Collaborative robots (or cobots) are robots that are capable of safely operating in a shared environment or interacting with humans. In recent years, cobots have become increasingly common. Compliant actuators are critical in the design of cobots. In real applications, this type of actuation system may be able to reduce the amount of damage caused by an unanticipated collision. As a result, elastic joints are expected to outperform stiff joints in complex situations. In this work, the control of a 2-DOF robot arm with elastic actuators is addressed by proposing a two-loop adaptive controller. For the outer control loop, an adaptive sliding mode controller (ASMC) is adopted to deal with uncertainties and disturbance on the load side of the robot arm. For the inner loops, model reference adaptive controllers (MRAC) are utilised to handle the uncertainties on the motor side of the robot arm. To show the effectiveness of the proposed controller, extensive simulation experiments and a comparison with the conventional sliding mode controller (SMC) are carried out. As a result, the ASMC has a 50.35% lower average RMS error than the SMC controller, and a shorter settling time (5% criterion) (0.44 s compared to 2.11 s).