Screw and Lie group theory in multibody kinematics
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Abstract
After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and userfriendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive \(O ( n ) \) algorithms, for which the socalled “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of treetopology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using bodyfixed joint frames, a formulation without joint frames, and a formulation without either joint or bodyfixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of treetopology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.
Keywords
Rigid bodies Multibody systems Kinematics Relative coordinates Recursive algorithms Screws Lie groups Frame invariance1 Introduction
Computational multibody system (MBS) dynamics aims at mathematical formulations and efficient computational algorithms for the kinetic analysis of complex mechanical systems. At the same time the modeling process is supposed to be intuitive and user friendly. Moreover, the efficiency of MBS algorithms and the complexity of the actual modeling process is largely determined by the way the kinematics is described. This concerns the core issues of representing rigid body motions and describing the kinematics of technical joints. Both issues can be addressed with concepts of screw and Lie group theory.
Spatial MBS perform complicated motions, and in general rigid bodies perform screw motions that form a Lie group. Although the theory of screw motions is well understood, screw theory has almost completely been ignored for MBS modeling with only a few exceptions. The latter can be grouped into two classes. The first class uses of the fact that the velocity of a rigid body is a screw, referred to as the twist. The propagation of twists within an MBS is thus described as a frame transformation of screw coordinates. This gave rise to the socalled “spatial vector” formulation introduced in [23, 24] and to the socalled “spatial operator algebra”, which was formalized in [64] and used for \(O ( n ) \) forward dynamics algorithms, for example, in [25, 32, 33, 38, 63, 65]. Screw notations are also used in the formulations presented in [5, 36, 37, 74]. Further MBS formulations were reported that use screw notations uncommon for the MBS community [26]. All these approaches only exploit the algebraic properties of screws as far as relevant for a compact handling of velocities, accelerations, wrenches, and inertia. The second class goes one step further by recognizing that finite motions form the Lie group \(\mathit{SE} ( 3 ) \) with the screw algebra as its Lie algebra \(\mathit{se} ( 3 ) \). Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and efficient modeling of rigid body mechanisms. Some of the first publications reporting Lie group formulations of the kinematics of an open kinematic chain are [15, 28, 29] and [19, 20]. In this context the term product of exponentials (POE) is being used since Brockett used it in [15]. Unfortunately, these publications have not reached the MBS community, presumably because of the used mathematical concepts that differ from classical MBS formalisms. The basic concept is the exponential mapping that determines the finite relative motion of two adjacent bodies connected by a lower pair joint in terms of a screw associated with the joint. The product of the exponential mappings for all consecutive joints determines the overall motion of the chain. Within this formulation, twists are naturally represented as screws, and joint motions are described in terms of screw coordinates.
Motivated by [15], Lie group formulations for MBS dynamics were reported in a few publications, for example, [44, 46, 56, 57, 58, 59, 60]. It should be mentioned that the basic elements of a screw formulation for MBS dynamics were already presented in [39] but did not receive due attention.
A crucial feature of these geometric approaches is their frame invariance, which allows for arbitrary representations of screws and for freely assigning reference frames, which drastically simplify the kinematics modeling and also provide a direct link to CAD models. Moreover, the POE, and thus the kinematics, can even be formulated without the use of any joint frame, which basically resembles the “zero reference” formulation reported for a robotic arm in [27]. On the other hand, classical approaches to the description of joint kinematics are the Denavit–Hartenberg (DH) [3, 22, 35] (in its different forms) and the Sheth–Uicker twoframe convention [69]. Such twoframe conventions are used in most of the current MBS dynamics simulations packages that use relative coordinates. The Lie group description, on the other hand, not only allows for arbitrary placement of joint frames but makes them dispensable altogether.
