# Reconfigurable Slip Vectoring Control in Four In-Wheel Drive Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

- Driving mode selection:By acting on the allocation matrix coefficients, the SV allows a software configuration in several driving mode settings (2WD/4WD, front/rear differentials).
- Equalization (steady-state):The SV allocation could be equalized at steady-state on the basis of online monitored four-wheels load torques. Several TV approaches treat the topic of equalized CA to provide a trade-off between the four-wheels on the basis of estimated wheel parameters, tire-road friction or wheel vertical load measurements [11,13,21,40,41,44,45].
- Automatic IWM selection/de-selection (steady-state/transient):The initial driving mode settings (2WD/4WD), which can be selected by the driver, can be automatically adjusted/switched online, to meet the safety requirements.

- Minimum slip vector norm allocation on four-wheels;
- Adaptive IWM controllers (w.r.t. wheel load torque estimation);
- Non-model based tire condition monitoring;
- Non-model based equalization;
- Online reconfigurability: switching among several powertrain/differential layouts.

## 2. Vehicle, Wheel and Actuator Dynamics

- (i)
- Vehicle chassis: longitudinal speed ${v}_{x}$, lateral speed ${v}_{y}$ and yaw-rate $r$ (roll, pitch and vertical dynamics are neglected);
- (ii)
- Wheels: the four-wheel angular speeds ${\omega}_{ij}$
- (iii)
- Induction motors: reduced-order models of four current-fed motors [53].

- small steering angles ($\delta \approx 0)$;
- high-cornering radius,

## 3. Preliminaries: Wheel Speed/Slip Control Motivations

#### 3.1. Matching Request on Force-Slip PMF Characteristics (Figure 3a)

#### 3.2. PMF-Curve Saturation Recognition (Figure 3b)

#### 3.3. Reconfigurability of References Vectoring

## 4. Reconfigurable Slip Vectoring Control

#### 4.1. Wheel Slip Control (WSC)

#### 4.2. Motion Planning (MP)

- (i)
- Motion control reference generation ${\overrightarrow{M}}_{c}^{*}$ (by driver commands); Section 5
- (ii)
- Slip/IWM selection (switch-ON/OFF commands ${s}_{ij},{\rho}_{{m}_{ij}}$); Section 4.2.1
- (iii)
- Reconfiguration Matrix (by measurement/switch/monitoring information); Section 4.2.1
- (iv)
- Driving Mode Selection (2WD/4WD, front/rear differentials ${\overrightarrow{d}}_{(\cdot )}$); Section 4.2.1
- (v)
- Wheel Status Monitoring (estimates/measurements of vehicle and WSC); Section 4.2.2
- (vi)
- Safety/Performance Driving (SPD) ${\mathrm{m}}_{\mathsf{\Psi}}^{*}$; Section 4.2.2
- (vii)
- SV equalization (equalization coefficients ${\tilde{\mathsf{\Psi}}}_{\mathrm{ij}}$); Section 4.2.3
- (viii)
- Fault-Detection and Isolation (FDI). Section 4.2.4

#### 4.2.1. Reconfiguration Matrix Design: Hard/Soft SV Configurations (Table 1)

#### 4.2.2. Wheel Status Monitoring (WSM)

#### 4.2.3. SV Equalization

#### 4.2.4. Fault-Detection and Isolation (FDI): IWM Switching-Off

## 5. Simulations

^{2}] is set on the basis of IWM power size, while ${a}_{yM}=$3.15 [m/s

^{2}] is found empirically via preliminary simulations with disabled differential control.

