workspace analysis of a hybrid kinematic machine tool with...

Research ArticleWorkspace Analysis of a Hybrid Kinematic Machine Tool withHigh Rotational Applications

Haiqiang Zhang1 Hairong Fang 12 Yuefa Fang12 and Bingshan Jiang1

1Robotics Research Center School of Mechanical Electronic and Control Engineering Beijing Jiaotong University Beijing China2Key Laboratory of Vehicle Advanced Manufacturing Measuring and Control Technology Ministry of Education Beijing China

Correspondence should be addressed to Hairong Fang hrfangbjtueducn

Received 14 April 2018 Accepted 22 May 2018 Published 26 June 2018

Academic Editor Carlos Llopis-Albert

Copyright copy 2018 Haiqiang Zhang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a novel parallel manipulator with one translational and two rotational (1T2R) degrees of freedom that can beemployed to form a five-degree-of-freedom hybrid kinematic machine tool for large heterogeneous complex structural componentmachining in aerospace field Compared with serial or parallel machine hybrid machine has the merits of high stiffness highspeed large workspace and complicated surface processing ability To increase stiffness three-degree-of-freedom redundantlyactuated and overconstrained 2PRU-PRPS parallel manipulator (P denotes the active prismatic joint) is proposed which is utilizedas the main body of hybrid machine By resorting to the screw theory the degree of freedom of the proposed mechanism isbriefly addressed including the initial configuration and general configuration and validated by Grubler-Kutzbach (G-K) equationNext kinematic inverse solution and parasitic motion of the parallel manipulator are deduced and the transformational relationsbetween the Euler angle and Tilt-Torsion (T-T) angle are identified Thirdly the performance evaluation index of orientationworkspace is introduced and the reachableworkspace and jointworkspace are formulatedThrough specific examples the reachableworkspace task workspace and joint workspace of the redundant actuation parallel manipulator are depicted Compared withoverstrained 2PRU-PRS parallel manipulator corresponding analyses illustrate that the proposed parallel manipulator owns muchbetter orientation capability and is very meaningful to the development of the five-axis hybrid machine tool

1 Introduction

Parallel kinematic manipulator tools were claimed to possessthe inherent advantages such as high stiffness high loadingcapability high precision low error accumulation quickresponse speed and high orientation capability Howeverparallel manipulators suffer inherently from the unfavorableworkspace Therefore it is of crucial significance to havea larger workspace so as to satisfy the working capability[1 2] While the three-axis NCmachine has more advantagesin large workspace and good dexterity however it is onlysuitable for simple surface or small parts processing [34] The five-axis series CNC machine adding two-degree-of-freedom rotating head attached to its mobile platformor two-degree-of-freedom rotary tables on the three-axismachine can maintain the favourable orientation but oftenscarified certain workspace and may cause poor precisionand stiffness by increasing the length of the actuator to

enlarge the workspace [5 6] So there is a contradictionbetween workspace and precision and stiffness for freesurface machining by using the traditional series or parallelmachine tool and they cannot be able to satisfy requirementsof the high speed milling for large heterogeneous complexsurface in aerospace At present many complicated freedomsurfaces are still milling manually which inevitably has highcost low efficiency and long cycle and the process notonly depends heavily on the expertise and experience ofthe operators but also requires much attention be given toprocessing what is more it is difficult to obtain high qualitymachining surface [7] So it is of importance to explore adesign approach for solving the required reachableworkspaceto envelope the task workspace which will offer an idealsolution for machining [8]

Hybrid kinematic machine is expected to integrate therespective merits of pure serial and parallel machine whichhas bigger workspace and better dynamic performance

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 2607497 12 pageshttpsdoiorg10115520182607497

2 Mathematical Problems in Engineering

higher precision and higher rigidity more complicated sur-face processing ability and more flexible orientation capa-bility and has been successfully employed as machine toolsand robots for high speed milling drilling and welding inaerospace and automotive industry for free surface process-ing as well as assembly operations of aluminum structuralparts [9] It has been demonstrated practically by verysuccessful applications such as a typical Sprint Z3 mecha-nism [10] Tricept hybrid machine tool [11] and Exechonhybrid machine tool [12] In many practical applicationsto increase the workspace of the three-degree-of-freedomparallel manipulator we can add one or two long tracksSimultaneously to improve the orientation adjusting abilityof the end effector one can attach two- or three-degree-of-freedom rotating head and then form a multi-degree-of-freedom hybrid machine tool with large workspace and highstiffness as well as high orientation capability [13 14]

Hybrid machine tool underwent fast improvements anddrew particular interests for numerous researchers sincethey satisfy the increasing demanding task requirements ofmany various applications such as inmachine tools assemblylines and high speed machining used in automotive railwayand construction industries For instance the German DSTechnologie launched five-axis machining center Ecospeedspindle for aircraft structure components with complexgeometries and the spindle head was mounted on the endeffector of parallel manipulator which can realize rotationabout the x- and y-axis and translation along z-axis andtranslation in X and Y direction can be realized by twovery long tracks [15] Wang et al [16] have proposed a 3-SPR parallel mechanism which forms the main body of a 5-DOF hybrid manipulator especially designed for high speedmachining in the aircraft industry Afive-axis hybridmachinehas been developed to realize lapping and milling for largecomplex structure component surface and it is generated byserially adding a 2-DOFACndashaxis head to the coupling three-degree-of-freedom 3RPS parallelmanipulator in terms of tworotations and one translation [17 18] Hao et al [19] cameup with a novel two-degree-of-freedom parallel manipulatorincorporated a two-degree-of-freedom rotating head withA and C axis arrangement and supplemented a mobileplatform which can form a gantry type five-axis hybridmachine tool to provide five-DOF movement capabilitiesHuang et al [20] put forward a practical 5-DOF hybridreconfigurable manipulator module called Trivariant andbuilt a variety of equipment to complete high speed millingfor large aluminum structural components and complexmolds Wu et al [21] have studied the three-degree-of-freedom redundant actuation parallel manipulator increasedthe freedom of movement and rotation in the worktable andapplied it to the five-degree-of-freedom hybrid machine toolto perform machining

The remainder of the paper is organized as follows Theresearch background of hybrid parallel machine tools andtraditional serial-parallel machines is presented firstly In thesubsequent section the required degree of freedom for highspeed milling machining of large heterogeneous complexfreedom surface is briefly addressed and a redundantly actu-ated and overconstrained 2PRU-PRPS parallel manipulator

XZ

Y

3000mm

8000mm 700mm

minusTast

Figure 1 Limit rotational angle about x-axis

configuration is presented and selected as the main body ofthe hybrid machine tool Afterward the degree of freedomof the proposed parallel manipulator was analyzed includinginitial configuration and general configuration based onthe screw theory and the modified Grubler-Kutzbach (G-K) equation was utilized to verify the correctness for thedegree of freedom Then the kinematic inverse position andtransformational relation between Z-Y-X Euler angle andTilt-Torsion (T-T) angle are performed Next based on theinverse position analysis and constraint conditions the limitboundary searching algorithms and flowchart are introducedin detail In the subsequent section the parasitic motionof the parallel manipulator was carried out by numericalexamples simultaneously the orientation workspace analysisof the parallel manipulator is performed and the reachableworkspace the taskworkspace and the joint workspace of theparallel manipulator are intuitively depicted by using com-puter code programming Finally this article is concluded inldquoConclusionsrdquo section

