Article Contents
Citation: H. R. Fang, P. F. Liu, H. Yang, B. S. Jiang. Design and Analysis of a novel 2T2R parallel mechanism with the closed-loop limbs. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1294-z doi:  10.1007/s11633-021-1294-z
Cite as: Citation: H. R. Fang, P. F. Liu, H. Yang, B. S. Jiang. Design and Analysis of a novel 2T2R parallel mechanism with the closed-loop limbs. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1294-z doi:  10.1007/s11633-021-1294-z

Design and Analysis of a Novel 2T2R Parallel Mechanism with the Closed-loop Limbs

Author Biography:
  • 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, 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, Beijing Jiaotong University, China. Her research interests include parallel mechanisms, digital control, robotics and automation, and machine tool equipment. E-mail: hrfang@bjtu.edu.cn (Corresponding author)ORCID iD: 0000-0001-7938-4737

    Peng-Fei Liu received the B. Eng. degree in mechanical engineering from Beijing Jiaotong University, China in 2018. Currently, he is a master student at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. His research interests include robotics in computer integrated manufacturing and parallel manipulator. E-mail: 18121311@bjtu.edu.cn

    Hui Yang received the B. Eng. degree in mechanical engineering from Shenyang Ligong University, China in 2014. She won the China Scholarship Council (CSC) Founding to Stony Brook University, the State University of New York as a joint Ph. D. student in 2018. Currently, she is a Ph. D. degree candidate at Beijing Jiaotong University, China. Her research interests include parallel mechanisms, hybrid perfusion manipulator and intelligent manufacturing. E-mail: 15116342@bjtu.edu.cn

    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. degree candidate 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

  • Received: 2020-11-24
  • Accepted: 2021-03-15
  • Published Online: 2021-04-13
通讯作者: 陈斌, bchen63@163.com
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Design and Analysis of a Novel 2T2R Parallel Mechanism with the Closed-loop Limbs

Abstract: This paper presents a novel four degrees of freedom (DOF) parallel mechanism with the closed-loop limbs, which includes two translational (2T) DOF and two rotational (2R) DOF. By connecting the proposed parallel mechanism with the guide rail in series, the 5-DOF hybrid robot system is obtained, which can be applied for the composite material tape laying in aerospace industry. The analysis in this paper mainly focuses on the parallel module of the hybrid robot system. First, the freedom of the proposed parallel mechanism is calculated based on the screw theory. Then, according to the closed-loop vector equation, the inverse kinematics and Jacobian matrix of the parallel mechanism are carried out. Next, the workspace stiffness and dexterity analysis of the parallel mechanism are investigated based on the constraint equations, static stiffness matrix and Jacobian condition number. Finally, the correctness of the inverse kinematics and the high stiffness of the parallel mechanism are verified by the kinematics and stiffness simulation analysis, which lays a foundation for the automatic composite material tape laying.

Citation: H. R. Fang, P. F. Liu, H. Yang, B. S. Jiang. Design and Analysis of a novel 2T2R parallel mechanism with the closed-loop limbs. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1294-z doi:  10.1007/s11633-021-1294-z
Citation: Citation: H. R. Fang, P. F. Liu, H. Yang, B. S. Jiang. Design and Analysis of a novel 2T2R parallel mechanism with the closed-loop limbs. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1294-z doi:  10.1007/s11633-021-1294-z
    • The wrapping, laying and forming of advanced composite materials are the key technologies for manufacturing core components of spacecraft. These key technologies play a pivotal role in the development of the aerospace industry[1-3]. The laying of composite materials is a key manufacturing technology for core components such as the shell of the solid rocket motor, the fuselage and wings of a large plane and wind turbine blades. These components have structural features, such as large dimensions and irregular shapes[2]. The tape laying head used for laying composite materials has a complex structure and heavy weight[4]. It is necessary to use a robot to grab the tape laying head for the tape laying operation. High stiffness and less space is required for the posture adjustment platform of the tape laying head. At present, the traditional laying method occupies a large space, and the posture adjustment platform of the tape laying head has low stiffness, which is difficult to meet the processing requirements. Therefore, it is necessary to design a new type of tape laying robot according to the characteristics of special-shaped components.

      In order to ensure the precision of composite material placement, the tape laying robot system should be able to complete the overall placement process with only one clamping. The special-shaped parts have a large size, and the laying head has characteristics of heavyweight. According to the above characteristics, the robot system is required to have high stiffness along the direction of the gravity and a larger workspace in the direction along the parts′ axis. Traditional series robots have the advantages of large workspace and better flexibility, but the disadvantages of low stiffness and carrying capacity also exist. Compared with series robots, parallel robots have the advantages of high stiffness and high precision[5]. There are a variety of parallel mechanisms used for aerospace industry projects[6, 7]. However, the traditional parallel mechanism also has shortcomings, such as small workspace and complicated control. The serial-parallel hybrid mechanism can combine the advantages of the two mechanisms so that the robot has a larger workspace while having greater stiffness. At present, a variety of serial-parallel hybrid mechanisms have been used in actual production, such as Tricept mechanism[8-10], Exechon hybrid robot[11,12], Z3 spindle head[13], etc. Also, in the configuration of parallel mechanisms, Cao et al.[14] used generalized function (GF) set theory to construct a series of completely decoupled parallel mechanisms. Sun et al.[15] used topology synthesis and classification to construct a series of mechanisms. Fan et al.[16,17] used the mechanism evolution method to introduce U-joints to the plane 6R mechanism and added unconstrained limbs to obtain the 2T2R mechanism. A new type of parallel mechanism can be obtained by adding constrained limb to the 6 degrees of freedom (DOF) mechanism. Yang et al.[18] and Fang et al.[19] added a passive constrained limb to the 6-DOF mechanism to obtain a new type of parallel mechanism. This paper proposes a new type of 2T2R parallel mechanism with closed-loop limb according to the requirements of adjusting the position of the placement head of the special-shaped workpiece for the composite material laying.

