In social environments, complex interactive scenes and various tasks bring great challenges to the motion planning of autonomous ground vehicles. Application scenarios typically require vehicles to plan a smooth trajectory in real time that takes the shortest amount of time and conforms to all constraints. The construction of an optimal control problem in the state space is a common approach in this field, however, this inevitably entails a compromise between the optimality of the trajectories and the computational efficiency. The proposed method formulates a trajectory optimization problem based on differential flatness theory, which realizes efficient obstacle avoidance while satisfying the nonholonomic constraints of the ground vehicles. The representation of trajectories is simplified and a trajectory planning problem is constructed in the differential flatness space of vehicles. Furthermore, safe driving corridors are utilized to achieve smooth obstacle avoidance. The output trajectories are tracked by a model predictive controller for deployment on autonomous ground vehicles. Experiments in both simulation and real-world are conducted to demonstrate the feasibility of the algorithms in complex scenarios.