^{1}

^{2}

^{1}

^{2}

If three or more GPS antennas are mounted properly on a platform and differences of GPS signals measurements are collected simultaneously, the baselines vectors between antennas can be determined and the platform orientation defined by these vectors can be calculated. Thus, the prerequisite for attitude determination technique based on GPS is to calculate baselines between antennas to millimeter level of accuracy. For accurate attitude solutions to be attained, carrier phase double differences are used as main type of measurements. The use of carrier phase measurements leads to the problem of precise determination of the ambiguous integer number of cycles in the initial carrier phase (integer ambiguity). In this work two algorithms (LSAST and LAMBDA) were implemented and tested for ambiguity resolution allowing accurate real-time attitude determination using measurements given by GPS receivers in coupled form. Platform orientation was obtained using quaternions formulation, and the results showed that LSAST method performance is similar to LAMBDA as far as the number of epochs which are necessary to resolve ambiguities is concerned, but with processing time significantly higher. The final result accuracy was similar for both methods, better than 0.1° to 0.2°, when baselines are considered in decoupled form.

Global Navigation Satellite Systems (GNSSs) are satellite-based radionavigation systems, providing to worldwide users precise position and timing. System satellites transmit radio-frequency signals containing information required for the user equipment to compute its navigation solution (position, velocity, and time). GNSS can also be used to determine attitude of a platform, in which three or more antennas are needed to calculate attitude parameters [

If three or more GNSS antennas properly mounted on a platform and differences of GNSS signals measurements are collected simultaneously, baselines vectors formed between antennas can be determined, and orientation of the platform defined by these vectors can be calculated. Thus, the prerequisite for the attitude determination technique based on GNSS systems is to calculate the baselines between the antennas.

Accurate attitude solutions can be obtained using carrier phase double difference observables as the main type of measurements, including all independent combinations of antenna positions. Baselines between antennas must be determined in millimeter level of accuracy. Typically, the distance between the antennas is a few meters or less, and all spatially correlated errors between the antennas are almost eliminated in differencing (single and double) process, including orbital, ionospheric, and tropospheric errors. Therefore, main error sources affecting attitude determination are the multipath, receiver internal noise, and antenna phase center variation [

The use of carrier phase measurements leads to the problem of determining precisely the ambiguous initial carrier phase integer number of cycles (integer ambiguity). To increase confidence and accelerate the process by limiting the search space, three types of restrictions can be established from prior knowledge of the antenna fixed geometry: (i) length of the baseline, (ii) angle between baselines, and (iii) knowledge of the geometry of the antennas as a network in which double difference ambiguities must satisfy a closed loop condition [

A common attitude representation is done by Euler angles. This parameterization has difficult computation in general, because of the use of trigonometric functions and the appearance of a singularity in the motion modeling. Another way to attitude parameterization is using quaternions. This parameterization has some advantages over Euler angles; it is computationally efficient, there is no singularity, and it does not depend on trigonometric functions [

So, in this work an implementation and analysis of algorithms for integer ambiguity resolution allowing accurate attitude determination in real time, using measurements provided by GPS receivers, will be tested. Algorithm tests, using quaternions for attitude representation, will be implemented with real data, collected at INPE and described in [

least-squares ambiguity decorrelation adjustment (LAMBDA) method is a procedure for integer ambiguity estimation in carrier phase measurements. This method executes the integer ambiguity estimation through a Z transform, in which ambiguities are decorrelated before the integer values search process. Then, a minimization problem is approached as a discrete search inside an ellipsoidal region defined by decorrelated ambiguities, which is smaller than original ones. As a result, integer least-squares estimates for the ambiguities are obtained. This method was introduced in [

Least-Squares Ambiguity Solution Technique (LSAST) method, also known as LSAST method, was proposed in [

Euler angles are a means of representing the spatial orientation of any coordinate system as a composition of rotations from a frame of reference. These angles uniquely determine the orientation of a rigid body in three-dimensional space. There are several conventions for defining the Euler angles, depending on the choice of axes and the order in which rotations about these axes are performed. A matrix expression can be found for any frame given its Euler angles, performing three rotations in sequence. Here, Euler angles are denoted as

When distance between receivers is short (until 10 km), ionospheric and tropospheric residuals are small compared to multipath and internal receiver noise errors. Thus, for a short baseline, carrier phase double-difference measurements (

Data used in this study were originally collected for use in [

3 AllStar CMC GPS receivers (Canadian Marconi Space Company) and

3 AllStar CMC model AT 575-70 GPS antennas.

Antenna configuration in data acquisition.

GPS measurements were processed to obtain a precise baseline length. Ambiguity resolution was made using LAMBDA and LSAST methods. Each baseline was independently determined to form a frame, whose attitude is calculated referred to the east-north-up reference system. Rotation matrix, obtained by quaternions, is transformed to Euler angles using (

In this test, same 8 satellites were kept in view (SV04, SV08, SV13, SV19, SV23, SV27, SV28 e SV31), resulting in 7 double difference measurements. SV13 was chosen as master satellite, because of its high elevation. Ambiguity resolution methods use a Kalman filter to obtain real-valued (float) ambiguities. Kalman filter needs some epochs to converge to an ambiguity solution, and thus to a given baseline. Data were free of cycle slips.

In this way, LAMBDA method presents a solution with correct values of ambiguities after 107 epochs, while LSAST method takes 157 epochs to reach the correct values. Graphs in Figure

Table

LSAST method sweeps a fixed number of cycles in satellite primary set, leading to a slower search. This method had a mean processing time of

Mean and standard-deviation (SD) for data set 1.

LAMBDA | LSAST | |||

Angle [°] | Mean | SD | Mean | SD |

Roll | 0.136 | 0.132 | 0.097 | 0.014 |

Pitch | 0.482 | 0.060 | 0.520 | 0.028 |

Yaw | −6.488 | 0.045 | −6.478 | 0.001 |

Euler angles for data set 1.

LAMBDA

LSAST

In this set, 7 satellites were visible (SV08, SV13, SV19, SV23, SV27, SV28, and SV31), resulting in 6 double difference measurements. SV13 was taken as master satellite. Data were free of cycle slips.

Graphs in Figure

In this test, mean processing time was

Mean and standard-deviation (SD) for data set 2.

LAMBDA | LSAST | |||

Angle [°] | Mean | SD | Mean | SD |

Roll | 1.839 | 0.190 | 1.839 | 0.190 |

Pitch | −0.820 | 0.079 | −0.820 | 0.079 |

Yaw | −5.608 | 0.056 | −5.608 | 0.056 |

Euler angles for data set 2.

LAMBDA

LSAST

This data set had the same visible satellites as Data Set 2. Both LAMBDA and LSAST methods present solutions for ambiguities after 320 epochs. Graphs in Figure

The average processing time for ambiguity resolution step, whenever a successful solution is obtained, is

Mean and standard deviation (SD) for data set 3.

LAMBDA | LSAST | |||

Angle [°] | Mean | SD | Mean | SD |

Roll | 1.049 | 0.306 | 1.049 | 0.306 |

Pitch | 0.557 | 0.116 | 0.557 | 0.116 |

Yaw | −5.612 | 0.178 | −5.612 | 0.178 |

Euler angles for data set 3.

LAMBDA

LSAST

Processing time for all data sets.

Data set 1

Data set 2

Data set 3

Due to short baseline (1 m), both methods have a tendency to solve ambiguities for the same number of epochs, as filter converges. Both LAMBDA and LSAST methods had a similar performance in number of epochs needed to give the correct solution; however, processing time of LSAST method was significantly longer. This is due to the fact that LSAST method performs a systematic search throughout the primary set of measurements, while LAMBDA optimizes the search space as a whole. Processing time for LAMBDA method is search space size dependant [

The accuracy of final result is also similar for both methods, better than 0.1° to 0.2°, once they find the same ambiguity set. The small variations are due to difference in number of epochs to obtain the solution. Data Set 2 and Data Set 3 showed same mean values and standard deviation for Euler angles. This occurs because the only difference in algorithms is resolution ambiguity routine. Once ambiguities were resolved to the same values, statistics must be equal. LAMBDA method also has an extensive series of studies documented in the literature.

Attitude estimation using quaternions led to the same values when using a rotation matrix based on Euler angles, as results showed in [

Using dual frequency measurements is not necessary for the mitigation of ionospheric errors since the baselines are short but can be useful for a faster ambiguity resolution, for example, using widelane technique. If compared with the use of GPS alone, measurements from two different GNSS systems can certainly improve the attitude determining accuracy as result of increased number of measurements and improved satellite geometry. This can be accomplished with the use of GPS/GLONASS and future GPS/Galileo receivers.