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)

\dot{C} = \hat{V}^{s} C, \dot{C} = C \hat{V}^{b},

1.1

where C(t) ∈ SE(3) describes the motion of a frame in E³ (regarded as Euclidean motion), and

\hat{V}^{s} (t) \in se (3)

is the velocity in spatial representation, and

\hat{V}^{b} (t) \in 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 $C (t) = \exp \hat{X} (t) C (0)$ in terms of the canonical coordinates $X \in R^{6}$ of first kind, then for small t, and with $\hat{X} \in se (3)$ defined in (2.24),

\hat{V}^{s} = {dexp}_{\hat{X}} (\dot{\hat{X}}) and \hat{V}^{b} = {dexp}_{- \hat{X}} (\dot{\hat{X}})

1.2

where the right-trivialized differential

{dexp}_{\hat{X}} : se (3) \to se (3)

is defined by

(D_{\hat{X}} \exp) (\hat{Y}) = {dexp}_{\hat{X}} (\hat{Y}) \exp (\hat{X}),

1.3

with

D_{\hat{X}} \exp : se (3) \to T_{\exp \hat{X}} SE (3)

, so that

(D_{\hat{X}} \exp) (\hat{Y}) := (d / d t) \exp (\hat{X} + t \hat{Y}) |_{t = 0}

is the directional derivative¹

1The directional derivative $(D_{\hat{X}} \exp) (\hat{Y})$ is also denoted in the literature with $D_{\hat{X}} \exp \cdot \hat{Y}$ .

of exp at

\hat{X}

in direction of

\hat{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

\hat{X}

and taking into account (1.1). Occasionally,

{dexp}_{- \hat{X}}

, which satisfies

(D_{\hat{X}} \exp) (\hat{Y}) = \exp (\hat{X}) {dexp}_{- \hat{X}} (\hat{Y}),

1.4

is referred to as the left-trivialized differential, and denoted

{dexp}_{\hat{X}}^{L} (\hat{Y}) := {dexp}_{- \hat{X}} (\hat{Y})

. Equations (1.2) will be called the local reconstruction equations.

The Cayley map $C (t) = cay \hat{X} (t) C (0)$ 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.

X \in R^{6}

. It was shown in [6] that a solution parameterized by the Cayley map satisfies, for small t, the following relation:

\hat{V}^{s} = {dcay}_{\hat{X}} (\dot{\hat{X}}) and \hat{V}^{b} = {dcay}_{- \hat{X}} (\dot{\hat{X}}),

1.5

where now the right-trivialized differential

{dcay}_{\hat{X}} : se (3) \to se (3)

is defined by

(D_{\hat{X}} cay) (Y) = {dcay}_{\hat{X}} (\hat{Y}) cay (\hat{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 (\hat{X} (t)) C (0)

of (1.1) satisfies

\hat{V}^{s} = d ψ_{\hat{X}} (\dot{\hat{X}})

, with

d ψ_{\hat{X}} (\hat{Y})

defined by

(D_{\hat{X}} ψ) (\hat{Y}) = d ψ_{\hat{X}} (\hat{Y}) ψ (\hat{X})

. Such alternative coordinate maps are the kth-order Cayley map [8,9] or the exponential map in terms of dual quaternions [10–12] 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

\dot{\hat{X}} = {dexp}_{\hat{X}}^{- 1} (\hat{V}^{s}) and \dot{\hat{X}} = {dcay}_{\hat{X}}^{- 1} (\hat{V}^{s}),

1.7

and accordingly with negative argument

- \hat{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 [21,23–25] 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 [30–33], 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^{'} = \hat{χ}^{s} C and C^{'} = C \hat{χ}^{b},

1.8

where

\hat{χ} : [0, L] \to 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

\hat{χ}^{s}

was used and referred to as the base-pole generalized curvature, and

\hat{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

\hat{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) \times R^{3}

for rigid body systems as well as geometrically exact beams and shells must be emphasized, which was discussed in [36,39–45]. The modelling of beam kinematics with (1.8) has recently been used to describe soft robots [46–48].

(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 $\exp : g \to G$ on a n-dimensional Lie group G admits the series expansion [49]

\exp (\tilde{x}) = \sum_{i = 0}^{\infty} \frac{1}{i!} {\tilde{x}}^{i} .

2.1

The Lie algebra

g

is assumed isomorphic to

R^{n}

, and

\tilde{x} \in g

denotes the matrix associated with the vector

x \in R^{n}

. For notational convenience, throughout the paper,

\tilde{x} \in so (3)

corresponds to

x \in R^{3}

, while

\hat{X} \in se (3)

is constructed from

X \in R^{6}

[1].

The right-trivialized differential ${dexp}_{\tilde{x}} : g \to g$ of the exponential map on a Lie group admits the series expansion

{dexp}_{\tilde{x}} (\tilde{y}) = \sum_{i = 0}^{\infty} \frac{1}{(i + 1)!} {ad}_{\tilde{x}}^{i} (\tilde{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

{dexp}_{\tilde{x}}^{- 1} (\tilde{y}) = \sum_{i = 0}^{\infty} \frac{B_{i}}{i!} {ad}_{\tilde{x}}^{i} (\tilde{y}),

2.3

with the Bernoulli numbers B_i.

In the following, when representing Lie algebra elements $\tilde{x} \in g$ as vectors $x \in R^{n}$ , the differential and inverse are represented by a matrix denoted ${dexp}_{x} : R^{n} \to R^{n}$ and ${dexp}_{x}^{- 1} : R^{n} \to R^{n}$ so that ${dexp}_{\tilde{x}} (\tilde{y}) = {({dexp}_{x} y)}^{\sim}$ and ${dexp}_{\tilde{x}}^{- 1} (\tilde{y}) = {({dexp}_{x}^{- 1} y)}^{\sim}$ , 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

\begin{aligned} α := sinc φ = \cos \frac{φ}{2} sinc \frac{φ}{2}, β := {sinc}^{2} \frac{φ}{2} \\ γ := \frac{α}{β} = \frac{\cos \frac{φ}{2}}{sinc \frac{φ}{2}} = \frac{\cos^{2} \frac{φ}{2}}{sinc φ} = \frac{sinc φ}{{sinc}^{2} \frac{φ}{2}}, δ := \frac{1 - sinc φ}{φ^{2}} = \frac{1 - α}{φ^{2}} . \end{aligned}

2.4

These terms allow minimizing the number of transcendental function evaluations.

A skew-symmetric matrix $\tilde{x} \in so (3)$ satisfies the characteristic equation ${\tilde{x}}^{3} + | | x | |^{2} \tilde{x} = 0$ . Denote with φ ≔ ‖x‖ the rotation angle. The series (2.1) gives rise to various closed forms [1,55–57]:

\begin{aligned} \exp \tilde{x} & = I + \frac{\sin | | x | |}{| | x | |} \tilde{x} + \frac{1 - \cos | | x | |}{| | x | |^{2}} {\tilde{x}}^{2} \end{aligned}

2.5

\begin{aligned} = I + α \tilde{x} + \frac{1}{2} β {\tilde{x}}^{2} \end{aligned}

2.6

\begin{aligned} \exp (φ \tilde{n}) & = I + \sin φ \tilde{n} + (1 - \cos φ) {\tilde{n}}^{2} \end{aligned}

2.7

\begin{aligned} = I + \sin φ \tilde{n} + \frac{1}{2} \sin^{2} \frac{φ}{2} {\tilde{n}}^{2}, \end{aligned}

2.8

where x ≔ φn, with rotation angle φ and the unit vector n along the rotation axis. The components of

x \in R^{3}

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 $\tilde{x} \in so (3)$ as vector $x \in R^{3}$ , the adjoint operator ${ad}_{\tilde{x}} : so (3) \to so (3)$ is simply $\tilde{x} : R^{3} \to R^{3}$ , and evaluating (2.2) along with ${\tilde{x}}^{3} = - | | x | |^{2} \tilde{x}$ yields the following different matrix forms ${dexp}_{x} : R^{3} \to R^{3}$ of the right-trivialized differential:

\begin{aligned} {d e x p}_{x} & = I + \frac{1 - \cos | | x | |}{| | x | |^{2}} \tilde{x} + \frac{| | x | | - \sin | | x | |}{| | x | |^{3}} {\tilde{x}}^{2} \end{aligned}

2.9

\begin{aligned} = \frac{1}{| | x | |^{2}} [(I - \exp \tilde{x}) \tilde{x} + {x x}^{T}] \end{aligned}

2.10

\begin{aligned} = \frac{1}{| | x | |^{2}} {x x}^{T} - α \tilde{x} - \frac{β}{2} {\tilde{x}}^{2} \end{aligned}

2.11

\begin{aligned} = I + \frac{1}{2} {sinc}^{2} \frac{| | x | |}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - \frac{1}{2} sinc \frac{| | x | |}{2} \cos \frac{| | x | |}{2}) {\tilde{x}}^{2} \end{aligned}

2.12

\begin{aligned} = I + \frac{β}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - α) {\tilde{x}}^{2} . \end{aligned}

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 lim_‖x‖→0dexp_x = I exists, and the relations (2.11) and (2.13) admit a numerically stable evaluation by replacing the term

{\tilde{x}}^{2} / | | x | |^{2}

with

{\tilde{n}}^{2}

, and xx^T/‖x‖² with nn^T.

The matrix form of the inverse of the right-trivialized differential, ${dexp}_{\tilde{x}}^{- 1} : so (3) \to so (3)$ , is found from the series (2.3) as

\begin{aligned} {d e x p}_{x}^{- 1} & = I - \frac{1}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - \frac{| | x | |}{2} \cot \frac{| | x | |}{2}) {\tilde{x}}^{2} \end{aligned}

2.14

\begin{aligned} = I - \frac{1}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - \frac{sinc | | x | |}{{sinc}^{2} \frac{| | x | |}{2}}) {\tilde{x}}^{2} \end{aligned}

2.15

\begin{aligned} = I - \frac{1}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - \frac{\cos \frac{| | x | |}{2}}{sinc \frac{| | x | |}{2}}) {\tilde{x}}^{2} = I - \frac{1}{2} \tilde{x} + \frac{1}{| | x | |^{2}} (1 - γ) {\tilde{x}}^{2} . \end{aligned}

2.16

The expression (2.14) was reported in [59–61], 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 ${dexp}_{- x} = {dexp}_{x}^{T}$ and ${dexp}_{- x}^{- 1} = {dexp}_{x}^{- T}$ , respectively. The spatial and body-fixed angular velocity are hence related to the time derivative of the rotation axis times angle via

ω^{s} = {d e x p}_{x} \dot{x}, \dot{x} = {d e x p}_{x}^{- 1} ω^{s}

2.17

and

ω^{b} = {d e x p}_{x}^{T} \dot{x}, \dot{x} = {d e x p}_{x}^{- T} ω^{b} .

2.18

Lemma 2.1.

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

\begin{aligned} \exp \tilde{x} & = {d e x p}_{x}^{- T} {d e x p}_{x} \end{aligned}

2.19

\begin{aligned} = {d e x p}_{x} {d e x p}_{x}^{- T} \end{aligned}

2.20

\begin{aligned} = I + \tilde{x} {d e x p}_{x} \end{aligned}

2.21

\begin{aligned} = I + {d e x p}_{x} \tilde{x} . \end{aligned}

2.22

Proof.

Equating the differential of $R = \exp \tilde{x}$ written in terms of the right- and left-trivialized differential (1.3) and (1.4) yields ${dexp}_{\tilde{x}} (\tilde{y}) R = R {dexp}_{- \tilde{x}} (\tilde{y})$ , and thus ${dexp}_{\tilde{x}} (\tilde{y}) = R {dexp}_{- \tilde{x}} (\tilde{y}) R^{T}$ , with arbitrary $y \in R^{3}$ . Using the matrix form of dexp, this can be written as ${({dexp}_{\tilde{x}} y)}^{\sim} = {({R dexp}_{- \tilde{x}} y)}^{^{\sim}}$ , which yields ${R dexp}_{- \tilde{x}} = {dexp}_{\tilde{x}}$ and thus (2.19). From the explicit form (2.14) follows that $\tilde{x} = {dexp}_{\tilde{x}}^{- T} - {dexp}_{\tilde{x}}^{- 1}$ . Multiplication with ${dexp}_{\tilde{x}}$ from the left yields $I + {dexp}_{\tilde{x}} \tilde{x} = {dexp}_{\tilde{x}} {dexp}_{\tilde{x}}^{- T}$ and noting (2.19) yields (2.22). The series expansion (2.3) shows that ${dexp}_{\tilde{x}} \tilde{x} = \tilde{x} {dexp}_{\tilde{x}}$ and hence (2.21). Finally, left-multiplication of $\tilde{x} = {dexp}_{\tilde{x}}^{- T} - {dexp}_{\tilde{x}}^{- 1}$ with ${dexp}_{\tilde{x}}$ , 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 $SE (3) = SO (3) ⋉ R^{3}$ , which is represented as subgroup of GL(4) with elements

