  In $f(R)$ gravity, the metric, presented in the form of the multipole
expansion, for the external gravitational field of a spatially compact
supported source up to $1/c^3$ order is provided, where $c$ is the velocity of
light in vacuum. The metric consists of General Relativity-like part and $f(R)$
part, where the latter is the correction to the former in $f(R)$ gravity. At
the leading pole order, the metric can reduce to that for a point-like or
ball-like source. For the gyroscope moving around the source without
experiencing any torque, the multipole expansions of its spin's angular
velocities of gravitoelectric-type precession, gravitomagnetic-type precession,
$f(R)$ precession, and Thomas precession are all derived. The first two types
of precession are collectively called General Relativity-like precession, and
the $f(R)$ precession is the correction in $f(R)$ gravity. At the leading pole
order, these expansions can recover the results for the gyroscope moving around
a point-like or ball-like source. If the gyroscope has a nonzero
four-acceleration, its spin's total angular velocity of precession up to
$1/c^3$ order in $f(R)$ gravity is the same as that in General Relativity.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory