Volume 16 Number 6
December 2019
Article Contents
Bing-Shan Jiang, Hai-Rong Fang and Hai-Qiang Zhang. Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles. International Journal of Automation and Computing, vol. 16, no. 6, pp. 775-785, 2019. doi: 10.1007/s11633-019-1192-9
Cite as: Bing-Shan Jiang, Hai-Rong Fang and Hai-Qiang Zhang. Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles. International Journal of Automation and Computing, vol. 16, no. 6, pp. 775-785, 2019. doi: 10.1007/s11633-019-1192-9

Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles

Author Biography:
  • Bing-Shan Jiang received the B. Eng. degree in mechanical electronic engineering from Liaoning Technical University, China in 2015, and the M. Eng. degree in mechanical engineering from Liaoning Technical University, China in 2017. He is currently a Ph. D. candidate in mechanical engineering at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. His research interests include synthesis, kinematics, dynamics and control of parallel robots. E-mail: 17116381@bjtu.edu.cn ORCID iD: 0000-0002-9471-8309

    Hai-Rong Fang received the B. Eng. degree in mechanical engineering from Nanjing University of Science and Technology, China in 1990, the M. Eng. degree in mechanical engineering from Sichuan University, China in 1996, and the Ph. D. degree in mechanical engineering from Beijing Jiaotong University, China in 2005. She worked as associate professor in Department of Engineering Mechanics at Beijing Jiaotong University, China from 2003 to 2011. She is a professor in School of Mechanical Engineering from 2011 and director of Robotics Research Center. Her research interests include parallel mechanisms, digital control, robotics and automation, machine tool equipment. E-mail: hrfang@bjtu.edu.cn (Corresponding author) ORCID iD: 0000-0001-7938-4737

    Hai-Qiang Zhang received the B. Eng. degree in mechanical design and theories from Yantai University, China in 2012, the M. Eng. degree in mechanical engineering from Hebei University of Engineering, China in 2015. He is a Ph. D. candidate in mechanical engineering at Beijing Jiaotong University, China. His research interests include robotics in computer integrated manufacturing, parallel kinematics machine tool, redundant actuation robots, over-constrained parallel manipulators, and multi-objective optimization design. E-mail: 16116358@bjtu.edu.cn ORCID iD: 0000-0003-4421-5671

  • Received: 2019-01-20
  • Accepted: 2019-06-19
  • Published Online: 2019-08-20
通讯作者: 陈斌, bchen63@163.com
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Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles

Abstract: Based on Lie group theory and the integration of configuration, a class of 3T2R (T denotes translation and R denotes rotation) parallel mechanisms with large output rotational angles is synthesized through a five degree of freedom single limb evolving into two five degree of freedom limbs and constraint coupling of each kinematics chain. A kind of 3T2R parallel mechanisms with large rotational angles was selected from type synthesis of 3T2R parallel mechanisms, inverse kinematics and velocity Jacobian matrix of the parallel mechanism are established. The performance indices including workspace, rotational capacity, singularity and dexterity of the parallel mechanism are analyzed. The results show that the parallel mechanism has not only large output rotational angles but also better dexterity.

Bing-Shan Jiang, Hai-Rong Fang and Hai-Qiang Zhang. Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles. International Journal of Automation and Computing, vol. 16, no. 6, pp. 775-785, 2019. doi: 10.1007/s11633-019-1192-9
Citation: Bing-Shan Jiang, Hai-Rong Fang and Hai-Qiang Zhang. Type Synthesis and Kinematics Performance Analysis of a Class of 3T2R Parallel Mechanisms with Large Output Rotational Angles. International Journal of Automation and Computing, vol. 16, no. 6, pp. 775-785, 2019. doi: 10.1007/s11633-019-1192-9
    • 3T2R (T denotes translation and R denotes rotation) 5-DOF (degree of freedom) parallel mechanisms has many advantages of high stiffness, dynamic response performance, high changing posture capability and high machining flexibility[1]. They are suitable for many tasks, such as five-axis machine tools, 3D printing and so on[2, 3]. At present, many researchers have studied 3T2R parallel mechanisms. Shayya et al.[4] presented a new form of 3T2R parallel mechanism, the mechanism is suitable for machining complex curved surfaces and has larger non-singular workspace and rotational capability. Masouleh et al.[58] investigated a new form of 3T2R 5-RPRRR (P denotes translation joint) parallel mechanism and analyzed forward kinematic problems and workspaces of 5-RPRRR parallel mechanism. Wang et al.[9] designed two kinds of articulated moving platforms and presented a class of 3T2R parallel mechanisms on the basis of Lie group theory with high rotational capability, it is suitable for 3D printing. Cheng et al.[1012] designed a new kind of 3T2R parallel mechanism with redundance actuation and with five UPS (U denotes hooke joint and S denotes spherical joint) limbs and one PRPU limb, it has high stiffness and better dynamic performance and is used for numerical control machine tools. Song et al.[3, 13] designed a T 5 3T2R parallel mechanism based on Lie group theory, the mechanism equipment can be applied to large aluminum alloy blank processing with compact structure and good dynamic response performance. Xie et al.[14, 15] designed a 3T2R parallel mechanism with SPR–2(R–|RP(R) RP(R)>S) and the structure of the icosahedron, the machine tool can reach more than 90 degrees and has larger orientational rotational capability.

