Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
You have accessResearch articles

Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the generalized Poisson equation and flexible multibody systems

Published:https://doi.org/10.1098/rspa.2021.0303

    Abstract

    The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized-α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized-α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

    1. Introduction

    This paper addresses the exponential and Cayley map used to express solutions of the generalized right and left Poisson equation on SE(3)

    C˙=V^sC,C˙=CV^b, 1.1
    where C(t) ∈ SE(3) describes the motion of a frame in E3 (regarded as Euclidean motion), and V^s(t)se(3) is the velocity in spatial representation, and V^b(t)se(3) in body-fixed representation [1,2]. This is used to describe the relative motion of rigid bodies as well as the displacement field of Cosserat continua.

    Equations (1.1) relate the curve V(t) in se(3), i.e. rigid body twist, to the corresponding curve in SE(3), i.e. the rigid body motion. In context of rigid body kinematics and dynamics, they are thus referred to as the kinematic reconstruction equations. The equations were originally derived by Poisson for pure rotations which is why (1.1) are referred to as generalized Poisson equations. Since the equations for pure rotations are also attributed to Darboux [3], they are occasionally called Poisson–Darboux equations [4].

    (a) Lie group integration and coordinate maps

    When seeking solutions of the form C(t)=expX^(t)C(0) in terms of the canonical coordinates XR6 of first kind, then for small t, and with X^se(3) defined in (2.24),

    V^s=dexpX^(X^˙)andV^b=dexpX^(X^˙) 1.2
    where the right-trivialized differential dexpX^:se(3)se(3) is defined by
    (DX^exp)(Y^)=dexpX^(Y^)exp(X^), 1.3
    with DX^exp:se(3)TexpX^SE(3), so that (DX^exp)(Y^):=(d/dt)exp(X^+tY^)|t=0 is the directional derivative1

    1The directional derivative (DX^exp)(Y^) is also denoted in the literature with D^expY^.

    of exp at X^ in direction of Y^. This was shown by Magnus [5] for a general Lie group. The second equation in (1.2) simply follows by changing the sign of X^ and taking into account (1.1). Occasionally, dexpX^, which satisfies
    (DX^exp)(Y^)=exp(X^)dexpX^(Y^), 1.4
    is referred to as the left-trivialized differential, and denoted dexpX^L(Y^):=dexp X^(Y^). Equations (1.2) will be called the local reconstruction equations.

    The Cayley map C(t)=cayX^(t)C(0) allows expressing the solution in terms of non-canonical coordinates2

    2For simplicity, the same symbol X is used for either coordinates as their meaning is clear from the context.

    XR6. It was shown in [6] that a solution parameterized by the Cayley map satisfies, for small t, the following relation:
    V^s=dcayX^(X^˙)andV^b=dcayX^(X^˙), 1.5
    where now the right-trivialized differential dcayX^:se(3)se(3) is defined by
    (DX^cay)(Y)=dcayX^(Y^)cay(X^). 1.6
    In [7], these results were generalized to arbitrary coordinate maps ψ on a Lie group. Adopted to the special Euclidean group, if ψ : se(3) → SE(3) is a (local) coordinate map, then a solution C(t)=ψ(X^(t))C(0) of (1.1) satisfies V^s=dψX^(X^˙), with dψX^(Y^) defined by (DX^ψ)(Y^)=dψX^(Y^)ψ(X^). Such alternative coordinate maps are the kth-order Cayley map [8,9] or the exponential map in terms of dual quaternions [1012] or the dual number formulation [13] of the vector parameterizations of motion [14,15]. This paper deals with the exponential and Cayley maps as they are the prevailing coordinate maps used for modelling and Lie group integration.

    Various Lie group integration methods depart from the local reconstruction equations, as discussed in [16]. Most of the time integration schemes assume an explicit ODE system, which amounts to express (1.2) and (1.5), respectively, as

    X^˙=dexpX^1(V^s)andX^˙=dcayX^1(V^s), 1.7
    and accordingly with negative argument X^ for the body-fixed representation of twists in (1.1). In the original paper [17,18], Munthe–Kaas used the exponential map along with (1.2). The Cayley map with (1.5) was later used, and the heavy top, with motion evolving on SO(3), served as a standard example for Lie group methods [19,20]. Lie group methods using general coordinate maps were presented in [7,16] and applied to rigid body systems [21]. An excellent overview can be found in [16,22]. While most integration methods [21,2325] adopt explicit integration schemes for solving the vector space ODE (1.2) or (1.5), the generalized-α method on vector spaces (originally derived from the semi-implicit Newmark schemes [26]) was reformulated to deal with ODE on Lie groups [27,28]. These Lie group generalized-α methods are applied to finite rotations [29] and to multibody systems comprising rigid as well as flexible bodies [3033], where rigid body motions as well as the deformation field of flexible beams are consistently represented as curves in SE(3).

    (b) Lie group modelling of flexible beam kinematics

    The kinematics of flexible beams can be modelled as Euclidean motions. Let C(s), s ∈ [0, L] describe the configuration of a beam cross section. The beam kinematics can then be described by

    C=χ^sCandC=Cχ^b, 1.8
    where χ^:[0,L]se(3) serve as deformation measure to define the strain field. Borri & Bottasso [34,35] were the first to model the deformation field in geometrically exact beam models as screw (also called helicoidal) motions, aiming at a strain invariant formulation. The spatial (right-invariant) deformation measure χ^s was used and referred to as the base-pole generalized curvature, and χ^b as the convected generalized curvature . Therefore, the approach was called base-pole (or fixed-pole) formulation. Using the base-pole formulation, an invariant conserving integration scheme was presented in [36,37]. The material (left-invariant) deformation measure χ^b was used by Sonneville [38,39] who extended the approach to geometrically exact beams and shells. The main differences of the right- and the left-invariant formulations are their invariance properties. Also note that in [34,35] the adjoint representation of SE(3) is used. With the helicoidal approximation, beam deformations are expressed in terms of the exponential map on SE(3), and reconstruction equations analogous to (1.3) and (1.4) apply. Since Euclidean motions are screw motions, this allows exact recovery of beam deformations with locally constant curvature, i.e. pure bending or helical deformation. In the general situation, the assumption of a helicoidal deformation field is an approximation. In this context, the importance of using SE(3) instead of SO(3)×R3 for rigid body systems as well as geometrically exact beams and shells must be emphasized, which was discussed in [36,3945]. The modelling of beam kinematics with (1.8) has recently been used to describe soft robots [4648].

