This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems, where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case by applying the Pontryagin maximum principle. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws, a minimum required value of the coefficient of sliding friction is defined, so that the considered system is moving in accordance with nonholonomic bilateral constraints

ln designing robots statical balancing is of great importance. It enables essential reduction of the required motor power. Elimination of significant reduction of the gravity load at a powered joint can also lead to a simpler and more effective control system. It is shown that this problem can be formulated as an optimization problem. As objective function the average force on the gripper in the working area is used. It is shown that those forces can easily be derived with the aid of a Computer Algebra System. The lengths of the links and angles between them as well as the stiffness of springs can be considered as design variables. It is shown that a general design program based on Monte Carlo methods and genetic algorithms, that has a graphical interface that allows the user to modify search intervals, is very appropriate to these kinds of problems. As an example an industrial robot with 6-DOF is treated. The robot has a spring balancing system that has to be optimized. The robot with the parameters of its balancing system found this way is evaluated with an existing multibody dynamics program SPACAR that simulates the movements of the robot and calculates the forces at its powered joints. With the aid of MATLAB the forces in the working area are plotted in a 3D figure. The method described in this article is general and very appropriate to solve problems in practice without simplifications.

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