The inner-product of each term in the square brackets with itself is the square of the length of that vector. For
example,
R31R2-112 R31R2112 12.12 = a22 + d22 = 122.
These inner-product manipulations represent a considerable algebraic simplification that requires little or no mental effort. Further, they provide a methodology and
considerable insight into how to find other algebraic reductions.
Some of the cross terms also reduce; for instance,
R3-113 R3-1R2 112 = 13 R2-112.
Complete expansion of Eq. (9.4) and application of the reduction techniques just discussed lead to
(p2 1423 _12 2)/2 = 14.[R3- 13 + R3-1R2-112] + 13.R2-112" For this manipulator, vectors
14= [0,0,d4] T = d4z,
R3-113 = [a3,d3,0]T, and
R27I12 = (a2' 0, 0]T a2 x
allow us to simplify the last equation to
(p2 142_13 2_12 2)/2 = d4z [R3-113 + a2 R3- x] + a213 x, (e.g., 14.R3-113 is obviously 0, which eliminates 84 from this equation).
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