The benefits of geometric modeling have been recognized already in robotics. Recently, at least in robotics, the text books [40, 52, 67] have reached a wider audience. Modern approaches to robotics make extensive use of screw and Lie group theoretical concepts. This is, also supported by the Universal Robot Description Format (URDF) that is used, for instance, in the Robot Operating System (ROS), rather than DH parameters. In MBS dynamics the benefits of geometric mechanics are slowly being recognized. Interestingly, this mainly applies to the modeling of MBS with flexible bodies undergoing large deformations [8, 72]. This is not surprising since geometrically exact formulations require correct modeling of the finite kinematics of a continua. The displacement field of a Cosserat beam, for instance, is a proper motion in \(E^{3}\) and thus modeled as motion in \(\mathit{SE} ( 3 ) \). This is an extension of the original work on geometrically exact beams and shells by Simo [70, 71], where the displacement field is modeled on \(\mathit{SO} ( 3 ) \times{\mathbb{R}}^{3}\). Another topic where Lie group theory is recently applied in MBS dynamics is the time integration. To this end, Lie group integration schemes were modified and applied to MBS models in absolute coordinate formulation [17], where the motions of individual bodies are described as a general screw motion that are constrained according to the interconnecting joints. It shall be remarked that, despite the current trend to emphasize the use of Lie group (basic) concepts, the basics formulations for nonlinear flexible MBS were already reported by Borri et al. [10, 11, 12].
The aim of this paper is to provide a comprehensive summary of the basic concepts for modeling MBS in terms of relative coordinates using joint screws and to relate them to existing formulations that are scattered throughout the literature. Without loss of generality the concepts are introduced for a kinematic chain within an MBS with arbitrary topology [34, 45]. It is also the aim to show that MBS can be modeled in a userfriendly way without having to follow restrictive modeling conventions and that this gives rise to \(O ( n ) \) formulations. The latter are not the topic of this paper.
The paper is organized as follows. In Sect. 2, the MBS configuration is described in terms of joint variables, used as generalized coordinates, with the joint geometry parameterized by joint screw coordinates. This classical approach of using bodyfixed joint frames to describe relative configurations is extended to a formulation that does not involve joint frames. The corresponding relations for the MBS velocity are derived in Sect. 3. A formulation is introduced for each of the four different definitions of rigid body twists found in the literature. The latter are called the bodyfixed, spatial, hybrid, and mixed twists. They differ by the reference point used to measure the velocity and by the frame in which the angular and translational velocities are resolved. The different twist representations are introduced in Sect. A.2. Recursive relations for the respective Jacobians are derived, and the computational aspects are discussed with emphasize on their decomposition. The presented formulation allows for an efficient modeling of the MBS kinematics in terms of readily available geometric data. Throughout the paper, only a few basic concepts from Lie group theory are required, which are summarized in Appendix A. The used nomenclature is summarized in Appendix B.
As for all Lie group formulations, the biggest hurdle for a reader (who may be already be familiar with MBS formulations) is the notation. The reader not familiar with screws and Lie group modeling may want to consult Sect. A.1 before reading Sect. 2 and Sect. A.2 before reading Sect. 3. This paper is aimed to provide a reference and cannot replace an introductory textbook like [40, 52, 67]. A beginner is recommended to consult [40]. Yet there is no text book that treats the topic from an MBS perspective. Readers not interested in the derivations could simply use the main relations that are displayed with a black border.
2 Configuration of a kinematic chain
In this section the kinematics modeling using joint screw coordinates is presented. For simplicity, a single open kinematic chain is considered comprising \(n\) moving bodies interconnected by \(n\) 1DOF lower pair joints. To simplify the formulation, but without loosing generality, higherDOF joints are modeled as combination of 1DOF lower pair joints. Bodies and joints are labeled with the same indices \(i=1,\ldots,n\), whereas the ground is indexed with 0. With the sequential numbering of bodies and joints of the kinematic chain, joint \(i\) connects body \(i\) to its predecessor body \(i1\). A bodyfixed reference frame (BFR) \(\mathcal{F}_{i}\) is attached to body \(i\) of the MBS. The body is then kinematically represented by this BFR.
