Electric vehicle traction control for optimal energy consumption

Int. J. Electric and Hybrid Vehicles, Vol. 5, No. 3, 2013

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Electric vehicle traction control for optimal energy consumption Kumeresan A. Danapalasingam Faculty of Electrical Engineering, Department of Control and Mechatronics Engineering, Universiti Teknologi Malaysia, UTM Skudai, 81310 Johor, Malaysia E-mail: [email protected] Abstract: An electric vehicle (EV) with four in-wheel motors (IWMs) offers several advantages over other types of EVs. Due to a limited electrical energy source and a long battery charging time, any way to minimise energy consumption in an EV has to be fully utilised. Undoubtedly one of the systems in an EV that drains the most amount of energy is the propulsion system. In this work the existence of an optimal slip ratio that enables energy saving in an EV propulsion system is investigated. A controller is designed to ensure the slip ratio of each wheel is limited by the optimal value. Simulation results demonstrate the effectiveness of the proposed traction control scheme in energy optimisation in an EV. Keywords: electric vehicle; energy optimisation; traction control; slip ratio. Reference to this paper should be made as follows: Danapalasingam, K.A. (2013) ‘Electric vehicle traction control for optimal energy consumption’, Int. J. Electric and Hybrid Vehicles, Vol. 5, No. 3, pp.233–252. Biographical notes: Kumeresan A. Danapalasingam received his B.Eng. (Electrical – Mechatronics) and M.Eng. (Electrical – Mechatronics and Automatic Control) degrees from Universiti Teknologi Malaysia (UTM) in 2003 and 2006, respectively. In 2010, he was awarded a PhD (Electrical and Electronic) by Aalborg University, Denmark. At present, he is a Senior Lecturer at the Department of Control and Mechatronics Engineering, Faculty of Electrical Engineering, UTM. His research interests include control engineering, autonomous helicopter, artificial intelligence and hybrid electric vehicle.

1 Introduction An electric vehicle (EV) has a propulsion system that is powered by one or more electric motors. While in conventional vehicles the mechanical power is solely provided by internal-combustion engines (ICEs), EVs are driven by clean and efficient electric motors such as DC, brushless DC, AC induction or AC permanent-magnet synchronous motors. Some of the advantages of an EV compared to a vehicle with an ICE are less Copyright © 2013 Inderscience Enterprises Ltd.

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environmental pollution, high torque generation at low speed, fast torque response, high efficiency over a wide range of speeds and improved traction control. Depending on the electrical energy delivery method, an EV can be categorised as the following (Faiz et al., 1996; Yoshimura et al., 2011; Hu et al., 2012): •

an EV with an off-board electrical energy supply, e.g., electric buses, electric trains

•

an EV with a battery pack, e.g., electric car

•

an EV with an on-board electrical energy generation system, e.g., series and parallel hybrid vehicles

One of the disadvantages of EVs is closely related to battery technology. Present advancements in batteries only allow shorter range than what is desired by vehicle drivers (Faiz et al., 1996). The limited electrical energy supply by a battery pack in an EV is exacerbated by a long battery charging time. Thus, improvements in electrical energy efficiency to maximise energy saving in an EV is crucial. In this research, a method to minimise the propulsion energy of a four-wheel drive (4WD) EV with four IWMs (see Figure 1) is proposed. An EV with 4 IWMs offers several benefits over an EV with chassis mounted motors such as independent drive torque control, additional space on the EV platform, better stability, among others (van Schalkwyk and Kamper, 2012; Fraser, 2012; Hori, 2004). Here, a traction control strategy is introduced to enable a reduced electrical energy consumption in the EV. Figure 1 A 4WD EV with 4 IWMs (see online version for colours)

