This paper presents the design and analysis of Pegasus, a quadrupedal wheeled robot grounded in biomimicry principles. Pegasus offers two distinct motion modes, including a wheeled motion and a hybrid wheeled-legged motion, enabling adaptability across various tasks and environmental conditions. The robot draws inspiration from the joint structures of quadruped animals and incorporates biomimetic features. At the robot’s ankle joint, we imitate the articulation of a radiusulna joint to enhance the wheeled motion’s agility. Additionally, we establish comprehensive mathematical models for adaptive dynamics model, providing a robust theoretical foundation for subsequent motion planning and high-precision control. A novel telescopic vehicle mode is also proposed for complex wheel-leg hybrid motion, offering optimized solutions for intricate robot locomotion. Furthermore, we employ parallel underactuated MPC controllers for each leg at the control level, contributing to heightened motion precision and stability. Extensive validation through physical platform experiments highlights the effectiveness and feasibility of the proposed controllers, offering substantial support for real-world applications in robotics.