2.1 Joint kinematics
\(n\)dimensional motion subgroups of \(\mathit{SE}(3)\)
n  Subgroup  Motion 

1  ℝ  1dim. translation along some axis 
1  SO(2)  1dim. rotation about arbitrary fixed axis 
1  \(H_{p}\)  screw motion about arbitrary axis with finite pitch 
2  \(\mathbb{R}^{2}\)  2dim. planar translation 
2  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}\)  translation along arbitrary axis & rotation along this axis 
3  \(\mathbb{R}^{3}\)  spatial translations 
3  SO(3)  spatial rotations about arbitrary fixed point 
3  \(H_{p}\ltimes\mathbb{R}^{2}\)  translation in a plane + screw motion ⊥ to this plane (pitch h) 
3  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}^{2}=\mathit{SE} ( 2 ) \)  planar motions 
4  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}^{3}=\mathit{SE} ( 2 ) \ltimes \mathbb{R}\)  spatial translations + rotation about axis with fixed orientation (Schönflies motion) 
6  SE(3)  spatial motion 
Mechanical generators of the \(n\)dimensional subgroups of \(\mathit{SE}(3)\). A motion subgroup can be generated by a lower pair or by a “macro joint”, i.e., a combination of joints with smaller DOF
n  Subgroup  Lower pair  Macro joint 

1  ℝ  Prismatic joint  × 
1  SO(2)  Revolute joint  × 
1  \(H_{p}\)  Screw joint  × 
2  \(\mathbb{R}^{2}\)  ×  combination of two nonparallel prismatic joints 
2  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}\)  Cylindrical joint  × 
3  \(\mathbb{R}^{3}\)  ×  combination of three nonparallel prismatic joints 
3  SO(3)  Spherical joint  × 
3  \(H_{p}\ltimes\mathbb{R}^{2}\)  ×  planar joint + screw joint with axis normal to plane 
3  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}^{2}=\mathit{SE} ( 2 ) \)  Planar joint  × 
4  \(\mathit{SO} ( 2 ) \ltimes\mathbb{R}^{3}=\mathit{SE} ( 2 ) \ltimes \mathbb{R}\)  ×  planar joint + prismatic joint with axis normal to plane 
6  SE(3)  ×  “free joint” 
Assumption 1
It is assumed throughout the paper that the two JFRs coincide in the reference configuration \(q_{i}=0\). This assumption can be easily relaxed if required.
Denote with \(q_{i}\) the joint variable (angle, translation). With Assumption 1, the configuration of the JFR \(\mathcal{J}_{i,i}\) on body \(i\) relative to the JFR \(\mathcal{J}_{i1,i}\) on body \(i1\) is given by the exponential in (69) as \(\mathbf{D}_{i} ( q_{i} ) :=\exp ({}{}{^{i1}}\mathbf{Z}_{i}q_{i})\).
Remark 1
2.2 Recursive kinematics using bodyfixed joint frames
The absolute configuration of body \(i\), that is, the configuration of its BFR \(\mathcal{F}_{i}\) relative to the inertial frame (IFR) \(\mathcal {F}_{0}\) is represented by \(\mathbf{C}_{i}\in \mathit{SE} ( 3 ) \). The relative configuration of body \(i\) relative to body \(i1\) is \(\mathbf {C}_{i1,i}:=\mathbf{C}_{i1}^{1}\mathbf{C}_{i}\). The configuration of a rigid body in the kinematic chain can be determined recursively by successive combination of the relative configurations of adjacent bodies as \(\mathbf {C}_{i}=\mathbf{C}_{0,1}\mathbf{C}_{1,2}\cdots\mathbf{C}_{i1,i}\).
