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Index Page qdq2av_c

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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

       qdq2av_c (Quaternion and quaternion derivative to a.v.)

       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 -Particulars below.

       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(0),
                   sin(theta/2) * a(1),
                   sin(theta/2) * a(2)  )

       while the engineering quaternions representing `m' are

          (+/-) ( -sin(theta/2) * a(0),
                  -sin(theta/2) * a(1),
                  -sin(theta/2) * a(2),
                   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   -n2   n1  |
                     |                 |
           omega  =  |   n2   0   -n0  |
                     |                 |
                     |  -n1   n0   0   |
                     `-               -'

       The vector `n' of matrix entries (n0, n1, n2) is the rotation axis
       of `m' and `theta' is m's rotation angle. Note that `n' and `theta'
       are not unique.

       Let

          cth = cos(theta/2)
          sth = sin(theta/2)

       Then the unit quaternions `q' corresponding to `m' are

          q = +/- ( cth, sth*n0, sth*n1, sth*n2 )

       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 `quat1' and `quat2' be SPICE quaternions with respective scalar
       and vector parts `s1', `s2' and `v1', `v2':

          quat1 = s1 + v1
          quat2 = 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

          quat1*quat2 = s1*s2 - <v1,v2>  + s1*v2 + s2*v1 + (v1 x v2)

       If `quat1' and `quat2' represent the rotation matrices `m1' and `m2'
       respectively, then the quaternion product

          quat1*quat1

       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  =  I * ( -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) * k                                             (2)

       where `k' 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 `k' to represent a coordinate
       transformation from frame `f1' to frame `f2'. We'll call `f1' the
       "base frame" of `k'. 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  * k                                   (5)
            dt   |t=0

       The rotation axis `a' associated with `omega' is defined by    (6)

          a(0) =  - omega(1,2)
          a(1) =    omega(0,2)
          a(2) =  - omega(0,1)

       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 `k'.

       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 `qk' be quaternions representing the
       time-dependent matrices r(t), m(t) and `k' 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) * qk                                          (8)

       Differentiating with respect to `t', then evaluating derivatives
       at t = 0, we have

          d(qr(t))|         d(qm(t))|
          --------|     =   --------|     * qk                        (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  *  qk                         (12)
             dt   |t=0

       or for short,

          dq = 1/2 * a * qk                                          (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 * qk                                     (14)

       Since

          avf2 = k * avf1                                            (15)

       we can apply the quaternion transformation formula
       (from the Rotation Required Reading)

                                   *
          avf2 =  qk  *  avf1  * qk                                  (16)

       Now we re-write (15) as

                                       *
          dq = - 1/2 * ( qk * avf1 * qk ) * qk

             = - 1/2 *   qk * avf1                                   (17)

       Then the angular velocity vector `avf1' is given by

                         *
          avf`  = -2 * qk  * 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 `qk' and derivative
       `dq' at a given time `t' have the same angular velocity at time `t',
       that angular velocity must be `avf1'.

#### Examples

       The numerical results shown for this example may differ across
       platforms. The results depend on the SPICE kernels used as
       input, the compiler and supporting libraries, and the machine
       specific arithmetic implementation.

       1) 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.

          Example code begins here.

          /.
             Program qdq2av_ex1
          ./

          #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"
                      " %19.14f %19.14f %19.14f\n"
                      "qdq2av_c's angular velocity:\n"
                      " %19.14f %19.14f %19.14f\n"
                      "xf2rav_c's angular velocity:\n"
                      " %19.14f %19.14f %19.14f\n",
                      expav[0],   expav[1],  expav[2],
                      av   [0],   av   [1],  av   [2],
                      avx  [0],   avx  [1],  avx  [2]    );

             return ( 0 );
          }

          When this program was executed on a Mac/Intel/cc/64-bit
          platform, the output was:

          Original angular velocity:
              1.00000000000000    2.00000000000000    3.00000000000000
          qdq2av_c's angular velocity:
              1.00000000000000    2.00000000000000    3.00000000000000
          xf2rav_c's angular velocity:
              1.00000000000000    2.00000000000000    3.00000000000000

#### Restrictions

       None.

#### Literature_References

       None.

#### Author_and_Institution

       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)

#### Version

       -CSPICE Version 1.0.2, 10-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard.

       -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

#### Link to routine qdq2av_c source file qdq2av_c.c

    Fri Dec 31 18:41:11 2021