Unit Generalized Quaternions in Spatial Kinematics

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1 Vol (07) - Pages 7-4 Unit Generalized Quaternions in Spatial Kinematics Mehdi Jafari, echnical Vocational University, Urmia, Iran, mj_msc@yahoo.com Abstract- After a review of some fundamental properties of the generalized uaternions, we apply a unit generalized uaternion to rotation in the -dimensional space. Also, the angular velocity of this motion is obtained. Keywords: Generalized uaternion, uasi-orthogonal matrix, rotation. I. INRODUCION he number system of uaternions is an extended one of the complex numbers. hey were first described by W.R. Hamilton in 84 applied to mechanics in three-dimensional space. he algebra of uaternions is often denoted by 0 H. It can also be given by the Clifford algebra classifications Cl (R) Cl (R). he algebra H holds a special place in 0,,0 analysis since, according to the Frobenius theorem, it is one of the only two finite-dimensional division rings containing the real numbers as a proper subring; the other being the complex numbers. Unit uaternions provide a convenient mathematical notation for representing orientations rotations of objects in the three dimensional space. he unit uaternions can therefore be thought of as a choice of a group structure on the -sphere that gives the group Spin (), which is isomorphic to SU() also to the universal cover of SO(). Kula Yayli [] showed that a unit split 4 uaternion determines a rotation in semi-uclidean space.in [], it is demonstrated how unit timelike split uaternions are used to perform rotations in the Minkowski -space.rotations in a complex -dimensional space are considered in [9] applied to the treatment of the Lorentz transformation in special relativity. A brief introduction of the generalized uaternions is provided in [4]. Recently, the work was done on the generalized uaternions by Jafari [5] there is given some of their algebraic properties. In this paper, we briefly recall some fundamental properties of the generalized uaternions, show that the set of all unit generalized uaternions with the group operation of uaternion multiplication is a Lie group of -dimension. A similar relation to the relationship between uaternions rotations in the uclidean space exists between generalized uaternions rotations in the -space. We demonstrate how a unit generalized uaternion is used to represent a rotation of a vector in -dimensional space. he angular velocity of this motion is obtained. Finally, we give some examples for more clarification. A. Reel split uaternions A real uaternion is defined as a ai ajak where a, a, a a are real numbers, i, j, k of may be interpreted as the four basic vectors of Cartesian set of coordinates; they satisfy the non-commutative multiplication rules i j k i jk, ij k ji, jk = i = kj, 7

2 Vol (07) - Pages 7-4 ki = j = ik. A uaternion may be defined as a pair ( S, V ), where S a R is scalar part V ai a j a k R is the vector part of. he uaternion product of two uaternions p is defined as p S S V, V S V S V V V where '', '' '' '' are the inner vector products in sum of the suares of its components: p p p p p N a a a a N R,respectively. he norm of a uaternion is given by the, R. It can also be obtained by multiplying the uaternion by its conjugate ( S V ), in either order since a uaternion of conjugated commute: N. very non-zero uaternion has a multiplicative inverse given by its conjugate divided by its norm: / N. he uaternion algebra H is a normed division algebra, meaning that for any two uaternions p, N N N, the p p norm of every non-zero uaternion is non-zero ( positive) therefore the multiplicative inverse exists for any nonzero uaternion. Of course, as is well known, multiplication of uaternions is not commutative, so that in general for any two uaternions p, p p. his can have subtle ramifications, for example: ( p) pp p. For detailed information about real uaternions, we refer the reader to [5]. he algebra H' of split uaternions is defined as the four-dimensional vector space over R having a basis {, i, j, k } with the following properties; i, j k, ij k ji, jk = i = kj, ki = j = ik. he uaternion product of two split uaternions p is defined as where '', l '' p S S V, V S V S V V V p p p p p l l '' '' are Lorentzian inner vector products, respectively. It is clear that H H are associative l non-commutative algebras []. Definition. Let u ( u, u, u), v ( v, v, v) R. If, R, the generalized inner product of u v is defined by g( uv, ) uv uv uv, It could be written 0 0 g( uv, ) u 0 0 vu Gv. 0 0 Also, if 0, 0, g( uv, ) is called the generalized Lorentzian inner product. We put (R, g(,)), he vector product in is defined by i j k uv u u u v v v ( uv uv ) i( uv uv ) j( uv uv ) k, where i j k, jk i, ki j. 8