      In many cases not only the corresponding performance of the parallel mechanism but also parallel mechanisms with large output rotational angles are required. So, many scholars have adopted different methods to achieve large angles, Jin et al.[16] designed two kinds of articulated moving platforms and a common parallelogram and an evolution parallelogram and presented a class of novel 3T2R parallel mechanisms with large decoupled output rotational angles based on Lie group theory, the mechanism can reach ±180 angle of rotation. Xie et al.[17] invented a redundant 3-DOF parallel mechanism, the motion of which can reach two translations and one rotation, the moving platform can reach 115°(25°~90°), the rotational angle of redundant parallel mechanism is 75° greater than that of the Sprint Z3 head. Lu et al.[18] proposed a 2T1R 3-DOF parallel mechanism with a rotational angle range of 130°(–40°~90°), the parallel mechanism consists of two primary limbs, one of which consists of two secondary limbs, the relative motion of the two secondary limbs controls the large angle of the parallel mechanism. Lin et al.[19] designed a 3T2R parallel mechanism for which the connection points between the branch chain and the moving platform are arranged in space, the moving platform can reach 90°. Sangveraphunsiri et al.[1] used a gear or amplification mechanism of a H-4 series parallel mechanism which is used to realize the idea of a large angle, and the relative motion is transformed into the large rotational angle of the parallel mechanism by changing the relative motion of different components on the moving platform. Krut et al.[20] designed a new redundantly actuated 2R3T parallel mechanism (PM), which can reach ±90°.

      According to researchers having studied 3T2R parallel mechanisms, the 3T2R PMs with large output rotational angles appear sporadically in some papers, such as Wang et al.[9], Jin et al.[16], Krut et al.[20]. Therefore, this paper will discuss how to synthesize a class of 3T2R parallel mechanisms with large output rotational angles on the basis of Lie group theory and the integration of configuration. Combined Lie group theory with the integration of configuration is more convenient and intuitive than Lie group theory and screw theory through in type synthesis of the parallel mechanism.

      According to the above studies, articulated moving platform and single rotational joint can realize larger rotational angle than general platform, U and S joint, respectively, which is used for configuration of parallel mechanisms. So the limb and moving platform of parallel mechanisms are designed by means of realizing a large angle to design a class of 3T2R parallel mechanisms with large output rotational angles. Section 2 designs a general configuration diagram and articulated moving platform of the parallel mechanism. A five degree of freedom single limb evolving into two five degree of freedom limbs on the basis of combining Lie group theory with the integration of configuration. Inverse kinematics and velocity Jacobian matrix are established for the parallel mechanism which was selected from type synthesis of 3T2R parallel mechanisms in Section 3. Section 4 analyzes kinematics performance including workspace singularity dexterity. Finally, conclusions are drawn in Section 5.

    • Lie group theory is a method of type synthesis of parallel mechanism. The set of 6-dimensional motion of rigid bodies can be endowed with the algebraic structures of a group represented by D as a Lie group. The three dimensional displacement subgroup G(w) is one of the 12 kinds of displacement subgroups, G(w)[21] represents translation perpendicular to vector w. The set of displacement subgroups and kinematic chains of G(w) are received on basis of closure and commutativity based on the Lie group theory in Table 1. G(w) is regarded as an indivisible set of kinematic chains and evolved into 3T2Ruv kinematic chains in Table 2, u, v and w are axis directions of the motion joint. N1, N2 and N3 are the position of motion joint.

      Set of displacement subgroupKinematic chainSet of displacement subgroupKinematic chain
      R(N1,w)R(N2,w)R(N3,w)wRwRwRR(N1,w)T(u)T(v)wRuPvP
      T(v)R(N1,w)R(N2,w)vPwRwRT(u)R(N1,w)T(v)uPwRvP
      R(N1,w)T(v)R(N2,w)wRvPwRT(u)T(v)R(N1,w)uPvPwR
      R(N1,w)R(N2,w)T(v)wRwRvP

      Table 1.  Kinematic chain of G(w)

      3T2Ruv displacement subgroupKinematic chain
      $\{$T(u)$\}$G(u)$\{$R(N1,v)$\}$uP[vPwPuR]vR uP[vPuRuR]vR uP uRuRuRvR
      G(u)$\{$T(u)$\}$$\{$R(N1,v)$\}$uRuRuR uPvR [vPuRuR] uPvR [vPwPuR]uPvR
      $\{$R(N1,v)$\}$$\{$R(N2,v)$\}$G(u)vRvRuRuRuR vRvR [vPuRuR] vRvR[vPwPuR]
      $\{$R(N1,v)$\}$G(u)$\{$R(N2,v)$\}$vRuRuRuRvR vR[vPuRuR]vR vR[vPwPuR]vR

      Table 2.  3T2Ruv equivalent limbs

    • Configuration evolution is the most practical and intuitive method for the synthesis of parallel mechanisms and obtains new parallel mechanism to meet specific requirements through different evolution methods. A 3T2Ruv kinematic chain is divided into three groups of motion joint: G(w), rotational joint R, prismatic joint P. Without changing the degree of freedom of kinematic chains, one kinematic chain is evolved into two kinematic chains. According to characteristics of the articulated moving platform and the spatial arrangement of constraint couple of each kinematic chain, a class of 3T2Ruv parallel mechanism is obtained by evolving kinematic chains.

