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On the Relation between Rotation Increments in Different Tangent Spaces S. Ghosh 1 D. Roy ∗ Structures Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India.

Abstract In computational mechanics, finite rotations are often represented by rotation vectors. Rotation vector increments corresponding to different tangent spaces are generally related by a linear operator, known as the tangential transformation T . In this note, we derive the higher order terms that are usually left out in linear relation. The exact nonlinear relation is also presented. Errors via the linearized T are numerically estimated. While the concept of T arises out of the nonlinear characteristics of the rotation manifold, it has been derived via tensor analysis in the context of computational mechanics [1] . We investigate the operator T from a Lie group perspective, which provides a better insight and a 1-1 correspondence between approaches based on tensor analysis and the standard matrix Lie group theory. Key words: Finite rotation; Rotation manifold; Tangent space; Lie group

1. Introduction Three dimensional finite rotations are important in studying rigid body and flexible body mechanics. 3D rotations are non-commutative and elements of the set of proper orthogonal tensors, also known as special orthogonal group SO(3); this is a nonlinear manifold. Over the years, several parametrizations of finite rotations have been used [2–5]. Different parametrizations have their own advantages and disadvantages depending on the application. A rotation vector parametrization, in particular, is easier to implement within the finite element framework, requires only three parameters and has a simple geometric interpretation. Moreover, every rotation has a rotation vector, notwithstanding the non-uniqueness. From the standpoint of the SO(3) manifold [1,6], the rotation-vector parametrization is a projection from the SO(3) manifold onto a specific tangent-space of SO(3). Vector addition of rotation vectors is strictly not admissible since they could be associated with different tangent spaces depending on the base point of rotation. However, Cardona and Geradin [1] have derived the linearized operator (call tangential transformation) relating two incremental rotation vectors corresponding to two different tangent spaces. The two rotational increments with two different rotations as base points are treated as small perturbations on these base rotations. Then one incremental rotation tensor is represented as a compound rotation of the other incremental rotation and the relative rotation between the base points. Finally the linear operator is obtained via the directional derivative of Argyris’ formula [7] for the rotational pseudo vector of compound rotation. This approach is followed in a number of papers that use the rotation vector parametrization and deal with rigid body dynamics [8–11] as well as flexible body mechanics [12–17]. However, it might be noted that the use of the linear operator, the tangential transformation, in the aforementioned works are justified, since in the variational formulation and in the linearized weak form of the governing nonlinear equation only linear part is sufficient. Given the non-admissibility of vectorial addition of two rotation vectors, the tangential transformation, T , provides a way of respecting the nonlinear nature of the rotation manifold. ∗ Corresponding author. Email addresses: [email protected] (S. Ghosh), [email protected] (D. Roy). 1 Current address:Laboratori de C` alcul Num`eric, Departament de Matem`atica Aplicada III. Universitat Polit`ecnica de Catalunya, Barcelona, Spain

Preprint submitted to Mechanics Research Communications

8 April 2010

T is a linear operator relating rotation vectors in two tangent spaces and this relation neglects higher order terms in the rotation increments. The higher order terms are obtained in section 3. This section also describes an exact transformation that relates rotation vector in one tangent space to another. While the existing formula of T is derived via simple tensor analysis, the concept of tangential transformation arises out of an inherently nonlinear nature of the rotation manifold. An objective of this study is thus to investigate T from the standpoint of matrix Lie groups and this is done in section 4. However, to begin with, we briefly recapitulate some basic facts about finite rotations in the next section. 2. Some Facts about Finite Rotations In what follows, we recap a few basic facts of rotations, rotation vector parametrization and the relation between elements corresponding to different tangent spaces ([18,1,9]). The special non-commutative Liegroup of proper orthogonal linear transformations is defined as S O(3) = {R : R3 → R3 |Rt R = I, det(R) = 1}

(1)

By Euler’s theorem, any rotation tensor R represents a rotation with respect to a vector ΨI through an angle ||ΨI ||. This rotation vector is related to rotation tensor via the exponential mapping: ˜2+ 1 Ψ ˜3 + ··· ˜ I) = I + Ψ ˜I+ 1Ψ R = exp(Ψ 2! I 3! I

(2)

˜ 3 = −kΨI k2 Ψ ˜ I , the last equation may be modified to derive Rodrigues formula: Using the identity Ψ I R=I+

 sin(kΨ k/2) 2 sin kΨI k ˜ I ˜2 Ψ ΨI + 2 I kΨI k kΨI k

(3)



