Research Article  Open Access
Jiejie Shao, Tong Ma, Wenyu Chen, Xin Fu, "Position Analysis of a Hybrid SerialParallel Manipulator in Immersion Lithography", Mathematical Problems in Engineering, vol. 2015, Article ID 573071, 14 pages, 2015. https://doi.org/10.1155/2015/573071
Position Analysis of a Hybrid SerialParallel Manipulator in Immersion Lithography
Abstract
This paper proposes a novel hybrid serialparallel mechanism with 6 degrees of freedom. The new mechanism combines two different parallel modules in a serial form. 3() parallel module is architecture of 3 degrees of freedom based on higher joints and specializes in describing two planes’ relative pose. 3SP parallel module is typical architecture which has been widely investigated in recent researches. In this paper, the directinverse position problems of the 3SP parallel module in the couple mixedtype mode are analyzed in detail, and the solutions are obtained in an analytical form. Furthermore, the solutions for the direct and inverse position problems of the novel hybrid serialparallel mechanism are also derived and obtained in the analytical form. The proposed hybrid serialparallel mechanism is applied to regulate the immersion hood’s pose in an immersion lithography system. Through measuring and regulating the pose of the immersion hood with respect to the wafer surface simultaneously, the immersion hood can track the wafer surface’s pose in realtime and the gap status is stabilized. This is another exploration to hybrid serialparallel mechanism’s application.
1. Introduction
Immersion lithography is a more advanced semiconductor technology compared with the traditional dry lithography. Immersion technology can improve the lithography resolution to 45 nm or even higher by replacing the air between the wafer and bottom lens with liquid of higher refractive index [1–3]. The efficient implementation of immersion technology must be based on the successful management of immersion fluid [4, 5]. While the wafer scans, a device called immersion hood (IH) is used to constrain and update the immersion fluid within the limited area below the lens [6, 7], as shown in Figure 1. In order to form a piece of immersion flow field without leakage and pressure fluctuation, a steady gap with specified parallelism and central distance should be formed between the IH and the wafer. Immersion hood pose regulation manipulator (IHPRM) is employed to precisely adjust and sustain the pose of the IH with respect to the wafer surface.
Commonly, the main structure of the IHPRM is a 3degree of freedom (DOF) parallel mechanism (PM), which adjusts the pose of the IH’s lower surface with respect to the upper surface of the wafer. The other three DOFs of the IH are actuated by three independent screws manually during the assembling process and then fixed, as shown in Figure 1. As the pose parameters of the tiny gap (about 0.1 mm) is extremely hard to obtain directly, the indirect pose measurement (with respect to the main frame) is usually applied in the immersion hood pose regulation, which performs insufficient precision in the pose regulating. On the other hand, the wafer stage adjusts the pose of the wafer with respect to the optical focus and scans simultaneously during the exposure process; thus, the pose of the immersion hood should be adjusted accordingly in the meantime. However, the kinematic problem of the parallel mechanism is often solved via numerical iterative methods which can provide only some of the possible solutions and usually need more time compared to the analytical method. Therefore, the general 3DOF parallel mechanism is difficult to be competent in high precision pose regulation and realtime tracking.
The hybrid serialparallel mechanism (HSPM) which combines high precision, high rigidity, and low inertia of fully parallel mechanism and extended workspace of serial mechanism have attracted increasing attention during the past several years. Waldron et al. [8] have analyzed a hybrid type of manipulator with 10 DOFs, which comprises a serial arm with 6 DOFs and a parallel micromanipulator with 4 DOFs in 1989. Naccarato et al. [9] raised the socalled variablegeometry truss manipulators which realize the combination of the parallel manipulators in a serial form in 1991. Shahinpoor [10] has studied a hybrid serialparallel manipulator formed by two serially connected parallel manipulators and analyzed the inverse kinematic problem in 1992. Romdhane [11] proposed a Stewart platformlike hybrid serialparallel mechanism, obtained a closedform solution of the forward analysis problem, and analyzed the orientation workspace in 1999. Tanev [12] presented a hybrid robot with 6 DOFs and acquired the closedform solutions for the direct and inverse position problems in 2000. Kanaan et al. [13] applied 5DOF serialparallel architecture in the VERNE machine tool and derived its inverseforward kinematic equations in 2009. Ibrahim and Khalil [14] discussed the dynamic models of the hybrid robots in 2010. Zeng et al. [15, 16] analyzed the hybrid serialparallel mechanisms with spatial multiloop kinematic chains in 2012 and put forward a design method of hybrid manipulators with kinematotropic property and deployability in 2014.
