Optimal design of the trivariant robot t

Optimal design of the TriVariant robot to achieve a nearlyaxial symmetry of kinematic performance H.T.. Chetwynd b a School of Mechanical Engineering, Tianjin University, Tianjin 300072,

Optimal design of the TriVariant robot to achieve a nearly

axial symmetry of kinematic performance

H.T Liu a, T Huang a,b,*, X.M Zhao a, J.P Mei a, D.G Chetwynd b

a

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

b

School of Engineering, Warwick University, Coventry CV4 7AL, UK Received 13 April 2006; received in revised form 23 November 2006; accepted 2 December 2006

Available online 7 February 2007

Abstract

This paper revisits the optimal kinematic design of a 3-DOF parallel mechanism that forms the main body of a 5-DOF reconfigurable hybrid robot named the TriVariant The dimensional synthesis of the 3-DOF parallel mechanism is carried out by introducing a tilt structure angle that enables to achieve a nearly axial symmetry of kinematic performance with respect to the configuration when the properly constrained active limb is vertically or horizontally placed The results are then compared with those of the Tricept robot via an example It concludes that for the same task workspace the mod-ified version of the TriVariant has a very similar kinematic performance to that of the Tricept provided that they both have similar dimensions

Ó 2006 Elsevier Ltd All rights reserved

Keywords: Dimensional synthesis; Parallel kinematic machines; Reconfigurable machines

1 Introduction

It is well known that the existing parallel kinematic machines (PKM) having fewer than six degrees of free-dom (DOF) can be classified into two categories associated with the number and type of DOF of the limbs[1] One class includes the PKM essentially composed of a number of identical constrained active limbs having fewer than six DOF (Delta[2], Sprint-Z3 Head[3], for example) Another class contains the PKM basically composed of a properly constrained passive/active limb and a number of identical 6-DOF active limbs (Tricept[4–8], George V[9]and SKM[10], for example) The most significant feature of the PKM in the sec-ond class lies in that the number and type of DOF of the properly constrained limb are exactly identical to those of the platform Enlightened by the noted Tricept design, Huang et al proposed a novel 5-DOF recon-figurable hybrid PKM named the TriVariant[11]which is a simplified version of the Tricept by integrating the

0094-114X/$ - see front matter Ó 2006 Elsevier Ltd All rights reserved.

doi:10.1016/j.mechmachtheory.2006.12.001

* Corresponding author Address: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Tel./fax: +86 22 27405280.

E-mail address: htiantju@public.tpt.tj.cn (T Huang).

Available online at www.sciencedirect.com

Mechanism and Machine Theory 42 (2007) 1643–1652

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Mechanism and

Machine Theory

functionality of one 6-DOF active limb into the passive one Kinematic and dynamic analyses of the TriVari-ant were also carried out in[12,13]

It is desirable for the PKM having a cylindrical workspace to possess axially symmetrical kinematic perfor-mance This has no problem for a PKM having symmetrical structure but it does for the TriVariant robot since its structure is merely plane symmetrical Therefore, in the previous design proposed by[12], an offset has to be set between the axis of the cylindrical task workspace and the origin of the reference coordinate sys-tem in order to achieve a nearly axially symmetrical kinematic performance However, this arrangement leads

to a tile angle of the properly constrained active limb when the reference point of the end-effector reaches the center of the task workspace

In order to overcome the drawback mentioned above, this paper proposed a modified architecture by intro-ducing a tilt structure angle that allows a nearly axially symmetrical kinematic performance to be achieved with respect to the configuration when the properly constrained active limb is vertically or horizontally placed (seeFig 1) The dimensional synthesis of the modified TriVariant is then carried out and the results are com-pared with those of the Tricept via an example

2 Kinematic equations

The conceptual and detailed mechanical design of the TriVariant robot can be found in[12] For simplicity,

we use the term ‘‘TriVariant’’ to denote the 3-DOF parallel mechanism of the robot hereafter.Fig 2shows the schematic diagram of the TriVariant Here, Bi ði ¼ 1; 2; 3Þ represents the center of the U joint connecting the limb i to the base For convenience, all Biare taken to lie within a plane that has a tilt angle / with respect to a horizontal plane represented by DB0

