It is unlikely that gimbal checks would be done on a running engine (at least in flight) due to the resultant effects on the vehicle.
In addition to the famous prelaunch gimbal test of the main engines prior to launch as described here, gimbal checks were also done on the Orbital Maneuvering Engines (OMEs) and the Solid Rocket Boosters (SRBs) thrust vector control systems.
- Orbital Maneuvering System
The OMEs were 6000 lbf pressure-fed engines burning nitrogen tetroxide and mono-methyl hydrazine. They were gimballed using electromechanical actuators.
On some missions, a prelaunch gimbal check was performed on the OMS at T-7 hours 30 minutes. The criteria were as follows
OMS GIMBAL PROFILE IS ONLY
REQUIRED ONE TIME PRIOR TO
LAUNCH MAKING THE REPEAT OF
THE TEST OPTIONAL FOR A
RECYCLE LAUNCH COUNTDOWN.
DO NOT PERFORM THIS TEST IF
IT HAS BEEN PERFORMED IN THE
PREVIOUS COUNTDOWN UNLESS
THERE IS CONCERN OVER ACTIVITY
IN THE GENERAL LRU AREA OR
CONCERN OVER POSSIBLE TREND
DATA OR CONCERN OVER THE TIME
ELAPSED SINCE THE LAST TEST.
(Apologies for all caps, that is how it is written in the OMI S0007 Shuttle Countdown)
In flight, a gimbal check was performed by executing an ITEM 34 on the Maneuver display.
The gimbal check flag caused the engines to be
commanded to plus 7° yaw and 6° pitch, then to
minus 7° yaw and 6° pitch, and back to zero°
yaw and pitch.
On a nominal mission, these gimbal checks were performed twice, once after the OMS-2 circularization burn using procedures in the Ascent Checklist, and once before the deorbit burn, using procedures in the Entry Checklist. The Entry Checklist procedure is shown.
If problems were suspected with the thrust vector control system, gimbal checks could be done as required.
The shuttle Solid Rocket Boosters thrust vector control system was a hydraulic system pressurized by two hydrazine-powered Hydraulic Power Units (HPU) in each booster. The HPUs were started at T-26 seconds and the gimbal test was started at T-21 seconds and nominally completed by T-16 seconds.