RESEARCH ARTICLE

Conceptual design and kinematic analysis of a novel parallel robot for high-speed pick-and-place operations

  • Qizhi MENG 1 ,
  • Fugui XIE , 1,2 ,
  • Xin-Jun LIU , 1,2
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  • 1. The State Key Laboratory of Tribology & Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
  • 2. Beijing Key Laboratory of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, China

Received date: 29 Jan 2017

Accepted date: 13 Jun 2017

Published date: 16 Mar 2018

Copyright

2018 The Author(s) 2017. This article is published with open access at link.springer.com and journal.hep.com.cn

Abstract

This paper deals with the conceptual design, kinematic analysis and workspace identification of a novel four degrees-of-freedom (DOFs) high-speed spatial parallel robot for pick-and-place operations. The proposed spatial parallel robot consists of a base, four arms and a 1½ mobile platform. The mobile platform is a major innovation that avoids output singularity and offers the advantages of both single and double platforms. To investigate the characteristics of the robot’s DOFs, a line graph method based on Grassmann line geometry is adopted in mobility analysis. In addition, the inverse kinematics is derived, and the constraint conditions to identify the correct solution are also provided. On the basis of the proposed concept, the workspace of the robot is identified using a set of presupposed parameters by taking input and output transmission index as the performance evaluation criteria.

Cite this article

Qizhi MENG , Fugui XIE , Xin-Jun LIU . Conceptual design and kinematic analysis of a novel parallel robot for high-speed pick-and-place operations[J]. Frontiers of Mechanical Engineering, 2018 , 13(2) : 211 -224 . DOI: 10.1007/s11465-018-0471-4

Introduction

As indispensable equipment in advanced manufacturing systems, industrial robots have become an important symbol to measure the level of a country’s manufacturing industry. Due to the increasing demand for cost efficiency, such robots have been widely used to implement high-speed manipulations of sorting and packaging in the food, pharmaceuticals, and electronics industries. As the most representative and most significant application category, pick-and-place operations have received extensive research attention. In fact, for most pick-and-place operations, at least four degrees-of-freedom (DOFs) are required, including three translations (3T) to move the object from one point to another and one rotation (1R) to adjust the posture in its final location [1]. This kind of DOFs is usually called SCARA motion [24] or Schönflies motion [57]. The motions are generally characterized by high flexibility and stiffness respectively in the horizontal and vertical directions.
In the production lines, the operational objectives are generally in small size and light weight. The first robot for pick-and-place operations was realized by serial kinematic mechanisms (SKMs), whose drive units were placed within the joints. This placement leads to a large volume and high inertia force. On the contrary, parallel kinematic mechanisms (PKMs) are advantageous over serial structures due to their high rigidity, high precision and better dynamic characteristics [8,9]. In relation to this, the most famous Delta robot [10,11] with only 3T for parallel architecture has been proposed by Clavel [12]. This Delta robot, a milestone for high-speed parallel robots, has become a real commercial success with a large workspace and compact structure. Based on the configuration of Delta, a passive RUPUR chain has been adopted to realize the rotary output. This scheme has a wide range of applications, such as the ABB IRB340 FlexPicker robot [13] and the Bosch SIGPack Systems XR22 robot. However, the serially added device may become a weak point and may lead to the lack of stiffness and short service life [1416]. To avoid this disadvantage, Pierrot [1,1720] proposed a new family of 4-DOF, double-platform parallel robots called H4. The rotary output of H4 is realized by the relative movement of the two platforms. Notably, the double-platform structure may lead to structural complexity and difficulty in manufacturing. Therefore, in this work, we present a parallel robot with concise and compact structure as well as good performance.
As shown in Fig. 1, the double-platform structure improved from H4, which has been adopted by Par4 [16], demonstrates good rotational performance; however, it also leads to a complex structure, increased difficulty in manufacturing, and higher accessory cost. Meanwhile, the single platform adopted by X4 [21,22], contributes a simple structure and no amplification mechanism, but has worse performance when the platform is close to the maximum output angle. Therefore, developing a 1½ mobile platform that carries the merits of both single and double platforms provides an entry point for new parallel robots. The design and development of a high-speed parallel robot with simple structure and high performance is still a hot issue in both the academic and industrial fields.
Fig.1 CAD models of different mobile platforms. (a) Double-platform structure of H4; (b) single-platform structure of X4

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The remainder of this paper is organized into sections. The conceptual design of the proposed parallel robot is introduced in Section 2. The mobility analysis is carried out in Section 3. The inverse kinematics is investigated in Section 4, and the workspaces are identified in Section 5. Finally, Section 6 concludes the paper.

