# Position Analysis of a Hybrid Serial-Parallel Manipulator in Immersion Lithography

Authors: Jie-jie Shao; Tong Ma; Wen-yu Chen; Xin Fu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper proposes a novel hybrid serial-parallel mechanism with 6 degrees of freedom. The new mechanism combines two different parallel modules in a serial form. 3-P̲(PH) parallel module is architecture of 3 degrees of freedom based on higher joints and specializes in describing two planes’ relative pose. 3-P̲SP parallel module is typical architecture which has been widely investigated in recent researches. In this paper, the direct-inverse position problems of the 3-P̲SP parallel module in the couple mixed-type 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 serial-parallel mechanism are also derived and obtained in the analytical form. The proposed hybrid serial-parallel 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 real-time and the gap status is stabilized. This is another exploration to hybrid serial-parallel mechanism’s application.


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## Body

## 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.Figure 1
The IHPRM in immersion lithography.Commonly, the main structure of the IHPRM is a 3-degree 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 Figure1. 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 3-DOF parallel mechanism is difficult to be competent in high precision pose regulation and real-time tracking.The hybrid serial-parallel mechanism (HS-PM) 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 so-called variable-geometry truss manipulators which realize the combination of the parallel manipulators in a serial form in 1991. Shahinpoor [10] has studied a hybrid serial-parallel manipulator formed by two serially connected parallel manipulators and analyzed the inverse kinematic problem in 1992. Romdhane [11] proposed a Stewart platform-like hybrid serial-parallel mechanism, obtained a closed-form 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 closed-form solutions for the direct and inverse position problems in 2000. Kanaan et al. [13] applied 5-DOF serial-parallel architecture in the VERNE machine tool and derived its inverse-forward 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 serial-parallel 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 HS-PM is generally composed of several parallel mechanisms (regarded as its parallel module) with lower mobility, whether analytical form solutions can be obtained for the HS-PM’s direct-inverse position problem mainly depends on whether its parallel modules can. 3-PSP is a typical parallel mechanism (PM) with 3 DOFs and has been widely discussed in recent years. In 2001, Gregorio and Parenti-Castelli [17] made the position analysis of a general 3-PSP parallel robot in an analytical form. Deng and Li [18] obtained the 3-PSP PM’s direct position solution in a numerical form by solving 9 independent equations in 2009. Hao et al. [19] analyzed the 3-PSP PM in 2009. But because the pose status was described by Tilt-Torsion angle, only the inverse position problem was solved in an analytical form. Zhang and Li [20] discussed the direct position problem of the 3-P_SP PM with the screw theory in 2013, which still needs to solve 9 independent equations. Rezaei et al. [21] investigated the 3-PSP PM systematically in 2013. The direct-inverse position problem was divided into two modes according to the different pose parameters: the coupled mixed-type mode with θ, φ, and zp and the nonpure translational mode with xp, yp, and zp. 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 mixed-type 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 direct-inverse position problems of 3-PSP PM in the coupled mixed-type mode are still absent.In this paper, according to the high precision regulation and real-time tracking demand of the IHPRM in the immersion lithography system, a new type of hybrid serial-parallel mechanism with 6 DOFs is proposed. The new mechanism is constituted by combining two different parallel modules (including a 3-P_SP module and a 3-P_(PH) module) in a serial form. The 3-PSP module is employed to regulate IH’s pose, while the 3-P_(PH) module is adopted to detect the pose status of the IH’s lower surface with respect to the wafer’s upper surface.The direct-inverse position problems of 3-P_SP PM in the coupled mixed-type 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 direct-inverse position analysis for the new hybrid serial-parallel 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 Serial-Parallel Manipulator

### 2.1. The Hybrid Serial-Parallel 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 of100±5 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 ε. 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 HS-PM 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-P_(PH) PM and a 3-P_SP PM, we call it 3-P_SP-P_(PH) HS-PM. Combining two Stewart’s platforms in a series form to adjust these three planes is obviously much more complex and redundant than the 3-P_SP-P_(PH) HS-PM.


#### 2.1.1. The 3-P(PH) Parallel Modular

The 3-P_(PH) 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 (PH) in sequence, where the prismatic joint P is the active joint. The translational axis (CiDi) of the prismatic joint P within the ith (i=1,2,3) kinematic chain is perpendicular to the base. The higher joints (PHi) are attached to the sliders moving along these axes (CiDi), respectively, with their apexes Di constrained within the moving plane.Figure 2
The schematic of the 3-P_(PH) parallel mechanism.
(a)
The kinematic model
(b)
The CAD modelHere, the Grübler-Kutzbach criterion [22, 23] is applied to analyze the DOF of the 3-P_(PH) PM. Consider(1)M=λn-j-1+∑i=1jfi=65-6-1+18=6,where λ indicates the dimension of task space, n is the number of links, j is the number of joints, and fi stands for the DOF of joint i.As shown in (1), 3-P_(PH) 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-P_(PH) PM has 3 DOFs.


#### 2.1.2. The 3-PSP Parallel Modular

The 3-P_SP 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 P_, a spherical joint S, and a passive joint P in sequence.Figure 3
The schematic of the 3-P_SP parallel mechanism.
(a)
The kinematic model
(b)
The CAD model


#### 2.1.3. The Hybrid Mechanism

The novel3-P_SP-P_(PH) HS-PM combines these two parallel modules in serial form, by attaching the base platform of the 3-P_(PH) PM to the moving platform of the 3-P_SP PM, as shown in Figure 4. Summing the DOFs of each parallel module obtains the DOFs of the HS-PM, which is 6.Figure 4
The schematic of the HS-PM.
(a)
The kinematic model
(b)
The CAD modelThe three platforms in the3-P_SP-P_(PH) HS-PM are named base platform, moving platform, and target platform which lie on the fixed plane τ, moving plane ρ, and target plane ε, respectively. The 1st prismatic joint (P1i with the actuate parameter d1i, i=1,2,3) and the 3rd prismatic joint (P3i, with the actuate parameter d3i, i=1,2,3) are active joints with their translational axes AiBi and CiDi perpendicular to the base platform and the moving platform, respectively. The 2nd prismatic joint (P2i, i=1,2,3) and the spherical joint (Si, i=1,2,3) as passive joints coincide with the plane ρ. The higher joint (PHi, i=1,2,3) has 5 DOFs related to the plane ε, as its apex Di lies on the plane ε.


### 2.2. The Physical Model of the IHPRM

The3-P_SP-P_(PH) HS-PM 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 (SI-F01, 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 (PH).Figure 5
The IHPRM in the immersion lithography system.


## 2.1. The Hybrid Serial-Parallel 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 of100±5 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 ε. 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 HS-PM 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-P_(PH) PM and a 3-P_SP PM, we call it 3-P_SP-P_(PH) HS-PM. Combining two Stewart’s platforms in a series form to adjust these three planes is obviously much more complex and redundant than the 3-P_SP-P_(PH) HS-PM.


### 2.1.1. The 3-P(PH) Parallel Modular

The 3-P_(PH) 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 (PH) in sequence, where the prismatic joint P is the active joint. The translational axis (CiDi) of the prismatic joint P within the ith (i=1,2,3) kinematic chain is perpendicular to the base. The higher joints (PHi) are attached to the sliders moving along these axes (CiDi), respectively, with their apexes Di constrained within the moving plane.Figure 2
The schematic of the 3-P_(PH) parallel mechanism.
(a)
The kinematic model
(b)
The CAD modelHere, the Grübler-Kutzbach criterion [22, 23] is applied to analyze the DOF of the 3-P_(PH) PM. Consider(1)M=λn-j-1+∑i=1jfi=65-6-1+18=6,where λ indicates the dimension of task space, n is the number of links, j is the number of joints, and fi stands for the DOF of joint i.As shown in (1), 3-P_(PH) 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-P_(PH) PM has 3 DOFs.


### 2.1.2. The 3-PSP Parallel Modular

The 3-P_SP 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 P_, a spherical joint S, and a passive joint P in sequence.Figure 3
The schematic of the 3-P_SP parallel mechanism.
(a)
The kinematic model
(b)
The CAD model


### 2.1.3. The Hybrid Mechanism

The novel3-P_SP-P_(PH) HS-PM combines these two parallel modules in serial form, by attaching the base platform of the 3-P_(PH) PM to the moving platform of the 3-P_SP PM, as shown in Figure 4. Summing the DOFs of each parallel module obtains the DOFs of the HS-PM, which is 6.Figure 4
The schematic of the HS-PM.
(a)
The kinematic model
(b)
The CAD modelThe three platforms in the3-P_SP-P_(PH) HS-PM are named base platform, moving platform, and target platform which lie on the fixed plane τ, moving plane ρ, and target plane ε, respectively. The 1st prismatic joint (P1i with the actuate parameter d1i, i=1,2,3) and the 3rd prismatic joint (P3i, with the actuate parameter d3i, i=1,2,3) are active joints with their translational axes AiBi and CiDi perpendicular to the base platform and the moving platform, respectively. The 2nd prismatic joint (P2i, i=1,2,3) and the spherical joint (Si, i=1,2,3) as passive joints coincide with the plane ρ. The higher joint (PHi, i=1,2,3) has 5 DOFs related to the plane ε, as its apex Di lies on the plane ε.