    • Fig. 1 shows the overall configuration of a 3T2R parallel mechanism which includes a moving platform, fixed platform and three first limbs. The three first limbs are named 1, 2, 3. The moving platform and the fixed platform are connected with the three first limbs, which includes two secondary limbs, four secondary limbs are named 1.1, 1.2, 2.1, 2.2. The two secondary limbs are initially connected to a fixed platform, the end of first limb is made up of the end of the secondary limb and consists of the R joint which connets to moving platform.

      Figure 1.  General configuration of parallel mechanism

    • Fig. 2 shows articulated moving platform, which obtains A, B, C, o-uvw is at center of moving platforms. R1, R2, R3 are axis of A, B, C and parallel to each other. O1, O2, O3 is at the center of A, B, C. The end of joint of each limb connets to the moving platform with R1, R2, R3. 3T2Ruv motion is outputted by the moving platform.

      Figure 2.  Articulated moving platform

    • In general, a 3T2R parallel mechanism should have 5 limbs and 5 driving pairs, each limb has only one driving pair. Displacement subgroups of the first limb LA, LB, LC are a subset of $ \{ $MA$\}$, $\{$MB$\}$, $\{$MC$\}$, the first limb LA contains two secondary limbs LA1, LA2, which are subsets of $\{$MA1$\}$, $\{$MA2$\}$. The first limb LB contains two secondary limbs LB1, LB2, which are subsets of $\{$MB1$\}$, $\{$MB2$\}$. The first limb LC only contains kinematic chain, which is a subset of $\{$MC$\}$, the end kinematic joint of the displacement subgroup $\{$MA$\}$, $\{$MB$\}$ and $\{$Mc$\}$ are R joint. In order to satisfy output motion of the moving platform, three first limbs should satisfy $\{$MA$\}$$\{$MB$\}$$\{$MC$\}$=3T2Ruv, the displacement subset of $\{$MA$\}$, $\{$MB$\}$, $\{$Mc$\}$ is a 3T2Ruv 5-D displacement submanifold or a 6-D displacement submanifold.

      We should divide each 3T2Ruv kinematic chain into three indivisible groups, there are G(w), rotational joint R, prismatic joint P. The uRuRuR kinematic chain is equal to G(w), uRuRuR represents G(w) in Fig. 3. uRuRuR is added to the 3T2Ruv kinematic chain and is parallel to another uRuRuR without changing the degrees of freedom of the 3T2Ruv kinematic chain in Fig. 4.

      Figure 3.  Four classes of five degrees of freedom limb

      Figure 4.  Evolution of limb

      Combined with characteristics of the moving platform and evolution of limbs (a), (b), (c) and (d) in Fig. 4, the evolution of limb in Fig. 4 is shown up in Table 3.

