 
Reference Frames
===========================================================================
 
   Last revised on 2017 FEB 01 by B. V. Semenov.
 
 
Abstract
--------------------------------------------------------
 
   The frames subsystem specifies the relationships of various kinds of
   reference frames supported by SPICE. This facilitates
   ``behind-the-scenes'' transformations between these frames.
 
 
Purpose
 
   This document describes how reference frames are treated within SPICE.
   The document includes a general discussion of reference frames, detailed
   information about various types of frames supported within SPICE, and
   instructions on defining additional reference frames to assist in a
   user's computations.
 
 
Intended Audience
 
   This document addresses the needs of several groups of users. Users
   looking for a basic discussion of reference frames and a list of the
   frames supported by the SPICE system should read the chapter ``Using
   Frames.'' Users desiring to customize their environment by adding new
   frames should read the chapter ``Creating a Frame Kernel.''
 
   This document assumes you have some familiarity with SPICE concepts and
   terminology. If you are new to the SPICE system, or just a bit rusty
   with it, you should consider reviewing ``An Overview of the SPICE
   System'' and ``An Introduction to SPICE.''
 
 
Using Frames
===========================================================================
 
 
Frame Functions in CSPICE
--------------------------------------------------------
 
   The SPICE frame subsystem facilitates ``behind-the-scenes'' frame
   transformations. This allows you to concentrate on questions more
   closely related to the problem you are trying to solve instead of the
   details of on how to get position or state vectors in the frame of
   interest.
 
 
Frame Transformation Functions
 
   Several user-level CSPICE functions require that the user supply the
   name of a reference frame as one of the inputs to the function. The most
   important of these is the function spkezr_c. This function returns the
   state (Cartesian position and velocity) of one object relative to
   another in a user specified reference frame. The choice of reference
   frame often makes a big difference in the usefulness of a returned
   state. If the state is given relative to the reference frame of interest
   to the user, computations involving that state can be much simpler than
   if the state is returned relative to some other reference frame.
 
   The two user-level interface functions that deal solely with frame
   transformations are sxform_c and pxform_c. sxform_c supports
   transformations of Cartesian state vectors (6 components) between
   reference frames while pxform_c supports transformations of Cartesian
   position vectors (3 components). pxform_c may be used when only position
   information is needed, or when the derivatives required for a state
   transformation are unavailable, for example when one frame is defined by
   a C-kernel that lacks angular velocity data.
 
   The calling sequences for these functions are
 
         sxform_c ( from, to, et, xform  );
         pxform_c ( from, to, et, rotate );
 
   The output of sxform_c, `xform', is a 6 by 6 matrix used to transform
   state vectors relative to a reference frame, the name of which is
   specified by the `from' input argument, to states relative to another
   reference frame, the name of which is specified by the `to' input
   argument, at the epoch `et' (specified in seconds past J2000).
 
   The output of pxform_c, `rotate', is a 3 by 3 transformation matrix
   equivalent to the upper left 3x3 block of `xform'. This matrix
   transforms position as opposed to state vectors.
 
 
Frame Information Functions
 
   The CSPICE frame subsystem contains a set of functions that enable
   applications to retrieve information about frames known to CSPICE,
   whether they are built-in or specified by means of frame kernels:
 
         frmnam_c        Convert frame ID code to frame name.
 
         namfrm_c        Convert frame name to frame ID code.
 
         frinfo_c        Return frame specification parameters: frame
                         center, frame class, and frame class ID.
 
         cidfrm_c        Map body ID code to the default frame centered on
                         the specified body. Both frame name and ID are
                         returned.
 
         cnmfrm_c        Map body name to the default frame centered on the
                         specified body. Both frame name and ID are
                         returned.
 
         ccifrm_c        Map frame class and class ID to frame
                         specification parameters: frame ID code, frame
                         name and frame center are returned.
 
                         This function provides a way to identify frames
                         referenced in CK and PCK files: in these files,
                         segment descriptors contain class IDs of frames.
 
   The calling sequences for the frame information functions are
 
      frmnam_c ( frcode, lenout, frname  )
      namfrm_c ( frname, &frcode )
      frinfo_c ( frcode, &cent,  &class,  &clssid, &found )
      cidfrm_c ( cent,   lenout, &frcode, frname,  &found )
      cnmfrm_c ( cname,  lenout, &frcode, frname,  &found )
      ccifrm_c ( class,  clssid, lenout,  &frcode, frname,
                 &cent,  &found )
 
   See the section ``Specifying a New Frame'' below for more information on
   frame specification parameters.
 
 
Frames Supported in SPICE
--------------------------------------------------------
 
   In both cases -- with the functions requiring specification of a
   reference frame as one of the inputs (for example spkezr_c), and with
   the functions computing transformation between two reference frames
   (sxform_c and pxform_c) -- you specify the frame or frames of interest
   using a character string that contains the name of the reference frame.
 
   In CSPICE function interfaces, frames are typically designated by C
   strings. In text kernel files, frame names are designated by strings
   delimited by single quotes, as in FORTRAN. Examples below showing
   single-quoted frame names exhibit the names as they appear in text
   kernels; these same names are double-quoted when referred to as literal
   strings in C source code.
 
   A number of names are automatically recognized by the frame subsystem
   because the definitions for these frames are ``built into'' CSPICE
   software. Among these frames are:
 
       --   inertial frames such as Earth mean equator and equinox of J2000
            frame ('J2000'), Mean ecliptic and equinox of J2000
            ('ECLIPJ2000'), Galactic System II frame ('GALACTIC'), Mars
            Mean Equator and IAU vector of J2000 frame ('MARSIAU'), etc.
            For the complete list of ``built in'' inertial reference frames
            refer to the appendix ``built in Inertial Reference Frames'' of
            this document.
 
       --   body-fixed frames based on IAU rotation models provided in text
            PCK files, such as Earth body-fixed rotating frame
            ('IAU_EARTH') and Mars body-fixed rotating frame ('IAU_MARS'),
            and body-fixed frames based on high precision Earth rotation
            models provided in binary PCK files such as 'ITRF93'. For the
            complete lists of ``built in'' body-fixed reference frames
            refer to the appendixes ``built in PCK-Based Reference Frames''
            and High Precision Earth Fixed Frames'' of this document.
 
   For all other frames the names are not ``built into'' SPICE. Instead,
   these names, as well as the parameters specifying the frames, are
   provided via keywords included in a text kernel file. The types of
   frames defined in text kernels include:
 
       --   CK-based frames, i.e. frames for which orientation is provided
            in CK files
 
       --   fixed offset frames, i.e. frames for which orientation is
            constant with respect to another frame and is specified as part
            of the frame definition stored in a text kernel
 
       --   Dynamic frames, i.e. frames for which orientation is based on
            dynamic directions computed based on SPICE kernel data (SPKs,
            CK, PCKs), on mathematical models implemented in CSPICE
            functions, or on formulas defined in frame kernels.
 
   You can find the names of these frames by examining the text kernel file
   that contains the frame definitions. Normally definitions of all frames
   specific for a given mission are stored in that mission's Frames Kernel
   (FK) file but they can also be provided in the Instrument Kernels (IK)
   or any other text kernels. In order to make frame definitions from the
   text kernels available to SPICE, these kernels need to be loaded via a
   call to furnsh_c. For example, to load an FK named ``myframe.tf'', call
   furnsh_c as follows:
 
         furnsh_c ( "myframe.tf" );
 
   Note that the SPICE data loading mechanism detects and prohibits loading
   text kernel files containing lines terminated with EOL character(s)
   non-native to the platform on which the Toolkit was compiled. If a
   non-native EOL terminator is detected in the first 132 characters of a
   text kernel, the execution is stopped and an error message is displayed.
   This feature does not work with files that are smaller that 132 bytes or
   have the first line longer that 132 characters.
 
 
Kernels Needed For Computing Frame Transformations
--------------------------------------------------------
 
   In many cases data needed to compute transformation of one frame
   relative to another is stored in SPICE kernels: PCK, CK, FK, and even
   SPK. The appropriate kernels must be loaded for the SPICE system to
   compute a frame transformation from a non-inertial frame to any other
   frame.
 
   The ``built in'' inertial frames are the only frames the transformations
   between which can be computed without loading any SPICE kernels.
 
   Since the body-fixed frames are tied to the rotation of planets,
   satellites, asteroids, etc, the information about how the orientation of
   these frames is changing with respect to inertial frames is stored in
   SPICE PCK files. It is important to note that although the names of
   these frames are ``built in'' their relationship to inertial frames is
   not. This information must be ``loaded'' into the SPICE system from a
   PCK file. Without loading this information you cannot compute the
   transformation to or from a body-fixed frame.
 
   As the name suggests, the orientation of CK-based frames is computed
   using data provided in CK files and cannot be computed without loading
   these. In addition to the CKs, an SCLK kernel establishing time
   correlation for the on-board clock that is used to tag data in the CKs
   must be loaded to support time conversion between that clock and
   ephemeris time.
 
   Because the fixed offset frame definitions stored in text kernels
   provide all information needed to determine their orientation relative
   to the frame with respect to which they are defined, only the text
   kernel containing the definition need be loaded.
 
   Depending on the particular family to which a dynamic frame belongs, no
   additional data may be needed in order to compute its orientation, or
   one or more types of SPICE kernels, including SPKs, PCKs, CKs, and SCLK,
   may have to be loaded.
 
 
Creating a Frame Kernel
===========================================================================
 
   To create a frame kernel you will need to understand the SPICE text
   kernel file format described in detail in the Kernel Required Reading
   document, kernel.req. When making a new frame kernel, make sure that the
   first line of the file contains the proper SPICE file identification
   word for the FK files -- ``KPL/FK'' -- left-justified, on a line by
   itself.
 
   You will also need to understand the concept of a frame class.
 
 
Frame Classes
--------------------------------------------------------
 
   The method by which a frame is related to some other frame is a function
   of the ``class'' of the frame. You describe the class of a frame with an
   integer called the frame's ``class number.'' The reference frame classes
   are enumerated below.
 
       1.   Inertial frames. These frames do not rotate with respect to the
            star background. They are the frames in which Newton's laws of
            motion apply. The class number associated with inertial frames
            is 1.
 
       2.   PCK (body-fixed) frames. PCK frames are reference frames whose
            orientation with respect to inertial frames is supplied through
            either binary or text PCK files. To determine a transformation
            to or from a PCK frame, you must load a PCK file that describes
            the orientation of the frame with respect to one of the
            inertial frames ``built into'' SPICE. The class number
            associated with PCK frames is 2.
 
       3.   CK frames. CK frames are reference frames whose orientation
            with respect to some other reference frame is supplied via a
            SPICE C-kernel. The other reference frame may be any of the
            four classes of frames described here. C-kernels use spacecraft
            clock ``ticks'' as their basic time unit. Consequently you need
            to load a spacecraft clock kernel appropriate for the C-kernel
            to determine the transformation from or to a C-kernel frame. In
            addition you will need to load a PCK, CK, or TK frame kernel if
            the ``other'' frame belongs to one of these classes. The class
            number associated with CK frames is 3.
 
       4.   Fixed offset frames. These frames are also called Text Kernel
            (TK) frames because they have a constant orientation with
            respect to some other reference frame and this orientation is
            included in the frame definition provided in a SPICE text
            kernel. They may be defined relative to a frame of any of the
            other classes of reference frames. The class number associated
            with TK frames is 4.
 
       5.   Dynamic frames. These are time-dependent reference frames
            defined via parameters or formulas specified in a frame kernel.
            The class number associated with dynamic frames is 5.
 
 
Specifying a New Frame
--------------------------------------------------------
 
   In addition to the data/model needed to specify the orientation of a
   frame with respect to some other reference frame, you must tell the
   SPICE system how to find the data or model. This specification requires
   five pieces of information:
 
       1.   the name of the frame,
 
       2.   the ID code for the frame,
 
       3.   the class number of the frame,
 
       4.   the SPK ID code or name for the frame center,
 
       5.   the internal ID code used by the class (CLASS_ID) to refer to
            the frame.
 
   The rules for selecting these items are given in the next section, but
   for the moment let's assume that the rules have been obeyed and we have
   arrived at the following values.
 
      Frame Name    :    'WALDO'
      Frame ID code :    1234567   (A number guaranteed to be suitable
                                    for private use)
      Frame Class   :          3   (C-kernel)
      Frame Center  :     -10001   (Waldo Spacecraft ID code)
      Frame Class_id:  -10000001   (ID code in C-kernel for Waldo)
 
   The frame kernel that specifies this frame is given below:
 
      \begindata
 
         FRAME_WALDO            =  1234567
         FRAME_1234567_NAME     = 'WALDO'
         FRAME_1234567_CLASS    =  3
         FRAME_1234567_CENTER   = -10001
         FRAME_1234567_CLASS_ID = -10000001
 
      \begintext
 
   Note that single quotes are used to delimit strings in SPICE text
   kernels.
 
 
Guidelines for Frame Specification
--------------------------------------------------------
 
 
Selecting a Name
 
   The name chosen for a frame must not exceed 26 characters taken from the
   set including uppercase letters, numbers, underscore, and plus and minus
   signs. It should have some mnemonic value so that users can recognize
   what the name means. Finally, it should not be the name of one of the
   ``built in'' frames listed above or the name of any other frame you wish
   to specify. If you try to use a ``built in'' name, the frame subsystem
   will ignore your frame specification. In the example given above, we
   chose the name 'WALDO' for the name of our reference frame. If ``Waldo''
   would be a lander and would need to specify a local level frame at its
   landing site, we could have named that frame 'WALDO_LOCAL_LEVEL'. A good
   name for a frame associated with the camera flown on ``Waldo'' would be
   'WALDO_CAMERA'.
 
 
Selecting a Frame ID
 
   What you choose for a frame ID depends upon the class of the frame.
 
   If the class is CK, you may use the same ID as you use for the CLASS_ID.
   In the previous example, we selected the Frame ID to be 1234567. (Since
   our example frame above is of class 3, a CK frame, we would normally use
   the same number for the frame ID as we used for the class ID. However,
   in this example, we have chosen a different value to illustrate the
   connection between the frame ID and the variables needed to define the
   frame.)
 
   For TK frames, the frame and class IDs must be identical. For TK frames
   associated with an instrument, the instrument ID is used for both frame
   ID and class ID. For topocentric TK frames at tracking station sites,
   both frame ID and class ID are created by ``combining'' the ID of the
   body on which the station is located with the station number (for
   example frame and class ID 1399012 is used for ``DSS-12'', with the
   formula used to arrive at this ID being 1000000 + ``Earth ID''*1000 +
   ``station ID''.) For local level and surface fixed TK frames at a
   landing site, both frame ID and class ID are based on the ID of the
   lander (for example frame and class ID of -222999 would be the natural
   choice for the lander with ID -222.)
 
   If the frame is a PCK frame or a dynamic frame and you are working
   without consultation with NAIF, select an integer in the range from
   1400000 to 2000000.
 
 
Selecting the Class
 
   This is usually the easiest part of specifying a frame. Presumably you
   know how the orientation of the frame with respect to some other frame
   will be computed. Simply choose the appropriate class number. In the
   example above, the class number is 3 because we are defining a CK-based
   frame.
 
