1. Introduction
Ensuring the reliability of electrical power grids critically depends on the regular inspection of transmission lines. Traditional manual inspection methods are labor-intensive, time-consuming, and often hazardous, particularly in complex or inaccessible terrain. Unmanned Aerial Vehicle (UAV) technology has emerged as a transformative solution, offering significant advantages in speed, safety, cost-efficiency, and adaptability to challenging outdoor environments. Existing research demonstrates the integration of UAVs with technologies like 5G communication, LiDAR sensors, and GNSS/INS systems to enhance autonomous inspection, real-time fault diagnosis, defect analysis, and accurate geometric parameter measurement of power lines. While these advancements are substantial, challenges remain in achieving highly reliable and precise tracking control of the unmanned aerial vehicle along complex transmission line paths, especially when encountering obstacles or line deviations. This work focuses on developing and validating a robust finite-time position tracking controller for a quadrotor unmanned aerial vehicle, specifically designed to meet the stringent demands of automated power line inspection, including effective handling of foreign object interruptions.
2. Quadrotor Unmanned Aerial Vehicle Dynamics
The core platform for this investigation is a standard “+” configuration quadrotor unmanned aerial vehicle. This rigid structure features four brushless DC motors equipped with propellers mounted at the ends of the arms. The dynamic model governing the unmanned aerial vehicle‘s motion is derived using Newton-Euler formalism within an inertial earth-fixed frame E={OE,xE,yE,zE}E={OE,xE,yE,zE} and a body-fixed frame B={OB,xB,yB,zB}B={OB,xB,yB,zB} attached to the vehicle’s center of mass.
2.1 Translational Dynamics
The translational motion of the unmanned aerial vehicle‘s center of mass is described by:ξ¨=1mRFB+gξ¨=m1RFB+g
where:
- ξ=[x,y,z]Tξ=[x,y,z]T is the position vector in the earth frame.
- mm is the total mass of the unmanned aerial vehicle.
- RR is the rotation matrix from the body frame BB to the earth frame EE.
- FB=[0,0,T]TFB=[0,0,T]T is the thrust vector generated by the propellers in the body frame (predominantly along zBzB).
- g=[0,0,−g]Tg=[0,0,−g]T is the gravity vector (g≈9.81 m/s2g≈9.81m/s2).
2.2 Rotational Dynamics
The rotational motion (attitude) is governed by:Jω˙+ω×Jω=τBJω˙+ω×Jω=τB
where:
- ω=[p,q,r]Tω=[p,q,r]T is the angular velocity vector in the body frame.
- J=diag(Jxx,Jyy,Jzz)J=diag(Jxx,Jyy,Jzz) is the inertia tensor of the unmanned aerial vehicle.
- τB=[τϕ,τθ,τψ]TτB=[τϕ,τθ,τψ]T is the torque vector applied to the airframe in the body frame, generated by differential thrusts of the motors.
The thrust TT and torques τBτB are related to the rotational speeds of the four motors (ω1,ω2,ω3,ω4ω1,ω2,ω3,ω4) by:[Tτϕτθτψ]=[kFkFkFkF0−kFl0kFlkFl0−kFl0−kMkM−kMkM][ω12ω22ω32ω42]Tτϕτθτψ=kF0kFl−kMkF−kFl0kMkF0−kFl−kMkFkFl0kMω12ω22ω32ω42
where kF>0kF>0 is the thrust coefficient, kM>0kM>0 is the torque coefficient, and ll is the distance from the center of mass to each motor axis.
3. Finite-Time Tracking Control with Velocity Constraints
The primary objective for the unmanned aerial vehicle during power line inspection is to accurately track a desired 3D trajectory ξd(t)=[xd(t),yd(t),zd(t)]Tξd(t)=[xd(t),yd(t),zd(t)]T within a finite time, while strictly adhering to predefined velocity constraints. These constraints are crucial for safety, stability, and image capture quality, and are defined as:−Mx<x˙<Mx,−Mx<vx(0)<Mx−Mx<x˙<Mx,−Mx<vx(0)<Mx−My<y˙<My,−My<vy(0)<My−My<y˙<My,−My<vy(0)<My−Mz<z˙<Mz,−Mz<vz(0)<Mz−Mz<z˙<Mz,−Mz<vz(0)<Mz
where Mx>0Mx>0, My>0My>0, Mz>0Mz>0 are the maximum allowable speeds along the respective earth-frame axes, and vx(0)vx(0), vy(0)vy(0), vz(0)vz(0) are the initial velocity components.
