Initial Attitude Determination and Correction of gyro INS free angular oreintation on the Basis of GPS linear navigation by ashi7790

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									   Initial Attitude Determination and C
                     tation on the Basis o
                                arameters
                        K i d S. Mostov              Andrey A. Soloviev               T.-K. John Koo

                          Department of Electrical Engineering and Computer Sciences
                                     University of California at Berkeley
  Abstract                                                           with an array of relatively inexpensive semiconductor
       The exclusion of gyroscopes from navigation system            accelerometers. Here, the determination of the linear and
  architecture achieves an essential reduction in systems cost.      angular parameters of motion is based on the weighted
  However, the absence of gyroscopes leads to the growth of          averaging of data from the accelerometers.
  errors in the determination of navigation system angular                However, an absence of gyroscopes leads to an
  orientation. This, in its turn, leads to the growth of errors in   accelerated increase in attitude errors. This in turn causes a
  the determination of vehicle coordinates. Therefore, one of the    rapid increase in errors of linear parameters of motion
  main problems in the construction of an integrated navigation      (coordinates and velocity) between GPS fixes. Therefore, in
  system which is based on a gyro-free Inertial Navigation           a gyro-free system, an abrupt degradation of navigation
  System (INS) and a single antenna Global Positioning System        system accuracy can be observed.
  (GPS) is the determination of the initial angular orientation           It is important to mention that a single antenna GPS
  and correction of gyro-free INS angular characteristics on the
                                                                     which is traditionally used in integrated navigation systems
  basis of linear navigation parameters (coordinates and
  velocity) that are obtained from GPS. This paper is devoted to     regards the vehicle as a material point with no angular
  the description of the proposed solution method for this           orientation. The use of a multiple-antenna GPS will allow
  problem.                                                           determination of the vehicle attitude, but will lead to an
                                                                     approximate twofold increase in the system cost. Thus, the
  Introduction                                                       critical point in the design of gyro-free navigation systems
                                                                     is the development of an algorithm for the initial attitude
       Currently, the rapid reduction of the cost of inertial        determination and the correction of attitude errors on the
  sensors and GPS receivers can be observed. A persistent            basis of GPS data. In this paper, the idea of a “virtual
  decrease in the price is predicted for the as well, due to the     gyro”, consisting of distributed accelerometers combined
  continual     improvements       in    semiconductor      and      with a single antenna GPS is presented. The paper has the
  micromachining technology. The drop of the cost of these           following structure:
  sensors leads to the possibility of employing integrated
                                                                           Section two is devoted to the brief analysis of the
  navigation systems in a wide range of commercial
                                                                     existing approaches to the correction of angular parameters
  applications including the navigation of cars, yachts,
                                                                     used in gyroscopic navigation systems. Since the attitude
  agricultural vehicles, etc. The traditional approach to the
                                                                     correction algorithm has been developed for the gyro-free
  integrated navigation system design is based on the
                                                                     INS, section three describes briefly the error inducing
  combination of the determination of absolute navigation            mechanism of the accelerometer based system.
  parameters (coordinates and velocity) which are obtained
                                                                           A formal problem statement and the basic idea of the
  from the GPS and the determination of the relative
                                                                     proposed approach are presented in section four. In section
  navigation parameters (relative changes in coordinates and         five, the realization of the proposed approach is presented.
  velocity) which are obtained from the INS. Here the INS
                                                                     The validity of the proposed approach is verified by a
  provides the required update rate of the navigation
                                                                     simulation, the results of which are presented in section six.
  parameters between the GPS fixes and also during loss of a
                                                                     In conclusion, the plan for future research is discussed.
  GPS signal.
       However, the INS errors in determination of linear and
  angular parameters of motion grow continuously with time.
                                                                     2. Related Work
  The correction of linear errors is based on the GPS data.               A review of publications devoted to the problem of
  For the precise determination of angular parameters                attitude correction has shown that currently there exist no
  accurate gyroscopes are used. The use of gyros in INS leads        published research results addressing the problem of the
  to:                                                                correction of gyro-free INS angular parameters. An analysis
       a) high system costs ($2500 - $40000) and                     of the publications devoted to the attitude correction of
       b) the inclusion of an additional operation mode              gyroscopic INS has shown that the proposed approaches are
  (gyroscope initialization) which takes additional time.            based on specifications of gyro-augmented systems [2], [ 3 ] ,
       Therefore, it is more economical to replace gyroscopes        [41.

