# Performance Analysis of NDT-based Graph SLAM for Autonomous Vehicle in Diverse Typical Driving Scenarios of Hong Kong

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## Abstract

**:**

## 1. Introduction

- (1)
- This paper proposes to generate the 3D building models of the tested area to define the degree of urbanization of the given scenario. The Skyplot is generated as an indicator of the degree of urbanization and the corresponding definition is presented. The classification criteria of different urban scenarios are proposed using Skyplot features.
- (2)
- (3)
- This paper qualitatively analyzes the relationship between the performance of NDT-based graph SLAM and the traffic conditions and degree of urbanization. The evaluated results related to the traffic and performance of LiDAR-based positioning can be a useful basic work for further mitigating the effects of traffic and urbanization to improve the performance of LiDAR-based positioning.

## 2. The Transformation from LiDAR-based Mapping

#### 2.1. Transformation Calculation

**R**and

**T**indicates the rotation and translation matrix respectively, to transform the input point cloud (${p}_{i}$) into the reference point cloud (${q}_{i}$). Objective function $C(\widehat{R},\widehat{T})$ indicates the transformation error.

- (1)
- Normal distribution transform: fetch all the points ${x}_{i=1\dots n}$ cotained in 3D the cell.Calculate the mean among all the points, $q=\frac{1}{n}{\displaystyle \sum}_{i}{x}_{i}$.Calculate the covariance matrix $\mathit{\mu}$,$$\text{}\mathit{\mu}=\frac{1}{n}{{\displaystyle \sum}}_{i}({x}_{i}-q){({x}_{i}-q)}^{T}\text{}$$
- (2)
- The matching score is modeled as:$$\text{}\mathrm{f}(\mathrm{p})=-\mathrm{score}(\mathrm{p})={{\displaystyle \sum}}_{i}\mathrm{exp}(-\frac{{({x}_{i}{}^{\prime}-{q}_{i})}^{T}{\mathit{\mu}}_{i}{}^{-1}({x}_{i}{}^{\prime}-{q}_{i})}{2})\text{}$$
- (3)
- Update the pose using the Quasi-Newton method using the objective function to minimize the score $\mathrm{f}(\mathrm{p})$.
- (4)
- Repeat the steps (2) and (3) until the convergence is achieved.

#### 2.2. Uncertainty Estimation of Transformation

- (1)
- The degree of matching between the two consecutive frames of point clouds.
- (2)
- The used time to complete the transformation calculation.
- (3)
- The iteration number used to make the Quasi-Newton method converge.

**I**is a unit matrix, ${\mathrm{C}}_{p}^{2}$ and ${\mathrm{C}}_{r}^{2}$ are heuristically determined coefficients for adjusting the covariance of translation and rotation, respectively. In this case, the covariance of the transformation calculation is correlated with the degree of matching, time used for the NDT convergence and times of iteration.

## 3. Graph-based SLAM

#### 3.1. Graph Generation

#### 3.2. Graph Optimization

## 4. Experimental Evaluation

- (1)
- Sparse area: (a) Sparse area with normal traffic. (b) The sparse area with dense traffic.
- (2)
- Sub-urban area: (a) Sub-urban area with normal traffic. (b) The sub-urban area with dense traffic. (presented in the Appendix A)
- (3)
- Dense urban area: (a) Dense urban area with normal traffic. (b) The dense urban area with dense traffic.

**Definition**

**1.**

**Definition**

**2.**

#### 4.1. Experimental Setup

#### 4.2. Experiment in Sparse Area

#### 4.2.1. Experiment 1: Performance Evaluation of NDT-based Graph SLAM in Sparse Area with Normal Traffic

#### 4.2.2. Experiment 2: Performance Evaluation of NDT-based Graph SLAM in Sparse Area with Dense Traffic

#### 4.3. Experiment in Dense Urban Area

#### 4.3.1. Experiment 3: Performance Evaluation of NDT-based Graph SLAM in Dense Urban Area with Normal Traffic

#### 4.3.2. Experiment 4: Performance Evaluation of NDT-based Graph SLAM in Dense Urban Area with Dense Traffic

## 5. Discussion, Conclusion and Future Work

**Traffic condition and accuracy of NDT-based graph SLAM**: The detailed analysis of the relationship between the traffic conditions and the accuracy of LiDAR-based positioning is shown in Figure 14, which shows the results in two different degrees of traffic conditions. According to the presented six experiments (including two experiments presented in the Appendix A); the accuracy of the SLAM is degraded with increased traffic density. For example, the mean 3D positioning error increased from 1.58 m (experiment 5 with normal traffic) to 1.91 m (experiment 6 with dense traffic). This phenomenon is also the same in the sparse area and dense urban areas. The main reason causing this degradation in SLAM performance is the moving objects in traffic, such as the double-decker bus, cars, and trucks. Our previous research [20] shows that the height of the double-decker bus can go up to 4.5 m in Hong Kong and, thus, it takes up the majority of the field of view (FOV) of 3D LiDAR. The double-decker bus is a moving object on the roads. In this case, the majority of the 3D point clouds are scanned from the moving objects. The points from moving objects can distort the mapping between two consecutive frames of point clouds, thus impairing the accuracy of SLAM. We can also see that the positioning error gradient increases with enhanced traffic density. Overall, traffic conditions have negative effects on the accuracy of NDT-based SLAM. In other words, more dynamic environments with more moving objects introduce more degradation in the positioning accuracy of NDT-based graph SLAM. The evaluated results related to the traffic and accuracy of NDT-based SLAM can be a good benchmark for further mitigating the effects of traffic to improve the accuracy of LiDAR-based positioning.