 
Selecting the Center
 
   A frame is used to specify the orientation of some object. The frame
   consists of a set of coordinate axes relative to some point -- the
   origin of the reference frame. When viewed from some other frame the
   axes rotate about the origin. The origin about which the rotation takes
   place is the center of the frame. For body-fixed frames this is the
   center of the body to which they are fixed. For C-kernel frames the
   center is often the spacecraft whose orientation is provided by the
   C-kernel. Simply find the SPK ID code or name for the object to which
   the frame is attached and use that as the value for the center. In our
   example, the SPK ID code for the ``Waldo'' spacecraft is -10001.
 
   Note that this center ID is used to look up the position of the frame
   origin when SPICE computes frame orientation adjusted for light time.
   Therefore, only centers for which supporting SPK data are expected to be
   available should be picked. It is usually an issue only for TK and CK
   frames associated with instruments because the positions of instruments
   are rarely available in SPKs. To get around the need to provide the
   instrument positions, it is appropriate to specify the ID of the
   spacecraft on which an instrument is mounted as the center of a TK or CK
   frame associated with it.
 
 
Selecting a Class ID
 
   A frame's ``CLASS_ID'' is an integer used internally by CSPICE software.
   It is the integer code used by the CSPICE reference frame subsystem to
   look up reference frame information.
 
   If your frame is a PCK class frame the CLASS_ID is the ID code for the
   body for which rotation constants are provided in the text PCK file or
   the ID associated with the orientation data provided in the binary PCK
   file.
 
   If your frame is a CK class frame, the CLASS_ID is the ID code used in
   the C-kernel to describe the orientation of the spacecraft.
 
   If the frame is a TK frame, the class ID must match the frame ID. For
   both ID codes you should use a positive integer in the range from
   1400000 to 2000000 (unless you are working in an official project
   capacity in which case you should ask NAIF to provide a CLASS_ID for
   you). In the example above, the CLASS_ID is the ID code for the C-kernel
   structure: -10000001.
 
   If the frame is a dynamic frame, the class ID must match the frame ID.
 
 
Frame IDs Reserved for Public Use
 
   The range 1400000 to 2000000 has been set aside by NAIF as ranges of
   Frame IDs that can be used freely by SPICE users without fear of
   conflict with ``officially recognized'' frames. However, if you and a
   colleague plan to create several such frames, you will need to
   coordinate your work to ensure that your definitions are not in conflict
   with one another.
 
 
Why have a Frame ID and a Class ID?
 
   When the CSPICE software receives a request to compute a frame
   transformation, it first translates the name of the frame to the
   corresponding frame ID. There is a one to one correspondence between
   frame names and frame IDs. Once the frame ID is in hand, the class of
   the frame can be located and an appropriate subsystem identified for
   carrying out the initial computations needed to construct a frame
   transformation matrix. However, the frame subsystem evolved to unify
   several distinct reference frame systems. In each of these systems,
   reference frames are identified by integer codes. Unfortunately, since
   these subsystems evolved independently, the numeric codes used to
   identify the reference systems overlapped from one system to the next.
   Moreover, to support backward compatibility, NAIF was not free to change
   the numeric codes used by the various systems or the meaning of the
   frame codes that were already present in existing data products.
 
   To support existing data products and allow extension of the SPICE
   system, NAIF needed to associate the old ID code with the new frame ID.
   The CLASS_ID fills this role. When the frame is identified, the ID code
   suitable for the frame class is located and passed onto the frame's
   class so that the initial portion of the frame transformation can be
   carried out.
 
 
Putting the Pieces Together
--------------------------------------------------------
 
   Once you've determined the name, ID code, center, class and class ID of
   your frame, you create the frame specification by filling in the frame
   template below. This should be stored in a text kernel called a Frame
   Specification Kernel or Frames Kernel (FK).
 
      FRAME_<name>             = <ID code>
      FRAME_<ID code>_NAME     = '<name>'
      FRAME_<ID code>_CLASS    = <class>
      FRAME_<ID code>_CLASS_ID = <classid>
      FRAME_<ID code>_CENTER   = <center>
 
   The example we used for the frame 'WALDO' illustrates this.
 
      \begindata
 
         FRAME_WALDO            =  1234567
         FRAME_1234567_NAME     = 'WALDO'
         FRAME_1234567_CLASS    =  3
         FRAME_1234567_CENTER   = -10001
         FRAME_1234567_CLASS_ID = -10000001
 
      \begintext
 
   Once you've completed the frame specification you tell the SPICE system
   about the frame by ``loading'' the frame kernel that contains it. As
   with all text kernels, you load it via the routine furnsh_c. For example
   if the frame kernel containing your frame specification is contained in
   the file ``myframe.tf'' you load the kernel via the call
 
         furnsh_c ( "myframe.tf" );
 
 
Connecting an Object to its Body-fixed Frame
--------------------------------------------------------
 
   Every extended object has both a position and orientation in space. The
   SPICE ephemeris subsystem (SPK) allows you to specify the location of
   such an object. The frame subsystem allows you to name the body-fixed
   frame that describes the orientation of the object, and to retrieve the
   orientation of the frame relative to some other frame as a function of
   time. Given the name or SPK ID code associated with an object we can
   locate its position through the SPK subsystem. Unfortunately, the
   body-fixed frame of the object cannot always be determined from the
   object's name or ID code. For example, we have already mentioned that
   there are two ``built in'' reference frames that describe the
   orientation of the Earth: 'IAU_EARTH' and 'ITRF93'. For other objects,
   such as the asteroid Simbad, there is no ``built in'' frame associated
   with the object. The body-fixed frame of Simbad must be defined through
   a text kernel. In both cases, the connection between the object and its
   body-fixed frame needs to be supplied via a kernel pool variable. There
   are two ways to do this.
 
      OBJECT_<name or spk_id>_FRAME =  '<frame name>'
 
   or
 
      OBJECT_<name or spk_id>_FRAME =  <frame ID code>
 
   You may use the ID codes for either the object, the frame or both. As
   example, four of the following assignments could serve to connect the
   Earth with the 'ITRF93' frame.
 
      OBJECT_399_FRAME   =  13000
      OBJECT_399_FRAME   = 'ITRF93'
      OBJECT_EARTH_FRAME =  13000
      OBJECT_EARTH_FRAME = 'ITRF93'
 
   Note: if you use the name of either the object or frame, you must use
   upper case letters.
 
   Of these four means of specifying an object's body-fixed frame the
   second (OBJECT_399_FRAME = 'ITRF93') is the most robust.
 
   For the sun, the planets and their satellites the frame subsystem
   maintains a default connection between the object and its body-fixed
   frame ``built into'' SPICE. The complete list of ``built in'' body-fixed
   frames is provided in the ``built in PCK-Based IAU Body-Fixed Reference
   Frames'' appendix of this document.
 
 
The rest of the frame information
--------------------------------------------------------
 
   The information supplied in the frame specification tells the SPICE
   system where to look for a particular frame model. However, the
   specification alone doesn't tell the SPICE system how to actually
   transform from the specified frame to some other frame of interest. To
   do this you need to supply other information. How this information is
   supplied depends upon the class of the frame.
 
 
Inertial Frames
===========================================================================
 
   Inertial frames are ``built into'' the SPICE system via the routine
   CHGIRF. Only the frames defined in that routine are available as
   inertial class frames. For this reason there is rarely a need to specify
   an inertial frame through a frame specification kernel. Essentially all
   you can do by creating a frame specification for an inertial frame is to
   supply a second name for one of the ``built in'' frames. For example you
   might define EME2000 as another name for the J2000 frame.
 
   NAIF recommends against creating inertial frame specifications. However,
   if you choose to do so anyway, you are done once you've defined the
   frame specification. (You may not be done explaining to your colleagues
   why you've decided to do this.)
 
   In the example cited earlier (EME2000) here's how you'd specify the
   frame.
 
      \begindata
 
         FRAME_EME2000          = 2000000
         FRAME_2000000_NAME     = 'EME2000'
         FRAME_2000000_CLASS    = 1
         FRAME_2000000_CENTER   = 0
         FRAME_2000000_CLASS_ID = 1
 
      \begintext
 
   Everything else about this frame is ``built into'' the SPICE system.
 
 
PCK Frames
===========================================================================
 
   If you specify a PCK frame, you will need to load either a text or
   binary PCK file for the body with which the frame is associated. The
   construction of PC kernels is discussed in the SPICE document PCK
   Required Reading (pck.req.)
 
 
CK Frames
===========================================================================
 
   If a frame is defined as a CK frame, you will need both a C-kernel for
   the structure identified by the FRAME_..._CLASS_ID variable and an SCLK
   kernel for converting ephemeris time to the ``ticks'' used to represent
   time in the C-kernel. Both the C-kernel(s) and SCLK kernel must be
   loaded prior to attempting to use the CK frame.
 
 
SCLK and SPK ID codes
--------------------------------------------------------
 
   For many C-kernels, the spacecraft clock and spacecraft ID codes can be
   determined by performing an integer division of the C-kernel ID code by
   1000. However, under some circumstances this numerical correspondence
   between C-kernel ID code and the associated SCLK or spacecraft ID may
   break down. When the numerical relationship fails you need to tell the
   SPICE system the ID code of the SCLK or spacecraft via two kernel pool
   variables.
 
      CK_<ck_ID code>_SCLK = <ID code of SCLK>
      CK_<ck_ID code>_SPK  = <SPK ID code>
 
   These variables are normally placed in either the SCLK kernel or in the
   frame specification kernel (FK).
 
   To illustrate how you would create a C-kernel frame, we shall suppose
   that we have a C-kernel for structure -100001 aboard the fictional
   spacecraft ``Waldo'' which has ID code -1001. Moreover we shall assume
   that the clock ID appropriate for this structure is -1002. Below is a
   frame specification together with the CK_..._SCLK and CK_..._SPK
   variable definitions for the 'WALDO' frame.
 
      \begindata
 
         FRAME_WALDO            = -100001
         FRAME_-100001_NAME     = 'WALDO'
         FRAME_-100001_CLASS    = 3
         FRAME_-100001_CLASS_ID = -100001
         FRAME_-100001_CENTER   = -1001
 
         CK_-100001_SCLK        = -1002
         CK_-100001_SPK         = -1001
 
      \begintext
 
 
TK Frames
===========================================================================
 
   The relationship between a constant offset Text Kernel (TK) frame and
   the frame it is offset from is given via a text kernel that can be
   loaded via the kernel pool routine furnsh_c. The first five kernel pool
   variables required for TK frame specification are the same as for any
   other frame defined via a text kernel:
 
      FRAME_<name>             = <ID code>
      FRAME_<ID code>_NAME     = '<name>'
      FRAME_<ID code>_CLASS    = 4
      FRAME_<ID code>_CLASS_ID = <ID code>
      FRAME_<ID code>_CENTER   = <center>
 
   You need to supply information that indicates the frame, RELATIVE, from
   which the TK frame is offset. It is done using this kernel pool
   variable:
 
      TKFRAME_<frame>_RELATIVE = '<name of relative frame>'
 
   where `frame' is the ID code or name you used in the frame
   specification.
 
   Because the rotation from the TK frame to the RELATIVE frame is fixed
   (time invariant) it can be specified in the FK along with the frame
   specification information described above. This rotation data can be
   provided in any of three ways:
 
       1.   as a 3 by 3 matrix, M, that converts vectors from the TK frame
            to the RELATIVE frame by left multiplication
 
                  V_relative = M * V_tkframe
 
       2.   as a set of 3 Euler angles and axes that can be used to produce
            M
 
       3.   as a SPICE-style quaternion representing M.
 
   You let the frame subsystem know which method you've chosen for
   representing the rotation via the kernel pool variable
 
      TKFRAME_<frame>_SPEC.
 
   To use a matrix to define the rotation, use the assignment:
 
      TKFRAME_<frame>_SPEC = 'MATRIX'
 
   To define the rotation via three Euler angles, use the assignment:
 
      TKFRAME_<frame>_SPEC = 'ANGLES'
 
   To define the rotation via a SPICE-style quaternion, use the assignment:
 
      TKFRAME_<frame>_SPEC = 'QUATERNION'
 
   Depending upon the value of the `SPEC' variable, you need to supply one
   of the following sets of kernel pool variables.
 
 
Defining a TK Frame Using a Matrix
--------------------------------------------------------
 
   If you've chosen to define the rotation using a matrix, supply the
   matrix using the kernel pool variable assignment below:
 
      TKFRAME_<frame>_MATRIX = ( matrix_value[0][0],
                                 matrix_value[1][0],
                                 matrix_value[2][0],
                                 matrix_value[0][1],
                                 matrix_value[1][1],
                                 matrix_value[2][1],
                                 matrix_value[0][2],
                                 matrix_value[1][2],
                                 matrix_value[2][2]  )
 
   For example, if the matrix defining your TK frame is
 
      0.4   -0.6   0.0
      0.6    0.4   0.0
      0.0    0.0   1.0
 
   and the ID code you've selected for the frame is 1234567, then you would
   supply the following information in a text kernel.
 
      TKFRAME_1234567_SPEC   = 'MATRIX'
 
      TKFRAME_1234567_MATRIX = (  0.4
                                  0.6
                                  0.0
                                 -0.6
                                  0.4
                                  0.0
                                  0.0
                                  0.0
                                  1.0 )
 
 
Defining a TK Frame Using Euler Angles
--------------------------------------------------------
 
   If you've chosen to define a TK frame as a sequence of three Euler angle
   rotations about specified coordinate axes, you need to supply the
   following pieces of information:
 
       1.   The values of the three Euler angles;
 
       2.   The axes about which the Euler rotations are performed;
 
       3.   The units associated with the three Euler angles. The
            recognized units are: 'DEGREES', 'RADIANS', 'ARCSECONDS',
            'ARCMINUTES' 'HOURANGLE', 'MINUTEANGLE', 'SECONDANGLE'.
 
   This information is supplied to the SPICE system using the kernel pool
   variables shown below.
 
      TKFRAME_<frame>_ANGLES = ( angle_1, angle_2, angle_3 )
      TKFRAME_<frame>_AXES   = ( axis_1,  axis_2,  axis_3  )
      TKFRAME_<frame>_UNITS  = 'units_of_angles'
 
   The units must be from the list given above. The axes must be chosen
   from the set of integers 1,2,3 where 1 stands for the x-axis, 2 for the
   y-axis, and 3 for the z-axis. If M is the matrix that converts vectors
   relative to the TK frame to the RELATIVE frame by left multiplication,
   then the angles and axes must satisfy the following relationship:
 
      M = [angle_1]      [angle_2]      [angle_3]
                   axis_1         axis_2         axis_3
 
 
   where the symbol
 
      [ A ]
           i
 
   stands for a rotation by the angle A about the i'th axis.
 
      +-                     -+
      |   1       0      0    |
      |   0     cos A   sin A |   =  [ A ]
      |   0    -sin A   cos A |           1
      +-                     -+
 
      +-                     -+
      |  cos A    0    -sin A |
      |   0       1      0    |   =  [ A ]
      |  sin A    0     cos A |           2
      +-                     -+
 
      +-                     -+
      |  cos A   sin A   0    |
      | -sin A   cos A   0    |   =  [ A ]
      |   0       0      1    |           3
      +-                     -+
 
   This method of definition is particularly well suited for defining
   topocentric frames on the surface of the Earth. For example, suppose you
   have an SPK (ephemeris) file that specifies the location of some surface
   point on the Earth, and that the SPK ID code of this point is 399100.
   Moreover suppose you have the geodetic co-latitude (COLAT) and longitude
   (LONG) measured in degrees for this point. (Note that the co-latitude is
   the complement of latitude: latitude + co-latitude = 90 degrees.)
 