2 Design Requirements and Configuration ofHybrid Machine

21 Function Requirement Analysis The purpose of thispaper is to design a hybridmachine tool used in the aerospacefield for a large heterogeneous free surface high speedmillingthe workpiece magnitude is shown in Figures 1 and 2 whosespan in x-axis is 3000mm and in y-axis is 3600mm thetotal length of z-axis is 9400mm and the feed stroke of theend face in Z direction is 700mm because the workpiececan only be placed flat and cannot be stood therefore thefirst condition of the overall layout selection of horizontalhybrid machine tool should be considered In order toachieve the desired machining effects the machine tool andsurface normal are kept reasonable in the process of surfacemilling so the machine tool should have at least five degreesof freedom including three translational degrees and two

Mathematical Problems in Engineering 3

8000mm

3600mm

Z

X

Y

700mm

Task

minusTask

Figure 2 Limit rotational angle about y-axis

rotational degrees The latter is much more important thanthe former when it comes to perform three-dimensionalmilling [22] Limit rotation angles (ie 120572119879119886119904119896 120573119879119886119904119896) betweenthe tool and the surface during the curved surface machiningare shown in Figures 1 and 2

According to the characteristics of the curve surfacefive relative motion between the surface and the cutter areprocessed the x- and y-axis motion are configured by theball screw motion unit in series to satisfy the requirementsof large workspace and translation along z-axis and tworotations that are perpendicular to the z-axis are completedby parallel manipulator to meet the stiffness and orientationrequirement of the tool If the movements are achieved viatraditional feed unit and orientation rotating head then itcannot guarantee stiffness and accuracy of themachine owingto its cantilever The specific design requirements are asfollows

(a) Absolute position accuracy parallelmanipulator con-figuration instead of traditional series configurationcan reduce error accumulation

(b) Normal precision the parallel manipulator has goodorientation capability to ensure the normal contactbetween the normal of free surface and machine toolpoint so as to keep its normal position precision

(c) Workspace workspace is generated by adoptingcompound spherical joint to increase orientationworkspace and serial X-Y long tracks to expand theposition workspace of the hybrid machine tool

(d) High stiffness hybrid machine tool will produceheavy cutting force in the high speed machiningprocess so as tomaintain highermachining accuracyand themachine tool should be able to bear the heavyforce and resist the external force deformation Thusthe parallel manipulator should have higher stiffnesscharacteristics

(e) High quality the high orientation capability of par-allel manipulator is employed instead of manualmilling which is beneficial to ensure the machiningquality

According to the above processing requirements theparallel manipulator tool requires high stiffness and goodorientation capability for the high speed machining of thefree surface Considering development trend of the hybridmachine tool a five-axis hybrid machine tool can be con-structed by adopting 1T2R three-degree-of-freedom parallelmanipulator with two long X-Y tracks which is the bestchoice to realize the machining task requirements

22 ConfigurationDesign of the 1T2RMechanism It is config-uration innovation of the 1T2R three degrees of freedom thatis the kernel of the hybrid machine tool In order to completethe surface process with high efficiency and high precisionit is of crucial importance and significance for novel 1T2Rlower-degree of freedom parallel manipulator with high stiff-ness large workspace and high orientation capability Thereis an abundance of research on 1T2R mechanism Kong andLi [23 24] divided the 1T2R parallel manipulator into threecategories one of which is PU configuration parallel manip-ulator second kind of which is UP configuration parallelmanipulator with coupling between rotation and movementand the third is RPR configuration parallel manipulatorwhich can eliminate the coupling of rotation and movementand have certain space axis Li et al [25] pointed out thata class of one translation and two rotation DOFs parallelmanipulator called [PP]S configuration mainly include 3-PRS 3-RPS 3-RRS and 3-PPSWang et al [26] presented the3-PUU parallel manipulator with rotational and translationalcoupling degrees of freedom the difference between pro-posed mechanism and the 3PRS parallel manipulator is thatthe former did not have the spherical joint but yet possessesmuch larger rotation angle and higher precision Cui et al[27] designed a 3RPS parallel manipulator with compoundspherical joint that can increase the rotation angle Li et al[28] proposed a novel overconstrained parallel manipulator2RPUampSPR the degree of freedom was analyzed based onthe screw theory and kinematic inverse position and Jacobianmatrix were derived Yan et al [29] introduced a comparisonstudy of the kinematics characteristics of two overconstrained2-RPUampSPR parallel manipulators Xie et al [30] conductedperformance comparison analysis including motion forcetransmission performance parasitic motion and orientationcapability of two overconstrained 2PRU-PRS and 2PRU-UPRparallel manipulator Pashkevich [31] demonstrated that theoverconstrained parallel manipulator can effectively improvethe stiffness characteristic of the mechanism

To increase the workspace of the parallel manipula-tor enhance the kinematics performance of the machinetool and improve the stiffness characteristic and dynamiccharacteristic this paper adopts the redundantly actuatedand overconstrained 2PRU-PRPS parallel manipulator withcompound spherical joint as the main body of the hybridmachine tool Simultaneously to obtain a high rotation angleusing single limb double redundantly actuated technology


Z

X

Y

O

Figure 3 A hybrid kinematic machine with five DOFs

with two long X-Y tracks to form a 5-axis hybrid machinetool it can be applied to machining for a large complexheterogeneous surface and the overall structural concept isshown in Figure 3

3 Mobility Analysis of the 2PRU-PRPSParallel Manipulator

31 Architecture Description of the 2PRU-PRPS ParallelManipulator A novel redundantly actuated and overcon-strained 2-PRU-PRPS parallel manipulator with compoundspherical joint has been proposed in this paper as shownin Figure 4 which is composed of a moving platform afixed base and two identical constrained PRU chains and onedouble actuated PRPS chain together connecting the movingand fixed base And two of PRU chains are symmetricallyarranged and are located in a plane furthermore two revoluteaxes are parallel with each other and the second revolute axesof two Hooke joints are coincident and perpendicular to thethird revolute axis of spherical joint The parasitic motionappears only in an axial direction The parallel manipulatoris actuated by four active prismatic joints and three actuatorsare fixed at the base which reduces the inertia of the parallelmanipulator tool The spindle head is mounted at the end ofthe moving platform to complete the high speed milling

