# Trajectory Planning for an Articulated Tracked Vehicle and Tracking the Trajectory via an Adaptive Model Predictive Control

^{*}

## Abstract

**:**

## 1. Introduction

- Using the Hybrid A-star path planning method to obtain a feasible kinematic trajectory.
- Using the minimum snap method to optimize the planned trajectory and obtain the reference vehicle kinematic states.
- Designing a kinematic controller based on the AMPC control scheme to achieve robust trajectory tracking control.

## 2. Autonomous Articulated Vehicle System

#### 2.1. Kinematic Vehicle Models

#### 2.2. Tracking Error Dynamics Model

#### 2.3. Kinematic LPV Modelling

## 3. Trajectory Planning

#### 3.1. Node Expansion

#### 3.2. Heuristics Cost

#### 3.3. Analytical Expansion

#### 3.4. Trajectory Optimized

## 4. Control Design

#### 4.1. Reference Trajectory

#### 4.2. Adaptive MPC Controller

#### 4.3. Track-Speed Control

## 5. Simulation and Discussion

#### 5.1. Simulation Setup

#### 5.2. Simulation of Path Planning

#### 5.3. Simulation of the Trajectory Tracking

#### 5.3.1. Simulation Result of Case 1

#### 5.3.2. Simulation Result of Case 2

#### 5.3.3. Simulation Result of Case 3

#### 5.3.4. Simulation Result of Case 4

## 6. Conclusions

- Although the adaptive model predictive control algorithm has been applied in the path-tracking of the mobile robot, its application in the articulation vehicle is not mature. The MPC algorithm has yet to be applied in the path-tracking control of the articulated tracked vehicle. Thus, our work has extended the application of the MPC algorithm in the field of ATVs.
- The ATVs have unique steering characteristics compared to the skid-steering tracked vehicles. The path tracking of the ATVs also needs to consider its kinematic characteristics, for example, the multi-input and multi-output for the ATV control system. Thus, it is challenging for the developed control methods to control the ATV in a complex maneuver accurately. To this end, our work provides a practical method for the path planning and path tracking of ATVs.
- The simulation of several path-tracking cases has demonstrated that the standard-MPC controller cannot accurately control the ATV to follow a path with varying curvature. However, the proposed AMPC controller outperforms the standard-MPC controller, while the AMPC controller can achieve the same level of tracking performance compared to the nonlinear MPC controller.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ASV | Articulated steering vehicle |

ATV | Articulated tracked vehicle |

AMPC | Adaptive model predictive control |

NMPC | Nonlinear model predictive control |

MPC | Model predictive control |

RS | Reeds Shepp |

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**Figure 1.**The modeling of an articulated tracked vehicle system in this work is divided into theoretical mathematical modeling and virtual multi-body dynamic modeling based on the real vehicle system. The modeling depicts the steering of the ATVs driven by the hydraulic cylinders, which results in the change in articulation angle $\gamma $, the articulation angular rate $\dot{\gamma}$, and the yaw-rate response of the front unit and rear unit $\dot{\theta}$ and $\dot{\psi}$.

**Figure 2.**Overall scheme of the path planning and tracking modules for the articulated tracked vehicle system.

**Figure 4.**The virtual model of the articulated tracked vehicle constructed on the multi-body dynamics software Recurdyn.

**Figure 5.**Original Hybrid A-star (solid line) and the proposed Hybrid A-star (dash-dot line) path planning in the simulation. (

**a**) Map A; (

**b**) map B.

**Figure 8.**The orientation angle error of the AMPC controller and the fuzzy-PID controller in Case 1.

**Figure 10.**The lateral position error of the AMPC controller and the standard-MPC controller in Case 2.

**Figure 11.**The orientation angle error of the AMPC controller and the standard-MPC controller in Case 2.

**Figure 12.**The articulation angle of the AMPC controller, the NMPC controller, and the switch-MPC controller in Case 3.

**Figure 13.**The lateral position error of the AMPC controller, the NMPC controller, and the switch-MPC controller in Case 3.