Conceptual design

The architecture of the proposed parallel robot, shown in Fig. 2(a), consists of a base (Part 1), four limbs (I, II, III, and IV), and a 1 ½ mobile platform (Assembly 8) with an end effector. The four limbs are connected to the base and mobile platform, and collectively, these components make up the spatial closed-loop mechanism. As presented in Fig. 2(b), the layouts of the four arms are symmetrically arranged in a circular manner.
Fig.2 Architecture of the proposed parallel robot. (a) Conceptual model; (b) kinematic model

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The four limbs have two kinds of structures and can be summarized as follows. Limbs I and IV have the same structure, represented by R-R-(Pa)-R, where R represents an active revolute joint, R indicates the revolute joint, and Pa denotes the planar parallelogram mechanism consisting of four revolute joints connected end to end. Compared with Limbs I and IV, Limbs II and III have an additional R and can be expressed as R-R-(Pa)-R-R. For better understanding, the joint-and-loop graph for the basic principle is illustrated in Fig. 3, where the lines and boxes represent the parts and the joints, respectively.
Fig.3 The joint-and-loop graph for basic principle

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Figure 4(a) shows the structure of the 1½ mobile platform, which consists of an upper platform, a lower platform, and an end effector; the upper and lower platforms are connected by a revolute Joint R. The meaning of the “1½ mobile platform” comes from the total contributions of platforms to output motions. As an auxiliary output platform, the lower platform generates incomplete output motions (3T). The upper platform providing the complete output motions (3T1R) is defined as the final output platform; therefore, they are treated as ½ and 1 mobile platforms, respectively, and together constitute a generalized mobile platform.
Fig.4 The architecture of the mobile platform. (a) Conceptual model, (b) kinematic model

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An equivalent kinematic model of the proposed parallel robot is developed, as shown in Fig. 2(b). The active and passive arms in each limb are represented by AiBi and BiCi, respectively. The junction points between the base and active arms are denoted as A i, whereas the central points of the passive revolute joints between the active and passive arms are denoted as Bi. The connective points between the passive arms and mobile platform are denoted as Ci. A fixed global reference frame :O-XYZ is located at the central point of the base. The X- and Z-axes are coincident with A3A1 andA4A2, respectively. Another reference frame, called the moving frame : O- XY Z, is located at the central point of the lower platform. The kinematic model in the initial position of the mobile platform is shown in Fig. 4(b), where C1C2C3C4 is considered as a rectangle (i.e., D2 OD 390°) to avoid singularity.

Mobility analysis

This section considers a line-graph method [23,24] based on Grassmann line geometry [25,26] to analyze the mobility of the presented spatial parallel mechanism. The principle for the articulated mobile platform to generate the 3T1R motions are described in detail.

Line-graph method

According to Grassmann line geometry, spatial line clusters can be divided into five types from 1 to 5, corresponding to their respective dimensions. In addition, an N-dimensional spatial line cluster, which consists of vector or couple in two different colors, represents a freedom space or constraint space for a mechanism. To depict the spatial line cluster, four basic elements with different mathematic and physical meanings are generated in Table 1. The manner of adopting line graphs to describe mobility is generally referred to as line-graph method.
Tab.1 Graphic expressions of the basic elements and their meanings
Graphic expressions Mathematical meanings Physical meanings Screw representation
Fig.5

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Vector Rotational motion (ω;r×ω)
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Vector Constraint force (f; r× f)
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Couple Translational motion (0;v)
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Couple Constraint couple (0;τ)

