Bifurcation analysis of A 2-DOF robot manipulator driven by constant torques

The bifurcations of a two-degree-of-freedom (2-DOF) robot manipulator with linear viscous damping and constant torques at joints are analyzed at three equilibrium points. We show that, provided some conditions on the parameters are satisfied, two of these equilibria have a Jacobian matrix with a double zero eigenvalue and a pair of pure imaginary eigenvalues, while the other one has a quadruple zero eigenvalue. We use the center manifold theorem and the normal form theory to show the presence of different kinds of local bifurcations, ranging from codimension one (Hopf and fold), three (cusp and degenerate zero-Hopf), and higher (double zero, double zero-Hopf, triple zero, quadruple zero and Hopf-Hopf).