Abstract

For the first time, a single article presents not only a complete justification for a method of deriving an analytical form for a nonlinear dynamic model and a step-by-step algorithm of deriving it for systems with geometric constraints, but also the results of its effective application to a comprehensive consideration for stability and stabilization problems for a specific technical device with parallel kinematics (Ball and Beam). The rigor and generality of abstract theoretical methods for deriving an analytical model in the form of vectormatrix equations for the first time for non-free systems with geometric constraints created the possibility for a justified application of the general nonlinear stability theory results and mathematical control theory. In particular, using the theory of critical cases, sufficient conditions for the solvability of the genera stability problem for a given system configuration are obtained. For the general stabilization problem of configuration under incomplete information, a theorem on sufficient conditions for stabilization to asymptotic stability with respect to all variables is proved. The coefficients of the linear stabilizing control and the estimation system are determined by solving the corresponding linear-quadratic problems using Krasovskii’s method. It is demonstrated how transient processes in actuators can be included in the analysis by significantly reducing the model’s dimensionality. The entire process of using the proposed approach is applied to a complete solution to the stabilizing problem of two different equilibrium positions for the two different selection of dependent coordinate options for the Ball and Beam rig as a mechatronic system with a nonlinear geometric constraints. An additional voltage on the armature winding of the actuator commutator motor is used as a control. Both analytical and numerical modeling of the controlled dynamics of a specific rig variant are performed. Graphs of the transient processes are provided.

Keywords

mathematical dynamics model, geometric constraints, stability, stabilization, linear-quadratic problem