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I'm studying for an exam and I came across these questions where you have to determine the number of possible inverse kinematic equations like these (qs from Craig Introduction To Robotics, 3rd Edition)

How exactly do you find the number of solutions? I know how to find the DH paramaters and then the inverse kinematics but is there some way to calculate the number of solutions quickly like this question suggests? Like, I can visualize it (4 for the second one) but is there a proper way?

Thanks.

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The number of solutions come from the multiple solutions to the inverse kinematics equations. You can easily develop an intuition for different groups of mechanisms and identify correctly the number of solutions, but this is based on experience and intuition, not on a mathematical formalism. This however is only the theoretical side of the problem. In practice, the joint limits play an important role in limiting the number of theoretical solutions practically achievable.

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