involved multiplying the A-matrices and simplifying the expressions obtained for the elements of 3t and 3p. Besides being lengthy, this method does not allow insight into the mechanisms that make the simplifications possible. The

approach presented here provides the same results with much less effort and greater insight by taking advantage of the properties of rotation transformations.

 By writing the product of two A matrices-in the form



 RiRj (Rilj + Ii)
 0 0 0 1



we divide Eq. (4.1) into a position equation


 p = RI(R2(R3(R4(R516+15)+14)+13)+12)+1I (4.2)


and an orientation equation


 R = R1 R2 R3 R4 R5 R6. (4.3)


With the frame assignment conventions discussed, 16=0 whenjoint 6 is revolute. Equation (4.2) then simplifies to

 .p =-RI(R2(R3(R415+l4)+3)+l2)+11. (4.4) .


 Three independent scalar equations for Px, Py, and Pz can be obtained from Eq. (4.4) and more equations can be selected out of the 9 scalar equations implied by Eq. (4.3).

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