Attitude Determination by Means of Dual Frequency GPS Receivers

Transcription

1 Attitude Determination by Means of Dual Frequency GPS Receivers Vadim Rokhlin and Gilad Even Tzur Department of Mapping and Geo Information Engineering Faculty of Civil and Environmental Engineering Technion Israel Institute of Technology FIG Working Week 2012 Rome, Italy, 6-10 May Introduction Attitude determination in air, sea or land with high accuracy and high reliability has always been a technological and engineering challenge. Attitude determination can contribute to improving civil and military systems. Using GPS technology is not requiring precalibration as needed when using IMU. 1

2 Introduction (cont.) The basic idea behind the attitude determination based on a GPS multi antenna system is to calculate first the baselines between the antennas using differential positioning and then to derive the attitude parameters. For high accuracy applications, carrier phase data should be employed after the integer cycle ambiguities are correctly resolved. Processing only the pseudorange measurements also yields the attitude parameters, but with lower accuracy. Purpose The purpose of the current research is to examine speed of solution from dual frequency GPS measurements against one frequency receiver and examine L1 and L2 combination on the solution of the attitude parameters. 2

3 Coordinate System Two coordinate Systems need to be distinguished: The local level coordinate frame LLF. The antenna body frame AF. Attitude Determination Attitude of a rigid body platform is determined by the orientation of the specified body frame coordinate system with respect to the reference coordinate system. y using a GPS multi-antenna system, the attitude of the GPS antenna body frame with respect to the local level frame can be precisely computed at each observation epoch. b x X y = R2() r R1( p) R3() h Y z Z AF l LLF 3

4 Attitude Determination (cont.) There are two methods for estimation of heading, pitch and roll : Direct computation method which does not require knowledge of the antenna's body frame coordinates and only uses the local level coordinates of the three antennas that define the platform. Least squares estimation procedure that make full use of all the position information of multiple antennas in the local level as well as in body frame. Direct Attitude Determination Approach Two antennas determine the Azimuth (h) and pitch (p) of the platform. 1 h= tan ( X / Y ) 2, l 2, l p = tan ( Z / X + Y ) , l 2, l 2, l Roll (r) parameter is obtained after rotation coordinates of third antenna in LLF system by heading and pitch angles and transforming them to AF system. r = tan ( Z / X ) 1 '' '' 3, l 3, l 4

5 Least Squares Estimation When we have more than three antennas LSE gives the best estimates based on all the position information contained in a multiple GPS antenna array. The rotation matrix R is solely defined by the three elements, h, p and r. For each antenna the following relation is obtained: b xi Xi, l y = R ( r) R ( p) R ( h) Y, when i= 2,3... n i i, l z i Z AF i, l LLF Double Difference Method of GPS Carrier Phase Double Difference equation: Φ () t = ρ () t + λ A, L / L A A, L / L I + T + noise A, L / L A 1 2 5

6 Double Difference Method of GPS Carrier Phase with Dual Frequency Receiver Four satellites j, k, l, m and two epochs t1, t2 and the system follows as: dx dy ( t ) ( t ) dz ( ) ( ) ( ) ( ) jl = ; = ; x = ( ) ( ) ( ) ( ) ( ) ( ) jl 1 1 jl jl t1 t1 t1 t1 L1 L2 t2 t2 jl jl t2 t2 t2 t2 L1 T 1 T = = Ax; x= ( A PA) A P L2 Double Difference Method of GPS Carrier Phase with Dual Frequency Receiver (cont.) jl dx L1, L2 = Ax = [ Aa Aλ] x = A a dy + Aλ dz jl A λ = [ Aλ1 Aλ2] Aa 1 Aa = A a2 6