Kinematic analysis is the basic issue for the parallel mechanism studies, and the position analysis is the first step of the kinematic analysis. As the HSPM is generally composed of several parallel mechanisms (regarded as its parallel module) with lower mobility, whether analytical form solutions can be obtained for the HSPM’s directinverse position problem mainly depends on whether its parallel modules can. 3PSP is a typical parallel mechanism (PM) with 3 DOFs and has been widely discussed in recent years. In 2001, Gregorio and ParentiCastelli [17] made the position analysis of a general 3PSP parallel robot in an analytical form. Deng and Li [18] obtained the 3PSP PM’s direct position solution in a numerical form by solving 9 independent equations in 2009. Hao et al. [19] analyzed the 3PSP PM in 2009. But because the pose status was described by TiltTorsion angle, only the inverse position problem was solved in an analytical form. Zhang and Li [20] discussed the direct position problem of the 3 PM with the screw theory in 2013, which still needs to solve 9 independent equations. Rezaei et al. [21] investigated the 3PSP PM systematically in 2013. The directinverse position problem was divided into two modes according to the different pose parameters: the coupled mixedtype mode with , , and and the nonpure translational mode with , , and . A geometrical approach was applied to the analysis of inverse position problem in the nonpure translational mode, and a solution in an analytical form was obtained. However, the coupled mixedtype mode which is more frequently used when describing a moving platform’s position and orientation was calculated in the numerical way. Currently, the analytical solutions for the directinverse position problems of 3PSP PM in the coupled mixedtype mode are still absent.
In this paper, according to the high precision regulation and realtime tracking demand of the IHPRM in the immersion lithography system, a new type of hybrid serialparallel mechanism with 6 DOFs is proposed. The new mechanism is constituted by combining two different parallel modules (including a 3 module and a 3 module) in a serial form. The 3PSP module is employed to regulate IH’s pose, while the 3 module is adopted to detect the pose status of the IH’s lower surface with respect to the wafer’s upper surface.
The directinverse position problems of 3 PM in the coupled mixedtype mode as the basis of the kinematic analysis are fully discussed in this paper. By choosing proper pose parameters and geometric constraints, the analytical solutions to both direct and inverse position problems are obtained. On this basis, the directinverse position analysis for the new hybrid serialparallel mechanism is also discussed in detail, which acquires an analytical solution as well. Finally, this new mechanism realizes tracking IH’s lower surface with respect to the wafer surface in real time. The variation of gap parameters caused by pose regulation during wafer scanning can be reduced effectively.
2. Structure Description of the Hybrid SerialParallel Manipulator
2.1. The Hybrid SerialParallel Mechanism
As a subsystem of immersion lithography system, IHPRM holds the IH in the proper pose and assures a stable gap formed between IH’s lower surface and the wafer surface which is also suitable for fluid management. In the present application, the thickness of the gap should be adjusted within the range of μm. The angle between IH’s lower surface and the wafer surface cannot exceed 0.00005 rad, and the tilt direction should be in accordance with wafer scanning direction. Therefore, IHPRM must adjust poses in 3 DOFs. As the pose of the wafer surface varies during wafer scanning, the pose of the IH’s lower surface should track wafer’s surface in real time to sustain a steady gap. It means a necessity of the detection of the gap’s status in real time. Therefore, IHPRM must satisfy 3 DOFs’ detecting requirements. The pose regulation process can be simplified as the movements among three planes: the base plane is fixed, the pose of a target plane with respect to base plane varies to keep the surface of the wafer within the focus depth of the lens, and a moving plane is adjusted to keep the parallelism and the distance with plane . The delta mechanism, Stewart’s platform, and other normal parallel mechanisms only adjust the pose between two platforms, which are not suitable for this application. In this paper, a novel HSPM is designed to realize both immersion hood pose’s regulation and the gap’s status detection. As the new mechanism is composed of a 3 PM and a 3 PM, we call it 3 HSPM. Combining two Stewart’s platforms in a series form to adjust these three planes is obviously much more complex and redundant than the 3 HSPM.
2.1.1. The 3 Parallel Modular
The 3 PM consists of a base platform, a moving platform, and three chains with identical kinematic architecture, as shown in Figure 2. The moving platform, actually a moving plane, is a virtual object without definite shape. Each chain connects the moving platform to the base platform through a prismatic joint P and a higher joint (P_{H}) in sequence, where the prismatic joint P is the active joint. The translational axis () of the prismatic joint P within the th () kinematic chain is perpendicular to the base. The higher joints () are attached to the sliders moving along these axes (), respectively, with their apexes constrained within the moving plane.
(a) The kinematic model
(b) The CAD model
Here, the GrüblerKutzbach criterion [22, 23] is applied to analyze the DOF of the 3 PM. Considerwhere indicates the dimension of task space, is the number of links, is the number of joints, and stands for the DOF of joint .
As shown in (1), 3 PM has 6 DOFs, which can be divided into two types. The 1 translational DOF and 2 rotational DOFs that the moving plane moves with respect to the base platform are regarded as the first type, while the 2 translational DOFs and 1 rotational DOF that the virtual moving platform moves in the moving plane are regarded as the second type. Since the moving platform is virtual and has no definite shape, its motion in the moving plane is meaningless, and the 3 PM has 3 DOFs.
2.1.2. The 3PSP Parallel Modular
The 3 PM consists of a base platform, a moving platform, and three chains with identical kinematic architecture and is a parallel mechanism with 3 DOFs [24]. With reference to Figure 3, the moving platform and the base platform are connected to each other by means of three independent kinematic chains which are constituted by an active prismatic joint , a spherical joint S, and a passive joint P in sequence.
(a) The kinematic model
(b) The CAD model
2.1.3. The Hybrid Mechanism
The novel HSPM combines these two parallel modules in serial form, by attaching the base platform of the 3 PM to the moving platform of the 3 PM, as shown in Figure 4. Summing the DOFs of each parallel module obtains the DOFs of the HSPM, which is 6.
(a) The kinematic model
(b) The CAD model
The three platforms in the HSPM are named base platform, moving platform, and target platform which lie on the fixed plane , moving plane , and target plane , respectively. The 1st prismatic joint ( with the actuate parameter , ) and the 3rd prismatic joint (, with the actuate parameter , ) are active joints with their translational axes and perpendicular to the base platform and the moving platform, respectively. The 2nd prismatic joint (, ) and the spherical joint (, ) as passive joints coincide with the plane . The higher joint (, ) has 5 DOFs related to the plane , as its apex lies on the plane .
2.2. The Physical Model of the IHPRM
The HSPM is applied in the IHPRM, with the main frame, immersion hood, and wafer corresponding to the base platform, moving platform, and target platform, respectively, as shown in Figure 5. Three laser displacement sensors (SIF01, Keyence) are installed on the IH’s lower surface, which detect the distance between the emission point and the reflection point, respectively, by emitting and receiving the laser. The three laser displacement sensors are applied to measure the pose of the IH’s lower surface with respect to the wafer surface, which integrate the behavior of both the 3rd prismatic joint and the higher joint (P_{H}).
3. Description of Reference Frames and Motion Parameters
To illustrate the relationship between different components more intuitively, three reference frames are defined in the three reference planes, respectively, as shown in Figure 6. Point () distributes on a fixed circle with radius of in the plane . A fixed reference frame () is assigned at the central point of the fixed circle, with the axis perpendicular to the base platform and the axis passing through point . The angle between and axis is . According to the Cartesian righthand rule, the axis can be determined. The axes of the 2nd prismatic joints intersect at point . The point is defined as the center of the platform. Then a moving reference frame () is assigned in the plane at point , with axis perpendicular to plane and the axis passing through point . The axis is also determined by the Cartesian righthand rule. The angle between and axis is . Point () distributes on a mobile circle which centered at point with radius of in the plane . Points , , and are collinear, and is . Another moving reference frame () is assigned in the plane at point , with axis perpendicular to plane , and the axis passing through point . The axis is also determined by the Cartesian righthand rule. The angle between and axis is . Point is the intersection of the axis and the plane . A target reference frame () is assigned in the plane at point , with axis perpendicular to plane and the axis passing through the apex of the higher joint (P_{H}). The axis is also determined by the Cartesian righthand rule.
Three parameters are required to describe the pose of the plane and plane , respectively, in frame , which includes the angular parameters and the position parameters.
3.1. Angular Parameters Definition
The fixed plane , moving plane , and target plane are moved together and intersect at point , which will not change the angular relationship between them, as shown in Figure 7.
Vector is the normal unit vector of the target plane ; vector is the normal unit vector of the moving plane . The angle between and the axis is defined as tilt angle . The angle between ’s projection on the fixed plane and the axis is defined as twist angle . Similarly, the angle between and the axis is defined as tilt angle . The angle between ’s projection on the fixed plane and the axis is defined as twist angle . The angle between the vector and vector is defined as tilt angle . The angle between ’s projection and ’s projection on the fixed plane is defined as twist angle . As defined above, these six angular parameters have the constraints as follows:
3.2. Position Parameters Definition
As the angular relationships of the fixed plane , moving plane , and target plane are specified, the position relationships between them should also be determined in frame .
As shown in Figure 8, the axis intersects the plane at point ; then the distance between point and point is defined as the central distance between the fixed plane and the target plane . Similarly, point is the projection of the moving platform’s central point in the plane . The distance between point and point is defined as the central distance between the fixed plane and the moving plane . Further, axis intersects the target plane at point ; the distance between point and point is defined as the central distance between the moving plane and the target plane .
The two out of six angular parameters can be obtained by the other arbitrarily specified four parameters, as shown in (2). Similarly, while the two out of three position parameters are specified, the other one position parameter can be calculated, as shown in (47) in Section 5. As a consequence, six couple mixedtype pose parameters (including 4 independent angular parameters and 2 independent position parameters) are required to fully describe the pose relationship between the three planes. The pose description mode based on the couple mixedtype is more significant to the pose control algorithm compared to the nonpure translational mode.
4. Position Analysis of the 3 Mechanism
The position analysis for mechanism is comprised of two parts: the direct position analysis and the inverse position analysis. The direct position analysis is to calculate the pose parameters of the platform relative to the base with the given actuated joint parameters, while the inverse position analysis aims to calculate the actuated joint parameters from the given pose of the platform relative to the base [17]. The inverse kinematic analysis of the most parallel mechanism is quite simple and usually has solutions in analytical form, while the direct kinematic analysis is extremely complicated, and only numerical or closedform solutions can be obtained. As the basis of the HSPM’s position analysis, the position analysis of the 3 mechanism is investigated firstly.
4.1. Direct Position Analysis of the 3 Mechanism
The architecture of the 3 PM and the reference frames are shown in Figures 3 and 6. The direct position analysis of the 3 PM aims to obtain the orientation parameters (, ) and position parameter () of the moving platform with respect to the base platform using the given actuated parameters () of the actuate prismatic joints .
4.1.1. Tilt Angle
The homogeneous coordinates of and in the fixed reference frame arewhere the additional right side subscripts of and indicate the reference frame in which the points are measured.
Since the spherical joint ’s central point () is in the moving plane , the normal vector of the plane can be generated from :
Then we have the normal unit vector of the plane as follows:where
The equation of the moving plane in the frame is
As defined above, the tilt angle , from the normal unit vector to the positive direction of axis, is limited within the range of , which can be expressed as
4.1.2. Twist Angle
Let the unit vector denote the direction of the normal unit vector ’s projection on the fixed plane
Calculating the angle between vector and the positive axis yields the twist angle , within the range of as defined above, which can be expressed aswhere function calculates the arc tangent of the two variables and , in which the signs of both arguments are used to determine the quadrant of the result.
4.1.3. Central Distance
The unit vector which denotes the direction of the prismatic joint ’s axis in the frame is
The homogeneous transformation matrices which transfers the coordinates in the frame to the coordinates in the frame can be obtained by the Euler’s transformation method:where denotes vector from the point to the point in the fixed reference frame . is the Euler matrix. denotes the counterclockwise rotation angle about the axis, indicates the counterclockwise rotation angle about the axis, and indicates the counterclockwise rotation angle about the axis. Consider the following: ; ; ; ; ; ; ; ; .
Assume the coordinate of the moving platform’s central point in the frame as
The coordinates of the vectors from point to point are
The unit vector of the axis in the frame is
As the vectors , , and are coplanar according to the geometrical constraint [19, 24], their mixed product equals zero:
Equation (16) can be rearranged to obtain three constraint equations:
The three constraint equations yield the constraint equation of the torsion angle wherewhen ,
The coordinates and of the central point in the frame can be expressed in the analytical formwhen , ; ; ; .
Equation (18) can be transformed as
The tangent of the torsion angle is
Then the and of the central point in the frame are
As the moving platform’s central point is in plane , the coordinate value in the direction can be obtained by (7). The distance between the point and the fixed plane is the central distance as defined before
4.2. Inverse Position Analysis of the 3 Mechanism
The inverse position analysis of the 3 PM is to obtain the actuated parameters () of the prismatic joints with the given orientation parameters (, ) and position parameter () of the moving platform relative to the base platform.
The coordinate values and in the frame can be obtained directly according to the direct position analysis in Section 4.1. The homogeneous coordinate of the moving platform’s central point iswhere .
The vector indicates the direction from the moving platform’s central point to the spherical joint’s central point in the plane
The normal unit vector of the plane in the frame can be transferred to in the frame with the given twist angle and tilt angle :where is the transformation matric. denotes the counterclockwise rotation angle about the axis; indicates the counterclockwise rotation angle about the axis. Consider
Since the vector in the plane is orthogonal with respect to the normal unit vector of the plane , the following equation can be obtained:
Overall, by choosing the proper pose parameters and constraints, both the direct and inverse position analyses of the 3 PM obtain the solutions in analytical form, which indicates the onetoone correspondent relationships between the actuated parameters and the pose parameters of the 3 PM.
5. Position Analysis of the Hybrid SerialParallel Mechanism
5.1. Direct Position Analysis of the Hybrid SerialParallel Mechanism
The direct position analysis of the HSPM aims to obtain the pose parameters of the target plane relative to the fixed plane () and the moving plane (), respectively, using the given actuated parameters and () of the prismatic joints and .
As the actuated parameters () are given, the parameters , , and which describe the pose of the plane with respect to the plane can be obtained directly as the direct position analysis of the 3 PM shown in Section 4.1.
The actuated parameters () of the prismatic joints equal the module of the vector . As shown in the Section 4, the normal unit vector of the target plane in frame iswhere can be transferred to the frame , as where can be transferred to the frame , aswhere
As defined above, the tilt angle , from the normal unit vector to the positive direction of axis, is limited within the range of , which can be expressed as
Let the unit vector denote the direction of the normal unit vector ’s projection on the fixed plane
Calculating the angle from vector to the positive axis yields the twist angle , within the range of as defined above, which can be expressed as
The tilt angle and twist angle can be calculated according to (2).
In order to obtain the central distance , an intermediate variable is introduced here, which indicates the distance between the points and :
The target plane in the frame is
Transfer the coordinate of point in the frame to the frame
By substituting formula (42) into (41), can be obtained as follows:
By transfer the coordinate of point in frame to frame with the homogeneous transformation matrices
The equation of the target plane is
Let and yield the central distance :
The relationship between the central distances , , and can be obtained with (44) and (46):
5.2. Inverse Position Analysis of the Hybrid SerialParallel Mechanism
The inverse position analysis of the HSPM is to obtain the actuated parameters and () of the prismatic joints and for the given pose parameters of the target plane relative to the fixed plane () and the moving plane (), respectively.
The homogeneous coordinate of the higher joint’s apex in frame is
As discussed before, the pose of the target plane with respect to the moving plane can be described by the point and the normal unit vector of the plane . Consider
Since and can be calculated by (2), the coordinate of vector can be transferred from the frame to the frame by Euler’s transformation aswhere ; ; ; ; ; ; ; ; and .
Since the vector in the plane is orthogonal with respect to the vector in frame , the following equation can be obtained:
Equation (51) can be rearranged to obtain
The coordinate of the moving platform’s central point in frame can be obtained as shown in Section 4.2. Further, the actuated parameters () of the prismatic joints are
With this, the whole actuated parameters of the HSPM in analytical form are obtained.
5.3. Numerical Example
Specify the architectural parameters of the HSPM according to the definition in Figure 6, as , , , , , , , = 25°, = 0°, = 90°, and = 180°, which are consistent with the prototype of the IHPRM. Several cases are calculated in both direct and inverse ways of the position analysis, as obtained in Table 1.

The left side of Table 1 shows the position solving in the direct way, which calculate the pose parameters , , , , , and with the given actuate parameters and (). The right side of Table 1 shows the position solving in the inverse way, which calculate the actuate parameters and () with the given pose parameters , , , , , and . In addition, the moving platform’s pose parameters and coordinate of the central point o are also obtained in both sides. The onetoone correspondent relationships between the actuated parameters and the pose parameters are obvious.
6. Experimental Results and Discussions
IHPRM regulates the pose of the IH with 3 DOFs and detects the gap status with another 3 DOFs simultaneously. The error caused by the manufacturing and assembly can be reduced efficiently by introducing the closedloop feedback of the gap status, which can be verified in the prototype.
A series of contrast experiments between the openloop control method and closedloop control method are carried out. As shown in Figure 9, the openloop control method (without the red line) shows the behavior of the single 3 PM and assumes that the pose between the wafer surface and the main frame is known and fixed, and the sensors are employed only for the error observation. The closedloop control method (with the red line) introduces the six pose parameters (, , , , , and ) as the feedback signals.
The experiments are divided into two parts according to the wafer status, as the yellow region and pink region shown in Figures 10 and 11.
The yellow region shows the regulating process of the gap with the wafer still. As shown in the IZone (openloop), due to the impact of the deviations generated from the manufacturing and assembly processes, when the IH’s lower surface is adjusted to be parallel to the fixed plane, the angle between the IH’s lower surface and the wafer surface is about 0.000205 rad and the central distance is about 0.271 mm, which is far beyond the installation requirement of the IH. By introducing the closedloop feedback of the gap status, the angle and the central distance are controlled within 0.00001 rad and mm, respectively, as shown in the IIZone (closedloop). This shows the beneficial effects of the HSPM in a closedloop control application.
The pink region shows the variations of the gap with the wafer scanning. The gap status changes violently while the pose of the IH’s lower surface remains unchanged during the wafer scanning. As shown in the IIIZone (openloop control without pose tracking), varies in the range of 0.000003~0.000035 rad and varies in the range of 0.091~0.105 mm. In order to stabilize the status of the gap, the pose of the IH’s lower surface should adjust with respect to the wafer surface in real time, which makes extremely great demands on the computational efficiency of the direct and inverse position analysis. As both the direct and inverse position solutions in analytical form are obtained, the computational efficiency has very big enhancement, making realtime pose tracking based on online calculation possible. As shown in the IVZone (closedloop control without pose tracking), by introducing the pose tracking in real time, the angle and the central distance are controlled within 0.000025 rad and mm, respectively, during the wafer scanning process, which meets the installation requirements of IH well.
7. Conclusions
In this paper, a novel hybrid serialparallel mechanism is presented to meet the pose regulation requirement of immersion hood in the immersion lithography system. The directinverse position analysis as the basic issue of the mechanical study is mainly discussed in this paper. The conclusions are drawn as follows.(1)By attaching the base platform of the 3 PM to the moving platform of the 3 PM, the novel HSPM combines these two parallel modules in serial form and has 6 DOFs.(2)By choosing the proper pose parameters and constraints, the solutions in analytical form of both 3 PM’s direct and inverse position analyses in the couple mixedtype mode are obtained, which improves the analytical analysis of the 3 PM. Based on the position analysis of the 3 PM, both the direct and the inverse position analyses of the HSPM are discussed, with the solutions presented in analytical form.(3)Since both the direct and inverse position solutions in analytical form are obtained, the computational efficiency obtains very big enhancement, which makes realtime pose tracking based on online calculation possible. The directinverse position model is applied in the realtime control of the IH’s lower surface pose tracking with respect to the wafer surface. The stability improvement of the gap is verified by the experiment, which indicates significant effect.(4)Currently, improving the parallel mechanism’s major drawback of the limited workspace while keeping the advantage (precision, rigidity, and loadtoweight ratio) is the main focus of many researchers on the hybrid serialparallel mechanism. The novel