qxq_c |

## Procedurevoid qxq_c ( ConstSpiceDouble q1 [4], ConstSpiceDouble q2 [4], SpiceDouble qout [4] ) ## AbstractMultiply two quaternions. ## Required_ReadingROTATION ## KeywordsMATH POINTING ROTATION ## Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- q1 I First SPICE quaternion factor. q2 I Second SPICE quaternion factor. qout O Product of `q1' and `q2'. ## Detailed_Inputq1 is a 4-vector representing a SPICE-style quaternion. See the discussion of "Quaternion Styles" in the Particulars section below. Note that multiple styles of quaternions are in use. This routine will not work properly if the input quaternions do not conform to the SPICE convention. q2 is a second SPICE-style quaternion. ## Detailed_Outputqout is 4-vector representing the quaternion product q1 * q2 Representing q(i) as the sums of scalar (real) part s(i) and vector (imaginary) part v(i) respectively, q1 = s1 + v1 q2 = s2 + v2 qout has scalar part s3 defined by s3 = s1 * s2 - <v1, v2> and vector part v3 defined by v3 = s1 * v2 + s2 * v1 + v1 x v2 where the notation < , > denotes the inner product operator and x indicates the cross product operator. ## ParametersNone. ## ExceptionsError free. ## FilesNone. ## ParticularsQuaternion Styles ----------------- There are different "styles" of quaternions used in science and engineering applications. Quaternion styles are characterized by - The order of quaternion elements - The quaternion multiplication formula - The convention for associating quaternions with rotation matrices Two of the commonly used styles are - "SPICE" > Invented by Sir William Rowan Hamilton > Frequently used in mathematics and physics textbooks - "Engineering" > Widely used in aerospace engineering applications CSPICE function interfaces ALWAYS use SPICE quaternions. Quaternions of any other style must be converted to SPICE quaternions before they are passed to CSPICE functions. Relationship between SPICE and Engineering Quaternions ------------------------------------------------------ Let M be a rotation matrix such that for any vector V, M*V is the result of rotating V by theta radians in the counterclockwise direction about unit rotation axis vector A. Then the SPICE quaternions representing M are (+/-) ( cos(theta/2), sin(theta/2) A(1), sin(theta/2) A(2), sin(theta/2) A(3) ) while the engineering quaternions representing M are (+/-) ( -sin(theta/2) A(1), -sin(theta/2) A(2), -sin(theta/2) A(3), cos(theta/2) ) For both styles of quaternions, if a quaternion q represents a rotation matrix M, then -q represents M as well. Given an engineering quaternion QENG = ( q0, q1, q2, q3 ) the equivalent SPICE quaternion is QSPICE = ( q3, -q0, -q1, -q2 ) Associating SPICE Quaternions with Rotation Matrices ---------------------------------------------------- Let FROM and TO be two right-handed reference frames, for example, an inertial frame and a spacecraft-fixed frame. Let the symbols V , V FROM TO denote, respectively, an arbitrary vector expressed relative to the FROM and TO frames. Let M denote the transformation matrix that transforms vectors from frame FROM to frame TO; then V = M * V TO FROM where the expression on the right hand side represents left multiplication of the vector by the matrix. Then if the unit-length SPICE quaternion q represents M, where q = (q0, q1, q2, q3) the elements of M are derived from the elements of q as follows: +- -+ | 2 2 | | 1 - 2*( q2 + q3 ) 2*(q1*q2 - q0*q3) 2*(q1*q3 + q0*q2) | | | | | | 2 2 | M = | 2*(q1*q2 + q0*q3) 1 - 2*( q1 + q3 ) 2*(q2*q3 - q0*q1) | | | | | | 2 2 | | 2*(q1*q3 - q0*q2) 2*(q2*q3 + q0*q1) 1 - 2*( q1 + q2 ) | | | +- -+ Note that substituting the elements of -q for those of q in the right hand side leaves each element of M unchanged; this shows that if a quaternion q represents a matrix M, then so does the quaternion -q. To map the rotation matrix M to a unit quaternion, we start by decomposing the rotation matrix as a sum of symmetric and skew-symmetric parts: 2 M = [ I + (1-cos(theta)) OMEGA ] + [ sin(theta) OMEGA ] symmetric skew-symmetric OMEGA is a skew-symmetric matrix of the form +- -+ | 0 -n3 n2 | | | OMEGA = | n3 0 -n1 | | | | -n2 n1 0 | +- -+ The vector N of matrix entries (n1, n2, n3) is the rotation axis of M and theta is M's rotation angle. Note that N and theta are not unique. Let C = cos(theta/2) S = sin(theta/2) Then the unit quaternions Q corresponding to M are Q = +/- ( C, S*n1, S*n2, S*n3 ) The mappings between quaternions and the corresponding rotations are carried out by the CSPICE routines q2m_c {quaternion to matrix} m2q_c {matrix to quaternion} m2q_c always returns a quaternion with scalar part greater than or equal to zero. SPICE Quaternion Multiplication Formula --------------------------------------- Given a SPICE quaternion Q = ( q0, q1, q2, q3 ) corresponding to rotation axis A and angle theta as above, we can represent Q using "scalar + vector" notation as follows: s = q0 = cos(theta/2) v = ( q1, q2, q3 ) = sin(theta/2) * A Q = s + v Let Q1 and Q2 be SPICE quaternions with respective scalar and vector parts s1, s2 and v1, v2: Q1 = s1 + v1 Q2 = s2 + v2 We represent the dot product of v1 and v2 by <v1, v2> and the cross product of v1 and v2 by v1 x v2 Then the SPICE quaternion product is Q1*Q2 = s1*s2 - <v1,v2> + s1*v2 + s2*v1 + (v1 x v2) If Q1 and Q2 represent the rotation matrices M1 and M2 respectively, then the quaternion product Q1*Q2 represents the matrix product M1*M2 ## Examples1) Let qid, qi, qj, qk be the "basis" quaternions qid = ( 1, 0, 0, 0 ) qi = ( 0, 1, 0, 0 ) qj = ( 0, 0, 1, 0 ) qk = ( 0, 0, 0, 1 ) respectively. Then the calls ## RestrictionsNone. ## Literature_ReferencesNone. ## Author_and_InstitutionN.J. Bachman (JPL) ## Version-CSPICE Version 1.0.1, 27-FEB-2008 (NJB) Updated header; added information about SPICE quaternion conventions. -CSPICE Version 1.0.0, 27-OCT-2005 (NJB) ## Index_Entriesquaternion times quaternion multiply quaternion by quaternion |

Wed Apr 5 17:54:41 2017