The problem of traction control in EVs has been addressed by many researchers. However, most of the traction control approaches only focus on tracking optimal slip ratios to maximise tyre-road friction for an effective vehicle control and safety (see e.g., Hu et al., 2012; Lee et al., 2004; Kataoka et al., 2001; Gu et al., 2011; Fujii and Fujimoto, 2007; Hori et al., 1998). EV propulsion energy optimisation is addressed by Yuan et al. (2012). By analysing the energy consumption distribution of an EV over the new European driving cycle (NEDC), a high level of energy consumption is detected in regions of high speed and low torque. In this region, Yuan et al. proposes to operate only one motor for an improved energy efficiency. In Li et al. (2012) the dynamic

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programming method is applied to find high-efficiency operating points of IWMs for obtaining an optimal torque distribution and vehicle velocity profile for energy saving. An energy optimisation strategy for EVs using induction motors is discussed in Li et al. (2006). By regulating flux level in the induction motor the proposed control scheme improves the EV efficiency in high speed region. From the above review (see also Li et al., 2008; Murphey et al., 2013; Chen and Wang, 2011; Hofman and Dai, 2010) it is evident that no research works have been exploring slip ratio control for propulsion energy optimisation. This paper presents a control approach to minimise the propulsion energy consumed by a 4WD EV with four IWMs. The traction control scheme involves the identification of an optimal slip ratio value of each wheel, and individual slip ratio control of the wheels. The objective of the slip ratio control is to ensure the slip ratio of each wheel does not exceed the optimal value. To evaluate the performance of the proposed traction control simulations of the EV energy consumption are carried out using Matlab software. It is shown that by limiting the slip ratio of each wheel to the optimal value precious electrical energy is not wasted. The ouline of this paper is as follows. In Section 2 the mathematical model of the EV is explained. The controller design is detailed in Section 3, followed by the simulation results that is presented in Section 4. A discussion of the proposed slip ratio control is done in Section 5, and Section 6 concludes the paper.

2 Mathematical model In this work a four-wheel drive (4WD) electric vehicle (EV) with four IWMs is considered (see Figure 1). An IWM (also called a wheel motor, wheel hub motor, wheel hub drive or hub mounted motor) is an integration of an electric motor, and control, brake, suspension and steering systems. An IWM is incorporated into the hub of a wheel that is driven either directly or through a gear system (McDonnell, 2011; Anderson and Harty, 2010; Fraser, 2012; Gerling et al., 2012; Perovic, 2012; van Schalkwyk and Kamper, 2012). Figure 2 shows an example of an IWM. The following subsections are dedicated to the derivation of the EV mathematical model. Figure 2 A IWM by Protean Electric Ltd. (see online version for colours)

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2.1 Rigid body dynamics Assuming that the passenger car is a rigid body the Newton–Euler equations of motion are given by (Bera et al., 2011) ˙ + Fsx m¨ xb = m(y˙ b ψ˙ − z˙b θ) ˙ + Fsy m¨ yb = m(z˙b ϕ˙ − x˙ b ψ) ˙ + Fsz m¨ zb = m(x˙ b θ˙ − y˙ b ϕ) ˙ y − Jz ) + Msx Jx ϕ¨ = θ˙ψ(J ˙ z − Jx ) + Msy Jy θ¨ = ψ˙ ϕ(J ˙ x − Jy ) + Msz . Jz ψ¨ = ϕ˙ θ(J The external forces and moments that act on the vehicle is shown in Figure 3. Figure 3 External forces and moments acting on the EV

2.2 Unsprung mass external forces The sum of external forces Fs = [Fsx Fsy Fsz ]⊤ acting on sprung mass of the vehicle is given by Fs = RFu where Fu = [Fux Fuy 0]⊤ . The x unsprung mass external force is given by fl fr rl rr Fux = Fux + Fux + Fux + Fux

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where fl fl fl Fux = Ftx cos(δ) − Fty sin(δ) − Rf l − Fa /4 fr fr fr Fux = Ftx cos(δ) − Fty sin(δ) − Rf r − Fa /4 rl rl Fux = Ftx − Rrl − Fa /4 rr rr Fux = Ftx − Rrr − Fa /4.

The tyre rolling resistance force is given by (Rajamani, 2012) i Ri = CR Fuz , i = f l, f r, rl, rr.

