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Procedure
Abstract
Required_Reading
Keywords
Brief_I/O
Detailed_Input
Detailed_Output
Parameters
Exceptions
Files
Particulars
Examples
Restrictions
Literature_References
Author_and_Institution
Version
Index_Entries

Procedure

   void qdq2av_c ( ConstSpiceDouble    q  [4],
                   ConstSpiceDouble    dq [4],
                   SpiceDouble         av [3]  ) 

Abstract

 
   Derive angular velocity from a unit quaternion and its derivative 
   with respect to time. 
    

Required_Reading

 
   ROTATION 
  

Keywords

 
   MATH 
   POINTING 
   ROTATION 
 

Brief_I/O

 
   VARIABLE  I/O  DESCRIPTION 
   --------  ---  -------------------------------------------------- 
   q          I   Unit SPICE quaternion. 
   dq         I   Derivative of `q' with respect to time. 
   av         O   Angular velocity defined by `q' and `dq'. 
 

Detailed_Input

 
   q              is a unit length 4-vector representing a SPICE-style
                  quaternion. See the discussion of "Quaternion Styles"
                  in the Particulars section below.
 
                  Note that multiple styles of quaternions are in use.
                  This routine will not work properly if the input
                  quaternions do not conform to the SPICE convention.
                  See the Particulars section for details.

 
   dq             is a 4-vector representing the derivative of `q' with
                  respect to time.
 

Detailed_Output

 
   av             is 3-vector representing the angular velocity defined
                  by `q' and `dq', that is, the angular velocity of the
                  frame defined by the rotation matrix associated with
                  `q'.  This rotation matrix can be obtained via the
                  CSPICE routine q2m_c; see the Particulars section for
                  the explicit matrix entries.
 
                  `av' is the vector (imaginary) part of the quaternion
                  product
 
                           *     
                     -2 * q  * dq   
 
                  This angular velocity is the same vector that could
                  be obtained (much less efficiently ) by mapping `q'
                  and `dq' to the corresponding C-matrix `r' and its
                  derivative `dr', then calling the CSPICE routine
                  xf2rav_c.
 
                  `av' has units of 
 
                     radians / T 
 
                  where 
 
                     1 / T 
 
                  is the unit associated with `dq'. 
 

Parameters

 
   None. 
 

Exceptions

 
   Error free. 
 
   1) A unitized version of input quaternion is used in the 
      computation.  No attempt is made to diagnose an invalid 
      input quaternion. 
 

Files

 
   None. 
 

Particulars

 
   Quaternion Styles
   -----------------

   There are different "styles" of quaternions used in
   science and engineering applications. Quaternion styles
   are characterized by

      - The order of quaternion elements

      - The quaternion multiplication formula

      - The convention for associating quaternions
        with rotation matrices

   Two of the commonly used styles are

      - "SPICE"

         > Invented by Sir William Rowan Hamilton
         > Frequently used in mathematics and physics textbooks

      - "Engineering"

         > Widely used in aerospace engineering applications


   CSPICE function interfaces ALWAYS use SPICE quaternions.
   Quaternions of any other style must be converted to SPICE
   quaternions before they are passed to CSPICE functions.


   Relationship between SPICE and Engineering Quaternions
   ------------------------------------------------------

   Let M be a rotation matrix such that for any vector V,

      M*V

   is the result of rotating V by theta radians in the
   counterclockwise direction about unit rotation axis vector A.
   Then the SPICE quaternions representing M are

      (+/-) (  cos(theta/2),
               sin(theta/2) A(1),
               sin(theta/2) A(2),
               sin(theta/2) A(3)  )

   while the engineering quaternions representing M are

      (+/-) ( -sin(theta/2) A(1),
              -sin(theta/2) A(2),
              -sin(theta/2) A(3),
               cos(theta/2)       )

   For both styles of quaternions, if a quaternion q represents
   a rotation matrix M, then -q represents M as well.

   Given an engineering quaternion

      QENG   = ( q0,  q1,  q2,  q3 )

   the equivalent SPICE quaternion is

      QSPICE = ( q3, -q0, -q1, -q2 )


   Associating SPICE Quaternions with Rotation Matrices
   ----------------------------------------------------

   Let FROM and TO be two right-handed reference frames, for
   example, an inertial frame and a spacecraft-fixed frame. Let the
   symbols

      V    ,   V
       FROM     TO

   denote, respectively, an arbitrary vector expressed relative to
   the FROM and TO frames. Let M denote the transformation matrix
   that transforms vectors from frame FROM to frame TO; then

      V   =  M * V
       TO         FROM

   where the expression on the right hand side represents left
   multiplication of the vector by the matrix.

