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The stability control analyzed by us, in this show, is based on our results in the domain of dynamical systems that depend of parameters. Any dynamical system can be considered as dynamical system that depends of parameters, without numerical particularization of them. All concrete dynamical systems, meted in the specialized literature, underline the property of separation between the stable and unstable zones, in sense of Liapunov, for two free parameters. This property can be also seen for one or more free parameters. Some mathematical conditions of separation between stable and unstable zones for linear dynamical systems are identified by us. For nonlinear systems, the conditions of separation may be identified using the linear system of first approximation attached to nonlinear system. A necessary condition of separation between stable and unstable zones, identified by us, is the sufficient order of differentiability or conditions of continuity for the functions that define the dynamical system. The property of stability zones separation can be used in defining the strategy of stability assurance and optimizing of the parameters, in the manner developed in the paper. The cases of dynamical systems that assure the separations of the stable and unstable zones, in your evolution, and permit the stability control, are analyzed in the paper.

The behavior of linear or nonlinear dynamic systems depends on different parameters (identifiable or free) that are involved in their definition. The stability analysis of such dynamical systems is realized by using a domain of selected free parameters. In this paper, we discuss specific theorems that concern the stability of linear dynamical systems, the stability of nonlinear dynamical systems in terms of "first linear approximations", and other stability criteria. We study the stable/unstable separation property in the free parameters domain and present a rigorous mathematical justification of this property with specific examples from various branches of science. Furthermore, we investigate specific conditions when the separation property is passed on to the nonlinear dynamical system from its first order linear approximation. The stable/unstable separation property is also emphasized as an important property of the environment that can contribute to its mathematical modeling.

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The exposure is dedicated in the first to mathematical modeling of the environment where the aspects on the walking robots evolution models are described. The environment’s mathematical model is defined through the models of kinematics or dynamic systems in the general case of systems that depend on parameters. The important property of the dynamic system evolution models that approach the phenomenon from the environment is property of separation between stable and unstable regions from the free parameters domain of the system. Some mathematical conditions that imply the separation of stable regions from the free parameters domain of the system are formulated.  In the second part is described our idea on walking robot kinematics and dynamic models with aspects exemplified on walking robot leg. An inverse method for identification of possible critical positions of the walking robot leg is established.