      In this paper, the research objective is mainly aimed at the establishing the kinematic model, kinematics performance analysis and the simulation analysis for the proposed 2T2R parallel tape laying mechanism. The remainder of this article is given as follows. In Section 2, the structure description and DOF analysis of the proposed 2T2R parallel manipulator is represented. The analysis of the kinematics and performance is conducted in Sections 3 and 4. In Section 5, the simulation research is carried out by utilizing the kinematics model, and the finite element simulation is compared with the model without the closed-loop limbs. Conclusions are given in Section 6.

    • As shown in Fig. 1(a), the series-parallel hybrid robot tape laying system is mainly composed of a tape laying head, a special-shaped workpiece, a 2UPU&R(2R(RR&RPR)R) parallel posture adjustment platform (R represents revolute pair, P represents prismatic pair, U represents universal pair) and a horizontal rail. The parallel mechanism and the horizontal rail are connected in series through a P joint, and the parallel mechanism can reciprocate along the horizontal rail. This article mainly focuses on the research and analysis of the novel 2UPU&R(2R(RR&RPR)R) parallel mechanism.

      Figure 1.  Serial-parallel tape laying robot system

      The 2UPU&R(2R(RR&RPR)R) parallel mechanism is composed of a moving platform, a fixed base and four limbs as shown in Fig. 1(b). The moving platform also includes a main platform, a sub-platform, and an R joint. The R joint connects the main and sub-platforms. The coordinate system of the main platform and the sub-platform coincide with each other, and the centers are at the same point P. The 2UPU&R(2R(RR&RPR)R) parallel mechanism contains two kinds of limbs. The two limbs on the diagonal are the same kind. One of them is the R(RR&RPR)R limb, as shown in Fig. 2. The R(RR&RPR)R limb is connected with the fixed base and the sub-platform, respectively. Another type of limb is the UPU limb, which is connected to the main platform of the fixed platform and the moving platform through the U joint at both ends of the limb. In the U joint, the axis of the rotating joint connected to the moving platform is the same direction as the x-axis. The axis of the first revolute joint of the upper U joint is perpendicular to that of the first revolute joint of the lower U joint. The axis of the second revolute joint of the upper U joint is perpendicular to that of the second revolute joint of the lower U joint. The first joint of the upper U pair and the second joint of lower U pair in the two UPU limbs are parallel to each other. The second joint of the upper U pair and the two UPU limbs are collinear, and the first joint of the lower U pairs are also collinear. The two UPU limbs are arranged in a diagonal position. Among them, each P joint Pi (i=1, 2, 3, 4) is the active P joint.

      Figure 2.  Twist of each joint

      Fig. 3 is a sketch of the 2UPU&R(2R(RR&RPR)R) parallel mechanism. The distance from C1 and C3 to the center point P on the moving platform is b. The distance from C2 and C4 to the center point P is a. The four connecting points of the moving platform are on the same plane and distributed in a diamond shape. The coordinate system P-xyz fixed on the main platform is established at the center point P. The x axis points to C3, and the z axis is vertical to the main platform and points to the workpiece. The y axis is determined by the right-hand system. The coordinate system P-x1y1z1 fixed on the sub-platform is established at the center point P of the moving platform, which coincides with the P-xyz coordinate system.

      Figure 3.  Schematic diagram of 2UPU&R(2R(RR&RPR)R) mechanism

      The distance from A1, A3 to the center point O on the fixed base is c. The distance from A2, A4 to the center point O is d, and the fixed base coordinate system O-XYZ is established at the center point O. The X axis points from point O to point A3, the Z axis points from point O to point P, and the Y axis is determined according to the right-hand rule. The length of the driving joint is li (i = 1, 2, 3, 4), and the vector of driving joint is qi (i = 1, 2, 3, 4).

    • In Fig. 3, the initial position of the mechanism is defined as the moving platform parallel to the fixed platform. Axis Z overlaps axis z. The Z axis of the coordinate system O-XYZ coincides with the Z axis of P-xyz. The mechanism has a main platform C2C4 and a sub-platform C1C3. The two limbs A1B1B2C1, A3B3B4C3 are connected to the sub-platform, and the two limbs A2C2 and A4C4 are connected to the main platform. The sub-platform and the main platform are connected by the R joint.

      The two limbs A1B1B2C1, A3B3B4C3 and the sub-platform are connected to the main platform as a whole, which can be considered as a 2R(RR&RPR)R limb. Therefore, three limbs need to be considered when calculating the DOF of this mechanism.

    • First, analyze the first limb composed of the limbs A1B1B2C1, A3B3B4C3 and the sub-platform. The 2R(RR&RPR)R limb is shown in Fig. 4. The coordinates of each point are: A1(–a1, 0, 0), A3(a1, 0, 0), B1(b1, 0, c1), B2(b2, 0, c2), B3(b3, 0, c3), B4(b4, 0, c4), C1(d1, 0, e1), C3(d3, 0, e3).