   Then if the unit-length SPICE quaternion q represents M, where

      q = (q0, q1, q2, q3)

   the elements of M are derived from the elements of q as follows:

        +-                                                         -+
        |           2    2                                          |
        | 1 - 2*( q2 + q3 )   2*(q1*q2 - q0*q3)   2*(q1*q3 + q0*q2) |
        |                                                           |
        |                                                           |
        |                               2    2                      |
    M = | 2*(q1*q2 + q0*q3)   1 - 2*( q1 + q3 )   2*(q2*q3 - q0*q1) |
        |                                                           |
        |                                                           |
        |                                                   2    2  |
        | 2*(q1*q3 - q0*q2)   2*(q2*q3 + q0*q1)   1 - 2*( q1 + q2 ) |
        |                                                           |
        +-                                                         -+

   Note that substituting the elements of -q for those of q in the
   right hand side leaves each element of M unchanged; this shows
   that if a quaternion q represents a matrix M, then so does the
   quaternion -q.

   To map the rotation matrix M to a unit quaternion, we start by
   decomposing the rotation matrix as a sum of symmetric
   and skew-symmetric parts:

                                      2
      M = [ I  +  (1-cos(theta)) OMEGA  ] + [ sin(theta) OMEGA ]

                   symmetric                   skew-symmetric


   OMEGA is a skew-symmetric matrix of the form

                 +-             -+
                 |  0   -n3   n2 |
                 |               |
       OMEGA  =  |  n3   0   -n1 |
                 |               |
                 | -n2   n1   0  |
                 +-             -+

   The vector N of matrix entries (n1, n2, n3) is the rotation axis
   of M and theta is M's rotation angle.  Note that N and theta
   are not unique.

   Let

      C = cos(theta/2)
      S = sin(theta/2)

   Then the unit quaternions Q corresponding to M are

      Q = +/- ( C, S*n1, S*n2, S*n3 )

   The mappings between quaternions and the corresponding rotations
   are carried out by the CSPICE routines

      q2m_c {quaternion to matrix}
      m2q_c {matrix to quaternion}

   m2q_c always returns a quaternion with scalar part greater than
   or equal to zero.


   SPICE Quaternion Multiplication Formula
   ---------------------------------------

   Given a SPICE quaternion

      Q = ( q0, q1, q2, q3 )

   corresponding to rotation axis A and angle theta as above, we can
   represent Q using "scalar + vector" notation as follows:

      s =   q0           = cos(theta/2)

      v = ( q1, q2, q3 ) = sin(theta/2) * A

      Q = s + v

   Let Q1 and Q2 be SPICE quaternions with respective scalar
   and vector parts s1, s2 and v1, v2:

      Q1 = s1 + v1
      Q2 = s2 + v2

   We represent the dot product of v1 and v2 by

      <v1, v2>

   and the cross product of v1 and v2 by

      v1 x v2

   Then the SPICE quaternion product is

      Q1*Q2 = s1*s2 - <v1,v2>  + s1*v2 + s2*v1 + (v1 x v2)

   If Q1 and Q2 represent the rotation matrices M1 and M2
   respectively, then the quaternion product

      Q1*Q2

   represents the matrix product

      M1*M2


   About this routine 
   ================== 
 
   Given a time-dependent SPICE quaternion representing the 
   attitude of an object, we can obtain the object's angular 
   velocity AV in terms of the quaternion Q and its derivative  
   with respect to time DQ: 
 
                        * 
      AV  =  Im ( -2 * Q  * DQ )                                  (1) 
    
   That is, AV is the vector (imaginary) part of the product 
   on the right hand side (RHS) of equation (1).  The scalar part  
   of the RHS is zero. 
 
   We'll now provide an explanation of formula (1). For any 
   time-dependent rotation, the associated angular velocity at a 
   given time is a function of the rotation and its derivative at 
   that time. This fact enables us to extend a proof for a limited 
   subset of rotations to *all* rotations:  if we find a formula 
   that, for any rotation in our subset, gives us the angular 
   velocity as a function of the rotation and its derivative, then 
   that formula must be true for all rotations. 
 
   We start out by considering the set of rotation matrices 
 
      R(t) = M(t)C                                                (2) 
 
   where C is a constant rotation matrix and M(t) represents a 
   matrix that "rotates" with constant, unit magnitude angular 
   velocity and that is equal to the identity matrix at t = 0. 
 
   For future reference, we'll consider C to represent a coordinate 
   transformation from frame F1 to frame F2.  We'll call F1 the 
   "base frame" of C.  We'll let AVF2 be the angular velocity of 
   M(t) relative to F2 and AVF1 be the same angular velocity 
   relative to F1. 
 
