Abstract
A multitude of flexure mechanism designs are set forth herein that furnish enhanced stiffness properties. The flexure mechanism designs can have various stages including ground stages, motion stages, flexure stages, and reinforcement stages, among other possible stages depending on the construction of the flexure mechanism designs. It has been determined that stages constituting twin stages can be connected via one or more inter-stage connections.
Claims
1. A flexure mechanism assembly, comprising: a first ground body and a second ground body; a first motion body and a second motion body; a first flexure blade extending in a first direction between said first ground body and said first motion body, said first flexure blade having a first reinforcement body; a second flexure blade extending in the first direction between said second ground body and said second motion body, said second flexure blade having a second reinforcement body; a first inter-stage connection extending in a second direction between said first reinforcement body and said second reinforcement body, said first inter-stage connection providing at least three degrees of constraint between said first reinforcement body and said second reinforcement body, wherein one of said at least three degrees of constraint between said first and second reinforcement bodies is in the first direction; a second rigid inter-stage connection extending in the second direction between said first ground body and said second ground body, said second rigid inter-stage connection providing six degrees of constraint between said first ground body and said second ground body; and a third rigid inter-stage connection extending in the second direction between said first motion body and said second motion body, said third rigid inter-stage connection providing six degrees of constraint between said first motion body and said second motion body.
2. The flexure mechanism assembly as set forth in claim 1, wherein said first inter-stage connection provides at least four degrees of constraint between said first reinforcement body and said second reinforcement body.
3. The flexure mechanism assembly as set forth in claim 1, wherein said first inter-stage connection is a rigid inter-stage connection that provides six degrees of constraint between said first reinforcement body and said second reinforcement body.
4. The flexure mechanism assembly as set forth in claim 1, further comprising: a third flexure blade extending in the first direction between said first ground body and said first motion body, said third flexure blade having a third reinforcement body, said third flexure blade being offset from said first flexure blade in a third direction; a fourth flexure blade extending in the first direction between said second ground body and said second motion body, said fourth flexure blade having a fourth reinforcement body, said fourth flexure blade being offset from said second flexure blade in the third direction; and a fourth inter-stage connection extending in the second direction between said third reinforcement body and said fourth reinforcement body, said fourth inter-stage connection providing at least three degrees of constraint between said third reinforcement body and said fourth reinforcement body.
5. The flexure mechanism assembly as set forth in claim 4, wherein said first inter-stage connection provides at least four degrees of constraint between said first reinforcement body and said second reinforcement body, and said fourth inter-stage connection provides at least four degrees of constraint between said third reinforcement body and said fourth reinforcement body.
6. The flexure mechanism assembly as set forth in claim 4, wherein said first inter-stage connection is a first rigid inter-stage connection that provides six degrees of constraint between said first reinforcement body and said second reinforcement body, and said fourth inter-stage connection is a fourth rigid inter-stage connection that provides six degrees of constraint between said third reinforcement body and said fourth reinforcement body.
7. The flexure mechanism assembly as set forth in claim 4, further comprising: a third motion body and a fourth motion body, a fifth rigid inter-stage connection extending in the second direction between said third motion body and said fourth motion body, said fifth rigid inter-stage connection providing six degrees of constraint between said third motion body and said fourth motion body; a fifth flexure blade extending in the first direction between said first motion body and said third motion body, said fifth flexure blade having a fifth reinforcement body, said fifth flexure blade being offset from said third flexure blade in the third direction; a sixth flexure blade extending in the first direction between said second motion body and said fourth motion body, said sixth flexure blade having a sixth reinforcement body, said sixth flexure blade being offset from said fourth flexure blade in the third direction; a sixth inter-stage connection extending in the second direction between said fifth reinforcement body and said sixth reinforcement body, said sixth inter-stage connection providing at least three degrees of constraint between said fifth reinforcement body and said sixth reinforcement body; a seventh flexure blade extending in the first direction between said first motion body and said third motion body, said seventh flexure blade having a seventh reinforcement body, said seventh flexure blade being offset from said fifth flexure blade in the third direction; an eighth flexure blade extending in the first direction between said second motion body and said fourth motion body, said eighth flexure blade having an eighth reinforcement body, said eighth flexure blade being offset from said sixth flexure blade in the third direction; and a seventh inter-stage connection extending in the second direction between said seventh reinforcement body and said eighth reinforcement body, said seventh inter-stage connection providing at least three degrees of constraint between said seventh reinforcement body and said eighth reinforcement body.
8. (canceled)
9. The flexure mechanism assembly as set forth in claim 7, wherein said first inter-stage connection is a first rigid inter-stage connection that provides six degrees of constraint between said first reinforcement body and said second reinforcement body, said fourth inter-stage connection is a fourth rigid inter-stage connection that provides six degrees of constraint between said third reinforcement body and said fourth reinforcement body, said sixth inter-stage connection is a sixth rigid inter-stage connection that provides six degrees of constraint between said fifth reinforcement body and said sixth reinforcement body, and said seventh inter-stage connection is a seventh rigid inter-stage connection that provides six degrees of constraint between said seventh reinforcement body and said eighth reinforcement body.
10. The flexure mechanism assembly as set forth in claim 1, wherein said first inter-stage connection has a distance in the second direction that is greater than a height of said first flexure blade or of said second flexure blade.
11. A flexure mechanism assembly, comprising: a first ground body; a first motion body having a first degree of freedom with respect to said first ground body in a first direction, and having a second degree of freedom with respect to said first ground body in a second direction; a plurality of first parallelogram flexure modules arranged between said first ground body and said first motion body in a first plane; at least one first intermediate body arranged among said plurality of first parallelogram flexure modules, with at least one flexure blade from one of said plurality of first parallelogram flexure modules extending between said first ground body and said at least one first intermediate body, and at least another flexure blade from another of said plurality of first parallelogram flexure modules extending between said first motion body and said at least one first intermediate body; a second ground body spaced from said first ground body in a third direction; a second motion body having a first degree of freedom with respect to said second ground body in the first direction, and having a second degree of freedom with respect to said second ground body in the second direction; a plurality of second parallelogram flexure modules arranged between said second ground body and said second motion body in a second plane, said plurality of second parallelogram flexure modules being spaced from said plurality of first parallelogram flexure modules in the third direction; at least one second intermediate body arranged among said plurality of second parallelogram flexure modules, with at least one flexure blade from one of said plurality of second parallelogram flexure modules extending between said second ground body and said at least one second intermediate body, and at least another flexure blade from another of said plurality of second parallelogram flexure modules extending between said second motion body and said at least one second intermediate body; a first rigid inter-stage connection extending in the third direction between said first ground body and said second ground body, said first rigid inter-stage connection providing six degrees of constraint between said first ground body and said second ground body; a second rigid inter-stage connection extending in the third direction between said first motion body and said second motion body, said second rigid inter-stage connection providing six degrees of constraint between said first motion body and said second motion body; and at least one third inter-stage connection extending in the third direction between said at least one first intermediate body and said at least one second intermediate body, said at least one third inter-stage connection providing at least three degrees of constraint between said at least one first intermediate body and said at least one second intermediate body, wherein said at least one third inter-stage connection has a distance in the third direction that is at least approximately twenty percent of a length of at least one flexure blade of said plurality of first parallelogram flexure modules or of said plurality of second parallelogram flexure modules.
12. The flexure mechanism assembly as set forth in claim 11, wherein said at least one third inter-stage connection is an at least one third rigid inter-stage connection providing six degrees of constraint between said at least one first intermediate body and said at least one second intermediate body.
13. The flexure mechanism assembly as set forth in claim 11, wherein said first rigid inter-stage connection has a distance in the third direction that is greater than the height of the at least one flexure blade of said plurality of first parallelogram flexure modules or of said plurality of second parallelogram flexure modules, and said second rigid inter-stage connection has a distance in the third direction that is greater than the height of the at least one flexure blade of said plurality of first parallelogram flexure modules or of said plurality of second parallelogram flexure modules.
14. The flexure mechanism assembly as set forth in claim 11, wherein said at least one first intermediate body and said at least one second intermediate body constitute twin stages of the flexure mechanism assembly.
15. The flexure mechanism assembly as set forth in claim 11, wherein said plurality of first parallelogram flexure modules is a plurality of first double parallelogram flexure modules, and said plurality of second parallelogram flexure modules is a plurality of second double parallelogram flexure modules.
16. The flexure mechanism assembly as set forth in claim 15, wherein said plurality of first double parallelogram flexure modules comprises a plurality of first secondary bodies, said plurality of second double parallelogram flexure modules comprises a plurality of second secondary bodies, and said at least one third inter-stage connection comprises at least one fourth inter-stage connection extending in the third direction between at least one of said plurality of first secondary bodies and at least one of said plurality of second secondary bodies, said at least one fourth inter-stage connection providing at least three degrees of constraint, wherein said plurality of first double parallelogram flexure modules comprises a plurality of first flexure blades, said plurality of second double parallelogram flexure modules comprises a plurality of second flexure blades, said plurality of first flexure blades extending from said plurality of first secondary bodies, said plurality of second flexure blades extending from said plurality of second secondary bodies, some of said plurality of first flexure blades extending to said first ground body and other of said plurality of first flexure blades extending to said first motion body, and some of said plurality of second flexure blades extending to said second ground body and other of said plurality of second flexure blades extending to said second motion body.
17. The flexure mechanism assembly as set forth in claim 15, wherein said plurality of first flexure blades have a plurality of first reinforcement bodies, said a plurality of second flexure blades have a plurality of second reinforcement bodies, and said at least one third inter-stage connection comprises at least one fourth inter-stage connection extending in the third direction between at least one of said plurality of first reinforcement bodies and at least one of said plurality of second reinforcement bodies, said at least one fourth inter-stage connection providing at least three degrees of constraint.
18. The flexure mechanism assembly as set forth in claim 11, wherein said plurality of first parallelogram flexure modules comprises a plurality of first flexure blades with a plurality of first reinforcement bodies, said plurality of second parallelogram flexure modules comprises a plurality of second flexure blades with a plurality of second reinforcement bodies, and said at least one third inter-stage connection comprises at least one fourth inter-stage connection extending in the third direction between at least one of said plurality of first reinforcement bodies and at least one of said plurality of second reinforcement bodies, said at least one fourth inter-stage connection providing at least three degrees of constraint.
19. A flexure mechanism assembly, comprising: a ground body; a motion body having a first degree of freedom with respect to said ground body in a first direction, and having a second degree of freedom with respect to said ground body in a second direction; a plurality of first double parallelogram flexure modules arranged between said ground body and said motion body, said plurality of first double parallelogram flexure modules comprising a plurality of first flexure blades with a plurality of first reinforcement bodies and comprising a plurality of first secondary bodies, said plurality of first flexure blades extending from said plurality of first secondary bodies, some of said plurality of first flexure blades extending to said ground body and other of said plurality of first flexure blades extending to said motion body; at least one first intermediate body arranged among said plurality of first double parallelogram flexure modules, some of said plurality of first flexure blades extending to said at least one first intermediate body; a plurality of second double parallelogram flexure modules arranged between said ground body and said motion body, said plurality of second double parallelogram flexure modules being spaced from said plurality of first double parallelogram flexure modules in a third direction, said plurality of second double parallelogram flexure modules comprising a plurality of second flexure blades with a plurality of second reinforcement bodies and comprising a plurality of second secondary bodies, said plurality of second flexure blades extending from said plurality of second secondary bodies, some of said plurality of second flexure blades extending to said ground body and other of said plurality of second flexure blades extending to said motion body; at least one second intermediate body arranged among said plurality of second double parallelogram flexure modules, some of said plurality of second flexure blades extending to said at least one second intermediate body; at least one first inter-stage connection extending in the third direction between at least one of said plurality of first reinforcement bodies and at least one of said plurality of second reinforcement bodies, said at least one first inter-stage connection providing at least three degrees of constraint between said at least one of said plurality of first reinforcement bodies and said at least one of said plurality of second reinforcement bodies; at least one second inter-stage connection extending in the third direction between at least one of said plurality of first secondary bodies and at least one of said plurality of second secondary bodies, said at least one second inter-stage connection providing at least three degrees of constraint between said at least one of said plurality of first secondary bodies and said at least one of said plurality of second secondary bodies; and at least one third inter-stage connection extending in the third direction between said at least one first intermediate body and said at least one second intermediate body, said at least one third inter-stage connection providing at least three degrees of constraint said at least one first intermediate body and said at least one second intermediate body.
20. The flexure mechanism assembly as set forth in claim 19, wherein said at least one first, second, and third inter-stage connections are rigid inter-stage connections that provide six degrees of constraint between the respective bodies.
