ELEKTROTEHNIŠKI VESTNIK 8(1-2): -11, 2011 ENGLISH EDITION Simultaneous Orthogonal Rotations Angle Sašo Tomažič 1, Sara Stančin 2 Facult of Electrical Engineering, Universit of Ljubljana 1 E-mail: saso.tomaic@fe.uni-lj.si 2 E-mail: sara.stancin@fe.uni-lj.si Abstract. Angular orientation refers to the position of a rigid bod intrinsic coordinate sstem relative to a reference coordinate sstem with the same origin. It is determined with a sequence of rotations needed to move the rigid-bod coordinate-sstem aes initiall aligned with the reference coordinate-sstem aes to their new position. In this paper we present a novel wa for representing angular orientation. We define the Simultaneous Orthogonal Rotations Angle (SORA) vector with components equal to the angles of three simultaneous rotations around the coordinate-sstem aes. The problem of non-commutativit is here avoided. We numericall verif that SORA is equal to the rotation vector the three simultaneous rotations it comprises are equivalent to a single rotation. The ais of this rotation coincides with the SORA vector while the rotation angle is equal to its magnitude. We further verif that if the coordinate sstems are initiall aligned, simultaneous rotations around the reference and rigid-bod intrinsic aes represent the same angular orientation. Considering the SORA vector, angular orientation of a rigid bod can be calculated in a single step thus avoiding the iterative infinitesimal rotation approimation computation. SORA can thus be a ver convenient wa for angular-orientation representation. Kewords: angular orientation, rotation, rotation ais, rotation angle 1 INTRODUCTION Angular orientation refers to the position of a rigid-bod intrinsic coordinate sstem relative to a reference coordinate sstem with the same origin. It is determined with a rotation needed to move the rigid-bod coordinate sstem initiall aligned with the reference coordinate sstem to its new position. The reference and the rigid-bod intrinsic coordinate sstem are considered Cartesian with aes,, and ', ', ', respectivel. Orientation of the sstem aes conforms to the right hand rule. Both coordinate sstems are illustrated in Fig. 1. According to the Euler s rotation theorem [1 p.83] an two independent coordinate sstems of the same origin can be related b a sequence of not more than three rotations around the coordinate aes, where no two successive rotations ma be about the same ais. The successive rotation ais distinction restriction from the Euler s theorem permits twelve different rotation aes sequences. Angular orientation can be specified as a sequence of rotations around the reference or rigid-bod intrinsic coordinate-sstem aes. This consideration doubles the number of possible rotation sequences. As rotation sequences are not commutative, each of these 24 rotation sequences in general specifies different angular orientations. Received 23 December 2010 Accepted 3 Februar 2011 Figure 1: Reference and rigid-bod intrinsic coordinate sstems. In this paper we present angular orientation with three simultaneous rotations around the coordinate-sstem aes. We define the Simultaneous Orthogonal Rotations Angle (SORA) vector with components equal to the angles of these three simultaneous rotations, thus avoiding the problem of non-commutativit. SORA represents a unique position of a rigid-bod intrinsic coordinate sstem relative to a reference coordinate sstem with the same origin.
8 TOMAŽIČ, STANČIN We numericall verif that SORA is equal to the rotation vector the three simultaneous rotations of the SORA notation are equivalent to a single rotation. The ais of this rotation coincides with the SORA vector while the rotation angle is equal to its magnitude. To allow for verification, we compare matri representations of angular orientation obtained b using SORA and b considering the total rotation being a consequence of a repeating sequence of approimatel infinitesimal rotations around three coordinate aes. The order of appling infinitesimal rotations is not important, as these are shown to be commutative [2]. 2 ANGULAR ORIENTATION REPRESENTATION Said above, angular orientation is determined with a rotation needed to move a rigid-bod coordinate sstem initiall aligned with the reference coordinate sstem to its new position. Man different notations are used to represent rotation. These include Euler angles, rotation matri, ais and angle, rotation vector, and quaternions. There eist also some other notations like the Rodrigues and Cale-Klein parameter, but are not widel used. 2.1 Euler angles The three successive rotation angles described in the Euler s rotation theorem are called Euler angles [1 p.83, 3, 4]. If the rigid-bod intrinsic and the reference coordinate sstems are initiall aligned, the Euler angles sequence ma be identified b specifing the aes of rotation of each of the three rotations. These are epressed in terms of the reference coordinate sstem and the two intermediate coordinate sstems [3]. The orientations of these two intermediate coordinatesstem aes are specified with the first and the second successive rotation. The first rotation ma be about an of the three orthogonal aes of the reference coordinate sstem. Considering the successive rotation-ais distinction restriction, the second and third rotation ma be about either of the two aes of the first and second intermediate coordinate sstem. This gives twelve Euler-angle sequence possibilities. Each of these twelve sequences is equivalent to one of the possible twelve rotation sequences around the reference coordinatesstem aes. This gives a total of 24 different notations. 2.2 Ais and angle The Euler s rotation theorem also states that an rotation about the three coordinate-sstem aes can be epressed as a single rotation about some new ais. The ais and angle [5, 6] is a pair comprising a unit vector representing a rotation ais and an angle of rotation around that ais: a ( a, ϕ) = a, ϕ (1) r a 2.3 Rotation vector The rotation ais and angle can be comprised in a single non-normalied rotation vector which has the same direction as the rotation ais while its magnitude is equal to the rotational angle: a ϕ r = a ϕ = a ϕ (2) r a ϕ It is important to note that successive rotations cannot be represented b rotation vector addition. 2.4 Rotation matri Angular orientation of the rigid-bod coordinate sstem can be completel specified with a sstem of three base vector equations. Each base vector of the rigid-bod intrinsic coordinate sstem is epressed as a linear combination of the reference base vectors: (3) Nine parameters from equations (3) form a 33 rotation matri []: r r r R = r r r (4) r r r Matri (4) represents a rotation needed to move the rigid-bod intrinsic coordinate sstem initiall aligned with the reference coordinate sstem to its new position. The rotation matri (4) is real and orthonormal: T T R R = R R = 1 (5) having a determinant equal to: R = 1 (6) A product of two or more rotation matrices is again a rotation matri, representing a rotation sequence, where the rotation order in a sequence is important.
