In transportation, autonomous driving, and road line marking operations, one of the crucial technologies is the trajectory tracking of steering wheels. Addressing the issue of trajectory tracking in quad-steering wheel mobile robots, this paper presents a method for tracking the trajectory of a quad-steering wheel chassis, which is based on a kinematic model and employs a Model Predictive Control (MPC) algorithm to study the trajectory tracking controller for the mobile robot. In this approach, the state error equation is first employed as the prediction model, which is then linearized and discretized to obtain a linear time-varying model. Terminal constraints are incorporated into the cost function, and appropriate weighting coefficients and terminal sets are designed to ensure that the reference trajectory is tracked within a finite time horizon. Subsequently, trajectory tracking simulations under varying prediction horizons and different paths are conducted for comparative analysis, leading to the determination of an optimal prediction horizon. Finally, the trajectory tracking experimental effectiveness is validated by having the mobile robot draw lines, with results demonstrating that the algorithm can effectively accomplish the trajectory tracking task for quad-steering wheel robots.