**Traffic condition and reliability estimation**: As presented in Experiment 4, it tends to overestimate the uncertainty of SLAM in a normal traffic scenario and underestimate that in dense traffic scene. In other words, the dense traffic scenes introduce larger uncertainty. This result again shows that the traffic has a bad impact on the performance of NDT-based graph SLAM. Proper methods to cope with the dynamic objects are needed to effectively estimate the uncertainty caused by dense traffic.

**The degree of urbanization and accuracy of NDT-based graph SLAM**: The detailed analysis of the relationship between the degree of urbanization and the accuracy of NDT-based graph SLAM is shown in Figure 15. According to the six experiments, three levels of areas classified based on the degree of urbanization are presented. We can see from Figure 15, the 3D gradient in sub-urban is similar to that in the sparse area. However, the 3D gradient in dense urban is significantly larger than that in both sparse areas and sub-urban. The main reason for this result is the environment features availability. In the sub-urban and sparse areas experiments, the main features are buildings, moving objects, and some trees, which means that abundant features are available. In the dense urban area, the main features are tall buildings and moving objects, which means less feature availability. Moreover, we can find that the 3D positioning error in altitude direction increases dramatically with an increased degree of urbanization which can be seen by comparing with Experiment 1 and 3. In total, the increased density of urbanization can degrade the accuracy of SLAM-based positioning, especially in the altitude direction. To effectively model the uncertainty of LiDAR-based positioning, the surrounding environment features are needed to be considered, for example, the degree of urbanization. The 3D building model generated in Figure 2 is a potential resource which contains the models of buildings to improve the accuracy of NDT-based graph SLAM. Inspired by this, we are going to employ the 3D building model to facilitate the effects estimation of urbanization on the performance of LiDAR-based positioning.

**The degree of urbanization and reliability estimation of NDT-based graph SLAM**: as discussed in experiment 4, the reliability estimation of NDT-based SLAM is highly related to the traffic condition. However, there are no obvious relationships between the reliability estimation and the degree of urbanization according to the presented experiments. In total, the sub-urban area has the smallest mean uncertainty and the dense urban area possesses the largest estimated uncertainty.

**Future work:**we aim to employ moving objects detection and 3D building models to improve the performance of NDT-based graph SLAM. Moreover, the uncertainty estimation of LiDAR-based SLAM will be conducted by considering both the traffic conditions and 3D building models.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Experiment in Sub-Urban Area

#### Appendix A.1.1. Experiment 5: Performance Evaluation of NDT-Based Graph SLAM in Sub-Urban Area with Normal Traffic

**Figure A1.**Experiment 5: the trajectory of NDT-based graph SLAM in a sub-urban area with normal traffic condition. The left panel indicates the generated points map and trajectory from SLAM. Right panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory.

**Figure A2.**Experiment 5: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions, respectively. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Table A1.**Experiment 5: Performance of NDT-based graph SLAM in a sub-urban area with normal traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 0.91 | 0.54 | 0.15 | 4.69 | 1.44 | 0.023 | 1.58 | 0.025 |

Std | 0.79 | 0.48 | 0.11 | 6.09 | 1.05 | 0.016 | 1.11 | 0.016 |

#### Appendix A.1.2. Experiment 6: Performance Evaluation of NDT-Based Graph SLAM in Sub-Urban Area with Dense Traffic

**Figure A3.**Experiment 6: the trajectory of NDT-based graph SLAM in a sub-urban area with dense traffic condition. The left panel indicates the generated points map and trajectory from SLAM. Top panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory.

**Figure A4.**Experiment 6: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions separately. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Table A2.**Experiment 4: Performance of NDT-based graph SLAM in a sub-urban area with dense traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 0.85 | 0.694 | 0.367 | 5.26 | 1.54 | 0.024 | 1.91 | 0.029 |

Std | 0.59 | 0.537 | 0.25 | 5.05 | 0.94 | 0.015 | 1.01 | 0.016 |

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**Figure 4.**Scenarios with different traffic conditions. The green boxes indicate the surrounding dynamic vehicles. The left figure shows the scenario with normal traffic and right figure with dense traffic.