      Figure 4.  Schematic diagram of 2R(RR&RPR)R) mechanism

      The screw of the A1B1B2C1 limb, shown in Fig. 2, can be expressed as

      $\left\{ \begin{aligned} &{{S\mkern-10.5mu/}_{\rm{1}}} = \left[ {{\rm{0,1,0;0,0, - }}{a_{\rm{1}}}} \right] \\ &{{S\mkern-10.5mu/}_{\rm{2}}} = \left[ {{\rm{0,1,0; - }}{c_{\rm{1}}}{\rm{,0,}}\;{b_{\rm{1}}}} \right] \\ &{{S\mkern-10.5mu/}_{3}} = \left[ {{\rm{0,1,0; - }}{c_{\rm{2}}}{\rm{,0,}}\;{b_{\rm{2}}}} \right] \\ &{{S\mkern-10.5mu/}_{\rm{4}}} = \left[ {{\rm{0,0,0;}}\;{d_{\rm{1}}}{\rm{ - }}{b_{\rm{2}}}{\rm{,0,}}\;{e_{\rm{1}}}{\rm{ - }}{c_{\rm{2}}}} \right] \\ &{{S\mkern-10.5mu/}_{\rm{5}}} = \left[ {{\rm{0,1,0; - }}{e_{\rm{1}}}{\rm{,0,}}\;{d_{\rm{1}}}} \right] . \end{aligned} \right.$

      (1)

      The constrain screw system of the limb can be described as

      $\left\{ \begin{aligned} & {S\mkern-10.5mu/}_1^{r1} = \left[ {0,1,0;0,0,0} \right] \\ &{S\mkern-10.5mu/}_1^{r2} = \left[ {0,0,0;1,0,0} \right] \\ & {S\mkern-10.5mu/}_1^{r3} = \left[ {0,0,0;0,0,1} \right] . \end{aligned} \right.$

      (2)

      The motion between the main-platform and the sub-platform can be written as

      ${S\mkern-10.5mu/}_C^4 = \left[ {0,0,0;1,0,0} \right].$

      (3)

      And the motion of the moving platform can be written as

      $\left\{ \begin{aligned} & {S\mkern-10.5mu/}_C^1 = \left[ {1,0,0;0,0,0} \right] \\ & {S\mkern-10.5mu/}_C^2 = \left[ {0,0,1;0,0,0} \right] \\ & {S\mkern-10.5mu/}_C^3 = \left[ {0,0,0;0,1,0} \right] \\ & {S\mkern-10.5mu/}_C^4 = \left[ {0,0,0;1,0,0} \right] . \end{aligned} \right.$

      (4)

      The 2R(RR&RPR)R limb has two constraints on the main platform, namely a constraint force along the y axis and a constraint couple around the z axis. The 2R(RR&RPR)R limb has 4-DOF, namely movement along the x and z axes and rotation around the x and y axes.

    • As described in Section 1, the axis of the first revolute joint of the upper U joint is perpendicular to that of the first revolute joint of the lower U joint. And the axis of the second revolute joint of the upper U joint is perpendicular to that of the second revolute joint of the lower U joint. The first joint of the upper U pair and the second joint of the lower U pair in the two UPU limbs are parallel to each other. The second joint of the upper U pair and the two UPU limbs are collinear, and the first joint of the lower U pairs are collinear too.

      The moving platform and the fixed platform herein are parallel at the initial position. According to Fig. 5, the screw system of the UPU limb can be expressed as

      Figure 5.  Schematic diagram of 2-R(RR&RPR)R) mechanism

      $\left\{ \begin{aligned} & {{S\mkern-10.5mu/}_{{\rm{21}}}} = \left[ {l{\rm{,0,0;0,0,0}}} \right] \\ & {{S\mkern-10.5mu/}_{{\rm{22}}}} = \left[ {{\rm{0,}}\;l{\rm{,0;0,0,0}}} \right] \\ & {{S\mkern-10.5mu/}_{{\rm{23}}}} = \left[ {{\rm{0,0,0;}}\;l,m,n} \right] \\ & {{S\mkern-10.5mu/}_{{\rm{24}}}} = \left[ {l{\rm{,0,0;0,}}\;n, - m} \right] \\ & {{S\mkern-10.5mu/}_{{\rm{25}}}} = \left[ {{\rm{0,}}\;l{\rm{,0;}} - n{\rm{,0,}}\;l} \right]. \end{aligned} \right.$

      (5)

      The twist system can be written as

      ${S\mkern-10.5mu/}_{\rm{2}}^r = \left[ {0,0,0;0,0,1} \right].$

      (6)

      Therefore, when the moving platform and the fixed base are parallel, the UPU limb restricts the rotation around the P joint.

      When the moving platform and the fixed base are not parallel, the UPU limb is in a general position, and the limb rotates θ2 and θ1 around the x and y axes, respectively, and the screw system also changes. The screw system of the UPU limb is

      $ \left\{\begin{aligned} &{S\mkern-10.5mu/}_{31}=\left[0,1,0;0,0,0\right]\\ &{S\mkern-10.5mu/}_{32}=\left[\cos{\theta }_{1},0,\sin{\theta }_{1};0,0,0\right]\\ &{S\mkern-10.5mu/}_{33}=\left[0,0,0;\sin{\theta }_{1}\cos{\theta }_{2},-\sin{\theta }_{2},\cos{\theta }_{1}\cos{\theta }_{2}\right]\\ &{S\mkern-10.5mu/}_{34}=\Bigg[\begin{array}{l}\cos{\theta }_{1},0,\sin{\theta }_{1};-{l}_{1}\sin{\theta }_{1}\sin{\theta }_{2},\\{l}_{1}{\cos}^{2}{\theta }_{1}\cos{\theta }_{2} -{l}_{1}{\sin}^{2}{\theta }_{1}\cos{\theta }_{2},\\{l}_{1}\cos{\theta }_{1}\sin{\theta }_{2}\end{array}\Bigg]\\ &{{S\mkern-10.5mu/}_{35}}=\Bigg[\begin{array}{l} a,b,c;-{l}_{1}c\sin{\theta }_{2}-{l}_{1}b\cos{\theta }_{1}\cos{\theta }_{2},\\{l}_{1}a\cos{\theta }_{1}\cos{\theta }_{2} -{l}_{1}c\sin{\theta }_{1}\cos{\theta }_{2},\\{l}_{1}b\sin{\theta }_{1}\cos{\theta }_{2}+{l}_{1}a\sin{\theta }_{2}\end{array}\Bigg].\end{aligned}\right.$

