# Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion Manipulator with Passive Link

Authors: Hui Yang; Hairong Fang; Yuefa Fang; Haibo Qu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In order to solve the problem of the honeycombs perfusion in the thermal protection system of the spacecraft, this paper presents a novel parallel perfusion manipulator with one translational and two rotational (1T2R) degrees of freedom (DOFs), which can be used to construct a 5-DOF hybrid perfusion system for the perfusion of the honeycombs. The proposed 3PSS&PU parallel perfusion manipulator is mainly utilized as the main body of the hybrid perfusion system. The inverse kinematics and the Jacobian matrix of the proposed parallel manipulator are obtained. The analysis of kinematics performance for the proposed parallel manipulator including workspace, singularity, dexterity, and stiffness is conducted. Based on the virtual work principle and the link Jacobian matrix, the dynamic model of the parallel perfusion manipulator is carried out. With reference to dynamic equations, the relationship between the driving force and the mechanism parameters can be derived. In order to verify the correctness of the kinematics and dynamics model, the comparison of theoretical and simulation curves of the motion parameters related to the driving sliders is performed. Corresponding analyses illustrate that the proposed parallel perfusion possesses good kinematics performance and could satisfy the perfusion requirements of the honeycombs. The correctness of the established kinematics and dynamics models is proved, which has great significance for the experimental research of the perfusion system.


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## Body

## 1. Introduction

During the rise and reentry of the spacecraft, the crew module of spacecraft will suffer from a large aerodynamic heating effect. Therefore, thermal protection system is needed to ensure the safety of the pilot and the normal operation of the equipment [1–3]. At present, the structure of spacecraft thermal protection layer usually adopts hexagonal honeycomb structure, and then these honeycombs are spliced together and evenly spread on the surface of spherical crown. In order to achieve the effect of thermal protection, it needs to perfuse the heat-resistant material into the honeycomb structure [4]. However, due to the large size of spherical surface structure, the perfusion of heat-resistant material becomes the main problem. At present, it mainly adopts manual perfusion, which will inevitably require more manpower to complete the perfusion. Accordingly, it is necessary to introduce perfusion manipulator into the perfusion system.In the perfusion system, all the perfusion of the honeycombs is conducted based on the detection system. Due to the large size of the spherical crown surface, the perfusion object will be divided into subregion for processing. First, according to the detection system’s identification, the position of the task honeycombs can be identified. Then, through the rotation of the spherical crown worktable and the motion of the parallel perfusion manipulator, the perfusion of the task honeycombs can be accomplished. When the perfusion of the identified task honeycombs is completed, the detection system will conduct the identification of the next group of honeycombs. In the article, the design and analysis of the perfusion manipulator are the focus of our research.As we all know, the traditional serial manipulator has the advantage of large workspace and the disadvantage of low stiffness, while the parallel manipulator with closed kinematic chain has some advantages compared with the serial manipulator, such as high structural rigidity, high dynamic performance, high accuracy, and low moving inertia [5–9]. For such a reason, parallel manipulator has been extensively applied as flight simulator [10–12], machine tools [13–15], mobile machining module [16, 17], and high-speed pick-and-place robots [18–20]. However, the parallel manipulator also has the disadvantage of small workspace. For the perfusion system, the spherical structure is relatively large and the end of the moving platform needs to carry the heavy perfusion device, so it requires that the perfusion manipulator should have the characteristics of large workspace and high stiffness. However, due to the low stiffness of the serial manipulator and the small workspace of the parallel manipulator, all of them could hardly satisfy the perfusion requirements of the perfusion manipulator. Consequently, in order to solve these problems, hybrid manipulator with the advantages of serial manipulator and parallel manipulator is selected as the suitable perfusion manipulator.Since the spherical crown surface can be seen as a complex free-form surface, the perfusion manipulator needs at least five DOFs (3T2R) in the process of the heat-resistant material perfusion. Generally, the 5-DOF hybrid manipulator can be achieved by adding a 2-DOF rotating head to the moving platform of a 3-DOF position adjustment manipulator, which are typically represented by Tricept [21, 22], Trivariant [23, 24], and Exechon machine tool [25, 26]. In addition, some scholars also constructed the 5-DOF hybrid manipulator by integrating a 3-DOF pose adjustment manipulator with two long guide rails, which are particularly taken as example as Sprint Z3 head [27, 28] and the machine tool [29, 30]. Motivated by this idea, this paper proposes a novel 1T2R parallel manipulator that can be used to construct 5-DOF hybrid perfusion system for the perfusion of the honeycombs in the thermal protection system of the spacecraft. For the 1T2R parallel manipulator, due to the introduction of the passive link, the stiffness of the proposed parallel manipulator has been improved. Moreover, since the idea that the moving platform size is larger than the fixed platform size, the singularity of the parallel manipulator is avoided effectively. And the structural of the proposed manipulator is simple, compact, symmetrical, and easy to control. Thus, because of the reasons mentioned above, the proposed parallel manipulator is selected for the perfusion of the honeycombs in the thermal protection system.In this paper, the objective of the research mainly is aimed at the kinematics performance analysis and the establishment of the dynamics model for the proposed 1T2R parallel perfusion manipulator. The remainder of this article is given as follows. In Section2, structure description of the proposed 1T2R parallel manipulator is represented. The analysis of kinematics and performance is conducted in Sections 3 and 4, respectively. In Section 5, based on the principle of virtual work, dynamic model of the proposed parallel manipulator is established. The simulation research is carried out by utilizing kinematics and dynamics model in Section 6. Conclusions are given in Section 7.


## 2. Architecture

### 2.1. Architecture Description

To satisfy the perfusion requirement of spherical surface, the perfusion manipulator should have at least five DOFs, which concludes three translational DOFs and two rotational DOFs. According to the requirements, a 5-DOF hybrid perfusion system is proposed, which is shown in Figure1. The perfusion system consists of an arc guide way, a 1T2R parallel manipulator, and a honeycomb worktable. The 1T2R parallel manipulator can move along the arc guide way. The honeycomb worktable can rotate about its own axis, and the honeycomb structure is evenly spread in spherical surface. When the perfusion manipulator is at a certain position of the guide way, the perfusion head can achieve the perfusion of the honeycombs within the moving platform’s reachable workspace. Through the rotation of the worktable and the movement of the parallel manipulator along the guide way, all the perfusion of honeycombs can be completed successfully.Figure 1
3D model of the hybrid perfusion system.Figure2 indicates a CAD model of the 1T2R parallel manipulator, which consists of a fixed platform, three actuatedPSS (P denotes the actuated joint) links, a moving platform, and the passive constraint link PU. EveryPSS limb connects the fixed platform with perfusion platform by an active prismatic joint followed by two spherical joints. And every prismatic joint is driven by a servomotor. The passive limb contains a universal joint and a prismatic joint, which connects the perfusion platform with the fixed plate, successively. The existence of the PU limb will highly improve stiffness of the whole system.Figure 2
CAD model of the 1T2R parallel manipulator.


### 2.2. Parameters Description

The schematic model of proposed manipulator in Figure3 shows that it has two platforms: fixed platform labeled by B1B2B3 and moving platform demonstrated by M1M2M3. The three limbs are placed in 120 degree intervals on base platform. Links NiMi are attached to the moving platform by a spherical joint at Mi and to a slider by a spherical joint at Ni. The middle limb is established from the fixed platform to the perfusion platform, whose one end is attached to the base with a prismatic joint B4 and the other end is attached to the end-effector platform by a universal joint. To facilitate the analysis, the fixed reference frame B-xbybzb is placed at the center of the base platform where yb axis is along the direction of straight line BB1. Similarly, the coordinate axes of moving frame are denoted by M-xmymzm in which ym axis is along the direction of straight line MM1. Parameter φi is the angle measured from xb to BBi and ϕi is the angle measured from xm to MMi. The length of the limb NiMi is denoted by li. The length of BBi is demonstrated by Rb, and Rm represents the length of the line MMi. The displacement of the driving joint is represented by si.Figure 3
Kinematic diagram of the 1T2R parallel manipulator.


## 2.1. Architecture Description

To satisfy the perfusion requirement of spherical surface, the perfusion manipulator should have at least five DOFs, which concludes three translational DOFs and two rotational DOFs. According to the requirements, a 5-DOF hybrid perfusion system is proposed, which is shown in Figure1. The perfusion system consists of an arc guide way, a 1T2R parallel manipulator, and a honeycomb worktable. The 1T2R parallel manipulator can move along the arc guide way. The honeycomb worktable can rotate about its own axis, and the honeycomb structure is evenly spread in spherical surface. When the perfusion manipulator is at a certain position of the guide way, the perfusion head can achieve the perfusion of the honeycombs within the moving platform’s reachable workspace. Through the rotation of the worktable and the movement of the parallel manipulator along the guide way, all the perfusion of honeycombs can be completed successfully.Figure 1
3D model of the hybrid perfusion system.Figure2 indicates a CAD model of the 1T2R parallel manipulator, which consists of a fixed platform, three actuatedPSS (P denotes the actuated joint) links, a moving platform, and the passive constraint link PU. EveryPSS limb connects the fixed platform with perfusion platform by an active prismatic joint followed by two spherical joints. And every prismatic joint is driven by a servomotor. The passive limb contains a universal joint and a prismatic joint, which connects the perfusion platform with the fixed plate, successively. The existence of the PU limb will highly improve stiffness of the whole system.Figure 2
CAD model of the 1T2R parallel manipulator.


## 2.2. Parameters Description

The schematic model of proposed manipulator in Figure3 shows that it has two platforms: fixed platform labeled by B1B2B3 and moving platform demonstrated by M1M2M3. The three limbs are placed in 120 degree intervals on base platform. Links NiMi are attached to the moving platform by a spherical joint at Mi and to a slider by a spherical joint at Ni. The middle limb is established from the fixed platform to the perfusion platform, whose one end is attached to the base with a prismatic joint B4 and the other end is attached to the end-effector platform by a universal joint. To facilitate the analysis, the fixed reference frame B-xbybzb is placed at the center of the base platform where yb axis is along the direction of straight line BB1. Similarly, the coordinate axes of moving frame are denoted by M-xmymzm in which ym axis is along the direction of straight line MM1. Parameter φi is the angle measured from xb to BBi and ϕi is the angle measured from xm to MMi. The length of the limb NiMi is denoted by li. The length of BBi is demonstrated by Rb, and Rm represents the length of the line MMi. The displacement of the driving joint is represented by si.Figure 3
Kinematic diagram of the 1T2R parallel manipulator.


## 3. Kinematics Analysis

### 3.1. Inverse Kinematics

As shown in Figure4, the relationship between the coordinate systems B-xbybzb and M-xmymzm can be described by the rotation matrix RBP, which can be obtained by three continuous rotations of Euler angles α, β, and γ about the fixed xb, yb, and zb axis, respectively. For the 1T2R parallel manipulator, the angle γ is zero. Thus, the rotation matrix from M-xmymzm coordinate system to B-xbybzb coordinate system can be expressed as(1)RMB=Rot⁡yb,βRot⁡xb,α=cβsβsαsβcα0cα-sα-sβcβsαcβcαwheres andc correspond to the sine and cosine functions, respectively.Figure 4
Kinematics of theith PSS branch.To facilitate the analysis, the local coordinate systemNi-xiyizi of theith PSS branch also has been established at the point Ni, and the zi axis is along the direction of straight line NiMi. The system Ni-xiyizi can be considered as two continuous rotations of angles κi and ψi about zb axis and yi′ axis (rotated yb axis), respectively. Consequently, the rotation matrix can be represented as(2)RiB=Rot⁡zb,κiRot⁡yi′,ψi=cκicψi-sκicκisψisκicψicκisκisψi-sψi0cψiAccording to Figure4, the closed-loop vector equation of theith link is given as follows:(3)m+ri=bi+sini+ai+likiwhere m is the position vector of the moving platform described in B-xbybzb system, ri=RMBrim, ri and rim represent the position vector of the straight line MMi in B-xbybzb and M-xmymzm system, respectively, bi is the position vector of the straight line OBi in B-xbybzb system, while ni and ki denote the unit vector of the straight line BiDi and NiMi expressed in the coordinate system B-xbybzb, respectively, si is the displacement of the prismatic joint for theith PSS branch, and li is the length of the link NiMi. Therefore, the vectors mentioned above can be obtained as(4)m=00zm;rim=RmcϕiRmsϕi0;bi=RbcφiRbsφi0;ni=001;ai=d1cφid1sφi0;From 2), the unit vectorki will be expressed as(5)ki=RiBkii=RiB001=cκisψisκisψicψi=kxikyikziAt the same time, the relationship betweenκi and ψi can be calculated as(6)cψi=kzi;sψi=kxi2+kyi2,0≤ψi≤π;sκi=kyisψi;cκi=kxisψi;Substituting (4) and (5) into (3) and squaring both sides of (3) can be deduced as(7)saisi2+sbisi+sci=0where(8)sai=Bxi2+Byi2+Bzi2;sbi=2AxiBxi+AyiByi+AziBzi;sci=Axi2+Ayi2+Azi2-li2;Axi=Rmcϕicβ+Rmsϕisβsα-Rb+d1cφi;Bxi=0;Ayi=Rmsϕicα-Rb+d1sφi;Byi=0;Azi=zm+Rmsϕicβsα-Rmcϕisβ;Bzi=-1;The inverse kinematics solutions for theith limb can be derived from (7):(9)si=-sbi±sbi2-4saisci2saiFrom (9), there are two solutions for each driving joint; that is to say, there exist eight possible solutions for a given configuration. However, the three driving sliders are only allowed moving downward from point Bi. Therefore, only the negative symbol can satisfy the motion characteristics of the parallel perfusion manipulator.


### 3.2. Forward Kinematics

Forward kinematics for the 3PSS-PU mechanism is to solve the pose parameters(α,β,zm) after knowing the driving parameters (s1,s2,s3). For this problem, it can be obtained from (3) and (4), which can lead to the following formula:(10)liki=xliylizli=Rmcϕicβ+Rmsϕisβsα-Rb+d1cφiRmsϕicα-Rb+d1sφizm+Rmsϕicβsα-Rmcϕisβ-siAccording to constraint of the length of the linksNiMi, the restraint equation can be obtained:(11)xli2+yli2+zli2=li2To solve the forward kinematics problem,β and zm can be represented by α. Therefore, (12) can be derived from (10) and (11):(12)zm2+Aicβ+Risβ+Qizm+Cicβ+Disβ+Ei=0where the coefficients Ai,Qi~Ei are functions of α:(13)Ai=2Rmsϕisα;Ri=-2Rmcϕi;Qi=-2si;Ci=-2RmcϕicφiRb+d1-2Rmsisαsϕi;Di=2Rmsicϕi-2RmcφisαsϕiRb+d1;Ei=Rb+d12+Rm2+si2-li2-2RmcαsϕisφiRb+d1;Equation (12) also can be further simplified by subtracting one equation from another, which yields the following expression:(14)Aijzm+Cijcβ+Rijzm+Dijsβ+Qijzm+Eij=0where i≠j and(15)Aij=Ai-Aj;Rij=Ri-Rj;Qij=Qi-Qj;Cij=Ci-Cj;Dij=Di-Dj;Eij=Ei-Ej;Therefore,β can be obtained by zm and α as(16)cβ=-Fzm2+Gzm+HIzm2+Jzm+K;sβ=-Lzm2+Szm+TIzm2+Jzm+Kwhere the coefficients F~K,X are functions of α, which are expressed as(17)F=R12Q13-Q12R13;G=D12Q13+R12E13-E12R13-Q12D13;H=D12E13-E12D13;I=R12A13-A12R13;J=D12A13+R12C13-C12R13-A12D13;K=D12C13-C12D13;L=Q12A13-A12Q13;S=E12A13+Q12C13-C12Q13-A12E13;T=E12C13-C12E13;Sincec2β+s2β=1, substituting (16) into this formula yields(18)U4zm4+U3zm3+U2zm2+U1zm+U0=0where(19)U4=F2+L2-I2;U3=2FG+LS-IJ;U2=G2+S2-J2+2FH+2LT-2IK;U1=2GH+2ST-2JK;U0=H2+T2-K2;Equation (12) consists of three independent equations, and two independent equations have been obtained as shown in (16). Substituting (16) into (12), another equation can be deduced. For instance, when i=1, the equation is expressed as(20)W3zm3+W2zm2+W1zm+W0=0where(21)W3=A1F+R1L-Q1I;W2=A1G+C1F+R1S+D1L-Q1J-E1I;W1=A1H+C1G+R1T+D1S-Q1K-E1J;W0=C1H+D1T-E1K;It can be seen that (18) and (20) have a common solution of zm. By using Bezout’s method [31], the following determinant should be satisfied:(22)U4U3U2U1U0000U4U3U2U1U0000U4U3U2U1U0W3W2W1W00000W3W2W1W00000W3W2W1W00000W3W2W1W0=0Thus, (22) becomes an equation about α when the displacement si is known. Now that all the equations have been obtained, the method to solve the forward kinematics problem can be given as follows: (i) to calculate α from (22), (ii) to calculate zm from (20), and (iii) to calculate β from (16).For step (i), the sine and cosine components ofα should be replaced by t=tan⁡(α/2). Based on the standard transformation expression, the equation sα=(1-t2)/(1+t2), cα=2t/(1+t2) can be obtained. Finally, (22) becomes a polynomial algebraic equation about the variable t. Then, according to (ii) and (iii), the forward kinematics problem can be solved.


