Volume 18 Number 1
January 2021
Article Contents
Bing-Shan Jiang, Hai-Rong Fang, Hai-Qiang Zhang. Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms. International Journal of Automation and Computing, 2021, 18(1): 96-109. doi: 10.1007/s11633-020-1255-y
Cite as: Bing-Shan Jiang, Hai-Rong Fang, Hai-Qiang Zhang. Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms. International Journal of Automation and Computing, 2021, 18(1): 96-109. doi: 10.1007/s11633-020-1255-y

Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms

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. 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

    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 an 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 the Robotics Research Center.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

    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. degree candidate in mechanical design and theory 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: 2020-05-19
  • Accepted: 2020-09-16
  • Published Online: 2020-12-23
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Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms

Abstract: This paper presents a five degree of freedom (5-DOF) redundantly actuated parallel mechanism (PM) for the parallel machining head of a machine tool. A 5-DOF single kinematic chain is evolved into a secondary kinematic chain based on Lie group theory and a configuration evolution method. The evolutional chain and four 6-DOF kinematic chain SPS (S represents spherical joint and P represents prismatic joint) or UPS (U represents universal joint) can be combined into four classes of 5-DOF redundantly actuated parallel mechanisms. That SPS-(2UPR)R (R represents revolute joint) redundantly actuated parallel mechanism is selected and is applied to the parallel machining head of the machine tool. All formulas of the 4SPS-(2UPR)R mechanism are deduced. The dynamic model of the mechanism is shown to be correct by Matlab and automatic dynamic analysis of mechanical systems (ADAMS) under no-load conditions. The dynamic performance evaluation indexes including energy transmission efficiency and acceleration performance evaluation are analyzed. The results show that the 4SPS-(2UPR)R mechanism can be applied to a parallel machining head and have good dynamic performance.

Bing-Shan Jiang, Hai-Rong Fang, Hai-Qiang Zhang. Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms. International Journal of Automation and Computing, 2021, 18(1): 96-109. doi: 10.1007/s11633-020-1255-y
Citation: Bing-Shan Jiang, Hai-Rong Fang, Hai-Qiang Zhang. Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms. International Journal of Automation and Computing, 2021, 18(1): 96-109. doi: 10.1007/s11633-020-1255-y
    • Parallel mechanisms (PMs) have begun to be manufactured into parallel machining heads and be applied to the high-precision machine tool, such as Sprint Z3[1], Tricept[2], Exechon[3]. The Sprint Z3, Tricept and Exechon are 3-degree of freedom (DOF) parallel machining heads. There are few five degrees of freedom (5-DOF) redundantly actuated parallel machining heads. The 5-DOF parallel machining head will be expected to complete the process task at a high velocity[4, 5], high-precision[6], high-security and high-performance[7] when the machine tool is processing workpieces. It is important to ensure high-security and high-performance of the 5-DOF parallel machining head, otherwise, it will be more destructive to work pieces. Some researchers try to add redundant actuation[811] to the parallel mechanism (PM) to improve fault tolerance, security, high-performance and stability. Contrasted with the traditional parallel mechanisms, the redundantly actuated parallel mechanisms can reduce singularity, increase effective workspace and improve security. It has high-security, high-performance, high fault tolerance and fewer singularities. Therefore, it is a worthwhile task to design the 5-DOF redundantly actuated parallel mechanism with good dynamic performance.

      Some researchers have researched the 5-DOF parallel mechanism. Qu[12] selected the single-loop mechanism or the multi-loop mechanism as an actuated unit and completed the constraint couple of the kinematic chain to obtain the non-over-constrained or low-over-constrained redundantly actuated parallel mechanism. Chen et al.[13] proposed a 4UPS (U represents universal joint, P represents prismatic joint, S represents spherical joint)-RPU (R represents revolute joint) 5-DOF parallel mechanism, and studied the influence of different performance indexes on a task workspace and optimized the parallel mechanism's parameters. Chen et al.[14] designed a 4UPS-UPU 5-DOF parallel mechanism and established a dynamic mathematical model of the mechanism by the principle of virtual work. Lu et al.[15, 16] designed two 5-DOF parallel mechanisms and deduced formulas of displacement, velocity and acceleration of the mechanism. The dynamic model with friction considered is established by the principle of virtual tests and the mathematical model is verified to be correct. Yao et al.[17] established a dynamic mathematical model of the 5UPS-PRPU redundantly actuated parallel mechanism through the Lagrange method and optimized the actuated moment. The dynamic mathematical model was verified to be correct by theory analysis and simulated experiments. Liu et al.[18] designed a 6PUS-UPU redundantly actuated parallel mechanism and established the generalized pseudo-inverse Jacobi matrix of global dynamic models by the principle of virtual work. The control model of the parallel mechanism is analyzed by the force-position hybrid control strategy. Song et al.[19] designed a T5 parallel mechanism and established the elastic dynamics model of the T5 parallel mechanism. Jiang et al.[20] designed a class of 5-DOF parallel mechanisms with large output rotational angles and analyzed kinematics performance of parallel mechanisms. Guo et al.[21] designed 4-CPS-RPS (C represents cylindrical joint) parallel mechanism and researched the proportional-integral-derivative (PID) control and the force-position redundant control. Masouleh et al.[22, 23] analyzed the kinematics of the 5-RPUR parallel mechanisms. Saadatzi et al.[24] used a geometric interpretation of the so-called vertex space and analyzed the workspace of the 5-PRUR parallel mechanisms. Xie et al.[25] designed a 5-DOF parallel mechanism and optimized parameters of the mechanism. Wang et al.[26] designed a 5UPS-RPS parallel mechanism and analyzed the velocity global performance index of this parallel mechanism. Jin et al.[27] proposed a novel method about the synthesis of generalized parallel mechanisms (GPMs) and designed a class of novel 5-DOF generalized parallel mechanisms with high performance.

      In the above research, there are mostly 5-DOF traditional parallel mechanisms and few 5-DOF redundantly actuated parallel mechanisms. Most researchers directly design 5-DOF traditional parallel mechanisms, but few researchers research dynamics performance evaluation of a class of 5-DOF redundantly actuated parallel mechanisms. But most traditional parallel mechanisms have low fault tolerance and many singularities. In order to design a feasible parallel machining head, this paper will research type synthesis of the redundantly actuated parallel mechanism and analyze dynamic performance of the 5-DOF redundantly actuated parallel mechanism as the machining head.

      In this paper, one 5-DOF single kinematic chain is evolved into one first-order kinematic chain including two secondary kinematic chains by Lie group theory method and configuration evolution method. The evolutional kinematic chain and four 6-DOF kinematic chain SPS or UPS are selected and combined into four classes of redundantly actuated parallel mechanisms in Section 2. The 4SPS-(2UPR)R redundantly actuated parallel mechanism comes from one of four classes of mechanisms and is applied to the parallel machining head. The position, velocity and acceleration of the 4SPS-(2UPR)R mechanism are deduced in Section 3. Dynamic mathematical equations of the 4SPS-(2UPR)R mechanisms are established by the principle of virtual work. A simplified dynamic mathematical model of the 4SPS-(2UPR)R mechanism is obtained and is shown to be correct by Matlab and automatic dynamic analysis of mechanical systems (ADAMS) under no-load conditions in Section 4. The fifth part introduces a dynamic acceleration evaluation index and energy transmission efficiency index. Combined with application numerical examples and parameters of the 4SPS-(2UPR)R mechanism, acceleration performance and transfer efficiency of the mechanism is analyzed. Finally, the article is summarized in Section 6.

