# A Least Squares Collocation Method for Accuracy Improvement of Mobile LiDAR Systems

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Definition of Coordinate Systems

- the geo-spatial coordinate system (GCS);
- the POS body coordinate system (PCS);
- the laser scanner coordinate system (LCS).

#### 2.2. Error Model of the MLS

_{G}is the coordinate vector of the LiDAR point in the GCS, T

_{G}= [T

_{E}, T

_{N}, T

_{H}]

^{T}is the coordinate vector of the POS navigation centre in the GCS, ${R}_{P}^{G}$ is the rotation matrix from the PCS to GCS constituted from the orientation o(o = [γ, ρ, ϕ]

^{T}, γ roll, ρ pitch, ϕ heading), X

_{L}is the coordinate vector of the laser scanner measurement in the LCS, ${T}_{L}^{P}={[{T}_{x},{T}_{y},{T}_{z}]}^{T}$ is the translation vector from the LCS to the PCS and ${R}_{L}^{P}$ is the rotation matrix constituted from the alignment angles (θ = [θ

_{x}, θ

_{y}, θ

_{z}]

^{T}). Owing to the fact that there are errors in every measurement and calibration parameter, Equation (1) becomes Equation (2) by adding error vectors:

_{G}and orientation error Δo; the second part represents calibration errors, i.e., misalignment, including $\mathrm{\Delta}{T}_{L}^{P}$ and Δθ; and the third part is the laser scanning error, including ranging and scanning angle errors.

#### 2.3. POS Error

_{G}changes with time, thus showing a kind of tendency variable. Ultimately, the positioning error contains a tendency error, as well as a random error.

#### 2.4. The Least Squares Collocation Model

#### 2.4.1. The Basic Principles of Least Squares Collocation

#### 2.4.2. The Least Squares Collocation Model for the Correction of POS Error

_{G}is denoted by ΔT, the random variable of ΔT

_{G}by Λ

_{T}and the random variable of orientation error Δo by Λ

_{o}. Then, Equation (5) can be modified to Equation (7):

_{T}= I is the coefficient matrix of Λ

_{T}and B

_{o}is the coefficient matrix of Λ

_{o}. ΔX

_{G}is the vector of the coordinate difference between the control points and their corresponding LiDAR points. To deduce the specific representation of B

_{o}, an intermediate variable $\widehat{G}$ is introduced:

_{o}is:

_{0}, a

_{1}, …, a

_{n}, b

_{0}, b

_{1}, …, b

_{n}, c

_{0}, c

_{1}, …, c

_{n}]

^{T}and C is the corresponding time-variant coefficient. Let B = [B

_{T}, B

_{o}] and Λ = [Λ

_{T}, Λ

_{o}]

^{T}; then Equation (7) becomes Equation (12):

_{Λ}is the priori covariance matrix of Λ and D

_{ω}is the covariance matrix of δ. The random variables of other epochs Λ′ are calculated by Equation (15),

_{Λ′ Λ}is the covariance matrix between the random variables of control epochs and the other epochs to be corrected (that is, the interpolated epoch).

#### 2.4.3. The Least Squares Collocation Model for Misalignment Calibration

_{G}and orientation error Δo being treated as random variables with zero means denoted by Λ. The LSC model for misalignment calibration is therefore as follows:

_{G}is also the difference between the control points and LiDAR measurements. Here, B is the same as Equation (12). To deduce the specific representation of A, another intermediate variable $\widehat{P}$ is introduced,

#### 2.5. Covariance Functions

_{Λ}, D

_{ω}and D

_{ΛΛ′}. As far as the diagonal elements of D

_{Λ}are concerned, their value is equal to the corresponding variance value v(Λ) directly proposed by POS solution software, with the other elements being set to zero.

_{ω}, is:

_{0}is a base value of covariance and is positive. The covariance along different directions is zero. Both δt and C

_{0}are adjustable. δt is relative to the condition of GNSS environments and the interval of neighbouring control epochs. In environments favourable to GNSS, δt is set big, while in environments that are harsh, its value is smaller. C

_{0}is relative to the polynomial order fitting, the condition of GNSS environments and the interval of neighbouring control epochs. The significance of δt and C

_{0}will be detailed in Section 3.1.3 and 3.1.4, respectively.

