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To expand on my comment, the last line of your DH table gives the transform between END_EFFECTOR-1 and END_EFFECTOR frames, right? I'm saying instead of <θ2, 0, 0, 0> to have <180+θ2, 180, 0, 0>. There's no movement between your end effector and the object you're attaching to the end effector, so there's no point in keeping the end effector frame....


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Do I understand correctly, that the problem is with the matrix equation you have posted and how to calculate $T_{sb}$ from that equation? You can solve the matrix equation by multiplying from the right with the inverse of the last matrix on the rhs, step by step. $$\require{cancel} $$ $$T_{se} = T_{sb} \times T_{b0} \times T_{0e}$$ $$T_{se} \times T_{0e}^{-1}...


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