#### 5.1. Double-Lane Change (DLC)

^{−4}(RL, RR in A, C and FL, FR in B). In the dual 4WD cases D, E, the front and rear propulsion wheels, respectively, move slightly around the steady-state value, while relevant variations are visible on the rear (D) and front (E) wheels, respectively, which perform the differential action. The configurations in Figure 13B,D,F have the active rear differential, so that the RL and RR oscillatory slip behaviors are replicated: in B around 5$\times $10

^{−3}, in D around 0 since they do not perform the propulsion. The two configurations in Figure 13C,E have the active front differential, so that the FL and FR behaviors are replicated: in C they move around 5$\times $10

^{−3}, in E around 0 since they do not perform the propulsion. Figure 13F shows that the slip values are halved when the propulsion and the differential actions are shared by all the 4WD, oscillating around 2.5$\times $10

^{−3}. The reference slips are regulated by the 4IWM torques (Figure 14), which are delivered continuously by a PI-control (14)–(15) and (16)–(21). In addition, Figure 14F shows that the torque values are halved in the fully configured setting.

#### 5.2. SV Equalization

#### 5.3. Bending On a Low-Friction Patch

^{−3}. At time $t$ = 6.35[s], the vehicle encounters the snow patch. Automatically, the opposite-side IWMs react with differential torques, to provide a correction on the yaw-rate which is affected by the patch impact. The monitoring task detects a transient variation of the average wheel angle under the SPD threshold (Figure 26), which indicates that a transient variation on at least one wheel occurred. Then, the “Transient alert” flag ${\mathsf{\Psi}}_{T}$ = ‘1’ is set (Figure 26). At 7.5 s, the sensor error occurs on the RL wheel sensor, and the RL wheel starts skidding. At time $t$ = 7.7[s], the “Slip-over” condition occurs on RL wheel-slip ${\lambda}_{RL}$ (Figure 24), then the flag ${\lambda}_{T}{}_{RL}$ = ‘1’ is set and the switching-off command ${s}_{OFF}{}_{RL}^{FDI}$ = ‘1’ is triggered (Figure 25). The fault is detected in 0.2[s], while according to (27)–(28) the corresponding IWM selector ${\rho}_{m}{}_{RL}$ (and slip selector ${s}_{RL}$) switch-off takes 4$\tau $ = 0.7[s] (Figure 27). Consequently, the whole process of fault detection-and-fault isolation ends in about 1[s] (the nominal IWM time constant is 0.4[s], Table 2). The switching-off of the slip selector ${s}_{RL}$=‘0’ produces a SV reconfiguration (${w}_{RL,L}$ ≡ 0 and ${w}_{RL,D}$ ≡ 0, Figure 25).

^{−3}) values are doubled to compensate the loss of two actuators (Figure 23 and Figure 24). The rear wheels drift free, with a slip value of −5$\times $10

^{−4}. The average angle lowers down-to 87.51[deg] value (Figure 26): this means that the same maneuver with two less actuators is more dangerous. The speed and yaw-rate tracking performance are not affected by the successive reconfigurations (Figure 21 and Figure 22b). From (38), the yaw-rate references recover the evolution of driver steering-angles (Figure 22), who provides about only 22[deg] during all the maneuver, being assisted by IWMs. In Figure 27, the rotor-flux tracking, the 4IWMs stator-currents and motor mechanical power (${P}_{{m}_{ij}}={T}_{e.m{.}_{ij}}\cdot {\omega}_{ij}$, according to [53]) are reported. The rotor flux and currents are determined analytically from the Equations (12) and (13). The flux is given by the second equation of (12), from the known initial condition ${\mathsf{\Phi}}_{rd}\left(0\right)=0.1\left[Wb\right]$.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