The equivalent aerodynamic drag force acting on the vehicle is given by (Rajamani, 2012) Fa =

1 ρCd AF (x˙ b + Vwind )2 2

where the vehicle frontal area is given by (Wong, 2001) AF = 1.6 + 0.00056(m − 765). The y unsprung mass external force is given by rr rl fr fl + Fuy + Fuy + Fuy Fuy = Fuy

where fl fl fl = Fty Fuy cos(δ) + Ftx sin(δ) fr fr fr Fuy = Fty cos(δ) + Ftx sin(δ) rl rl Fuy = Fty rr rr . = Fty Fuy

The z unsprung mass external forces are given by fl Fuz =

fr Fuz =

˙ mg(lr + dx ) (dr − dy ) m(¨ xb + z˙b θ˙ − y˙ b ψ)h dr − lf + lr dl + dr lf dl + dr ˙ ˙ m(¨ yb + x˙ b ψ − z˙b ϕ)h lr Mϕ l r Mθ dr + − − dl lf + lr dl lf + lr lf dl + dr ˙ xb + z˙b θ˙ − y˙ b ψ)h dl mg(lr + dx ) (dl + dy ) m(¨ − lf + lr dl + dr lf dl + dr ˙ ˙ m(¨ yb + x˙ b ψ − z˙b ϕ)h lr Mϕ l r Mθ dl − + − dr lf + lr dr lf + lr lf dl + dr

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rl Fuz =

rr Fuz =

˙ mg(lf − dx ) (dr − dy ) m(¨ xb + z˙b θ˙ − y˙ b ψ)h dr + lf + lr dl + dr lr dl + dr ˙ m(¨ yb + x˙ b ψ˙ − z˙b ϕ)h lf Mϕ lf Mθ dr + − + dl lf + lr dl lf + lr l r dl + dr ˙ mg(lf − dx ) (dl + dy ) m(¨ xb + z˙b θ˙ − y˙ b ψ)h dl + lf + lr dl + dr lr dl + dr ˙ m(¨ yb + x˙ b ψ˙ − z˙b ϕ)h lf Mϕ lf Mθ dl − + + dr lf + lr dr lf + lr l r dl + dr

where [dx dy (·)]⊤ = R⊤ [0 0 − h]⊤ . The suspension rolling and pitching moments are given by Mϕ = (cfϕ ϕ˙ + kϕf ϕ) + (crϕ ϕ˙ + kϕr ϕ) Mθ = (cfθ θ˙ + kθf θ) + (crθ θ˙ + kθr θ) respectively (Zheng et al., 2006).

2.3 Tyre forces The longitudinal and lateral tyre forces are given by (Hahn et al., 2002) σxi i F σi t σyi = i Fti , i = f l, f r, rl, rr, σ

i Ftx = i Fty

respectively, where { Fti

= 2µ

i

i Fuz

( )2 ( )3 } 1 Cf σ i 1 Cf σ i Cf σ i − + , i i i µi Fuz 3 µi Fuz 27 µi Fuz

Cf σ i ≤3 i µi Fuz Cf σ i i Fti = 2µi Fuz , if i i > 3, i = f l, f r, rl, rr. µ Fuz if

The total slip is given by σi =

√ (σxi )2 + (σyi )2 , i = f l, f r, rl, rr

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where σxi =

ω i r − x˙ b ωi r

σyi =

x˙ b tan(αi ) i = f l, f r, rl, rr. ωi r

(1)

The tyre slip angles (see Figure 4) are given by αf l = δ − tan−1

( y˙ + l ψ˙ ) b f x˙ b + dl ψ˙

αf r = δ − tan−1

( y˙ + l ψ˙ ) b f x˙ b − dr ψ˙

αrl = − tan−1

( y˙ − l ψ˙ ) b r x˙ b + dl ψ˙

αrr = − tan−1

( y˙ − l ψ˙ ) b r . x˙ b − dr ψ˙

Figure 4 Front-left tyre slip angle

The tyre-road friction coefficient is given by ( ) i i µi (σ i , x) ˙ = C1 (1 − e−C2 σ ) − C3 σ i e−C4 σ x˙ b where C1 , C2 , C3 , and C4 are parameters that depend on tyre and road conditions, determined from experiments (Bera et al., 2011) (see Table 1).

