Inertia reference member

The coefficient matrix of ao in Equation 6.51 rq>resoits the effective simple closed-chain mechanism as seen by the reference member at the origin of its own coordinate system. The operational space inertia of the reference member is just its spatial inertia matrix, lo. Note that the operational space inertias of the augmented chains (acting in parallel on the reference member) add in a simple sum. This is a general rule for inertia matrices. For actuated chains connected in series, the combination rule is not as simple. In this case, extended versions of the recursive algorithms of Chapter 4 may be applied. 

The vector ao refers to the motion of the coordinate oigin of frame 0. The spatial inertia matrix, lo, is also defined at this point, and it is known and constant Because wq is givra fw the present state, the velocity-dependent term, bo, may be computed directly. If we combine Equations 6.6 and 6.7, we finally obtain the following dynamic equation for the refnence member. 

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