## 2.1.1. The 3-P(PH) Parallel Modular

The 3-P_(PH) 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 (PH) in sequence, where the prismatic joint P is the active joint. The translational axis (CiDi) of the prismatic joint P within the ith (i=1,2,3) kinematic chain is perpendicular to the base. The higher joints (PHi) are attached to the sliders moving along these axes (CiDi), respectively, with their apexes Di constrained within the moving plane.Figure 2
The schematic of the 3-P_(PH) parallel mechanism.
(a)
The kinematic model
(b)
The CAD modelHere, the Grübler-Kutzbach criterion [22, 23] is applied to analyze the DOF of the 3-P_(PH) PM. Consider(1)M=λn-j-1+∑i=1jfi=65-6-1+18=6,where λ indicates the dimension of task space, n is the number of links, j is the number of joints, and fi stands for the DOF of joint i.As shown in (1), 3-P_(PH) 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-P_(PH) PM has 3 DOFs.


## 2.1.2. The 3-PSP Parallel Modular

The 3-P_SP 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 P_, a spherical joint S, and a passive joint P in sequence.Figure 3
The schematic of the 3-P_SP parallel mechanism.
(a)
The kinematic model
(b)
The CAD model


## 2.1.3. The Hybrid Mechanism

The novel3-P_SP-P_(PH) HS-PM combines these two parallel modules in serial form, by attaching the base platform of the 3-P_(PH) PM to the moving platform of the 3-P_SP PM, as shown in Figure 4. Summing the DOFs of each parallel module obtains the DOFs of the HS-PM, which is 6.Figure 4
The schematic of the HS-PM.
(a)
The kinematic model
(b)
The CAD modelThe three platforms in the3-P_SP-P_(PH) HS-PM are named base platform, moving platform, and target platform which lie on the fixed plane τ, moving plane ρ, and target plane ε, respectively. The 1st prismatic joint (P1i with the actuate parameter d1i, i=1,2,3) and the 3rd prismatic joint (P3i, with the actuate parameter d3i, i=1,2,3) are active joints with their translational axes AiBi and CiDi perpendicular to the base platform and the moving platform, respectively. The 2nd prismatic joint (P2i, i=1,2,3) and the spherical joint (Si, i=1,2,3) as passive joints coincide with the plane ρ. The higher joint (PHi, i=1,2,3) has 5 DOFs related to the plane ε, as its apex Di lies on the plane ε.


## 2.2. The Physical Model of the IHPRM

The3-P_SP-P_(PH) HS-PM 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 (SI-F01, 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 (PH).Figure 5
The IHPRM in the immersion lithography system.


## 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 Figure6. Point Ai (i=1,2,3) distributes on a fixed circle with radius of R in the plane τ. A fixed reference frame O{X,Y,Z} (ΣO) is assigned at the central point O of the fixed circle, with the OZ-axis perpendicular to the base platform and the OX-axis passing through point A1. The angle between OAi→ and OX-axis is φi. According to the Cartesian right-hand rule, the OY-axis can be determined. The axes of the 2nd prismatic joints intersect at point o. The point o is defined as the center of the platform. Then a moving reference frame o{x,y,z} (Σo) is assigned in the plane ρ at point o, with oz-axis perpendicular to plane ρ and the ox-axis passing through point B1. The oy-axis is also determined by the Cartesian right-hand rule. The angle between oBi→ and ox-axis is ψi. Point Ci (i=1,2,3) distributes on a mobile circle which centered at point t(p,q) with radius of r in the plane ρ. Points o, t, and C1 are collinear, and ∠B1  o  t is λ. Another moving reference frame t{x,y,z} (Σt) is assigned in the plane ρ at point t, with tz-axis perpendicular to plane ρ, and the tx-axis passing through point C1. The ty-axis is also determined by the Cartesian right-hand rule. The angle between tCi→ and tx-axis is ϕi. Point t3 is the intersection of the tz-axis and the plane ε. A target reference frame t3{u,v,w} (Σt3) is assigned in the plane ε at point t3, with t3w-axis perpendicular to plane ε and the t3u-axis passing through the apex D1 of the higher joint (PH). The t3v-axis is also determined by the Cartesian right-hand rule.Figure 6
The hybrid serial-parallel mechanism with the three reference frames.Three parameters are required to describe the pose of the planeτ and plane ρ, respectively, in frame ΣO, which includes the angular parameters and the position parameters.


### 3.1. Angular Parameters Definition

The fixed planeτ, moving plane ρ, and target plane ε are moved together and intersect at point O, which will not change the angular relationship between them, as shown in Figure 7.Figure 7
The angular relationships between the three planes.Vectorj→ is the normal unit vector of the target plane ε; vector i→ is the normal unit vector of the moving plane ρ. The angle between j→ and the OZ-axis is defined as tilt angle β. The angle between j→’s projection on the fixed plane τ and the OX-axis is defined as twist angle α. Similarly, the angle between i→ and the OZ-axis is defined as tilt angle β1. The angle between i→’s projection on the fixed plane τ and the OX-axis is defined as twist angle α1. The angle between the vector j→ and vector i→ is defined as tilt angle β2. The angle between j→’s projection and i→’s projection on the fixed plane τ is defined as twist angle α2. As defined above, these six angular parameters have the constraints as follows:(2)α2=α-α1cos⁡β1cos⁡β+sinβ1sinβcos⁡α2=cos⁡β2.


### 3.2. Position Parameters Definition

As the angular relationships of the fixed planeτ, moving plane ρ, and target plane ε are specified, the position relationships between them should also be determined in frame ΣO.As shown in Figure8, the OZ-axis intersects the plane ε at point t1; then the distance between point O and point t1 is defined as the central distance H between the fixed plane τ and the target plane ε. Similarly, point g1 is the projection of the moving platform’s central point o in the plane τ. The distance between point o and point g1 is defined as the central distance H1 between the fixed plane τ and the moving plane ρ. Further, oz-axis intersects the target plane ε at point t2; the distance between point o and point t2 is defined as the central distance H2 between the moving plane ρ and the target plane ε.Figure 8
The position relationships between the three planes.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 mixed-type 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 mixed-type is more significant to the pose control algorithm compared to the nonpure translational mode.


## 3.1. Angular Parameters Definition

The fixed planeτ, moving plane ρ, and target plane ε are moved together and intersect at point O, which will not change the angular relationship between them, as shown in Figure 7.Figure 7
The angular relationships between the three planes.Vectorj→ is the normal unit vector of the target plane ε; vector i→ is the normal unit vector of the moving plane ρ. The angle between j→ and the OZ-axis is defined as tilt angle β. The angle between j→’s projection on the fixed plane τ and the OX-axis is defined as twist angle α. Similarly, the angle between i→ and the OZ-axis is defined as tilt angle β1. The angle between i→’s projection on the fixed plane τ and the OX-axis is defined as twist angle α1. The angle between the vector j→ and vector i→ is defined as tilt angle β2. The angle between j→’s projection and i→’s projection on the fixed plane τ is defined as twist angle α2. As defined above, these six angular parameters have the constraints as follows:(2)α2=α-α1cos⁡β1cos⁡β+sinβ1sinβcos⁡α2=cos⁡β2.


## 3.2. Position Parameters Definition

As the angular relationships of the fixed planeτ, moving plane ρ, and target plane ε are specified, the position relationships between them should also be determined in frame ΣO.As shown in Figure8, the OZ-axis intersects the plane ε at point t1; then the distance between point O and point t1 is defined as the central distance H between the fixed plane τ and the target plane ε. Similarly, point g1 is the projection of the moving platform’s central point o in the plane τ. The distance between point o and point g1 is defined as the central distance H1 between the fixed plane τ and the moving plane ρ. Further, oz-axis intersects the target plane ε at point t2; the distance between point o and point t2 is defined as the central distance H2 between the moving plane ρ and the target plane ε.Figure 8
The position relationships between the three planes.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 mixed-type 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 mixed-type is more significant to the pose control algorithm compared to the nonpure translational mode.


## 4. Position Analysis of the 3-P_SP 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 closed-form solutions can be obtained. As the basis of the 3-P_SP-P_(PH) HS-PM’s position analysis, the position analysis of the 3-P_SP mechanism is investigated firstly.


### 4.1. Direct Position Analysis of the 3-P_SP Mechanism

The architecture of the 3-P_SP PM and the reference frames are shown in Figures 3 and 6. The direct position analysis of the 3-P_SP PM aims to obtain the orientation parameters (α1, β1) and position parameter (H1) of the moving platform with respect to the base platform using the given actuated parameters d1i (i=1,2,3) of the actuate prismatic joints P1i.


#### 4.1.1. Tilt Angleβ1

The homogeneous coordinates ofAi and Bi in the fixed reference frame ΣO are(3)Ai∑O=Rcos⁡φiRsinφi01T(i=1,2,3)Bi∑O=Rcos⁡φiRsinφid1i1Ti=1,2,3,where the additional right side subscripts of Ai and Bi indicate the reference frame in which the points are measured.Since the spherical jointSi’s central point Bi (i=1,2,3) is in the moving plane ρ, the normal vector of the plane ρ can be generated from B1B2⃑∑O×B1B3⃑∑O:(4)B1B2⃑∑O=Rcos⁡φ2-cos⁡φ1Rsinφ2-sinφ1d12-d110TB1B3⃑∑O=Rcos⁡φ3-cos⁡φ1Rsinφ3-sinφ1d13-d110T.Then we have the normal unit vectork⃑1∑O of the plane ρ as follows:(5)k⃑1∑O=B1B2⃑∑O×B1B3⃑∑OB1B2⃑∑O×B1B3⃑∑O=1k112+k122+k132k11k12k130T,where(6)k11=d12-d13sinφ1+d13-d11sinφ2+d11-d12sinφ3,k12=d13-d12cos⁡φ1+d11-d13cos⁡φ2+d12-d11cos⁡φ3,k13=Rsinφ1-φ3+sinφ2-φ1+sinφ3-φ2.The equation of the moving planeρ in the frame ΣO is(7)k11x-Rcos⁡φ1+k12y-Rsinφ1+k13z-d11=0.As defined above, the tilt angleβ1, from the normal unit vector k⃑1∑O to the positive direction of OZ-axis, is limited within the range of [0,π/2), which can be expressed as(8)β1=arctank112+k122k13.


#### 4.1.2. Twist Angleα1

Let the unit vectorw⃑1∑O denote the direction of the normal unit vector k⃑1∑O’s projection on the fixed plane τ(9)w⃑1∑O=1k112+k122k11k1200T.Calculating the angle between vectorw⃑1∑O and the positive OX-axis yields the twist angle α1, within the range of (-π,π] as defined above, which can be expressed as(10)α1=atan2k12,k11,where atan2 function calculates the arc tangent of the two variables k12 and k11, in which the signs of both arguments are used to determine the quadrant of the result.


#### 4.1.3. Central DistanceH1

The unit vector which denotes the direction of the prismatic jointP2i’s axis in the frame Σo is(11)n2i⃑∑o=cos⁡ψisinψi00Ti=1,2,3.The homogeneous transformation matricesToO which transfers the coordinates in the frame Σo to the coordinates in the frame ΣO can be obtained by the Euler’s transformation method:(12)ToO=RP01R=Rotz,α1·Roty,β1·Rotz,γ1=R11R12R13R21R22R23R31R32R33,where P denotes vector from the point O to the point o in the fixed reference frame ΣO. R is the Euler matrix. Rot(z,α1) denotes the counterclockwise rotation angle α1 about the oz-axis, Rot(y,β1) indicates the counterclockwise rotation angle β1 about the oy-axis, and Rot(z,γ1) indicates the counterclockwise rotation angle γ1 about the oz-axis. Consider the following:R11=cos⁡α1cos⁡β1cos⁡γ1-sinα1sinγ1;R12=-cos⁡α1cos⁡β1sinγ1-sinα1cos⁡γ1;R13=cos⁡α1sinβ1;R21=sinα1cos⁡β1cos⁡γ1+cos⁡α1sinγ1;R22=-sinα1cos⁡β1sinγ1+cos⁡α1cos⁡γ1;R23=sinα1sinβ1;R31=-sinβ1cos⁡γ1;R32=sinβ1sinγ1;R33=cos⁡β1.Assume the coordinate of the moving platform’s central pointo in the frame ΣO as(13)o∑O=oxoyoz1T.The coordinates of the vectors from pointo to point Ai are(14)oAi⃑∑O=Rcos⁡φi-oxRsinφi-oy-oz0Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.The unit vector of theOZ-axis in the frame ΣO is(15)τ⃑∑O=0010T.As the vectorsni⃑∑O, oAi⃑∑O, and τ⃑∑O are coplanar according to the geometrical constraint [19, 24], their mixed product equals zero:(16)ni⃑∑OoAi⃑∑Oτ⃑∑O=0i=1,2,3.Equation (16) can be rearranged to obtain three constraint equations:(17)(R21ox-R11oy)cos⁡ψi+(R22ox-R12oy)sinψi+R·R11cos⁡ψisinφi-R·R22cos⁡φisinψi+R·R12sinφisinψi-R·R21cos⁡φicos⁡ψi=0(i=1,2,3).The three constraint equations yield the constraint equation of the torsion angleγ1(18)aR11+bR12+cR21+dR22=0,where(19)a=sinψ1cos⁡ψ2cos⁡ψ3sinφ3-sinψ1cos⁡ψ2cos⁡ψ3sinφ2+cos⁡ψ1sinψ2cos⁡ψ3sinφ1-cos⁡ψ1sinψ2cos⁡ψ3sinφ3-cos⁡ψ1cos⁡ψ2sinψ3sinφ1+cos⁡ψ1cos⁡ψ2sinψ3sinφ2;b=sinψ1sinψ2cos⁡ψ3sinφ1-sinψ1sinψ2cos⁡ψ3sinφ2-sinψ1cos⁡ψ2sinψ3sinφ1+sinψ1cos⁡ψ2sinψ3sinφ3+cos⁡ψ1sinψ2sinψ3sinφ2-cos⁡ψ1sinψ2sinψ3sinφ3;c=cos⁡ψ2sin(ψ1-ψ3)cos⁡φ2-cos⁡ψ3sin(ψ1-ψ2)cos⁡φ3-cos⁡ψ1sinψ2cos⁡ψ3cos⁡φ1+cos⁡ψ1cos⁡ψ2sinψ3cos⁡φ1;d=sinψ2sin(ψ1-ψ3)cos⁡φ2-sinψ1sinψ2cos⁡ψ3cos⁡φ1+sinψ1sinψ3cos⁡ψ2cos⁡φ1-sinψ3sinψ1-ψ2cos⁡φ3;when cos⁡γ1=0,(20)γ1=-α1α1π2α1≠0π2α1=0.The coordinatesox and oy of the central point o in the frame ΣO can be expressed in the analytical form(21)ox=-Rcos⁡φ1R21cos⁡ψ1+R22sinψ1·R11cos⁡ψ2+R12sinψ2·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)(sinφ1-sinφ2)·(R11cos⁡ψ2+R12sinψ2)·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)cos⁡φ2·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1,oy=Rsinφ1R11cos⁡ψ1+R12sinψ1·R21cos⁡ψ2+R22sinψ2·R12R21-R11R22sinψ1-ψ2-1-Rsinφ2R11cos⁡ψ2+R12sinψ2·R21cos⁡ψ1+R22sinψ1·R12R21-R11R22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(R21cos⁡ψ1+R22sinψ1)·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1;when cos⁡γ1≠0, K11=R11/cos⁡γ1=cos⁡α1cos⁡β1-sinα1tanγ1; K12=R12/cos⁡γ1=-cos⁡α1cos⁡β1tanγ1-sinα1; K21=R21/cos⁡γ1=sinα1cos⁡β1+cos⁡α1tanγ1; K22=R22/cos⁡γ1=-sinα1cos⁡β1tanγ1+cos⁡α1.Equation (18) can be transformed as(22)aK11+bK12+cK21+dK22=0.The tangent of the torsion angleγ1 is(23)tanγ1=acos⁡α1cos⁡β1-bsinα1+csinα1cos⁡β1+dcos⁡α1asinα1+bcos⁡β1cos⁡α1-ccos⁡α1+dsinα1cos⁡β1.Then theox and oy of the central point o in the frame ΣO are(24)ox=-Rcos⁡φ1K21cos⁡ψ1+K22sinψ1·K11cos⁡ψ2+K12sinψ2·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)(sinφ1-sinφ2)·(K11cos⁡ψ2+K12sinψ2)·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)cos⁡φ2·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1,oy=Rsinφ1K11cos⁡ψ1+K12sinψ1·K21cos⁡ψ2+K22sinψ2·K12K21-K11K22sinψ1-ψ2-1-Rsinφ2K11cos⁡ψ2+K12sinψ2·K21cos⁡ψ1+K22sinψ1·K12K21-K11K22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(K21cos⁡ψ1+K22sinψ1)·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1.As the moving platform’s central pointo is in plane ρ, the coordinate value oz in the OZ direction can be obtained by (7). The distance between the point o and the fixed plane τ is the central distance H1 as defined before(25)H1=oz=d11-k11ox-Rcos⁡φ1+k12oy-Rsinφ1k13.


### 4.2. Inverse Position Analysis of the 3-P_SP Mechanism

The inverse position analysis of the 3-P_SP PM is to obtain the actuated parameters d1i (i=1,2,3) of the prismatic joints P1i with the given orientation parameters (α1, β1) and position parameter (H1) of the moving platform relative to the base platform.The coordinate valuesox and oy in the frame ΣO can be obtained directly according to the direct position analysis in Section 4.1. The homogeneous coordinate of the moving platform’s central point o is(26)o∑O=oxoyoz1T,where oz=H1.The vectoroBi⃑ indicates the direction from the moving platform’s central point o to the spherical joint’s central point Bi in the plane ρ(27)oBi⃑∑O=Rcos⁡φi-oxRsinφi-oyd1i-oz0Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.The normal unit vectork⃑1∑o=0010Tof the plane ρ in the frame Σo can be transferred to k⃑1∑O in the frame ΣO with the given twist angle α1 and tilt angle β1:(28)k⃑1∑O=R001·k⃑1∑oR=Rotz,α1Roty,β1=cos⁡α1cos⁡β1-sinα1cos⁡α1sinβ1sinα1cos⁡β1cos⁡α1sinα1sinβ1-sinβ10cos⁡β1,where R is the transformation matric. Rot(z,α1) denotes the counterclockwise rotation angle α1 about the oz-axis; Rot(y,β1) indicates the counterclockwise rotation angle β1 about the oy-axis. Consider (29)k⃑1∑O=cos⁡α1sinβ1sinα1sinβ1cos⁡β10T.Since the vectoroBi⃑∑O in the plane ρ is orthogonal with respect to the normal unit vector k⃑1∑O of the plane ρ, the following equation can be obtained:(30)oBi⃑∑O·k⃑1∑O=0(i=1,2,3)d1i=oz-Rsinφi-oyRcos⁡φi-oxcos⁡α1+Rsinφi-oysinα1tanβ1.Overall, by choosing the proper pose parameters and constraints, both the direct and inverse position analyses of the 3-P_SP PM obtain the solutions in analytical form, which indicates the one-to-one correspondent relationships between the actuated parameters and the pose parameters of the 3-P_SP PM.


## 4.1. Direct Position Analysis of the 3-P_SP Mechanism

The architecture of the 3-P_SP PM and the reference frames are shown in Figures 3 and 6. The direct position analysis of the 3-P_SP PM aims to obtain the orientation parameters (α1, β1) and position parameter (H1) of the moving platform with respect to the base platform using the given actuated parameters d1i (i=1,2,3) of the actuate prismatic joints P1i.


### 4.1.1. Tilt Angleβ1

The homogeneous coordinates ofAi and Bi in the fixed reference frame ΣO are(3)Ai∑O=Rcos⁡φiRsinφi01T(i=1,2,3)Bi∑O=Rcos⁡φiRsinφid1i1Ti=1,2,3,where the additional right side subscripts of Ai and Bi indicate the reference frame in which the points are measured.Since the spherical jointSi’s central point Bi (i=1,2,3) is in the moving plane ρ, the normal vector of the plane ρ can be generated from B1B2⃑∑O×B1B3⃑∑O:(4)B1B2⃑∑O=Rcos⁡φ2-cos⁡φ1Rsinφ2-sinφ1d12-d110TB1B3⃑∑O=Rcos⁡φ3-cos⁡φ1Rsinφ3-sinφ1d13-d110T.Then we have the normal unit vectork⃑1∑O of the plane ρ as follows:(5)k⃑1∑O=B1B2⃑∑O×B1B3⃑∑OB1B2⃑∑O×B1B3⃑∑O=1k112+k122+k132k11k12k130T,where(6)k11=d12-d13sinφ1+d13-d11sinφ2+d11-d12sinφ3,k12=d13-d12cos⁡φ1+d11-d13cos⁡φ2+d12-d11cos⁡φ3,k13=Rsinφ1-φ3+sinφ2-φ1+sinφ3-φ2.The equation of the moving planeρ in the frame ΣO is(7)k11x-Rcos⁡φ1+k12y-Rsinφ1+k13z-d11=0.As defined above, the tilt angleβ1, from the normal unit vector k⃑1∑O to the positive direction of OZ-axis, is limited within the range of [0,π/2), which can be expressed as(8)β1=arctank112+k122k13.


### 4.1.2. Twist Angleα1

Let the unit vectorw⃑1∑O denote the direction of the normal unit vector k⃑1∑O’s projection on the fixed plane τ(9)w⃑1∑O=1k112+k122k11k1200T.Calculating the angle between vectorw⃑1∑O and the positive OX-axis yields the twist angle α1, within the range of (-π,π] as defined above, which can be expressed as(10)α1=atan2k12,k11,where atan2 function calculates the arc tangent of the two variables k12 and k11, in which the signs of both arguments are used to determine the quadrant of the result.


### 4.1.3. Central DistanceH1

The unit vector which denotes the direction of the prismatic jointP2i’s axis in the frame Σo is(11)n2i⃑∑o=cos⁡ψisinψi00Ti=1,2,3.The homogeneous transformation matricesToO which transfers the coordinates in the frame Σo to the coordinates in the frame ΣO can be obtained by the Euler’s transformation method:(12)ToO=RP01R=Rotz,α1·Roty,β1·Rotz,γ1=R11R12R13R21R22R23R31R32R33,where P denotes vector from the point O to the point o in the fixed reference frame ΣO. R is the Euler matrix. Rot(z,α1) denotes the counterclockwise rotation angle α1 about the oz-axis, Rot(y,β1) indicates the counterclockwise rotation angle β1 about the oy-axis, and Rot(z,γ1) indicates the counterclockwise rotation angle γ1 about the oz-axis. Consider the following:R11=cos⁡α1cos⁡β1cos⁡γ1-sinα1sinγ1;R12=-cos⁡α1cos⁡β1sinγ1-sinα1cos⁡γ1;R13=cos⁡α1sinβ1;R21=sinα1cos⁡β1cos⁡γ1+cos⁡α1sinγ1;R22=-sinα1cos⁡β1sinγ1+cos⁡α1cos⁡γ1;R23=sinα1sinβ1;R31=-sinβ1cos⁡γ1;R32=sinβ1sinγ1;R33=cos⁡β1.Assume the coordinate of the moving platform’s central pointo in the frame ΣO as(13)o∑O=oxoyoz1T.The coordinates of the vectors from pointo to point Ai are(14)oAi⃑∑O=Rcos⁡φi-oxRsinφi-oy-oz0Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.The unit vector of theOZ-axis in the frame ΣO is(15)τ⃑∑O=0010T.As the vectorsni⃑∑O, oAi⃑∑O, and τ⃑∑O are coplanar according to the geometrical constraint [19, 24], their mixed product equals zero:(16)ni⃑∑OoAi⃑∑Oτ⃑∑O=0i=1,2,3.Equation (16) can be rearranged to obtain three constraint equations:(17)(R21ox-R11oy)cos⁡ψi+(R22ox-R12oy)sinψi+R·R11cos⁡ψisinφi-R·R22cos⁡φisinψi+R·R12sinφisinψi-R·R21cos⁡φicos⁡ψi=0(i=1,2,3).The three constraint equations yield the constraint equation of the torsion angleγ1(18)aR11+bR12+cR21+dR22=0,where(19)a=sinψ1cos⁡ψ2cos⁡ψ3sinφ3-sinψ1cos⁡ψ2cos⁡ψ3sinφ2+cos⁡ψ1sinψ2cos⁡ψ3sinφ1-cos⁡ψ1sinψ2cos⁡ψ3sinφ3-cos⁡ψ1cos⁡ψ2sinψ3sinφ1+cos⁡ψ1cos⁡ψ2sinψ3sinφ2;b=sinψ1sinψ2cos⁡ψ3sinφ1-sinψ1sinψ2cos⁡ψ3sinφ2-sinψ1cos⁡ψ2sinψ3sinφ1+sinψ1cos⁡ψ2sinψ3sinφ3+cos⁡ψ1sinψ2sinψ3sinφ2-cos⁡ψ1sinψ2sinψ3sinφ3;c=cos⁡ψ2sin(ψ1-ψ3)cos⁡φ2-cos⁡ψ3sin(ψ1-ψ2)cos⁡φ3-cos⁡ψ1sinψ2cos⁡ψ3cos⁡φ1+cos⁡ψ1cos⁡ψ2sinψ3cos⁡φ1;d=sinψ2sin(ψ1-ψ3)cos⁡φ2-sinψ1sinψ2cos⁡ψ3cos⁡φ1+sinψ1sinψ3cos⁡ψ2cos⁡φ1-sinψ3sinψ1-ψ2cos⁡φ3;when cos⁡γ1=0,(20)γ1=-α1α1π2α1≠0π2α1=0.The coordinatesox and oy of the central point o in the frame ΣO can be expressed in the analytical form(21)ox=-Rcos⁡φ1R21cos⁡ψ1+R22sinψ1·R11cos⁡ψ2+R12sinψ2·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)(sinφ1-sinφ2)·(R11cos⁡ψ2+R12sinψ2)·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)cos⁡φ2·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1,oy=Rsinφ1R11cos⁡ψ1+R12sinψ1·R21cos⁡ψ2+R22sinψ2·R12R21-R11R22sinψ1-ψ2-1-Rsinφ2R11cos⁡ψ2+R12sinψ2·R21cos⁡ψ1+R22sinψ1·R12R21-R11R22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(R21cos⁡ψ1+R22sinψ1)·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1;when cos⁡γ1≠0, K11=R11/cos⁡γ1=cos⁡α1cos⁡β1-sinα1tanγ1; K12=R12/cos⁡γ1=-cos⁡α1cos⁡β1tanγ1-sinα1; K21=R21/cos⁡γ1=sinα1cos⁡β1+cos⁡α1tanγ1; K22=R22/cos⁡γ1=-sinα1cos⁡β1tanγ1+cos⁡α1.Equation (18) can be transformed as(22)aK11+bK12+cK21+dK22=0.The tangent of the torsion angleγ1 is(23)tanγ1=acos⁡α1cos⁡β1-bsinα1+csinα1cos⁡β1+dcos⁡α1asinα1+bcos⁡β1cos⁡α1-ccos⁡α1+dsinα1cos⁡β1.Then theox and oy of the central point o in the frame ΣO are(24)ox=-Rcos⁡φ1K21cos⁡ψ1+K22sinψ1·K11cos⁡ψ2+K12sinψ2·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)(sinφ1-sinφ2)·(K11cos⁡ψ2+K12sinψ2)·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)cos⁡φ2·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1,oy=Rsinφ1K11cos⁡ψ1+K12sinψ1·K21cos⁡ψ2+K22sinψ2·K12K21-K11K22sinψ1-ψ2-1-Rsinφ2K11cos⁡ψ2+K12sinψ2·K21cos⁡ψ1+K22sinψ1·K12K21-K11K22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(K21cos⁡ψ1+K22sinψ1)·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1.As the moving platform’s central pointo is in plane ρ, the coordinate value oz in the OZ direction can be obtained by (7). The distance between the point o and the fixed plane τ is the central distance H1 as defined before(25)H1=oz=d11-k11ox-Rcos⁡φ1+k12oy-Rsinφ1k13.


## 4.1.1. Tilt Angleβ1

The homogeneous coordinates ofAi and Bi in the fixed reference frame ΣO are(3)Ai∑O=Rcos⁡φiRsinφi01T(i=1,2,3)Bi∑O=Rcos⁡φiRsinφid1i1Ti=1,2,3,where the additional right side subscripts of Ai and Bi indicate the reference frame in which the points are measured.Since the spherical jointSi’s central point Bi (i=1,2,3) is in the moving plane ρ, the normal vector of the plane ρ can be generated from B1B2⃑∑O×B1B3⃑∑O:(4)B1B2⃑∑O=Rcos⁡φ2-cos⁡φ1Rsinφ2-sinφ1d12-d110TB1B3⃑∑O=Rcos⁡φ3-cos⁡φ1Rsinφ3-sinφ1d13-d110T.Then we have the normal unit vectork⃑1∑O of the plane ρ as follows:(5)k⃑1∑O=B1B2⃑∑O×B1B3⃑∑OB1B2⃑∑O×B1B3⃑∑O=1k112+k122+k132k11k12k130T,where(6)k11=d12-d13sinφ1+d13-d11sinφ2+d11-d12sinφ3,k12=d13-d12cos⁡φ1+d11-d13cos⁡φ2+d12-d11cos⁡φ3,k13=Rsinφ1-φ3+sinφ2-φ1+sinφ3-φ2.The equation of the moving planeρ in the frame ΣO is(7)k11x-Rcos⁡φ1+k12y-Rsinφ1+k13z-d11=0.As defined above, the tilt angleβ1, from the normal unit vector k⃑1∑O to the positive direction of OZ-axis, is limited within the range of [0,π/2), which can be expressed as(8)β1=arctank112+k122k13.


## 4.1.2. Twist Angleα1

Let the unit vectorw⃑1∑O denote the direction of the normal unit vector k⃑1∑O’s projection on the fixed plane τ(9)w⃑1∑O=1k112+k122k11k1200T.Calculating the angle between vectorw⃑1∑O and the positive OX-axis yields the twist angle α1, within the range of (-π,π] as defined above, which can be expressed as(10)α1=atan2k12,k11,where atan2 function calculates the arc tangent of the two variables k12 and k11, in which the signs of both arguments are used to determine the quadrant of the result.


## 4.1.3. Central DistanceH1

The unit vector which denotes the direction of the prismatic jointP2i’s axis in the frame Σo is(11)n2i⃑∑o=cos⁡ψisinψi00Ti=1,2,3.The homogeneous transformation matricesToO which transfers the coordinates in the frame Σo to the coordinates in the frame ΣO can be obtained by the Euler’s transformation method:(12)ToO=RP01R=Rotz,α1·Roty,β1·Rotz,γ1=R11R12R13R21R22R23R31R32R33,where P denotes vector from the point O to the point o in the fixed reference frame ΣO. R is the Euler matrix. Rot(z,α1) denotes the counterclockwise rotation angle α1 about the oz-axis, Rot(y,β1) indicates the counterclockwise rotation angle β1 about the oy-axis, and Rot(z,γ1) indicates the counterclockwise rotation angle γ1 about the oz-axis. Consider the following:R11=cos⁡α1cos⁡β1cos⁡γ1-sinα1sinγ1;R12=-cos⁡α1cos⁡β1sinγ1-sinα1cos⁡γ1;R13=cos⁡α1sinβ1;R21=sinα1cos⁡β1cos⁡γ1+cos⁡α1sinγ1;R22=-sinα1cos⁡β1sinγ1+cos⁡α1cos⁡γ1;R23=sinα1sinβ1;R31=-sinβ1cos⁡γ1;R32=sinβ1sinγ1;R33=cos⁡β1.Assume the coordinate of the moving platform’s central pointo in the frame ΣO as(13)o∑O=oxoyoz1T.The coordinates of the vectors from pointo to point Ai are(14)oAi⃑∑O=Rcos⁡φi-oxRsinφi-oy-oz0Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.The unit vector of theOZ-axis in the frame ΣO is(15)τ⃑∑O=0010T.As the vectorsni⃑∑O, oAi⃑∑O, and τ⃑∑O are coplanar according to the geometrical constraint [19, 24], their mixed product equals zero:(16)ni⃑∑OoAi⃑∑Oτ⃑∑O=0i=1,2,3.Equation (16) can be rearranged to obtain three constraint equations:(17)(R21ox-R11oy)cos⁡ψi+(R22ox-R12oy)sinψi+R·R11cos⁡ψisinφi-R·R22cos⁡φisinψi+R·R12sinφisinψi-R·R21cos⁡φicos⁡ψi=0(i=1,2,3).The three constraint equations yield the constraint equation of the torsion angleγ1(18)aR11+bR12+cR21+dR22=0,where(19)a=sinψ1cos⁡ψ2cos⁡ψ3sinφ3-sinψ1cos⁡ψ2cos⁡ψ3sinφ2+cos⁡ψ1sinψ2cos⁡ψ3sinφ1-cos⁡ψ1sinψ2cos⁡ψ3sinφ3-cos⁡ψ1cos⁡ψ2sinψ3sinφ1+cos⁡ψ1cos⁡ψ2sinψ3sinφ2;b=sinψ1sinψ2cos⁡ψ3sinφ1-sinψ1sinψ2cos⁡ψ3sinφ2-sinψ1cos⁡ψ2sinψ3sinφ1+sinψ1cos⁡ψ2sinψ3sinφ3+cos⁡ψ1sinψ2sinψ3sinφ2-cos⁡ψ1sinψ2sinψ3sinφ3;c=cos⁡ψ2sin(ψ1-ψ3)cos⁡φ2-cos⁡ψ3sin(ψ1-ψ2)cos⁡φ3-cos⁡ψ1sinψ2cos⁡ψ3cos⁡φ1+cos⁡ψ1cos⁡ψ2sinψ3cos⁡φ1;d=sinψ2sin(ψ1-ψ3)cos⁡φ2-sinψ1sinψ2cos⁡ψ3cos⁡φ1+sinψ1sinψ3cos⁡ψ2cos⁡φ1-sinψ3sinψ1-ψ2cos⁡φ3;when cos⁡γ1=0,(20)γ1=-α1α1π2α1≠0π2α1=0.The coordinatesox and oy of the central point o in the frame ΣO can be expressed in the analytical form(21)ox=-Rcos⁡φ1R21cos⁡ψ1+R22sinψ1·R11cos⁡ψ2+R12sinψ2·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)(sinφ1-sinφ2)·(R11cos⁡ψ2+R12sinψ2)·R12R21-R11R22sinψ1-ψ2-1+R(R11cos⁡ψ1+R12sinψ1)cos⁡φ2·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1,oy=Rsinφ1R11cos⁡ψ1+R12sinψ1·R21cos⁡ψ2+R22sinψ2·R12R21-R11R22sinψ1-ψ2-1-Rsinφ2R11cos⁡ψ2+R12sinψ2·R21cos⁡ψ1+R22sinψ1·R12R21-R11R22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(R21cos⁡ψ1+R22sinψ1)·(R21cos⁡ψ2+R22sinψ2)·R12R21-R11R22sinψ1-ψ2-1;when cos⁡γ1≠0, K11=R11/cos⁡γ1=cos⁡α1cos⁡β1-sinα1tanγ1; K12=R12/cos⁡γ1=-cos⁡α1cos⁡β1tanγ1-sinα1; K21=R21/cos⁡γ1=sinα1cos⁡β1+cos⁡α1tanγ1; K22=R22/cos⁡γ1=-sinα1cos⁡β1tanγ1+cos⁡α1.Equation (18) can be transformed as(22)aK11+bK12+cK21+dK22=0.The tangent of the torsion angleγ1 is(23)tanγ1=acos⁡α1cos⁡β1-bsinα1+csinα1cos⁡β1+dcos⁡α1asinα1+bcos⁡β1cos⁡α1-ccos⁡α1+dsinα1cos⁡β1.Then theox and oy of the central point o in the frame ΣO are(24)ox=-Rcos⁡φ1K21cos⁡ψ1+K22sinψ1·K11cos⁡ψ2+K12sinψ2·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)(sinφ1-sinφ2)·(K11cos⁡ψ2+K12sinψ2)·K12K21-K11K22sinψ1-ψ2-1+R(K11cos⁡ψ1+K12sinψ1)cos⁡φ2·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1,oy=Rsinφ1K11cos⁡ψ1+K12sinψ1·K21cos⁡ψ2+K22sinψ2·K12K21-K11K22sinψ1-ψ2-1-Rsinφ2K11cos⁡ψ2+K12sinψ2·K21cos⁡ψ1+K22sinψ1·K12K21-K11K22sinψ1-ψ2-1+Rcos⁡φ2-cos⁡φ1(K21cos⁡ψ1+K22sinψ1)·(K21cos⁡ψ2+K22sinψ2)·K12K21-K11K22sinψ1-ψ2-1.As the moving platform’s central pointo is in plane ρ, the coordinate value oz in the OZ direction can be obtained by (7). The distance between the point o and the fixed plane τ is the central distance H1 as defined before(25)H1=oz=d11-k11ox-Rcos⁡φ1+k12oy-Rsinφ1k13.


## 4.2. Inverse Position Analysis of the 3-P_SP Mechanism

The inverse position analysis of the 3-P_SP PM is to obtain the actuated parameters d1i (i=1,2,3) of the prismatic joints P1i with the given orientation parameters (α1, β1) and position parameter (H1) of the moving platform relative to the base platform.The coordinate valuesox and oy in the frame ΣO can be obtained directly according to the direct position analysis in Section 4.1. The homogeneous coordinate of the moving platform’s central point o is(26)o∑O=oxoyoz1T,where oz=H1.The vectoroBi⃑ indicates the direction from the moving platform’s central point o to the spherical joint’s central point Bi in the plane ρ(27)oBi⃑∑O=Rcos⁡φi-oxRsinφi-oyd1i-oz0Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.The normal unit vectork⃑1∑o=0010Tof the plane ρ in the frame Σo can be transferred to k⃑1∑O in the frame ΣO with the given twist angle α1 and tilt angle β1:(28)k⃑1∑O=R001·k⃑1∑oR=Rotz,α1Roty,β1=cos⁡α1cos⁡β1-sinα1cos⁡α1sinβ1sinα1cos⁡β1cos⁡α1sinα1sinβ1-sinβ10cos⁡β1,where R is the transformation matric. Rot(z,α1) denotes the counterclockwise rotation angle α1 about the oz-axis; Rot(y,β1) indicates the counterclockwise rotation angle β1 about the oy-axis. Consider (29)k⃑1∑O=cos⁡α1sinβ1sinα1sinβ1cos⁡β10T.Since the vectoroBi⃑∑O in the plane ρ is orthogonal with respect to the normal unit vector k⃑1∑O of the plane ρ, the following equation can be obtained:(30)oBi⃑∑O·k⃑1∑O=0(i=1,2,3)d1i=oz-Rsinφi-oyRcos⁡φi-oxcos⁡α1+Rsinφi-oysinα1tanβ1.Overall, by choosing the proper pose parameters and constraints, both the direct and inverse position analyses of the 3-P_SP PM obtain the solutions in analytical form, which indicates the one-to-one correspondent relationships between the actuated parameters and the pose parameters of the 3-P_SP PM.


## 5. Position Analysis of the Hybrid Serial-Parallel Mechanism

### 5.1. Direct Position Analysis of the Hybrid Serial-Parallel Mechanism

The direct position analysis of the3-P_SP-P_(PH) HS-PM aims to obtain the pose parameters of the target plane ε relative to the fixed plane τ (α,β,H) and the moving plane ρ (α2,β2,H2), respectively, using the given actuated parameters d1i and d3i (i=1,2,3) of the prismatic joints P1i and P3i.As the actuated parametersd1i (i=1,2,3) are given, the parameters α1, β1, and H1 which describe the pose of the plane ρ with respect to the plane τ can be obtained directly as the direct position analysis of the 3-P_SP PM shown in Section 4.1.The actuated parametersd3i (i=1,2,3) of the prismatic joints equal the module of the vector CiDi⃑. As shown in the Section 4, the normal unit vector k⃑2∑o of the target plane ε in frame Σt is(31)k⃑2∑t=1k21′′2+k22′′2+k23′′2k21′′k22′′k23′′0T,where(32)k21′′=d32-d33sinϕ1+d33-d31sinϕ2+d31-d32sinϕ3,k22′′=d33-d32cos⁡ϕ1+d31-d33cos⁡ϕ2+d32-d31cos⁡ϕ3,k23′′=rsinϕ1-ϕ3+sinϕ2-ϕ1+sinϕ3-ϕ2.k⃑2∑t can be transferred to the frame Σo, as (33)k⃑2∑o=Tto·k⃑2∑t=1k21′2+k22′2+k23′2k21′k22′k23′0T,where (34)Tto=cos⁡λ-sinλ0psinλcos⁡λ0q00100001,k21′=k21′′cos⁡λ-k22′′sinλ,k22′=k21′′sinλ+k22′′cos⁡λ,k23′=k23′′.k⃑2∑o can be transferred to the frame ΣO, as(35)k⃑2∑O=R001·k⃑2∑o=1k212+k222+k232k21k22k230,where(36)k21=k21′cos⁡α1cos⁡β1-k22′sinα1+k23′cos⁡α1sinβ1,k22=k21′sinα1cos⁡β1+k22cos⁡α1+k23′sinα1sinβ1,k23=-k21′sinβ1+k23′cos⁡β1.As defined above, the tilt angleβ, from the normal unit vector k⃑2∑O to the positive direction of OZ-axis, is limited within the range of [0,π/2), which can be expressed as(37)β=arctank212+k222k23.Let the unit vectorw⃑2∑O denote the direction of the normal unit vector k⃑2∑O’s projection on the fixed plane τ(38)w⃑2∑O=1k212+k222k21k2200T.Calculating the angle from vectorw⃑2∑O to the positive OX-axis yields the twist angle α, within the range of (-π,π] as defined above, which can be expressed as(39)α=atan2k22,k21.The tilt angleβ2 and twist angle α2 can be calculated according to (2).In order to obtain the central distanceH2, an intermediate variable dc is introduced here, which indicates the distance between the points t and t3:(40)dc=-2d32cos⁡ϕ32+cscϕ32·ϕ32d33sinϕ2-d31sinϕ2-ϕ3·2cos⁡ϕ2-ϕ32-cos⁡ϕ32-1.The target planeε in the frame Σt is(41)k21′′u-0+k22′′v-0+k23′′w-dc=0.Transfer the coordinate of pointt2 in the frame Σo to the frame Σt(42)t2∑t=Tto-1·00H21T=-pcos⁡λ-qsinλ-qcos⁡λ+psinλH21T.By substituting formula (42) into (41), H2 can be obtained as follows:(43)H2=k21′′pcos⁡λ+qsinλ+k22′′qcos⁡λ-psinλk23′′+dc.By transfer the coordinate of pointt2 in frame Σo to frame ΣO with the homogeneous transformation matrices ToO(44)t2∑O=t2x∑Ot2y∑Ot2z∑O1T=ToO·00H21T.The equation of the target planeε is(45)k21x-t2x∑O+k22y-t2y∑O+k23z-t2z∑O=0.Letx=0 and y=0 yield the central distance H:(46)H=z=k21t2x∑O+k22t2y∑Ok23+t2z∑O.The relationship between the central distancesH, H1, and H2 can be obtained with (44) and (46):(47)H1=oz=H-k21t2x∑O+k22t2y∑Ok23-H2R33.


### 5.2. Inverse Position Analysis of the Hybrid Serial-Parallel Mechanism

The inverse position analysis of the3-P_SP-P_(PH) HS-PM is to obtain the actuated parameters d1i and d3i (i=1,2,3) of the prismatic joints P1i and P3i for the given pose parameters of the target plane ε relative to the fixed plane τ (α,β,H) and the moving plane ρ (α2,β2,H2), respectively.The homogeneous coordinate of the higher joint’s apexDi in frame Σo is(48)Di∑o=rcos⁡ϕi+λ+prsinϕi+λ+qd3i1Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.As discussed before, the pose of the target planeε with respect to the moving plane ρ can be described by the point t2 and the normal unit vector t2w⃑ of the plane ε. Consider(49)t2∑o=00H21T,t2w⃑∑O=cos⁡αsinβsinαsinβcos⁡β0T.Sinceα1 and β1 can be calculated by (2), the coordinate of vector t2w⃑ can be transferred from the frame ΣO to the frame Σo by Euler’s transformation as(50)t2w⃑∑o=R-1001·t2w⃑∑OR-1=R11R12R13R21R22R23R31R32R33-1=r11r12r13r21r22r23r31r32r33,where r11=cos⁡γ1cos⁡β1cos⁡α1-sinγ1sinα1; r12=cos⁡γ1cos⁡β1sinα1+sinγ1cos⁡α1; r13=-cos⁡γ1sinβ1; r21=-cos⁡γ1sinα1-sinγ1cos⁡β1cos⁡α1; r22=-sinγ1cos⁡β1sinα1+cos⁡γ1cos⁡α1; r23=sinγ1sinβ1; r31=sinβ1cos⁡α1; r32=sinβ1sinα1; and r33=cos⁡β1.Since the vectort2Di⃑ in the plane ε is orthogonal with respect to the vector t2w⃑ in frame Σo, the following equation can be obtained:(51)t2w⃑∑o·t2Di⃑∑o=0i=1,2,3.Equation (51) can be rearranged to obtain(52)d3i=H2-p·r13+q·r23+p·r11+q·r12cos⁡α+p·r12+q·r22sinαtanβ·r33+r31cos⁡α+r32sinαtanβ-1-rcos⁡ϕi+λr13+r11cos⁡α+r12sinαtanβr33+r31cos⁡α+r32sinαtanβ+rsinϕi+λr23+r21cos⁡α+r22sinαtanβr33+r31cos⁡α+r32sinαtanβ(i=1,2,3).The coordinate of the moving platform’s central pointo in frame ΣO can be obtained as shown in Section 4.2. Further, the actuated parameters d1i (i=1,2,3) of the prismatic joints P1i are(53)d1i=oz-Rcos⁡φi-oxcos⁡α1Rsinφi-oysinα1+Rsinφi-oysinα1tanβ1,i=1,2,3.With this, the whole actuated parameters of the3-P_SP-P_(PH) HS-PM in analytical form are obtained.


### 5.3. Numerical Example

Specify the architectural parameters of the3-P_SP-P_(PH) HS-PM according to the definition in Figure 6, as R=150, r=55, φ1=ψ1=0, φ2=ψ2=2π/3, φ3=ψ3=10π/9, p=0, q=0, λ = 25°, ϕ1 = 0°, ϕ2 = 90°, and ϕ3 = 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.Table 1
Direct-inverse position analysis of the 3-PSP-P(PH) HS-PM measured in frame ΣO (R=150, r=55,  φ1=ψ1=0,  φ2=ψ2=2π/3, φ3=ψ3=10π/9, p=0, q=0, λ=25°, ϕ1=0°, ϕ2=90°, and ϕ3=180°).
Direct
1
2
3
Inverse
4
5
6
d
11
2.500000
2.200000
1.400000
α
1.000000
3.141593
0.500000
d
12
2.000000
2.500000
2.200000
β
0.001000
0.010000
0.020000
d
13
1.500000
1.700000
2.400000
H
2.000000
2.500000
2.900000
d
31
0.200000
0.700000
0.400000
α
2
1.000000
0.300000
0.010000
d
32
0.500000
0.100000
0.350000
β
2
0.002000
0.005000
0.001000
d
33
0.100000
0.400000
0.500000
H
2
0.100000
0.800000
0.080000
α
−1.735114
2.832609
0.489347
d
11
1.546799
3.647094
0.043040
β
0.007869
0.007279
0.003988
d
12
2.076600
0.204843
2.925715
H
2.177085
2.599314
2.369710
d
13
2.231900
0.076335
5.936067
α
2
0.933895
4.646567
0.534729
d
31
0.170882
0.744787
0.134401
β
2
0.006428
0.008624
0.002033
d
32
0.015882
1.069403
0.071905
H
2
0.150000
0.550000
0.450000
d
33
0.029118
0.855213
0.025599
α
1
−2.669009
−1.813957
−0.045381
α
1
0.000000
2.841593
0.490000
β
1
0.003541
0.004172
0.003468
β
1
0.002355
0.013587
0.020979
H
1
2.027085
2.049327
1.919709
H
1
1.900000
1.699856
2.819845
o
x
0.000113
0.001052
−0.000589
o
x
−0.000251
−0.011449
0.004906
o
y
0.000157
0.000031
0.000040
o
y
0.000025
−0.000597
0.005566
o
z
2.027085
2.049327
1.919709
o
z
1.900000
1.699856
2.819845The left side of Table1 shows the position solving in the direct way, which calculate the pose parameters α, β, H, α2, β2, and H2 with the given actuate parameters d1i and d3i (i=1,2,3). The right side of Table 1 shows the position solving in the inverse way, which calculate the actuate parameters d1i and d3i (i=1,2,3) with the given pose parameters α, β, H, α2, β2, and H2. In addition, the moving platform’s pose parameters and coordinate of the central point o are also obtained in both sides. The one-to-one correspondent relationships between the actuated parameters and the pose parameters are obvious.


## 5.1. Direct Position Analysis of the Hybrid Serial-Parallel Mechanism

The direct position analysis of the3-P_SP-P_(PH) HS-PM aims to obtain the pose parameters of the target plane ε relative to the fixed plane τ (α,β,H) and the moving plane ρ (α2,β2,H2), respectively, using the given actuated parameters d1i and d3i (i=1,2,3) of the prismatic joints P1i and P3i.As the actuated parametersd1i (i=1,2,3) are given, the parameters α1, β1, and H1 which describe the pose of the plane ρ with respect to the plane τ can be obtained directly as the direct position analysis of the 3-P_SP PM shown in Section 4.1.The actuated parametersd3i (i=1,2,3) of the prismatic joints equal the module of the vector CiDi⃑. As shown in the Section 4, the normal unit vector k⃑2∑o of the target plane ε in frame Σt is(31)k⃑2∑t=1k21′′2+k22′′2+k23′′2k21′′k22′′k23′′0T,where(32)k21′′=d32-d33sinϕ1+d33-d31sinϕ2+d31-d32sinϕ3,k22′′=d33-d32cos⁡ϕ1+d31-d33cos⁡ϕ2+d32-d31cos⁡ϕ3,k23′′=rsinϕ1-ϕ3+sinϕ2-ϕ1+sinϕ3-ϕ2.k⃑2∑t can be transferred to the frame Σo, as (33)k⃑2∑o=Tto·k⃑2∑t=1k21′2+k22′2+k23′2k21′k22′k23′0T,where (34)Tto=cos⁡λ-sinλ0psinλcos⁡λ0q00100001,k21′=k21′′cos⁡λ-k22′′sinλ,k22′=k21′′sinλ+k22′′cos⁡λ,k23′=k23′′.k⃑2∑o can be transferred to the frame ΣO, as(35)k⃑2∑O=R001·k⃑2∑o=1k212+k222+k232k21k22k230,where(36)k21=k21′cos⁡α1cos⁡β1-k22′sinα1+k23′cos⁡α1sinβ1,k22=k21′sinα1cos⁡β1+k22cos⁡α1+k23′sinα1sinβ1,k23=-k21′sinβ1+k23′cos⁡β1.As defined above, the tilt angleβ, from the normal unit vector k⃑2∑O to the positive direction of OZ-axis, is limited within the range of [0,π/2), which can be expressed as(37)β=arctank212+k222k23.Let the unit vectorw⃑2∑O denote the direction of the normal unit vector k⃑2∑O’s projection on the fixed plane τ(38)w⃑2∑O=1k212+k222k21k2200T.Calculating the angle from vectorw⃑2∑O to the positive OX-axis yields the twist angle α, within the range of (-π,π] as defined above, which can be expressed as(39)α=atan2k22,k21.The tilt angleβ2 and twist angle α2 can be calculated according to (2).In order to obtain the central distanceH2, an intermediate variable dc is introduced here, which indicates the distance between the points t and t3:(40)dc=-2d32cos⁡ϕ32+cscϕ32·ϕ32d33sinϕ2-d31sinϕ2-ϕ3·2cos⁡ϕ2-ϕ32-cos⁡ϕ32-1.The target planeε in the frame Σt is(41)k21′′u-0+k22′′v-0+k23′′w-dc=0.Transfer the coordinate of pointt2 in the frame Σo to the frame Σt(42)t2∑t=Tto-1·00H21T=-pcos⁡λ-qsinλ-qcos⁡λ+psinλH21T.By substituting formula (42) into (41), H2 can be obtained as follows:(43)H2=k21′′pcos⁡λ+qsinλ+k22′′qcos⁡λ-psinλk23′′+dc.By transfer the coordinate of pointt2 in frame Σo to frame ΣO with the homogeneous transformation matrices ToO(44)t2∑O=t2x∑Ot2y∑Ot2z∑O1T=ToO·00H21T.The equation of the target planeε is(45)k21x-t2x∑O+k22y-t2y∑O+k23z-t2z∑O=0.Letx=0 and y=0 yield the central distance H:(46)H=z=k21t2x∑O+k22t2y∑Ok23+t2z∑O.The relationship between the central distancesH, H1, and H2 can be obtained with (44) and (46):(47)H1=oz=H-k21t2x∑O+k22t2y∑Ok23-H2R33.


## 5.2. Inverse Position Analysis of the Hybrid Serial-Parallel Mechanism

The inverse position analysis of the3-P_SP-P_(PH) HS-PM is to obtain the actuated parameters d1i and d3i (i=1,2,3) of the prismatic joints P1i and P3i for the given pose parameters of the target plane ε relative to the fixed plane τ (α,β,H) and the moving plane ρ (α2,β2,H2), respectively.The homogeneous coordinate of the higher joint’s apexDi in frame Σo is(48)Di∑o=rcos⁡ϕi+λ+prsinϕi+λ+qd3i1Tiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,2,3.As discussed before, the pose of the target planeε with respect to the moving plane ρ can be described by the point t2 and the normal unit vector t2w⃑ of the plane ε. Consider(49)t2∑o=00H21T,t2w⃑∑O=cos⁡αsinβsinαsinβcos⁡β0T.Sinceα1 and β1 can be calculated by (2), the coordinate of vector t2w⃑ can be transferred from the frame ΣO to the frame Σo by Euler’s transformation as(50)t2w⃑∑o=R-1001·t2w⃑∑OR-1=R11R12R13R21R22R23R31R32R33-1=r11r12r13r21r22r23r31r32r33,where r11=cos⁡γ1cos⁡β1cos⁡α1-sinγ1sinα1; r12=cos⁡γ1cos⁡β1sinα1+sinγ1cos⁡α1; r13=-cos⁡γ1sinβ1; r21=-cos⁡γ1sinα1-sinγ1cos⁡β1cos⁡α1; r22=-sinγ1cos⁡β1sinα1+cos⁡γ1cos⁡α1; r23=sinγ1sinβ1; r31=sinβ1cos⁡α1; r32=sinβ1sinα1; and r33=cos⁡β1.Since the vectort2Di⃑ in the plane ε is orthogonal with respect to the vector t2w⃑ in frame Σo, the following equation can be obtained:(51)t2w⃑∑o·t2Di⃑∑o=0i=1,2,3.Equation (51) can be rearranged to obtain(52)d3i=H2-p·r13+q·r23+p·r11+q·r12cos⁡α+p·r12+q·r22sinαtanβ·r33+r31cos⁡α+r32sinαtanβ-1-rcos⁡ϕi+λr13+r11cos⁡α+r12sinαtanβr33+r31cos⁡α+r32sinαtanβ+rsinϕi+λr23+r21cos⁡α+r22sinαtanβr33+r31cos⁡α+r32sinαtanβ(i=1,2,3).The coordinate of the moving platform’s central pointo in frame ΣO can be obtained as shown in Section 4.2. Further, the actuated parameters d1i (i=1,2,3) of the prismatic joints P1i are(53)d1i=oz-Rcos⁡φi-oxcos⁡α1Rsinφi-oysinα1+Rsinφi-oysinα1tanβ1,i=1,2,3.With this, the whole actuated parameters of the3-P_SP-P_(PH) HS-PM in analytical form are obtained.


## 5.3. Numerical Example

Specify the architectural parameters of the3-P_SP-P_(PH) HS-PM according to the definition in Figure 6, as R=150, r=55, φ1=ψ1=0, φ2=ψ2=2π/3, φ3=ψ3=10π/9, p=0, q=0, λ = 25°, ϕ1 = 0°, ϕ2 = 90°, and ϕ3 = 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.Table 1
Direct-inverse position analysis of the 3-PSP-P(PH) HS-PM measured in frame ΣO (R=150, r=55,  φ1=ψ1=0,  φ2=ψ2=2π/3, φ3=ψ3=10π/9, p=0, q=0, λ=25°, ϕ1=0°, ϕ2=90°, and ϕ3=180°).
Direct
1
2
3
Inverse
4
5
6
d
11
2.500000
2.200000
1.400000
α
1.000000
3.141593
0.500000
d
12
2.000000
2.500000
2.200000
β
0.001000
0.010000
0.020000
d
13
1.500000
1.700000
2.400000
H
2.000000
2.500000
2.900000
d
31
0.200000
0.700000
0.400000
α
2
1.000000
0.300000
0.010000
d
32
0.500000
0.100000
0.350000
β
2
0.002000
0.005000
0.001000
d
33
0.100000
0.400000
0.500000
H
2
0.100000
0.800000
0.080000
α
−1.735114
2.832609
0.489347
d
11
1.546799
3.647094
0.043040
β
0.007869
0.007279
0.003988
d
12
2.076600
0.204843
2.925715
H
2.177085
2.599314
2.369710
d
13
2.231900
0.076335
5.936067
α
2
0.933895
4.646567
0.534729
d
31
0.170882
0.744787
0.134401
β
2
0.006428
0.008624
0.002033
d
32
0.015882
1.069403
0.071905
H
2
0.150000
0.550000
0.450000
d
33
0.029118
0.855213
0.025599
α
1
−2.669009
−1.813957
−0.045381
α
1
0.000000
2.841593
0.490000
β
1
0.003541
0.004172
0.003468
β
1
0.002355
0.013587
0.020979
H
1
2.027085
2.049327
1.919709
H
1
1.900000
1.699856
2.819845
o
x
0.000113
0.001052
−0.000589
o
x
−0.000251
−0.011449
0.004906
o
y
0.000157
0.000031
0.000040
o
y
0.000025
−0.000597
0.005566
o
z
2.027085
2.049327
1.919709
o
z
1.900000
1.699856
2.819845The left side of Table1 shows the position solving in the direct way, which calculate the pose parameters α, β, H, α2, β2, and H2 with the given actuate parameters d1i and d3i (i=1,2,3). The right side of Table 1 shows the position solving in the inverse way, which calculate the actuate parameters d1i and d3i (i=1,2,3) with the given pose parameters α, β, H, α2, β2, and H2. In addition, the moving platform’s pose parameters and coordinate of the central point o are also obtained in both sides. The one-to-one 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 closed-loop feedback of the gap status, which can be verified in the prototype.A series of contrast experiments between the open-loop control method and closed-loop control method are carried out. As shown in Figure9, the open-loop control method (without the red line) shows the behavior of the single 3-P_SP 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 closed-loop control method (with the red line) introduces the six pose parameters (α, β, H, α2, β2, and H2) as the feedback signals.Figure 9
The block diagram for the open-loop control method (without the red line) and closed-loop control method (with the red line).The experiments are divided into two parts according to the wafer status, as the yellow region and pink region shown in Figures10 and 11.Figure 10
The angular parameters during the pose regulating.Figure 11
The position parameters during the pose regulating.The yellow region shows the regulating process of the gap with the wafer still. As shown in the I-Zone (open-loop), 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 surfaceβ2 is about 0.000205 rad and the central distance H2 is about 0.271 mm, which is far beyond the installation requirement of the IH. By introducing the closed-loop feedback of the gap status, the angle β2 and the central distance H2 are controlled within 0.00001 rad and 0.1±0.0005 mm, respectively, as shown in the II-Zone (closed-loop). This shows the beneficial effects of the HS-PM in a closed-loop 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 III-Zone (open-loop control without pose tracking),β2 varies in the range of 0.000003~0.000035 rad and H2 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 real-time pose tracking based on online calculation possible. As shown in the IV-Zone (closed-loop control without pose tracking), by introducing the pose tracking in real time, the angle β2 and the central distance H2 are controlled within 0.000025 rad and 0.1±0.002 mm, respectively, during the wafer scanning process, which meets the installation requirements of IH well.


## 7. Conclusions

In this paper, a novel3-P_SP-P_(PH) hybrid serial-parallel mechanism is presented to meet the pose regulation requirement of immersion hood in the immersion lithography system. The direct-inverse 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-P_(PH) PM to the moving platform of the 3-P_SP PM, the novel 3-P_SP-P_(PH) HS-PM 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-P_SP PM’s direct and inverse position analyses in the couple mixed-type mode are obtained, which improves the analytical analysis of the 3-P_SP PM. Based on the position analysis of the 3-P_SP PM, both the direct and the inverse position analyses of the 3-P_SP-P_(PH) HS-PM 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 real-time pose tracking based on online calculation possible. The direct-inverse position model is applied in the real-time 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 load-to-weight ratio) is the main focus of many researchers on the hybrid serial-parallel mechanism. The novel3-P_SP-P_(PH) HS-PM is applied in the immersion hood pose regulation in immersion lithography system. By measuring and regulating the pose of the immersion hood with respect to the wafer surface simultaneously, the immersion hood tracks the pose of the wafer surface in real time, which is another exploration in the applications of the hybrid serial-parallel mechanism.



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*Source: 573071-2015-04-05.xml*