In robotics, pneumatic drives are an interesting alternative to classic electric drives, as their physical properties enable the design of safe, lightweight and intuitively operable robots. In this work, a nonlinear model-based feedback control concept for a robot with 4 compressed-air-driven rotary joints is presented. The robot model comprising kinetics and pressure dynamics is formulated. For this drive type, significant nonlinear friction effects are present, which is known to be challenging for controller design, especially since there is no output-side measurement of the effective drive torque after subtracting friction. A control concept based on differential flatness, here with the particular case of feedback linearization, is derived, which is particularly well suited for a system with such effects. The algorithm contains both kinetics and pressure dynamics in one central controller, and its working principle and practical implications are discussed. With experimental results, the effectiveness of the trajectory tracking controller is demonstrated and insights into the components of this controller are provided.