# The Integration of GPS/BDS Real-Time Kinematic Positioning and Visual–Inertial Odometry Based on Smartphones

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Implementation of RTK

#### 2.1.1. The Single Difference and the Double Difference

#### 2.1.2. RTK’s Kalman Filter

#### 2.2. The Structure of VINS-Mono

#### 2.3. The Integration Strategy of VINS-Fusion

#### 2.4. The Improved Integration Strategy for Smartphones

#### 2.5. Field Testing

#### 2.5.1. Data Collection and Processing

#### 2.5.2. Description of Devices and Scenarios

#### 2.6. Differences between This Work and Our Previous Work

## 3. Results

#### 3.1. The Validity of GNSS Measurements Collected by the Smartphone

#### 3.2. The Advantage of the Introduction of BDS Satellites

#### 3.3. The Performance of a Standalone VIO

#### 3.4. Performances of the RTK/VIO Integration

#### 3.5. The Performance of the Improved Integration

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notations

Matrix | |

$\mathit{C}$ | The covariance matrix of the observation noise |

${\mathit{D}}_{{}_{\mathrm{G}}}$ | The SD transition matrix for GPS |

${\mathit{D}}_{{}_{\mathrm{B}}}$ | The SD transition matrix for BDS |

$\mathit{F}\left(t\right)$ | The state transition matrix in RTK |

$\mathit{H}\left(t\right)$ | The observation matrix in RTK |

$\mathit{I}$ | The identity matrix |

${\mathit{L}}_{{}_{\mathrm{G}}}\left(t\right)$ | The matrix comprised of LOS vectors of GPS |

${\mathit{L}}_{{}_{\mathrm{B}}}\left(t\right)$ | The matrix comprised of LOS vectors of BDS |

$\mathit{O}$ | The zero matrix |

$\mathit{Q}\left(t\right)$ | The covariance matrix of the process noise |

${\mathit{R}}_{{}_{\mathrm{ENU}}}^{{}^{\mathrm{ECEF}}}\left(t\right)$ | The coordinates rotation matrix from the ENU coordinate to the ECEF coordinate |

${\mathit{R}}_{\mathrm{w}}^{\mathrm{g}}\left(t\right)$ | The rotation matrix from the world frame to the global frame |

${\mathit{T}}_{\mathrm{w}}^{\mathrm{g}}\left(t\right)$ | The transformation matrix between the global frame and the world frame |

${\mathit{T}}_{\mathrm{b}}^{\mathrm{g}}\left(t\right)$ | The transformation matrix between the body frame and the global frame |

${\mathit{T}}_{\mathrm{b}}^{\mathrm{w}}\left(t\right)$ | The transformation matrix between the body frame and the world frame |

Vector | |

${\mathit{a}}_{\mathrm{u}}\left(t\right)$ | The user’s acceleration |

${\mathit{d}}_{\mathrm{u}}\left(t\right)$ | The user’s position |

${\mathit{d}}_{{S}_{{\mathrm{G}}_{{\mathrm{k}}_{1}}}}\left(t\right)$ | The position of the ${k}_{1}$th GPS satellite |

${\mathit{d}}_{\mathrm{ref}}$ | The position of the reference station |

${\mathit{e}}_{{}_{\mathrm{G}}}\left(t\right)$ | The SD phase ambiguities of GPS satellites |

${\mathit{e}}_{{}_{\mathrm{B}}}\left(t\right)$ | The SD phase ambiguities of BDS satellites |

$\mathit{h}\left({\mathbf{\chi}}_{{}_{\mathrm{RTK}}}\left(t\right)\right)$ | The measurement function for the update step in RTK |

${\mathit{l}}_{\mathrm{w}}^{\mathrm{g}}\left(t\right)$ | The translation from the world frame to the global frame |

$\mathit{o}\left(t\right)$ | The LOS vector |

${\mathit{p}}_{\mathrm{cam}}^{\mathrm{b}}$ | The translation from the camera frame to the body frame |