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Table 1 Tyre-road friction parameters Surface condition Asphalt, dry Asphalt, wet Snow Source:

C1

C2

C3

C4

1.029 0.857 0.1946

17.16 33.822 94.129

0.523 0.347 0.0646

0.03 0.03 0.03

Bera et al. (2011)

2.4 External moments The sum of external moments acting on sprung mass of the vehicle about x, y and z axes is given by Msx = −m¨ yb h − Mϕ + Mgx + Msf x Msy = m¨ xb h − Mθ + Mgy + Msf y Msz = Msf z respectively, where [Msf x Msf y Msf z ]⊤ = R [0 0 Muf z ]⊤ fl rl fr rr fl fr rl rr Muf z = (Fux + Fux )dl − (Fux + Fux )dr + (Fuy + Fuy )lf − (Fuy + Fuy )lr

and [−Mgy Mgx (·)]⊤ = R [0 0 mgh]⊤ .

2.5 Wheel dynamics The wheel dynamics are given by i Jw ω˙ i = Twi − Ftx r − Tbi , i = f l, f r, rl, rr.

(2)

In this section a mathematical model of the EV in consideration is explored. The equations that are presented here govern the dynamics of motion of the EV. It is from these equations that one could understand the relationship between the EV motion and electrical energy consumption. An analysis to determine the factors that contribute to excessive energy consumption by the EV propulsion system is described in the next section.

3 Controller design To study the energy consumption of the electric vehicle (EV), the car has to be steered according to a predetermined velocity profile. Subsequently by analysing the torque produced by each IWM to enable the prescribed motion, electrical energy consumed by the propulsion system can be understood. In this section a driver controller is developed to allow the EV to track a desired path. Next, a method to reduce electrical energy wastage by the EV propulsion system is discussed.

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3.1 Driver controller In equation (2), Twi is the torque produced by each IWM to rotate the respective wheel of the EV. In our study an electric motor unit (EMU) is attached to each wheel. The EMU is composed of a brushless DC IWM, a motor controller, an inverter and a bearing. In the simulations a black box model of the EMU that is built using actual experimental data, is implemented in Matlab. The parameters, inputs and outputs of the EMU black box model are listed in Table 2. Given the built-in motor controller the torque produced i by each IWM, Twi is equivalent to the torque demand Twd . In cases where the EMU is not able to deliver a desired torque the maximum available torque is produced. Table 2 Parameters, inputs and outputs of the EMU black box model Parameter Initial coil temperature Coolant specific heat capacity Coolant density Input EMU enable Torque demand Rotor speed DC supply voltage Rate of coolant flow Coolant inlet temperature Output Torque delivered Coil temperature Current Coolant outlet temperature Over temperature flag

Unit ◦ C J/kg/K kg/m3

Unit 0 or 1 Nm rad/s V m3 /s ◦ C Unit Nm ◦ C A ◦ C 0 or 1

Note that from the vehicle dynamics presented in Section 2 the control inputs to our system are wheel torque Twi and steering angle δ. Thus a driver controller is designed for the generation of Twi and δ to achieve a desired vehicle trajectory. The control architecture is depicted in Figure 5. In our setting simple PID controllers are tuned and a simulation of the vehicle motion for a simple trajectory tracking is shown in Figure 6. The figures show the EV x and y reference and actual positions. Despite the complexity of the mathematical model of the car, the PID controllers perform well in the straight line trajectory tracking. These controllers are utilised in Section 4 for more complicated trajectories.