To facilitate analysis a fixed coordinate system B-xyz islocated at the center of the fixed base and a moving coor-dinate system A-uvw is attached at the center of the movingplatform respectively Let the middle point of hypotenuseB1B2 be B x-axis is perpendicular to B1B2 y-axis coincideswith B1B2 and z-axis is perpendicular to the fixed baseupward Similarly let the middle point of hypotenuse A1A2be A u-axis is perpendicular to A1A2 v-axis coincides withA1A2 and w- axis is perpendicular to the moving platformupward Without loss of generality Δ119861111986121198613 and Δ119860111986021198603are both isosceles right triangle ang119861111986131198612 = ang119860111986031198602 = 90∘and their circumradii are nominated as a and b respectivelyBB1=BB2=BB3=b andAA1=AA2=AA3=aWith respect to B-xyz the position vectors of points 119861i and 119860 i (i=123) are as

z

y

x

uAw

B

v

b

a

P

R

U

S

PRPS

P

R

PPRU

PRU

PR

U$42

$32

$43$33

$12

$22$23

$13

$11

$21

$31$53

$63

$41 $c1

$c2$c5

$c4

$c3

l220 l330 l110

A1

A2

A3

s220

s330

s110

B1

B2

B3

Figure 4 The schematic diagram of the 2-PRU-PRPS parallelmanipulator

follows The coordinates of B1 B2 and B3 are (0 b 0) (0 -b0) and (b 0 0) respectively The coordinates of A1 A2 andA3 are (x1 y1 z1) (x2 y2 z2) and (x3 y3 z3) respectively Thelayout angle of the fixed actuators is defined as 120579 and the anglebetween Bi joint and x-axis of the coordinate system B-xyz is12057911989432 Mobility Analysis of Initial Configuration Lower-mobility parallel manipulators whose independent degreesof freedom of the end effector are usually less than sixcan be implemented in many applications In order todetermine the motion pattern of the redundantly actuatedand overconstrained 2PRU-PRPS parallel manipulatormobility analysis is indispensable whereas it is necessary toanalyze the constraint screw provided by each chain to themoving platform and comprehensively analyze the constrainttype of the moving platform

It is assumed that the axes of the fixed coordinate systemand the moving coordinate system are parallel to each otherin the initial pose Firstly the first PRU limb is analyzed in thefixed coordinate system and the twist screw of the chain canbe expressed as

$11 = (0 0 0 0 minusc120579 s120579)$21 = (1 0 0 0 0 minus119887)$31 = (1 0 0 0 1199111 minus1199101)$41 = (0 minus1 0 1199111 0 minus1199091)

(1)

where s and c are the abbreviation of sine and cosinerespectively Employing reciprocal screw theory the wrenchsystem of (1) is obtained as


$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1) (2)

where $1198881 represents a constraint force passing A1 pointand parallel to x-axis and $1198882 represents a constraint coupleperpendicular to the fixed base

Similarly the twist screw of the second PRU limb is givenby

$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 0 0 119887)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 1 0 minus1199112 0 1199092)

(3)

The wrench screw of (3) is obtained as

$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1) (4)

where $1198883 represents a constraint force passing point A2 andparallel to the x-axis of the fixed coordinate system and $1198884represents a constraint couple perpendicular to the fixed base

The twist screw of the third PRPS limb is given by

$13 = (0 0 0 minusc120579 0 s120579)$23 = (0 1 0 0 0 119887)$33 = (0 0 0 1198973 0 1198993)$43 = (minus1198973 0 minus1198993 minus11989931199103 11989931199093 minus 11989731199113 11989731199103)$53 = (0 1 0 minus1199113 0 1199093)$63 = (1 0 0 0 1199113 minus1199103)

(5)

where l3 and n3 are direction cosines of the three prismaticjoints in limb 3


$1198885 = (0 1 0 minus1199113 0 1199093) (6)

where $1198885 represents a constraint force passing point A3 andparallel to y-axis of the fixed coordinate system

The constraint screw of all the limbs acting on themovingplatform can be obtained as

$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1)$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1)$1198885 = (0 1 0 minus1199113 0 1199093)

(7)

It is worth noting that z1=z2=z3=z and when the mech-anism is initial configuration the twist screw of the movingplatform can be obtained by adapting the reciprocity of (7)

$1198981 = (0 1 0 minus119911 0 0)$1198982 = (1 0 0 0 119911 0)$1198983 = (0 0 0 0 0 1)

(8)

where (8) represents the moving platform that is capable ofrotation about x- and y-axes and translation along the z-axis

33 Mobility Analysis of General Configuration Similarlythe twist screw of the first PRU limb is given in the fixedcoordinate system by

$11 = (0 0 0 0 minusc120579 s120579)$21 = (1 0 0 0 1199041119904120579 1199041119888120579 minus 119887)$31 = (1 0 0 0 1199111 minus1199101)$41 = (0 1198901 1198911 11989111199101 minus 11989011199111 minus11989111199091 11989011199091)

(9)

where si denotes the twist of the P joint and ei and f i representthe direction cosine of the second revolute axis of the Hookejoint of i limb

Then the wrench screw of limb 1 is easily obtained as

$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901) (10)

where $1198881 represents a constraint force passing point A1 andparallel to $21 $1198882 represents a constraint couple perpendic-ular to v and $31

As previously mentioned the twist screw of the secondPRU limb can be expressed as

$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 1199042119904120579 0 119887 minus 1199042119888120579)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 minus1198901 minus1198911 11989011199112 minus 11989111199102 11989111199092 minus11989011199092)

(11)


$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901) (12)



The twist screw of the third PRPS limb can be expressedas

$13 = (0 0 0 minusc120579 0 s120579)$23 = (0 1 0 minus1199043119904120579 0 119887 minus 1199043119888120579)$33 = (0 0 0 1198973 0 1198993)$43 = (minus1198973 0 minus1198993 minus11989931199103 11989931199093 minus 11989731199113 11989731199103)$53

= (minus1198991 1198903 1198971 11989711199103 minus 11989031199113 minus11989711199093 minus 11989911199113 11989031199093 + 11989911199103)$63

= (1198921 minus1198911 1198901 11989011199103 + 11989111199113 11989211199113 minus 11989011199093 minus11989111199093 minus 11989211199103)

(13)

where 1198921 = (11989011198971 minus 11989111198903)1198991 e3 is a constant and l3 and n3represent the direction cosine of the third joint axis

Subsequently the wrench screw of (13) can be expressedas

$1198885 = (0 1 0 minus1199113 0 1199093) (14)

where $1198885 represents a constraint force passing point 1198603 andparallel to $23

So far the constraint screw of the parallel manipulator canbe expressed as

$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901)$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901)$1198885 = (0 1 0 minus1199113 0 1199093)

(15)

It is noteworthy that the direction vector A1A2 coincideswith the axes of the second joint axis of the Hooke joint in thetwo PRU limbs so the relation can be obtained as

1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)

By combining (16) and rearranging (15) the twist screwof the moving platform yields the following

$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)

In (17) $1198981 represents one rotational degree of freedompassing point A3 and parallel to x-axis $1198982 represents onerotational degree of freedom of the moving platform passingpoint A2 and parallel to the v-axis and $1198983 represents atranslational degree of freedom that is perpendicular to thefixed base There are two instantaneous rotation axes one ofwhich is a straight line passing point A3 and parallel to the

x-axis and another is v-axis located in the moving platformand they change with the motion of the moving platform Itis worth noting that the mechanism has the same constraintscrew and the degree of freedom with the 2PRU-PRS parallelmanipulator

Generally the degree of freedom of parallel manipulatorcan be calculated by the modified Grubler-Kutzbach (G-K)equations that is

