## 1 Introduction  
According to the linear vibration theory, the isolation of micro-vibration can only be achieved when the excitation frequency is greater than \(\sqrt{2}\mathrm{w}_{\mathrm{n}}\), where \(\mathrm{w}_{\mathrm{n}}\) is the natural frequency of a vibration isolator. The vibration isolation bandwidth of a vibration isolator increases when its natural frequency decreases. The natural frequency of a vibration isolator can be reduced by either increasing its mass of payload or decreasing its stiffness. However, it is usually necessary to increase the stiffness of the isolator to maintain its stability when the mass of the payload increases. Negative-stiffness vibration isolators are isolators that use various kinds of restoring force, whether passive [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21], semi-active [22; 23; 24; 25; 26; 27; 28], or active [29; 30], to produce negative-stiffness characteristics, thus partially counteracting the positive stiffness within a micro-vibration stroke. And they have shown effective to produce high-static-low-dynamic stiffness characteristics. Much work has been performed on passive negative stiffness vibration isolators and their application has been summarized by [PERSON] [1]. For example, pre-stressed bars or beams [2; 3], inclined springs with axial preload [4; 5; 6], cam-roller-spring mechanisms [7; 8], and scissor-like structures [9] combined with a vertical spring in parallel, can produce high-static-low-dynamic stiffness. In addition, cubic permanent magnets in attractive [10; 11], repulsive [12; 13], or a combination of attractive and repulsive configuration [14; 15], permanent magnet rings magnetized axially [16; 17] or radially [18; 19] were well proposed to produce negative stiffness with non-contact characteristics. The negative stiffness of the above vibration isolators was not easy to adjust, so their optimal performance can only be achieved when they are with a certain payload and geometric parameters. In addition, these vibration isolators only show advantages for small- and micro-vibration, and the resonance peak was noticeably large and its inherent cubic nonlinearity may lead to bifurcation and a chaotic response for vibration with large amplitude [20; 21]. Semi-active negative stiffness vibration isolators were proposed to realize the variable stiffness characteristic and to cope with the variation of excitation frequency or system parameters. [PERSON] [22] used linear pneumatic actuators instead of inclined springs to obtain adjustable negative stiffness, and the natural frequency was reduced by up to 58%. [PERSON] [23] proposed a fast and effective dynamically tunable stiffness structure using piezoelectric actuators. [PERSON] [24] proposed a tunable negative-stiffness spring composed of a pair of electromagnets and a permanent magnet. [PERSON] [25] used an electromagnetic asymmetric tooth structure to obtain variable negative stiffness, which can be adjusted through tuning the magnitude of input current. [PERSON] [26] proposed an electromagnetic negative stiffness vibration isolator composed of multilayer magnets and coils. [PERSON] [27] presented an electromagnetic negative stiffness spring containing with two coaxial annular electromagnets, whose negative stiffness was adjusted by changing the current in rectangular cross-section coils. [PERSON] [28] realized the stiffness adjustment by introducing a regulation mechanism into a negative stiffness structure composed of a pair of coaxially magnetized coaxial magnetic rings. The semi-active negative stiffness isolator was an open-loop control system, which had low control accuracy and cannot compensate the deviation caused by external interference. Active negative stiffness vibration isolators with high control accuracy also show advantages in generating variable stiffness characteristics and better low-frequency vibration isolation performance.