1B0

2B3 Aiði ¼ 1; 2Þ is the center of the spherical joint of limb i (UPS limb)

A3is the intersection of the axial axis of the limb 3 (UP limb) and its normal plane in which all Aiare placed And A4is defined as the reference point located at the intersection of the two orthogonal axes of the 2-DOF rotating head The terms U, P and S represent respectively universal, prismatic and spherical joints and under-lined P denotes an active prismatic joint driven by a servomotor Establish the reference coordinate system B3–

x3y3z3with y3axis being the rotation axis of the outer ring of the U joint and the z3axis being placed vertically downwards as shown Similarly, the reference coordinate system Bi–xiyiziassociated with limb iði ¼ 1; 2Þ is placed with the zi axis being vertically downwards and the yi axis being parallel to B3B0i Meanwhile, the body-fixed coordinate systems Bi–uiviwi(i = 1,2,3) are also placed where the ui ði ¼ 1; 2; 3Þ axis is coincident with the inner ring’s rotational axis of the U joint and the wiaxis is coincident with the axial axis of the limb

In order to evaluate the kinematic performance, it is necessary to carry out the inverse position and velocity analyses The position vector of A4is given by

which satisfies the constraint equation

Fig 1 The redesigned TriVariant.

where d is the fixed length of A3A4; qi and wi are the length and unit vector of the axial axis of limb

i ði ¼ 1; 2; 3Þ; ai¼ R3ai0; ai0 and biare the constant position vectors of Ai and Bi ði ¼ 1; 2Þ measured in the

A3–u3v3w3 and the B3–x3y3z3, R3is the orientation matrix of the A3–u3v3w3 with respect to the B3–x3y3z3, respectively In order to achieve symmetrical kinematic performance with respect to the x3–z3plane, geometry

of the base and platform, i.e DA1A2A3and DB1B2B3are designed as isosceles triangles with a and b being the side lengths, and aa, aband / being the structural angles as shown inFig 2 Hence

bi¼ b^bi

^

bi¼  cos / cos ab ð1Þiþ1sin ab sin / cos ab

ai0¼ a^ai0

^

ai0¼  cos a a ð1Þiþ1sin aa 0T

; i¼ 1; 2

ð3Þ

where ^ai0 and ^bi are the unit vectors of ai0and bi, respectively Meanwhile, the orientation matrix of the

Bi–uiviwi ði ¼ 1; 2; 3Þ with respect to the B3–x3y3z3can be generated by three sets of angles associated with the rotations about the axes of the U joints

where Ri¼ Ri1Ri2with

Ri1¼

cos ai  sin ai 0

sin ai cos ai 0

2

6

3 7 5; Ri2¼

cos hi sin hisin wi sin hicos wi

0 cos wi  sin wi

 sin hi cos hisin wi cos hicos wi

2 6

3 7

ai¼

p=2 a0

p=2þ a0

8

>

0

b¼ arctanðtan ab=cos /Þ

and ui, viand wirepresent the unit vectors of ui, viand wiaxes, respectively Noted that wi¼ wð ix wiy wizÞT, the rotation angles relate directly to the unit vectors along the limbs by

UPS limb

(i=1)

3

A

3

x

3

y

3

ψ

3

B

3

z

3

θ

1

1w

UP limb

(i=3)

1

A

2

A

3

u

b

α

a

α

1

B

1

u

1

y

1

x

1

z

1 ψ

1 θ

1

w

2

B

φ

1

B′

2

B′

4

A

t

W

H

r

Fig 2 Schematic diagram of the TriVariant.