C = (\begin{array}{cc} R & r \\ 0 & 1 \end{array}) \in SE (3),

2.23

where R ∈ SO(3) describes the rotation of a body-fixed frame, and

r \in R^{3}

is the position vector of the frame origin w.r.t. a reference frame. The Lie algebra se(3) consists of matrices

\hat{X} = (\begin{array}{cc} \tilde{x} & y \\ 0 & 0 \end{array}) \in se (3),

2.24

with

\tilde{x} \in so (3)

and

y \in R^{3}

. There is an obvious correspondence of

\hat{X} \in se (3)

and

X = (x, y) \in R^{6}

, which renders se(3) isomorphic to

R^{6}

The Chasles theorem [64,65] asserts that any finite rigid body displacement C ∈ 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 $\exp (t \hat{X}), t \in [0, 1]$ and $C = \exp \hat{X}$ . 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 $C (t) = \exp \hat{X} (t) C (0)$ . 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

\begin{aligned} \exp (\hat{X}) & = (\begin{array}{cc} \sum_{i = 0}^{\infty} \frac{1}{i!} {\tilde{x}}^{i} & \sum_{i = 0}^{\infty} \frac{1}{(i + 1)!} {\tilde{x}}^{i} y \\ 0 & 1 \end{array}) \\ = (\begin{array}{cc} \exp \tilde{x} & {d e x p}_{x} y \\ 0 & 1 \end{array}) . \end{aligned}

2.25

The dexp mapping on SO(3) occurs since SE(3) is the semidirect product with

R^{3}

as normal subgroup, and se(3) is the semidirect sum with

R^{3}

as ideal (i.e. translations do not affect rotations). Inserting (2.10) into (2.25) yields

\begin{aligned} \exp (\hat{X}) & = (\begin{array}{cc} R & \frac{1}{| | x | |^{2}} (I - R) \tilde{x} y + h y \\ 0 & 1 \end{array}), for x \neq 0 \\ = (\begin{array}{cc} I & y \\ 0 & 1 \end{array}), for x = 0, \end{aligned}

2.26

with rotation matrix

R = \exp \tilde{x}

. The term h ≔ x^Ty/‖x‖² 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

n \in R^{3}

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 (\hat{X}) = (\begin{array}{cc} R & - (α \tilde{x} + \frac{β}{2} {\tilde{x}}^{2}) y + {n n}^{T} y \\ 0 & 1 \end{array}), for x \neq 0,

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 $\hat{V}^{s} = \dot{C} C^{- 1} \in se (3)$ , where the twist vector $V^{s} = (ω^{s}, v^{s}) \in R^{6}$ comprises the spatial angular velocity defined by ${\tilde{ω}}_{i}^{s} = {\dot{R}}_{i} R_{i}^{T}$ and spatial velocity $v_{i}^{s} = {\dot{r}}_{i} + r_{i} \times ω_{i}^{s}$ [1,2,69]. The velocity in body-fixed representation, called body-fixed twist, is defined by $\hat{V}^{b} = C^{- 1} \dot{C} \in se (3)$ , and the twist vector $V^{b} = (ω^{b}, v^{b}) \in R^{6}$ consists of the body-fixed angular velocity ${\tilde{ω}}_{i}^{b} = R_{i}^{T} {\dot{R}}_{i}$ and the velocity $v_{i}^{b} = R_{i}^{T} {\dot{r}}_{i}$ .

Expressing the twist in terms of canonical (screw) coordinates X, according to (1.2), involves the differential ${dexp}_{\hat{X}} : se (3) \to se (3)$ . When representing twists and screws as vectors, the spatial and body-fixed twist is determined as $V^{s} = {dexp}_{X} \dot{X}$ and $V^{b} = {dexp}_{- X} \dot{X}$ , respectively, where ${dexp}_{X} : R^{6} \to R^{6}$ is the matrix form of the right-trivialized differential so that ${dexp}_{\hat{X}} (\hat{Y}) = {({dexp}_{X} Y)}^{\land}$ . The differential can be determined with the general relation (2.2) in terms of ${ad}_{\hat{X}} : se (3) \to se (3)$ . The adjoint operator on se(3) defines the Lie bracket as matrix commutator ${ad}_{{\hat{X}}_{1}} ({\hat{X}}_{2}) = [{\hat{X}}_{1}, {\hat{X}}_{2}] = {\hat{X}}_{1} {\hat{X}}_{2} - {\hat{X}}_{2} {\hat{X}}_{1}$ . In vector representation $X = (x, y) \in R^{6}$ , the matrix form of the adjoint operator is [1,68]

{a d}_{X} = (\begin{array}{cc} \tilde{x} & 0 \\ \tilde{y} & \tilde{x} \end{array}),

2.28

so that

\hat{Z} = {ad}_{{\hat{X}}_{1}} ({\hat{X}}_{2})

is equivalent to

Z = {ad}_{X_{1}} X_{2}

. 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 coordinates X = (x, y),

{d e x p}_{X} = (\begin{array}{cc} {d e x p}_{x} & 0 \\ (D_{x} d e x p) (y) & {d e x p}_{x} \end{array}),

2.29

where dexp_x is the matrix of the right-trivialized differential on so(3) and its directional derivative is given explicitly as

\begin{aligned} (D_{x} d e x p) (y) & = \frac{1}{| | x | |^{2}} (1 - \cos | | x | |) \tilde{y} + \frac{1}{| | x | |^{3}} (| | x | | - \sin | | x | |) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) \\ + \frac{x^{T} y}{| | x | |^{4}} (| | x | | \sin | | x | | + 2 \cos | | x | | - 2) \tilde{x} + \frac{x^{T} y}{| | x | |^{5}} (3 \sin | | x | | - | | x | | \cos | | x | | - 2 | | x | |) {\tilde{x}}^{2} \end{aligned}

2.30

\begin{aligned} = \frac{1}{2} {sinc}^{2} \frac{| | x | |}{2} \tilde{y} + \frac{1}{| | x | |^{2}} (1 - \cos \frac{| | x | |}{2} sinc \frac{| | x | |}{2}) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) \\ + \frac{x^{T} y}{| | x | |^{2}} (\cos \frac{| | x | |}{2} sinc \frac{| | x | |}{2} - {sinc}^{2} \frac{| | x | |}{2}) \tilde{x} \\ + \frac{x^{T} y}{| | x | |^{2}} (\frac{1}{2} {sinc}^{2} \frac{| | x | |}{2} - \frac{3}{| | x | |^{2}} (1 - \cos \frac{| | x | |}{2} sinc \frac{| | x | |}{2}) {\tilde{x}}^{2} \end{aligned}