    • Lie group theory can describe collections of all rigid body motions. Jin[28] gives twelve classes of displacement subgroups including G(x) in space. G(x) represents that two-dimensional movement is perpendicular to the line x and one-dimensional rotation is rotated to the line x. G(x) is shown in Table 1. 5-DOF kinematic chains can be obtained by adding translational joints and revolute joints when G(x) is considered as one group. 5-DOF kinematic chains are shown in Table 2. T(x) (T represents translational joint), G(x) and R(N, y) can be exchanged as one group, but they are not internally exchanged. x, y and z are axis directions of motion joints. N1, N2 and N3 are position of motion joints.

      Displacement subgroupsKinematic chainsDisplacement subgroupsKinematic chains
      R(N1,x)R(N2,x)R(N3,x)xRxRxRR(N1,x)T(x)T(y)xRxPyP
      T(y)R(N1,x)R(N2,x)yPxRxRT(x)R(N1,x)T(y)xPxRyP
      R(N1,x)T(y)R(N2,x)xRyPxRT(x)T(y)R(N1,x)xPyPxR
      R(N1,x)R(N2,x)T(y)xRxRyP

      Table 1.  G(x) kinematic chains

      5-DOF displacement subgroupsKinematic chains
      {T(x)}G(x){R(N1,y)}xP[yPzPxR]vR xP[yPxRxR]vR xP[xRxRxR]yR
      G(x){T(x)}{R(N1,y)}[xRxRxR]xPyR [yPxRxR]xPyR [yPzPxR]xPyR
      {R(N1,y)}{R(N2,y)}G(x)yRyR[xRxRxR] yRyR[yPxRxR] yRyR[yPzPxR]
      {R(N1,y)}G(x){R(N2,y)}yR[xRxRxR]yR yR[yPxRxR]yR yR[yPzPxR]yR

      Table 2.  5-DOF kinematic chains

    • Configuration evolution is an effective method for designing the redundantly actuated parallel mechanism and obtaining new redundantly actuated parallel mechanisms to satisfy specific requirements. Fan et al.[29] have evolved a planar 6R mechanism into 4-DOF parallel mechanisms.

      In order to configurate type synthesis of the redundantly actuated parallel mechanism, each 5-DOF kinematic chain is shown in Table 2. G(x) is replaced with the xRxRxR, 5-DOF kinematic chain in Table 2 can be expressed as Fig. 1. Without changing the freedom of the 5-DOF kinematic chain, T(x), G(x), R(N, y) can be exchanged as one group, but they are not internally exchanged. Two G(x) are connected in parallel, the 3R open-loop chain is developed into a 6R closed-loop chain, each kinematic chain in Fig. 1 is evolved into a kinematic chain in Fig. 2. One kinematic chain is developed into one first-order kinematic chain including two secondary kinematic chains. In order to maintain large rotational angles and workspaces, the evolutional kinematic chain in Table 2 is shown in Table 3.

      Figure 1.  Four classes of 5-DOF kinematic chains

      Figure 2.  5-DOF kinematic chains

      Kinematic chainsEvolutionary kinematic chainsKinematic chainsEvolutionary kinematic chains
      (a)xP-{xRxRxR/xRxRxR}-yR {xPyPxRxR/xPyPxRxR}-yR (b){ xRxRxR/xRxRxR}-xP-yR {yPzPxC/xRzPxC}-yR
      {xPzPxRxR/xPzPxRxR}-yR {xRxRxC/xRxRxC}-yR
      {xPyPzPxR/xPyPzPxR}-yR {xRyPxC/xRyPxC}-yR
      {xCxRxR/xCxRxR}-yR {yPzPxC/yPzPxC}-yR
      {xCyPxR/xCyPxR}-yR {xRyPxRxP/xRyPxRxP}-yR
      {xCzPxR/xCzPxR}-yR {xRzPxRxP/xRzPxRxP}-yR
      { xCzPyP/xCzPyP}-yR {xRyPzPxP/xRyPzP xP}-yR
      (c) yR-{ xRxRxR/xRxRxR}-yR {yUxxRxR/yUxxRxR}-yR (d){ xRxRxR/xRxRxR}-yR-yR {yPxRxRyR/yPxRxRyR}-yR
      {yUxzPxR/yUxzPxR}-yR {zPxRxRyR/zPxRxRyR}-yR
      {yUxyPxR/yUxyPxR}-yR {yPzPxRyR/yPzPxRyR}-yR
      {yUxzPyP/yUxzPyP}-yR {yPxRxUy/yPxRxUy}-yR
      {yUxyPxR/yUxyPxR}-yR {zPxRxUy/zPxRxUy}-yR
      {yCxxRxR/yCxxRxR}-yR {zPyPxUy/zPyPxUy}-yR
      {yCxzPxR/yCxzPxR}-yR {xRxRxUy/xRxRxUy}-yR
      {zPxRyC /zPxRyC}-yR
      {xRxRyC/xRxRyC}-yR
      {zPxRyC/zPxRyC}-yR

      Table 3.  5-DOF kinematic chains

      In general, each kinematic chain of the parallel mechanism has only one actuated joint, it is mounted on the base or the adjacent joint. The 5-DOF redundantly actuated parallel mechanism should have six kinematic chains to place six actuated joints. Each kinematic chain in Table 3 is selected as one first-order kinematic chain, and each kinematic chain includes two secondary kinematic chains, then four kinematic chains SPS or UPS are selected as the third, fourth, fifth and sixth kinematic chains. The above six kinematic chains connect the moving platform with the fixed platform, respectively. Four kinematic chains SPS or UPS are symmetrically arranged in the center, and evolutional kinematic chains are axially symmetric. Because the 6-DOF kinematic chain is an unconstrained kinematic chain, the evolutional kinematic chain determines DOF of the redundantly actuated parallel mechanism. Each kinematic chain in Table 3 can be selected, each kinematic chain and four unconstrained kinematic chains can be combined into a 5-DOF redundantly actuated parallel mechanism. The new 5-DOF redundantly actuated parallel mechanism is shown in Table 4.