   Referring to the axis-and-angle decomposition of M(t) 
 
                                              2 
      M(t) = I + sin(t)OMEGA + (1-cos(t))OMEGA                    (3) 
 
   (see the Rotation Required Reading for a derivation) we  
   have 
 
      d(M(t))| 
      -------|     = OMEGA                                        (4) 
        dt   |t=0 
 
   Then the derivative of R(t) at t = 0 is given by 
 
 
      d(R(t))| 
      -------|     = OMEGA  * C                                   (5) 
        dt   |t=0 
 
 
   The rotation axis A associated with OMEGA is defined by        (6) 
 
      A(1) =  - OMEGA(2,3) 
      A(2) =    OMEGA(1,3) 
      A(3) =  - OMEGA(1,2) 
      
   Since the coordinate system rotation M(t) rotates vectors about A 
   through angle t radians at time t, the angular velocity AVF2 of 
   M(t) is actually given by 
 
      AVF2  =  - A                                                (7) 
 
   This angular velocity is represented relative to the image 
   frame F2 associated with the coordinate transformation C. 
 
   Now, let's proceed to the angular velocity formula for 
   quaternions. 
    
   To avoid some verbiage, we'll freely use 3-vectors to represent 
   the corresponding pure imaginary quaternions. 
 
   Letting QR(t), QM(t), and QC be quaternions representing the 
   time-dependent matrices R(t), M(t) and C respectively, where 
   QM(t) is selected to be a differentiable function of t in a 
   neighborhood of t = 0, the quaternion representing R(t) is 
 
      QR(t) = QM(t) * QC                                          (8) 
 
   Differentiating with respect to t, then evaluating derivatives 
   at t = 0, we have 
 
      d(QR(t))|         d(QM(t))| 
      --------|     =   --------|     * QC                        (9) 
         dt   |t=0         dt   |t=0 
 
 
   Since QM(t) represents a rotation having axis A and rotation 
   angle t, then (according to the relationship between SPICE 
   quaternions and rotations set out in the Rotation Required 
   Reading), we see QM(t) must be the quaternion (represented as the 
   sum of scalar and vector parts): 
    
      cos(t/2)  +  sin(t/2) * A                                  (10) 
 
   where A is the rotation axis corresponding to the matrix 
   OMEGA introduced in equation (3).  By inspection 
 
      d(QM(t))| 
      --------|     =   1/2 * A                                  (11) 
         dt   |t=0 
 
   which is a quaternion with scalar part zero.  This allows us to  
   rewrite the quaternion derivative   
 
      d(QR(t))|           
      --------|     =   1/2  *  A  *  QC                         (12) 
         dt   |t=0           
 
   or for short, 
 
      DQ = 1/2 * A * QC                                          (13) 
 
   Since from (7) we know the angular velocity AVF2 of the frame 
   associated with QM(t) is the negative of the rotation axis 
   defined by (3), we have 
 
      DQ = - 1/2 * AVF2 * QC                                     (14) 
 
   Since  
 
      AVF2 = C * AVF1                                            (15) 
 
   we can apply the quaternion transformation formula 
   (from the Rotation Required Reading) 
 
                               * 
      AVF2 =  QC  *  AVF1  * QC                                  (16) 
                    
   Now we re-write (15) as 
 
                                   * 
      DQ = - 1/2 * ( QC * AVF1 * QC ) * QC                       
  
         = - 1/2 *   QC * AVF1                                   (17) 
 
   Then the angular velocity vector AVF1 is given by 
 
                     * 
      AVF1  = -2 * QC  * DQ                                      (18) 
 
   The relation (18) has now been demonstrated for quaternions 
   having constant, unit magnitude angular velocity.  But since 
   all time-dependent quaternions having value QC and derivative 
   DQ at a given time t have the same angular velocity at time t,  
   that angular velocity must be AVF1.   
 

Examples

 
   The following test program creates a quaternion and quaternion 
   derivative from a known rotation matrix and angular velocity 
   vector.  The angular velocity is recovered from the quaternion 
   and quaternion derivative by calling qdq2av_c and by an 
   alternate method; the results are displayed for comparison. 
 

      #include <stdio.h>
      #include "SpiceUsr.h"
      #include "SpiceZfc.h"

      int main()
      {
         /.
         Local constants 
         ./

         /.
         Local variables
         ./
         SpiceDouble             angle  [3];
         SpiceDouble             av     [3];
         SpiceDouble             avx    [3];
         SpiceDouble             dm     [3][3];
         SpiceDouble             dq     [4];
         SpiceDouble             expav  [3];
         SpiceDouble             m      [3][3];
         SpiceDouble             mout   [3][3];
         SpiceDouble             q      [4];
         SpiceDouble             qav    [4];
         SpiceDouble             xtrans [6][6];

         SpiceInt                i;