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                                                                                           0-7803-4269-0/97/S10.00 0 1998 l€EE
    For this reason, the direct application of traditional      the errors (correction, it is necessary to perform joint
methods to gyro-free navigation systems will lead to the        correction of’deterministic and stochastic error components.
increase in methodical errors. A comparison of the existing         b) In c<asesof initial angular alignment (with initial
approaches to the correction of angular orientation with the    angular uncertainty & E ) , and after the loss of GPS signal
proposed attitude correction approach for gyro-free
                                                                (for example while in a tunnel or while moving in the
navigation systems will be presented in detail in section 5.
                                                                vicinity of skyscrapers, etc.), the components 2b) and 2c)
                                                                grow rapidly with time and their magnitudes are essentially
3. Gyro-Free INS Error Analysis                                 larger than magnitudes of components 1 and 2a).
     As mentioned above, an attitude correction algorithm            Therefore, based on the conducted research and
has been developed taking into consideration a gyro-free        analysis, the development of attitude correction algorithm
INS error analysis. This section presents the main results of   will be performed in two successive stages:
this analysis.                                                  Stage 1. The development of an algorithm for the
     An analysis of the growth of errors in the                 correction of the deterministic components of total error -
determination of angular and linear parameters of motion        as the basic part of the algorithm.
has shown that:                                                 Stage 2. A combination of an algorithm of deterministic
     1. The noise of accelerometers is the major source of      error correclion with methods of stochastic error filtering
errors in angular parameters (error component 1). The           (particularly on the basis of adaptive Kalman filtering) - as
errors show continuous and rather regular growth                a modernization of the algorithm.
characterized by a constant parameter.                               This paper is devoted to the description of the basic
     2. The sources of errors in linear parameters (errors in   part of the algorithm.
the determination of coordinates and velocity) are the
following:                                                      4. Formall Problem Statement and the Basic
     a) accelerometer noise. The character of growth of this
component (error component 2a) of error approximately
                                                                Idea of the Proposed Solution Approach
                                                                     The formal statement of an angular characteristics
coincides with the character of attitude error growth;
                                                                correction problem is the basis for the proposed approach.
     b) a mismatch between angular characteristics of body
                                                                A graphical interpretation used for the formal problem
frame and an Earth-referenced frame which leads to an
                                                                statement is presented in Fig. 1. For simplification, the
incorrect determination of the vehicle’s direction of motion.
                                                                graphical interpretation is initially presented for the two-
This component (error component 2b) is characterized by
                                                                dimensional case.
continuous growth. Here, with the increase of mismatching
angles’ magnitudes, an increase in the parameter which             zEarth
determines error growth rate can be observed.
     c) an incorrect compensation of gravity due to an                                    zmov.
angular mismatch between the body frame and the Earth-
referenced frame. The growth of this error component                                     Moving coordinate
(error component 2c) is more rapid than that of the previous                             frame (related with axes
component and has an abrupt character. This phenomenon
can be explained by the fact that for the 2b) component, the
accelerations have, in general, a sign-changing character.
For the 2c) component the direction of gravity is always the
same in the Earth-referenced frame.
     Conducted simulations and an analysis of simulation
results have shown the possibility of classification of the
error components. First, the errors can be categorized as
either stochastic or deterministic:
     a) components 1 and 2a) have a stochastic nature;
     b) components 2b) and 2c) have a deterministic nature
                                                                 Figure 1. !$paceOrientation of the Body Frame,
and are determined by the misalignment angles.
     On the other hand, errors can be sorted according to
                                                                  Earth-referenced Frame and Fictitious Frame
their magnitude:                                                     As illustrated, there is an Earth-referenced coordinate
     a) In cases when the correction of angular parameters      frame. Accelerations, measured by accelerometers, are
is performed at the frequency equal to the frequency of         determined in the moving frame or the body frame. Spacial
updates of the linear parameters (or inversely proportional     orientation of the body frame is determined by the spacial
to the time between the two successive GPS fixes),              orientation of INS accelerometers’ sensitive axes. Suppose
components 1, 2a), 2b), 2c) are commensurable. Here, for        that the anglle between the body frame axes and the Earth-