119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)

where F represents the degree of freedom of the mechanism119899 represents the number of the components 119892 represents thenumber of the kinematic joints 119889 = 6minus120582 represents the orderof the mechanism 119891i represents the degree of freedom of theith kinematic joint v represents the redundant constraints ofthe mechanism and 120589 represents the local degree of freedom

There was neither constraint couple in the same directionnor constraint force in collinearity among the constraintscrew in the parallel manipulator therefore there is nocommon constraint that is 120582 = 0 Because there are onlythree linearly independent variables in the five-constraintscrew of the parallel manipulator therefore the parallelmanipulator has two redundant constraints that is V =2 Because the parallel manipulator has no local degree offreedom so 120589 = 0 We can see from the schematic of themechanism that the number of the components is 6 thenumber of the kinematic joints is 10 and the relative freedomof all the kinematic joints in the mechanism is 14 Due tothe introduction of redundant actuation there are five linearcorrelations between the six-twist screw so there is a localdegree of freedom that is 120589 = 1 [32 33]

Thus based on the revisedGrubler-Kutzbach (G-K) equa-tions the degree of freedom of the 2PRU-PRPS parallelmanipulator can be recalculated as follows

119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589= 6 times (9 minus 10 minus 1) + 14 + 2 minus 1 = 3

(19)

In summary the redundantly actuated and overcon-strained 2PRU-PRPS parallel manipulator has three degreesof freedom ie two rotational degrees of freedom about x-axis and v-axis and one translational degree of freedom alongz-axis

4 Inverse Kinematics of theParallel Manipulator

41 Position Inverse Analysis The inverse kinematics solutionis based on the determination of the structural parameters ofthe parallel manipulator when the position and orientationof the moving platform are given so as to solve the inputdisplacement of the prismatic joints

Z-Y-X Euler angles are adopted to describe orientationmatrix of the moving coordinate system with respect to thefixed coordinate system first rotating the moving coordinate


about z-axis by angle 120574 then about y-axis of the newcoordinate system by angle 120573 and finally about x-axis ofthe new coordinate system by angle 120572 Thus the orientationmatrix R can be expressed as

R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[

119888120573119888120574 119904120572119904120573119888120574 minus 119888120572119904120574 119888120572119904120573119888120574 + 119904120572119904120574119888120573119904120574 119904120572119904120573119904120574 + 119888120572119888120574 119888120572119904120573119904120574 minus 119904120572119888120574minus119904120573 119904120572119888120573 119888120572119888120573

]]]

(20)

p = [119909 119910 119911]119879 represents the position vector of theoriginal point A in the fixed coordinate system 119861 minus 119909119910119911 119886119894and 119887119894 represent the position vector in the fixed coordinate ofjoints Ai and Bi and the coordinate of each joint in the fixedcoordinate system can be respectively expressed in matrixform as

a1 = 119877 (0 119886 0)119879a2 = 119877 (0 minus119886 0)119879a3 = 119877 (119886 0 0)119879 b1 = (0 119887 0)119879b2 = (0 minus119887 0)119879b3 = (119887 0 0)119879

(21)

Because of the arrangement of revolute joint in PRUand PRPS limbs the center of Hooke joints and sphericaljoint cannot move along the axis of the revolute joint so thefollowing constraint conditions can be structured as

(p + a1)119879 sdot (1 0 0) = 0(p + a2)119879 sdot (1 0 0) = 0(p + a3)119879 sdot (0 1 0) = 0

(22)

Selecting parameters 120572 120573 119911 as three independent param-eters parasitic motion can be arranged as

119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)

The close-loop vector method is used to establish theequation of vector AiBi in the fixed coordinate system B-xyz

L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)

Equation (24) squares on both sides we arrange andobtain

1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)

Position inverse solution of the 2PRU-PRPS parallelmanipulator about si in (25) can be expressed as follows [34]

119904119894 = L119879119894 s1198941 plusmn radic(L119879119894 s1198941)2 minus L119879119894 sdot L119894 + 1198971198942 (26)

where s1198941 = [minus119888120579119894119888120579 minus119904120579119894119888120579 119904120579]119879 l1=l2=l and l3 is anextensible and compressible link

42 Orientation Description of T-T Angle To better describethe orientation capability of the parallel manipulator Liu andBonev [35] pointed out that the [PP]S mechanism is threedegrees of freedom with zero-torsion angle and systemicallystudied the relationships between different Euler angles andTilt-Torsion (T-T) angle that is the orientation of themovingplatform can be easily described as two variables the azimuthangle and tilt angle Therefore T-T angle is a new orientationdescription method which is usually more concise andmore efficient to reflect the orientation capability of a classof 3-[PP]S mechanism compared with the description ofEuler angles method When the torsion angle is zero theorientation matrix of T-T angle can be expressed as

119879minus119879R (120593 120601 0) = R119911 (120593)R119910 (120601)R119911 (minus120593)R119911 (0)

= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)

Combining (20) and (27) the T-T angles with zero-torsioncan be converted to the Z-Y-X Euler angles via the followingequation

120573 = 119886 sin (119888120593119904120601)120572 = 119886 sin(minus119904120593119904120601119888120573 )120574 = 119886 sin(119904120593119888120601 (119888120601 minus 1)119888120573 )

(28)

Figures 5 6 and 7 show the relations between (120572 120573 120574)and (120593 120601) separately The circle coordinate denotes azimuth120593 and radial coordinate denotes the tilt angle 1206015 Orientation Workspace Analysis

The orientation workspace of parallel manipulator tool isan important performance index which is the set of allpractically feasible orientation of the moving platform Byanalyzing theworkspace we can deduce the problemwhetherthe expected machining range can be realized The mainanalysis methods contain analytical method and numericalmethod in this paper the limit boundary searching methodof reachable workspace was adopted and the mechanism isdivided into several single limbs and the boundary of singlelimb space is obtained by using the surface enveloping theoryfinally the whole workspace of the mechanism is obtained byusing the surface intersection technique The difficulty of the


150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘

minus60 minus30 0 30 60

minus60

minus40

minus20

0

20

40

60

(∘

)

Figure 5 The relation of 120573 referring to (120593 120601)

0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)


hybrid machine tool workspace is to solve the workspace ofparallel manipulator while the workspace analysis of parallelmanipulator is mainly to investigate the reachable workspaceof the end effector For 1T2R parallel manipulator theworkspace of themanipulatormainly considers the rotationalcapability about the x-axis and the v-axis and the translationalcapability along z-axis Over the past decade workspace wasinvestigated by many scholars with different methods andalgorithms Referring to [36] a discrete boundary searchingmethodwas implemented to calculate theworkspace of paral-lel manipulator considering the driving constraint and jointsconstraint in the polar coordinate system Pond [37] investi-gated reachable workspace and dexterous workspace of threeparallel manipulators (including 3-PRS 3-RPS and Tricept)based on homogeneous Jacobian matrix condition numberand conducted quantitative analysis on the workspace Her-rero et al [38] solve the workspace of 2PRU-1PRS parallelmanipulator based on the calculation of inverse kinematicsand geometrical constraint and singularity constraint whatis more they pointed out that maximum inscribed ball canbe utilized to measure the workspace Fu and Gao et al [39]obtained the position workspace and orientation workspaceof the parallel manipulator with decoupling three branchesof six degrees of freedom by using a numerical searchingmethod and chose the maximum inscribed cylinder andsphere envelope as the specified shade in the workspace

