Biomimetic quadruped robots, inspired by the musculoskeletal systems of animals, employ pneumatic artificial muscles (PAMs) as compliant actuators to achieve flexible, efficient, and adaptive locomotion. This study focuses on a pneumatic artificial muscle (PAM)-driven biomimetic leg joints system. First, its kinematic and dynamic models are established. Next, to address the challenges posed by the strong nonlinearities and complex time-varying uncertainties inherent in PAMs, an adaptive robust control algorithm is proposed by employing the Udwadia controller. Rigorous theoretical analysis of the adaptive robust control algorithm is verified via the Lyapunov stability method. Finally, numerical simulations and hardware experiments are conducted on the PAM-driven biomimetic leg joints system under desired trajectories, where the adaptive robust control algorithm is systematically compared with three conventional control algorithm to evaluate its control performance. The experimental results show that the proposed controller achieves a maximum tracking error of within 0.05 rad for the hip joint and within 0.1 rad, highlighting its strong potential for practical deployment in real-world environments.