   Given this information we can easily define a topocentric reference
   frame at the point such that the x-axis points north along the local
   meridian, the y-axis points west along the local latitude and the z-axis
   points up from the reference spheroid.
 
   The transformation from Earth body-fixed frame to topocentric frame is
   given by
 
      BF2TP = [180] [COLAT] [LONG]
                   3       2      3
 
   Consequently the transformation from the topocentric frame to the
   body-fixed frame is given by
 
      M = TP2BF = [-LONG] [-COLAT] [180]
                         3        2     3
 
   Let 1234567 be the ID code for the topocentric frame; let the name of
   this frame be 'MYTOPO'; and define this relative to the IAU frame for
   the Earth (one of the ``built in'' frames). The topocentric frame at the
   ephemeris point 399100 is specified as shown below:
 
      \begindata
 
         FRAME_MYTOPO             = 1234567
         FRAME_1234567_NAME       = 'MYTOPO'
         FRAME_1234567_CLASS      = 4
         FRAME_1234567_CLASS_ID   = 1234567
         FRAME_1234567_CENTER     = 399100
 
         TKFRAME_1234567_SPEC     = 'ANGLES'
         TKFRAME_1234567_RELATIVE = 'IAU_EARTH'
         TKFRAME_1234567_ANGLES   = ( <-long>, <-colat>, 180 )
         TKFRAME_1234567_AXES     = (       3,        2,   3 )
         TKFRAME_1234567_UNITS    = 'DEGREES'
 
      \begintext
 
   As we'll see a bit later, we can make a more flexible definition for
   this topocentric frame.
 
 
Defining a TK Frame Using a SPICE-style Quaternion
--------------------------------------------------------
 
   If you've chosen to define a TK frame using a SPICE-style quaternion,
   supply the quaternion using the kernel pool variable assignment below:
 
      TKFRAME_<frame>_Q = ( q_0, q_1, q_2, q_3 )
 
   where component zero is the so-called ``real'' component of the
   quaternion (the ``cosine'' component of the quaternion). The last 3
   components (components 1 through 3) are the ``axis'' components of the
   quaternion -- the i, j, and k components respectively of the quaternion.
   The quaternion must be a unit quaternion.
 
           2        2        2        2
      (q_0)  + (q_1)  + (q_2)  + (q_3)  = 1
 
   A more detailed discussion of quaternions is available in the reference
   document ``Rotations Required Reading'' (rotation.req), and in a
   ``Quaternions White Paper'' available from NAIF.
 
 
Gaining Flexibility via TK Frames
--------------------------------------------------------
 
   The use of non-inertial frames gives you an easy means of creating
   ephemerides for points on the surface of a body such as the Earth, Moon
   or Mars. The ephemeris is simply the body-fixed location of the object
   relative to a body-fixed frame for the same object. However, the model
   used to relate the body-fixed frame to other reference frames may not be
   fixed. Indeed, for the Earth there are several different methods with
   varying degrees of accuracy that give the orientation of the Earth with
   respect to inertial space. Each of these different realizations may have
   a different frame ID code. This ability to ``plug in'' different
   orientations is one of the strengths of the SPICE system. However, if
   you create an ephemeris relative to one of these specific models, you
   won't be able to use it unless you've loaded the correct model. To make
   the ephemeris usable regardless of the orientation model you happen to
   have at your disposal, you should define the body-fixed ephemeris
   relative to a TK frame. Then define the TK frame so that rotation from
   the TK frame to the PCK frame is the identity matrix. For example, you
   can define a lunar body-fixed frame as shown below.
 
      \begindata
 
         FRAME_MOONFIXED          = 3010000
         FRAME_3010000_NAME       = 'MOONFIXED'
         FRAME_3010000_CLASS      = 4
         FRAME_3010000_CLASS_ID   = 3010000
         FRAME_3010000_CENTER     = 301
 
         TKFRAME_3010000_SPEC     = 'MATRIX'
         TKFRAME_3010000_RELATIVE = '<name of base frame>'
         TKFRAME_3010000_MATRIX   = ( 1,
                                      0,
                                      0,
                                      0,
                                      1,
                                      0,
                                      0,
                                      0,
                                      1 )
 
      \begintext
 
   By editing this definition you can make the MOONFIXED frame be the IAU
   MOON frame or some other model if one is available. Or you can create
   several such definitions and, at run-time, load the file that best fits
   your current environment.
 
   Using this indirect method of defining the various frames for which more
   than one orientation model may be available, you can avoid limiting how
   various kernels can be used.
 
 
Dynamic Frames
===========================================================================
 
   In CSPICE documentation, the term ``dynamic frame'' designates a
   time-dependent reference frame defined via a frame kernel.
 
   A ``parameterized dynamic frame'' is a dynamic frame defined by a
   formula implemented in CSPICE code and having user-selectable parameters
   set via a frame kernel. The formula defining a dynamic frame may rely on
   data from other SPICE kernels, for example state vectors provided by SPK
   files or rotation matrices from C-kernels or PCK files.
 
   An example of a parameterized dynamic frame is a nadir-pointing
   reference frame for a spacecraft orbiting a planet, where the
   spacecraft's nadir direction and velocity vector define the frame. Using
   a frame kernel, a CSPICE user may specify the planet and spacecraft, the
   relationship between the nadir and velocity vectors and the frame's
   axes, and a small set of additional parameters required to define the
   frame.
 
   Currently parameterized dynamic frames are the only type of dynamic
   frame supported by CSPICE. Other types of dynamic frames, such as frames
   defined by complete formulas (as opposed to parameters) provided in
   frame kernels, may be implemented in future versions of CSPICE.
 
   Below we'll discuss the various types of supported dynamic frames, how
   to create frame kernels that define dynamic frames, and dynamic frame
   implementation considerations. The appendix ``Frame Definition
   Examples'' contains frame definition templates for a variety of popular
   dynamic frames.
 
 
Parameterized Dynamic Frame Families
===========================================================================
 
   The ``family'' to which a parameterized dynamic frame belongs indicates
   the underlying mathematical formula by which the frame is defined.
   Currently there are four parameterized dynamic frame families:
 
       --   Two-vector frames: a reference frame is defined by two vectors.
            The first vector is parallel to one axis of the frame; the
            component of the second vector orthogonal to the first is
            parallel to another axis of the frame, and the cross product of
            the two vectors is parallel to the remaining axis.
 
       --   Mean equator and equinox of date frames: these use mathematical
            precession models to define orientation of a body's equatorial
            plane and location of the frame's x-axis. Currently these
            frames are supported only for the earth.
 
       --   True equator and equinox of date frames: these use mathematical
            precession and nutation models to define orientation of a
            body's equatorial plane and location of the frame's x-axis.
            Currently these frames are supported only for the earth.
 
       --   Mean ecliptic and equinox of date frames: these use
            mathematical precession and mean obliquity models to define
            orientation of a body's orbital plane and location of the
            frame's x-axis. Currently these frames are supported only for
            the earth.
 
       --   Euler frames: polynomial coefficients, a reference epoch, and
            an axis sequence are used to specify time-dependent Euler
            angles giving the orientation of the frame relative to a
            second, specified frame as a function of time.
 
 
Notation
 
   A lower case letter `x' is used to designate the cross product operator,
   as in
 
      C = A x B
 
   Double vertical bars bracketing the name of a vector indicate the norm
   of the vector:
 
      ||A||
 
   Throughout this discussion we'll use text enclosed in angle brackets to
   indicate values to be filled in by the creator of a frame kernel.
   Examples are:
 
         Token                 Replacement Value
      -------------            -----------------------------------------
      <vec_ID>                 'PRI' or 'SEC' [See discussion of
                               two-vector frames below.]
      <frame_name>             SPICE frame name, .e.g. 'J2000'
      <frame_ID>               Integer frame ID code
      <observer_ID>            NAIF integer ID for the observing body
      <aberration correction>  String indicating aberration correction,
                               e.g.:  'NONE', 'LT', 'XLT', 'LT+S'
 
 
Required Keywords for Parameterized Dynamic Frames
--------------------------------------------------------
 
   All parameterized dynamic frame kernel definitions contain the
   assignments shown here:
 
      FRAME_<frame_name>                  =  <frame_ID>
      FRAME_<frame_ID>_NAME               =  <frame_name>
      FRAME_<frame_ID>_CLASS              =  5
      FRAME_<frame_ID>_CLASS_ID           =  <frame_ID>
      FRAME_<frame_ID>_CENTER             =  <center_ID>
 
      FRAME_<frame_ID>_RELATIVE           =  <base_frame_name>
      FRAME_<frame_ID>_DEF_STYLE          =  'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY             =  <frame_family>
 
   These first five of the assignments are common to all CSPICE frame
   definitions; the class code 5 indicates that the frame is dynamic. See
   the section ``Guidelines for Frame Specification'' in the chapter
   ``Creating a Frame Kernel'' above for a detailed discussion of these
   assignments.
 
   The sixth assignment (for keyword FRAME_<frame_ID>_RELATIVE) is the
   ``base frame'' specification; this indicates the frame the
   transformation defined by the frame kernel ``maps to'': starting with an
   epoch ET and a state vector S specified relative to the defined frame
 
      <frame name>
 
   the frame definition determines the 6x6 state transformation matrix
   XFORM such that the product
 
      XFORM * S
 
   yields the equivalent state specified relative to the base frame at ET.
 
   The seventh assignment (for keyword FRAME_<frame_ID>_DEF_STYLE) is used
   to simplify future implementation of other dynamic frame definition
   styles. Only the value
 
      'PARAMETERIZED'
 
   is currently supported.
 
   The last assignment indicates the frame family. The possible values are
 
      'TWO-VECTOR'
      'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
      'EULER'
 
   Additional, required frame kernel assignments are a function of the
   family to which a dynamic frame belongs. These are discussed below.
 
 
Conditional Keywords for Parameterized Dynamic Frames
--------------------------------------------------------
 
 
Rotation State
 
   A parameterized dynamic frame definition can specify a frame's
   ``rotation state'' as ``rotating'' or ``inertial.'' Rotating frames are
   nominally time-dependent, although it is possible for them to be
   constant (an Euler frame with all Euler angles constant is an example).
 
   When a parameterized dynamic frame is specified as ``inertial,'' the
   derivative with respect to time of the transformation between the frame
   and any inertial frame, for example the J2000 frame, is zero. The
   rotation between the frame and any inertial frame is still treated as
   time-dependent. For such a frame F, the call
 
      sxform_c ( "F", "J2000", t, xform );
 
   yields a 6x6 state transformation matrix `xform' having the structure
 
      +-----+-----+
      | R(t)|  0  |
      +-----+-----+
      |  0  | R(t)|
      +-----+-----+
 
   where R(t) is the 3x3 rotation matrix that transforms vectors from frame
   F to the J2000 frame at time `t'. By contrast, when the rotation state
   of F is ``rotating,'' `xform' has the structure
 
      +-----+-----+
      | R(t)|  0  |
      +-----+-----+
      |dR/dt| R(t)|
      +-----+-----+
 
   So, when the rotation state of frame F is ``inertial,'' velocities are
   transformed from frame F to J2000 by left-multiplication by R(t); the
   time derivative of the rotation from F to J2000 is simply ignored.
 
   Normally the inertial rotation state makes sense only for slowly
   rotating frames such as the earth mean equator and equinox of date
   frame.
 
   A parameterized dynamic frame's rotation state is specified via the
   assignment
 
      FRAME_<frame_ID>_ROTATION_STATE     =  <state>
 
   where
 
      <state>
 
   is one of
 
      'ROTATING'
      'INERTIAL'
 
   For frames belonging to the parameterized dynamic frame families
 
      'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
 
   either the rotation state must be specified, or the frame must be frozen
   (see ``Frozen Frames'' below).
 
   For two-vector and Euler frames, the rotation state specification is
   optional; these frames are considered to be rotating by default.
 
   When the rotation state of a parameterized frame is specified, the frame
   cannot be frozen; these options are mutually exclusive.
 
 
Freeze Epoch
 
   A parameterized dynamic frame definition can specify a frame as
   ``frozen'' at a particular epoch. The rotation between a frozen frame
   and its base frame is constant; the derivative with respect to time of
   this rotation is zero.
 
   A frozen frame whose base frame is time-varying is still time-varying:
   it is the relationship between the frozen frame and the base frame that
   is time-independent.
 
   A frame is declared frozen by specifying a ``freeze epoch.'' This is
   done via the assignment:
 
      FRAME_<frame_ID>_FREEZE_EPOCH       =  <time_spec>
 
   where
 
      <time_spec>
 
   is a TDB calendar date whose format conforms to the SPICE text kernel
   date format specification. These dates
 
       --   are unquoted
 
       --   start with the character
 
               @
 
       --   contain no embedded blanks
 
   An example of a template for these calendar strings is
 
      @YYYY-MON-DD/HR:MN.SEC.###
 
   Literal examples include
 
      @7-MAR-2005
      @March-7-2005-3:10:39.221
      @2005-MAR-07/3:10:39.221
 
   Note that unlike time strings supported by the CSPICE function str2et_c,
   time system tokens such as
 
      UTC
      TDT
      TDB
 
   are not supported; times are always assumed to be TDB.
 
   For frames belonging to the parameterized dynamic frame families
 
      'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
 
   either the frame must be frozen or the rotation state must be specified,
   (see ``Rotation State'' above).
 
   For two-vector and Euler frames, the freeze epoch specification is
   optional; these frames are considered to be time-varying relative to
   their base frames by default.
 
   When a parameterized frame is frozen, the rotation state of the frame
   cannot be specified; these options are mutually exclusive.
 
 
Two-Vector Frames
===========================================================================
 
   Two-vector frames use two user-specified, non-parallel vectors to define
   the mutually orthogonal axes of a right-handed reference frame.
 
   In a two-vector frame definition, one defining vector is parallel to a
   specified axis of the reference frame; this vector is called the
   ``primary vector.'' The other vector, called the ``secondary vector,''
   defines another axis: the component of the secondary vector orthogonal
   to the primary vector is parallel to a specified axis of the reference
   frame. The secondary vector itself need not be, and typically is not,
   aligned with an axis of the defined frame.
 
   Below, we'll call the primary and secondary defining vectors PRI and
   SEC, and we'll name the axes of the right-handed frame X, Y, and Z. The
   unit +Z vector is the cross product of the unit +X and +Y vector.
 
   In a two-vector frame definition, the vectors PRI and SEC are specified
   geometrically; for example, PRI could be the position of the earth
   relative to a spacecraft, and SEC could be defined by the right
   ascension and declination of a given star in a specified reference
   frame.
 
   In a frame kernel, the vectors PRI and SEC are associated with two
   members of the set of unit vectors
 
      { X, -X, Y, -Y, Z, -Z }
 
   An example: in this case PRI is associated with -Z and SEC is associated
   with +X. SEC itself is not parallel to the X axis, but the component of
   SEC orthogonal to PRI points in the +X direction.
 