      LimbsEquivalent kinematic chainsLimbsEquivalent kinematic chains
      (a) uP–${\{}$ uR uR uR/ uR uR uR${\}}$– vR uP–${\{}$ vP uR uR/ vP uR uR${\}}$– vR(b) ${\{}$ uR uR uR/ uR uR uR${\}}$– uP vR${\{}$ uR vP/ uR vP${\}}$– uC vR
      uP–${\{}$ wP uR uR/ wP uR uR${\}}$– vR${\{}$ uR wP/ uR wP${\}}$– uC vR
      uP–${\{}$ vP wP uR/ vP wP uR${\}}$– vR${\{}$ vP wP/ vP wP${\}}$– uC vR
      ${\{}$ uP vP uR uR/ uP vP uR uR${\}}$– vR${\{}$ vP wP uC/ uR wP uC${\}}$– vR
      ${\{}$ uP wP uR uR/ uP wP uR uR${\}}$– vR${\{}$uR uR uC/ uR uR uC${\}}$– vR
      ${\{}$ uP vP wP uR/ uP vP wP uR${\}}$– vR${\{}$ uR vP uC/ uR vP uC${\}}$– vR
      ${\{}$ uC uR uR/ uC uR uR${\}}$– vR${\{}$ vP wP uC/ vP wP uC${\}}$– vR
      ${\{}$ uC vP uR/ uC vP uR${\}}$– vR${\{}$ uR vP uR/ uR vP uR${\}}$– uP vR
      ${\{}$ uC wP uR/ uC wP uR${\}}$– vR${\{}$ uR wP uR/ uR wP uR${\}}$– uP vR
      ${\{}$ uC wP vP/ uC wP vP${\}}$– vR${\{}$ uR vP wP/ uR vP wP${\}}$– uP vR
      ${\{}$ vP uR uR/ vP uR uR${\}}$– vR vR
      ${\{}$ wP uR uR/ wP uR uR${\}}$– vR vR
      (c) vR–${\{}$ uR uR uR/ uR uR uR${\}}$– vR vR–${\{}$ vP uR uR/ vP uR uR${\}}$– vR(d) ${\{}$ uR uR uR/ uR uR uR${\}}$– vR vR${\{}$ vP wP uR/ vP wP uR${\}}$– vR vR
      vR–${\{}$ wP uR uR/ wP uR uR${\}}$– vR${\{}$ vP uR/ vP uR${\}}$– uU v vR
      vR–${\{}$ vP wP uR/ vP wP uR${\}}$– vR${\{}$ wP uR/ wP uR${\}}$– uU v vR
      vR–${\{}$ vP wP uR/ vP wP uR${\}}$– vR${\{}$ wP vP/ wP vP${\}}$– uU v vR
      ${\{}$ vUuuR uR/ vUuuR uR${\}}$– vR${\{}$ uR uR uU v/ uR uR uU v${\}}$– vR
      ${\{}$ vUuwP uR/ vUuwP uR${\}}$– vR${\{}$ vP uR uU v/ vP uR uU v${\}}$– vR
      ${\{}$ vUuvP uR/ vUuvP uR${\}}$– vR${\{}$ wP uR uU v/ wP uR uU v${\}}$– vR
      ${\{}$ vUuwP vP/ vUuwP vP${\}}$– vR${\{}$ wP vP uU v/ wP vP uU v${\}}$– vR
      ${\{}$ vUuvP uR/ vUuvP uR${\}}$– vR${\{}$ wP uR/ wP uR${\}}$– vC vR
      ${\{}$ vCuuR uR/ vCuuR uR${\}}$– vR${\{}$ uR uR vC/ uR uR vC${\}}$– vR
      ${\{}$ vCuwP uR/ vCuwP uR${\}}$– vR${\{}$ wP uR vC/ wP uR vC${\}}$– vR
      ${\{}$ vP uR uR vR/ vP uR uR vR${\}}$– vR
      ${\{}$ wP uR uR vR/ wP uR uR vR${\}}$– vR
      ${\{}$ vP wP uR vR/ vP wP uR vR${\}}$– vR

      Table 3.  Evolution of five degrees of freedom equivalent limbs

    • Any limb in Table 3 without (c) is selected twice as the first limb 1, 2. The kinematic joint vR of evolutional chains of the first limb 1, 2 is connected to A, B of the moving platform, the other end of evolutional chains of the first limb 1, 2 is connected to A′, B′ of fixed platform. The pre-evolution limb in Table 3 as the first limb 3 connects C of moving platform with C′ of the fixed platform. When the limbs (c) is selected and connect A, B of the moving platform with A′, B′ of fixed platform for type synthesis of parallel mechanisms, then, the constraint couple of each kinematic chain isn′t parallel, the parallel mechanism isn′t 3T2R parallel mechanisms. When the kinematic chains (a), (b), and (d) as limbs are selected and connects A, B of moving platform with A′, B′ of fixed platform for type synthesis of parallel mechanisms, the constraint couple of each kinematic chain is parallel to each other, the parallel mechanisms are 3T2R parallel mechanisms.

      Using the above method, PMs are synthesized as shown in Table 4.

      PM typeThe first limb 1, 2The first limb 3PM
      (a) uP–${\{}$ uR uR uR/ uR uR uR${\}}$– vR uP vP uR uR vR uP vP uR uR vR–${\{}$2–[ uP–( uR uR uR/ uR uR uR)– vR]${\}}$
      ${\{}$ uP uR uR uR/ uP uR uR uR${\}}$– vR uP uR uR uR vR uP uR uR uR vR–${\{}$2–[( uP uR uR uR/ uP uR uR uR)– vR]${\}}$
      ${\{}$ uC uR uR/ uC uR uR${\}}$– vR uC uR uR vR uC uR uR vR–${\{}$2– [( uC uR uR/ uC uR uR)– vR]${\}}$
      ${\{}$ uCwP vP/ uC wP vP${\}}$– vRuCwP vP vRuCwP vP vR–${\{}$2–[( uCwP vP/ uCwP vP)– vR]${\}}$
      (b)${\{}$ uR uR/ uR uR${\}}$– uC vR uR uR uC vR uR uR uC vR–${\{}$2–[( uR uR/ uR uR)– uC vR]${\}}$
      ${\{}$ uR uR uR/ uR uR uR${\}}$– uP vR uR uR uR uP vRuR uR uR uP vR–${\{}$2–[( uR uR uR/ uR uR uR)– uP vR]${\}}$
      ${\{}$ uR uRC/ uR uR uC${\}}$– vR uR uR uC vR uR uR uC vR–${\{}$2–[( uR uRC/ uR uR uC)– vR]${\}}$
      (d1)${\{}$ uR uR uR/ uR uR uR${\}}$– vR vR uR uR uR vR vR uR uR uR vR vR–${\{}$2–[( uR uR uR/ uR uR uR)– vR vR]${\}}$
      ${\{}$ uR uR/ uR uR${\}}$– uU v vR uR uR uUvvR uR uR uUvvR–${\{}$2–[( uR uR/ uR uR)– uU v vR]${\}}$
      ${\{}$ uR uR uU v/ uR uR uU v${\}}$– vR uR uR uUvvR uR uR uUvvR–${\{}$2–[( uR uR uU v/ uR uR uU v)– vR]${\}}$
      ${\{}$ uR uR uR vR/ uR uR uR vR${\}}$– vR uR uR uR vR vR uR uR uR vR vR–${\{}$2–[( uR uR uR vR/ uR uR uR vR)– vR]${\}}$
      ${\{}$ uR uR/ uR uR${\}}$– vC vR uR uR vC vR uR uR vC vR–${\{}$2–[( uR uR/ uR uR)– vC vR]${\}}$
      (d2) uR–( uR uR vR vR/ uR uR vR vR) uR uR uR vR vR uR uR uR vR vR–${\{}$2–[ uR–( uR uR vR vR/ uR uR vR vR)]${\}}$

