Analyzing dH
Equation (5.27) shows that the value of dH depends on the seven robot parameters 0l, U2, 03, al, a2, a3, and d2 as well as the pose quantities wl, w2, w3, and w4. However,
for certain robot structures dH is equal to zero no matter what the end-effector pose is. The expression of dH above
provides us with a way to find all such 4-DOF robot
geometries. Due to our link frames assignment, the only robot parameter in the expression of dh that can be negative is d2. By expanding Eq. (5.27) we get dH = ao2 a3 2 wI + C 1 032 a1 a2 w2
+ 2 0I 2 23 a1 a3 d2 w3 + a12 C3 a22 a3 w4
+20223 a3d2 w4+022
1 a 2 2d22 4 22 3 a1 a3 w4 (5.28)
where only the quantities d2, w3, and w4 can be negative. If an arm structure is such that dH is zero for every possible end-effector pose, then dH will be zero even for a pose with positive w3 and w4.
If we assume w3, and w4 non negative, then with d2negative, dH can be zero if the equality
-2 01 02 03 a1 a3 d2 w3 =
01 02 al a2 a32 wI + 01 02 032 al a2 w2
+U12 3 a22 a3 w4 + 0l2 022 03 a3 d22 w4 + 022 03 aI2 a3 w4 holds. However, such an equality is actually a condition on pose quantities wl, w2, w3, and w4. We conclude that robot
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