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referenced frame axes is equal to a . The determination of          ( t G P S ) the change in vehicle coordinates is characterized
vehicle coordinates is carried out by INS (taking into              by the vector of relative displacement r. The coordinates of
consideration the current value of a ) on the basis of the          this vector are known in:
relationships presented in [ 2 ] . However, as noted above,              a) Earth-referenced frame, based on GPS data;
accelerometer noise leads to the accumulation of error in                b) fictitious frame, based on INS data.
                                                                         When the coordinates frame coincide with each other
determination of a angle. Therefore, INS determines the
                                                                    (for example, after initial alignment) the components of the
coordinates in some fictitious coordinate frame, rather than        relative displacement vector will be equal in both
in an Earth-referenced frame. The angle between the Earth-          coordinate systems. However, in presence of angular
referenced frame axes and the fictitious frame axes is equal        mismatching the components of the relative displacement
to a o . Thus, at the initial moment of time (after initial         vector will be different in different coordinate systems.
alignment) the fictitious frame and the Earth-referenced            This can be explained by the rotation of relative
frame coincide with each other. Thereafter, with                    displacement vector (due to the rotation of the fictitious
accumulation of error in alpha determination, the fictitious        frame) and incorrect compensation of gravity. Thus, it is
frame rotates relative to the Earth-referenced frame and the        necessary to rotate the fictitious frame for providing the
angle between these coordinate frames becomes equal to              coincidence of the relative displacement vector components
                                                                    in both coordinate systems. Obtained values of the rotation
ao.    Thus,   the   attitude   correction     algorithm    must
                                                                    angles will determine angular mismatching, and thus, will
periodically determine the values of angular mismatching            determine the correction to the direct cosine matrix - Ccorr.
(for the two dimension case - it is          a o ;for   the three        Note that even in the absence of displacement (when
                                                                    the value of the displacement vector in the Earth-referenced
dimensional case there are three mismatching angles:         <Po,   frame is equal to zero) the INS measures a non-zero vector
                                                                     of displacement which has the following components:
e,, vo), provide the alignment of the fictitious frame
       and
to make it collinear to the Earth-referenced frame. Taking
into consideration a gyro-free INS structure, attitude
correction algorithm must periodically define the correction             Thus, this approach provides the possibility of
to the direct cosine matrix Ccorr. which depends on                 realizing both the initial alignment (for example, when the
misalignment of angles values. The multiplication of the            vehicle is at rest) and the correction of attitude errors during
current value of the direct cosine matrix by the Ccorr. must        vehicle motion.
be equivalent to the alignment of the fictitious frame with               For the realization of the attitude correction algorithm
the Earth-referenced frame.                                         on the basis of the above-formulated idea, the following
     The formal problem statement and the basic idea of the         steps must be taken:
proposed algorithm are presented below. The graph of the                  1. Prove the criteria of collinearity of the fictitious and
basic idea is presented in Figure 2.                                the Earth-referenced frames: coincidence of relative
                                                                    displacement vector components in both coordinate
                                                                    systems.
                                                                          2. Choose the trajectory of rotation of the fictitious
                                                                    coordinate system (the sequence of fictitious system
                                                                    rotations) that will provide the criteria fulfilment:
                                                                    coincidence of displacement vector components in both
                                                                    coordinate systems.
                                                                          3. Determine the formal transformation of relative
                                                                    displacement vector components and the correction to the
                                                                    direct cosine matrix Ccorr that occur during the rotation of
                                                                     the fictitious coordinate system.

