# Performance Analysis of Localization Algorithms for Inspections in 2D and 3D Unstructured Environments Using 3D Laser Sensors and UAVs

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Methodology

#### 3.1. Maps

#### 3.2. 2D Localization

- X, current set of particles.
- ${X}_{t-1}$, the previous set of particles.
- ${U}_{t}$, last motion and odometry measurements.
- ${Z}_{t}$, last laser rangefinder measurements.
- m, max number of particles.

- Motion kinematics: e.g., differential or omnidirectional.
- Uncertainty of the robot odometry, which determines the error in translation ${\alpha}_{1},{\alpha}_{2}$ or rotation ${\alpha}_{3},{\alpha}_{4},{\alpha}_{5}$.
- Observation model: e.g., beam model or probability field model.
- Measurements errors: e.g., measurement noise ${z}_{hit}$, unexpected objects ${z}_{short}$, object detection failures ${z}_{max}$, unexplained random noise ${z}_{rand}$.
- Number of random particles, defined by the probabilities of long-term ${\alpha}_{slow}$ and short-term ${\alpha}_{fast}$ measurements.

#### 3.3. 3D Localization

- Subsampling: Reduction of the number of points in the sample using Voxel grid filter and Pass-through Filter techniques [49] to optimize processing time.
- Pose estimation with nonlinear ICP: Based on the ICP technique presented in [50]. It takes as inputs a source and a target point cloud matched under the nearest-neighbor criterion. Singular Value Decomposition (SVD) is applied to obtain an estimate of the transformation matrix that aligns them. This process is repeated until a termination criterion is met, removing outliers and redefining the correspondences.
- Unscented Kalman Filter-based localization: This is an improvement of EKF for application to highly nonlinear systems. This approach uses the unscented transform to take a set of samples called sigma points, which are propagated by nonlinear functions and used to calculate the mean and variance. Unlike EKF, UKF eliminates the need for a Jacobian, facilitating calculations on complex functions.

Algorithm 1: Localization 3D (${X}_{t-1}$, ${V}_{t}$, ${\Delta}_{qt}$, $3D\_sensor\_points$, $3D\_map$) |

## 4. Results

#### 4.1. Software

#### 4.2. Hardware

- Personal Computer (PC) Intel core i7 2.70 GHz, 4 Gb RAM.
- Hummingbird UAV provided by Rotors (Figure 6a).
- Odometry measurements from a configurable odometry sensor (see Appendix A for configuration details).
- Velodyne VLP16 3D laser sensor [55] provides a point cloud with 300,000 points per second and ±3 cm accuracy, 100 m range, 360° horizontal and 30° vertical field of view (see Appendix B for setting other parameters). Due to computational limitations in the simulation, we worked with 5120 points per second.

- Manifold Intel i7 1.8 Ghz, 2 GB RAM.
- DJI Matrice 100 UAV (Figure 6c).
- Odometry measurements were provided by a sensor fusion algorithm [56] that merges the onboard DJI sensors: Altimeter, Velocity and IMU, plus the 3D Light Detection and Ranging (LIDAR) IMU.
- 3D LIDAR Ouster OS1-64, providing a point cloud with 327,680 points per second and ±1 cm accuracy 100 m range, 0.3 cm resolution, 360° horizontal and 45° vertical field of view.

#### 4.3. Error Metric

- ${T}_{i}$, is the magnitude of the Euclidean distance along the horizontal plane between the estimated and ground-truth poses at frame i.
- n, number of frames.

