In this paper we give presentations for the monoid $\DP_n$ of all partial isometries on $\{1,\ldots,n\}$ and for its submonoid $\ODP_n$ of all order-preserving partial isometries.
http://arxiv.org/abs/1502.04829
http://link.springer.com/article/10.1007/s00233-015-9759-4