                                                                    5 Realization of Attitude Correction Algorithm
                                                                    5.1 Analysis of Existing Approaches
                                                                          Traditionally, the angular orientation correction has
 Figure 2. Graphical Interpretation of Proposed Basic               been approached through the comparison of the relative
                         Idea                                       displacement vector components in an Earth-referenced and
                                                                    fictitious frames; a number of existing works discuss the
     During the time between two successive GPS fixes               algorithms of attitude correction using such comparison [2].
                                                                          These papers propose determining the correction to the
direct cosine matrix on the basis of three vectors, obtained
in the Earth-referenced and fictitious frames during three
successive GPS fixes [2]. Three 3-dimensional vectors are
necessary to determine the nine components of the direct
cosine matrix. This method is based on the proposition that
during three successive GPS fixes the angular mismatching
between the coordinate systems remains constant. Thus, the
nine components of the correction matrix also remain
constant. However, conducted research has shown, that in
the case of gyro-free navigation system this proposition
leads to the essential error growth in determining
misalignment angles. Therefore, this method is not
applicable for a gyro-free system. In addition, there are
cases for which the system of linear equations for
determining the correction matrix components has multiple
solutions. For example, this can be observed when the
direction of vehicle motion remains constant during three
successive GPS fixes.
      Therefore, for the realization of the attitude correction
algorithm in gyro-free navigation systems it was proposed
to match coordinate frames on the basis of a single GPS fix.
Thus, the angular orientation correction will be performed
on the basis of a single GPS fix rather than three GPS fixes.
This will lead to a qualitative decrease in errors in attitude
determination, since the frequency of attitude correction
will be equal to the frequency of the linear parameters
correction: the correction will be performed after each GPS
fix. The obtained errors will be commensurable with the
random error components presented above (components 1
and 2a). Random errors will be decreased using Kalman                          Figure 3. Proof of Alignment Criteria
filtering during the second step of algorithm development.

5.2 Proof of Alignment Criteria Correctness                       5.3 Fictitious Frame Rotation Trajectory Choice
      The next logical step is to prove the possibility of             The next step in the attitude correction algorithm
angular matching of the two coordinate frames on the basis        development is to determine the trajectory of the fictitious
of a single vector of relative displacement.                      frame rotation, which provides the fulfilment of the
      As outlined above, the criteria of collinearity between     alignment criteria: coincidence of the relative displacement
the fictitious and the Earth-referenced frames is the             vector components in both coordinate frames.
coincidence of the relative displacement vector components             The traditional approach to solving this problem is
in both coordinate frames. The formal proof of this criteria      based on the analytical approach: definition of the relative
is presented in Fig. 3.                                           displacement vector components in a fictitious frame
      The proof is based on determining the values of the         (obtained from INS) as a function of the relative
mismatched angles; the components of the relative                 displacement vector components in the Earth-referenced
displacement vector are the same in both coordinate               frame (obtained from GPS or other external reference
systems. The only possible value of mismatched angles is          points) and angle misalignment.
zero. Thus, the only incidence of orientation of the
fictitious coordinate frame for which the component of the
relative displacement vector in this frame coincides with
the components of the relative displacement vector in the
                                                                      ‘Eurrh
                                                                                                s i n g s i n 0 s i n p cosgsinp + s i n g s i n 0 c o s p -singcos8
                                                                                                                               cos 0cosp
                                                                                    singcosrp + cosyrsinesinq s i n g s i n 9 - cosy sine cosp c o s g c o s e
                                                                                                                                                              sine
                                                                                                                                                                   I
Earth-referenced frame is the collinearity of the Earth-
referenced and the fictitious coordinate frames. An
exception   is   possible   only   with   absolutely   vertical
movement of a vehicle. However, such a scenario is highly         This approach leads to the necessity of solving a system of
unlikely and can be virtually disregarded.                        nonlinear equations for determining the misalignment of
                                                                  angles. The traditional approach is the linearization of this