21. The flexure mechanism assembly as set forth in claim 11, wherein the distance of said at least one inter-stage connection in the third direction ranges between approximately twenty percent and eighty percent of a length of at least one flexure blade of said plurality of first parallelogram flexure modules or of said plurality of second parallelogram flexure modules.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0006] Exemplary embodiments will hereinafter be described in conjunction with the appended drawings, wherein like designations denote like elements, and wherein:
[0007] FIG. 1 is a perspective view of a double parallelogram flexure mechanism (DPFM), showing degrees of freedom (DoFs) and degrees of constraint (DoCs);
[0008] FIG. 2 is a graph demonstrating in a qualitative manner DoF stiffness and DoC stiffness of the DPFM in a Y direction, with motion stage displacement (y) plotted on an x-axis and stiffness plotted on a y-axis;
[0009] FIG. 3 is a perspective view of an embodiment of a flexure joint in the form of a flexure blade connecting a ground stage and a motion stage together;
[0010] FIG. 4 is a perspective view of an embodiment of a flexure module having a pair of flexure blades and three bodies (i.e., a ground body/stage, a reinforcement body/stage, and a motion body/stage);
[0011] FIG. 5 is a schematic demonstrating parasitic compliance of a secondary stage in an X direction of a DPFM (i.e., undeformed secondary stage is presented in broken lines, and deformed secondary stage is presented in solid lines);
[0012] FIG. 6 is an abstract representation of a multi-degree-of-freedom (MDoF) serial kinematic (SK) mechanism;
[0013] FIG. 7 is a schematic of an embodiment of a MDoF SK mechanism;
[0014] FIG. 8 is an abstract representation of a multi-degree-of-freedom (MDoF) parallel kinematic (PK) mechanism;
[0015] FIG. 9 is a schematic of an embodiment of a MDoF PK XY flexure mechanism having four DPFMs;
[0016] FIG. 10 is a schematic of an embodiment of a MDoF PK XY flexure mechanism having eight DPFMs;
[0017] FIG. 11 shows an intermediate stage of the MDoF PK XY flexure mechanism of FIG. 10 being displaced in the X direction;
[0018] FIG. 12 shows another intermediate stage of the MDoF PK XY flexure mechanism of FIG. 10 being displaced in the Y direction;
[0019] FIG. 13 is a perspective view of an embodiment of a flexure blade connecting a ground stage and a motion stage together;
[0020] FIG. 14 is a schematic demonstration of parasitic error motion in an axial direction (X direction; DoC) of a motion stage of a parallelogram flexure mechanism (PFM) when the motion stage is displaced in a transverse direction (Y direction; DoF);
[0021] FIG. 15 is a schematic demonstration of the parasitic error motion of FIG. 14 of a motion stage of a DPFM being cancelled due to the geometric design of the DPFM;
[0022] FIG. 16 is a graph demonstrating in a qualitative manner X direction stiffness with respect to Y direction displacement of motion stages of a PFM and of a DPFM, with motion stage displacement plotted on an x-axis and X translation DoC stiffness plotted on a y-axis;
[0023] FIG. 17a is a graph demonstrating in a qualitative manner Z direction stiffness with respect to Y direction displacement of a motion stage of a DPFM, with motion stage Y displacement plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis;
[0024] FIG. 17b is a schematic demonstration of deformation occurring at a secondary stage and at flexure blades of a DPFM at nominal position when a force is applied in the Z direction at a motion stage;
[0025] FIG. 17c is a schematic demonstration of torsional deformation occurring at the secondary stage and at the flexure blades of the DPFM of FIG. 17b at non-nominal position and upon displacement of the motion stage along its motion axis when a force is applied in the Y direction and when a force is applied in the Z direction at the motion stage;
[0026] FIG. 17d is a schematic demonstration of torsional deformation occurring at the secondary stage and at the flexure blades of the DPFM of FIG. 17b when a .sub.x moment is applied along the X axis on the motion stage;
[0027] FIG. 18a is a perspective view of an embodiment of a DPFM having reinforcement stages at flexure blades;
[0028] FIG. 18b is a graph demonstrating in a qualitative manner Z direction stiffness of reinforcement stages of flexure blades based on length (L.sub.R) of the reinforcement stages, with reinforcement length (L.sub.R) plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis;
[0029] FIG. 18c is a graph demonstrating in a qualitative manner Y direction stiffness of reinforcement stages of flexure blades based on length (L.sub.R) of the reinforcement stages, with reinforcement length (L.sub.R) plotted on an x-axis and Y translation DoF stiffness plotted on a y-axis;
[0030] FIG. 19a is a top view of an embodiment of a PK XY flexure mechanism assembly having twelve DPFMs, the PK XY flexure mechanism assembly being in its nominal position;
[0031] FIG. 19b is a sectional view of the PK XY flexure mechanism assembly of FIG. 19a demonstrating deformation at inner and peripheral DPFMs due to a force applied in the Z direction at a motion stage, the motion stage and intermediate stages caused to move in the Z direction;
[0032] FIG. 19c is a graph demonstrating in a qualitative manner Z direction stiffness of the PK XY flexure mechanism assembly of FIG. 19a with respect to X direction translation of a motion stage thereof, with motion stage displacement in X direction plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis;
[0033] FIG. 20a1 is a perspective view of an embodiment of a sandwich flexure module;
[0034] FIG. 20a2 is a perspective view of another embodiment of a sandwich flexure module;
[0035] FIG. 20a3 is a perspective view of another embodiment of a sandwich flexure module;
[0036] FIG. 20a4 is a perspective view of another embodiment of a sandwich flexure module;
[0037] FIG. 20b is a perspective view of an embodiment of a sandwich parallelogram flexure mechanism (PFM);
[0038] FIG. 20c is a perspective view of an embodiment of a sandwich double parallelogram flexure mechanism (DPFM);
[0039] FIG. 20d is a perspective view of another embodiment of a sandwich DPFM;
[0040] FIG. 20e is a perspective view of another embodiment of a sandwich DPFM;
[0041] FIG. 20f is a graph demonstrating in a qualitative manner Z direction stiffness with respect to Y direction displacement of motion stages of a sandwich DPFM and a DPFM having reinforcement stages at flexure blades, with motion stage displacement in Y direction plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis;
[0042] FIG. 20g is a graph demonstrating in a qualitative manner Z direction stiffness with respect to separation between first and second DPFMs of an embodiment of a sandwich DPFM, with separation between first DPFM and second DPFM (d.sub.R) plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis;
[0043] FIG. 20h is a graph demonstrating in a qualitative manner Y translation DoF stiffness with respect to separation between first and second DPFMs of an embodiment of a sandwich DPFM, with separation between first and second DPFMs (d.sub.R) plotted on an x-axis and Y translation DoF stiffness plotted on a y-axis;
[0044] FIG. 21a is a perspective view of an embodiment of a sandwich XY parallel kinematic flexure mechanism;
[0045] FIG. 21b is a top view of another embodiment of a sandwich XY parallel kinematic flexure mechanism;
[0046] FIG. 21c is a perspective view of the sandwich XY parallel kinematic flexure mechanism of FIG. 21b;
[0047] FIG. 21d is a top view of another embodiment of a sandwich XY parallel kinematic flexure mechanism;
[0048] FIG. 21e is an enlarged view of a portion of the sandwich XY parallel kinematic flexure mechanism of FIG. 21d taken in isolation and shown in perspective view;
[0049] FIG. 21f is a top view of another embodiment of a sandwich XY parallel kinematic flexure mechanism;
[0050] FIG. 21g is an enlarged view of a portion of the sandwich XY parallel kinematic flexure mechanism of FIG. 21f taken in isolation and shown in perspective view;
[0051] FIG. 21h is a top view of another embodiment of a sandwich XY parallel kinematic flexure mechanism;
[0052] FIG. 21i is an enlarged view of a portion of the sandwich XY parallel kinematic flexure mechanism of FIG. 21h taken in isolation and shown in perspective view;
[0053] FIG. 22 is a perspective view of an embodiment of an XY parallel kinematic flexure mechanism having reinforcement stages at flexure blades; and
[0054] FIG. 23 is a graph demonstrating in a qualitative manner Z direction stiffness with respect to X direction displacement of motion stages of a sandwich XY parallel kinematic flexure mechanism and an XY parallel kinematic flexure mechanism having reinforcement stages at flexure blades, with motion stage displacement in X direction plotted on an x-axis and Z translation DoC stiffness plotted on a y-axis.
DETAILED DESCRIPTION
[0055] Initially, numbered definitions are presented for certain terms and phrases used in this detailed description and in the claims.
[0056] 1) Body. A body can be part of an assembly, sub-assembly, or mechanism, etc. A body can be a rigid body or a flexible body. But nothing is ever truly rigid. Rather, what this term means is that the intent for the rigid body is to be rigid. In general, a rigid body will have some undesired compliance, referred to parasitic compliance, as subsequently described. A flexible body, on the other hand, is flexible by design. In other words, it has intentional (as opposed to undesired) compliance.
[0057] Elements, components, stages, etc., are all examples of rigid or flexible bodies. In this patent, stages are typically rigid bodies, while flexure elements (e.g., beams, strips, wires) are flexible bodies.
[0058] A flexible body may be intentionally flexible in certain portions and intentionally rigid in other portions. The portions can be monolithic or structurally assembled. A flexible body may also be referred to as a compliant body. Further, a body may be of homogeneous material composition or heterogeneous material composition.
[0059] As subsequently set forth, a rigid body or a flexible body can be a monolith or a structural assembly of multiple bodies. Structural assembly is defined as an interface between two bodies that eliminates all relative degrees of freedom between the two bodies.
[0060] 2) The displacement(s) (used interchangeably with motion(s); a general term that includes both translations and rotations) of a body can generally be defined with respect to another body by describing motion or lack thereof along six directions (three translations and three rotations).
[0061] Of the six directions, the directions along which relative displacement is allowed between two bodies are called the Degrees of Freedom (DoF) between the two bodies. DoF is a technical term to capture or convey these allowed motions. Other terms that can be used interchangeably with degrees of freedom are directions of freedom, and motion directions.
[0062] Of the six directions, the directions along which relative displacement is not allowed are called Degrees of Constraint (DoC) between the two bodies. Displacements and loads (a general term that includes both force and moments) are transmitted from one body to the other along the DoC. Since relative motion is constrained, these are directions along which motions (i.e., translations or rotations) and loads (i.e., forces or moments) can be transmitted from one body to the other body. Since the joint/interface between the two bodies does not allow relative motion between the two bodies in the DoC direction, if one body moves in the DoC direction, it drives along with it the other body along that direction. In other words, motions are transmitted from one body to another body in the DoC directions. Consequently, loads are also transmitted from one body to the other in the DoC directions. Therefore, these directions are referred to as constraint directions, directions of constraint, load bearing directions, bearing directions, transmission directions, or non-motion directions.
[0063] In general, DoF plus DoC equals six in total. The number of relative DoF between two bodies is determined by the joint or interface between the two bodies.
[0064] 3) Joint. Joints are interfaces between two bodies. When a joint or an interface is intended to eliminate all relative motions, or DoF, between two bodies, it is a structural joint. The two bodies interconnected by a structural joint may be referred to as a structural assembly. Examples of structural joints include, but are not limited to, a screwed joint, a bolted joint, a riveted joint, a brazed joint, a welded joint, a glued joint, and the like. When a joint or interface between two bodies is intended to allow one or more DoF(s), the joint is a kinematic joint. Examples of kinematic joints include, but are not limited to, a pin/pivot joint, a slider joint, a universal joint, a ball and socket joint, a flexure joint, and the like.
[0065] In all, there are six independent directions along which motions, or DoFs, are possible between two rigid bodies when there is no joint between them: three translations and three rotations. A joint allows anywhere between zero and six DoF(s) between two bodies. For the case when the joint allows zero DoF, this effectively becomes a structural joint as described above. In this case, the two bodies become rigid extensions of each other. While a structural joint is intended to be ideali.e., the stiffness of the joint or interface is desired to be infinitythat is never the case. There is always some undesired joint stiffness that leads to parasitic compliance in a structural assembly.
[0066] Where a joint allows six DoFs, this effectively means that there is no joint, or that the joint does not constrain any motions between the two bodies. In other words, the motions of the two bodies are entirely independent of each other. Any practical kinematic joint for the purpose this patent allows one, two, three, four, or five DoF(s) between two bodies. If a joint allows one DoF, then the remaining five possible motions (DoCs) are constrained by the joint. If a joint allows two DoFs, then the remaining four possible motions (DoCs) are constrained by the joint. If a joint allows three DoFs, then the remaining three possible motions (DoCs) are constrained by the joint. If a joint allows four DoFs, then the remaining two possible motions (DoCs) are constrained by the joint. If a joint allows five DoFs, then the remaining one possible motion (DoC) is constrained by the joint.
TABLE-US-00001 TABLE 1 below sets forth certain nomenclature as used in this patent: Dimensions Lower case, italicized Example: t for flexure blade thickness Direction Upper case Example: X, Y, Z, .sub.x, .sub.y, .sub.z directions Motion/ Lower case Displacement Example: x, y, z, .sub.x, .sub.y, .sub.z displacements/translations/rotations Axis Upper case Example: X1, X2, X3, Y1, Y2, Y3 axes
[0067] Descriptions of certain embodiments are now presented with references to the figures.
Flexure Mechanism
[0068] A mechanism is an assembly of bodies that may be interconnected via interfaces/joints, and serves to offer relative motions (or DoF) between the bodies. In particular, a flexure mechanism is one that has at least one flexible body. Further, a flexure mechanism is one that provides at least some of these relative motions via intentional elastic deformation of one or more of its constituent flexible bodies. A flexure mechanism may also provide all of these relative motions via elastic deformation of its constituent flexible bodies.
[0069] In general, while a flexure mechanism can have multiple bodies, two bodies are often identifieda motion stage or body, and a ground stage or body. Typically, relative motions of the motion stage are described with respect to the ground stage. A flexure mechanism employs elastic deformation to provide relative motion between the motion stage and ground stage in one or more motion directions, or DoF; and, provides high stiffness and load bearing capability in the remaining non-motion directions, or bearing directions or DoC. Further, in the context of a mechanism like the flexure mechanism, the ground body or reference ground serves as a kinematic reference for the other bodies. The ground body is not necessarily an absolute ground in the sense that it is attached or bolted for the actual ground, although that is a possibility. The ground body serves as a mechanical reference with respect to which motions of other bodies are described or investigated.
[0070] Referring now to FIG. 1, an embodiment of a double parallelogram flexure mechanism (DPFM) 10 is presented. The DPFM 10 provides relative translation of a first body 12 in the Y direction with respect to a second body 14. The DPFM 10 constrains the first body 12 with respect to the second body 14 in the remaining five directions, i.e., X, Z, .sub.x, .sub.y, and .sub.z. Therefore, the DPFM 10 provides one DoF and five DoCs, according to this embodiment. The first and second bodies 12, 14, or their respective portions, are also considered a part of the DPFM 10. In a particular context or application, and still referring to FIG. 1, the first body 12 is a motion body 12 or a motion stage of the DPFM 10, and the second body 14 is a ground body 14 or a ground stage of the DPFM 10. In another context or application, this arrangement could be reversed and the first body 12 would then be the ground body or ground stage of the DPFM 10, and the second body 14 would be the motion body or motion stage of the DPFM 10. Furthermore, in this embodiment, the DPFM 10 has a third body 16. Here, the third body 16 is a secondary body 16 or secondary stage of the DPFM 10.
[0071] The DPFM 10 also includes flexible or compliant bodies 18 in the form of flexure blades, per this embodiment. The elastic deformation of the flexible bodies 18 due to their decreased thickness (t) provides the Y direction translational motion of the motion body 12 relative to the ground body 14, according to this embodiment. The geometry of the DPFM 10 and its constituent bodiesi.e., motion body 12, ground body 14, secondary body 16, and flexible bodies 18provides high stiffness in the remaining five directions, i.e., X, Z, .sub.x, .sub.y, and .sub.z, making these five directions the load bearing or DoC directions.
[0072] The parallel geometric arrangement of the four flexible bodies 18 in the DPFM 10 of FIG. 1 and the increased separation among them, given by d.sub.i (d.sub.i>>t) and d.sub.o (d.sub.o>>t), along the Y direction provides high stiffness that constrains the displacement of the motion body 12 relative to the ground body 14 along the .sub.x and .sub.z directions. The increased height of the flexible bodies 18, given by h (h>>t), along the Z direction provides high stiffness that constrains the displacement of the motion body 12 relative to the ground body 14 along Z and .sub.y directions. Similarly, the increased stiffness of the flexible bodies 18 along the X direction constrains the displacement of the motion body 12 relative to the ground body 14 along the X direction.
[0073] Furthermore, as used herein, there is a certain equivalence among the terms flexure mechanism, flexure module, and flexure joint. All of these can be viewed as allowing certain motion(s) along certain degree(s) of freedom (DoF) between two bodies and constraining the remaining motions. A flexure joint, in particular, can include a single flexible body or a few flexible bodies; a flexure module can include more flexible bodies; and a flexure mechanism generally includes many flexible bodies. Typically, a joint is of simpler construction, a module if more complex and can include multiple joints, and a mechanism can include multiple joints and/or multiple modules.
[0074] With reference to FIG. 3, the single flexible body 18in the form of a flexure blade 20is an example of a flexure joint 22 that allows relative displacements between the motion body 12 and ground body 14 along the Y, .sub.x, and .sub.z directions, and restricts relative displacements along the .sub.y, Z, and X directions. In FIG. 4, an example of a flexure module 24 is shown. The flexure module 24 has two flexible joints. A first flexible joint is made between the ground body 14 and another body. The first flexible joint here is a first flexible body 26 in the form of a first flexure blade 28. The other body in this example is a reinforcement body 30 or reinforcement stage. A second flexible joint is made between the motion body 12 and the reinforcement body 30. The second flexible joint, like the first flexible joint, is a second flexible body 32 in the form of a second flexure blade 34. The bodies that constitute the flexure module 24 can be monolithic or can be formed via the structural assembly of two or more bodies. In the example of FIG. 4, the motion body 12 and ground body 14 are monolithic bodies, while the reinforcement body 30 is an assemblage of two bodies joined structurally together. A structural joint 36 of the reinforcement body 30 can be carried out in various ways including, but not limited to, adhesives, bolting, and/or riveting.