Basic rules

We can extract the motion line graphs of each limb from the architecture of the presented mechanism. Unlike the traditional SKM, the motions of the end effector of PKM cannot be obtained by the simple superposition of the motions of each limb. In other words, the motion line graphs of the mobile platform cannot be directly accessed from the motion line graphs of the branched chains.
Furthermore, the branched chain provides all the necessary constraints for the mobile platform, and these constraints are superimposed. Therefore, finding a way to establish an equivalent conversion between motion and constraint line graphs is quite indispensable. Based on the above analysis, a generalized Blanding rule [27], which states that there is no power between the constraint forces and the allowable motions as a foundation, is introduced and illustrated in Fig. 5. This is further summarized below.
Basic rule A. Every vector of the motion/constraint line graphs intersects with all vectors of the corresponding constraint/motion line graphs (in projective theory, parallel lines are seen as a special intersection). Hence, (ω; r× ω) (f; r× f)=0, but only when f and ω intersect at one point, where the symbol represents reciprocal product between two screws in the screw theory.
Basic ruleB. Every vector of the motion/constraint line graphs is orthogonal to all couples of the corresponding constraint/motion line graphs. Hence, (f; r× f)( 0;v)=0 or ( 0;τ) (ω; r× ω)=0, but only when f and v or τand ω are perpendicular to each other.
Basic ruleC. The directions between the couple of motion/constraint line graphs and the corresponding constraint/motion line graphs are arbitrary. Obviously, (0;τ) (0; v)=0.
Fig.9 Graphical representations of the basic rules

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Analysis results

Four steps are necessary to investigate the mobility of the mechanism. First, the motion line graph of each limb is extracted. Second, the corresponding constraint line graph of each limb are derived using the basic rules. Third, all the constraint line graphs of the four limbs to the mobile platform are added. Finally, the corresponding motion line graph of the mobile platform are derived.
Four limbs can be classified into two categories according to their structural characteristics. One group embraces Limbs I and IV, and the other group includes Limbs II and III. The mobility analysis of the proposed mechanism can be carried out based on the four steps described below.
1) Limb I is taken as a case study because of the same configuration of Limbs I and IV. As shown in Fig. 6(a), the motion line graphs consist of three rotations ( ω I-1,ωI-2, ωI-3) and a translation (vI 1) located in the plane of the parallelogram and orthogonal to B1C1. Therefore, the motion line graph has four dimensions, and the corresponding constraint line graph must be 2D. According to basic rule B, every couple in the constraint line graph should be orthogonal to all the vectors ( ω I-1,ωI-2,ωI-3) of the motion line graph. In this way, τI- 1 and τI-2, which lie at Point C1 and are parallel to the X- and Z-axes, respectively, can be derived. According to basic rules A and B, a vector in the constraint line graph should intersect all the vectors (ωI- 1, ω I-2,ωI-3) and must be orthogonal to all the couples ( vI1) of the motion line graph; hence, no vector ( f I1) can be found. Together, the constraint line graph of Limb I is a 2D couple constraint (τI-1 and τI-2).
Fig.10 Motion and constraint line graphs for two groups of limbs. (a) Limbs I and IV; (b) Limbs II and III