Here •˜ denotes the skew symmetric tensor (matrix) corresponding to the axial vector • . The subscript I indicates that the rotation vector Ψ is represented in the non-rotated material frame, i.e. when R = I. That this parametirzation is non-unique is evidenced from the fact that Ψ = [2kπ, 0, 0]t , k = 0, 1, 2, · · · yields the same rotation R = I. We recall that the set of skew symmetric matrices constitutes the Lie algebra: ˜ 3 3 ˜ ˜t so(3) = Ψ I : R → R | ΨI + ΨI = 0 2.1. Left and Right Translations in SO(3) and Their Invariant Vector Fields Variations of rotation (in the form of compound rotations) are often represented through material and spatial descriptions. The ‘left translation map’ (material description) and the ‘right translation map’ (spatial description) of the Lie-group are defined as: ˜ R) Rnew = RRle f t = R exp(Θ

Rnew = Rright R = exp(θ˜ R ) R

(4)

Here ΘR and θR are the incremental material and spatial rotation vectors respectively with respect to the base orientation R; Rright = RRle f t Rt . The material and spatial rotation vectors are mutually related by θ = RΘ. ˜ with respect to t at t = 0 gives the tangent space at R. This left-invariant Differentiating R exp(tΘ) (material) vector field on S O(3) at R is defined as: ˜ ˜ ˜ ˜ (5) le f t T R S O(3) = ΘR = (R, Θ) | with left translation RΘ, R ∈ S O(3), Θ ∈ so(3) 2

˜ R is an element of material vector field le f t T R S O(3), i.e. the skew-symmetric vector field with R repreΘ senting the base orientation. Similarly the right invariant (spatial) vector field may be defined as: ˜ ˜ ) | with right translation θ˜ R, R ∈ S O(3), θ˜ ∈ so(3) (6) right T R S O(3) = θ R = (R, θ

For notational simplicity, we may sometimes drop the subscript R if the base orientation is clear from the context. For a geometric interpretation of these translations, see [9]. The material vector space on the rotation manifold is presently defined as:  ˜ ˜ le f t T R = ΘR | ΘR ∈ le f t T R S O(3), R = exp(ΨI )

2.2. Relation between Rotation Increments with Two Different Base Points We note that elements of the material vector field are amenable to vector addition only if the associated skew-symmetric tensors belong to the same tangent space of rotation manifold. The above definition of material vector space is in line with equation (5). Besides it is particularly useful in the present work, since rotations are parametrized by rotation vectors. Now consider the material form of the compound rotation as described in equation (4). It is important to identify the tangent space to which the incremental rotation corresponds (see figure 1(a)). Note that the elements of the material vector field may be added (just like vectors) provided that the corresponding skew-symmetric tensors belong to the same tangent space of the manifold. In the context of engineering mechanics, this has first been pointed out in [1]. In order to facilitate an intuitive understanding of the relation between two different incremental rotations from a geometrical perspective, Figure 1(a) is useful; this figure is due to [1]. A relation between two incremental rotation vectors is derived in [1]. As shown in Figure 1(b), let RA, RB be two given ˜ Let RB be perturbed to obtain RBǫ such that: rotations such that RB = RA exp(Ψ). ˜ B ) with Ψǫ = Ψ + ǫΘA RBǫ = RB exp(ǫ Θ ˜ exp(ǫ Θ ˜ B ) = RA exp((Ψ + ǫΘA )e ) i.e. RA exp(Ψ)

(7)

ΘB = T (Ψ) ΘA ! "  1 − cos θ  # sin θ sin θ ˜ Ψ I+ 1− e⊗e− T (Ψ) = θ θ θ

(8)

