Kinematics and statics analysis of a novel 5-DoF parallel manipulatorwith two composite rotational/linear active legs

Yi Lu a,b,n, Peng Wang a, Shaohua Zhao a, Bo Hu a, Jianda Han c, Chunping Sui c

a College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR Chinab Parallel Robot and Mechatronic System Laboratory of Hebei Province, Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministryof National Education, PR Chinac State Key Laboratory of Robotics, Shenyan Institute of Automation, Liaoning 110016, PR China

a r t i c l e i n f o

Article history:Received 12 May 2013Received in revised form21 July 2013Accepted 27 July 2013Available online 7 September 2013

Keywords:Parallel manipulatorKinematicsStaticsComposite active legs

a b s t r a c t

A novel 5-DoF parallel manipulator (PM) with two composite rotational/linear active legs is proposedand its kinematics and statics are studied systematically. First, a prototype of this PM is constructed andits displacement is analyzed. Second, the formulas are derived for solving the linear/angular velocity andacceleration of UPS composite active leg. Third, the Jacobian and Hessian matrices are derived andformulas for solving the velocity, statics and acceleration of this PM are derived. Third, a reachable workspace is constructed using a CAD variation geometric approach. Finally, the kinematics and statics of thisPM are illustrated and solved. The solved results are veried by the simulation results.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Currently, various lower-mobility parallel manipulators (PMs)have been studied and applied widely due to their very goodperformances in terms of accuracy, rigidity, ability to manipulatelarge loads, simple in structure, actuators close to base, and easy tocontrol [1]. Generally, a PM with ve degree of freedoms (DoFs) isthe lower-mobility PM and has attracted much attentions due toits potential applications for parallel machine tool normal machin-ing 3D freedom- surface, arm of robot, legs of walking robots,medicine surgical operator, the tunnel borer, the barbette of warship, and the satellite surveillance platform [1,2]. In this aspect,Gao et al. synthesized some 5-DoF PMs with sub-chain structurelimbs [3,4]. Fang and Tsai [5] synthesized a class of over-constrained 5-DoF PMs with identical limbs by screw theory.Zhu and Huang proposed 18 fully symmetrical 5-DoF PMs with3R2T [6] and analyzed the singularity of six PMs using screwtheory and Grassmann geometry [7]. Wang and Wu et al. [8,9]proposed a redundantly actuated PM of a 5-DoF hybrid machinetool and studied its inverse dynamics by virtual work principle.Piccin [10] and Mehdi presented the architecture of some 5-DoFPMs with 3T2R for semi-spherical workspace, Gosselin [11] et al.proposed a 5-RPUR PM with identical limb structures and studiedtheir inverse/forward displacement.

Sangveraphunsiri et al. [12] designed a unique hybrid 5-DoFPM based on an H-4 family PM with 3T1R and a single axisrotating table. Zhang and Gosselin [13] established kinetostaticmodeling of N-DoF PMs with passive constraining leg. Li andHuang [14] designed a 5-DoF hybrid serial-parallel machine toolbased on TriVariant PM and analyzed its kinematics. Lu and Hu[15,16] proposed a 5-DoF 4SPSSPR PM and a 5SPSUPU PM andstudied their kinematics. Above mentioned 5-DoF PMs include amoving platform and xed base, ve active legs with one actuator,or a passive leg, and relative studies of kinematics basically focuson inverse/forward displacement, velocity, acceleration, or singu-larity analysis.