      Table 4.  Feasible limbs and the 3T2R PMs

    • The constraint couple of every secondary limb is parallel, the moving platform of all parallel mechanisms has no rotation around Z. The moving platform of all parallel mechanisms includes moving along the axis of X, Y, Z and rotating along the axis of X, Y axes on the basis of the above the method.

    • The 3T2R parallel mechanism shown in Table 4 (d1) is more suitable for five-axis machine tools. Therefore, this section uses uRuRuRvRvR–$\{$2–[(uRuRuRvR/uRuRuRvR)–vR]$\}$ PM as an example to illustrate the kinematical properties of the family, the uRuRuRvRvR$\{$2–[(uRuRuRvR/uRuRuRvR)–vR]$\}$ PM can be written 5-PRUR PM. In Fig. 5 the parallel mechanism contains a moving platform, fixed platform and five PRUR kinematic chains. The P joint of each PRUR kinematic chain can move on the rack, the second R joint axis is parallel to the first R joint axis of U in the PRUR kinematic chain, the second R joint axis of U is parallel to the last R joint axis of the PRUR kinematic chain. The parallel mechanism realizes 3T2Rxy motion by moving the P joint.

      Figure 5.  Kinematic model of the 5-PRUR mechanism

    • In Fig. 6, the static coordinate system O-XYZ and moving coordinate system o-uvw is built in fixed platform and moving platform, respectively. O and o are in the center of square E1E2E4E3 and the moving platform, respectively. The coordinate of O is (0, 0, 0)T, the coordinate of o is oo=(x, y, z)T in O-XYZ. The moving frame o-uvw is obtained by translating and rotating O-XYZ, its rotational matrix is R, R = R(α)R(β).

      Figure 6.  Fixed posture workspace

      $R = \left[\!\!\!{\begin{array}{*{20}{c}} {\cos \beta }&{\sin \alpha \sin \beta }&{\cos \alpha \sin \beta } \\ 0&{\cos \alpha }&{ - \sin \alpha } \\ { - \sin \beta }&{\sin \alpha \cos \beta }&{\cos \alpha \cos \beta } \end{array}}\!\!\!\right].$

      (1)

      The coordinates of D1, D2, D3 represented in o-uvw can be written as

      $\left\{ \begin{aligned} & {}^o{D_1} = \left(\dfrac{\sqrt 3 r}{2}, - \dfrac{r}{2},0\right)\\ & {}^o{D_2} = \left( -\dfrac{\sqrt 3 r}{2}, -\dfrac{r}{2},0\right)\\ & {}^{o}{D_3} = \left( {0,\;r,\;0} \right). \end{aligned} \right. $

      (2)

      The coordinates of Di represented in O-XYZ can be written as

      $ ^{\rm{o}}{{{D}}_{{i}}}{ = ^{\rm{o}}}{\rm{o}} + {{{R}}^{\rm{o}}}{{{D}}_{{i}}}_{\rm{}}. $

      (3)

      Bi and Ci are expressed in O-XYZ as

      $ \begin{split} & \left\{ \begin{aligned} & {}^o{B_i} = {\left( {{B_{ix}},\;{B_{iy}},\;{p_i}} \right)^{\rm{T}}}^{}\\ & {}^o{C_i} = {\left( {{C_{ix}},\;{C_{iy}},\;{C_{iz}}} \right)^{\rm{T}}}. \end{aligned} \right. \end{split} $

      (4)

      The relationship between oBi, oCi, length L1 and L2 is expressed as

      $ \left\{ \begin{aligned} & \left( {^o{C_i}{ - ^o}{B_i}} \right)\cdot\left( {^o{C_i}{ - ^o}{B_i}} \right) = {L_1^2} \\ & \left( {^o{C_i}{ - ^o}{D_i}} \right)\cdot\left( {^o{C_i}{ - ^o}{D_i}} \right) = {L_2^2} \\ & \left( {{Y_{Di}} - {Y_{Ci}}} \right)\cos\alpha - \left( {{Z_{Di}} - {Z_{Ci}}} \right)\sin\alpha = 0. \end{aligned} \right. $

      (5)

      L1, L2 are known, oBi, oCi, oDi are determined by geometry conditions. Equation (5) is expressed as

