# Three-Dimensional Path-Following Control Method for Flying–Walking Power Line Inspection Robot Based on Improved Line of Sight

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- An adaptive acceptance circle strategy is developed in response to the inspection principle of the FPLIR and the complex working conditions of power lines. Based on the path angle and current flight speed, the acceptance circle radius is adaptively adjusted to improve the following accuracy during transitions between paths.
- (2)
- An adaptive heading control strategy is developed based on the characteristics of the robot’s ascent process and the accuracy requirements for following. The strategy involves using a scaling factor that is based on the distance between the robot and the path in order to adjust the priority of parallel and perpendicular path following. This approach effectively addresses issues related to significant path-following errors and slow convergence speed, achieving fast, stable, and accurate path following.
- (3)
- The error equation for 3D path following is analyzed using the proposed improved LOS method. A state-feedback-based control law is designed using the following error as the control objective. Stability analysis is carried out to ensure that the designed control law can satisfy the accuracy and reliability requirements of path following.

## 2. Problem Description

## 3. Path-following Control Based on Improved LOS

#### 3.1. Traditional LOS

#### 3.2. Improved LOS

Algorithm 1: Improved LOS |

#The forward control vector, ${n}^{t}$, is continuously adjusted based on the distance from the robot’s position, P, to the path in order to achieve the fast convergence of the robot’s trajectory.function PathFollowing#Calculate the path segment on which the reference point, ${P}_{LOS}$, is located: ${P}_{i-1}{P}_{i}$$/{P}_{i}{P}_{i+1}$ $\hspace{1em}\mathbf{While}\mathbf{i}\mathbf{}\mathbf{n}\mathbf{do}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}({n}_{\perp}^{t},{n}_{\parallel}^{t})\u27f8\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\left(\right);$ $\hspace{1em}\hspace{1em}{\theta}_{d}\u27f8\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{A}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\left({n}^{t}\right)$$;(\#{\theta}_{d}$ is theoretical angle) $\hspace{1em}\hspace{1em}\theta \u27f8\mathrm{A}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{A}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\left(\right);$(# Real angle) $\hspace{1em}\hspace{1em}u\u27f8\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{F}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{C}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\left({\theta}_{d},\theta \right)$; # The mass center arrives at the end neighborhood and then moves to the next control segment. The next control phase starts when ${P}_{LOS}\in {P}_{i}{P}_{i+1.}$ $\hspace{1em}\hspace{1em}{R}_{0,i}\u27f8\mathrm{A}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{R}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}({R}_{0},{\theta}_{i})$; if $P\in ({R}_{0,i},{P}_{i})$ then$\hspace{1em}\hspace{1em}{P}_{i-1}\u27f5{P}_{i},{P}_{i}\u27f5{P}_{i+1}$; end ifreturn $({n}^{t}$, u); end whileend function |

- (1)
- Assess whether the FPLIR enters the vicinity of the acceptance circle radius of the corner.
- (2)
- If it does enter the vicinity, update the current following path segment, and update the projection point, ${P}^{`}$, of the FPLIR position, P, within the new path segment.
- (3)
- Based on the forward-looking distance, $\mathsf{\Delta}$, locate a new target point, ${P}_{LOS}$, on the new path segment starting from the projection point, ${P}^{`}$.
- (4)
- Calculate and update the following angle and following inclination according to the new target point, ${P}_{LOS}$.
- (5)
- Calculate the control volume according to the new following angle and following inclination.

#### 3.2.1. Adaptive Acceptance Circle Strategy

#### 3.2.2. Adaptive Heading Control Strategy

- (1)
- The start and end points of the current path segment are designated as ${P}_{i-1}$ and ${P}_{i}$, respectively. The robot’s path following involves closed-loop control of the current path segment. The next path segment is then controlled in a similar way.
- (2)
- The point P represents the robot’s centroid. ${P}^{`}$ is defined as the projection of P onto the linear path. ${d}_{l}$ refers to the distance between the robot and the planned path.
- (3)
- The vector ${n}^{t}$ represents the forward direction of the robot at time t. The vector ${n}_{\parallel}^{t}$ is the parallel projection of ${n}^{t}$ onto the segment ${P}_{i-1}{P}_{i}$, while ${n}_{\perp}^{t}$ is the orthogonal vector to ${n}_{\parallel}^{t}$, thus representing the vertical vector of the segment, ${P}_{i-1}{P}_{i}$.
- (4)
- The letter m is defined as the scaling coefficient of the distance, ${d}_{l}$, between the robot centroid, P, and the planned path; thus, the forward direction vector of the control robot is$${n}^{t}=\frac{m{d}_{l}}{1+m{d}_{l}}{n}_{\perp}^{t}+\frac{1}{1+m{d}_{l}}{n}_{\parallel}^{t}$$

