as given in Eq. (2.7). Equation (2.14) is then equivalent to
A1 A2 A3 A4 A5 A6 =P B6-1
where the right hand side of this last equation is seen to be a constant pose matrix for a manipulator described by the left hand side (i.e. one for which d6=a6=a6=0 so that A6 =A6)"
When joint 1 is not prismatic, dI is constant and the origin of the base frame F0 can be positioned so that dI is equal to 0.
The Reduced System of Equations
For a 6-DOF arm, Eq. (2.14) becomes
A1 A2 A3 A4 A5 A6 = P (4.1)
and it yields twelve non trivial scalar equations in the six unknown variables. It is desirable to reduce-this system to a minimal number of equations involving as few of the joint variables as possible. For all-revolute, 6-DOF
manipulators, Tsai and Morgan (1984) have establishe that with respect to frame F3, the z-component of the position vector 3p and that of vector 3t along with the inner products (3t.3p) and (3 p. 3p) provide 4 equations in only 4 of the unknowns, thereby reducing the complexity of the problem. The process of obtaining these four equations
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