2.31

\begin{aligned} = \frac{1}{2} {sinc}^{2} \frac{| | x | |}{2} \tilde{y} + \frac{1}{| | x | |^{2}} (1 - sinc | | x | |) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) \\ + \frac{x^{T} y}{| | x | |^{2}} (sinc | | x | | - {sinc}^{2} \frac{| | x | |}{2}) \tilde{x} + \frac{x^{T} y}{| | x | |^{2}} (\frac{1}{2} {sinc}^{2} \frac{| | x | |}{2} - \frac{3}{| | x | |^{2}} (1 - sinc | | x | |)) {\tilde{x}}^{2} \end{aligned}

2.32

\begin{aligned} = \frac{β}{2} \tilde{y} + δ (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) + \frac{x^{T} y}{| | x | |^{2}} (α - β) \tilde{x} + \frac{x^{T} y}{| | x | |^{2}} (\frac{β}{2} - 3 δ) {\tilde{x}}^{2} . \end{aligned}

2.33

Proof.

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

{a d}_{X}^{i} = (\begin{array}{cc} {\tilde{x}}^{i} & 0 \\ P_{i} & {\tilde{x}}^{i} \end{array}) with P_{i} (\tilde{x}, \tilde{y}) = \sum_{j = 0}^{i - 1} {\tilde{x}}^{j} \tilde{y} {\tilde{x}}^{i - j - 1}, i \geq 1

2.34

and P₀ = 0. The matrix P_i can be written as directional derivative of the ith power

{\tilde{x}}^{i}

, considered as a function of

\tilde{x}

, in the direction of

\tilde{y}

\begin{aligned} P_{i} (\tilde{x}, \tilde{y}) & = (D_{\tilde{x}} {\tilde{x}}^{i}) (\tilde{y}) = \frac{d}{d t} {(\tilde{x} + t \tilde{y})}^{i} |_{t = 0} \\ = {(\tilde{x} + t \tilde{y})}^{i - 1} \tilde{y} + {(\tilde{x} + t \tilde{y})}^{i - 2} \tilde{y} (\tilde{x} + t \tilde{y}) + \dots + (\tilde{x} + t \tilde{y}) {\tilde{y}}^{i - 1} |_{t = 0} . \end{aligned}

2.35

The series expansion (2.2) for the differential thus leads to

\begin{aligned} {d e x p}_{X} & = \sum_{i = 0}^{\infty} \frac{1}{(i + 1)!} {a d}_{X}^{i} = (\begin{array}{cc} {d e x p}_{x} & 0 \\ \sum_{i = 0}^{\infty} \frac{1}{(i + 1)!} (D_{\tilde{x}} {\tilde{x}}^{i}) (\tilde{y}) & {d e x p}_{x} \end{array}) \\ = (\begin{array}{cc} {d e x p}_{x} & 0 \\ (D_{x} d e x p) (y) & {d e x p}_{x} \end{array}), \end{aligned}

2.36

with the directional derivative of the right-trivialized differential on SO(3)

(D_{x} d e x p) (y) = \frac{d}{d t} d e x p (\tilde{x} + t \tilde{y}) |_{t = 0} .

2.37

This is evaluated using the following expressions:

(D_{x} | | x | |) (y) = \frac{(x^{T} y)}{| | x | |}, (D_{x} \tilde{x}) (\tilde{y}) = \tilde{y}, (D_{x} {\tilde{x}}^{2}) (y) = \tilde{x} \tilde{y} + \tilde{y} \tilde{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 $\tilde{x} (x^{T} y) / | | x | |^{2}$ can be computed robustly as $\tilde{n} (n^{T} y)$ . 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

{d e x p}_{X}^{- 1} = (\begin{array}{cc} {d e x p}_{x}^{- 1} & 0 \\ (D_{x} {d e x p}^{- 1}) (y) & {d e x p}_{x}^{- 1} \end{array}),

2.39

with canonical coordinates X = (x, y), the matrix in (2.14)–(2.16), and

(D_{x} {d e x p}^{- 1}) (y) = - \frac{1}{2} \tilde{y} + \frac{1}{| | x | |^{2}} (1 - γ) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) + \frac{x^{T} y}{| | x | |^{4}} (\frac{1}{β} + γ - 2) {\tilde{x}}^{2} .

2.40

Proof.

Invoking the series expansion (2.3), along with the power of $\hat{X}$ in (2.34), yields

\begin{aligned} {d e x p}_{X}^{- 1} & = \sum_{i = 0}^{\infty} \frac{B_{i}}{i!} {a d}_{X}^{i} = (\begin{array}{cc} {d e x p}_{x}^{- 1} & 0 \\ \sum_{i = 0}^{\infty} \frac{B_{i}}{i!} (D_{\tilde{x}} {\tilde{x}}^{i}) (\tilde{y}) & {d e x p}_{x}^{- 1} \end{array}) \end{aligned}

2.41

\begin{aligned} = (\begin{array}{cc} {d e x p}_{x}^{- 1} & 0 \\ (D_{x} {d e x p}^{- 1}) (y) & {d e x p}_{x}^{- 1} \end{array}), \end{aligned}

2.42

where the relation (2.35) along with (2.3) has been used to identify D_xdexp⁻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 ${a d}_{X}^{6} = - | | x | |^{4} {a d}_{X}^{2} - 2 | | x | |^{2} {a d}_{X}^{4}$ . In terms of the parameters (2.4), they are

\begin{aligned} {d e x p}_{X} & = I + \frac{1}{2 | | x | |^{2}} (4 - | | x | | \sin | | x | | - 4 \cos | | x | |) {a d}_{X} \\ + \frac{1}{2 | | x | |^{3}} (4 | | x | | - 5 \sin | | x | | + | | x | | \cos | | x | |) {a d}_{X}^{2} \\ + \frac{1}{2 | | x | |^{4}} (2 - | | x | | \sin | | x | | - 2 \cos | | x | |) {a d}_{X}^{3} \\ + \frac{1}{2 | | x | |^{5}} (2 | | x | | - 3 \sin | | x | | + | | x | | \cos | | x | |) {a d}_{X}^{4} \end{aligned}

2.43

\begin{aligned} = I + (β - \frac{α}{2}) {a d}_{X} + \frac{1}{2} (5 δ - \frac{β}{2}) {a d}_{X}^{2} + \frac{1}{2 | | x | |^{2}} (β - α) {a d}_{X}^{3} \\ + \frac{1}{2 | | x | |^{2}} (3 δ - \frac{β}{2}) {a d}_{X}^{4} \end{aligned}