   The diagram below shows the relationship between PRI, SEC, X, Y, and Z:
 
 
         Component of SEC orthogonal to PRI
                         |
                         |      ^
                         v      |
                       <-----+--+
                        \    |  |
                         \   +--+
                          \     |
                       SEC \    |  +Z  = - PRI / ||PRI||
                            \   |
                             \  |
                              \ +--+
                               \|  |
      +X = Y x Z  <---------+---+--+
                           /   /|
                          +---/ |
                             /| /
                            / |/|
                           /  + |  -Z  =   PRI / ||PRI||
                          /     |
                         /      |
                        v       v  PRI
 
               Z x SEC
        +Y = -----------
             ||Z x SEC||
 
         =   Z x X
 
   By defining PRI and SEC we can create a concrete frame definition.
   Continuing the above example, we can define a nadir-pointing frame for
   the Mars Global Surveyor (MGS) spacecraft as follows:
 
      PRI  =  Vector from MGS to nearest point on Mars reference
              ellipsoid
 
      Z    =  -PRI / ||PRI||
 
      SEC  =  Inertially referenced velocity of MGS relative to Mars
 
      Y    =  Z x SEC / ||Z x SEC||
 
      X    =  Y x Z
 
   For this nadir-pointing frame, -Z is the nadir direction, X points
   roughly in the direction of the inertially referenced spacecraft
   velocity, and Y is aligned with the orbital angular velocity vector.
 
   By converting the above definition into the frame kernel
   ``keyword=value'' format, we can make the definition usable by the
   CSPICE system. Above, for brevity, we've glossed over a few aspects of
   the vector definitions. Below we'll discuss in detail all of the
   elements of two-vector frame specifications.
 
 
Defining a Two-Vector Frame in a Frame Kernel
--------------------------------------------------------
 
 
Kernel Availability
 
   In the following discussion, for brevity, we will use the term
   ``computable'' to describe frames whose definitions are known to CSPICE
   and for which kernels have been loaded sufficient to enable computation
   of the transformations between these frames and their base frames.
 
   We'll also call a frame transformation between frames F1 and F2
   ``computable'' if both frames F1 and F2 are computable and kernels have
   been loaded sufficient to enable computation of the transformation
   between F1 and F2. For example, the transformation between the J2000 and
   IAU_TITAN frames is computable once a PCK containing rotational elements
   for TITAN has been loaded.
 
 
Specifying the Base Frame
 
   When a two-vector frame F is defined with a base frame F_BASE, and when
   the necessary kernels are loaded, the transformation between F and
   F_BASE (in both directions) becomes computable by the CSPICE frame
   subsystem. In addition, for any frame F2 such that the transformation
   between F2 to F_BASE is computable, the transformation from F2 to F (in
   both directions) becomes computable.
 
   For a two-vector frame, the base frame may be any frame F_BASE such that
   the transformation between F_BASE and the J2000 reference frame is
   computable at the time the two-vector frame definition is referenced.
 
   Normally for two-vector frames the base frame should be set to 'J2000';
   this choice yields optimal run-time efficiency. The assignment is made
   as follows.
 
      FRAME_<frame_ID>_RELATIVE           =  'J2000'
 
   Base frame specifications are part of the two-vector frame definition
   because the base frame can be used to control how CSPICE chains together
   two-vector frames with other frames. However, from a mathematical point
   of view, two-vector frames are fully defined without reference to a base
   frame. For example, suppose the two-vector frame F1 is defined by the
   earth-moon position vector and the earth-sun position vector, and the
   base frame for F1 is IAU_EARTH. Suppose that the two-vector frame F2 is
   defined by the same vectors and that the base frame of F2 is J2000.
   Then, ignoring small round-off errors, the transformation between F1 and
   F2 is the identity transformation.
 
   Base frames should not be confused with other frames occurring in
   two-vector frame definitions: constant vectors and velocity vectors have
   associated frames which are also specified by keyword assignments. See
   the discussion below under the heading ``Constant Vectors'' and
   ``Velocity Vectors'' for details.
 
 
Specifying the Frame Family
 
   Definitions of two-vector frames include the frame family specification:
 
      FRAME_<frame_ID>_FAMILY             =  'TWO-VECTOR'
 
   Further assignments (discussed below) define the primary and secondary
   vectors and relate these vectors to the frame's axes.
 
 
Specifying the Rotation state or Freeze Epoch
 
   These specifications are optional for two-vector frames. See the section
   above titled ``Conditional Keywords for Parameterized Dynamic Frames''
   for details.
 
 
Specifying the Angular Separation Tolerance
 
   This specification applies only to two-vector frames and is optional. To
   diagnose near-degenerate geometry, specifically cases where the defining
   vectors have angular separation too close to zero or pi radians, users
   can specify a limit on these angular separations. This is done via the
   keyword assignment
 
      FRAME_<frame_ID>_ANGLE_SEP_TOL      = <tolerance>
 
   where <tolerance> is the separation limit in radians. If the angular
   separation of the defining vectors differs from zero or pi radians by
   less than the specified tolerance, an error will be signaled at run
   time.
 
   When a two-vector frame definition omits specification of an angular
   separation tolerance, CSPICE uses a default value of one milliradian.
 
 
Frame Axis Labels
 
   The primary defining vector is associated with a frame axis via the
   assignment
 
      FRAME_<frame_ID>_PRI_AXIS           = <label>
 
   Here
 
      <label>
 
   may be any of
 
      { 'X',  '-X',  'Y',  '-Y',  'Z',  '-Z' }
 
   Blanks and case in the label are not significant. Unsigned axis
   designations are treated as positive; optionally '+' signs may be used
   to prefix positive axis designations. The primary vector is aligned with
   the indicated axis and has the sense indicated by the implied or
   explicit sign.
 
   The secondary defining vector is associated with a frame axis via the
   assignment
 
      FRAME_<frame_ID>_SEC_AXIS           = <label>
 
   where the axis labels are as above. The assignment means that the
   component of the secondary vector orthogonal to the primary vector is
   aligned with the indicated axis and has the sense indicated by the
   implied or explicit sign.
 
 
Vector Specifications
--------------------------------------------------------
 
   The vectors used to define a two-vector frame are specified by geometric
   means. Each defining vector may be any of:
 
       --   The position of one ephemeris object relative to another
 
       --   The vector from an observer to the nearest point on an extended
            body to the observer
 
       --   The velocity of one ephemeris object relative to another in a
            specified reference frame
 
       --   A constant vector in a specified reference frame
 
   The frames (explicit or implicit) associated with the two defining
   vectors need not match each other or the base frame. CSPICE will map the
   defining vectors to a common frame before performing vector arithmetic
   to derive the axes of the defined frame.
 
   All keywords comprising the primary vector definition start with the
   prefix
 
      FRAME_<frame_ID>_PRI_
 
   All keywords for the second defining vector are prefixed by
 
      FRAME_<frame_ID>_SEC_
 
   Here <frame_ID> is the integer ID code for the frame being defined.
 
   Both the primary and secondary vectors are specified using the sets of
   keywords described below.
 
 
Observer-Target Position Vectors
 
   An observer-target position vector is simply the position of one
   ephemeris object relative to another. These vectors are defined by an
   observer, a target, an aberration correction, a reference frame, and an
   epoch. In the frame kernel, there is no need to specify the reference
   frame or epoch: the CSPICE frame subsystem will determine which frame to
   use, and the epoch is supplied by the calling application at run time.
 
   The observer and target are specified by name or ID code. The aberration
   correction may be any value accepted by spkezr_c.
 
   The frame kernel assignments used to define an observer-target position
   vector are:
 
      FRAME_<frame_ID>_<vec_ID>_VECTOR_DEF = 'OBSERVER_TARGET_POSITION'
      FRAME_<frame_ID>_<vec_ID>_OBSERVER   = <observer name or ID code>
      FRAME_<frame_ID>_<vec_ID>_TARGET     = <target name or ID code>
      FRAME_<frame_ID>_<vec_ID>_ABCORR     = <aberration correction>
 
   where <vec_ID> may be either PRI or SEC, and <frame_ID> is the ID code
   of the frame established by the generic assignments described above.
 
   In order for a two-vector frame using a position vector as part of its
   definition to be computable, kernel data must be loaded that enable
   computation of the specified position vector with respect to the J2000
   frame.
 
   For an example of a two-vector frame definition using an observer-target
   position vector, see the subsection titled ``Geocentric Solar Ecliptic
   (GSE) Frame'' in the appendix ``Frame Definition Examples.''
 
 
Target Near point Vectors
 
   Target near point vectors point from an observer to the closest point on
   an extended target body to the observer.
 
   Target near point vectors are defined by an observer, a target, an
   aberration correction, a frame, and an epoch. As with position vectors,
   the frame and epoch are not specified in the frame kernel.
 
   The observer and target are specified by name or ID code. Aberration
   corrections may be any supported by the CSPICE function subpt_c. Light
   time corrections are applied both to the observer- target center vector
   and to the rotation of the target body. The stellar aberration
   correction, if specified, is applied to the observer-target center
   vector.
 
   The frame kernel assignments used to define a target near point position
   vector are:
 
      FRAME_<frame_ID>_<vec_ID>_VECTOR_DEF = 'TARGET_NEAR_POINT'
      FRAME_<frame_ID>_<vec_ID>_OBSERVER   = <observer name or ID code>
      FRAME_<frame_ID>_<vec_ID>_TARGET     = <target name or ID code>
      FRAME_<frame_ID>_<vec_ID>_ABCORR     = <aberration correction>
 
   In order for a two-vector frame using a target near point vector as part
   of its definition to be computable, kernel data must be loaded that
   enable computation of the target near point vector with respect to the
   J2000 frame.
 
   For an example of a two-vector frame definition using a target near
   point vector, see the subsection titled ``Nadir Frame for Mars Orbiting
   Spacecraft'' in the appendix ``Frame Definition Examples.''
 
 
Observer-Target Velocity Vectors
 
   An observer-target velocity vector is the velocity portion of the state
   of one ephemeris object relative to another. These vectors are defined
   by an observer, a target, an aberration correction, a reference frame,
   and an epoch. Of these, only the epoch is not specified in the frame
   kernel. Unlike observer-target position vectors, velocity vectors
   require a user-supplied frame specification. The specified frame (we'll
   call this the ``velocity frame'') will be used to look up the velocity
   vector from the CSPICE SPK subsystem.
 
   When the velocity frame is non-inertial and aberration corrections are
   used, the epoch at which the velocity frame is evaluated will be
   adjusted by the one-way light time between the observer and the frame's
   center---just as is done by spkezr_c (see the header of that function
   for details).
 
   The reason the velocity frame specification is crucial is that, (unlike
   rotations) state transformations between non-inertial frames don't
   preserve geometric properties of velocity vectors. Example: compare the
   specific angular momentum vector of a geosynchronous satellite (obtained
   by taking the cross product of the satellite's geocentric position and
   velocity vectors) in both the J2000 frame and in the earth body-fixed
   frame. In the latter frame, the specific angular momentum is zero. A
   valid two-vector frame could be defined using the satellite's position
   and velocity in the J2000 frame, while using the position and velocity
   in the earth body-fixed frame gives rise to a degenerate case for which
   the two-vector frame is undefined.
 
   The observer and target defining the velocity vector are specified by
   name or ID code. The aberration correction may be any value accepted by
   spkezr_c. The velocity frame may be any computable by CSPICE, including
   a dynamic frame, as long as the transformation between the velocity
   frame and the J2000 frame doesn't require multiple levels of simulated
   recursion (see the discussion of recursion in the chapter ``Dynamic
   Frame Implementation Considerations'' below for details).
 
   The frame kernel assignments used to define an observer-target velocity
   vector are:
 
      FRAME_<frame_ID>_<vec_ID>_VECTOR_DEF = 'OBSERVER_TARGET_VELOCITY'
      FRAME_<frame_ID>_<vec_ID>_OBSERVER   = <observer name or ID code>
      FRAME_<frame_ID>_<vec_ID>_TARGET     = <target name or ID code>
      FRAME_<frame_ID>_<vec_ID>_FRAME      = <frame_name>
      FRAME_<frame_ID>_<vec_ID>_ABCORR     = <aberration correction>
 
   In order for a two-vector frame using a velocity vector as part of its
   definition to be computable, kernel data must be loaded that enable
   computation of the velocity vector with respect to both the velocity
   frame and the J2000 frame.
 
   For an example of a two-vector frame definition using an observer-target
   velocity vector, see the subsection titled ``Geocentric Solar Ecliptic
   (GSE) Frame'' in the appendix ``Frame Definition Examples.''
 
 
Constant Vectors
 
   Constant vectors are defined by specifying a reference frame and a
   vector expressed relative to that frame. Optionally, aberration
   corrections may be specified.
 
   The coordinates of a constant vector may be specified in any of the
   rectangular, latitudinal, or RA/DEC (right ascension and declination)
   systems. If the coordinates are angular, the associated angular units
   must be specified; any angular units supported by the CSPICE function
   convrt_c may be used.
 
   All constant vectors require the frame kernel assignments
 
      FRAME_<frame_ID>_<vec_ID>_VECTOR_DEF = 'CONSTANT'
      FRAME_<frame_ID>_<vec_ID>_SPEC       = <coordinate_system>
      FRAME_<frame_ID>_<vec_ID>_FRAME      = <frame_name>
 
   where <coordinate_system> is one of
 
      'RECTANGULAR'
      'LATITUDINAL'
      'RA/DEC'
 
   and the frame is any computable by CSPICE, including a dynamic frame, as
   long as the transformation between the constant vector's frame and the
   J2000 frame doesn't require multiple levels of simulated recursion (see
   the discussion of recursion in the chapter ``Dynamic Frame
   Implementation Considerations'' below for details).
 
   When the coordinate system is rectangular, the vector is specified by
   the frame kernel assignment
 
      FRAME_<frame_ID>_<vec_ID>_SPEC   = 'RECTANGULAR'
      FRAME_<frame_ID>_<vec_ID>_VECTOR = ( <X component>,
                                           <Y component>,
                                           <Z component>  )
 
   When the coordinate system is latitudinal, the vector is specified by
   the frame kernel assignments
 
      FRAME_<frame_ID>_<vec_ID>_SPEC      = 'LATITUDINAL'
      FRAME_<frame_ID>_<vec_ID>_UNITS     = <angular_units>
      FRAME_<frame_ID>_<vec_ID>_LONGITUDE = <longitude>
      FRAME_<frame_ID>_<vec_ID>_LATITUDE  = <latitude>
 
   where <angular_units> designates one of the units supported by the
   CSPICE function convrt_c. The set of supported units includes
 
      'RADIANS'
      'DEGREES'
      'ARCSECONDS'
 
   When the coordinate system is RA/DEC, the vector is specified by the
   frame kernel assignments
 
      FRAME_<frame_ID>_<vec_ID>_SPEC      = 'RA/DEC'
      FRAME_<frame_ID>_<vec_ID>_UNITS     = <angular_units>
      FRAME_<frame_ID>_<vec_ID>_RA        = <RA>
      FRAME_<frame_ID>_<vec_ID>_DEC       = <DEC>
 
   where <angular_units> are as described above.
 
   Aberration corrections are optional for constant vectors. The set of
   available corrections is unique to this application: either light time
   correction or stellar aberration correction may be applied, but both
   cannot be applied together.
 
   Light time corrections adjust the orientation of the constant vector's
   frame for the one-way light time between the center of the frame and a
   specified observer. The application to the frame of light time
   correction is identical to that performed by the CSPICE function
   spkezr_c when it is asked to compute a light-time corrected state
   relative to a non-inertial reference frame. Supported light time
   corrections are any of those supported by spkezr_c that don't include
   stellar aberration correction.
 