      (7)

      The twist system can be written as

      ${S\mkern-10.5mu/}_{3}^r = \left[ {c,0, - a;\sin {\theta _{\rm{1}}},0,{\rm{ - }}\cos {\theta _{\rm{1}}}} \right].$

      (8)

      In (8), acosθ1+csinθ1=0. Therefore, when the moving platform and the fixed platform are not parallel, the constraint given by the UPU limb is a constraint couple rotating around the z axis and a constraint force along the y axis. Under normal working conditions, the UPU limb restricts the rotational DOF around the z axis and translational DOF along the y direction. According to the configuration of the two UPU limbs mentioned in the previous article, when the mechanism is in general position, the two UPU branches together provide a constraint couple along the Y axis and a constraint couple around the Z axis. When the mechanism is in general position, the two UPU limbs together provide a constraint couple around the Z axis.

      In summary, when the moving platform and the fixed platform are parallel, the constraint given by the UPU to the moving platform is a constraint couple rotating around the Z axis. When the moving platform and the fixed platform are not parallel, the constraint given by the UPU limb to the moving platform is a constraint couple rotating around the Z axis and a constraint force along y direction. Thus, considering the three limbs′ constraints, the DOF of the 2UPU&R(2R(RR&RPR)R) mechanism is 2T2R, which are the translational DOF along X and Z and the rotational DOF around the X and Y axes.

    • Based on the initial position of the mechanism, as shown in Fig. 3, the parameters of the moving platform are given as follows: x, z are the distances of the moving platform origin point P moving along the x and z axes, α is the angle that the moving platform has rotated around the x axis, and β is the angle that the moving platform has rotated around the y axis. Next, solve the position of the driving joint according to the above parameters. Since the mechanism involves two complex limbs and two simple limbs, it is necessary to solve the simplified mechanism of the limb first. The limb A1B1B2C1 can be simplified into a P joint, as shown by the dotted line in Fig. 6.

      Figure 6.  Schematic diagram of closed-loop limb

      According to Fig. 3, the closed-loop vector equation of the i (i = 1, 2, 3, 4) limb can be established as

      ${{OP}}{\rm{ + }}{{P}}{{{C}}_i}{\rm{ = }}{{O}}{{{A}}_i}{\rm{ + }}{{{A}}_i}{{{C}}_i}$

      (9)

      where OP can be expressed as OP = (X, 0, Z)T. If PCi in the formula is expressed in the P-xyz coordinate system: PCi = Tci. The coordinates of ci point are c1 = (−b, 0, 0)Tc2 = (0, −a, 0)T, c3 = (b, 0, 0)T, c4 = (0, a, 0)T. OAi in (9) is OA1 = (−c, 0, 0)T, OA2 = (0, −d, 0)T, OA3 = (c, 0, 0)T, OA4 = (0, d, 0)T. T is the conversion matrix from P-xyz to O-XYZ.

      ${{T}} = {{{T}}_\alpha } \times {{{T}}_\beta }$

      (10)

      ${{{T}}_\alpha } = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{ - \sin \alpha } \\ 0&{\sin \alpha }&{\cos \alpha } \end{array}} \right]$

      (11)

      ${{{T}}_\beta } = \left[ {\begin{array}{*{20}{c}} {\cos \beta }&0&{\sin \beta } \\ 0&1&0 \\ { - \sin \beta }&0&{\cos \beta } \end{array}} \right].$

      (12)

      The closed vector (9) can be expressed as

      ${{{A}}_i}{{{C}}_i} = {{OP}} + {{P}}{{{C}}_i} - {{O}}{{{A}}_i}.$

      (13)

      Use li to represent the length of each limb after the equivalent transformation:

      ${l_i} = \sqrt {{{{A}}_i}{{{C}}_i}^2}. $

      (14)

      The closed-loop limb can be equivalent to a P joint. In Fig. 6, the dotted line draws the moving joint after the equivalent transformation, and the solid line is the closed-loop link before the equivalent transformation. There is a corresponding relationship between $p_i$ and $l_i$. The angle $\gamma $ can be obtained from the law of cosines:

      $\begin{split} & \cos {\gamma _1} = \frac{{{l_1}^2 + {e^2} - {g^2}}}{{2e{l_1}}}{\kern 1pt} \\ & \cos {\gamma _3} = \frac{{{l_3}^2 + {e^2} - {g^2}}}{{2e{l_3}}}. \end{split} $

      (15)

      The driving joint pi (i=1, 3) is

      $\left\{ \begin{aligned} & {p_1} = \sqrt {{f^2} + {l_1}^2 + 2f \times {l_1} \times \cos {\gamma _1}} \\ & {p_2} = {l_2} \\ & {p_3} = \sqrt {{f^2} + {l_3}^2 + 2f \times {l_3} \times \cos {\gamma _3}} \\ & {p_4} = {l_4}. \end{aligned} \right.$