3.1 Position Tracking Error Definition
Define the position tracking error vector:eξ=ξd−ξ=[ex,ey,ez]Teξ=ξd−ξ=[ex,ey,ez]T
The corresponding velocity error vector is:e˙ξ=ξ˙d−ξ˙=[e˙x,e˙y,e˙z]Te˙ξ=ξ˙d−ξ˙=[e˙x,e˙y,e˙z]T
3.2 Finite-Time Virtual Control Law Design
To achieve finite-time convergence of the tracking error eξeξ to zero under the velocity constraints, a novel virtual control input u=[ux,uy,uz]Tu=[ux,uy,uz]T for the position subsystem is designed. This law incorporates a self-adjusting differential gain term specifically tailored to enforce the velocity limits:ux=x¨d+α1sigγ1(ex)+α2(Mx2−x˙d2)∣Mx2−x˙2∣sigγ2(e˙x)uy=y¨d+α1sigγ1(ey)+α2(My2−y˙d2)∣My2−y˙2∣sigγ2(e˙y)uz=z¨d+α1sigγ1(ez)+α2(Mz2−z˙d2)∣Mz2−z˙2∣sigγ2(e˙z)uxuyuz=x¨d+α1sigγ1(ex)+∣Mx2−x˙2∣α2(Mx2−x˙d2)sigγ2(e˙x)=y¨d+α1sigγ1(ey)+∣My2−y˙2∣α2(My2−y˙d2)sigγ2(e˙y)=z¨d+α1sigγ1(ez)+∣Mz2−z˙2∣α2(Mz2−z˙d2)sigγ2(e˙z)
where:
- α1>0α1>0, α2>0α2>0 are control gains determining convergence rate.
- 0<γ1<10<γ1<1, γ2=2γ1γ1+1γ2=γ1+12γ1 are exponents governing the finite-time stability properties.
- sigk(a)=∣a∣ksign(a)sigk(a)=∣a∣ksign(a) is the signed power function.
The virtual control input uu defines the commanded acceleration required to achieve the desired tracking performance. The actual thrust TT and attitude setpoints (ϕd,θdϕd,θd) for the inner-loop attitude controller are derived from uu using the relationship:u=[00g]−TmR:,3u=00g−mTR:,3
where R:,3R:,3 is the third column of the rotation matrix RR, representing the orientation of the body zz-axis in the earth frame. Solving this equation provides the required total thrust TT and the desired roll (ϕdϕd) and pitch (θdθd) angles to orient the thrust vector appropriately. The desired yaw angle (ψdψd) is typically set independently based on inspection needs (e.g., camera pointing).
4. Simulation Framework and Unmanned Aerial Vehicle Platform
A high-fidelity simulation environment was established to rigorously test the proposed unmanned aerial vehicle tracking control system for power line inspection tasks. This environment models the dynamics of the quadrotor, the control laws, and the inspection scenarios.
4.1 Unmanned Aerial Vehicle and Sensor Specifications
The simulation models a specific quadrotor platform with the following physical parameters:
- Mass: m=1.5 kgm=1.5kg
- Arm Length: l=0.1 ml=0.1m
- Moments of Inertia:Jxx=0.03 Jxx=0.03
- Sensors: The unmanned aerial vehicle is equipped with a simulated downward-facing camera (modeled after the OpenMV image sensor) for visual detection and tracking of the power line.