**Figure 5.**Sensors setup of the vehicle: 3D LiDAR sensor is installed on the top of the vehicle. GNSS RTK/INS integrated navigation system is installed on the top of the vehicle next to 3D LiDAR. The GNSS RTK/INS in ENU coordinate system is used to provide the ground truth of vehicle’s position.

**Figure 6.**Experiment 1: the trajectory of NDT-based graph SLAM in a sparse area with normal traffic condition. Top panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory. The bottom panel indicates the generated points map and trajectory from SLAM and ground truth.

**Figure 7.**Experiment 1: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions, respectively. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Figure 8.**Experiment 2: the trajectory of NDT-based graph SLAM in a sparse area with dense traffic condition. Top panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory. The bottom panel indicates the generated points map and trajectory from SLAM.

**Figure 9.**Experiment 2: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions, respectively. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Figure 10.**Experiment 3: the trajectory of NDT-based graph SLAM in a dense urban area with normal traffic condition. Top panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory. The bottom panel indicates the generated points map and trajectory from SLAM.

**Figure 11.**Experiment 3: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions, respectively. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Figure 12.**Experiment 4: the trajectory of NDT-based graph SLAM in a dense urban area with dense traffic condition. Top panel represents the snapshot in Google Maps. The black curve indicates the ground truth of the vehicle’s trajectory. The bottom panel indicates the generated points map and trajectory from SLAM.

**Figure 13.**Experiment 4: positioning error and reliability estimation result. The top panel represents the positioning error in lateral, longitudinal and altitude directions, respectively. The bottom panel represents the estimated reliability and 3D positioning error of SLAM.

**Figure 14.**The relationship between the traffic conditions and the performance of NDT-based graph SLAM. The x-axis indicates the traffic condition (including normal traffic and dense traffic). The y-axis on the left side represents the value of positioning error and the y-axis on the right side indicates the positioning error gradient.

**Figure 15.**The relationship between the degree of urbanization and the performance of LiDAR-based positioning. The x-axis indicates the degree of urbanization. The y-axis on the left side represents the value of positioning error and the y-axis on the right side indicates the positioning error gradient.

Sparse Area | Sub-urban Area | Dense Urban Area | |
---|---|---|---|

${\beth}_{urban}$ | $0\xb0~15\xb0$ | $15\xb0~46\xb0$ | $>46\xb0$ |

**Table 2.**Experiment 1: Performance of NDT-based graph SLAM in a sparse area with dense traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 3.44 | 3.19 | 3.05 | 7.52 | 6.64 | 0.017 | 9.69 | 0.024 |

Std | 1.88 | 1.79 | 1.02 | 2.75 | 3.29 | 0.008 | 4.12 | 0.010 |

**Table 3.**Experiment 2: Performance of NDT-based graph SLAM in a sparse area with dense traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 6.31 | 4.91 | 0.77 | 5.93 | 11.21 | 0.028 | 11.99 | 0.03 |

Std | 5.26 | 4.36 | 0.84 | 2.87 | 5.18 | 0.013 | 5.60 | 0.014 |

**Table 4.**Experiment 3: Performance of NDT-based graph SLAM in a dense urban area with normal traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 6.73 | 4.81 | 11.9 | 18.9 | 11.54 | 0.094 | 14.85 | 0.121 |

Std | 4.39 | 2.31 | 9.01 | 9.61 | 6.01 | 0.049 | 9.75 | 0.079 |

**Table 5.**Experiment 4: Performance of NDT-based graph SLAM in dense urban area with dense traffic condition.

Error | Lateral (m) | Longitudinal (m) | Altitude (m) | Reliability (m) | 2D (m) | 2D Gradient (m/s) | 3D (m) | 3D Gradient (m/s) |
---|---|---|---|---|---|---|---|---|

Mean | 11.25 | 2.77 | 19.07 | 14.38 | 14.02 | 0.114 | 23.22 | 0.189 |

Std | 5.09 | 2.06 | 9.60 | 10.25 | 4.80 | 0.039 | 10.25 | 0.083 |

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## Share and Cite

**MDPI and ACS Style**

Wen, W.; Hsu, L.-T.; Zhang, G. Performance Analysis of NDT-based Graph SLAM for Autonomous Vehicle in Diverse Typical Driving Scenarios of Hong Kong. *Sensors* **2018**, *18*, 3928.
https://doi.org/10.3390/s18113928

**AMA Style**

Wen W, Hsu L-T, Zhang G. Performance Analysis of NDT-based Graph SLAM for Autonomous Vehicle in Diverse Typical Driving Scenarios of Hong Kong. *Sensors*. 2018; 18(11):3928.
https://doi.org/10.3390/s18113928

**Chicago/Turabian Style**

Wen, Weisong, Li-Ta Hsu, and Guohao Zhang. 2018. "Performance Analysis of NDT-based Graph SLAM for Autonomous Vehicle in Diverse Typical Driving Scenarios of Hong Kong" *Sensors* 18, no. 11: 3928.
https://doi.org/10.3390/s18113928