  The recent flight experiments with Neutron Star Interior Composition Explorer
(\textit{NICER}) and \textit{Insight}-Hard X-ray Modulation Telescope
(\textit{Insight}-HXMT) have demonstrated the feasibility of X-ray pulsar-based
navigation (XNAV) in the space. However, the current pulse phase estimation and
navigation methods employed in the above flight experiments are computationally
too expensive for handling the Crab pulsar data. To solve this problem, this
paper proposes a fast algorithm of on-orbit estimating the pulse phase of Crab
pulsar called X-ray pulsar navigaTion usIng on-orbiT pulsAr timiNg (XTITAN).
The pulse phase propagation model for Crab pulsar data from
\textit{Insight}-HXMT and \textit{NICER} are derived. When an exposure on the
Crab pulsar is divided into several sub-exposures, we derive an on-orbit timing
method to estimate the hyperparameters of the pulse phase propagation model.
Moreover, XTITAN is improved by iteratively estimating the pulse phase and the
position and velocity of satellite. When applied to the Crab pulsar data from
\textit{NICER}, XTITAN is 58 times faster than the grid search method employed
by \textit{NICER} experiment. When applied to the Crab pulsar data from
\textit{Insight}-HXMT, XTITAN is 180 times faster than the Significance
Enhancement of Pulse-profile with Orbit-dynamics (SEPO) which was employed in
the flight experiments with \textit{Insight}-HXMT. Thus, XTITAN is
computationally much efficient and has the potential to be employed for onboard
computation.