3 Vol (07) - Pages 7-4 Special cases:. If, then is an uclidean -space.. If,, then is a Minkowski -space []. Proposition. For, R, the inner vector products satisfy the following properties;. uv vu.. g( uv, w) g( vw, u) g( uw, v).. u( vw) g( uw) v g( u, v) w. Definition. A matrix A is called a uasi-orthogonal matrix if A A det A where he set Q() of all uasi-orthogonal matrices with the operation of matrix multiplication constitutes, R 0. a group, called rotation group, in the -space. Definition. Consider rotational motion of a rigid body, i.e., a rigid body motion leaving one point fixed. Let r (0) be the position of a point in the body at t 0, rt () its position at time t, Rt () the rotation operator which carries r(0) to rt () rt () Rt () r(0). R() t thus traces the orientation of the rigid body in configuration space SO(). he velocity of the point is r Rr (0) RR ror r r, RR. he operator is the angular velocity operator [5]. B. Generalized uaternions algebra A generalized uaternion is an expression of form a0 ai a jak where a0, a, a a are real numbers i, j, k are uaternionic units which satisfy the eualities i, j, k, ij k ji, jk = i = kj, ki = j = ik,, R. he set of all generalized uaternions are denoted by H. A generalized uaternion is a sum of a scalar a vector, called scalar part,, S a vector part V 0 ai a jak. herefore, H is form a 4-dimensional real space which contains the real axis R a -dimensional real linear space, so that, H R. If ( a0, V ) p ( b0, V p ) are two uaternions, their sum is defined as p ( a0 b0, V Vp) their product (non-commutative) as p ( a0b0 g( V, Vp), a0v p b0v VpV), here "g(,)" " " are the inner vector products in, respectively. he conjugate uaternion of is defined as ( a0, V ) the length or norm as N a a a a R. 9

4 Vol (07) - Pages 7-4 Note that N N N. very non-zero uaternion has a multiplicative inverse given by its conjugate divided by its p p norm: / N. he generalized uaternion with a norm of one, N, is a unit generalized uaternion. If a generalized uaternion is looked at as a four-dimensional vector, the generalized uaternion product can be described by a matrix-vector product as a0 a a ab0 a a0 a a b p. a a a0 a b a a a a0 b II. ROAION OPRAORS IN H In uclidean -space, there are a lot of methods used to represent rotations like orthogonal matrices, uler angles uaternions. Quaternionic representation is the most useful one for the purpose. If we compare to orthogonal matrices, there are some constraints as each column of an orthogonal matrix must be unit vector they must be perpendicular to each other. hese constraints make it difficult to construct an orthogonal matrix using nine numbers. But we can construct easily a rotation orthogonal matrix using a unit uaternion. hat is, only four numbers are enough to represent a rotation such that there is only one constraint which is that the norm of the uaternion must be eual to one. Here, we show that every unit generalized uaternion represents a rotation in the -space. Let a Lie group G be given, to its every element the mapping C : G G, x x for all x G is assigned. he mapping C is a differentiable isomorphism (inner automorphism) of the group G, the differential of C at the point e, i.e. ( dc) e ad maps G () e into G (). e hen, the mapping ad is called the adjoint representation of the group G [8]. If we associate the generalized uaternion x (0, x) with the three-dimensional vector x define the operation, with the unit generalized uaternion, as ad( x) x ' x x, () then this transformation, from x to x ', represents a rotation in -space. he resulting uaternion x ' is a vectorial uaternion with the same length as x, since N N N N N N N x ' x x. x heorem. For any unit generalized uaternion, the matrix corresponding to map ad is Special cases: a0 a a a ( aa aa 0 ) ( aa aa 0 ) ad ( aa a0a) a0 a a a ( aa a0a). ( aa aa 0 ) ( aa 0 aa ) a0 a a a ) For the case, we get ad for real uaternion H, denote it by M. M is a orthogonal matrix, then the map ad corresponds to a rotation in. If we take the rotation axis to be S ( s, s, s), then we have where S is a skew-symmetric matrix, M I S S sin ( cos ), 0 s s S s 0 s. s s 0 40