#### 4.4. 2D Localization

#### 4.4.1. Simulation Tests

#### 4.4.2. Real Tests

#### 4.5. 3D Localization

#### 4.5.1. Simulation Tests

#### 4.5.2. Real Tests

## 5. Discussions

#### 5.1. 2D Localization

#### 5.2. 3D Localization

## 6. Conclusions

#### 6.1. 2D Localization

#### 6.2. 3D Localization

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

UAV | Unmanned Aerial Vehicle |

CAD | Computer-Aided Design |

3D | Three Dimensions |

2D | TWO Dimensions |

UKF | Unscented Kalman Filter |

EKF | Extended Kalman Filter |

IMU | Inertial Measurement Units |

GPS | Global Positioning System |

ICP | Iterative Closest Point Algorithm |

ICP-NL | Noun lineal Iterative Closest Point Process |

NDT | Normal Transformation Distribution |

SLAM | Simultaneous Localization and Mapping) |

MCL | Monte Carlo Localization |

AMCL | Adaptive Monte Carlo Localization |

ROS | Robot Operative System |

DAE | Digital Asset Exchange |

URDF | Unified Robot Description Format |

ATE | Absolute Trajectory Error |

MSE | Mean Square Error |

PCL | Point Cloud Laser Library |

GT | Ground Truth |

PC | Personal Computer |

SVD | Singular Value Decomposition |

LIDAR | Light Detection and Ranging |

## Appendix A. Hummingbird Simulated Odometry Sensor

file: /rotors_simulator/rotors_description/urdf/mav_generic_odometry_sensor.gazebo mass_odometry_sensor = 0.00001 measurement_divisor = 1 measurement_delay = 0 unknown_delay = 0.0 noise_normal_position = 0 0 0 noise_normal_quaternion = 0 0 0 noise_normal_linear_velocity = 0 0 0 noise_normal_angular_velocity = 0 0 0 noise_normal_position = 0.01 0.01 0.01 noise_normal_quaternion = 0.017 0.017 0.017 noise_uniform_linear_velocity = 0 0 0 noise_uniform_angular_velocity = 0 0 0 enable_odometry_map=false inertia ixx = 0.00001 ixy = 0.0 ixz = 0.0 iyy = 0.00001 iyz = 0.0 izz = 0.00001 [kg m^2] origin xyz=0.0 0.0 0.0 rpy=0.0 0.0 0.0

## Appendix B. Velodyne VLP16 Simulated 3D Laser Sensor

update rate in hz = 10 samples = 512 minimum range value in meters = 0.9 maximum range value in meters = 130 noise Gausian in meters = 0.008 minimum horizontal angle in radians = -3.14 maximum horizontal angle in radians = 3.14

## Appendix C. AMCL Simulations Parameters

odom_model_type value=omni-corrected

laser_max_beams value=30 min_particles value=200 max_particles value=3000 kld_err value=0.05 kld_z value=0.99

odom_alpha1 value=0.2 odom_alpha2 value=0.2 odom_alpha3 value=0.8 odom_alpha4 value=0.2 odom_alpha5 value=0.2

laser_likelihood_max_dist value=2 laser_z_hit value=0.5 laser_z_short value=0.05 laser_z_max value=0.05 laser_z_rand value=0.5 laser_sigma_hit value=0.2 laser_lambda_short value=0.1 laser_model_type value=likelihood_field

update_min_d value=0.2 update_min_a value=0.5 resample_interval value=1 transform_tolerance value=1.0 recovery_alpha_slow value=0.001 recovery_alpha_fast value=0.1

initial_cov_xx value=0.5 initial_cov_yy value=0.5 initial_cov_aa value=0.15

## Appendix D. AMCL Real Fligts Parameters

odom_model_type value=omni-corrected

laser_max_beams value=50 min_particles value=200 max_particles value=3000 kld_err value=0.01 kld_z value=0.8

odom_alpha1 value=0.3 odom_alpha2 value=0.3 odom_alpha3 value=0.05 odom_alpha4 value=0.05 odom_alpha5 value=0.3

laser_likelihood_max_dist value=0.5 laser_z_hit value=0.8 laser_z_short value=0.05 laser_z_max value=0.05 laser_z_rand value=0.5 laser_sigma_hit value=0.2 laser_lambda_short value=0.1 laser_model_type value=likelihood_field

update_min_d value=0.1 update_min_a value=0.1 resample_interval value=1 transform_tolerance value=1.0 recovery_alpha_slow value=0.001 recovery_alpha_fast value=0.1

initial_cov_xx value=25 initial_cov_yy value=25 initial_cov_aa value=0.15

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**Figure 1.**Kind of maps to be used. (

**a**) Three-dimensional Occupancy grid Map (Octomap). (

**b**) Two-dimensional Occupancy grid Map. (

**c**) Graph SLAM map.

**Figure 2.**Example of 2D localization using AMCL algorithm (red), 2D laser Scan (green) and an Occupancy grid map (black).

**Figure 4.**Example of 3D localization using nonlinear ICP algorithm (white), 3D laser Scan (green) and a 3D Occupancy grid map (red).

