  Direct imaging and spectroscopy of Earth-like planets and young Jupiters
require contrasts up to 10^6-10^10 at angular separations of a few dozen
milliarcseconds. To achieve this goal, one of the most promising approaches
consists of using large segmented primary mirror telescopes with coronagraphic
instruments. However, coronagraphs are highly sensitive to wavefront errors.
The segmentation itself is responsible for phasing errors and segment
vibrations to be controlled at a subnanometric accuracy. We propose an
innovative method for a coronagraph design that allows a consequent relaxation
of the segment phasing constraints for low segment-count mirrors and generates
an instrument that is more robust to segment-level wavefront errors. It is
based on an optimization of the coronagraph that includes a segment-level
apodization. This is repeated over the pupil to match the segmentation
redundancy and improves the contrast stability beyond the minimum separation
set by the single-segment diffraction limit. We validate this method on a
GMT-like pupil for two coronagraph types: apodized pupil Lyot coronagraphs
(APLC) and apodizing phase plate coronagraphs (APP). For the APLC, redundant
apodization enables releasing the piston phasing constraints by a factor of 5
to 20 compared to classical designs. For the APP, the contrast remains almost
constant up to 1 radian RMS of the phasing errors. We also show that redundant
apodizations increase the robustness of the coronagraph to segment tip-tilt
errors, as well as to missing segments. This method cannot be applied to
higher-segment count mirrors such as the ELT or the TMT, but it is particularly
suitable for low segment-count mirrors (fewer than 20 segments) such as the GMT
aperture. These mirrors aim for high-contrast imaging of debris disks or
exoplanets down to 100 mas.
