An Alternate Form of the Simulation Algorithm

In the develc ment of the simulation algorithm in the previous section, the objective was to decouple the simple closed-chain mechanism by computing the spatial force vectors exerted by the chains on the reference membo. The spatial tip forces computed in that algorithm are real, measurable forces, associated with the general jdnts which connect the reference membo and each chain tip. Once these forces are known, the chains are effectively decoupled from the refnence member, and the general joint accelerations may be computed for each chain separately. 

Und these conditions, the spatial tip acceleration of augmented chain k, XJ, is exactly equal to the spatial accel tion of the reference member. That is. 

If we combine Equation 6.43 and the dynamic equation for chain k given in Equation 6.S, we may write 

the augmented chain has at least six degrees of freedom, and if it is not in a singular position, the matrix MJ will be exactly equal to the operational space inertia mauix of the augmented chain. That is. 

For convenience, let us assume that this condition is true for each chain. Then, we may write  

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