Optimization based method

The basic idea of optimization based method is to take a look at the primary equation $\theta = f^{-1} (X)$ as minimization problem [Wel93]. Thus, the previous equation could be transformed into

$$E(\theta) = \left( P - X(\theta) \right)^2 \eqno (9)$$

After that, standard iterative non-linear optimization techniques could be applied to minimize error function $E(\theta)$, where $P$ is the goal position of end effector and $X(\theta)$ is the current position of the end effector [PFTV90].

Gradient-based optimization could be also used. This may increase the computation cost in each iteration step, but the convergence rate should be better and the number of iteration steps should decrease.