Compliant control of a two-link flexible manipulator by constraint Hamiltonian system
The paper attempts to describe the control of a two-link flexible manipulator. But unfortunately a lot of confusion takes place. For example Hamiltonian equations are valid for holonomic systems, so the constraints do not depend on impulses. The first group of equations \(\dot q={\partial H\over\partial p_S}\) give the definition of the impulses and another group give the result of the transformation of a general dynamical equation. And only in the next group of equations the generalized forces: \(\dot q= -{\partial H\over\partial q_S}- Q_S\) are used. But here the generalized forces are entered in both groups of equations. The D'Alembert principle has another signification, namely: The dynamical equations may be rewritten in the form of static equations if one adds the inertia forces to the applied forces. The investigation of stability by the Lyapunov method is incorrect because, for example, the derivative of one term of the Lyapunov function \({\partial\over\partial t}(e^2_1)=2e_1\cdot\dot e_1\), where \(e_1= \theta_1- \theta^*_1\) is of variable sign. So it is impossible to guarantee the negative definiteness of the derivative of the Lyapunov function and the statement about stability makes no sense. Finally, it is relevant to remember that the description on how to account for the flexibility of deformable bodies is contained in the book of \textit{A. I. Lur'e} [Analytic Mechanics (Russisan), Moscow: FizMatLit (1961; Zbl 0103.39803)].
scientific article; zbMATH DE number 1331544
scientific article