^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Although GPS kinematic relative positioning can provide high accuracy, GPS observables, like any other kind of measurement, are not free of errors. Indeed, they have several kinds of errors. In this paper, we show how to construct a functional mathematical model within the context of a Kalman Filter in order to eliminate most of these errors. Furthermore, we discuss how the multipath effect, a kind of error not modeled in the functional model, can be corrected using the proposed wavelet method. The behavior of the double difference functional model in the kinematic mode is also demonstrated and analyzed aiming to provide better insight into the problem. The results obtained from the multipath experiments were very promising and are presented here.

In relative positioning, single and double differences (DDs) of GPS observables are commonly used to construct the functional model, as they can eliminate or reduce most GPS errors. This functional model reduces most errors if the baseline is short, that is, no more than 20 km. However, the multipath error is the only one that is not eliminated even for short baselines. This is because the multipath depends on the geometry and environment of each point where the GPS antenna is collecting the signals (observables). Therefore, multipath is a major residual error source in double-differenced GPS observables. In kinematic positioning, the nonstationary behavior of the multipath effect is worse, and it is very difficult to remove it from the data.

Although many studies have attempted to mitigate multipath for kinematic positioning, this effect remains a challenge for research.

Multipath effects and signal blockage in GPS navigation in the vicinity of the International Space Station (ISS) were analyzed by [

Receiver [

Multiple proposals have been put forward with regard to processing techniques, such as those using code-minus-phase measurements that accurately separate or eliminate the multipath signals [

Spectral analysis has a powerful technique to analyze this kind of nonstationary signal: the wavelet transform [

In summary, we show how errors are accounted for in the functional model construction in relative positioning as well as how the remaining multipath error is detected and corrected by the proposed method using wavelet techniques. Furthermore, the functional models are described, and their behaviors in the kinematic mode are demonstrated and analyzed.

To perform GPS relative positioning, the Kalman Filter (KF) is usually applied. KF deals with two important components: a mathematical functional model and a stochastic model. The functional model describes the mathematical relationship between the GPS observables and the unknown parameters, while the stochastic model describes the statistical characteristics of the GPS observables. The latter therefore depends on the choice of the functional model. The DD technique is commonly used to construct the functional model. In stochastic models, it is usually assumed that all measurements of each observable (PR or CP) have equal variance, and that they are statistically independent [

In the next subsections, models of PR and CP observables are described, together with how the errors involved in these observables are eliminated in the functional model.

There are two important GPS observables: PR and CP. Measuring PRs and CPs involves advanced techniques in electronics and signal processing [

PR is related to the distance between the satellite and the receiver’s antenna, implied by the instants of emission and reception of the PRN codes. PR measurements are obtained from correlation of the code generated by the satellite in the transmission instant (^{t}

^{t}_{r}

_{r}

The CP measurement at a nominal frequency ^{s}^{t}_{r}_{r}

_{r}^{s}

For receivers at stations _{r}_{1} and _{r}_{2,} one can write two PR equations like (

Single difference.

In the single difference equation, the errors related to the satellite (like orbit and satellite clock errors) are assumed the same for the observations collected from

As the multipath depends on the geometry and the environment of each station, it is not eliminated in the SDs.

If two receivers

Double difference.

The equations of PR and CP DDs can be written as

An important feature of the DD is that the receiver clock errors (

The classical KF processing of GPS measurements generates residuals, which contain the signature of both nonmodeled errors and random measurement noise. If the functional mathematical model is adequate, the residuals obtained from the KF solution should be randomly distributed [

The detection and correction of the multipath from DD residuals using wavelet spectral analysis were described for static applications by [

The multipath detection and mitigation using wavelet spectral analysis is based on four steps.

(1)

(2)

(3)

(4)

At this step the KF is performed again, and the remaining residuals are now characterized just by the measurement noise.

Unlike in static applications, in kinematic multipath detection it is necessary to use data windows to process the data. Therefore, the four previous steps are performed for each window. Regarding step 1, in each instant

In this section, the DD functional model is analyzed in order to get insight on what it actually represents in practice. It is shown how the DD functional model, which involves differences of distances between receivers and satellites, describes the movement of the rover receiver. Therefore, only using DD observations time series one can have information about the rover movement.

To simplify the problem, we use graphic operations with vectors for a generic case. However, the process can be realized algebraically by finding the components of each vector, combining them to form the components of the resultant vector, and then converting to polar form.