    (c) Contribution of this paper

    Key elements of most Lie group integration schemes are the closed form expressions for the coordinate maps and their trivialized differentials. Moreover, the (semi-)implicit generalized-α method necessitates the directional derivative of the trivialized differentials in order to construct the Hessian for the iteration steps involved. Closed form expressions for relevant relations of the exponential map were already reported in various publications, and it is difficult to find the corresponding proofs. While closed form expressions were also reported for the basic relations of the Cayley map, the differential and its derivative seem not be published. The motivation of this paper is to provide a comprehensive reference including all relevant proofs. The reader is referred to the seminal papers of Bottasso and co-workers [42,44], where several of the presented relations were already reported using different notions and approaches. These papers seem not have found their due recognition.

    The paper consists of two main parts. Section 2 addresses the parameterization of motions using the exponential map, while §3 deals with the motion representation and parameterization using the Cayley map. Section 2a recalls well-known expressions for rotation parameterization in terms of canonical coordinates (axis/angle), and its differential along with several relations that are crucial for structure preserving integration schemes. Section 2b presents closed-form relations for the exponential of Euclidean motions in terms of screw coordinates, and the related expressions for the trivialized differential and their directional derivatives are derived. The complete list of closed-form relations and the corresponding proofs is the contribution of §2. For completeness, the adjoint representation of SE(3) is discussed, which is used for geometrically exact beam modelling. The Cayley map for representing rotations and Euclidean motions is recalled in §3a,b, respectively, and the trivialized differential and its directional derivative are derived. Finally, the adjoint representation is considered, and it is shown that the differentials of the Cayley map of SE(3) and of its adjoint representation are different. The closed form expressions for the differential of the Cayley map and its derivative are the main contribution of §3, which provides the basis for a generalized-α integration method in terms of the Cayley map. The paper closes with a short conclusion in §4.

    2. Motion parameterization via the exponential map

    The exponential map exp:gG on a n-dimensional Lie group G admits the series expansion [49]

    exp(x~)=i=01i!x~i. 2.1
    The Lie algebra g is assumed isomorphic to Rn, and x~g denotes the matrix associated with the vector xRn. For notational convenience, throughout the paper, x~so(3) corresponds to xR3, while X^se(3) is constructed from XR6 [1].

    The right-trivialized differential dexpx~:gg of the exponential map on a Lie group admits the series expansion

    dexpx~(y~)=i=01(i+1)!adx~i(y~). 2.2
    This was shown by Hausdorff [50, pp. 26, 36ff], and a proof can be found in [51]. The series expansion of the left-trivialized differential is obtained by inserting −x in (2.2), which was also shown in [50] and a proof is given in [52, theorem 2.14.3]. Hausdorff [50, pp. 27, 36ff] further showed that the inverse of the right-trivialized differential can be represented by the series
    dexpx~1(y~)=i=0Bii!adx~i(y~), 2.3
    with the Bernoulli numbers Bi.

    In the following, when representing Lie algebra elements x~g as vectors xRn, the differential and inverse are represented by a matrix denoted dexpx:RnRn and dexpx1:RnRn so that dexpx~(y~)=(dexpxy) and dexpx~1(y~)=(dexpx1y), respectively.3

    3In computational mechanics, in particular in context of numerical integration of multibody systems, the matrix representation of dexp is frequently called tangent operator.

    (a) Spatial rotations: SO(3)

    (i) Exponential map: Euler–Rodrigues formula

    The exponential map on SO(3) is the analytic form of the classical result in rigid body kinematics, according to which the rotation about a given axis can be expressed in closed form, which is attributed to Euler [53] and Rodrigues [54]; see also [55]. For later use introduce the abbreviations

    α:=sincφ=cosφ2sincφ2,β:=sinc2φ2γ:=αβ=cosφ2sincφ2=cos2φ2sincφ=sincφsinc2φ2,δ:=1sincφφ2=1αφ2. 2.4
    These terms allow minimizing the number of transcendental function evaluations.

    A skew-symmetric matrix x~so(3) satisfies the characteristic equation x~3+||x||2x~=0. Denote with φ ≔ ‖x‖ the rotation angle. The series (2.1) gives rise to various closed forms [1,5557]:

    expx~=I+sin||x||||x||x~+1cos||x||||x||2x~2 2.5
    =I+αx~+12βx~2 2.6
    exp(φn~)=I+sinφn~+(1cosφ)n~2 2.7
    =I+sinφn~+12sin2φ2n~2, 2.8
    where x ≔ φn, with rotation angle φ and the unit vector n along the rotation axis. The components of xR3 serve as canonical coordinates of first kind. The form (2.6) along with the parameters α, β, γ were reported in [58].