      (16)
    • The speed vector of the driving pair is $\dot {{\mathit{\boldsymbol{q}}}}$, expressed as ${\dot{\mathit{\boldsymbol{q}}}} = [{{{\mathit{\boldsymbol{q}}}}_{1{\kern 1pt} }}{\kern 1pt} {{{\mathit{\boldsymbol{q}}}}_2}{\kern 1pt} {{{\mathit{\boldsymbol{q}}}}_3}{\kern 1pt} {{{\mathit{\boldsymbol{q}}}}_4}]$. The speed vector output by the moving platform is $\dot {{\mathit{\boldsymbol{x}}}}$, expressed as ${\dot{\mathit{\boldsymbol{x}}}} = [{{{\mathit{\boldsymbol{x}}}}_{1{\kern 1pt} }}{\kern 1pt} {{{\mathit{\boldsymbol{x}}}}_2}{\kern 1pt} {{{\mathit{\boldsymbol{x}}}}_3}{\kern 1pt} {{{\mathit{\boldsymbol{x}}}}_4}]$. The inversely solved constraint equation can be written as (17). AiCi is the scalar length and ji is the direction vector:

      ${{OP}} + {{P}}{{{C}}_i} = {{O}}{{{A}}_i} + {{{A}}_i}{{{C}}_i} \times {{{j}}_i}.$

      (17)

      Take the derivative of (17) and multiply ji on both sides of (17) to obtain (18), ω13 is the angular velocity of the rotation of the secondary platform, ω24 is the angular velocity of the rotation of the main platform, and AiCi is the length of the equivalent P joint for the sub-platform.

      $\left\{ \begin{aligned} &{{{j}}_i} \times {{V}} + \left( {{{P}}{{{C}}_i} \times {{{j}}_i}} \right){\omega _{13}} = \mathop {{l_i}}\limits^. (i = 1,3)\\ &{{{j}}_{{i}}} \times {{V}} + \left( {{{P}}{{{C}}_{{i}}} \times {{{j}}_{{i}}}} \right){\omega _{13}} = \mathop {{l_i}}\limits^. (i = 2,4). \end{aligned} \right.$

      (18)

      In (18), $\mathop {{l_i}}\limits^. $ is the speed of the equivalent P joint, and the Jacobian matrix must be converted to the actual P joint speed $\mathop {{p_i}}\limits^. = \mathop {{l_i}}\limits^. (i = {\rm{2}},{\rm{4}})$ of the P joint.

      Derivatives on both sides of (15) can obtain:

      ${\left( {\cos {\gamma _1}} \right)^\prime } = \frac{{{l_1}^2 + {g^2} - {e^2}}}{{2e{l_1}^2}}{l_1}^\prime. $

      (19)

      According to the law of cosines:

      ${p_1}^2 = {f^2} + {l_1}^2 + 2f \times {l_1} \times \cos {\gamma _1}.$

      (20)

      Take the derivation on both sides of (19) and substitute it into (20) to get:

      $\begin{split}2{p_1} \times {p_1}^\prime = &2{l_1} \times {l_1}^\prime + 2{l_1}^\prime \times f\cos {\gamma _1} + 2{l_1} \times f{(\cos {\gamma _1})^\prime } = \\ &{l_1}^\prime \left[ {{l_1} + \frac{{f{l_1}}}{e}} \right].\\[-10pt]\end{split}$

      (21)

      After solving (21), (22) can be obtained:

      ${p_1}^\prime = {l_1}^\prime \left[ {\frac{{e{l_1} + f{l_1}}}{{{p_1}e}}} \right].$

      (22)

      It can be obtained in the same way:

      ${p_3}^\prime = {l_3}^\prime \left[ {\frac{{e{l_3} + f{l_3}}}{{{p_3}e}}} \right].$

      (23)

      Substituting (22) and (23) into (18), we can get:

      ${{{j}}_i} \times {{V}} + \left( {{{P}}{{{C}}_i} \times {{{j}}_i}} \right){\omega _{13}} = \mathop {{p_i}}\limits^. \frac{{{p_i}e}}{{e{l_i} + f{l_i}}}\;\;(i = 1,3).$

      (24)

      Therefore, the corresponding relationship can be drawn:

      ${{{J}}_x}{{\dot x}} = {{{j}}_q}{{\dot q}}$

      (25)

      ${{{J}}_{{x}}} = [{{{j}}_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{P}}{{{C}}_i} \times {{{j}}_i}].$

      (26)

      Substituting ${{{j}}_i}$ and ${{P}}{{{C}}_i}$ into (26), we get (27)−(30).

      ${\begin{split} {}\\{}\\ \left\{ \begin{aligned} & {j_{11}} = \frac{{c + X - b\cos \beta }}{{\sqrt {{{\left( {c + X - b\cos \beta } \right)}^2} + {b^2}\sin {\alpha ^2}\sin {\beta ^2} + {{\left( {Z + b\cos \alpha \sin \beta } \right)}^2}} }} \\ & {j_{12}} = \frac{{Z + b\cos \alpha \sin \beta }}{{\sqrt {{{\left( {c + X - b\cos \beta } \right)}^2} + {b^2}\sin {\alpha ^2}\sin {\beta ^2} + {{\left( {Z + b\cos \alpha \sin \beta } \right)}^2}} }} \\ & {j_{13}} = \frac{{ - bZ\sin \beta \sin \alpha }}{{\sqrt {{b^2} + {{\left( {c + X} \right)}^2} + {Z^2} - 2b\left( {c + X} \right)\cos \beta + 2bZ\cos \alpha \sin \beta } }} \\ & {j_{14}} = \frac{{bZ\cos \beta + b\left( {c + X} \right)\cos \alpha \sin \beta }}{{\sqrt {{b^2} + {{\left( {c + X} \right)}^2} + {Z^2} - 2b\left( {c + X} \right)\cos \beta + 2bZ\cos \alpha \sin \beta } }} \end{aligned} \right.\end{split}}$

      (27)

      ${\left\{ \begin{aligned} & {j_{21}} = \frac{X}{{\sqrt {{X^2} + {{\left( {d - a\cos \alpha } \right)}^2} + {{\left( {Z - a\sin \alpha } \right)}^2}} }} \\ & {j_{22}} = \frac{{Z - a\sin \alpha }}{{\sqrt {{X^2} + {{\left( {d - a\cos \alpha } \right)}^2} + {{\left( {Z - a\sin \alpha } \right)}^2}} }} \\ & {j_{23}} = \frac{{a\left( { - Z\cos \alpha + d\sin \alpha } \right)}}{{\sqrt {{a^2} + {d^2} + {X^2} + {Z^2} - 2a\left( {d\cos \alpha + Z\sin \alpha } \right)} }} \\ & {j_{24}} = - \frac{{aX\sin \alpha }}{{\sqrt {{a^2} + {d^2} + {X^2} + {Z^2} - 2a\left( {d\cos \alpha + Z\sin \alpha } \right)} }} \end{aligned} \right.}$

      (28)

      ${\left\{ \begin{aligned} & {j_{31}} = \frac{{ - c + X + b\cos \beta }}{{\sqrt {{{\left( { - c + X + b\cos \beta } \right)}^2} + {b^2}\sin {\alpha ^2}\sin {\beta ^2} + {{\left( {Z - b\cos \alpha \sin \beta } \right)}^2}} }} \\ & {j_{3{\rm{2}}}} = \frac{{Z - b\cos \alpha \sin \beta }}{{\sqrt {{{\left( { - c + X + b\cos \beta } \right)}^2} + {b^2}\sin {\alpha ^2}\sin {\beta ^2} + {{\left( {Z - b\cos \alpha \sin \beta } \right)}^2}} }} \\ & {j_{3{3}}} = \frac{{Z\sin \alpha \left( { - b\sin \beta } \right)}}{{\sqrt {{b^2} + {{\left( {c + X} \right)}^2} + {Z^2} - 2b\left( {c + X} \right)\cos \beta + 2bZ\cos \alpha \sin \beta )} }} \\ & {j_{3{\rm{4}}}} = \frac{{bZ\cos \beta + b\left( {c + X} \right)\cos \alpha \sin \beta )}}{{\sqrt {{b^2} + {{\left( {c + X} \right)}^2} + {Z^2} - 2b\left( {c + X} \right)\cos \beta + 2bZ\cos \alpha \sin \beta )} }} \end{aligned} \right.}$

      (29)

      $ \left\{ \begin{aligned} & {j_{41}} = \frac{X}{{\sqrt {{X^2} + {{\left( { - d + a\cos \alpha } \right)}^2} + {{\left( {Z + a\sin \alpha } \right)}^2}} }} \\ &{j_{42}} = \frac{{Z - a\sin \alpha }}{{\sqrt {{X^2} + {{\left( {d - a\cos \alpha } \right)}^2} + {{\left( {Z - a\sin \alpha } \right)}^2}} }} \\ & {j_{43}} = \frac{{a\left( {Z\cos \alpha + d\sin \alpha } \right)}}{{\sqrt {{a^2} + {d^2} + {X^2} + {Z^2} - 2ad\cos \alpha + 2aZ\sin \alpha } }} \\ & {j_{44}} = \frac{{aX\sin \alpha }}{{\sqrt {{a^2} + {d^2} + {X^2} + {Z^2} - 2ad\cos \alpha + 2aZ\sin \alpha } }} . \end{aligned} \right. $

      (30)

      The Jacobian matrix Jx is as (31):

      ${{{J}}_x} = \left[ {\begin{array}{*{20}{c}} {{j_{11}}}&{{j_{12}}}&{{j_{13}}}&{{j_{14}}} \\ {{j_{21}}}&{{j_{22}}}&{{j_{23}}}&{{j_{24}}} \\ {{j_{31}}}&{{j_{32}}}&{{j_{33}}}&{{j_{34}}} \\ {{j_{41}}}&{{j_{42}}}&{{j_{43}}}&{{j_{44}}} \end{array}} \right].$

      (31)

      The elements of Jq are as (32):

      ${{{J}}_q} = \left[ \begin{array}{*{20}{c}} {\dfrac{{{p_1}e}}{{e{l_1} + f{l_1}}}\;}&0&0&0 \\ 0&1&0&0 \\ 0&0&{\dfrac{{{p_3}e}}{{e{l_3} + f{l_3}}}\;}&0 \\ 0&0&0&1 \end{array} \right].$

      (32)

      The complete Jacobian matrix of the mechanism is

      ${{J}} = {{J}}_q^{ - {{1}}} \times {{{J}}_x}.$

      (33)
    • During the tape laying process, the posture adjustment platform needs to adjust the rotation around the X axis and Y axis and move along the X and Z axis. The tape laying process requires the posture adjustment platform to have a rotation angle adjustment of ±10° around the x and y axes and a working space of 300 mm along the x and z axes. Therefore, this paper needs to analyze the position workspace and orientation workspace of the mechanism.

      The Monte Carlo method is used when calculating the workspace. That is, a series of points are generated in advance with a certain step length, and it is calculated point by point whether the point meets the inverse solution of the mechanism. If the solution satisfies the constraints of each joint variable, the point is in the workspace, and the point is retained. Otherwise, the position is not within its workspace, and the point is removed. The main factors affecting the workspace are the relative distance of each driving joint and the relative angular displacement limit of each R and U joints.

      Assuming that the moving distance of Pi (i = 1, 3) is l13:

      ${l_{{\rm{13}}\min }} \le {l_{{\rm{13}}}} \le {l_{{\rm{13}}\max }}.$

      (34)

      Assuming that the moving distance of Pi (i = 2, 4) is l24:

      ${l_{24\min }} \le {l_{24}} \le {l_{24\max }}.$

      (35)

      Assuming that the angle between the moving platform and each limb is θ1:

      ${\theta _{1\min }} \le {\theta _1} \le {\theta _{1\max }}.$

      (36)

      Assuming that the angle between the fixed base and each limb is θ2:

      ${\theta _{2\min }} \le {\theta _2} \le {\theta _{2\max }}.$

      (37)

      Assign values to the size parameters of the mechanism, the displacement limit of the driving joint, and the range of rotation angle constraints, as shown in Table 1.

      ParameterValueParameterValue
      a/mm280l13min/mm500
      b/mm330l24max/mm750
      c/mm480l24min/mm450
      d/mm280${\theta _{1\min }}$/(°)−45
      e/mm150${\theta _{{\rm{1}}\max }}$/(°)45
      f/mm150${\theta _{{\rm{2}}\min }}$/(°)−45
      g/mm600${\theta _{{\rm{2}}\max }}$/(°)45
      l13max/mm800

      Table 1.  Dimension parameters of the parallel mechanism

      Since the four DOF of the mechanism are composed of 2T DOF and 2R DOF, it is necessary to calculate the orientation workspace and the position workspace when calculating the workspace.

      First, calculate the orientation workspace when the position is fixed. Take X = 0 and Z = 680. The orientation workspace is shown in Fig. 7(a). The orientation workspace at a fixed position is generally distributed in a rectangular shape. The rotation angle around the x axis in the entire workspace is slightly larger than the rotation angle around the y axis. The reachable range of the α angle hardly changes with the β angle.

      Figure 7.  Workspace of parallel mechanism

      Then calculate the position workspace. Take α = 0 and β = 0. The workspace is shown in Fig. 7(b), the reachable range of the x axis decreases as the Z value increases, and the position workspace is symmetric about x = 0.

      The orientation workspace changes with the Z axis coordinates as shown in Fig. 7(c). Fig. 7(c) shows that the maximum orientation workspace appears when the Z axis is about 620 mm. When the value of the Z coordinate increases or decreases relative to 620 mm, the orientation workspace will decrease.

    • The maximum stiffness that the moving platform of the mechanism can withstand is the standard for testing whether the mechanism has a high stiffness. Usually, in order to solve the output stiffness of the moving platform, it is necessary to establish the functional relationship between the stiffness of the driving joint and the stiffness of the moving platform. At present, the most commonly used method is to solve the static stiffness matrix of the mechanism[20]. This paper uses the stiffness matrix to analyze the rationality of the mechanism configuration.

      ${{K}} = {{{J}}^{{{\rm{T}}}}}{{\chi J}}.$

      (38)

      In (38), $\chi $ is the equivalent elastic coefficient of each joint. The indicators for analyzing the stiffness matrix include calculating the determinant, condition number, and eigenvalue of the stiffness matrix. This paper uses the method of calculating the minimum eigenvalue of the stiffness matrix to analyze the stiffness matrix and the equivalent elastic coefficient of each drive joint as (39).

      ${{\chi }} = \left[ {\begin{array}{*{20}{c}} {2\;000}&0&0&0 \\ 0&{1\;000}&0&0 \\ 0&0&{2\;000}&0 \\ 0&0&0&{1\;000} \end{array}} \right].$

      (39)

      With the given parameters and different variables, the relationship between stiffness index and different independent variables can be drawn. Calculating the stiffness change at a fixed position means that when the position of the moving platform of the mechanism is fixed, X = 0, Z = 620, the distribution of the stiffness of the moving platform with its rotation angles α and β is as shown in Fig. 8(a). It shows that the stiffness increases as the absolute value of α increases and decreases with the increase of β. When the stiffness change under the fixed attitude is calculated, the attitude of the movable platform of the mechanism is fixed. When α = 0, β = 0, the stiffness of the movable platform along with its displacement x, y, z distribution is shown in Fig. 8(b). As shown in Fig. 8(b), the stiffness decreases as the absolute value of X increases and decreases as the Z coordinate increases. The stiffness of the mechanism changes smoothly throughout the workspace without sudden changes.

      Figure 8.  Distribution of stiffness

    • The dexterity of the mechanism reflects the mechanism′s ability to move in a specified direction in a certain posture state. The Jacobian condition number is used to evaluate the dexterity of the mechanism[21,22].

      The Jacobian condition number ranges from 1 to infinity. The smaller the value of the condition number, the better the sensitivity of the mechanism. When the Jacobian condition number is 1, the mechanism has the highest sensitivity and is isotropic. When the Jacobian condition number tends to infinity, the organization is in a strange position. The Jacobian matrix is more suitable for evaluating the dexterity of a mechanism with only rotation or movement. For a mechanism with both translational and rotational DOF, local condition indicators should be used for analysis.

      The output speed ${{\dot x}}$ is represented by linear speed V and angular speed ω.

      ${{\dot x}} = [\begin{array}{*{20}{c}} {{{\dot x}}}&{{{\dot z}}}&{{{\dot \alpha }}}&{{{\dot \beta }}} \end{array}] = [\begin{array}{*{20}{c}} {{V}}&{{\omega }} \end{array}].$

      (40)

      Equation (25) can be rewritten as

      ${{\dot x}} = \left[ {\begin{array}{*{20}{c}} {{{{J}}_{{v}}}} \\ {{{{J}}_{{\omega }}}} \end{array}} \right]{{\dot q}}.$

      (41)

      In (41), Jv and Jω are the local matrices related to the Jacobian matrix J and movement and rotation, respectively. Based on this, the condition numbers of the position and orientation parts of the Jacobian matrix can be obtained, denoted as κv and κω. κv and κω are referred to the linear velocity and angular velocity isotropic indicators of the mechanism, which can be used to evaluate the dexterity of the linear velocity and angular velocity of the mechanism.