The sum of two vectors,

(a) Graphic representation of the SD. (b) Illustration of the vector sum. (c) Illustration of the vector subtraction

Independent of the receiver and satellite positions, it is possible to take a plane passing by three points, similarly to that in Figure

When a DD is computed, a difference is performed between two SD vectors, as illustrated in Figure

Graphic representation of the DD = SD_{1} – SD_{2}.

We note that the proceeding refers to an analysis of the DD as a vector. However, the DD is used as a scalar to calculate the coordinates of the

In relative kinematic positioning, the receiver _{i}_{i+k}

Graphic representation of the DD when the rover (

Some experiments, presented in the next section, confirm this theoretical analysis.

An experiment was conducted at the “Luiz de Queiroz” College of Agriculture of São Paulo University, Piracicaba, Brazil, in November 9, 2007. This experiment was carried out in a kinematic mode with a controlled vehicle movement. The vehicle (small tractor) moved anchored around a pivot (Figure

Experiment scheme.

In order to cause the multipath, a bus was placed near (2 m) the receiver (Figure

Experiment scenario with the reflector (bus).

The data were collected at a sample rate of 1 per second. The baseline reference length (14.98 m) was computed by topographic techniques, that is, by independent means (not GPS). This allows us to evaluate the accuracy of the proposed method in relation to the reference value.

Since the baseline length is short, errors resulting from ionosphere, troposphere, and orbits are assumed to be insignificant. Therefore, the DD estimated residuals for the PR and CP should exhibit mainly multipath and observable noise.

The L1 PR and CP DDs were processed using the software GPSeq, which is under development at UNESP, Presidente Prudente [

At the beginning of the experiment, the tractor remained stopped near the bus for about 20 minutes in order to solve the integer ambiguities. Then, the tractor started to move around the pivot. The static (pivot) antenna also rotates with respect to an axis through its center (spinning axis), thus experiencing the same phase wind-up effect as the slave. This common error is removed after performing an SD. The repeated loops provide ways of testing the results more than once.

PRN 14 with the highest elevation angle, as illustrated in Table

PRN elevation angles (degrees) from the initial to the final instant.

PRN | Elevation angles | |
---|---|---|

Initial | Final | |

22 | 57° | 51° |

14 | 63° | 68° |

3 | 55° | 59° |

1 | 36° | 44° |

18 | 23° | 18° |

19 | 28° | 34° |

The results and analyses are presented in the next section.

The kinematic relative positioning was performed using two strategies. In the first one, hereafter referred to as the Standard, no multipath mitigation method was applied in the KF. In the second, the wavelet method (WAV) described in Section

Not all collected data were processed in one run due to implementation restrictions. The processing is shown for a period of 1000 seconds of data, where the tractor remained stationary during the first 400 seconds and then started its movement, performing about 5 loops. Thus, it is possible to see the performance of the method for a static and kinematic processing as well as the transition between both cases. For other datasets, the results were similar.

In each instant

The first analysis performed is related to the aspects discussed in Section

Figures

Pseudorange DD 14-01 observations in the Standard and WAV strategies.

Pseudorange DD 14-03 observations in the Standard and WAV strategies.

Pseudorange DD 14-18 observations in the Standard and WAV strategies.

Pseudorange DD 14-19 observations in the Standard and WAV strategies.

Pseudorange DD 14-22 observations in the Standard and WAV strategies.

Figures

For the CP observables, the DDs are plotted in Figures

CP DD 14-01 observations in the Standard and WAV strategies.

CP DD 14-03 observations in the Standard and WAV strategies.

CP DD 14-18 observations in the Standard and WAV strategies.

CP DD 14-19 observations in the Standard and WAV strategies.

CP DD 14-22 observations in the Standard and WAV strategies.

As the measurement noise for the CP is much smaller than for the PR, it is difficult to see differences between the Standard and WAV processing. But for either the PR or the CP DDs, it was possible to verify that the DD functional model represents the rover receiver movement.

To analyze the improvement in the functional model, the DD residuals obtained from Standard and WAV results were also compared. The estimated PR residuals are plotted in Figures

PR DD 14-01 residuals in the Standard and WAV strategies.

PR DD 14-03 residuals in the Standard and WAV strategies.

PR DD 14-18 residuals in the Standard and WAV strategies.

PR DD 14-19 residuals in the Standard and WAV strategies.

PR DD 14-22 residuals in the Standard and WAV strategies.