Up to now, it is has been a signicant issue to develop the5-DoF PMs with less oscillating active legs for avoiding interference,with larger reachable workspace and dexterous workspace forincreasing dexterity or avoiding singularity, and with more actuatorsattached on or close to base for decreasing vibration and increasingprecision. Meanwhile, it is has been a signicant and challengingissue to establish their mathematic model of velocity, acceleration,statics and dynamics. Lu et al. [17] proposed a 3-leg 5-DoF PMincluding two SPS-type active legs and a PRRPR composite leg withthree actuators (the bold symbol represents active joint) and a 3-leg5-DoF PM including two SPS-type active legs and one UPU-typecomposite active leg with three actuators [18], and studied theirkinematics/statics and workspace. However, both the PRRPR-typeand the UPU-type composite active legs are quite complicatedin structure due to one composite leg including three actuators.In addition, the precision, stability, and load-bearing capability may

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Robotics and Computer-Integrated Manufacturing

0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.rcim.2013.07.003

n Corresponding author at: College of Mechanical Engineering, Yanshan University,Qinhuangdao, Hebei 066004, PR China. Tel.: 86 15243477660; fax: 86335857031.

E-mail address: luyi@ysu.edu.cn (Y. Lu).

Robotics and Computer-Integrated Manufacturing 30 (2014) 2533

be decreased due to the vibration of the actuator being far frombase in composite active legs. Our motivation is to develop a 5-DoFPM with three more simple active legs and a larger workspace.Therefore, this paper is focuses on a novel 5-DoF 2UPSSPR PMwith two UPS-type composite rotational/linear active legs and aSPR-type linear active leg, and establishing the inverse/forwarddisplacement, velocity, acceleration and statics of this PM in orderto simplify the structure of composite leg and increase its capabilityof load-bearing and stability. It is has been found by pre-simulationanalysis that when rotating the rst revolute joints of universaljoints attached on base, the stability and the capability of loadbearing of this PM can be increased and the force situations can beimproved, reachable and orientation workspace can be increasedand is symmetry. Thus, comparing with other 5-DoF PMs, this PMpossesses following advantages: (1) less (only 3) oscillating activelegs for avoiding interference between legs easily, (2) larger work-space for avoiding singularity easily, (3) better dexterity for normalmachining 3D freedom-surface and complex operating, (4) moreactuators attached on or close to the base for decreasing vibrationand increasing precision. Therefore, this PM has potential applica-tions for the 5-DoF parallel machine tools, legs of walk robots, handof surgical manipulator, the micro manipulators, the sensor, thetunnel borer, the barbette of war ship, rescues robot, and the satellitesurveillance platform.

2. Displacement analysis of 5-DoF PM with two compositerotational/linear active legs

2.1. Characteristics and DoF of this PM

A novel 5-DoF PM with two composite rotational/linear activelegs is shown in Fig. 1a. It includes a xed base B, a movingplatform m, 2 UPS (active universal joint-active prismatic joint-spherical joint) composite rotational/linear active legs ri (i1, 3)and 1 SPR (spherical joint-active prismatic joint-revolute joint)linear leg r2. Here, m is an equilateral ternary link b1b2b3 withthree sides li l, 3 vertices bi, and a central point o. B is anequilateral ternary link B1B2B3 with three sides LiL, three verticesBi, and a central point O. Each of the UPS-type compositerotational/linear active legs ri (i1, 3) connects m to B by an activeuniversal joint U attached to B at Bi, an active leg ri with a activeprismatic joint P, and a spherical joint S at bi. The SPR-type

active leg r2 connects B to m by a spherical joint S attached to Bat B2, an active leg r2 with a active prismatic joint P, and a revolutejoint R attached to m at b2. U is composed of two cross revolutejoints Ri1 and Ri2. Let, {m} be a coordinate frame o-xyz xed on mat central point o, and {B} be a coordinate frame O-XYZ xed on Bat central point O, || be parallel constraint,?be a perpendicularconstraint. The geometric constrains (Ri1?Ri2, Ri2?ri, R?r2, R||l2, x||l2, Ri1 being coincident with OBi, X||L2, i1, 3) are satised in thisPM.In addition, Ri1 is connected with rotational actuator motor.

