## 1 Introduction
Moreover, according to the Euler bending beam principle, the Euler spring boundary condition was optimized to achieve QZS with a LaCoste spring linkage [15]. With structural improvement, the LaCoste spring linkage principle yielded excellent ultra-low frequency vibration isolation and allowed highly precise gravitational wave measurements [16]. A dual-chamber air spring, rubber membrane, and viscoelastic materials also played important roles in the field of low-frequency vibration isolation [17; 18]. Single-pendulum principle-based vibration isolators always perform well at low frequencies. Folded pendulum isolators were first proposed, and here, a reasonable geometric arrangement and mass distribution could achieve an approximate central stiffness of zero [19]. Outstanding vibration isolation was observed at low frequencies because an isolator with a compliant structure (e.g., folded pendulum) could possess a large bandwidth, without influence from revolute joint friction [20]. An inverted pendulum has negative stiffness, and the presence of long rocking rods could decrease the natural frequency of a pendulum to 30 mHz. Attenuation could be as high as 65 dB under a disturbance frequency of 1 Hz [21]. A high-performance material and seismic attenuation system were designed to reduce the size of the inverted pendulum [22]. As Roberts linkage movement does not have a vertical component when the mechanism is displaced, the load exhibits QZS at the central position. The experiment demonstrated the strong effect of precision on vibration isolation, as well as the benefits of reasonably designed precision to ultra-low frequency vibration isolation [23]. For example, a long-period conical pendulum with Scott-Russel isolation has advantages such as a high loading capacity. Furthermore, vibration reduction could reach 75 dB within 0.5 Hz [24; 25]. A friction rolling-pendulum is usually applied to the vibration isolation of a specific structure (e.g., earthquake isolator) [26]. Here, a ball, or rolling pendulum, moves on a selected three-dimensional (3D) rolling surface, the center of which changes as the curvature of the rolling surface transforms. When the load places positive pressure on the ball, friction can be ignored at the equilibrium position, where a QZS system is constituted. Experimentation revealed that the load could afford low-frequency disturbances from two directions. Studies of QZS vibration isolation have focused on translational and vertical vibration. Regarding non-linear vibration isolation, harmonic response, jump phenomena, chaotic motion, resonance, and transmissibility have been deeply explored [27; 28; 29; 30]. Recently, QZS vibration isolation has been introduced into the field of torsional vibration isolation through the constitution of a high-static-low-dynamic torsional stiffness unit. [PERSON] et al. constructed a quasi-zero torsional stiffness mechanism by distributing four cylindrical cams symmetrically around a shaft [31]. Recently, [PERSON] et al. proposed a HSLDS shaft and deeply studied the influence of stiffness and nonlinear damping on vibration isolation. The results proved the outstanding capability of this high-static-low-dynamic isolator [32]. However, reports of vibration isolation during rocking motions are rare, even though such low frequency vibrations are commonly encountered in situations such as sea waves and astronaut motion. Still, [PERSON] optimized the parameters of a tuned liquid column damper to suppress pitching vibration [33], and [PERSON] et al. focused on the influence of an anti-rolling tank on the rocking motions of ships, finding that the amplitude of rocking could be reduced by using optimized parameters [34]. This paper describes the design of a novel, dual-parallelogram, passive rocking vibration isolator with QZS for the attenuation of low-frequency vibration. The scheme of the vibration isolator is presented in Section 2, and the QZS of the mechanism and four different management approaches are described in Section 3.