5 Vol (07) - Pages 7-4 ) For the case,, we get ad for split uaternion H', denote it by N. N is a semi-orthogonal matrix, then the map ad corresponds to a rotation in -Minkowski space C ( c, c, c ), then we have be where C is a skew-symmetric matrix, i.e. N I C C sinh ( cosh ), C C. If we take the rotation axis to 0 c c 0 0 C c 0 c, 0 0 c c heorem 4. For every G, R, ad is a uasi-orthogonal matrix. []. heorem 5. Let be a unit generalized uaternion, R, then the matrix ad can be written as ad I sins ( cos )S, where cos a0, si sin ai, i,,. 0 s s S s 0 s s s 0 is a skew-symmetric matrix, i.e., S S. Proof: very unit generalized uaternion a0ai a jak can be written in polar form cos S sin, then the matrix ad can be written as cos ( s s s) sin ( ss sin scos sin ) ( ss sin scos sin ) ad ( sssin scos sin ) cos ( s s s)sin ( sssin scos sin ) ( ss sin scos sin ) ( ss sin scos sin ) cos ( sss)sin ( s s s )sin sssin ssin sssin ssin I4 sssin ssin ( s s s )sin sssin ssin ss sin ssin ss sin ssin ( sss)sin with used of sin cos s s s, we have 0 s s s s ss ss M Isin s 0 s ( cos ) ss s s ss, s s 0 ss ss s s so, proof is complete. xample. Let x such that x i 0 j k. We want to rotate point x about the unit vector s i j at an angle of. Using theorem, we find a unit generalized uaternion that will accomplish such a rotation. 4

6 Vol (07) - Pages 7-4 cos Ssin / / cos,,0 sin i j. Now we have defined a unit generalized uaternion such that x x' where x ' vector s at an angle of. Now that is known, the operator ad is applied. is the point x rotated about x' x ( i j)(i 0 j k) ( i j) (,, ). Corollary: Let cos S sin be a unit generalized uaternion. If, R, the linear map ( ) represents a rotation of the original vector by an angle around the axis S in -space map represents a rotation in Lorentzian manner (boost).. Also, if 0, 0, the above III. ANGULAR VLOCIY OF H MOION Suppose that the orientation of a body is represented as a uaternion rotation so that x ' x where x is a vector in fixed space coordinates x ' is a vector in body coordinates. By differentiational of the rotational transformation () as in x' x x, expressing it in terms of fixed coordinates gives x' ( ) x ( ) x ( ) ( ), x' x'. he scalar part of happens to be zero because is a unit generalized uaternion, so applying this into the above gives ( ), x' x' x'. which is just the antisymmetric product of x '. So the velocity x ' has a zero scalar part a vectorial part, x ' where (0, ). We conclude that the angular velocity w expressed in the space fixed reference in terms of the uler parameters time derivatives as follows or w 4

7 Vol (07) - Pages w a0 a a a a 0 a a a a a a a a0 a a a a a a a 0. 0 Special Case: If, then w is the angular velocity for real uaternion rotation [7]. ACKNOWLDGMN he author is very grateful to referees for the useful suggestions remarks for the revised version. RFRNCS [] 4 L. Kula., Y. Yayli, Split uaternions rotations in semi-uclidean space, Journal of Korean Math. Soc., vol. 44(6).-7, 007. [] M. Ozdemir, A.A. rgin, Rotations with unit timelike uaternions in Minkowski -space, Journal of geometry physics 56, pp [] J.P. Ward, Quaternions Cayley numbers algebra applications, Kluwer Academic Publishers, London, 997, pp. 5. [4] H. Pottman J. Wallner, Computational line geometry, Springer-Verlag Berlin Heidelberg New York, 000, pp. 50. [5] M. Jafari, Generalized Hamilton operators Lie groups, Ph.D. thesis, Ankara University, Ankara, urkey, 0. [6] M. Jafari Y. Yayli, Rotation in four dimensions via generalized Hamilton operators, Kuwait journal of science, vol. 40() pp , 0. [7] W. B. Heard, Rigid body mechanics, Alexria, Wiley-vch Verlag GmbH, 006. [8] A. Karger J. Novak, Space kinematics Lie groups. Gordon Breach science publishers, Now York,