This paper presents a framework for trajectory planning that explicitly considers robotic traversability based on a quasi-dynamic vehicle model of a mobile robot in loose soil. The quasi-dynamic model estimates the slip effect due to wheel-terrain interaction forces regardless of solving complicated multibody dynamics. Therefore, our proposed model is computationally efficient for quantifying how the robot safely traverses each trajectory segment generated by a planning algorithm. The trajectory planning in our framework exploits a sampling-based incremental search algorithm, i.e., Closed-Loop Rapidly-Exploring Random Trees (CL-RRT). In the tree extension process of the CL-RRT, the traversability assessment based on the quasi-dynamic vehicle model excludes the trajectory segment associated with a hazardous wheel slip ratio. As a result, a trajectory generated from the proposed framework is safely traversable for the robot even in high slip terrain. Simulation results show that the proposed vehicle model can run 57K times faster than the dynamic model and predict the robot motion 3 times more accurately than the kinematic model. Multiple trials of the trajectory planning simulation show that our proposed framework incorporated with the quasi-dynamic model reduces a wheel slip ratio by about 40 % as compared with the kinematic model.