Dielectric elastomer transducer (DET) is a smart device that converts electrical energy into mechanical energy, and usually works as a promising candidate for artificial muscles. Currently, the high-precision motion control of the DET is still challenging due to its complex input-output characteristics. This letter proposes a feedforward-cascaded adaptive generalized predictive control approach for the DET to accomplish its high-precision trajectory tracking control. Firstly, a control-oriented dynamic model is built to characterize the complex input-output characteristics of the DET, in which a square function, a Bouc-Wen model (BWM) and a linear dynamic model are connected in series. Secondly, by connecting the inverse of the BWM and a square root function, a feedforward compensator is built to mitigate the asymmetric hysteresis characteristic and the square input characteristic of the DET. Thirdly, an adaptive generalized predictive controller is devised to deal with external disturbances and model uncertainties, so as to accomplish the control goal. Finally, a series of experiments are carried out on an actual platform. Since the root-mean-square errors of all tracking results are lower than 1.26%, the proposed approach is remarkable from the practical application point of view.