List of Acronyms | |

CA | Control Allocation |

CoG | Centre of Gravity |

DLC | Double-Lane-Change |

EM | Electric Motor |

EP | Electric Powertrain |

EV | Electric Vehicle |

FD | Fault-Detection |

FDI | Fault-Detection and Isolation |

FI | Fault Isolation |

FT | Fault Tolerance |

IM | Induction Motor |

MP | Motion Planning |

PI | Proportional-Integral |

PMF | Pacejka Magic Formula |

SC | Slip Control |

SMC | Sliding Mode Control |

SPD | Safety/Performance Driving |

SV | Slip Vectoring |

TC | Traction Control |

TV | Torque Vectoring |

WD | Wheel-Drive |

WSC | Wheel Slip Control |

WSM | Wheel Status Monitoring |

List of Symbols | |

${v}_{x},{v}_{y}$ | longitudinal/lateral speeds |

$r$ | yaw-rate |

$\omega ,{\omega}_{ij}$ | wheel angular speed |

$\lambda ,{\lambda}_{ij}$ | wheel-slip ratio |

${T}_{e.m.},{T}_{L}$ | motor and load torques |

$F\left(\lambda \right),{F}_{x},{F}_{y}$ | wheel forces |

$\delta ,{\delta}_{sw}$ | steering/steering wheel angle |

${r}_{eq}$ | equivalent wheel radius |

${f}_{L},{f}_{D}$ | overall traction/differential forces |

${\overrightarrow{M}}_{c}$ | overall motion control |

${i}_{s}$ | motor stator current |

${\mathsf{\Phi}}_{r}$ | motor rotor flux |

$\widehat{F},{\widehat{T}}_{L}$ | estimated wheel force/load torque |

$\mathsf{\Psi},{\mathsf{\Psi}}_{ij}$ | wheel angle on Pacejka curve |

$\mathsf{\Psi}\left(0\right)$,${\mathsf{\Psi}}_{SAT}$ | origin/saturation wheel angle |

$\overline{\mathsf{\Psi}}$ | average wheel angle |

${K}_{p},{K}_{I}$ | proportional/integral gain |

$\left({\overline{\gamma}}_{ij},{\overline{\sigma}}_{ij}\right)$ | reconfiguration coefficients |

${s}_{ij},{\rho}_{m}{}_{ij}$ | slip/IWM selectors |

${\tilde{\mathsf{\Psi}}}_{ij}$ | equalization coefficients |

${m}_{\mathsf{\Psi}}^{*},{\overrightarrow{d}}_{(\cdot )}$ | SPD/Driving mode selection |

$W$ | Reconfiguration Matrix |

$\left[{a}_{x},{a}_{y}\right]$ | longitudinal/lateral accelerations |

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**Figure 3.**Wheel Speed/Slip control: (

**a**) torque (TV) vs. slip control (SV); (

**b**) Force-Slip $F\left(\lambda \right)-\lambda $ operating point (SV case).

**Figure 5.**Hard configuration: Motor selection/de-selection.

^{1,2,3}(See Table 1):

^{1}“soft” configurations ${\overrightarrow{d}}_{\left(a\right)},{\overrightarrow{d}}_{\left(b\right)}$;

^{2}“soft” configurations ${\overrightarrow{d}}_{\left(e\right)},{\overrightarrow{d}}_{\left(g\right)}$;

^{3}“soft” configurations ${\overrightarrow{d}}_{\left(i\right)-\left(l\right)}$.

**Figure 6.**Soft configuration: Driving Mode Selection. (

**a**–

**d**) Rear (half) propulsion; (

**e**–

**h**) Front (half) propulsion; (

**i**–

**l**) Four-wheels (full) propulsion.

^{1,2,3}(See Table 1):

^{1}2WD “hard” configuration ${\overrightarrow{s}}_{\left(a\right)}=\left(0,0,1,1\right)$;

^{2}2WD “hard” configuration ${\overrightarrow{s}}_{\left(f\right)}=\left(1,1,0,0\right)$;

^{3}differential ON/OFF command ${d}_{D}\equiv 0$.