#### 3.2.3. Design of the State-Feedback-following Controller

## 4. Simulation Experiments and Analysis

#### 4.1. Experiments on Heading Control Strategy

- (1)
- For smaller angles, the adaptive navigation control shows a smaller following error and an improved following effect. At point a, the maximum following error decreases from 0.238 m to 0.183 m, indicating a reduction in the following error of 0.055 m. At point b, there is little or no significant reduction in the maximum following error. At point c, the maximum following error decreases from 0.152 m to 0.148 m, indicating a reduction in the following error of 0.004 m. This improvement also enhances the oscillatory convergence phenomenon.
- (2)
- The convergence time of the adaptive heading control is faster. At point a, the convergence time is reduced from 8.1 s (from 33.7 s to 41.8 s) to 5.1 s (from 33.7 s to 38.8 s), and the time taken to converge is reduced by 3 s. At point b, the convergence time is reduced from 9.3 s (from 67.9 s to 77.2 s) to 2.8 s (from 66.6 s to 69.4 s), and the time consumed for convergence is reduced by 6.5 s. Similarly, at point c, the convergence time is reduced from 5.8 s (from 102.1 s to 107.9 s) to 4.7 s (from 100.0 s to 104.7 s), and the time taken for convergence is reduced by 1.1 s. Convergence time is determined when the following curve intersects the 5% error margin.

#### 4.2. Acceptance Circle Comparison Experiments

- (1)
- The adaptive acceptance circle strategy effectively solves the issue of switching ahead or lagging and maintains a controllable following error even when the reference path switches ahead. Figure 13 shows that the following trajectory, based on the adaptive acceptance circle, switches to the reference path segment earlier because of the larger acceptance circle radius, resulting in faster convergence to the next path segment. Figure 14 shows that the following trajectory based on the adaptive acceptance circle switches closer to the reference path because of the smaller acceptance circle radius, resulting in a reduction in the maximum following error from 0.156 m to 0.126 m.
- (2)
- The convergence of the adaptive acceptance circle strategy is faster. For point a, where ${\theta}_{a}<\pi /2$, the overall convergence time using AAC-LOS slightly increases compared with FC-LOS, as shown in Figure 13. However, during the xoy path following, the AAC-LOS can follow the reference path faster compared with FC-LOS. For point c, where ${\theta}_{c}>\pi /2$, as shown in Figure 14, the total convergence time is reduced from 1.4 s (from 40.6 s to 42 s) to 1.4 s (from 41.2 s to 42 s), a reduction of about 0.6 s.

## 5. Discussion

#### 5.1. Comparison of Improved LOS and Traditional LOS

#### 5.2. Selection of Experimental Parameters for Improved LOS

#### 5.3. Limitations and Future Work

## 6. Conclusions

- (1)
- Under the condition of a fixed acceptance circle, compared with the traditional LOS method, the average path-following error of the improved LOS method with adaptive heading control is reduced by 0.167 m, and the average convergence time is reduced by 2.05 s for a path angle of $\theta \in (0,\pi /2)$. However, when $\theta \in (\pi /2,\mathsf{\pi})$, the following error remains almost constant, and the convergence time is reduced by 0.6 s. The proposed adaptive heading control strategy significantly improves following accuracy and the convergence speed of the path following, reducing following errors caused by large switching angles and effectively following the reference path.
- (2)
- When keeping the reference path and the adaptive heading control scale coefficient unchanged, the adaptive acceptance circle strategy improves the problem of switching the reference path too early or lagging behind. This strategy fully follows the reference path and reduces the following error. When the path angle is between $\theta \in (0,\pi /2)$, the following trajectory based on the adaptive acceptance circle switches the reference path segment earlier due to the acceptance circle’s larger radius. However, this increases the following error while maintaining the controllable state and reduces the convergence time by 0.8 s. When the path angle is between $\theta \in (\pi /2,\mathsf{\pi})$, the following trajectory based on the adaptive acceptance circle is closer to the reference path because of the smaller radius. This approach reduces the maximum following error by 0.3 m and the convergence time by approximately 0.4 s, leading to shorter convergence times and smaller following errors.
- (3)
- Using the improved LOS method, the state feedback path-following control method satisfies the following requirements of convergence, reliability, and accuracy and also has advantages in following accuracy and convergence speed.
- (4)
- The proposed method is highly reliant on the selection of the scaling coefficient, m, which must be determined via testing under various conditions. The method is effective for following a path consisting of straight lines, but further research is needed to solve the problem of following a path under different types of paths.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Schematic of LOS-following control. (