      $ \left\{ \begin{aligned} & {\left( {{B_{ix}} - {C_{ix}}} \right)^2} + {\rm{ }}{\left( {{B_{iy}} - {C_{iy}}} \right)^2} + {\left( {{\rm{ }}{p_i} - {C_{iz}}} \right)^2} = {L_1^2} \\ & {\left( {{D_{ix}} - {C_{ix}}} \right)^2} + {\left( {{\rm{ }}{D_{iy}} - {C_{iy}}} \right)^2} + {\left( {{\rm{ }}{D_{iz}} - {C_{iz}}} \right)^2} = {L_2^2} \\ & \left( {{Y_{Di}} - {Y_{Ci}}} \right)\cos\alpha - \left( {{Z_{Di}} - {Z_{Ci}}} \right)\sin\alpha = 0. \end{aligned} \right. $

      (6)

      The position pi is obtained as a slide block in Z direction:

      $\begin{split} {p_i} = &{D_{iz}} \pm \sqrt{\dfrac{({l_2} - {{({D_{iy}} - {C_{iy}})}^2})}{(1 + {{\tan }^2}(\alpha ))}}\pm \\ &\sqrt {({l_1} - {{({B_{iy}} - {D_{iy}})}^2})}. \end{split}$

      (7)
    • The velocity of the central origin of the moving platform is expressed as

      $ V = {\left( {{v_x}\;{v_y}\;{v_z}\;{w_x}\;{w_y}\;0} \right)^{\rm{T}}}. $

      (8)

      The motion screw of kinematic chain 1.1 is expressed as

      $\left\{ \begin{aligned} & {{\textit{\$}}_{11}} = \left( {0\;\;0\;\;0;0\;\;0\;\;{q_1}} \right) \\ & {{\textit{\$}}_{12}} = \left( {1\;\;0\;\;0;0\;\;{q_1} - {y_{B1}}} \right) \\ & {{\textit{\$}}_{13}} = \left( {1\;\;0\;\;0;0\;\;{z_{C1}} - {y_{C1}}} \right) \\ & {{\textit{\$}}_{14}} = \left( {0\;\;{b_1}\;\;{c_1};\;{d_1}\;{e_1}\;{f_1}} \right) \\ & {{\textit{\$}}_{15}} = \left( {0\;\;{b_1}\;\;{c_1};\;{d_2}\;{e_2}\;{f_2}} \right). \end{aligned}\right. $

      (9)

      The Jacobian matrix of kinematic chain 1.1 is written as

      $ {{{J}}_1} = {\left( {{{\textit{\$}}_{11}}\;{{\textit{\$}}_{12}}\;{{\textit{\$}}_{13}}\;{{\textit{\$}}_{14}}\;{{\textit{\$}}_{15}}} \right)^{\rm{T}}}. $

      (10)

      Similarly, the Jacobian matrix of kinematic chain 1.2, 2.1, 2.2, limb 3 are written as J2, J3, J4, J5. Ji=[$i1 $i2 $i3 $i4 $i5]T which is 6×5 Jacobian matrix, the velocity equation of five kinematic chains and moving platform are established.

      $ V= \left[ {{J_i}} \right]\dot \theta _k^i. $

      (11)

      The second and third lines of [Ji] are identical, [Ji′] is a 5×5 Jacobian matrix obtained from [Ji] when the second or third lines are cutted out. The inverse matrix of [Ji′] is Gi. Because the P joint of PRUR is an actuated joint, the first line of Gi is extracted and composed of a new 5×5 Jacobian matrix G. The relationship between actuated joint P and the output of the moving platform is written as

      $\dot q = GV.$

      (12)

      The inverse solution of (12) is written as

      $V = {G^{ - 1}}\dot q$

      (13)

      where [G–1] is a 5×5 Jacobian matrix in nonsingular configuration.

    • In this paper, the large-angle parallel mechanism is designed and measured by fixed posture workspace and fixed position workspace. The mechanism parameters are shown in Table 5.

      ParameterDescriptionValuesUnit
      qi,minMinimum value of input0mm
      qi,maxMaximum value of input800mm
      rThe radius of the moving platform500mm
      E1E2The distance between E1 and E21 000mm
      E1E3The distance between E1 and E31 000mm
      E3E4The distance between E3 and E41 000mm
      E2E4The distance between E2 and E41 000mm
      OE5The distance between O and E51 000mm
      l1Length of l1600mm
      l2Length of l2600mm
      RminMinimum rotational angle of R joint${\dfrac{-{\pi}}{2} }$rad
      RmaxMaximum rotational angle of R joint${\dfrac{{\pi}}{2}}$rad

      Table 5.  Related geometrical parameters about the model of 5-PRUR mechanisms

    • Based on the Gosselin and Angeles[22] geometric method, the inverse kinematics is used to calculate the fixed posture workspace of parallel mechanism. By considering length pi and the angle of R joint, the constraint inequalities of the reachable workspace can be written as

      $\left\{ {\begin{aligned} & {q_{i,\;\min} < {p_i} < q_{i,\;\max}} \\ & {{R_{\min}} < R < {R_{\max}}} \end{aligned}} \right.$

      (14)

      where qi,min, qi,max represent pi as the highest and lowest positions, respectively.