   The user may also correct the constant vector for stellar aberration;
   this correction is a function of the constant vector and the velocity of
   an observer relative to the solar system barycenter. A typical
   application would be to correct an inertially referenced star direction
   vector for the stellar aberration induced by motion of an observing
   spacecraft. The supported stellar aberration corrections are
 
      'S'      {correct for stellar aberration, reception case}
      'XS'     {correct for stellar aberration, transmission case}
 
   In the application above, one would correct the apparent observer-star
   direction by selecting the 'S' option. See the discussion in the header
   of the CSPICE function spkezr_c for a description of the ``reception''
   and ``transmission'' aberration correction cases.
 
   When aberration corrections are desired, the observer and the correction
   are specified by the frame kernel assignments
 
      FRAME_<frame_ID>_<vec_ID>_OBSERVER  = <observer name or ID code>
      FRAME_<frame_ID>_<vec_ID>_ABCORR    = <aberration correction>
 
   In order for a two-vector frame using a constant vector as part of its
   definition to be computable, kernel data must be loaded that enable
   computation of the specified vector with respect to both the constant
   vector's frame and the J2000 frame.
 
   For examples of two-vector frame definitions using constant vectors, see
   the subsections titled ``Geocentric Solar Magnetospheric (GSM) Frame''
   and ``Mercury Solar Equatorial (MSEQ) Frame'' in the appendix ``Frame
   Definition Examples.''
 
 
Mean Equator and Equinox of Date Frames
===========================================================================
 
   Mean Equator and Equinox of Date Frames are defined for a solar system
   body (for example, a planet) using mathematical models of the
   orientation of the body's mean equatorial and orbital planes. The term
   ``mean equator'' indicates that orientation of the equatorial plane is
   modeled accounting for precession only. The ``mean equinox'' is the
   intersection of the body's mean orbital plane with the mean equatorial
   plane. The X-axis of such a frame is aligned with the cross product of
   the north-pointing vectors normal to the body's mean equator and mean
   orbital plane of date. The Z-axis is aligned with the first of these
   normal vectors. The Y axis is the cross product of the Z and X axes. The
   resulting reference frame is time-varying; the term ``of date'' means
   this frame is evaluated at a specified epoch.
 
   The mathematical model for a mean equator and equinox of date frame is
   typically called a ``precession model''; CSPICE adopts this usage.
 
   The CSPICE frame subsystem supports mean equator and equinox of date
   frames via precession models built into CSPICE. In principle, for any
   body, a frame kernel definition for a mean equator and equinox of date
   frame identifies which precession model to use for that body. Currently
   CSPICE supports only one precession model: the 1976 IAU precession model
   for the earth.
 
 
Defining a Mean Equator and Equinox of Date Frame in a Frame Kernel
--------------------------------------------------------
 
 
Specifying the Base Frame
 
   The base frame of a mean equator and equinox of date frame is a function
   of the precession model. For the 1976 IAU earth precession model the
   base frame is J2000. This association is made via the assignment:
 
      FRAME_<frame_ID>_RELATIVE           =  'J2000'
 
 
Specifying the Frame Family
 
   A mean equator and equinox of date frame is identified by frame family
   specification:
 
      FRAME_<frame_ID>_FAMILY = 'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
 
 
Specifying the Precession Model
 
   The 1976 IAU precession model is ``selected'' via the assignment:
 
      FRAME_<frame_ID>_PREC_MODEL   = 'EARTH_IAU_1976'
 
 
Specifying a Rotation State or Freeze Epoch
 
   Although mean equator and equinox of date frames are, strictly speaking,
   non-inertial, their time variation may be very slow. In some cases it
   may be desirable to treat them as inertial (specifically, non-rotating),
   perhaps in order to simplify computations or to ensure compatibility
   with computations from another source.
 
   Users can instruct the CSPICE frame subsystem to treat a mean equator
   and equinox of date frame as either inertial or rotating by making a
   ``rotation state'' assignment. Users can also direct the frame subsystem
   to treat a mean equator and equinox of date frame as though it were
   ``frozen'' at a specified epoch. See the section above titled
   ``Conditional Keywords for Parameterized Dynamic Frames'' for
   instructions on how to make these assignments.
 
   Definitions of mean equator and equinox of date frames require either,
   but not both, the rotation state or a freeze epoch to be specified.
 
   For examples of Mean Equator and Equinox of Date frame definitions, see
   the subsection titled ``Earth Mean Equator and Equinox of Date Frames''
   in the appendix ``Frame Definition Examples.''
 
 
True Equator and Equinox of Date Frames
===========================================================================
 
   True Equator and Equinox of Date Frames may be viewed as a refinement of
   mean equator and equinox of date frames. The term ``true equator''
   indicates that orientation of a body's equatorial plane is modeled
   accounting for precession and nutation. The ``true equinox'' is the
   intersection of the body's mean orbital plane with the true equatorial
   plane. The X-axis of such a frame is aligned with the cross product of
   the north-pointing vectors normal to the body's true equator and mean
   orbital plane of date. The Z-axis is aligned with the first of these
   normal vectors. The Y axis is the cross product of the Z and X axes. The
   term ``of date'' means that these axes are evaluated at a specified
   epoch.
 
 
Defining a True Equator and Equinox of Date Frame in a Frame Kernel
--------------------------------------------------------
 
   True Equator and Equinox of date frame definitions are nearly identical
   to those for mean of date frames (see above): the only differences are
   the frame family specification and the addition of an assignment
   identifying the nutation model.
 
 
Specifying the Base Frame
 
   The base frame of a true equator and equinox of date frame is a function
   of the precession model. For the 1976 IAU earth precession model the
   base frame is J2000. This association is made via the assignment:
 
      FRAME_<frame_ID>_RELATIVE           =  'J2000'
 
 
Specifying the Frame Family
 
   A true equator and equinox of date frame is identified by frame family
   specification:
 
      FRAME_<frame_ID>_FAMILY = 'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
 
 
Specifying the Precession Model
 
   Currently CSPICE supports only one precession model: the 1976 IAU
   precession model for the earth.
 
   The 1976 IAU precession model is ``selected'' via the assignment:
 
      FRAME_<frame_ID>_PREC_MODEL   = 'EARTH_IAU_1976'
 
 
Specifying the Nutation Model
 
   The choice of nutation model is specified by the assignment:
 
      FRAME_<frame_ID>_NUT_MODEL     = <nutation_model>
 
   Currently the only available nutation model is the 1980 IAU nutation
   model for the earth. An assignment specifying this model has the form:
 
      FRAME_<frame_ID>_NUT_MODEL     = 'EARTH_IAU_1980'
 
 
Specifying a Rotation State or Freeze Epoch
 
   Although true equator and equinox of date frames are, strictly speaking,
   non-inertial, their time variation may be very slow. In some cases it
   may be desirable to treat them as inertial (specifically, non-rotating),
   perhaps in order to simplify computations or to ensure compatibility
   with computations from another source.
 
   Users can instruct the CSPICE frame subsystem to treat a true equator
   and equinox of date frame as either inertial or rotating by making a
   ``rotation state'' assignment. Users can also direct the frame subsystem
   to treat a true equator and equinox of date frame as though it were
   ``frozen'' at a specified epoch. See the section above titled
   ``Conditional Keywords for Parameterized Dynamic Frames'' for
   instructions on how to make these assignments.
 
   Definitions of true equator and equinox of date frames require either,
   but not both, the rotation state or a freeze epoch to be specified.
 
   For examples of True Equator and Equinox of Date frame definitions, see
   the subsection titled ``Earth True Equator and Equinox of Date Frames''
   in the appendix ``Frame Definition Examples.''
 
 
Mean Ecliptic and Equinox of Date Frames
===========================================================================
 
   Mean Ecliptic and Equinox of Date Frames are closely related to mean
   equator and equinox of date frames: for a given body, the former is
   obtained by rotating the latter about the X-axis by the mean obliquity
   of date.
 
   The term ``mean equator'' indicates that orientation of a body's
   equatorial plane is modeled accounting for precession. The ``mean
   equinox'' is the intersection of the body's mean orbital plane with the
   mean equatorial plane. The X-axis of such a frame is aligned with the
   cross product of the north-pointing vectors normal to the body's mean
   equator and mean orbital plane of date. The Z-axis is aligned with the
   second of these normal vectors. The Y axis is the cross product of the Z
   and X axes. The term ``of date'' means that these axes are evaluated at
   a specified epoch.
 
 
Defining a Mean Ecliptic and Equinox of Date Frame in a Frame Kernel
--------------------------------------------------------
 
   Mean Ecliptic and Equinox of date frame definitions are nearly identical
   to those for mean of date frames (see above): the only differences are
   the frame family specification and the addition of an assignment
   identifying the mean obliquity model.
 
 
Specifying the Base Frame
 
   The base frame of a mean ecliptic and equinox of date frame is a
   function of the precession model. For the 1976 IAU earth precession
   model the base frame is J2000. This association is made via the
   assignment:
 
      FRAME_<frame_ID>_RELATIVE           =  'J2000'
 
 
Specifying the Frame Family
 
   A mean ecliptic and equinox of date frame is identified by frame family
   specification:
 
      FRAME_<frame_ID>_FAMILY = 'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
 
 
Specifying the Precession Model
 
   Currently CSPICE supports only one precession model: the 1976 IAU
   precession model for the earth.
 
   The 1976 IAU precession model is ``selected'' via the assignment:
 
      FRAME_<frame_ID>_PREC_MODEL   = 'EARTH_IAU_1976'
 
 
Specifying the Mean Obliquity Model
 
   The choice of mean obliquity model is specified by the assignment:
 
      FRAME_<frame_ID>_OBLIQ_MODEL     = <obliquity_model>
 
   Currently the only available mean obliquity model is the 1980 IAU
   obliquity model for the earth. An assignment specifying this model has
   the form:
 
      FRAME_<frame_ID>_OBLIQ_MODEL     = 'EARTH_IAU_1980'
 
 
Specifying a Rotation State or Freeze Epoch
 
   Although mean ecliptic and equinox of date frames are, strictly
   speaking, non-inertial, their time variation may be very slow. In some
   cases it may be desirable to treat them as inertial (specifically,
   non-rotating), perhaps in order to simplify computations or to ensure
   compatibility with computations from another source.
 
   Users can instruct the CSPICE frame subsystem to treat a mean ecliptic
   and equinox of date frame as either inertial or rotating by making a
   ``rotation state'' assignment. Users can also direct the frame subsystem
   to treat a mean ecliptic and equinox of date frame as though it were
   ``frozen'' at a specified epoch. See the section above titled
   ``Conditional Keywords for Parameterized Dynamic Frames'' for
   instructions on how to make these assignments.
 
   Definitions of mean ecliptic and equinox of date frames require either,
   but not both, the rotation state or a freeze epoch to be specified.
 
   For examples of Mean Ecliptic and Equinox of Date frame definitions, see
   the subsection titled ``Earth Mean Ecliptic and Equinox of Date Frames''
   in the appendix ``Frame Definition Examples.''
 
 
Euler Frames
===========================================================================
 
   An Euler frame is defined by a sequence of rotation axes and
   corresponding time-dependent Euler angles. Each angle is defined by a
   set of polynomial coefficients. A reference epoch must be provided in
   the frame definition; the independent variable of each polynomial
   represents ephemeris seconds past the J2000 TDB epoch.
 
   The rotation defined by the Euler angles maps position vectors via left
   multiplication from the defined Euler reference frame to the base frame:
 
      V           = r(t) * V
       base_frame           Euler_frame
 
   This rotation can be considered to be a time-dependent matrix
 
      r(t)
 
   where r(t) represents the composition of the rotations defined by the
   input angle-axis pairs. Naming the axis indices and angles of the Euler
   angle sequence
 
      axindx_i, angle_i,  i = 1, 2, 3
 
   r(t) is
 
      r(t) = [ angle_1(t) ]      [ angle_2(t) ]      [ angle_3(t) ]
                        axindx_1            axindx_2            axindx_3
 
   The axis indices axindx_i, for i = 1, 2, 3, are in the set { 1, 2, 3 };
   axindx_2 cannot equal axindx_1 or axindx_3. For example, we could have
 
      axindx_1 = 3
      axindx_2 = 1
      axindx_3 = 3
 
   Here the notation
 
      [ A ]
           j
 
   stands for a frame rotation by the angle A radians about the jth axis of
   a right-handed frame, where we assign the axes {X, Y, Z} the indices {1,
   2, 3} respectively:
 
      +-                     -+
      |   1       0      0    |
      |   0     cos A   sin A |   =  [ A ]
      |   0    -sin A   cos A |           1
      +-                     -+
 
      +-                     -+
      |  cos A    0    -sin A |
      |   0       1      0    |   =  [ A ]
      |  sin A    0     cos A |           2
      +-                     -+
 
      +-                     -+
      |  cos A   sin A   0    |
      | -sin A   cos A   0    |   =  [ A ]
      |   0       0      1    |           3
      +-                     -+
 
   The base frame can be constructed from the Euler frame via a sequence of
   Euler angle rotations as follows:
 
       1.   Rotate the axes of the Euler frame by angle_3 about the axis
            indexed by axindx_3.
 
       2.   Rotate the axes of the frame resulting from the first rotation
            by angle_2 about the axis indexed by axindx_2.
 
       3.   Rotate the axes of the frame resulting from the second rotation
            by angle_1 about the axis indexed by axindx_1.
 
   The resulting set of axes are those of the base frame.
 
   The rotation angles are defined as follows: letting t0 represent the
   reference epoch, and letting
 
      c   ,  i = 1, 2, 3;   j = 0, ... , ni
       i,j
 
   be the polynomial coefficients for the ith angle, we have
 
                                                           n1
      angle_1(t) = c   + c   * (t-t0) + ... + c    * (t-t0)
                    1,0   1,1                  1,n1
 
                                                           n2
      angle_2(t) = c   + c   * (t-t0) + ... + c    * (t-t0)
                    2,0   2,1                  2,n2
 
                                                           n3
      angle_3(t) = c   + c   * (t-t0) + ... + c    * (t-t0)
                    3,0   3,1                  3,n3
 
   See the Rotation Required Reading, rotation.req, or the header of the
   CSPICE function eul2m_c for details concerning definition of rotations
   via Euler angles. Note however that the referenced document and source
   code use a different convention for labeling Euler angles and their
   rotation axes: here the elements of the rotation sequence are numbered
   left to right; in those documents the order is that in which rotations
   are performed, namely right to left.
 
 
Defining an Euler Frame in a Frame Kernel
--------------------------------------------------------
 
 
Specifying the Base Frame
 
   The base frame of an Euler frame specified via the assignment:
 
      FRAME_<frame_ID>_RELATIVE           =  '<frame_name>'
 
 
Specifying the Frame Family
 
   An Euler frame is identified by frame family specification:
 
      FRAME_<frame_ID>_FAMILY = 'EULER'
 
 
Specifying the Epoch
 
   The zero epoch for the independent variable of the polynomials is
   defined using the SPICE text kernel calendar ephemeris time syntax. A
   sample template is shown below:
 
      FRAME_<frame_ID>_EPOCH           =  @YYYY-MON-DD/HR:MN.SEC.###
 
   A concrete example is:
 
      FRAME_<frame_ID>_EPOCH           =  @2000-JAN-1/12:00:00.000
 
   The calendar time string is assumed to represent a TDB epoch.
 
   See the discussion in the section ``Freeze Epoch'' above or the Kernel
   Required Reading, kernel.req, for further information.
 