                                                                                                                                                                1037
  nonlinear system and determination of misalignment angles          second stage of the attitude correction algorithm
  as solutions of the linear system. However, the provided           development - when the stochastic errors are taken into
  estimation demonstrates that this method remains valid             consideration - current criteria is substituted by the least
  only for a small range of misalignment angles. In practice it      mean squares criteria, realized through Kalman filtering.
  is possible for the misalignment angles to be out of this
  range - for example, after GPS signal loss or during the           5.3 Determining Formal Transformations for the
  initial alignment. In such cases the error of this method          Fictitious Frame Rotations
  grows dramatically.                                                      In accordance with the chosen sequence of the
       In order to eliminate the outlined disadvantages it was       fictitious frame rotations (steps 1 and 2 ) it is necessary
  proposed to include an intermediate criteria of alignment:         determine the formal transformations of the relative
  with precise rotation around the vector of relative                displacement vector components and the correction of the
  displacement. This allows for separation of the alignment          direct cosine matrix which takes place during rotations.
  procedure into two independent steps, and significantly                  Formal transformations for the rotation 1 are as follows:
  simplifies the determination of misalignment angles for an               Formal transformation of the correction to the direct
  arbitrary range of these angles.                                   cosine matrix rotational axis:
       The successive alignment steps are as follows:
       SteD 1 (Fig. 4):
                                                                     p=       RGPS         RINS

       Rotation of the fictitious frame allows for the                            RGPSRINS
  coincidence of the angular characteristics of the relative         rotational angle:
  displacement vector in both coordinate frames. After this          r            i
  procedure only the absolute values of vectors may differ.

    ZfiC                                                             (5.3) correction matrix transformation:


                                                                                 2
                                                                           casa+p *(I-cosa)          p p ( I - c o s a ) + p sina p p (1-eosa)-p sina
                                                                                                      X Y                   z      x z          Y
                                                                         b p (l-cosa)-pzsina
                                                                          X   Y
                                                                                                       cosa+pZy(l -cosa) p p (I-cosa)+pxsina
                                                                                                                                  Y Z
                                                                                                                                                         (5.4)
                                                                                                                                        2
                                                                         p X p Z ( 1 - c i ) s a ) + b sina p p (l-cosa)-pxsina   casa+p y(l-cosa)
                                                           1 fict.                                   Y       Y z
       /
  Xfict.
             'XEarth              yr
                                      XEarth

           Before Step 1                    After Step 1
        Figure 4. First Step of Fictitious Frame Rotation
       Stea 2 (Fie. 5 ) :
       Rotation of the fictitious frame around the vector of                                                           L            J   L            J


  relative displacement allows for equality of vector absolute       It was proposed to perform step 2 as an iteration
  values in both coordinate systems. When vector                     procedure: where the fictitious coordinate frame is rotated
  components coincide, the coordinate frames are collinear.          around the displacement vector in this frame. The
                                                                     coincidence of the displacement vector components in the
                                                                     fictitious frame and the Earth-referenced frame is the
                                                                     criteria for ending the iterations. The iterational procedure
                                                                     for step 2 is presented in Fig. 7.

