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Research Article Volume 12 Issue 3
An analytical form for a rigorous nonlinear model of the dynamics for systems with geometric constraints and its use by applying the results of control theory to a complete solution of stabilization problems
Aleksandr Ya Krasinskiy, Andrey A Yuldashev
Received: July 20, 2026 | Published: September 29, 2026
Citation: Krasinskiy AY,Yuldashev AA. An analytical form for a rigorous nonlinear model of the dynamics for systems with geometric constraints and its use by applying the results of control theory to a complete solution of stabilization problems. Int Rob Auto J. 2026;12(3):98-120. DOI: 10.15406/iratj.2026.12.00323
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
©2026 Krasinskiy, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.