At present, questions related to the synthesis of asymptotic observers for nonlinear dynamic objects are of considerable interest. This article presents two approaches to converting the right-hand side of a nonlinear system of differential equations describing the dynamics of a robot manipulator to a form convenient for the practical construction of a nonlinear asymptotic observer. At the same time, non-measurable terms that satisfy the Lipschitz condition are singled out as functions. The first approach provides global fulfillment of the Lipschitz condition, however, the corresponding mathematical model is significantly simplified in relation to the original one. In the second case, only local fulfillment of the Lipschitz condition in a limited domain of variable variables is ensured, but for the initial non-simplified model. The article also compares the dynamics of objects represented by these mathematical models.