Unified formulation of dynamics for serial rigid multibody systems

Abstract

There has been a growing interest in the development of new and efficient algorithms for multibody dynamics in recent years. Serial rigid multibody systems form the basic subcomponents of general multibody systems, and a variety of algorithms to solve the serial chain forward dynamics problem have been proposed. In this paper, the economy of representation and analysis tools provided by the spatial operator algebra are used to clarify the inherent structure of these algorithms, to identify those that are similar, and to study the relationships among the ones that are distinct. For the purposes of this study, the algorithms are categorized into three classes: algorithms that require the explicit computation of the mass matrix, algorithms that are completely recursive in nature, and algorithms of intermediate complexity. In addition, alternative factorizations for the mass matrix and closed form expressions for its inverse are derived. These results provide a unifying perspective, within which these diverse dynamics algorithms arise naturally as a consequence of a progressive exploitation of the structure of the mass matrix. Serial Chain oft €(R 9lx9l n .91 r p (k) r v (k) Link/ Joint Properties Nomenclature! = mass matrix for the serial chain = number of links in the serial chain = total number of motion DOF for the serial chain = number of positional DOF of the kth joint = number of motion DOF of the kth joint = joint matrix for the kth joint = moment of inertia of the Ath linkabout G k = vector from 0^ to 0^_ i = spatial inertia of the Ath link about G k = mass of the Ath link = reference location of the Ath joint on the Ath link = reference location of the Arth joint on the (k + l)th link = vector from 0* to the center of mass of the Arth link = joint motion parameters for Ath joint = configuration variables for Ath joint Forces and Velocities a(k) €(R 6 = Coriolis and centrifugal spatial acceleration at 0* b(k) € (ft 6 = gyroscopic spatial force at G k F(k) € (R 3 = linear force of interaction between the (k + l)th and kth links at 0* l(k,k-\) M(Ar) m(k)

Forward dynamics algorithms for multibody chains and contact

“…We note thatM k is precisely the articulated body inertia introduced by Featherstone [10]; the bias force z(k) in [12] is ourb k +M k a k . For instance, in the n = 3 case, we obtain the following structure of non-zero terms …”

“…Suppose the links of the robot are numbered, as in [22,12,5], starting from the free end (note that this is the opposite of the convention used elsewhere in the robotics literature). Further, we will assume that if the coordinate frame is not specified, a quantity for body k is expressed in frame k coordinates.…”

“…In the following we consider a chain of rigid bodies; we use a notation that is similar to [11,22,12,18,19]. The motion of a rigid body i = 1, .…”

“…The derivation of ABM algorithms, as the name implies, hinges on the concept of Articulated Body Inertias, whose connection to fast algorithms is subtle to non-experts in dynamics. An alternate approach us-ing spatial operator algebras was proposed by [21,12], who recognized the structural similarity of the ABM recursions to those arising in Kalman filtering. The operator algebra approach provides new insights into these algorithms and is more intuitive to those familiar with Kalman filtering.…”

A reusable, real-time spacecraft dynamics simulator

“…The Darts dynamics compute engine [3] implements a fast and ecient spatial algebra recursive algorithm [4,5] for solving the dynamics of exible, multibody, tree-topology systems. It is also used for nonspacecraft applications, such as molecular dynamics [6].…”