Hopf bifurcation calculation in neutral delay differential equations: Nonlinear robotic arms subject to delayed acceleration feedback control

This study gives the derivation of the Hopf bifurcation calculation for neutral delay differential equations using the centre manifold reduction theorem and normal form calculation. The whole concept was inspired by a case study where a robotic arm subjected to nonlinear stiffness with delayed acceleration feedback controller is modelled. Two different configurations are distinguished depending on the location of the acceleration sensor, a collocated and a non-collocated one. After a brief investigation of the linear stability, the bifurcation occurring at the loss of stability is calculated with the presented analytic equations for neutral delay differential equations. Then, a nonlinear term is introduced in the control law that can modify the occurring subcritical behaviour and improve the robustness of the system. The analytic results are carefully analysed and validated via a numerical continuation software, which also provided useful information about the global behaviour of the bifurcations in addition to the locally valid analytic results.