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2) Limb II is taken as a case study because of the same configuration of limbs II and III. As shown in Fig. 6(b), the motion line graph consists of four rotations ( ω II-1,ωII-2,ω II-3,ω II-4) located at Points A2,B2,C2, andD 2, respectively, and a translation ( vII-1) lying on the plane of the parallelogram and is orthogonal to B2C2. Therefore, the motion line graph has five dimensions, and the corresponding constraint line graph must be 1D. According to basic rule B, every couple in the constraint line graph should be orthogonal to all the vectors ( ω II-1,ωII-2, ωII-3, ωII-4) of the motion line graph, and τII-1 must lie at Point D2 and is parallel to the Y-axis. According to basic rules A and B, a vector in the constraint line graph should be intersected to all the vectors (ωII-1, ωII-2, ωII-3, ωII-4) and orthogonal to all the couples (vII-1) of the motion line graph; hence, no vector (fII-1) can be found. Together, the constraint line graph of Limb II is a 1D couple constraint (τII-1).
3) For the 1½ mobile platform, on the one hand, the constraints imposed on the lower platform can be obtained through the superposition of the constraints of Limb I ( τ I-1 and τI-2), Limb IV (τIV-1 and τIV-2), and the upper platform ( τ U-1 and τU-2, which resulted from τII-1 and τIII-3 provided by Limbs II and III). The constraint line graph of the lower platform can be derived by processes ①, ②, and ③ shown in Fig. 7. The six couple constraints ( τ I-1, τI-2,τIV-1, τIV-2,τ U-1, τU-2) together construct a 3D couple constraint (equivalent to τL-1, τL-2, and τL-3). Therefore, three DOFs with 3T motions of the lower platform are identified. On the other hand, the constraints imposed on the upper platform can be obtained through the superposition of the constraints of Limb II (τII-1), Limb III ( τIII-1), and the lower platform ( τ L-1, τL-2, and τL-3, which resulted from τ I-1, τI-2,τIV-1 and τIV-2 provided by Limbs I and IV). The constraint line graph of the upper platform can be derived by processes ②, ①, and ④ shown in Fig. 7. Because of the revolute joint between the lower and upper platforms, the five couple constraints ( τ II-1,τ III-1, τL-1, τL-2, τL-3) together construct a 2D couple constraint (equivalent to τU-1 and τU-2). Therefore, four DOFs with 3T and 1R motions of the upper platform are clarified.
Fig.11 Constraint line graphs for the lower and upper platforms

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Above all, the output motion spaces of the presented mechanism are 3T and 1R, which can be represented by a 4D atlas. In addition, the constraint and motion spaces of different robot configurations are summarized in Table 2. For better understanding, the CAD model with 3T and 1R motions is shown in Fig. 8.
Tab.2 Constraint and motion spaces of different configurations to the robot
Configurations Constraint line graphs Motion line graphs
Fig.12

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Fig.17

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Fig.27

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Fig.28

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Fig.29

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Fig.30 Three translational motions and one rotational motion for the proposed parallel robot. (a) 3T motions; (b) 1R motion

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Inverse kinematic investigation

