US20070284502A1

US20070284502A1 - Hexapod kinematic mountings for optical elements, and optical systems comprising same

Info

Publication number: US20070284502A1
Application number: US11/644,467
Authority: US
Inventors: Yi-Ping Hsin; Hideyuki Hashimoto; Jin Nishikawa; Scott Coakley; Kunitomo Fukai; Wen-Hou Ma; Bausan Yuan
Original assignee: Nikon Corp
Current assignee: Nikon Corp
Priority date: 2006-06-13
Filing date: 2006-12-21
Publication date: 2007-12-13

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

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No. 60/813,481, filed Jun. 13, 2006, incorporated herein by reference in its entirety.

FIELD

This disclosure pertains to, inter alia, adjustable mounting structures for optical elements, to assemblies of optical elements comprising at least one optical element mounted on an adjustable mounting structure, and to optical systems comprising one or more such assemblies. More specifically, the mounting structure includes a “Stewart Platform,” which is a platform mounted on six legs. The mounting structures and assemblies are especially advantageous when used for mounting mirrors in a manner providing active mirror adjustment in multiple degrees of freedom of motion. An advantageous application of such an assembly is in a projection-optical system as used for performing microlithography, such as extreme ultraviolet (EUV) lithography (EUVL).

BACKGROUND

Imaging technology in many fields has advanced greatly in recent years, aided especially by substantial advancements in the art and science of optical design and by the debut of “adaptive optics,” in which real-time changes can be deliberately made to an optical property of an individual optical element. Consequently, various optical imaging systems have become available that exhibit performance levels considered impossible only a short time ago. One field in which such advancements have been achieved is astronomy, in which sophisticated telescopes have recently been developed that produce exceptionally good resolution of astronomical objects. Another field is terrestrial imaging from space, in which sophisticated and powerful imaging satellites provide extremely well-resolved images of the surface of the earth or other celestial body. Yet another field is microlithography, a technology used for imaging microcircuit patterns and the like on the surfaces of silicon wafers or other substrates.
Since the beginning of microlithography, so-called “optical microlithography” (projection lithography performed using deep ultraviolet light transmitted through optical systems that are at least partially refractive) has been the workhorse imaging technology for forming microcircuits on silicon wafers and other substrates. As critical dimensions of microcircuits have become increasingly miniaturized (currently less than 100 nm), accompanied by prodigious increases in the density of active circuit elements being formed in microcircuits, the resolution limitations of optical microlithography have become apparent. This has fueled an urgent need for a practical “next-generation” lithography (NGL) capable of resolving smaller pattern features than can be imaged using optical microlithography.
Extreme-ultraviolet lithography (EUVL) is currently regarded as a viable candidate next-generation lithography offering good prospects of substantially finer pattern resolution than currently obtainable using conventional optical lithography. The expectations of increased resolution from EUVL stem largely from the fact that, whereas current optical lithography is performed using an imaging wavelength in the range of 150-250 nm, EUVL is performed using an imaging wavelength in the range of 11-5 nm, which is at least ten times shorter than the shortest conventional “optical” wavelengths. Generally, the shorter the wavelength of light being used for pattern imaging in microlithography, the finer the obtainable resolution.
In view of the extremely small pattern elements (critical dimensions of less than 70 nm) that can be resolved using EUVL, the accuracy and precision with which pattern imaging is performed using this technology must be extremely high to ensure proper resolution, placement, and registration of multiple pattern layers on a substrate and to ensure that the pattern elements are transferred to the substrate with high fidelity. To obtain such high accuracy and precision, extreme efforts are being expended in the design and configuration of optical systems used in EUVL systems.