      When the posture is fixed to take α = β = 0°. Plot the change of condition number as shown in Fig. 9(b).

      Figure 9.  Distribution of condition numbers when α = 0, β = 0

      Fig. 9 shows that the overall value of the condition number in the entire workspace is small, ensuring that the organization has a good dexterity index in the entire workspace. When the absolute value of the sum is greater, the dexterity of the organization decreases. Therefore, when the absolute value of the sum is smaller, the dexterity in the center is better.

      When setting the position, take X = 0, Z = 620, the change of κv is shown in Fig. 10(a) and the change of κω is shown in Fig. 10(b).

      Figure 10.  Distribution of condition numbers when x = 0, z = 620

      Figs. 9 and 10 show that the overall value of the condition number is small in the entire workspace, and the mechanism has good dexterity in the entire workspace.

    • In this section, in order to verify the correctness of the kinematics analysis of the mechanism, it is necessary to perform a motion simulation analysis on the virtual prototype. First, use UG NX software to build a 3D model based on the schematic diagram, and then use UG′s kinematics simulation function to solve the displacement of the driving pair. The kinematic simulation data was exported and taken to Matlab software to plot the simulation curve. Calculate the curve obtained by the inverse kinematics solution in Matlab, and then compare the two mechanisms.

      The initial position of the moving platform is X = 0, Z = 620, α = β = 0°. In the kinematic simulation, the moving platform is given a displacement with a speed of 5 mm/s along both X and Z direction. Calculate the relative length of each P joint as the moving platform moves. The relative value of each limb′s length variation is derived. Then import these data into Matlab to plot the rod length change curve as shown in Fig. 11. Based on the inverse kinematics, the variation of the length of each limb is calculated. Use the inverse solution to find the displacement of the four drive joints, as shown in Fig. 12.

      Figure 11.  Simulation displacement curve of the driving joint when the moving platform moves along the X and Z directions

      Figure 12.  Inverse kinematic displacement change curve of the driving joint when the movable platform moves along the X and Z directions

      The initial position of the moving platform is X = 0, Z = 680, α = β = 0°. In the kinematic simulation, an angular velocity of 1°/s around the x and y axes is given to the moving platform. The changes of each driving joint are calculated as the moving platform moves. Then import these data into Matlab to plot the curves of each limbs length change as shown in Fig. 13. Matlab is used to calculate the relative motion of the four driving pairs based on the inverse kinematics solution, as shown in Fig. 14.

      Figure 13.  Simulation displacement curve of the driving joint when the moving platform rotates around the X and Y axes

      Figure 14.  Inverse solution displacement curve of the driving joint when the moving platform rotates around the X and Y axes

      According to the simulation results and inverse solution calculation, the correctness of the inverse solution calculation can be verified.

    • In order to verify the feasibility of the new 2UPU&R(2R(RR&RPR)R) 4-DOF parallel mechanism and whether its closed-loop limb structure has better stiffness compared to the general mechanism. Select the similar 2UPU&R (2RPR) mechanism shown in Fig. 15 for finite element simulation and comparative analysis.

      Figure 15.  2UPU&R(2RPR) parallel mechanism

      The 4-DOF parallel mechanism mainly bears the gravity from the tape laying head along x-axis during work. The difference between the two mechanisms is whether they contain closed-loop limbs. Therefore, in this paper, the R(2R(2R(RR&RPR)R) limb and R(2RPR) limb (Fig. 16) are taken out for finite element analysis. A1 and A3 points are given fixed constraints, and a force of 10000 N along the x axis is applied to the C1C2 limb. Under the same size conditions, they are substituted into the Abaqus finite element analysis software for finite element simulation analysis. The simulation results are shown in Fig. 17. It shows that the maximum stress in the R(2RPR) limb is 15.61 MPa under the condition of the drive, and the size of the limbs unchanged. The maximum stress in the limb R(2R(2R(RR&RPR)R) is 9.464 MPa, which reduces the stress by 39.4% compared to the mechanism without the closed-loop limb. It can be concluded that under the condition of the same limb thickness, adding a closed-loop structure to the limb can significantly increase the stiffness of the mechanism. Adding a closed-loop structure can reduce the stress in the driving joint and ensure the structural strength of the posture adjustment platform.

      Figure 16.  Sketch of R(2-RPR) limb

      Figure 17.  Von Mises stress nephogram

      The difference between the 2UPU&R(2R(RR&RPR)R) novel parallel mechanism and the 2UPU&R(2RPR) mechanism is that the former replaces the limb connected with the U joint into a closed-loop limb. After finite element simulation analysis, it can be concluded that the closed-loop limb structure increases the stiffness of the mechanism along the direction of the converging rotation axis.

    • 1) A novel 4-DOF 2UPU&R(2R(RR&RPR)R) parallel mechanism is proposed, which can be regarded as the parallel module of the 5-DOF hybrid robot processing system. The proposed hybrid robot system can meet the laying requirements of the large-scale and special-shaped workpieces.

      2) Through the degree of freedom calculation and the kinematics analysis, the movement of two translational and two rotational of the proposed parallel mechanism has been proved. Furthermore, according to the established inverse kinematics model and Jacobian matrix, the correctness of the kinematic analysis is verified by numerical simulation.

      3) The analysis of workspace, stiffness, and dexterity shows that the parallel mechanism has good stiffness and dexterity in the workspace. In addition, the finite element simulation analysis has been carried out to verify the closed-loop limb can significantly enhance the stiffness of the parallel mechanism.

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

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