    (ii) Differential of the exponential map

    When representing x~so(3) as vector xR3, the adjoint operator adx~:so(3)so(3) is simply x~:R3R3, and evaluating (2.2) along with x~3=||x||2x~ yields the following different matrix forms dexpx:R3R3 of the right-trivialized differential:

    dexpx=I+1cos||x||||x||2x~+||x||sin||x||||x||3x~2 2.9
    =1||x||2[(Iexpx~)x~+xxT] 2.10
    =1||x||2xxTαx~β2x~2 2.11
    =I+12sinc2||x||2x~+1||x||2(112sinc||x||2cos||x||2)x~2 2.12
    =I+β2x~+1||x||2(1α)x~2. 2.13
    The form (2.9) was reported in [59] and later in [60,61], but was already derived before in [62]. Expression (2.13) was reported in [58]. These explicit expressions reveal the singularity at ‖x‖ = 0, which reflects the fact that the three-dimensional group SO(3) is not simply connected [55], and thus does not admit a global singularity-free parameterization in terms of three canonical coordinates. A singularity-free parameterization needs at least four (dependent) parameters. Using unit quaternions (Euler parameters) is a common choice, which implies using SU(2) to represent rotations [55,63]. However, this singularity is removable, i.e. the limit limx‖→0dexpx = I exists, and the relations (2.11) and (2.13) admit a numerically stable evaluation by replacing the term x~2/||x||2 with n~2, and xxT/‖x2 with nnT.

    The matrix form of the inverse of the right-trivialized differential, dexpx~1:so(3)so(3), is found from the series (2.3) as

    dexpx1=I12x~+1||x||2(1||x||2cot||x||2)x~2 2.14
    =I12x~+1||x||2(1sinc||x||sinc2||x||2)x~2 2.15
    =I12x~+1||x||2(1cos||x||2sinc||x||2)x~2=I12x~+1||x||2(1γ)x~2. 2.16
    The expression (2.14) was reported in [5961], and (2.16) was used in [58]. In the form (2.15) and (2.16), the inverse can be numerically evaluated without difficulties.

    The matrix of the left-trivialized differential and its differential enjoy the property dexpx=dexpxT and dexpx1=dexpxT, respectively. The spatial and body-fixed angular velocity are hence related to the time derivative of the rotation axis times angle via

    ωs=dexpxx˙,x˙=dexpx1ωs 2.17
    and
    ωb=dexpxTx˙,x˙=dexpxTωb. 2.18

    Lemma 2.1.

    The exponential map on SO(3) and its differential are related via

    expx~=dexpxTdexpx 2.19
    =dexpxdexpxT 2.20
    =I+x~dexpx 2.21
    =I+dexpxx~. 2.22

    Proof.

    Equating the differential of R=expx~ written in terms of the right- and left-trivialized differential (1.3) and (1.4) yields dexpx~(y~)R=Rdexpx~(y~), and thus dexpx~(y~)=Rdexp~(y~)RT, with arbitrary  yR3. Using the matrix form of dexp, this can be written as (dexpx~y)=(R dexpx~y), which yields R dexpx~=dexpx~ and thus (2.19). From the explicit form (2.14) follows that x~=dexpx~Tdexpx~1. Multiplication with dexpx~ from the left yields I+dexpx~x~=dexpx~dexpx~T and noting (2.19) yields (2.22). The series expansion (2.3) shows that dexpx~x~=x~dexpx~ and hence (2.21). Finally, left-multiplication of ~=dexpx~Tdexpx~1 with dexpx~, noting (2.21), shows (2.20).

    The relations (2.19)–(2.22) were reported in [62] and served as key relations for deriving conservative integration schemes for flexible systems. Relations (2.19), (2.20) and a slightly different form of (2.21), (2.22) were derived in [60] with help of computer algebra software.

    (b) Euclidean motions: SE(3)

    (i) Exponential map

    The motion of a rigid body evolves on the Lie group SE(3)=SO(3)R3, which is represented as subgroup of GL(4) with elements

    C=(Rr01)SE(3), 2.23
    where R ∈ SO(3) describes the rotation of a body-fixed frame, and rR3 is the position vector of the frame origin w.r.t. a reference frame. The Lie algebra se(3) consists of matrices
    X^=(x~y00)se(3), 2.24
    with x~so(3) and yR3. There is an obvious correspondence of X^se(3) and X=(x,y)R6, which renders se(3) isomorphic to R6.

    The Chasles theorem [64,65] asserts that any finite rigid body displacementC ∈ SE(3) can be achieved by a screw motion about a fixed screw axis, i.e. there is a constant X so that the screw motion is exp(tX^),t[0,1] and C=expX^. According to a theorem by Mozzi [66] and Cauchy [67], for any given motion in time C(t), there is an instantaneous screw axis, i.e. there is a X(t) so that C(t)=expX^(t)C(0). The components of X are the screw coordinates [1,68]. Solving the kinematic reconstruction equations thus amounts to determining X(t) for given twist.

    Evaluating the series (2.1) with matrices (2.24) yields

    exp(X^)=(i=01i!x~ii=01(i+1)!x~iy01)=(expx~dexpxy01). 2.25
    The dexp mapping on SO(3) occurs since SE(3) is the semidirect product with R3 as normal subgroup, and se(3) is the semidirect sum with R3 as ideal (i.e. translations do not affect rotations). Inserting (2.10) into (2.25) yields
    exp(X^)=(R1||x||2(IR)x~y+hy01),forx0=(Iy01),forx=0, 2.26
    with rotation matrix R=expx~. The term h ≔ xTy/‖x2 is the instantaneous pitch of the screw motion. For pure rotation h = 0, and a pure translation corresponds to X = (0, y), i.e. x = 0, for which h = ∞. The translation is then described by the vector y = φn, where nR3 is the unit vector in direction of the translation and φ is the amount of translation.

    Using (2.6), the relation (2.26) for x ≠ 0 can be written as

    exp(X^)=(R(αx~+β2x~2)y+nnTy01),forx0, 2.27
    with n = x/x‖, which admit robust numerical evaluation. The exponential can be expressed in terms of geometric attributes (direction, position, magnitude, pitch) of the instantaneous screw motion[1,2,63], which is relevant for modelling mechanisms and multibody systems in terms of joint variables and joint screw coordinates.