3.2 Optimal slip ratio For the tracking of a straight line, the relationship between front-left slip ratio and (scaled down) longitudinal tyre force is depicted in Figure 7. Note that initially as the slip ratio increases the longitudinal tyre force increases accordingly. However starting from around 0.7 s an increment in the slip ratio causes the tyre force to drop. The value

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of slip ratio where the tyre force attains a peak value before it starts to decrease, is i defined as the optimal slip ratio σxo (see Figure 7). Figure 5 Driver control architecture

Figure 6 EV straight line trajectory tracking (see online version for colours)

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From equations (1) and (2) we can deduce that the higher the wheel torque Twi is (and thus electrical energy consumption), the higher the slip ratio σxi gets. Referring to i Figure 7 a slip ratio higher than σxo only reduces the longitudinal tyre force. In essence i if one drives the wheels past σxo , electrical energy is wasted for a reduced traction force. Therefore to optimise the energy consumed by the EV propulsion system the slip i ratio of each wheel has to be limited to σxo . The control architecture to realise this concept is given in Figure 8. Notice in the figure that an additional PID controller is required to adjust the signal generated by the driver controller for the stated purpose. In the next section the control design is tested for various velocity profiles and road conditions. Figure 7 Front-left slip ratio and (scaled down) longitudinal tyre force (see online version for colours)

Figure 8 Torque control architecture

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4 Simulation results In Section 3 it is discovered that the generation of slip ratio of each wheel above its i optimal value σxo only leads to a waste of precious electrical energy in the electric i vehicle (EV). This is because a slip ratio higher than σxo demands more wheel torque but produces a decreasing longitudinal tyre force. A control strategy is proposed and is implemented here in this section to test its effectiveness in energy saving. Simulations are performed using the NEDC and US06 driving cycles (see Karavalakis et al., 2009; Wiese et al., 1999; Borup et al., 2006) for the road conditions listed in Table 1. Vehicle parameters that are used in the simulations are listed in Table 3. Table 3 Vehicle parameters Parameter m Jx Jy Jz Cf CR Cd ρ h df dr lf lr r Jw cfϕ crϕ kϕf kϕr cfθ crθ kθf kθr

Value

Unit

1700 500 1000 2100 100, 000 0.015 0.35 1.225 0.6 1 1 1.5 1.5 0.26 8.2 2100 2100 65, 590 65, 590 2100 2100 65, 590 65, 590

kg kg m2 kg m2 kg m2 N rad−1

kg m−3 m m m m m m kg m2 Nm s rad−1 Nm s rad−1 Nm rad−1 Nm rad−1 Nm s rad−1 Nm s rad−1 Nm rad−1 Nm rad−1

With reference to Figure 8 and equation (1) measurements of ω i and x˙ b are needed to implement the slip ratio controller. In an actual vehicle these measurements can be obtained from odometer and inertial measurement unit (IMU) installations. In our simulations, for realistic measurements an IMU Simulink block is utilised to generate required readings. Note that the IMU block permits the addition of measurement noise. Some simulation results using the NEDC and US06 driving cycles are shown in Figures 9 and 10 respectively. In each figure plots of the EV motion, front-left slip ratio and longitudinal tyre force, with and without the proposed slip ratio control are given. Note that the simulation results in Figures 9 and 10 are for snow road surface.

Electric vehicle traction control for optimal energy consumption Figure 9 EV NEDC trajectory tracking, front-left slip ratio and longitudinal tyre force in snow (see online version for colours)

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Figure 10 EV US06 trajectory tracking, front-left slip ratio and longitudinal tyre force in snow (see online version for colours)

