A novel Gaussian state estimator named Chebyshev Polynomial Kalman Filter is proposed that exploits the exact and closed-form calculation of posterior moments for polynomial nonlinearities. An arbitrary nonlinear system is at first approximated via a Chebyshev polynomial series. By exploiting special properties of the Chebyshev polynomials, exact expressions for mean and variance are then provided in computationally efficient vector-matrix notation for prediction
and measurement update. Approximation and state estimation are performed in a black-box fashion without the need of manual operation ormanual inspection. The superior performance of the Chebyshev Polynomial Kalman Filter compared to state-of-the-art Gaussian estimators is demonstrated by means of numerical simulations and a real-world application.