In this paper the notion of robot-manipulators in the Euclidean space is generalized to the case in a general homogeneous space with the Lie group $G$ of motions. Some kinematic subspaces of the Lie algebra $𝒢$ (the subspaces of velocity operators, of Coriolis acceleration operators, asymptotic subspaces) are introduced and by them asymptotic and geodesic motions are described.

The external derivative $d$ on differential manifolds inspires graded operators on complexes of spaces ${\Lambda }^r{g}^{*}$, ${\Lambda }^r{g}^{*}\otimes g$, ${\Lambda }^r{g}^{*}\otimes g^{*}$ stated by $g^{*}$ dual to a Lie algebra $g$. Cohomological properties of these operators are studied in the case of the Lie algebra $g=se\left(3\right)$ of the Lie group of Euclidean motions.

On donne des conditions larges sur un champ de normes symplectiques sur ${\mathbf{R}}^{2n}$ pour que les opérateurs d’ordre zéro associés opèrent sur $L^2({\mathbf{R}}^n)$; les éléments de cet espace se laissent alors écrire comme somme d’états propres, de niveau d’énergie bornée, de la famille d’oscillateurs harmoniques associée.