**Figure 6.**Components used in the simulation (

**a**,

**b**), and real flights (

**c**,

**d**). (

**a**) Velodyne sensor and Hummingbird UAV. (

**b**) Aircraft Airbus A330 model. (

**c**) Three-dimensional LIDAR Ouster and DJI Matrice 100 UAV. (

**d**) Airbus A330 [58].

**Figure 7.**2D occupancy grip map creation process. (

**a**) Airbus A330 Gazebo Model. (

**b**) Octomap generated with 0.1 cm voxel resolution. (

**c**) Octomap horizontal projection. (

**d**) Modified 2D occupancy grip map.

**Figure 8.**AMCL convergence process. Two-dimensional laser scan (blue), AMCL particle cloud (red). (

**a**) UAV takes off, particles are distributed all over the map. (

**b**) UAV moves forward, particle scattering begins to decrease The 2D laser scan does not match the map. (

**c**) AMCL converges, particle scattering is small, and the laser data matches the map.

**Figure 9.**(

**a**) AMCL simulation test with the lowest ATE. AMCL (blue), ground-truth (green), take-off position (blue circle), error (red). (

**b**) AMCL errors.

**Figure 10.**(

**a**) AMCL simulation test with the highest ATE. AMCL (blue), ground-truth (green), take-off position (blue circle), error (red). (

**b**) AMCL errors.

**Figure 11.**Proposed initial flight (green), Aircraft (cyan), take-off point (red). (

**a**) Squared path. (

**b**) Circular path.

**Figure 12.**Two-dimensional occupancy grip map creation process. (

**a**) Truck Gazebo Model. (

**b**) Octomap generated with 0.1 cm voxel resolution. (

**c**) Octomap horizontal projection. (

**d**) Modified 2D occupancy grip map.

**Figure 14.**(

**a**) AMCL simulation test with the lowest ATE. AMCL (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) AMCL errors.

**Figure 15.**(

**a**) AMCL simulation test with the highest ATE. AMCL (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) AMCL errors.

**Figure 16.**Proposed trajectories for real flights. Take-off and landing point (red circle). (

**a**) Path 1. (

**b**) Path 2. (

**c**) Path 3.

**Figure 18.**Wall removal. Three-dimensional LIDAR point Cloud (blue), 2D laser scan (red). (

**a**) Original 2D laser scan data and 3D LIDAR point cloud. (

**b**) Wall segmentation by RANSAC PCL. (

**c**) Wall segmentation by limiting 2D laser scan data. Range [0.1, 15] meters, angular field of view [−135, 135] degrees.

**Figure 19.**Convergence tests on Path 1 (Figure 16a) real flight with limited laser ranges. Three-dimensional LIDAR point cloud (blue), 2D laser scans (red), AMCL particles (green). (

**a**) UAV position before algorithm convergence. (

**b**) UAV position when the algorithm converged at 37 iterations.

**Figure 20.**Convergence tests on Path 1 (Figure 16a) real flight with RANSAC PCL segmentation. Three-dimensional LIDAR point cloud (blue), 2D laser scans (red), AMCL particles (green). (

**a**) UAV position before algorithm convergence. (

**b**) UAV position when the algorithm converges after 250 iterations.

**Figure 21.**Convergence tests on Path 2 (Figure 16b) real flight with RANSAC PCL segmentation. Three-dimensional LIDAR point cloud (blue), 2D laser scans (red), AMCL particles (green). (

**a**) UAV position before algorithm convergence. (

**b**) UAV position when the algorithm converges after 71 iterations.

**Figure 22.**Convergence tests on Path 3 (Figure 16c) real flight with RANSAC PCL segmentation. Three-dimensional LIDAR point cloud (blue), 2D laser scans (red), AMCL particles (green). (

**a**) UAV position before algorithm convergence. (

**b**) The algorithm does not converge during the flight.

**Figure 24.**Position estimate by the algorithms Non-linear ICP (

**a**) and NDT (

**b**). Three-dimensional graph SLAM map (red), position estimate by the algorithm (white), ground truth (green).

**Figure 25.**ICP-NL simulation test with the lowest ATE. (

**a**) Trajectories. ICP-N (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) ICP-N errors.

**Figure 26.**ICP-NL simulation test with the highest ATE. (

**a**) Trajectories. ICP-NL (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) ICP-N errors.

**Figure 28.**Pose estimation by ICP-NL. Three-dimensional map (red), ground truth path (green), ICP-NL path (blue).

**Figure 29.**ICP-NL simulation test with the lowest ATE. (

**a**) Trajectories. ICP-N (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) ICP-N errors.

**Figure 30.**ICP-NL simulation test with the highest ATE. (

**a**) Trajectories. ICP-NL (blue), ground truth (green), take-off position (blue circle), error (red). (

**b**) ICP-N errors.

Altitude (m) | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

Mean ATE (m) | 0.3460 | 0.3287 | 0.3127 | 0.3376 | 0.4077 | 2.1437 |

Tests | ATE Min (m) | Ate Max (m) | Mean ATE (m) |
---|---|---|---|

81 | 0.18 | 5.63 | 0.34 |

ATE | GT Initial Pos (x,y) (m) | AMCL Initial Pos (x,y) (m) | Initial Covariance (xx,yy) (m) | Initial Error (x,y) (m) |
---|---|---|---|---|

