The skylight polarization pattern, which is a result of the scattering of unpolarized sunlight by particles in the atmosphere, can be used by many insects for navigation. Inspired by insects, several polarization navigation sensors have been designed and combined with various heading determination methods in recent years. However, up until now, few of these studies have fully considered the influences of different meteorological conditions, which play key roles in navigation accuracy, especially in cloudy weather. Therefore, this study makes a major contribution to the study on bio-inspired heading determination by designing a skylight compass method to suppress cloud disturbances. The proposed method transforms the heading determination problem into a binary classification problem by segmentation, connected component detection, and inversion. Considering the influences of noise and meteorological conditions, the binary classification problem is solved by the soft-margin support vector machine. In addition, to verify this method, a pixelated polarization compass platform is constructed that can take polarization images at four different orientations simultaneously in real time. Finally, field experimental results show that the designed method can more effectively suppress the interference of clouds compared with other methods.