 
Specifying the Euler Angles
 
   Euler angles are specified by an axis sequence, a set of polynomial
   coefficients, and associated units. The axes are specified by an
   assignment of the form:
 
      FRAME_<frame_ID>_AXES            =  ( <index of axis 1>
                                            <index of axis 2>
                                            <index of axis 3> )
 
   The axis indices must be taken from the set
 
      { 1, 2, 3 }
 
   and the middle value must differ from its neighbors. The first integer
   listed is the axis index for angle 1, the second for angle 2, and the
   last for angle 3, where the role of the angles is as shown in the
   equation for r(t) above.
 
   Let n1, n2, and n3 represent the maximum degrees of the polynomials for
   angles 1, 2, and 3 respectively. Then the polynomial coefficients are
   defined by the assignments
 
      FRAME_<frame_ID>_ANGLE_1_COEFFS = ( <order 0 coefficient>
                                          <order 1 coefficient>
                                                 ...
                                          <order n1 coefficient>  )
 
      FRAME_<frame_ID>_ANGLE_2_COEFFS = ( <order 0 coefficient>
                                          <order 1 coefficient>
                                                 ...
                                          <order n2 coefficient>  )
 
      FRAME_<frame_ID>_ANGLE_3_COEFFS = ( <order 0 coefficient>
                                          <order 1 coefficient>
                                                 ...
                                          <order n3 coefficient>  )
 
   Angular units are specified by the frame kernel assignment
 
      FRAME_<frame_ID>_UNITS     = <angular_units>
 
   where <angular_units> designates one of the units supported by the
   CSPICE function convrt_c. The set of supported units includes
 
      'RADIANS'
      'DEGREES'
      'ARCSECONDS'
 
   For an example of an Euler frame definition, see the subsection titled
   ``Euler Frames'' in the appendix ``Frame Definition Examples.''
 
 
Dynamic Frame Implementation Considerations
===========================================================================
 
 
Introduction
--------------------------------------------------------
 
   This chapter discusses issues affecting implementation of dynamic
   frames:
 
       --   Simulated Recursion
 
       --   Frame Derivative Accuracy
 
       --   Degenerate Geometry
 
       --   Efficiency
 
   By necessity, this chapter presents some aspects of the implementation
   of the CSPICE parameterized dynamic frame subsystem. The implementation
   described here is not considered part of the CSPICE API specification.
   Although unlikely, this implementation could be changed in a future
   version of the CSPICE Toolkit.
 
 
Simulated Recursion
--------------------------------------------------------
 
 
The Need for Recursion in the CSPICE Frame Subsystem
 
   In the following discussion, we'll use the graph notation below to
   indicate that function A calls function B:
 
      A -> B
 
   A function R_0 is ``recursive'' if it calls itself
 
      R_0 -> R_0
 
   or if some sequence of calls initiated in the function R_0 results in a
   call to R_0:
 
      R_0 -> R_1-> ... -> R_0
 
   ANSI standard Fortran 77 doesn't permit recursive calls. However, the
   implementation of two-vector frames requires sequences of calls that at
   face value are recursive. For example, to look up a state vector in the
   GSE frame (see the appendix ``Frame Definition Examples''), the function
   SPKEZ must initiate the sequence of calls (ellipses indicate omitted
   portions of the call graph)
 
      SPKEZ -> ... -> FRMGET -> ... -> SPKEZ -> ... -> FRMGET
 
   Both SPKEZ and FRMGET are called recursively in this graph.
 
   This issue affects not only SPICELIB but CSPICE and Icy as well because
   these products rely on the SPICELIB (Fortran) implementation of the
   frame subsystem.
 
 
Implementation of Limited Simulated Recursion
 
   SPICELIB solves the recursion problem by providing renamed duplicates of
   routines that must be called recursively. For example, the invalid call
   graph
 
      SPKEZ -> ... -> FRMGET -> ... -> SPKEZ -> ... -> FRMGET
 
   is implemented in (valid) ANSI standard Fortran 77 using the call graph
 
      SPKEZ -> ... -> FRMGET -> ... -> ZZSPKEZ0 -> ... -> ZZFRMGT0
 
   To a limited extent, two levels of simulated recursion are supported in
   the frame subsystem, so call graphs of the form
 
      SPKEZ -> ... -> FRMGET    -> ... -> ZZSPKEZ0    -> ... -> ZZFRMGT0
            -> ... -> ZZSPKEZ1  -> ... -> ZZFRMGT1
 
   are possible.
 
   For brevity, when we refer to recursion in the following discussion,
   we'll omit the qualifier ``simulated.''
 
 
Limits on Recursion in Frame Definitions
 
   We say a reference frame is ``evaluated'' when the transformation from
   the frame to its base frame is computed for some epoch. A parameterized
   dynamic frame normally is evaluated each time it is referenced in a
   function call. For example, the calls
 
      spkezr_c ( moon, et, "GSE", "NONE", "EARTH", state, &lt );
 
      sxform_c ( "GSE", "J2000", et, xform );
 
   both cause the GSE parameterized dynamic frame to be evaluated at ET.
 
   When the definition of a parameterized dynamic frame F1 refers to a
   second frame F2 as
 
       --   the base frame
 
       --   the frame relative to which a constant vector is specified
 
       --   the frame relative to which a velocity vector is specified
 
   the referenced frame F2 may be dynamic, but F2 must not make reference
   to any dynamic frame. If deeper recursion is required to evaluate the
   referenced frame F2, an error will occur at run time.
 
   If F2 is not dynamic but its evaluation requires evaluation of a dynamic
   frame F3, the same restrictions apply to F3.
 
   When a dynamic frame is used as a base frame in either an SPK or CK
   segment, evaluation of data from that segment may result in a call to
   the dynamic frame subsystem. That call may result in lookup of another
   segment whose base frame is dynamic, and so on: the original kernel
   lookup could easily exhaust the dynamic frame subsystem's ability to
   handle recursive calls.
 
   Clearly use of dynamic frames in SPK and CK files requires caution.
   However, there are some ``reasonable'' applications that call for
   dynamic base frames in kernels, for example: representing ephemerides of
   earth orbiters expressed relative to the earth true equator and equinox
   of date frame.
 
 
Frame Derivative Accuracy
--------------------------------------------------------
 
   Transformation of state vectors between frames F1 and F2 via a
   time-dependent rotation R(t) requires the derivative with respect to
   time of R(t): d(R(t))/dt. The accuracy of the velocity portion of a
   transformed state is limited by the accuracy of d(R(t))/dt. When either
   frame F1 or F2 is dynamic, loss of accuracy in d(R(t))/dt can occur for
   a number of reasons, including but not limited to:
 
       --   R(t) depends on CK data. Often angular rates in C-kernels have
            low accuracy. (This issue applies to non-dynamic frames as
            well.)
 
       --   R(t) is defined via a two-vector frame using position vectors,
            and the velocities associated with those vectors have low
            accuracy. This can happen for SPK data types for which position
            and velocity are represented independently, for example SPK
            types 3 or 9.
 
       --   R(t) is defined via a two-vector frame using
            aberration-corrected position vectors. Even if the geometric
            velocities of the vectors are accurate, the
            aberration-corrected velocities associated with those vectors
            will probably have low accuracy due to accuracy limitations of
            the aberration corrections applied to velocity vectors by the
            SPK subsystem.
 
       --   R(t) is defined via a two-vector frame using a velocity vector.
            The acceleration associated with the velocity vector is
            required to compute d(R(t))/dt, and this acceleration must be
            computed numerically. The results are likely to have at best
            single precision validity.
 
 
Degenerate Geometry
--------------------------------------------------------
 
   Two-vector frame definitions can suffer from singularities: the defining
   vectors may, in some cases, become extremely close to parallel. In such
   cases the frame evaluation may generate meaningless results.
 
   Because two-vector frame definitions may be perfectly valid for some
   epochs and yield degenerate geometry for others, testing can easily fail
   to reveal problems with these definitions. Careful frame design is the
   best defense.
 
   As a backup measure, setting the angular separation tolerance in
   two-vector frame definitions can enable the frame subsystem to diagnose
   at run time degenerate or near-degenerate geometry. See the section
   ``Specifying the Angular Separation Tolerance'' above for details.
 
 
Efficiency Concerns
--------------------------------------------------------
 
   In many cases, when recursion is required by a frame evaluation, that
   evaluation requires a relatively large amount of computation. For
   example, when an SPK call results in a two-vector frame evaluation,
   several additional SPK calls may be required to support the original
   call. The original call may be many times slower than a call requiring
   only non-dynamic frame evaluation.
 
   To minimize the performance degradation imposed by recursion, avoid
   unnecessary references to dynamic frames in frame definitions. When
   possible, use J2000 or another inertial frame as the base frame, or as
   the frame relative to which constant or velocity vectors are defined.
   When it is not possible to use an inertial frame, prefer non-dynamic,
   non-inertial frames to dynamic frames.
 
 
Appendix. ``Built in'' Inertial Reference Frames
===========================================================================
 
 
Complete List of ``Built in'' Inertial Reference Frames
 
   SPICE software includes the definitions of several inertial reference
   frames. The numeric IDs and names of the inertial frames defined in
   SPICE software are:
 
         ID  Name        Description
      -----  --------    -------------------------------------------
          1  J2000       Earth mean equator, dynamical equinox of J2000.
                         The root reference frame for SPICE.
 
          2  B1950       Earth mean equator, dynamical equinox of B1950.
                         The B1950 reference frame is obtained by
                         precessing the J2000 frame backwards from
                         Julian year 2000 to Besselian year 1950, using
                         the 1976 IAU precession model.
 
                         The rotation from B1950 to J2000 is
 
                         [ -z ]  [ theta ]  [ -zeta ]
                                3          2          3
                         The values for z, theta, and zeta are taken
                         directly from formulas given in table 5 of [5].
 
                         z     =  1153.04066200330"
                         theta = -1002.26108439117"
                         zeta  =  1152.84248596724"
 
          3  FK4         Fundamental Catalog (4). The FK4 reference
                         frame is derived from the B1950 frame by
                         applying the equinox offset determined by
                         Fricke.
 
                         [ 0.525" ]
                                   3
 
          4  DE-118      JPL Developmental Ephemeris (118). The DE-118
                         reference frame is nearly identical to the FK4
                         frame. It is also derived from the B1950 frame.
                         Only the offset is different
 
                         [ 0.53155" ]
                                     3
 
                         In [2], Standish uses two separate rotations,
 
                         [ 0.00073" ]  P [ 0.5316" ]
                                      3              3
 
                         (where P is the precession matrix used above to
                         define the B1950 frame). The major effect of the
                         second rotation is to correct for truncating the
                         magnitude of the first rotation. At his
                         suggestion, we will use the untruncated value,
                         and stick to a single rotation.
 
 
                         Most of the other DE historical reference frames
                         are defined relative to either the DE-118 or
                         B1950 frame. The values below are taken
                         from [4].
 
                         DE number  Offset from DE-118  Offset from B1950
                         ---------  ------------------  -----------------
                                96            +0.1209"           +0.4107"
                               102            +0.3956"           +0.1359"
                               108            +0.0541"           +0.4775"
                               111            -0.0564"           +0.5880"
                               114            -0.0213"           +0.5529"
                               122            +0.0000"           +0.5316"
                               125            -0.0438"           +0.5754"
                               130            +0.0069"           +0.5247"
 
          5  DE-96       JPL Developmental Ephemeris ( 96).
 
          6  DE-102      JPL Developmental Ephemeris (102).
 
          7  DE-108      JPL Developmental Ephemeris (108).
 
          8  DE-111      JPL Developmental Ephemeris (111).
 
          9  DE-114      JPL Developmental Ephemeris (114).
 
         10  DE-122      JPL Developmental Ephemeris (122).
 
         11  DE-125      JPL Developmental Ephemeris (125).
 
         12  DE-130      JPL Developmental Ephemeris (130).
 
         13  GALACTIC    Galactic System II. The Galactic System II
                         reference frame is defined by the following
                         rotations:
                              o          o            o
                         [ 327  ]  [ 62.6  ]  [ 282.25  ]
                                 3          1            3
 
                         In the absence of better information, we
                         assume the rotations are relative to the
                         FK4 frame.
 
         14  DE-200      JPL Developmental Ephemeris (200).
 
         15  DE-202      JPL Developmental Ephemeris (202).
 
         16  MARSIAU     Mars Mean Equator and IAU vector of
                         J2000. The IAU-vector at Mars is the point
                         on the mean equator of Mars where the equator
                         ascends through the earth mean equator.
                         This vector is the cross product of Earth
                         mean north with Mars mean north.
 
         17  ECLIPJ2000  Ecliptic coordinates based upon the
                         J2000 frame.
 
                         The value for the obliquity of the
                         ecliptic at J2000 is taken from page 114
                         of [7] equation 3.222-1. This agrees with the
                         expression given in [5].
 
         18  ECLIPB1950  Ecliptic coordinates based upon the B1950
                         frame.
 
                         The value for the obliquity of the ecliptic at
                         B1950 is taken from page 171 of [7].
 
         19  DE-140      JPL Developmental Ephemeris. (140)
                         The DE-140 frame is the DE-400 frame rotated:
 
              0.9999256765384668  0.0111817701197967  0.0048589521583895
             -0.0111817701797229  0.9999374816848701 -0.0000271545195858
             -0.0048589520204830 -0.0000271791849815  0.9999881948535965
 
                         The DE-400 frame is treated as equivalent to
                         the J2000 frame.
 
         20  DE-142      JPL Developmental Ephemeris. (142)
                         The DE-142 frame is the DE-402 frame rotated:
 
              0.9999256765402605  0.0111817697320531  0.0048589526815484
             -0.0111817697907755  0.9999374816892126 -0.0000271547693170
             -0.0048589525464121 -0.0000271789392288  0.9999881948510477
 
                         The DE-402 frame is treated as equivalent to
                         the J2000 frame.
 
         21  DE-143      JPL Developmental Ephemeris. (143)
                         The DE-143 frame is the DE-403 frame rotated:
 
              0.9999256765435852  0.0111817743077255  0.0048589414674762
             -0.0111817743300355  0.9999374816382505 -0.0000271622115251
             -0.0048589414161348 -0.0000271713942366  0.9999881949053349
 
                         The DE-403 frame is treated as equivalent to
                         the J2000 frame.
 
 
Inertial Reference Frame References
 
      [1] Jay Lieske, ``Precession Matrix Based on IAU (1976)
          System of Astronomical Constants,'' Astron. Astrophys.
          73, 282-284 (1979).
 
      [2] E.M. Standish, Jr., ``Orientation of the JPL Ephemerides,
          DE 200/LE 200, to the Dynamical Equinox of J2000,''
          Astron. Astrophys. 114, 297-302 (1982).
 
      [3] E.M. Standish, Jr., ``Conversion of Ephemeris Coordinates
          from the B1950 System to the J2000 System,'' JPL IOM
          314.6-581, 24 June 1985.
 
      [4] E.M. Standish, Jr., ``The Equinox Offsets of the JPL
          Ephemeris,'' JPL IOM 314.6-929, 26 February 1988.
 
      [5] Jay Lieske, ``Expressions for the Precession  Quantities
          Based upon the IAU (1976) System of Astronomical
          Constants'' Astron. Astrophys. 58, 1-16 (1977).
 
      [6] Laura Bass and Robert Cesarone "Mars Observer Planetary
          Constants and Models" JPL D-3444 November 1990.
 
