62 Basilio Bona - Dynamic Modelling

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1 6 Basilio Bona - Dynamic Modelling Only 6 multiplications are required compared to the 7 that are involved in the product between two 3 3 rotation matrices...6 Quaternions Quaternions were introduced by Hamilton who discovered them in 843, in order to generalize in three dimensions the complex numbers and their characteristic of being plane rotators. For an detailed history of the problems and discussion raised by the quaternions in the scientific community in the middle of XIX century, read the interesting monograph by Crowe [7]. The quaternion algebra, with its definitions and operators used for representing rotations, is described in details in Appendix D. The symbol for the generic quaternion is h...7 Quaternions and Rotations In order to use a quaternion u = (u 0, u, u u 3 ) to represent a rotation, we assign to each element u i, an Euler parameter v i u0 = v4 The quaternion so defined is therefore u = v u = v u 3 = v 3 (.88) u = (u 0, u, u, u 3 ) = (cos θ, u sin θ, u sin θ, u 3 sin θ ) and has unit norm. Since a biunivocal correspondence exists between the Euler parameters and the unit quaternions, any unit quaternion represents a rotation in the 3D space, as any unit complex number represents a rotation in the D plane. We write R(u) to indicate that for every rotation there is a corresponding a unit quaternion u and vice versa. In order to convert a unit quaternion u = (u 0, u v ) into the corresponding matrix R(u), we use the following relation, that is equal to (.84), valid for Euler parameters: R(u) = (u 0 u T v u v )I + u v u T v u 0 S(u v ) = u 0 + u u u 3 (u u u 3 u 0 ) (u u 3 + u u 0 ) (u u + u 3 u 0 ) u 0 u + u u 3 (u u 3 u u 0 ) (.89) (u u 3 u u 0 ) (u u 3 + u u 0 ) u 0 u u + u 3

2 Basilio Bona - Dynamic Modelling 63 Conversely, to compute the quaternion u starting from the elements r ij of the rotation matrix R(h)we use a relatione that is equal to (.85), valid for Euler parameters: u 0 = ± ( + r + r + r 33 ) u = (r 3 r 3 ) u = (r 3 r 3 ) u 3 = (r r ) When u 0 = 0, i.e., θ/ = π/ θ = π = 80, we use a different formula (.90) u 0 = ( + r + r + r 33 ) u = sgn(r 3 r 3 ) ( + r r r 33 ) u = sgn(r 3 r 3 ) ( r + r r 33 ) (.9) where sgn(x) is the sign function of x u 3 = sgn(r r ) ( r r + r 33 ) sgn(x) = + for x > 0 sgn(x) = 0 for x = 0 sgn(x) = for x < 0 According to (.9) (.3), elementary rotations R(i, α), R(j, β) e R(k, γ) correspond to the following elementary quaternions: R(i, α) u = (cos α, sin α, 0, 0) R(j, β) u = (cos β, 0, sin β, 0) (.9) R(k, γ) u 3 = (cos γ, 0, 0, sin γ ) therefore the vectorial base of the quaternions corresponds to the three elementary rotations by 80 angles around the principal axes: i = (0,, 0, 0) R(i, π) j = (0, 0,, 0) R(j, π) k = (0, 0, 0, ) R(k, π) (.93)

3 64 Basilio Bona - Dynamic Modelling Observe that, while the product of two equal elementary quaternions gives iı = jj = kk = ijk = (, 0, 0, 0), the analog product of two equal elementary rotation matrices gives the identity matrix, i.e., a rotation of kπ: R(i, π)r(i, π) = R(j, π)r(j, π) = R(k, π)r(k, π) = R(i, π)r(j, π)r(k, π) = I (.94) corresponding to the quaternion (, 0, 0, 0); this strange property is related to a new entity, called spinor, that will not be further discussed here; the interested reader can find more details in [], [8] and [38, Ch. ]. The quaternion operations that are related to the computation of rotations are the following:. Given n rotations R, R,, R n and the corresponding unit quaternions u, u,, u n, the product of rotations corresponds to the quaternion product in the same order; R(u) = R(u )Ru ) R(u n ) u = u u u n. given the rotation R(u) and the corresponding unit quaternion u, the transpose matrix R T corresponds to the conjugate unit quaternion u R(u) R T (u ) 3. given a generic vector x, that corresponds to a generic quaternion consisting on the vectorial part only x = (0, x T ) = (0, x, x, x 3 ) and given the rotation R(u) corresponding to the unit quaternion u, the rotated vector x = R(u)x coincides with the quaternion product, that has always a zero real part x = R(u)x x = uxu 4. the product of two or more rotation matrices product R (u )R (u ) is computed adopting the following identity: R (u )R (u )x u (u xu )u = (u u )x(u u ) = (u u )x(u u ). As a last comment, it should be taken into consideration the fact that in space applications and aerospace textbooks often the quaternions are organized in a list theta is different from the one adopted here: namely the real part is the fourth term of the quaternion, not the first as in our notations.

4 Basilio Bona - Dynamic Modelling 65 Example.. Given the elementary rotation R(j, 90 ), find the relative Euler parameters and the corresponding unit quaternion u Since 0 0 R(j, 90 ) = applying relations (.85) and (.90), we have u 0 = v 4 = u = v = 0 u = v = u 3 = v 3 = 0 that are the same as those obtained applying (.8). Example.. Find the quaternion and the Euler parameters that represent the rotation R = R (i, 90 ) R (j, 90 ), then compute the rotation axis u and the angle θ. The two quaternions are: ( ) R (i, 90 ) u =,, 0, 0 ( ) R (j, 90 ) u =, 0, 0 from which we obtain the product quaternion R (i, 90 ) R (j, 90 ) u u = }{{} i j, 0 i + j + (i }{{ j } ) = k (,,, ) Therefore the Euler parameters are v 4 = h 0 = ; v = h = ; v = h = ; v 3 = h 3 =

5 66 Basilio Bona - Dynamic Modelling The angle is θ ( ) = arccos θ = 0 and the axis is u = [ ] T ; since u norm is not unit we compute its norm u = 3 and after that, applying (.8) we obtain the same result = sin θ 3 sin θ = 3 θ = 60 θ = 0..8 Cayley-Klein Parameters Another parameterizations of rotations was introduced by Felix Klein (849-95), with the aim to make easier the integration of differential equations in complex gyroscopic problems [6]. This parameterizations, as well as that with quaternions or Euler parameters, has the advantage of not requiring the computation of trigonometric functions. The Cayley-Klein Parameters (CK parameters) are represented by complex matrices [ ] α β Q = (.95) γ δ where { α β γ δ } are complex variables. Q must be unitary, hence the CK parameters obeys to the following constraints: α = δ β = γ and the matrix Q can be written as [ ] α β Q = β α with an additional constraint αα + ββ = In Q there are three free parameters and they can be used to characterize the rotations: indeed from the CK parameters it is possible to compute the rotation matrix (α β γ + δ j ) ( α β + γ + δ ) γδ αβ R = j (α β + γ δ ) (α + β + γ + δ ) j(αβ + γ + δ) (.96) βδ αγ j(αγ + βδ) αδ + βγ