Development of the General Simulation Algorithm

The method outlined above is quite straightforward and illustrates some features of a parallel computational structure as discussed previously. Of course, the illustrated example represents a special case. We will now develop a similar approach fex a general simple closed-chain mechanism which, although the notation becomes a bit more complicated, also exhibits the parallel computational characteristics we expect. 

Consider a mechanism with m chains, each with an arbitrary number of degrees of freedom, N. The interaction between each chain tip and the reference member is arbitrary and will be modelled using the general joint model of Chapter 2. To begin, we will derive an explicit relationship between the spatial acceleration of each chain tip and the spatial acceleration of the reference member. The spatial acceleration of the tip of chain k is denoted by xt. The relative spatial acceleration between the tip of chain k and the reference member, x, resolved in the orthogonal vector spaces of the general joint between them, may be written  

We may also ex]xess in terms of the spatial acceleradon of the reference member, ao, as follows  

This equation matches the spatial accelerations at the coupling point between chain k and the reference member, giving an explicit relationship between jtk and ao when the coiqtling is arbitrary. The unknowns in Equation 6.19 are x , Ok, and ao. All other vectors and matrices are known from the initial information given for the simulation problem. 

To decouple the chains and the reference member, we need an explicit mathematical relationship between the spatial force exerted by chain ib on the reference member, f t and the spatial acceleration of the reference member, ao. Equation 6.19 relates the spatial acceleration of the refnence member and the spatial acceleration of the tip of chain k. We may eliminate Ok, the unknown components of the relative acceleration, by projecting Equation 6.19 onto the constraint space of the corresponding general joint as follows  

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