    (ii) Differential of the exponential map

    The velocity of a rigid body in spatial representation, called spatial twist, is defined by V^s=C˙C1se(3), where the twist vector Vs=(ωs,vs)R6 comprises the spatial angular velocity defined by ω~is=R˙iRiT and spatial velocity vis=r˙i+ri×ωis [1,2,69]. The velocity in body-fixed representation, called body-fixed twist, is defined by V^b=C1C˙se(3), and the twist vector Vb=(ωb,vb)R6 consists of the body-fixed angular velocity ω~ib=RiTR˙i and the velocity vib=RiTr˙i.

    Expressing the twist in terms of canonical (screw) coordinates X, according to (1.2), involves the differential dexpX^:se(3)se(3). When representing twists and screws as vectors, the spatial and body-fixed twist is determined as Vs=dexpXX˙ and Vb=dexpXX˙, respectively, where dexpX:R6R6 is the matrix form of the right-trivialized differential so that dexpX^(Y^)=(dexpXY). The differential can be determined with the general relation (2.2) in terms of adX^:se(3)se(3). The adjoint operator on se(3) defines the Lie bracket as matrix commutator adX^1(X^2)=[X^1,X^2]=X^1X^2X^2X^1. In vector representation X=(x,y)R6, the matrix form of the adjoint operator is [1,68]

    adX=(x~0y~x~), 2.28
    so that Z^=adX^1(X^2) is equivalent to Z=adX1X2. The result Z of this operation is also called the screw product of X1 and X2 [57,68].

    Lemma 2.2.

    The right-trivialized differential of the exponential map on SE(3) possesses the explicit matrix form, with canonical coordinatesX = (x, y),

    dexpX=(dexpx0(Dxdexp)(y)dexpx), 2.29
    wheredexpx is the matrix of the right-trivialized differential on so(3) and its directional derivative is given explicitly as
    (Dxdexp)(y)=1||x||2(1cos||x||)y~+1||x||3(||x||sin||x||)(x~y~+y~x~)+xTy||x||4(||x||sin||x||+2cos||x||2)x~+xTy||x||5(3sin||x||||x||cos||x||2||x||)x~2 2.30
    =12sinc2||x||2y~+1||x||2(1cos||x||2sinc||x||2)(x~y~+y~x~)+xTy||x||2(cos||x||2sinc||x||2sinc2||x||2)x~+xTy||x||2(12sinc2||x||23||x||2(1cos||x||2sinc||x||2)x~2 2.31
    =12sinc2||x||2y~+1||x||2(1sinc||x||)(x~y~+y~x~)+xTy||x||2(sinc||x||sinc2||x||2)x~+xTy||x||2(12sinc2||x||23||x||2(1sinc||x||))x~2 2.32
    =β2y~+δ(x~y~+y~x~)+xTy||x||2(αβ)x~+xTy||x||2(β23δ)x~2. 2.33

    Proof.

    Evaluating the series (2.2) with (2.28) involves the powers

    adXi=(x~i0Pix~i)withPi(x~,y~)=j=0i1x~jy~x~ij1,i1 2.34
    and P0 = 0. The matrix Pi can be written as directional derivative of the ith power x~i, considered as a function of x~, in the direction of y~:
    Pi(x~,y~)=(Dx~x~i)(y~)=ddt(x~+ty~)i|t=0=(x~+ty~)i1y~+(x~+ty~)i2y~(x~+ty~)++(x~+ty~)y~i1|t=0. 2.35
    The series expansion (2.2) for the differential thus leads to
    dexpX=i=01(i+1)!adXi=(dexpx0i=01(i+1)!(Dx~x~i)(y~)dexpx)=(dexpx0(Dxdexp)(y)dexpx), 2.36
    with the directional derivative of the right-trivialized differential on SO(3)
    (Dxdexp)(y)=ddtdexp(x~+ty~)|t=0. 2.37
    This is evaluated using the following expressions:
    (Dx||x||)(y)=(xTy)||x||,(Dxx~)(y~)=y~,(Dxx~2)(y)=x~y~+y~x~. 2.38
    Starting from (2.9) yields (2.30), using (2.12) yields (2.31), while using (2.13) yields (2.32), which can both be expressed as (2.33). The closed form (2.29) for the differential on SE(3) can now be written in terms of these expressions.

    The expression (2.33) was presented in [58] without proof in terms of the parameters α and β. Prior to this, it was reported in [61,62] in almost the same form. The singularity at ‖x‖ = 0 is inherited from the dexp map on SO(3). The term x~(xTy)/||x||2 can be computed robustly as n~(nTy). The following expression was also presented in [58].

    Lemma 2.3.

    The inverse of the right-trivialized differential of the exponential map on SE(3) possesses the matrix form

    dexpX1=(dexpx10(Dxdexp1)(y)dexpx1), 2.39
    with canonical coordinatesX = (x, y), the matrix in (2.14)–(2.16), and
    (Dxdexp1)(y)=12y~+1||x||2(1γ)(x~y~+y~x~)+xTy||x||4(1β+γ2)x~2. 2.40

    Proof.

    Invoking the series expansion (2.3), along with the power of X^ in (2.34), yields

    dexpX1=i=0Bii!adXi=(dexpx10i=0Bii!(Dx~x~i)(y~)dexpx1) 2.41
    =(dexpx10(Dxdexp1)(y)dexpx1), 2.42
    where the relation (2.35) along with (2.3) has been used to identify Dxdexp−1 as the directional derivative of dexp−1 on SO(3). The statement follows with the directional derivatives of (2.15) or (2.16) calculated using (2.38) in lemma 2.2.

