On-Manifold Unit Quaternion Trajectory Representation via Chebyshev Polynomials and Its Application to Attitude Estimation

Abstract

Continuous-time attitude estimation methods are becoming increasingly popular because they can be evaluated at any query time and offer many other advantages by expressing the attitude as a continuous function of time. Consequently, the representation of the continuous-time attitude trajectory becomes a critical issue. This article introduces an attitude representation method using Chebyshev polynomials defined directly on the unit quaternion manifold. The attitude trajectory is parameterized as a cumulative product of exponential-map rotational components using Lie algebra coefficients of increasing frequency. This on-manifold formulation inherently guarantees unit-norm consistency and avoids the singularities associated with minimal three-parameter representations. A recursive strategy is adapted to derive the trajectory’s time derivatives (angular velocity) and its Jacobians with respect to the coefficients, including extensions for navigation frame kinematics considering Earth’s rotation. The representation is applied within a polynomial optimization framework for batch or window-based attitude estimation. Comprehensive simulations and experiments evaluate the method in the inertial-magnetic attitude estimation scenario. Results demonstrate that the proposed method exhibits superior convergence and robustness compared to the standard multiplicative Extended Kalman Filter and achieves superior or comparable accuracy while offering substantial improvements in computational efficiency over other polynomial optimization methods.