Min ATE 0.18 | 32.53, −27.83 | 32.91, −27.5 | 0.5, 0.5 | −0.38, −0.33 |

Max ATE 5.63 | 33.72, −29 | 36.01, −26.56 | 0.5, 0.5 | −2.29, −2.44 |

**Table 4.**Relationship between the ATE, covariance in x and y directions, and matches of occupied grids cells and 2D laser scan points.

Error between AMCL and Ground Truth (m) | Mean Covariance(x,y) | Mean Occupancy Grip Map | Mean 2D Laser Scan | Mean Relation Matches/Laser-Scan |
---|---|---|---|---|

<=0.3 | 0.0557 | 41,820 | 245,917.5 | 0.17005 |

ATE | GT Initial Pos (x,y) (m) | AMCL Initial Pos (x,y) (m) | Initial Covariance (xx,yy) (m) | Initial Error (x,y) (m) |
---|---|---|---|---|

Min ATE 0.86 | 20.66, 7.16 | 20.77 , 5.56 | 0.5, 0.5 | −0.11, −1.6 |

Max ATE 1.4 | −4.82, −5 | −4.77, −4.6 | 0.5, 0.5 | −0.05, −0.4 |

**Table 6.**Relationship between the ATE, covariance in x and y, and matches of occupied grid cells and 2D laser scan points in the truck inspection.

Error between AMCL and Ground Truth (m) | Mean Covariance (x,y) | Mean Occupancy Grip Map | Mean 2D Laser Scan | Mean Relation Matches/Laser-Scan |
---|---|---|---|---|

<=0.3 | 0.0115 | 18,037 | 76,847 | 0.2347 |

Algorithm | ATE |
---|---|

ICP-NL | 0.68 |

NDT | 289.93 |

**Table 8.**Aircraft simulation position estimation errors for ICP-NL and NDT algorithms for different altitudes.

Algorithm | Height (m) | ATE (m) | Convergence Time (s) |
---|---|---|---|

ICP-NL | 3.5 | 0.7265 | 2.3 |

4.5 | 0.7077 | 2.8 | |

5.5 | 0.7442 | 4 | |

6.5 | 0.6857 | 3.1 | |

7.5 | 0.6334 | 1.6 | |

8.5 | 0.552 | 4.1 | |

NDT | 3.5 | 149.6588 | 181.4 |

4.5 | 511.6744 | 57.7 | |

5.5 | 363.2043 | 103.3 | |

6.5 | 133.5604 | 50.9 | |

7.5 | 0.6597 | 7 | |

8.5 | 1.6485 | 21.5 |

Algorithm | Height (m) | ATE (m) | Convergence Time (s) |
---|---|---|---|

ICP-NL | 2.5 | 0.3801 | 2.3 |

3.5 | 0.5552 | 2.8 | |

4.5 | 0.5442 | 4 | |

5.5 | 0.6857 | 3.1 |

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## Share and Cite

**MDPI and ACS Style**

Espinosa Peralta, P.; Luna, M.A.; de la Puente, P.; Campoy, P.; Bavle, H.; Carrio, A.; Cruz Ulloa, C. Performance Analysis of Localization Algorithms for Inspections in 2D and 3D Unstructured Environments Using 3D Laser Sensors and UAVs. *Sensors* **2022**, *22*, 5122.
https://doi.org/10.3390/s22145122

**AMA Style**

Espinosa Peralta P, Luna MA, de la Puente P, Campoy P, Bavle H, Carrio A, Cruz Ulloa C. Performance Analysis of Localization Algorithms for Inspections in 2D and 3D Unstructured Environments Using 3D Laser Sensors and UAVs. *Sensors*. 2022; 22(14):5122.
https://doi.org/10.3390/s22145122

**Chicago/Turabian Style**

Espinosa Peralta, Paul, Marco Andrés Luna, Paloma de la Puente, Pascual Campoy, Hriday Bavle, Adrián Carrio, and Christyan Cruz Ulloa. 2022. "Performance Analysis of Localization Algorithms for Inspections in 2D and 3D Unstructured Environments Using 3D Laser Sensors and UAVs" *Sensors* 22, no. 14: 5122.
https://doi.org/10.3390/s22145122