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5 Discussion A PID slip ratio controller is designed and applied to control the slip ratio of each i wheel of the electric vehicle (EV) to be within the optimal value σxo . Simulations are carried out using the NEDC and US06 driving cycles for various road conditions such as asphalt dry, asphalt wet and snow. In this section the feasibility of the proposed control technique in optimisation of energy consumption in the EV propulsion system is discussed. Simulation results for snow road surface are presented in Figures 9 and 10. From the plots of the EV motion it can be observed that the driver controller developed in Section 3 is able to track the NEDC and US06 driving cycles satisfactorily. For both of the driving cycles applying the slip ratio controller significantly reduces the slip ratio. In Figure 9 without the slip ratio controller, a drop in the longitudinal tyre force as the slip ratio increases can be noticed at around 808 s. This problem is rectified by applying i the slip ratio controller to keep the slip ratio below σxo . Analysing the tyre force plots in Figure 10, even with the help of the slip ratio controller the tyre force still drops at around 12 s. But since the slip ratio is noticeably smaller compared to the value without the slip ratio controller, a reduction in electrical energy consumption is still obtained. In fact in both cases the proposed control strategy minimises energy consumption of the EV propulsion system due to the reduction of slip ratio. Tables 4 and 5 contain data from the simulations. In the tables error is the average of the difference between EV desired and actual positions. Average values of electrical current sunk by the IWMs are included as measures of electrical energy consumption of the EV propulsion system. From Table 4 for all the road surface types i.e., asphalt dry, asphalt wet and snow, the proposed slip ratio control enables electrical energy saving. The highest reduction in energy consumption is obtained in snow where the improvement is 7.25%. For asphalt dry and asphalt wet, the savings in energy consumption are 1.82% and 1.73% respectively. For the US06 driving cycle, the slip ratio controller enables energy minimisation as the improvements made in asphalt dry, asphalt wet and snow road surfaces are 7.01%, 9.10% and 11.26% respectively (see Tables 5). Table 4 Simulation results using NEDC Without control

With control

Improvement (%)

1.39 20.85

1.24 20.47

11.02 1.82

1.39 20.83

1.24 20.47

10.95 1.73

1.49 22.08

1.24 20.48

17.20 7.25

Asphalt, dry Error (m) Current (A) Asphalt, wet Error (m) Current (A) Snow Error (m) Current (A)

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Table 5 Simulation results using US06 Without control

With control

Improvement (%)

0.71 74.47

0.63 69.25

10.90 7.01

0.71 74.82

0.62 68.01

12.57 9.10

8.60 69.72

8.45 61.87

1.68 11.26

Asphalt, dry Error (m) Current (A) Asphalt, wet Error (m) Current (A) Snow Error (m) Current (A)

In both driving cycles while the highest reductions in energy consumption are obtained for snow road surface, lowest improvements in energy saving are made for asphalt dry road surface. The reason for this is mainly because in slippery roads, more wheel slip and thus electrical energy wastage occurs naturally. Since the proposed slip ratio controller limits wheel slip, the control technique for energy saving is more effective in slippery roads. Note that for both NEDC and US06 driving cycles, improvements in EV trajectory error are also made. This means that the slip ratio controller not only improves the energy consumption of the EV propulsion system, but also makes the handling of the car better.

6 Conclusion In this research the existence of an optimal value of slip ratio of each wheel of a four-wheel drive electric vehicle (EV) with four IWMs is shown. Since exceeding the optimal slip ratio only results in a reduction of longitudinal tyre force, it is considered as a waste of the electrical energy. A controller is developed to keep the slip ratio of each wheel below the optimal value. Simulation results demonstrate the capability of the proposed control algorithm in optimisation of energy consumption in the EV propulsion system. As a bonus it is discovered that handling of the EV is also improved with the application of the slip ratio control. As a future work, the feasibility of the control approach to save energy in an actual EV will be studied.

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Nomenclature xb yb zb ϕ θ ψ Msx Msy Msz m g Jx Jy Jz R Fu Fux fl Fux fr Fux rl Fux rr Fux Fuy fl Fuy fr Fuy rl Fuy rr Fuy

Vehicle position in x direction Vehicle position in y direction Vehicle position in z direction Vehicle roll angle Vehicle pitch angle Vehicle yaw angle x sprung mass external moment y sprung mass external moment z sprung mass external moment Vehicle mass Gravitational acceleration Principal moment of inertia about x axis Principal moment of inertia about y axis Principal moment of inertia about z axis Rotation matrix Sum of external forces acting on unsprung mass x unsprung mass external force Front-left x unsprung mass external force Front-right x unsprung mass external force Rear-left x unsprung mass external force Rear-right x unsprung mass external force y unsprung mass external force Front-left y unsprung mass external force Front-right y unsprung mass external force Rear-left y unsprung mass external force Rear-right y unsprung mass external force

