front
$J_{t} = 2.5744$ kg m²	Inertia

Table 3.

Bounded parameters.

Minimum	Estimated	Maximum	Deviation
Value	Value	Value
$e_{D, min} = 0.31$	${\hat{e}}_{D} = 0.36$	$e_{D, max} = 0.41$	${\tilde{e}}_{D} = 0.05$
$e_{ro, min} = . 009$	${\hat{e}}_{ro} = 0.155$	$e_{ro, max} = 0.301$	${\tilde{e}}_{ro} = 0.146$
$m_{min} = 450$	$\hat{m} = 750$	$m_{max} = 1050$	$\tilde{m} = 300$ kg
$r_{min} = 0.26$	$\hat{r} = 0.31$	$r_{max} = 0.36$	$\tilde{r} = 0.05$ m

The simulation specifications are explained here. Two test sets are considered for simulation. First, one is composed of asphalt dry–snow–concrete dry and the second one is made of cobblestone dry–ice–cobblestone wet. The allocated simulation time is $10$ s. Every test set is a combination of three periods. From $0$ s to $3$ s the car proceeds on the dry asphalt in the first phase. In the second phase, the car moves forward on snow road, from $3$ s to $6$ s. Then in the last phase, during $6$ s to $10$ s the car advances on a concrete dry road. For the first phase of the second test set, the car proceeds on the dry cobblestone in the time $0$ s to $3$ s. Then in the second period, from $3$ s to $6$ s, the car moves forward on ice road condition. In the last period, during $6$ s to $10$ s the car advances on wet cobblestone road. The proposed control law in Equation (51) can be predetermined with design parameters $G_{i}$ and $σ$ . Then values of both parameters are set to $G_{i} = 8$ and $σ = 1$ to assert the potency of the controller. The chosen value of the width of the boundary layer is $β = 0.025$ .

The mass of the car and road surface conditions are altered together to demonstrate the robustness performance with the parameter uncertainties. The mass of the car deviates as $m = 450$ kg, $m = 600$ kg, $m = 800$ kg, and $m = 1050$ kg. For the first and second test sets, four different masses are used, as shown in Figures 4 –11.

Figure 4.

Slip value when $m = 450 kg$ , first test set.

Figure 5.

Slip value when $m = 600 kg$ , first test set.

Figure 6.

Slip value when $m = 800 kg$ , first test set.

Figure 7.

Slip value when $m = 1050 kg$ , first test set.

Figure 8.

Slip value when $m = 450 kg$ , second test set.

Figure 9.

Slip value when $m = 600 kg$ , second test set.

Figure 10.

Slip value when $m = 800 kg$ , second test set.

Figure 11.

Slip value when $m = 1050 kg$ , second test set.

The slip ratio decreases from $1$ , as error extinguished by a resemblance between the given reference signal and generated output signal for $0$ s to $3$ s as the vehicle advances on asphalt dry road condition for the first set and dry cobblestone for the second set. In $3$ s to $6$ s when the car moves forward on snow and ice road surface condition in open loop, the slip ratio becomes excessively larger and causes instability. But in the closed-loop, it becomes stable by tracking the reference slip value. A stable system is achieved using a closed-loop for $6$ s to $10$ s, when the car moves on the dry concrete and wet cobblestone road surface condition, but due to huge steady-state error, open control stands unstable.

5. Four wheel model using Catia and Adams

5.1. CAD model using Catia

Catia (computer aided three-dimensional interactive application) is a platform which allows the generation of three-dimensional (3D) parts and helps them to assemble. In case of design, only four wheels and a chassis are needed. To simulate vehicle characteristics in Adams/View a 3D CAD model of the car is needed, which is being designed here using Part Design in Catia and then assembled using Assembly Design. Now, after designing the assembled car, joints have to be defined using DMU Kinematics under the section Digital Mockup. The Digital Mockup workbench is moderately vast, but we only used the DMU Kinematics module. After that material is chosen as wanted from the Material Library. In case of selection of material in this car model aluminum used for chassis parts and steel used for the inner portion of wheel and rubber used for the top layer of the wheel. To design road first three blocks are designed then it assembled to make one road. The final version of road and vehicle model after assembly is given in Figure 12.

Figure 12.

Vehicle and road model in Catia V5.

