  With the temperature power spectrum of the cosmic microwave background (CMB)
at least four orders of magnitude larger than the B-mode polarisation power
spectrum, any instrumental imperfections that couple temperature to
polarisation must be carefully controlled and/or removed. Here we present two
new map-making algorithms that can create polarisation maps that are clean of
temperature-to-polarisation leakage systematics due to differential gain and
pointing between a detector pair. Where a half wave plate is used, we show that
the spin-2 systematic due to differential ellipticity can also by removed using
our algorithms. The algorithms require no prior knowledge of the imperfections
or temperature sky to remove the temperature leakage. Instead, they calculate
the systematic and polarisation maps in one step directly from the time ordered
data (TOD). The first algorithm is designed to work with scan strategies that
have a good range of crossing angles for each map pixel and the second for scan
strategies that have a limited range of crossing angles. The first algorithm
can also be used to identify if systematic errors that have a particular spin
are present in a TOD. We demonstrate the use of both algorithms and the ability
to identify systematics with simulations of TOD with realistic scan strategies
and instrumental noise.
