# Trajectory Tracking Model Predictive Controller Design for Autonomous Vehicles with Updating Constrains of Tire Characteristics

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## Abstract

**:**

## 1. Introduction

- Proposing a MPC controller by considering the tires’ dynamics and the road curvature for the trajectory tracking control;
- Designing a tire force estimator based on the radial basis function (RBF) neural network to estimate the vehicle driving states, which are used to update the slip constraint.
- Conducting the co-simulation (CarSim/Simulink) and Hardware-in-loop (HIL) platform to validate the performance of the proposed trajectory tracking control system.

## 2. Proposed Trajectory Tracking Control System

## 3. Trajectory Tracking Model

- The influence of road slope is not taken into account;
- The coupling relationship between longitudinal and lateral forces is ignored; only the pure cornering characteristics of the tire is considered;
- The load transfer of the tire is ignored;
- The air resistance is ignored;

_{yr}are functions of the slip angle of the tire, which can be calculated in the following ways:

^{5}is the state quantity of vehicle trajectory tracking system.

## 4. Control Law Allocation

#### 4.1. Control Objectives

- The target of the trajectory tracking behavior can be regarded as the vehicle’s accurate tracking of the desired trajectory:$$Objective\left(A\right):\{\begin{array}{c}{e}_{y}\left(k\right)\to 0\\ {e}_{\phi}\left(k\right)\to 0\\ {e}_{v}\left(k\right)\to 0\end{array},ask\to \infty ,$$
- The controller proposed uses the envelope suggested by Ref. [35], and the vehicle stability is judged by the vehicle’s velocity state: lateral velocity $\left({v}_{y}\right)$ and the yaw rate ($r$), and we estimate the lateral force under different driving conditions through the RBF neural network. Using the lateral force estimated by the RBF neural network, the threshold of the slip angle α under different driving conditions is calculated from the tire model, and the lateral velocity $\left({v}_{y}\right)$ limit is defined by limiting the slip angle of the rear tire. The maximum slip angle limit of the rear tire can be transformed into the constraint on the lateral velocity $\left({v}_{y}\right)$ and the yaw rate $\left(r\right)$ of the vehicle by Equation (7):$$\left|\frac{{v}_{y}-{l}_{r}r}{{v}_{x}}\right|\le {\alpha}_{t},$$$$\left|r+\frac{g}{{v}_{x}}\phi \right|\le min\left(\frac{{F}_{yf}\left(1+\frac{{l}_{f}}{{l}_{r}}\right)}{m{v}_{x}},\frac{{F}_{yr}\left(1+\frac{{l}_{r}}{{l}_{f}}\right)}{m{v}_{x}}\right)$$The constraints (16) and (17) form a closed envelope, as shown in Figure 5. Equation (16) limits the control boundary of the yaw rate, and the Equation (17) limits the control boundary of the lateral velocity. When all states are within the envelope, the vehicle stability can be guaranteed.
- In order to achieve the required lateral tracking accuracy and control stability, the steering angle and the physical characteristics of the driving and braking should be considered. Apply the following constraints to the vehicle’s variables:$$\{\begin{array}{l}\Delta {u}_{min}\le \Delta u\left(k\right)\le \Delta {u}_{max}\hfill \\ {u}_{min}\le u\left(k\right)\le {u}_{max}\hfill \end{array},$$

#### 4.2. Trajectory Tracking Controller Based on MPC

_{x}is the dimension of the state quantity and N

_{u}means the dimension of the control quantity, and the change of the control input is $\Delta u\left(k\right)=u\left(k\right)-u\left(k-1\right)$. The predicted output performance vector and the sequence of the future incremental inputs at time step $k$ are denoted as $Y\left(\mathrm{k}\right)$ and $\Delta U\left(k\right)$, respectively.