In this PM, the number of links is g08 including one platform,three cylinders, three piston-rods, and one base; the number ofjoints is g9 including one revolute joint R, three prismatic jointsP, two universal joint U, and three spherical joints S; the locatedDoFs of the joints are Mk1312233, passive DoFM00. Based on revised Kutzbach Grbler equations in [1,2], DoFof this PM is calculated as below

M 6g0g1 g

k 1MkM0

6 89113 12 23 30 5 1

2.2. Displacement of this PM

The position vectors Bi (i1, 2, 3) of Bi on B in {B}, the positionvectors bim and bi of bi on m in {m} and {B} can be expressed asfollows:

Bi XBiYBiZBi

264

375; bim

xbiybizbi

264

375; bi

XbiYbiZbi

264

375; o

XoYoZo

264

375;

RBm xl yl zlxm ym zmxn yn zn

264

375; bi RBmbimo: 2

here, o is the position vector of point o on m in {B}, (Xo, Yo, Zo) arethe components of o; RBm is a rotational transformation matrixfrom {m} to {B}; (xl, xm, xn, yl, ym, yn, zl, zm, zn) are nine orientationparameters of m, their constraint equations can be obtained fromRefs. [1,15].

Symbol list

PM parallel manipulatorM DOF (degree of freedom) of PMB xed base; m moving platformR, P revolute joint and prismatic jointU, S universal joint and spherical joint{m} a coordinate frame o-xyz xed on m at o{B} a coordinate frame O-XYZ xed on B at OBi position vectors of Bi on B in {B}bim, bi position vectors of bi on m in {m} and {B}RBm rotational transformation matrix from {m} to {B}xl, xm, xn, yl, ym, yn, zl, zm, zn nine orientation parameters of m, , three Euler angles of mri, i the ith linear leg, its unit vectorei, ei the distance from bi to o, its unit vectorEi the distance from Bi to Ov, a linear velocity and acceleration of m, angular velocity and acceleration of m

vin, V general inverse/forward velocityain, A general inverse/forward accelerationi, i angular velocity and acceleration of rii1, i1 angular velocity and acceleration of ri about Ri1i2 angular velocity of ri about Ri2vbi the velocity of m at point bi||, ? parallel and perpendicular constraintJ a 66 Jacobian matrixH a 6 layer 66 Hessian matrixF, T a concentrated force and a concentrated torqueFai active force applied onto ri and along riTaj active torque applied onto ri and about Rj1Fc constrained force exerted on r2 at B2hri a 66 sub-Hessian matrix corresponding to Fai

(i1,2,3)hj1 a 66 sub-Hessian matrix corresponding to Taj

(i1,3)hc a 66 sub-Hessian matrix corresponding to Fc

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 253326

bim, Bi and bi can be obtained from Eq. (2) as follows:

bm1 e2

q

10

264

375; b2m

0e

0

264

375; bm3 e2

q10

264

375;

B1 E2

q

10

264

375; B2

0E

0

264

375; B3 E2

q10

264

375;

b1 12

qexleyl2Xoqexmeym2Yoqexneyn2Zo

264

375; b2

eylXoeymYoeynZo

264

375;

b3 12

qexleyl2Xoqexmeym2Yoqexneyn2Zo

264

375; q 3p : 3

here, e is the distance from bi (i1, 2, 3) to o, E is the distance fromBi to O.

Let , , be three Euler angles of m. RBm is formed by threeEuler rotations of (ZY1X2), namely a rotation of about Z, followedby a rotation of about Y, and a rotation of about X2. Here, Y1 isformed by Y rotating about Z by , X2 is formed by Z1 rotatingabout Y1 by . Let be one of the (, , ), set ccos , ssin ,ttan , ctc tan . Thus, RBm is derived as below

