# Research on the Trajectory Tracking of Adaptive Second-Order Sliding Mode Control Based on Super-Twisting

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## Abstract

**:**

## 1. Introduction

## 2. Construction of the Vehicle Dynamics Model

_{yf}is the lateral force of the front wheel, F

_{yr}is the lateral force of the rear wheel, v

_{x}is the longitudinal velocity, v

_{y}is the transverse velocity, ω is the angular velocity of the transverse pendulum, β is the lateral deflection of the center of mass, v

_{f}is the front wheel speed, v

_{r}is the rear wheel speed.

_{z}is the rotational inertia of the vehicle at the center of mass, $\text{}{\dot{v}}_{y}$ is the lateral acceleration, and $\text{}\dot{\omega}$ is the angular acceleration of the transverse pendulum.

## 3. Adaptive Preview Model

#### 3.1. Optimum Curvature Single Point Preview

_{G}-Y

_{G}is body coordinate system; point G is the position of the center of mass of the vehicle; point C is the predicted position after t

_{p}time; point P is the preview point on the target trajectory; Δf is the lateral deviation; x

_{GC}and y

_{GC}are the displacements in the X

_{G}and Y

_{G}directions after t

_{p}time; φ is the vehicle heading angle; β is the mass lateral deviation angle; and v is the vehicle travel speed.

#### 3.2. Adaptive Preview Time

_{p}, generally in the range of 0.3 to 1.5 s.

## 4. Design of a Second Order Sliding Mode Controller

#### 4.1. Design of Low-Pass Filters

#### 4.2. Design of the Super-Twisting Controller

_{r}and the ideal transverse pendulum angular velocity ω

_{d}is chosen as the tracking error of this system:

**Theorem**

**1.**

_{1}> 0, k

_{2}> 0, the system state can converge steadily in finite time and reach the origin point (0,0). At this time, $\dot{s}=s=0$.

**Proof of Theorem 1.**

_{1}> 0, k

_{2}> 0, then matrix is the Hurwitz. For any positive definite matrix P that satisfies the Lyapunov equation:

_{0}, we can get:

_{sw}is the angular transmission ratio of the steering wheel angle and the wheel angle.

#### 4.3. Phase Diagram Analysis of the Vehicle Dynamic Stability

#### 4.4. Low-Pass Filter Validation

_{dx}is the gradient of the steering wheel angle, S is the standard deviation, of the gradient and defines S as smoothness, μ is the mean of the gradient.

_{steer0}= 0.7107, and the smoothness of the unfiltered steering wheel angle S

_{steer1}= 0.0418. The range of the gradient of unfiltered angles is [−13.8526, 13.2884], the average of it is 7.9913 × 10

^{−6}; the range of the gradient of filtered angles is [−0.1192, 0.1825], the average of it is 1.1592 × 10

^{−5}. As can be seen from these values, the mean values of the gradients before and after filtering are approximately equal, however, the maximum value of the gradient before filtering is 72.8131 times greater than after filtering, and the gradient reflects the rate of change of the value. This means that the value of the steering wheel angle before filtering varies quite sharply compared to after filtering, and a large value of the gradient indicates the presence of a large jitter in the curve, and this shows the smoother curve of the steering wheel corner after filtering. Also, S

_{steer0}is 17.0024 times larger than S

_{steer1}, and this means that the gradient of the filtered angle is much less discrete than before filtering, and the gradient values of the filtered angle are closer to the average. In summary, it can be concluded that the low-pass filter works well and the filtered curve is smoother.

