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

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

## References

- Richalet, J.; Rault, A.; Papon, J. Model Predictive Heuristic Control. Automatica
**1978**, 15, 413–428. [Google Scholar] [CrossRef] - Giorgetti, N.; Ripaccioli, G.; Bamporad, A.; Kolmanovsky, I.; Hrovat, D. Hybrid Model Predictive Control of Direct Injection Stratified Charge Engines. IEEE/ASME Trans. Mechatron.
**2006**, 11, 499–506. [Google Scholar] [CrossRef] - Giorgetti, N.; Bamporad, A.; Tseng, H.E.; Hrovat, D. Traction control Hybrid model predictive control application towards optimal semi-active suspension. Int. J. Control
**2005**, 79, 521–533. [Google Scholar] [CrossRef] - Liu, Y.; Fan, K.; Ouyang, Q. Intelligent Traction Control Method Based on Model Predictive Fuzzy PID Control and Online Optimization for Permanent Magnetic Maglev Trains. IEEE Access
**2021**, 9, 29032–29046. [Google Scholar] [CrossRef] - Jhang, J.H.; Lian, F.L. An autonomous parking system of optimally integrating bidirectional rapidly-exploring random trees* and parking-oriented model predictive control. IEEE Access
**2020**, 8, 163502–163523. [Google Scholar] [CrossRef] - Ma, H.; Chu, L.; Guo, J.; Wang, J.; Guo, C. Cooperative Adaptive Cruise Control Strategy Optimization for Electric Vehicles Based on SA-PSO With Model Predictive Control. IEEE Access
**2020**, 8, 225745–225756. [Google Scholar] [CrossRef] - Salt Ducajú, J.M.; Salt Llobregat, J.J.; Cuenca, Á.; Tomizuka, M. Autonomous ground vehicle lane-keeping LPV model-based control: Dual-rate state estimation and comparison of different real-time control strategies. Sensors
**2021**, 21, 1531. [Google Scholar] [CrossRef] - Gray, A.; Gao, Y.; Hedrick, J.K.; Borelli, F. Robust Predictive Control for semi-autonomous vehicles with an uncertain driver model. In Proceedings of the 2013 IEEE Intelligent Vehicle Symposium (IV), Gold Coast, QLD, Australia, 23–26 June 2013; pp. 208–213. [Google Scholar]
- Zhang, H.; Heng, B.; Zhao, W. Path tracking control for active rear steering vehicles considering driver steering characteristics. IEEE Access
**2020**, 8, 98009–98017. [Google Scholar] [CrossRef] - González, D.; Pérez, J.; Milanés, V.; Nashashibi, F. A review of motion planning techniques for automated vehicles. IEEE Trans. Intell. Transp. Syst.
**2015**, 17, 1135–1145. [Google Scholar] [CrossRef] - Falcone, P.; Borrelli, F.; Asgari, J.; Tseng, H.E.; Hrovat, D. A model predictive control approach for combined braking and steering in autonomous vehicles. In Proceedings of the 2007 Mediterranean Conference on Control & Automation, Athens, Greece, 27–29 June 2007; pp. 1–6. [Google Scholar]
- Mayne, D.Q.; Rawlings, J.B.; Rao, C.V.; Scokaert, P.O.M. Constrained model predictive control: Stability and optimality. Automatica
**2000**, 36, 789–814. [Google Scholar] [CrossRef] - Gao, Y.; Gray, A.; Tseng, H.E.; Borrelli, F. A tube-based robust nonlinear predictive control approach to semiautonomous ground vehicles. Veh. Syst. Dyn.
**2013**, 52, 802–823. [Google Scholar] [CrossRef] - Zhang, B.; Zong, C.; Chen, G.; Li, G. An adaptive-prediction-horizon model prediction control for path tracking in a four-wheel independent control electric vehicle. Proc. Inst. Mech. Eng.
**2019**, 233, 3246–3262. [Google Scholar] [CrossRef] - Hoffmann, N.; Andresen, M.; Fuchs, F.W.; Asiminoaei, L.; Thøgersen, P.B. Variable sampling time finite control-set model predictive current control for voltage source inverters. In Proceedings of the 2012 IEEE Energy Conversion Congress and Exposition (ECCE), Raleigh, NC, USA, 15–20 September 2012; pp. 2215–2222. [Google Scholar]
- Taherian, S.; Halder, K.; Dixit, S.; Fallah, S. Autonomous Collision Avoidance Using MPC with LQR-Based Weight Transformation. Sensors
