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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In the field of Global Navigation Satellite System (GNSS) attitude determination, the constraints usually play a critical role in resolving the unknown ambiguities quickly and correctly. Many constraints such as the baseline length, the geometry of multi-baselines and the horizontal attitude angles have been used extensively to improve the performance of ambiguity resolution. In the GNSS/Inertial Navigation System (INS) integrated attitude determination systems using low grade Inertial Measurement Unit (IMU), the initial heading parameters of the vehicle are usually worked out by the GNSS subsystem instead of by the IMU sensors independently. However, when a rotation occurs, the angle at which vehicle has turned within a short time span can be measured accurately by the IMU. This measurement will be treated as a constraint, namely the rate-gyro-integral constraint, which can aid the GNSS ambiguity resolution. We will use this constraint to filter the candidates in the ambiguity search stage. The ambiguity search space shrinks significantly with this constraint imposed during the rotation, thus it is helpful to speeding up the initialization of attitude parameters under dynamic circumstances. This paper will only study the applications of this new constraint to land vehicles. The impacts of measurement errors on the effect of this new constraint will be assessed for different grades of IMU and current average precision level of GNSS receivers. Simulations and experiments in urban areas have demonstrated the validity and efficacy of the new constraint in aiding GNSS attitude determinations.

Ambiguity resolution is a core technique in GNSS relative positioning and attitude determination. There is no essential difference between the two applications in terms of this technique. Unlike the relative positioning, however, the distances between antennas are constant and short in attitude determination, generally ranging from a few meters to dozens of meters. Therefore, more constraints exist in attitude determination and the double-difference (DD) carrier phase measurement is more precise than relative positioning. These constraints have become very important for ambiguity resolution in attitude determination.

There exist many constraints applied to ambiguity resolution. The baseline length is a readily available and widely used constraint. The ambiguity search space can be reduced from 3D to 2D by using this constraint [

Platform rotations can produce more information used in ambiguity resolution. This idea was proposed by [

In this paper, a new constraint is proposed for the ambiguity resolution in GNSS attitude determination applications. For land vehicle applications, the baseline approximately lies in the plane of local level during a rotational motion [

The rest part of this paper is organized as follows: Section 2 summarizes the basic principle and mathematic model for the rate-gyro-integral constraint. Section 3 presents a specific implementation method and its geometric analysis. Section 4 discusses on the error factors of the proposed implementation method. Section 5 gives some simulation results. Section 6 shows the results of processing actual field data. Section 7 concludes this paper and gives some suggests for future research.

The rotational motions of vehicle can be measured by gyroscopes that are part of the IMU. In this section, our work is to find the proper mathematical models for explicating how this measurement improves the efficiency of ambiguity search.

It is assumed that an IMU is attached to the vehicle with two GNSS antennas. The b-frame has its origin at the IMU reference point. The longitudinal axis of the vehicle is the X axis of the b-frame and the transvers axis of the vehicle is the Y axis of the b-frame. The Z axis obeys right-handed rule and points downwards. The baseline vector ^{(b)} is the 3D vector between the carrier phase centers of the two antennas, with three known constants being its coordinates in the b-frame. The origin of the n-frame (North, East, Down) is consistent with that of the b-frame. If
^{(n)} is the baseline vector in the n-frame and _{0} is the starting time of rotation.

Updating
^{(n)}(_{0}) and ^{(n)} (_{k}_{k}_{0}, _{k}_{k}_{k}_{0}) has no effect on _{k}^{(b)} is the unit vector of ^{(b)} and _{k}^{(n)} (_{0}) and ^{(n)} (_{k}

The GNSS DD carrier phase observation equations are formulated as follows [_{i}^{i}^{i}

Four satellites with minimum Geometry Dilution of Precision (GDOP) value are selected from all visible satellites, and the satellite with the largest elevation is chosen as the “reference” satellite, the other three satellites are recognized as the “master satellites”. Thus there are three independent DD carrier phase measurement equations in ^{(n)′}. Thus, the inner product of the GNSS baseline vector solutions at _{0} and _{k}^{(n)} (_{0})′ and ^{(n)} (_{k}

Hence, if the true integer ambiguity combination, which is denoted as ∇Δ_{m}^{(n)} (_{0})′ _{m}^{(n)}(_{k}_{m}^{(n)} (_{0})′_{m}^{(n)}(_{k}_{m}

For land vehicles, the baseline approximately lies in the local level. Thus,
_{0} to _{k}_{k}_{m}_{k}

In

An implementation method for the rate-gyro-integral constraint is proposed in this section. By comparing the testing objectives with a properly selected threshold, the unacceptable testing objectives can be found out and the corresponding candidates are filtered out from the ambiguity search space.

The mathematical description of a rotational motion usually consists of a rotation axis and a rotation angle. The projections of ^{(n)}(_{0})′ and ^{(n)}(_{k}^{(n)} (_{o})′ and ^{(n)} (_{k}^{(n)} (_{o})′ and ^{(n)} (_{k}^{(n)}(_{0})′_{m}^{(n)}(_{k}_{m}^{(n)}(_{0})′_{m}^{(n)}(_{k}_{m}

Assuming that the rotation angle can be measured by the IMU sensors, thus it is easy to verify that
_{k}_{k,m}_{k,m}_{threshold} is selected properly for the implementation method.