H.T Liu et al / Mechanism and Machine Theory 42 (2007) 1643–1652 1645

hi¼ arctan wix cos a i þw iy sin a i

w iz

;

wi¼ arcsinðwixsin ai wiycos aiÞ;

Given a specified r, w3and q3can be completely determined by(1)

Hence, R3is fully defined in terms of w3and h3in Eq.(5) So, aiis also known Taking norms on both sides

of re-arranged Eq.(2)leads to

qi¼ jr þ ai bi dw3j;

This allows wiand hiði ¼ 1; 2Þ to be found explicitly via Eq.(5)

Taking the derivatives of Eqs.(1) and (2)with respect to time yields

_r¼ _qiwiþ qiðxi wiÞ  aðx3 ^aiÞ þ dðx3 w3Þ; i¼ 1; 2 ð9Þ where _r is the velocity of point A4, xiis the angular velocity of limb iði ¼ 1; 2; 3Þ, and _qiis the joint velocity of limb iði ¼ 1; 2Þ Taking dot products with wion both sides of(8) and (9)gives

_q3¼ wT

_qi¼ wT

In order to establish the relationship between x3and _r3, take the cross product with w3on both sides of Eq

(8)and note that wT

3x3¼ 0 This results in

x3¼ðw3 _rÞ

Substituting Eq.(12)into Eq.(11)and rewriting in matrix form, finally results in the velocity mapping func-tion of the TriVariant

where J is the Jacobian matrix

J ¼

q3w T

1 aw T

1 w 3 ^ T

1 þdw T

1 w 3 w T 3

q3þd

q3w T

2 aw T

2 w 3 ^ T

2 þdw T

2 w 3 w T 3

q3þd

wT 3

2

6

6

3 7

3 Optimal kinematic design

In this section, dimensional synthesis of the TriVariant will be carried out in order to obtain a set of kine-matic parameters that allows an optimized global kinekine-matic performance to be achieved Firstly, the design variables and a global conditioning index are defined Secondly, the number of design variables is reduced through monotonic analysis Finally, two independent geometric parameters are determined by solving two nonlinear equations representing the requirements of the minimum stroke/UPS limb-length ratio and the nearly axial symmetry of the kinematic performance All the computations are carried out using Matlab 3.1 Workspace and design variables

As depicted in[12], the task workspace Wtof the TriVariant is defined as a cylinder of radius R and height h with H being the distance from B to the upper bound of W as shown inFig 2 Then, define the configuration

when A4is located at the axial axis of Wtas the reference configuration (seeFig 2) At this configuration, we can easily obtain the extreme length of the UP limb in Wt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðH þ hÞ2þ R2

q

where q3 minand q3 max represent the minimum and maximum lengths of the UP limb, respectively

Referring back to Eqs.(1) and (3), we can define aa, ab, /, a, b and H as the design variables of the Tri-Variant provided that R and h have been specified Note that the extreme lengths of the UPS limb, i.e

qimin and qimaxði ¼ 1; 2Þ, are also functions of aa, ab, /, a, b, d, H, R and h, although it is difficult to derive useful explicit relationships for them Thus, it is more convenient to determine qimin and qimaxby inverse dis-placement analysis Without losing generality, let a, d, H, R be normalized by b such that

where kR/b can be considered as a factor representing the ratio of workspace/footprint Meanwhile, let

kh=R¼ h=R represent the ratio of height/radius of Wt For a particular problem, kR/band kh/Rmay be specified according to the requirement of Wt, and kd/bmay be treated as a constant Then, the scope of the dimensional synthesis of the TriVariant can be stated as follows Given kd/b, kR/b, kh/Rand a set of appropriate constrains, determine aa, ab, /, ka/b and kH/bin order to achieve good kinematic performance throughout Wt

3.2 Global conditioning index

It has been well accepted that the condition number j of J is one of the most suitable local conditioning index for evaluating the kinematic performance of a manipulator [14–17] A global conditioning index [14]

which represents the mean of j over Wtwill be used as the global performance index to be minimized



j¼

R

where V denotes the volume of Wt

Given kR=b¼ 0:9, kh=R¼ 0:8 and kd=b¼ 0:4,Fig 3shows the variations of jvs aa¼ 0–90and ab¼ 20–70

for a specified set of ka=b¼ 0:15 and 0:3, kH =b¼ 2, and / ¼ 10 and 20 It can be seen that given kR/b and

kh/R, jtakes the minimum value when ab¼ 30and decreases with the decrease of ka/b Meanwhile, the value

of jis nearly unchanged with respect to aa Without losing generality, aa¼ 30can be specified Furthermore, given ab¼ 30and ka/b, smaller jfor /¼ 20can be achieved in comparison with that for /¼ 10 Computer simulations also show that these observations hold true for other reasonable values in the ranges 1:5 6 kH =b62:5 and 06/ 630, respectively

