xf2eul_c |

Table of contents## Procedurexf2eul_c ( State transformation to Euler angles ) void xf2eul_c ( ConstSpiceDouble xform [6][6], SpiceInt axisa, SpiceInt axisb, SpiceInt axisc, SpiceDouble eulang [6], SpiceBoolean * unique ) ## AbstractConvert a state transformation matrix to Euler angles and their derivatives, given a specified set of axes. ## Required_ReadingPCK ROTATION ## KeywordsANGLES DERIVATIVES STATE ## Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- xform I A state transformation matrix. axisa I Axis A of the Euler angle factorization. axisb I Axis B of the Euler angle factorization. axisc I Axis C of the Euler angle factorization. eulang O An array of Euler angles and their derivatives. unique O Indicates if `eulang' is a unique representation. ## Detailed_Inputxform is a state transformation matrix from some frame FRAME1 to another frame FRAME2. Pictorially, `xform' has the structure shown here. .- -. | | | | r | 0 | | | | |-------+-------| | | | | dr/dt | r | | | | `- -' where `r' is a rotation matrix that varies with respect to time and dr/dt is its time derivative. More specifically, if `s1' is the state of some object in FRAME1, then `s2', the state of the same object relative to FRAME2 is given by s2 = xform * s1 where "*" denotes the matrix vector product. axisa, axisb, axisc are the axes desired for the factorization of `r'. All must be in the range from 1 to 3. Moreover it must be the case that `axisa' and `axisb' are distinct and that `axisb' and `axisc' are distinct. Every rotation matrix can be represented as a product of three rotation matrices about the principal axes of a reference frame. r = [ alpha ] [ beta ] [ gamma ] axisa axisb axisc The value 1 corresponds to the X axis. The value 2 corresponds to the Y axis. The value 3 corresponds to the Z axis. ## Detailed_Outputeulang is the set of Euler angles corresponding to the specified factorization. If we represent `r' as shown here: r = [ alpha ] [ beta ] [ gamma ] axisa axisb axisc then eulang[0] = alpha eulang[1] = beta eulang[2] = gamma eulang[3] = dalpha/dt eulang[4] = dbeta/dt eulang[5] = dgamma/dt The range of `alpha' and `gamma' is (-pi, pi]. The range of `beta' depends on the exact set of axes used for the factorization. For factorizations in which the first and third axes are the same, the range of `beta' is [0, pi]. For factorizations in which the first and third axes are different, the range of `beta' is [-pi/2, pi/2]. For rotations such that `alpha' and `gamma' are not uniquely determined, `alpha' and dalpha/dt will always be set to zero; `gamma' and dgamma/dt are then uniquely determined. unique is a logical that indicates whether or not the values in `eulang' are uniquely determined. If the values are unique then `unique' will be set to SPICETRUE. If the values are not unique and some components ( eulang[0] and eulang[3] ) have been set to zero, then `unique' will have the value SPICEFALSE. ## ParametersNone. ## Exceptions1) If any of `axisa', `axisb', or `axisc' do not have values in { 1, 2, 3 } an error is signaled by a routine in the call tree of this routine. 2) If `axisb' is equal to `axisc' or `axisa', an error is signaled by a routine in the call tree of this routine. An arbitrary rotation matrix cannot be expressed using a sequence of Euler angles unless the second rotation axis differs from the other two. 3) If the input matrix `xform' is not a rotation matrix, an error is signaled by a routine in the call tree of this routine. 4) If eulang[0] and eulang[2] are not uniquely determined, eulang[0] is set to zero, and eulang[2] is determined. ## FilesNone. ## ParticularsA word about notation: the symbol [ x ] i indicates a coordinate system rotation of x radians about the ith coordinate axis. To be specific, the symbol [ x ] 1 indicates a coordinate system rotation of x radians about the first, or x-, axis; the corresponding matrix is .