7 Least square AMiguity Decorrelation Adjustment (LAMDA) method The parameter estimation is carried out in three steps: 1. float solution. 2. integer ambiguity estimation. 3. fix solution. Least squares principle used to compute the integer difference ambiguities and the baseline coordinates: n min y A a A b, a Z, b R ab, λ α 2 Q y p RECURSIVE LEAST SQUARE ADJUSTMET ew measurements are added after every new epoch. The measurements are included into the least square along with the previous solution, while taking into account the cofactor matrix of the previous solution. Every new epoch is contributing for more accurate value of the variables. 7

8 Linear Combination Of The Frequencies The most commonly used Linear Combinations when working with the original L 1 and L 2 frequencies: Symbol a1 a2 1, 2 Remark L Original L₁ signal L Original L₂ signal Lc 1 fl2 / fl Ionosphere free, floating ambiguities Lw Wide lane Ln arrow lane c λ a a λa 1, a2 = ; fa 1, a2 = = a1 fl 1 + a2 fl2 a1 fl 1+ a2 fl2 λa 1, a2 c Linear Combination (cont.) Combined double difference phase measurement equation : Φ = a Φ + a Φ ij ij ij A a a AL AL [ 1, 2] 1 [ 1] 2 [ 2] Ambiguity associated with the linearly combined phase measurement: = a + a ij ij ij A a a AL AL [ 1, 2] 1 [ 1] 2 [ 2] 8

9 GUI lock Diagram 9

10 Test with two Dual Frequency Antennas Two dual frequency Ashtech Z Surveyor GPS receivers were used to collect 80 minutes of data at 1 second interval simultaneously in semistatic mode. 8 satellites were used for the solution of the observation equation, and the Pdop values ranged between 1.8 and 2.1. Test Results Static Mode L1: Azimuth mean = Std = 0.11 º º umber of epochs = 200 L1L2: Azimuth mean = º Std = 0.07 º umber of epochs = 40 10

11 Test Results Static Mode Lw: Azimuth mean = Std = 0.6 º º umber of epochs = 100 Ln: Azimuth mean = º Std = 0.07 º umber of epochs =225 Test Results Static Mode L1Ln: Azimuth mean = Std = 0.09 º º umber of epochs = 80 L1Lw: Azimuth mean = Std = 0.33 º º umber of epochs = 20 11

12 aseline Rotation Method Antenna swap method: First Position I Second Position II rotation by180 deg Antenna A Antenna Antenna Antenna A The vector is: The vector is: ( dx, dy, dz ) ( ) I dx, dy, dz = II ( dx, dy, dz ) I aseline Rotation Method (cont.) First position I: Ι L1, L2 Second position II: jl dx = A a dy Ι + Aλ dz jl = + jl dx II A L1, L2 aii dy Aλ dz jl 12

13 aseline Rotation Method (cont.) Difference between equations: dx = ( A + A ) dy dz II I aii ai As a result, the true vector can be obtained without solving for the IA variables. aseline Rotation Method (cont.) Antenna swap by θ deg : First Position I rotation by θ deg Second Position II Antenna A Antenna A Antenna θ In this case: ( dx, dy, dz ) ( dx, dy, dz ) I II Antenna 13

14 aseline Rotation Method (cont.) Antenna swap by θ deg: dx 1 T II I = ( AaII AaI PI R3 ( θ ) PII ) dy dz II R 3 (θ) rotation matrix about z axis. P transformation matrix between Cartesian and LLF coordinate systems. L1: Test Results with Antenna Swap Azimuth mean = Std = 0.08 º º Azimuth mean = Std = 0.11 º º umber of epochs = = 230 sec 14

15 L1 L2: Test Results with Antenna Swap Azimuth mean = Std = 0.09 º º Azimuth mean = Std = 0.12 º º umber of epochs = = 70 sec Conclusions Combining different frequencies significantly shortens the solution s time of the integer number of the wavelengths. In less than 30 seconds a solution can be obtained once the receiver starts picking up the satellites signals. umber of frequencies have no dramatic influence on the solution's accuracy. 15

16 Attitude Determination by Means of Dual Frequency GPS Receivers Thank you! FIG Working Week 2012 Rome, Italy, 6-10 May 16