      PM typesEvolutionary kinematic chainsFour kinematic chainsPMs
      (a) {xPyPxRxR/xPyPxRxR}-yR SPS or UPS 4SPS+ {xPyPxRxR/xPyPxRxR}-yR
      {xPzPxRxR/xPzPxRxR}-yR 4SPS+ {xPzPxRxR/xPzPxRxR}-yR
      {xPyPzPxR/xPyPzPxR}-yR 4SPS+ {xPyPzPxR/xPyPzPxR}-yR
      {xCxRxR/xCxRxR}-yR 4SPS+ {xCxRxR/xCxRxR}-yR
      {xCyPxR/xCyPxR}-yR 4SPS+ {xCyPxR/xCyPxR}-yR
      {xCzPxR/xCzPxR}-yR 4SPS+ {xCzPxR/xCzPxR}-yR
      { xCzPyP/xCzPyP}-yR 4SPS+ { xCzPyP/xCzPyP}-yR
      (b) {yPzPxC/xRzPxC}-yR SPS or UPS 4SPS+ {yPzPxC/xRzPxC}-yR
      {xRxRxC/xRxRxC}-yR 4SPS+ {xRxRxC/xRxRxC}-yR
      {xRyPxC/xRyPxC}-yR 4SPS+ {xRyPxC/xRyPxC}-yR
      {yPzPxC/yPzPxC}-yR 4SPS+ {yPzPxC/yPzPxC}-yR
      {xRyPxRxP/xRyPxRxP}-yR 4SPS+ {xRyPxRxP/xRyPxRxP}-yR
      {xRzPxRxP/xRzPxRxP}-yR 4SPS+ {xRzPxRxP/xRzPxRxP}-yR
      {xRyPzPxP/xRyPzP xP}-yR 4SPS+ {xRyPzPxP/xRyPzP xP}-yR
      (c) {yUxxRxR/yUxxRxR}-yR SPS or UPS 4SPS+{yUxxRxR/yUxxRxR}-yR
      {yUxzPxR/yUxzPxR}-yR 4SPS+{yUxzPxR/yUxzPxR}-yR
      {yUxyPxR/yUxyPxR}-yR 4SPS+{yUxyPxR/yUxyPxR}-yR
      {yUxzPyP/yUxzPyP}-yR 4SPS+{yUxzPyP/yUxzPyP}-yR
      {yUxyPxR/yUxyPxR}-yR 4SPS+{yUxyPxR/yUxyPxR}-yR
      {yCxxRxR/yCxxRxR}-yR 4SPS+{yCxxRxR/yCxxRxR}-yR
      {yCxzPxR/yCxzPxR}-yR 4SPS+{yCxzPxR/yCxzPxR}-yR
      (d) {yPxRxRyR/yPxRxRyR}-yR SPS or UPS 4SPS+{yPxRxRyR/yPxRxRyR}-yR
      {zPxRxRyR/zPxRxRyR}-yR 4SPS+{zPxRxRyR/zPxRxRyR}-yR
      {yPzPxRyR/yPzPxRyR}-yR 4SPS+{yPzPxRyR/yPzPxRyR}-yR
      {yPxRxUy/yPxRxUy}-yR 4SPS+{yPxRxUy/yPxRxUy}-yR
      {zPxRxUy/zPxRxUy}-yR 4SPS+{zPxRxUy/zPxRxUy}-yR
      {zPyPxUy/zPyPxUy}-yR 4SPS+{zPyPxUy/zPyPxUy}-yR
      {xRxRxUy/xRxRxUy}-yR 4SPS+{xRxRxUy/xRxRxUy}-yR
      {zPxRyC/zPxRyC}-yR 4SPS+{zPxRyC/zPxRyC}-yR
      {xRxRyC/xRxRyC}-yR 4SPS+{xRxRyC/xRxRyC}-yR
      {zPxRyC/zPxRyC}-yR 4SPS+{zPxRyC/zPxRyC}-yR

      Table 4.  Feasible limbs and 5-DOF PMs

    • The redundantly actuated parallel mechanism is selected as the parallel machining head according to the following criteria: 1) The actuated joint is mounted on the base or near the base. 2) In order to maintain fast response movement of the moving platform, the prismatic joint should be selected as the actuated joint. 3) The translation workspace in the vertical direction may be limited by the range of linear actuators. The 4SPS-(2UPR)R parallel mechanism is more suitable for the machining head. The model of 4SPS-(2UPR)R is shown in Fig. 3. The 4SPS-(2UPR)R is composed of five first-order kinematic chains, the fixed platform and the moving platform. Five first-order kinematic chains include four first-order kinematic chains SPS and (2UPR)R kinematic chains. The (2UPR)R includes two secondary kinematic chains SPR and a single rotated joint R, the single rotated joint R is named as H. The axis of the single rotation joint R is perpendicular to the axis of the rotated joint R of the SPR. The first-order kinematic chain SPS is an unconstrained kinematic chain. The first-order kinematic chain (2UPR)R is a 5-DOF kinematic chain. 4SPS-(2UPR)R is a 3t2r (t denotes translation and r denotes rotation) redundantly actuated parallel mechanism when H is parallel to the fixed platform. 4SPS-(2UPR)R is a 2t3r[30] redundantly actuated parallel mechanism when H is not parallel to the fixed platform.

      Figure 3.  3D model of 4SPS-(2UPR)R

      All Ai coordinates are set on the fixed platform, all Bi coordinates are set on the moving platform in Fig. 4, i=1,···, 6. The fixed coordinate system O-XYZ is in the center of the fixed platform. A1 and A2 are on the OX-axis, A1 and A2 are symmetric along OY-axis. The moving coordinate system o-uvw is in the center of the moving platform. The o point is in the center of the moving platform. B1 and B2 are on the ou-axis, B1 and B2 are symmetric along ov-axis. The O point of O-XYZ to Ai is $\dfrac{D}{2} $, i=1, 2. The O point of O-XYZ to the point Ai is D, i=3,···, 6. The o point of o-uvw to Bi is $\dfrac{d}{2} $, i=1, 2. The o point of o-uvw to the point Bi is d, i=3,···,6. The initial length of each kinematic chain is li0, i=1,···,6. The P joint of each kinematic chain is selected as the actuated joint.

      Figure 4.  Schematic of the mechanism

    • The O coordinate is (0, 0, 0)T in O-XYZ, it is expressed as OO=(0, 0, 0)T. The o coordinate is (x, y, z)T in O-XYZ, it is expressed as Oo=(a, b, c)T. A1B1B2A1 is considered as the plane closed-loop mechanism. Four R joints of the plane closed-loop mechanism correspond to three Euler angles, there are α, β, γ. Rα, Rβ1, Rβ2, Rγ is the vector axis of Rα, Rβ1, Rβ2, Rγ, respectively. The rotation matrix ${}_o^OR$[15, 16] of the mechanism is expressed by YXY Euler angle

      ${}_o^OR = \left[ {\begin{array}{*{20}{c}} { - s\alpha c\beta s\gamma + c\alpha c\gamma }&{\operatorname{s} \alpha \operatorname{s} \beta }&{s\alpha c\beta c\gamma + c\alpha s\gamma } \\ {s\beta s\gamma }&{c\beta }&{ - \operatorname{s} \beta c\gamma } \\ { - c\alpha c\beta s\gamma - s\alpha c\gamma }&{c\alpha s\beta }&{c\alpha c\beta c\gamma - s\alpha s\gamma } \end{array}} \right]$

      (1)

      where s=sin, c=cos, cγ=$\dfrac{{{D}} \times c \alpha +2z\times s\alpha -2x\times c \alpha }{{{d}}}$.