    It was shown in [68, p. 77] that the matrix of the right-trivialized differential and its inverse can be expressed directly in terms of the adjoint operator matrix on se(3) in (2.28), which satisfies adX6=||x||4adX22||x||2adX4. In terms of the parameters (2.4), they are

    dexpX=I+12||x||2(4||x||sin||x||4cos||x||)adX+12||x||3(4||x||5sin||x||+||x||cos||x||)adX2+12||x||4(2||x||sin||x||2cos||x||)adX3+12||x||5(2||x||3sin||x||+||x||cos||x||)adX4 2.43
    =I+(βα2)adX+12(5δβ2)adX2+12||x||2(βα)adX3+12||x||2(3δβ2)adX4 2.44
    dexpX1=I12adX+(2||x||2+||x||+3sin||x||4||x||(cos||x||1))adX2+(1||x||4+||x||+sin||x||4||x||3(cos||x||1))adX4 2.45
    =I12adX+1||x||2(21+3α2β)adX2+1||x||4(11+α2β)adX4. 2.46
    The expression (2.45) was already presented in [60] with negative sign so as to express the left-trivialized differential. Using different abbreviations, these expressions were reported in [42], where it arose naturally using the adjoint representation (see §3d). In contrast to (2.40), the expression (2.46) can be evaluated numerically stable without separating the case when x = 0. If efficiently implemented, it provides a computationally efficient alternative to (2.39).

    (c) Directional derivative of the right-trivialized differential

    The computational procedure of the (semi)implicit generalized-α Lie group method [3033] employs an iteration matrix (called the tangent matrix), which involves the directional derivative of the trivialized differential of the respective coordinate map. This is necessary also when using other implicit integration schemes.

    To simplify notation, denote the directional derivative (2.33) of the matrix dexpx in (2.13) with Ddexp(X) ≔ (Dxdexp)(y), and the derivative (2.40) of dexpx1 in (2.16) with Ddexp−1(X) ≔ (Dxdexp−1) (y), where X = (x, y). The second directional derivative of dexp is then written as (DXDdexp) (U) with U = (u, v), and similarly for its inverse.

    Lemma 2.4.

    The directional derivative of the matrixdexp in (2.29) is

    (DXdexp)(U)=(Dxdexp)(u)0(DXDdexp)(U)(Dxdexp)(u)), 2.47
    whereU = (u, v), and the directional derivative of matrixDdexp possesses the explicit form
    (DXDdexp)(U)=β2v~+αβ||x||2((xTu)y~+(xTv+yTu)x~+(xTy)u~)β2||x||2(xTu)(xTy)x~+δ(x~v~+v~x~+y~u~+u~y~)+β/23δ||x||2((xTy)(x~u~+u~x~)+(xTu)(x~y~+y~x~)+(xTv+yTu)x~2)+1||x||4(xTy)(xTu)((15α+4β)x~+(α72β+15δ)x~2) 2.48
    =β2v~+αβ||x||2(xTu)y~+β/23δ||x||2((xTu)(x~y~+y~x~)+(xTy)(x~u~+u~x~)+δ(x~v~+v~x~+y~u~+u~y~))+1||x||2((αβ)(xTv+yTu)12β(xTu)(xTy)+15α+4β||x||2(xTy)(xTu))x~+1||x||2((12β3δ)(xTv+yTu)+1||x||2(α72β+15δ)(xTy)(xTu))x~2. 2.49

    Proof.

    The derivatives of the parameters (2.4) are readily found with (2.38) to be

    (Dxα)(u)=(δ12β)xTu,(Dxβ)(u)=2xTu||x||2(αβ),(Dxδ)(u)=xTu||x||2(12β3δ). 2.50
    A straightforward manipulation using (2.50) yields the directional derivative of (2.33) in the form (2.48), and rearranging yields (2.49). The statement follows with the matrix of dexpX in (2.29).

    The directional derivative (2.47) can be evaluated along with one of the expressions in (2.30)–(2.33). A slightly different expression for the second directional derivative of the left-trivialized differential was presented in [39].

    Lemma 2.5.

    The directional derivative of the matrixdexp−1 in (2.39) is

    (DXdexp1)(U)=((Dxdexp1)(u)0(DXDdexp1)(U)(Dxdexp1)(u)), 2.51
    whereU = (u, v), and the directional derivative of matrixDdexp−1 possesses the explicit form
    (DXDdexp1)(U)=12v~+1γ||x||2(x~v~+v~x~+y~u~+u~y~+14(xTu)(x~y~+y~x~))1||x||4((1γ)(xTu)(2+γ)(x~y~+y~x~)+(2γ1β)(xTy)(x~u~+u~x~)+(xTv+yTu)×(2+14(xTy)(xTu)γ1β)x~2)+1||x||6(xTy)(xTu)(83γγ22β2(α+β))x~2 2.52
    =12v~+1γ||x||2(x~v~+v~x~+y~u~+u~y~)+(1γ)||x||2(xTu)(141||x||2(2+γ))(x~y~+y~x~)+1||x||4(γ+1β2)(xTy)(x~u~+u~x~)+1||x||4((γ+1β2)(xTv+yTu)(xTy)(xTu)(14(xTv+yTu)+1||x||2(γ(3+γ)+2β2(α+β)8)))x~2. 2.53

    Proof.

    The directional derivative of (2.40) involves the derivative of parameter γ

    (Dxγ)(u)=xTu||x||2(γγ2||x||24). 2.54
    The derivative of the coefficient of x~2 in (2.40) is then found to be, along with (2.50),
    (DX(xTy||x||4(1β+γ2)))(U)=1||x||4(xTv+yTu)(γ+1β214(xTy)(xTu))+(xTy)(xTu)||x||6(γ(1γ)+2β2(βα)4(γ+1β2)). 2.55
    The expressions and (2.52) and (2.53) are obtained after some algebraic manipulation.

    An equivalent expression for the directional derivative of the matrix of the left-trivialized differential, i.e. with negative argument X, was derived in [39, appendix A.1]. The directional derivative is thus available in closed form as (2.51) along with (2.40).

    The above relations for the derivative of the matrices representing the directional derivatives of the right-trivialized differential and its inverse are crucial for numerical integration using implicit Lie group integration methods, such as the generalized-α scheme. They can be implemented so as to cope with ‖x‖ = 0. An alternative formulation could be obtained using the expressions (2.44) and (2.46), respectively.