The geometric parameters of the PKM presented in Fig. 2 are defined as follows: OAi =R1, OCi=R2, AiBi=L1, BiCi=L2. The distance between o and C 2(C4) along the Z-axis direction is denoted by L3. The rotational input of the four arms and output of the mobile platform are represented by αi (i= 1,2,3,4) and θ, respectively.
As shown in Fig. 2(b), the coordinates of Ai (i= 1,2,3,4) under the global frame can be respectively expressed by
{A1= [ R1, 0,0] TA2= [0, R1,0] TA3= [R1,0,0] TA4= [0, R 1,0]T,
The coordinates of B i( i=1,2,3,4)can be respectively derived as
{B1= [ R1L1cos α1, 0,L1sinα1]T B2= [0, R1 L 1cos α 2, L1sinα2]T B3= [R1+ L1cos α3, 0,L1 sinα 3] TB4= [0, R 1+L 1cosα 4, L1sin α4 ]T.
Under the local frame , the coordinates of C i( i=1,2,3,4) can be respectively expressed by
{ C1= [ R2cosϕ , R2sinϕ ,0]T C2= [R2sin (θ+ ϕ) ,R2cos(θ+ϕ),0] TC3=[ R2 cos( θϕ),R2 sin(θϕ), 0 ]T C4 =[ R2sinϕ,R2cosϕ,0]T.
In the coordinates above, ϕ= 90° ( 180° D2O D3)2= 15°, and the displacement matrix can be expressed as
T=[x,y,z]T.
Based on the vector equation, we arrive at
Ci =C i+T,
where the coordinates of Ci( i=1,2,3,4)under the global frame can be respectively derived as
{C1= [x+R 2cos ϕ, yR2 sinϕ,z]T C2 =[ xR2 sin(θ+ϕ), y+R2 cos( θ+ϕ ), z]T C3 =[ xR2 cos( θϕ),y R2sin(θϕ),z]T C4 =[ x+R2 sinϕ, yR2cosϕ,z]T.
Considering the constraints of BiCi=L 2 (i =1,2,3,4 ), the inverse kinematics can be obtained according to Eqs. (2) and (6).
1) The inverse kinematic solution of Limb I is given by
αI-1 =cos 1vI + or αI-2 = 2π cos1 vI+orαI-3 = cos 1vIor α I-4 =2π cos1 vI,
where
vI ±=p11 p12±4L12z1 2(z12+p122) p112z122 L1(z12+p122),
p11=L2 2L12(R 1x1)2y1 2z12,
p12=R1x1.
2) The inverse kinematic solution of Limb II is expressed as
αII-1 =cos 1vII+ or α II-2 =2 πcos 1v II+ or αII-3 =cos 1vIIorα II-4 =2π cos1 vII,
where
vII±= p21p22± 4L12z22(z22+p222) p212z222 L1(z22+p222),
p21=L22L12x22(R1y2)2z22,
p22=( R1y2).
3) The inverse kinematic solution of Limb III is given by
αIII-1 =cos 1vIII+or αIII-2 = 2π cos1 vIII+or α III-3 =cos1v IIIor αIII-4 =2πcos 1vIII,
where
vIII±= p31p32± 4L12z32(z32+p322)p 312z322L1(z32+ p322),
p31=L22L12(R1+x3)2y32z32,
p32=R1+x3.
4) The inverse kinematic solution of Limb IV is expressed as
αIV-1 =cos 1vIV+ or αIV-2 =2πcos 1vIV+ or αIV-3 =cos 1vIVorαIV-4 =2 πcos 1v IV,
where
vIV±= p41p42± 4L12z42(z42+p422) p412z422 L1(z42+p422),
p41=L22L12x42(R1+y4)2z42,
p42=( R1+y4) .
By observing Eqs. (7)–(22), we can find four possible solutions for each rotational input, but only one solution is correct. Therefore, identifying the accurate solution from the possible solutions is necessary at this point. To eliminate the solution corresponding to the given configuration, two constraint conditions combined with Fig. 9 are presented below.
Fig.31 Inverse solution identification process (where i = I, II, III, IV).

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① The restricted conditions of the configuration are given by
αimin<αi<αi max, i= 1,2,3,4.
For Limb I
{α 1min= π2 tan1 (x1R1 z) α1max =2tan1( x1R1z),
For Limb II
{α 2min= π2 tan1 (y2R1 z) α2max =2tan1( y2R1z),
For Limb III
{α 3min= π2+ tan1 (x3+R1 z) α3max =2+tan1( x3+R1z),
For Limb IV
{α 4min= π2+ tan1 (y4+R1 z) α4max =2+tan1( y4+R1z).
② The fixed length of the BiCi bars can be solved using
|BiCi |=L2, i=1,2,3,4.
According to Eqs. (23)–(28), the inverse kinematics of this mechanism can be uniquely identified.

Workspace identification

In this section, we investigate the workspace of the proposed parallel robot. The constraint conditions are both the physical dimensions and the motion/force transmission index. As shown in Fig. 10, the accelerator can be added as an attachment between the lower and upper platforms to extend the rotary output. A realization of the accelerator can be achieved by gear trains. Especially, the planetary gear trains are chosen because of their high accuracy and compact structure. The simple internal principle of a typical planetary gear train is shown in Fig. 10(b), where the axles O1 and OH are the input and output ends, respectively. The central Gear 3 is fixed with the lower platform, whereas the central Gear 1 is connected with the upper platform. The transmission ratio between axle O1 and OH can be represented by
n1-H =1+ Z3Z1,
where Z3and Z1 denote the tooth number of central Gears 3 and 1, respectively. If we set 2 as the transmission ratio of the accelerator, the workspace with a rotational capability of ±45°should be reached, and this ability is required in most cases.
Fig.32 The architecture of the mobile platform with accelerator. (a) Conceptual model; (b) internal principle of accelerator

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Performance evaluation index

The four active rotational inputs are respectively expressed as
SITS1=( 0,1,0;0,0,R1)SITS 2= ( 1, 0 ,0;0,0,R1)SITS 3= (0,1, 0;0, 0,R1)SITS 4= (1,0,0;0,0,R1).
The transmission wrench screw of the ith (i=1, 2,3,4) limb is derived using
STWSi = (BiCi;OBi×BiCi)| BiCi|,i= 1,2,3,4.
Assuming that the respective output twist screws of the lower and upper platforms are
SOTS l =( 0; v),
SOTS u =( k;r×k+ v),
where k and r are identified as
{k =(0, 0,1)r=(x,y ,z),
then the output twist screw of the upper platform for the i th limb can be represented by SOTS li=( 0; v i) and the output twist screw of the lower platform for the i th limb can be represented by SOTSui=(k;r×k+vi). Considering that the power between the transmission wrench screw of Limbs II, III, and IV and the output twist screw of Limb I as a foundation, we arrive at
{SOTSl1 STWS4=0SOTSu1 STWS2=0SOTSu1 STWS3=0.
Because of the connecting of Limb I and the lower platform, the output twist screw of Limb I can be obtained by
SOTS1=SOTSl1=( 0; v 1) .
Similarly, the respective output twist screws of Limbs II, III, and IV are obtained using
SOTS2=SOTSu2=(k;r×k+v2),
SOTS3=SOTSu3=(k;r×k+v3),
SOTS4=SOTSl4=( 0; v 4) .
All of the SOTS2, SOTS3, and SOTS4 can be respectively obtained through the equations
{SOTSu2 STWS3=0SOTSl2 STWS1=0SOTSl2 STWS4=0,
{SOTSu3 STWS2=0SOTSl3 STWS1=0SOTSl3 STWS4=0,
{SOTSl4 STWS1=0SOTSu4 STWS2=0SOTSu4 STWS3=0.
According to the motion/force transmission index [28], an input transmission index (ITI) for the ith ( i=1 ,2,3,4) limb can be derived by
ξi= | SITS iSTWSi|| SITS iSTWSi|max,
where SITS iSTWSi is the reciprocal product between the unit input twist screw S ITSi and the unit transmission wrench screw STWSi, ξi falls into the range [0, 1]; in addition, the input transmission singularity occurs if ξi= 0, whereas the mechanism is farthest away from singularity if ξi=1. The presented robot with parameters R1=275 mm, R2= 100 mm,L1= 365 mm, and L2= 805 mm is taken as a presupposed condition. In the given positions (0, 0, −600 mm), the input singularity indices of the four limbs are investigated when the rotational angle changes from 180° to 180°. Figure 11 shows the ξi distribution. As can be seen, ξ1 and ξ4 are constant values, whereas ξ2 and ξ 3 are symmetrical around θ=0°. The definite physical meaning can be described by stating that, in a fixed position, the lower platform has no translation. This phenomenon causes the unit input twist screw SITS i and unit transmission wrench screw STWSi for Limbs I and IV to remain unchanged. The upper platform has a symmetrical rotation relative to θ =0°. These symmetrical relationships also apply to Limbs II and III.
Fig.33 Distribution of input singularity indices at point (0, 0, −600 mm).

