# 3 Forward kinematics (FK) for a planar robot manipulator

The forward kinematics (FK) function of a planar robot manipulator is defined as

 \displaystyle\bm{f} \displaystyle=\begin{bmatrix}\textstyle\bm{\ell}^{\scriptscriptstyle\top}\cos(\bm{L}\bm{x})\\ \textstyle\bm{\ell}^{\scriptscriptstyle\top}\sin(\bm{L}\bm{x})\\ \textstyle\bm{1}^{\scriptscriptstyle\top}\bm{x}\end{bmatrix} \displaystyle=\begin{bmatrix}\textstyle\ell_{1}\!\cos(x_{1})\!+\!\ell_{2}\!\cos(x_{1}\!+\!x_{2})\!+\!\ell_{3}\!\cos(x_{1}\!+\!x_{2}\!+\!x_{3})\!+\!\ldots\\ \textstyle\ell_{1}\sin(x_{1})\!+\!\ell_{2}\!\sin(x_{1}\!+\!x_{2})\!+\!\ell_{3}\!\sin(x_{1}\!+\!x_{2}\!+\!x_{3})\!+\!\ldots\\ \textstyle x_{1}+x_{2}+x_{3}+\ldots\end{bmatrix}\!\!,

with \bm{x} the state of the robot (joint angles), \bm{f} the position of the robot end-effector, \bm{\ell}\, a vector of robot links lengths, \bm{L} a lower triangular matrix with unit elements, and \bm{1} a vector of unit elements, see Fig. 6 .

The position and orientation of all articulations can similarly be computed with the forward kinematics function