We note that the DD residuals involving PRN 18 with the lowest (55°–59°) elevation angles present the largest errors (Figure

In the same way, DD residuals obtained from the Standard and WAV strategies were also compared for CP residuals in Figures

CP DD 14-01 residuals in the Standard and WAV strategies.

CP DD 14-03 residuals in the Standard and WAV strategies.

CP DD 14-18 residuals in the Standard and WAV strategies.

CP DD 14-19 residuals in the Standard and WAV strategies.

CP DD 14-22 residuals in the Standard and WAV strategies.

One can observe from Figures

_{i}

Statistics for DD PR residuals, including the RMS maximum error and reduction factor.

DD PR | RMS | Maximum error | ||||

Stand | Wav | Reduction factor | Stand | Wav | Reduction factor | |

14-01 | 0.959 | 0.35 | 2.7 | 2.497 | 0.497 | 5.0 |

14-03 | 0.622 | 0.276 | 2.3 | 2.033 | 0.562 | 3.6 |

14-18 | 1.176 | 0.385 | 3.1 | 4.419 | 0.335 | 13.2 |

14-19 | 0.887 | 0.357 | 2.5 | 3.018 | 0.583 | 5.2 |

14-22 | 0.779 | 0.292 | 2.7 | 2.247 | 0.272 | 8.3 |

Statistics for DD CP residuals, including the RMS maximum error and reduction factor.

DD Phase | RMS | Maximum error | ||||

Stand | Wav | Reduction factor | Stand | Wav | Reduction factor | |

14-01 | 0.007 | 0.004 | 1.8 | 0.028 | 0.009 | 3.1 |

14-03 | 0.017 | 0.004 | 4.3 | 0.026 | 0.02 | 1.3 |

14-18 | 0.007 | 0.003 | 2.3 | 0.016 | 0.01 | 1.6 |

14-19 | 0.008 | 0.002 | 4.0 | 0.023 | 0.007 | 3.3 |

14-22 | 0.012 | 0.005 | 2.4 | 0.038 | 0.013 | 2.9 |

Tables

To compare the quality of the PR and CP DD observables, with and without the multipath correction, the

LOM statistical test.

Figure

In relation to the ambiguities solution, they could not be solved in the Standard processing because of the multipath effect. After correcting for this effect by the WAV method, the ambiguities could be reliably fixed at the beginning of the processing (less than 7 minutes).

In order to evaluate the accuracy of the baseline estimation, the baseline length estimated for each instant was compared with the reference value. For this, the discrepancies between the baseline estimative and the known baseline value were computed for standard and WAV processing. The RMS of the discrepancies for standard processing was 0.296 m. After applying the WAV method, the RMS was reduced to 0.029 m, representing a 90% improvement. Thus, one can verify that after the multipath correction, the results were better than in the standard processing.

In this paper, we show how to construct a functional mathematical model in order to eliminate most GPS errors.

The behavior of the functional model in kinematic mode is also demonstrated and analyzed in terms of vector operations. The experiment verifies the discussed principles, concluding that the DD functional model, which involves the differences in distances between receivers and satellites, describes the movement of the rover receiver.

We also present how the multipath effect, the only significant error not eliminated in the functional model, can be corrected by a proposed method using wavelets in kinematic positioning.

The results obtained in the experiments were very promising, showing that the proposed wavelet method appears to be a powerful method for mitigating multipath effects in kinematic GPS applications.

The multipath trend in the DD residuals was significantly corrected. The LOM statistical test also showed improvements in the quality of the data once the wavelet method was applied.

It was possible to observe the performance of the method for static and kinematic processing as well as the transition between both cases. The method demonstrated the potential for adaptation to the experimental condition. Other challenges must be addressed, such as more reflectors and different distances from the antenna, but due to the principles and features of the wavelet method, these can probably be accommodated.

Furthermore, the experiment was carried out with the rover varying smoothly. The next step is to evaluate how the solution is affected if the motion of the rover is highly dynamic. Free movements can also be experimented with, but the circular and controlled movement shown in this paper is very important for correct evaluation of the results.

The data processing and analyses have been restricted to the L1 observables but can also be extended to jointly process L1 and L2 observables.

The authors thank FAPESP for financial support to the first author (03/12770-3) and the Thematic Project (06/04008-2). They also thank Dr. José Paulo Molin and Thiago Martins Machado (“Luiz de Queiroz’’ College of Agriculture of São Paulo University) who made the kinematic experiment possible.