˜ B ∈le f t T RB S O(3) and Ψ, ˜ Ψ ˜ ǫ, Θ ˜ A ∈le f t T RA S O(3). The symbol (•)edenotes (•). ˜ We need to obtain Here Θ a functional relationship between ΘA and ΘB , representing the rotation increments corresponding to two different tangent spaces. Using the formula of composition of rotation vectors and taking derivative with respect to ǫ at ǫ = 0, the following relation has been obtained in [1]:

where,

θ = kΨk ; e = Ψ/kΨk

2.3. Relation between Angular Velocity Vectors with Different Base Points ˜ Let R(t), RA be two given rotations such that R(t) = RA exp(Ψ(t)) with RA being a constant rotation. Let 2 2 θ = kΨk, α = sin(θ)/θ, β = (1 − cos(θ))/θ , γ = θ . From Rodrigues formula (equation 3), one can write ˜ R = Rt R′ ) as: the material angular velocity tensor (Ω ˙˜ Ψ ˜ R = α˙ Ψ ˜ + αΨ ˜˙ + (βα˙ − αβ˙ )Ψ ˜ 3 − αβΨ ˜Ψ ˜ Ω ˙˜ + (β˙ + ββ˙ Ψ ˙˜ Ψ ˙˜ Ψ ˜ 2 )Ψ ˜ + (β − αα˙ − α2 + β2 Ψ ˜ ˜ 2 )Ψ ˜Ψ ˜ 2 + β2 Ψ ˜ 2Ψ +βΨ ˙˜ = Ψ. ˜˙ Now, using the identities Note that Ψ 3

Exponential map RΑ

RΑ

T SO(3) RA

~ Ψ

Α

T R SO(3) A

~

Ψ SO(3)

~

Ψε

SO(3)

~

ε ΘΑ

RΒ

RΒ

~

ΨΑ

~

ε ΘΒ

T SO(3) RB

T SO(3) RB

RΒ

ε

(a)

(b)

˜ A ∈ T R S O(3) and Ψ ˜ B ∈ T R S O(3)); Fig. 1. Schematic representation of : (a) rotation increments corresponding to different tangent spaces (Ψ A B (b) projection of rotation increments

˜ 2Ψ ˜ 1 = −(Ψ1 , Ψ2 )Ψ ˜ 2 ∈ so(3) ˜ 1; Ψ ˜ 1Ψ ˜ 1 = −γΨ ˜ 1; Ψ ˜ 1Ψ ˜ 31 = −(Ψ1 , Ψ1 )Ψ Ψ ˜ 1Ψ ˜2 −Ψ ˜ 1 = Ψ] ˜ 2Ψ ˙ Ψ β2 γ + α2 = 2β; β˙ − αα˙ − ββ˙ γ = β2 (Ψ, Ψ) 1 ×Ψ2 ;

(9)

we get: ˙ ΩR = T (Ψ)Ψ

(10)

˙˜ ∈ T S O(3). Thus ˜ ∈ le f t T R S O(3) and Ψ Here, (·, ·) denotes standard inner product. We observe that Ω le f t RA the material linear transformation T (Ψ) (i.e., the tangential transformation) is a linear mapping between the left invariant vector fields T (Ψ) : le f t T RA → le f t T R . Thus the tangential transformation shifts the base point of angular velocity. This is also clear from the fact that T (Ψ) → I only when Ψ → 0. Generally, the same transformation is used for transforming rotation increments (equation 8) and angular velocities (equation 10). Nevertheless, it should be noted that time derivative of equation 8 does not lead to the angular velocity relation (10), since the first one is an linear (i.e. exact only upto first order) relation while the second one is exact. ˜ ˜ ∈ right T R S O(3). Similar to the case of In spatial representation, we may write R(t) = exp(ψ(t))R A, ψ material angular velocity, the spatial angular velocity vector ωR may be written as: " !  1 − cos θ  # sin θ sin θ ωR = ψ˜ ψ˙ or, ωR = T t (ψ)ψ˙ (11) I+ 1− e⊗e+ θ θ θ We will sometimes drop the subscript R for notational convenience. The spatial tangential transformation T t is a mapping between vector spaces on the rotation manifold, i.e., T t (ψ) : right T RA → right T R 3. Exact Relation between Two Rotation Increments Corresponding to Two Different Tangent Spaces and Errors through Linear Relation The relation between the angular velocity and the time derivative of rotation vector, i.e. equations (10) and (11), is directly obtained from the definition of the angular velocity tensor and hence it is exact. This may be contrasted with the linear relation between two rotation increments corresponding to two different tangent spaces (i.e. the equation 8), which is only linear in the rotation increment and neglects higher order terms. Indeed, the relation (8) is only a first order approximation to the exact one. With this in background, the neglected terms in equation (8) are obtained in the present section and the exact 4

transformation for incremental rotations (not necessarily small) belonging to two different tangent spaces is presented. Subsequently the possible errors through equation (8) are investigated numerically. In order to obtain the the additional (error) terms in equation (8), let us consider equation (7) and account for the terms left out in equation (8). The associated rotations are defined by equation (7). Using Taylor’s expansion, we have: i h ˜ A) ˜ + ǫΘ ˜ A ) = exp(Ψ) ˜ + D exp(Ψ) ˜ · ǫΘ ˜ A + g(Ψ, ǫ Θ (12) exp(Ψ

