# Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Multi-GNSS PPP Functional and Stochastic Model

- $\lambda $ is the wavelength on selected frequency signal;
- $\phi $ is the phase observable in cycles;
- $P$ is the pseudorange in meters;
- l and i represent station and satellite, respectively;
- $\rho $ is the geometric range between satellite and station;
- ${t}_{l}$ and ${t}^{i}$ are the receiver and satellite clock corrections in seconds, respectively;
- $c$ is the speed of light in meters per second;
- ${b}_{l}$ and ${b}^{i}$ denote the frequency-dependent receiver and satellite phase delays in cycles, respectively, which include initial and hardware phase biases;
- ${d}_{l}$ and ${d}^{i}$ are the frequency-dependent receiver and satellite code biases in meters, respectively;
- α refers to the troposphere mapping function coefficient;
- ZTD denotes the zenith tropospheric delay;
- $I$ denotes ionospheric delay;
- $N$ is the phase ambiguity term;
- $\u03f5$ denotes the observational noise.

#### 2.2. Filtration of PPP Coordinate Time Series for the Extraction of the High-Rate Dynamic Displacements

## 3. Experiment Design and Results

#### 3.1. Shake-Table System

#### 3.2. Experiment Design and Data Collection

#### 3.3. Initial Evaluation of the High-Rate PPP Displacement Time Series Noise

#### 3.4. Applicability of High-Rate PPP to the Detection of Dynamic Displacements

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**A scheme of the in-house developed shake-table system. The components of the shake-table system are as follows: (

**1**) main GNSS antenna, (

**2**) auxiliary GNSS antenna, (

**3**) main trolley of GNSS antenna, (

**4**) auxiliary trolley of GNSS antenna, (

**5**) main GNSS antenna linear motion guide, (

**6**) auxiliary GNSS antenna linear motion guide, (

**7**) motion wheel with engine (stepper motor), (

**8**) rigid driving tie, (

**9**) control system built of a PC and dedicated drivers and software, (

**10**) driving cord, (

**11**) and (

**12**) driving rolls, and (

**13**) high-quality rigid frame.

**Figure 3.**Number of satellites and corresponding PDOP during the data collection time span. Solid blue, red, green, and magenta lines correspond to BDS, GPS, Galileo, and total number of satellites, respectively. Dotted line visualizes PDOP of multi-constellation solution.

**Figure 5.**Histograms of coordinate residuals obtained with precise point positioning (PPP) G (

**first row**) and PPP G + E + C (

**second row**).

**Figure 6.**Displacement time series of the N–S component obtained from multi-GNSS 100 Hz PPP and 50 Hz RTK solutions during the session with simulated oscillations of 9 mm amplitude.

**Figure 7.**Displacement time series of the W–E component obtained from multi-GNSS 100 Hz PPP and 50 Hz RTK solutions during the session with simulated oscillations of 9 mm amplitude.

**Figure 8.**Histograms of differences between multi-GNSS PPP and RTK displacements given for the N (

**left panel**) and E (

**right panel**) components.

**Figure 9.**Frequency responses extracted from GNSS–PPP displacement time series when simulating an antenna oscillation of 3.80 Hz frequency and amplitudes of 9 mm (

**left panel**), 6 mm (

**middle panel**), and 1.5 mm (

**right panel**). Top and bottom panels correspond to the north and east coordinate component, respectively.

Option | Setting |
---|---|

Signals | Phase and code GPS L1/L2, BDS B1/B2, and Galileo E1/E5a |

Sampling rate | 0.01 s (100 Hz) |

Observable combination | Undifferenced ionosphere-free linear combination (IF-LC) |

Weighting scheme | Elevation-dependent weighting |

Elevation cutoff angle | 10° |

Troposphere delay modelling | Estimation of the residual ZTD, a priori modified Hopfield model + global mapping function |

Ionosphere delay modelling | First order of ionospheric delay eliminated taking advantage of IF-LC |

Type of the solution | Float kinematic |

Stochastic modelling: a priori standard deviation of observations | 0.3 m/0.003 m for code/phase of GPS and Galileo signals, respectively; BDS signals down-weighted by the value of 20%. |

Satellite orbits and clocks | Precise CODE (interval: orbits 5 min; clocks 30 s) |

Parameter estimation method | Sequential least squares with a priori parameter constraining and equivalent elimination |

Feature | Value |
---|---|

Dimensions | 1.34 m length, 0.23 width, 0.18 m height (without antenna) |

Total mass | 10 kg |

Number of GNSS antennas that can be set in motion | 2 |

Maximum load weight | 4 kg |

Number of motion axes | single |

Characteristics of the motion | Linear, periodical in selected horizontal direction |

Maximum frequency of the oscillations | Up to 25 Hz |

Amplitude range | 1–85 mm |

Precision of dedicated amplitude | −0.5 mm |

Control system | dedicated software running under a PC |

Engine | Stepper motor |

**Table 3.**Evaluation of the coordinate time series noise—statistics of coordinate residuals. RTK, real-time kinematics.

Type of Solution | std_{N} [mm] | std_{E} [mm] | std_{U} [mm] |
---|---|---|---|

RTK G + E + C | 1.6 | 1.3 | 3.1 |

PPP G | 2.8 | 2.4 | 4.5 |

PPP G + E + C | 3.4 | 3.4 | 5.7 |

**Table 4.**Frequency response extracted from N, E displacement time series when simulating an antenna oscillation of 9 mm amplitude.

Detected Values | Benchmark Values | ||||
---|---|---|---|---|---|

F_{N} [Hz] | F_{E} [Hz] | A [mm] | F [Hz] | A [mm] | |

RTK G + E + C | 3.775 | 3.775 | 9.1 | 3.800 | 9.0 |

PPP G | 3.778 | 3.778 | 10.0 | ||

PPP G + E + C | 3.778 | 3.778 | 10.0 |

**Table 5.**Frequency response extracted from N, E displacement time series when simulating an antenna oscillation of 6 mm amplitude.

Detected Values | Benchmark Values | ||||
---|---|---|---|---|---|

F_{N} [Hz] | F_{E} [Hz] | A [mm] | F [Hz] | A [mm] | |

RTK G + E + C | 3.777 | 3.777 | 7.6 | 3.800 | 6.0 |

PPP G | 3.777 | 3.777 | 6.9 | ||

PPP G + E + C | 3.777 | 3.777 | 6.9 |

**Table 6.**Frequency response extracted from N, E displacement time series when simulating an antenna oscillation of 1.5 mm amplitude.

Detected Values | Benchmark Values | ||||
---|---|---|---|---|---|

F_{N} [Hz] | F_{E} [Hz] | A [mm] | F [Hz] | A [mm] | |

RTK G + E + C | 3.773 | 3.773 | 3.1 | 3.800 | 1.5 |

PPP G | 3.774 | 3.774 | 2.0 | ||

PPP G + E + C | 3.774 | 3.774 | 2.8 |

© 2019 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Paziewski, J.; Sieradzki, R.; Baryla, R.
Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation. *Sensors* **2019**, *19*, 4832.
https://doi.org/10.3390/s19224832

**AMA Style**

Paziewski J, Sieradzki R, Baryla R.
Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation. *Sensors*. 2019; 19(22):4832.
https://doi.org/10.3390/s19224832

**Chicago/Turabian Style**

Paziewski, Jacek, Rafal Sieradzki, and Radoslaw Baryla.
2019. "Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation" *Sensors* 19, no. 22: 4832.
https://doi.org/10.3390/s19224832