Screw Theory – A forgotten Tool in Multibody Dynamics

Abstract

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.

How to Manipulate? Kinematics, Dynamics and Architecture of Robot Arms

“…Also, (Angeles, 2014) introduced an alternative method to solve the inverse dynamics of a robotic system, the natural orthogonal complement (NOC). Regarding mathematical tools useful for the analysis of the mobility, kinematics and dynamics of robotic systems and mechanisms, you can take a look into group theory (Angeles, 2014), screw theory (Davidson, 2004;Müller, 2017), where the twist and wrench concept originate, and dual-numbers algeabra, useful to combine a translation and a rotation into one single variable. Finally, you can also look into the concept of constraint singularities for parallel mechanisms (Zlatanov et al, 2002).…”

Phase-Space Modeling and Control of Robots in the Screw Theory Framework Using Geometric Algebra

“…In addition, the development is tedious for some robots. Hence, the new proposed method expands the Hamiltonian control approach using screw theory, where this powerful mathematical tool has been used in recent years for the analysis of spatial mechanisms and some works have expressed it as "the forgotten tool in multibody dynamics" [12,13].…”

Kinematics and Dynamics Analysis of a 3UPS-UPU-S Parallel Mechanism

“…As Pennock has discussed the geometric relationship between velocity screws and momentum screws [19], researchers such as Gallardo [17,20] have presented kinematic and dynamic models for parallel mechanisms utilizing screw coordinates. Müller has pointed out that besides the contribution to the kinematic analysis made by screw theory, screw theory and the Lie group can also be applied to the dynamic analysis of multi-body systems with high efficiency [21,22]. Zhao [5] has investigated an approach to dynamic analysis of multi-body systems by expressing acceleration in screw form using screw and Lie products, but the dynamic equations are established using the velocity expressed in screw coordinates as a global variable according to the Newton-Euler method.…”