Electric vehicle traction control for optimal energy consumption fl Fuz fr Fuz rl Fuz rr Fuz Fs Fsx Fsy Fsz Rf l Rf r Rrl Rrr δ CR Fa ρ Cd AF Vwind h dl dr lf lr Mϕ cfϕ crϕ kϕf kϕr Mθ cfθ crθ kθf kθr Ftf l Ftf r Ftrl Ftrr fl Ftx fr Ftx rl Ftx rr Ftx fl Fty fr Fty rl Fty rr Fty σf l σf r σ rl σ rr σxf l

Front-left z unsprung mass external force Front-right z unsprung mass external force Rear-left z unsprung mass external force Rear-right z unsprung mass external force Sum of external forces acting on sprung mass x sprung mass external force y sprung mass external force z sprung mass external force Front-left tyre rolling resistance force Front-right tyre rolling resistance force Rear-left tyre rolling resistance force Rear-right tyre rolling resistance force Front wheel steering angle Rolling resistance coefficient Equivalent longitudinal aerodynamic drag force Air mass density Aerodynamic drag coefficient Vehicle frontal area Wind velocity Distance of vehicle c.g. from rotation centre Distance of left wheel from rotation centre Distance of right wheel from rotation centre Distance of front axle from rotation centre Distance of rear axle from rotation centre Suspension rolling moment Front roll damping coefficient Rear roll damping coefficient Front roll stiffness Rear roll stiffness Suspension pitching moment Front pitch damping coefficient Rear pitch damping coefficient Front pitch stiffness Rear pitch stiffness Front-left tyre force Front-right tyre force Rear-left tyre force Rear-right tyre force Front-left longitudinal tyre force Front-right longitudinal tyre force Rear-left longitudinal tyre force Rear-right longitudinal tyre force Front-left lateral tyre force Front-right lateral tyre force Rear-left lateral tyre force Rear-right lateral tyre force Front-left total slip Front-right total slip Rear-left total slip Rear-right total slip Front-left slip ratio

251

252 σxf r σxrl σxrr σyf l σyf r σyrl σyrr αf l αf r αrl αrr µf l µf r µrl µrr C1 , . . . , C 4 r Cf Mgx Mgy Muf z Msf x Msf y Msf z ωf l ωf r ω rl ω rr Jw Twf l Twf r Twrl Twrr Tbf l Tbf r Tbrl Tbrr fl Twd fr Twd rl Twd rr Twd xbr ybr fl σxo fr σxo rl σxo rr σxo fl σxr fr σxr rl σxr rr σxr

K.A. Danapalasingam Front-right slip ratio Rear-left slip ratio Rear-right slip ratio Front-left lateral slip Front-right lateral slip Rear-left lateral slip Rear-right lateral slip Front-left tyre slip angle Front-right tyre slip angle Rear-left tyre slip angle Rear-right tyre slip angle Front-left tyre-road friction coefficient Front-right tyre-road friction coefficient Rear-left tyre-road friction coefficient Rear-right tyre-road friction coefficient Tyre-road friction parameters Effective tyre radius Tyre cornering stiffness x gravity external moment y gravity external moment z unsprung mass external moment x sprung mass external moment y sprung mass external moment z sprung mass external moment Front-left wheel angular velocity Front-right wheel angular velocity Rear-left wheel angular velocity Rear-right wheel angular velocity Wheel moment of inertia Front-left wheel torque Front-right wheel torque Rear-left wheel torque Rear-right wheel torque Front-left braking torque Front-right braking torque Rear-left braking torque Rear-right braking torque Front-left wheel torque demand Front-right wheel torque demand Rear-left wheel torque demand Rear-right wheel torque demand Reference vehicle position in x direction Reference vehicle position in y direction Front-left optimal slip ratio Front-right optimal slip ratio Rear-left optimal slip ratio Rear-right optimal slip ratio Front-left reference slip ratio Front-right reference slip ratio Rear-left reference slip ratio Rear-right reference slip ratio