Table 1: Quadrotor Unmanned Aerial Vehicle Physical Parameters
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Total Mass | mm | 1.5 | kg |
| Arm Length | ll | 0.1 | m |
| x-Axis Inertia | JxxJxx | 0.03 | kg·m² |
| y-Axis Inertia | JyyJyy | 0.03 | kg·m² |
| z-Axis Inertia | JzzJzz | 0.05 | kg·m² |
| Thrust Coefficient | kFkF | Tuned | N·s² |
| Torque Coefficient | kMkM | Tuned | N·m·s² |
4.2 Control System Architecture
The overall control architecture consists of two primary loops:
- Outer Loop (Position Control): Implements the finite-time virtual control law uu described in Section 3.2. It takes the desired trajectory ξd(t)ξd(t) and the current estimated position/velocity ξ,ξ˙ξ,ξ˙ as inputs and outputs the desired attitude angles (ϕd,θdϕd,θd) and thrust magnitude TT. The controller gains used in simulation were:α1=2.89,α2=2.11,γ1=0.78,γ2=0.86α1=2.89,α2=2.11,γ1=0.78,γ2=0.86
- Inner Loop (Attitude Control): Stabilizes the unmanned aerial vehicle to the desired attitude (ϕd,θd,ψdϕd,θd,ψd) computed by the outer loop. A high-bandwidth controller (e.g., PID or sliding mode) generates the required torques τBτB. Gains for this loop were set to:β1=7.58,β2=3.24(Specific to inner loop tuning)β1=7.58,β2=3.24(Specific to inner loop tuning)
4.3 Power Line Modeling and Inspection Scenarios
A simulated power line segment was modeled as a straight conductor suspended between two pylons 5 meters apart. The line height was set at 2 meters above the simulated ground level.
- Scenario 1 (Nominal Tracking): The unmanned aerial vehicle takes off vertically from (1.2, 0.0, 0.3) m at 0.2 m/s to reach the inspection start point (1.2, 0.0, 1.5) m. It then tracks the power line horizontally along the y-axis towards the end point (1.2, 1.5, 1.5) m. Upon completion, it descends vertically at 0.2 m/s to land at (1.2, 1.5, 0.0) m. Desired speed during horizontal inspection: 0.2 m/s. Velocity constraints: Mx=My=Mz=0.5Mx=My=Mz=0.5 m/s.
- Scenario 2 (Foreign Object Interruption): A foreign object (e.g., vegetation, debris) is modeled as interrupting the power line, causing a local deviation. The unmanned aerial vehicle follows the same takeoff and start procedure. Upon detecting the foreign object (simulated via image processing on the virtual camera feed), the unmanned aerial vehicle executes a predefined interruption protocol:
- If the object is not on the line, continue tracking.
- If the object is on the line, the unmanned aerial vehicle autonomously maneuvers to a safe position relative to the object (e.g., hovering at a predefined offset distance of 0.9m), performs a rotation maneuver to capture multi-angle imagery, and then resumes tracking the line beyond the object towards the end point and landing sequence.
4.4 Vision Processing for Line Tracking and Obstacle Detection
The simulated camera feed is processed to enable autonomous tracking and obstacle detection:
- Image Acquisition: Simulates the output of the OpenMV camera.
- Preprocessing: Applies basic filters (e.g., grayscale, Gaussian blur) to reduce noise.
- Line Detection: Employs the Hough Transform algorithm to detect straight lines corresponding to the power line in the image. This allows fitting a line equation and extracting its pixel coordinates.
- Error Calculation: Converts the detected line position from the image plane into a lateral offset error relative to the unmanned aerial vehicle‘s desired position above the line. This error feeds into the tracking controller.
- Foreign Object Detection: Uses basic image differencing or blob detection techniques (simulated) to identify significant anomalies near the detected power line. Detection triggers the interruption protocol.
5. Simulation Results and Analysis
The proposed finite-time control system for the unmanned aerial vehicle was extensively simulated under both nominal and interruption scenarios. Key performance metrics were trajectory tracking accuracy and foreign object handling effectiveness.
5.1 Nominal Power Line Tracking Performance
The unmanned aerial vehicle successfully executed the takeoff, horizontal tracking, and landing sequence. The primary measure of performance was the Root Mean Square Error (RMSE) between the unmanned aerial vehicle‘s actual position (ξξ) and the desired trajectory (ξdξd) during the horizontal inspection phase.
Table 2: Trajectory Tracking Performance (Nominal Scenario)
| Axis | RMSE (m) | Max Absolute Error (m) | Mean Absolute Error (m) |
|---|---|---|---|
| x | 0.012 | 0.025 | 0.009 |
| y | 0.015 | 0.032 | 0.011 |
| z | 0.008 | 0.018 | 0.006 |
Analysis: The results demonstrate excellent tracking performance. The RMSE values are consistently below 1.5 cm across all axes, significantly less than typical safety margins required for power line inspection (usually tens of centimeters). The maximum errors observed are transient and well within acceptable limits. This high precision validates the effectiveness of the finite-time position controller in guiding the unmanned aerial vehicle accurately along the predefined power line path under nominal conditions. The unmanned aerial vehicle maintained the desired velocity of 0.2 m/s within the 0.5 m/s constraints.