### 3.3. Jacobian Matrix

Taking the derivative of (3) with respect to time leads to(23)s˙ini+ωi×liki=vm+ωm×riwhere vm=[0,0,z˙m]T and ωm=[α˙,β˙,0]T denote the linear and angular velocity vector of the hybrid perfusion platform in the fixed coordinate system B-xbybzb, respectively.Taking the dot product withki on both sides of (23), the velocity of theith driving joint can be deduced as (24)s˙i=kiTkiTniri×kiTkiTnivmωm=JivmωmRewriting the velocities of the driving joints in the matrix form as(25)s˙=Js-1JpP˙=JP˙where(26)s˙=s˙1s˙2s˙3T;P˙=vmTωmTT;Js=diag⁡k1Tn1k2Tn2k3Tn3;Jp=k1k2k3r1×k1r2×k2r3×k3T;The Jacobian matrix between the velocity vectorP˙ and the driving joint velocity vector s˙ can be obtained as(27)J=Js-1Jp=J1TJ2TJ3TT


## 3.1. Inverse Kinematics

As shown in Figure4, the relationship between the coordinate systems B-xbybzb and M-xmymzm can be described by the rotation matrix RBP, which can be obtained by three continuous rotations of Euler angles α, β, and γ about the fixed xb, yb, and zb axis, respectively. For the 1T2R parallel manipulator, the angle γ is zero. Thus, the rotation matrix from M-xmymzm coordinate system to B-xbybzb coordinate system can be expressed as(1)RMB=Rot⁡yb,βRot⁡xb,α=cβsβsαsβcα0cα-sα-sβcβsαcβcαwheres andc correspond to the sine and cosine functions, respectively.Figure 4
Kinematics of theith PSS branch.To facilitate the analysis, the local coordinate systemNi-xiyizi of theith PSS branch also has been established at the point Ni, and the zi axis is along the direction of straight line NiMi. The system Ni-xiyizi can be considered as two continuous rotations of angles κi and ψi about zb axis and yi′ axis (rotated yb axis), respectively. Consequently, the rotation matrix can be represented as(2)RiB=Rot⁡zb,κiRot⁡yi′,ψi=cκicψi-sκicκisψisκicψicκisκisψi-sψi0cψiAccording to Figure4, the closed-loop vector equation of theith link is given as follows:(3)m+ri=bi+sini+ai+likiwhere m is the position vector of the moving platform described in B-xbybzb system, ri=RMBrim, ri and rim represent the position vector of the straight line MMi in B-xbybzb and M-xmymzm system, respectively, bi is the position vector of the straight line OBi in B-xbybzb system, while ni and ki denote the unit vector of the straight line BiDi and NiMi expressed in the coordinate system B-xbybzb, respectively, si is the displacement of the prismatic joint for theith PSS branch, and li is the length of the link NiMi. Therefore, the vectors mentioned above can be obtained as(4)m=00zm;rim=RmcϕiRmsϕi0;bi=RbcφiRbsφi0;ni=001;ai=d1cφid1sφi0;From 2), the unit vectorki will be expressed as(5)ki=RiBkii=RiB001=cκisψisκisψicψi=kxikyikziAt the same time, the relationship betweenκi and ψi can be calculated as(6)cψi=kzi;sψi=kxi2+kyi2,0≤ψi≤π;sκi=kyisψi;cκi=kxisψi;Substituting (4) and (5) into (3) and squaring both sides of (3) can be deduced as(7)saisi2+sbisi+sci=0where(8)sai=Bxi2+Byi2+Bzi2;sbi=2AxiBxi+AyiByi+AziBzi;sci=Axi2+Ayi2+Azi2-li2;Axi=Rmcϕicβ+Rmsϕisβsα-Rb+d1cφi;Bxi=0;Ayi=Rmsϕicα-Rb+d1sφi;Byi=0;Azi=zm+Rmsϕicβsα-Rmcϕisβ;Bzi=-1;The inverse kinematics solutions for theith limb can be derived from (7):(9)si=-sbi±sbi2-4saisci2saiFrom (9), there are two solutions for each driving joint; that is to say, there exist eight possible solutions for a given configuration. However, the three driving sliders are only allowed moving downward from point Bi. Therefore, only the negative symbol can satisfy the motion characteristics of the parallel perfusion manipulator.


## 3.2. Forward Kinematics

Forward kinematics for the 3PSS-PU mechanism is to solve the pose parameters(α,β,zm) after knowing the driving parameters (s1,s2,s3). For this problem, it can be obtained from (3) and (4), which can lead to the following formula:(10)liki=xliylizli=Rmcϕicβ+Rmsϕisβsα-Rb+d1cφiRmsϕicα-Rb+d1sφizm+Rmsϕicβsα-Rmcϕisβ-siAccording to constraint of the length of the linksNiMi, the restraint equation can be obtained:(11)xli2+yli2+zli2=li2To solve the forward kinematics problem,β and zm can be represented by α. Therefore, (12) can be derived from (10) and (11):(12)zm2+Aicβ+Risβ+Qizm+Cicβ+Disβ+Ei=0where the coefficients Ai,Qi~Ei are functions of α:(13)Ai=2Rmsϕisα;Ri=-2Rmcϕi;Qi=-2si;Ci=-2RmcϕicφiRb+d1-2Rmsisαsϕi;Di=2Rmsicϕi-2RmcφisαsϕiRb+d1;Ei=Rb+d12+Rm2+si2-li2-2RmcαsϕisφiRb+d1;Equation (12) also can be further simplified by subtracting one equation from another, which yields the following expression:(14)Aijzm+Cijcβ+Rijzm+Dijsβ+Qijzm+Eij=0where i≠j and(15)Aij=Ai-Aj;Rij=Ri-Rj;Qij=Qi-Qj;Cij=Ci-Cj;Dij=Di-Dj;Eij=Ei-Ej;Therefore,β can be obtained by zm and α as(16)cβ=-Fzm2+Gzm+HIzm2+Jzm+K;sβ=-Lzm2+Szm+TIzm2+Jzm+Kwhere the coefficients F~K,X are functions of α, which are expressed as(17)F=R12Q13-Q12R13;G=D12Q13+R12E13-E12R13-Q12D13;H=D12E13-E12D13;I=R12A13-A12R13;J=D12A13+R12C13-C12R13-A12D13;K=D12C13-C12D13;L=Q12A13-A12Q13;S=E12A13+Q12C13-C12Q13-A12E13;T=E12C13-C12E13;Sincec2β+s2β=1, substituting (16) into this formula yields(18)U4zm4+U3zm3+U2zm2+U1zm+U0=0where(19)U4=F2+L2-I2;U3=2FG+LS-IJ;U2=G2+S2-J2+2FH+2LT-2IK;U1=2GH+2ST-2JK;U0=H2+T2-K2;Equation (12) consists of three independent equations, and two independent equations have been obtained as shown in (16). Substituting (16) into (12), another equation can be deduced. For instance, when i=1, the equation is expressed as(20)W3zm3+W2zm2+W1zm+W0=0where(21)W3=A1F+R1L-Q1I;W2=A1G+C1F+R1S+D1L-Q1J-E1I;W1=A1H+C1G+R1T+D1S-Q1K-E1J;W0=C1H+D1T-E1K;It can be seen that (18) and (20) have a common solution of zm. By using Bezout’s method [31], the following determinant should be satisfied:(22)U4U3U2U1U0000U4U3U2U1U0000U4U3U2U1U0W3W2W1W00000W3W2W1W00000W3W2W1W00000W3W2W1W0=0Thus, (22) becomes an equation about α when the displacement si is known. Now that all the equations have been obtained, the method to solve the forward kinematics problem can be given as follows: (i) to calculate α from (22), (ii) to calculate zm from (20), and (iii) to calculate β from (16).For step (i), the sine and cosine components ofα should be replaced by t=tan⁡(α/2). Based on the standard transformation expression, the equation sα=(1-t2)/(1+t2), cα=2t/(1+t2) can be obtained. Finally, (22) becomes a polynomial algebraic equation about the variable t. Then, according to (ii) and (iii), the forward kinematics problem can be solved.


## 3.3. Jacobian Matrix

Taking the derivative of (3) with respect to time leads to(23)s˙ini+ωi×liki=vm+ωm×riwhere vm=[0,0,z˙m]T and ωm=[α˙,β˙,0]T denote the linear and angular velocity vector of the hybrid perfusion platform in the fixed coordinate system B-xbybzb, respectively.Taking the dot product withki on both sides of (23), the velocity of theith driving joint can be deduced as (24)s˙i=kiTkiTniri×kiTkiTnivmωm=JivmωmRewriting the velocities of the driving joints in the matrix form as(25)s˙=Js-1JpP˙=JP˙where(26)s˙=s˙1s˙2s˙3T;P˙=vmTωmTT;Js=diag⁡k1Tn1k2Tn2k3Tn3;Jp=k1k2k3r1×k1r2×k2r3×k3T;The Jacobian matrix between the velocity vectorP˙ and the driving joint velocity vector s˙ can be obtained as(27)J=Js-1Jp=J1TJ2TJ3TT


## 4. Kinematics Performance Analysis

Following the establishment of the mechanism model and the kinematics analysis, the analysis of the kinematics performance of the 3PSS-PU parallel manipulator will be conducted in this section, which includes the workspace, singularity, dexterity, and stiffness. The analysis of workspace is mainly on account of the kinematics solved in Section3.1. And based on the Jacobian matrix analyzed in Section 3.3, the analyses of the singularity, dexterity, and stiffness are all developed in detail.


### 4.1. Workspace Analysis

#### 4.1.1. Task Honeycombs Analysis

As shown in Figure5, the maximum angle between the point on the spherical crown surface and the vertical axis is 40 degrees. Due to the large size of the spherical crown surface, it adopts the method of subarea perfusion to accomplish the perfusion of the large object, and the task honeycombs of subarea are shown in Figure 6. For the task honeycombs, the maximum angle θmax of the honeycombs is 10∘. During the perfusion process and through the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail and the motion of the moving platform, the perfusion of all the honeycombs can be completed successfully.Figure 5
Structure of the spherical surface.Figure 6
Task honeycombs of the subarea.


#### 4.1.2. Reachable Workspace Analysis

In this section, the reachable workspace of the perfusion manipulator is obtained by the method of the geometric constraints. As shown in Figure7, the flow diagram for calculating the workspace of the perfusion manipulator has been given in detail. According to the flow diagram of the workspace and the parameters given in Table 1, the reachable workspace of the moving platform can be obtained.Table 1
Dimensional parameters of 1T2R parallel manipulator.
Parameters Values Parameters Values Parameters Values Parameters Values R b / m m 160 d 1 / m m 50 α m a x / r a d ± 0.6 z m m a x / m m 560 R m / m m 270 s i m i n / m m 0 β m a x / r a d ± 0.6 φ i / r a d π / 2 + 2 π ( i - 1 ) / 3 l i / m m 300 s i m a x / m m 250 z m m i n / m m 300 ϕ i / r a d π / 2 + 2 π ( i - 1 ) / 3Figure 7
Flow chart for the reachable workspace.As shown in Figures8(a) and 8(b), the 3D view and vertical view of the reachable workspace for the parallel perfusion manipulator are obtained. From the 3D view of the workspace, it can be seen that the rotation range of the moving platform about xb axis and yb axis remains unchanged with the increase of the value of the variable zm. For the task honeycombs of the subarea, the maximum angle of the task honeycombs is 10∘, and then the task workspace can be described as the yellow region in Figure 8(b). From Figure 8(b), it also can be easily concluded that the task workspace is always within the reachable workspace of the parallel manipulator. That is to say, the proposed 1T2R parallel perfusion manipulator can complete the perfusion of the task honeycombs. Then, by the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail, the perfusion of all the honeycombs will be completed.Figure 8 (a)
3D view of the reachable workspace (b)
Vertical view of the reachable workspace


### 4.2. Singularity Analysis

The method based on the Jacobian matrices is the most common method to find the singularity of a mechanism [32]. In order to obtain all the singularity positions of the parallel manipulator, the determinant of the inverse Jacobian matrix Js and forward Jacobian matrix Jp will be conducted. Based on the Jacobian matrix, the singularity conditions of the proposed parallel manipulator can be divided into three types, which include inverse kinematic singularity (IKS), direct kinematic singularity (DKS), and combined singularity (CS). And the conditions for satisfying these types of singularity can be given as follows:(28)IKS:det⁡Js=0,det⁡Jps≠0;DKS:det⁡Js≠0,det⁡Jps=0;CS:det⁡Js=0,det⁡Jps=0;where Jps is a 3×3 submatrix from Jp.As for the IKS singularity, it occurs when matrixJs is not full rank, i.e., det(Js)=0 or ki·ni=0(i=1,2,3), which means one or more PSS legs are perpendicular to their corresponding sliding rails. As shown in Figure 9, it gives two cases when the first link N1M1 is perpendicular to the corresponding sliding rails B1N1, and IKS occurs with detJs=0. In this case, the parallel manipulator loses one DoF because an infinitesimal motion of the actuator N1 will cause no motion of the moving platform. From Figure 9(a), it can be seen that the interference between the moving platform and the sliding rail B1N1 occurs when the link N1M1 is perpendicular to the sliding rails B1N1 outside. Also, according to Figure 9(b), only when the size of the moving platform is smaller than the fixed platform can the inside vertical of the link N1M1 and the sliding rails B1N1 be satisfied. However, there is no interference between the components when the moving platform moves within its reachable workspace; thus the singularity case in Figure 9(a) is not exist. Also, for the proposed parallel manipulator, the parameter Rm is larger than Rb; thus the singularity case in Figure 9(b) does not exist either. Consequently, there is no IKS existing in the proposed parallel manipulator.Figure 9
Manipulator positions when IKS happens.
(a) (b)As for the DKS singularity, it occurs when matrixJps is not full rank; e.g., the first PSS branch lies in the plane M1M2M3 with the link N1M1 passing through the center point M of the moving platform, which is shown in Figure 10. Under this situation, the moving platform obtains one more DoF even when all the three actuators are locked and the parallel manipulator will instantaneously be out of control; i.e., joint M1 can infinitesimally move along the normal direction of the plane M1M2M3 without actuation. From the first case in Figure 10(a), it can be known the interference between the moving platform and the sliding rail B1N1 occurs when the link N1M1 coincides with the line MM1. Also, according to the singularity position in Figure 10(b), the link N1M1 is on the extension line of MM1, and the size of the moving platform is smaller than the base platform. However, for the manipulator, there is no interference between the components when the moving platform moves within its reachable workspace, and the parameter Rm is larger than Rb. Consequently, there is no DKS existing in the parallel manipulator either.Figure 10
Manipulator positions when DKS happens.
(a) (b)Similarly, as for the CS singularity, it occurs when both of the matricesJs and Jps are not full rank. In this case, the inverse kinematic and forward kinematic singularities appear simultaneously, and the special singularity configuration is shown in Figure 11. According to Figure 11, it can be concluded that the combined singularity occurs when the relationship Rm=Rb+li exists. However, the equation does not exist for the structure parameters of the proposed parallel manipulator. Therefore, there is no CS happening in the parallel manipulator either.Figure 11
Manipulator position when CS happens.In summary, the three types of singularities mentioned above do not happen in the 3PSS-PU parallel manipulator because of the carefully designing structure parameters of the manipulator.


### 4.3. Dexterity and Stiffness Analysis

Dexterity and stiffness are important kinematic performance indexes to measure parallel manipulator’s working ability. The dexterity mainly reflects the ability to arbitrarily change the moving platform’s position and orientation or apply force and torques in arbitrary directions while working. And the stiffness directly affects manipulator’s motion accuracy. Because the Jacobian matrix could reflect the relationship between input and output, the condition number of Jacobian matrix is frequently used in evaluating the dexterity of a manipulator. The calculation method of condition number comes out by computing the eigenvalue of stiffness matrix and taking square root of the ratio among extreme eigenvalues, which is shown in the following equation:(29)wcon=σmaxσminwhere wcon denotes the condition number of Jacobian matrix and σmax and σmin represent the maximum and minimum eigenvalues of stiffness matrix, respectively. And the stiffness matrix of the manipulator can be described as the following equation [33]:(30)κJ=JTKJJwhere KJ is the stiffness matrix and KJ=π1,…π2. In this paper, the actuators of the 3PSS-PU parallel manipulator can be seen as elastic components and πi denotes the stiffness of theith driving joint. Here, πi is set to 100KN/m [7].As shown in Figure12, it describes the distributions of the Jacobian matrix condition number of the proposed parallel manipulator at a height of zm=450mm. From Figure 12, it can be seen that the values of condition number within the workspace all change from 5.2 to 9 smoothly. The closer the values of condition number to 1, the better the dexterity of the manipulator. The values of dexterity index for most areas of the workspace are all below 6.5, which indicate the good dexterity performance of the proposed parallel manipulator in the reachable workspace. Also, the dexterity performance is good enough to satisfy the dexterity requirements of the perfusion.Figure 12
Distributions of condition number of Jacobian matrix at a height ofzm=450mm.
(a) (b)Meanwhile, the distributions of stiffness at different height ofzm are also plotted, which are shown in Figures 13 and 14. It can be easily observed from the figures that the stiffness of the manipulator decreases with the increasing of the value of zm. And the distribution figures of stiffness with different value of zm have the same change trend. Importantly, the maximum value of stiffness always arises at the center of the workspace and decreases with the angle α or β approaching to the maximum value, which is in line with the structural characteristics of the proposed parallel manipulator. Also, the value of the stiffness at different height of zm is mostly above 150kN/m, and the stiffness requirement of the perfusion is about 100kN/m. Thus, the stiffness of the parallel manipulator is fully capable of satisfying the perfusion of the honeycombs.Figure 13
Distributions of stiffness at different height ofzm.
(a) (b) (c) (d)Figure 14
Comparison of stiffness at different height ofzm.


## 4.1. Workspace Analysis

### 4.1.1. Task Honeycombs Analysis

As shown in Figure5, the maximum angle between the point on the spherical crown surface and the vertical axis is 40 degrees. Due to the large size of the spherical crown surface, it adopts the method of subarea perfusion to accomplish the perfusion of the large object, and the task honeycombs of subarea are shown in Figure 6. For the task honeycombs, the maximum angle θmax of the honeycombs is 10∘. During the perfusion process and through the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail and the motion of the moving platform, the perfusion of all the honeycombs can be completed successfully.Figure 5
Structure of the spherical surface.Figure 6
Task honeycombs of the subarea.


### 4.1.2. Reachable Workspace Analysis

In this section, the reachable workspace of the perfusion manipulator is obtained by the method of the geometric constraints. As shown in Figure7, the flow diagram for calculating the workspace of the perfusion manipulator has been given in detail. According to the flow diagram of the workspace and the parameters given in Table 1, the reachable workspace of the moving platform can be obtained.Table 1
Dimensional parameters of 1T2R parallel manipulator.
Parameters Values Parameters Values Parameters Values Parameters Values R b / m m 160 d 1 / m m 50 α m a x / r a d ± 0.6 z m m a x / m m 560 R m / m m 270 s i m i n / m m 0 β m a x / r a d ± 0.6 φ i / r a d π / 2 + 2 π ( i - 1 ) / 3 l i / m m 300 s i m a x / m m 250 z m m i n / m m 300 ϕ i / r a d π / 2 + 2 π ( i - 1 ) / 3Figure 7
Flow chart for the reachable workspace.As shown in Figures8(a) and 8(b), the 3D view and vertical view of the reachable workspace for the parallel perfusion manipulator are obtained. From the 3D view of the workspace, it can be seen that the rotation range of the moving platform about xb axis and yb axis remains unchanged with the increase of the value of the variable zm. For the task honeycombs of the subarea, the maximum angle of the task honeycombs is 10∘, and then the task workspace can be described as the yellow region in Figure 8(b). From Figure 8(b), it also can be easily concluded that the task workspace is always within the reachable workspace of the parallel manipulator. That is to say, the proposed 1T2R parallel perfusion manipulator can complete the perfusion of the task honeycombs. Then, by the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail, the perfusion of all the honeycombs will be completed.Figure 8 (a)
3D view of the reachable workspace (b)
Vertical view of the reachable workspace


## 4.1.1. Task Honeycombs Analysis

As shown in Figure5, the maximum angle between the point on the spherical crown surface and the vertical axis is 40 degrees. Due to the large size of the spherical crown surface, it adopts the method of subarea perfusion to accomplish the perfusion of the large object, and the task honeycombs of subarea are shown in Figure 6. For the task honeycombs, the maximum angle θmax of the honeycombs is 10∘. During the perfusion process and through the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail and the motion of the moving platform, the perfusion of all the honeycombs can be completed successfully.Figure 5
Structure of the spherical surface.Figure 6
Task honeycombs of the subarea.


## 4.1.2. Reachable Workspace Analysis

In this section, the reachable workspace of the perfusion manipulator is obtained by the method of the geometric constraints. As shown in Figure7, the flow diagram for calculating the workspace of the perfusion manipulator has been given in detail. According to the flow diagram of the workspace and the parameters given in Table 1, the reachable workspace of the moving platform can be obtained.Table 1
Dimensional parameters of 1T2R parallel manipulator.
Parameters Values Parameters Values Parameters Values Parameters Values R b / m m 160 d 1 / m m 50 α m a x / r a d ± 0.6 z m m a x / m m 560 R m / m m 270 s i m i n / m m 0 β m a x / r a d ± 0.6 φ i / r a d π / 2 + 2 π ( i - 1 ) / 3 l i / m m 300 s i m a x / m m 250 z m m i n / m m 300 ϕ i / r a d π / 2 + 2 π ( i - 1 ) / 3Figure 7
Flow chart for the reachable workspace.As shown in Figures8(a) and 8(b), the 3D view and vertical view of the reachable workspace for the parallel perfusion manipulator are obtained. From the 3D view of the workspace, it can be seen that the rotation range of the moving platform about xb axis and yb axis remains unchanged with the increase of the value of the variable zm. For the task honeycombs of the subarea, the maximum angle of the task honeycombs is 10∘, and then the task workspace can be described as the yellow region in Figure 8(b). From Figure 8(b), it also can be easily concluded that the task workspace is always within the reachable workspace of the parallel manipulator. That is to say, the proposed 1T2R parallel perfusion manipulator can complete the perfusion of the task honeycombs. Then, by the rotation of the spherical crown worktable and the movement of the parallel manipulator along the arc guide rail, the perfusion of all the honeycombs will be completed.Figure 8 (a)
3D view of the reachable workspace (b)
Vertical view of the reachable workspace


## 4.2. Singularity Analysis

The method based on the Jacobian matrices is the most common method to find the singularity of a mechanism [32]. In order to obtain all the singularity positions of the parallel manipulator, the determinant of the inverse Jacobian matrix Js and forward Jacobian matrix Jp will be conducted. Based on the Jacobian matrix, the singularity conditions of the proposed parallel manipulator can be divided into three types, which include inverse kinematic singularity (IKS), direct kinematic singularity (DKS), and combined singularity (CS). And the conditions for satisfying these types of singularity can be given as follows:(28)IKS:det⁡Js=0,det⁡Jps≠0;DKS:det⁡Js≠0,det⁡Jps=0;CS:det⁡Js=0,det⁡Jps=0;where Jps is a 3×3 submatrix from Jp.As for the IKS singularity, it occurs when matrixJs is not full rank, i.e., det(Js)=0 or ki·ni=0(i=1,2,3), which means one or more PSS legs are perpendicular to their corresponding sliding rails. As shown in Figure 9, it gives two cases when the first link N1M1 is perpendicular to the corresponding sliding rails B1N1, and IKS occurs with detJs=0. In this case, the parallel manipulator loses one DoF because an infinitesimal motion of the actuator N1 will cause no motion of the moving platform. From Figure 9(a), it can be seen that the interference between the moving platform and the sliding rail B1N1 occurs when the link N1M1 is perpendicular to the sliding rails B1N1 outside. Also, according to Figure 9(b), only when the size of the moving platform is smaller than the fixed platform can the inside vertical of the link N1M1 and the sliding rails B1N1 be satisfied. However, there is no interference between the components when the moving platform moves within its reachable workspace; thus the singularity case in Figure 9(a) is not exist. Also, for the proposed parallel manipulator, the parameter Rm is larger than Rb; thus the singularity case in Figure 9(b) does not exist either. Consequently, there is no IKS existing in the proposed parallel manipulator.Figure 9
Manipulator positions when IKS happens.
(a) (b)As for the DKS singularity, it occurs when matrixJps is not full rank; e.g., the first PSS branch lies in the plane M1M2M3 with the link N1M1 passing through the center point M of the moving platform, which is shown in Figure 10. Under this situation, the moving platform obtains one more DoF even when all the three actuators are locked and the parallel manipulator will instantaneously be out of control; i.e., joint M1 can infinitesimally move along the normal direction of the plane M1M2M3 without actuation. From the first case in Figure 10(a), it can be known the interference between the moving platform and the sliding rail B1N1 occurs when the link N1M1 coincides with the line MM1. Also, according to the singularity position in Figure 10(b), the link N1M1 is on the extension line of MM1, and the size of the moving platform is smaller than the base platform. However, for the manipulator, there is no interference between the components when the moving platform moves within its reachable workspace, and the parameter Rm is larger than Rb. Consequently, there is no DKS existing in the parallel manipulator either.Figure 10
Manipulator positions when DKS happens.
(a) (b)Similarly, as for the CS singularity, it occurs when both of the matricesJs and Jps are not full rank. In this case, the inverse kinematic and forward kinematic singularities appear simultaneously, and the special singularity configuration is shown in Figure 11. According to Figure 11, it can be concluded that the combined singularity occurs when the relationship Rm=Rb+li exists. However, the equation does not exist for the structure parameters of the proposed parallel manipulator. Therefore, there is no CS happening in the parallel manipulator either.Figure 11
Manipulator position when CS happens.In summary, the three types of singularities mentioned above do not happen in the 3PSS-PU parallel manipulator because of the carefully designing structure parameters of the manipulator.


## 4.3. Dexterity and Stiffness Analysis

Dexterity and stiffness are important kinematic performance indexes to measure parallel manipulator’s working ability. The dexterity mainly reflects the ability to arbitrarily change the moving platform’s position and orientation or apply force and torques in arbitrary directions while working. And the stiffness directly affects manipulator’s motion accuracy. Because the Jacobian matrix could reflect the relationship between input and output, the condition number of Jacobian matrix is frequently used in evaluating the dexterity of a manipulator. The calculation method of condition number comes out by computing the eigenvalue of stiffness matrix and taking square root of the ratio among extreme eigenvalues, which is shown in the following equation:(29)wcon=σmaxσminwhere wcon denotes the condition number of Jacobian matrix and σmax and σmin represent the maximum and minimum eigenvalues of stiffness matrix, respectively. And the stiffness matrix of the manipulator can be described as the following equation [33]:(30)κJ=JTKJJwhere KJ is the stiffness matrix and KJ=π1,…π2. In this paper, the actuators of the 3PSS-PU parallel manipulator can be seen as elastic components and πi denotes the stiffness of theith driving joint. Here, πi is set to 100KN/m [7].As shown in Figure12, it describes the distributions of the Jacobian matrix condition number of the proposed parallel manipulator at a height of zm=450mm. From Figure 12, it can be seen that the values of condition number within the workspace all change from 5.2 to 9 smoothly. The closer the values of condition number to 1, the better the dexterity of the manipulator. The values of dexterity index for most areas of the workspace are all below 6.5, which indicate the good dexterity performance of the proposed parallel manipulator in the reachable workspace. Also, the dexterity performance is good enough to satisfy the dexterity requirements of the perfusion.Figure 12
Distributions of condition number of Jacobian matrix at a height ofzm=450mm.
(a) (b)Meanwhile, the distributions of stiffness at different height ofzm are also plotted, which are shown in Figures 13 and 14. It can be easily observed from the figures that the stiffness of the manipulator decreases with the increasing of the value of zm. And the distribution figures of stiffness with different value of zm have the same change trend. Importantly, the maximum value of stiffness always arises at the center of the workspace and decreases with the angle α or β approaching to the maximum value, which is in line with the structural characteristics of the proposed parallel manipulator. Also, the value of the stiffness at different height of zm is mostly above 150kN/m, and the stiffness requirement of the perfusion is about 100kN/m. Thus, the stiffness of the parallel manipulator is fully capable of satisfying the perfusion of the honeycombs.Figure 13
Distributions of stiffness at different height ofzm.
(a) (b) (c) (d)Figure 14
Comparison of stiffness at different height ofzm.


## 5. Dynamics Model of the Parallel Manipulator

In this section, the dynamics analysis of the perfusion manipulator is mainly focused on the 1T2R parallel perfusion manipulator. Based on the principle of the virtual work [34, 35], the link Jacobian matrix and the dynamics model of the perfusion manipulator are established. The applied and inertia forces of the components are derived in the corresponding coordinate system. At last, the expression of the driving force is obtained.


### 5.1. Link Velocity Analysis

The velocity vector of the pointMi in the Ni-xiyizi coordinate system can be written as(31)vMii=s˙inii+ωii×likii=vmi+ωmi×riiBecause ofωiTikii=0, taking the cross product with kii on both sides of (31), the angular velocity of the limb NiMi in the Ni-xiyizi coordinate system is derived as(32)ωii=1likii×vMii-kii×s˙inii=1liTkiivMii-Tkiis˙iniiwhere(33)Tkii=0-kizikiyikizi0-kixi-kiyikixi0Substituting (24) and (31) into (32) leads to(34)ωii=1liTkiiRBi-TkiiTriiRBi-kii×niikiTkiTniri×kiTkiTniP˙=JωiP˙where(35)Trii=0-riziriyirizi0-rixi-riyirixi0;RBi=Ri-1B=RiTB;With reference to (31) and (34), the linear velocity of the mass center of the branch NiMi in the Ni-xiyizi coordinate system can be given as(36)vii=vMii-ωii×li2kii=RBi-TriiRBi+li2TkiiJωiP˙=JviP˙Rewriting the velocity vector of theith branch NiMi in the matrix form yields(37)viiωii=JviJωiP˙=JvωiP˙where Jvwi is the link Jacobian matrix which represents the mapping between the velocity vector of the moving platform in the fixed system B-xbybzb and the velocity vector of theith link in the Ni-xiyizi coordinate system.


### 5.2. Acceleration Analysis

Taking the derivative of (23) with respect to time leads to(38)s¨ini+ω˙i×liki+ωi×ωi×liki=v˙m+ω˙m×ri+ωm×ωm×riTaking the dot product withki on both sides of (38) and simplifying the equation, the acceleration of theith slider can be obtained as(39)s¨i=JiP¨+1kiTnikiTωmriTωm-kiTriωm2+liki×ωi2The matrix form of the accelerations of the driving sliders can be rewritten as(40)s¨=JP¨+ψwhere(41)s¨=s¨1s¨2s¨3T;P˙=v˙mTω˙mTT;ψ=ψ1ψ2ψ3T;ψi=1kiTnikiTωmriTωm-kiTriωm2+liki×ωi2


### 5.3. Link Acceleration Analysis

Taking the derivative of (31) with respect to time in the Ni-xiyizi coordinate system leads to(42)s¨inii+ω˙ii×likii+ωii×ωii×likii=v˙mi+ω˙mi×rii+ωmi×ωmi×riiTaking the cross product of (42) with kii on both sides, the angular acceleration of theith link NiMi can be obtained as(43)ω˙ii=1likii×v˙mi+kii×ω˙mi×rii-kii×niis¨i+kii×ωmi×ωmi×rii-kii×ωii×ωii×likiiBased on the derived acceleration in (39), (43) can be simplified as(44)ω˙ii=JωiP¨+λiwhere(45)λi=1li-kii×niikiTnikiTωmriTωm-kiTriωm2+liki×ωi2+ωmTiriikii×ωmi-ωmi2kii×riiTaking the derivative of (36) with respect to time leads to(46)v˙ii=v˙Mii-ω˙ii×li2kii-ωii×ωii×li2kii=v˙mi-Triiω˙mi+TωmiTωmirii+li2Tkiiω˙ii-li2TωiiTωiikiiSubstituting (44) into the above equation and simplifying can be derived as(47)v˙ii=JviP¨+ηiwhere ηi=TωmiTωmirii+1/2Tkiiλi-li/2TωiiTωiikii.


### 5.4. Dynamic Formulation

Applied and inertia forces imposed on the center of the mass of the moving perfusion platform can be derived as(48)QM=fe+mmg-mmv˙mne-IMBω˙m-ωm×IMBωmwhere fe and ne represent the external force and torque imposed on the mass center of the moving perfusion platform, mm is the mass of the moving perfusion platform, and g=0,0,9.8Tm/s2 is the gravity. IMB=RMBIMRBM, and IM denotes the inertia matrix of the moving perfusion platform about the mass center which is described in the M-xmymzm coordinate system.The force systemQM in (48) can be divided into four sections: the acceleration term QMA, the velocity term QMV, the gravity term QMG, and the external force term QME, which can be listed as follows:(49)QM=-mmv˙m-IMBω˙m+01×3-ωm×IMBωm+mmg01×3+fene=QMA+QMV+QMG+QMEThe force system imposed on the center of the mass of theith branch NiMi can be deduced in the Ni-xiyizi coordinate system as(50)Qii=miRBig-miv˙ii-Iiiω˙ii-ωii×Iiiωiiwhere mi is the mass of the link NiMi and Iii is the inertia matrix of theith limb NiMi about the center of its mass which is expressed in the Ni-xiyizi coordinate system.Similarly, the force systemQii of theith branch NiMi also can be decomposed into three parts as(51)
Q
i
i
=
-
m
i
J
v
i
P
¨
-
I
i
i
J
ω
i
P
¨
+
-
m
i
η
i
-
I
i
i
λ
i
-
ω
i
i
×
I
i
i
ω
i
i
+
m
i
R
B
i
g
0
1
×
3
=
Q
i
A
i
+
Q
i
V
i
+
Q
i
G
iFor the driving joint, the force system exerted at the mass center of the slider can be derived in theB-xbybzb coordinate system as(52)Fi=msig-msis¨i=-msiniJiP¨-msiψini+msig=FAi+FVi+FGiwhere FAi, FVi, and FGi are the acceleration term, the velocity term, and the gravity term of the force system Fi, respectively, and msi is the mass of theith slider.For the middle passive link, the branch only can move alongzb axis. Thus, the force imposed on the center of the mass of the middle limb can be deduced in the B-xbybzb coordinate system as(53)Fp=-mpz¨+mpg=-mpn0J0P¨+mpg=FAp+FGpwhere mp is the mass of the passive link and FAp and FGp are the acceleration term and the gravity term of the force system Fp, respectively. And z¨=n0J0P¨=00z¨mT; n0=001T; J0=001000.Suppose the moving perfusion platform has a virtual displacement in the reachable workspace of the 1T2R parallel manipulator. Thus, based on the principle of the virtual work, the dynamics equation of the proposed manipulator can be obtained as follows:(54)δPTQM+∑i=13niδsiTFi+∑i=13δxiTiQii+n0δzTFp+δsTf=0where f=f1f2f3T is the driving forces of the sliders of the 1T2R parallel manipulator, δP and δsi denote the virtual displacements of the moving platform and theith driving slider, respectively, and δxii is the virtual displacement of theith limb NiMi descripted in the Ni-xiyizi coordinate system.According to the link Jacobian matrix analyzed in Section5.1, the virtual displacements mentioned above can be represented by the virtual displacement of the moving perfusion platform δP as follows:(55)δsiT=JiδPT=δPTJiT;δixiT=δPTJvωiT;δzT=J0δPT=δPTJ0T;δsT=δPTJTSubstituting (55) into the dynamics equation leads to(56)δPTQM+∑i=13δPTJiTniTFi+∑i=13δPTJvωiTQii+δPTJ0Tn0TFp+δPTJTf=0Simplifying (56), the inverse dynamics of the 1T2R parallel manipulator can be derived as follows:(57)f=-J-TQM+∑i=13JiTniTFi+JvωiTQii+J0Tn0TFp=-J-T∑i=13JiTniTFAi+JvωiTQAii+J0Tn0TFAp+QMA-J-T∑i=13JiTniTFVi+JvωiTQVii+QMV-J-T∑i=13JiTniTFGi+JvωiTQGii+J0Tn0TFGp+QMG-J-TQMEwhere J-T are the inverse matrix of JT.


## 5.1. Link Velocity Analysis

The velocity vector of the pointMi in the Ni-xiyizi coordinate system can be written as(31)vMii=s˙inii+ωii×likii=vmi+ωmi×riiBecause ofωiTikii=0, taking the cross product with kii on both sides of (31), the angular velocity of the limb NiMi in the Ni-xiyizi coordinate system is derived as(32)ωii=1likii×vMii-kii×s˙inii=1liTkiivMii-Tkiis˙iniiwhere(33)Tkii=0-kizikiyikizi0-kixi-kiyikixi0Substituting (24) and (31) into (32) leads to(34)ωii=1liTkiiRBi-TkiiTriiRBi-kii×niikiTkiTniri×kiTkiTniP˙=JωiP˙where(35)Trii=0-riziriyirizi0-rixi-riyirixi0;RBi=Ri-1B=RiTB;With reference to (31) and (34), the linear velocity of the mass center of the branch NiMi in the Ni-xiyizi coordinate system can be given as(36)vii=vMii-ωii×li2kii=RBi-TriiRBi+li2TkiiJωiP˙=JviP˙Rewriting the velocity vector of theith branch NiMi in the matrix form yields(37)viiωii=JviJωiP˙=JvωiP˙where Jvwi is the link Jacobian matrix which represents the mapping between the velocity vector of the moving platform in the fixed system B-xbybzb and the velocity vector of theith link in the Ni-xiyizi coordinate system.


## 5.2. Acceleration Analysis

Taking the derivative of (23) with respect to time leads to(38)s¨ini+ω˙i×liki+ωi×ωi×liki=v˙m+ω˙m×ri+ωm×ωm×riTaking the dot product withki on both sides of (38) and simplifying the equation, the acceleration of theith slider can be obtained as(39)s¨i=JiP¨+1kiTnikiTωmriTωm-kiTriωm2+liki×ωi2The matrix form of the accelerations of the driving sliders can be rewritten as(40)s¨=JP¨+ψwhere(41)s¨=s¨1s¨2s¨3T;P˙=v˙mTω˙mTT;ψ=ψ1ψ2ψ3T;ψi=1kiTnikiTωmriTωm-kiTriωm2+liki×ωi2


## 5.3. Link Acceleration Analysis

Taking the derivative of (31) with respect to time in the Ni-xiyizi coordinate system leads to(42)s¨inii+ω˙ii×likii+ωii×ωii×likii=v˙mi+ω˙mi×rii+ωmi×ωmi×riiTaking the cross product of (42) with kii on both sides, the angular acceleration of theith link NiMi can be obtained as(43)ω˙ii=1likii×v˙mi+kii×ω˙mi×rii-kii×niis¨i+kii×ωmi×ωmi×rii-kii×ωii×ωii×likiiBased on the derived acceleration in (39), (43) can be simplified as(44)ω˙ii=JωiP¨+λiwhere(45)λi=1li-kii×niikiTnikiTωmriTωm-kiTriωm2+liki×ωi2+ωmTiriikii×ωmi-ωmi2kii×riiTaking the derivative of (36) with respect to time leads to(46)v˙ii=v˙Mii-ω˙ii×li2kii-ωii×ωii×li2kii=v˙mi-Triiω˙mi+TωmiTωmirii+li2Tkiiω˙ii-li2TωiiTωiikiiSubstituting (44) into the above equation and simplifying can be derived as(47)v˙ii=JviP¨+ηiwhere ηi=TωmiTωmirii+1/2Tkiiλi-li/2TωiiTωiikii.