                                                                     6. Simulation Results
                                                                         The effectiveness of the proposed algorithm has been
  xfi                                                                confirmed by the simulations of the integrated navigation
                                                                     system performance. The characteristics of NovAtel RTK-2
                                                                     were used as accuracy characteristics of a GPS receiver (the
        Before Step 2                     After Step 2               frequency of the GPS is 2 Hz; the standard deviation of
        Figure 5. Second Step of Fictitious Frame Rotation           error in the coordinates is 4 cm2). Accelerometers with
      It is important to mention that in this basic part of the      standard deviation of noise of 25sm21s4 were used in the
  algorithm the vector component coincidence is considered           simulations. Comparison of performance of an integrated
  as a criteria of the coordinate system matching. During the        navigation system with and without the attitude correction


1038
algorithm is presented in Fig. 6. Here the character of error                         determination of the formal transformations of attitude
                                                                                       correction parameters and the relative displacement
             Initial Conditions: {           R fici     Clcorr.}                       vector, which take place during fictitious frame
                                                                                      rotations.
                                         v                                         Correctness of the proposed method was confirmed by


 Zi+ 1
         =    [cosgcoscp cosysincpcose+ sinysine cosysincpsine-
                 -sincp          cOscpcase               cos cp sin 0
               sinycoscp sinysincpcose- cosysine sinysincpsine+
                                                                                   simulation results.
                                                                                        Future work will be devoted to the combination of the
                                                                                   deterministic error correction algorithm with filtering of
                                                                                   random eirors.

                                                                                                 a) without anitude correction


         cosy coscp cosy sincp cos0 + sin y sine cosy sincpsine - s h y cos0
                             cos cp cos 0                coscp sine         Jxci
             sinvcoscp sinvsincp cos 8 - cosy sin0 sin yr sincpsine + cosy cos0




                  I   Verification of Criteria Fulfilment             I
                                                                                                                              t, seconds
     Figure 7. Iteration procedure of Second Step of                                                b) with attitude correction
              Fictitious Framer Alignment                                              Ay, meters

in determination of y-coordinate ( A y ) for the gyro-free
integrated navigation system is shown with and without
attitude correction. The y-coordinate error increase with
and without attitude correction is similar to the x-
coordinate and z-coordinate error behavior.
     It is important to note that for the navigation system,
which uses the attitude correction algorithm, the initial
angular orientation was unknown. Thus, both initial
alignment and attitude correction have been tested.                                                                            t, seconds
Simulation results have shown that using the attitude                                Figure 6. Comparison of coordinate error growth for
correction algorithm allows to: a) define initial angular                                   intlegrated gyro-free navigation system
orientation of gyro-free INS; b) change qualitatively the
character of errors in determining linear navigation                               References:
parameters. Thus, a compensation of abrupt degradation of                          [l] J.H. Chen, S.C. Lee, D.B. DeBra “Gyroscope Free
navigation system accuracy was achieved.                                           Strapdown Inertial Measurement Unit by Six Linear
                                                                                   Accelerometers”, Journal of Guidance, Control, and
Conc1usion                                                                         Dynamics, Vo1.17, No. 2. March-April 1994, p. 286-290.
      This paper presented the algorithm of gyro-free INS                          [2] I.Y. Bar-Itzhack, J. Reiner “Recursive Attitude
angular parameters correction on the basis of single antenna                       Determination from Vector Observation: Direct Cosine
                                                                                   Matrix ldenlification ”, Journal of Guidance, Control and
GPS. The algorithm is based on the alignment of a fictitious
coordinate frame with an Earth-referenced frame through                            Dynamics, vol. 7, no. 1, Jan-Feb, 1984, p.51-6
                                                                                   [ 3 ] R.E. Phillips, G.T. Schmidt “GPSANS Integration ”,
fictitious frame rotation, with rotational axes and angles of
rotation defining angular orientation correction parameters.                       AGARD 1996, p 9/1-18, vi+ 158 pp.
      The realization of attitude correction algorithm was                         [4] K.R. Britting “Inertial Navigation Systems Analysis ”,
                                                                                   John Wiley, 1972
considered. This realization includes:
    choice of criteria of alignment of a fictitious frame with
    Earth-referenced frame and a proof of criteria
    correctness;
    choice of trajectory of fictitious frame rotation, which
    provides for the fulfilment of the alignment criteria;



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