  In this paper, a new framework for continuous-time maximum a posteriori
estimation based on the Chebyshev polynomial optimization (ChevOpt) is
proposed, which transforms the nonlinear continuous-time state estimation into
a problem of constant parameter optimization. Specifically, the time-varying
system state is represented by a Chebyshev polynomial and the unknown Chebyshev
coefficients are optimized by minimizing the weighted sum of the prior,
dynamics and measurements. The proposed ChevOpt is an optimal continuous-time
estimation in the least squares sense and needs a batch processing. A recursive
sliding-window version is proposed as well to meet the requirement of real-time
applications. Comparing with the well-known Gaussian filters, the ChevOpt
better resolves the nonlinearities in both dynamics and measurements. Numerical
results of demonstrative examples show that the proposed ChevOpt achieves
remarkably improved accuracy over the extended/unscented Kalman filters and
extended batch/fixed-lag smoother, closes to the Cramer-Rao lower bound.