## 5. Design of the MPC Controller

#### 5.1. MPC Output Function Derivation

#### 5.2. MPC Algorithm Objective Function Design

#### 5.3. MPC Algorithm Constraint Design

#### 5.4. MPC Simulation Verification

## 6. Design of the Conventional Sliding Mode Control (CSMC) Controller

## 7. Control System Simulation Verification

#### 7.1. Construction of a Joint Simulation Platform

#### 7.2. ST Controller Simulation without Considering the Uncertainty of the Parameters

#### 7.3. Comparison Tests of ST Controller and MPC Controller

#### 7.4. Comparison Tests of ST Controller and CSMC Controller

_{STUF}= 0.7107, the smoothness of steering wheel angle of the CSMC controller is S

_{CSMC}= 0.7990, the smoothness of steering wheel angle of the ST controller with filtering (STF) is S

_{STF}= 0.0418. In numerical terms, S

_{STUF}is 17.00 times larger than S

_{STF}, S

_{CSMC}is 19.11 times larger than S

_{STF}, which shows the smooth curve of the angle compared to the other two. The range of the gradient of unfiltered angles of the STUF controller is [−13.8526, 13.2884], the range of the CSMC controller is [−7.8851, 11.8837], and the range of the STF controller is [−0.1192, 0.1825]. The gradient reflects the rate of change of the value, therefore, in terms of the range of the gradient, the variation range of the angles of the STUF controller is smaller than that of the CSMC controller, while that of STF controller is smaller than that of STUF controller. This indicates that the STF controller has a smoother angular profile with less undulation and less chatter than the other two. The average gradient of angles of STUF is 7.9913 × 10

^{−6}, the average of angles of CSMC is −8.1337 × 10

^{−6}, and the average of angles of STF is 1.1592 × 10

^{−5}. Numerically, the values of all three averages are close to 0, the smoothness reflects the degree of dispersion of the gradient, and consequently, the degree of dispersion of STF is the smallest of the three. This further illustrates that the filtered ST controller produces only a litter chattering compared to the other two.

#### 7.5. Simulation with Considering the Uncertainty of the Parameters

## 8. Conclusions

- This article designs a second-order sliding mode controller based on the ST algorithm, which combines an adaptive preview controller with the second-order sliding mode algorithm. The adaptive preview can take into account the trajectory deviation, road boundary, and overall vehicle motion response characteristics to solve for a suitable preview time and set a matching preview distance to make the controller more in line with the driver’s habits so that it can sense the road ahead in advance to make the corresponding control strategy. Finally, the Lyapunov function and phase plane analysis methods are used to prove its convergence and stability respectively;
- In designing the ST second-order sliding mode control algorithm, the chattering is also further reduced by combining a low-pass filter with the ST algorithm. This paper also proposes a method based on the standard deviation of the gradient to calculate the smoothness of the curve, using this parameter to evaluate the chattering of the curve, the standard deviation of the gradient was used to evaluate smoothness, and the smoothness after filtering is one-seventeenth of that before filtering;
- The ideal yaw rate can be obtained by adaptive preview control. The difference between the ideal yaw rate and the actual yaw rate is fed into the designed ST second-order sliding mode controller to solve the required steering wheel angle as the target for tracking control. To demonstrate the effectiveness of this controller, an MPC algorithm was designed for comparison experiments. Compared to the MPC controller, the tracking accuracy of the ST controller has improved to 64.42% and 51.02% at 36 and 54 km/h, respectively. At the same time, it was also compared with conventional sliding mode control and the results showed that the tracking accuracy of the ST controller has improved to 41.78%, and the smoothness of the ST controller is one-nineteenth that of the CSMC. This means the ST controller can produce inputs with weaker chattering; and
- Simulations are carried out on parameter uncertainties in this article, where parameter uncertainties include system parameter uptake and external disturbances, and Gaussian white noise is used to replace these uncertainties. With simulation at 36 and 54 km/h, the simulation results show that despite the effect of Gaussian white noise, the trajectory of the ST controller still fits the ideal trajectory and the tracking error does not exceed 0.3 m. Although there are slight fluctuations in the steering wheel angle and transverse angular velocity, they are still within the acceptable range and the actuator still works properly.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Phase plane at different speeds and front wheel angles. (