The second term on the right hand side denotes the directional derivative and the third term the remainder ˜ that goes to zero faster than ǫ Θ. ˜ Since R(Ψ ˜ + ǫΘ ˜ A ) = exp(Ψ)exp(ǫ Θ ˜ B ) (see equation 7), function of ǫ Θ we get: ] ˜ B ) = I + exp(−Ψ) ˜ d exp(Ψ + ˜ g1 (Ψ, ǫ Θ ˜ A) exp(ǫ Θ δǫΘA ) + exp(−Ψ) (13) dδ δ=0 ˜ g(Ψ, ǫ Θ ˜ A) = I + ǫ Tg ΘA + exp(−Ψ) ] ˜ d exp(Ψ + δǫΘA ) = ǫ Tg ΘA may be proved following similar steps used to derive The identity exp(−Ψ) dδ δ=0 equation (10). Let the right hand side of the equation (13) be denoted by the orthogonal tensor U. Using the formula for the natural logarithm of the rotation matrix [19], we get: ˜B = ǫΘ

sin−1 kyk y˜ kyk

where

1 y˜ = (U − U t ) 2

˜ g(Ψ, ǫ Θ ˜ A ). By Maclaurin series We may also write y˜ = ǫ Tg ΘA + 12 (V − V t ), where V = exp(−Ψ) −1 expansion of sin kyk, it follows that:    ] A + 1 (V − V t ) ˜ B = 1 + 1 kyk2 + 3 kyk4 + · · · ǫ T (Ψ)Θ (14) ǫΘ 6 40 2 ] A + 1 kyk2 (V − V t ) ˜ B = T (Ψ)Θ ] A + 1 (V − V t ) + 1 kyk2 T (Ψ)Θ or Θ 2ǫ 6 12ǫ 2

Since g1 is of order O(kǫΘk2 ), we observe that V is of order O(kǫΘk2 ). Hence all the terms on the right hand side excepting the first one have been neglected in equation (8). Note that ǫ could have been discarded in equations (12) through (14) by just requiring that ΘA is a small vector. However ǫ is retained in order to make the present case precisely correspond to what is done in [1], enabling an easier comparison between equations (8) and(14). Thus the neglected terms of equation (8) are readily obtained. In the following, we will present the exact relation. Let exp(w) ˜ = exp(˜u)exp(˜v), where u˜ , v˜ , w ˜ ∈ so(3). The first few terms of the BCH (Baker-CampbellHausdorff) formula are given below, with all higher order terms involving [˜u, v˜ ] and commutator nestings (in so(3)) thereof. 1 1 1 1 1 w ˜ = u˜ + v˜ + [˜u, v˜ ] + [˜u, [˜u, v˜ ]] − [˜v, [˜u, v˜ ]] − [˜v, [˜u, [˜u, v˜ ]]] − [˜u, [˜v, [˜u, v˜ ]]] + · · · 2 12 12 48 48

No closed-form expression for the above relation is generally available for an arbitrary Lie-algebra. However, as an exception for the so(3), there is a closed-from expression provided recently in [20]. This gives: w ˜ = λ˜u + σ˜v + µ[˜u, v˜ ];

in axial vector form:

w = λu + σv + µ u × v

(15)

2 sin−1 is implemented in MATLAB such that, in case of all positive real parts of the eigenvalues of U = exp(ǫ Θ ˜ B ), sin−1 ≡ arcsin. However, for the case of two equal negative parts, sin−1 ≡ π − arcsin.