      Rmin, Rmax represent the maximum and minimum angle of the R joint, respectively.

      Based on inverse kinematics and constraint (14), the fixed posture workspace of the parallel mechanism is shown in Fig. 6. The range of the reachable workspace can be obtained as follows: X $\in$ [–533, 533] mm, Y $\in $ [150, 350] mm, Y $\in $ [–920, 0] mm. The workspace of the parallel mechanism is symmetry about the X0Z plane.

    • Similarly, fixed position workspace is obtained. From fixed position workspace[23, 24], we know when the P joint is moving, the moving platform center can move from oo=(0, 250, –920)T to oo=(0, 250, –30)T. Fig. 7 shows the fixed position workspace when the moving platform center is in oo=(0, 250, –900)T, oo=(0, 250, –600)T, oo=(0, 250, –300)T, oo=(0, 250, –60)T. The parallel mechanism can reach ±94.5° in the Y direction and ±180° in the X direction, respectively. Therefore, when length l1, l2, r and R are taken to the appropriate size, designing a parallel mechanism can achieve a large-angle.

      Figure 7.  Fixed position workspace

    • The rotational angle of the parallel mechanism in different positions is constantly changing from the fixed position workspace. Parallel mechanisms generally have singular positions that have a negative effect on rotational movement. The Jacobian matrix singularity of parallel mechanisms include the forward kinematics singularity and inverse kinematics singularity. If det(Jx) = 0 or det(Jt) = 0, it′s forward kinematics singularity or inverse kinematics singularity, respectively.

      The instantaneous velocity of the i(i = 1, 2, 3, 4, 5) kinematic chain on the basis of screw theory can be expressed as

      $ \begin{split} {{\textit{\$}} _{{p}}} = & {\left[ {{v_x}\;{v_y}\;{v_{{z}}}\;{{{w}}_x}\;{{{w}}_y}\;{{{w}}_{{z}}}} \right]^{\rm{T}}} = \\ & {{\dot q}_{1,i}}{{\textit{\$}}_{1,i}} + {\theta _{2,i}}{{\textit{\$}}_{2,i}} + {\theta _{3,i}}{{\textit{\$}}_{3,i}} + {\theta _{4,i}}{{\textit{\$}}_{4,i}} + {\theta _{5,i}}{{\textit{\$}}_{5,i}} \end{split} $

      (15)

      where w and v represent angular velocity and linear velocity, respectively.

      θji represents angular velocity of the j(j = 1, 2, 3, 4, 5) R joint of the $i$ kinematic chain.

      $\dot q$ represents P joint linear velocity.

      The motion screw of joint of each kinematic chain of the parallel mechanism is expressed as

      $ \left\{ \begin{aligned} & {{\textit{\$}}_{1,i}} = [0;\;{p_i}] \\ & {{\textit{\$}}_{2,i}} = [{s_{2,i}};\;O{B_i} \times {s_{2,i}}] \\ & {{\textit{\$}}_{j,i}} = [{s_{j,i}};\;O{C_i} \times {s_{j,i}}]\begin{array}{*{20}{c}} {} \end{array}\;\;\;j = 3,4 \\ & {{\textit{\$}}_{5,i}} = [{s_{5,i}};O{D_i} \times {s_{5,i}}]. \end{aligned} \right. $

      (16)

      The lock P joint of the i(i = 1, 2, 3, 4, 5) kinematic chain, reverse screw of kinematic chain is expressed as

      $ {{\textit{\$}}_i^r}= \left[ {{{{S}}_{\rm{i}}};\;{{{S}}_i^0}} \right] $

      (17)

      where

      $\begin{aligned} {S_i} = & \left[ {\left( {{x_{Ci}} - {x_{Di}}} \right)\left( {c{y_{Ci}} - c{y_{Bi}} + b{z_{Bi}} - b{z_{Ci}}} \right),\left( {{y_{Bi}} - {y_{Ci}}} \right)} \right.\\ &\left( {b{z_{Ci}} - b{z_{Di}} - c{y_{Ci}} + c{y_{Di}}} \right),\left( {{z_{Bi}} - {z_{Ci}}} \right)\\ &\left.{\left( {b{z_{Ci}} - b{z_{Di}} - c{y_{Ci}} + c{y_{Di}}} \right)}\right].\\ {S_i}^{\rm{0}} = &[\left( {{y_{Ci}}{z_{Bi}} - {y_{Bi}}{z_{Ci}}} \right)\left( {b{z_{Ci}} - b{z_{Di}} - c{y_{Ci}} + c{y_{Di}}} \right),\\ &(b(c{y_{Ci}} - c{y_{Bi}} + b{z_{Bi}} - b{z_{Ci}})\left( {{x_{Ci}}{z_{Di}} - {x_{Di}}{z_{Ci}}} \right) + \\ &c\left( {{x_{Ci}}{y_{Di}} - {x_{Di}}{y_{Ci}}} \right)\left( {c{y_{Bi}} - c{y_{Ci}} + b{z_{Bi}} - b{z_{Ci}}} \right)),0]. \end{aligned}$

