Hexapod kinematic mountings for optical elements, and optical systems comprising same

Abstract

“Hexapod” mountings are disclosed for use with optical elements. An exemplary mounting includes a base, a platform that is movable relative to the base, and six legs having nominally identical length. Three pairs of legs, having substantially equal stiffness, extend between the base and platform and support the platform relative to the base. In each pair of legs, respective first ends are coupled together in a Λ-shaped manner forming a respective apex. Respective second ends are splayed relative to the apex, desirably forming an angle of substantially 109.5° at the apex. The apices are mounted equidistantly from each other on a circle on the platform. The respective second ends of the pairs of legs are mounted at respective locations on a circle on the base. The axes of each pair of legs define a respective leg plane substantially perpendicular to the base plane. Each leg has an actuator that, when energized, changes a length of the respective leg. Coordinated energization of the actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of certain relationships of an embodiment of a hexapod mounting, showing the locations at which the legs are attached to base and platform and the two coordinate axes of the system.

FIG. 2 is a plan view of FIG. 1, showing locations at which legs are attached and the relationships of those locations.

FIG. 3(A) is a plan schematic diagram showing certain geometric relationships of the points at which the legs are attached, and the designations of key geometric variables.

FIG. 3(B) is a vertical schematic diagram (orthogonal to FIG. 3(A)) showing certain other geometric relationships and variable designations.

FIG. 4(A) is a perspective view of a hexapod mounting according to a representative embodiment.

FIG. 4(B) is a plan view of hexapod mounting shown in FIG. 4(A).

FIG. 5 is a perspective view of a pair of legs used in the embodiment of FIGS. 4(A) and 4(B).

FIG. 6 is a schematic diagram of a projection-optical system including one optical element on a hexapod mounting as described herein.

FIG. 7 is a schematic optical diagram of a conventional extreme-ultraviolet (EUV) lithography system, including EUV source, illumination-optical system, and projection-optical system.

FIG. 8 is a perspective view showing the arrangement of legs of a conventional hexapod configured as a Stewart platform.

DETAILED DESCRIPTION

This disclosure is set forth in the context of a representative embodiment that is not intended to be limiting in any way. Also, even though the embodiment is described in the context of holding an optical mirror, it will be understood that the mirror alternatively can be another type of optical element or other object. Hence, an “active-mirror-adjustment” (AMA) mechanism as described below is not limited to use with a mirror. In addition, positional terms such as “above,” below,” “upper,” “lower,” “over,” “under,” “horizontal,” and “vertical” are used to facilitate comprehension of spatial relationships, but are not intended to be limited to their literal meanings in the context of a terrestrial environment.

The particular AMA mechanism that is the subject of this disclosure is a so-called “hexapod” mount configured as a Stewart platform. At least one optical element of an EUVL system or other optical system is mounted on the platform, which is mounted by six legs to a base and is movable relative to the base in six degrees of freedom (DOF). Movement of the platform is feedback-controlled in a manner that provides a high bandwidth (very rapid response time) and extremely high accuracy and precision, which are advantageous for use in an EUVL system. Another key advantage of the instant AMA mechanism is its exhibited absence of coupling effects (zero coupling stiffness) and high axial stiffness. Both these characteristics are discussed in more detail later.

A representative embodiment of an AMA mechanism is depicted in FIGS. 1 and 2. The mechanism comprises a platform 12, a base 14, and six legs L₁-L₆ arranged into three pairs L₁ and L₂, L₃ and L₄, and L₅ and L₆. The position of the base is typically fixed, achieved by mounting the base to a rigid frame, in an optical “barrel” or “column,” or analogous structure. The legs L₁-L₆ support the platform 12 relative to the base 14 in a manner allowing movement of the platform relative to the base. The “upper” ends of each pair of legs L₁-L₂, L₃-L₄, L₅-L₆ converge at a respective apex attached at a respective location B₁₂, B₃₄, B₅₆ on the undersurface of the platform 12, and the “lower” ends of each pair of legs are attached to the upper surface of the base 14 at respective locations A₁ and A₂, A₃ and A₄, A₅ and A₆. The locations B₁₂, B₃₄, B₅₆ are located equally spaced from each other on a circle 16 (on the underside of the platform 12) having a center at O_B, and the locations A₁-A₆ are located on a circle 18 (on the upper side of the base 14) having a center at O_A.

In a “null” condition as shown, all the legs L₁-L₆ nominally have identical length, the platform 12 is exactly parallel to the base 14, the moving coordinates (u, v, w) of the platform are coincident with fixed coordinates (x, y, z) of the base, and the center O_A of the base and center O_B of the platform are on a vertical axis A_x. Also, the “vertical” axis w of the moving coordinates and the “vertical” axis z of the fixed coordinates are on the vertical axis A_x, and the centroid of the platform 12 is at an elevation h= O_AO_B above the base 14. Starting from the null condition, a change in length of any one or more of the legs L₁-L₆ causes a shift of the moving coordinates (u,v,w) relative to the fixed coordinates (x,y,z). The shift can reflect a respective change in one or more degrees of freedom (x, y, z, θ_x, θ_y, θ_z) of motion of the platform 12 relative to the base 14.

In the following discussion, a, b, c, and L are design parameters that are defined as follows (see FIGS. 3(A) and 3(B)):

A_i is any of A₁-A₆,

B_i is any of B₁-B₆,

B_ij is any of B₁₂, B₃₄, B₅₆,

a is the radius from O_A to any point A_i on the base: a= O_AA_i,

b is the radius from O_B to any point B_i on the platform: b= O_BB_i,

c is the “leg interval”, wherein 2c= A₁A₂= A₃A₄= A₅A₆, and

L is the length of a leg.

The points A_i and B_ij have respective coordinates (e.g., x, y, z), as follows:

A ₁ =[−c,√{square root over (a²−c²)},0]  (1)

A ₂ =[c,√{square root over (a²−c²)},0]  (2)

$\begin{array}{cc}{A}_3=\left[\frac{c}{2}+\frac{\sqrt{3}}{2}\sqrt{a^2-c^2},\frac{\sqrt{3}}{2}c-\frac{1}{2}\sqrt{a^2-c^2},0\right]& \left(3\right)\\ A_4=\left[\frac{-c}{2}+\frac{\sqrt{3}}{2}\sqrt{a^2-c^2},\frac{-\sqrt{3}}{2}c-\frac{1}{2}\sqrt{a^2-c^2},0\right]& \left(4\right)\\ A_5=\left[\frac{c}{2}-\frac{\sqrt{3}}{2}\sqrt{a^2-c^2},\frac{-\sqrt{3}}{2}c-\frac{1}{2}\sqrt{a^2-c^2},0\right]& \left(5\right)\\ A_6=\left[\frac{-c}{2}-\frac{\sqrt{3}}{2}\sqrt{a^2-c^2},\frac{\sqrt{3}}{2}c-\frac{1}{2}\sqrt{a^2-c^2},0\right]& \left(6\right)\end{array}$

B₁₂=[o,b,h]  (7)

$\begin{array}{cc}{B}_{34}=\left[\frac{-\sqrt{3}}{2}b,\frac{-1}{2}b,h\right]& \left(8\right)\\ B_{56}=\left[\frac{-\sqrt{3}}{2}b,\frac{-1}{2}b,h\right]& \left(9\right)\end{array}$

To determine h, consider that L= A₁B₁₂. If A₁=[−c, √{square root over (a²−c²)}, 0] and B₁₂=[0, b, h], then L²= A₁B₁₂²=c²+(b−√{square root over (a²−c²)})²+h². Solving for h² yields:

h ² =L ² −b ²+2b√{square root over (a²−c²)}−a².  (10)

Exemplary coordinates are as follows (dimensions are in mm):

	Point
	x	y	z
	A1	75.6329	−179.0000	20.0000
	A2	−75.6329	−179.0000	20.0000
	A3	−192.8350	24.0000	20.0000
	A4	−117.2021	155.0000	20.0000
	A5	117.2021	155.0000	20.0000
	A6	192.8350	24.0000	20.0000
	B12	0.0000	−179.0000	151.0000
	B34	−155.0185	89.5000	151.0000
	B56	155.0185	89.5000	151.0000

From these coordinates, the following exemplary values can be obtained:

• • a=((−179)²+(75.6329)²)^1/2=194.32 (from coordinate A₁)
  • b=179 (from coordinate B₁₂)
  • c=75.6329 (from coordinate A₁ or A₂)

The vector d_i is a corresponding elongation vector for the respective leg. Since there are six legs, i=1, 2, 3, . . . , 6. In other words, d₁={right arrow over (A₁B₁₂)}, d₂={right arrow over (A₂B₁₂)}, d₃={right arrow over (A₃B₃₄)}, d₄={right arrow over (A₄B₃₄)}, d₅={right arrow over (A₅B₅₆)}, and d₆={right arrow over (A₆B₅₆)}. The elongation vectors can be collectively denoted by the vector q=[d₁, d₂, d₃, d₄, d₅, d₆]^T. If the position of the platform is denoted by the vector x, the kinematical constraints imposed by the legs can be expressed in the general form:

f(x,q)=0  (1)

Differentiating this expression with respect to time yields a relationship between leg-elongation velocity ({dot over (q)}) and an output-velocity vector ({dot over (x)}) for the platform:

J_x{dot over (x)}=J_q{dot over (q)}  (12)

where

$J_x=\frac{\partial f}{\partial x}$

and

$J_q=-\frac{\partial f}{\partial q}.$

The derivation leads to two separate Jacobian matrices. The overall Jacobian matrix, J, can be written:

{dot over (q)}=J{dot over (x)},  (13)

thus, J=J_q⁻¹J_x. Note that, in general, the Jacobian matrix maps output velocities (leg-joint velocities) to leg-elongation velocities. The output-velocity vector {dot over (x)} can be described by the velocity (v_P) of the centroid P and the angular velocity (ω_B) of the platform, thus:

$\begin{array}{cc}\stackrel{.}{x}=\left[\begin{array}{c}{v}_P\\ {\omega }_B\end{array}\right].& \left(14\right)\end{array}$

A loop-closure equation for each leg can be written as:

OP+ PB _i = OA _i+ A_iB_i  (15)

Differentiating this equation with respect to time yields:

v _P+ω_B×b_i=d_iω_i×s_i+d_is_i  (16)

where b_i denotes the vector {right arrow over (PB)}_i, s_i is the unit vector along {right arrow over (A_iB)}_i (i.e.,

$s_i=\frac{\overrightarrow{A_iB_i}}{\overrightarrow{A_iB_i}}$

), and ω_i is the angular velocity of the i^th leg with respect to the fixed reference frame A. To eliminate ω_i, both sides of equation (16) are dot-multiplied by s_i:

s _i ·v _p+(b_i×s_i)·ω_b={dot over (d)}_i  (17)

Rewriting equation (17) for each leg yields J_x(x_i)=J_q(q_i), where J_x=[s_i^T(b_i×s_i)^T] and J_q=I. The kinematic Jacobian then can be computed using the relation J=J_q⁻¹J_x.

Equation (17) can be assembled as equation (12) with the vector {dot over (x)} (equation (14)), where:

$\begin{array}{cc}{J}_x=\left[\begin{array}{cc}{s}_1^T& {\left(b_1\times s_1\right)}^T\\ \cdots & \cdots \\ s_6^T& {\left(b_6\times s_6\right)}^T\end{array}\right]& \left(18\right)\end{array}$

and

J _q =I(a 6×6 identity matrix)  (19)

Note again that J=J_q⁻¹J_x.

Based on the principle of virtual work, at equilibrium:

δW=τ^Tδq−F^Tδx=0  (20)

where F=[f, n]′ is the applied force (a vector) to the platform to move the platform, and τ=[τ₁, τ₂, . . . , τ₆]′ (or [f₁, f₂, . . . , f₆]′) represents the vector force applied by the actuated legs.

Equation (13) can be written as a virtual displacement, or kinematic Jacobian, relationship:

δq=Jδx  (21)

wherein δq is incremental leg movement, which for small changes can be expressed:

$\begin{array}{cc}\Delta \left[\begin{array}{c}{q}_1\\ q_2\\ q_3\\ q_4\\ q_5\\ q_6\end{array}\right]=J\Delta \left[\begin{array}{c}x\\ y\\ z\\ {\theta }_x\\ {\theta }_y\\ {\theta }_z\end{array}\right]& \left(22\right)\end{array}$

and δx is incremental displacement of the platform. From equations (20) and (21) can be obtained:

F=J^Tτ  (23)

where J, from the relationship J=J_q⁻¹J_x and from equation (18), is as follows:

$\begin{array}{cc}J=J_q^{-1}J_x=\left[\begin{array}{cc}{s}_1^T& {\left(b_1\times s_1\right)}^T\\ \cdots & \cdots \\ s_6^T& {\left(b_6\times s_6\right)}^T\end{array}\right]& \left(24\right)\end{array}$

where b_i denotes the vector {right arrow over (PB)}_i, and s_i is a unit vector as defined above.

Equation (23) becomes:

$\begin{array}{cc}\left[\begin{array}{c}f\\ n\end{array}\right]=\left[\begin{array}{cccc}{s}_1& s_2& \dots & s_6\\ b_1\times s_1& b_2\times s_2& \cdots & b_6\times s_6\end{array}\right]\left[\begin{array}{c}{f}_1\\ f_2\\ ⋮\\ f_6\end{array}\right]& \left(25\right)\end{array}$

where f represents translational forces, n denotes the moment torques, and f_i is the respective force generated by an actuated leg.

Equation (25) can also be obtained in the following manner. The force acting on the moving platform by each leg can also be written as:

f_i=f_iS_i,  (26)

for i=1, 2, . . . , 6, and S_i=d_i/d_i, defined as above. Summing all the forces acting on the moving platform,

$\begin{array}{cc}\sum _{i=1}^6f_iS_i=f& \left(27\right)\end{array}$

and summing the moments contributed by all forces about the centroid P of the moving platform yields:

$\begin{array}{cc}\sum _{i=1}^6f_ib_i\times S_i=n& \left(28\right)\end{array}$

Equation (25) is obtained by combining equations (26) and (27).

The vector force τ applied by actuated legs can be defined as:

τ=χΔq  (29)

where χ=diag [k₁, k₂, . . . , k₆], and k_i is the stiffness (such as a spring constant) of each leg. From equations (23), (26), and (21),

F=J^TχJΔx=KΔx  (30)

This is a Hooke's Law relationship in which K is a stiffness factor that, in this instance is a matrix (“stiffness matrix”) due to the multiple degrees of freedom of motion of the platform. The stiffness matrix is symmetric, positive semi-determinative, and configuration-dependent. The values of k_i desirably are equal to alleviate cross-coupling.

In equation (30), the Jacobian matrix is as expressed in equation (18), namely:

$J=\left[\begin{array}{cc}{s}_1^T& {\left(b_1\times s_1\right)}^T\\ \cdots & \cdots \\ s_6^T& {\left(b_6\times s_6\right)}^T\end{array}\right]$

(See also equation (24).) The unit stiffness can be obtained from:

K=J^TJ.  (31)

The height, h, cannot be zero. The expression for h is as set forth in equation (10). Based on coordinate expressions as set forth earlier above, the 6×6 Jacobian matrix can be computed as follows:

$J=\left[\begin{array}{cc}\frac{c}{L}& \frac{-\left(-b+\sqrt{a^2-c^2}\right)}{L}\\ \frac{-c}{L}& \frac{-\left(-b+\sqrt{a^2-c^2}\right)}{L}\\ \frac{-\left(-\sqrt{3}b+c+\sqrt{3}\sqrt{a^2-c^2}\right)}{2L}& \frac{-\left(b+\sqrt{3}c-\sqrt{a^2-c^2}\right)}{2L}\\ \frac{\sqrt{3}b+c-\sqrt{3}\sqrt{a^2-c^2}}{2L}& \frac{-b+\sqrt{3}c+\sqrt{a^2-c^2}}{2L}\\ \frac{-\left(\sqrt{3}b+c-\sqrt{3}\sqrt{a^2-c^2}\right)}{2L}& \frac{-b+\sqrt{3}c+\sqrt{a^2-c^2}}{2L}\\ \frac{-\sqrt{3}b+c+\sqrt{3}\sqrt{a^2-c^2}}{2L}& \frac{-\left(b+\sqrt{3}c-\sqrt{a^2-c^2}\right)}{2L}\end{array}\begin{array}{cc}\frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}\\ \frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}\\ \frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{-{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}\\ \frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{-{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}\\ \frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{-{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}\\ \frac{{\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{L}& \frac{-{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}\end{array}\begin{array}{cc}0& \frac{-\mathrm{bc}}{L}\\ 0& \frac{\mathrm{bc}}{L}\\ \frac{-\sqrt{3}{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}& \frac{-\mathrm{bc}}{L}\\ \frac{-\sqrt{3}{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}& \frac{\mathrm{bc}}{L}\\ \frac{\sqrt{3}{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}& \frac{-\mathrm{bc}}{L}\\ \frac{\sqrt{3}{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}{2L}& \frac{\mathrm{bc}}{L}\end{array}\right]$

The 6×6 stiffness matrix K is as follows:

$K=\hspace{1em}[\begin{array}{cc}\frac{-3\left(-b^2+2b\sqrt{a^2-c^2}-a^2\right)}{L^2}& 0\\ 0& \frac{-3\left(-b^2+2b\sqrt{a^2-c^2}-a^2\right)}{L^2}\\ 0& 0\\ 0& \frac{-3\left(-b+\sqrt{a^2-c^2}\right)}{L^2{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}\\ \frac{3\left(-b+\sqrt{a^2-c^2}\right)}{L^2{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}& 0\\ 0& 0\end{array}\hspace{1em}\begin{array}{cc}0& 0\\ 0& 0\\ \frac{6\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}{L^2}& \frac{-3\left(-b+\sqrt{a^2-c^2}\right)}{L^2{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}\\ 0& \frac{3b^2\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}{L^2}\\ 0& 0\\ 0& 0\end{array}\begin{array}{cc}\frac{3\left(-b+\sqrt{a^2-c^2}\right)}{L^2{b\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}^{1/2}}& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ \frac{3b^2\left(L^2-b^2+2b\sqrt{a^2-c^2}-a^2\right)}{L^2}& 0\\ 0& \frac{6b^2c^2}{L^2}\end{array}]$

In the stiffness matrix, the upper-left 3×3 submatrix represents the translational (x, y, z) stiffness, the lower-right 3×3 submatrix represents the torsional (θ_x, θ_y, θ_z) stiffness, and the other submatrices represent cross-coupling effects between forces and moments, and between rotations and translations, respectively.

By way of example, consider a configuration in which:

	Point
	x	y
	A1	−70	230.6
	A2	70	230.6
	A3	234.7	−54.7
	A4	164.7	−175.9
	A5	−164.7	−175.9
	A6	−234.7	−54.7
	B12	0	170
	B34	147.2	−85
	B56	−147.2	−85

a=[(230.6)²+(70)²]^1/2=240.99 mm

b=170.00 mm

c=70.00 mm

L=156.7 mm

$J=\left[\begin{array}{cccccc}0.4467& -0.3867& 0.8068& 137.1522& 0& -75.9413\\ -0.4467& -0.3867& 0.8068& 137.1522& 0& 75.9413\\ -0.5583& -0.1935& 0.8068& -68.5761& -118.7773& -75.9413\\ -0.1116& 0.5802& 0.8068& -68.5761& -118.7773& 75.9413\\ 0.1116& 0.5802& 0.8068& -68.5761& 118.7773& -75.9413\\ 0.5583& -0.1935& 0.8068& -68.5761& 118.7773& 75.9413\end{array}\right]$ $\mathrm{and}K=\left(1.0\times {10}^4\right)\left[\begin{array}{cccccc}0.0001& 0& 0& 0& 0.0159& 0\\ 0& 0.0001& 0& -0.0159& 0& 0\\ 0& 0& 0.0004& 0& 0& 0\\ 0& -0.0159& 0& 5.6432& 0& 0\\ 0.0159& 0& 0& 0& 5.6432& 0\\ 0& 0& 0& 0& 0& 3.4602\end{array}\right]$

As noted above in equation (10), h²=L²−b²+2b√{square root over (a²−c²)}−a². If we define M=−b+√{square root over (a²−c²)} and N=a²+b²−2b√{square root over (a²−c²)}, then the stiffness matrix above reduces to:

$K=\left[\begin{array}{cccccc}\frac{3N}{L^2}& 0& 0& 0& \frac{3\mathrm{Mbh}}{L^2}& 0\\ 0& \frac{3N}{L^2}& 0& \frac{-3\mathrm{Mbh}}{L^2}& 0& 0\\ 0& 0& \frac{6h^2}{L^2}& 0& 0& 0\\ 0& \frac{-3\mathrm{Mbh}}{L^2}& 0& \frac{3b^2h^2}{L}& 0& 0\\ \frac{3\mathrm{Mbh}}{L^2}& 0& 0& 0& \frac{3b^2h^2}{L^2}& 0\\ 0& 0& 0& 0& 0& \frac{3b^2c^2}{L^2}\end{array}\right]$

Again, the upper-left 3×3 submatrix represents the translational (x, y, z) stiffness, the lower-right 3×3 submatrix represents the torsional (θ_x, θ_y, θ_z) stiffness, and the other submatrices represent cross-coupling effects between forces and moments, and between rotations and translations, respectively. Note that the cross-coupling terms have the same magnitude, in this instance,

$\frac{3\mathrm{Mbh}}{L^2}.$

These terms become zero whenever M=0. In other words, there are no cross-coupling terms in the stiffness matrix if the following relationship is satisfied (see FIG. 3(B)):

a ² =b ² +c ²  (32)

Under such a situation, the stiffness matrix reduces further to:

$K=\left[\begin{array}{cccccc}\frac{3c^2}{L^2}& 0& 0& 0& 0& 0\\ 0& \frac{3c^2}{L^2}& 0& 0& 0& 0\\ 0& 0& \frac{6\left(L^2-c^2\right)}{L^2}& 0& 0& 0\\ 0& 0& 0& \frac{3b^2\left(L^2-c^2\right)}{L^2}& 0& 0\\ 0& 0& 0& 0& \frac{3b^2\left(L^2-c^2\right)}{L^2}& 0\\ 0& 0& 0& 0& 0& \frac{3b^2c^2}{L^2}\end{array}\right]$

which is representative of a configuration in which the plane of each pair of legs is perpendicular to the plane of the base. Note that, in this instance, all the cross-coupling terms have been reduced to zero.

From the foregoing formulation of the stiffness matrix K, a change in leg interval c (or the included angle θ) will affect axial stiffness. For example, increasing c will increase x, y, and θ_z stiffness while decreasing z, θ_x, and θ_y stiffness. Consequently, the leg interval c is the design parameter that determines the stiffness of the platform. For equal stiffness at the translational-axis condition (x, y, and z axes), the leg interval is:

$\begin{array}{cc}c=\sqrt{\frac{2}{3}}L& \left(33\right)\end{array}$

and the included angle (θ) (see FIG. 3(B)) of this embodiment is substantially 109.5°. It is noted that equal stiffness is not a requirement for achieving zero cross-coupling. Rather, the key to achieving zero cross-coupling, as stated earlier above, is M=0. It is also noted that the non-zero terms in the preceding matrix need not always be equal to the respective values listed.

In an exemplary mechanical and control system for achieving 6 DOF motion, two different coordinate-transformation matrices are used. For converting global-sensor location to top-mirror position, a geometric-coordinate transformation matrix T can be obtained by:

dŷ _x =T·dŷ _s  (34)

wherein the matrix T reflects a conversion of sensor-position data to mirror position. Also, the 6-DOF control output needs to be converted to each individual leg command. The derived inverse of the Jacobian matrix will suffice for this conversion. For small displacements, in an ideal case in which plant and sensor dynamics are ignored:

d{circumflex over (l)} _q =J·dŷ _x  (35)

A representative embodiment of the hexapod AMA mechanism is shown in FIGS. 4(A)-4(B). Turning first to FIG. 4(A), shown are a base 14 and a platform 12, and an optical element (e.g., a mirror) 32 mounted to the underside of the platform 12 by holds 34. The platform 12 is supported relative to the base by three pairs of legs L₁-L₂, L₃-L₄, and L₅-L₆, of which legs L₁ and L₂ largely visible in the figure. The legs L₁-L₆ are nominally all of identical length, but each leg comprises a respective actuator (discussed later below) that produces, when actuated, a desired amount of elongation of the respective leg. The discussion below concerning the legs L₁-L₂ is applicable to the other two pairs of legs L₃-L₄ and L₅-L₆.

A distal end of the leg L₁ is mounted to the base 14 by a block 30a, and a distal end of the leg L₂ is mounted to the base by a block 30b (also visible in FIG. 4(A) is the block 30c by which a distal end of the leg L₃ is mounted to the base). At or near the respective distal end of each leg L₁-L₂ is a respective distal flexure 36a, 36b mounted to the respective block 30a, 30b. The distal flexures 36a, 36b provide flexibility of the respective legs L₁, L₂ in the desired degrees of freedom (DOF) relative to the base 14 as required to accommodate extensions and retractions of the respective legs. For example, each distal flexure 36a, 36b is configured to provide a respective two DOF of motion (but not all six DOF) to the respective leg L₁, L₂ relative to the base 14.

The proximal end of each leg L₁-L₂ has a respective proximal flexure 38a, 38b (only the flexure 38b is visible). The proximal flexures 38a, 38b of the legs are connected to an apex block 40. The proximal flexures 38a, 38b provide flexibility of the respective legs L₁, L₂ in the desired DOF relative to the platform 12 as required to accommodate extensions and retractions of the respective legs. For example, each proximal flexure 38a, 38b is configured to provide at least two DOF of motion (but not all six DOF) to the respective leg L₁, L₂ relative to the platform 12. The proximal flexures 38a, 38b work in coordination with the distal flexures 36a, 36b in this regard.

FIG. 4(A) shows particularly the legs L₁ and L₂ of this embodiment. Note the splayed arrangement of the legs L₁, L₂, giving the pair an inverted “V” configuration in which the apex of the V corresponds to the proximal ends conjoined at the apex block 40. The other two pairs of legs L₃, L₄ and L₅, L₆ are similarly configured. A plan view of the embodiment is shown in FIG. 4(B), showing all the legs L₁-L₆ and blocks 30a-30f. The respective apices of the three pairs of legs are situated equidistantly on a circle on the platform 12 (see FIG. 3(A), showing the circle 16 on which the points B₁₂, B₃₄, B₅₆, corresponding to respective apices, are located). Note also that the distal ends of the legs (at the respective blocks 30a-30f) are situated on a circle on the base 14 (see FIG. 1, showing the circle 18 on which the points A₁-A₆, corresponding to respective distal ends, are located).

Each leg L₁-L₆ includes a respective length monitor 42a-42f that measures and monitors the length (including changes in length) of the respective leg. The leg-length monitors 42a-42f can be highly accurate encoders utilizing, for example, laser scales. The leg-length monitors 42a-42f are especially advantageous when used in feedback-control systems for controlling extension and retraction of the legs, including in real time.

To effect changes in their length, each leg L₁-L₆ comprises a respective piezoelectric (e.g., PZT-based piezo-ceramic) actuator 44a-44f. In FIG. 5 only one piezoelectric actuator 44a can be seen. As previously noted, all the legs L₁-L₆ have nominally the same length. The piezoelectric actuators 44a-44f impart very small changes in length to the respective legs L₁-L₆ as required to perform a fine positional adjustment of the platform 12 (and optical element mounted on it) relative to the base 14. For example, in this particular embodiment, the range of leg-length change achievable by the actuators is tens of micrometers. Leg extension (increasing the length of the leg) is achieved by energizing the respective actuator sufficiently to cause a desired increase in length of the actuator. Leg retraction (decreasing the length of the leg) is achieved by reducing the degree of energization of the respective actuator sufficiently to cause a desired reduction in length of the actuator.

For measuring height (and changes in height) of the platform 12 relative to the base 14 (or other fixed reference) accompanying a particular extension or retraction of one or more of the legs, each pair of legs L₁-L₂, L₃-L₄, L₅-L₆ has an associated height monitor 46a, 46b, 46c mounted to the base 14. The height monitors 46a-46c measure height along a respective line that is perpendicular to the base and that passes through the apex of the respective pair of legs. Additional positional monitoring is performed by monitors 48a, 48b, 48c situated midway between pairs of legs and mounted to the base 14. The height monitors 46a-46c and the position monitors 48a-48c can be highly accurate encoders or interferometers that measure displacement relative to a stationary frame of reference such as a lens barrel holding the optical elements of the system. If an encoder is used, it can be based upon reflection or transmission of light.

Whereas the base 14 can be used as a positional reference for the monitors, use of the base as a reference may not be practical in certain instances. For example, movements of the optical element 32 relative to the base 14 can generate vibrations in the base, thereby creating a moving reference.

Turning now to FIG. 5, the legs L₁ and L₂ are shown, including the blocks 30a, 30b; the distal flexures 36a, 36b, the proximal flexures 38a, 38b, and the apex block 40. Note that each leg L₁, L₂ has a respective leg axis Ax₁, Ax₂, and that the angle θ between the leg axes is as noted in FIG. 3(B). The axes Ax₁, Ax₂ collectively define a “leg plane” that, when the legs L₁, L₂ are mounted between the base 14 and platform 12, is perpendicular to the plane of the base 14. This perpendicularity of the leg planes is apparent in the view of FIG. 4(B), which depicts the leg plane P₁ for the legs L₁, L₂, the leg plane P₂ for the legs L₃, L₄, and the leg plane P₃ for the legs L₅, L₆. The existence of these leg planes P₁-P₃ relies upon satisfaction of the relationship a²=b²+c², as noted in equation (32), wherein a, b, and c are as shown in FIG. 3(A).

Returning to FIG. 5, and using the leg L₁ as an example (and referring to components associated with the leg L₁ as exemplary of corresponding components on the other legs), each leg has a respective piezoelectric actuator 44a, which exhibits extension when the actuator is electrically energized. The piezoelectric actuator 44a includes a fine-motion actuator 50a and a coarse-motion actuator 50b. In this embodiment the fine-motion actuator 50a comprises one piezoelectric element, and the coarse-motion actuator 50b comprises multiple piezoelectric elements arranged in tandem along the leg axis Ax₁. The fine-motion actuator 50a and coarse-motion actuator 50b also are arranged in tandem along the leg axis Ax₁. Each of the actuators 50a, 50b is separately energized by a respective driver (not shown). The piezoelectric actuator 44a is situated in a yoke 52 that allows the piezoelectric actuator 44a to apply an extension force (one DOF of motion) strictly along the longitudinal axis A₁ of the respective leg L₁. Compliance along the axes A₁, A₂ of the legs L₁, L₂ to accommodate leg extensions is provided by respective flexures 56a, 56b.

By way of example, the coarse-motion actuator 50b is configured to provide an accuracy of actuation performance in the micrometer range, and the fine-motion actuator 50a is configured to provide an accuracy of actuation performance in the nanometer range. For some applications, one of the portions (the fine-motion actuator 50a) of the piezoelectric actuator can be omitted if the particular application does not require it.

The optical system with which a hexapod as described above can be associated can be any of various reflective, catadioptric, refractive, and other types of optical systems including combinations of these specific systems. In general, the optical system can be any such system that is used under conditions requiring adjustability in the nm range as well as 6 DOF of movement. An example system is shown in FIG. 6, in which the system includes six mirrors PM1-PM6, as exemplary optical elements of the system, all mounted to a “frame” F (e.g., an optical “barrel” or “column”). The depicted system is particularly suitable for use as a projection-optical system for performing EUV microlithography. In the depicted system the second mirror PM2 is mounted on a hexapod mounting as described above.

Whereas optical-element mountings have been described above in the context of representative embodiments, it will be understood that the subject mountings are not limited to those representative embodiments. On the contrary, the subject optical-element mountings are intended to encompass all modifications, alternatives, and equivalents as may be included within the spirit and scope of the following claims.

Claims

1. A hexapod kinematic mounting, comprising: a base defining a base plane;a platform situated relative to the base and movable relative to the base; andsix legs each having nominally identical length and a respective leg axis, the legs having substantially equal stiffness and being arranged in three pairs of legs extending between the base and platform and supporting the platform relative to the base, each pair of legs having first and second ends, the first ends of each pair being coupled together in a Λ-shaped manner forming a respective apex and the respective second ends being splayed relative to the apex, the apices being situated substantially equidistantly from each other at respective locations on a circle on the platform, and the respective second ends of the pairs of legs being mounted at respective locations on a circle on the base such that the respective axes of each pair of legs define a respective leg plane that is substantially perpendicular to the base plane, each leg comprising an actuator serving, when energized, to change a length of the respective leg such that a coordinated energization of the respective actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.

2. The mounting of claim 1, wherein the respective legs of each pair form an angle of substantially 109.5° at the respective apex.

3. The mounting of claim 1, wherein the leg actuators are respective piezoelectric actuators.

4. The mounting of claim 3, wherein each piezoelectric actuator comprises a respective coarse actuator and a respective fine actuator.

5. The mounting of claim 4, wherein the coarse actuator and the fine actuator are arranged in tandem along the respective leg axis.

6. The mounting of claim 1, wherein each leg further comprises a respective leg-extension flexure situated relative to the leg actuator to provide at least one, but not all six, degrees of freedom of motion accompanying leg extension and retraction caused by the respective leg actuator.

7. The mounting of claim 1, wherein each leg further comprises a respective leg-length monitor.

8. The mounting of claim 1, wherein: the first ends of each pair of legs comprise respective flexures providing the respective end with at least two, but not all six, degrees of freedom of motion; andthe second ends of each pair of legs comprise respective flexures providing the respective end with at least two, but not all six, degrees of freedom of motion.

9. The mounting of claim 1, further comprising at least one height monitor situated and configured to measure and monitor position of the platform relative to a fixed reference.

