The Zeta Filter: Attitude Estimation Using Von Mises–Fisher Concentration Dynamics on S3

This paper presents an attitude filter that encodes both orientation and uncertainty in a single four-dimensional vector, requiring no covariance propagation or normalization constraints. The filter state is the natural parameter of the von Mises–Fisher (vMF) distribution on S3, whose exponential family structure reduces measurement updates to vector addition. Prediction is governed by a continuous-time ODE (Ordinary Differential Equation) that couples rotational kinematics with concentration decay. The QUEST-based construction of measurement natural parameters with a Fisher-information-matched concentration, an antipodal switching mechanism for the quaternion double cover, and a global exponential convergence analysis of the attitude error are described. The filter construction is left-invariant: it commutes with rotations of the reference frame, making the error dynamics trajectory-independent. The result is a filter with the computational simplicity of a complementary filter and the statistical grounding of Bayesian vMF fusion, operating entirely in unconstrained ℝ4 space. The filter is validated in simulation, on two recorded flights—including an evaluation against an EFIS attitude reference—and its computational cost is measured down to on-target microcontroller cycle counts. Gyroscope bias estimation is not included and is left to future work.