Negative stiffness system for gravity compensation of micropositioner

A negative stiffness system for gravity compensation of a micropositioner, wherein the negative stiffness system includes at least three sets of quasi-zero stiffness units, each of the sets of quasi-zero stiffness units comprises a pair of negative stiffness springs and a positive stiffness spring, the positive stiffness spring is vertically positioned, the pair of negative stiffness springs are obliquely and symmetrically positioned at two sides of the positive stiffness spring, upper ends of the negative stiffness springs and the positive stiffness spring are connected together and fixed to a bottom surface of a rotor of the micropositioner, and lower ends of the negative stiffness springs and the positive stiffness spring are connected to a base, respectively,

• wherein an initial angle θ0 of each of the negative stiffness springs with respect to a horizontal plane satisfies the following specific stiffness ratio αQZS of each of the negative stiffness springs to the positive stiffness spring at a quasi-zero stiffness point;
• α
• QZS
• =
• γ
• 2
• ⁢
• (
• 1
• -
• γ
• )
• ,
• wherein, the stiffness ratio αQZS is k0/kv;
• L0:
• an initial length of each of the negative stiffness springs;
• L:
• a length of each of the negative stiffness springs that is deformed;
• γ:
• cosine of the initial angle θ0 of each of the negative stiffness springs, defined as γ=cos θ0;
• α:
• a length of projection of each of the negative stiffness springs in a horizontal direction, defined as α=L cos θ0;
• k0:
• a stiffness of each of the negative stiffness springs;
• and
• kv:
• a stiffness of the positive stiffness spring,
• wherein the base includes a first sidewall portion, a second sidewall portion, and a middle portion, the middle portion being disposed between the first sidewall portion and the second sidewall portion to connect the first and second sidewall portions,
• wherein each of the lower ends of the negative stiffness springs are connected to an inner surface of a corresponding one of the first and second sidewall portions, and
• wherein the lower end of the positive stiffness spring is connected to a top surface of the middle portion.