This paper proposes a new robust trajectory control scheme for a two-link rigid/flexible robot. First, the energy dynamics of the flexible link is derived based on a distributed-parameter dynamic model (a partial differential equation). Second, a coupled sliding surface is defined based on the energy dynamics. Third, a new trajectory control scheme is designed based on the coupled sliding surface, and then extended to an adaptive scheme to cope with parametric uncertainties, where the Lyapunov stability theorem is used as a mathematical tool. The proposed control is a collocated control designed based on the distributed-parameter dynamic model and is free from the so-called spillover instability. Using only two joint actuators, the proposed control guarantees stability throughout the entire trajectory control and asymptotic stability at desired goal positions. Furthermore, the proposed control guarantees zero steady-state joint-tracking errors even in the presence of low-frequency disturbances due to unavoidable mechanical inaccuracies in application. The theoretical results have also been proven by control experiments.