2.44

\begin{aligned} {d e x p}_{X}^{- 1} & = I - \frac{1}{2} {a d}_{X} + (\frac{2}{| | x | |^{2}} + \frac{| | x | | + 3 \sin | | x | |}{4 | | x | | (\cos | | x | | - 1)}) {a d}_{X}^{2} + (\frac{1}{| | x | |^{4}} + \frac{| | x | | + \sin | | x | |}{4 | | x | |^{3} (\cos | | x | | - 1)}) {a d}_{X}^{4} \end{aligned}

2.45

\begin{aligned} = I - \frac{1}{2} {a d}_{X} + \frac{1}{| | x | |^{2}} (2 - \frac{1 + 3 α}{2 β}) {a d}_{X}^{2} + \frac{1}{| | x | |^{4}} (1 - \frac{1 + α}{2 β}) {a d}_{X}^{4} . \end{aligned}

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 [30–33] 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 dexp_x in (2.13) with Ddexp(X) ≔ (D_xdexp)(y), and the derivative (2.40) of ${d e x p}_{x}^{- 1}$ in (2.16) with Ddexp⁻¹(X) ≔ (D_xdexp⁻¹) (y), where X = (x, y). The second directional derivative of dexp is then written as (D_XDdexp) (U) with U = (u, v), and similarly for its inverse.

Lemma 2.4.

The directional derivative of the matrix dexp in (2.29) is

(D_{X} d e x p) (U) = (\begin{array}{cc} D_{x} d e x p) (u) & 0 \\ (D_{X} D d e x p) (U) & (D_{x} d e x p) (u) \end{array}),

2.47

where U = (u, v), and the directional derivative of matrix Ddexp possesses the explicit form

\begin{aligned} (D_{X} D d e x p) (U) & = \frac{β}{2} \tilde{v} + \frac{α - β}{| | x | |^{2}} ((x^{T} u) \tilde{y} + (x^{T} {v + y}^{T} u) \tilde{x} + (x^{T} y) \tilde{u}) \\ - \frac{β}{2 | | x | |^{2}} (x^{T} u) (x^{T} y) \tilde{x} + δ (\tilde{x} \tilde{v} + \tilde{v} \tilde{x} + \tilde{y} \tilde{u} + \tilde{u} \tilde{y}) \\ + \frac{β / 2 - 3 δ}{| | x | |^{2}} ((x^{T} y) (\tilde{x} \tilde{u} + \tilde{u} \tilde{x}) + (x^{T} u) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) + (x^{T} v + y^{T} u) {\tilde{x}}^{2}) \\ + \frac{1}{| | x | |^{4}} (x^{T} y) (x^{T} u) ((1 - 5 α + 4 β) \tilde{x} + (α - \frac{7}{2} β + 15 δ) {\tilde{x}}^{2}) \end{aligned}

2.48

\begin{aligned} = \frac{β}{2} \tilde{v} + \frac{α - β}{| | x | |^{2}} (x^{T} u) \tilde{y} \\ + \frac{β / 2 - 3 δ}{| | x | |^{2}} ((x^{T} u) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) + (x^{T} y) (\tilde{x} \tilde{u} + \tilde{u} \tilde{x}) + δ (\tilde{x} \tilde{v} + \tilde{v} \tilde{x} + \tilde{y} \tilde{u} + \tilde{u} \tilde{y})) \\ + \frac{1}{| | x | |^{2}} ((α - β) (x^{T} {v + y}^{T} u) - \frac{1}{2} β (x^{T} u) (x^{T} y) + \frac{1 - 5 α + 4 β}{| | x | |^{2}} (x^{T} y) (x^{T} u)) \tilde{x} \\ + \frac{1}{| | x | |^{2}} ((\frac{1}{2} β - 3 δ) (x^{T} {v + y}^{T} u) + \frac{1}{| | x | |^{2}} (α - \frac{7}{2} β + 15 δ) (x^{T} y) (x^{T} u)) {\tilde{x}}^{2} . \end{aligned}

2.49

Proof.

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

(D_{x} α) (u) = (δ - \frac{1}{2} β) x^{T} u, (D_{x} β) (u) = \frac{2 x^{T} u}{| | x | |^{2}} (α - β), (D_{x} δ) (u) = \frac{x^{T} u}{| | x | |^{2}} (\frac{1}{2} β - 3 δ) .

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 dexp_X 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 matrix dexp⁻¹ in (2.39) is

(D_{X} {d e x p}^{- 1}) (U) = (\begin{array}{cc} (D_{x} {d e x p}^{- 1}) (u) & 0 \\ (D_{X} {D d e x p}^{- 1}) (U) & (D_{x} {d e x p}^{- 1}) (u) \end{array}),

2.51

where U = (u, v), and the directional derivative of matrix Ddexp⁻¹ possesses the explicit form

\begin{aligned} (D_{X} {D d e x p}^{- 1}) (U) & = - \frac{1}{2} \tilde{v} + \frac{1 - γ}{| | x | |^{2}} (\tilde{x} \tilde{v} + \tilde{v} \tilde{x} + \tilde{y} \tilde{u} + \tilde{u} \tilde{y} + \frac{1}{4} (x^{T} u) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x})) \\ - \frac{1}{| | x | |^{4}} ((1 - γ) (x^{T} u) (2 + γ) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) \\ + (2 - γ - \frac{1}{β}) (x^{T} y) (\tilde{x} \tilde{u} + \tilde{u} \tilde{x}) + (x^{T} v + y^{T} u) \\ \times (2 + \frac{1}{4} (x^{T} y) (x^{T} u) - γ - \frac{1}{β}) {\tilde{x}}^{2}) \\ + \frac{1}{| | x | |^{6}} (x^{T} y) (x^{T} u) (8 - 3 γ - γ^{2} - \frac{2}{β^{2}} (α + β)) {\tilde{x}}^{2} \end{aligned}

2.52

\begin{aligned} = - \frac{1}{2} \tilde{v} + \frac{1 - γ}{| | x | |^{2}} (\tilde{x} \tilde{v} + \tilde{v} \tilde{x} + \tilde{y} \tilde{u} + \tilde{u} \tilde{y}) \\ + \frac{(1 - γ)}{| | x | |^{2}} (x^{T} u) (\frac{1}{4} - \frac{1}{| | x | |^{2}} (2 + γ)) (\tilde{x} \tilde{y} + \tilde{y} \tilde{x}) \\ + \frac{1}{| | x | |^{4}} (γ + \frac{1}{β} - 2) (x^{T} y) (\tilde{x} \tilde{u} + \tilde{u} \tilde{x}) + \frac{1}{| | x | |^{4}} ((γ + \frac{1}{β} - 2) (x^{T} v + y^{T} u) \\ - (x^{T} y) (x^{T} u) (\frac{1}{4} (x^{T} v + y^{T} u) + \frac{1}{| | x | |^{2}} (γ (3 + γ) + \frac{2}{β^{2}} (α + β) - 8))) {\tilde{x}}^{2} . \end{aligned}

2.53

Proof.