      [7] "Explanatory Supplement to the Astronomical Almanac"
           edited by P. Kenneth Seidelmann. University Science
           Books, 20 Edgehill Road, Mill Valley, CA 94941 (1992)
 
 
Low Level Inertial Reference Frame Functions
 
   You may obtain the rotation between any two ``built in'' inertial frames
   using the CSPICE function irfrot_c and supplying the IDs for the frames
   of interest. The module header for irfrot_c, and this document, always
   contain the definitive list of recognized frames.
 
   This example shows how to rotate a position vector from FK4 coordinates
   to J2000 coordinates (the ID for the FK4 frame is 3, the ID for the
   J2000 frame is 1);
 
         SpiceInt from = 3;
         SpiceInt to   = 1;
 
         irfrot_ ( &from, &to, rot );
         mxv_c   ( rot  , old, new );
 
   (`rot' is a 3-by-3 matrix, `old' and `new' are 3-vectors; subroutine
   mxv_c multiplies a matrix and a vector to produce a vector.)
 
   Two additional subroutines can be used to convert a frame name to ID and
   vice versa. This example shows how to find the index of the DE-125
   frame:
 
         irfnum_ ( "DE-125", frid, strlen("DE-125") );
 
   This example shows how to find the name corresponding to ID 11:
 
         SpiceInt frid = 11;
 
         irfnam_ ( &fride, frname, strlen(frname) );
 
 
Appendix. ``Built in'' PCK-Based IAU Body-Fixed Reference Frames
===========================================================================
 
   SPICE software includes the definitions of body-fixed frames for all
   natural bodies -- planets, satellites, and some asteroids -- listed in
   International Astronomical Union (IAU) reports on cartographic
   constants. These frames are fixed to and do not move with respect to
   ``surface'' features of a natural object, but they do move with respect
   to inertial frames as the object rotates. The complete list of
   body-fixed frames ``built into'' SPICE is given below. Each name is
   constructed by adding the prefix ``IAU_'' to the name of the body. The
   prefix ``IAU_'' indicates that the orientation of this frame is
   typically determined from the IAU model for the body in question. The
   constants associated with this model are stored in one or more text PCK
   files, which, therefore, must be loaded in order for orientation of
   these frames to be computed.
 
      IAU_ADRASTEA
      IAU_AMALTHEA
      IAU_ANANKE
      IAU_ARIEL
      IAU_ATLAS
      IAU_BELINDA
      IAU_BENNU
      IAU_BIANCA
      IAU_BORRELLY
      IAU_CALLIRRHOE
      IAU_CALLISTO
      IAU_CALYPSO
      IAU_CARME
      IAU_CERES
      IAU_CHALDENE
      IAU_CHARON
      IAU_CORDELIA
      IAU_CRESSIDA
      IAU_DAVIDA
      IAU_DEIMOS
      IAU_DESDEMONA
      IAU_DESPINA
      IAU_DIONE
      IAU_EARTH
      IAU_ELARA
      IAU_ENCELADUS
      IAU_EPIMETHEUS
      IAU_ERINOME
      IAU_EROS
      IAU_EUROPA
      IAU_GALATEA
      IAU_GANYMEDE
      IAU_GASPRA
      IAU_HARPALYKE
      IAU_HELENE
      IAU_HIMALIA
      IAU_HYPERION
      IAU_IAPETUS
      IAU_IDA
      IAU_IO
      IAU_IOCASTE
      IAU_ISONOE
      IAU_ITOKAWA
      IAU_JANUS
      IAU_JULIET
      IAU_JUPITER
      IAU_KALYKE
      IAU_LARISSA
      IAU_LEDA
      IAU_LUTETIA
      IAU_LYSITHEA
      IAU_MAGACLITE
      IAU_MARS
      IAU_MERCURY
      IAU_METIS
      IAU_MIMAS
      IAU_MIRANDA
      IAU_MOON
      IAU_NAIAD
      IAU_NEPTUNE
      IAU_NEREID
      IAU_OBERON
      IAU_OPHELIA
      IAU_PALLAS
      IAU_PAN
      IAU_PANDORA
      IAU_PASIPHAE
      IAU_PHOBOS
      IAU_PHOEBE
      IAU_PLUTO
      IAU_PORTIA
      IAU_PRAXIDIKE
      IAU_PROMETHEUS
      IAU_PROTEUS
      IAU_PUCK
      IAU_RHEA
      IAU_ROSALIND
      IAU_SATURN
      IAU_SINOPE
      IAU_STEINS
      IAU_SUN
      IAU_TAYGETE
      IAU_TELESTO
      IAU_TEMPEL_1
      IAU_TETHYS
      IAU_THALASSA
      IAU_THEBE
      IAU_THEMISTO
      IAU_TITAN
      IAU_TITANIA
      IAU_TRITON
      IAU_UMBRIEL
      IAU_URANUS
      IAU_VENUS
      IAU_VESTA
 
 
Appendix. High Precision Earth Fixed Frames
===========================================================================
 
   In addition to the text PCK based IAU body-fixed frame for Earth,
   'IAU_EARTH', these two body-fixed frames for Earth are also ``built
   into'' the SPICE system:
 
      ITRF93
      EARTH_FIXED
 
   'ITRF93' is a frame ``fixed'' to the Earth's crust. It provides a high
   precision model for the orientation of the Earth with respect to J2000.
   In SPICE this is also a PCK type frame but its orientation is provided
   in a binary PCK file.
 
   'EARTH_FIXED' is a ``generic frame'' that gives the orientation of the
   Earth with respect to some other frame (usually 'IAU_EARTH' or 'ITRF93')
   via a constant rotational offset. Such frames are called Text Kernel
   (TK) frames. See the subsection `` Gaining Flexibility via TK Frames''
   for a discussion of the use of TK frames.
 
 
Appendix. Frame Identifiers Reserved for Earth Fixed Frames
===========================================================================
 
   NAIF has set aside a range of frame ID codes for Earth fixed frames to
   be added in the future when/if additional high precision Earth
   orientation model become available and are implemented in SPICE. This
   reserved range is from 13000 to 13999. The ID assigned to 'ITRF93',
   which is only currently implemented frame of this kind, is 13000. All of
   these frames are PCK based frames. They model the orientation of the
   Earth with respect to an inertial reference frame such as the J2000
   frame. Since the primary customer of these frames is NASA's Deep Space
   Network (DSN), we shall refer to any frame with ID code in this reserved
   range as a DSN Earth Fixed frame or simply DSN frame.
 
   The class ID to associate with any DSN frame is the frame ID minus
   10000. For example, the class ID associated with frame 13003 is 3003. It
   is this class ID that should be placed in the PCK file that implements
   the relationship between the DSN frame and the corresponding inertial
   frame.
 
   The center of any DSN frame is the center of mass of the Earth, which
   has SPK ID code 399.
 
   These frames are partially ``built in''. Given a frame ID in the range
   from 13001 to 13999, the frame subsystem ``knows'' that the frame is a
   PCK frame, the center of the frame is 399 and the class ID of the frame
   is the frame ID - 10000. This knowledge cannot be overridden. However,
   the frame subsystem does not ``know'' the relationship between the names
   of these frames and their ID codes. The relationship must be specified
   via the appropriate kernel pool frame definition.
 
         FRAME_<name>              = <DSN Frame-ID>
         FRAME_<DSN Frame-ID>_NAME = '<name>'
         OBJECT_EARTH_FRAME        = <DSN Frame-ID>
 
   Note that this specification leaves out the items below
 
      FRAME_<DSN Frame-ID>_CENTER   = 399
      FRAME_<DSN Frame-ID>_CLASS    = 2
      FRAME_<DSN Frame-ID>_CLASS_ID = <DSN Frame-ID  - 10000>
 
   You may supply these values if you like, but they have no effect on the
   frame subsystem's recognition and interpretation of the frame with the
   specified frame ID.
 
 
Appendix. Frame Definition Examples
===========================================================================
 
   Below are examples that you can modify to create frame specifications
   for similar situations.
 
 
Inertial Frame
 
   Inertial frames can be defined only as aliases for the ``built in''
   inertial frames. This example defines EME50 to be the same frame as
   B1950.
 
      \begindata
 
         FRAME_EME50            = 2000000
         FRAME_2000000_NAME     = 'EME50'
         FRAME_2000000_CLASS    =  1
         FRAME_2000000_CLASS_ID =  2
         FRAME_2000000_CENTER   =  0
 
      \begintext
 
 
PCK Frame
 
   This definition shows how you create a frame definition for the asteroid
   Eros. Note we also define which frame is associated with the asteroid
   Eros.
 
      \begindata
 
         FRAME_EROS_FIXED       =  2000433
         FRAME_2000433_NAME     = 'EROS_FIXED'
         FRAME_2000433_CLASS    =  2
         FRAME_2000433_CLASS_ID =  2000433
         FRAME_2000433_CENTER   =  2000433
 
         OBJECT_2000433_FRAME   = 'EROS_FIXED'
 
      \begintext
 
 
CK Frames
 
   This definition shows how you create a frame definition for the MGS
   spacecraft. Note this frame definition includes the appropriate SCLK
   definition as well as which frame to attach to the MGS spacecraft.
 
      \begindata
 
         FRAME_MGS_SPACECRAFT   = -94000
         FRAME_-94000_NAME      = 'MGS_SPACECRAFT'
         FRAME_-94000_CLASS     =  3
         FRAME_-94000_CLASS_ID  = -94000
         FRAME_-94000_CENTER    = -94
 
         CK_-94000_SCLK         = -94
         CK_-94000_SPK          = -94
 
         OBJECT_-94_FRAME       = 'MGS_SPACECRAFT'
 
      \begintext
 
 
TK frame --- Alias
 
   This example shows how you can make up an alias for a frame using a TK
   frame. Note we make the reference frame to associate with Mars the
   MARS_FIXED frame.
 
      \begindata
 
         FRAME_MARS_FIXED       =  1400499
         FRAME_1400499_NAME     = 'MARS_FIXED'
         FRAME_1400499_CLASS    =  4
         FRAME_1400499_CLASS_ID =  1400499
         FRAME_1400499_CENTER   =  499
 
         OBJECT_499_FRAME       = 'MARS_FIXED'
 
      \begintext
 
      To make this point to another frame just replace
      'IAU_MARS' below with the name of that frame.
 
      \begindata
 
         TKFRAME_1400499_RELATIVE = 'IAU_MARS'
         TKFRAME_1400499_SPEC     = 'MATRIX'
         TKFRAME_1400499_MATRIX   = ( 1   0   0
                                      0   1   0
                                      0   0   1 )
      \begintext
 
 
TK frame --- Topographic
 
   This example shows how you could create a topographic frame for the DSN
   Station DSS-17.
 
      \begindata
 
         FRAME_DSS-17_TOPO      =  1399017
         FRAME_1399017_NAME     = 'DSS-17_TOPO'
         FRAME_1399017_CLASS    =  4
         FRAME_1399017_CLASS_ID =  1399017
         FRAME_1399017_CENTER   =   399017
 
         OBJECT_399017_FRAME    = 'DSS-17_TOPO'
 
      \begintext
 
      Note that the geodetic longitude and co-latitude of the DSS-17
      tracking station are: 243.126496675 and 54.657822839 respectively.
 
      \begindata
 
         TKFRAME_DSS-17_TOPO_RELATIVE = 'EARTH_FIXED'
         TKFRAME_DSS-17_TOPO_SPEC     = 'ANGLES'
         TKFRAME_DSS-17_TOPO_UNITS    = 'DEGREES'
         TKFRAME_DSS-17_TOPO_AXES     = ( 3, 2, 3 )
         TKFRAME_DSS-17_TOPO_ANGLES   = ( -243.126496675,
                                           -54.657822839,
                                           180.0 )
      \begintext
 
      Recall that the frame `EARTH_FIXED' is a TK frame. As a result
      its relationship to other frames must be specified via
      a kernel pool variable. We make that specification here.
 
      If the ITRF93 PCK kernel is not available we can simply rename the
      "RELATIVE" frame to be IAU_EARTH and still have the topocentric
      frame well defined.
 
      \begindata
 
         TKFRAME_EARTH_FIXED_RELATIVE = 'ITRF93'
         TKFRAME_EARTH_FIXED_SPEC     = 'MATRIX'
         TKFRAME_EARTH_FIXED_MATRIX   = ( 1   0   0
                                          0   1   0
                                          0   0   1 )
 
      \begintext
 
 
TK frame --- Instrument
 
   This example shows how you could create a TK frame for the Medium
   Resolution Imager (MRI) instrument on-board the Deep Impact Flyby (DIF)
   spacecraft.
 
   The rotation from the DIF spacecraft frame to the MRI instrument frame
   determined from the in-flight calibration data can be represented by the
   following rotation angles:
 
       mri
      M    = |0.129539306414|  * |-45.006884881185|  * |0.004898709285|
       sc                    Z                     Y                   X
 
   The frame definition contains the opposite of these rotation angles --
   with the angle order reversed and the angle signs changed to the
   opposite ones -- because the angles specified in it define the
   transformation from the MRI frame to the spacecraft frame.
 
      \begindata
 
         FRAME_DIF_MRI             = -140200
         FRAME_-140200_NAME        = 'DIF_MRI'
         FRAME_-140200_CLASS       = 4
         FRAME_-140200_CLASS_ID    = -140200
         FRAME_-140200_CENTER      = -140
         TKFRAME_-140200_SPEC      = 'ANGLES'
         TKFRAME_-140200_RELATIVE  = 'DIF_SPACECRAFT'
         TKFRAME_-140200_ANGLES    = ( -0.004898709285,
                                       45.006884881185,
                                       -0.129539306414 )
         TKFRAME_-140200_AXES      = ( 1,    2,   3   )
         TKFRAME_-140200_UNITS     = 'DEGREES'
 
      \begintext
 
 
Examples of Two-Vector Parameterized Dynamic Frames
--------------------------------------------------------
 
 
Geocentric Solar Ecliptic (GSE) Frame
 
   Definition of the Geocentric Solar Ecliptic frame:
 
            All vectors are geometric: no aberration corrections are used.
 
            The position of the sun relative to the earth is the primary
            vector: the X axis points from the earth to the sun.
 
            The inertially referenced velocity of the sun relative to the
            earth is the secondary vector: the Y axis is the component of
            this velocity vector orthogonal to the X axis.
 
            The Z axis is X cross Y, completing the right-handed reference
            frame.
 
   The GSE frame can be defined using the following assignments, where
   <frame_ID> must be replaced by an integer ID code.
 
      FRAME_GSE                       =  <frame_ID>
      FRAME_<frame_ID>_NAME           = 'GSE'
      FRAME_<frame_ID>_CLASS          =  5
      FRAME_<frame_ID>_CLASS_ID       =  <frame_ID>
      FRAME_<frame_ID>_CENTER         =  399
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'TWO-VECTOR'
      FRAME_<frame_ID>_PRI_AXIS       = 'X'
      FRAME_<frame_ID>_PRI_VECTOR_DEF = 'OBSERVER_TARGET_POSITION'
      FRAME_<frame_ID>_PRI_OBSERVER   = 'EARTH'
      FRAME_<frame_ID>_PRI_TARGET     = 'SUN'
      FRAME_<frame_ID>_PRI_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_AXIS       = 'Y'
      FRAME_<frame_ID>_SEC_VECTOR_DEF = 'OBSERVER_TARGET_VELOCITY'
      FRAME_<frame_ID>_SEC_OBSERVER   = 'EARTH'
      FRAME_<frame_ID>_SEC_TARGET     = 'SUN'
      FRAME_<frame_ID>_SEC_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_FRAME      = 'J2000'
 
 
Geocentric Solar Magnetospheric (GSM) Frame
 
   Definition of the Geocentric Solar Magnetospheric frame:
 
            All vectors are geometric: no aberration corrections are used.
 