- -. | 1 0 0 | | | | 0 cos(x) sin(x) | | | | 0 -sin(x) cos(x) | `- -' Remember, this is a COORDINATE SYSTEM rotation by x radians; this matrix, when applied to a vector, rotates the vector by -x radians, not x radians. Applying the matrix to a vector yields the vector's representation relative to the rotated coordinate system. The analogous rotation about the second, or y-, axis is represented by [ x ] 2 which symbolizes the matrix .- -. | cos(x) 0 -sin(x) | | | | 0 1 0 | | | | sin(x) 0 cos(x) | `- -' and the analogous rotation about the third, or z-, axis is represented by [ x ] 3 which symbolizes the matrix .- -. | cos(x) sin(x) 0 | | | | -sin(x) cos(x) 0 | | | | 0 0 1 | `- -' The input matrix is assumed to be the product of three rotation matrices, each one of the form .- -. | 1 0 0 | | | | 0 cos(r) sin(r) | (rotation of r radians about the | | x-axis), | 0 -sin(r) cos(r) | `- -' .- -. | cos(s) 0 -sin(s) | | | | 0 1 0 | (rotation of s radians about the | | y-axis), | sin(s) 0 cos(s) | `- -' or .- -. | cos(t) sin(t) 0 | | | | -sin(t) cos(t) 0 | (rotation of t radians about the | | z-axis), | 0 0 1 | `- -' where the second rotation axis is not equal to the first or third. Any rotation matrix can be factored as a sequence of three such rotations, provided that this last criterion is met. This routine is related to the routine eul2xf_c which produces a state transformation from an input set of axes, Euler angles and derivatives. The two function calls shown here will not change `xform' except for round off errors. ## ExamplesThe numerical results shown for this example may differ across platforms. The results depend on the SPICE kernels used as input, the compiler and supporting libraries, and the machine specific arithmetic implementation. 1) Determine the rate of change of the right ascension and declination of the pole of the moon, from the state transformation matrix that transforms J2000 states to object fixed states. Recall that the rotation component of the state transformation matrix is given by [w] [halfpi_c-dec] [ra+halfpi_c] 3 1 3 Use the meta-kernel shown below to load the required SPICE kernels. KPL/MK File name: xf2eul_ex1.tm This meta-kernel is intended to support operation of SPICE example programs. The kernels shown here should not be assumed to contain adequate or correct versions of data required by SPICE-based user applications. In order for an application to use this meta-kernel, the kernels referenced here must be present in the user's current working directory. The names and contents of the kernels referenced by this meta-kernel are as follows: File name Contents --------- -------- pck00010.tpc Planet orientation and radii naif0012.tls Leapseconds \begindata KERNELS_TO_LOAD = ( 'pck00010.tpc', 'naif0012.tls' ) \begintext End of meta-kernel Example code begins here. /. Program xf2eul_ex1 ./ #include <stdio.h> #include "SpiceUsr.h" int main( ) { /. Local parameters. ./ #define META "xf2eul_ex1.tm" #define UTCSTR "May 15, 2007" /. Local variables. ./ SpiceDouble eulang [6]; SpiceDouble et; SpiceDouble ftmtrx [6][6]; SpiceBoolean unique; /. Load SPICE kernels. ./ furnsh_c ( META ); /. Convert the input time to seconds past J2000 TDB. ./ str2et_c ( UTCSTR, &et ); /. Get the transformation matrix from J2000 frame to IAU_MOON. ./ sxform_c ( "J2000", "IAU_MOON", et, ftmtrx ); /. Convert the transformation matrix to Euler angles (3-1-3). ./ ## RestrictionsNone. ## Literature_ReferencesNone. ## Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) W.L. Taber (JPL) ## Version-CSPICE Version 1.0.2, 01-NOV-2021 (JDR) Edited the header to comply with NAIF standard. Added complete code example based on existing example. -CSPICE Version 1.0.1, 05-MAR-2008 (NJB) Fixed typo (missing double quote character) in code example. Corrected order of header sections. -CSPICE Version 1.0.0, 15-JUN-1999 (WLT) (NJB) ## Index_EntriesEuler angles and derivatives from state transformation |

Fri Dec 31 18:41:15 2021