${\mathit{p}}_{\mathrm{b},i}^{\mathrm{w}}\left(t\right)$ | The position of the IMU in the world frame when the ith frame is captured |

${\mathit{p}}_{\mathrm{b},i}^{\mathrm{g}}\left(t\right)$ | The position of the IMU in the global frame when the ith frame is captured |

${\mathit{p}}_{i}^{\mathrm{GNSS}}\left(t\right)$ | The position given by GNSS when the ith frame is captured |

${\mathit{q}}_{\mathrm{cam}}^{\mathrm{b}}$ | The rotation from the camera frame to the body frame |

${\mathit{q}}_{\mathrm{b},i}^{\mathrm{w}}\left(t\right)$ | The orintation of the IMU in the world frame when the ith frame is captured |

${\mathit{q}}_{\mathrm{b},i}^{\mathrm{g}}\left(t\right)$ | The orintation of the IMU in the global frame when the ith frame is captured |

${\mathit{v}}_{\mathrm{u}}\left(t\right)$ | The user’s velocity |

${\mathit{v}}_{\mathrm{b},i}^{\mathrm{w}}\left(t\right)$ | The velocity of the IMU in the world frame when the ith frame is captured |

${\mathit{v}}_{\mathrm{b},i}^{\mathrm{g}}\left(t\right)$ | The velocity of the IMU in the global frame when the ith frame is captured |

${\mathit{x}}_{i}\left(t\right)$ | The IMU state vector when the ith frame is captured |

${\mathbf{\chi}}_{{}_{\mathrm{RTK}}}\left(t\right)$ | The state vector of the Kalman filter in RTK |

${\mathbf{\chi}}_{{}_{\mathrm{V}-\mathrm{mono}}}\left(t\right)$ | The state vector of VINS-mono |

${\mathbf{\chi}}_{{}_{\mathrm{VIO}}}\left(t\right)$ | The state vector of VINS-fusion |

$\mathit{y}\left(t\right)$ | The measurements vector of the Kalman filter in RTK |

$\nabla \Delta {\mathbf{\varphi}}_{{}_{\mathrm{G}}}\left(t\right)$ | The DD carrier phase vectors of GPS satellites |

$\nabla \Delta {\mathbf{\rho}}_{{}_{\mathrm{G}}}\left(t\right)$ | The DD pseudo-range vectors of GPS satellites |

$\nabla \Delta {\mathbf{\varphi}}_{{}_{\mathrm{B}}}\left(t\right)$ | The DD carrier phase vectors of BDS satellites |

$\nabla \Delta {\mathbf{\rho}}_{{}_{\mathrm{B}}}\left(t\right)$ | The DD pseudo-range vectors of BDS satellites |

${\mathbf{\u03f5}}_{\mathrm{acc}}\left(t\right),{\mathbf{\u03f5}}_{\mathrm{gyro}}\left(t\right)$ | The IMU biases |

Scalar | |

$\rho \left(t\right)$ | The pseudo-range |

$\varphi \left(t\right)$ | The carrier phase |

$r\left(t\right)$ | The geometric distance between the receiver and the satellite |

$\nabla \Delta \rho \left(t\right)$ | The DD pseudo-range |

$\nabla \Delta \varphi \left(t\right)$ | The DD carrier phase |

$\nabla \Delta r\left(t\right)$ | The DD geometric distance |

$\nabla \Delta {r}^{{S}_{{\mathrm{B}}_{1}},{S}_{{\mathrm{B}}_{{k}_{2}}}}\left(t\right)$ | The DD geometric range between the 1st BDS satellite and the ${k}_{2}$th BDS satellite |