    (d) Adjoint representation: the ‘configuration tensor’

    Representing frame transformations by matrices C ∈ SE(3), which describe transformations of homogeneous point coordinates, is useful to compute relative configurations of rigid bodies by means of matrix multiplication. This is not relevant when C describes the absolute configuration of a rigid body or the displacement field of Cosserat beam, for instance. Moreover, storing rotation matrix and position vector in C is merely a means of bookkeeping. In this context, a more relevant operation is the frame transformation of twists, which is described by the adjoint operator AdC : se(3) → se(3). In vector representation of twists, this is expressed by the matrix [1,68]

    AdC=(R0r~RR)Ad(SE(3)). 2.56
    This adjoint representation is used in the geometrically exact beam formulation by Borri et al. [36,37,44,62] and Bauchau [70]. In this context, the adjoint transformation matrix AdC is referred to as the configuration tensor [14,70], and used to describe frame transformations, where Ad(SE(3)) was denoted SR(6). The expression (1.1) for the twists, and (1.8) for the deformation tensor defining the strain field, are then replaced by
    Ad˙C=adVsAdC,AdC˙=AdCadVb 2.57
    and
    AdC=adχsAdC,AdC=AdCadχb, 2.58
    where adV and adχ serve to represent rigid body twists and strain measures, with ad in (2.28).

    The adjoint representation is canonically parameterized in terms of instantaneous screw coordinates XR6 using the relation

    AdexpX^=expadX, 2.59
    where exp is the exponential map of the adjoint representation. Clearly, the canonical (screw) coordinates XR6 in the exponential map (2.25) and (2.59) are identical. An important consequence is that the local reconstruction equations (1.2) remain valid, and hence the closed form expressions (2.29) and (2.39) apply. Closed form expressions for the exponential (2.59) and its derivatives were reported in [71]. This will not be considered here as recent modelling approaches are using the SE(3) representation of Euclidean motions [38,39]. Moreover, the adjoint representation (2.56) is redundant and does not directly yield the displacement vector r.

    For completeness, the following relations, which were central for developing integration schemes using the base-pole formulation [36,37], and reported without proof, are summarized.

    Lemma 2.6.

    Let C=expX^SE(3), the adjoint operator matrix (2.56) and right-trivialized differential of exp : se(3) → SE(3) satisfy the relations

    AdC=dexpX^1dexpX^ 2.60
    =dexpX^dexpX^1 2.61
    =I+adX^dexpX^ 2.62
    =I+dexpX^adX^. 2.63

    Proof.

    The proof is similar to that of lemma 2.1. From (1.3) and (1.4) it follows that dexpX^(Y^)=CdexpX^(Y^)C1=AdC(dexpX^(Y^)), and hence the vector representation of se(3) yields dexpXY = AdCdexpXY and thus (2.60). The explicit form (2.45) shows that adX=dexpX1dexpX1, and hence I+dexpXadX=dexpXdexpX1 in (2.63). From (2.14) it follows that adXdexpX = dexpXadX, which proves (2.62).

    The relations (2.60)–(2.63) were exploited in [36,37] to construct invariant conserving, respectively dissipating, integration schemes for multibody systems with flexible members. Note that (2.60)–(2.63) hold true for general matrix Lie groups. The relations (2.19)–(2.22) are special cases with adx~=x~,AdR=R.

    3. Motion parameterization via the Cayley map

    The vectorial parameterizations of motion admit an algebraic description without transcendental functions. Parameterizations in terms of non-redundant parameters provide a computationally efficient alternative to canonical coordinates within Lie group integration schemes. The simplest of such are the Rodrigues parameters. The corresponding coordinate maps are obtained via the Cayley map.

    The Cayley map on a quadratic Lie group provides an approximation of the exponential map [72]. Aiming at computationally efficient Lie group integration schemes, the Cayley map has been widely used for general systems [6,7375], and for the rigid body motion described on SO(3) in particular (e.g. [16,21]). A symplectic integration scheme for rotating rigid bodies, where the Cayley map is used, was presented in [74]. The Cayley map for the adjoint representation of SE(3) along with the base-pole formulation was used to derive integration schemes for geometrically exact beams [44] that preserve certain invariants. The necessary explicit relations for the Cayley map and its differentials are scattered in the literature partly without proof. Moreover, to the author’s knowledge, the differential of the Cayley map on SE(3) and its directional derivative are not present in the literature. They will be presented in this section.

    The Cayley map cay:gG on a quadratic matrix Lie group can be defined as

    cay(x~)=(Ix~)1(I+x~) 3.1
    =(I+x~)(Ix~)1 3.2
    =I+2(x~+x~2+x~3+). 3.3
    The components of xRn serve as local (non-canonical) coordinates around the identity in G. The directional derivative is
    (Dx~cay)(y~):=ddtcay(X^+tY^)|t=0=(Ix~)1y~(I+cay(x~))=2(Ix~)1y~(Ix~)1 3.4
    and thus, with (3.1), the right-trivialized differential dcayx~:gg in (1.6) is [21,72]
    dcayx~(y~)=2(Ix~)1y~(I+x~)1. 3.5
    Note that the Cayley transform is occasionally defined with x~ in (3.1) divided by 2. Then the factor 2 in (3.5) disappears. A motivation for doing so is that then (3.5) is the identity mapping for x = 0.

    The inverse of the right trivialized differential is immediately found from (3.4) [21,72]

    dcayx~1(y~)=12(Ix~)y~(I+x~)=12(y~[x~,y~]x~y~x~), 3.6
    with the matrix commutator [x~,y~]=x~y~y~x~ as Lie bracket.

