Singular perturbation model of robots with elastic joints and elastic links constrained by rigid environment
The contribution of this paper lies in the modelling methodology of robotic manipulators with flexible joints and links, subject to a contact with a stiff environment. The model of the dynamics of such a manipulator is provided in terms of a finite number of rigid and flexible coordinates describing the link dynamics, while the joint flexibility is accounted for classically according to Spong's simplified model. Eventually, the following model is derived: \[ M(q_r,q_f)\left[\begin{matrix} \ddot q_r\\ \ddot q_f\end{matrix}\right]+ \left[\begin{matrix} h_r(q_r, \dot q_r, q_f, \dot q_f)\\ h_f(q_r, \dot q_r, q_f,\dot q_f)\end{matrix}\right]+ \left[\begin{matrix} g_r(q_r, q_f)\\ g_f(q_r, q_f)\end{matrix}\right]+ \left[\begin{matrix} K_J(q_m- q_r)\\ K_f q_f\end{matrix}\right]= \left[\begin{matrix} A^T_r(q_r, q_f)\\ A^T_f(q_r, q_f)\end{matrix}\right] \lambda, \] \[ J_m\ddot q_m- K_J(q_m- q_r)= u, \] where \(q_r\), \(q_f\), \(q_m\) denote, respectively, the rigid, the flexible, and the actuator coordinates, and the matrices \(A^T_r\), \(A^T_f\) contain suitable partial derivatives of the constraint equation \(\Phi(q_r, q_f)= 0\). Under the assumption that the constraint equation can be solved with respect to a part of the rigid vector, the model is further reduced resulting in the elimination of the Lagrange multipliers \(\lambda\). Finally, a singularly perturbed version of the dynamics model is introduced. A model corresponding to the general singularly perturbed model with specific cases of rigid joints and flexible links as well as of elastic joints and rigid links is obtained.
scientific article; zbMATH DE number 1204422