**2021**, 21, 4296. [Google Scholar] [CrossRef] - Wang, H.; Liu, B.; Ping, X.; An, Q. Path tracking control for autonomous vehicles based on an improved MPC. IEEE Access
**2019**, 7, 161064–161073. [Google Scholar] [CrossRef] - Rajamani, R. Vehicle Dynamics and Control, 2nd ed.; Springer Science & Business Media: New York, NY, USA, 2011; pp. 27–31. [Google Scholar]
- Marzbani, H.; Khayyam, H.; To, C.N.; Quoc, Đ.V.; Jazar, R.N. Autonomous vehicles: Autodriver algorithm and vehicle dynamics. IEEE Trans. Veh. Technol.
**2019**, 68, 3201–3211. [Google Scholar] [CrossRef] - Ji, J.; Khajepour, A.; Melek, W.W.; Huang, Y. Path Planning and Tracking for Vehicle Collision Avoidance Based on Model Predictive Control with Multiconstraints. IEEE Trans. Veh. Technol.
**2017**, 66, 952–964. [Google Scholar] [CrossRef] - Kong, J.; Pfeiffer, M.; Schildbach, G.; Borrelli, F. Kinematic and dynamic vehicle models for autonomous driving control design. In Proceedings of the 2015 IEEE Intelligent Vehicles Symposium (IV), Seoul, Korea, 28 June–1 July 2015; pp. 1094–1099. [Google Scholar]
- Shieh, L.S.; Wang, H.; Yates, R.E. discrete continuous model conversion. Appl. Mathmatical Model.
**1980**, 4, 449–455. [Google Scholar] [CrossRef] - Borrelli, F.; Falcone, P.; Keviczky, T.; Asgari, J.; Hrovat, D. MPC-based approach to active steering for autonomous vehicle systems. Int. J. Veh. Auton. Syst.
**2005**, 3, 265–291. [Google Scholar] [CrossRef] - Wang, L. Model Predictive Control System Design and Implementation Using MATLAB
^{®}; Springer Science & Business Media: New York, NY, USA, 2009; pp. 47–50, 53–68. [Google Scholar] - Cao, Y.; Chen, W.H. Variable sampling-time nonlinear model predictive control of satellites using magneto-torquers. Syst. Sci. Control Eng. Open Access J.
**2014**, 2, 593–601. [Google Scholar] [CrossRef] [Green Version] - Wang, L. Recognition of human activities using continuous autoencoders with wearable sensors. Sensors
**2016**, 16, 189. [Google Scholar] [CrossRef] - Giurgică, G.; Florescu, R.D. A case study for modeling autonomous driving systems. In Proceedings of the 2020 24th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 8–10 October 2020; pp. 745–750. [Google Scholar]
- Bokare, P.S.; Maurya, A.K. Acceleration-deceleration behaviour of various vehicle types. Transp. Res. Procedia
**2017**, 25, 4733–4749. [Google Scholar] [CrossRef] - Din, Z.M.U.; Razzaq, W.; Arif, U.; Ahmad, W.; Muhammad, W. Real Time Ackerman Steering Angle Control for Self-Driving Car Autonomous Navigation. In Proceedings of the 2019 4th International Conference on Emerging Trends in Engineering, Sciences and Technology (ICEEST), Karachi, Pakistan, 10–11 December 2019; pp. 1–4. [Google Scholar]
- Li, S.; Li, K.; Rajamani, R.; Wang, J. Model predictive multi-objective vehicular adaptive cruise control. IEEE Trans. Control Syst. Technol.
**2010**, 19, 556–566. [Google Scholar] [CrossRef] - Kayacan, E.; Saeys, W.; Ramon, H.; Belta, C.; Peschel, J.M. Experimental validation of linear and nonlinear MPC on an articulated unmanned ground vehicle. IEEE/ASME Trans. Mechatron.
**2018**, 23, 2023–2030. [Google Scholar] [CrossRef] - Lin, X.; Görges, D.; Weißmann, A. Simplified energy-efficient adaptive cruise control based on model predictive control. IFAC-PapersOnLine
**2017**, 50, 4794–4799. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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