5

−1

−1

−1

Here λ, σ, µ are real constants given by λ = sind d ϑa , σ = sind d φb , µ = sind kvk, ∠(u, v) =angle between two vectors u and v. 3 a, b, c and d are defined as:

d c ϑφ ,

with ϑ = kuk, φ =

a = sin ϑ cos2 φ/2 − sin φ sin2 ϑ/2 cos(∠(u, v)) b = sin φ cos2 ϑ/2 − sin ϑ sin2 φ/2 cos(∠(u, v)) 1 c = sin ϑ sin φ − 2 sin2 ϑ/2 sin2 φ/2cos(∠(u, v)) 2p d = a2 + b2 + 2ab cos(∠(u, v)) − c2 sin(∠(u, v))2 Here, cos (∠(u, v)) =

p (u, v) , sin (∠(u, v)) = 1 − cos2 (∠(u, v)) kuk kvk

˜ B) = In order to obtain the exact transformation for incremental rotation vectors, we compare exp(Θ ˜ ˜ +Θ ˜ A ) with equation (15). This leads to: exp(−Ψ)exp( Ψ h i ˜ ΘA ΘB = (σ − λ)Ψ + σ I − µ Ψ (16) λ, σ and µ represent the same functions as used in equation (15) for vectors ΘB , −Ψ and Ψ + ΘA (instead of u, v, w). At this stage, further simplification of the above expression appears to be difficult. The exact transformation for incremental rotation vectors (equation 16) is rather involved compared to the linear one. The linear relation is exact upto first order term but the magnitude of the higher order terms could be quite large and increasing with Ψ as evident from the equation (16). It is worth mentioning that the Spurrier algorithm [21] is widely used to extract rotation vector from rotation matrix. To emphasize that there is no ill-conditioning in the exact analytical relation (equation 16), in figure 2 we have plotted the norm of difference between ΘB obtained via Spurrier algorithm and the exact relation for very large random rotation vectors. Note that one has to choose a rotation vector or its complement [1] (to choose the rotation vector with length ≤ π) while comparing with Spurrier algorithm and using equation (16). It is clear that the two methods match exactly except the round-off errors. 0

0

10

−2

10

−2

10

−2

10

−4

10

−4

10

−4

10

−6

10

−6

10 Error

0

10

−6

10

−8

10

−8

10

−8

10

−10

10

−10

10

−10

10

−12

10

−12

10

−12

10

−14

10

−14

10

−14

10

−16

10

−16

10

−16

10

0

5

10

0

||Ψ||

||ΘA||

5

0

2 ||ΘB||

Fig. 2. Norm of difference between Θ B obtained via Spurrier algorithm and equation 16, for different kΨk, kΘA k and kΘ B k.

3

See footnote 2

6

4. Tangential Transformation via Matrix Lie Group Theory In this section the tangential transformation is derived from matrix Lie group theory. At the onset, the matrix Lie group and Lie algebra and the well known formula for derivative of exponential map are touched upon very briefly. Detailed accounts of these may be found in [22,23,19,24]. 4.1. A Brief Review of Matrix Lie Group Theory A matrix Lie group is a smooth subset G ⊆ Rn×n , closed under matrix products and matrix inversion. The Lie algebra Lie group G is the linear subspace g ⊆ Rn×n consisting of all matrices of o n g of a matrix dρ the form g = A ∈ Rn×n : A = dt |t=0 , where ρ(t) ∈ G is a smooth curve such that ρ(0) = I. The space g is closed under matrix addition, scalar multiplication and the Lie bracket (or commutator) [A, B] = AB − BA where, A, B ∈ g Consider a matrix Lie group G and its Lie algebra g. The exponential mapping, exp : g → G, is defined as: ∞ X Aj exp(A) = where, A ∈ g; exp(A) ∈ G j! j=0 Hence, for A ∈ g, we have exp(A) ∈ G. Moreover, exp is a local diffeomorphism in a neighbourhood of A = 0. Here exp(A = 0) is the identity I of G. The adjoint representation, Ad, and its derivative, ad : For each P ∈ G, a linear map AdP : g → g is defined by the formula AdP (A) = PAP−1

where, A ∈ g, P ∈ G

For each P ∈ G, AdP is an invertible linear transformation of g with inverse AdP−1 , and the map P → AdP is a group homomorphism of G into GL(g), where GL(g) denotes the group of all invertible linear transformations of g. For A ∈ g, a linear map adA : g → g is defined by: adA (B) = AB − BA = [A, B] where, A, B ∈ g

(17)

Therefore, the map A → adA (i.e. ad) may be viewed as a linear map from g into gl(g), where gl(g) denotes the space of linear operators from g to g. The ad notation is useful in some situations, for example in writing [A, [A, [A, [A, B]]]] as ad4A B. The derivative of the exponential mapping dtd exp(A(t)) is a tangent to the Lie group G at point P(t) = exp(A(t)). The differential of exponential mapping is defined as the right trivialized tangent to the exponential map, i.e. as a function dexp:g × g → g such that: d exp(A(t)) = dexpA(t) (A′ (t)) exp(A(t)) dt