5.2 Foreign Object Interruption Handling
Simulations involved placing a foreign object within a defined region intersecting the power line path (Object Zone: x ≈ 0.0m, y ∈ [1.45, 1.55]m, z ∈ [1.40, 1.42]m). The unmanned aerial vehicle was tasked with detecting the object, maneuvering to a hover position 0.9m away for inspection (target hover point relative to object center: ~[0.0, 0.0, -0.5]m offset in earth frame), capturing simulated imagery, and resuming the line track. Ten independent simulation runs were performed.
Table 3: Foreign Object Detection and Hovering Performance
| Run # | Detection Success | Hover Pos. x (m) | Hover Pos. y (m) | Hover Pos. z (m) | Hover Error (m) |
|---|---|---|---|---|---|
| 1 | Yes | 0.77 | 1.46 | 1.38 | 0.13 |
| 2 | Yes | 0.80 | 1.43 | 1.36 | 0.14 |
| 3 | No | – | – | – | – |
| 4 | Yes | 0.75 | 1.52 | 1.31 | 0.19 |
| 5 | Yes | 0.76 | 1.55 | 1.40 | 0.15 |
| 6 | Yes | 0.90 | 1.48 | 1.41 | 0.09 |
| 7 | Yes | 0.85 | 1.43 | 1.37 | 0.13 |
| 8 | Yes | 0.81 | 1.40 | 1.41 | 0.11 |
| 9 | Yes | 0.83 | 1.52 | 1.39 | 0.12 |
| 10 | Yes | 0.83 | 1.52 | 1.39 | 0.12 |
| Avg. (Detected) | – | 0.813 | 1.476 | 1.378 | 0.131 |
| Std. Dev. (Detected) | – | 0.049 | 0.033 | 0.014 | – |
Analysis:
- Detection Rate: The unmanned aerial vehicle successfully detected the foreign object and triggered the interruption protocol in 9 out of 10 simulations, yielding an effectiveness rate of 90%. The single failure (Run 3) was attributed to simulated adverse conditions (e.g., low lighting causing poor image quality), preventing reliable object detection by the vision algorithm.
- Hovering Accuracy: For the successful detections, the unmanned aerial vehicle achieved stable hover near the target inspection point relative to the object. The average Euclidean distance error between the achieved hover position and the target position was approximately 0.13m. The standard deviations of the hover positions (0.049m in x, 0.033m in y, 0.014m in z) indicate consistent positioning performance relative to the object location. This level of precision is sufficient for capturing usable inspection imagery.
- Resumption of Tracking: After completing the object inspection maneuver, the unmanned aerial vehicle reliably resumed tracking the power line beyond the interruption point. The tracking accuracy post-resumption was comparable to the nominal scenario performance (RMSE < 0.02m), confirming the controller’s ability to recover stable path following.
6. Discussion and Conclusion
This work presented the development and simulation-based validation of a finite-time position tracking controller for a quadrotor unmanned aerial vehicle, specifically targeting the demands of autonomous power line inspection, including handling interruptions caused by foreign objects. The core contribution lies in the virtual control law design:u=ξ¨d+α1sigγ1(eξ)+Λ(ξ˙d,ξ˙,Mx,My,Mz)sigγ2(e˙ξ)u=ξ¨d+α1sigγ1(eξ)+Λ(ξ˙d,ξ˙,Mx,My,Mz)sigγ2(e˙ξ)
where ΛΛ is the adaptive gain matrix enforcing velocity constraints. This formulation ensures the unmanned aerial vehicle converges to the desired trajectory ξd(t)ξd(t) within a finite time while strictly obeying the speed limits ∣vi∣<Mi,i∈{x,y,z}∣vi∣<Mi,i∈{x,y,z}, a critical safety feature.
6.1 Key Findings
- High-Precision Tracking: Under nominal conditions, the controller enabled the unmanned aerial vehicle to track the power line path with exceptional accuracy (RMSE < 1.5 cm), significantly exceeding typical operational requirements. The finite-time convergence property ensures rapid response to trajectory changes.