EUVL differs substantially from optical microlithography in that no known materials are sufficiently transmissive and refractive to EUV light to be useful for making EUV lenses. Consequently, EUVL optical systems consist of reflective optical elements, namely EUV-reflective mirrors. In one type of conventional EUVL system currently under development, a reflective “illumination-optical system” (comprising multiple mirrors) is used for illuminating a pattern-defining reticle with a beam of EUV light, and a reflective “projection-optical system” (also comprising multiple mirrors) is used for projecting an imaging beam from the reticle to the wafer or other substrate. Most, if not all, of the mirrors in the illumination-optical system and projection-optical system are “multilayer-film” mirrors, which are the only known types of mirrors (besides grazing-incidence mirrors) that exhibit useful levels of reflectivity to incident EUV light.
A conventional EUVL system 100 is shown schematically in FIG. 7. The depicted system 100 comprises a laser-plasma EUV source ES, an illumination-optical system IL comprising five mirrors IM₁-IM₅, and a projection-optical system PL comprising six mirrors PM₁-PM₆. The source ES comprises a high-power pulsed laser 111 and a convex lens 112 that converges the beam produced by the laser 111 to a point 113. The source ES also includes a conduit 114 for a suitable target material that is discharged at the point 113 where the incident laser beam converts the target material to a plasma. The plasma generates various wavelengths of light, including EUV light, that is “collected” by an elliptical mirror 115 and delivered convergently to the illumination-optical system IL. A vacuum conduit 116 removes debris produced by the plasma.
The illumination-optical system IL is situated between the source ES and a pattern-defining reflective reticle M, and the projection-optical system PL is situated optically between the reticle and a wafer W or other substrate onto which the projection-optical system projects the pattern. Although the depicted projection-optical system PL comprises six multilayer-film mirrors PM₁-PM₆, other configurations for EUVL projection-optical systems have four mirrors or as few as two mirrors. The mirrors are arranged so as to conserve space between the reticle M and wafer W, with the reticle and wafer being located on opposing ends of the projection-optical system PL to allow sufficient room for a reticle stage RS and a wafer stage WS and to satisfy other practical considerations.
To achieve high imaging resolution in EUVL, extreme demands are imposed on the configuration and operation of the optical systems, especially the projection-optical system PL. For example, the reflective surfaces of the mirrors (e.g., PM₁-PM₆) must have extremely low figure errors (e.g., <0.25 nm), the surficial multilayer films must be formed with extremely high accuracy and precision, and the mirrors must be mounted with extremely high positional accuracy and stability in a rigid frame or “barrel” that does not deform and that isolates the mirrors from external vibrations. Experience has shown that even these measures do not produce optimal optical performance, especially over time and/or from one optical system to the next. For example, significant variations in imaging performance can exist from one exposure die to the next on the surface of the wafer. In addition, the mirrors and their supporting structures can experience thermal deformations, vibrations, and other stresses that change with time. Also, even though movements of the reticle stage RS and wafer stage WS are synchronized and independently controlled to achieve very accurate positioning of these components from die to die of exposure, these controls are insufficient for achieving the level of optical performance currently being demanded from EUVL systems.
To improve optical performance of EUVL systems further, adaptive-optical schemes such as active mirror adjustment (AMA) schemes are being considered. In one type of AMA scheme one or more of the mirrors (especially of the projection-optical system PL) are mounted in a manner allowing positional adjustments of the respective mirror(s) to be made during actual use of the optical system. Certain projection-optical systems currently in use employ, for at least one mirror of the system, a mounting that provides three degrees of freedom (DOF) of motion. Whereas such adjustability provides some benefits, it falls short of the ideal full six DOF (x, y, z, θ_x, θ_y, θ_z) of adjustability.
A type of kinematic mounting that has found some utility, particularly in certain types of astronomical telescopes (notably having a catadioptric configuration), is the so-called “hexapod” mounting conventionally known as a “Stewart Platform” or “Stewart-Gough Platform.” Stewart platforms also are being used for mounting mirrors and other optical elements used in space-borne optical systems. As used herein, “hexapod” refers to a mounting structure by which a platform is supported relative to a base by six legs, in the general manner of a Stewart Platform or Stewart-Gough Platform, but does not encompass various six-legged, self-propelled robotic devices (also referred to in the literature as “hexapods”) that walk about or otherwise perambulate using their legs in a walking-insect manner.
A conventional hexapod mount (Stewart Platform) 120 is depicted in FIG. 8. A mirror 122 (as an exemplary payload) is mounted to a platform 124 by multiple holds 126. The platform 124 is supported relative to a stationary base 128 by three pairs of legs 130. The legs 130 of each pair have upper ends 132 that are mounted via respective spherical bearings 134 to the underside of the platform 124 where the upper ends of the legs converge. The spherical bearings 134 allow changes in the angles of the legs 130 relative to the platform 124. The spherical bearings 134 are equally spaced from one another on the underside of the platform 124. From the platform 124 to the base 128 the legs 130 of each pair are divergent (splayed), and the bottom ends 138 of the legs are attached to the base by respective spherical bearings 140 that allow changes in the angles of the legs relative to the base. Each leg 130 includes a respective actuator (not detailed) that is operable to change the length of the leg. Each actuator is independently operable, and the actuators typically are operated in a coordinated manner to achieve motions of the platform 124 (and hence of the mirror 122) relative to the base 128 in all six DOF. Thus, whereas each leg 130 when actuated exhibits only one DOF of translational motion (by increasing or decreasing in length), coordinated actuations of the legs provide the full six DOF of movement of the platform 124 and mirror 122 relative to the base 128.
Various types of leg actuators are used in conventional Stewart platforms. One type of actuator is a hydraulic cylinder. Another type is a motor connected to a rotatable screw threaded into a bearing such as a ball-screw. Yet other types are linear servo motors and voice-coil motors.
Whereas a conventional hexapod mount as summarized above is satisfactory for certain applications (such as conventional astronomical telescopes and other systems requiring accuracy and precision in the micrometer range), it is unsatisfactory for use in EUVL systems and other optical systems requiring accuracy and precision in the nanometer range (i.e., three or more orders of magnitude greater accuracy and precision than in conventional systems). In other words, conventional hexapod mounts have certain limitations that have prevented them from being used satisfactorily in EUVL systems and other demanding applications.
One limitation of many types of conventional hexapods is their use of spherical bearings for coupling the upper and lower ends of the legs to the platform and base, respectively. Spherical bearings inherently exhibit friction and backlash that usually do not cause significant problems whenever the required accuracy and precision of the hexapod is in the micrometer range, but are intolerable whenever the required accuracy and precision are in the nanometer range or less. Spherical bearings also tend to transmit vibrations, to and from the platform (and optical element supported thereby), that degrade optical performance.
Another limitation of many types of conventional hexapods is the particular types of actuators used for increasing or decreasing the lengths of the legs. Conventional hexapods that support large masses (such as large telescope mirrors) use hydraulic cylinders or linear motors for actuating the legs. Smaller conventional hexapods generally use micromotors, servomotors, or voice-coil motors. These types of actuators are too imprecise for use in hexapods for EUVL systems and other systems demanding precision in the nanometer range.
Yet another limitation of conventional hexapods is their tendency to exhibit “cross-coupling.” Cross-coupling is an unintended change in one or more movement parameters (x, y, z, θ_x, θ_y, θ_z) accompanying a change in another movement parameter. One example is an unwanted y-direction and/or z-direction shift accompanying actuation of the hexapod to achieve a change in x-direction position. Another example is an unwanted change in θ_x, θ_y, and/or θ_zaccompanying actuation of the hexapod to achieve a change in x-, y-, and/or z-direction position. Whereas significant cross-coupling can be offset using feedback control of the platform position, achieving the offsetting actuations of the legs requires massive control software and consumes time that would substantially decrease throughput of an EUVL system.
Therefore, there is a need for improved hexapod mountings, useful for mounting mirrors and other types of optical elements for active adjustment, that provide an operational precision in the nanometer range and that exhibit substantially no cross-coupling errors.

SUMMARY

The needs articulated above, as well as other needs, are satisfied by hexapod mountings, and optical systems comprising same, as disclosed herein.
A first aspect is directed to “hexapod” mountings. These mountings are generally configured in the manner generally known as “Stewart” or “Stewart-Gough” platforms, but have certain distinctive features compared to conventional platforms. An embodiment of such a mounting comprises a base, a platform, and six legs. The base defines a base plane. The platform is situated relative to the base and is movable relative to the base. The legs each have nominally identical length and a respective leg axis. The legs have substantially equal stiffness and are arranged in three pairs that extend between the base and platform and that support the platform relative to the base. Each pair of legs has first and second ends. The first ends of each pair are coupled together in a Λ-shaped manner forming a respective apex. The respective second ends are splayed relative to the apex The apices are mounted equidistantly from each other at respective locations 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 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 comprises 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 (namely, x, y, z, θ_x, θ_y, θ_z).
Desirably, the respective legs of each pair form an angle of substantially 109.5° at the apex.
The leg actuators desirably are respective piezoelectric actuators, but alternatively can be any of various other types of actuators such as voice-coil motor, pneumatic, linear motor, etc., or combinations of these. Each actuator desirably comprises a respective coarse actuator and a respective fine actuator, a configuration that is especially amenable to piezoelectric actuators, and can be used with other types of actuators. The coarse actuator and the fine actuator desirably are arranged in tandem along the respective leg axis, or in any other suitable arrangement.
In one embodiment each leg comprises at least one respective leg-extension flexure(s) 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. Each leg also desirably comprises a respective leg-length monitor. The leg-length monitors are useful in, for example, a feed-back system for ensuring that the platform moves in a desired manner and amount accompanying extension or retraction of one or more of the legs.
In another embodiment 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. Also, the 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. Each leg also desirably comprises a respective leg-length monitor.
Any of the embodiments summarized above can further comprise at least one height monitor that is situated and configured to measure and monitor position (height) of the platform relative to a fixed reference. The fixed reference can be the base. The height monitor is useful in, for example, a feed-back system for ensuring that the platform moves in a desired manner and amount accompanying extension or retraction of one or more of the legs.
According to another aspect, kinematically mounted optical elements are provided. An embodiment of the same comprises an optical element, a base, a platform, at least one hold affixing the optical element to the platform, and a hexapod situated between the base and the platform. The base defines a base plane, and the platform is movable relative to the base. The hexapod supports the platform relative to the base and comprises six legs having nominally identical length. Each leg has a respective leg axis. The legs have substantially equal stiffness and are arranged in three pairs each having first and second ends. The first ends of each pair are coupled together in a Λ-shaped manner forming a respective apex. (Desirably, the respective legs of each pair form an angle of substantially 109.5° at the apex.) The respective second ends are splayed relative to the apex. The apices are mounted equidistantly from each other at respective locations 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 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 comprises an actuator that serves, 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.
The optical element can be any of various types and configurations, such as (but not limited to) a mirror or mirror group, a lens or lens group, a filter or filter group, or other type of optical element(s), or combinations of the same.
According to another aspect, optical systems are provided. An embodiment of such a system comprises a frame, a base, a platform, at least one optical element mounted to the platform, and a hexapod situated between the base and the platform. The base is mounted to the frame and defines a base plane, and the platform is movable relative to the base. The at least one optical element is mounted to the platform. The hexapod is situated between the base and the platform so as to support the platform relative to the base. The hexapod comprises six legs having nominally identical length and substantially equal stiffness. Each leg has a respective leg axis, and the legs are arranged in three pairs each having first and second ends. The first ends of each pair are coupled together in a Λ-shaped manner forming a respective apex, and the respective second ends are splayed relative to the apex. Desirably, the respective legs of each pair form an angle of substantially 109.5° at the apex. The apices are mounted equidistantly from each other at respective locations 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 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 comprises an actuator that serves, 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.
The leg actuators desirably are respective piezoelectric actuators, but alternatively can be any of various other types of piezoelectric actuators, as noted above. Each piezoelectric actuator can comprise a respective coarse actuator and a respective fine actuator, as note above.
Each leg can further comprise at least one respective leg-extension flexure(s) 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. As noted above, each leg can further comprise a respective leg-length monitor.
In another embodiment the first ends of each pair of legs comprise respective flexures that provide the respective end with at least two, but not all six, DOF of motion. Similarly, the 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.
Various embodiments of an optical system as summarized above can further comprise at least one monitor situated and configured to measure and monitor position of the optical element relative to a fixed reference.
The optical element can be a reflective optical element or any of various other types of optical elements, as noted above. A particularly advantageous optical system incorporating reflective optical elements is an EUVL optical system, such as (but not limited to) an EUVL projection-optical system.
The optical system can be one that comprises multiple optical elements. In such a system, at least one optical element can be mounted to the frame by a respective base, platform, and hexapod.
Yet another aspect is directed to kinematically mounted optical elements. An embodiment of the same comprises an optical element, a base, a platform, at least one hold affixing the optical element to the platform, and a hexapod. The base defines a base plane, and the platform is movable (via the hexapod) relative to the base. The hexapod is situated between the base and the platform so as to support the platform relative to the base. The hexapod comprises six legs having nominally identical length. Each leg has a respective leg axis, and the legs are arranged in three pairs each having first and second ends. The first ends of each pair are coupled together in a Λ-shaped manner forming a respective apex, and the respective second ends are splayed relative to the apex. The apices are mounted equidistantly from each other at respective locations 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 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 comprises an actuator that serves, 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.
Desirably, the legs have substantially equal thickness. Further desirably, the legs of each pair form an angle of substantially 109.5° at the apex.
According to another aspect, an optical system is provided that comprises an optical element such as that summarized above in the preceding paragraph.
The foregoing and additional features and advantages of the invention will be more readily apparent from the following detailed description, which proceeds with reference to the accompanying drawings.

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-L₆arranged into three pairs L₁and L₂, L₃and L₄, and L₅and 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-L₆support 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-L₆converge at a respective apex attached at a respective location B₁₂, B₃₄, B₅₆on 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 A₁and A₂, A₃and A₄, A₅and A₆. The locations B₁₂, B₃₄, B₅₆are 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-A₆are 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-L₆nominally 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_Aof the base and center O_Bof 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-L₆causes 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_iis any of A₁-A₆,
B_iis any of B₁-B₆,
B_ijis any of B₁₂, B₃₄, B₅₆,
a is the radius from O_Ato any point A_ion the base: a= O_AA_i ,
b is the radius from O_Bto any point B_ion 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_iand B_ijhave respective coordinates (e.g., x, y, z), as follows:
A ₁ =[−c,√{square root over (a ² −c ²)},0] (1)
A ₂ =[c,√{square root over (a ² −c ²)},0] (2)
$\begin{matrix} A_{3} = [\frac{c}{2} + \frac{\sqrt{3}}{2} \sqrt{a^{2} - c^{2}}, \frac{\sqrt{3}}{2} c - \frac{1}{2} \sqrt{a^{2} - c^{2}}, 0] & (3) \\ A_{4} = [\frac{- c}{2} + \frac{\sqrt{3}}{2} \sqrt{a^{2} - c^{2}}, \frac{- \sqrt{3}}{2} c - \frac{1}{2} \sqrt{a^{2} - c^{2}}, 0] & (4) \\ A_{5} = [\frac{c}{2} - \frac{\sqrt{3}}{2} \sqrt{a^{2} - c^{2}}, \frac{- \sqrt{3}}{2} c - \frac{1}{2} \sqrt{a^{2} - c^{2}}, 0] & (5) \\ A_{6} = [\frac{- c}{2} - \frac{\sqrt{3}}{2} \sqrt{a^{2} - c^{2}}, \frac{\sqrt{3}}{2} c - \frac{1}{2} \sqrt{a^{2} - c^{2}}, 0] & (6) \end{matrix}$
B₁₂=[o,b,h] (7)
$\begin{matrix} B_{34} = [\frac{- \sqrt{3}}{2} b, \frac{- 1}{2} b, h] & (8) \\ B_{56} = [\frac{- \sqrt{3}}{2} b, \frac{- 1}{2} b, h] & (9) \end{matrix}$

To determine h, consider that L= A₁B₁₂ . If A₁=[−c, √{square root over (a²−c²)}, 0] and B₁₂=[0, b, h], then L²= A₁B₁₂ ²=c²+(b−√{square root over (a²−c²)})²+h². Solving for h²yields:

h ² =L ² −b ²+2b√{square root over (a ² −c ²)}−a ². (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 A₁or A₂)

The vector d_iis 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:
$\begin{matrix} \dot{x} = [\begin{matrix} v_{P} \\ ω_{B} \end{matrix}] . & (14) \end{matrix}$
A loop-closure equation for each leg can be written as:
OP+ PB _i = OA _i+ A _i B _i (15)
Differentiating this equation with respect to time yields:
v _P+ω_B ×b _i =d _iω_i ×s _i +d _i s _i (16)
where b_idenotes the vector {right arrow over (PB)}_i, s_iis the unit vector along {right arrow over (A_iB)}_i(i.e.,
$s_{i} = \frac{\vec{A_{i} B_{i}}}{\langle \vec{A_{i} B_{i}} \rangle}$
), and ω_iis the angular velocity of the i^thleg 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:
$\begin{matrix} J_{x} = [\begin{matrix} s_{1}^{T} & {(b_{1} \times s_{1})}^{T} \\ \dots & \dots \\ s_{6}^{T} & {(b_{6} \times s_{6})}^{T} \end{matrix}] & (18) \end{matrix}$
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 ^T δq−F ^T δx=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:
$\begin{matrix} Δ [\begin{matrix} q_{1} \\ q_{2} \\ q_{3} \\ q_{4} \\ q_{5} \\ q_{6} \end{matrix}] = J Δ [\begin{matrix} x \\ y \\ z \\ θ_{x} \\ θ_{y} \\ θ_{z} \end{matrix}] & (22) \end{matrix}$
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_xand from equation (18), is as follows:
$\begin{matrix} J = J_{q}^{- 1} J_{x} = [\begin{matrix} s_{1}^{T} & {(b_{1} \times s_{1})}^{T} \\ \dots & \dots \\ s_{6}^{T} & {(b_{6} \times s_{6})}^{T} \end{matrix}] & (24) \end{matrix}$
where b_idenotes the vector {right arrow over (PB)}_i, and s_iis a unit vector as defined above.
Equation (23) becomes:
$\begin{matrix} [\begin{matrix} f \\ n \end{matrix}] = [\begin{matrix} s_{1} & s_{2} & \dots & s_{6} \\ b_{1} \times s_{1} & b_{2} \times s_{2} & \dots & b_{6} \times s_{6} \end{matrix}] [\begin{matrix} f_{1} \\ f_{2} \\ ⋮ \\ f_{6} \end{matrix}] & (25) \end{matrix}$
where f represents translational forces, n denotes the moment torques, and f_iis 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,
$\begin{matrix} \sum_{i = 1}^{6} f_{i} S_{i} = f & (27) \end{matrix}$
and summing the moments contributed by all forces about the centroid P of the moving platform yields:
$\begin{matrix} \sum_{i = 1}^{6} f_{i} b_{i} \times S_{i} = n & (28) \end{matrix}$

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_iis 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_idesirably are equal to alleviate cross-coupling.
In equation (30), the Jacobian matrix is as expressed in equation (18), namely:
$J = [\begin{matrix} s_{1}^{T} & {(b_{1} \times s_{1})}^{T} \\ \dots & \dots \\ s_{6}^{T} & {(b_{6} \times s_{6})}^{T} \end{matrix}]$
(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 = [\begin{matrix} \frac{c}{L} & \frac{- (- b + \sqrt{a^{2} - c^{2}})}{L} \\ \frac{- c}{L} & \frac{- (- b + \sqrt{a^{2} - c^{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{\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{- (\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{- \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} \end{matrix} \begin{matrix} \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{{b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} \\ \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{{b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} \\ \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{- {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} \\ \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{- {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} \\ \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{- {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} \\ \frac{{(L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{L} & \frac{- {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} \end{matrix} \begin{matrix} 0 & \frac{- bc}{L} \\ 0 & \frac{bc}{L} \\ \frac{- \sqrt{3} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} & \frac{- bc}{L} \\ \frac{- \sqrt{3} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} & \frac{bc}{L} \\ \frac{\sqrt{3} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} & \frac{- bc}{L} \\ \frac{\sqrt{3} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}}{2 L} & \frac{bc}{L} \end{matrix}]$
The 6×6 stiffness matrix K is as follows:
$K = [\begin{matrix} \frac{- 3 (- b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}{L^{2}} & 0 \\ 0 & \frac{- 3 (- b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}{L^{2}} \\ 0 & 0 \\ 0 & \frac{- 3 (- b + \sqrt{a^{2} - c^{2}})}{L^{2} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}} \\ \frac{3 (- b + \sqrt{a^{2} - c^{2}})}{L^{2} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}} & 0 \\ 0 & 0 \end{matrix} \begin{matrix} 0 & 0 \\ 0 & 0 \\ \frac{6 (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}{L^{2}} & \frac{- 3 (- b + \sqrt{a^{2} - c^{2}})}{L^{2} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}} \\ 0 & \frac{3 b^{2} (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}{L^{2}} \\ 0 & 0 \\ 0 & 0 \end{matrix} \begin{matrix} \frac{3 (- b + \sqrt{a^{2} - c^{2}})}{L^{2} {b (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}^{1 / 2}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \frac{3 b^{2} (L^{2} - b^{2} + 2 b \sqrt{a^{2} - c^{2}} - a^{2})}{L^{2}} & 0 \\ 0 & \frac{6 b^{2} c^{2}}{L^{2}} \end{matrix}]$
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
$J = [\begin{matrix} 0.4467 & - 0.3867 & 0.8068 & 137.1522 & 0 & - 75.9413 \\ - 0.4467 & - 0.3867 & 0.8068 & 137.1522 & 0 & 75.9413 \\ - 0.5583 & - 0.1935 & 0.8068 & - 68.5761 & - 118.7773 & - 75.9413 \\ - 0.1116 & 0.5802 & 0.8068 & - 68.5761 & - 118.7773 & 75.9413 \\ 0.1116 & 0.5802 & 0.8068 & - 68.5761 & 118.7773 & - 75.9413 \\ 0.5583 & - 0.1935 & 0.8068 & - 68.5761 & 118.7773 & 75.9413 \end{matrix}]$ $and K = (1.0 \times 10^{4}) [\begin{matrix} 0.0001 & 0 & 0 & 0 & 0.0159 & 0 \\ 0 & 0.0001 & 0 & - 0.0159 & 0 & 0 \\ 0 & 0 & 0.0004 & 0 & 0 & 0 \\ 0 & - 0.0159 & 0 & 5.6432 & 0 & 0 \\ 0.0159 & 0 & 0 & 0 & 5.6432 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3.4602 \end{matrix}]$
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:
$K = [\begin{matrix} \frac{3 N}{L^{2}} & 0 & 0 & 0 & \frac{3 Mbh}{L^{2}} & 0 \\ 0 & \frac{3 N}{L^{2}} & 0 & \frac{- 3 Mbh}{L^{2}} & 0 & 0 \\ 0 & 0 & \frac{6 h^{2}}{L^{2}} & 0 & 0 & 0 \\ 0 & \frac{- 3 Mbh}{L^{2}} & 0 & \frac{3 b^{2} h^{2}}{L} & 0 & 0 \\ \frac{3 Mbh}{L^{2}} & 0 & 0 & 0 & \frac{3 b^{2} h^{2}}{L^{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{3 b^{2} c^{2}}{L^{2}} \end{matrix}]$
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,
$\langle \frac{3 Mbh}{L^{2}} \rangle .$

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:
$K = [\begin{matrix} \frac{3 c^{2}}{L^{2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{3 c^{2}}{L^{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{6 (L^{2} - c^{2})}{L^{2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3 b^{2} (L^{2} - c^{2})}{L^{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{3 b^{2} (L^{2} - c^{2})}{L^{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{3 b^{2} c^{2}}{L^{2}} \end{matrix}]$
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_zstiffness while decreasing z, θ_x, and θ_ystiffness. 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:
$\begin{matrix} c = \sqrt{\frac{2}{3}} L & (33) \end{matrix}$
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 L₁and L₂largely visible in the figure. The legs L₁-L₆are 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-L₂is applicable to the other two pairs of legs L₃-L₄and L₅-L₆.
A distal end of the leg L₁is mounted to the base 14 by a block 30 a, and a distal end of the leg L₂is mounted to the base by a block 30 b (also visible in FIG. 4(A) is the block 30 c by which a distal end of the leg L₃is mounted to the base). At or near the respective distal end of each leg L₁-L₂is a respective distal flexure 36 a, 36 b mounted to the respective block 30 a, 30 b. The distal flexures 36 a, 36 b provide flexibility of the respective legs L₁, L₂in 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 36 a, 36 b is configured to provide a respective two DOF of motion (but not all six DOF) to the respective leg L₁, L₂relative to the base 14.
The proximal end of each leg L₁-L₂has a respective proximal flexure 38 a, 38 b (only the flexure 38 b is visible). The proximal flexures 38 a, 38 b of the legs are connected to an apex block 40. The proximal flexures 38 a, 38 b provide flexibility of the respective legs L₁, L₂in the desired DOF relative to the platform 12 as required to accommodate extensions and retractions of the respective legs. For example, each proximal flexure 38 a, 38 b is configured to provide at least two DOF of motion (but not all six DOF) to the respective leg L₁, L₂relative to the platform 12. The proximal flexures 38 a, 38 b work in coordination with the distal flexures 36 a, 36 b in this regard.
FIG. 4(A) shows particularly the legs L₁and L₂of 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, L₄and L₅, L₆are similarly configured. A plan view of the embodiment is shown in FIG. 4(B), showing all the legs L₁-L₆and blocks 30 a-30 f. 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 30 a-30 f) 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-L₆includes a respective length monitor 42 a-42 f that measures and monitors the length (including changes in length) of the respective leg. The leg-length monitors 42 a-42 f can be highly accurate encoders utilizing, for example, laser scales. The leg-length monitors 42 a-42 f 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-L₆comprises a respective piezoelectric (e.g., PZT-based piezo-ceramic) actuator 44 a-44 f. In FIG. 5 only one piezoelectric actuator 44 a can be seen. As previously noted, all the legs L₁-L₆have nominally the same length. The piezoelectric actuators 44 a-44 f impart very small changes in length to the respective legs L₁-L₆as 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-L₆has an associated height monitor 46 a, 46 b, 46 c mounted to the base 14. The height monitors 46 a-46 c 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 48 a, 48 b, 48 c situated midway between pairs of legs and mounted to the base 14. The height monitors 46 a-46 c and the position monitors 48 a-48 c 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 L₁and L₂are shown, including the blocks 30 a, 30 b; the distal flexures 36 a, 36 b, the proximal flexures 38 a, 38 b, and the apex block 40. Note that each leg L₁, L₂has 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, Ax₂collectively define a “leg plane” that, when the legs L₁, L₂are 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 P₁for the legs L₁, L₂, the leg plane P₂for the legs L₃, L₄, and the leg plane P₃for the legs L₅, L₆. The existence of these leg planes P₁-P₃relies 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 L₁as an example (and referring to components associated with the leg L₁as exemplary of corresponding components on the other legs), each leg has a respective piezoelectric actuator 44 a, which exhibits extension when the actuator is electrically energized. The piezoelectric actuator 44 a includes a fine-motion actuator 50 a and a coarse-motion actuator 50 b. In this embodiment the fine-motion actuator 50 a comprises one piezoelectric element, and the coarse-motion actuator 50 b comprises multiple piezoelectric elements arranged in tandem along the leg axis Ax₁. The fine-motion actuator 50 a and coarse-motion actuator 50 b also are arranged in tandem along the leg axis Ax₁. Each of the actuators 50 a, 50 b is separately energized by a respective driver (not shown). The piezoelectric actuator 44 a is situated in a yoke 52 that allows the piezoelectric actuator 44 a to apply an extension force (one DOF of motion) strictly along the longitudinal axis A₁of the respective leg L₁. Compliance along the axes A₁, A₂of the legs L₁, L₂to accommodate leg extensions is provided by respective flexures 56 a, 56 b.
By way of example, the coarse-motion actuator 50 b is configured to provide an accuracy of actuation performance in the micrometer range, and the fine-motion actuator 50 a is configured to provide an accuracy of actuation performance in the nanometer range. For some applications, one of the portions (the fine-motion actuator 50 a) 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; and

six 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; and

the 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; and

a 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; and

a 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; and

the 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; and

at 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; and

a 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; and

six 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.