  Cooperative positioning with multiple low earth orbit (LEO) satellites is
promising in providing location-based services and enhancing
satellite-terrestrial communication. However, positioning accuracy is greatly
affected by inter-beam interference and satellite-terrestrial topology
geometry. To select the best combination of satellites from visible ones and
suppress inter-beam interference, this paper explores the utilization of
flexible beam scheduling and beamforming of multi-beam LEO satellites that can
adjust beam directions toward the same earth-fixed cell to send positioning
signals simultaneously. By leveraging Cram\'{e}r-Rao lower bound (CRLB) to
characterize user Time Difference of Arrival (TDOA) positioning accuracy, the
concerned problem is formulated, aiming at optimizing user positioning accuracy
under beam scheduling and beam transmission power constraints. To deal with the
mixed-integer-nonconvex problem, we decompose it into an inner beamforming
design problem and an outer beam scheduling problem. For the former, we first
prove the monotonic relationship between user positioning accuracy and its
perceived signal-to-interference-plus-noise ratio (SINR) to reformulate the
problem, and then semidefinite relaxation (SDR) is adopted for beamforming
design. For the outer problem, a heuristic low-complexity beam scheduling
scheme is proposed, whose core idea is to schedule users with lower channel
correlation to mitigate inter-beam interference while seeking a proper
satellite-terrestrial topology geometry. Simulation results verify the superior
positioning performance of our proposed positioning-oriented beamforming and
beam scheduling scheme, and it is shown that average user positioning accuracy
is improved by $17.1\%$ and $55.9\%$ when the beam transmission power is 20
dBw, compared to conventional beamforming and beam scheduling schemes,
respectively.
