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R = S1S1 + a12a12 + S2S2 + a23a23 (A.6)
where S1 and S2 are the translation along the ,1 and S2
axes, respectively, and al2 and a23 are the perpendicular
distances between successive joint axes.
We can rewrite Eq. (A.6) by using the Table A.1 (set 1
of direction cosines spatial heptagon) as follows:
1 r r r
c[ -s1 0 0 1 0
R = s1 C 0 { S1 0 + a212 + S2 -s12
0 0 1 1 0 c12
c2
+ a23 U2 } (A.7)
U21
where U21 = s2c12 and U21 = s2s12*
Therefore, the coordinates of joint C with respect to the
local coordinate system can be expressed as follows:
C, = al2c1 + S2sl2S1 + a23c1 (A.8)
Cy = al2s1 S2s12c1 + a23s1 (A.9)
C, = S1 + S2c12 (A.10)
The above expression are the same as those in Eq. (2.13)
except for different notations for the elements of the
subchain.