RBm cc csssc cscsssc ssscc ssccss cs cc

264

375;

xl cc; xm sc; xn s;yl csssc ; ym ssscc ; yn cs ;zl cscss ; zm ssccs ; zn cc : 4

Thus, each of (xl, xm, xn, yl, ym, yn, zl, zm, zn) in Eq. (2) isexpressed by (, , ) from Eq. (4). Let ei be the distance from bi too, Ei be the distance from Bi to O. The length ri (i1, 2, 3) and theunit vectors i of the active linear legs, and the vectors ei of lines eican be solved from Eqs. (1)(3) as follows:

ri jbiBij; i biBi=ri; ei bio 5

1 12r1

qexleyl2XoqEqexmeym2YoEqexneyn2Zo

264

375; 2 1r2

eylXoeymYoEeynZo

264

375;

3 12r3

qexleyl2XoqEqexmeym2YoEqexneyn2Zo

264

375; e1 e2

qxlylqxmymqxnyn

264

375;

e2 eylymyn

264

375; e1 e2

qxlylqxmymqxnyn

264

375; x

xlxmxn

264

375; q 3p : 6

Based on the structure constraints R||x and R?r2, x20 issatised. Thus, a constraint equation is derived from x20 andEq. (6) as below:

eylXoxleymYoExmeynZoxn 0) XoxlYoxmZoxn Exm 7

Let Ris be the xed line of Ri2 at started position, Ris||Li are satised.Let i1 be the rotational angular of Ri2 about Ri1. Thus, based onstructure constraints Ri1?Ri2 and Ri2?ri, the formulae for solvingcos i1 are derived as the following:

R11 12

q

10

264

375; R31 12

q10

264

375; R1s 12

1q

0

264

375;R3s 12

1q0

264

375;

Ri2 i Ri1ji Ri1j

ci1 R1 URi2i 1;3: 8

When given ve pose parameters (Xo, Yo, , , ), the inversedisplacements of this PM are solved from Eqs. (4), (5), (7), and (8)as follows:

r21 G2e2qEXoEYoqeExmeG1eEqylqxm3xlym=2;r22 Ge22eG12EeymYo;r23 G2e2qEXoEYoqeExmeG1eEqylqxm3xlym=2;

G X2oY2oZ2oE2; G1 ylXoymYoynZo;i1 acrcosRis URi2 i 1;3;

Zo ExmXoxlYoxm

xn XocEYosct: 9

When given ve input parameters (r1, r2, r3, 11, 31), theforward displacement kinematics formulae of this PM can bederived from Eqs. (4), (6), (7), and (9).

P, r3

U, B3

S, b1

P, r1

b2, R S, b3

m

motor R11

B2, S

B

R12

motor R31

R32

U, B1

P, r1

U, B1 O

S, b1 o

va

e1

1

vb

11

X Y

Z

x y

z

P, r3

r2, P

FT

b2, R

b3, S

vr1

vr3

B2, S

2 vr2

3

Zo

Xo

Yo

11,R11

R12, 12r1 1211

31

323 3 B3, U

R||Fc

FaFa1

Ta1

Ta3 Fa3

11

l1

L1

l2

R1s

R3s

Fig. 1. A prototype of 5-DoF PM with two UPS-type composite rotational/linearactive legs and one SPR-type linear active leg (a) and its force situations (b).

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 2533 27

3. Velocity and statics of this PM

Suppose there are a vector and its skew-symmetric matrix ^.They must satisfy

xyz

264

375; ^

0 z yz 0 xy x 0

264

375;

E1 0 00 1 00 0 1

264

375 ; ^; ^

2 ^^;^T ^:

10

Here may be one of the vectors i, ei, Ri2, (i1, 3).Let V be a general forward velocity of m; v and be the linear

and angular velocities of m at o; A be a general forward accelera-tion of m, a and be the linear and angular accelerations of m at o.They are expressed as follows:

V v

; v

vxvyvz

264

375;

x

y

z

264

375;

A a

; a

axayaz

264

375;

x

y

z

264

375: 11

The scalar velocities vri of ri along ri (i1, 2, 3) have been derivedin [10] as follows:

vri JriV ; Jri Ti ei iT 16 : 12

A velocity vbi of m at point bi is solved as below

vbi vriii rii vriii ri v ei 13

here i is angular velocity of ri.In these linear UPS legs ri (i1, 3), each of the U includes two

crossed revolute joints Ri1 and Ri2, Ri1 is xed on the base B, someconstraints (Ri1?Ri2 and Ri2?ri) are satised. Thus, i can beexpressed as below

i i1Ri1i2Ri2 14

here, i1 is the angular velocities of ri about Ri1 (i1, 3), i2 is theangular velocities of ri about Ri2.

Cross-multiplying both the sides of Eq. (14) by ri, from Eqs. (10)and (13), it leads to

i1Ri1 rii2Ri2 ri i ri

vbivrii i Uivbivbi Uii

i i vbi ^2i vbi

^2i v ei ^2i v ^

2i e^i 15

Dot-multiplying both the sides of Eq. (15) by Ri2, it leads to

i1Ri1 riURi2 vbivriiURi2 RTi2vbi RTi2v ei 16

Dot-multiplying both the sides of Eq. (15) by Ri1, it leads to

i2Ri2 riURi1 RTi1^2i v e^i 17

From the structure constraints (Ri1?Ri2, Ri2?ri, R?r2, R||l2, x||l2, Ri1being coincident with OBi, X||L2, i1, 3) and Eq. (10), some

formulae are derived as follows:

Ri2 i Ri1ji Ri1j

; RTi2 RTi1^iji Ri1j

;

Ri1 Ri2 iRi1 UiRi1

ji Ri1j;

ri U Ri1 Ri2 ri1Ri1 Ui2ji Ri1j

;

v ei ve^i E e^i V ;

v eiT VTEe^i

" #i 1;3

eiURi2 eiU Ri2

T e^iR^i2 VT033 033033 e^iR^i2

" #V :

18

From Eqs. (16)(18), the following equations are derived:

i1 RTi2v eiri U Ri1 Ri2

Ji1V i 1; 3

Ji1 RTi2 E e^i

h iri URi1 Ri2

RTi1^iE e^i

ririRi1 Ui2;

i2 RTi1^

2i v e^i

Ri2 riUR1i

RTi1^2i E e^ih i

ri URi1 Ri2V : 19

The angular velocity i of UPS-type linear legs ri is derived fromEqs. (14), (10), (18) and (19) as follows:

i 2

j 1ijRij

Ri1RTi2Ri2RTi1^

2i

ri U Ri1 Ri2E e^i

h iV

Ri1RTi1^iRi1RTi1^i^i

ririRi1 Ui2E e^i

h iV 20

The force situation of the 2UPSSPR PM is shown in Fig. 1b.The workloads can be simplied as a wrench (F, T) applied onto mat o. Here, F is a concentrated force and T is a concentrated torque.(F T) are balanced by three active forces Fai (i1, 2, 3) applied onand along active leg ri, two active torques Taj (j1, 3) applied onactive leg rj and about Rj1, a constraint force Fc exerted on r2 atpoint B2. Let c be the unit vector of Fc, and d be the arm vectorfrom point o to Fc. Since Fc do not do any work during themovement of m, there must be

FccUvd FccU 0 )cT c dT

h iV 0;

0 JcV ; Jc cT c dTh i

:

21

Let vr2 be a translation velocity along r2, there must beFc vr20, i.e. Fc?r2. Let Fc be a torque of Fc, there must beR ( Fc)0. In this case, Fc must intersect with spherical joint Son r2 and Fc||R. Thus, the unit vector c of Fc is the same as x, i.e.cx.

