I am looking at closed form algebraic solutions to inverse kinematics problems (in this case for robot manipulators).
The correct solution found the transform matrices via forward kinematics (using Denavit Hartenberg method). These were multiplied together to give the compound transform, then each side of the transform equation was multiplied by an inverse transform $[T^0_1]^{-1}T^0_3 = T_3^1 $. The desired joint transform matrix $T_d$, was subbed in place of $T^0_3$ and coefficients were equated.
Correct solution (as labeled) on left, my incorrect attempt on the right
I see why this makes sense algebraically, and practically in terms of separating out variables. However, I was hoping someone could point out what it is that makes my solution incorrect.
I attempted to find a solution for a joint variable by finding the compound transform matrix using forward kinematics (Denavit Hartenberg method), then equating coefficients of this to the desired transform matrix $ T^0_3 = T_d $.
Would anyone be able to explain why my method yields an incorrect answer? I am unsure when to use the method of multiplying by an inverse transform.
Many thanks in advance!