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					Attitude Determination

          - Using GPS
                  Table of Contents

                  • Definition of Attitude
                  • Attitude and GPS
                  • Attitude Representations
                  • Least Squares Filter
                  • Kalman Filter
                  • Other Filters
                  • The AAU Testbed
                  • Results
                  • Conclusion
20/12-2000 (MJ)            Danish GPS Center   2
                  What is Attitude?

                   Z                    Orientation of a coordinate
                       V
                                        system (u,v,w) with
                                U       respect to some reference
                                        system (x,y,z)
                       W




                                    Y




      X




20/12-2000 (MJ)            Danish GPS Center                          3
   When is Attitude information needed?


           • Controlling an Aircraft, Boat or Automobile
           • Onboard Satellites
           • Pointing of Instruments
           • Pointing of Weapons
           • Entertainment industri (VR)
           • Etc...




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                  Attitude sensors

       Currently used sensors include:


                   • Gyroscopes
                   • Rate gyros (+integration)
                   • Star trackers
                   • Sun sensors
                   • Magnetometers
                   • GPS




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                  Advantages of GPS

      • Adding new functionality to existing equipment
      • No cost increase
      • No weight increase
      • No moving parts (solid-state)
      • Measures the absolute attitude

                  Disadvantages
      • Mediocre accuracy (0.1 - 1º RMS error)
      • Low bandwidth (5-10 Hz maximum)
      • Requires direct view of satellites

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             Interferometric Principle
                         Carrier Wave (from GPS satellite)




                        Integer
                      Component

             Master
            Antenna        k                      Fractional
                                                Component, 

                                           Baseline, b(3x1)
                                                                 Slave
                                                                Antenna



                                           e(3x1)
                                         Unit Vector to GPS Satellite




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             Interferometric Principle

     Measurement equation:




      The full phase difference is the projection of the
      baseline vector onto the LOS vector:




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                      Attitude Matrix
                  Z
                       V                  9 parameters needed:
                           U



                       W




                               Y



                                          When (x,y,z) is a reference
     X                                    system:




20/12-2000 (MJ)                Danish GPS Center                   9
                  Properties of “A”

      “A” rotates a vector from the reference system
      to the body system




      The transpose of “A” rotates in the opposite
      direction (back again)




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                  Properties of “A”

     Rotation does not change the size of the vectors:




     Every rotation has a rotation-axis (and a rotation-
     angle)



     The rotation-angle is the eigenvalue of “A”


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                  Euler sequences

      A sequence of rotations by the angles (,,) about
      the coordinate axes of the reference system



      Single axis:




      Multiple axes:


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                  Quaternions

    A quaternion consists of four composants



    Where i,j and k are hyperimaginary numbers




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                  Quaternions

    A quaternion can be thought of as a 4 dimensional
    vector with unit length:




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                  Quaternions

    Quaternions represent attitude as a rotation-axis
    and a rotation-angle




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                  Quaternions

    Quaternions can be multiplied using the special
    operator            defined as:




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                  Quaternions

    The attitude matrix can be formed from the
    quaternion as:




     Where




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          Least Squares Solution

     Including attitude information into the
     measurement equation




      Linearization of the attitude matrix




20/12-2000 (MJ)          Danish GPS Center     18
          Least Squares Solution

     Forming the phase residual




20/12-2000 (MJ)        Danish GPS Center   19
          Least Squares Solution




20/12-2000 (MJ)    Danish GPS Center   20
          Least Squares Solution

     Estimate update




20/12-2000 (MJ)        Danish GPS Center   21
         Extended Kalman Filter

     A Kalman filter consists of a model equation




     and a measurent equation




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         Extended Kalman Filter

     And their linearized counterparts….




     And




20/12-2000 (MJ)         Danish GPS Center   23
          Extended Kalman Filter

      Algorithm

         z(tk)                    x(tk-), P(tk-)


                     Estimate
                      Update
     Estimate of

     x(tk), P(tk)
                          x(tk+), P(tk+)           k=k+1


                       Time
                    Propagation

                          x(tk+1-), P(tk+1-)


20/12-2000 (MJ)                         Danish GPS Center   24
         Extended Kalman Filter
    Tuning of the filter




     Noise variance determined experimentally




20/12-2000 (MJ)            Danish GPS Center    25
         Extended Kalman Filter
    Tuning of the filter




     Noise variance determined by „trial-and-error‟




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         Extended Kalman Filter
    Determining the system model

             N    Dynamic                      Kinematic   q
                   model                          model




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                             Other Filters
                  Raw phase measurements                                    Non-iterative                                                          Iterative


                     Integer resolution

             Extended phase measurements


                   Line bias calibration

             Corrected phase measurements




                                                                                                               New Algorithm
                                                                                                 Bar-Itzhack




                                                                                                                                         Geometric Descent
                   Vector Determination



                                                                                 Least Squares




                                                                                                                                                             Bar-Itzhack 2
                                               Kalman filter

                                                               ALLEGRO

                                                                         Cohen
                      Vector estimate




                                                                                                                               Euler-q
                  Attitude Determination

                      Attitude estimate

20/12-2000 (MJ)                            Danish GPS Center                                                                                                                 28
                                  Testbed