For joint \(i\), denote by \({}\mathbf{S}_{i1,i}\) the constant transformation from JFR \(\mathcal{J}_{i1,i}\) to the RFR \(\mathcal{F}_{i1}\) on body \(i1\), and by \(\mathbf{S}_{i,i}\) the constant transformation from JFR \(\mathcal {J}_{i,i}\) to the RFR \(\mathcal{F}_{i}\) on body \(i\) (Fig. 2). Then the relative configuration is \(\mathbf {C}_{i1,i}={}\mathbf{S}_{i1,i}\mathbf{D}_{i} ( q_{i} ) {}\mathbf{S}_{i,i}^{1}\). Denote by \(\mathbf{q}\in{\mathbb{V}}^{n}\) the vector of joint variables that serve as generalized coordinates of the MBS. The joint space manifold is \({\mathbb{V}}^{n}={\mathbb{R}}^{n_{\text{P}}}\times \mathbb{T}^{n_{\text{R}}}\) for an MBS model comprising \(n_{\text{P}}\) prismatic and \(n_{\text{R}}\) revolute/screw joints (\(n_{\text{P}}+n_{\text{R}}=n\)).

Introduction of bodyfixed JFR \(\mathcal{J}_{i,i}\) at body \(i\) with relative configuration \(\mathbf{S}_{i,i}\);

Introduction of bodyfixed JFR \(\mathcal{J}_{i1,i}\) at body \(i1\) with relative configurations \(\mathbf{S}_{i1,i}\);

The screw coordinate vector \({^{i1}}\mathbf{Z}_{i}\) of joint \(i\) represented in JFR \(\mathcal{J}_{i1,i}\) at body \(i1\).
Expression (3) is the standard MBS formulation for the kinematics of an open chain in terms of relative coordinates, that is, joint angles or translations. For 1DOF joints, the JFR is usually oriented so that its 3axis points along the joint axis (as in Fig. 2). Then the screw coordinates are \({^{i1}}\mathbf{Z}_{i}=(0,0,1s_{i},0,0,s_{i}+h_{i} ( 1s_{i} ) )^{T}\), where \(s_{i}\,=1\) for prismatic joint, and \(s_{i}=0\) for a screw joint with finite pitch \(h_{i}\) (for revolute joints, \(h_{i}=0\)).
Remark 2
The matrix \(\mathbf{C}_{i}\) is used to represent the configuration of body \(i \); hence the symbol. Frequently, the symbol \(\mathbf{T}_{i}\) is used [40, 74], which refers to the fact that these matrices describe the transformation of point coordinates (Sect. A.1).
Remark 3
It is important to emphasize that the Lie group formulation (3) is merely another approach to the standard matrix formulation of MBS kinematics aiming at compact expressions that simplify the implementation without compromising the efficiency. It also includes various conventions used to describe the joint kinematics. An excellent overview of classical matrix methods (also with emphasis on how they can be employed for synthesis) can be found in [74]. For instance, \(\mathbf {S}_{i1,i}\) and \(\mathbf{S}_{i,i}\) can be parameterized in terms of the constant part of the Denavit–Hartenberg (DH) parameters [74]. Formulation (3) in particular resembles the Sheth–Uicker convention (which was introduced to eliminate the ambiguity of the DH parameters) [69, 74]. In that notation the matrices \(\mathbf{S}_{i1,i}\) and \(\mathbf{S}_{i,i}\) are called the shape matrices of joint \(i\). However, the Sheth–Uicker convention still presumes certain alignment of joint axes. For example, a revolute axis is supposed to be parallel to the 3axis of the JFRs. A recent discussion of these notations can be found in [9]. An expression similar to (3) was also presented in [54], where no restriction on the joint axis is imposed. A recursive formulation of the MBS motion equations using homogeneous transformation matrices was also presented in [36, 37].
