Articulated-body recursive equation

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 ... 

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 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. 

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... 

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 ... 

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