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Similarly, an output transmission index (OTI) for the ith ( i=1 ,2,3,4) limb can be derived by
σi= | STWSi SOTSi| |STWSi SOTSi|max,
where STWSi SOTSi is the reciprocal product between the unit transmission wrench screw STWSi and the unit output twist screw SOTSi, σi falls into the range [0, 1]; in addition, the output transmission singularity occurs if σi= 0, whereas the mechanism is farthest away from singularity if σi=1. The output transmission singularity occurs if σi= 0, whereas the mechanism is farthest away from singularity if σi=1. In the given positions (0, 0, −600 mm), the output singularity indices of the four limbs are investigated when the rotational angle changes from – 180°to 180°. Figure 12 shows the σi distribution. As can be seen, σ 1 and σ4 as well as σ2 and σ 3 are symmetrical around θ=0°. The definite physical meaning can be described by stating that, for Limbs I and IV, the unit output twist screw SITS i is changed symmetrically, and the unit transmission wrench screw STWSi remains unchanged. For Limbs II and III, the unit output twist screw S ITSi and the unit transmission wrench screw STWSi are symmetrically and simultaneously change.
Fig.34 Distribution of the output singularity indices at point (0, 0, –600 mm)

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Two indices are defined to evaluate the rotation ability. One is the symmetrical rotational range index
θsrr=min{|θmin |,θ max},
and the other is the maximum rotation range index
θmrr=θmax θmin .

Identification results

By the constraint of σi 0.05 and ξi0.3, the rotation capacity of the upper platform when z= −600 mm is fully investigated. According to the physical dimensions, the size of a cross-section defined by x=r cosω, y=r sinω, where r[0 ,600] (unit: mm) and ω[0 ,2π]. The distributions of θmax, θ min, θsrr, and θmrr are plotted in Fig. 13, which clearly presents the rotation capacity. Notably, θsrr and θmrr are exactly symmetrical with respect to ω=0.75π.
Fig.35 Rotational capability when z=600mm. (a) Distribution of θmax; (b) distribution of θmin; (c) distribution of θsrr; (d) distribution of θmrr

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To meet the requirements of practical application, different θ srr values and the corresponding atlases under different values of z are shown in Fig. 14. Assuming the increasing ratio of the accelerator is 2, the area where θsrr45° should be focused on. Therefore, the blue segments are sub-planes of the workspace. Two types of workspaces are identified and demonstrated in the following sections.
Fig.36 Distributions of θsrr under different values of z. (a) z=450 mm; (b) z=500 mm; (c) z=550 mm; (d) z=600 mm

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Maximum inscribed circle workspace

The outlines of θsrr45° under different values of z are plotted. In their intersection, a maximal inscribed circle ϕ: 950 mm can be delineated as the high rotational output area in Fig. 15. Combining the analysis results presented in Fig. 14, a maximal inscribed cylinder ϕ×h (950 mm×150 mm) can be identified as the workspace, which is shown in Fig. 16.
Fig.37 Cross-sections when z= –450, –500, –550, and –600 mm

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Fig.38 The maximal inscribed circle workspace

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Maximum symmetrical circle workspace

Considering workspace symmetry, a maximal inscribed circle ϕ: 730 mm can be delineated as the high rotational output area in Fig. 17. Combining the height information, a maximum symmetrical circle ϕ×h (730 mm×150 mm) can be delineated as the workspace, as shown in Fig. 18.
Fig.39 Cross-sections when z=–450, –500, –550, and –600 mm

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Fig.40 The maximal symmetrical circle workspace

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Conclusions

In this paper, the concept of a novel parallel robot for pick-and-place operations with four limbs and a 1½ mobile platform has been proposed. Furthermore, its kinematic issues, including mobility analysis, kinematic modeling, and workspace identification have also been investigated. A line-graph method based on Grassmann line geometry was introduced to derive the freedom space, and the results indicate that the parallel robot shows the exact DOFs (3T and 1R) to undertake Schönflies motion. Then, the inverse kinematics has been derived according to the principle of mechanism, and the identification conditions to pick out the practical solution have been provided. Based on the motion/force transmission index, an initial workspace far from output transmission singularity has been identified with a set of presupposed parameters. Under this precondition, the rotational capability can reach at least ±45° without using an additional device. Two emblematic sub-workspaces were selected as final workspace by a different definition. The analysis results presented in this paper are conducive to the parallel robot’s practical application.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 51425501), and by the Beijing Municipal Science & Technology Commission (Grant No. Z171100000817007). The second author wishes to acknowledge the support provided by the Alexander von Humboldt (AvH) Foundation.

Open Access

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