      It can be seen from (15), (17) ${{\textit{\$}}_i^r} \circ {{\textit{\$}}_p} ={\dot q} {{\textit{\$}}_i^r} \circ {{\textit{\$}}_{1,i}}$ is written as

      ${J_x}\dot x = {J_t}\dot q $

      (18)

      where

      $\begin{aligned} \dot x = &\; {\left[ {{v_x}\;{v_y}\;{v_z}\;{w_x}\;{w_y}\;{w_z}} \right]^{\rm{T}}}\\ {J_x} = &\; {\left[ {{{\textit{\$}}_1^r}\;{{\textit{\$}}_2^r}\;{{\textit{\$}}_3^r}\;{{\textit{\$}}_4^r}\;{{\textit{\$}}_5^r}} \right]^{\rm{T}}}\\ {J_t} = &\; {\rm{diag}} [ \left( {{z_{B1}} - {z_{C1}}} \right)\left( {b{z_{C1}} - b{z_{D1}} - c{y_{C1}} + c{y_{D1}}} \right),\\ & \left( {{z_{B2}} - {z_{C2}}} \right)(b{z_{C2}} - b{z_{D2}} - c{y_{C2}} + c{y_{D2}}),\\ & \left( {{z_{B3}} - {z_{C3}}} \right)\left( {b{z_{C3}} - b{z_{D3}} - c{y_{C3}} + c{y_{D3}}} \right),\\ & \left( {{z_{B4}} - {z_{C3}}} \right)\left( {b{z_{C3}} - b{z_{D4}} - c{y_{C3}} + c{y_{D4}}} \right),\\ & \left( {{z_{{B5}}} - {z_{C5}}} \right)\left( {b{z_{C5}} - b{z_{{D5}}} - c{y_{C5}} + c{y_{{D5}}}} \right)]. \end{aligned}$

    • Jt is the diagonal matrix, when the Z coordinate of B, C, D is the same, (zBizCi)×(bzCibzDicyCi+cyDi)) = 0, the inverse kinematics singularity of the parallel mechanism occurs.

    • We know from (18), Jt is 5×5 Jacobian matrix, $\dot x$=[ vx vy vz wx wy wz]T is a 6×1 matrix, Jx is a 5×6 matrix, but, the sixth column of Jx is 0. Jx is translated into $J_x' $ when the sixth column of Jx is deleted. Det($J_x'$) = 0 when the direction vectors AB, BC and CD are co-linear, the forward kinematics singularity of parallel mechanism occurs.

    • A Jacobian condition number can be used to measure the input-output relationship of parallel mechanisms with pure movement or pure rotation, the motion transmission performance is best when the condition number of the Jacobian matrix is equal to 1. However, for the parallel mechanism with both movement and rotation, the Jacobian condition number cannot guarantee its accuracy, therefore, the local condition number is used to analyze the output linear velocity and angular velocity of the parallel mechanism. From (13) and $\dot x$, [G–1] is written as local moving matrix Jv and local rotational matrix Jw, with corresponding condition numbers kJv and kJw, respectively, kJv and kJw can evaluate the isotropy and dexterity of linear and angular velocities of parallel mechanisms. Fig. 8 is the distribution of kJv of parallel mechanisms, when the attitude angle of the parallel mechanism is that α = 0 and β = 0, kJv of the parallel mechanism is best in α = 0, β = 0, Z = –60 mm. Fig. 9 shows kJw of parallel mechanism, dexterity of the parallel mechanism is best in α = 0, β = 0 when the center of moving platform is in X = 0 mm, Y = 250 mm, Z = –30 mm.

      Figure 8.  Distribution of kJv in α=0, β=0

      Figure 9.  Distribution of kJw in x=0, y=250

    • 1) This paper focused on type synthesis and kinematics performance analysis of a family of 3T2R parallel mechanisms with large output rotational angles. Articulated moving platforms are designed to guarantee large rotational angles for the PMs, the kinematic chain is evolved into the limb1 in Section 2.1, a class of 3T2R parallel mechanisms with large output rotational angles is obtained on the basis of Lie group theory and configuration evolution. The correctness of the above method is verified through analyzing constraint couple of the secondary Kinematic chain in Sections 1.1, 1.2, 2.1, 2.2 and the limb in Section 3.

      2) The new parallel mechanism is selected and analyzed with inverse kinematics and velocity. According to analyzing workspace, the parallel mechanism can achieve large rotational angles. The parallel mechanism can reach ±94.5° in the Y direction and ±180° in the X direction.

      3) The fixed position workspace, the singular angle workspace of the mechanism are found, and singularity of the forward kinematics and inverse kinematics of the mechanism is analyzed, the new mechanism is shown to have good dexterity. It proves that type synthesis of the parallel mechanism on the basis of Lie group theory and the integration of configuration is more convenient and intuitive than Lie group theory and screw theory through Lie group theory and the integration of configuration. It provides a theoretical basis for the later control.

    • This work was supported by Fundamental Research Funds for the Central Universities (No. 2018JBZ007).

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