The principle of the traditional attitude solution approach based on GPS (Global Position System) is to get the attitude matrix according to the relationship of coordinates. During the progress, the error of baseline position assumed in ECEF (Earth-Centered Earth-Fixed) and the error of coordinate transform between body frame and reference frame (ENU, East-North-Up) have been included in the result, and finally the precision of attitude determination is reduced. This contribution presents a new approach of attitude determination, in which the attitude angles are calculated by the double-difference carrier phase equation of GPS according to the relationship of attitude matrix and attitude angles through least-squares estimate method. The new approach predigests the procedure of attitude determination which reduces the error assumed. According to the analysis the precision of attitude determination is higher than that of traditional method. It is shown it gets a precise attitude result with the direct attitude determination method in the simulation. A novel algorithm is proposed to solve some problems. Simulation results show the effectiveness of the proposed algorithm.
GPS-based attitude determination is an important research field in recent years, where the key technologies involve the detection and reparation of cycle skip, integer ambiguities resolution, attitude solution, and so forth. At present, the common methods of attitude solution can be divided into two types [
One type is to get the attitude matrix. Based on the relationship between baseline coordinates in the local orthogonal frame and the body frame, this type includes the TRIAD algorithm which estimates optimally according to the orthogonal property of the attitude matrix, the algorithms based on Wahba problem, for example, QUEST (QUaternion ESTimator), SVD (Singular Value Decomposition), FOAM (Fast Optimal Attitude Matrix), Euler-q, and so forth, and the least-square method using multiple epoch baseline coordinate [
The other type is to get the attitude angles. Setting one baseline along the azimuth axis of vehicle, the yaw and pitch angles can be obtained first, and then according to the rotating relationship of the second baseline, the roll angle can be calculated [
The method of getting attitude matrix includes two steps that estimations of baseline coordinate and attitude matrix parameters, and each of them will lead estimation errors in final results. On the other hand, the attitude matrix derived by the observed equations directly is usually not orthogonal. While, in the method of getting attitude angles the estimations of baseline coordinate and attitude matrix are dropt, and procedure errors are except. But, when the
This contribution presents a new approach estimating the attitude angles by double-difference carrier phase equation of GPS, which predigests the process calculating attitude angles by attitude matrix. The new method improves the precision and efficiency of estimation and avoids the problem of possible ambiguous values caused by traditional attitude-angle methods.
Assume that attitude determination system consists of three antennas, namely, antenna 0 which is set as main antenna and antennas 1 and 2 which are set as slave antennas. All of the three antennas are tracking
Single difference between the two slave receivers can be operated to eliminate many errors, such as the delays of ionospheric and tropospheric, the delays of instrumental and errors of satellite clock. Since the length of baseline is rather less than the distance between receiver and satellite, it is considered that the line-of-sight vectors of two endpoints of baseline are the same [
Single-difference observation of carrier phase measurements.
According to the phase observation equation, when two antennas track the same satellite I, the single-difference equation can be derived as follows:
Aiming to eliminate receiver clock error the differences between the observations of two satellites taken at the same time and different receivers are formed as follows:
The purpose of relative positioning is to get the baseline vector
Therefore, the relationship between double-difference distance and baseline vector can be expressed as
The double-difference equation of carrier phase is transformed into
In traditional process, the attitude determination needs two steps including the estimation of baseline coordinates and attitude matrix.
Aiming to simplify the calculations, avoiding the error during estimation, and improving the precision of solution, a direct method of gaining attitude angles using double-difference equation is introduced in the following [
Assume that GPS attitude determination system is connected rigidly with vehicle; that is, antenna coordinates in body frame are invariant and known which are, respectively,
The relative position of antenna in the body frame.
According to the transformation relationship of coordinates, baseline vector can be expressed in body frame. Therefore, model (
After the double-difference integer ambiguity is determined, there are three attitude angles unknown in model (
M-1 double-difference equations can be established for every baseline:
The form of rotation matrix
Because of the nonlinear relation between the model and attitude angle, attitude angle is determined by nonlinear least-square estimation. Assuming that state variables are three attitude angles as follows:
Firstly, with the high-order terms being neglected, the nonlinear equation is linearized through first-order Taylor expansion of every equation in the model. The nonlinear equations can be approximately transformed into linear equations expressed by the following matrix:
Observation matrix
Then the least-square solution is
According to the equation of nonlinear least-square estimation and the initial attitude angle given above, the estimation of unknown parameters can be obtained. Namely, the attitude angle is estimated by double-difference observation equation of carrier phase.