Remark 4
(MultiDOF joints)
The description for 1DOF joints in terms of a screw coordinate vector \(\mathbf{Z}_{i}\) can be generalized to joints with more than one DOF. For a joint with DOF \(\nu\), the relative configuration of the JFRs can alternatively be described in terms of \(\nu\) joint variables \(q_{i_{1}},\ldots,q_{i_{\nu}}\) by \(\mathbf{D}_{i} ( q_{i_{1}},\ldots ,q_{i_{\nu}} ) :=\exp({}{}{^{i1}}\mathbf {Z}_{i_{1}}q_{i_{1}}+\cdots+{^{i1}}\mathbf{Z}_{i_{\nu}}q_{i_{\nu }})\) or \(\mathbf{D}_{i} ( q_{i_{1}},\ldots,q_{i_{\nu}} ) :=\exp({}{}{^{i1}}\mathbf{Z}_{i_{1}}q_{i_{1}})\cdot\ldots\cdot\exp({}{}{^{i1}}\mathbf{Z}_{i_{\nu}}q_{i_{\nu}})\). For a spherical joint, for instance, the variables in the first form are the components of the rotation axis times angle in (64), and in the second form, these are three angles corresponding to the order of 1DOF rotations (e.g. Eulerangles). For lower pair joints, in the first case, \(q_{i_{1}},\ldots,q_{i_{\nu}}\) are canonical coordinates of the first kind on the joint motion subgroup, and in the second case, they are canonical coordinates of the second kind [52]. The \(\mathbf{Z}_{i_{1}},\ldots,\mathbf{Z}_{i_{\nu}}\) form a basis on the subalgebra of the motion subgroup generated by the joint.
2.3 Recursive kinematics without bodyfixed joint frames
The introduction of joint frames is a tedious step within the MBS kinematics modeling. Moreover, it is desirable to minimize the data required to formulate the kinematic relations. In this regard the frame invariance of screws is beneficial.

The relative reference configuration \(\mathbf{B}_{i}\) of the adjacent bodies connected by joint \(i\) for \(q_{i}=0\);

The screw coordinates \({^{i}}\mathbf{X}_{i}\) of joint \(i\) represented in the BFR \(\mathcal{F}_{i}\) at body \(i\), or alternatively the screw coordinates \({^{i1}}\bar{\mathbf{X}}_{i}\) represented in the BFR \(\mathcal{F}_{i1}\) at body \(i1\).
The form (8) simplifies the expression for the joint kinematics. Its main advantage is that it only involves the reference configuration \(\mathbf{B}_{i}\) of BFRs.
2.4 Recursive kinematics without bodyfixed joint frames and screw coordinates

Absolute reference configurations \(\mathbf{A}_{i}=\mathbf {C}_{i} ( \mathbf{0} ) \), that is, the reference configuration of body \(i\) with respect to the IFR \(\mathcal{F}_{0}\) for \(\mathbf{q}=\mathbf{0}\).

Joint screw coordinates \(\mathbf{Y}_{i}\equiv\)\({^{0}}\mathbf {Y}_{i}^{0}\) in spatial representations, that is, measured and resolved in the IFR \(\mathcal{F}_{0}\) for \(\mathbf{q}=\mathbf{0}\).
The result (10) is remarkable since it allows for formulating the MBS kinematics without bodyfixed joint frames. From a modeling perspective this has proven very useful since no joint transformations \(\mathbf{S}_{i,i},\mathbf{S}_{i1,i}\) or \(\mathbf{B}_{i}\) are needed. Only the absolute reference configurations \(\mathbf{A}_{i}\) with respect to the IFR and the reference screw coordinates (12), that is, \(\mathbf{e}_{i}\) and \(\mathbf{p}_{i}\), resolved in the IFR, are required. This is in particular advantageous when processing CAD data. Moreover, if in the reference (construction) configuration the RFR of the bodies coincide with the IFR (global CAD reference system), that is, all parts are designed with respect to the same RFR, then \(\mathbf{A}_{i}=\mathbf{I}\) and \(\mathbf{Y}_{j}={^{j}}\mathbf{X}_{j}\).