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

(D_{x} γ) (u) = \frac{x^{T} u}{| | x | |^{2}} (γ - γ^{2} - \frac{| | x | |^{2}}{4}) .

2.54

The derivative of the coefficient of

{\tilde{x}}^{2}

in (2.40) is then found to be, along with (2.50),

\begin{aligned} (D_{X} (\frac{x^{T} y}{| | x | |^{4}} (\frac{1}{β} + γ - 2))) (U) = \frac{1}{| | x | |^{4}} (x^{T} v + y^{T} u) (γ + \frac{1}{β} - 2 - \frac{1}{4} (x^{T} y) (x^{T} u)) \\ + \frac{(x^{T} y) (x^{T} u)}{| | x | |^{6}} (γ (1 - γ) + \frac{2}{β^{2}} (β - α) - 4 (γ + \frac{1}{β} - 2)) . \end{aligned}

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 Ad_C : se(3) → se(3). In vector representation of twists, this is expressed by the matrix [1,68]

{A d}_{C} = (\begin{array}{cc} R & 0 \\ \tilde{r} R & R \end{array}) \in Ad (S E (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 Ad_C 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

{\dot{A d}}_{C} = {a d}_{V^{s}} {A d}_{C}, \dot{{A d}_{C}} = {A d}_{C} {a d}_{V^{b}}

2.57

and

{A d}_{C}^{'} = {a d}_{χ^{s}} {A d}_{C}, {A d}_{C}^{'} = {A d}_{C} {a d}_{χ^{b}},

2.58

where ad_V and

{a d}_{χ}

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 $X \in R^{6}$ using the relation

{A d}_{\exp \hat{X}} = \exp {a d}_{X},

2.59

where exp is the exponential map of the adjoint representation. Clearly, the canonical (screw) coordinates

X \in R^{6}

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 $C = \exp \hat{X} \in SE (3)$ , the adjoint operator matrix (2.56) and right-trivialized differential of exp : se(3) → SE(3) satisfy the relations

\begin{aligned} {A d}_{C} & = {d e x p}_{- \hat{X}}^{- 1} {d e x p}_{\hat{X}} \end{aligned}

2.60

\begin{aligned} = {d e x p}_{\hat{X}} {d e x p}_{- \hat{X}}^{- 1} \end{aligned}

2.61

\begin{aligned} = I + {a d}_{\hat{X}} {d e x p}_{\hat{X}} \end{aligned}

2.62

\begin{aligned} = I + {d e x p}_{\hat{X}} {a d}_{\hat{X}} . \end{aligned}

2.63

Proof.

The proof is similar to that of lemma 2.1. From (1.3) and (1.4) it follows that ${dexp}_{\hat{X}} (\hat{Y}) = C {dexp}_{- \hat{X}} (\hat{Y}) C^{- 1} = {Ad}_{C} ({dexp}_{- \hat{X}} (\hat{Y}))$ , and hence the vector representation of se(3) yields dexp_XY = Ad_Cdexp_−XY and thus (2.60). The explicit form (2.45) shows that ${a d}_{X} = {d e x p}_{- X}^{- 1} - {d e x p}_{X}^{- 1}$ , and hence $I + {d e x p}_{X} {a d}_{X} = {d e x p}_{X} {d e x p}_{- X}^{- 1}$ in (2.63). From (2.14) it follows that ad_Xdexp_X = dexp_Xad_X, 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 ${a d}_{\tilde{x}} = \tilde{x}, {A d}_{R} = R$ .

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 [6,73–75], 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 $cay : g \to G$ on a quadratic matrix Lie group can be defined as

\begin{aligned} cay (\tilde{x}) & = {(I - \tilde{x})}^{- 1} (I + \tilde{x}) \end{aligned}

3.1

\begin{aligned} = (I + \tilde{x}) {(I - \tilde{x})}^{- 1} \end{aligned}

3.2

\begin{aligned} = I + 2 (\tilde{x} + {\tilde{x}}^{2} + {\tilde{x}}^{3} + \dots) . \end{aligned}

3.3

The components of

x \in R^{n}

serve as local (non-canonical) coordinates around the identity in G. The directional derivative is

\begin{aligned} (D_{\tilde{x}} cay) (\tilde{y}) & := \frac{d}{d t} cay (\hat{X} + t \hat{Y}) |_{t = 0} \\ = {(I - \tilde{x})}^{- 1} \tilde{y} (I + cay (\tilde{x})) = 2 {(I - \tilde{x})}^{- 1} \tilde{y} {(I - \tilde{x})}^{- 1} \end{aligned}

3.4

and thus, with (3.1), the right-trivialized differential

{dcay}_{\tilde{x}} : g \to g

in (1.6) is [21,72]

{dcay}_{\tilde{x}} (\tilde{y}) = 2 {(I - \tilde{x})}^{- 1} \tilde{y} {(I + \tilde{x})}^{- 1} .

3.5

Note that the Cayley transform is occasionally defined with

\tilde{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]

\begin{aligned} {dcay}_{\tilde{x}}^{- 1} (\tilde{y}) & = \frac{1}{2} (I - \tilde{x}) \tilde{y} (I + \tilde{x}) \\ = \frac{1}{2} (\tilde{y} - [\tilde{x}, \tilde{y}] - \tilde{x} \tilde{y} \tilde{x}), \end{aligned}

3.6

with the matrix commutator

[\tilde{x}, \tilde{y}] = \tilde{x} \tilde{y} - \tilde{y} \tilde{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 ${\tilde{x}}^{3} = - | | x | |^{2} \tilde{x}$ , the Cayley map attains the well-known closed form [56,57]

\begin{aligned} cay (\tilde{x}) & = {(I - \tilde{x})}^{- 1} (I + \tilde{x}) = (I + \tilde{x} + {\tilde{x}}^{2} + {\tilde{x}}^{3} + \dots) (I + \tilde{x}) \\ = I + σ (\tilde{x} + {\tilde{x}}^{2}), \end{aligned}

3.7

with

σ := \frac{2}{1 + | | x | |^{2}} .