SIMULTANEOUS ORTHOGONAL ROTATIONS ANGLE 9 2.5 Quaternion Quaternions [1 p.106-112] form a four-dimensional vector space with base elements denoted with 1, i, j, and k: q = a + i b + j c + k d () where: 2 2 i = j 2 = k = i j k = 1 (8) In epression (), a refers to quaternion q real part and b, c, and d to its imaginar parts. A thorough insight into different quaternion operations can be found in [1 p.106-112]. It has been shown that a normalied quaternion: 2 2 2 2 a + b + c + d = 1 (9) represents a rotation of angle ϕ around an oriented vector [ 0 0 0 ]. The rotation angle and ais are defined with the components of the normalied quaternion: q = cos ϕ + i sin ϕ + j sin ϕ + k sin ϕ (10) 2 2 2 2 Rotation of arbitrar vector v is epressed as a quaternion product: vr = q v q (11) where the oriented rotation vector and angle are given with quaternion q components (10). q denotes the comple conjugate of q. Vectors v and vr in equation (11) appear as quaternions with real parts equal to 0. Angular orientation of the rigid-bod intrinsic coordinate-sstem aes using quaternions is then represented according to the following: (12) where imaginar parts of quaternions qu, qu, qu, qu ', qu ', and qu ' are equal to the respective coordinate-sstem base vectors while their real part is equal to ero. A product of the quaternions representing rotations is also a quaternion representing the sequence of these rotations where, as is the case with the rotation matrices, the rotation order in a sequence is important. 3 SIMULTANEOUS ORTHOGONAL ROTATIONS ANGLE 3.1 Definition In the chapter above, our focus was on sequential rotations enabling us to represent angular orientation of a rigid-bod coordinate sstem. Let us now represent angular orientation with constant simultaneous rotations. Let,, represent angular and velocities of the rigid-bod intrinsic coordinate ais rotating around the reference aes and let further,, φ represent the respective angular φ φ and displacements. We can then write: dφ dφ dφ = ; = ; =. (13) dt dt dt Let us further consider angular velocities as vectors: 0 0 Ω = 0 ; Ω = ; Ω = 0 (14) 0 0 Angular velocit vectors (14) are aligned with rotation aes and their magnitudes are equal to angular velocities. Their sum is a 3D angular velocit vector: Ω = Ω + Ω + Ω = (15) Considering (13)-(15) we define SORA, as a new notation used in representing angular orientation: φ = Ω t = t = φ (16) φ where t is the total time of rotation. The components of the SORA vector (16) are the angles of the three simultaneous rotations around the coordinate-sstem aes. According to the Euler s rotational theorem, as mentioned above, an rotation about the three coordinate-sstem aes can be epressed as a single rotation about some new ais. Let us now assume that the three simultaneous rotations comprised in the SORA vector can be considered as one single rotation, the ais and angle of which are equal to the SORA vector orientation and magnitude, respectivel:
10 TOMAŽIČ, STANČIN u = = φ + φ + φ 2 2 2 2 2 2 (1) ϕ = = φ + φ + φ (18) For the time being we are unable to prove this assumption. However, we can verif it numericall. Our verification is presented in the net section. 3.2 Numerical verification Finite rotation denoted with rotation matri R can be represented as an infinite sequence of infinitesimal rotations denoted with rotation matri dr. Let us consider this representation and approimate the three simultaneous rotations around the reference coordinatesstem aes described in the previous section with n consecutive sequences of rotations around the coordinate-sstem aes for small angles φ, φ, and φ where φ φ φ φ = ; φ = ; φ = (19) n n n Each of these rotation sequences can be represented with rotation matri R : R = R( φ, u ) R( φ, u ) R( φ, u ) (20) where rotation matrices R( φ, u ), R( φ, u ), and R(, u ) represent individual rotations around the φ coordinate-sstem aes. To rotate the rigid-bod intrinsic coordinate sstem φ the above sequence of small for angles φ, φ, and rotations must be repeated n times. The rotation matri of the infinite sequence of infinitesimal rotations th approimation R ( n) is thus given as the n power of rotation matri R : n R ( n ) = R (21) We can obtain an other of the 12 possible sequences simpl b changing the matri multiplication order in the above procedure. As infinitesimal rotations are commutative [2], we epect, for large n, to obtain the same result with different rotation sequences. The above rotation matri represents rotation around the reference coordinate-sstem aes. If the rigid bod rotates around its intrinsic coordinate-sstem aes, the corresponding approimation rotation matri denoted with R ( n) ' can be obtained b performing n steps of the following recursive