2.5 Example
3 Velocity of a kinematic chain
In this section, recursive relations are derived for the four forms of twists introduced in Sect. A.2, namely the bodyfixed, spatial, hybrid, and mixed twists [18].
3.1 Bodyfixed representation of rigid body twists
3.1.1 Bodyfixed twists
3.1.2 Bodyfixed Jacobian and recursive relations
Remark 5
(Dependence on joint variables)
With (8), respectively (10), it is clear that the Jacobian \(\mathbf{J}_{i}^{\text{b}}\) of body \(i\) can only depend on \(q_{1},\ldots ,q_{i}\). Moreover, noting in (16) that \(\mathbf{C}_{i,j}=\mathbf {C}_{i}^{1}\mathbf{C}_{j}\) is independent from \(q_{1}\), it follows that \(\mathbf{J}_{i}^{\text{b}}\) depends on \(q_{2},\ldots,q_{i}\), that is, it is independent from the first joint in the chain. This is obvious from a kinematic perspective since \(\mathbf{V}_{i}^{\text{b}}\) is the sum of twists of the preceding bodies in the chain expressed in the BFR on body \(i\). This only depends on the configuration of the bodies relative to body \(i\) but not on the absolute configuration of the overall chain, which is determined by \(q_{1}\).
Remark 6
(Required data)
The second form in (16) in conjunction with (10) allows for computation of the bodyfixed Jacobian without introducing bodyfixed JFRs. The only information needed is the joint screw coordinates \(\mathbf{Y}_{j}\) represented in the IFR and the reference configurations \(\mathbf{A}_{j}\).
Remark 7
(Change of reference frame)
Remark 8
(Application of bodyfixed representation)
The recursive relations for bodyfixed twist and Jacobian are the basis for the MBS dynamics algorithms in [1, 2, 7, 25, 30, 38, 39, 56, 57, 60, 75]. In [1, 2] the adjoint transformation matrix in (14) was called the “shift matrix”, and \(\mathbf{X}_{i}\) was called the “motion map matrix”. However, the geometric background was rarely exploited as in [56, 57, 60] and [39]. Remarkably, Liu [39] already presented all relevant formulations in terms of screws.
3.1.3 Bodyfixed system Jacobian and its decomposition
Remark 9
(Overall inverse kinematics solution)
Although solution (26) seems straightforward, it should be remarked that there is no frame invariant inner product on \(\mathit{se} ( 3 ) \), that is, no norm of screws that is invariant under a change of reference frame can be defined [67]. The correctness of (26) follows by regarding the transposed joint screw coordinates as coscrews, and \({^{i}\mathbf{X}}_{i}^{T}{^{i}\mathbf{X}}_{i}\) is the pairing of screw and coscrew coordinates rather than an inner product.
3.2 Spatial representation of rigid body twists
3.2.1 Spatial twists
3.2.2 Spatial Jacobian and recursive relations
Remark 10
The spatial representation has remarkable advantages. The velocity recursion (30) is the simplest possible since the twists of individual bodies can simply be added without any coordinate transformation. An important observation is that \(\mathbf{J}_{j}^{\text{s}}\) is intrinsic to joint \(j\). The nonzero screw vectors in the Jacobian (29) are thus the same for all bodies. This is a consequence of using a single spatial reference frame.
3.2.3 Spatial system Jacobian and its decomposition
Remark 11
(Dependence on joint variables)
Similarly to the bodyfixed twist, since \(\mathbf{J}_{i}^{\text{s}}=\mathbf{Ad}_{\mathbf{C}_{i}}{^{i}}\mathbf{X}_{i}=\mathbf {Ad}_{\mathbf{C}_{i1}}\mathbf{Ad}_{\mathbf{B}_{i}\exp{^{i}}\mathbf {X}_{i}q_{i}}{^{i}}\mathbf{X}_{i}=\mathbf{Ad}_{\mathbf {C}_{i1}}{^{i}}\mathbf{X}_{i}\) is independent from \(q_{i}\), it follows that the spatial Jacobian of body \(i\) only depends on \(q_{1},\ldots,q_{i1}\). Indeed, the motion of joint \(i\) does not change its screw axis about which body \(i\) is moving.
