In response to the issue of violation in trajectory tracking of single leg dynamics for quadruped robots due to the incompatibility between initial conditions and constraint equations, as well as the violation caused by numerical integration on low-order constraints, a trajectory tracking control method based on the Udwadia-Kalaba method is proposed. This method utilizes the Lagrangian equation to establish unconstrained dynamic equations, considers the predetermined trajectory of the system as a constraint relationship, and establishes constraint equations based on the end position obtained from direct kinematics. Firstly, to address the problem of initial value violation, non-zero initial angular velocities of each joint are set. It is found that the trajectory tracking error based on the Udwadia-Kalaba method is relatively large. To mitigate this issue, the Baumgarte stability method is employed to reduce system errors, and a comparison with the end trajectory obtained under PID control is conducted, validating the superiority of the Baumgarte stability method in correcting initial value violations. Subsequently, considering the violation caused by numerical integration on low-order constraints, one of the constraint trajectories is set as a time-dependent linear equation. Trajectory tracking control is performed using the Udwadia-Kalaba method, which results in significant trajectory deviation. Then, the Baumgarte stability method is utilized to correct the end trajectory, achieving the goal of aligning the end trajectory with the trajectory planning.