5.2. Catia to Adams transport

Sim Designer develops a linear and nonlinear model from MSC Nastran, Marc, and Adams, which is available within the Catia V5 CAD environment. Using the Sim Designer extension in Catia V5 one can make CAD model in Windows Command Script(.cmd) file extension, which can be opened in Adams/View and necessary analysis could be done there. To import car file in Adams first open Adams/View. It opens the import dialogue box in which selects Adams/View Command File (.cmd) in file type and gives the location of (.cmd) in files to read, and then select continue command. Upon completion vehicle model opens in Adams/View, if any error happens in between then it appears in the message window.

5.3. Adams interface

Four revolute joints which are defined previously are modified first. In this software, each revolute joint is assigned to each pair of wheel and the chassis. Then a dummy is created between wheel and road in front of the vehicle. As we needed longitudinal motion only then, we need to restrict the motion for longitudinal direction. So we have to create two translational joints. Two translational joints created for this model, one between chassis and dummy and another between dummy and road. Torque is defined for joints between each wheel and chassis. Four torques are created in this model for four revolute joints in wheels. To design road surfaces for different road conditions, three different friction coefficient is defined for the different blocks of road. The complete Adams/View vehicle consists of the parts below and has six degrees of freedom (DoF):

6 Gruebler count (approximate DoF);

6 moving parts (not including ground);

4 revolute joints;

2 translational joints.

The complete vehicle model developed in Adams/View is shown in Figure 13.

Figure 13.

Adams/View model of the vehicle.

6. Setting up co-simulation

In the case of co-simulation, two or multiple discrete simulation software is used during the simulation. During the run time, this software communicates with each other and affect each other’s output for the simulation of the whole system. In this case, the car model is generated in Adams/View, and the controller is designed in Simulink. Then a co-simulation is set up to drive the car model in Adams/View using the controller model in Simulink. Figure 14 gives a simple schematic of the procedure of co-simulation. The output variables generated from the Adams/View car model are transported to the controller model in Simulink. Then the control torque is evaluated from these car parameters and is fed back again into Adams/View.

Figure 14.

Schematic of co-simulation.

6.1. Selection of input and output variables

Input and output variables from Adams required by the controller are listed in Tables 4 and 5.

Table 4.

Adams input variables.

Variable name	Description
$Torque_1$	Rear left control torque
$Torque_2$	Rear right control torque
$Torque_3$	Front left control torque
$Torque_4$	Front right control torque

Table 5.

Adams output variables.

Variable Name	Description
$Vel_1$	Rear left angular velocity
$Vel_2$	Rear right angular velocity
$Vel_3$	Front left angular velocity
$Vel_4$	Front right angular velocity
chesis	Chassis velocity

6.2. Steps of co-simulation

Steps required to establish a co-simulation between Adams/View and Simulink are as follows:

load Adams/Controls;

select input and output variables;

set references for input variables;

Adams block exporting to MATLAB/Simulink;

linking Adams plant model and the controller model in Simulink;

starting co-simulation;

checklist before co-simulation.

The vehicle model is opened in Adams/View. Adams and controls application such as Simulink communicate using state variables. So we have to define all input and output variables in Adams state variables. After defining the input and output variables, the input variable values that are obtained from Simulink have to be applied/referenced at the proper location in the Adams model. In this case, the values of the control torques obtained from Simulink have to be referenced. Once the input variables are referenced properly in Adams, then the Adams model is complete for transporting to Simulink as a control system block. To transport the model, $Controls > PlantExport$ is selected, which opens the Adams/Controls Plant Export dialogue box, as shown in Figure 15. The proper input and output pins must be connected to the proper control system blocks for the actual functioning of the system.

Figure 15.

Adams/Controls Plant Export dialogue box.

To configure the controller with the Adams/View model, the Adams exported model is opened in MATLAB. Then the working directory of MATLAB is altered to the working directory of Adams/View. Then run the .m file. This generates the input and output variables of the Adams model as variables in MATLAB. Next, to the command, $Adam_sys$ is typed at the prompt, which opens a window containing Adams model in it, as shown in Figure 16. After opening the $adams_sub$ block, an Adams subsystem appears in a new window, as shown in Figure 17. The Animation mode field is set to interactive so that the simulation is shown graphically as it is being computed. Alternatively, it can be set to batch. The $adams_sub$ block is then copied and pasted onto a new model window along with the slip control system. Figure 18 shows how the simulation works in Simulink.

Figure 16.

Adams model in Simulink.

Figure 17.

Adams model subsystem.

Figure 18.

Block diagram of Adams model and controller in Simulink.

7. Co-simulation results and analysis

To execute the co-simulation in Adams and Matlab/Simulink first parameters of both should be adequately matched, as shown in Table 6.