**a**) Phase plane of 36 km/h and δ = 0.5 rad; (

**b**) phase plane of 36 km/h and δ = −0.5 rad; (

**c**) phase plane of 54 km/h and δ = 0.5 rad; (

**d**) phase plane of 54 km/h and δ = −0.5 rad.

**Figure 5.**Steering wheel angle before and after filtering. (

**a**) Unfiltered steering wheel Angle; (

**b**) filtered steering wheel angle.

**Figure 8.**Comparison of different parameters at different speeds. (

**a**) Comparison of trajectories at different speeds; (

**b**) comparison of tracking error at different speeds; (

**c**) comparison of steering angle at different speeds; (

**d**) comparison of lateral acceleration at different speeds.

**Figure 9.**(

**a**) Comparison of trajectories at 36 km/h; (

**b**) comparison of the tracking error at 36 km/h; (

**c**) comparison of trajectories at 54 km/h; (

**d**) comparison of the tracking error at 54 km/h.

**Figure 10.**(

**a**) Comparison of trajectories at 54 km/h; (

**b**) comparison of the tracking error at 54 km/h.

**Figure 13.**(

**a**) Comparison of trajectories at different speeds; (

**b**) comparison of tracking error at different speeds; (

**c**) comparison of steering angle at different speeds; (

**d**) comparison of lateral acceleration at different speeds.

Parameters | Values | Comments |
---|---|---|

T | 0.5 | Time associated with vehicle steering response characteristics |

λ | 60 | Parameter of ST algorithm |

k_{1} | 0.2 | Parameter of ST algorithm |

k_{2} | 0.1 | Parameter of ST algorithm |

ξ | 6 | The cut-off frequency of a low-pass filter |

a | 1.016 | Distance from center of mass to front wheel |

b | 1.562 | Distance from center of mass to rear wheel |

C_{f} | 108,861 | Lateral stiffness of front wheel |

C_{r} | 108,861 | Lateral stiffness of rear wheel |

I_{z} | 1523 | Yaw inertia |

i_{sw} | 19.562 | Angular velocity ratio |

Parameters | Values | Comments |
---|---|---|

N_{p} | 30 | Predicted step size |

N_{c} | 60 | Control step |

Row | 10 | Relaxation factor weight |

Parameters | Input/Output Channels | Comments |
---|---|---|

Lead distance to driver preview point 1 | IMP_LX_SEN_1 | m |

Steering wheel angle | IMP_STEER_SW | deg |

Lateral distance to target point 2 | L_Drv_2 | m |

Longitudinal speed | Vx_SM | km/h |

Yaw rate | AV_Y | deg/s |

Slip angle | Beta | deg |

Lateral distance to target point 2 | L_Drv_1 | s |

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## Share and Cite

**MDPI and ACS Style**

Bei, S.; Hu, H.; Li, B.; Tian, J.; Tang, H.; Quan, Z.; Zhu, Y.
Research on the Trajectory Tracking of Adaptive Second-Order Sliding Mode Control Based on Super-Twisting. *World Electr. Veh. J.* **2022**, *13*, 141.
https://doi.org/10.3390/wevj13080141

**AMA Style**

Bei S, Hu H, Li B, Tian J, Tang H, Quan Z, Zhu Y.
Research on the Trajectory Tracking of Adaptive Second-Order Sliding Mode Control Based on Super-Twisting. *World Electric Vehicle Journal*. 2022; 13(8):141.
https://doi.org/10.3390/wevj13080141

**Chicago/Turabian Style**

Bei, Shaoyi, Hongzhen Hu, Bo Li, Jing Tian, Haoran Tang, Zhenqiang Quan, and Yunhai Zhu.
2022. "Research on the Trajectory Tracking of Adaptive Second-Order Sliding Mode Control Based on Super-Twisting" *World Electric Vehicle Journal* 13, no. 8: 141.
https://doi.org/10.3390/wevj13080141