The general inverse and forward velocities vin and V can bederived from Eqs. (12), (19) and (21) as follows:

vin J66V ; V J1vin;vin vr1 vr2 vr3 11 31 0

T;

J Jr1 Jr2 Jr3 J11 J31 Jch iT

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 253328

T1 e1 1TT2 e2 2T

T3 e3 3T

RT11^1r1r1RT1112

RT11^1e^1r1r1RT1112

RT31^3r3r3RT3132

RT31^3e^3r3r3RT3132

cT d cT

26666666666666664

37777777777777775

: 22

here, J is a 66 Jacobian matrix.Based on the principle of virtual work, three active forces Fai

(i1, 2, 3), two active torques Tj (j1, 3), and a constrained forceFc can be solved from Eqs. (21) and (22) as follows:

Fa1Fa2Fa3Ta1Ta2Fc

26666666664

37777777775

Tvr1vr2vr311

310

2666666664

3777777775 F

T

TV 0)

Fa1Fa2Fa3Ta1Ta2Fc

26666666664

37777777775JT 1 F

T

23

4. Acceleration of this PM

4.1. Angular acceleration of UPS legs rj about Rj1

Since rotational actuators are connected with Ri1, the angularacceleration i1 of ri about Ri1 must be solved. Based on theconstraints (Rj1?Ri2 and Rj2?rj), j1 is derived from Eqs. (18)(20), and (24) by differentiating j1 in Eq. (19) with respect to timeas follows:

j1 _i1 v ejURj2rj U Rj1 Rj2

v ejURj2rj URj1 Rj2

v ejURj2rj U Rj1 Rj2rj U Rj1 Rj22

v ejURj2v ejU_Rj2

rj U Rj1 Rj2

v ejURj2_rj U Rj1 Rj2rj U Rj1 Rj2

rj U Rj1 Rj22

RTj2a ej ejv ejUj Rj2

rj U Rj1 Rj2

RTj2v ejRj1 Rj2T v ejrj URj1 j Rj2

rj URj1 Rj22

RTj2a ej ejv ejT Rj2 j

rj U Rj1 Rj2

v ejTRj2Rj1 Rj2T v ejrj U Rj1 Rj22

1rj URj1 Rj2

(RTj2 E e^j

h iA

VT033 033033 e^jR^j2

" #VVT

Ee^j

" #C E e^jh i

V

)

Jj116AVT hj166V 24a

here hj1 is a 66 sub-Hessian matrix corresponding to Taj (j1, 3).It can be solved as follows:

hj1 1

rj U Rj1 Rj2C Ce^je^jC e^jR^j2Ce^j

" #;

C R^j2Rj1RTj1^jRj1RTj1^j^j^jRj1Rj1 Rj2T

rjrjRj1 Uj2: 24b

4.2. Acceleration kinematics of this PM

The scalar accelerations ari of ri along ri (i1, 2, 3) have beenderived for the lower-mobility PMs with linear active legs in [15].The general inverse/forward accelerations ain and A of this PM arederived from Eq. (22) and, (24) as follows:

ain JAVTHV ; A J1ainVTHV;ain ar1 ar2 ar3 11 31 0

T;

H hr1 hr2 hr3 h11 h31 hc T

;

hri 1ri

^2i ^2i e^i

e^i^2i rie^i^i e^i^

2i e^i

" #66

; hc 033 c^c^ c^d^

66

;

hj1 1

rj U Rj1 Rj2C Ce^je^jC e^jR^j2Ce^j

66

; i 1;2;3; j 1;3:25

here J is a 66 Jacobian matrix and can be solved from Eq. (22). His a 6 layer 66 Hessian matrix and is composed of six sub-Hessian matrices. Each of the hri (i1, 2, 3) is a 66 sub-Hessianmatrix corresponding to Fai, and can be obtained from Ref. [15].Each of the hj1 (j1, 3) is a 66 sub-Hessian matrix correspond-ing to Taj, and can be solved by Eq. (24b). hc is a 66 sub-Hessianmatrix corresponding to Fc, and can be obtained from Ref. [15].