Table 6.

Co-simulation parameters.

Parameters	Values
Length between two wheels of vehicle, $L$	$2.4 m$
Width between two wheels, $l_{w}$	$1.1 m$
Radius of wheel, $r$	$0.28 m$
Mass of vehicle, $m$	$668.47 kg$
Drag coefficient, $C_{d}$	$0.37$
Rolling resistance coefficient, $C_{roll}$	$0.1$

During the co-simulation, the controller tracks the reference slip in MATLAB/Simulink, which is described in Figure 19 and 20. Figure 19 shows that the vehicle runs on three different road conditions, i.e., during $0$ – $6$ s on asphalt dry road, during $6$ – $9$ s on a snowy road, and $9$ – $10$ s on a concrete dry road. For the second set, the car also runs of three different road conditions i.e., during $0$ – $6$ s on cobblestone dry, during $6$ – $9$ s on the ice road, and during $9$ – $10$ s on wet cobblestone road, which is shown in Fig. 20. At these road conditions, the controller effectively tracks the reference slip. As the system operates in closed-loop control, the starting torque is initially given by controller and then diminishes error to procure reference value of slip ratio. On the other hand, the slip value reaches below the reference values in open-loop control, as shown in the figures. In snowy and icy road conditions, the wheel velocity escalates due to skidding, therefore the slip ratio value increments. So, more torque is required to preserve the slip ratio. Hence the usefulness of controller gets highlighted in those road conditions. Initially, due to high starting torque, slip value rises above desired value and stables after $4$ s in a $10$ s co-simulation in case of the first set. In the case of the second test set slip value increases above the desired value and stables after $1$ s in co-simulation. Asphalt wet, ice, and snowy roads are the most critical conditions identified and controlled at closed-loop control, in which open-loop control fails to show the desired result. Hence, it can be said that the controller will work for other road conditions also which are generated for Matlab simulation, but for that testing contact in Adams should be appropriately changed. In the simulation during $0$ s to $3$ s peak value of $slip = 1$ occurs because of the high value of starting torque given by the controller as there is no torque limiter designed for this model and oscillations occur as the chosen values of integral gain $G_{i}$ and design parameter $σ$ is not optimized. It is proven that an EV requires a traction controller to run on different road conditions.

Figure 19.

Slip controller reference tracking for Set-1.

Figure 20.

Slip controller reference tracking for Set-2.

8. Conclusion

This paper presents a novel method for longitudinal slip ratio control using sliding mode. Design of a robust control law using the SMC technique has been described. The proposed technique is examined and then simulated in the Simulink environment, and the results are verified. In the simulation, the vehicle runs on two different roads, each constructed using three distinct friction coefficients for a certain interval of time. The proposed method exhibits much better performance in the slippery road, thus reducing energy cost. Then the validation leads to the development of four-wheeler in-hub EV using Catia and transportation to Adams. A road model has been designed and constructed for three different friction coefficients. The co-simulations have been achieved between Adams/View and Matlab/Simulink to test the robustness of the controller. In the case of closed-loop control after 4 s slip-based traction, the controller follows the reference slip in a 10 s co-simulation between Adams and Matlab/Simulink. Undesirable torque production and oscillations are the shortcomings of the controller. The design of the controller needs to be improved so that the oscillations can be avoided and torque should be in the desired limit.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

ORCID iDs

Sudipta Saha

Syed Muhammad Amrr

Author biographies

Sudipta Saha received the bachelor’s degree in Electrical Engineering from the West Bengal University of Technology (WBUT), Kolkata, India in 2012 and his M. Tech degree in Mechatronics & Robotics from School of Mechatronics & Robotics, IIEST, Shibpur, India in 2015. He is currently pursuing his PhD degree with the Control and Automation Group, Department of Electrical Engineering, Indian Institute of Technology Delhi (IITD), India. His research interests include Model order reduction and Sliding mode control, Robust Control, Optimal control, Electric vehicle, Active magnetic Bearing.

Syed Muhammad Amrr received the bachelor’s degree in Electrical Engineering and the master’s degree in Instrumentation and Control in 2014 and 2016, respectively from the Department of Electrical Engineering, Aligarh Muslim University (AMU), Aligarh, India. He is currently pursuing his PhD. degree with the Control and Automation Group, Department of Electrical Engineering, Indian Institute of Technology Delhi (IITD), New Delhi, India. His research interests include nonlinear control, sliding mode control, robust control, spacecraft attitude control, renewable energy, and power electronics.

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