The interplay of slow CC loop and AA loop depends on the parallel-kinematics design of the 3-axial actuator, i.e. mod. S-325.30L from PI. The tip-tilt motion is not independent on the axial motion. In the attached plots a test with HD cameras monitoring the position of LO beam while moving the M4 mirror along the horizontal tilt angle (M4_x) and along the axial position (M4_Z). In the first plot, we apply a slow ramp on M4_X and a fast modulation on M4_Z. In the second plot, we apply a slow ramp on M4_Z and a fast modulation on M4_X.
The axial modulation induces an angular modulation which is seen on the position of the beam on the cameras. The horizontal modulation is only null when the horizontal angular position of the actuator is centered to zero. The vertical modulation is never zero during the test, as the vertical angular position (M4_Y) was not centered to zero.
In principle the effect of axial motion on angular position can be canceled by properly combining the M4_Z command with the M4_X and M4_Y commands, provided that the calibration given by the first plot is constant with time.
This effect is absent with the actuator of the fast CC, ie. S-303.CD from PI. See third attachment, where the axial modulation only induces a tiny horizontal displacement as expected on a mirror at 45 deg incidence.