#### 4.3. Lower-level Controller Design

## 5. Experimental Platform and Testing

#### 5.1. Simulation

#### 5.1.1. Accuracy of RBF Neural Network in Estimating Lateral Force

#### 5.1.2. Influence of Slip Constraints

#### 5.1.3. Effect of Nonlinear Envelope Constraints

#### 5.1.4. Comparison of Simulation Result

#### 5.2. Hardware-in-loop Platform and Test

#### 5.2.1. Desired Trajectory Setting

#### 5.2.2. Hardware-in-loop Platform Setup

#### 5.2.3. Hardware-in-loop Platform Results and Analysis

#### 5.2.4. Comparative Analysis of Co-simulation and Hardware-in-loop Platform Experiments

## 6. Conclusions

- In order to improve the tracking accuracy of the vehicle and the stability of the vehicle, a control method, which considers the influence of road curvature and tire nonlinear dynamic characteristics on the trajectory tracking performance, is developed by the control law based on the MPC. Furthermore, the lateral force is estimated based on the RBF neural network, which is incorporated into the boundary conditions of the vehicle envelope constraint to reduce the slip phenomenon of vehicles;
- The trajectory tracking control system with the control law is simulated by Matlab/Simulink and Raspberry Pi combined with the CarSim-based vehicle model. Meanwhile, the development technology route was expounded;
- In co-simulation (CarSim/Simulink), the accuracy of the RBF neural network in estimating the lateral force is verified under actual driving conditions; the slip phenomenon can be effectively reduced after the vehicle is applied to the envelope constraint; and the influence of the vehicle on the driving state after the linear and nonlinear envelope constraint is analyzed. Finally, compared with the Ref. [34], the results show that the proposed controller is better than the controller proposed in the Ref. [34];
- In the HIL platform, we verify the performance of the proposed MPC controller in three different scenarios. Finally, in comparing the co-simulation results with the HIL platform, the results show that the co-simulation is better than the HIL platform;
- In the future research, the study of the tire deflection characteristics based on experimental data would be considered. In addition, on the basis of the establishment of the HIL platform, the ROS-based verification platform is built to verify the effectiveness of the proposed algorithm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Description |

$x/y\left(\mathrm{m}\right)$ | Coordinates of center of gravity (CG) |

$\phi $ (rad) | Yaw angle of vehicle body |

$r$ (rad/s) | Yaw rate of vehicle body |

${v}_{x}/{v}_{y}$ (m/s) | Longitudinal/lateral velocity |

${e}_{y}\left(\mathrm{m}\right)$ | Offset of CG from the reference point |

${e}_{\phi}$ (rad) | Orientation error of yaw angle with respect to the desired trajectory |

${e}_{v}\left(\mathrm{m}/\mathrm{s}\right)$ | Error between the current and the desired longitudinal velocity |

$m$ (kg) | Vehicle mass |

${I}_{z}\left(\mathrm{kg}\xb7{\mathrm{m}}^{2}\right)$ | Yaw moment of inertia of the vehicle |

${l}_{f}/{l}_{r}$ (m) | Distance from CG to the front/real axle |

${F}_{xf}/{F}_{xr}\left(\mathrm{N}\right)$ | Longitudinal tire force of the front/rear wheel |

${F}_{yf}/{F}_{yr}\left(\mathrm{N}\right)$ | Lateral tire force of the front/rear wheel |

${F}_{z}\left(\mathrm{N}\right)$ | Vertical load |

${C}_{f}\left(\mathrm{N}/\mathrm{rad}\right)$ | Cornering stiffness of the front wheel |

${C}_{r}\left(\mathrm{N}/\mathrm{rad}\right)$ | Cornering stiffness of the real wheel |

${\alpha}_{f}/{\alpha}_{r}\left(\mathrm{rad}\right)$ | Slip angle of the front/ rear wheel |

${a}_{x}\left(\mathrm{m}\xb7{\mathrm{s}}^{-2}\right)$ | Longitudinal acceleration of vehicle |

${\delta}_{f}$ (rad) | Front-wheel steering angle |

${C}_{R}\left({\mathrm{m}}^{-1}\right)$ | Road curvature |

$\mu $ | Road friction coefficient |

${\beta}_{j}$ | Center width of the hidden layer |

X | Input vector for RBF neural network |

${C}_{j}$ | Central unit of the jth radial basis function |

${w}_{i}$ | Weight for RBF neural network |

${T}_{s}$ (s) | Sampling time |

$Q/R$ | Weight matrix for MPC |

$\epsilon $ | Coefficient of relaxation |

${a}_{thdes}\left(\mathrm{MPa}\right)$ | Desired throttle opening |

${P}_{bdes}\left(\mathrm{MPa}\right)$ | Desired braking master cylinder pressure |

${a}_{t{h}_{sat}}\left(\mathrm{MPa}\right)$ | Throttle opening threshold |

${p}_{pdes}\left(\mathrm{MPa}\right)$ | Expected braking main cylinder pressure |

${y}_{i}$ | Predicted value for RESM |

${Y}_{i}$ | True value for RESM |

MPC | Model prediction control |

RBF | Radial basis function |

PID | Proportional-integrated-differential |

LQR | Linear quadratic regulator |

SMC | Sliding mode control |

ZOH | Zero-order retention |

CarSim | A simulation software specifically for vehicle dynamics |

Simulink | A modular diagram environment for multidomain simulation as well as model-based design |

RMSE | Root mean square error |

UKF | Unscented Kalman Filter |

PWA | Piecewise affine |

HIL | Hardware-in-loop |

ROS | Robot Operating System |

CG | Center of gravity |

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**Figure 10.**Estimation results based on RBF neural network tire lateral force under ice-covered pavement; (

