## 1 Introduction  
The vibration caused by a vehicle running, an engine operating, a person walking or an earthquake will have non-negligible and negative effects on precise engineering, such as in measurement experiments and on machining instruments. Generally, in most of the vibration isolation system used in engines, vehicles, etc., the frequency range of an active VIS is from tens to hundreds of Hz [1]. Various types of microscopes, laser frequency stabilization devices and gravitational wave detection devices usually operate in relatively quiet experimental environments, and these devices will be exciting by low-frequency ground vibrations from a few Hz to several Hz. Therefore, vibration isolation at these low frequencies is of importance to improve the precision of measurement experiments. The amplitude of ground vibration is very small, generally at the level of urad, so it will not harm people and cannot spread far. The six-degree-of-freedom AVI system in a molecular measuring machine (M\({}^{3}\)) uses five PZTs as the lateral-constraint links in a Mallock suspension system to control the vertical system, which improves the M\({}^{3}\) image resolution and measurement speed [2]. A schematic of the lumped model of the atomic force microscope (AFM) system was derived and the vibrational influences of the AFM components were experimentally investigated, and a reconfigured AFM system was constructed and its effects were compared with a conventional system through a series of simulations and experiments [3]. An improved mounting configuration for a passive optical cavity used for laser frequency stabilization was reported, and the associated reduction in the vibration sensitivity of the effective cavity length lead to a simple and compact reference cavity system for laser stabilization at the line-width of 1 Hz [4]. A feed-forward vibration isolation method on a suspended Fabry-Perot interferometer, using Wiener filtering, was reported, with a variant of the least mean square (LMS) adaptive filter algorithm, and the experimental results were compared with theoretical estimations of the attenuation efficiency [5]. To determine the influence of the Raman phase noise, as well as of parasitic vibrations, on the interferometer phase, the sensitivity function of a three-pulse atom interferometer was calculated theoretically and measured experimentally, which determined that the vibration noise would be, by far, the dominant limitation to the sensitivity of the gravimeter [6]. A compact and stable low-frequency AVI system with a commercial voice coil actuator and seismometer was designed and demonstrated to attenuate the vibration noise caused by reflected Ramon beams, and the vertical vibration with a frequency bandwidth (0.01-10 Hz), measured by the in-loop seismometer, was reduced by up to 500, as assessed by the PVI platform [7]. For laser interferometric gravitational wave detectors, the vibration isolation of the test mass from ground motions or seismic noise is very significant, especially as regards large, low-frequency motion, due to the resonant modes of some vibration isolators. A laser interferometer combined with a long-period folded pendulum reference was used to measure the 0.3-3 Hz frequency range horizontal seismic spectrum, and the experiment showed that the horizontal seismic motion of the laboratory was about \(10^{-9}\) m\(/\sqrt{\mathrm{Hz}}\)[8]. To weaken the low-frequency residual tilt vibration, some tilt detectors and servo-loop controllers have been reported and developed. A high-frequency tiltmeter, based on a dumbbell suspended near its center mass, was used to place a preliminary limit on the magnitude of tilts in the laboratory of \(1\) nrad\(/\sqrt{\mathrm{Hz}}\), in the frequency range 3-10 Hz [9]. The sensitivity obtainable from a kind of optical angular sensor, using the walk-off of a laser between two parallel mirrors, was analyzed theoretically and investigated experimentally [10].