A geometrical formulation of the dynamical equations describing kinematic chains

Brockett, Roger W.; Stokes, A.; Park, F.

doi:10.1109/robot.1993.291887

Cited by 25 publications

(20 citation statements)

References 5 publications

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“…If the forward kinematics are specified using the product of exponentials formula, then it is possible to get more explicit formulas for the inertia and Coriolis matrices. In this section we derive these formulas, based on the treatments given by Brockett et al [15] and Park et al [87].…”

Section: Robot Dynamics and The Product Of Exponentials Formulamentioning

confidence: 99%

“…Its application to the dynamics of a robot manipulator can be found in most standard textbooks on robotics, for example [2,21,35,52,110]. The geometric formulation of the equations of motion for kinematic chains presented in Section 3.3 is based on the recent work of Brockett, Stokes, and Park [15,87]. This is closely related to the spatial operator algebra formulation of Rodriguez, Jain, and Kreutz-Delgado [45,98], in which the tree-like nature of the system is more fully exploited in computing inertial properties of the system.…”

Section: Bibliographymentioning

confidence: 99%

See 1 more Smart Citation

A Mathematical Introduction to Robotic Manipulation

Murray¹,

Sastry²,

Li³

2017

2,718

2,002

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Section: Robot Dynamics and The Product Of Exponentials Formulamentioning

confidence: 99%

Section: Bibliographymentioning

confidence: 99%

A Mathematical Introduction to Robotic Manipulation

Murray¹,

Sastry²,

Li³

2017

2,718

2,002

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“…The instantaneous joint screws (105) and thus their derivatives (16) and (21) are independent of a particular body. The closed form of the νth-order partial derivative has been reported in [54]:…”

Section: Spatial Representationmentioning

confidence: 99%

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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“…Without resorting the exponential mapping, a 'zero reference formulation' was already reported by Gupta [72] in terms of frame transformation matrices. The POE formulae (55) was first used for MBS modeling in [30]. The main advantages of the spatial POE formulation is that all required data are represented in the spatial inertial frame.…”

Section: Relative Coordinate Modeling Of Multibody Systems In Lie Gromentioning

confidence: 99%

“…The expression (86) for the Christoffel symbols was reported in [30,115] using Lie group notation. Prior to these publications, an equivalent expression was presented in [94] using tensor notation, where the bodyfixed Jacobians were called 'kinematic basic functions.'…”

Section: Motion Equations In Closed Formmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

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A geometrical formulation of the dynamical equations describing kinematic chains

Cited by 25 publications

References 5 publications

A Mathematical Introduction to Robotic Manipulation

A Mathematical Introduction to Robotic Manipulation

Screw and Lie group theory in multibody dynamics

Geometric methods and formulations in computational multibody system dynamics

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