Mech. Macb. Theory Vol. 28. No. 4, pp. $53-561, 1993 0094-114X/93 $6.00 + 0.00 Pnnted in Great Britain. All rights m~-rved Copyright ~ 1993 Perllamoa Pres Lid

E C H E L O N F O R M S O L U T I O N O F D I R E C T

K I N E M A T I C S F O R T H E G E N E R A L

F U L L Y - P A R A L L E L S P H E R I C A L W R I S T

CARLO INNOCENTI Dipartimento di lngegneria deUe Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia,

Facolt~ di lngegneria, Universit:i di Bologna-Viale Risorgimento, 2-40136 Bologna, Italy

VINCENZO PARENTI-CASTELLI lstituto di lngegneria Meccanica, Facolt:i di Ingegneria, Universit:i di Salerno, 84081 Baronissi,

Salerno, Italy

(Received 17 September 1991: received for publication 9 July 1992)

Abstract--This paper presents the echelon form direct position analysis of a class of fully in-parallel actuated mechanisms for the orientation of a rigid body with a fixed point. The mechanisms have a structure which is the most general one for manipulator spherical wrists with three degrees of freedom and fully-parallel arrangement. The analysis results in a two-equation system in echelon form; the first equation is of 8th order and the remaining is linear. As a consequence, when a set of actuator displacements is given, eight configurations of the mechanism are possible. A numerical example confirms the new theoretical result.

I. INTRODUCTION

The need to orient a rigid body with a fixed point, i.e. to perform a pure orientation of the body in three-dimensional space, is common to many fields. Solar panels, telescopes, radar antennas, mirrors for laser beams, and manipulator end-effectors are some of the most outstanding applications.

The ability to orient a body and keep it steady asks for devices that can be challenging from the viewpoint of mechanical design and control. Serial mechanisms, which are open chains with all pairs actively controlled, have been used to this purpose. Recently, however, new kinematic structures based on parallel arrangements, which consist of one or more closed kinematic chains where only some pairs are actuated, have focused an increasing attention. Indeed, when precise position control, high stiffness and favourable load capacity to mechanism weight ratios are the major requirements with respect to the working space volume and manoeuvrability, parallel mechanisms can provide higher performances than serial ones [I, 2].

A reliable and easy to build mechanism, together with the availability of an efficient mathematical model on which the control algorithm must be based, are also items that play an important role when an optimal mechanism is sought.

For parallel mechanisms the direct position analysis (DPA), that is to find the location (position and orientation) of the output link (platform) when a set of actuator displacements is given, is a difficult problem. What makes the DPA challenging is the non-linear nature of the equations involved which provide more than one solution.

Numerical iterative methods, the most common way to solve the DPA, consider the system of closure equations as a whole and try to converge to as many solutions as possible, missing information about their number.

Considerably more significant, if achievable, is the echelon form arrangement of the closure equation system, which provides a set of new equations organized as follows: the first equation contains only one unknown, the second equation contains, in addition to the unknown of the first equation, one more unknown, and so on. Therefore a sort of decoupling exists among the equations of the echelon form system, and the solution of the system itself can be accomplished, through successive steps, by finding at each step the roots of one equation in one unknown.

Generally, the DPA of parallel mechanisms leads to an algebraic system of closure equations; when echelon form is possible, and every equation is of degree 4 at most, a closed form solution

553

5.54 CARLO INNO~NTI and V[NCENZO PARENTI-CASTELLI

of the DPA can be obtained. More frequently the first equation of the echelon form system is of an order higher than 4, and the remaining equations are linear; hence the order of the first equation--to be numerically solved---directly provides the number of possible assembly configur- ations of the parallel mechanism. As a rule, an echelon form solution allows a deeper insight into the kinematic behaviour of the mechanisms and the implementation of more efficient control strategies. Echelon form solutions for various classes of mechanisms have been obtained [2-6].

