TRAJECTORY ESTIMATION WITH NATURAL SPLINE AND EXPONENTIAL PARAMETERIZATION

We discuss the problem of fitting reduced data M_n formed by the ordered set of points x_i = γ(t_i) in arbitrary Euclidean space (here γ is a regular curve). The unknown interpolation knots T_n are compensated by their substitutes T̂_n based on exponential parameterization (which depends on M_n and parameter λ ∈ [0, 1]). A natural spline is applied here to interpolate M_n based on guessed knots T̂_n. The linear (quadratic) convergence orders in γ estimation are tested for λ ∈ [0, 1) (λ = 1) with n getting larger. The latter together with its sharpness is verified numerically in affirmative. In addition, the conducted experiments confirm the necessity of more-or-less uniformity in samplings T_n and of the regularity of γ to retain the examined asymptotics.