qdq2av_c |

## Procedurevoid qdq2av_c ( ConstSpiceDouble q [4], ConstSpiceDouble dq [4], SpiceDouble av [3] ) ## AbstractDerive angular velocity from a unit quaternion and its derivative with respect to time. ## Required_ReadingROTATION ## KeywordsMATH POINTING ROTATION ## Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- q I Unit SPICE quaternion. dq I Derivative of `q' with respect to time. av O Angular velocity defined by `q' and `dq'. ## Detailed_Inputq is a unit length 4-vector representing a SPICE-style quaternion. See the discussion of "Quaternion Styles" in the Particulars section below. Note that multiple styles of quaternions are in use. This routine will not work properly if the input quaternions do not conform to the SPICE convention. See the Particulars section for details. dq is a 4-vector representing the derivative of `q' with respect to time. ## Detailed_Outputav is 3-vector representing the angular velocity defined by `q' and `dq', that is, the angular velocity of the frame defined by the rotation matrix associated with `q'. This rotation matrix can be obtained via the CSPICE routine q2m_c; see the Particulars section for the explicit matrix entries. `av' is the vector (imaginary) part of the quaternion product * -2 * q * dq This angular velocity is the same vector that could be obtained (much less efficiently ) by mapping `q' and `dq' to the corresponding C-matrix `r' and its derivative `dr', then calling the CSPICE routine xf2rav_c. `av' has units of radians / T where 1 / T is the unit associated with `dq'. ## ParametersNone. ## ExceptionsError free. 1) A unitized version of input quaternion is used in the computation. No attempt is made to diagnose an invalid input quaternion. ## 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 About this routine ================== Given a time-dependent SPICE quaternion representing the attitude of an object, we can obtain the object's angular velocity AV in terms of the quaternion Q and its derivative with respect to time DQ: * AV = Im ( -2 * Q * DQ ) (1) That is, AV is the vector (imaginary) part of the product on the right hand side (RHS) of equation (1). The scalar part of the RHS is zero. We'll now provide an explanation of formula (1). For any time-dependent rotation, the associated angular velocity at a given time is a function of the rotation and its derivative at that time. This fact enables us to extend a proof for a limited subset of rotations to *all* rotations: if we find a formula that, for any rotation in our subset, gives us the angular velocity as a function of the rotation and its derivative, then that formula must be true for all rotations. We start out by considering the set of rotation matrices R(t) = M(t)C (2) where C is a constant rotation matrix and M(t) represents a matrix that "rotates" with constant, unit magnitude angular velocity and that is equal to the identity matrix at t = 0. For future reference, we'll consider C to represent a coordinate transformation from frame F1 to frame F2. We'll call F1 the "base frame" of C. We'll let AVF2 be the angular velocity of M(t) relative to F2 and AVF1 be the same angular velocity relative to F1. Referring to the axis-and-angle decomposition of M(t) 2 M(t) = I + sin(t)OMEGA + (1-cos(t))OMEGA (3) (see the Rotation Required Reading for a derivation) we have d(M(t))| -------| = OMEGA (4) dt |t=0 Then the derivative of R(t) at t = 0 is given by d(R(t))| -------| = OMEGA * C (5) dt |t=0 The rotation axis A associated with OMEGA is defined by (6) A(1) = - OMEGA(2,3) A(2) = OMEGA(1,3) A(3) = - OMEGA(1,2) Since the coordinate system rotation M(t) rotates vectors about A through angle t radians at time t, the angular velocity AVF2 of M(t) is actually given by AVF2 = - A (7) This angular velocity is represented relative to the image frame F2 associated with the coordinate transformation C. Now, let's proceed to the angular velocity formula for quaternions. To avoid some verbiage, we'll freely use 3-vectors to represent the corresponding pure imaginary quaternions. Letting QR(t), QM(t), and QC be quaternions representing the time-dependent matrices R(t), M(t) and C respectively, where QM(t) is selected to be a differentiable function of t in a neighborhood of t = 0, the quaternion representing R(t) is QR(t) = QM(t) * QC (8) Differentiating with respect to t, then evaluating derivatives at t = 0, we have d(QR(t))| d(QM(t))| --------| = --------| * QC (9) dt |t=0 dt |t=0 Since QM(t) represents a rotation having axis A and rotation angle t, then (according to the relationship between SPICE quaternions and rotations set out in the Rotation Required Reading), we see QM(t) must be the quaternion (represented as the sum of scalar and vector parts): cos(t/2) + sin(t/2) * A (10) where A is the rotation axis corresponding to the matrix OMEGA introduced in equation (3). By inspection d(QM(t))| --------| = 1/2 * A (11) dt |t=0 which is a quaternion with scalar part zero. This allows us to rewrite the quaternion derivative d(QR(t))| --------| = 1/2 * A * QC (12) dt |t=0 or for short, DQ = 1/2 * A * QC (13) Since from (7) we know the angular velocity AVF2 of the frame associated with QM(t) is the negative of the rotation axis defined by (3), we have DQ = - 1/2 * AVF2 * QC (14) Since AVF2 = C * AVF1 (15) we can apply the quaternion transformation formula (from the Rotation Required Reading) * AVF2 = QC * AVF1 * QC (16) Now we re-write (15) as * DQ = - 1/2 * ( QC * AVF1 * QC ) * QC = - 1/2 * QC * AVF1 (17) Then the angular velocity vector AVF1 is given by * AVF1 = -2 * QC * DQ (18) The relation (18) has now been demonstrated for quaternions having constant, unit magnitude angular velocity. But since all time-dependent quaternions having value QC and derivative DQ at a given time t have the same angular velocity at time t, that angular velocity must be AVF1. ## ExamplesThe following test program creates a quaternion and quaternion derivative from a known rotation matrix and angular velocity vector. The angular velocity is recovered from the quaternion and quaternion derivative by calling ## 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, 31-OCT-2005 (NJB) ## Index_Entriesangular velocity from quaternion and derivative |

Wed Apr 5 17:54:41 2017