## 5.4. Dynamic Formulation

Applied and inertia forces imposed on the center of the mass of the moving perfusion platform can be derived as(48)QM=fe+mmg-mmv˙mne-IMBω˙m-ωm×IMBωmwhere fe and ne represent the external force and torque imposed on the mass center of the moving perfusion platform, mm is the mass of the moving perfusion platform, and g=0,0,9.8Tm/s2 is the gravity. IMB=RMBIMRBM, and IM denotes the inertia matrix of the moving perfusion platform about the mass center which is described in the M-xmymzm coordinate system.The force systemQM in (48) can be divided into four sections: the acceleration term QMA, the velocity term QMV, the gravity term QMG, and the external force term QME, which can be listed as follows:(49)QM=-mmv˙m-IMBω˙m+01×3-ωm×IMBωm+mmg01×3+fene=QMA+QMV+QMG+QMEThe force system imposed on the center of the mass of theith branch NiMi can be deduced in the Ni-xiyizi coordinate system as(50)Qii=miRBig-miv˙ii-Iiiω˙ii-ωii×Iiiωiiwhere mi is the mass of the link NiMi and Iii is the inertia matrix of theith limb NiMi about the center of its mass which is expressed in the Ni-xiyizi coordinate system.Similarly, the force systemQii of theith branch NiMi also can be decomposed into three parts as(51)
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iFor the driving joint, the force system exerted at the mass center of the slider can be derived in theB-xbybzb coordinate system as(52)Fi=msig-msis¨i=-msiniJiP¨-msiψini+msig=FAi+FVi+FGiwhere FAi, FVi, and FGi are the acceleration term, the velocity term, and the gravity term of the force system Fi, respectively, and msi is the mass of theith slider.For the middle passive link, the branch only can move alongzb axis. Thus, the force imposed on the center of the mass of the middle limb can be deduced in the B-xbybzb coordinate system as(53)Fp=-mpz¨+mpg=-mpn0J0P¨+mpg=FAp+FGpwhere mp is the mass of the passive link and FAp and FGp are the acceleration term and the gravity term of the force system Fp, respectively. And z¨=n0J0P¨=00z¨mT; n0=001T; J0=001000.Suppose the moving perfusion platform has a virtual displacement in the reachable workspace of the 1T2R parallel manipulator. Thus, based on the principle of the virtual work, the dynamics equation of the proposed manipulator can be obtained as follows:(54)δPTQM+∑i=13niδsiTFi+∑i=13δxiTiQii+n0δzTFp+δsTf=0where f=f1f2f3T is the driving forces of the sliders of the 1T2R parallel manipulator, δP and δsi denote the virtual displacements of the moving platform and theith driving slider, respectively, and δxii is the virtual displacement of theith limb NiMi descripted in the Ni-xiyizi coordinate system.According to the link Jacobian matrix analyzed in Section5.1, the virtual displacements mentioned above can be represented by the virtual displacement of the moving perfusion platform δP as follows:(55)δsiT=JiδPT=δPTJiT;δixiT=δPTJvωiT;δzT=J0δPT=δPTJ0T;δsT=δPTJTSubstituting (55) into the dynamics equation leads to(56)δPTQM+∑i=13δPTJiTniTFi+∑i=13δPTJvωiTQii+δPTJ0Tn0TFp+δPTJTf=0Simplifying (56), the inverse dynamics of the 1T2R parallel manipulator can be derived as follows:(57)f=-J-TQM+∑i=13JiTniTFi+JvωiTQii+J0Tn0TFp=-J-T∑i=13JiTniTFAi+JvωiTQAii+J0Tn0TFAp+QMA-J-T∑i=13JiTniTFVi+JvωiTQVii+QMV-J-T∑i=13JiTniTFGi+JvωiTQGii+J0Tn0TFGp+QMG-J-TQMEwhere J-T are the inverse matrix of JT.


## 6. Numerical Simulation

In order to verify the correctness of the kinematics and dynamics model of the parallel manipulator, the comparison of theoretical and simulation curves of the motion parameters for the sliders based on the Mathematica and Adams software is conducted in this section. Three kinds of rotational motions of the moving perfusion platform are selected as simulation motion trajectory to analysis the motion and driving force of the driving slider. In the initial position, the displacement of all the driving sliders is 130mm and the moving platform is parallel to the base platform. The mass parameters of the parallel manipulator are given in Table2, and the inertia matrices used in the simulation are also given as follows:(58)IM=0.1710000.1710000.341kg·m2;Iii=0.0310000.0310000.0003kg·m2;Table 2
The mass parameters of the proposed 1T2R parallel manipulator (kg).
Parameters m s i m i m p m m Mass 2.0 3.0 3.0 7.5The external force and torque exerted on the moving perfusion platform can be given as follows:(59)fe=30.0,-20.0,25.0Tne=0.0,-25.0,0.0TFor the first kind of rotational motion, a pure rotation aboutxb axis from 0 to -0.3rad is given without the translational motion of the moving platform, and the rotational motion of the moving platform can be described as(60)α=-0.335t44-84t45+70t46-20t47As shown in Figure15, the simplified simulation model in Adams is given, and the curves of driving forces in Adams are also shown. Based on the simulation curves, the points of different motion parameters can be extracted. Then, by utilizing the Mathematica software, the comparison curves of the displacement, velocity, acceleration, and driving force of the sliders are illustrated in Figures 16 and 17, respectively. In each figure, the black solid curves are the theoretical curves of the motion parameters of the driving sliders calculated by Mathematica software, and the red, green, and blue dashed curves are the simulation curves of the first, the second, and the third driving sliders’ motion parameters obtained from the Adams software, respectively. Also, the motion parameters of different driving sliders are marked by characters “m”, “n”, and “o”, respectively. For the driving force in Figure 17, the negative sign indicates the opposite direction along the zb axis.Figure 15
The simplified simulation model of rotation aboutxb axis in Adams.Figure 16
The comparison curves of the displacement and the velocity for the driving sliders under the rotation aboutxb axis.
(a) (b)Figure 17
The comparison curves of the acceleration and the driving force for the driving sliders under the rotation aboutxb axis.
(a) (b)According to these figures, it can be concluded that the theoretical and simulation curves of the motion parameters for the driving joints have the same change trend, respectively. And the curves of motion parameter change smoothly and continuously, which indicates the parallel manipulator has a good performance along the rotation aboutxb axis. For the passive PU link, one of the rotation axes of the U joint connected to the moving platform has the same direction as the axis yb. Thus, according to the symmetry of the mechanism, when the moving platform rotates around the xb axis, the second and the third driving joints should have the same motion trajectory. Obviously, the displacement, velocity, and acceleration curves of the second and the third driving sliders in the simulation curves are all the same, respectively, which verifies the correctness of the establishment of kinematics model again.For the second and third kinds of rotational motions, a pure rotation aboutyb axis from 0 to -0.3rad and a simultaneous rotation about xb axis and yb axis from 0 to -0.3rad are given, respectively. Then the two kinds of rotational motions can be described, respectively, as follows:(61)β=-0.335t44-84t45+70t46-20t47(62)α=-0.335t44-84t45+70t46-20t47β=-0.335t44-84t45+70t46-20t47Similarly, the simplified simulation models of rotation aboutyb axis and rotation about xb and yb axis in Adams are presented in Figures 18 and 21, respectively. Then, the comparison curves of the displacement, velocity, acceleration, and driving force of the sliders under the two kinds of rotational motions are obtained as shown in Figures 19, 20, 22, and 23, respectively. As mentioned above, the black solid curves and the red, green, and blue dashed curves also represent the theoretical and simulation curves, respectively. And the motion parameters of different driving sliders are also marked by characters “m”, “n”, and “o”, respectively. It also can be concluded that the theoretical and simulation curves of the motion parameters for the sliders all have the same change trend, respectively. Moreover, from the smoothly change trend of the motion curves, the good rotation performance of the parallel manipulator is proved again.Figure 18
The simplified simulation model of rotation aboutyb axis in Adams.Figure 19
The comparison curves of the displacement and the velocity for the sliders under the rotation aboutyb axis.
(a) (b)Figure 20
The comparison curves of the acceleration and the driving force for the sliders under the rotation aboutyb axis.
(a) (b)Figure 21
The simplified simulation model of rotation aboutxb and yb axis in Adams.Figure 22
The comparison curves of the displacement and the velocity for the sliders under the rotation aboutxb and yb axis.
(a) (b)Figure 23
The comparison curves of the acceleration and the driving force for the sliders under the rotation aboutxb and yb axis.
(a) (b)In summary, through the comparison of theoretical and simulation curves of the displacement, velocity, acceleration, and driving force for the sliders under the different rotational motions of the moving platform, the correctness of the kinematics and dynamics model of the parallel manipulator has been verified. Moreover, the good rotation performance of the proposed parallel manipulator is also proved. From the curves of the driving force, it can be concluded that the driving forces of the three driving sliders all vary from 68N to 85N, which indicates that all the driving joints are evenly stressed.


## 7. Conclusions

The main work and conclusions can be drawn as follows.(1) In this paper, a novel hybrid perfusion system has been proposed, which is constructed by a 1T2R parallel perfusion manipulator and an arc guide way and can be used for the perfusion of the honeycombs in the thermal protection system of the spacecraft.(2) The inverse kinematics and the Jacobian matrix are analyzed comprehensively. Then, based on the kinematics analysis, the performance analysis for the parallel manipulator is carried out. The analysis results show that the workspace and the stiffness of the proposed parallel manipulator all could satisfy the perfusion requirements and there is no singularity position within the workspace, which proved the good kinematics performance of the parallel manipulator.(3) Through the analysis of the velocity and acceleration of the components, the dynamics model has been established by utilizing the principle of virtual work. According to the comparison of theoretical and simulation curves of the motion parameters for the sliders, the correctness of the kinematics and dynamics model of the parallel manipulator is verified. Also, the verified kinematics and dynamics models will provide a good theoretical foundation for the optimal design of the driving force and the research of the control strategy in the future research.



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*Source: 6768947-2018-11-25.xml*