## 3. Experiments

_{0}for roll was 1.21 × 10

^{−8}arc

^{2}, with 1.21 × 10

^{−8}arc

^{2}for pitch and 6 × 10

^{−8}arc

^{2}for azimuth, according to the orientation variance generated by the POS solution software. However, the C

_{0}for east (C

_{0}

_{E}), for north (C

_{0}

_{N}) and for height (C

_{0}

_{H}) varies from experiment to experiment. δt also varies with the number of control points. The significance of C

_{0}and δt will be discussed respectively in detail.

#### 3.1. Validation for the LSC-Based Accuracy Improvement Method for POS

#### 3.1.1. Comparison between LSC and LS and the Effect of the Order of Polynomials

#### 3.1.2. Effect of the Number of Control Points

#### 3.1.3. Effect of C_{0}

_{0}is one of the two key factors determining the value of random variables in the LSC. As C

_{0}increases, the value of random variables becomes greater. A correct setting for C

_{0}is therefore of paramount importance. To simplify the discussion, but without becoming too general, we will take the north direction as an example. We will try several values for C

_{0}for an LSC with 21, 15, 9 control points each and check the RMS. The results of this experiment are given in Table 3 and shown in Figures 7 and 8a.

_{0}has an optimal range for each LSC with a different number of control points. Within this range, the random variables are given an appropriate value, and the LSC achieves an optimal performance. If C

_{0}is set too far below this range, the random variables become too small, and the performance of the LSC will thus be close to that of the LS. Conversely, if C

_{0}is set too far beyond this range, the random variables become too great, and the LSC will overly correct the positioning error. Secondly, as the number of control points decreases, the optimal value of C

_{0}and the lower end of the optimal range both increase.

_{0}and the polynomial order is also examined. Figure 8b shows the change in the optimal value of C

_{0}with the polynomial order. Here, the number of control points is 21. As is clearly shown in Figure 8b, the optimal value of C

_{0}increases when the order increases. The strict relationship between C

_{0}and the number of control points, the polynomial order and the conditions affecting the GNSS remains an interesting issue. In practice, we will initialize C

_{0}with the variance computed by the adopted POS solution software and then adjust it according to its correction performance at check epochs. Thus, for engineering applications, check points are required, as well as control points.

#### 3.1.4. Effect of δt

#### 3.1.5. Validation Using Boat-Borne Data

#### 3.2. Validation for the LSC-Based Calibration Method

## 4. Conclusions

_{0}were also discussed in detail. Generally, the performance of LSC became better when the polynomial order increased. The fourth-order and the fifth-order LSC were recommended for engineering applications. When the number of control points was decreased from 21 to 15 and 9, the performance of the LSC method did not show significant decrease. Fewer control points mean less cost. When applying a fifth-order LSC for a road section, the recommended number of control points ranges from nine to 15, depending on the environment. C

_{0}and δt were the two key factors determining the value of random variables in the LSC. We found that δt had an optimal range varying with the number of control points, and C

_{0}also had an optimal range varying with the polynomial order, as well as the number of control points. Furthermore, experiments on a real vehicle-borne dataset, which suffered from a severe multi-path effect, indicated that the proposed LSC method outperformed the traditional coordinate transformation method.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Coordinate systems related to the mobile LiDAR system (MLS). LCS, laser scanner coordinate system; PCS, positioning and orientation system (POS) body coordinate system; GCS, geo-spatial coordinate system.

**Figure 6.**Coordinate difference of control points between MLS measurements and total station measurements.

**Figure 7.**The correction effect of least squares collocation (LSC) with different C

_{0}. (

**a**) Control points number = 21; (

**b**) control points number = 15; (

**c**) control points number = 9.

**Figure 8.**The relationship between C

_{0}and the number of control points and the polynomial order. (

**a**) Relationship between C

_{0}and the control points number; (

**b**) relationship between C

_{0}and the polynomial order.