**Figure 9.**DLC Maneuver: path following in six driving configurations. [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$},C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 10.**DLC Maneuver: driver steering effort in six driving configurations. [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$},C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 11.**DLC Maneuver: yaw-rate tracking ($r$): (

**A**) 2WD (front) NO DIFF.{${\overrightarrow{d}}_{\left(e\right)}$ }; (

**B**) 2WD (rear) active DIFF.{${\overrightarrow{d}}_{\left(b\right)}$ }; (

**C**) 2WD (front) active DIFF. {${\overrightarrow{d}}_{\left(g\right)}$ }; (

**D**) 4WD front prop.+rear DIFF. {${\overrightarrow{d}}_{\left(f\right)}$ }; (

**E**) 4WD rear prop.+front DIFF. {${\overrightarrow{d}}_{\left(c\right)}$ }; (

**F**) 4WD full prop.+full DIFF.{${\overrightarrow{d}}_{\left(l\right)}$ }. [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$ },C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 12.**DLC Maneuver: speed tracking (${v}_{x}$). [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$ },C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 13.**DLC Maneuver: four-slips coordination: (

**A**) 2WD (front) NO DIFF.{${\overrightarrow{d}}_{\left(e\right)}$}; (

**B**) 2WD (rear) active DIFF.{${\overrightarrow{d}}_{\left(b\right)}$ }; (

**C**) 2WD (front) active DIFF. {${\overrightarrow{d}}_{\left(g\right)}$ }; (

**D**) 4WD front prop.+rear DIFF. {${\overrightarrow{d}}_{\left(f\right)}$ }; (

**E**) 4WD rear prop.+front DIFF. {${\overrightarrow{d}}_{\left(c\right)}$ }; (

**F**) 4WD full prop.+full DIFF.{${\overrightarrow{d}}_{\left(l\right)}$ }. [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$ },C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 14.**DLC Maneuver: 4IWM torques coordination: (

**A**) 2WD (front) NO DIFF.{${\overrightarrow{d}}_{\left(e\right)}$}; (

**B**) 2WD (rear) active DIFF.{${\overrightarrow{d}}_{\left(b\right)}$ }; (

**C**) 2WD (front) active DIFF. {${\overrightarrow{d}}_{\left(g\right)}$ }; (

**D**) 4WD front prop.+rear DIFF. {${\overrightarrow{d}}_{\left(f\right)}$ }; (

**E**) 4WD rear prop.+front DIFF. {${\overrightarrow{d}}_{\left(c\right)}$ }; (

**F**) 4WD full prop.+full DIFF.{${\overrightarrow{d}}_{\left(l\right)}$ }. [

^{1}(B{${\overrightarrow{d}}_{\left(b\right)}$ },C{${\overrightarrow{d}}_{\left(g\right)}$ }) and

^{2}(D{${\overrightarrow{d}}_{\left(f\right)}$ },E{${\overrightarrow{d}}_{\left(c\right)}$ }) are dual layouts].

**Figure 18.**SV Equalization: Pseudoinverse matrix ${\mathsf{\Theta}}_{\left(ij,1-2\right)}$(32), ${\tilde{\mathsf{\Psi}}}_{ij}$ (33) equalization coefficients: (

**a**) Centre “heavy” extra-load (first-row); (

**b**) RL “light”, RR “heavy” extra-load (second-row).

**Figure 22.**Bending on Low-Friction Patch: (

**a**) steering wheel angle ${\delta}_{sw}$; (

**b**) yaw-rate tracking $r$.

**Figure 25.**Bending on Low-Friction Patch: Pseudoinverse matrix ${\mathsf{\Theta}}_{\left(ij,1-2\right)}$ and IWM selectors ${\rho}_{{m}_{ij}}$.