    (a) Spatial rotations: SO(3)

    (i) Gibbs–Rodrigues parameters

    Rodrigues [54,55] derived the rotation of a vector as combination of two half-rotations about the rotation axis, which leads to the corresponding rotation matrix. The latter can be constructed by means of the Cayley transform on so(3), as a formalization of Cayley’s original result [76]. Invoking again the relation x~3=||x||2x~, the Cayley map attains the well-known closed form [56,57]

    cay(x~)=(Ix~)1(I+x~)=(I+x~+x~2+x~3+)(I+x~)=I+σ(x~+x~2), 3.7
    with
    σ:=21+||x||2. 3.8
    Vector xR3 is the Gibbs–Rodrigues vector while its components are the Rodrigues parameters. The Gibbs–Rodrigues vector is given in terms of the rotation angle φ and the unit vector n along the rotation axis as x=tanφ2n~. A singularity is encountered at φ → ±π for which ‖x‖ → ∞. Apparently rotations can be represented with φ ∈ (− π, π) only, i.e. the Cayley map cannot be used to describe full turn or even multiple turn rotations. This issue can be tackled by using higher-order Cayley transformation [8,9,77], where the singularities are shifted to multiple full turns. The conformal rotation vector, also introduced by Milenkovic [78,79] and Wiener [80, p. 69], is a special case, which allows describing double rotations. It should be recalled that Lie group integration methods using the Cayley map solve (1.5), respectively, the right equation in (1.7), for the incremental rotation within a time step with initial value X = 0, assuming that no full rotation takes place within a time step.

    The Gibbs–Rodrigues parameterization is related to the axis-angle description via

    exp(φn~)=cay(tanφ2n~)=I+21+tan2(φ/2)(n~+n~2). 3.9
    It can also be related to the description in terms of unit quaternions. If Q = (q0, q) ∈ Sp(1) denotes a unit quaternion, then the Gibbs–Rodrigues vector, describing the same rotation, is obtained as x = q/q0. Unit quaternions and Gibbs–Rodrigues vectors are thus related via the gnomic projection of R3 onto the unit sphere S3 ≅ Sp(1).

    (ii) Differential of the Cayley map

    Closed form expressions of the differential of the Cayley map on SO(3) were reported in [21,71]. A proof of these relations is given in the following.

    Lemma 3.1.

    The right-trivialized differential of cay:so(3) → so(3) and its inverse possess the matrix form

    dcayx=σ(I+x~) 3.10
    dcayx1=1σI+12(x~2x~) 3.11
    =12σ(I+cay(x~))=12σ(I+RT) 3.12
    with R=cay(x~). They satisfy the properties dcayx=dcayxT and dcayx1=dcayxT.

    Proof.

    The last term in (3.6) can be expressed as x~y~x~=x~2y~x~x~y~=xTxy~x~x~y~||x||2y~=x~y~x~x~x~y~||x||2y~=[x~,[x~,y~]]||x||2y~, and hence dcayx~1(y~)=12((1+||x||2)y~[x~,y~]+[x~,[x~,y~]]). With [x~,y~]=x~y~ and [x~,[x~,y~]]=x~x~y~ follows the matrix (3.11) so that z=dcayx1y when z~=dcayx~1(y~), and noting (3.7) yields (3.12). It is easy to show (3.10) by multiplication with (3.11). The closed form expression (3.10) shows that dcayx=dcayxT.

    The spatial and body-fixed angular velocity and the time derivative of the Gibbs–Rodrigues vector (also called Cayley quasi-velocities [81]) are thus related as

    ωs=dcayxx˙,x˙=dcayx1ωsandωb=dcayxTx˙,x˙=dcayxTωb. 3.13

    (b) Euclidean motions: SE(3)

    (i) Cayley map on SE(3)

    The Cayley map for Euclidean motions is obtained with the general relation (3.1) applied to matrices (2.24). The explicit form of the power of X^se(3), shown in (2.25), yields

    cay(X^)=(cay(x~)(I+cay(x~))y01) 3.14
    =(cay(x~)2σdcayx~Ty01), 3.15
    where (3.15) is obtained from (3.14) using (3.12). The form (3.14) was presented in [82]. There is no established name for the parameter vector XR6 in cay(X^).

    (ii) Differential of the Cayley map

    The local reconstruction equations (1.5) in terms of the Cayley map, with Rodrigues parameters as local coordinates, can be written in vector form as Vs=dcayXX˙ and Vb=dcayXX˙, respectively, where dcayX:R6R6 is the coefficient matrix of this linear relation.

    Lemma 3.2.

    The coefficient matrix of the right-trivialized differential of cay:se(3) → SE(3) and its inverse in vector representation possess the closed form expressions

    dcayX=(dcayx~0σy~(I+x~)2I+σ(x~+x~2))=(σ(I+x~)0σy~(I+x~)2I+σ(x~+x~2)) 3.16
    and
    dcayX1=(dcayx~1012(x~I)y~12(Ix~))=12(2σI+x~2x~0(x~I)y~Ix~). 3.17

    Proof.

    Inserting matrices of the form (2.24) corresponding to X = (x, y) and U=(u,v)R6 into (3.6) yields

    dcayX^1(U^)=(dcayx~1(u~)12(Ix~)(vy~u)00)se(3). 3.18
    Expressing Z^=dcayX^1(U^) in vector form as Z=dcayX^1U, along with (3.11), yields (3.17). The inverse of (3.17) is readily obtained as
    dcayX^=21+||x||2((I+x~)0y~(I+x~)(1+||x||2)I+x~+x~2), 3.19
    and along with (3.10) yields (3.16).

    The closed form expressions for the right-trivialized differential and its inverse were reported in [83,84] without proof.

    (c) Directional derivative of the right-trivialized differential

    If the Cayley map is to be used in generalized-α Lie group schemes, the directional derivative of decay is needed. This is derived next.

    Lemma 3.3.