**Figure 9.**The correction effect of LSC with different δt. (

**a**) Control points number = 21; (

**b**) control points number = 15; (

**c**) control points number = 9.

**Figure 10.**The experimental scenario of boat-borne dataset. (

**a**) iScan on a boat; (

**b**) points cloud of the river scenario.

**Figure 11.**Comparison between the LSC method and coordinate transformation method using the boat-borne dataset. The raw residuals are big. Considering that the GNSS visibility is good along the river, these big POS errors are generated by the multi-path effect. (

**a**) East residual; (

**b**) north residual; (

**c**) height residual.

**Figure 12.**The residuals of check points. (

**a**) East residual; (

**b**) north residual; (

**c**) height residual.

Method | East | North | Height | ||||||
---|---|---|---|---|---|---|---|---|---|

RMS | Min | Max | RMS | Min | Max | RMS | Min | Max | |

Raw | 0.5476 | −1.2570 | 0.7410 | 2.4528 | −3.3000 | −0.125 | 0.2900 | 0.1520 | 0.5140 |

LS-2 | 0.2009 | −0.2868 | 0.3054 | 0.4975 | −0.8918 | 0.6479 | 0.0227 | −0.0504 | 0.0190 |

LSC-2 | 0.0576 | −0.0723 | 0.1409 | 0.1764 | −0.4720 | 0.0656 | 0.0140 | −0.0424 | 0.0060 |

LS-3 | 0.1343 | −0.3060 | 0.2571 | 0.2964 | −0.5496 | 0.3562 | 0.0168 | −0.0335 | 0.0114 |

LSC-3 | 0.0625 | −0.2781 | 0.0100 | 0.1469 | −0.2848 | 0.3125 | 0.0149 | −0.0320 | 0.0073 |

LS-4 | 0.0695 | −0.0877 | 0.1440 | 0.2278 | −0.4710 | 0.1916 | 0.0171 | −0.0334 | 0.0128 |

LSC-4 | 0.0344 | −0.0842 | 0.0959 | 0.1234 | −0.2305 | 0.0362 | 0.0142 | −0.0311 | 0.0071 |

LS-5 | 0.0392 | −0.0551 | 0.0661 | 0.1618 | −0.3538 | 0.1381 | 0.0155 | −0.0342 | 0.0108 |

LSC-5 | 0.0278 | −0.0391 | 0.0485 | 0.1092 | −0.2526 | 0.0422 | 0.0143 | −0.0339 | 0.0086 |

_{0}

_{E}= 0.030 m

^{2}, C

_{0}

_{N}= 0.035 m

^{2}, C

_{0}

_{H}= 0.060 m

^{2}, δt = 30 s

_{0}

_{E}= 0.045 m

^{2}, C

_{0}

_{N}= 0.055 m

^{2}, C

_{0}

_{H}= 0.070 m

^{2}, δt = 30 s

_{0}

_{E}= 0.045 m

^{2}, C

_{0}

_{N}= 0.070 m

^{2}, C

_{0}

_{H}= 0.080 m

^{2}, δt = 30 s

_{0}

_{E}= 0.060 m

^{2}, C

_{0}

_{N}= 0.090 m

^{2}, C

_{0}

_{H}= 0.090 m

^{2}, δt = 30 s

Number | East | North | Height | ||||||
---|---|---|---|---|---|---|---|---|---|

RMS | Min | Max | RMS | Min | Max | RMS | Min | Max | |

Raw | 0.5476 | −1.2570 | 0.7410 | 2.4528 | −3.3000 | −0.125 | 0.2900 | 0.1520 | 0.5140 |

21 | 0.0278 | −0.0391 | 0.0485 | 0.1092 | −0.2526 | 0.0422 | 0.0143 | −0.0339 | 0.0086 |

15 | 0.0265 | −0.0288 | 0.0440 | 0.1195 | −0.2136 | −0.0184 | 0.0148 | −0.0321 | 0.0147 |

9 | 0.0336 | −0.0474 | 0.0677 | 0.1238 | −0.2128 | 0.0176 | 0.0160 | −0.0403 | 0.0137 |

_{0}

_{E}= 0.060 m

^{2}, C

_{0}

_{N}= 0.090 m

^{2}, C

_{0}

_{H}= 0.090 m

^{2}, δt = 30 s

_{0}

_{E}= 0.160 m

^{2}, C

_{0}

_{N}= 0.250 m

^{2}, C

_{0}

_{H}= 0.250 m

^{2}, δt = 60 s

_{0}

_{E}= 0.800 m

^{2}, C

_{0}

_{N}= 0.800 m

^{2}, C

_{0}

_{H}= 0.800 m

^{2}, δt = 120 s

21 | C_{0}(m^{2}) | 0.01 | 0.07 | 0.08 | 0.09 | 0.1 | 0.11 | 0.2 |

RMS (m) | 0.1523 | 0.1122 | 0.1099 | 0.1091 | 0.1101 | 0.1125 | 0.1834 | |

15 | C_{0}(m^{2}) | 0.05 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.5 |