**Figure 26.**Bending on Low-Friction Patch: average wheel angle monitoring $\overline{\mathsf{\Psi}}$.

**Figure 27.**Bending on Low-Friction Patch: 4IWM fluxes, currents and powers (Equations (12) and (13)).

$\mathbf{Soft}\mathbf{Configuration}\overrightarrow{\mathit{d}}=\left({\mathit{d}}_{\mathit{L}\mathit{F}},{\mathit{d}}_{\mathit{L}\mathit{R}},{\mathit{d}}_{\mathit{D}\mathit{F}},{\mathit{d}}_{\mathit{D}\mathit{R}}\right)$ $\{\mathbf{LF}=\mathbf{Long}.\mathbf{Front},\mathbf{LR}=\mathbf{Long}.\mathbf{Rear},\mathbf{DF}=\mathbf{Diff}.\mathbf{Front},\mathbf{DR}=\mathbf{Diff}.\mathbf{Rear}\}$ | ${\mathit{d}}_{\mathit{D}}$ | $\mathbf{Hard}\mathbf{Configuration}\overrightarrow{\mathit{s}}=\left({\mathit{s}}_{\mathit{F}\mathit{L}},{\mathit{s}}_{\mathit{F}\mathit{R}},{\mathit{s}}_{\mathit{R}\mathit{L}},{\mathit{s}}_{\mathit{R}\mathit{R}}\right)$ | |||
---|---|---|---|---|---|

REAR propulsion | 1} ${\overrightarrow{d}}_{\left(a\right)}=$ (0, 1, 0, 0) | no differential$({\mathit{d}}_{\mathit{D}}\mathbf{\equiv}\mathbf{0})$ | $0$ | 1–2} ${\overrightarrow{s}}_{\left(a\right)}=\left(0,0,1,1\right)$ | 2WD (rear) |

2} ${\overrightarrow{d}}_{\left(b\right)}=$ (0,1,0,1) | rear differential | $1$ | |||

3} ${\overrightarrow{d}}_{\left(c\right)}=$ (0, 1, 1, 0) | front differential | $1$ | 13} ${\overrightarrow{s}}_{\left(c\right)}=\left(0,1,1,1\right)$ | 3WD (motor 1 off) | |

14} ${\overrightarrow{s}}_{\left(e\right)}=\left(1,0,1,1\right)$ | 3WD (motor 2 off) | ||||

4} ${\overrightarrow{d}}_{\left(d\right)}=\left(\mathbf{0},\mathbf{1},1,1\right)$ | full differential | $1$ | 15} ${\overrightarrow{s}}_{\left(g\right)}=\left(1,1,0,1\right)$ | 3WD (motor 3 off) | |

16} ${\overrightarrow{s}}_{\left(h\right)}=\left(1,1,1,0\right)$ | 3WD (motor 4 off) | ||||

FRONT propulsion | 5} ${\overrightarrow{d}}_{\left(e\right)}=\left(\mathbf{1},\mathbf{0},0,0\right)$ | no differential$({\mathit{d}}_{\mathit{D}}\mathbf{\equiv}\mathbf{0})$ | $0$ | 5–6} ${\overrightarrow{s}}_{\left(f\right)}=\left(1,1,0,0\right)$ | 2WD (front) |

6} ${\overrightarrow{d}}_{\left(g\right)}=\left(\mathbf{1},\mathbf{0},1,0\right)$ | front differential | $1$ | |||

7} ${\overrightarrow{d}}_{\left(f\right)}=\left(\mathbf{1},\mathbf{0},0,1\right)$ | rear differential | $1$ | 17} ${\overrightarrow{s}}_{\left(b\right)}=\left(0,1,1,0\right)$ | 2WD (crossed I) | |

8} ${\overrightarrow{d}}_{\left(h\right)}=\left(\mathbf{1},\mathbf{0},1,1\right)$ | full differential | $1$ | 18} ${\overrightarrow{s}}_{\left(d\right)}=\left(1,0,0,1\right)$ | 2WD (crossed II) | |

4-WHEELS propulsion | 9} ${\overrightarrow{d}}_{\left(i\right)}=\left(\mathbf{1},\mathbf{1},0,0\right)$ | no differential$({\mathit{d}}_{\mathit{D}}\mathbf{\equiv}\mathbf{0})$ | $0$ | 9–12} ${\overrightarrow{s}}_{\left(i\right)}=\left(1,1,1,1\right)$ | 4WD (full) |

10} ${\overrightarrow{d}}_{\left(j\right)}=\left(\mathbf{1},\mathbf{1},0,1\right)$ | rear differential | $1$ | |||