51 Parameters Constraint Condition In the actual machin-ing process of parallel manipulator tool the following limita-tions should be considered

(1) Limitations of the Actuated Joints si

119904min le 119904119894 le 119904max (29)

where 119904min and 119904max represent the minimum and maximumstroke of active joint respectively Here 119904min = 0 and 119904max =650mm

(2) Limitation of Redundantly Actuated Joint q4

119902min le 1199024 le 119902max (30)

where 119902min and 119902max represent the minimum and maximumdisplacement of the second-stage active joint of the thirdlimb respectively Here 119902min = 900mm and 119902max =1200mm

(3) Joint Angle Constraints

(a) The Rotation Angle R2i of the Rotation Joint R Should BeSatisfied

1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)


where 1198772min and 1198772max represent the minimum and max-imum limit angles of the rotation joint R of the ith limbrespectively Here 1198772min = 0 and 1198772max = 1205876(b) The Rotation Angle R3i of the First Revolute Axis of theHooke Joint

1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)

where 1198773min and 1198773max represent the minimum and maxi-mum angle of the first revolute axis of the Hooke joint of theith limb respectively Here 1198773min = 1205876 and 1198773max = 120587(c) The Rotation Angle R53 of the Second Revolute Axis of theCompound Spherical Joint Should Be Satisfied

1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)

where 1198774min and 1198774max represent the minimum and maxi-mum angles of the second revolute axis of the compoundspherical joint respectively Here1198774min = 1205879 and1198774max = 120587(4) Interference Constraints of the Kinematic Limb Becausethe parallel manipulator once determines limitation of therevolute joint R there is no interference between adjacentlinks so the interference constraint cannot be taken intoaccount

(5) Singularity Configuration Constraint The singularity con-figuration of the parallel manipulator will seriously affectthe kinematic performance of the parallel manipulator andthose configurations should be avoided So the rank of theconstraint screw system is bound to be three that is

119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)

52 Workspace Algorithm The orientation workspace of themanipulator tool is generated according to the structureparameters and constraint conditions of the parallel manipu-lator aforementioned Based on the inverse position solutionthe constraint conditions of each step are judged respectivelyand the detailed search algorithm is seen in the flowchart asshown as Figure 8 If the condition is satisfied the feasiblepoints are recorded on the contrary the others are discardedThe singularity-free reachable workspace and task workspaceof the 2PRU-PRPS parallel manipulator have been generatedby adopting Matlab programming Simultaneously a seriesof displacement of the active prismatic joint can be obtainedwhose set is called joint workspace

6 Application Examples andPerformance Analysis

Numerical examples are presented to verify the validity of thetheory and its main structure parameters of the redundantactuation parallel manipulator are as follows (unitmm) The

Start

Define geometry parameters

Position Inverse

Joint Workspace

Task Workspace Reachable Workspace

End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ

si le sＧＲ q4 le qＧＲ Ri le RＧＲ Rank = 3

Record (s1 s2 s3) Record ( z)

le Task Task

z = zＧＣＨ

z le zＧＣＨ

Figure 8 Flowchart of singularity-free workspace calculation

fixed base radius is b=600 the moving platform radius isa=300 the length of limbs is l1=l2=1045 900 le 1198973 le 1200and the search range of the workspace is minus41205879 le 120572 le 41205879minus1205872 le 120573 le 1205872 900 le 119911 le 1600 minus1205874 le 120572119879119886119904119896 le 0 andminus1205876 le 120573119879119886119904119896 le 1205876 in terms of the relationships betweenEuler angle and T-T angle the range of T-T angle is derivedas 0 le 120593 le 2120587 0 le 120601 le 90∘ and the figures are drawn in thecylindrical coordinate system for convenience

61 Kinematic Analysis Because the redundantly actuatedand overconstrained parallel manipulator has a parasiticmotion in 119910 direction the relation mapping between |119910| and(120593 120601) is drawn in Figure 9 by (23)

62 Performance Analysis of Orientation Workspace Toillustrate the analysis process of the orientation workspacecomprising reachable workspace task workspace and jointworkspace the novel parallel manipulator proposed has beentaken as an example By adopting a numerical searchingmethod aforementioned the reachable workspace and taskworkspace of the redundantly actuated and overconstrainedparallel manipulator are generated as shown in Figure 10in other words the machine tool will possess a workspaceencasing a specified task workspaceThe results show that themanipulator can translate 900mm to 1600 mm along z-axisrotate 0∘ to 360∘ about 120593 and rotate 0∘ to 90∘ about 120601


30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)

Figure 9 Parasitic motion relation between |119910| and (120593 120601) withz=1300

Reachable workspace

Task workspace

Figure 10 Singularity-free reachable and task workspace

By specifying the value of the q4 (that is link length ofl3) we can get a surface in three-dimensional joint workspace(s1s2s3) formed by the three prismatic joints And the surfacecontains all the possible actuated parameters which affect themotion of the moving platform Figure 11 shows the jointworkspacewhen q4=1150mm which gives us information thatvarious constraints can result in a set of unreachable pointsand implies that the capability of motions can be efficientlyutilized

The reachable workspace of the proposed parallel manip-ulator and the overconstrained 2PRU-PRS parallel manip-ulator with the same degree of freedom is illustrated inFigure 12 To facilitate analysis the same search space is usedfor comparison of the overconstrained 2PRU-PRS parallelmanipulator and the proposed parallel manipulator The

Figure 11 Joint workspace of the parallel manipulator

2PRU-PRS2PRU-PRPS

Figure 12 Singularity-free workspace comparison of two parallelmanipulators

result shows that the redundant actuation parallel manipu-lator has a larger workspace and much higher orientationrotation capability than the 2PRU-PRS parallel manipulatorTherefore it is feasible to select the redundantly actuatedand overconstrained 2PRU-PRPS parallel manipulator as themain body of the hybridmachine tool and is verymeaningfulto the development of the five-axis hybrid machine tool

7 Conclusions

In this paper a novel serial-parallel hybrid kinematicmachine tool with high stiffness high orientation capabilityand large workspace has been presented which is a com-bination of 1T2R parallel manipulator and two long X-Ytracks and can be applied to high speed machining of a largeheterogeneous complex freedom surface in the aerospacefield

The mobility of proposed redundantly actuated andoverconstrained 2PRU-PRPS parallel manipulator has been


detailed and analyzed based on the screw theory andGrubler-Kutzbach (G-K) equations and the correctness of analysismethod is verified The inverse kinematic position and par-asitic motion of the derived parallel manipulator have beensequentially analyzed and the relationship between Z-Y-XEuler angle and Tilt-Torsion (T-T) angle is briefly expressed

The orientation workspace the performance evaluationindex can serve as an alternative for the description of theorientation capability Compared with overstrained 2PRU-PRS parallel manipulator the corresponding results illus-trate that the redundant actuation parallel manipulator caneffectively increase the workspace and potentially improveits orientation capability which lays a theoretical foundationfor the stiffness analysis and optimal design of the parallelmanipulator in the future work