3.8

Vector

x \in R^{3}

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 = \tan \frac{φ}{2} \tilde{n}

. 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 (φ \tilde{n}) = cay (\tan \frac{φ}{2} \tilde{n}) = I + \frac{2}{1 + \tan^{2} (φ / 2)} (\tilde{n} + {\tilde{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

R^{3}

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

\begin{aligned} {d c a y}_{x} & = σ (I + \tilde{x}) \end{aligned}

3.10

\begin{aligned} {d c a y}_{x}^{- 1} & = \frac{1}{σ} I + \frac{1}{2} ({\tilde{x}}^{2} - \tilde{x}) \end{aligned}

3.11

\begin{aligned} = \frac{1}{2 σ} (I + cay (- \tilde{x})) = \frac{1}{2 σ} (I + R^{T}) \end{aligned}

3.12

with

R = cay (\tilde{x})

. They satisfy the properties

{d c a y}_{- x} = {d c a y}_{x}^{T}

and

{d c a y}_{- x}^{- 1} = {d c a y}_{x}^{- T}

Proof.

The last term in (3.6) can be expressed as $\tilde{x} \tilde{y} \tilde{x} = {\tilde{x}}^{2} \tilde{y} - \tilde{x} \tilde{\tilde{x} y} = x^{T} x \tilde{y} - \tilde{x} \tilde{\tilde{x} y} - | | x | |^{2} \tilde{y} = \tilde{\tilde{x} y} \tilde{x} - \tilde{x} \tilde{\tilde{x} y} - | | x | |^{2} \tilde{y} = [\tilde{x}, [\tilde{x}, \tilde{y}]] - | | x | |^{2} \tilde{y}$ , and hence ${dcay}_{\tilde{x}}^{- 1} (\tilde{y}) = \frac{1}{2} ((1 + | | x | |^{2}) \tilde{y} - [\tilde{x}, \tilde{y}] + [\tilde{x}, [\tilde{x}, \tilde{y}]])$ . With $[\tilde{x}, \tilde{y}] = \tilde{\tilde{x} y}$ and $[\tilde{x}, [\tilde{x}, \tilde{y}]] = \tilde{\tilde{x} \tilde{x} y}$ follows the matrix (3.11) so that $z = {d c a y}_{x}^{- 1} y$ when $\tilde{z} = {dcay}_{\tilde{x}}^{- 1} (\tilde{y})$ , 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 ${d c a y}_{- x} = {d c a y}_{x}^{T}$ .

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

\begin{aligned} ω^{s} = {d c a y}_{x} \dot{x}, \dot{x} = {d c a y}_{x}^{- 1} ω^{s} \\ and & ω^{b} = {d c a y}_{x}^{T} \dot{x}, \dot{x} = {d c a y}_{x}^{- T} ω^{b} . \end{aligned}

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 $\hat{X} \in se (3)$ , shown in (2.25), yields

\begin{aligned} cay (\hat{X}) & = (\begin{array}{cc} cay (\tilde{x}) & (I + cay (\tilde{x})) y \\ 0 & 1 \end{array}) \end{aligned}

3.14

\begin{aligned} = (\begin{array}{cc} cay (\tilde{x}) & 2 σ {d c a y}_{\tilde{x}}^{- T} y \\ 0 & 1 \end{array}), \end{aligned}

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

X \in R^{6}

cay (\hat{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 $V^{s} = {d c a y}_{X} \dot{X}$ and $V^{b} = {d c a y}_{- X} \dot{X}$ , respectively, where ${d c a y}_{X} : R^{6} \to R^{6}$ 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

{d c a y}_{X} = (\begin{array}{cc} {d c a y}_{\tilde{x}} & 0 \\ σ \tilde{y} (I + \tilde{x}) & 2 I + σ (\tilde{x} + {\tilde{x}}^{2}) \end{array}) = (\begin{array}{cc} σ (I + \tilde{x}) & 0 \\ σ \tilde{y} (I + \tilde{x}) & 2 I + σ (\tilde{x} + {\tilde{x}}^{2}) \end{array})

3.16

and

{d c a y}_{X}^{- 1} = (\begin{array}{cc} {d c a y}_{\tilde{x}}^{- 1} & 0 \\ \frac{1}{2} (\tilde{x} - I) \tilde{y} & \frac{1}{2} (I - \tilde{x}) \end{array}) = \frac{1}{2} (\begin{array}{cc} \frac{2}{σ} I + {\tilde{x}}^{2} - \tilde{x} & 0 \\ (\tilde{x} - I) \tilde{y} & I - \tilde{x} \end{array}) .

3.17

Proof.

Inserting matrices of the form (2.24) corresponding to X = (x, y) and $U = (u, v) \in R^{6}$ into (3.6) yields

{dcay}_{\hat{X}}^{- 1} (\hat{U}) = (\begin{array}{cc} {dcay}_{\tilde{x}}^{- 1} (\tilde{u}) & \frac{1}{2} (I - \tilde{x}) (v - \tilde{y} u) \\ 0 & 0 \end{array}) \in se (3) .

3.18

Expressing

\hat{Z} = {dcay}_{\hat{X}}^{- 1} (\hat{U})

in vector form as

Z = {d c a y}_{\hat{X}}^{- 1} U

, along with (3.11), yields (3.17). The inverse of (3.17) is readily obtained as

{d c a y}_{\hat{X}} = \frac{2}{1 + | | x | |^{2}} (\begin{array}{cc} (I + \tilde{x}) & 0 \\ \tilde{y} (I + \tilde{x}) & (1 + | | x | |^{2}) I + \tilde{x} + {\tilde{x}}^{2} \end{array}),

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 of dcay_x:so(3) → so(3) and its inverse is given in closed form as

(D_{x} d c a y) (\tilde{y}) = σ \tilde{y} - σ^{2} (x^{T} y) (I + \tilde{x})

3.20

and

(D_{x} {d c a y}^{- 1}) (\tilde{y}) = (x^{T} y) I + \frac{1}{2} (\tilde{x} \tilde{y} + \tilde{y} \tilde{x} - \tilde{y}) .

3.21

The directional derivative of matrix dcay_X of the right-trivialized differential in (3.16) and its inverse admit the following explicit forms, where X = (x, y) and U = (u, v):

(D_{X} d c a y) (U) = (\begin{array}{cc} σ \tilde{u} - σ^{2} (x^{T} u) (I + \tilde{x}) & 0 \\ σ (\tilde{v} + \tilde{v} \tilde{x} + \tilde{y} \tilde{u}) - σ^{2} (x^{T} u) (\tilde{y} + \tilde{y} \tilde{x}) & σ (\tilde{u} + \tilde{x} \tilde{u} + \tilde{u} \tilde{x}) - σ^{2} (x^{T} u) (\tilde{x} + {\tilde{x}}^{2}) \end{array})

3.22

and

(D_{X} {d c a y}^{- 1}) (U) = \frac{1}{2} (\begin{array}{cc} 2 (x^{T} u) I + \tilde{u} \tilde{x} + \tilde{x} \tilde{u} - \tilde{u} & 0 \\ \tilde{x} \tilde{v} + \tilde{u} \tilde{y} - \tilde{v} & - \tilde{u} \end{array}) .