11} ${\overrightarrow{d}}_{\left(k\right)}=\left(\mathbf{1},\mathbf{1},1,0\right)$ | front differential | $1$ | |||

12} ${\overrightarrow{d}}_{\left(l\right)}=\left(\mathbf{1},\mathbf{1},1,1\right)$ | full differential | $1$ | |||

${m}_{\mathsf{\Psi}}^{*}$ | $\mathbf{2}$ [deg] | safety | $\mathbf{1}$ [deg] | performance |

Vehicle | Wheels (Pacejka 5.2) | IWM [10] ^{1}(37 kW, 50 Hz, 111 A) | |||

$m$ | 1005 [kg] | ${m}_{w}$ | 20.75 [kg] | ${m}_{IWM}$ | 23 [kg] |

${J}_{z}$ | 750 [kg$\cdot $m^{−2}] | ${J}_{t}$ | 1.177 [kg$\cdot $m^{−2}] | ${J}_{IWM}$ | 0.255 [kg$\cdot $m^{−2}] |

${l}_{F}$ | 1.10 [m] | ${r}_{eq}$ | 0.298 [m] | $\left(M,{L}_{r}\right)$ | (29.1, 29.1) [mH] |

${l}_{R}$ | 1.25 [m] | $\left({C}_{ij,x},{E}_{ij,x},{\overline{C}}_{f\lambda}\right)$ | (1.62, 0.50, 19.4) | ${R}_{s}$ | 85.1 [mΩ] |

${l}_{w}$ | 1.39 [m] | ${F}_{z0}$ | 4100 [N] | ${R}_{r}$ | 68.5 [mΩ] |

Control parameters | |||||

PI gains (14)–(21),(37)–(38) | SV Equalization (29), (33) | FDI-switching (27)–(28), (31), (34)–(36) | |||

$\left({K}_{{P}_{\omega}},{K}_{{I}_{\omega}}\right)$ | (−1.56, −750)$\times $10^{3} | $\left({k}_{\gamma},{k}_{\sigma}\right)$ | (0.85, 0.101) | $\tau $ | 0.175 [s] |

${K}_{P{\mathsf{\Phi}}_{rd}}$ | −10 | $\left({a}_{xM},{a}_{yM}\right)$ | (0.282, 0.321) [g’s] | ${T}_{A}$ | 2 [s] |

$\left({K}_{{P}_{v}},{K}_{{I}_{v}}\right)$ | (−0.39, −0.4) | $\left({k}_{\mathsf{\Psi}},{\epsilon}_{\mathsf{\Psi}}\right)$ | (−40, 0.05 [%]) | ${k}_{\lambda}$ | 2.5 |

$\left({K}_{{P}_{r}},{K}_{{I}_{r}}\right)$ | (−5.62, −42.13) | ${T}_{c}$ | 50 [ms] | ${\mathsf{\Psi}}_{SAT}$ | 82.56 [deg] |

**{${\alpha}_{m}={R}_{r}/{L}_{r}$, ${\mu}_{mot}=M/\left({J}_{IWM}{L}_{r}\right)$}.**

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## Share and Cite

**MDPI and ACS Style**

Amato, G.; Marino, R.
Reconfigurable Slip Vectoring Control in Four In-Wheel Drive Electric Vehicles. *Actuators* **2021**, *10*, 157.
https://doi.org/10.3390/act10070157

**AMA Style**

Amato G, Marino R.
Reconfigurable Slip Vectoring Control in Four In-Wheel Drive Electric Vehicles. *Actuators*. 2021; 10(7):157.
https://doi.org/10.3390/act10070157

**Chicago/Turabian Style**

Amato, Gerardo, and Riccardo Marino.
2021. "Reconfigurable Slip Vectoring Control in Four In-Wheel Drive Electric Vehicles" *Actuators* 10, no. 7: 157.
https://doi.org/10.3390/act10070157