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors would like to acknowledge the financial supportof the Fundamental Research Funds for the Central Univer-sities under Grant no 2018JBZ007 no 2018YJS136 and no2017YJS158 and the National Natural Science Foundation ofChina (NSFC) under Grants 51675037 and 51505023

References

[1] M Terrier A Dugas and J-Y Hascoet ldquoQualification ofparallel kinematics machines in high-speed milling on freeform surfacesrdquo The International Journal of Machine Tools andManufacture vol 44 no 7-8 pp 865ndash877 2004

[2] Y Lu and J Xu ldquoComputer simulation machining a 3Dfree surface by using a 3-RPRU parallel machine toolrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 33 no 7-8 pp 782ndash792 2007

[3] Y D Chen J Ni and S M Wu ldquoReal-time CNC tool pathgeneration formachining IGES surfacesrdquo Journal of Engineeringfor Industry vol 115 no 4 pp 480ndash486 1993

[4] D C H Yang and Z Han ldquoInterference detection and optimaltool selection in 3-axis NC machining of free-form surfacesrdquoComputer-Aided Design vol 31 no 5 pp 303ndash315 1999

[5] C-C Lo ldquoReal-time generation and control of cutter path for5-axis CNC machiningrdquo The International Journal of MachineTools and Manufacture vol 39 no 3 pp 471ndash488 1999

[6] Y H Jung D W Lee J S Kim and H S Mok ldquoNC post-processor for 5-axis milling machine of table-rotatingtiltingtyperdquo Journal of Materials Processing Technology vol 130-131pp 641ndash646 2002

[7] P Xu C-F Cheung B Li L-THo and J-F Zhang ldquoKinematicsanalysis of a hybrid manipulator for computer controlled

ultra-precision freeform polishingrdquo Robotics and Computer-Integrated Manufacturing vol 44 pp 44ndash56 2017

[8] Y Lu Y Shi and B Hu ldquoSolving reachable workspace ofsome parallel manipulators by computer-aided design variationgeometryrdquo Proceedings of the Institution of Mechanical Engi-neers Part C Journal of Mechanical Engineering Science vol222 no 9 pp 1773ndash1781 2008

[9] J Tlusty J Ziegert and S Ridgeway ldquoFundamental comparisonof the use of serial and parallel kinematics for machines toolsrdquoCIRP Annals - Manufacturing Technology vol 48 no 1 pp 351ndash356 1999

[10] N Hennes and D Staimer ldquoApplication of PKM in aerospacemanufacturing-high performance machining centersECOSPEED ECOSPEED-F and ECOLINERrdquo in Proceedingsof the 4th Chemnitz Parallel Kinematics Seminar pp 557ndash577 Zwickau Verlag Wissenschaftliche Scripten ChemnitzGermany 2004

[11] Y Wang H Liu T Huang and D G Chetwynd ldquoStiffnessmodeling of the tricept robot using the overall jacobianmatrixrdquoJournal of Mechanisms and Robotics vol 1 no 2 pp 1ndash8 2009

[12] J Zhang Y Q Zhao and Y Jin ldquoElastodynamic Modeling andAnalysis for an Exechon Parallel Kinematic Machinerdquo Journalof Manufacturing Science and Engineering vol 138 no 3 2016

[13] B Gherman D Pisla C Vaida and N Plitea ldquoDevelopment ofinverse dynamic model for a surgical hybrid parallel robot withequivalent lumped massesrdquo Robotics and Computer-IntegratedManufacturing vol 28 no 3 pp 402ndash415 2012

[14] M Weck and D Staimer ldquoParallel kinematic machine tools -Current state and future potentialsrdquo CIRP Annals - Manufac-turing Technology vol 51 no 2 pp 671ndash683 2002

[15] N Hennes ldquoEcospeedan innovative machinery concept forhigh-performance 5-axis machining of large structural compo-nents in aircraft engineeringrdquo inProceedings of the 3rdChemnitzParallel Kinematics Seminar pp 763ndash774 Zwickau Germany2002

[16] MWang H Liu T Huang and D G Chetwynd ldquoComplianceanalysis of a 3-SPR parallel mechanism with consideration ofgravityrdquo Mechanism and Machine Theory vol 84 pp 99ndash1122015

[17] X Yang J Zhao L Zhang D Li and R Li ldquoA novel surface self-adapting parallelmachine tool for blademachiningrdquo inProceed-ings of the 2009 IEEE International Conference on Mechatronicsand Automation ICMA 2009 pp 3921ndash3926 August 2009

[18] Y G Li H T Liu X M Zhao T Huang and D G ChetwyndldquoDesign of a 3-DOF PKM module for large structural compo-nent machiningrdquo Mechanism and Machine Theory vol 45 no6 pp 941ndash954 2010

[19] Q Hao L Guan and L Wang ldquoIntelligent accelerationdeceleration control algorithm for drive force for a heavyduty hybrid machine toolrdquo Qinghua Daxue XuebaoJournal ofTsinghua University vol 49 no 11 pp 1770ndash1778 2009

[20] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOF module of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[21] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine Theory vol 44 no 2 pp 289ndash3052009

[22] J Fan and A Ball ldquoQuadric method for cutter orientationin five-axis sculptured surface machiningrdquo The International


Journal of Machine Tools and Manufacture vol 48 no 7-8 pp788ndash801 2008

[23] X Kong and M Gosselin C ldquoType synthesis of three-DOF up-equivalent parallel manipulators using a virtual-chainapproachrdquo Advances in Robot Kinematics pp 123ndash132 2006

[24] Q Li and JMHerve ldquoType synthesis of 3-DOFRPR-equivalentparallel mechanismsrdquo IEEE Transactions on Robotics vol 30no 6 pp 1333ndash1343 2017

[25] Q Li Z Chen Q Chen C Wu and Z Huang ldquoStructural con-dition for [PP]S parallel mechanism without parasitic motionrdquoJournal of Mechanical Engineering vol 46 no 15 pp 31ndash352010

[26] L Wang H Xu and L Guan ldquoOptimal design of a 3-PUUparallel mechanismwith 2R1TDOFsrdquoMechanism andMachineTheory vol 114 pp 190ndash203 2017

[27] X Cui X Han and W Chen ldquoContinuous stiffness modelingof 3-RPS parallel kinematic machine with special compos-ite spherical jointsrdquo Beijing Hangkong Hangtian Daxue Xue-baoJournal of Beijing University of Aeronautics and Astronau-tics vol 36 no 11 pp 1275ndash1280 2010

[28] B Li Y Li and X Zhao ldquoKinematics analysis of a novel over-constrained three degree-of-freedom spatial parallel manipu-latorrdquo Mechanism and Machine Theory vol 104 pp 222ndash2332016

[29] F Xie X Liu andT Li ldquoAComparison Study on theOrientationCapability and Parasitic Motions of Two Novel ArticulatedTool Heads with Parallel Kinematicsrdquo Advances in MechanicalEngineering vol 4 pp 249103-249103 2015

[30] Q Yan B Li Y Li and X Zhao ldquoKinematics comparative studyof two overconstrained parallel manipulatorsrdquo MathematicalProblems in Engineering vol 2016 Article ID 5091405 12 pages2016

[31] A Pashkevich D Chablat and P Wenger ldquoStiffness analy-sis of overconstrained parallel manipulatorsrdquo Mechanism andMachine Theory vol 44 no 5 pp 966ndash982 2009

[32] H Qu S Guo and Y Zhang ldquoA novel relative degree-of-freedom criterion for a class of parallel manipulators withkinematic redundancy and its applicationsrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 231 no 22 pp 4227ndash42402017

[33] Z Huang J Liu and Y Li On mobility of mechanisms SciencePress Beijing China 2011

[34] G Cui H Zhang D Zhang and F Xu ldquoAnalysis of thekinematic accuracy reliability of a 3-DOF parallel robot manip-ulatorrdquo International Journal of Advanced Robotic Systems vol12 pp 1ndash11 2015

[35] X LIU ldquoAttitude Description Method of [PP]S Type ParallelRobotic Mechanismsrdquo Chinese Journal of Mechanical Engineer-ing vol 44 no 10 p 19 2008

[36] Z Z Chi D Zhang L Xia and Z Gao ldquoMulti-objectiveoptimization of stiffness and workspace for a parallel kinematicmachinerdquo International Journal of Mechanics and Materials inDesign vol 9 no 3 pp 281ndash293 2013

[37] G Pond and J A Carretero ldquoQuantitative dexterous workspacecomparison of parallel manipulatorsrdquoMechanism and MachineTheory vol 42 no 10 pp 1388ndash1400 2007

[38] S Herrero C Pinto O Altuzarra and M Diez ldquoWorkspacestudy of the 2PRU-1PRS parallel manipulatorrdquo in Proceedingsof the 14th International Federation for the Promotion of Mech-anism and Machine Science World Congress IFToMM 2015October 2015

[39] J Fu and F Gao ldquoOptimal design of a 3-leg 6-DOF paral-lel manipulator for a specific workspacerdquo Chinese Journal ofMechanical Engineering vol 29 no 4 pp 659ndash668 2016

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higher precision and higher rigidity more complicated sur-face processing ability and more flexible orientation capa-bility and has been successfully employed as machine toolsand robots for high speed milling drilling and welding inaerospace and automotive industry for free surface process-ing as well as assembly operations of aluminum structuralparts [9] It has been demonstrated practically by verysuccessful applications such as a typical Sprint Z3 mecha-nism [10] Tricept hybrid machine tool [11] and Exechonhybrid machine tool [12] In many practical applicationsto increase the workspace of the three-degree-of-freedomparallel manipulator we can add one or two long tracksSimultaneously to improve the orientation adjusting abilityof the end effector one can attach two- or three-degree-of-freedom rotating head and then form a multi-degree-of-freedom hybrid machine tool with large workspace and highstiffness as well as high orientation capability [13 14]

Hybrid machine tool underwent fast improvements anddrew particular interests for numerous researchers sincethey satisfy the increasing demanding task requirements ofmany various applications such as inmachine tools assemblylines and high speed machining used in automotive railwayand construction industries For instance the German DSTechnologie launched five-axis machining center Ecospeedspindle for aircraft structure components with complexgeometries and the spindle head was mounted on the endeffector of parallel manipulator which can realize rotationabout the x- and y-axis and translation along z-axis andtranslation in X and Y direction can be realized by twovery long tracks [15] Wang et al [16] have proposed a 3-SPR parallel mechanism which forms the main body of a 5-DOF hybrid manipulator especially designed for high speedmachining in the aircraft industry Afive-axis hybridmachinehas been developed to realize lapping and milling for largecomplex structure component surface and it is generated byserially adding a 2-DOFACndashaxis head to the coupling three-degree-of-freedom 3RPS parallelmanipulator in terms of tworotations and one translation [17 18] Hao et al [19] cameup with a novel two-degree-of-freedom parallel manipulatorincorporated a two-degree-of-freedom rotating head withA and C axis arrangement and supplemented a mobileplatform which can form a gantry type five-axis hybridmachine tool to provide five-DOF movement capabilitiesHuang et al [20] put forward a practical 5-DOF hybridreconfigurable manipulator module called Trivariant andbuilt a variety of equipment to complete high speed millingfor large aluminum structural components and complexmolds Wu et al [21] have studied the three-degree-of-freedom redundant actuation parallel manipulator increasedthe freedom of movement and rotation in the worktable andapplied it to the five-degree-of-freedom hybrid machine toolto perform machining

The remainder of the paper is organized as follows Theresearch background of hybrid parallel machine tools andtraditional serial-parallel machines is presented firstly In thesubsequent section the required degree of freedom for highspeed milling machining of large heterogeneous complexfreedom surface is briefly addressed and a redundantly actu-ated and overconstrained 2PRU-PRPS parallel manipulator

XZ

Y

3000mm

8000mm 700mm

minusTast

Figure 1 Limit rotational angle about x-axis

configuration is presented and selected as the main body ofthe hybrid machine tool Afterward the degree of freedomof the proposed parallel manipulator was analyzed includinginitial configuration and general configuration based onthe screw theory and the modified Grubler-Kutzbach (G-K) equation was utilized to verify the correctness for thedegree of freedom Then the kinematic inverse position andtransformational relation between Z-Y-X Euler angle andTilt-Torsion (T-T) angle are performed Next based on theinverse position analysis and constraint conditions the limitboundary searching algorithms and flowchart are introducedin detail In the subsequent section the parasitic motionof the parallel manipulator was carried out by numericalexamples simultaneously the orientation workspace analysisof the parallel manipulator is performed and the reachableworkspace the taskworkspace and the joint workspace of theparallel manipulator are intuitively depicted by using com-puter code programming Finally this article is concluded inldquoConclusionsrdquo section

2 Design Requirements and Configuration ofHybrid Machine

21 Function Requirement Analysis The purpose of thispaper is to design a hybridmachine tool used in the aerospacefield for a large heterogeneous free surface high speedmillingthe workpiece magnitude is shown in Figures 1 and 2 whosespan in x-axis is 3000mm and in y-axis is 3600mm thetotal length of z-axis is 9400mm and the feed stroke of theend face in Z direction is 700mm because the workpiececan only be placed flat and cannot be stood therefore thefirst condition of the overall layout selection of horizontalhybrid machine tool should be considered In order toachieve the desired machining effects the machine tool andsurface normal are kept reasonable in the process of surfacemilling so the machine tool should have at least five degreesof freedom including three translational degrees and two


8000mm

3600mm

Z

X

Y

700mm

Task

minusTask













Z

X

Y

O






z

y

x

uAw

B

v

b

a

P

R

U

S

PRPS

P

R

PPRU

PRU

PR

U$42

$32

$43$33

$12

$22$23

$13

$11

$21

$31$53

$63

$41 $c1

$c2$c5

$c4

$c3

l220 l330 l110

A1

A2

A3

s220

s330

s110

B1

B2

B3





(1)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1) (2)



$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 0 0 119887)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 1 0 minus1199112 0 1199092)

(3)


$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1) (4)




(5)



$1198885 = (0 1 0 minus1199113 0 1199093) (6)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1)$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1)$1198885 = (0 1 0 minus1199113 0 1199093)