    The directional derivative ofdcayx:so(3) → so(3) and its inverse is given in closed form as

    (Dxdcay)(y~)=σy~σ2(xTy)(I+x~) 3.20
    and
    (Dxdcay1)(y~)=(xTy)I+12(x~y~+y~x~y~). 3.21
    The directional derivative of matrixdcayX of the right-trivialized differential in (3.16) and its inverse admit the following explicit forms, whereX = (x, y) andU = (u, v):
    (DXdcay)(U)=(σu~σ2(xTu)(I+x~)0σ(v~+v~x~+y~u~)σ2(xTu)(y~+y~x~)σ(u~+x~u~+u~x~)σ2(xTu)(x~+x~2)) 3.22
    and
    (DXdcay1)(U)=12(2(xTu)I+u~x~+x~u~u~0x~v~+u~y~v~u~). 3.23

    Proof.

    The expressions (3.20) and (3.21) follow with (2.38) directly from (3.10) and (3.11), respectively. Noting that the block entries of (3.16) and (3.17) depend on X, the expressions (3.22) and (3.23) are readily found with (2.38).

    The expressions (3.20)–(3.23) seem not to be present in the current literature. They will be crucial for constructing iteration matrices within numerical time stepping schemes such as the generalized-α method.

    (d) Adjoint representation: the ‘configuration tensor’

    Explicit forms of the Cayley map for the adjoint representation (2.56) and its derivative were reported in [71], were it is referred to as the configuration tensor. In the following, the Cayley map for the adjoint representation is presented to emphasize that the involved parameters are different from those of the Cayley map on SE(3). The Cayley map for the adjoint representation is formally introduced as

    cay(adX)=(IadX)1(I+adX). 3.24

    Lemma 3.4.

    The Cayley map of the adjoint representation of SE(3) admits the following closed form expressions:

    cay(adX)=(cay(x~)0A(x,y)cay(x~)), 3.25
    with
    A(x~,y~)=12(I+R)y~(I+R) 3.26
    =2(Ix~)1y~(Ix~)1 3.27
    =(Dx~cay)(y~) 3.28
    =dcayx~(y~)R 3.29
    =y~dcayx+σx~y~R, 3.30
    with R:=cay(x~).

    Proof.

    The first term in the product (3.24) is readily obtained with adX in (2.28) as

    (IadX)1=I+adX+adX2+adX3+=i=0(x~i0Pi(x~,y~)x~i), 3.31
    with Pi in (2.35). The expression (3.3) shows that (Dx~cay)(y~)=2i=0P(x~,y~)=2i=0(Dx~x~i)(y~), and hence
    (IadX)1=i=0((Ix~)1012(Dx~cay)(y~)(Ix~)1). 3.32
    Post-multiplication with I + adX, according to (3.24), yields (3.25) with
    A(x~,y~)=12(Dx~cay)(y~)(I+x~)+(Ix~)1y~=(Ix~)y~(I+(Ix~)1(I+x~)) 3.33
    =(Ix~)y~(I+R), 3.34
    where (3.33) is obtained invoking (3.5). Since R and x~ are related via the Cayley transform, they satisfy I+R=2(Ix~)1. Inserting this, and its inverse, into (3.34) yields (3.26) and (3.27), respectively. Relation (3.4) shows the identity of (3.27) and (3.28). Inserting (3.2) and (3.5) into (3.27) leads to (3.29), which is merely restating the definition dcayx~(y~)cay(x~)=(Dx~cay)(y~).

    Now (3.29) can be written, with (3.7) and (3.10), as

    dcayx~(y~)cay(x~)=(dcayxy)cay(x~)=σ((I+x~)y)(I+σ(x~+x~2))=σ(y~+x~y~y~x~)(I+σ(x~+x~2))=σ(y~+x~y~y~x~)+σ2((1+||x||2)y~x~+x~y~(x~+x~2))=σ(y~(I+x~)+x~y~(I+σ(x~+x~2)))=y~dcayx+σx~y~cay(x~), 3.35
    which proves (3.30).

    The expression (3.27) was derived in [82], and (3.29) was presented in [71]. Relation (3.26) was reported without proof in [14], and elements of the parameter vector XR6 were called Cayley–Gibbs–Rodrigues motion parameters.

    Equation (3.29) along with (2.56) reveals that the position determined by the Cayley map for the adjoint representation determines the position r = dcayxy, which shows a striking similarity to the exponential map (2.25) on SE(3). On the other hand, the position vector determined by (3.14) and (3.15) is r=(I+cay(x~))y=2σdcayx~Ty. This reveals that the parameters XR6 describing C ∈ SE(3) and those describing the ‘configuration tensor’ AdC are different. Consequently, the right-trivialized differential of the Cayley map on SE(3) and of the adjoint representation are different. This is due to the use of non-canonical coordinates. The differential of the exponential map is indeed the same for both representations (2.23) and (2.28) in terms of canonical coordinates.

    4. Conclusion

    Closed-form relations for the exponential and the Cayley map, their right-trivialized differentials, and the directional derivative of these differentials play a crucial part in most Lie group integration schemes. This includes the Munthe–Kaas methods as well the Lie group generalized-α method. The latter has become an alternative to classical integration methods for multibody systems, in particular for systems comprising flexible bodies undergoing large deformations. The Lie group generalized-α method was originally derived in terms of canonical coordinates, and thus involved the derivatives of the exponential on SE(3). Evidently, the Cayley map is computationally more efficient, and can equally be used as coordinate map within this method. This paper presents a comprehensive summary of all relevant closed form expressions. The presented explicit relations for the Cayley map of Euclidean motions, as well as its differential and directional derivative, will facilitate the development of computationally efficient Lie group generalized-α integration schemes in terms of non-canonical local coordinates. Besides these original relations, a novel derivation of the closed form expressions of the trivialized derivative of the exponential on SE(3) was given.

    Data accessibility

    This article has no additional data.

    Competing interests

    I declare I have no competing interests.

    Funding

    This work was supported by the LCM-K2 Center within the framework of the Austrian COMET-K2 programme.

    Footnotes

    Published by the Royal Society. All rights reserved.

    References