equations: P = R( φ, R ( n) ' ) i i Q = R( φ, P u ) i i R ( n) ' = R( φ, Q u ) i+ 1 i where the initial value of rotation matri R ( n) ' is: (22) R ( n ) ' 0 = I (23) and I is a 3 3 identit matri. Let us now sa that our approimation is good enough provided the resulting rotation matrices for the two considered rotation sequences around the reference coordinate-sstem aes R ( n) and R ( n) match up to the 6 th decimal place. We calculated the rotation matrices according to the above procedure using Mathematica [8] for numerous eamples. For all, the demanded precision was achieved when n = 10. We further calculated rotation matri R ( n) ' representing rotations around the rigid-bod intrinsic coordinate aes when n = 10 for the same eample set. Comparing R ( n) ' with respective R ( n) and R ( n) results, these also matched up to the 6 decimal place. This shows that rotations around the reference and rigid-bod intrinsic coordinate-sstem aes give the same result. Finall, we calculated rotation matri R SORA representing a single rotation around an ais defined with the SORA vector orientation (16) and for an angle defined with the SORA vector magnitude (1) as specified in the previous section. The R results SORA matched all considered approimations of rotation matrices R ( n), R ( n), and R ( n) ' up to the demanded precision for all considered eamples. This verifies that the SORA vector is equal to the rotation vector. As said above, a number of numerical eamples were considered. Here we present onl an illustrational eample where angular orientation of a rigid-bod coordinate sstem is represented with angular velocit angles φ = 0.5, φ = 0.90, andφ = 0.60. Our results obtained b using Mathematica [8] for matched up to the 6 th decimal place: n = 10 all
SIMULTANEOUS ORTHOGONAL ROTATIONS ANGLE 11 R = R ( n) = R ( n) = R ( n) ' = SORA 0.49430 0.149651 0.856065 = 0.32655 0.601614 0.318240 0.46395 0.84643 0.40280 (24) Sašo Tomažič is a Full Professor at the Facult of Electrical Engineering of the Universit of Ljubljana. He is the Head of the Laborator of Communication Devices and the Chair of the Telecommunication Department. His work includes research in the field of signal processing, securit in telecommunications, electronic commerce and information sstems. 4 CONCLUSION In this paper we defined the Simultaneous Orthogonal Rotations Angle (SORA) vector with components equal to the angles of three simultaneous orthogonal rotations around the rigid-bod intrinsic coordinate aes. We numericall verified that SORA is equal to the rotation vector the three simultaneous rotations it comprises are equivalent to a single rotation around the SORA vector and for an angle equal to its magnitude. We further showed that simultaneous rotations around the reference and rigid-bod intrinsic coordinate-sstem aes represent the same angular orientation. Considering the SORA vector, angular orientation of a rigid bod can be calculated in a single step thus avoiding the iterative infinitesimal rotation approimation computation. SORA can thus be a ver convenient wa for angular orientation representation. In our further research, we will tr to derive the presented results analticall. Sara Stančin graduated from the Facult of Electrical Engineering of the Universit of Ljubljana, Slovenia, in 200. She is currentl emploed as a researcher of the National Young-Researcher Scheme in the Laborator of Communication Devices at the same facult. Her research focuses on sstems and protocols for mobile communication and wireless sensors. ACKNOWLEDGEMENTS The research presented in this paper was performed under the scope of the research program 'Algorithms and optimiation procedures in telecommunications' financed b the Slovenian Research Agenc. REFERENCES [1] Kuipers J.B., Quaternions and Rotation Sequences, Princeton Universit Press, Princeton, New Jerse, 1999 [2] WolframMathWorld, Infinitesimal Rotation, last accessed November 2010 http://mathworld.wolfram.com/infinitesimalrotation.html [3] Pio R.L., Euler angles transformations, IEEE Transactions on automatic control, Vol AC-11, No. 4, October 1966 [4] Absolute Astronom, Euler angles, last accessed November 2010 http://www.absoluteastronom.com/topics/euler_angles [5] Absolute Astronom, Ais angle, last accessed November 2010 http://www.absoluteastronom.com/topics/ais_angle [6] Euclidian Space, Phsics Kinematics Angular Velocit, last accessed November 2010 http://www.euclideanspace.com/phsics/kinematics/angularveloci t/inde.htm#aisangle [] Euclidian Space, Maths Rotation Matrices, last accessed November 2010 http://www.euclideanspace.com/maths/algebra/matri/orthogonal/ rotation/inde.htm [8] Wolfram Research, Inc., Mathematica, Version.0, Champaign, IL (2009)