3.3 Constraints

On the basis of the above analysis, three design variables, i.e ka/b, kH/band / remain to be determined since

aa¼ ab¼ 30 have been optimized

Note that the structure of the TriVariant is symmetrical with respect to the x3–z3plane Hence, j must take maximum values, maxiðjÞ ði ¼ 1; 2; 3Þ at three points located on the boundary of Wtand max1ðjÞ ¼ max2ðjÞ

as shown inFig 4 In order to achieve a nearly axially symmetrical distribution of j with respect to the z axis, the following equality constraint can be set:

This leads to / being the function of ka/band kH/bprovided that other design variables have been specified Given kd=b¼ 0:4, kh=R¼ 0:8 and aa¼ ab¼ 30, Fig 5 shows j vs ka=b¼ 0:15–0:3, kR=b¼ 0:7–1:1 and

kH =b¼ 1:5–2:5 while satisfying Eq.(18) For the time being, the limb length constrains and the joint interfer-ence are not taken into consideration It is easy to see that jincreases with the increase of kR/b, meaning that achieving a large workspace/footprint ratio is at the cost of reducing the kinematic performance Therefore, a compromise should be made to choose k Referring to the Tricept design, k ¼ 0:7–1:1 is recommended It

H.T Liu et al / Mechanism and Machine Theory 42 (2007) 1643–1652 1647

is also seen that a smaller ka/bis helpful to enhance the kinematic performance However, it should remain sufficient room to situate the 2-DOF rotating head and at the same time the UP limb should be as strong

as possible to resist against the torsional moment about its symmetrical axis Therefore, ka/bcan be predeter-mined accordingly

As the reference configuration has been already defined, the maximum joint angles of the U limb can be directly obtained by

0 15

30 45

60 75 90

20 30 40 50 60

705 10 15 20

κ

0 15

30 45

60 75 90

20 30 40 50 60

705 10 15 20

κ

a

( )

b

( )

b

( )

a( )

0 15.

a b

0 30.

a b

0 15.

a b

0 30.

a b

Fig 3  j vs k a/b , a a and a b with k d=b ¼ 0:4, k H =b ¼ 2, k R=b ¼ 0:9, k h=R ¼ 0:8 (a) / = 10°; (b) / = 20.

0

κ

3

x

( ) 3

max κ

( )( )

1 2

max κ

1

φ

2

φ

3

φ

Fig 4 The effect of / on the distribution of j, where /3> /2> /1.

Similarly, the maximum joint angles of the UPS limb should satisfyjw1ð2Þjmax6w0, where w0is its allowable value to avoid mechanical interference It should be pointed out that the constraint over h1(2)is not considered since there is no limit for the current joint axis arrangement

The significant constraint for the detailed mechanical design is the ratio of the UPS stroke to its minimum length This is because the span between two support bearings of the limb should be large enough to provide sufficient lateral stiffness [12] Thus, the corresponding constraint should be set by

l¼q1ð2Þ max q1ð2Þ min

where l0is the maximum allowable value of l; q1ð2Þ maxand q1ð2Þ minare the maximum and minimum lengths of the UPS limb Practical mechanical design would allow l0¼ 0:7–0:8 as recommended by[12]

3.4 Optimal design

Given aa¼ ab¼ 30and ka/b, kd/b, kR/band kh/R, the optimal kinematic design of the TriVariant can the-oretically be formulated as the following constrained nonlinear programming problem:



jðxÞ

x2R

! min

x¼ ðkH =b /Þ

s:t: jw1ð2Þjmax6w0

l 6 l0

max1ð2ÞðjÞ  max3ðjÞ ¼ 0

ð21Þ

Fig 6 shows the variations of j, jw1ð2Þjmax, l and / vs kH =b¼ 1:5–2:5, kR=b¼ 0:7–1:1 given aa¼ ab¼ 30,

ka=b¼ 0:2, kd=b¼ 0:4 and kh=R¼ 0:8 It can be seen that j is monotonically increasing functions of kH/b, and jw1ð2Þjmaxand l are all monotonically decreasing functions of kH/b, meaning that a small kH/bwould be helpful to improve the kinematic performance, but it requires a larger stroke/length ratio and larger joint an-gles Meanwhile, given a level of j, larger kR/brequires smaller kH/band largerjw1ð2Þjmax Moreover, if we as-sume that w0¼ 50the limb length constraint l k H =b