At the ambiguity search stage, if a candidate ∇Δ_{m}_{k,m}_{k,m}_{threshold}, thus the testing objective is verified to be unacceptable and ∇Δ_{m}

Assuming that the baseline vector rotates a whole round, thus, in _{k,m}′ lie inside the interval of [0°,360°], and the other notations will be illustrated later.

In ^{+}). A planar Cartesian coordinate, namely the p-frame, is defined on the rotation plane. The X-axis of the p-frame is consistent with the baseline vector at _{0}, and the Y-axis of the p-frame vertically points to the right of X-axis. _{0}_{0},_{m}^{(n)}(_{0})′_{m}^{−1} (•) is defined as follows:

Taking the derivative of the right side of

During the whole rotation procedure, there are two _{k,m}_{k}_{1} and _{k}_{2} respectively. Then, just denote the larger one of the abstract values of the two peak values as:

Finally, the explicit expression of

According to _{m}

In brief, there are two important elements of the implementation method for the rate-gyro-integral constraint. One is to generate a set of testing objectives for each candidate, the other is to select a proper threshold. For the former, it is necessary to investigate two aspects, the distribution range and density of the testing objectives on the rotation process. For the latter, the success rate and shrinking efficiency are analyzed under different conditions of measurement scenarios and threshold settings.

Contributors to the inaccuracy of testing objective involve the IMU measurement errors, especially those associated with angle rate, GNSS carrier phase measurement errors and the actual rotational axis offsets.

The angle rate measurement errors are largely responsible for the inaccuracy of testing objective by IMU measurement effects. In the strapdown mechanization, the measurement model of angle rate with respect to the n-frame can be expressed as [

Note that the second formula in

Since the attitude of vehicle is unknown,
_{0}to _{k}

For land vehicle rotational motions, the unit vector of local gravitational vector, which has three constant coordinates in the n-frame and is denoted as ^{(n)}′ = [0 0 1]^{T}^{(n)}, always offsets from its assuming observation, thus an error model is constructed for ^{(n)} as follows:
^{(n)} and ^{(n)}′, and it follows a normal probability distribution. ^{(n)} with respect to the true north, and it follows a uniform probability distribution in the interval of [0,2

In short-baseline cases, the dominant errors of DD carrier phase measurements include multipath, which can be considered noise-like, and receiver thermal noise [^{i}^{(n)}(_{k}^{(n)}(_{k}_{ϕ}_{∇Δb⃑} . Hence, the errors in ^{(n)}′ and

^{(n)}(_{k}

Basing on the implementation method and the models constructed for various measurement errors, some simulations are conducted. For simplicity, the measurement errors of IMU and GNSS receivers, plus the actual rotational axis offsets are addressed as the 1st, 2nd and 3rd type measurement error, respectively. For different simulation scenarios, the results were assessed on two aspects,

^{(b)} = [3 0 0]^{T}

with the actual locations and GPS constellation imposed, select a reference satellite and three master satellites, then compute _{i}

set the parameters for rotation axis ^{(n)}, the complete rotational angle ^{(n)} (_{k}

with the selected satellites in Step 2 imposed, generate the true ambiguity vector ∇Δ_{k}

for each candidate lies inside ℤ at _{k}_{k,m}

test all the Δ_{k,m} s for each _{k}_{k,m}_{threshold} from ℤ.

In Step 2, the actual GNSS data was collected on 11 June 2011, at N 29.5650°, E 106.2197°. From the satellites in view, the satellite with maximal elevation is chosen as the reference satellite, then three master satellites are selected based on the minimal GDOP principle. In Step 4, the parameter of search radius,

In the first simulation experiment, except for the 1st type measurement error, both the 2nd and 3rd type measurement errors are considered.

In the second one, all the three types of measurement errors are all taken into account. The simulation parameters for

To make

Shrinking efficiency herein is explained by the size of steady ambiguity search space, which is concluded by averaging and rounding figures obtained from a lot of simulations for the successful shrinking procedures.

In

The GNSS/INS integrated attitude determination system used herein primarily consists of a tactic grade FOG-IMU, an array of three GNSS antennas with receivers connected individually and a navigation computer. The concerned technical characteristic of the FOG-IMU is the equivalent bias of gyroscope denoted as

The actual field data was collected on 16 April 2008 in the Chong Qing urban area, China. The GPS measurements were available at a rate of 1 Hz and the output rate of IMU was 200 Hz. After a successful static initialization, the vehicle was driven into the dense urban area where it followed the trajectory shown in

Each data section selected is processed following the scheme shown in

The initial ambiguity search space (3D) is determined by an float ambiguity estimation vector denoted as

In this subsection, the feasibility of the rate-gyro-integral constraint in actual applications is tested with data section C, E and F. During the collecting period of data section F, the vehicle was driven along a turntable road shown in

When the threshold value is chosen to be 0.5°, 1° and 3 °, respectively, the first group of results of feasibility test can be obtained by processing data section F and given by

_{threshold}=1° and |Δ_{threshold}=3°,respectively. If |Δ_{threshold}=0.5° is chosen, for data section F, the true ambiguity combination will be locked only by using the rate-gyro-integral constraint.