${\tau}_{\mathrm{iono}}\left(t\right)$ | The ionospheric delay |

${\tau}_{\mathrm{tropo}}\left(t\right)$ | The tropospheric delay |

$\lambda $ | The wavelength of the GNSS signal |

$\delta {}_{\mathrm{rec}}\left(t\right)$ | The receiver clock bias |

$\delta {}_{\mathrm{sat}}\left(t\right)$ | The satellite clock bias |

${\omega}_{\rho}\left(t\right)$ | The measurement noise in the pseudo-range |

${\omega}_{\varphi}\left(t\right)$ | The measurement noise in the carrier phase |

$\nabla \Delta {\omega}_{\rho}\left(t\right)$ | The DD measurement noise in the pseudo-range |

$\nabla \Delta {\omega}_{\varphi}\left(t\right)$ | The DD measurement noise in the carrier phase |

${\sigma}_{ve}$ | The standard deviation of the east component of the velocity |

${\sigma}_{vn}$ | The standard deviation of the north component of the velocity |

${\sigma}_{vu}$ | The standard deviation of the up component of the velocity |

${\sigma}_{\varphi}$ | The standard deviation of the phase measurement error |

${\sigma}_{\rho}$ | The standard deviation of the pseudo-range measurement error |

$\beta \left(t\right)$ | The RTK weight in the integration |

$\mu \left(t\right)$ | The positioning RMSE in the sliding window |

${\mu}_{0}$ | The threshold |

$\gamma \left(t\right)$ | The position covariance given by RTK |

c | The speed of light |

${E}^{{S}_{{\mathrm{G}}_{1}}}\left(t\right)$ | The SD phase ambiguity state variable of the 1st GPS satellite |

i | The keyframe index |

j | The feature index |

${k}_{1}$ | The index of GPS satellites |

${k}_{2}$ | The index of BDS satellites |

${m}_{{}_{\mathrm{G}}}$ | The number of the visible GPS satellite |

${m}_{{}_{\mathrm{B}}}$ | The number of the visible BDS satellite |

${n}_{1}$ | The number of keyframes in the sliding window |

${n}_{2}$ | The number of features in the sliding window |

N | The integer ambiguity |

${s}_{j}\left(t\right)$ | The inverse distance of the jth feature |

${S}_{{\mathrm{G}}_{{k}_{1}}}$ | The ${k}_{1}$th GPS satellite |

${S}_{{\mathrm{B}}_{{k}_{2}}}$ | The ${k}_{2}$th BDS satellite |

t | The time |

$\Delta t$ | The time interval |

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**Figure 2.**SD measurements and DD measurements. (

**a**) Two receivers track the signals of two satellites. (

**b**) Steps to calculate SD measurements and DD measurements.

**Figure 10.**The visible satellites with a valid pseudo-range and the visible satellites with a valid carrier phase. (

**a**) The visible satellites with a valid pseudo-range. (

**b**) The visible satellites with a valid carrier phase.

**Figure 11.**The number of the visible satellites with a valid pseudo-range and the number of the visible satellites with a valid carrier phase at different moments. (