6l0¼ 0:7–0:8 will take in effect This means that the

‘optimized’ kH =b and /*can directly be obtained by solving the nonlinear algebraic equations as follows:

l l0¼ 0

For example, given ka=b¼ 0:2; kd=b¼ 0:4, kR=b¼ 0:8, kh=R¼ 0:8 and l0¼ 0:8, solution of Eq (22) will produce k ¼ 2:1860, /¼ 17:77, j¼ 6:2095 and jw j ¼ 41:7 as clearly depicted inFig 6

4 5 6 7 8

H b

λ

κ

1 2 3

0 30

a b

λ =

0 15

a b

λ =

Fig 5 Variations of  j vs k H/b , k a/b and k R/b with k d=b ¼ 0:4, k h=R ¼ 0:8, a a ¼ a b ¼ 30  (1) k R=b ¼ 0:7, (2) k R=b ¼ 0:9, (3) k R=b ¼ 1:1.

H.T Liu et al / Mechanism and Machine Theory 42 (2007) 1643–1652 1649

1.5 1.7 1.9 2.1 2.3 2.5 10

15 20 25

30

1 3 4

H b

λ

( )

φ °

2

*

φ

0.4 0.5 0.6 0.7 0.8 0.9 1

H b

λ

μ

1 3 4

2

*

H b

λ

4 4.5 5 5.5 6 6.5 7 7.5

1

3 4

H b

λ

κ

2

30 35 40 45 50

1 3 4

max ( )

i

2

H b

λ

0 0 8.

μ =

Fig 6 Variations of  j vs k H/b and k R/b with k d=b ¼ 0:4, k h=R ¼ 0:8, k a=b ¼ 0:2, a a ¼ a b ¼ 30  (1) k R=b ¼ 0:7, (2) k R=b ¼ 0:8, (3) k R=b ¼ 0:9, (4) k ¼ 1:1.

4 Example

Referring to the design parameters of Tricept 605[5], an example is taken to illustrate the procedure of the kinematic design for the TriVariant, and its kinematic performance is then compared with that of the Tricept

As shown in Fig 2, assume that the task workspace Wtis a cylinder of radius R¼ 500 mm and height

h¼ 400 mm, resulting in kh=R¼ 0:8 Then, given aa¼ ab¼ 30, we take ka=b¼ 0:1818, kd=b¼ 0:4711, kR=b¼ 0:8264 and l0¼ 0:8 for a feasible mechanical design Then solution to Eq.(22)gives the geometric parameters and the global conditioning index as shown inTables 1 and 2

Utilizing the parameters given inTable 1, the extremities of limb lengths and jof the Tricept are listed in

Table 2for comparison For the Tricept, a represents the length of the equilateral triangle of the moving plat-form.Fig 7a and b shows the distributions of j in the planes associated with the upper, middle and bottom layers of Wt It can be seen that there is little difference in terms of the global conditioning index jbetween the TriVariant and Tricept, although the kinematic performance of the TriVariant are not exactly axially symmet-rical As seen in Table 2, the minimum and maximum lengths of the UP limb of the TriVariant and the UP limb of the Tricept are identical However, the minimum length of the UPS limb of the TriVariant is smaller than that of the Tricept, but its stroke is a bit longer than that of the Tricept Therefore, it can be concluded the TriVariant would have a competitive kinematic performance compared with that of the Tricept provided that the both have similar dimensional parameters

Table 1

Dimensional parameters of the TriVariant (mm)

Table 2

Comparison of limb lengths and kinematic performance of the TriVariant and Tricept (mm)



a q min , q max – the minimum and maximum lengths of the UPS limb.

b r min , r max – the minimum and maximum lengths of the UP(UP) limb.

-500 0 500

4

5

6

7

8

9

κ

( )mm

-500 0

500 -500 0

500 4

5 6 7 8 9

κ

( )mm

Fig 7 Distributions of j in W : (a) TriVariant, (b) Tricept.