Compared with traditional approach of attitude determination, the solution error of model is analyzed.
Assume that the result is obtained from double-difference equation of carrier phase in both methods. In traditional attitude determination method, the baseline coordinates in local frame can be calculated through the equation firstly.
The phase observation error of all satellites is assumed as
There is only one least-square solution expressed as follows:
In (
According to the rule of error transmitting, the error of least-square solution is
So another least-square estimation is needed to solve the overdetermined equation. Then the estimation error increases to
Many different theories are used to solve attitude matrix in the other methods, but the baseline coordinates in the local frame should be estimated at first. And the estimation error also exists in the process of attitude matrix determination. So the error of traditional attitude determination approach is in a high level.
Due to only one step of least-square estimation being operated in the direct attitude determination approach, the error of state vector estimated is as follows:
Observation errors are mutually independent, whose variance is
Dimensionless matrix
Therefore, the error of attitude angle estimated is concerned with the measurement error, measurement matrix, and length of baseline. Ignoring other errors, the longer the baseline is, the higher the precision is; according (
Through setting the geodetic coordinate of main antenna, the body coordinate of baseline, and the true attitude angles, attitude determination algorithm can be verified by simulation system with the known satellite positions.
Simulation system consists of the following sections.
With the almanac files supplied by the website of navcen.uscg.gov and actually broadcasted by GPS satellites, the real-time coordinate of all satellites in orbit are calculated; then the coarse precision satellites’ tracks can be obtained. After all elevations are calculated in the setting user position, visible satellites information can be obtained.
After positions of main antenna and reference antenna are set, coordinates of all antennas in the local frame can be obtained according to the predetermined attitude angles. Namely, distance from satellite to antenna is obtained. With certain measurement noise added, carrier signal can be simulated. And the equation of phase observation can be established (assuming that integer ambiguity is determined).
Parameters of simulation are known information, and according to different attitude solution approaches, equation is built to solve the attitude angles which will be compared with predetermined attitude angles to verify the algorithm.
Installing the array of the three antennas as Figure
The distribution of antenna in the simulation model.
Assume that reference antenna is at the rotation center of vehicle, phase error is set at (4, 5, 6) mm, and the other simulation conditions are set as Table
Settings of simulation parameter.
Parameter | Value |
---|---|
Reference antenna coordinates in geographic frame | 45°N, 120°E, 0 m |
Antenna 1 coordinates in the body frame | (3, 0, 0) m |
Antenna 2 coordinates in the body frame | (0, 3, 4) m |
Predetermined attitude angles of simulation | Yaw: 90°, roll: 30°, pitch: 30° |
Elevation angle limited is set at 5° [
The visible satellites in the simulation position.
Sequence number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
The visible satellite number | 4 | 14 | 18 | 20 | 22 | 24 | 25 | 29 |
Because that attitude solution of traditional approach needs multiepochs information, simulation is done for 10 minutes. Namely, 20 epochs is sampled. At the same time, the new approach is simulated. The two attitude determination results are compared.
In traditional attitude determination approach, the results estimated by least-square method once are listed as yaw is 90.0404°, roll is 30.0088°, and pitch is 29.9516°.
Attitude results estimated by nonlinear least-square iteration in the direct double-difference approach are listed as: yaw is 90.1092°, roll is 29.9925°, and pitch is 29.9831°. The estimation process is shown at Figure
The process of attitude angles estimated using the direct approach based on carrier double-difference equation.
According to the result of direct approach, attitude angles become steady after 5 times cyclic calculations within only single epoch. While traditional method need more epochs for obtaining the final attitude solution. Precision of estimation is
The approach of solving attitude angles based on double-difference carrier phase measurements is proposed in this contribution, which is analyzed and compared with traditional approach, and some estimation procedures are avoided. The accuracy and low error in calculation of the new attitude determination model are verified. According to the attitude verification program based on MATLAB simulation, reality and reliability are certified and a certain attitude solution precision is achieved.
The authors declare there is no conflict of interests regarding the publication of this paper.
Funding for this work was provided by the National Nature Science Foundation of China under Grant no. 61374007 and no. 61104036. The authors would like to thank all the editors and anonymous reviewers for improving this paper.