Boundary cooperative control by flexible Timoshenko arms

Abstract

This paper discusses a cooperative control problem by two one-link flexible Timoshenko arms. The goal is to control a grasping force to collect an object with the two flexible arms, and to simultaneously suppress the vibrations of the arms. To solve this problem, we propose a boundary controller that is based on a dynamic model represented by a hybrid PDE-ODE model; the exponential stability of the closed-loop system is then proven by the frequency domain method. Finally, several numerical simulations are carried out to investigate the validity of the proposed boundary cooperative controller.

ISS Property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems

“…In contrast, the boundary perturbations appear in the boundary conditions of the PDEs. This second type of perturbation naturally appears in numerous boundary control problems such as heat equations [8], transport equations [19], diffusion or diffusive equations [2], and vibration of structures [8] with numerous practical applications, e.g., in robotics [12,13] and aerospace engineering [22]. While many results have been reported regarding the ISS property with respect to distributed disturbances [1,9,10,24,25,26,27,31], the establishment of ISS properties with respect to boundary disturbances remains challenging [2,18,19,20].…”

Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances

“…As P −k−1 = P k , the study for k ∈ Z reduces to k ∈ N. The discriminant of P k is given by disc(P k ) =k 2 π 2 (k 2 β 2 π 2 − 4α). Based on Assumption 3.1, disc(P k ) = 0, thus disc(P k ) > 0 for k ≥ k0 and disc(P k ) < 0 for 0 ≤ k ≤ k0 − 1, providing the eigenvalues λ k,ǫ given by (9). Finally the associated eigenvectors are characterized by x1(ξ) = a(e ikπξ − e −ikπξ ) = 2ai sin(kπξ), and x2 = λ k,ǫ x1, providing (10).…”

“…We need to check that γ 2 k,ǫ is a summable sequence. To do so, considering k ≥ k0, Re λ k,ǫ = λ k,ǫ , which gives along with (9) and (11) γ k,ǫ = √ 2 kπ k2 β 2 π 2 − 4α 1 +k 2 απ 2 |λ k,ǫ | 2 .…”

“…The second type concerns boundary perturbations, i.e., perturbations acting on the boundary of the domain. This second type of perturbation naturally appears in numerous boundary control problems such as heat equations [5], transport equations [22], diffusion or diffusive equations [2], and vibration of structures [5] with practical applications, e.g., in robotics [9], [14], aerospace engineering [3], [25], [26], and additive manufacturing [8], [13].…”

Exponential stability of dual flexible arms for grasping and orientation control

“…However, these studies have been treated only the vibration suppression problem. On the other hand, several studies have investigated the grasping by flexible arms based on the hybrid PDE-ODE model [20][21][22][23]. These studies discussed the cooperative control by two one-link flexible arms and realized stable grasping of an object.…”