Data section C was collected on a crossroad (

From

The three groups of results demonstrate that if adequate common visual satellites are available, as well as the turning angle is large enough, the rate-gyro-integral constraint is practicable in land navigation applications.

In this subsection, some characteristics of the rate-gyro-integral constraint will be carried out by use of data section F. It is not difficult to know that in those successful shrinking procedures, lower threshold value can promote the shrinking efficiency. To verify this characteristic in practical cases, two zooms to

In both

After each data element with turning angle larger than 190° was excluded from data section F, an approximate 1/2 cycle turning is forced. From this preserved data section, four subsequences will be picked out and each of them has a relative even distribution on turning angle. _{threshold}=1° and approximate 1/2 cycle turning size of the search space □ .

From

In this paper, a new constraint for use in GNSS attitude determination in land vehicle applications has been developed and analyzed. To speed up the initialization of attitude parameters under dynamic rotation circumstances, the new method of applying a constraint to vehicle turning motion has been assessed. It has been demonstrated that this new constraint can result in high ambiguity search success rates and efficiency for shrinking the ambiguity search space. The contributing factors to the actual performance of the new rate-gyro-integral constraint can be divided into two categories, namely the systematical factors and the measurement factors respectively. The systematical factors, which contain the length of baseline and the amplitude of turning angle, determine the attained peak value of the testing objective, while the accuracy of computed testing objective depends on the measurement factors, which include the IMU measurement errors, especially those associated with angle rate, GNSS carrier phase measurement errors and the actual turning axis offsets. Additionally, the threshold has an important effect on the performance of the new constraint as well.

The proposed constraint has been verified with simulations as well as actual field data. In the simulations, both the successful rate and shrinking efficiency have been analyzed in different measurement scenarios, especially with different grades of inertial sensors considered. Our vehicle test, with the data collected in the urban area of Chong Qing, covers three typical curves in urban areas. From the obtained results, we can conclude that the rate-gyro-integral constraint is practical in land navigation applications. Although only using a tactic grade IMU in our vehicle test, it is emphasized that the new constraint is generally applicable and thus also applicable to the GNSS/INS integrated system by use of low grade MIMU.

The idea behind the proposed new rate-gyro-integral constraint for GNSS attitude determination is the utilization of the platform dynamic information measured by affordable inertial sensors. Hence, further research may focus on the use of the rate-gyro-integral constraint to enhance the strength of GNSS model under the poor observation circumstances which are very common for the operations of dynamic platforms.

If both _{0} ∈ [0, 2

However, it should be note that once

This value cannot reflect the actual rotational angle between ^{(n)} (_{0})′_{m}^{(n)} (_{k}_{m}_{k,m}′ should be given by:

Finally, together with Δ_{k,m}_{k,m}^{′} −

This research is supported by the National Science Foundation of China (Grant No. 61104201). We would like to thank Liangqing Lu for his assistance on constructing the experimental system and collecting the experimental data, as well as the Google company for providing free digital map online.

The authors declare no conflict of interest.

Geometric depiction of the rate-gyro-integral constraint.

Projection vectors and angles on the rotation plane.

Success rate varies with threshold at different accuracy levels of gyroscope.

Shrinking efficiency varies with threshold at different accuracy of gyroscope.

Testing vehicle is set up: (

Testing trajectory.

Scheme for actual field data processing.

Turntable road and locations of 23 GPS epochs for data section F.

ζ s for data section F.

Vehicle orientations for data section F.

The first group of results of feasibility test (data section F).

(

The second group of the results for the feasibility test (data section C).

The third group of the results for the feasibility test (data section E).

Size of ℤ contracts to less than 100 for each |Δ_{threshold}.

Size of the search space □ obtains a steady status for each |Δ_{threshold}.

Comparison of shrinking efficiencies of Baselines 1 and 2.

Success rates with the 2nd or 3rd type measurement error considered individually.

2nd Type | 100% | 100% | 82.46% | 5.19% | 0% |

3rd Type | 100% | 99.98% | 95.05% | 68.18% | 1.46% |

“Current” accuracy levels of gyroscopes.

Middle grade | 0.1 | <0.03 |

Tactic grade | 1∼100 | 0.03∼0.1 |

Automotive grade | >100 | >1 |

Shrinking processes of the search space □ from different subsequences.

14p | 1331 | 665 | 306 | 166 | 83 | 44 | 29 |

7p | 1331 | – | 306 | – | 88 | – | 30 |

5p | 1331 | – | – | 179 | – | – | 31 |

3p | 1331 | – | – | – | – | – | 79 |

100° | 112.4° | 126° | 140.5° | 155° | 169.5° | 183.9° | |

14p | 17 | 14 | 12 | 10 | 8 | 7 | 6 |

7p | – | 14 | – | – | 8 | – | 6 |

5p | – | – | – | 11 | – | – | 7 |

3p | – | – | – | – | – | – | 7 |