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 ({a d}_{X}) = {(I - {a d}_{X})}^{- 1} (I + {a d}_{X}) .

3.24

Lemma 3.4.

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

cay ({a d}_{X}) = (\begin{array}{cc} cay (\tilde{x}) & 0 \\ A (x, y) & cay (\tilde{x}) \end{array}),

3.25

with

\begin{aligned} A (\tilde{x}, \tilde{y}) & = \frac{1}{2} (I + R) \tilde{y} (I + R) \end{aligned}

3.26

\begin{aligned} = 2 {(I - \tilde{x})}^{- 1} \tilde{y} {(I - \tilde{x})}^{- 1} \end{aligned}

3.27

\begin{aligned} = (D_{\tilde{x}} cay) (\tilde{y}) \end{aligned}

3.28

\begin{aligned} = {dcay}_{\tilde{x}} (\tilde{y}) R \end{aligned}

3.29

\begin{aligned} = \tilde{y} {d c a y}_{x} + σ \tilde{x} \tilde{y} R, \end{aligned}

3.30

with

R := cay (\tilde{x})

Proof.

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

\begin{aligned} {(I - {a d}_{X})}^{- 1} & = I + {a d}_{X} + {a d}_{X}^{2} + {a d}_{X}^{3} + \dots \\ = \sum_{i = 0}^{\infty} (\begin{array}{cc} {\tilde{x}}^{i} & 0 \\ P_{i} (\tilde{x}, \tilde{y}) & {\tilde{x}}^{i} \end{array}), \end{aligned}

3.31

with P_i in (2.35). The expression (3.3) shows that

(D_{\tilde{x}} cay) (\tilde{y}) = 2 \sum_{i = 0}^{\infty} P (\tilde{x}, \tilde{y}) = 2 \sum_{i = 0}^{\infty} (D_{\tilde{x}} {\tilde{x}}^{i}) (\tilde{y})

, and hence

{(I - {a d}_{X})}^{- 1} = \sum_{i = 0}^{\infty} (\begin{array}{cc} {(I - \tilde{x})}^{- 1} & 0 \\ \frac{1}{2} (D_{\tilde{x}} cay) (\tilde{y}) & {(I - \tilde{x})}^{- 1} \end{array}) .

3.32

Post-multiplication with I + ad_X, according to (3.24), yields (3.25) with

\begin{aligned} A (\tilde{x}, \tilde{y}) & = \frac{1}{2} (D_{\tilde{x}} cay) (\tilde{y}) (I + \tilde{x}) + {(I - \tilde{x})}^{- 1} \tilde{y} \\ = (I - \tilde{x}) \tilde{y} (I + {(I - \tilde{x})}^{- 1} (I + \tilde{x})) \end{aligned}

3.33

\begin{aligned} = (I - \tilde{x}) \tilde{y} (I + R), \end{aligned}

3.34

where (3.33) is obtained invoking (3.5). Since R and

\tilde{x}

are related via the Cayley transform, they satisfy

I + R = 2 {(I - \tilde{x})}^{- 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

{dcay}_{\tilde{x}} (\tilde{y}) cay (\tilde{x}) = (D_{\tilde{x}} cay) (\tilde{y})

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

\begin{aligned} {dcay}_{\tilde{x}} (\tilde{y}) cay (\tilde{x}) & = {({d c a y}_{x} y)}^{\sim} cay (\tilde{x}) \\ = σ {((I + \tilde{x}) y)}^{\sim} (I + σ (\tilde{x} + {\tilde{x}}^{2})) \\ = σ (\tilde{y} + \tilde{x} \tilde{y} - \tilde{y} \tilde{x}) (I + σ (\tilde{x} + {\tilde{x}}^{2})) \\ = σ (\tilde{y} + \tilde{x} \tilde{y} - \tilde{y} \tilde{x}) + σ^{2} ((1 + | | x | |^{2}) \tilde{y} \tilde{x} + \tilde{x} \tilde{y} (\tilde{x} + {\tilde{x}}^{2})) \\ = σ (\tilde{y} (I + \tilde{x}) + \tilde{x} \tilde{y} (I + σ (\tilde{x} + {\tilde{x}}^{2}))) \\ = \tilde{y} {d c a y}_{x} + σ \tilde{x} \tilde{y} cay (\tilde{x}), \end{aligned}

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 $X \in R^{6}$ 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 = dcay_xy, 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 $r = (I + cay (\tilde{x})) y = 2 σ {d c a y}_{\tilde{x}}^{- T} y$ . This reveals that the parameters $X \in R^{6}$ describing C ∈ SE(3) and those describing the ‘configuration tensor’ Ad_C 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.

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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.

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

September 2021

Volume 477Issue 2253

Article Information

DOI:https://doi.org/10.1098/rspa.2021.0303
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received06/04/2021
Manuscript accepted03/08/2021
Published online01/09/2021
Published in print29/09/2021

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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

Abstract

1. Introduction

(a) Lie group integration and coordinate maps

(b) Lie group modelling of flexible beam kinematics

(c) Contribution of this paper

2. Motion parameterization via the exponential map

(a) Spatial rotations: SO(3)

(i) Exponential map: Euler–Rodrigues formula

(ii) Differential of the exponential map

Lemma 2.1.

Proof.

(b) Euclidean motions: SE(3)

(i) Exponential map

(ii) Differential of the exponential map

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

(c) Directional derivative of the right-trivialized differential

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

(d) Adjoint representation: the ‘configuration tensor’

Lemma 2.6.

Proof.

3. Motion parameterization via the Cayley map

(a) Spatial rotations: SO(3)

(i) Gibbs–Rodrigues parameters

(ii) Differential of the Cayley map

Lemma 3.1.

Proof.

(b) Euclidean motions: SE(3)

(i) Cayley map on SE(3)

(ii) Differential of the Cayley map

Lemma 3.2.

Proof.

(c) Directional derivative of the right-trivialized differential

Lemma 3.3.

Proof.

(d) Adjoint representation: the ‘configuration tensor’

Lemma 3.4.

Proof.

4. Conclusion

Data accessibility

Competing interests

Funding

Footnotes

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