Articulated-body

Recursive dynamic equations fw a single c n chain are derived by Featherstone in [9] and later presented again by Brandi, et al. in [3]. The formulation of these equations is based on the concept of treating a chain of rigid bodies connected by joints as a single articulated body . The recursive equations allow the dynamics of the entire chain to be resolved to a single link called the handle of the articulated body. All interactions with the articulated body are assumed to... 

Feathmtone defines the articulated-body inertia of a link as the 6 x 6 matrix which relates a spatial force applied to the link and the spatial acceleration of the link, taking into account the dynamics of the rest of the articulated body. This relationship is linear, and for link t, it may be written as follows [9] ... 

A number of important recurrence relations in the articulated-body fwmula-tion may be derived by analyzing a simple two-link articulated body as shown in Figure 4.3. This articulated body is made up of links 1 and 2 with spatial link inertias of Ii and I2, respectively. The two links are connected by a general joint, joint 2, which is characterized by its motion space, < 2> Recall from Chapter 2 that 2 represents the free modes of this joint In this analysis, it is assumed that the two bodies are initially at rest and experience no applied joint forces or torques on or about 2 (t2 = 0). The tip force, f, is zero b ause the chain is unconstrained. The only spatial force vector exerted on the articulated body is fi, which is applied to link 1, the handle of the articulated body. For the moment, we will assume that the physical parameters describing the links and joints, including the spatial link inertias, are given in absolute coordinates. 

The development of the recursive articulated-body dynamic equations begins with the simple free-body dynamic equations of the two individual links. In the notation of this book, we may write the free-body equation for link 1 as follows ... 

This is a simple recursive acceleration equation fw the unconstrained articulated body. Equation 4.37 may be combined with Equations 4.31 and 4.32 to give the following dynamic equation for link 1, the handle of the unconstrained articulated body ... 

The equations given above may be generalized for the case of an unconstrained articulated body with an arbitrary number of links. The dynamic equation for link i as the handle of the unconstrained articulated body, ignoring bias terms, is written in Equation 4.30, repeated here for convenience ... 

In Equation 4.51, L acts as a spatial transformation which prqjagates the spatial accel tion vector, a,- i, across joint i. We will call a matrix which transforms spatial vectcs s across actuated joint structures a spatial articulated tran ormation. In general, an articulated transformation is a nonlinear function of the articulated-body inertia and is a dimensionless 6x6 matrix. Featherstone calls the articulated transformation, L,, the acceleration propagator [9]. It relates the spatial acceleration of one link of an articulated body to the spatial acceleration of a neighbraing link in the same articulated body (ignoring bias... 

We will now extend the recursive articulated-body dynamic equations for an open chain to describe the dynamics of a chain which is constrained at the tip. For this task, we will refer again to Figure 4.3. Now, however, we will assume that f, the spatial force exerted by the tip, is nonzero. 

Note that I], the articulated-body inertia of link 1, is the same whether the chain is constrained or unconstrained. The force vector, f, defined by the equation ... 

It appears that (A]) is propagated across joint 2 via the pre- and postmultiplication by the acceleration and force propagatos, (L ) and (Lp, respectively. This propagated matrix is then combined with K2, which is associated with the inverse articulated-body inotia of link 2. 

Note the position of the coordinate transformations in this equation compared to the link coordinate version of the articulated-body inertia equation given in Thble 4.S. This difference in position is due to a difference in the order of tq)plication of the transformations across joints and links in the two algorithms. As a chain is traversed in the recursion for A, the ( rational space inertia matrix of a link is first transformed across the present link (via X) and then transfomed across the next joint (via L). In contrast, as a chain is traversed in the recursion for I. the articulated-body in a is first transfomed across a joint and then transfnmed back across the preceding link. 

Brandi, Johanni, and Otter [3] computes the articulated-body inertia of each link in the chain, starting at the tip and moving back to the base. This same recursion is the first recursion in the Force Propagation Method for computing A. That is, there is an overlap of computations between the solution for the q)en-chain acceleration terms, tjopen and Xopen. and the calculation of the inverse ( rational space inertia matrix, A for this case. This fact was taken into account when the operations were tabulated. The ( rations listed for SI and A in Table 5.2 include only the second recursion for A and the additional opoations needed for SI. The recursion which computes the articulated-body inertias is included in the computatimis for open and x<,pe . 

R. Featherstone. The Calculation of Robot Dynamics Using Articulated-Body Inertias. The International Journal of Robotics Research, 2(l) 13-30, Spring 1983. 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

Featherstone R (1983) The calculation of robot dynamics using articulated-body inertias. Int J Rob Res 2(l) 13-30... 

---