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Suppose I have a 3 link(1 dimensional) chain in which all the joints are revolutes, the axis of first revolute joint is along Z-axis(global) and axis of second joint is along X-axis(global). The first link is along X-axis(global) and second link is along Z-axis(global).

Now in order to use DH representation I introduced a local frame for link 1 at joint 1(z axis along Z and x axis along X) and another frame at joint 2.Here z-axis is along axis of rotation(global X) and here I am clueless how to determine x-axis for joint 2 because the two z axis are intersecting.(standard procedure is to find common normal between two z axis)3-link chain Thanks for your time.

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Check out this tutorial video on DH Parameters - it's amazing!.

Regarding your question, I believe if you try crossing your z-axes, $z_1 \times z_2$, you'll get the x-axis you're looking for (orthogonal to both axes). Because the two z axes intersect, your $r$ or $a$ value, depending on your notation, will be zero because the new x-axis lies on your previous z-axis.

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The right hand rule will help you here. For a given joint i: your thumb will correspond to the Zi-1 axis, your index will correspond to the Zi axis and your middle finger will correspond to Xi axis.

Here is the wikipedia page on the right hand rule, I linked it so people can look at the image if ever they forgot what the rule was. https://en.wikipedia.org/wiki/Right-hand_rule

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