Screw and Lie group theory in multibody kinematics

Müller, Andreas

doi:10.1007/s11044-017-9582-7

Cited by 64 publications

(75 citation statements)

References 81 publications

(302 reference statements)

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“…Notice that solving (109) forq i leads to the result in Remark 9 of [60]. Solving (66c) forq i yields (12).…”

Section: Remarkmentioning

confidence: 97%

“…In this section, results for the acceleration and jerk of a kinematic chain are presented for the body-fixed, spatial, and hybrid representation. The corresponding relations for the mixed representation are readily found from either one of these using the relations in Table 3 of [60].…”

Section: Acceleration Jerk and Partial Derivatives Of Jacobiansmentioning

confidence: 99%

“…The kinematic forward recursion, together with the factorization in Sect. 3.1.3 of [60], was used to derive O(n) forward dynamics algorithms in [30,44], where the Lie group concept is regarded as spatial operator algebra. Other O(n) forward dynamics algorithms were presented in [2,7,34,35].…”

Section: Remarkmentioning

confidence: 99%

“…A central concept is the representation of velocities (twists) as screws. Four different variants were recalled in [60]. In this paper, their application to dynamics modeling is reviewed.…”

Section: Introductionmentioning

confidence: 99%

“…2. The Newton-Euler equations for the four different representations of twists introduced in [60] are then recalled in Sect. 3.…”

Section: Introductionmentioning

confidence: 99%

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Screw and Lie group theory in multibody dynamics

Müller

2017

Multibody Syst Dyn

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Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS), and at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper, recursive O(n) Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formulation: the so-called Euler-Jourdain or "projection" equations, of which Kane's equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

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“…Notice that solving (109) forq i leads to the result in Remark 9 of [60]. Solving (66c) forq i yields (12).…”

Section: Remarkmentioning

confidence: 97%

Section: Acceleration Jerk and Partial Derivatives Of Jacobiansmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

“…A central concept is the representation of velocities (twists) as screws. Four different variants were recalled in [60]. In this paper, their application to dynamics modeling is reviewed.…”

Section: Introductionmentioning

confidence: 99%

“…2. The Newton-Euler equations for the four different representations of twists introduced in [60] are then recalled in Sect. 3.…”

Section: Introductionmentioning

confidence: 99%

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Screw and Lie group theory in multibody dynamics

Müller

2017

Multibody Syst Dyn

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Screw Theory – A forgotten Tool in Multibody Dynamics

Müller

2017

Proc Appl Math and Mech

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Mechanism theory, synthesis and control of robots, rigid body dynamics, geometrically exact formulations of the kinematics of continua, structure preserving numerical integration methods, and in part geometric mechanics -they all have a common denominator, namely the theory of screws, for which Lie group theory forms the mathematical foundation. This paper is an attempt to provide a short survey identifying several scientific areas where screw theory is already (sometimes implicitly) used and such where its systematic application could lead to new formulations and computational algorithms.In the fourth volume of the ZAMM, published in 1924, Richard von Mises introduced the 'Motorrechnung' (Motor Calculus) [13-15] building on the earlier work of Robert Stawell Ball [2] from the late 19th century on screw theory, Eduard Sudy's work on 'Dynamen' [25], and Julius Plücker's work on line geometry [23] together with the projective geometry of Grassmann [8] and Cayley [4]. Since then screw theory has become a corner stone of modern kinematics. Moreover, screw theory has established itself as a core concept in the theory of mechanisms and machines. In this context, the geometric and algebraic classification of screw systems in [7] was a crucial step toward a geometric analysis of the instantaneous kinematics of linkages with far reaching implications for the systematic analysis of mechanisms and robotic manipulators [11,12]. While being wellaccepted in kinematics, screw theory has not been applied to dynamics until the second half of the 20th century. Moreover, there is an apparent methodological disconnection of the screw theoretic treatment of space kinematics and the algorithmic approaches in MBS dynamics. Besides a few contributions, e.g. [3,5,16,17,[20][21][22] screw and Lie group theory is almost absent in computational MBS dynamics.Kurt Magnus dedicated his Habilitation thesis [10] to the application of motor calculus to the equations of motion (EOM) of rigid body systems. He made extensive use of the frame invariance of screws in order to derive the EOM represented in an arbitrary reference frame. In the last two decades, recognizing that screws form the algebra se (3) of the Lie group SE (3) of rigid body motions gave rise to frame invariant, compact and computationally efficient models for the kinematics and dynamics of (rigid and flexible) multibody systems (MBS). This also provides a link to geometric mechanics and allows for application of geometric numerical integration methods on Lie groups. The coordinate invariance of Lie group formulations allows to derive various formulations that can be employed for different purposes. Yet screw theory has been largely ignored in MBS dynamics.

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Modular Control and Services to Operate Lineless Mobile Assembly Systems

Kluge-Wilkes

Baier

Kunze

et al. 2023

Interdisciplinary Excellence Accelerator Series

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The increasing product variability and lack of skilled workers demand for autonomous, flexible production. Since assembly is considered a main cost driver and accounts for a major part of production time, research focuses on new technologies in assembly. The paradigm of Line-less Mobile Assembly Systems (LMAS) provides a solution for the future of assembly by mobilizing all resources. Thus, dynamic product routes through spatiotemporally configured assembly stations on a shop floor free of fixed obstacles are enabled. In this chapter, we present research focal points on different levels of LMAS, starting with the macroscopic level of formation planning, followed by the mesoscopic level of mobile robot control and multipurpose input devices and the microscopic level of services, such as interpreting autonomous decisions and in-network computing. We provide cross-level data and knowledge transfer through a novel ontology-based knowledge management. Overall, our work contributes to future safe and predictable human-robot collaboration in dynamic LMAS stations based on accurate online formation and motion planning of mobile robots, novel human-machine interfaces and networking technologies, as well as trustworthy AI-based decisions.

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Screw and Lie group theory in multibody kinematics

Cited by 64 publications

References 81 publications

Screw and Lie group theory in multibody dynamics

Screw and Lie group theory in multibody dynamics

Screw Theory – A forgotten Tool in Multibody Dynamics

Modular Control and Services to Operate Lineless Mobile Assembly Systems

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