         /.
         Pick some Euler angles and form a rotation matrix.
         ./
         angle[0] =  -20.0 * rpd_c();
         angle[1] =   50.0 * rpd_c();
         angle[2] =  -60.0 * rpd_c();

         eul2m_c ( angle[2], angle[1], angle[0], 3, 1, 3, m );

         m2q_c ( m, q );

         /.
         Choose an angular velocity vector.  
         ./
         expav[0] = 1.0;
         expav[1] = 2.0;
         expav[2] = 3.0;

         /.
         Form the quaternion derivative.
         ./
         qav[0]  =  0.0;
         vequ_c ( expav, qav+1 );

         qxq_c ( q, qav, dq );

         vsclg_c ( -0.5, dq, 4, dq );

         /.
         Recover angular velocity from `q' and `dq' using qdq2av_c.
         ./
         qdq2av_c ( q, dq, av );

         /.
         Now we'll obtain the angular velocity from `q' and
         `dq' by an alternate method.

         Convert `q' back to a rotation matrix.
         ./
         q2m_c ( q, m );

         /.
         Convert `q' and `dq' to a rotation derivative matrix.  This
         somewhat messy procedure is based on differentiating the
         formula for deriving a rotation from a quaternion, then
         substituting components of `q' and `dq' into the derivative
         formula.
         ./

         dm[0][0]  =  -4.0  * (   q[2]*dq[2]  +  q[3]*dq[3]  );

         dm[0][1]  =   2.0  * (   q[1]*dq[2]  +  q[2]*dq[1]
                                - q[0]*dq[3]  -  q[3]*dq[0]  );

         dm[0][2]  =   2.0  * (   q[1]*dq[3]  +  q[3]*dq[1]
                                + q[0]*dq[2]  +  q[2]*dq[0]  );

         dm[1][0]  =   2.0  * (   q[1]*dq[2]  +  q[2]*dq[1]
                                + q[0]*dq[3]  +  q[3]*dq[0]  );

         dm[1][1]  =  -4.0  * (   q[1]*dq[1]  +  q[3]*dq[3]  );

         dm[1][2]  =   2.0  * (   q[2]*dq[3]  +  q[3]*dq[2]
                                - q[0]*dq[1]  -  q[1]*dq[0]  );

         dm[2][0]  =   2.0  * (   q[3]*dq[1]  +  q[1]*dq[3]
                                - q[0]*dq[2]  -  q[2]*dq[0]  );

         dm[2][1]  =   2.0  * (   q[2]*dq[3]  +  q[3]*dq[2]
                                + q[0]*dq[1]  +  q[1]*dq[0]  );

         dm[2][2]  =  -4.0  * (   q[1]*dq[1]  +  q[2]*dq[2]  );

         /.
         Form the state transformation matrix corresponding to `m'
         and `dm'.
         ./

         /.
         Upper left block: 
         ./
         for ( i = 0;  i < 3;  i++ )
         {
            vequ_c ( m[i], xtrans[i] );
         }

         /.
         Upper right block: 
         ./
         for ( i = 0;  i < 3;  i++ )
         {
            vpack_c ( 0.0, 0.0, 0.0, xtrans[i]+3 );
         }

         /.
         Lower left block: 
         ./
         for ( i = 0;  i < 3;  i++ )
         {
            vequ_c ( dm[i], xtrans[3+i] );
         }

         /.
         Lower right block: 
         ./
         for ( i = 0;  i < 3;  i++ )
         {
            vequ_c ( m[i], xtrans[3+i]+3  );
         }

         /.
         Now use xf2rav_c to produce the expected angular velocity.
         ./
         xf2rav_c ( xtrans, mout, avx );

         /.
         The results should match to nearly full double precision.
         ./
         printf ( "Original angular velocity:   \n"
                  "   %24.16e, %24.16e, %24.16e \n"
                  "qdq2av_c's angular velocity: \n"
                  "   %24.16e, %24.16e, %24.16e \n"
                  "xf2rav's angular velocity:   \n"
                  "   %24.16e, %24.16e, %24.16e \n",
                  expav[0],   expav[1],  expav[2], 
                  av   [0],   av   [1],  av   [2],
                  avx  [0],   avx  [1],  avx  [2]    );


         return ( 0 );
      }

Restrictions

 
   None. 
 

Literature_References

 
   None. 
 

Author_and_Institution

 
   N.J. Bachman    (JPL) 
 

Version

 
   -CSPICE Version 1.0.1, 27-FEB-2008 (NJB)

      Updated header; added information about SPICE 
      quaternion conventions.

   -CSPICE Version 1.0.0, 31-OCT-2005 (NJB)

Index_Entries

 
   angular velocity from  quaternion and derivative 
Wed Apr  5 17:54:41 2017