  We demonstrate a closed-loop light-pulse atom interferometer inertial sensor
that can realize continuous decoupled measurements of acceleration and rotation
rate. The sensor operates with double-loop atom interferometers, which share
the same Raman light pulses in a spatially separated Mach-Zehnder configuration
and use continuous cold atomic beams propagating in opposite directions from
two 2D$^+$ magneto-optical trappings. Acceleration and the rotation rate are
decoupled and simultaneously measured by the sum and difference of dual
atom-interferometer signals, respectively. The sensitivities of inertial
measurements are also increased to be approximately 1.86 times higher than that
of a single atom interferometer. The acceleration phase shift is compensated in
real time by phase-locking these interferometers via the Raman laser phases
from the sum interferometer signal, and the gyroscope perfomance is improved.
We achieve long-term stabilities of $6.1 \ \mu g$ and 840 nrad/s for the
acceleration and the rotation rate, respectively, using a short interrogation
time of 0.87ms (interference area $A=0.097$ mm$^2$). This work provides a
building block for an atomic interferometer based inertial measurement unit for
use in field applications that require a high data-rate and high stability.
