In recent years, the robotics research community has focused on variable impedance actuation for its potential in safe physical interaction. Despite many advantages such as safety, efficiency, and dynamic adaptation, these systems usually have a low motion bandwidth due to the presence of impedance elements between the joints and the links. Presumably, reaction wheels, frequently employed in spacecraft attitude control for high bandwidth actuation, can be employed to improve the motion control performance of variable impedance robots. In order to test this hypothesis, in this work, we present the control of a dual-axis compliant inverted pendulum using reaction wheels. Two controller alternatives are considered. The first relies on the approximation of an offline optimal controller using deep neural networks (DNNs), and the second one is based on nonlinear model predictive control (NMPC). Both simulation and experimental results show successful control performance of both the DNN and NMPC controllers. However, the DNN control law can be executed in a much shorter time period than the NMPC one (0.4 ms versus 2.68 ms on average). This proves the feasibility of using approximate optimal controllers based on DNNs at high sampling rates for the control of variable impedance robots.