# Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors

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## Abstract

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## Featured Application

**The methodology presented in this paper can be applied to the design and evaluation of torque vectoring systems in hybrid electric vehicles. Comparison of different controllers is helpful in making design choices in these applications.**

## Abstract

## 1. Introduction

## 2. Vehicle Modelling and Maneuvers

#### 2.1. Vehicle Dynamics

- 6 DoF describe the free body motion of the vehicle sprung mass;
- 2 DoF for each of the four wheels represent the generalized position with respect to the body of the vehicle ζ (derived from the suspension kinematics) and the longitudinal slip of the tire.

#### 2.2. Model and Torque Allocation of the Hybrid Powertrain

#### 2.3. Steering Maneuver

**step steering**. In this standard open loop maneuver, the driver quickly passes from a null steering condition to a fixed value, as described in Table 3. The step steering is repeated in two conditions: the first one far from the limit conditions with a steering angle of 50° (equivalent to a lateral acceleration of around 0.5 g) and the second one with a higher steering (80°) condition in which the actuators are saturated and the tires near the maximum grip.

## 3. Control Systems and Performance Evaluation

#### 3.1. PID Controller

_{P}, K

_{I}, K

_{D}, and F are the gains of the proportional, integral, and derivative components, as well as the filtering frequency of the derivative term, respectively; γ

_{i}is the ideal reference yaw rate; and γ

_{a}is the actual yaw rate.

#### 3.2. First-Order Sliding Mode (FOSM)

#### 3.2.1. FOSM with a Low-Pass Filter

#### 3.2.2. FOSM with Continuous Approximation of the Sign Function

#### 3.3. Linear-Quadratic Regulator (LQR)

#### 3.4. Second-Order Sliding Mode (SOSM)

#### 3.4.1. SOSM Twisting Algorithm

#### 3.4.2. SOSM Suboptimal Algorithm

_{Mk}is defined as the last moment in which $\dot{S}=0$. For the practical application of the simulation environment, in which the timesteps are small yet discrete, t

_{Mk}is updated when there is a change of sign in the variation in S between two subsequent timesteps: S(t) − S(t − 1). The simulation begins with t

_{Mk}= t

_{M0}= 0, and thus S

_{1}(t

_{M0}) equals the initial value of $\left({\gamma}_{i}-{\gamma}_{a}\right)$.

_{r}must be sufficiently high to guarantee stability [23]. In the studied application, the chosen value to satisfy such a condition is ${k}_{r}=28.8$. To further reduce some of the residual chattering, known as the “ringing effect”, especially during the step steering maneuvers, the sign function was substituted with the continuous equivalent with a value of $\varnothing =4.5$, as in (5).

#### 3.5. Performance Evaluation

**performance factor (PF)**is composed of three indexes:

- The
**control penalty (CP)**is related to the effort of the controller in applying the control law, meaning that control systems that require more torque bias to stabilize the vehicle must be negatively evaluated. - The
**error penalty (EP)**directly evaluates the distance between the actual and target yaw rate during the maneuver, in such a way that less precise controllers are the most penalized. - The
**timed error penalty (TEP)**also takes into consideration the evolution of the error in time, meaning that controllers can afford a higher overshoot or initial displacement relative to the ideal behavior, if they are able to quickly eliminate the error. Steady-state error is heavily penalized in this indicator.

## 4. Simulation Results

- The baseline HEV without TV (
**OFF**); - The PID controller (
**PID**); - The FOSM controller with the low-pass filter (
**FOSM lowpass**); - The FOSM controller with continuous function (
**FOSM continuous**); - The LQR (
**LQR**); - The SOSM controller with the twisting algorithm (
**SOSM twisting**); - The SOSM controller with the suboptimal algorithm (
**SOSM suboptimal**).

#### 4.1. 50° Step Steering

#### 4.2. 80° Step Steering

#### 4.3. Ramp Steering

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Input steering angle ($\delta $) versus lateral acceleration (${a}_{y}$ ). The graph is changed by TV.

**Figure 4.**Simulink model organized in different blocks and subsystems [15].

**Table 1.**Parameters of the vehicle [15].

Property | Symbol | Value |
---|---|---|

Total mass | M | 1006 kg |

Wheelbase | l | 2.3 m |

Distance between CoG and front axle | a | 0.805 m |

Distance between CoG and rear axle | b | 1.495 m |

Track | t | 1.413 m |

CoG height | h_{cg} | 0.537 m |

Yaw inertia | I_{z} | 965.6 kg·m^{2} |

Tire radius (no load) | r | 0.291 m |

Lateral stiffness (front tires) | C_{f} | 21,094 N/rad |

Lateral stiffness (rear tires) | C_{r} | 14,556 N/rad |

Motor peak torque | T_{EM} | 103 N·m |

Motor peak power | P_{EM} | 25 kW |

Property | Value |
---|---|

Longitudinal velocity | 15 m/s |

Initial steering angle | 0° |

SWA increment rate $\theta $ | 8°/s |

Turning direction | Left |

Total maneuver time | 25 s |

Steering start | 1 s |

Steering end | 22 s |

Total maneuver time | 25 s |

Transmission gear | Fixed 3rd |

Turning direction | Left |

Property | Value |
---|---|

Longitudinal velocity | 15 m/s |

Initial steering angle | 0° |

Final SWA | 50° and 80° |

Turning direction | Left |

Total maneuver time | 5 s |

Steering start | 1 s |

Step duration | 1 s |

Total maneuver time | 5 s |

Transmission gear | Fixed 3rd |

Turning direction | Left |

Property | Unit | Value |
---|---|---|

K_{P} | - | 40 |

K_{I} | - | 10 |

K_{D} | - | 0.01 |

F | Hz | 100 |

**Table 5.**Numerical comparison of the control systems in the three conditions studied using the proposed PF.

Controller | Step 50° | Step 80° | Ramp Steering |
---|---|---|---|

OFF | 3.740 | 7.258 | 58.543 |

PID | 1.000 | 1.469 | 35.981 |

FOSM lowpass | 0.926 | 1.577 | 35.445 |

FOSM continuous | 0.641 | 1.009 | 35.011 |

LQR | 0.608 | 1.104 | 35.042 |

SOSM twisting | 0.728 | 2.084 | 34.776 |

SOSM suboptimal | 0.525 | 1.491 | 34.802 |

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**MDPI and ACS Style**

de Carvalho Pinheiro, H.; Carello, M.; Punta, E.
Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors. *Appl. Sci.* **2023**, *13*, 8109.
https://doi.org/10.3390/app13148109

**AMA Style**

de Carvalho Pinheiro H, Carello M, Punta E.
Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors. *Applied Sciences*. 2023; 13(14):8109.
https://doi.org/10.3390/app13148109

**Chicago/Turabian Style**

de Carvalho Pinheiro, Henrique, Massimiliana Carello, and Elisabetta Punta.
2023. "Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors" *Applied Sciences* 13, no. 14: 8109.
https://doi.org/10.3390/app13148109