10. The mounting of claim 9, wherein the fixed reference is the base.

11. A kinematically mounted optical element, comprising: an optical element;a base defining a base plane;a platform movable relative to the base;at least one hold affixing the optical element to the platform; anda hexapod situated between the base and the platform so as to support the platform relative to the base, the hexapod comprising six legs each having a respective leg axis, the legs having nominally identical length and substantially equal stiffness and being arranged in three pairs each having first and second ends, the first ends of each pair being coupled together in a Λ-shaped manner forming a respective apex and the respective second ends being splayed relative to the apex, the apices being situated substantially equidistantly from each other at respective locations on a circle on the platform, and the respective second ends of the pairs of legs being mounted at respective locations on a circle on the base such that the respective axes of each pair of legs define a respective leg plane that is substantially perpendicular to the base plane, each leg comprising an actuator serving, when energized, to change a length of the respective leg such that a coordinated energization of the respective actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.

12. The optical element of claim 11, wherein the optical element is a mirror.

13. The optical element of claim 11, wherein the respective legs of each pair form an angle of substantially 109.5° at the apex.

14. An optical system, comprising: a frame;a base mounted to the frame and defining a base plane;a platform movable relative to the base;an optical element mounted to the platform; anda hexapod situated between the base and the platform so as to support the platform relative to the base, the hexapod comprising six legs each having a respective leg axis, the legs having nominally identical length and substantially equal stiffness and being arranged in three pairs each having first and second ends, the first ends of each pair being coupled together in a Λ-shaped manner forming a respective apex and the respective second ends being splayed relative to the apex, the apices being mounted equidistantly from each other at respective locations on a circle on the platform, and the respective second ends of the pairs of legs being mounted at respective locations on a circle on the base such that the respective axes of each pair of legs define a respective leg plane that is substantially perpendicular to the base plane, each leg comprising an actuator serving, when energized, to change a length of the respective leg such that a coordinated energization of the respective actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.

15. The optical system of claim 14, wherein the respective legs of each pair form an angle of substantially 109.5° at the apex.

16. The optical system of claim 14, wherein the leg actuators are respective piezoelectric actuators.

17. The optical system of claim 16, wherein each piezoelectric actuator comprises a respective coarse actuator and a respective fine actuator.

18. The optical system of claim 14, wherein each leg further comprises a respective leg-extension flexure situated relative to the leg actuator to provide at least one, but not all six, DOF of motion accompanying leg extension and retraction caused by the respective leg actuator.

19. The optical system of claim 14, wherein each leg further comprises a respective leg-length monitor.

20. The optical system of claim 14, wherein: the first ends of each pair of legs comprise respective flexures providing the respective end with at least two, but not all six, DOF of motion; andthe second ends of each pair of legs comprise respective flexures providing the respective end with at least two, but not all six, DOF of motion.

21. The optical system of claim 14, further comprising at least one monitor situated and configured to measure and monitor position of the optical element relative to a fixed reference.

22. The optical system of claim 14, wherein the optical element is a reflective optical element.

23. The optical system of claim 14, wherein the optical system is an EUVL optical system.

24. The optical system of claim 23, wherein the optical system is an EUVL projection-optical system.

25. The optical system of claim 14, wherein: the optical system comprises multiple optical elements; andat least one optical element is mounted to the frame by a respective base, platform, and hexapod.

26. A kinematically mounted optical element, comprising: an optical element;a base defining a base plane;a platform movable relative to the base;at least one hold affixing the optical element to the platform; anda hexapod situated between the base and the platform so as to support the platform relative to the base, the hexapod comprising six legs each having a respective leg axis, the legs having nominally identical length and being arranged in three pairs each having first and second ends, the first ends of each pair being coupled together in a Λ-shaped manner forming a respective apex and the respective second ends being splayed relative to the apex, the apices being mounted equidistantly from each other at respective locations on a circle on the platform, and the respective second ends of the pairs of legs being mounted at respective locations on a circle on the base such that the respective axes of each pair of legs define a respective leg plane that is substantially perpendicular to the base plane, each leg comprising an actuator serving, when energized, to change a length of the respective leg such that a coordinated energization of the respective actuators in selected legs produces a desired movement of the platform, with substantially no cross-coupling, relative to the base in all six degrees of freedom of motion.

27. The optical system of claim 26, wherein the legs have substantially equal stiffness.

28. The optical system of claim 26, wherein the respective legs of each pair form an angle of substantially 109.5° at the apex.

29. An optical system, comprising an optical element as recited in claim 26.

30. A hexapod kinematic mounting, comprising: a base defining a base plane;a platform situated relative to the base and movable relative to the base; andsix legs each having nominally identical length and a respective leg axis, the legs being arranged in three pairs of legs extending between the base and platform and supporting the platform relative to the base, each pair of legs having first and second ends, the first ends of each pair being coupled together in a Λ-shaped manner forming a respective apex and the respective second ends being splayed relative to the apex such that the respective legs of the pair form an angle of substantially 109.5° at the apex, the apices being mounted equidistantly from each other at respective locations on a circle on the platform, and the respective second ends of the pairs of legs being mounted at respective locations on a circle on the base such that the respective axes of each pair of legs define a respective leg plane that is substantially perpendicular to the base plane, each leg comprising an actuator serving, when energized, to change a length of the respective leg such that a coordinated energization of the respective actuators in selected legs produces a desired movement of the platform relative to the base in all six degrees of freedom of motion.