Remark 12
(Change of reference frame)
Remark 13
(Application of spatial representation)
The spatial twist is used almost exclusively in mechanism kinematics (often without mentioning it) but is becoming accepted for MBS modeling since it was introduced in [23, 24]. For kinematic analysis of mechanisms, it is common practice to (instantaneously) locate the global reference frame so that it coincides with the frame where kinetostatic properties (twists, wrenches) are observed, usually at the endeffector. For a serial robotic manipulator, the endeffector frame is located at the terminal link of the chain, so that \(\mathbf {A}_{n}=\mathbf{I} \), and \(\mathbf{V}_{n}^{\text{s}}\) is then the spatial endeffector twist. From their definition it follows that the spatial and hybrid twist (see next section) of body \(i\) are numerically identical when the BFR \(\mathcal{F}_{i}\) overlaps with the IFR \(\mathcal{F}_{0}\).
The most prominent use of the spatial representation in dynamics is the \(O ( n ) \) forward dynamics method by Featherstone [23, 24]. This has not yet been widely applied in MBS dynamics. This may be due to use of an uncommon choice of reference point (the IFR origin) at which the spatial entities are measured, so that results and interaction wrenches must be transformed to bodyfixed reference frames. The spatial representation of twists must not be confused with the “spatial vector” notation proposed in [23, 24]. The latter is a general expression of twists as 6vectors (like bodyfixed and spatial) but without reference to a particular frame in which the components are resolved. This allows for abstract derivation of kinematic relations, but these relations must eventually be resolved in a particular frame, and this eventually determines the computational effort.
A notable application of the spatial twist is the modeling and numerical integration of nonlinear elastic MBS, where it is called the base pole velocity [10] or fixed pole velocity [13], and the intrinsic coupling of translational and angular velocity (according to the screw motion) was discussed. The corresponding momentum balance and conservation properties are discussed in [11, 12] (see also [50]).
Remark 14
As in Remark 9, relation (34) gives rise to an overall inverse kinematics solution. For given spatial twists \(\mathbf {V}_{i}^{\text{s}}\), this reads in components as \(\dot {q}_{i}={^{i}\mathbf{X}}_{i}^{T}\mathbf{Ad}_{\mathbf {C}_{i}}^{1}(\mathbf{V}_{i}^{\text{s}}\mathbf{V}_{i1}^{\text{s}})/ \Vert {^{i}\mathbf{X}}_{i} \Vert ^{2}\).
3.3 Hybrid form of rigid body twists
3.3.1 Hybrid twists
3.3.2 Hybrid Jacobian and recursive relations
Remark 15
(Application of hybrid representation)
The hybrid form was used in [76] for forward kinematics calculation of serial manipulators and in [4, 5] to compute the motion equations and the inverse dynamics solution. It is used in many recursive \(O ( n ) \) forward dynamics algorithms such as [6, 38, 53, 54, 64], where relations (42) and (40) play a central role. In the socalled “spatial operator algebra” [64], hybrid screw entities are called “spatial vectors”. The hybrid form is deemed computationally efficient since the transformations only involve translations. The actual configuration of the chain is not discussed in these publications, but it enters via the vectors \(\mathbf{e}_{i} ( \mathbf{q} ) \) and \(\mathbf{r}_{i} ( \mathbf{q} )\), respectively \(\mathbf{d}_{i,j} ( \mathbf{q} ) \). In [38] the inverse transformation \(\mathbf{Ad}_{\mathbf{r}_{i,j}}^{1}\) was denoted by \({}^{j}\mathbf{X}_{i}\) (not to be confused with (5)), and the screw coordinate vector \({^{0}}\mathbf{X}_{j}^{j}\) in (38) by \(\phi_{j}\). In [64], \(\mathbf{Ad}_{\mathbf{r}_{i,j}}^{1}\) was denoted by \(\phi_{i,j}^{T}\), and \({^{0}}\mathbf{X}_{j}^{j}\) with \(\mathbf {H}_{j}^{T}\). The transposed matrices appear since they arise from the transformation of wrenches.