[0075] But what is simple and what is complex depends on context. A mechanism under consideration may appear simple or small in the context of a much larger mechanism, in which case the particular mechanism under consideration may be referred to as a module or even as a joint. For example, the DPFM 10 of FIG. 1 can be considered a flexure mechanism when employed by itself. But when the DPFM 10 of FIG. 1 is employed to build a more complex mechanism like the parallel kinematic (PK) XY flexure mechanism of FIG. 9, then in that context the DPFM can be considered a flexure module (i.e., double parallelogram flexure module). In FIG. 9, the arrangement of four double parallelogram flexure modules results in an XY flexure mechanism that has two DoFs, a DoF in the X direction and a DoF in the Y direction.
Ideal Bearing Behavior
[0076] Flexure mechanisms are commonly used as bearings to provide guided motion along the associated DoF in various motion control applications. When a flexure mechanism employs only elastic deformation of its accompanying flexible bodies to provide such guided motion, then the flexure mechanism is free of the friction and backlash associated with conventional rolling and sliding joints with sliding guideways or rolling element bearings. Examples of some general applications in which flexure mechanisms are utilized include, but are not limited to, micro-electro mechanical system (MEMS) devices such as actuators and sensors, ultra-precision motions stages for fine alignment and adjustments, scanning probe microscopy and metrology, scanning probe nanolithography, memory storage, hard-drive and semiconductor inspection, semiconductor packaging, biological imaging, and non-contact macro scale actuators and sensors. In these applications as well as others, it is highly desirable to achieve ideal bearing behavior in the flexure mechanisms.
[0077] A mechanism employed as a bearing in a motion control application exhibits ideal bearing behavior if it can effect: i) infinite motion and zero constraint (i.e., zero stiffness or resistance) along the motion directions (DoF) and over the entire range of motion in the DoF directions, and ii) zero error motion and infinite constraint (i.e., infinite stiffness or load bearing capability) along the non-motion or bearing directions (DoC) over the entire range of motions in the DoF directions. But a flexure mechanism, it is thought, can never achieve ideal bearing behavior because it produces motion due to the elastic deformation of its flexible/compliant bodies. It is impractical for the flexible/compliant bodies to have zero stiffness (i.e., infinite compliance) in some directions and infinite stiffness (i.e., zero compliance) in the other directions. Therefore, flexure mechanisms will have small but non-zero stiffness along their DoFs and large but finite stiffness along their DoCs. The large but finite stiffness of a flexure mechanism along its DoCs implies small but non-zero motion along its DoCs in the presence of applied force. This motion is referred to as elastic motion along the DoC because it arises due to the finite compliance of the mechanism along its DoCs. Another kind of motion is possible along the DoCs which is attributed to the kinematics of the flexure mechanism (and not its elastic behavior). Such motion is referred to as kinematic motion along the DoC.
[0078] Apart from the flexible/compliant bodies, there are also rigid bodies in a flexure mechanism, as depicted in the DPFM 10 of FIG. 1. While ideally rigid bodies should have zero compliance (i.e., infinite stiffness), in practice there is always some compliance exhibited by them which is referred to as parasitic compliance (described in more detail below). The parasitic compliance of these rigid bodies leads to further deviation of the flexure mechanisms from the ideal bearing behavior. Accordingly, the aim in flexure mechanism design is to approach ideal bearing behavior, while knowing it can never truly be achieved.
[0079] Consider the DPFM 10 of FIG. 1. FIG. 2 provides a graphical illustration of stiffness of the DPFM 10 of FIG. 1 along its DoF (i.e., Y direction) and any one of its DoCs (i.e., X, Z, .sub.x, .sub.y, .sub.z directions) as a function of the Y direction translation (y) of the motion body 12. Although FIG. 2 illustrates the DoC stiffness along only one of the load bearing directions, the DoC stiffness curve of the graph is representative of the stiffness along the remaining load bearing directions as well. It is shown in FIG. 2 that the DPFM 10 exhibits non-zero stiffness (i.e., finite compliance) along its DoF, the Y direction. Similarly, the DPFM 10 exhibits finite stiffness along its DoC, and not infinite stiffness which is required for ideal bearing behavior. But even when exhibiting non-ideal bearing behavior, there is a large difference between the DoF and DoC stiffness, with the DoC stiffness being several orders of magnitude higher than the DoF stiffness when Y translation (y) of the motion body 12 is zero (i.e., y=0). The stiffness along the DoC are not only finite, but further decrease as the Y direction translation (y) of the motion body 12 is increased. This means that the DPFM 10 provides limited load bearing capability in the non-motion directions, or DoCs, which further worsens as the Y direction translation (y) of the motion body 12 is increased. The DoF stiffness can also increase as the Y direction translation (y) of the motion body 12 is increased. Although it is shown here that a DPFM does not achieve ideal bearing behavior, this holds true for any flexure mechanism that consists of flexible/compliant bodies. Certain previously-known flexure mechanism designs have been purported to achieve close to ideal bearing behavior. But these previously-known flexure mechanisms suffer from a trade-off between the DoF and DoC stiffnessany design that increases the DoC stiffness concomitantly increases the DoF stiffness. Likewise, any flexure mechanism design that decreases the DoF stiffness concomitantly decreases the DoC stiffness. This trade-off between DoF and DoC stiffness is described below.
Parasitic Compliance
[0080] Rigid bodies are intended to be infinitely stiff in all six directions (i.e., X, Y, Z, .sub.x, .sub.y, .sub.z directions). But no rigid body can be infinitely stiff in practice. Even supposedly rigid bodies have some small finite compliance. Such unintended and often undesired compliance of the rigid bodies in any direction is referred to as parasitic compliance. As an example, and with reference again to FIG. 1, the DPFM 10 per this embodiment has three rigid bodies: the ground body 14, the secondary body 16, and the motion body 12. All of these rigid bodies should ideally have infinite stiffness in all of the six directions. But depending on their geometry and dimensions, the rigid bodies exhibit varying amounts of parasitic compliance in every one of the six directions. Referring now to FIG. 5, a top view of the DPFM 10 is shown to demonstrate parasitic compliance of the secondary stage 16 in the X direction. The top view of FIG. 5 is viewed from the Z direction. When force F.sub.x is applied on the motion body 12 along the X direction, the secondary body 16 is deformed in the XY plane as a consequence and as shown. The undeformed secondary stage is shown in broken lines in the figure, while the deformed secondary stage is shown in solid lines, both for demonstrative purposes. Such deformation of the secondary body 16 upon application of the force F.sub.x is due to the finite stiffness, or non-zero compliance, of the secondary body 16 in the X direction. This is the parasitic compliance of the secondary body 16 and it is less-than-ideal stiffness in the X direction which is a bearing direction of the DPFM 10. Although lacking depiction in FIG. 5, forces applied to the motion body 12 in the other five directions (i.e., Y, Z, .sub.x, .sub.y, .sub.z directions) will similarly cause deformations of the secondary body 16 due to the parasitic compliance of the secondary body 16 in those directions. Such parasitic compliances of rigid bodiesdemonstrated here for the secondary body 16lead to less than desired stiffnesses in the DoC directions.
Types of Flexure Mechanisms
[0081] Single degree of freedom flexure mechanism. Flexure mechanisms that provide guided relative motion between two bodies only along one direction, and therefore constrain the relative motion along the remaining five directions, are called single degree of freedom (SDoF) flexure mechanisms. Single DoF flexure mechanisms provide one DoF and five DoCs. The DPFM 10 in FIG. 1 is an example of a SDoF flexure mechanism. The DPFM 10 provides relative translational motion between the motion body 12 and the ground body 14 only along the Y direction, which is the single DoF, and constrains the relative motions of the motion body 12 with respect to the ground body 14 along the remaining five directions, making these five directions the DoCs.
[0082] Multi degree of freedom flexure mechanism. Flexure mechanisms that provide guided relative motion between two bodies along more than one direction are called multi degree of freedom (MDoF) flexure mechanisms. Multi DoF flexure mechanisms provide more than one DoF and less than five DoC. The flexure joint 22 of FIG. 3 is an XY flexure mechanism, and is an example of a MDoF flexure mechanism. The flexure joint 22 provides relative translational motion between the ground body 14 and the motion body 12 along the X direction and the Y direction. Specifically, the flexure joint 22 is an example of a two DoF flexure mechanismit constrains the relative motions of the motion body 12 with respect to the ground body 14 along the remaining four directions, making these four directions the DoCs.
[0083] In general, in MDoF flexure mechanisms the DoFs can be effected in one of two ways: via serial design, also known as serial kinematic (SK) design or chain or mechanism, or via parallel design, also known as parallel kinematic (PK) design or chain or mechanism. The technical term kinematics refers to the geometric study and description of motion of bodies relative to other bodies. FIG. 6 shows an abstract representation of a serial design. A first body 38 is connected to a second body 40 via a serial chain of intermediate bodies 42. If one traces or scribbles a line from the first body 38 to the second body 40, there is only one mechanical path or line, making this a serial design. The first body 38, second body 40, and intermediate bodies 42 are rigid in this representation (but with parasitic compliances, as previously described). Connectors 44 between the first body 38, second body 40, and intermediate bodies 42 are flexure modules, which themselves are composed of rigid and flexible/compliant bodies. The flexure modules can be as simple as the flexure blade 20 of FIG. 3 or as complex as the DPFM 10 of FIG. 1. FIG. 7 presents an example of a serial kinematic XY flexure mechanism. In the figure, an intermediate stage or body 46 is connected to a ground stage or body 48 by way of a first DPFM 50. The first DPFM 50 only allows relative X translation, per this example. A motion stage or body 52 is connected to the intermediate stage 46 by way of a second DPFM 54. The second DPFM 54 only allows relative Y translation, per this example. In a deformed state of the serial kinematic XY flexure mechanism of FIG. 7, the intermediate stage 46 will only have an X translation with respect to the ground stage 48. The X translation of the intermediate stage 46 is inherited by the motion stage 52. Since the motion stage 52 has a Y translation with respect to the intermediate stage 46 via the second DPFM 54, the motion stage 52 exhibits both the X and Y translations with respect to the ground stage 48.
[0084] Referring now to FIG. 8, an abstract representation of a parallel design is shown. A first body 56 is connected to a second body 58 via multiple independent chains of intermediate bodies. A first chain of intermediate bodies 60 has a first intermediate body 62 and first connectors 64; and a second chain of intermediate bodies 66 has first and second intermediate bodies 68, 70 and second connectors 72. If one traces or scribbles a line from the first body 56 to the second body 58, there is more than one mechanical path or line, making this a parallel design. The mechanical paths are not necessarily parallel in a geometric sense (e.g., two straight lines being parallel such as the opposing sides of a rectangle). Rather, parallel as used here implied multiple independent, non-overlapping chains or paths between the first and second bodies 56, 58.
[0085] FIG. 9 presents an example of a parallel kinematic XY flexure mechanism. In the figure, the parallel kinematic XY flexure mechanism has four rigid bodies: a ground stage or body 74, a first intermediate stage or body 76, a second intermediate stage or body 78, and a motion stage or body 80. The first intermediate stage 76 is connected to the ground stage 74 by way of a first DPFM 82. The first DPFM 82 only allows relative X translation, per this example. The motion stage 80 is connected to the first intermediate stage 76 by way of a second DPFM 84. The second DPFM 84 only allows relative Y translation, per this example. Further, the motion stage 80 is connected to the second intermediate stage 78 by way of a third DPFM 86. The third DPFM 86 only allows relative X translation, per this example. Lastly, the second intermediate stage 78 is connected to the ground stage 74 by way of a fourth DPFM 88. The fourth DPFM 88 only allows relative Y translation, per this example. In the deformed state of the parallel kinematic XY flexure mechanism of FIG. 9, the first intermediate stage 76 will only have an X translation with respect to the ground stage 74, and the second intermediate stage 78 will only have a Y translation with respect to the ground stage 74. The motion stage 80 inherits the X translation of the first intermediate stage 76 and inherits the Y translation of the of the second intermediate stage 78, thus acquiring two translation DoFs.
[0086] Without causing over-constraint, the parallel design of FIG. 9 is further enhanced by making use of symmetry, which involves adding a third and fourth intermediate stage. The resulting parallel kinematic XY flexure mechanism is presented in FIG. 10 and is expected to exhibit superior performance. In the embodiment of FIG. 10, the parallel kinematic XY flexure mechanism includes a ground stage or body 90, a motion stage or body 92, a first intermediate stage or body 94, a second intermediate stage or body 96, a third intermediate stage or body 98, a fourth intermediate stage or body 100, a first DPFM 102, a second DPFM 104, a third DPFM 106, a fourth DPFM 108, a fifth DPFM 110, a sixth DPFM 112, a seventh DPFM 114, and an eighth DPFM 116. The use of symmetry and the addition of DPFMs leads to improved stiffness of the parallel kinematic XY flexure mechanism of FIG. 10 in its DoCs, and also makes the flexure mechanism design insensitive to thermal distortions and manufacturing errors. Interestingly, classical mobility analysis of mechanisms suggests that the parallel kinematic XY flexure mechanism of FIG. 10 should be immobile due to over-constraint. In rigid-link mechanisms, geometric imperfections arising from manufacture and assembly can cause otherwise redundant constraints to become independent, leading to over-constraint and thereby restricting the mobility of the mechanism. Adding redundant constraints in rigid-link mechanisms leads to reductions in the associated DoF(s) of the mechanisms. But in flexure mechanismsand particularly distributed-compliance topologiesit has been found that elastic averaging plays an important role in ensuring that redundant constraints remain unaffected despite small geometric variations. Elastic averaging is a consequence of finite stiffness of the flexible bodies along the DoC(s) of flexure mechanisms. Thus, while the lack of ideal bearing behavior in flexure mechanisms results in performance compromises, on the one hand, it is also responsible for permitting certain special geometries in mechanism topology design, on the other hand. The concept of elastic averaging and symmetry can hence be exploited to add redundant constraints to a flexure mechanism in such a way that its DoF(s) remains conserved. As examples, the parallel kinematic XY flexure mechanisms of FIGS. 9 and 10 exhibit two DoFs and the DoFs are in the same directions, X direction and Y direction.
[0087] For a flexure mechanism consisting of multiple rigid bodies, there may be two or more rigid bodies that possess the exact same DoFs and DoCs. In the example parallel kinematic XY flexure mechanism of FIG. 10, for instance, the first intermediate stage 94 and the third intermediate stage 98 have the same DoF (i.e., the X direction), and have the same DoCs (i.e., Y, Z, .sub.x, .sub.y, .sub.z directions). Such rigid bodies can be referred to as familial stages. Under the umbrella term familial stages, there are three categories: [0088] 1) Twin Stages. If two or more familial stagesunder quasi static loading along their DoF(s)have kinematically identical motions, the stages are referred to as twin stages. With reference now to FIG. 20c, subsequently described, corresponding stages in a top layer or plane and a bottom layer or plane of the sandwich DPFM design of the figure constitute twin stages. For example, top and bottom secondary stages are twin stages in the sandwich DPFM design of FIG. 20c. [0089] 2) Sister Stages. If two or more familial stagesunder the same quasi static loading along their DoF(s)have kinematically compatible motion, the stages are referred to as sister stages. The motion amongst stages is said to be kinematically compatible if they have substantially the same motion along their DoF(s) without a conflict amongst their kinematic motions along their DoC(s). Note that unlike sister stages, twin stages have kinematically identical motion. In the example parallel kinematic XY flexure mechanism of FIG. 10, for instance, the first intermediate stage 94 and the third intermediate stage 98 are sister stages. With reference now to FIG. 11, when a positive X direction quasi static force (X Force) is applied at the first intermediate stage 94, both the first and third intermediate stages 94, 98 translate in the positive X direction by substantially the same amount, as shown. Here, the first and third intermediate stages 94, 98 do not have kinematic motion along their DoCs (may still have small elastic motion; refer to the Section on Ideal bearing behavior), and hence there is no possibility of a conflict arising between their kinematic motions. In a similar way, when an X direction quasi static force is applied to the third intermediate stage 98, both the third and first intermediate stages 98, 94 translate in the X direction by substantially the same amount, and neither have kinematic motion along their DoCs. Further, the second and fourth intermediate stages 96, 100 are sister stages, and thus would have substantially the same Y direction translation if a Y direction quasi static force is applied to either of them. [0090] 3) Cousin Stages. If two or more familial stagesunder quasi static loading along their DoF(s)have substantially the same motion along their DoF(s) with a kinematic conflict along their DoC(s), the stages are referred to as cousin stages. Still referring to FIG. 11, in this example parallel kinematic XY flexure mechanism, a fourth secondary stage or body 118 of the fourth DPFM 108 and an eighth secondary stage or body 120 of the eighth DPFM 116 are cousin stages. Upon application of the X Force, Y translations (y) of the fourth and eighth secondary stages 118, 120 are in opposite directions. The motions of the fourth and eighth secondary stages 118, 120 is composed of X translation (x) and Y translation (y), which are the DoF and DoC, respectively, of the fourth and eighth secondary stages 118, 120. This motion of a secondary stage of a DPFM is the result of beam arc length conservation, which is a kinematic constraint that arises from the elastic deformation of the flexure blades along their DoFs (here, the X direction). As the flexure blades undergo elastic deformation to allow the associated secondary stage to translate along the X direction, the beam arc length conservation (i.e., kinematic constraint) forces the end of the flexure bladeswhich is attached to the secondary stageto also translate in the Y direction, leading to the Y translation (y) of the secondary stage in addition to the X translation (x). Again, taking the fourth and eighth secondary stages 118, 120 of FIG. 11 as examples, the fourth secondary stage 118 and the eighth secondary stage 120 have the same X translation (x) but their Y translations (y) are in opposite directions. This leads to a kinematic conflict along their DoCs. Hence, the fourth and eighth secondary stages 118, 120 are cousin stages.
[0091] In a similar way, other secondary stages of other DPFMs in the parallel kinematic XY flexure mechanism of FIG. 11 are cousin stages. A first secondary stage or body 122 of the first DPFM 102 and a fifth secondary stage or body 124 of the fifth DPFM 110 are cousin stages. This relationship is evident in FIG. 11, where the Y translations (y) of the first and fifth secondary stages 122, 124 are in opposite directions with respect to each other upon application of the X force, while their X translations (x) are the same. Furthermore, a second secondary stage or body 126 of the second DPFM 104 and a sixth secondary stage or body 128 of the sixth DPFM 112 are cousin stages. And a third secondary stage or body 130 of the third DPFM 106 and a seventh secondary stage or body 132 of the seventh DPFM 114 are cousin stages. These latter cousin stages are evidenced in FIG. 12. In FIG. 12, upon application of a Y direction quasi static force (Y Force) at the second intermediate stage 96, the X translations (x) of the second and sixth secondary stages 126, 128 are in opposite directions with respect to each other, while their Y translations (y) are the same. And the X translations (x) of the third and seventh secondary stages 130, 132 are in opposite directions with respect to each other, while their Y translations (y) are the same.
Limitations of Prior Art
[0092] In many applications, it has been an aim of flexure mechanism design to make flexure mechanisms approach ideal bearing behavior, while knowing it can never truly be achieved. In general, there are three conditions often simultaneously sought in order to approach ideal bearing behavior in flexure mechanism design: i) infinite motion along the DoF(s) and infinite compliance (i.e., zero stiffness) along the DoF(s) over the entire range of motion; ii) zero motion along the DoC(s) over the entire range of motion; and iii) infinite stiffness (i.e., zero compliance) along the DoC(s) over the entire range of motion.
i) Infinite Motion Along the DoF(s) and Infinite Compliance (i.e., Zero Stiffness) Along the DoF(s) Over the Entire Range of Motion
[0093] As described, in general, flexure mechanisms are made up of flexible bodies and rigid bodies. The rigid bodies undergo displacement along their DoFs due to elastic deformation of the flexible bodies. When a flexible body undergoes elastic deformation, the flexible body builds stress within itself. The stress leads to the generation of internal loads within the flexible body that need to be overcome through the application of external loads in order to deform the flexible body. The larger the deformations, the more stress is built and larger external loads are needed to overcome it. Since the flexible bodies in a flexure mechanism require finite external loads for the rigid bodies to be displaced along their DoFs, the flexure mechanism cannot have infinite compliance (i.e., zero stiffness). Similarly, the flexible bodies have an upper limit on the amount of deformation they can undergo. This limit is determined by the yield stress of the material(s) used for manufacturing the flexible bodies. Therefore, the smaller the stiffness of the flexure mechanism along its DoFs, the larger will be its range of motion because it will take larger deformations in the flexible body to reach its yield stress point.
[0094] No flexure mechanism can achieve zero stiffness and infinite motion along its DoFs. However, the DoF stiffness can be made smaller by selecting the appropriate dimensions for the flexible bodies of the flexure mechanism so that the elastic deformation of the flexible bodies that leads to the motion of the rigid bodies along the DoFs are as large as possible for unit loads applied along the DoFs. With reference now to FIG. 13, the flexure blade 20 connects the ground stage 14 and the motion stage 12. The flexure blade 20 can undergo six different types of elastic deformationsaxial, in-plane bending, in-plane shear, out of plane bending, out of plane shear, and torsional deformations. The axial deformation of the flexure blade 20 occurs along the X direction and results in either stretching or compression of the flexure blade 20. This leads to the X translation (x) of the motion stage 12 relative to the ground stage 14. The in-plane bending and the in-plane shear deformation of the flexure blade 20 occur in the XY plane and lead to the Y translation (y) and .sub.z rotation (.sub.z) of the motion stage 12 relative to the ground stage 14. The out of plane bending and the out of plane shear deformation of the flexure blade 20 occur in the XZ plane and lead to the Z translation (z) and .sub.y rotation (.sub.y) of the motion stage 12 relative to the ground stage 14. The torsional deformation of the flexure blade 20 results in the twisting of the flexure blade 20 in the .sub.x direction. Therefore, it leads to the .sub.x rotation (.sub.x) of the motion stage 12 relative to the ground stage 14. The elastic deformations of the flexure blade 20 are associated with corresponding stiffness. The axial deformation of the flexure blade 20 is associated with the axial stiffness of flexure blade 20. Large axial stiffness of the flexure blade 20 implies small axial deformation of the flexure blade 20 when a force is applied to it along the X direction. This means small X translation (x) of the motion stage 12 relative to the ground stage 14. Similarly, the combination of in-plane shear and in-plane bending deformations of the flexure blade 20 is associated with the in-plane stiffness of the flexure blade 20. The combination of out of plane bending and out of plane shear deformations of the flexure blade 20 is associated with the out of plane stiffness of the flexure blade 20. The torsional deformation of the flexure blade 20 is associated with the torsional stiffness of the flexure blade 20. The distinction between in-plane and out of plane elastic deformations exist due to the rectangular cross-section of the flexure blade 20. In a conventional flexure bladelike the flexure blade 20 of FIG. 13one of the dimensions of its cross-section is much larger than the other dimension, i.e., h>>t. The elastic deformations of the flexure blade 20 in the plane that contains the larger dimension (h) are referred to as out of plane deformations, and the larger dimension (h) is referred to as the out of plane thickness of the flexure blade 20. The elastic deformations of the flexure blade 20 in the plane that contains the smaller dimension (t) are referred to as in-plane deformations, and the smaller dimension (t) is referred to as in-plane thickness of the flexure blade 20.
[0095] The flexure blade 20 of FIG. 13 can also be a flexible body in a DPFM, such as the DPFM 10 of FIG. 1. The DPFM 10 has one DoF, i.e., Y direction between the motion stage 12 and the ground stage 14. Therefore, in order to reduce its DoF stiffness, one needs to reduce the in-plane stiffness of the flexure blades 20. This can be achieved by increasing the length of the flexure blade 20, given by l in FIG. 13. The in-plane stiffness of the flexure blade 20 is very sensitive to the length (l) of the flexure blade 20. Hence, for a small increment in the length of the flexure blade 20, a large reduction in the DoF stiffness can be achieved. But the out of plane stiffness of the flexure blade 20 is also equally sensitive to the length (l) of the flexure blade 20. Therefore, a small increment in the flexure blade 20 length (l) will also lead to large reduction in the out of plane stiffness of the flexure blade 20 and hence a large reduction in the DoC stiffness of the associated DPFM. Therefore, the reduction in DoF stiffness comes at the cost of reduction in DoC stiffness. Another way to reduce the in-plane stiffness of the flexure blade 20 is to reduce the out of plane thickness of the flexure blade 20, given by h in FIG. 13. But this again leads to a similar trade-off as discussed above, where the reduction in DoF stiffness is accompanied by reduction in DoC stiffness. In this case, the trade-off is worse because the out of plane stiffness of the flexure blade 20 is more sensitive to variations in the out of plane thickness (h) as compared to the in-plane stiffness of the flexure blade 20. Hence, reducing the out of plane thickness (h) of the flexure blade 20 leads to higher percentage reduction in the DoC stiffness as compared to the DoF stiffness. Yet another way to reduce the in-plane stiffness of the flexure blade 20 is to reduce in plane thickness of the flexure blade 20, given by t in FIG. 13. In this case, there is still a trade-off where reduction in DoF stiffness comes at the cost of reduction in DoC stiffness. But this trade-off is better than the last two trade-offs discussed above. The in-plane stiffness of the flexure blade 20 is more sensitive to variation in the in-plane thickness (t) of the flexure blade 20 as compared to the out of plane stiffness. Hence, reduction in the in-plane thickness (t) of the flexure blade 20 will lead to higher percentage reduction in the DoF stiffness as compared to the DoC stiffness. Therefore, reducing the flexure blade in-plane thickness (t) is the most effective way to reduce the DoF stiffness and increase the range of motion of the associated flexure mechanisms that employ flexure blades such as DPFMs. However, there are practical constraints such as reliable manufacturing of ultra-thin flexure blades and the risk of buckling under loading, which place a lower limit on the in-plane thickness of flexure blades.
ii) Zero Motion Along the DoC(s) Over the Entire Range of Motion
[0096] There are certain flexure mechanism designs that have high DoC stiffness, yet exhibit finite error motion along the DoC due to the geometric placement of the flexible bodies and the kinematic constraint that is imposed by them. With reference now to FIG. 14, an embodiment of a parallelogram flexure mechanism (PFM) 134 is presented. The PFM 134 consists of two parallel flexure bodies or blades 136 connecting a motion body or stage 138 to a ground body or stage 140. The PFM 134 has one translational DoF in the Y direction between the motion stage 138 and the ground stage 140. However, when the motion stage 138 is displaced in the Y direction, it also translates in the X direction. This occurs because of the arc length conservation of the flexure blades 136. The arc length conservation of the flexure blade 136 is a kinematic constraint (because the flexure blade 136 is considered inextensible in the X direction) which dictates that the length of the flexure blade 136 should remain constant as it translates in the Y direction. In order to keep the flexure blade length constant, the end of the flexure blade 136 which is connected to the motion stage 138 translates in both X and Y directions, as shown in FIG. 14. The motion of the motion stage 138 in the X direction (DoC) is its kinematic motion along its DoC. To eliminate the finite error motion in the X direction when the motion stage 138 translates in the Y direction, a DPFM should be used instead of a PFM. An embodiment of a DPFM 142 is present in FIG. 15. The DPFM 142 is made up of two PFMs that are placed in geometrically reversed configuration. The DPFM 142, per this embodiment, has a ground body or stage 144, a motion body or stage 146, four flexure bodies or blades 148, and a secondary body or stage 150. Because the DPFM 142 is composed of two PFMs, the parasitic error motion of the two PFMs cancels each other such that the motion stage 146 has zero translation along the X direction. But this comes at the cost of reduced stiffness of the DPFM 142 along its DoC, i.e., X direction. FIG. 16 compares the stiffness of a PFM and of a DPFM along the X direction as a function of motion stage displacement along the Y direction. The graph of FIG. 16 illustrates that the stiffness of a DPFM is lower than that of a PFM at the nominal configuration (i.e., when motion stage Y displacement is zero) and continues to drop much faster as the motion stage Y displacement is increased. Therefore, here again, trade-offs between the competing requirements for ideal bearing behavior are illustrated.
iii) Infinite Stiffness (Zero Compliance) Along the DoC(s) Over the Entire Range of Motion
[0097] Finite compliance of a flexure mechanism along its DoC is due to the compliance of the associated flexible bodies and the parasitic compliance of the associated rigid bodies that are used to construct the flexure mechanism. Furthermore, the drop in the DoC stiffness of the flexure mechanism as its motion stage is displaced along the DoF is due to a combination of both the kinematic design of the flexure mechanism and the compliance of the flexible bodies along the DoCthis is referred to as the elastokinematic effect. The graph of FIG. 16 illustrates the rapid decrease in the X direction (DoC) stiffness of a DPFM as the motion stage displacement is increased in the Y direction (DoF). This decrease in DoC stiffness is attributed to the kinematically under-constrained secondary stage of the DPFM. However, the stiffness can be drastically improved by controlling the motion of the secondary stage such that its Y translation (y) remains approximately one-half of the Y translation (y) of the motion stage over the entire range of Y translation (y) of the motion stage. Therefore, the secondary stage is kinematically constrained to the motion stage. Certain flexure mechanism designs, such as those described in U.S. Pat. No. 9,200,689, have been shown to achieve this.
[0098] Cases in which the finite compliance of the flexible bodies along the DoC leads to loss of stiffness of the flexure mechanism are now described. In general, the DoC stiffness of a flexure mechanism can be classified as nominal DoC stiffness or non-nominal DoC stiffness. The DoC stiffness of the flexure mechanism, when its rigid body displacement along its DoF is zero, is called nominal DoC stiffness. The DoC stiffness of the flexure mechanism, when its rigid body has been displaced by finite amount along its DoF, is called non-nominal DoC stiffness. The graph of FIG. 17a shows the nominal and non-nominal Z direction (DoC) stiffness of a DPFM. The nominal DoC stiffness of the flexure mechanism occurs at y equal to zero, and the non-nominal DoC stiffness occurs at y not equal to zero. The nominal DoC stiffness can never be infinite due to the parasitic compliance of the rigid bodies and the finite compliance of the flexible bodies along the DoC. With reference now to FIG. 17b, an embodiment of a DPFM 152 has a ground body or stage 154, a motion body or stage 156, four flexure bodies or blades 158, and a secondary body or stage 160. FIG. 17b compares a deformed DPFM (represented in solid lines) with an undeformed DPFM (represented in broken lines) when Z direction force (F) is applied on the motion stage 156 which has not been displaced along its DoF in the Y direction. The Z direction is a DoC for the DPFM 152, which means it should exhibit infinite stiffness along that direction. But FIG. 17b illustrates a finite Z translation (z) and .sub.y rotation (.sub.y) of the motion stage 156 relative to the ground stage 154. These finite motions of the motion stage 156 along the DoCs are the result of the out of plane elastic deformations of the flexure blades 158, i.e., out of plane bending and out of shear deformation and the elastic deformations of the secondary stage 160 due to its parasitic compliances. These elastic deformations can be visualized in FIG. 17b.
[0099] Another observation from FIG. 17a is that the non-nominal DoC stiffness is smaller than the nominal stiffness and continues to drop as the motion stage is displaced along its DoF. FIG. 17c presents the DPFM 152 of FIG. 17b, and compares a deformed DPFM (represented in solid lines) with an undeformed DPFM (represented in broken lines) when Z direction force (F1) is applied on the motion stage 156 which has been displaced along its DoF in the Y direction (force F2). Since the flexure blades 158 have been deformed in the Y direction (DoF), the Z direction force on the motion stage 156 creates a torque that acts on the DPFM 152 which leads to the additional torsional deformation of the flexure blades 158 apart from the already existing out of plane elastic deformations. This causes additional displacement of the motion stage 156 in the Z direction. Therefore, the non-nominal stiffness of the DPFM 156 is lower than the nominal stiffness. As the motion stage Y translation (y) increases, the moment arm of the Z direction force F1 from the X axis increases, leading to more torque being applied on the DPFM 152 about the X axis. Higher torque leads to higher torsional deformation of the flexure blades 158 and hence the non-nominal stiffness keeps dropping as the motion stage displacement is increased along its DoF.
[0100] Furthermore, the .sub.y direction stiffness graph will behave the same way as the Z direction stiffness graph shown in FIG. 17a. When .sub.y direction moment is applied to the motion stage 156 in its nominal configuration, it will lead to finite Z translation (z) and .sub.y rotation (.sub.y) of the motion stage 156 relative to the ground stage 154. But unlike the case when only Z direction force is applied on the motion stage 156, this finite motion of the motion stage 156 along its DoCs will only be the result of out of plane bending deformation of the flexure blades 158 and the elastic deformations of the secondary stage 160 due to its parasitic compliances when only .sub.y direction moment is applied. When .sub.y direction moment is applied to the motion stage 156 in its non-nominal configuration, it leads to additional torsional deformation of the flexure blades 158 which leads to lower non-nominal .sub.y direction stiffness as compared to nominal .sub.y direction stiffness. Yet further, the .sub.x direction stiffness graph will behave the same way as the Z direction stiffness graph shown in FIG. 17a. When .sub.x direction moment is applied to the motion stage 156 in its nominal configuration, it will lead to finite .sub.x rotation (.sub.x) of the motion stage 156 relative to the ground stage 154, as shown in FIG. 17d. The finite .sub.x rotation (.sub.x) of the motion stage 156 along its DoC is the result of the out of plane elastic deformations of the flexure blades 158, i.e., out of plane bending and out of shear deformations and the elastic deformations of the secondary stage 160 due to its parasitic compliances. The flexure blades 158 connected to the motion stage 156 undergo the out of plane elastic deformation in opposite directions, as shown in FIG. 17d, which leads to the .sub.x rotation (.sub.x) of the motion stage 156 relative to the ground stage 154. When .sub.x direction moment is applied to the motion stage 156 in its non-nominal configuration, it leads to additional torsional deformation of the flexure blades 158 which leads to further .sub.x rotation (.sub.x) of the motion stage 156 and lower non-nominal .sub.x direction stiffness as compared to nominal .sub.x direction stiffness.
[0101] In FIGS. 17b, 17c, and 17d, the elastic deformations of the flexure blades 158 are much more pronounced than the elastic deformation of the secondary stage 160 due to its parasitic compliances. This occurs because the parasitic compliances of the secondary stage 160 are, generally, very small compared to the compliances of the flexure blades 158 along the DoCs. Accordingly, focus is on mitigation strategies to reduce the compliances of the flexure blades 158 in order to increase the DoC stiffness of the flexure mechanism.
[0102] One way to increase the nominal and non-nominal stiffness of a DPFM along the associated Z direction is to increase flexure blade stiffness in the Z direction and its torsional stiffness. This could be achieved by choosing appropriate dimensions of the flexure blades (e.g., flexure blades 158). In order to increase the Z direction and torsional stiffness, the out of plane thickness (e.g., h in FIG. 13) of the flexure blades can be increased. However, as described above, increasing the out of plane thickness of the flexure blades has been found to increase the DoF stiffness of the DPFM. Similarly, any change in the flexure blade dimension to increase its Z direction and torsional stiffness leads to increase in the DoF stiffness of the flexure mechanism.
[0103] Another way to reduce the nominal and non-nominal compliance of a DPFM is to replace simple flexure blades with reinforced flexure blades, such as the flexure module 24 with the reinforcement body 30 shown and described with reference to FIG. 4. FIG. 18a illustrates an embodiment of a DPFM 162 design with reinforced flexure blades. The DPFM 162, according to this embodiment, has a ground body or stage 164, a motion body or stage 166, four reinforced flexure bodies or blades 168, and a secondary body or stage 170. The reinforced flexure blades 168 each include a first flexure blade 172, a second flexure blade 174, and a reinforcement body or stage 176 therebetween. The use of the reinforcement stages 176 in the flexure blades 168 reduces the torsional and the Z direction compliance of the flexure blades 168. Hence it leads to substantial improvement in the nominal and non-nominal stiffness of the DPFM 162. However, this comes at the cost of the reduced DoF compliance (i.e., increased DoF stiffness). In the graph of FIG. 18b, as a reinforcement length (L.sub.R in FIG. 18a) of the reinforcement stages 176 is increased, the nominal DoC stiffness of the DPFM 162 increases as well. The graph of FIG. 18c, on the other hand, shows that as the reinforcement length (L.sub.R) is increased, the DoF stiffness of the DPFM 162 increases. Hence, the use of the reinforced flexure blades 168 is also not able to overcome the trade-off between the DoF and DoC stiffness of a DPFM. The trade-off between the DoF and DoC stiffness is a limitation of the known prior art on the design of flexure mechanisms.
[0104] When DPFMs are employed as flexure modules to build parallel kinematic XY flexure mechanisms, the deficiencies of a DPFM that prohibit it from achieving ideal bearing behavior are inherited by the parallel kinematic XY flexure mechanisms. FIG. 19a presents an embodiment of a parallel kinematic XY flexure mechanism that is constructed using twelve DPFMs as flexure modules and has two DoFs between a motion stage or body 178 and a ground stage or body 180, i.e., X direction and Y direction. This embodiment includes a first intermediate stage or body 182, a second intermediate stage or body 184, a third intermediate stage or body 186, a fourth intermediate stage or body 188, a first DPFM 190, a second DPFM 192, a third DPFM 194, a fourth DPFM 196, a fifth DPFM 198, a sixth DPFM 200, a seventh DPFM 202, an eighth DPFM 204, a ninth DPFM 206, a tenth DPFM 208, an eleventh DPFM 210, and a twelfth DPFM 212. The DPFMs that connect the intermediate stages to the ground stage 180 are referred to as peripheral DPFMs, while the DPFMs that connect the intermediate stages to the motion stage 178 are referred to as inner DPFMs. In the embodiment of FIG. 19a the first, third, fourth, sixth, seventh, ninth, tenth, and twelfth DPFMS 190, 194, 196, 200, 202, 206, 208, 212 constitute peripheral DPFMs; and the second, fifth, eighth, and eleventh DPFMs 192, 198, 204, 210 constitute inner DPFMs.
[0105] In FIG. 19a, the parallel kinematic XY flexure mechanism is in its nominal configuration, i.e., the motion stage 178 has no displacement along its DoFs. In this configuration, if a Z direction force is applied at the center of the motion stage 178, then the motion stage 178 undergoes finite Z translation (z) along its DoC due to finite Z direction compliance of the flexure mechanism. In order to understand the source of the finite Z direction compliance of the flexure mechanism, FIG. 19b shows a sectional view (taken along the plane X-X denoted in FIG. 19a) of the nominal configuration of the flexure mechanism when Z direction force (F) is applied at the center of the motion stage 178. FIG. 19b illustrates that the Z direction force at the motion stage 178 causes the flexure blades of the inner second, fifth, and eighth DPFMs 192, 198, 204 to undergo out of plane bending and shear deformations which leads to Z translation (z) of the motion stage 178 relative to the first, fourth, and third intermediate stages 182, 188, 186. This out of plane deformation of the inner second, fifth, and eighth DPFMs 192, 198, 204 in their nominal configurations is similar to the deformation of the DPFM 152 shown in FIG. 17b.
[0106] The motion of the first, third, and fourth intermediate stages 182, 186, 188 relative to the ground stage 180, per this embodiments, generally has two components. A first component is the Z translation (z) of the first, third, and fourth intermediate stages 182, 186, 188, which is caused by the out of plane shear and bending deformations of the accompanying flexure blades of the peripheral third, fourth, sixth, and seventh DPFMs 194, 196, 200, 202. This out of plane of deformations of these peripheral DPFMs 194, 196, 200, 202 is similar to the deformation of the DPFM 152 illustrated in FIG. 17b. The peripheral third, fourth, sixth, and seventh DPFMs 194, 196, 200, 202 connecting the first, third, and fourth intermediate stages 182, 186, 188 to the ground stage 180 undergo rotation about the Y1, Y3, and X4 axes (FIG. 19b) respectively leading to .sub.y rotation (.sub.y) of the first and third intermediate stages 182, 186 and leading to .sub.x rotation (.sub.x) of the fourth intermediate stage 188 relative to the ground stage 180. This out of plane deformation of the peripheral DPFMs 194, 196, 200, 202, which is also caused by the out of plane bending and shear deformation of the accompanying flexure blades, is similar to the deformation of the DPFM 152 illustrated in FIG. 17d. These rotations constitute a second component of the motion of the first, third, and fourth intermediate stages 182, 186, 188 relative to the ground stage 180. The sectional view of the parallel kinematic XY flexure mechanism in the embodiment of FIG. 19b only shows the deformation of inner second, fifth, and eighth DPFMs 192, 198, 204 and peripheral third, fourth, sixth, and seventh DPFMs 194, 196, 200, 202. But the remaining inner eleventh DPFM 210 and peripheral first, ninth, tenth, and twelfth DPFMs 190, 206, 208, 212 (not shown in FIG. 19b) would also experience and undergo similar out of plane deformations as the inner DPFMs 192, 198, 204 and peripheral DPFMs 194, 196, 200, 202 shown in FIG. 19b. Furthermore, in FIG. 19b, the first and third intermediate stages 182, 186 experience and undergo .sub.y rotation about their respective axes, i.e., Y1 and Y3 axes in opposite directions. Therefore, it has been found that a design intervention that restricts the relative rotation between the first and third intermediate stages 182, 186 will lead to higher Z direction stiffness for the parallel kinematic XY flexure mechanism. Such a design intervention is described below.
[0107] If the parallel kinematic XY flexure mechanism of FIG. 19a is displaced along one of its DoFsfor example, the motion stage 178 is displaced in the X directionthen the inner fifth and eleventh DPFMs 198, 210 and peripheral first, third, seventh, and ninth DPFMs 190, 194, 202, 206 are in their non-nominal configuration. When Z direction force (denoted Fin FIG. 19b) is applied at the motion stage 178, the accompanying flexure blades of the inner fifth and eleventh DPFMs 198, 210 and peripheral first, third, seventh, and ninth DPFMs 190, 194, 202, 206 in their non-nominal configuration experience and undergo additional torsional deformation. This deformation of the inner and peripheral DPFMs 198, 210, 190, 194, 202, 206 in their non-nominal configuration is similar to the deformation of the DPFM 152 in FIG. 17c. The reduction in the non-nominal stiffness of the inner and peripheral DPFMs 198, 210, 190, 194, 202, 206 due to the additional torsional deformation of their accompanying flexure blades leads to the reduction of the non-nominal Z direction stiffness of the parallel kinematic XY flexure mechanism. The nominal and non-nominal Z direction stiffness of the parallel kinematic XY flexure mechanism as a function of the motion stage displacement in the X direction is shown in the graph of FIG. 19c. The graph demonstrates finite DoC stiffness and decreasing non-nominal stiffness as the motion stage displacement is increased. When the motion stage 178 is displaced in the Y direction it leads to similar Z direction stiffness graph as shown in FIG. 19c.
Structural Reinforcements for Improved Static and Dynamic Performance
[0108] The known prior art has hence been shown to have limitations in designing flexure mechanisms when it comes to approaching ideal bearing behavior. It has been found that there can always be a significant trade-off between the DoF and DoC stiffness that can be achieved through any of the known prior art flexure mechanism designs.
[0109] Therefore, certain embodiments of flexure mechanism designs are set forth in this description that have been shown to mitigate or altogether resolve these trade-offs, and thereby approach the ideal bearing behavior more closely than has previously been shown. One or more inter-stage connections can be implemented in flexure mechanism designs, per the embodiments herein, in order to mitigate or altogether resolve the trade-offs and provide improved static and dynamic performance of the flexure mechanism. Inter-stage connectionsas the term and grammatical variations thereof are used hereinare connections that restrict one or more relative motions in one or more directions between two rigid bodies or stages that are otherwise not directly connected to each other within a flexure mechanism. An inter-stage connection referred to as a rigid inter-stage connection herein restricts and constrains relative motion along all six directions of motion between two rigid bodies. Other types of inter-stage connections restrict and constrain relative motion between two rigid bodies in one, two, three, four, or five directions of motion. Such inter-stage connections are referred to as 1 DoC, 2 DoC, 3 DoC, 4 DoC, and 5 DoC inter-stage connections. In the embodiments of 1 to 5 DoC inter-stage connections, the connections are rigid in directions along which relative motion is restricted between two rigid bodies, and flexible in directions along which relative motion is allowed between two rigid bodies.
[0110] In the flexure mechanism embodiments described herein with redundant constraints and symmetrical layouts, there are familial stages or rigid bodies, as set forth above: twin stages, sister stages, and cousin stages. In any flexure mechanism design with twin stages, sister stages, and/or cousin stages, it has been determined that rigid inter-stage connection(s) can only be made between two or more twin stages or two or more sister stages, while a 1 DoC, 2 DoC, 3 DoC, 4 DoC, and/or 5 DoC inter-stage connection(s) can be made either between two or more twin stages, two or more sister stages, and/or two or more cousin stages.
[0111] With reference now to FIG. 20a1, an embodiment of a sandwich flexure module 216 is illustrated. The sandwich flexure module 216 employs out of plane rigid inter-stage connections (sometimes referred to as simply rigid inter-stage connection(s) or just inter-stage connection(s)) to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) compared to known prior art flexure mechanism designs. The phrase out of plane, as used herein with the phrase inter-stage connection(s), is not intended to mean that the inter-stage connection(s) is itself strictly configured out of a particular plane and orthogonally relative to a particular plane; rather, the phrase out of plane is in reference to the stages subject to connection being arranged in different planes or layers with respect to each other. In this embodiment, the sandwich flexure module 216 is made-up of two identical flexure modules with reinforced blades (like the flexure module 24 of FIG. 4, for example) spaced apart from each other in a third direction, which in this embodiment is the Z direction (a first direction in this embodiment is the X direction, and a second direction in this embodiment is the Y direction; still, these directions could differ in other embodiments): a first or upper flexure module 218 and a second or lower flexure module 220. The first flexure module 218, per this embodiment, includes three rigid bodies: a first ground stage or body 222, a first motion stage or body 224, and a first reinforcement stage or body 226. Similarly, the second flexure module 220 includes three rigid bodies in this embodiment: a second ground stage or body 228, a second motion stage or body 230, and a second reinforcement stage or body 232. Further, the first flexure module 218 has a first flexure body or blade 225 reinforced via the first reinforcement stage 226; and, similarly, the second flexure module 220 has a second flexure body or blade 231 reinforced via the second reinforcement stage 232.
[0112] Here, the first and second ground stages 222, 228 constitute twin stages, the first and second motion stages 224, 230 constitute twin stages, and the first and second reinforcement stages 226, 232 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via rigid inter-stage connections, as depicted in FIG. 20a1. The broken lines in FIG. 20a1 are intended to denote the rigid inter-stage connection of the first and second reinforcement stages 226, 232. A first or ground rigid inter-stage connection 234 connects the first and second ground stages 222, 228, and extends in the Z direction therebetween. A second or motion rigid inter-stage connection 236 connects the first and second motion stages 224, 230, and extends in the Z direction therebetween. Lastly, a third or reinforcement rigid inter-stage connection 238 connects the first and second reinforcement stages 226, 232, and extends in the Z direction therebetween. In general, the construction of the rigid inter-stage connections described here and elsewhere in this description can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures. The embodiment of the flexure sandwich module 216 depicted in FIG. 20a1 constitutes the simplest sandwich flexure module that serves as a building block for constructing subsequent sandwich flexure module assemblies set forth herein.
[0113] With reference now to FIG. 20a2, an embodiment of a sandwich flexure module 217 is illustrated. As before, the sandwich flexure module 217 employs out of plane rigid inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) compared to known prior art flexure mechanism designs. This embodiment is similar to that of FIG. 20a1, and hence some descriptions may not be repeated here. A first or upper flexure module 219, per this embodiment, includes three rigid bodies: a first ground stage or body 221, a first motion stage or body 223, and a first reinforcement stage or body 227. Similarly, a second flexure module 229 includes three rigid bodies in this embodiment: a second ground stage or body 233, a second motion stage or body 235, and a second reinforcement stage or body 237. Further, the first flexure module 219 has a first flexure body or blade 239 reinforced via the first reinforcement stage 227; and, similarly, the second flexure module 229 has a second flexure body or blade 241 reinforced via the second reinforcement stage 237.
[0114] Here, the first and second ground stages 221, 233 constitute twin stages, the first and second motion stages 223, 235 constitute twin stages, and the first and second reinforcement stages 227, 237 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via out of plane rigid inter-stage connections, as depicted in FIG. 20a2. A first or ground rigid inter-stage connection 243 connects the first and second ground stages 221, 233, and extends in the Z direction therebetween. A second or motion rigid inter-stage connection 245 connects the first and second motion stages 223, 235, and extends in the Z direction therebetween. Lastly, a third or reinforcement rigid inter-stage connection 247 connects the first and second reinforcement stages 227, 237, and extends in the Z direction therebetween. Furthermore, and unlike the previous embodiment of FIG. 20a1, the first and second reinforcement stages 227, 237, as well as the reinforcement rigid inter-stage connection 247 are furnished via a single monolithic body 249. In other words, the monolithic body 249 establishes and constitutes the first and second reinforcement stages 227, 237 and the reinforcement rigid inter-stage connection 247. In this embodiment, the monolithic body 249 is secured to the first and second flexure blades 239, 241 via bolts and nuts 253. Still, the securement can involve adhesives, rivets, welds, monolithic construction and geometry, and/or other types of attachments and structures.
[0115] With reference now to FIG. 20a3, an embodiment of a sandwich flexure module 255 is illustrated. The sandwich flexure module 255 employs out of plane inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) compared to known prior art flexure mechanism designs. This embodiment is similar in some regards to that of FIG. 20a2, and hence some descriptions may not be repeated here. A first or upper flexure module 257, per this embodiment, includes three rigid bodies: a first ground stage or body 261, a first motion stage or body 263, and a first reinforcement stage or body 265. Similarly, a second flexure module 267 includes three rigid bodies in this embodiment: a second ground stage or body 269, a second motion stage or body 271, and a second reinforcement stage or body 273. Further, the first flexure module 257 has a first flexure body or blade 275 reinforced via the first reinforcement stage 265; and, similarly, the second flexure module 267 has a second flexure body or blade 277 reinforced via the second reinforcement stage 273.
[0116] Here, the first and second ground stages 261, 269 constitute twin stages, the first and second motion stages 263, 271 constitute twin stages, and the first and second reinforcement stages 265, 273 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via out of plane inter-stage connections, as depicted in FIG. 20a3. A first or ground rigid inter-stage connection 279 connects the first and second ground stages 261, 269, and extends in the Z direction therebetween. A second or motion rigid inter-stage connection 283 connects the first and second motion stages 263, 271, and extends in the Z direction therebetween. Lastly, a third or reinforcement inter-stage connection 285 connects the first and second reinforcement stages 265, 273, and extends in the Z direction therebetween. Unlike the reinforcement rigid inter-stage connections of previous embodiments, the third reinforcement inter-stage connection 285 is not rigid in all six directions (i.e., X, Y, Z, .sub.x, .sub.y, .sub.z directions). Rather, the third reinforcement inter-stage connection 285 is rigid in four directions, namely, the X, Z, .sub.x, and .sub.y, directions. The third reinforcement inter-stage connection 285 constrains relative motion between the first and second reinforcement stages 265, 273 in these four directions. The third reinforcement inter-stage connection 285 allows relative motion between the first and second reinforcement stages 265, 273 along the remaining two directions, namely, the Y and .sub.z directions. In this embodiment, a thickness dimension t.sub.2 of the third reinforcement inter-stage connection 285 can be approximately equivalent to a thickness dimension t.sub.1 of the first and second flexure blades 275, 277, enabling the relative motion between the first and second reinforcement stages 265, 273 along the Y and .sub.z directions. Further, the thickness dimensions t.sub.1, t.sub.2 in this embodiment are much less than a thickness dimension t.sub.R of the first and second reinforcement stages 265, 273 themselves (i.e., t.sub.1t.sub.2<<t.sub.R). In this embodiment, the third reinforcement inter-stage connection 285 is secured to the first and second reinforcement stages 265, 273 via bolts and nuts 287. Still, the securement can involve adhesives, rivets, welds, monolithic construction and geometry, and/or other types of attachments and structures. Moreover, the third reinforcement inter-stage connection 285 could have other designs and constructions in other embodiments for constraining relative motion between the first and second reinforcement stages 265, 273 in the four directions, and such an out of plane inter-stage connection that constrains relative motion in the four directions could be implemented in other flexure mechanism designs and constructions depicted and described herein. For example, such an out of plane inter-stage connection that constrains relative motion in the four directions could be implemented between secondary stages in a sandwich double parallelogram flexure mechanism (DPFM), like those described herein, and/or could be implemented between intermediate stages in a parallel kinematic sandwich XY flexure mechanism, like those described herein.
[0117] With reference now to FIG. 20a4, an embodiment of a sandwich flexure module 289 is illustrated. The sandwich flexure module 289 employs out of plane inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) compared to known prior art flexure mechanism designs. This embodiment is similar in some regards to that of FIG. 20a3, and hence some descriptions may not be repeated here. A first or upper flexure module 291, per this embodiment, includes three rigid bodies: a first ground stage or body 293, a first motion stage or body 297, and a first reinforcement stage or body 299. Similarly, a second flexure module 301 includes three rigid bodies in this embodiment: a second ground stage or body 303, a second motion stage or body 305, and a second reinforcement stage or body 307. Further, the first flexure module 291 has a first flexure body or blade 309 reinforced via the first reinforcement stage 299; and, similarly, the second flexure module 301 has a second flexure body or blade 311 reinforced via the second reinforcement stage 307.
[0118] Here, the first and second ground stages 293, 303 constitute twin stages, the first and second motion stages 297, 305 constitute twin stages, and the first and second reinforcement stages 299, 307 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via out of plane inter-stage connections, as depicted in FIG. 20a4. A first or ground rigid inter-stage connection 313 connects the first and second ground stages 293, 303, and extends in the Z direction therebetween. A second or motion rigid inter-stage connection 315 connects the first and second motion stages 297, 305, and extends in the Z direction therebetween. Lastly, a third or reinforcement inter-stage connection 317 connects the first and second reinforcement stages 299, 307, and extends in the Z direction therebetween. Unlike the reinforcement rigid inter-stage connections of some previous embodiments, the third reinforcement inter-stage connection 317 is not rigid in all six directions (i.e., X, Y, Z, .sub.x, .sub.y, .sub.z directions). Rather, the third reinforcement inter-stage connection 317 is rigid in three directions, namely, the X, Z, and .sub.y, directions. The third reinforcement inter-stage connection 317 constrains relative motion between the first and second reinforcement stages 299, 307 in these three directions. The third reinforcement inter-stage connection 317 allows relative motion between the first and second reinforcement stages 299, 307 along the remaining three directions, namely, the Y, .sub.x, and .sub.z directions. In this embodiment, a thickness dimension t.sub.2 of the third reinforcement inter-stage connection 317 can be approximately equivalent to a thickness dimension t.sub.1 of the first and second flexure blades 309, 311, enabling the relative motion between the first and second reinforcement stages 299, 307 along the Y, .sub.x, and .sub.z directions. Further, the thickness dimensions t.sub.1, t.sub.2 in this embodiment are much less than a thickness dimension t.sub.R of the first and second reinforcement stages 299, 307 themselves (i.e., t.sub.1t.sub.2<<t.sub.R). In this embodiment, the third reinforcement inter-stage connection 317 is a monolithic construction with the first and second reinforcement stages 299, 307. Moreover, the third reinforcement inter-stage connection 317 could have other designs and constructions in other embodiments for constraining relative motion between the first and second reinforcement stages 299, 307 in the three directions, and such an out of plane inter-stage connection that constrains relative motion in the three directions could be implemented in other flexure mechanism designs and constructions depicted and described herein. For example, such an out of plane inter-stage connection that constrains relative motion in the three directions could be implemented between secondary stages in a sandwich double parallelogram flexure mechanism (DPFM), like those described herein, and/or could be implemented between intermediate stages in a parallel kinematic sandwich XY flexure mechanism, like those described herein.
[0119] With reference now to FIG. 20b, an embodiment of a sandwich parallelogram flexure mechanism (PFM) 240 is illustrated. In this embodiment, a pair of the sandwich flexure modules 216 from FIG. 20a1 are used to construct the sandwich PFM 240. The sandwich PFM 240 employs out of plane rigid inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) compared to known prior art PFM designs. In this embodiment, the sandwich PFM 240 is made-up of two identical parallel flexure modules with reinforced blades (similar to the PFM 134 of FIG. 14 but with reinforcements, for example) spaced apart from each other in the Z direction: a first or upper PFM 242 and a second or lower PFM 244. The separation of the first and second PFMs 242, 244 along the Z direction is denoted in FIG. 20b by d.sub.R. The first PFM 242, per this embodiment, includes four rigid bodies: a first ground stage or body 246, a first reinforcement stage or body 248, a second reinforcement stage or body 250, and a first motion stage or body 252. Similarly, the second PFM 244 includes four rigid bodies in this embodiment: a second ground stage or body 254, a third reinforcement stage or body 256, a fourth reinforcement stage or body 258, and a second motion stage or body 260. Further, the first PFM 242 has first flexure bodies or blades 251 reinforced via the first and second reinforcement stages 248, 250; and, similarly, the second PFM 244 has second flexure bodies or blades 259 reinforced via the third and fourth reinforcement stages 256, 258. In an example, the distance d.sub.R can range between approximately twenty percent (20%) to eighty percent (80%) of a length l of the flexure blades 251, 259; still, other ranges and values are possible in other examples. Further, an out of plane thickness and height h of the ground stages 246, 254 and motion stages 252, 260 can range between approximately twenty percent (20%) to fifty percent (50%) of the length l of the flexure blades 251, 259; still, other ranges and values are possible in other examples. Still further, in at least some embodiments, the distance d.sub.Rwhich may be referred to as the out of plane extent of the out of plane inter-stage connectionsis greater in value than the height h of the ground stages 246, 254 and motion stages 252, 260. Satisfying these conditions, it has been observed, may facilitate and enable the beneficial performances described herein, according to at least some embodiments.
[0120] In FIG. 20b, the first and second ground stages 246, 254 constitute twin stages, the first and third reinforcement stages 248, 256 constitute twin stages, the second and fourth reinforcement stages 250, 258 constitute twin stages, and the first and second motion stages 252, 260 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via out of plane inter-stage connections, as depicted in the figure. The out of plane inter-stage connections could constrain relative motion in three directions, four directions, or in all six directions, as described, per various embodiments. The broken lines in FIG. 20b are intended to denote such inter-stage connections for the reinforcement stages. A first or ground rigid inter-stage connection 262 connects the first and second ground stages 246, 254, and extends in the Z direction therebetween. In this embodiment, the first rigid inter-stage connection 262 is in the form of a pair of fasteners or bolts. A second or first reinforcement rigid inter-stage connection 264 connects the first and third reinforcement stages 248, 256, and extends in the Z direction therebetween. In this embodiment, the second rigid inter-stage connection 264 is in the form of a monolithic construction. Further, a third or second reinforcement rigid inter-stage connection 266 connects the second and fourth reinforcement stages 250, 258, and extends in the Z direction therebetween. In this embodiment, the third rigid inter-stage connection 266 is in the form of a monolithic construction. Lastly, a fourth or motion rigid inter-stage connection 268 connects the first and second motions stages 252, 260, and extends in the Z direction therebetween. In this embodiment, the fourth rigid inter-stage connection 268 is in the form of a pair of fasteners or bolts. Again here, in alternative embodiments, the construction of the rigid inter-stage connections of the sandwich PFM 240 can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures.
[0121] With reference now to FIG. 20c, an embodiment of a sandwich double parallelogram flexure mechanism (DPFM) 270 is illustrated. The sandwich DPFM 270 employs out of plane inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) for a given Y translation DoF stiffness compared to known prior art DPFM designs. In this embodiment, the sandwich DPFM 270 is made-up of two identical double parallelogram flexure mechanisms with reinforced flexure blades (like the DPFM 162 in FIG. 18a, for example) spaced apart from each other in the Z direction: a first or upper DPFM 272 and a second or lower DPFM 274. The separation of the first and second DPFMs 272, 274 along the Z direction is denoted in FIG. 20c by distance d.sub.R. The first DPFM 272, per this embodiment, includes multiple rigid bodies: a first secondary stage or body 276 (also considered a motion stage or body), a first ground stage or body 278, a first motion stage or body 280, a first reinforcement stage or body 282, a second reinforcement stage or body 284, a third reinforcement stage or body 286, and a fourth reinforcement stage or body 288. Similarly, the second DPFM 274 includes multiple rigid bodies in this embodiment: a second secondary stage or body 290 (also considered a motion stage or body), a second ground stage or body 292, a second motion stage or body 294, a fifth reinforcement stage or body 296, a sixth reinforcement stage or body 298, a seventh reinforcement stage or body 300, and an eighth reinforcement stage or body 302. Further, the first DPFM 272 has first flexure bodies or blades 281 reinforced via the reinforcement stages 282, 284, 286, 288; and, similarly, the second DPFM 274 has second flexure bodies or blades 295 reinforced via the reinforcement stages 296, 298, 300, 302. In an example, and similar to the embodiment of FIG. 20b, the distance d.sub.R of FIG. 20c can range between approximately twenty percent (20%) to eighty percent (80%) of the length l (FIG. 20b) of the flexure blades 281, 295; still, other ranges and values are possible in other examples. Further, and as before, the height h (FIG. 20b) of the ground stages 278, 292 and motion stages 280, 294 can range between approximately twenty percent (20%) to fifty percent (50%) of the length l of the flexure blades 281, 295; still, other ranges and values are possible in other examples.
[0122] In FIG. 20c, the first and second secondary stages 276, 290 constitute twin stages, the first and second ground stages 278, 292 constitute twin stages, the first and second motion stages 280, 294 constitute twin stages, the first and fifth reinforcement stages 282, 296 constitute twin stages, the second and sixth reinforcement stages 284, 298 constitute twin stages, the third and seventh reinforcement stages 286, 300 constitute twin stages, and the fourth and eighth reinforcement stages 288, 302 constitute twin stages. Accordingly, it has been determined that these twin stages can be connected together via out of plane inter-stage connections, as depicted in the figure. The out of plane inter-stage connections could constrain relative motion in three directions, four directions, or in all six directions, as described, per various embodiments. The broken lines in FIG. 20c are intended to denote such out of plane inter-stage connections of the reinforcement stages. A first or secondary rigid inter-stage connection 304 connects the first and second secondary stages 276, 290, and extends in the Z direction therebetween. In this embodiment, the first rigid inter-stage connection 304 is in the form of a pair of fasteners or bolts. A second or ground rigid inter-stage connection 306 connects the first and second ground stages 278, 292, and extends in the Z direction therebetween. In this embodiment, the second rigid inter-stage connection 306 is in the form of a pair of fasteners or bolts. A third or motion rigid inter-stage connection 308 connects the first and second motion stages 280, 294, and extends in the Z direction therebetween. In this embodiment, the third rigid inter-stage connection 308 is in the form of a pair of fasteners or bolts. A fourth or first reinforcement rigid inter-stage connection 310 connects the first and fifth reinforcement stages 282, 296, and extends in the Z direction therebetween. In this embodiment, the fourth rigid inter-stage connection 310 is in the form of a monolithic construction. A fifth or second reinforcement rigid inter-stage connection 312 connects the second and sixth reinforcement stages 284, 298, and extends in the Z direction therebetween. In this embodiment, the fifth rigid inter-stage connection 312 is in the form of a monolithic construction. A sixth or third reinforcement rigid inter-stage connection 314 connects the third and seventh reinforcement stages 286, 300, and extends in the Z direction therebetween. In this embodiment, the sixth rigid inter-stage connection 314 is in the form of a monolithic construction. Lastly, a seventh or fourth reinforcement rigid inter-stage connection 316 connects the fourth and eighth reinforcement stages 288, 302, and extends in the Z direction therebetween. In this embodiment, the seventh rigid inter-stage connection 316 is in the form of a monolithic construction. Again here, in alternative embodiments, the construction of the rigid inter-stage connections of the sandwich DPFM 270 can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures.
[0123] In the embodiments of the DPFM 162 and the sandwich DPFM 270 presented in FIGS. 18a and 20c, the dimensions of the first and second DPFMs 272, 274 (FIG. 20c) and of the DPFM 162 (FIG. 18a) are precisely the same, with the exception that the out of plane thicknesses (i.e., along the Z direction) of the individual first and second DPFMs 272, 274 of the sandwich DPFM 270 are one-half the corresponding out of plane thickness of the DPFM 162. This means that the total out of plane thickness of the respective flexure blades of the DPFM 162 and of the DPFMs 272, 274 are equivalent relative to each other. That is, h.sub.1 in FIG. 18a is equal to h.sub.2 and h.sub.3 in FIG. 20c added together (h.sub.1=h.sub.2+h.sub.3). Since the twin stages of the first and second DPFMs 272, 274 of FIG. 20c exhibit kinematically identical motion, it has been determined that rigidly connecting them via rigid inter-stage connections in the out of plane manner as depicted in FIG. 20c does not increase the associated DoF stiffness. Accordingly, it has been found that the sandwich DPFM 270 of FIG. 20c and the DPFM 162 of FIG. 18a will possess similar Y translation DoF stiffness between the respective motion stages 280, 294 and 166 and ground stages 278, 292 and 164.
[0124] Moreover, it has been found that due to the separation of the first and second DPFMs 272, 274 of the sandwich DPFM 270 along the Z direction (i.e., distance d.sub.R), the out of plane direction DoC stiffness of the sandwich DPFM 270i.e., Z translation, .sub.x rotation, and .sub.y rotation direction stiffnessesbecomes significantly higher than the corresponding out of plane direction DoC stiffnesses of the DPFM 162 of FIG. 18a. This is evidenced by the graph presented in FIG. 20f. The graph shows a comparison of the nominal and non-nominal Z translation direction stiffnesses of the sandwich DPFM 270 and the DPFM 162, according to the embodiments set forth. Graphs of the .sub.x and .sub.y rotation direction stiffnesses of the same comparison as the graph in FIG. 20f would yield the same outcome shown and described in connection with FIG. 20f. The nominal Z translation direction stiffness is at zero motion stage displacement in the graph, and the non-nominal Z translation direction stiffness is at non-zero motion stage displacements. The sandwich DPFM 270 is represented by line L.sub.1 in the graph, and the DPFM 162 is represented by line L.sub.2. Without intending to be confined to a particular theory of causation, it has been determined that the nominal Z translation direction stiffness of the sandwich DPFM 270 (line L.sub.1) is higher than that of the DPFM 162 (line L.sub.2) because the Z direction separation (i.e., distance d.sub.R) of the first flexure blades 281 (first DPFM 272) and of the second flexure blades 295 (second DPFM 274) restrict rotation of the reinforcement stages 282, 284, 286, 288, 296, 298, 300, 302 about the .sub.y rotation direction when Z direction force is applied at the motion stages 280, 294.
[0125] Furthermore, it has been found that as the separation between the first DPFM 272 and the second DPFM 274 increasesand hence distance d.sub.R increasesthe nominal Z translation direction stiffness also increases, while the Y translation DoF stiffness of the sandwich DPFM 270 remains constant. This is evidenced by the graphs presented in FIGS. 20g and 20h. Compare this performance of the sandwich DPFM 270 to that of the DPFM 162 of FIG. 18a for demonstrative purposes. As evidenced by FIG. 18b for the DPFM 162, as the reinforcement length L.sub.R of the reinforcement stages 176 is increased, the nominal DoC stiffness of the DPFM 162 increases too. But along with those more desirable increases comes a more undesirable increase for the DPFM 162. As evidenced by FIG. 18c, as the reinforcement length L.sub.R of the reinforcement stages 176 is increased, the DoF stiffness of the DPFM 162 increases too. A comparison of the performances of the sandwich DPFM 270 (FIG. 20c) versus the DPFM 162 (FIG. 18a) shows that the sandwich DPFM 270 is able to overcome the observed trade-off between the DoF and DoC stiffness in known prior art DPFM designs. Moreover, and with reference again to the graph of FIG. 20f, it shows that the non-nominal DoC stiffness of the sandwich DPFM 270 (FIG. 20c) decreases at a slower rate than that of the DPFM 162 (FIG. 18a). This decrease in non-nominal DoC stiffness is attributed to the torsional stiffness of the associated flexure blades. The Z direction separation (i.e., distance d.sub.R) between the first DPFM 272 and the second DPFM 274 of the sandwich DPFM 270, and the rigid inter-stage connections 304, 306, 308, 310, 312, 314, 316 between them increases the torsional stiffness of the individual flexure blades of the first and second flexure blades 281, 295 in the sandwich DPFM 270.
[0126] In addition to FIG. 20c, further embodiments of the sandwich DPFM 270 are provided in FIGS. 20d and 20e. The sandwich DPFM 270 in the figures share similar overall structures and functionalities, but differ in the manner of their fabrication and assemblage. As set forth elsewhere in this description, the construction of the rigid inter-stage connections of the sandwich DPFM 270 in various embodiments can involve adhesives, fasteners, welds, monolithic construction and geometry, and/or other types of attachments and structures. In the further embodiment of FIG. 20d, in lieu of the fasteners or bolts of the embodiment of FIG. 20c, adhesive or weldments are employed in order to construct the rigid inter-stage connections 304 between the first and second secondary stages 276, 290, the rigid inter-stage connections 306 between the first and second ground stages 278, 292, and the rigid inter-stage connections 308 between the first and second motion stages 280, 294. In FIG. 20d, the rigid inter-stage connections are in the form of columnlike structures, but could be structures of other shapes in other embodiments. And in the further embodiment of FIG. 20e, the sandwich DPFM 270 is fabricated monolithically and hence monolithic structures are employed in order to construct the rigid inter-stage connections 304 between the first and second secondary stages 276, 290, the rigid inter-stage connections 306 between the first and second ground stages 278, 292, and the rigid inter-stage connections 308 between the first and second motion stages 280, 294. In this monolithic embodiment of FIG. 20e, bodies previously depicted and described as discrete may be constituted by a single monolithic body; for example, the first and second motion stages 280,294 and third rigid inter-stage connection 308 are, in a sense, a single motion body and may be referred to as just a motion body. This is true of the first and second ground stages 278, 292 and second rigid inter-stage connection 306 which may collectively be referred to as just a ground body, as well as monolithic constructions described elsewhere herein.
[0127] The sandwich DPFM 270 and its componentsas well as embodiments thereofcan be employed to build and form more complex flexure mechanism assemblies. For example, one or more of the DPFMs 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212 of the parallel kinematic XY flexure mechanism of FIG. 19a can be replaced with the sandwich DPFM 270 or an embodiment thereof.
[0128] With reference now to FIG. 21a, an embodiment of a parallel kinematic sandwich XY flexure mechanism 318 is presented. The parallel kinematic sandwich XY flexure mechanism 318 could have various designs, constructions, and components in various embodiments depending in part or more on the intended application and performance demands. In the embodiment of FIG. 21a, the parallel kinematic sandwich XY flexure mechanism 318 has a total of twelve sandwich DPFMs. Still, in other embodiments, a quantity of more or less sandwich DPFMs can be provided in a parallel kinematic sandwich XY flexure mechanism such as those described elsewhere in this patent; for example, only four or eight DPFMs could be in the form of sandwich DPFMs, while the remaining DPFMs could lack the design and construction and components of sandwich DPFMs. Yet further, in other embodiments, the parallel kinematic sandwich XY flexure mechanism could have one or more or all sandwich PFMs in its design and constructions. The parallel kinematic sandwich XY flexure mechanism 318 employs out of plane rigid inter-stage connections to achieve significantly higher DoC out of plane stiffness in the Z, .sub.x, and .sub.y directions between its ground and motion stages (subsequently introduced) for a given Y translation DoF stiffness compared to known prior art DPFM designs. In this embodiment, the parallel kinematic sandwich XY flexure mechanism 318 is made-up of two identical parallel kinematic XY flexure mechanisms with reinforced flexure blades spaced apart from each other in the Z direction: a first or upper parallel kinematic XY flexure mechanism 320 and a second or lower parallel kinematic XY flexure mechanism 322. The separation of the first and second parallel kinematic XY flexure mechanisms 320, 322 along the Z direction is denoted in FIG. 21a by distance d.sub.R. The first and second parallel kinematic XY flexure mechanism 320, 322 arranges a multitude of DPFMs in a first plane and a second plane that are parallel with respect to each and that are separated from each other in the Z direction. The first plane can be defined and reside at the first parallel kinematic XY flexure mechanism 320, and the second plane can likewise be defined and reside at the second parallel kinematic XY flexure mechanism 322.
[0129] The first parallel kinematic XY flexure mechanism 320, per this embodiment, includes: a first motion stage or body 324, a first ground stage or body 326, a first intermediate stage or body 328, a second intermediate stage or body 330, a third intermediate stage or body 332, a fourth intermediate stage or body 334, a first DPFM 336, a second DPFM 338, a third DPFM 340, a fourth DPFM 342, a fifth DPFM 344, a sixth DPFM 346, a seventh DPFM 348, an eighth DPFM 350, a ninth DPFM 352, a tenth DPFM 354, an eleventh DPFM 356, and a twelfth DPFM 358. In this embodiment, the DPFMs that connect the first, second, third, and fourth intermediate stages 328, 330, 332, 334 to the first ground stage 326 are referred to as peripheral DPFMs, while the DPFMs that connect the first, second, third, and fourth intermediate stages 328, 330, 332, 334 to the first motion stage 324 are referred to as inner DPFMs. In FIG. 21a then, the first, third, fourth, sixth, seventh, ninth, tenth, and twelfth DPFMs 336, 340, 342, 346, 348, 352, 354, 358 constitute peripheral DPFMs; and the second, fifth, eighth, and eleventh DPFMs 338, 344, 350, 356 constitute inner DPFMs. The second parallel kinematic XY flexure mechanism 322, per this embodiment, possesses the same bodies and stages as described for the first parallel kinematic XY flexure mechanism 320. While not all are visible in FIG. 21a due to its perspective view, the second parallel kinematic XY flexure mechanism 322 includes: a second motion stage or body, a second ground stage or body 360, a fifth intermediate stage or body, a sixth intermediate stage or body 362, a seventh intermediate stage or body 364, an eighth intermediate stage or body, a thirteenth DPFM, a fourteenth DPFM, a fifteenth DPFM, a sixteenth DPFM, a seventeenth DPFM, an eighteenth DPFM, a nineteenth DPFM 366, a twentieth DPFM 368, a twenty-first DPFM 370, a twenty-second DPFM 372, a twenty-third DPFM 374, and a twenty-fourth DPFM 376. As before, according to this embodiment, the thirteenth, fifteenth, sixteenth, eighteenth, nineteenth, twenty-first, twenty-second, and twenty-fourth DPFMs 366, 370, 372, 376 constitute peripheral DPFMs; and the fourteenth, seventeenth, twentieth, and twenty-third DPFMs 368, 374 constitute inner DPFMs.
[0130] In FIG. 21a, twin stages are established between corresponding bodies and stages of the first and second parallel kinematic XY flexure mechanism 320, 322. Accordingly, the first and second motion stages 324 constitute twin stages, the first and second ground stages 326, 360 constitute twin stages, the first and fifth intermediate stages 328 constitute twin stages, the second and sixth intermediate stages 330, 362 constitute twin stages, the third and seventh intermediate stages 332, 364 constitute twin stages, the fourth and eighth intermediate stages 334 constitute twin stages, the first and thirteenth DPFMs 336 constitute twin stages, the second and fourteenth DPFMs 338 constitute twin stages, the third and fifteenth DPFMs 340 constitute twin stages, the fourth and sixteenth DPFMs 342 constitute twin stages, the fifth and seventeenth DPFMs 344 constitute twin stages, the sixth and eighteenth DPFMs 346 constitute twin stages, the seventh and nineteenth DPFMs 348, 366 constitute twin stages, the eighth and twentieth DPFMs 350, 368 constitute twin stages, the ninth and twenty-first DPFMs 352, 370 constitute twin stages, the tenth and twenty-second DPFMs 354, 372 constitute twin stages, the eleventh and twenty-third DPFMs 356, 374 constitute twin stages, and the twelfth and twenty-fourth DPFMs 358, 376 constitute twin stages. Moreover, in this embodiment, corresponding and individual bodies or stages of twin DPFM stages constitute twin stages with respect to each other. For example, corresponding secondary stages 378, 380 of the eighth and twentieth DPFMs 350, 368 constitute twin stages; and corresponding reinforcement stages 382, 384 (not all readily visible) of the tenth and twenty-second DPFMs 354, 372 constitute twin stages.