**a**) The lateral force; (

**b**) Discrepancy between the estimated lateral force and the true value.

**Figure 11.**Simulation results of ice-covered pavement trajectory tracking at $50\mathrm{km}\xb7{\mathrm{h}}^{-1}$; (

**a**) Trajectory tracking results; (

**b**) Lateral tracking error; (

**c**) The front-wheel steering angle; (

**d**) Comparison of the lateral velocity and yaw rate under whether slip constraints are applied.

**Figure 12.**Simulation results of wet asphalt pavement trajectory tracking at $75\mathrm{km}\xb7{\mathrm{h}}^{-1}$; (

**a**) Trajectory tracking results; (

**b**) Lateral tracking error; (

**c**) The front-wheel steering angle; (

**d**) Comparison of the lateral velocity and yaw rate under whether slip constraints are applied.

**Figure 13.**Trajectory tracking results compared to the algorithm proposed at 36$\mathrm{km}\xb7{\mathrm{h}}^{-1}$; (

**a**) Comparison of driving paths; (

**b**) Comparison of lateral errors.

**Figure 14.**Trajectory tracking results compared to the algorithm proposed at $45\mathrm{km}\xb7{\mathrm{h}}^{-1}$; (

**a**) Comparison of driving paths; (

**b**) Comparison of lateral errors.

**Figure 15.**Trajectory tracking results at $55\mathrm{km}\xb7{\mathrm{h}}^{-1}$; (

**a**) Driving path; (

**b**) Lateral tracking error.

**Figure 17.**The simulation results of trajectory tracking: (

**a**) Driving paths; (

**b**) Lateral error; (

**c**) The front-wheel steering angle; (

**d**) The heading error; (

**e**) Longitudinal acceleration; (

**f**) Longitudinal velocity error.

**Figure 18.**Trajectory tracking simulation results at Scenario D: (

**a**) Driving paths; (

**b**) Lateral error; (

**c**) The front-wheel steering angle; (

**d**) The heading error; (

**e**) Longitudinal acceleration; (

**f**) Longitudinal velocity error.

Definition | Symbol | Value |
---|---|---|

Vehicle mass | m (kg) | 1723 |

Inertia moment of the vehicle about yaw axis | ${I}_{z}\left(\mathrm{kg}\xb7{\mathrm{m}}^{2}\right)$ | 4175 |

Distance of the front/rear axle from CG | ${l}_{f}/{l}_{r}$(m) | 1.232/1.464 |

Minimal/maximal yaw rate | ${r}_{min}/{r}_{max}\left(rad/s\right)$ | −0.5/0.5 |

Regardless of Constraints | Considering the Constraint |
---|---|

5.0016 | 0.9640 |

Linear Region | Nonlinear Regions |
---|---|

1.1974 | 1.0886 |

Proposed Controller | MPC-Ref Controller | Improvement | |
---|---|---|---|

36$\mathrm{km}\xb7{\mathrm{h}}^{-1}$ | 0.0574 | 0.0653 | 12.09% |

45$\mathrm{km}\xb7{\mathrm{h}}^{-1}$ | 0.0490 | 0.2774 | 82.33% |

55 $\mathrm{km}\xb7{\mathrm{h}}^{-1}$ | 0.0527 | - | - |

Initial Velocity | Center Point Velocity | Time of Lane Change | |
---|---|---|---|

Scenario A | 36 km/h | 45 km/h | 8.3 s |

Scenario B | 72 km/h | 80 km/h | 4.8 s |

Scenario C | $108$ km/h | 112 km/h | 3.3 s |

Scenario D | $144$ km/h | 147.6 km/h | 2.2 s |

Scenario A | Scenario B | Scenario C |
---|---|---|

0.0081 | 0.0191 | 0.0430 |

HIL | CarSim/Simulink | Improvement | |

Scenario D | 0.0705 | 0.0667 | −5.69% |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Yuan, T.; Zhao, R.
Trajectory Tracking Model Predictive Controller Design for Autonomous Vehicles with Updating Constrains of Tire Characteristics. *World Electr. Veh. J.* **2023**, *14*, 54.
https://doi.org/10.3390/wevj14020054

**AMA Style**

Liu Y, Yuan T, Zhao R.
Trajectory Tracking Model Predictive Controller Design for Autonomous Vehicles with Updating Constrains of Tire Characteristics. *World Electric Vehicle Journal*. 2023; 14(2):54.
https://doi.org/10.3390/wevj14020054

**Chicago/Turabian Style**

Liu, Yingjie, Tengfei Yuan, and Rongchen Zhao.
2023. "Trajectory Tracking Model Predictive Controller Design for Autonomous Vehicles with Updating Constrains of Tire Characteristics" *World Electric Vehicle Journal* 14, no. 2: 54.
https://doi.org/10.3390/wevj14020054