A $p$-parametric robot manipulator is a mapping $g$ of ${ℝ}^p$ into the homogeneous space $P=(C_6\times C_6)/\mathrm{Diag}(C_6\times C_6)$ represented by the formula $g(u_1,u_2,\cdots ,u_p)=exp\left(u_1{X}^1\right)·\cdots ·exp\left(u_p{X}^p\right)$, where $C_6$ is the Lie group of all congruences of $E_3$ and $X^1,X^2,\cdots ,X^p$ are fixed vectors from the Lie algebra of $C_6$. In this paper the $3$-parametric robot manipulator will be expressed as a function of rotations around its axes and an invariant of the motion of this robot manipulator will be given. Most of the results presented here have been obtained during the author’s stay at Charles University in Prague....