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This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality functional, while the uncertainty has a limited impact on the change in the value of the functional. The article introduces the concept of feasibility of the mathematical model of the object, which is associated with the contraction property of mappings if we consider the model of the object as a one-parameter mapping. It is shown that this property is sufficient for the development of stable practical systems. To find a solution to the stated problem, which would ensure the feasibility of the system, the synthesized optimal control method is proposed. This article formulates the theoretical foundations of the synthesized optimal control. The method consists in making the control object stable relative to some point in the state space and to control the object by changing the position of the equilibrium points. The article provides evidence that this approach is insensitive to the uncertainties of the mathematical model of the object. An example of the application of the method for optimal control of a group of robots is given. A comparison of the synthesized optimal control method with the direct method on the model without disturbances and with them is presented.

When machining free-form surfaces with a ball-end milling tool, the working tool diameter is constantly changing even if the tool path is constant. The reason is that the surface normal of the milled surface is continuously changing along the milling path. When the working diameter is changing, the cutting parameters also change. This variation effects the roughness homogeneity of the smoothed surface. Simultaneous five-axis milling solves this problem; however, the price and complexity of this technology can be a problem for some industrial sectors. In the paper, the geometrical background of a solution to this problem is presented for a 3-axis ball-end milling process for machining a free form surface. The paper provides the deduction of the theory by the use of homogeneous transformations. The geometrical problem of the cutting process is treated locally where the general machined surface is substituted at every point by its tangent plane. From the result of the presented method, a milling strategy can be formulated for ball-end milling that minimises the change in the momentary working diameter thus providing a more constant cutting parameter.

The literature solely uses the homogeneous transformation matrix method for solving the problems of rotations and translations, for gear contact problems. In this paper a different approach is introduced, using complex algebra for designing the profiles of gear teeth. We intend to present the general theory and technique of designing the generating gear rack and the meshing counter profile of arbitrary profile gears. For the illustration of the practical use of his method we shall present the design steps of a gear, with a cosine profile, a general involute gear and the profile of the lobes of a Root blower pump. These examples will properly illustrate the applicability of the presented design process.

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