Lagrange Polynomial Interpolation for Satellite Precise Ephemeris

Lagrange polynomial interpolation is a classical numerical approximation method widely employed for high-precision interpolation of satellite precise ephemeris. Its core principle involves constructing a smooth polynomial function from known discrete ephemeris data points (such as position, velocity, and other state vectors), enabling continuous description of satellite states at arbitrary time instances. In code implementation, this typically involves defining a Lagrange basis function L_i(t) = Π_{j≠i} (t-t_j)/(t_i-t_j) and combining them with weighted ephemeris values.

In satellite navigation systems, precise ephemeris data is typically provided at fixed intervals (e.g., 15 minutes) as discrete orbital parameters. Lagrange interpolation employs specific-order polynomials (commonly 8th or 10th order) to fill gaps between ephemeris points while maintaining accuracy. This method is particularly suitable for real-time positioning calculations in GNSS receivers due to its moderate computational complexity and effective suppression of Runge's phenomenon. Algorithm implementation often utilizes time normalization within interpolation windows to enhance numerical stability.

Practical applications require attention to two critical aspects: First, the selection of interpolation windows, typically implemented through sliding window strategies that dynamically adjust data points participating in the fit. Second, order optimization where excessive polynomial orders may cause oscillations while insufficient orders compromise accuracy. Key functions in implementation include window size determination and order adaptation based on ephemeris characteristics. Additionally, derived quantities like velocity/acceleration can be obtained by analytically differentiating the interpolation polynomial, providing continuous solutions without additional computational overhead.

Compared to linear interpolation or spline methods, Lagrange interpolation offers advantages through uniform formula structure that facilitates hardware implementation and better compatibility with compressed storage formats of GNSS broadcast ephemeris. Modern satellite orbit determination software frequently adopts it as a standardized interpolation tool in post-processing stages, where optimized algorithms often incorporate Chebyshev node selection or barycentric interpolation forms to improve computational efficiency.