            The position of the sun relative to the earth is the primary
            vector: the X axis points from the earth to the sun.
 
            The earth's geomagnetic centered north dipole vector is
            secondary: the Z axis is the component of this vector
            orthogonal to the X axis. For the purpose of this definition,
            we treat the dipole vector as constant in the IAU_EARTH
            body-fixed frame. Note that in an earth-fixed reference frame,
            the north geomagnetic centered dipole is actually time-varying;
            the values shown here may be unsuitable for your application.
 
            The Y axis direction is the cross product of the Z-axis and the
            X-axis.
 
   The GSM frame can be defined using the following assignments, where
   <frame_ID> must be replaced by an integer ID code.
 
      FRAME_GSM                       =  <frame_ID>
      FRAME_<frame_ID>_NAME           = 'GSM'
      FRAME_<frame_ID>_CLASS          =  5
      FRAME_<frame_ID>_CLASS_ID       =  <frame_ID>
      FRAME_<frame_ID>_CENTER         =  399
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'TWO-VECTOR'
      FRAME_<frame_ID>_PRI_AXIS       = 'X'
      FRAME_<frame_ID>_PRI_VECTOR_DEF = 'OBSERVER_TARGET_POSITION'
      FRAME_<frame_ID>_PRI_OBSERVER   = 'EARTH'
      FRAME_<frame_ID>_PRI_TARGET     = 'SUN'
      FRAME_<frame_ID>_PRI_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_AXIS       = 'Z'
      FRAME_<frame_ID>_SEC_VECTOR_DEF = 'CONSTANT'
      FRAME_<frame_ID>_SEC_FRAME      = 'IAU_EARTH'
      FRAME_<frame_ID>_SEC_SPEC       = 'LATITUDINAL'
      FRAME_<frame_ID>_SEC_UNITS      = 'DEGREES'
      FRAME_<frame_ID>_SEC_LONGITUDE  =  288.43
      FRAME_<frame_ID>_SEC_LATITUDE   =   79.54
 
 
Mercury Solar Equatorial (MSEQ) Frame
 
   Definition of the Mercury Solar Equatorial Frame:
 
            All vectors are geometric: no aberration corrections are used.
 
            The sun's north spin axis direction is primary: the Z axis of
            the MSEQ frame is aligned with this spin axis.
 
            The position of the Sun relative to Mercury is secondary: the Y
            axis is aligned with the component of this position orthogonal
            to the Z axis.
 
            The X axis direction is the cross product of the Y axis and Z
            axis.
 
   All vectors are geometric: no aberration corrections are used.
 
   The MSEQ frame can be defined using the following assignments, where
   <frame_ID> must be replaced by an integer ID code.
 
      FRAME_MSEQ                      =  <frame_ID>
      FRAME_<frame_ID>_NAME           = 'MSEQ'
      FRAME_<frame_ID>_CLASS          =  5
      FRAME_<frame_ID>_CLASS_ID       =  <frame_ID>
      FRAME_<frame_ID>_CENTER         =  199
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'TWO-VECTOR'
      FRAME_<frame_ID>_PRI_AXIS       = 'Z'
      FRAME_<frame_ID>_PRI_VECTOR_DEF = 'CONSTANT'
      FRAME_<frame_ID>_PRI_FRAME      = 'IAU_SUN'
      FRAME_<frame_ID>_PRI_SPEC       = 'RECTANGULAR'
      FRAME_<frame_ID>_PRI_VECTOR     =  ( 0, 0, 1 )
      FRAME_<frame_ID>_SEC_AXIS       = 'X'
      FRAME_<frame_ID>_SEC_VECTOR_DEF = 'OBSERVER_TARGET_POSITION'
      FRAME_<frame_ID>_SEC_OBSERVER   = 'MERCURY'
      FRAME_<frame_ID>_SEC_TARGET     = 'SUN'
      FRAME_<frame_ID>_SEC_ABCORR     = 'NONE'
 
 
Example: Nadir Frame for Mars Orbiting Spacecraft
 
   Definition of the nadir frame:
 
            All vectors are geometric: no aberration corrections are used.
 
            The Z axis points from the spacecraft to the closest point on
            Mars.
 
            The component of inertially referenced spacecraft velocity
            vector orthogonal to Z is aligned with the -X axis.
 
            The Y axis is the cross product of the Z axis and the X axis.
 
   This nadir frame can be defined using the following assignments, where
 
         <frame_name>        should be replaced by an actual frame name
 
         <orbiter_ID>        must be replaced with the integer ID code of
                             the orbiter
 
         <orbiter_ID/name>   must be replaced with either the integer ID
                             code of the orbiter or the name of the orbiter
 
         <frame_ID>          must be replaced by an integer ID code
 
      FRAME_<frame_name>              = <frame_ID>
      FRAME_<frame_ID>_NAME           = <frame_name>
      FRAME_<frame_ID>_CLASS          = 5
      FRAME_<frame_ID>_CLASS_ID       = <frame_ID>
      FRAME_<frame_ID>_CENTER         = <orbiter_ID>
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'TWO-VECTOR'
      FRAME_<frame_ID>_PRI_AXIS       = 'Z'
      FRAME_<frame_ID>_PRI_VECTOR_DEF = 'TARGET_NEAR_POINT'
      FRAME_<frame_ID>_PRI_OBSERVER   = <orbiter_ID/name>
      FRAME_<frame_ID>_PRI_TARGET     = 'MARS'
      FRAME_<frame_ID>_PRI_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_AXIS       = '-X'
      FRAME_<frame_ID>_SEC_VECTOR_DEF = 'OBSERVER_TARGET_VELOCITY'
      FRAME_<frame_ID>_SEC_OBSERVER   = <orbiter_ID/name>
      FRAME_<frame_ID>_SEC_TARGET     = 'MARS'
      FRAME_<frame_ID>_SEC_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_FRAME      = 'J2000'
 
 
Example: Roll-Celestial Spacecraft Frame
 
   There are a variety of roll-celestial frames in use. This example may
   not match frame definitions used for any specific flight project; it is
   intended to demonstrate how to define this category of frame.
 
   Definition of the roll-celestial frame:
 
            The Z axis points from the spacecraft to the earth. This vector
            is geometric (uncorrected).
 
            The component of an inertially referenced star direction vector
            orthogonal to the Z axis is the X axis. The star direction is
            provided by a specified star catalog in the form of right
            ascension and declination relative to the J2000 frame. If
            necessary, the RA/DEC coordinates should be adjusted for proper
            motion and parallax. This star direction vector is corrected
            for stellar aberration using the spacecraft as the observer.
 
            The Y axis is the cross product of the Z axis and the X axis.
 
   This roll-celestial frame can be defined using the following
   assignments, where
 
         <frame_name>           should be replaced by an actual frame name
 
         <spacecraft_ID>        must be replaced with the integer ID code
                                of the spacecraft
 
         <spacecraft_ID/name>   must be replaced with either the integer ID
                                code of the spacecraft or the name of the
                                spacecraft
 
         <frame_ID>             must be replaced by an integer ID code
 
      FRAME_<frame_name>              = <frame_ID>
      FRAME_<frame_ID>_NAME           = <frame_name>
      FRAME_<frame_ID>_CLASS          = 5
      FRAME_<frame_ID>_CLASS_ID       = <frame_ID>
      FRAME_<frame_ID>_CENTER         = <spacecraft_ID>
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'TWO-VECTOR'
      FRAME_<frame_ID>_PRI_AXIS       = 'Z'
      FRAME_<frame_ID>_PRI_VECTOR_DEF = 'OBSERVER_TARGET_POSITION'
      FRAME_<frame_ID>_PRI_OBSERVER   = <spacecraft_ID/name>
      FRAME_<frame_ID>_PRI_TARGET     = 'EARTH'
      FRAME_<frame_ID>_PRI_ABCORR     = 'NONE'
      FRAME_<frame_ID>_SEC_AXIS       = 'X'
      FRAME_<frame_ID>_SEC_VECTOR_DEF = 'CONSTANT'
      FRAME_<frame_ID>_SEC_FRAME      = 'J2000'
      FRAME_<frame_ID>_SEC_SPEC       = 'RA/DEC'
      FRAME_<frame_ID>_SEC_UNITS      = 'DEGREES'
      FRAME_<frame_ID>_SEC_RA         = <star RA in degrees>
      FRAME_<frame_ID>_SEC_DEC        = <star DEC in degrees>
      FRAME_<frame_ID>_SEC_ABCORR     = 'S'
      FRAME_<frame_ID>_SEC_OBSERVER   = <spacecraft_ID/name>
 
 
Examples of Mean Equator and Equinox of Date Frames
--------------------------------------------------------
 
 
Earth Mean Equator and Equinox of Date Frames
 
   Definition of a non-inertial Earth Mean Equator and Equinox of Date
   frame using 1976 IAU precession model. Here <frame_name> must be
   replaced by a string containing the name of the frame, and <frame_ID>
   must be replaced by an integer ID code:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_ROTATION_STATE= 'ROTATING'
 
   Definition for the inertial version of the above frame:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_ROTATION_STATE= 'INERTIAL'
 
   Definition for the frozen version of the above frame, where the ``freeze
   epoch'' is B1950 TDB. The resulting frame should match the inertial
   frame B1950 to round-off level:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_FREEZE_EPOCH  = @1949-DEC-31/22:09:46.861901
 
 
Examples of True Equator and Equinox of Date Frames
--------------------------------------------------------
 
   Definition of the Earth True Equator and Equinox of Date frame:
 
            The earth precession model is the 1976 IAU model.
 
            The earth nutation model is the 1980 IAU model.
 
   Here <frame_name> must be replaced by a string containing the name of
   the frame, and <frame_ID> must be replaced by an integer ID code:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_NUT_MODEL     = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_ROTATION_STATE= 'ROTATING'
 
   Definition for the inertial version of the above frame:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_NUT_MODEL     = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_ROTATION_STATE= 'INERTIAL'
 
   Definition for the frozen version of the above frame, where the ``freeze
   epoch'' is B1950 TDB.
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'TRUE_EQUATOR_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_NUT_MODEL     = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_FREEZE_EPOCH  = @1949-DEC-31/22:09:46.861901
 
 
Example of a Mean Ecliptic and Equinox of Date Frame
--------------------------------------------------------
 
   Definition of the Earth Mean Ecliptic and Equinox of Date frame:
 
            The earth precession model is the 1976 IAU model.
 
            The earth mean obliquity model is the 1980 IAU model.
 
   Here <frame_name> must be replaced by a string containing the name of
   the frame, and <frame_ID> must be replaced by an integer ID code:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_OBLIQ_MODEL   = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_ROTATION_STATE= 'ROTATING'
 
   Definition for the inertial version of the above frame:
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_OBLIQ_MODEL   = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_ROTATION_STATE= 'INERTIAL'
 
   Definition for the frozen version of the above frame, where the ``freeze
   epoch'' is B1950 TDB.
 
      FRAME_<frame_name>             =  <frame_ID>
      FRAME_<frame_ID>_NAME          =  <frame_name>
      FRAME_<frame_ID>_CLASS         =  5
      FRAME_<frame_ID>_CLASS_ID      =  <frame_ID>
      FRAME_<frame_ID>_CENTER        =  399
      FRAME_<frame_ID>_RELATIVE      = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE     = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY        = 'MEAN_ECLIPTIC_AND_EQUINOX_OF_DATE'
      FRAME_<frame_ID>_PREC_MODEL    = 'EARTH_IAU_1976'
      FRAME_<frame_ID>_OBLIQ_MODEL   = 'EARTH_IAU_1980'
      FRAME_<frame_ID>_FREEZE_EPOCH  = @1949-DEC-31/22:09:46.861901
 
 
Example of an Euler Frame
--------------------------------------------------------
 
   As an example, we construct an Euler frame called IAU_MARS_EULER. Frame
   IAU_MARS_EULER is mathematically identical to the PCK frame IAU_MARS.
 
   The PCK data defining the underlying IAU_MARS frame are:
 
      BODY499_POLE_RA          = (  317.68143   -0.1061      0.  )
      BODY499_POLE_DEC         = (   52.88650   -0.0609      0.  )
      BODY499_PM               = (  176.630    350.89198226  0.  )
 
   These values are from:
 
      Seidelmann, P.K., Abalakin, V.K., Bursa, M., Davies, M.E., Bergh, C.
      de, Lieske, J.H., Oberst, J., Simon, J.L., Standish, E.M.,
      Stooke, P., and Thomas, P.C. (2002). "Report of the IAU/IAG Working
      Group on Cartographic Coordinates and Rotational Elements of the
      Planets and Satellites: 2000," Celestial Mechanics and Dynamical
      Astronomy, v.82, Issue 1, pp. 83-111.
 
   Here pole RA/Dec terms in the PCK are in degrees and degrees/century;
   the rates here have been converted to degrees/sec. Prime meridian terms
   in the PCK are in degrees and degrees/day; the rate here has been
   converted to degrees/sec.
 
   The 3x3 transformation matrix M defined by the angles is
 
      M = [angle_1]   [angle_2]   [angle_3]
                   3           1           3
 
   Vectors are mapped from the J2000 base frame to the IAU_MARS frame via
   left multiplication by M.
 
   The relationship of these Euler angles to RA/Dec/PM for the J2000-to-IAU
   Mars body-fixed transformation is as follows:
 
      angle_1 is        PM  * (radians/degree)
      angle_2 is pi/2 - Dec * (radians/degree)
      angle_3 is pi/2 + RA  * (radians/degree)
 
   Since when we define the IAU_MARS_EULER frame we're defining the
   *inverse* of the above transformation, the angles for our Euler frame
   definition are reversed and the signs negated:
 
      angle_1 is -pi/2 - RA  * (radians/degree)
      angle_2 is -pi/2 + Dec * (radians/degree)
      angle_3 is       - PM  * (radians/degree)
 
   Then our frame definition is:
 
      FRAME_IAU_MARS_EULER            =  <frame_ID>
      FRAME_<frame_ID>_NAME           = 'IAU_MARS_EULER'
      FRAME_<frame_ID>_CLASS          =  5
      FRAME_<frame_ID>_CLASS_ID       =  <frame_ID>
      FRAME_<frame_ID>_CENTER         =  499
      FRAME_<frame_ID>_RELATIVE       = 'J2000'
      FRAME_<frame_ID>_DEF_STYLE      = 'PARAMETERIZED'
      FRAME_<frame_ID>_FAMILY         = 'EULER'
      FRAME_<frame_ID>_EPOCH          =  @2000-JAN-1/12:00:00
      FRAME_<frame_ID>_AXES           =  ( 3  1  3 )
      FRAME_<frame_ID>_UNITS          = 'DEGREES'
      FRAME_<frame_ID>_ANGLE_1_COEFFS = (  -47.68143
                                             0.33621061170684714E-10 )
      FRAME_<frame_ID>_ANGLE_2_COEFFS = (  -37.1135
                                            -0.19298045478743630E-10 )
      FRAME_<frame_ID>_ANGLE_3_COEFFS = ( -176.630
                                            -0.40612497946759260E-02 )
 