(7)


$1198981 = (0 1 0 minus119911 0 0)$1198982 = (1 0 0 0 119911 0)$1198983 = (0 0 0 0 0 1)

(8)




(9)



$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901) (10)




(11)


$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901) (12)







(13)



$1198885 = (0 1 0 minus1199113 0 1199093) (14)




(15)


1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)


$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1


(19)







R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









8000mm

3600mm

Z

X

Y

700mm

Task

minusTask













Z

X

Y

O






z

y

x

uAw

B

v

b

a

P

R

U

S

PRPS

P

R

PPRU

PRU

PR

U$42

$32

$43$33

$12

$22$23

$13

$11

$21

$31$53

$63

$41 $c1

$c2$c5

$c4

$c3

l220 l330 l110

A1

A2

A3

s220

s330

s110

B1

B2

B3





(1)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1) (2)



$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 0 0 119887)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 1 0 minus1199112 0 1199092)

(3)


$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1) (4)




(5)



$1198885 = (0 1 0 minus1199113 0 1199093) (6)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1)$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1)$1198885 = (0 1 0 minus1199113 0 1199093)

(7)


$1198981 = (0 1 0 minus119911 0 0)$1198982 = (1 0 0 0 119911 0)$1198983 = (0 0 0 0 0 1)

(8)




(9)



$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901) (10)




(11)


$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901) (12)







(13)



$1198885 = (0 1 0 minus1199113 0 1199093) (14)




(15)


1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)


$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1


(19)







R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









Z

X

Y

O






z

y

x

uAw

B

v

b

a

P

R

U

S

PRPS

P

R

PPRU

PRU

PR

U$42

$32

$43$33

$12

$22$23

$13

$11

$21

$31$53

$63

$41 $c1

$c2$c5

$c4

$c3

l220 l330 l110

A1

A2

A3

s220

s330

s110

B1

B2

B3





(1)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1) (2)



$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 0 0 119887)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 1 0 minus1199112 0 1199092)

(3)


$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1) (4)




(5)



$1198885 = (0 1 0 minus1199113 0 1199093) (6)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1)$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1)$1198885 = (0 1 0 minus1199113 0 1199093)

(7)


$1198981 = (0 1 0 minus119911 0 0)$1198982 = (1 0 0 0 119911 0)$1198983 = (0 0 0 0 0 1)

(8)




(9)



$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901) (10)




(11)


$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901) (12)







(13)



$1198885 = (0 1 0 minus1199113 0 1199093) (14)




(15)


1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)


$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1


(19)







R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1) (2)



$12 = (0 0 0 0 c120579 s120579)$22 = (1 0 0 0 0 119887)$32 = (1 0 0 0 1199112 minus1199102)$24 = (0 1 0 minus1199112 0 1199092)

(3)


$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1) (4)




(5)



$1198885 = (0 1 0 minus1199113 0 1199093) (6)



$1198881 = (1 0 0 0 1199111 0)$1198882 = (0 0 0 0 0 1)$1198883 = (1 0 0 0 1199112 0)$1198884 = (0 0 0 0 0 1)$1198885 = (0 1 0 minus1199113 0 1199093)

(7)


$1198981 = (0 1 0 minus119911 0 0)$1198982 = (1 0 0 0 119911 0)$1198983 = (0 0 0 0 0 1)

(8)




(9)



$1198881 = (1198901 0 0 0 11989011199111 minus 11989111199101 0)$1198882 = (0 0 0 0 minus1198911 1198901) (10)




(11)


$1198883 = (1198901 0 0 0 11989011199112 minus 11989111199102 0)$1198884 = (0 0 0 0 minus1198911 1198901) (12)







(13)



$1198885 = (0 1 0 minus1199113 0 1199093) (14)




(15)


1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)


$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1


(19)







R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018













(13)



$1198885 = (0 1 0 minus1199113 0 1199093) (14)




(15)


1199112 minus 11991111199102 minus 1199101 =11989111198901 (16)


$1198981 = (1 0 0 0 1199113 0)$1198982 = (0 1198901 1198911 11989111199102 minus 11989011199112 0 0)$1198983 = (0 0 0 0 0 1)

(17)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1

119891119894 + V minus 120589 (18)




119865 = 119889 (119899 minus 119892 minus 1) + 119892sum119894=1


(19)







R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018










R = R (120574 119911)R (120573 119910)R (120572 119909)

= [[[


]]]

(20)



(21)



(22)


119909 = 0119910 = minus119886c120573s120574120574 = arctan(119904120572119904120573119888120572 )

(23)


L119894 = a119894 + p minus b119894 = 119897119894l1198940 + 119904119894s1198941 (24)


1199042119894 minus 2L119879119894 s1198941119904119894 + L119879119894 L119894 minus 1198971198942 = 0 (25)






= [[[[

1199042120593 + 1198882120593119888120601 119904120593119888120601 (119888120601 minus 1) 119888120593119904120601119904120593119888120601 (119888120601 minus 1) 1198882120593 + 1199042120593119888120601 119904120593119904120601

minus119888120593119904120601 minus119904120593119888120601 119888120601]]]]

(27)



(28)




150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









150∘330∘

180∘

120∘

300∘

270∘

240∘210∘

0∘


minus60

minus40

minus20

0

20

40

60

(∘

)


0∘

120∘

330∘

300∘

270∘

240∘210∘

180∘

150∘

0∘30∘60∘60∘

40∘

20∘

minus20 0 20

minus40

minus20

0

20

40

(∘

)


60∘

120∘

330∘

300∘

270∘240∘210∘

180∘

150∘

0∘30∘60∘

40∘

minus3 0 3

minus10

minus5

0

5

10

(∘

)





119904min le 119904119894 le 119904max (29)



119902min le 1199024 le 119902max (30)




1198772min le 1198772119894 = 119886 cos( 1199041119894 ∙ 1198971198940100381610038161003816100381611990411198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198772max (31)



1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018










1198773min le 1198773119894 = 119886 cos( 1199044119894 ∙ 1198971198940100381610038161003816100381611990441198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198773max (32)


1198774min le 11987753 = 119886 cos( 1199046119894 ∙ 1198971198940100381610038161003816100381611990461198941003816100381610038161003816 100381610038161003816100381611989711989401003816100381610038161003816) le 1198774max (33)



119877119886119899119896 ($1198881 $1198882 $1198883 $1198884 $1198885) = 3 (34)




Start


Position Inverse

Joint Workspace


End

Y

Y

Y

Y

Y

N

N

N

N

z = z + Δz

= 0

= 0

le amx

= + Δ

le ＧＲ

= + Δ



le Task Task

z = zＧＣＨ

z le zＧＣＨ






30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









30∘330∘

0

10

20

30

40

50

60

70

300∘

270∘

240∘

210∘180∘ 150∘

120∘

90∘

0∘

60∘60∘

40∘20∘

0

20

40

60

y

para

sitic

mot

ion)


Reachable workspace

Task workspace





2PRU-PRS2PRU-PRPS



7 Conclusions






Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018











Data Availability




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018


































Journal of












Journal of







Volume 2018






Volume 2018








workspace analysis of a hybrid kinematic machine tool with...

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