A large amount of hemiplegic survivors are suffering from motor impairment. Ankle rehabilitation exercises act an important role in recovering patients' walking ability after stroke. Currently, patients mainly perform ankle exercise to reobtain range of motion (ROM) and strength of the ankle joint under a therapist's assistance by manual operation. However, therapists suffer from high work intensity, and most of the existed rehabilitation devices focus on ankle functional training and ignore the importance of neurological rehabilitation in the early hemiplegic stage. In this paper, a new robotic ankle rehabilitation platform (RARP) is proposed to assist patients in executing ankle exercise. The robotic platform consists of two three-DOF symmetric layer-stacking mechanisms, which can execute ankle internal/external rotation, dorsiflexion/plantarflexion, and inversion/eversion exercise while the rotation center of the distal zone of the robotic platform always coincides with patients' ankle pivot center. Three exercise modes including constant-speed exercise, constant torque-impedance exercise, and awareness exercise are developed to execute ankle training corresponding to different rehabilitation stages. Experiments corresponding to these three ankle exercise modes are performed, the result demonstrated that the RARP is capable of executing ankle rehabilitation, and the novel awareness exercise mode motivates patients to proactively participate in ankle training.