RMS (m) | 0.1572 | 0.1304 | 0.1226 | 0.1196 | 0.1217 | 0.1288 | 0.1714 | |

9 | C_{0}(m^{2}) | 0.1 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.5 |

RMS (m) | 0.1679 | 0.1301 | 0.1254 | 0.1233 | 0.1238 | 0.1269 | 0.1856 |

21 | δt(s) | 10 | 20 | 30 | 40 | 50 | 80 |

RMS (m) | 0.1158 | 0.1122 | 0.1092 | 0.1118 | 0.1163 | 0.1283 | |

15 | δt(s) | 20 | 40 | 60 | 80 | 100 | 160 |

RMS (m) | 0.1858 | 0.1287 | 0.1196 | 0.1237 | 0.1296 | 0.1429 | |

9 | δt(s) | 60 | 90 | 120 | 150 | 180 | 270 |

RMS (m) | 0.1654 | 0.1278 | 0.1233 | 0.1269 | 0.1320 | 0.1444 |

Method | East | North | Height | ||||||
---|---|---|---|---|---|---|---|---|---|

RMS | Min | Max | RMS | Min | Max | RMS | Min | Max | |

Raw | 0.1071 | 0.0574 | 0.1637 | 0.0264 | −0.0629 | 0.0331 | 0.2608 | 0.1742 | 0.3754 |

LSC | 0.0164 | −0.0258 | 0.0460 | 0.0213 | −0.0512 | 0.0382 | 0.0422 | −0.0514 | 0.0882 |

Transformation | 0.0310 | −0.0644 | 0.0431 | 0.0256 | −0.0529 | 0.0417 | 0.0616 | −0.0818 | 0.1195 |

_{0}

_{E}= 0.400 m

^{2},C

_{0}

_{N}= 0.010 m

^{2},C

_{0}

_{H}= 0.800 m

^{2}, δt = 120 s

Method | ΔT_{x} (m) | ΔT_{y} (m) | ΔT_{z} (m) | Δθ_{x} (Degree) | Δθ_{y} (Degree) | Δθ_{z} (Degree) |
---|---|---|---|---|---|---|

LS | 0.0005 | 0.0145 | −0.0645 | −0.0307 | −0.0550 | 0.2077 |

LSC | 0.0004 | 0.0145 | −0.0642 | −0.0307 | −0.0561 | 0.2077 |

Method | East | North | Height |
---|---|---|---|

Raw | 0.0775 | 0.0544 | 0.0909 |

LS | 0.0256 | 0.0227 | 0.0303 |

LSC | 0.0141 | 0.0114 | 0.0233 |

_{0}

_{E}= C

_{0}

_{N}= C

_{0}

_{H}= 0.0004 m

^{2}, δt = 120 s

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mao, Q.; Zhang, L.; Li, Q.; Hu, Q.; Yu, J.; Feng, S.; Ochieng, W.; Gong, H.
A Least Squares Collocation Method for Accuracy Improvement of Mobile LiDAR Systems. *Remote Sens.* **2015**, *7*, 7402-7424.
https://doi.org/10.3390/rs70607402

**AMA Style**

Mao Q, Zhang L, Li Q, Hu Q, Yu J, Feng S, Ochieng W, Gong H.
A Least Squares Collocation Method for Accuracy Improvement of Mobile LiDAR Systems. *Remote Sensing*. 2015; 7(6):7402-7424.
https://doi.org/10.3390/rs70607402

**Chicago/Turabian Style**

Mao, Qingzhou, Liang Zhang, Qingquan Li, Qingwu Hu, Jianwei Yu, Shaojun Feng, Washington Ochieng, and Hanlu Gong.
2015. "A Least Squares Collocation Method for Accuracy Improvement of Mobile LiDAR Systems" *Remote Sensing* 7, no. 6: 7402-7424.
https://doi.org/10.3390/rs70607402