# A Variable-Sampling Time Model Predictive Control Algorithm for Improving Path-Tracking Performance of a Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Design of Model Predictive Control Algorithm

#### 2.1. Dynamic Bicycle Model of a Vehicle

- (1)
- The longitudinal velocity of the vehicle is constant.
- (2)
- The left and right wheels (front and rear wheels) are considered single wheels.
- (3)
- Suspension movement and slippage aerodynamic effects are approximately zero.
- (4)
- The steering angle of the rear wheel is zero.

#### 2.2. Discrete State-Space Vehicle Model for MPC

#### 2.3. Cost Function of MPC

#### 2.4. Constraints of the MPC Algorithm

_{m}, the constraint must be expressed in terms of ΔU

_{m}. The constraint of the control input vector is as follows:

## 3. A Variable Sampling-Time Model Predictive Control (VST-MPC) Algorithm

#### 3.1. Proposed VST-MPC Algorithm Using Optimal Input Sequence

Algorithm 1 The Pseudo Code of VST-MPC. |

Step 1: Set the initial sampling time ${\mathrm{T}}_{\mathrm{S}}=0.2$ |

Step 2: Set the MPC parameter and calculate the optimal input with MPC |

Step 3: Predict ${\mathrm{X}}_{\mathrm{a}}\left(\mathrm{K}+1\right)={\mathrm{A}}_{\mathrm{a}}\xb7{\mathrm{X}}_{\mathrm{a}}\left(\mathrm{K}\right)+{\mathrm{B}}_{\mathrm{a}}\xb7\mathrm{u}\left(\mathrm{K}\right)$ |

Step 4: Update X_{past} = X_{a} (K + 1), u_{past} = u(K) |

Step 5: Calculate the next sampling time using the following equation. |

${\mathrm{T}}_{\mathrm{S}}\left(\mathrm{K}+1\right)={\mathrm{T}}_{\mathrm{S}}\left(\mathrm{K}\right)+\mathrm{sign}\left(\mathrm{Z}-\mathrm{C}\right)\xb7\mathrm{min}\left(\mathrm{Z},\mathrm{C}\right)$ |

where $\mathrm{Z}=-\mathsf{\lambda}\xb7\left|\frac{\partial}{\partial {\mathrm{T}}_{\mathrm{S}}}\left(\mathsf{\delta}\left(\mathrm{K}\right)\xb7\frac{\xb7{\mathrm{V}}_{\mathrm{y}}\left(\mathrm{K}\right)}{{\mathrm{T}}_{\mathrm{S}}\left(\mathrm{K}\right)}\right)\right|,C=-0.01$ |

Step 6: Set ${T}_{S}$ = ${\mathrm{T}}_{\mathrm{S}}\left(\mathrm{K}+1\right)$ |

Step 7: If T_{s} > ${\mathrm{T}}_{\mathrm{S},\mathrm{maximum}}$${\mathrm{T}}_{\mathrm{S}}$ = ${\mathrm{T}}_{\mathrm{S},\mathrm{maximum}}$ else if ${\mathrm{T}}_{\mathrm{S}}$ < ${\mathrm{T}}_{\mathrm{S},\mathrm{minimum}}$ ${\mathrm{T}}_{\mathrm{S}}$ = ${\mathrm{T}}_{\mathrm{S},\mathrm{minimum}}$ end |

Step 8: Go to Step 2 until MPC iteration is over. |

#### 3.2. Design a Function of VST-MPC for Sampling Time Variation

## 4. Configuration Driving Scenario Using MATLAB and Simulations

- CPU: AMD Ryzen 7 3800XT 8-Core Processor 3.90 GHz
- RAM: 32.0 GB
- GPU: NVIDIA GeForce RTX 3070
- RAM: 32.0 GB
- Tool: MATLAB 2020b, Automated Driving Toolbox

^{2}, the empirical maximum acceleration value of a petrol car driving at 20 m/s. The minimum acceleration constraint was set using 3.97 m/s

^{2}, the empirical maximum deceleration value of a petrol car driving at 20 m/s [28]. The maximum steering angle constraint was set by using Ackerman Jeantaud geometry in that the minimum radius of gyration is set to 6 m [29].

#### 4.1. Scenario Description

#### 4.2. Simulation Results of Scenario 1

#### 4.3. Simulation Result of Scenario 2

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**(

**a**) Lateral acceleration input of VST-MPC in scenario 2, (

**b**) steering angle input of VST-MPC in scenario 2.

Symbol | Description | Value [units] |
---|---|---|

m | Vehicle mass | 2020 [kg] |

l_{f} | C.g. distance to front wheel | 1.40 [m] |

l_{r} | C.g. distance to rear wheel | 1.65 [m] |

I_{z} | Yaw moment of inertia | 3234 [kg · m^{2}] |

C_{αf} | Front wheel cornering stiffness | 1420$\xb7\frac{180}{\pi}$ [N] |

C_{αr} | Rear wheel cornering stiffness | 1420$\xb7\frac{180}{\pi}$ [N] |

V | Velocity of vehicle | 20 [m/s] |

a_{y,max} | Maximum acceleration constraint | 2.24 [m/s^{2}] |

a_{y,min} | Minimum acceleration constraint | −3.97 [m/s^{2}] |

δ_{f,max} | Maximum steering angle constraint | 0.4864 [rad] |

δ_{f,min} | Minimum steering angle constraint | −0.4864 [rad] |

The MPC Algorithm | Average Tracking Error (m) | Computation Time (s) |
---|---|---|

The MPC algorithm with sampling time 0.1 | 0.1617 | 0.2880 |

The MPC algorithm with sampling time 0.2 | 0.6407 | 0.1292 |

The MPC algorithm with sampling time 0.05 | 0.1344 | 0.4170 |

VST-MPC | 0.1420 | 0.3267 |

The MPC Algorithm | Average Tracking Error (m) | Computation Time (s) |
---|---|---|

The MPC algorithm with sampling time 0.1 | 0.1395 | 0.1430 |

The MPC algorithm with sampling time 0.2 | 0.4079 | 0.0883 |

The MPC algorithm with sampling time 0.05 | 0.0556 | 0.3121 |

VST-MPC | 0.0570 | 0.2033 |

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**MDPI and ACS Style**

Choi, Y.; Lee, W.; Kim, J.; Yoo, J. A Variable-Sampling Time Model Predictive Control Algorithm for Improving Path-Tracking Performance of a Vehicle. *Sensors* **2021**, *21*, 6845.
https://doi.org/10.3390/s21206845

**AMA Style**

Choi Y, Lee W, Kim J, Yoo J. A Variable-Sampling Time Model Predictive Control Algorithm for Improving Path-Tracking Performance of a Vehicle. *Sensors*. 2021; 21(20):6845.
https://doi.org/10.3390/s21206845

**Chicago/Turabian Style**

Choi, Yoonsuk, Wonwoo Lee, Jeesu Kim, and Jinwoo Yoo. 2021. "A Variable-Sampling Time Model Predictive Control Algorithm for Improving Path-Tracking Performance of a Vehicle" *Sensors* 21, no. 20: 6845.
https://doi.org/10.3390/s21206845