Angular Path Integration by Projection Filtering with Increment Observations

Angular path integration is the ability of a system to estimate its own heading direction from potentially noisy angular velocity (or increment) observations. Despite its importance for robot and animal navigation, current algorithms for angular path integration lack probabilistic descriptions that take into account the reliability of such observations, which is essential for appropriately weighing one's current heading direction estimate against incoming information. In a probabilistic setting, angular path integration can be formulated as a continuous-time nonlinear filtering problem (circular filtering) with increment observations. The circular symmetry of heading direction makes this inference task inherently nonlinear, thereby precluding the use of popular inference algorithms such as Kalman filters and rendering the problem analytically inaccessible. Here, we derive an approximate solution to circular continuous-time filtering, which integrates increment observations while maintaining a fixed representation through both state propagation and observational updates. Specifically, we extend the established projection-filtering method to account for increment observations and apply this framework to the circular filtering problem. We further propose a generative model for continuous-time angular-valued direct observations of the hidden state, which we integrate seamlessly into the projection filter. Applying the resulting scheme to a model of probabilistic angular path integration, we derive an algorithm for circular filtering, which we term the circular Kalman filter. Importantly, this algorithm is analytically accessible, interpretable, and outperforms an alternative filter based on a Gaussian approximation.