**4. Inverse kinematics**

The solution exists only if the given end-effector position and orientation are in dexterous workspace of the solved mechanism. While the forward kinematic model is expressed as

$$\mathbf{x} = \mathbf{f}(\mathbf{q})\tag{19}$$

where **f** is a function defined between joint space *<sup>n</sup>* and workspace *<sup>m</sup>*, which maps the joint position variables **q**∈ *<sup>n</sup>* to the position/orientation of the endeffector of mechanism, the inverse kinematic model is based on

$$\mathbf{q} = \mathbf{f}^{-1}(\mathbf{x}) \tag{20}$$

where **q**∈ *<sup>n</sup>* and **x**∈ *<sup>m</sup>*. In the case of the forward kinematic model, endeffector position and orientation are computed for various kinds of mechanisms like manipulators, in a unique manner, for example, by above-mentioned transformation matrices. The inverse kinematic problem is more complex and finding the solution could be in many cases very complicated. While forward kinematics has a closed-form solution, an inverse kinematics in most cases does not have a closedform solution. A forward kinematic model has a unique solution, while an inverse

kinematic model may have multiple solutions or infinite number of solutions, especially for kinematically redundant mechanisms. In order to obtain a closedform solution, there are two main approaches, namely algebraic approach and geometric approach.