**Figure 14.**The orientation angle error of the AMPC controller, the NMPC controller, and the switch-MPC controller in Case 3.

**Figure 16.**The articulation angles of the AMPC controller and the standard-MPC controller in Case 4.

**Figure 17.**The articulation angle rates of the AMPC controller and the standard-MPC controller in Case 4.

**Figure 18.**The lateral position errors of the AMPC controller and the standard-MPC controller in Case 4.

**Figure 19.**The orientation angle errors of the AMPC controller and the standard-MPC controller in Case 4.

Symbol | Description | Value | Unit |
---|---|---|---|

B | Width of ATVs | 2.1 | [m] |

D | Length of ATVs | 4.8 | [m] |

${L}_{f}$ | Distance from the hitch point to front unit | 2.6 | [m] |

${L}_{f}$ | Distance from the hitch point to rear unit | 2.2 | [m] |

$\gamma $ | Articulation angle | [−0.75, 0.75] | [rad] |

$\dot{\gamma}$ | Articulation angular rate | [−0.18, 0.18] | [rad/s] |

$\upsilon $ | Vehicle longitudinal speed | [−1, 4] | [m/s] |

Description | Value | Unit |
---|---|---|

Minimum turn radius | 10.4 | [m] |

Maximum velocity | 5 | [m/s] |

Maximum acceleration | 2 | [m/s${}^{2}$] |

Maximum steering angle | 0.5 | [rad] |

Maximum steering rate | 0.15 | [rad/s] |

Grid resolution in distance | 2 | [m] |

Grid resolution in yaw angle | 15 | [degree] |

Motion step size | 1 | [m] |

Number of steering angle candidate | 20 | |

Steer angle change weighting coefficient | 2 | |

Steer angle weighting coefficient | 1 | |

Heuristic weighting coefficient | 2 |

Symbol | Description | Value |
---|---|---|

${T}_{s}$ | The sample time of controller | 0.2 [s] |

${H}_{p}$ | Length of the prediction horizon | 10 |

${H}_{c}$ | Length of the control horizon | 5 |

$\mathit{Q}$ | Weighting coefficient for states | diag(0.5 0.5 1 0.1 0) |

$\mathit{R}$ | Weighting coefficient for control input | diag(0.1 0.2) |

$\mathit{P}$ | Terminal cost coefficient | diag(0.1 0.1 1 1 0) |

Map | Method 1 | Method 2 | ||||||
---|---|---|---|---|---|---|---|---|

Curvature ^{1} | Number ^{2} | Length ^{3} | Time ^{4} | Curvature | Number | Length | Time | |

Map A | 0.095 | 4 | 88.46 | 15.5 | 0.059 | 1 | 96.06 | 23.3 |

Map B | 0.098 | 5 | 128.94 | 26.4 | 0.096 | 2 | 116.09 | 40.6 |

^{1}

**Curvature**denotes the maximum curvature of the planned path.

^{2}

**Number**denotes the number of the steering direction change.

^{3}

**Length**denotes the overall length of the planned path.

^{4}

**Time**denotes the computation time of the planning method to obtain the planned trajectory.

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## Share and Cite

**MDPI and ACS Style**

Hu, K.; Cheng, K.
Trajectory Planning for an Articulated Tracked Vehicle and Tracking the Trajectory via an Adaptive Model Predictive Control. *Electronics* **2023**, *12*, 1988.
https://doi.org/10.3390/electronics12091988

**AMA Style**

Hu K, Cheng K.
Trajectory Planning for an Articulated Tracked Vehicle and Tracking the Trajectory via an Adaptive Model Predictive Control. *Electronics*. 2023; 12(9):1988.
https://doi.org/10.3390/electronics12091988

**Chicago/Turabian Style**

Hu, Kangle, and Kai Cheng.
2023. "Trajectory Planning for an Articulated Tracked Vehicle and Tracking the Trajectory via an Adaptive Model Predictive Control" *Electronics* 12, no. 9: 1988.
https://doi.org/10.3390/electronics12091988