3.3.3 Hybrid system Jacobian and its decomposition
3.4 Mixed form of rigid body twists
3.4.1 Mixed twists
3.4.2 Mixed Jacobian and recursive relations
3.4.3 Mixed system Jacobian and its decomposition
3.5 Relation of different forms
Transformation of the different representations of twists and joint screw coordinates
It should be finally mentioned that the screw coordinates \({^{i}}\mathbf {X}_{i}\) and \({^{0}}\mathbf{X}_{i}^{i}\) are just different coordinates for the same geometric object, namely of the instantaneous joint screw of joint \(i\) measured in the BFR at body \(i\) but resolved either in this BFR or in the IFR. The vector \(\mathbf{Y}_{i}\) on the other hand is a snapshot of the joint screw coordinates of joint \(i\) in spatial representation at the reference \(\mathbf{q}=\mathbf{0}\).
Remark 16
(Computational efficiency)
It is clear from (14), (30), (42), and (52) that the number of numerical operations differ between the four different representations of twists. This allows for selecting the most efficient one when a kinematic analysis is envisaged. In [55] the problem of determining the twists of the terminal body in a kinematic chain (robot endeffector) was analyzed for bodyfixed, spatial, and hybrid forms. This study suggests that the spatial representation is computationally most efficient. A conclusive analysis of all four forms has not yet been reported. Moreover, the general situation includes the dynamic analysis. This was partly addressed in [73, 79].
3.6 Example (continued)
4 Conclusions and outlook
Screw and Lie group theory gives rise to compact formulations of the equations governing the MBS kinematics in terms of relative (joint) coordinates. This is beneficial for the actual modeling process and for the implementation of MBS algorithms and their computational properties. The frame invariance of these concepts allows for expressing the relevant modeling objects as suited best for a particular application. In particular, the MBS kinematics can be formulated without introduction of bodyfixed joint frames. This is a central result that gives rise to maximal flexibility as opposed to the use of modeling conventions like Denavit–Hartenberg parameters. These results have been published over the last two decades, but they have not been presented within a uniform MBS framework. In this paper, screw and Lie group theory has been employed to provide such a framework. Decisive for the computational efficiency is the actual representation of rigid body twists and accelerations. Four commonly used forms were recalled, and the recursive algorithms for MBS kinematics where presented. The corresponding recursive algorithms for evaluation of the motion equations are presented in the accompanying paper [50].
The reader used to work with the classical bodyfixed twists should be able to directly apply the presented modeling paradigm for MBS kinematics using relation (10) to determine the body configurations and (16) to determine the Jacobian while having the freedom to choose arbitrary BFR and IFR. This applies likewise to the spatial, hybrid, and mixed twists.
The full potential of Lie group formulations is yet to be explored in future research. This regards the modeling steps and the computational properties, in particular, given a current trend in computational MBS dynamics to put more emphasis on userfriendly modeling and on tailored simulation codes. A forthcoming paper will address MBS with general topology. To this end, the loop closure constraints are formulated in the form of a POE. Redundant loop constraints are still a major challenge. It is already known that the loop constraints can be concisely formulated in terms of joint screws, but even more that they can be reduced to a nonredundant constraint system by means of simple operations on the joint screw system [47].
Notes
Acknowledgements
Open access funding provided by Johannes Kepler University Linz. The author acknowledges that this work has been partially supported by the Austrian COMETK2 program of the Linz Center of Mechatronics (LCM).
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