H.T Liu et al / Mechanism and Machine Theory 42 (2007) 1643–1652 1651

5 Conclusions

In this paper, a kinematic design methodology of the TriVariant is revisited The conclusions are drawn as follows:

(1) The tilt angle / plays an important role for achieving a nearly axial symmetry of kinematic performance and the UPS limb length constraint should be taken into account in minimizing the cost function These considerations allow the ‘optimized’ kH =b and /*to be determined by solving two nonlinear algebraic equations

(2) For the same task workspace, the TriVariant has a similar kinematic performance compared with the Tricept

Acknowledgements

This research work is supported by the National Natural Science Foundation of China (NSFC) under Grants 50535010, 50375106 and 50328506, and Tianjin Science and Technology Commission under Grant 043103711

References

[1] T Huang, M Li, M.L Wu, et al., The criteria for conceptual design of reconfigurable PKM modules—theory and application, Chinese J Mech Eng 41 (8) (2005) 36–41.

[2] R Clavel, Delta, a fast robot with parallel geometry, in: 18th Int Symp on Industrial Robot, Lausanne, April, 1998, pp 91–100 [3] N Hennes, Ecospeed: an innovative machining concept for high performance 5-axis-machining of large structural component in aircraft engineering, in: 3rd Chemnitz Parallel kinematics Seminar, Chemnitz, April, 2002, pp 763–774.

[4] K.E Neumann, Robot, US Patent 4,732,525, 1988.

[5] <http://www.pkmtricept.com>

[6] D Zhang, C.M Gosselin, Kinetostatic analysis and design optimization of the Tricept machine tool family, ASME J Manuf Sci Eng 124 (3) (2002) 725–733.

[7] S.A Joshi, L.W Tsai, A comparison study of two 3-DOF parallel manipulators: one with three and the other with four supporting legs, IEEE Trans Rob Autom 19 (2) (2003) 200–209.

[8] S.A Joshi, L.W Tsai, The kinematics of a class of 3-DOF, 4 legged parallel manipulators, ASME J Mech Des 125 (1) (2003) 52–58 [9] H.K Tonshoff, H.K.R Grendel, Structure and characteristics of the hybrid manipulator George V, in: C.R Boer, L Molinari-Tosatti, K.S Smith (Eds.), Parallel Kinematic Machines, Springer-Verlag, London, 1999, pp 365–376.

[10] E Schoppe, A Po¨nisch, V Maier, T Puchtler, S Ihlenfeldt, Tripod machine SKM 400, design, calibration and practical applications, in: 3rd Chemnitz Parallel kinematics Seminar, Chemnitz, April, 2002, pp 579–594.

[11] T Huang, M Li, Z.X Li, A 5-DOF hybrid robot, Patent Cooperation Treaty (PCT), Int Appl PCT/CN2004/000479, 2004 [12] T Huang, M Li, X.M Zhao, J.P Mei, D.G Chetwynd, S.J Hu, Conceptual design and dimensional synthesis for a 3-DOF module

of the TriVariant—a novel 5-DOF reconfigurable hybrid robot, IEEE Trans Rob 21 (3) (2005) 449–456.

[13] M Li, T Huang, J.P Mei, X.M Zhao, D.G Chetwynd, S.J Hu, Dynamic formulation and performance comparison of the 3-DOF modules of two reconfigurable PKMs—the Tricept and the TriVariant, ASME J Mech Des 127 (6) (2005) 1129–1136.

[14] C.M Gosselin, J Angeles, A globe performance index for the kinematic optimization of robotic manipulators, ASME J Mech Des.

113 (3) (1991) 220–226.

[15] J.R Singh, J Rastegar, Optimal synthesis of robot manipulators based on global kinematic parameters, Mech Mach Theory 30 (4) (1995) 569–580.

[16] J Angeles, Fundamentals of robotic mechanical systems, theory, methods, and algorithms, second ed., Springer-Verlag, New York, 2003.

[17] T Huang, D.J Whitehouse, J.S Wang, Local dexterity, optimal architecture and design criteria of parallel machine tools, Annals CIRP 47 (1) (1998) 347–351.