5. Reachable workspace and orientation of PM

Aworkspace is a critical index for evaluating the characteristics ofPM. A reachable workspaceW of a PM is all the positions that can bereached by the center of m in the limited extension of active legs.Generally, W is formed by a family of similar spatial surfaces, whichare cascaded from a lower boundary surface Sl to an upper boundarysurface Su using the loft command. Each of the similar spatialsurfaces is formed by a family of similar spatial curves ck (k0, 1,,n1) using the loft command [19]. When set L, l, the maximumextension rmax and the minimum extension rmin of active legs ri (i1,2, 3), a W of the 2UPSSPR PM is constructed by varying ri in therange (rminrmax) using CAD variation geometric approach [19], seeFig. 2. The construction procedures are explained as follows:

Step 1: set L100 cm, l60 cm, rmax130 cm, rmin100 cm,r10 cm, n1(rmaxrmin)/r, 1101801 and 3101.Step 2: set r1rmin, r2rmax and r3rminkr (k0, 1,,n11).Step 3: set k0, and increase 11 from 0 by each time, andinspect the interferences among ri and m in the simulationmechanism. When the interferences occur, stop increasing 11.Step 4: solve the position components (Xo, Yo, Zo) of m usingauto-solving function of solid works, and insert them into thesimulation mechanism.Step 5: construct a spatial curve c0 using the curve pass throughXYZ command.Step 6: repeat the steps 3 through 5 above, except that set k1,, n1, respectively. Thus, other curves ck are created.Step 7: construct the rst upper boundary surface Sul1 fromn11 curves ck by the surface loft command.Step 8: construct the second upper boundary surface Sul2 usingsimilar construction procedures of Sul1, except that settingr1rmax.

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 2533 29

Step 9: construct the two lower boundary surfaces Sll1 and Sll2using similar construction procedures of Sul1 and Sul2, exceptthat setting r2rmin.Step 10: construct Sul from Sul1 and Sul2, construct Sll from Sll1and Sll2.Step 11: repeat steps 2 through 10, construct other surfaces kSuand kSl (k2, 3,,n1) except that set 3101801, respectively.Step 12: construct W from kSu and kSl using the loft command.

The results show that W of the 2UPSSPR PM is larger thanthat of the 4SPSSPR PM [16] and that of the 3-leg 5-DoF PM witha UPU-type composite active leg [18]. When set central point of mat Xo1.14, Yo2.03, Zo11.27 cm, three Euler angles (, , ) canbe varied in 1001 or more by giving r210, 11, 12, 13, 14, 14.5 cm,and varying ri, (i1, 3), see Fig. 3.

It is shown from simulation results that this PM has a largereach-workspace and a good dexterity.

6. Numerical results of kinematics/statics of PM

Let L120 cm, l60 cm. When given (Xo, Yo, , , ), see Fig. 4aand b, the inverse displacements of this PM are solved, see Fig. 4gand h; the inverse velocities and accelerations of this PM aresolved, see Fig. 4il.

When given ve input parameters (r1, r2, r3, 11, 31), see Fig. 4gand h, the forward displacements of this PM are solved, see Fig. 4aand b. The forward velocities and accelerations of this PM aresolved, see Fig. 4cf. When given workloads F[20,30,60]T N,T[30, 30, 100]T N m, the three active forces (Fa1, Fa2, Fa3), two

active torques (Ta1, Ta3), and a constrained force Fc of this PM aresolved, see Fig. 4m and n. The solved results are veried by itssimulation mechanism in Matlab/SimMachinc, see Appendix A.

7. Conclusions

A novel 5-DoF parallel manipulator (PM) with two compositerotational/linear active legs is designed. This PM has the followingmerits: (1) a fewer (only three) active legs for avoiding inter-ference between oscillating legs easily; (2) larger workspace foravoiding singularity easily; (3) better dexterity for normal machin-ing 3D freedom-surface and complex operating; (4) more

209.

07

127.

39207.71

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88

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15.2

7

Fig. 2. A reach workspace of 2UPSSPR PM: (a) the tope view and (b) the frontview.

Fig. 3. Orientations of m for 2UPSSPR PM at a xed point (X11.4, Y20.3,Z112.7 mm) of m.