**a**) Schematic of following in horizontal plane; (

**b**) schematic of following in vertical plane.

**Figure 9.**Results of 3D path following. (

**a**) 3D path following trajectory; (

**b**) projection of 3D path-following trajectory onto the xoy plane; (

**c**) projection of 3D path-following trajectory onto the xoz plane; (

**d**) comparison of following errors in 3D path-following trajectories.

**Figure 10.**Comparison of following error at point a. (

**a**) xoy plane projection; (

**b**) xoz plane projection; (

**c**) following error of trajectory.

**Figure 11.**Comparison of following error at point b. (

**a**) xoy plane projection; (

**b**) xoz plane projection; (

**c**) following error of trajectory.

**Figure 12.**Comparison of following error at point c. (

**a**) xoy plane projection; (

**b**) xoz plane projection; (

**c**) following error of trajectory.

**Figure 13.**Path following results for point a under different acceptance circles. (

**a**) xoy plane projection; (

**b**) xoz plane projection; (

**c**) following error of trajectory.

**Figure 14.**Path following results for point c under different acceptance circles. (

**a**) xoy plane projection; (

**b**) xoz plane projection; (

**c**) following error of trajectory.

Types | Description | Value |
---|---|---|

FPLIR design parameters | Mass, m/kg | 38 |

Dimensions, L × W × H/m | 1.76 × 1.76 × 1.1 | |

Distance from rotor center to fuselage center, r/m | 0.88 | |

FPLIR model parameters | Body x-axis moment of inertia, ${I}_{x}$/kg·m^{2} | 5.8 |

Body y-axis moment of inertia, ${I}_{y}$/kg·m^{2} | 5.8 | |

Body z-axis moment of inertia, ${I}_{z}$/kg·m^{2} | 8.5 | |

Atmospheric drag coefficient, ${k}_{f}$ | 1.6865 × 10^{–6} | |

Rotor torque coefficient, ${k}_{c}$ | 2.9250 × 10^{–8} |

Parameter/Unit | Value |
---|---|

$\mathrm{Fixed}\mathrm{forward}-\mathrm{looking}\mathrm{distance}\mathsf{\Delta}$/(m) | 0.5 |

$\mathrm{Fixed}\mathrm{acceptance}\mathrm{circle}\mathrm{radius}{R}_{0}$/(m) | 0.5 |

Start velocity/(m/s) | 0 |

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## Share and Cite

**MDPI and ACS Style**

Feng, T.; Lei, J.; Zeng, Y.; Qin, X.; Wang, Y.; Wang, D.; Jia, W.
Three-Dimensional Path-Following Control Method for Flying–Walking Power Line Inspection Robot Based on Improved Line of Sight. *Aerospace* **2023**, *10*, 945.
https://doi.org/10.3390/aerospace10110945

**AMA Style**

Feng T, Lei J, Zeng Y, Qin X, Wang Y, Wang D, Jia W.
Three-Dimensional Path-Following Control Method for Flying–Walking Power Line Inspection Robot Based on Improved Line of Sight. *Aerospace*. 2023; 10(11):945.
https://doi.org/10.3390/aerospace10110945

**Chicago/Turabian Style**

Feng, Tianming, Jin Lei, Yujie Zeng, Xinyan Qin, Yanqi Wang, Dexin Wang, and Wenxing Jia.
2023. "Three-Dimensional Path-Following Control Method for Flying–Walking Power Line Inspection Robot Based on Improved Line of Sight" *Aerospace* 10, no. 11: 945.
https://doi.org/10.3390/aerospace10110945