**a**) The number of the visible satellites with a valid pseudo-range at different moments. (

**b**) The number of the visible satellites with a valid carrier phase at different moments.

**Figure 12.**The performance comparison of GPS RTK, GPS/BDS RTK, and the reference. (

**a**) Horizontal positioning results of GPS RTK. (

**b**) Horizontal positioning results of GPS/BDS RTK. (

**c**) The reference.

**Figure 13.**The performance of the standalone visual–inertial odometry (VIO). (

**a**) The trajectory given by the standalone VIO. (

**b**) The rotation of VIO’s trajectory.

**Figure 14.**The performance of the integration of real-time kinematic (RTK) technique and VIO. (

**a**) Horizontal positioning results. (

**b**) Vertical positioning results.

**Figure 15.**The performance comparison of RTK, RTK/VIO, and the reference. (

**a**) Valid positioning results of RTK. (

**b**) Valid positioning results of RTK/VIO. (

**c**) The reference.

**Figure 17.**Horizontal positioning results. (

**a**) The positioning results in east. (

**b**) The positioning results in north.

**Figure 19.**The differences between the reference and the integration results of different strategies. (

**a**) The differences between the reference and the integration results in east. (

**b**) The differences between the reference and the integration results in north. (

**c**) The differences between the reference and the integration results in vertical.

**Figure 20.**The performances in continuity of different algorithms. (

**a**) The valid positioning results of RTK. (

**b**) The valid positioning results of the pre-improved integration. (

**c**) The valid positioning results of the improved integration. (

**d**) The reference.

Techniques | RTK | INS | VO | VIO | |
---|---|---|---|---|---|

Features | |||||

Absolute positioning results | √ | × | × | × | |

Long-term drifts | × | √ | √ | √ | |

Scale ambiguity | × | × | √ | × | |

Constraints | Signal blockage | IMU biases | Light Ambient textures Device’s speed | Light Ambient textures IMU biases |

Algorithms | Characteristics | Devices |
---|---|---|

VINS-fusion | Loose couple of GNSS/VIO | Stereo cameras + DJI A3 controller |

MSF-EKF | Loose couple of GNSS/VIO | An Asctec Firefly micro aerial vehicle |

R.S.’s algo | NHC | Novatel receiver + STIM300+ CCD camera |

T.L.’s algo | Tight couple of GNSS/VIO | Trimble receiver+MEMS IMU+ CCD camera |

GVINS | Tight couple of GNSS/VIO | Ublox receiver+VI-Sensor |

App. | GEO++ Logger | MARSLogger | CIGRLogger | |
---|---|---|---|---|

Sensors | ||||

GNSS | GPS/BDS/Galileo/GLONASS | × | GPS/BDS | |

Camera | × | √ | √ | |

IMU | × | √ | √ | |

Magnetometer | × | × | √ | |

Barometer | × | × | √ |

Team | Sensors | Techniques |
---|---|---|

NSL | GNSS chipset | PPP/RTK |

WHU | IMU | ZUPT + ZARU |

IST | GNSS chipset + IMU | Loose couple of RTK/INS |

P. L. | IMU + camera | VINS estimator |

Y. W. | IMU + camera | VIO |

H. Y. | IMU | SVM/deep learning |

**Table 5.**The average difference between the first and final position of the VIO and RTK trajectories.

Techniques | The Average Value (m) |
---|---|

VIO | 13.57 |

Reference | 0.08 |

Techniques | Average Deviation (m) | Average Percentage |
---|---|---|

RTK | 3.23 | 92% |

RTK+VIO | 2.80 | 100% |

Strategies | Vertical Deviation (m) | East Deviation (m) | North Deviation (m) |
---|---|---|---|

VINS-fusion strategy | 2.52 | 0.86 | 1.14 |

Improved strategy | 1.21 | 0.84 | 1.12 |

Strategies | Average Percentage |
---|---|

The pre-improved strategy | 100% |

The improved strategy | 100% |

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## Share and Cite

**MDPI and ACS Style**

Niu, Z.; Guo, F.; Shuai, Q.; Li, G.; Zhu, B.
The Integration of GPS/BDS Real-Time Kinematic Positioning and Visual–Inertial Odometry Based on Smartphones. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 699.
https://doi.org/10.3390/ijgi10100699

**AMA Style**

Niu Z, Guo F, Shuai Q, Li G, Zhu B.
The Integration of GPS/BDS Real-Time Kinematic Positioning and Visual–Inertial Odometry Based on Smartphones. *ISPRS International Journal of Geo-Information*. 2021; 10(10):699.
https://doi.org/10.3390/ijgi10100699

**Chicago/Turabian Style**

Niu, Zun, Fugui Guo, Qiangqiang Shuai, Guangchen Li, and Bocheng Zhu.
2021. "The Integration of GPS/BDS Real-Time Kinematic Positioning and Visual–Inertial Odometry Based on Smartphones" *ISPRS International Journal of Geo-Information* 10, no. 10: 699.
https://doi.org/10.3390/ijgi10100699