Negative stiffness system for gravity compensation of micropositioner

Abstract

A negative stiffness system for gravity compensation of a micropositioner of wafer table in lithography machine, characterized in that, 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 the 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. The system reduces the stiffness in vertical direction and prevents the influence of permanent magnet on its surroundings, while improving the bearing capacity.

Claims

1. 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 .sub.0 of each of the negative stiffness springs with respect to a horizontal plane satisfies the following specific stiffness ratio .sub.QZS of each of the negative stiffness springs to the positive stiffness spring at a quasi-zero stiffness point; ${}_{\mathrm{QZS}}=\frac{}{2\left(1-\right)},$ wherein, the stiffness ratio .sub.QZS is k.sub.0/k.sub.v; L.sub.0: an initial length of each of the negative stiffness springs; L: a length of each of the negative stiffness springs that is deformed; : cosine of the initial angle .sub.0 of each of the negative stiffness springs, defined as =cos .sub.0; : a length of projection of each of the negative stiffness springs in a horizontal direction, defined as =L cos .sub.0; k.sub.0: a stiffness of each of the negative stiffness springs; and k.sub.v: 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.

2. The negative stiffness system according to claim 1, wherein when the negative stiffness system comprises three sets of quasi-zero stiffness units, the three sets of quasi-zero stiffness units are arranged in a triangle, and wherein when the negative stiffness system comprises four sets of quasi-zero stiffness units, the four sets of quasi-zero stiffness units are arranged in a rectangle.

3. The negative stiffness system according to claim 1, wherein the upper ends of the negative stiffness springs and the upper end of the positive stiffness spring are directly connected together.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a 3D diagram of the negative stiffness system for gravity compensation of a micropositioner according to the present invention;

(2) FIG. 2 is a graph illustrating the characteristic curve of negative stiffness;

(3) FIG. 3 illustrates a combined spring composed of the positive and negative stiffness springs;

(4) FIG. 4 is a graph illustrating the characteristic curve of the positive, negative and the combined springs;

(5) FIG. 5 is a diagram of the simplified model of the quasi-zero stiffness unit of the negative stiffness system according to the present invention.

(6) Wherein, 1rotor of micropositioner; 2negative stiffness unit; 3positive stiffness unit; 4base.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(7) The specific structures, principles and operations of the present invention will be described in detail with reference to the drawings.

(8) A negative stiffness system for gravity compensation of a micropositioner is provided, characterized in that, 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 2 and a positive stiffness spring 3, the positive stiffness spring 3 is vertically positioned, the pair of negative stiffness springs 2 are obliquely and symmetrically positioned at two sides of the positive stiffness spring 3, upper ends of the negative stiffness springs 2 and the positive stiffness spring 3 are connected together and fixed to the bottom surface of a rotor 1 of the micropositioner, and lower ends of the negative stiffness springs 2 and the positive stiffness spring 3 are connected to a base 4, respectively, as shown in FIG. 1.

(9) A negative stiffness system for gravity compensation of a micropositioner is provided, wherein the negative stiffness system is arranged in triangle when the negative stiffness system comprising three sets of quasi-zero stiffness units, and the negative stiffness system is arranged in rectangle when the negative stiffness system comprising four sets of quasi-zero stiffness units.

(10) The quasi-zero stiffness unit refers to a combination of linear elastic elements (such as springs or struts) in such a way that the stiffness of the whole unit can be 0 or near to 0 when the unit under external load is at its balanced position (quasi-zero stiffness point). The negative stiffness system of the present invention is based on the following theory.

(11) Stiffness is generally defined as a change rate of a load exerted on an elastic unit on the deformation caused by the load, represented by K, that is:

(12) $\begin{array}{cc}K=\frac{\mathrm{dP}}{d}& \left(1\right)\end{array}$

(13) Wherein, P represents load, and represents the deformation.

(14) If the load of the elastic unit increases as the deformation increases, stiffness is positive; if it does not change with the deformation increased, stiffness is zero; and if it does not increase but decrease as the deformation increases, stiffness is negative. FIG. 2 illustrates a load-deformation curve of an elastic unit in negative stiffness section and such stiffness unit can be referred as a negative stiffness spring. In physics, the negative stiffness spring is an unsteady elastic unit while the vibration system is required to be a steady system. Thus, in a vibration system, the negative stiffness spring cannot be used independently, and it has to be used in parallel with a positive stiffness spring. FIG. 3 illustrates a combined spring formed by connecting a positive stiffness spring A and a negative stiffness spring B in parallel. Supposing that a deformation of is generated under an external force F, and the elastic forces of the positive and negative stiffness springs are P.sub.v() and P.sub.0(), respectively, the elastic force of the combined spring is:
P()=P.sub.v()+P.sub.0()=F(2)

(15) The relation between the elastic force and deformation of the spring can be deduced from equation (2), as shown in FIG. 4.

(16) Taking a derivative on the two sides of equation (2) with respect to , the following equation (3) can be deduced:

(17) $\begin{array}{cc}K=\frac{\mathrm{dP}}{d}=\frac{{\mathrm{dP}}_0}{d}+\frac{{\mathrm{dP}}_v}{d}=k_0+k_v& \left(3\right)\end{array}$

(18) Wherein, the stiffnesses of the positive stiffness spring A and the negative stiffness spring B are k.sub.0 and k.sub.v, and the stiffness of the combined spring is K.

(19) It can be seen from FIG. 4 and equation (3) that, in the negative stiffness section of the negative stiffness spring, since k.sub.2<0, the stiffness of the combined spring is algebraic sum of stiffnesses of the positive and negative stiffness springs. That is, in the negative stiffness section of the negative stiffness spring, the total stiffness of the combined spring connected in parallel is less than the stiffness of the positive stiffness spring. The reduced component in the total stiffness of the combined spring is offset by the negative stiffness spring due to the principle of stiffness offset with the positive and negative stiffness in parallel. Thus, the stiffness of the system can be reduced by connecting the positive and negative stiffness springs in parallel. It can be also seen form FIG. 4 that, in the negative stiffness section of the negative stiffness spring, the total elastic force, i.e., the total load capacity of the combined spring is larger than the positive stiffness spring when used alone, although the total stiffness of the combined spring is less than the stiffness of the positive stiffness spring when used alone. Therefore, an elastic element having a low stiffness and a large load capacity can be composed by connecting the positive and negative stiffness springs in parallel.

(20) In addition, the restoring force f of the spring can be deduced from the geometric relationship in FIG. 5, that is:
f=k.sub.vx+2k.sub.0(L.sub.0L)sin (4)

(21) Wherein, L.sub.0={square root over (h.sub.0.sup.2+.sup.2)}, L={square root over ((h.sub.0x).sup.2+.sup.2)}, in which, L: length of the negative stiffness spring that is deformed; .sub.0: initial angle of the negative stiffness spring with respect to the horizontal plane, i.e. the initial dip angle of the negative stiffness spring; : angle of the negative stiffness spring with respect to the horizontal plane after deformation, i.e. the dip angle of the negative stiffness spring after deformation; : cosine of the initial dip angle of the negative stiffness spring, defined as =cos .sub.0; : length of projection of the negative stiffness spring in horizontal direction, defined as =L cos , and it is a constant as all movements are in vertical direction in the system; h.sub.0: length of projection of the negative stiffness spring in vertical direction in the initial state, defined as h.sub.0=L.sub.0 sin .sub.0; x: displacement of the quasi-zero stiffness unit in vertical direction when under an external force; k.sub.0: stiffness of the negative stiffness spring; k.sub.v: stiffness of the positive stiffness spring; K: total stiffness of the quasi-zero stiffness unit; : stiffness ratio, defined as =k.sub.0/k.sub.v; f: external load of the quasi-zero stiffness unit, generally in vertical direction.

