## 1 Introduction  
With the rapid development of space technology, there is a growing demand for more efficient and reliable in-orbit launch systems [1, 2]. In-orbit launch systems allow spacecraft to be launched directly in earth orbit or deep space, and they offer advantages over traditional ground-based launch methods, such as high cost effectiveness, low environmental impact, and short launch preparation time [3, 4]. However, how to improve the launch speed while ensuring effective vibration isolation to protect sensitive payloads from damage remains a key challenge in the design of in-orbit launch systems. Traditional vibration isolation systems are designed mostly based on linear dynamics design and face problems of low efficiency when dealing with large-amplitude vibrations and dynamic payload changes [5, 6]. These systems often fail to provide ideal vibration isolation performance across a wide frequency range, especially in terms of low-frequency vibration. The limitations of this linear design approach have prompted researchers to explore the characteristics of nonlinear dynamics, especially the applications of nonlinear vibration isolation systems in the hope of achieving better vibration isolation performance. Nonlinear vibration isolation technology in the form of quasi-zero stiffness (QZS) isolators has been proven to effectively reduce the system's natural frequency and isolate vibrations across a wider frequency range [7, 8]. This particularly applies to the low frequency vibrations commonly encountered in in-orbit launch systems. For example, [PERSON] et al. [9] proposed an optimized QZS vibration isolation system with a multi-cam-roller mechanism, which demonstrated superior low-frequency vibration isolation performance. [PERSON] et al. [10] designed an advanced QZS isolator with three pairs of oblique springs, which offered enhanced design flexibility and superior performance over conventional one-pair and linear isolators. [PERSON] et al. [11] presented a QZS system with an oblique linear spring structure, which showed enhanced low-frequency vibration isolation ability by expanding the zero stiffness range. [PERSON] et al. [12; 13] designed an X-shaped-spring suspension nonlinear isolator, which could achieve QZS and a broader isolation frequency band through a quintic characteristic. [PERSON] et al. [14] proposed a bi-directional horizontal QZS isolation system utilizing pre-compressed oblique spring mechanisms, showing superior low-frequency vibration isolation performance. [PERSON] et al. [15] designed a five-spring QZS vibration isolator, in which four oblique springs provided negative stiffness and one vertical spring provided positive stiffness. The results show that the initial frequency of isolation could be reduced to 2.82 Hz. [PERSON] et al. [16] proposed a new QZS vibration isolator composed of a link-oblique spring negative stiffness mechanism and a linear spring, which had a significantly better low-frequency isolation effect while maintaining high-frequency isolation ability compared with linear isolators. [PERSON] et al. [17] designed a new \"X\" type QZS isolator using a spring arrangement method of tunable contact mechanism based on an X-shaped anti-vibration structure. This isolator was better than a linear isolator in the ultra-low frequency range with lower resonance frequency and wider isolation band. [PERSON] et al. [18] combined the Miura origami unit to propose a QZS vibration isolator composed of four rods and two springs. The structure could effectively reduce the formant and expand the effective vibration isolation band due to its nonlinear stiffness and nonlinear vertical damping. [PERSON] et al. [19] proposed a new type of passive asymmetric QZS isolator consisting of two linear springs in parallel with a negative stiffness element. Experimental results showed that the isolator had a smaller initial frequency and lower displacement transmissibility than the linear model. [PERSON] et al.