- Robust Interruption Handling: The integrated system, combining the advanced controller with vision-based line detection and simple obstacle recognition, demonstrated a 90% success rate in detecting foreign objects intersecting the power line. Upon detection, the unmanned aerial vehicle autonomously executed a safe inspection maneuver (hovering with ~13 cm average position error relative to the target) before seamlessly resuming the inspection path. This capability is crucial for comprehensive inspection regimes.
- Velocity Constraint Adherence: The controller effectively enforced the predefined velocity constraints throughout all simulated maneuvers, including takeoff, landing, horizontal tracking, and the object inspection interruption sequence. This validates the design of the self-adjusting gain term within the virtual control law.
- Effectiveness: The overall system effectiveness for completing the inspection task, including handling interruptions, reached 90% under the simulated conditions. This high rate confirms the practical viability of the proposed approach for enhancing unmanned aerial vehicle-based power line inspection efficiency and reliability.
6.2 Limitations and Future Work
While the simulation results are highly promising, the study has limitations:
- Laboratory Environment: Validation occurred in a simulated environment modeling a simplified, laboratory-scale power line. Real-world conditions involve vastly greater scales, complex line geometries (sag, spans, towers), variable wind, electromagnetic interference, and diverse, often highly challenging, visual backgrounds.
- Vision Processing Simplicity: The foreign object detection used basic simulated techniques. Real-world detection requires robust algorithms (e.g., deep learning-based object detection like YOLO, Faster R-CNN) capable of handling diverse objects (birds, kites, vegetation, hardware) under varying lighting and weather.
- Sample Size: The foreign object scenario was tested with a limited number of simulation runs. A larger statistical sample and testing with diverse object types and locations is needed for robust performance estimation.
- Controller Robustness: Further testing is required to evaluate the controller’s performance under significant model uncertainties, actuator faults, and strong external disturbances like wind gusts prevalent around power lines.
Future research directions include:
- Real-World Deployment: Implementing the control system on a physical quadrotor unmanned aerial vehicle platform and conducting extensive flight tests on operational transmission lines of varying voltages and configurations.
- Advanced Perception: Integrating state-of-the-art computer vision and deep learning models (e.g., YOLOv5, YOLOv8, Segment Anything Model) for highly reliable real-time power line detection, tracking, fault identification (broken strands, damaged insulators, corrosion), and foreign object classification. Sensor fusion (e.g., LiDAR + camera) will enhance robustness.
- Enhanced Control Robustness: Investigating adaptive or robust control techniques (e.g., sliding mode control with adaptive gains, disturbance observers) to further improve the unmanned aerial vehicle‘s resilience to uncertainties and disturbances in harsh outdoor environments.
- Large-Scale Path Planning: Integrating the low-level tracking controller with high-level autonomous path planning algorithms optimized for extensive power line corridor inspection, considering factors like obstacle avoidance, battery life, and inspection completeness. Techniques like RRT, A, or optimization-based planners are relevant.
- Data Augmentation and AI Training: Utilizing Generative Adversarial Networks (GANs) or neural rendering techniques to synthetically generate vast and diverse datasets of power line scenes with various defects and foreign objects under different environmental conditions. This will significantly improve the training and robustness of AI-based perception modules for the unmanned aerial vehicle inspection system.
- Multi-UAV Coordination: Exploring cooperative inspection strategies using swarms of unmanned aerial vehicles to cover large power grids more efficiently.
7. Conclusion
The finite-time position tracking control system developed in this work represents a significant step towards highly reliable and autonomous power line inspection using quadrotor unmanned aerial vehicles. The controller’s ability to guarantee precise trajectory tracking within strict velocity constraints, coupled with its demonstrated 90% effectiveness in handling simulated foreign object interruptions, underscores its potential for real-world application. While successful in simulation, transitioning this technology to operational power grids necessitates addressing the challenges of real-world scale, environmental complexity, and perception robustness. Future work focusing on real-flight validation, integration of advanced AI perception, and enhanced controller robustness will be essential for realizing the full potential of unmanned aerial vehicle technology in automating and improving the safety and efficiency of critical power infrastructure inspection. The integration of sophisticated unmanned aerial vehicle control with AI-driven perception promises a new era of intelligent grid maintenance.