(22) Since =k.sub.0/k.sub.v, substitute L.sub.0, L into equation (4), and in combination with sin =(h.sub.0x)/L, it can be derived that:

(23) $\begin{array}{cc}f=k_{v}x+2k_v\left(h_0-x\right)\left(\frac{\sqrt{h_0^2+a^2}}{\sqrt{{\left(h_0-x\right)}^2+a^2}}-1\right)& (\left(5\right)\end{array}$

(24) Both sides of equation (5) divided by L.sub.0k.sub.v to be dimensionless, that is,
{circumflex over (f)}={circumflex over (x)}+2({square root over (1.sup.2)}{circumflex over (x)}){[{circumflex over (x)}.sup.22{circumflex over (x)}{square root over (1.sup.2)}+1].sup.1/21}(6)

(25) Both sides of equation (6) divided by displacement x, the stiffness in a dimensionless form can be derived:

(26) $\begin{array}{cc}\hat{K}=1+2\left[1-\frac{{}^2}{{\left({\hat{x}}^2-2\hat{x}\sqrt{1-{}^2}+1\right)}^{3\text{/}2}}\right]& \left(7\right)\end{array}$

(27) It can be derived form equation (7) that when the stiffness of the system {circumflex over (K)} is minimal (equal to 0), {circumflex over (x)}=={square root over (1.sup.2)}. Here, when the initial dip angle is given, let the above equation to be zero, the unique stiffness ratio .sub.QZS of the quasi-zero stiffness can be solved, that is:

(28) ${}_{\mathrm{QZS}}=\frac{}{2\left(1-\right)}$

(29) And, the quasi-zero stiffness point is located at {circumflex over (x)}=={square root over (1.sup.2)}, that is, at {circumflex over (x)}={square root over (1.sup.2)}.

(30) Also, it can be known that the initial dip angle of negative stiffness spring can be solved when the stiffness ratio of the system is given, and the relationship is represented as:

(31) ${}_{\mathrm{QZS}}=\frac{2}{2+1}$

(32) And, the quasi-zero stiffness point is located at {circumflex over (x)}=={square root over (1.sub.QZS.sup.2)}

(33) In the present embodiment, the simplified model is shown in FIG. 5. When it is designated that =0.5, it can be solved that .sub.QZS=0.5, {circumflex over (x)}={square root over (1.sup.2)}=0.866. Supposing that the mass of a ball is 4 kg, and the acceleration of gravity is 9.8 N/Kg, then the force of gravity on the ball is 39.2 N, and substitute this value into equation (6), it can be solved that k.sub.vL.sub.0 is 45.26 N. On the assumption that the length of the spring is 0.2 m, it can be solved that the stiffness of the vertical spring is 226.3 N/m, and the stiffness of the oblique spring is 113.15 N/m.

(34) A simulation is performed based on the design parameters utilizing ABAQUS. Considering the ball as a rigid body, the stiffness of the vertical spring is 226.3 N/m, the stiffness of the oblique spring is 113.15 N/m, and the length of each of the springs is 0.2 m, and the dip angle of the oblique spring is 60. A concentrated load of 100 N is applied at the reference point of the rigid body. Through static force analysis, a force-time curve and a displacement-time curve are calculated and obtained, and further in combination with a time-force curve, a force-displacement curve is derived. From the force-displacement curve it can be seen that when the load is increased to 39 N, the slope of the curve tends to be 0 which means that the stiffness is getting close to 0 while the displacement of the spring is 0.152 m. When the load is increased to 40 N, the displacement of the ball is 0.207 m, that is, when the load increases by 1 N, the displacement increases by 0.055 m. It can be determined that the stiffness of the system is rather small when the displacement is in a range of 0.1520.207 m, and the analytic solution x.sub.l.sub.0=0.173 m also falls in this range, verifying the correctness of the simulation, and then it can be determined that the ball can be in a quasi-zero stiffness state when only under gravity. The curves of restoring force of the negative stiffness spring and the positive stiffness spring can be calculated and obtained which show that, at t=0.39 s, the restoring force of the negative stiffness spring decreases rapidly and the restoring force of the positive stiffness spring increases rapidly and the static force balance of the ball can be achieved, thus the vertical displacement increases sharply at t=0.39 s. Then, it can be determined that the above structure can be applied in the gravity balancing structure of the micropositioner.