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 253330

actuators attached on or close to the base for decreasing vibrationand increasing precision.

The Jacobian and Hessian matrices are derived and the mathe-matic formulae are derived for solving inverse/forward displace-ments, velocities, accelerations of platform and UPS legs. The

mathematic formulae are derived solving active forces, activetorques and constrained force.

When SPR-type active leg is exchanged with other type activelegs, such as UPU, RPS, or PRS-type legs, only formulae for solvingconstrained forces are changed, other formulae can be reused.

180160140120100

806040200

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Fig. 4. Solved results of kinematics and statics of 2UPSSPR PM.

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 2533 31

When the inertia wrenches and gravity of the legs are trans-formed into a part of the dynamic workload applied on platform atits center, its dynamics can be solved.

This PM has potential applications for the 5-DoF parallelmachine tools for normal machining 3D freedom-surface, themicro manipulators, the sensor, the surgical manipulator, dexter-ous leg of walk robots, the tunnel borer, the barbette of war ship,and the satellite surveillance platform, etc.

Acknowledgments

The authors would like to acknowledge (1) Project (51175447)supported by National Natural Science Foundation of China (NSFC),(2) key planned project of Hebei China Application Foundation No.11962127D, and (3) Project 2011-002 of State Key Laboratory ofRobotics.

Appendix A

The simulation processes of solving kinematics and statics ofthe 2UPSSPR PM in Matlab/SimMachincs are explained asfollows:

Step 1: generate an M le in Matlab, and set initial parametersof joints and body as in Table A1.Step 2: construct a block diagram of the UPS-type active leg, seeFig. A1.Step 3: construct a block diagram of SPR-type active leg, see Fig. A2.Step 4: construct a block diagram of the 2UPSSPR PM fromthe block diagrams of UPS-type active leg and SPR-type activeleg, see Fig. A3.Step 5: generate a simulation mechanism of the 2UPSSPR PMfrom the block diagram of 2UPSSPR PM, see Fig. A4.

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Table A1Block diagram parameters of 2UPSSPR PM.

ri block r1 (SPR) r2 (UPS) r3 (UPS)

Ground(Ai) A1T A2T A3T

Spherical joint S1 S1T

Universal joint R21T, R22T R31T, R32T

U1, U3 vs. (U1) vs. (U3)Body1(bottom) A1T A2T A3T

Body1(top) a1 T a2 T a2 T

Body1(CG) (a1TA1T)/2 (a2TA2T)/2 (a2TA3T)/2Prismatic P r1T/2 r2T/2 r3T/2Body2(bottom) A1T A2T A3T

Body2(top) a1T a2 T a2 T

Body2(CG) (a1TA1T)/2 (a2TA2T)/2 (a2TA3T)/2revolute joint R1T

Spherical joint S2, S3 S2(N/A) S3(N/A)

ground1constant

Joint actuator2

Ramp1 Joint actuator1

B31 universal

B32 S3

r3

P1

Fig. A1. A block diagram of the UPS-type active leg.

Joint Actuator

ground constant

S1 B11

P1 B12 R1

r1

Fig. A2. A block diagram of SPR-type active leg.

body sensor

scope to workspace

RRPS2RRPS1SPR

to workspace1

Upper platform

simout3

r3r1 r2

t

clock

Fig. A3. A block diagram of 2UPSSPR PM.

Fig. A4. A simulation mechanism of 2UPSSPR PM.

Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 253332

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Y. Lu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 2533 33

Kinematics and statics analysis of a novel 5-DoF parallel manipulator with two composite rotational/linear active legsIntroductionDisplacement analysis of 5-DoF PM with two composite rotational/linear active legsCharacteristics and DoF of this PMDisplacement of this PM

Velocity and statics of this PMAcceleration of this PMAngular acceleration of UPS legs rj about Rj1Acceleration kinematics of this PM

Reachable workspace and orientation of PMNumerical results of kinematics/statics of PMConclusionsAcknowledgmentsAppendix AReferences