      The coordinates of Ai and Bi can be expressed in Table 5, i=1,···,6. The coordinates of Bi in O-XYZ are expressed as

      ParametersDescriptionsValuesUnits
      DRadius of the fixed platform1m
      dRadius of the moving platform0.5m
      A1Coordinate of A1 in O-XYZ$(\dfrac{D}{2},\;0,\;0)$m
      A2Coordinate of A2 in O-XYZ$(-\dfrac{D}{2},\;0,\;0)$m
      A3Coordinate of A3 in O-XYZ$(D \;\dfrac{\sqrt 2}{2},\;D \;\dfrac{\sqrt 2 }{2},\;0)$m
      A4Coordinate of A4 in O-XYZ$(D \;\dfrac{\sqrt 2}{2},\;-D \;\dfrac{\sqrt 2 }{2},\;0)$m
      A5Coordinate of A5 in O-XYZ$(-D \;\dfrac{\sqrt 2}{2},\;-D \;\dfrac{\sqrt 2 }{2},\;0)$m
      A6Coordinate of A6 in O-XYZ$(-D \;\dfrac{\sqrt 2}{2},\;D \;\dfrac{\sqrt 2 }{2},\;0)$m
      B1Coordinate of B1 in o-uvw$(\dfrac{d}{2},\;0,\;0) $m
      B2Coordinate of B2 in o-uvw$(-\dfrac{d}{2},\;0,\;0) $m
      B3Coordinate of B3 in o-uvw$(d \;\dfrac{\sqrt 2}{2},\;d \;\dfrac{\sqrt 2 }{2},\;0)$m
      B4Coordinate of B4 in o-uvw$(d \;\dfrac{\sqrt 2}{2},\;-d \;\dfrac{\sqrt 2 }{2},\;0)$m
      B5Coordinate of B5 in o-uvw$(-d \;\dfrac{\sqrt 2}{2},\;-d \;\dfrac{\sqrt 2 }{2},\;0)$m
      B6Coordinate of B6 in o-uvw$(-d \;\dfrac{\sqrt 2}{2},\;d \;\dfrac{\sqrt 2 }{2},\;0)$m
      l1SiLength of l1Si0.5m
      l2SiLength of l2Si0.5m
      IMBiInertia matrix of the telescopic roddiag(0.017, 0.000471, 0.376)kg•m2
      IMAiInertia matrix of the oscillating roddiag(0.02, 0.000838, 0.019)kg•m2
      IoInertia matrix of the moving platformdiag(0.378, 0.746, 0.376)kg•m2
      gGravitational acceleration[0 0−9.807]Tm/s2
      mMBiMass of the telescopic rod1.14kg
      mMAiMass of the oscillating rod1.3kg
      moMass of the moving platform2.8kg
      tTiming of the mechanism′s movements4s

      Table 5.  Structural and physical parameters

      $ ^{ O}{{B}_{ i}} = {}^{ O}{ o} + {}_{\;\,{ o}}^{\rm O}{ R}\; { - ^{ o}}{ {B}_{ i}} $

      (2)

      where oBi is a description of Bi in o-uvw, OBi is a description of Bi in O-XYZ.

      The length vector of the kinematic chain of the mechanism in O-XYZ is expressed as

      $ {{{l}}_i}{ = ^O}{A_i}{ - ^O}{B_i} $

      (3)

      where OAi is the description of Ai in O-XYZ, li is vector length of the chain i.

      The displacement of the actuated joint of the mechanism is expressed as

      $ {{{S}}_i} = {{{l}}_i} - {{{l}}_i}_0 $

      (4)

      where Si is the variable length of the chain i.

      The OoBiAi closed-loop constraint equation in O-XYZ is written as

      $ {{Oo}} + {{o}}{{{B}}_i} = {{O}}{{{A}}_i} + {{{A}}_i}{{{B}}_i} $

      (5)

      where Oo=Oo, ${B_i}{{o}} = - {}_{\;\,o}^O{R^o}{B_i},$ OAi is the vector length of O to Ai, AiBi is the vector length of Ai to Bi.

      $ {{{A}}_i}{{{B}}_i} = {{{l}}_i} = {q_i}{w_i} $

      (6)

      where qi is the length of the chain i, wi is unit vector direction of the chain i.

      The equation of each kinematic chain based on (5) and Oo is written as

      $\left\{ \begin{array}{l} {q_i} = \left| {{}^O{{o}} + {{o}}{{{B}}_i} - {{O}}{{{A}}_i}} \right| \\ {w_i} = \dfrac{{{}^O{{o}} + {{o}}{{{B}}_i} - {{O}}{{{A}}_i}}}{{{q_i}}} \\ \end{array} \right.,\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{i = 1,\cdots,6} . \end{array}} \end{array}$

      (7)
    • In order to establish a relationship between output velocity of the o point and velocity of the P actuated joint, (5) is differentiated by time and the velocity of Bi is written as

      $ {v}_{Bi}=v+\omega \times (_{o}^{O}R\cdot {}^{o}B{}_{i})={\dot{q}}_{i}{w}_{i}+{q}_{i}{\omega }_{i}\times {w}_{i}$

      (8)

      where v is the linear velocity of o, ω is angular velocity of o, ωi is angular velocity of the kinematic chain i, ${\dot q_i}$ is velocity of the P actuated joint.

      Multiply wi on both sides of (8), (8) is separately obtained because of wiT(ωi $ \times $ wi).

      $ {w}_{i}^{\rm T}v+{((_{\;o}^{O}R\cdot {}^{o}B{}_{i})\times {w}_{i})}^{{\rm{T}}}\omega ={\dot{q}}_{i},\;\;\begin{array}{c}\left(i=1,\mathrm{\cdots},6\right).\end{array}$

      (9)

      Equation (9) is written as

      $ {\dot{q}}_{i}\!=\!\left[\begin{array}{cc}{w}_{i}^{{\rm{T}}} {((_{\;o}^{O}R\cdot {}^{o}B{}_{i})\times {w}_{i})}^{{\rm{T}}}\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right]\!=\!{J}_{qi}\left[\begin{array}{c}v\\ \omega \end{array}\right]\!=\!{J}_{qi}{v}_{s}$

      (10)

      where Jqi is the Jacobi matrix of the actuated mechanism.

      Equation (10) is the relation between velocity of P actuated joint and output velocity of the moving platform. Translational velocity of the moving platform in the Cartesian coordinates system is coincident as translational velocity in a generalized coordinate system. But the rotational velocity of the moving platform in the Cartesian coordinates system is different from the rotational velocity in the generalized coordinate system.

      The rotational velocity of the moving platform can be expressed by the derivative of the Euler angle.

      $\begin{split} & \omega \;{\rm{ = }}\;R(X,\alpha )\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right]\dot \alpha \!+\! R(X,\alpha )R(Y,\,\beta )\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \\ 0 \end{array}} \right]\dot \beta \!\;+ \\ & \quad\quad\quad \! R(X,\alpha )\times R(Y,\,\beta )R(Z,\gamma )\left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \end{array}} \right]\dot \gamma \;\; \!\!=\!\! \\ &\quad\quad\quad \left[ {\begin{array}{*{20}{c}} 1&0&{\sin \beta } \\ 0&{\cos \alpha }&{ - \sin \alpha } \\ 0&{\sin \alpha }&{\cos \alpha \cos \beta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\dot \alpha } \\ {\dot \beta } \\ {\dot \gamma } \end{array}} \right]. \\ \end{split} $

      (11)

      Equation (11) is represented as

      $\omega \;{\rm{ = }}\left[ {\begin{array}{*{20}{c}} 1&0&{\sin \beta } \\ 0&{\cos \alpha }&{ - \sin \alpha } \\ 0&{\sin \alpha }&{\cos \alpha \cos \beta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\dot \alpha } \\ \begin{array}{l} {\dot \beta } \\ {\dot \gamma } \\ \end{array} \end{array}} \right] = {J_\omega }{v_s}$

      (12)

      when the mechanism is 3t2r, the six-dimensional velocity of the moving platform is represented as

      $\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right] = {J_{01}}\left[ {\begin{array}{*{20}{c}} v \\ {\dot \alpha } \\ {\dot \beta } \end{array}} \right]$

      (13)

      where ${J_{01}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&{{c_\alpha }} \\ 0&0&0&0&{{s_\alpha }} \end{array}} \right]$.

      When the mechanism is 2t3r, the mechanism does not have Y translation. The six-dimensional velocity of the moving platform is represented as

      $\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right] = {J_{02}}\left[ {\begin{array}{*{20}{c}} {\dot X} \\ {\dot Z} \\ \omega \end{array}} \right]$

      (14)

      where ${J_{02}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0 \\ 0&0&0&0&0&0 \\ 0&0&1&0&0&0 \\ 0&0&0&1&0&{\sin \beta } \\ 0&0&0&0&{\cos \alpha }&{ - \sin \alpha } \\ 0&0&0&0&{\sin \alpha }&{\cos \alpha \cos \beta } \end{array}} \right]$.

      The Jacobi matrix between the velocity of the P joint and output independent parameter velocity of the mechanism is written as

      $J = {J_{qi}}{J_0}$

      (15)

      where J0 is J01 or J02.

      The angular velocity of the kinematic chain i is obtained by (8) which crosses ${w_i}$ on both sides,

      ${\omega _i} = \frac{{{w_i} \times {v_B}_i}}{{{q_i}}} = {J_{wi}}\left[ \begin{array}{l} v \\ \omega \\ \end{array} \right].$

      (16)

      The linear velocity of Bi is written as

      $\begin{split} {v}_{B}{}_{i}=&v+\omega \times (_{\;o}^{O}R\cdot {}^{o}B{}_{i})=\\ &\left[\begin{array}{cc}{I}_{3\times3}& -S({B}_{i})\end{array}\right]\left[\begin{array}{l}v\\ \omega \end{array}\right]={J}_{ri}\left[\begin{array}{l}v\\ \omega \end{array}\right]\end{split}$

      (17)

      where $ {J_{ri}} = \left[ {\begin{array}{*{20}{c}} {{I_{3\times3}}}&{ - S({B_i})} \end{array}} \right]\;. $

      The Jacobi matrix of the angular velocity of the kinematic chain i is written as

      ${J_{wi}} = \frac{{S({w_i})}}{{{q_i}}}{J_{ri}}$

      (18)

      where S(wi) and S(Bi) are anti-symmetric matrices.

      $S({w_i}) = \left[ {\begin{array}{*{20}{c}} 0&{ - {w_{iz}}}&{{w_{iy}}} \\ {{w_{iz}}}&0&{ - {w_{ix}}} \\ { - {w_{iy}}}&{{w_{ix}}}&0 \end{array}} \right]$

      $S({B_i}) = \left[ {\begin{array}{*{20}{c}} 0&{ - {B_{iz}}}&{{B_{iy}}} \\ {{B_{iz}}}&0&{ - {B_{ix}}} \\ { - {B_{iy}}}&{{B_{ix}}}&0 \end{array}} \right].$

      The kinematic chain i includes the oscillating rod and telescopic rod in Fig. 5, all oscillating rod and telescopic rods connect the fixed platform with the moving platform. The mass center of the oscillating rod is MAi, the distance from MAi to Si of fixed platform is l1Si. The mass center of the telescopic rod is MBi, the distance from MBi to the Si of the moving platform is l2Si.

      Figure 5.  Schematic of the chain

      The linear velocity of MAi is written as

      $\begin{split}{v_{MAi}} = {l_{1Si}}{\omega _i} \times {w_i} & = - {l_{1Si}}S({w_i}){J_{wi}} \left[ \begin{array}{l} v \\ \omega \\ \end{array} \right] = {J_{vAi}} \left[ \begin{array}{l} v \\ \omega \\ \end{array} \right] \end{split}$

      (19)

      where ${J_{vAi}} = - {l_{1Si}}S({w_i}){J_{wi}}$.

      The linear velocity of MBi is written as

      ${v_{MBi}} = {\dot w_i}{\dot q_i} + \left( {{l_i} - {l_{2Si}}} \right){\omega _i} \times {w_i} = {J_{vBi}}\left[ \begin{array}{l} v \\ \omega \\ \end{array} \right]$

      (20)

      where

      ${J_{vBi}} = \left[ {\begin{array}{*{20}{c}} {I + \dfrac{{{l_{2Si}}}}{{{q_i}}}S{{({w_i})}^2}}{ - \left(S({B_i}) + \dfrac{{{l_{2Si}}S{{({w_i})}^2}S({B_i})}}{{{q_i}}}\right)} \end{array}} \right]\!\!.$

      Angular velocities of the oscillating rod and telescopic rod are the same and can be written as

      ${\omega _i} = \frac{{{w_i} \times {v_B}_i}}{{{q_i}}} = {J_{wi}}\left[ \begin{array}{l} v \\ \omega \\ \end{array} \right].$

      (21)

      Jacobi matrices of the velocity of the oscillating rod and telescopic rod are written as

      $ {J_{v\omega Ai}} = \left[ {\begin{array}{*{20}{c}} {{J_{vAi}}} \\ {{J_{wi}}} \end{array}} \right],\;\;{J_{v\omega Bi}} = \left[ {\begin{array}{*{20}{c}} {{J_{vBi}}} \\ {{J_{wi}}} \end{array}} \right]. $

    • Equation (5) is differentiated by time, the acceleration of Bi is obtained.

      $ {\dot{v}}_{Bi}=\dot{v}+\dot{\omega }\times (R\cdot {}^{o}B{}_{i})+\omega \times (\omega \times (R\cdot {}^{o}B{}_{i})). $

      (22)

      $\begin{split}\quad\quad{\dot v_{Bi}} =\; &{\ddot q_i}{w_i} + {\dot q_i}{\omega _i} \times {w_i} + {\dot q_i}{\omega _i} \times {w_i} +\\ &{q_i}{\dot \omega _i} \times {w_i} + {q_i}{\omega _i} \times ({\omega _i} \times {w_i}).\end{split}$

      (23)

      Dot product both sides of (23) by ${w_i}$, the angular acceleration of P joint is obtained

      ${\dot \omega _i} = \dfrac{{{w_i} \times {{\dot v}_{Bi}} - 2{{\dot q}_i}{\omega _i}}}{{{q_i}}} = {J_{wi}}{[\begin{array}{*{20}{c}} a&\varepsilon \end{array}]^{\rm{T}}} + {K_{wi}}{[\begin{array}{*{20}{c}} v&\omega \end{array}]^{\rm{T}}}$

      (24)

      where ${K_{wi}} = \dfrac{{S({w_i}){{\dot J}_{ri}} - 2{J_{qi}}{J_{wi}}{{[v\begin{array}{*{20}{c}} {} \end{array}w]}^{\rm{T}}}}}{{{q_i}}}$.

      Equation (8) multiplying ${w_i}$ is additionally written as

      $ {\dot{q}}_{i}={w}_{i}^{{\rm{T}}}\cdot {v}_{Bi}.$

      (25)

      Equation (25) is differentiated by time. The linear acceleration of the P joint is obtained as

      $ {\dot q_i} = w_i^{\rm{T}}{\dot v_{Bi}} + v_{Bi}^{\rm{T}}({\omega _i} \times {w_i}) ={J_{qi}}{[a \quad \varepsilon]^{\rm{T}}} + {K_{li}}{[v\quad \omega ]^{\rm{T}}} $

      (26)

      where ${K}_{li}={w}_{i}^{\rm T} {\times{J}}_{r}-{[{J}_{r}{[v\begin{array}{c}\end{array}w]}^{{\rm{T}}}]}^{{\rm{T}}}\times S({w}_{i}){J}_{wi}$.

      Because ${\dot \omega _i}$ and ${\omega _i}$ are known, linear acceleration of the oscillating rod is additionally written as

      $\begin{split}{\dot v_{MAi}} =\; & {l_{1Si}}{\dot \omega _i} \times {w_i} + {l_{1Si}}{\omega _i} \times ({\omega _i} \times {w_i}) =\\ &{J_{vAi}}{[\begin{array}{*{20}{c}} a&\varepsilon \end{array}]^{\rm{T}}} + {J_{Ai}}{[\begin{array}{*{20}{c}} v&\omega \end{array}]^{\rm{T}}}\end{split}$

      (27)

      where ${J_{Ai}} \!=\! - {l_{1Si}}S({w_i}){K_{wi}} \!-\! {l_{1Si}}{w_i} \times {({J_{wi}}{[\begin{array}{*{20}{c}} v&w \end{array}]^{\rm{T}}})^{\rm{T}}}{J_{wi}}$.

      Linear acceleration of the telescopic rod is additionally written as

      $\begin{split} & {{\dot v}_{MBi}} = {{\ddot q}_i}{w_i} + 2\dot q({\omega _i} \times {w_i}) + ({l_i} - {l_{2Si}}){{\dot \omega }_i} \times {w_i} + \\ &\;\;\;\; ({l_i} - {l_{2Si}}){\omega _i} \times ({\omega _i} \times {w_i}) = {J_{vBi}}[\begin{array}{*{20}{c}} a& \varepsilon \end{array}]+{J_{Bi}}[\begin{array}{*{20}{c}} v&w \end{array}] \end{split} $

      (28)

      where

      $ \begin{split}{J}_{Bi}=&{w}_{i}{K}_{li}-2{J}_{qi}[\begin{array}{cc}v& w\end{array}]S({w}_{i}) {J}_{wi}+\\ &({l}_{2Si}-l)[S({w}_{i}){K}_{wi}+{w}_{i}{(}{{J}_{wi}}[\begin{array}{cc}v& w\end{array}]^{\rm{T}})^{\rm{T}}{J}_{wi}].\end{split}$

    • The gravity of the kinematic chain i is Gmi, the inertia force and inertia moment of the kinematic chain i are fmi and Tmi, respectively. Gravities of the fixed platform and moving platform are GO and Go, respectively. External force and the external torque of the moving platform are 0, respectively. The masses of the kinematic chain, the fixed platform and the moving platform are mmi, mO and mo, respectively.

      $\left\{ \begin{aligned} &{G_{mi}} = {\rm{ }}{m_{mi}}g\\ &{f_{mi}} = - {m_{mi}}{a_{mi}}\\ &{G_O} = {m_O}g\\ &{G_o} = {m_o}g. \end{aligned} \right.$

      (29)

      The inertial force and inertial moment of the moving platform in O-XYZ can be written as

      $\left[ {\begin{array}{*{20}{c}} {{f_o}} \\ {{T_o}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {m_o}a} \\ { - {}^O{I_o}\varepsilon - \omega \times ({}^O{I_o}\omega )} \end{array}} \right]$

      (30)

      where OIo is the inertia matrix of the moving platform in O-XYZ.

      The inertial force and inertial moment of the oscillating rod of the kinematic chain i in O-XYZ can be written as

      $\left[ {\begin{array}{*{20}{c}} {{f_{MAi}}} \\ {{T_{MAi}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {m_{MAi}}a} \\ { - {}^O{I_{MAi}}\varepsilon - \omega \times ({}^O{I_{MAi}}\omega )} \end{array}} \right]$

      (31)

      where OIMAi is the inertia matrix of the oscillating rod in O-XYZ, mMAi is mass of the oscillating rod of the kinematic chain i in O-XYZ.

      The inertial force and the inertial moment of the telescopic rod of the kinematic chain i in O-XYZ can be written as

      $\left[ {\begin{array}{*{20}{c}} {{f_{MBi}}} \\ {{T_{MBi}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {m_{MBi}}a} \\ { - {}^O{I_{MBi}}\varepsilon - \omega \times ({}^O{I_{MBi}}\omega )} \end{array}} \right]$

      (32)

      where OIMBi is the inertia matrix of the telescopic rod in O-XYZ, mMBi is mass of the telescopic rod of the kinematic chain i in O-XYZ.

      OIMBi, OIMAi and OIo can be written as

      $\left\{ \begin{aligned} & {}^O{I_o} = {}_{\;o}^OR{}^o{I_o}{({}_o^OR)^{\rm{T}}} \\ & {}^O{I_{MAi}} = {}^O{R_i}{}^i{I_{Ai}}{({}^O{R_i})^{\rm{T}}} \\ & {}^O{I_{MBi}} = {}^O{R_i}{}^i{I_{Bi}}{({}^O{R_i})^{\rm{T}}} \end{aligned} \right.$

      (33)

      where oIo is the inertial matrix of the moving platform in o-uvw, ORi is the inertial matrix of the kinematic chain i in O-XYZ, iIAi is the inertial matrix of the telescopic rod in the chain local coordinate system and iIBi is the inertial matrix of the oscillating rod in the chain local coordinate system.

    • When the mechanism is under no-load, the force and moment of the moving platform in O-XYZ are written as

      ${F_o} = \left[ {\begin{array}{*{20}{c}} {{G_o} - {f_o}} \\ {{T_o}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{m_o}g - {m_o}a} \\ { - {}^O{I_o}\varepsilon - \omega \times ({}^O{I_o}\omega )} \end{array}} \right].$

      (34)

      The force and moment of the telescopic rod of the kinematic chain i in O-XYZ are written as

      $\begin{split} {F}_{MBi}=&\left[\begin{array}{l}{G}_{mBi}+{f}_{MBi}\\ {T}_{MBi}\end{array}\right]=\\ &\left[\begin{array}{c}{m}_{MBi}g-{m}_{MBi}{a}_{B}\\ -{}^{O}I{}_{MBi}{\varepsilon }_{B}-{\omega }_{i}\times ({}^{O}I{}_{MBi}{\omega }_{i})\end{array}\right].\end{split}$

      (35)

      The force and moment of the oscillating rod of the kinematic chain i in O-XYZ are written as

      $\begin{split} {F}_{MAi}=&\left[\begin{array}{l}{G}_{mAi}+{f}_{MAi}\\ {T}_{MAi}\end{array}\right]=\\ &\left[\begin{array}{c}{m}_{MAi}g-{m}_{MAi}{a}_{A}\\ -{}^{O}I{}_{MAi}{\epsilon }_{A}-{\omega }_{i}\times ({}^{O}I{}_{MAi}{\omega }_{i})\end{array}\right].\end{split}$

      (36)

      The dynamic equilibrium equation of the mechanism based on the principle of virtual work is obtained under ideal state.

      ${v_i}^{\rm{T}}F + v_s^{\rm{T}}{F_o} + \sum\limits_i {(v_{MAi}^{\rm{T}}} {F_{MAi}} + v_{MBi}^{\rm{T}}{F_{MBi}}) = 0$

      (37)

      where F is actuated force, vi=Jvs=JqJovs, vMAi=JvωAivs, vMBi=JvωBivs.

      Equation (37) can also be written as

      $\tag{37a}(J_q^{\rm{T}}F + {F_o} + \sum\limits_i {J_{v\omega A}^{\rm{T}}} {F_{MAi}} + \sum\limits_i {J_{v\omega B}^{\rm{T}}} {F_{MBi}}) = 0$

      $\begin{split} \quad\quad F = & - {[J_o^{\rm{T}}J_q^{\rm{T}}]^ + }J_o^{\rm{T}}\times\\ &\left({F_o} + \sum\limits_i {J_{v\omega A}^{\rm{T}}} {F_{MAi}} + \sum\limits_i {J_{v\omega B}^{\rm{T}}} {F_{MBi}}\right).\end{split}\tag{37b}$

      Equation (37b) can also be written as

      $\tag{37c}F = M\ddot q + C\dot q + G - {[J_o^{\rm{T}}J_q^{\rm{T}}]^ + }J_o^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{F_o}} \\ {{T_o}} \end{array}} \right]$

      where M is the inertia matrix, G is the gravity matrix, C is the velocity matrix.

      $\begin{array}{l} M = {[J_o^{\rm{T}}J_q^{\rm{T}}]^ + }J_o^{\rm{T}}\left\{ {\left[ {\begin{array}{*{20}{c}} {{m_o}{J_v}}&o \\ o&{{I_o}{J_\omega }} \end{array}} \right] + } \right. \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_i {J_{v\omega Ai}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{MAi}}{J_{vAi}}}&o \\ o&{{I_{MAi}}{J_{wi}}} \end{array}} \right]} + \\ \left. {\begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_i {J_{v\omega Bi}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{MBi}}{J_{vBi}}}&o \\ o&{{I_{MBi}}{J_{wi}}} \end{array}} \right]} } \right\} \\ C={[}{{J}_{o}^{{\rm{T}}}}{J}_{o}^{{\rm{T}}}\Bigg\{\left[\begin{array}{l}0\\ \omega \times I\omega \end{array}\right]\!+\!{ \displaystyle\sum\limits _{i}{J}_{v\omega A}^{{\rm{T}}}}\left[\begin{array}{l}0\\ {\omega }_{i}\times ({I}_{MAi}{\omega }_{i})\end{array}\right]\!+\!\\ \begin{array}{cc}& \end{array}{ \displaystyle\sum\limits _{i}{J}_{v\omega B}^{{\rm{T}}}}\left[\begin{array}{l}0\\ {\omega }_{i}\times ({I}_{MBi}{\omega }_{i})\end{array}\right]\Bigg\}\\ G = - {[J_o^{\rm{T}}J_q^{\rm{T}}]^ + }J_o^{\rm{T}}\left\{ {\left[ \begin{array}{l} {m_o}g \\ 0 \\ \end{array} \right] + \displaystyle\sum\limits_i {J_{v\omega A}^{\rm{T}}} \left[ \begin{array}{l} {m_{MAi}}g \\ 0 \\ \end{array} \right] + } \right. \\ \left. {\begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_i {J_{v\omega B}^{\rm{T}}} \left[ \begin{array}{l} {m_{MBi}}g \\ 0 \\ \end{array} \right]} \right\}. \end{array} $

    • The structural parameters of the redundantly actuated parallel mechanism are given in Table 5. The motion trajectory of o is circular motion in the XY plane. The motion trajectory equation of o is given as

      $\left\{ {\begin{array}{*{20}{c}} {X = 0.05\cos (2t)}\;\;\; \\ \begin{array}{l} Y = - 0.05\sin (2t) \\ Z = 1.2\;{\rm m} \\ \alpha = {0^\circ } \\ \beta = {0^\circ } . \end{array} \end{array}} \right.$

      (38)

      The actuated force of the six actuated joints of the mechanism under a no-load operation can be obtained. Fig. 6 shows the theoretical result of the actuated force. The change rule of the six actuated forces of six kinematic chains is sine or cosine. The actuated forces of the Chain 1 and Chain 2 are bigger the other four chains. F3 and F6, F1 and F2, F4 and F5 are equal in size when the mechanism is in the initial position, respectively. F3 and F4, F1 and F2, F5 and F6 are symmetrical and differ by one cycle, respectively. Fig. 7 shows the simulation result of the actuated force. Theoretical results by Matlab are basically consistent with simulation results by ADAMS. The dynamic mathematical model of the 4SPS-(2UPR)R parallel mechanism is shown to be correct by Matlab and ADAMS under no-load conditions. So the simulation results and theoretical results are consistent with the described mechanism.

      Figure 6.  Theoretical value of the mechanism. Colored figures are available in the online version.

      Figure 7.  Simulation value of the mechanism

    • The dynamic acceleration performance index can evaluate the acceleration and deceleration characteristics of the mechanism. In (37c), the inertia factor mainly affects the acceleration and deceleration characteristics, the latter three terms are not considered. Equation (37c) is simplistically obtained as

      $F \approx M \ddot q$

      (39)

      where

      $\begin{split}M =& {[J_o^{\rm{T}}J_q^{\rm{T}}]^ + }\left\{ {J_o^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_o}{{E}}}&o \\ o&{{I_o}} \end{array}} \right]{J_o} + } \right.\\ &\begin{array}{*{20}{c}} {\displaystyle\sum\limits_i {J_o^{\rm{T}}J_{v\omega Ai}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{MAi}}{{E}}}&o \\ o&{{I_{MAi}}} \end{array}} \right]{J_{v\omega Ai}}{J_o}} } + \end{array}\\ &\left.{\displaystyle\sum\limits_i {J_o^{\rm{T}}J_{v\omega Bi}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{MBi}}{{E}}}&o \\ o&{{I_{MBi}}} \end{array}} \right]} {J_{v\omega Bi}}{J_o}} \right\}\end{split}$

      Equation (39) can also be written as

      $\ddot q \approx {[M]^ + }F$

      (40)

      where [M]+ is the generalized pseudo-inverse matrix of M.

      At present, the acceleration performance evaluation index can take advantage of the harmonic average harmonic mean (HMIM)[31] as the dynamic performance evaluation index in (41). The value of HMIM will be larger when both the maximum singular value and the minimum singular value are larger. To ensure that the mechanism has good isotropy and achieve good acceleration performance, the value of HMIM of the mechanism should be getting bigger and be expressed as

      ${\eta _{HMIM}} = \dfrac{2}{{\dfrac{1}{{{\sigma _{\max }}}} + \dfrac{1}{{{\sigma _{\min }}}}}}$

      (41)

      where σmax and σmin are the maximum and minimum singular values of [M]+, respectively.

      Because the mechanism has translation and rotation, its dimension is not uniform, translational acceleration performance and rotational acceleration performance should be analyzed, respectively.

      Fig. 8 shows the translational acceleration performance of the mechanism from different views. The translational acceleration performance of the mechanism is approximately symmetric along the X-axis and Y-axis, respectively. The mechanism has the best translational acceleration performance in Chain 1 or Chain 2 local vicinity, respectively.

      Figure 8.  Different angles of translational acceleration perfor-mance

      Fig. 9 shows the rotational acceleration performance of the mechanism from different views. The mechanism has the best rotational acceleration performance in the middle of the Chain 1 and Chain 2. Compared with [31], the numerical results of HMIM are relatively large in Figs. 8 and 9. The results show that the mechanism has good acceleration performance.

      Figure 9.  Different angles of rotational acceleration performance

    • Because the mechanism has translation and rotation, its dimension is not uniform. The energy transfer efficiency[32] can be used to avoid dimensional inconsistency of the mechanism and evaluate dynamic performance.

      The kinetic energy of the moving platform is expressed as

      ${E_o} = \frac{1}{2}{m_o}{v^2} + \frac{1}{2}{I_o}{\omega ^2} = \frac{1}{2}{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{m_o}{{E}}}&0 \\ 0&{{I_o}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right].$

      (42)

      The oscillating rod only has rotational motion, and the kinetic energy of the oscillating rod i is expressed as

      ${E_{Ai}} = \frac{1}{2}{I_i}\omega _i^2 = \frac{1}{2}{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]^{\rm{T}}}J_{v\omega A}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} o&o \\ o&{{I_{MAi}}} \end{array}} \right]{J_{v\omega A}}\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right].$

      (43)

      Each telescopic rod has both translation and rotation motion, and the kinetic energy of the telescopic rod i is expressed as

      $ \begin{split} {E_{Bi}} = &\dfrac{1}{2}{m_{Bi}}{v_{mBi}}^2 + \dfrac{1}{2}{I_i}{\omega _i}^2 = \\ &\dfrac{1}{2}{\left[ {\begin{array}{*{20}{c}} {{v_{mBi}}}\\ {{\omega _i}} \end{array}} \right]^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{m_{Bi}}{{E}}}&0\\ 0&{{I_i}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{v_{mBi}}}\\ {{\omega _i}} \end{array}} \right]=\\ \begin{array}{*{20}{c}} {}&{} \end{array} & \dfrac{1}{2}{\left[ {\begin{array}{*{20}{c}} v\\ \omega \end{array}} \right]^{\rm{T}}}J_{v\omega B}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{Bi}}E}&o\\ o&{{I_{MBi}}} \end{array}} \right]{J_{v\omega B}}\left[ {\begin{array}{*{20}{c}} v\\ \omega \end{array}} \right]. \end{split} $

      (44)

      The total kinetic energy of the mechanism is expressed as

      $ {E_{all}} = {E_o} + {E_{Ai}} + {E_{Bi}}. $

      (45)

      The energy proportion of the moving platform in the mechanism is expressed as

      ${\eta _{KE}} = \frac{{{E_o}}}{{{E_{all}}}} \times 100\%. $

      (46)

      The kinetic energy of the moving platform is considered as the effective energy of the mechanism, ηKE represents efficiency of the mechanism

      Combined (42)–(46), ηKE is expressed as (47).

      Fig. 10 shows the energy transfer efficiency of the mechanism from different views. The energy transfer efficiency of the mechanism is approximately symmetric along the X-axis and Y-axis, respectively. The mechanism has high energy transfer efficiency. The energy transfer efficiency of the mechanism is the best in the initial central position. Compared with [33], the numerical results of ηKE are relatively large in Fig. 10. The results show that the mechanism has good energy transfer efficiency.

      Figure 10.  Different angles of energy transfer efficiency

    • This paper focuses on type synthesis and dynamics performance evaluation of a family of 5-DOF redundantly actuated parallel mechanisms, which are suitable for the parallel machining heads of a machine tool. This paper mainly includes three aspects: 1) This paper presents a 5-DOF redundantly actuated parallel mechanism on the basis of Lie group theory and a configuration evolution method for the parallel machining head of the machine tool. This method is more intuitive and effective when the 5-DOF redundantly actuated parallel mechanism is designed. 2) A 4SPS-(2UPR)R mechanism suitable for the parallel machining head is selected. The kinematics and dynamics models of the parallel mechanism are established and verified to be correct by Matlab and ADAMS under no-load conditions. 3) By analyzing the translational acceleration, rotational acceleration performance and energy transfer efficiency of the mechanism, we find that 4SPS-(2UPR)R redundantly actuated parallel mechanism has not only good translational acceleration and rotational acceleration performance but also good energy transfer efficiency. This paper provides a practical parallel mechanism for the machining head of the machine tool, which will provide a foundation for control of the parallel mechanism.

      ${\begin{split}\\{}{\eta _{KE}} = \dfrac{{\dfrac{1}{2}\Bigg({{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]}^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{m_o}}&0 \\ 0&{{I_o}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]\Bigg)}}{{\dfrac{1}{2}{{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]}^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{m_o}}&0 \\ 0&{{I_o}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right] + \dfrac{1}{2}{{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]}^{\rm{T}}}J_{v\omega A}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&{{I_o}} \end{array}} \right]J_{v\omega A}^{}\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right] + \dfrac{1}{2}{{\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]}^{\rm{T}}}J_{v\omega B}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} {{m_{Bi}}E}&0 \\ 0&{{I_{MBi}}} \end{array}} \right]J_{v\omega A}^{}\left[ {\begin{array}{*{20}{c}} v \\ \omega \end{array}} \right]}} \times 100\%. \end{split}}$

      (47)
    • This work was supported by the Fundamental Research Funds for the Central Universities (No. 2018JBZ007).

Reference (33)

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