                                                        Controller box
                                      LON network


                  Labtop PC
                                                   Motor                    Encoder
                                                 commands                   readings
                              RS232


                                                                           Antenna array
                                  GPS receiver




                                                             Surveyor platform



20/12-2000 (MJ)                          Danish GPS Center                                 29
                                       Software
                                                       Start


                                              Load simulated
                                                 attitude                     angles.txt



                                            Start User Interface
                                            + Initialize Testbed




                        Proces 1                  Proces 2                      Proces 3
                  Store phases and                                            Send angle
                                            Store angle-values
                  LOS vectors from                                         commands to LON
                                           from encoders in file
                    receiver in file                                            nodes




                                                                                             Stop
               GPS            phase.out   angles.out           LON nodes
             receiver          los.out




20/12-2000 (MJ)                             Danish GPS Center                                       30
                  Motor Control




20/12-2000 (MJ)       Danish GPS Center   31
                                                                                                               Motor Angles
                                                            Angle of Antenna Array
                                200



                                150



                                100



                                  50




              Angle [degrees]
                                   0



                                -50



                                -100



                                -150



                                -200
                                       0   50   100   150   200       250        300   350   400   450   500
                                                                  Time [seconds]




20/12-2000 (MJ)                                                                                                    Danish GPS Center   32
        Local Horizontal System




20/12-2000 (MJ)   Danish GPS Center   33
                      Results

     Based on actual and simulated data, the following
     performance parameters were evaluated


                  • Accuracy
                  • Computational efficiency
                  • Ability to converge




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                  Accuracy




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                                                       4
                                                    x 10                            Speed Analysis with three baselines
                                                6
                                                           Kalman filter w. bias est.
                                                5          Kalman filter




                                                                                                                                  Speed
                                                           Geometric Descent
                                                4          Bar Itzhack 2

                                                3




          Number of Floating Point Operations
                                                2

                                                1

                                                0
                                                 2                          3                      4                      5   6


                                                       3
                                                    x 10
                                                5
                                                           ALLEGRO
                                                           New Algorithm + Euler-q
                                                4
                                                           New Algorithm + SVD
                                                           Single-point
                                                3          Cohen
                                                           Bar Itzhack + SVD

                                                2

                                                1


                                                0
                                                 2                          3                     4                       5   6
                                                                                           Number of satellites




20/12-2000 (MJ)                                                                                                                   Danish GPS Center   36
                                                                                                                                           Convergence
                                               Kalman Filter                                                Single-point Algorithm
                              6000                                                            6000
                                                Number of baselines = 3                                          Number of baselines = 3
                              5000              Convergence = 100%                            5000               Convergence = 100%




          Number of Samples




                                                                          Number of Samples
                                                Average = 5.75                                                   Average = 4.25
                              4000                                                            4000

                              3000                                                            3000

                              2000                                                            2000

                              1000                                                            1000

                                0                                                               0
                                     1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
                                         Convergence Time (Samples)                                      Convergence Time (Samples)

                                            ALLEGRO Algorithm                                            Geometric Descent Algorithm
                              6000                                                            6000
                                                Number of baselines = 3                                          Number of baselines = 3
                              5000              Convergence = 100%                            5000               Convergence = 100%




          Number of Samples




                                                                          Number of Samples
                                                Average = 4.56                                                   Average = 5.33
                              4000                                                            4000

                              3000                                                            3000

                              2000                                                            2000

                              1000                                                            1000

                                0                                                               0
                                     1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
                                         Convergence Time (Samples)                                      Convergence Time (Samples)




20/12-2000 (MJ)                                                                                                                               Danish GPS Center   37
                                               Kalman Filter                                                Single-point Algorithm




                                                                                                                                           Convergence
                              6000                                                            6000
                                                Number of baselines = 2                                          Number of baselines = 2
                              5000              Convergence = 96.61%                          5000               Convergence = 69.34%




          Number of Samples




                                                                          Number of Samples
                                                Average = 7.87                                                   Average = 4.64
                              4000                                                            4000

                              3000                                                            3000

                              2000                                                            2000

                              1000                                                            1000

                                0                                                               0
                                     1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
                                         Convergence Time (Samples)                                      Convergence Time (Samples)

                                            ALLEGRO Algorithm
                              6000
                                                Number of baselines = 2
                              5000              Convergence = 69.75%




          Number of Samples
                                                Average = 4.76
                              4000

                              3000

                              2000

                              1000

                                0
                                     1 2 3 4 5 6 7 8 9 1011 1213 1415
                                         Convergence Time (Samples)




20/12-2000 (MJ)                                                                                                                               Danish GPS Center   38
                  Conclusion

     • Kalman filter is by far most accurate, but also
     computationally very heavy

     • Single-point (LSQ) offers good accuracy + high speed

     • Vector matching algorithms has the lowest accuracy
     but does not suffer from convergence problems

     • Performance depend on satellite constellation

     • Results were affected by mechanical problems with
     levelling of the testbed
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