(18)

Note that AdP (•), adA (•) and dexpA (•) are linear in their second argument, (•), for fixed A and P. Hence they may be considered as matrices acting on g. dexpA is an analytic function of the matrix transformation adA : dexpA =

exp(adA ) − I adA

This can be written as the following series: 7

1 1 1 [A, C] + [A, [A, C]] + [A, [A, [A, C]]] + · · · 2! 3! 4! ∞ X 1 j adA C = ( j + 1)! j=0

dexpA (C) = C +

(19)

Similarly using the relation Adexp(A) = expm(adA ), one can write the derivative of the exponential map based on left trivialisation as: d exp(A(t)) = exp(A(t)) dexp−A(t) (A′ (t)) (20) dt Substituting A(t) by A + tC, where A, C ∈ g, and evaluating at t = 0: ( ) I − exp(−adA ) d (C) exp(A + tC) = exp(A) dexp−A (C) = exp(A) dt adA t=0 1 1 1 (21) = C − [A, C] + [A, [A, C]] − [A, [A, [A, C]]] + − · · · 2! 3! 4! 4.2. Derivation of the Tangential Transformation In arriving at the tangential transformation, we employ the matrix Lie group theory for rotations be˜ ∈ so(3), R(s) = longing to the special orthogonal group SO(3) and its Lie algebra so(3). Thus, for Ψ(s) ˜ exp(Ψ(s)) ∈ SO(3) and using equation 20, the derivative of R(s) is given by d ˜ ˜ ˜ ′ (s)) = exp(Ψ(s)) dexp−Ψ(s) R˙ = exp(Ψ(s)) ˜ (Ψ dt Therefore the material bending curvature tensor is: ˜ ′ (s)) K˜ = Rt R′ = dexp−Ψ(s) ˜ (Ψ " # exp(ad−Ψ(s) ˜ )−I ˜ ′ (s) = Ψ ad−Ψ(s) ˜

(22) (23)

Using equation (19), we have:   ∞ X  1 j  ˜ ′ K˜ =  ad−Ψ(s) ˜  Ψ (s) ( j + 1)!

(24)

j=0

j ˜ ˜ · · ·]]. Using [˜a, b] ˜ = a] From equation we have: ada˜ (b) = [˜a, [˜a, · · · [˜a, [˜a, b]] × b repeatedly, we get: h i(17), ∼ j ˜ ∼ j e ada˜ (b) = [˜a ]b ; where [·] denotes [·]. Thus one gets:   ∼ ∞  X  j (− 1 ) ˜ j  Ψ′ (s) Ψ(s) K˜ =    ( j + 1)! j=0

˜ we arrive at: ˜ 3 = −θ2 Ψ, Considering its axial vector and using the relation Ψ   ∞  X (−1) j−1 2( j−1) ˜ 2  ′ (−1) j 2( j−1) ˜ θ Ψ+ θ Ψ  Ψ K = I + (2 j)! (2 j + 1)! j=1

P P∞ (−1)n 2n (−1)n 2n −1= ∞ n=1 (2n+1)! x and cos x − 1 = n=1 (2n)! x , the above can be simplified to: ! "  # cos θ − 1  ˜ sin θ 1 ˜ 2 ′ Ψ+ 1− K= I+ Ψ Ψ = T (Ψ)Ψ′ θ θ2 θ2

Using

sin x x

8

(25)

This equation is exactly the same as equation 10 derived by Cardona and Geradin [1]. Now, we consider the relation between two rotation increments belonging to two different tangent ˜ + ǫΘ ˜ A ) = exp(Ψ) ˜ exp(ǫ Θ ˜ B ), as in [1]. Now, spaces in the light of these Lie group techniques. Let exp(Ψ taking derivative with respect to ǫ at ǫ = 0 and using the relation (21), we have: ˜ dexp−0˜ (Θ ˜ B ) = exp(Ψ) ˜ dexp−Ψ˜ (Θ ˜ A) exp(Ψ)

or

˜ B = dexp−Ψ˜ (Θ ˜ A) Θ

(26)

