This paper addresses the control design problem of stabilizing both joint-angle and spring-torque control objectives in robots with series elastic actuators (SEAs). The proposed method is a hierarchical control scheme which employs rapidly exponentially stabilizing control Lyapunov functions (RES-CLFs) to obtain controllers for each tier in the scheme. The main result of the paper demonstrates that for a proper choice of controller parameters, applying the proposed controller to the SEA control system results in simultaneous exponential stability of the joint-angle and the spring-torque control objectives. Furthermore, it is shown that for a locally exponentially stable periodic orbit in the zero dynamics of the control system considered, the controller renders a corresponding orbit in the full SEA dynamics locally exponentially stable.