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With the increasing number of urban cars, parking has become the primary problem that people face in daily life. Therefore, many scholars have studied the automatic parking system. In the existing research, most of the path planning methods use the combined path of arc and straight line. In this method, the path curvature is not continuous, which indirectly leads to the low accuracy of path tracking. The parking path designed using the fifth-order polynomial is continuous, but its curvature is too large to meet the steering constraints in some cases. In this paper, a continuous-curvature parking path is proposed. The parking path tracker based on Model Predictive Control (MPC) algorithm is designed under the constraints of the control accuracy and vehicle steering. Firstly, in order to make the curvature of the parking path continuous, this paper superimposes the fifth-order polynomial with the sigmoid function, and the curve obtained has the continuous and relatively small curvature.