Following the same procedure used to derive equation (25), we get ΘA = T (Ψ)ΘB , which is exactly the same as equation (8). 5. Conclusions Rotation vectors in different tangent spaces may be approximately related via a linear operator called the tangential transformation. In the context of geometrically exact beam or shell theories as well as in numerical time integrations of systems undergoing finite rotations, the tangential transformation remains very important wherever rotation is parametrized by the rotation vector. The study presented in this note has been motivated out of these observations. In particular, the neglected higher order terms in the linear relation between two rotation vectors are presented. An exact transformation between incremental rotation vectors associated with different tangent spaces is obtained via the BCH formula in so(3). From our deduction it is evident that the linear relation may produce only crude approximations to rotations of large magnitudes. Expression of the operator T for transforming angular velocity vectors and rotation increments from one tangent space to another is obtained from a Lie group perspective. Though the operator T has been used in the engineering literature quite often, a Lie group approach is probably far more insightful. References [1] A. Cardona, M. G´eradin, A beam of the finite element nonlinear theory with finite rotations, International Journal for Numerical Methods in Engineering 26 (1988) 2403–2438. [2] P. E. Nikravesh, Computer Aided Analysis of Mechanical Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1988. [3] M. G´eradin, A. Cardona, Flexible Multibody Dynamics: A Finite Element Approach, J. Wiley and Sons, New York, ISBN 0-471-48990-5, 2001. [4] K. Spring, Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review, Mechanism Machine Theory 21 (1986) 365–373. [5] M. G´eradin, D. Rixen, Parametrization of finite rotations in computational dynamics: a review, Revue europ´eenne des e´ l´ements finis 4 (1995) 497–553. [6] A. Ibrahimbegovi´c, F. Frey, I. Kozar, Computational aspects of vector-like parametrization of three-dimensional finite rotations, International Journal for Numerical Methods in Engineering 38 (1995) 3653–3673. [7] J. Argyris, An excursion into large rotations, Computer methods and applied mechanics in engineering 32 (1982) 85155. [8] A. Cardona, M. G´eradin, Time integration of the equations of motion in mechanism analysis, Computers and Structures 33 (3) (1988) 801–820. [9] J. M¨akinen, Critical study of newmark-scheme on manifold of finite rotations, Computer Methods in Applied Mechanics and Engineering 191 (2001) 817–828. [10] P. Krysl, L. Endres, Explicit newmark/verlet algorithm for time integration of the rotational dynamics of rigid bodies, International Journal for Numerical Methods in Engineering 62 (15) (2005) 2154–2177. [11] P. Krysl, Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm, International Journal for Numerical Methods in Engineering 63 (2005) 2171–2193. [12] J. Simo, L. Vu-Quoc, On the dynamics in space of rods undergoing large motions: A geometrically exact approach, Computer Methods in Applied Mechanics and Engineering 66 (1988) 125–161. [13] A. Ibrahimbegovi´c, Finite element implementation of reissner’s geometrically nonlinear beam theory: three dimensional curved beam finite element, Computer Methods in Applied Mechanics and Engineering 122 (1995) 10–26.

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[14] A. Ibrahimbegovi´c, On the choice of finite rotation parameters, Computer Methods in Applied Mechanics and Engineering 149 (1997) 49–71. [15] A. Ibrahimbegovi´c, A. Mamouri, On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model, Computer Methods in Applied Mechanics and Engineering 188 (2000) 805–831. [16] J. M¨akinen, Total lagrangian reissner’s geometrically exact beam element without singularities, International Journal of Numerical Methods in Engineering 70 (2007) 1009–1048. [17] A. Ibrahimbegovi´c, M. Al Mikdad, Finite rotations in dynamics of beams and implicit time-stepping schemes, International Journal for Numerical Methods in Engineering 41 (1998) 781–814. [18] Y. Choquet-Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics, North-Holland, Amsterdam, 1987. [19] A. Iserles, H. Z. Munthe-Kaas, S. P. Norsett, A. Zanna, Lie-group methods, Acta Numerica (2000) 215–365. [20] K. Engø, On the bch-formula in so(3), BIT Numerical Mathematics 41 (3) (2001) 629–632. [21] R. A. Spurrier, Comments on singularity-free extraction of a quaternion from a director-cosine matrix, J. Spacecraft 15 (1978) 255–256. [22] V. S. Varadarajan, Lie Groups, Lie Algebras and Their Representation, Graduate Texts in Math. 102, Springer-Verlag, 1984. [23] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, SpringerVerlag, New York, 1993. [24] B. C. Hall, Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Springer, 2003.

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