  Inertial-based navigation refers to the navigation methods or systems that
have inertial information or sensors as the core part and integrate a spectrum
of other kinds of sensors for enhanced performance. Through a series of papers,
the authors attempt to explore information blending of inertial-based
navigation by a polynomial optimization method. The basic idea is to model
rigid motions as finite-order polynomials and then attacks the involved
navigation problems by optimally solving their coefficients, taking into
considerations the constraints posed by inertial sensors and others. In the
current paper, a continuous-time attitude estimation approach is proposed,
which transforms the attitude estimation into a constant parameter
determination problem by the polynomial optimization. Specifically, the
continuous attitude is first approximated by a Chebyshev polynomial, of which
the unknown Chebyshev coefficients are determined by minimizing the weighted
residuals of initial conditions, dynamics and measurements. We apply the
derived estimator to the attitude estimation with the magnetic and inertial
sensors. Simulation and field tests show that the estimator has much better
stability and faster convergence than the traditional extended Kalman filter
does, especially in the challenging large initial state error scenarios.
