Effects of joint flexibility on the motion of a flexible-arm robot

Once flexibility is introduced into the arm of the robot severe problems in the accuracy and stability are likely to occur which make control a critical issue. These problems could amplify drastically once the joint flexibility is also added to the system. A more successful control algorithm can be developed if the nonlinear dynamics associated with the flexible joint-arm is properly accounted for. In this paper we study the behaviour of a three degree of freedom flexible joint-arm robot with quadratic and cubic nonlinearities. The symbolic manipulator language MAPLE is used to develop a simplified mathematical model of the system and to perform the analytical study. The two-variable expansion perturbation method is used to show the existence of various nonlinear resonances. If the system natural frequencies are defined as ω₁, ω₂, and ω₃, internal resonance occurs when ω₂ = 2ω₁ω₃/(ω²₁ + ω²₃)^½. Moreover, once subjected to harmonic forcing with frequency Ω, a nonlinear forced resonance occurs at Ω ≈ ω₂; the system undergoes a jump phenomenon. Numerical simulation concurs with the analytical results for small motions. However, at higher amplitudes of forcing, numerical studies indicate the existence of a chaotic solution in the resonance region. The route to chaos contains subharmonic bifurcations.