  Mutual localization provides a consensus of reference frame as an essential
basis for cooperation in multirobot systems. Previous works have developed
certifiable and robust solvers for relative transformation estimation between
each pair of robots. However, recovering relative poses for robotic swarm with
partially mutual observations is still an unexploited problem. In this paper,
we present a complete algorithm for it with optimality, scalability and
robustness. Firstly, we fuse all odometry and bearing measurements in a unified
minimization problem among the Stiefel manifold. Furthermore, we relax the
original non-convex problem into a semi-definite programming (SDP) problem with
a strict tightness guarantee. Then, to hold the exactness in noised cases, we
add a convex (linear) rank cost and apply a convex iteration algorithm. We
compare our approach with local optimization methods on extensive simulations
with different robot amounts under various noise levels to show our global
optimality and scalability advantage. Finally, we conduct real-world
experiments to show the practicality and robustness.