[0131] As in previous embodiments set forth, it has been determined that twin stages of the first and second parallel kinematic XY flexure mechanisms 320, 322 can be connected together via inter-stage connections. The out of plane inter-stage connections could constrain relative motion in three directions, four directions, or in all six directions, as described, per various embodiments. In the embodiment of FIG. 21a, all of the twin stages of the first and second parallel kinematic XY flexure mechanisms 320, 322 are connected together via rigid inter-stage connections. Still, in other embodiments, not all of the twin stages of the parallel kinematic sandwich XY flexure mechanism 318 are connected together via rigid inter-stage connections, and rather only some of the twin stages of the parallel kinematic sandwich XY flexure mechanism 318 could be connected together via rigid inter-stage connections. In FIG. 21a, a first or motion rigid inter-stage connection 386 (partially represented via broken lines) connects the first and second motion stages 324, and extends in the Z direction therebetween. In this embodiment, the first rigid inter-stage connection 386 is in the form of four columnlike structures adhered or welded to the first and second motions stages 324. A second or ground rigid inter-stage connection 388 connects the first and second ground stages 326, 360, and extends in the Z direction therebetween. In this embodiment, the second rigid inter-stage connection 388 is in the form of four columnlike structures adhered or welded to the first and second ground stages 326, 360. A third or first intermediate rigid inter-stage connection (not visible in FIG. 21a) connects the first and fifth intermediate stages 328, and extends in the Z direction therebetween. In this embodiment, the third rigid inter-stage connection is in the form of four columnlike structures adhered or welded to the first and fifth intermediate stages 328. A fourth or second intermediate rigid inter-stage connection 392 (only two visible) connects the second and sixth intermediate stages 330, 362, and extends in the Z direction therebetween. In this embodiment, the fourth rigid inter-stage connection 392 is in the form of four columnlike structures adhered or welded to the second and sixth intermediate stages 330, 362. A fifth or third intermediate rigid inter-stage connection 394 (only two visible) connects the third and seventh intermediate stages 332, 364, and extends in the Z direction therebetween. In this embodiment, the fifth rigid inter-stage connection 394 is in the form of four columnlike structures adhered or welded to the third and seventh intermediate stages 332, 364. A sixth or fourth intermediate rigid inter-stage connection (not visible) connects the fourth and eighth intermediate stages 334, and extends in the Z direction therebetween. In this embodiment, the sixth rigid inter-stage connection is in the form of four columnlike structures adhered or welded to the fourth and eighth intermediate stages 334.
[0132] Furthermore, with continued reference to the embodiment of FIG. 21a, a seventh or first DPFM rigid inter-stage connection (not visible) connects the first and thirteenth DPFMs 336, and extends in the Z direction therebetween. In this embodiment, the seventh rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the first and thirteenth DPFMs 336, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the first and thirteenth DPFMs 336. Indeed, in this regard and per this embodiment, the twin DPFM stages of the first and second parallel kinematic sandwich XY flexure mechanism 320, 322 establish sandwich DPFMs that resemble the sandwich DPFM 270 of FIG. 20d. An eighth or second DPFM rigid inter-stage connection (not visible) connects the second and fourteenth DPFMs 338, and extends in the Z direction therebetween. In this embodiment, the eighth rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the second and fourteenth DPFMs 338, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the second and fourteenth DPFMs 338. A ninth or third DPFM rigid inter-stage connection (not visible) connects the third and fifteenth DPFMs 340, and extends in the Z direction therebetween. In this embodiment, the ninth DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of third and fifteenth DPFMs 340, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the third and fifteenth DPFMs 340. A tenth or fourth DPFM rigid inter-stage connection (not visible) connects the fourth and sixteenth DPFMs 342, and extends in the Z direction therebetween. In this embodiment, the tenth DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the fourth and sixteenth DPFMs 342, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the fourth and sixteenth DPFMs 342. An eleventh or fifth DPFM rigid inter-stage connection (not visible) connects the fifth and seventeenth DPFMs 344, and extends in the Z direction therebetween. In this embodiment, the eleventh DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the fifth and seventeenth DPFMs 344, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the fifth and seventeenth DPFMs 344.
[0133] Still further, a twelfth or sixth DPFM rigid inter-stage connection (not visible) connects the sixth and eighteenth DPFMs 346, and extends in the Z direction therebetween. In this embodiment, the twelfth DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the sixth and eighteenth DPFMs 346, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the sixth and eighteenth DPFMs 346. A thirteenth or seventh DPFM rigid inter-stage connection 410 (partially visible) connects the seventh and nineteenth DPFMs 348, 366, and extends in the Z direction therebetween. In this embodiment, the thirteenth DPFM rigid inter-stage connection 410 is in the form of a pair of columnlike structures 409 adhered or welded to the accompanying and corresponding secondary stages of the seventh and nineteenth DPFMs 348, 366, and is in the form of monolithic constructions 411 spanning between accompanying and corresponding reinforcement stages of the seventh and nineteenth DPFMs 348, 366. A fourteenth or eighth DPFM rigid inter-stage connection (not readily visible) connects the eighth and twentieth DPFMs 350, 368, and extends in the Z direction therebetween. In this embodiment, the fourteenth DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages 378, 380 of the eighth and twentieth DPFMs 350, 368, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the eighth and twentieth DPFMs 350, 368. A fifteenth or nineth DPFM rigid inter-stage connection 414 (partially visible) connects the ninth and twenty-first DPFMs 352, 370, and extends in the Z direction therebetween. In this embodiment, the fifteenth DPFM rigid inter-stage connection 414 is in the form of a pair of columnlike structures 413 adhered or welded to the accompanying and corresponding secondary stages of the ninth and twenty-first DPFMs 352, 370, and is in the form of monolithic constructions 415 spanning between accompanying and corresponding reinforcement stages of the ninth and twenty-first DPFMs 352, 370.
[0134] A sixteenth or tenth DPFM rigid inter-stage connection 416 (partially represented via solid circles) connects the tenth and twenty-second DPFMs 354, 372, and extends in the Z direction therebetween. In this embodiment, the sixteenth DPFM rigid inter-stage connection 416 is in the form of a pair of columnlike structures 417 adhered or welded to the accompanying and corresponding secondary stages of the tenth and twenty-second DPFMs 354, 372, and is in the form of monolithic constructions 419 spanning between accompanying and corresponding reinforcement stages 382, 384 of the tenth and twenty-second DPFMs 354, 372. A seventeenth or eleventh DPFM rigid inter-stage connection (not readily visible) connects the eleventh and twenty-third DPFMs 356, 374, and extends in the Z direction therebetween. In this embodiment, the seventeenth DPFM rigid inter-stage connection is in the form of a pair of columnlike structures adhered or welded to the accompanying and corresponding secondary stages of the eleventh and twenty-third DPFMs 356, 374, and is in the form of monolithic constructions spanning between accompanying and corresponding reinforcement stages of the eleventh and twenty-third DPFMs 356, 374. Lastly, an eighteenth or twelfth DPFM rigid inter-stage connection 420 (partially visible) connects the twelfth and twenty-fourth DPFMs 358, 376, and extends in the Z direction therebetween. In this embodiment, the eighteenth DPFM rigid inter-stage connection 420 is in the form of a pair of columnlike structures 421 adhered or welded to the accompanying and corresponding secondary stages of the twelfth and twenty-fourth DPFMs 358, 376, and is in the form of monolithic constructions 423 spanning between accompanying and corresponding reinforcement stages of the twelfth and twenty-fourth DPFMs 358, 376. Again here, in alternative embodiments, the construction of the rigid inter-stage connections of the first and second parallel kinematic XY flexure mechanisms 320, 322 can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures.
[0135] Since the parallel kinematic sandwich XY flexure mechanism 318 is built with the use of sandwich DPFMs, the parallel kinematic sandwich XY flexure mechanism 318 inherits certain properties of a sandwich DPFM. With reference now to FIG. 22, an embodiment of an XY parallel kinematic flexure mechanism 422 with reinforced flexure blades is depicted. The XY parallel kinematic flexure mechanism 422, according to this embodiment, has a ground body or stage 424, a motion body or stage 426, a first intermediate stage or body 428, a second intermediate stage or body 430, a third intermediate stage or body 432, a fourth intermediate stage or body 434, a first DPFM 436, a second DPFM 438, a third DPFM 440, a fourth DPFM 442, a fifth DPFM 444, a sixth DPFM 446, a seventh DPFM 448, an eighth DPFM 450, a ninth DPFM 452, a tenth DPFM 454, an eleventh DPFM 456, and a twelfth DPFM 458. Each of the twelve DPFMs have four reinforced flexure bodies or blades 460. The dimensions of the XY parallel kinematic flexure mechanism 422 of FIG. 22 are mostly equivalent to those of the parallel kinematic sandwich XY flexure mechanism 318 of FIG. 21a, with the exception that the out of plane thickness in the Z direction of the individual flexure blades of the parallel kinematic sandwich XY flexure mechanism 318 of FIG. 21a is one-half the out of plane thickness in the Z direction of the individual flexure blades of the XY parallel kinematic flexure mechanism 422 of FIG. 22. But the total out of plane thickness in the Z direction of the respective flexure blades of the parallel kinematic sandwich XY flexure mechanism 318 (i.e., those at both first and second parallel kinematic XY flexure mechanisms 320, 322) and of the XY parallel kinematic flexure mechanism 422 are equivalent relative to each other. It has been found thatunder these conditions and with the rigid inter-stage connections between twin stages of the parallel kinematic sandwich XY flexure mechanism 318both of the flexure mechanism designs 318, 422 will possess the same DoF stiffness in the X direction and in the Y direction between the respective motion and ground stages.
[0136] Moreover, it has been found that the parallel kinematic sandwich XY flexure mechanism 318 of FIG. 21a provides a significantly higher nominal Z translation direction DoC stiffness than that provided by the XY parallel kinematic flexure mechanism 422 of FIG. 22. Further, the parallel kinematic sandwich XY flexure mechanism 318 provides a slower reduction in non-nominal Z translation direction DoC stiffness than that provided by the XY parallel kinematic flexure mechanism 422. This is evidenced by the graph presented in FIG. 23. The graph shows a comparison of the nominal and non-nominal Z translation direction DoC stiffnesses of the parallel kinematic sandwich XY flexure mechanism 318 and the XY parallel kinematic flexure mechanism 422, according to the embodiments set forth. Graphs of the .sub.x and .sub.y rotation direction stiffnesses of the same comparison as the graph in FIG. 23 would yield the same outcome shown and described in connection with FIG. 23. The nominal Z translation direction DoC stiffness is at zero motion stage displacement in the graph, and the non-nominal Z translation direction DoC stiffness is at non-zero motion stage displacements. The parallel kinematic sandwich XY flexure mechanism 318 is represented by line L.sub.1 in the graph, and the XY parallel kinematic flexure mechanism 422 is represented by line L.sub.2.
[0137] As described, not all of the twin stages of a parallel kinematic sandwich XY flexure mechanism like that similar to the embodiment of FIG. 21a need to be connected together via rigid inter-stage connections. While performance may not precisely conform, it may be suitable in a particular application. Example embodiments are presented in FIGS. 21b-21i. In the embodiment of FIGS. 21b and 21c, a parallel kinematic sandwich XY flexure mechanism 462 has a first or motion rigid inter-stage connection 464 that connects the associated motion stages of first or upper and second or lower parallel kinematic XY flexure mechanisms 466, 468, and a second or ground rigid inter-stage connection 470 that connects the associated ground stages. Rigid inter-stage connections are absent at other twin stages in this embodiment. Further, as before, the construction of the rigid inter-stage connections can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures. In the embodiment of FIGS. 21d and 21e, in addition to motion and ground rigid inter-stage connections 464, 470, a parallel kinematic sandwich XY flexure mechanism 472 has intermediate rigid inter-stage connections 474 that connect all of the associated intermediate stages. Rigid inter-stage connections are absent at other twin stages in this embodiment. Further, as before, the construction of the rigid inter-stage connections can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures. Further, in the embodiment of FIGS. 21f and 21g, in addition to motion and ground rigid inter-stage connections 464, 470, a parallel kinematic sandwich XY flexure mechanism 476 has secondary rigid inter-stage connections 478 that connect all of the associated secondary stages. Rigid inter-stage connections are absent at other twin stages in this embodiment. As before, the construction of the rigid inter-stage connections can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures. Lastly, in the embodiment of FIGS. 21h and 21i, in addition to motion and ground rigid inter-stage connections 464, 470, a parallel kinematic sandwich XY flexure mechanism 480 has reinforcement rigid inter-stage connections 482 that connect all of the associated reinforcement stages. Rigid inter-stage connections are absent at other twin stages in this embodiment. Further, as before, the construction of the rigid inter-stage connections can involve adhesives, fasteners or bolts, welds, monolithic construction and geometry, and/or other types of attachments and structures.
[0138] The out of plane inter-stage connections described herein are often shown and described as extending in the Z direction or in a direction orthogonal to the X and Y directions. But in embodiments not specifically depicted in the figures, the out of plane inter-stage connections can extend in a direction that is non-orthogonal relative to the X and Y directions, and is rather slanted with respect to the X and Y directions. Further, while embodiments of the sandwich DPFM designs presented have corresponding and analogous stages at the top layer or plane and at the bottom layer or plane relative to each other, in other embodiments the stages residing at the top and bottom layers/planes need not necessarily correspond and could differ relative to each other.
[0139] Still further, the out of plane inter-stage connections described herein can be implemented in flexure mechanisms of varying designs, constructions, and components, and apart from those depicted and described herein. Examples of such possibilities are disclosed in U.S. Pat. No. 6,688,183 to Massachusetts Institute of Technology, and in U.S. Pat. No. 9,200,689 to The Regents of the University of Michigan.
[0140] As used herein, the terms general and generally and substantially are intended to account for the inherent degree of variance and imprecision that is often attributed to, and often accompanies, any design and manufacturing process, including engineering tolerancesand without deviation from the relevant functionality and outcomesuch that mathematical precision and exactitude is not implied and, in some instances, is not possible. In other instances, the terms general and generally and substantially are intended to represent the inherent degree of uncertainty that is often attributed to any quantitative comparison, value, and measurement calculation, or other representation.
[0141] It is to be understood that the foregoing description is of one or more preferred exemplary embodiments of the invention. The invention is not limited to the particular embodiment(s) disclosed herein, but rather is defined solely by the claims below. Furthermore, the statements contained in the foregoing description relate to particular embodiments and are not to be construed as limitations on the scope of the invention or on the definition of terms used in the claims, except where a term or phrase is expressly defined above. Various other embodiments and various changes and modifications to the disclosed embodiment(s) will become apparent to those skilled in the art. All such other embodiments, changes, and modifications are intended to come within the scope of the appended claims.
[0142] As used in this specification and claims, the terms for example, e.g., for instance, and such as, and the verbs comprising, having, including, and their other verb forms, when used in conjunction with a listing of one or more components or other items, are each to be construed as open-ended, meaning that the listing is not to be considered as excluding other, additional components or items. Other terms are to be construed using their broadest reasonable meaning unless they are used in a context that requires a different interpretation.
