Consider a robot arm with \(n\)-DOF.
Consider a 2D-planar robot arm with 3 rotational joints. By choosing the absolute angle measured from the \(X\) axis as the generalized coordinate of this system, the end-effector position of this 3-DOF robot can be expressed as follows: \[ \mathbf{x} = \begin{bmatrix} p_x \\ p_y \end{bmatrix} = \begin{bmatrix} l_1 \cos q_1 + l_2 \cos q_2 + l_3 \cos q_3 \\ l_1 \sin q_1 + l_2 \sin q_2 + l_3 \sin q_3 \end{bmatrix} \]