  Imaging exo-Earths is an exciting but challenging task because of the 10^-10
contrast ratio between these planets and their host star at separations
narrower than 100 mas. Large segmented aperture space telescopes enable the
sensitivity needed to observe a large number of planets. Combined with
coronagraphs with wavefront control, they present a promising avenue to
generate a high-contrast region in the image of an observed star.
  Another key aspect is the required stability in telescope pointing, focusing,
and co-phasing of the segments of the telescope primary mirror for
long-exposure observations of rocky planets for several hours to a few days.
These wavefront errors should be stable down to a few tens of picometers RMS,
requiring a permanent active correction of these errors during the observing
sequence. To calibrate these pointing errors and other critical low-order
aberrations, we propose a wavefront sensing path based on Zernike
phase-contrast methods to analyze the starlight that is filtered out by the
coronagraph at the telescope focus. In this work we present the analytical
retrieval of the incoming low order aberrations in the starlight beam that is
filtered out by an Apodized Pupil Lyot Coronagraph, one of the leading
coronagraph types for starlight suppression. We implement this approach
numerically for the active control of these aberrations and present an
application with our first experimental results on the High-contrast imager for
Complex Aperture Telescopes (HiCAT) testbed, the STScI testbed for Earth-twin
observations with future large space observatories, such as LUVOIR and HabEx,
two NASA flagship mission concepts.