In spite of their very attractive performances, few parallel mechanisms for pure orientation of a rigid body have been proposed in the literature. In particular, a fully-parallel three-degree-of- freedom spherical wrist with coaxial actuators has been introduced in Ref. [7]. The design of a similar wrist from the viewpoint of the optimization of some kinematic performances has been considered in Ref. [8]. Two parallel spherical wrist architectures with actuated and passive revolute pairs have been proposed in Ref. [9]. The direct kinematics of parallel spherical wrists with simplified geometry, i.e. with base and platform both planar, and/or multiple spherical kinematic pairs, has been addressed in Refs [10, I l]. Once the centre of the wrist has been positioned, the DPA is reduced to the successive solution of two second-order algebraic equations, hence obtaining four solutions at most. Only in Ref. [5] the DPA of the general-geometry fully-parallel spherical wrist has been afforded, and the upper bound for the number of possible solutions was estimated to be 16. In Ref. [12] the DPA of a planar Assur group has been presented and, for the extension to the spherical case--which could represent a spherical wrist--an 8th order polynomial was foreseen.

This paper presents the echelon form DPA of a class of fully-parallel mechanisms for the pure orientation of a rigid body; every mechanism of the class, when the actuators are all locked, reduces to the structure shown in Fig. I. The class collects all the mechanisms characterized by the same kinematic features with regard to the DPA: indeed, when the input is given a mechanism becomes a statically determined structure, and the solution of the DPA of the mechanism is equivalent to finding all the possible closures of the structure itself.

The mechanisms of the class have all spherical pairs distinct one from another, a feature that makes the design very attractive from a mechanical point of view. The common structure they all reduce to when actuators are locked, represents the most general fully-parallel constraint for a rigid body with one fixed point. The platform (see Fig. I) is connected to the base through a spherical pair centred at point Q, while three legs meet both base and platform at three distinct points A, and Bj (j = I, 3), where spherical pairs are centred. The rotational freedom of each leg about the line through its two terminal spherical pairs does not affect the platform location; however, it can be eliminated by substituting a universal joint for one spherical pair. When actuated pairs and various leg arrangements are introduced to vary the leg lengths, a mechanism for the pure orientation of the platform is obtained. Examples are shown in Fig. 2 and Fig. 3, where different leg arrangements with actuated prismatic (P) and revolute (R) pairs have been introduced.

zp B l~~p la t fo rm - B 2

O B 3

L 1

\ \ \ \ \ \

Fi 8. I. The sphcrical wrist structure.

General fuly-parallel spherical wrist

/ y

Fig. 2. The fully parallel spherical wrist mechanism.

555

In the paper the DPA of the spherical wrist structure is solved in echelon form, leading to a final 8th order equation with only one unknown. Hence, for a given set of leg lengths eight closures of the structure in the complex field are possible.

Finally a numerical example is reported which confirms the new theoretical result.

2. DIRECT POSITION ANALYSIS

2. I. Closure equations The geometry of the spherical wrist structure, schematically shown in Fig. I, is given. In

particular, the lengths Lj of legs AjBj ( j = 1, 3) are known and the position of points Q, Ai, A2 and A~ are given in an arbitrary reference system W b fixed to the base. Without loss of generality, system Wb is chosen with origin coincident with point Q and ,- axis, zb, directed from point Q to point At; axis x~ has an arbitrary direction perpendicular to :~. Moreover, the position of points Q, BI, B2, and B 3 are given in an arbitrary reference system Wp fixed to the platform. Again without loss of generality, system Wp is chosen with origin coincident with point Q and z axis, zp, directed from point Q to point B,; axis xp has an arbitrary direction perpendicular to ,'p.

~B1 B 2

R

\\\\\\ Fig. 3. A parallel spherical wrist mechanism.

55~ CARLO INNOCENTI and VtNCENZO PARENT1-CA.s"UELLI

Zp

"--L--. / J

\ \ \ \ \ x

Fig. 4. Auxiliary open chain.

The location of the platform with respect to the base can be parameterized as follows. When the leg lengths are given, the triangle A, QB~ remains defined and can be considered as a binary link t connected to base and platform through two revolute pairs with axes zb and z~ respectively. Indeed, if legs Lz and L3 are momentarily removed from the platform and the base, the two-degree-of-freedom auxiliary open chain formed by base, link t and platform can be considered (see Fig. 4). The body t can rotate about zb and the platform can rotate about zp showing that the location of the platform (thus the closure of the structure) can be uniquely parameterized by the two angles 0, and 0: that define the relative position of link t with respect to the base, and of the platform with respect to link t. Angle 0~ is measured counterclockwise from the half plane (x~,zb), xb positive. Angle 02 measures the counterclockwise rotation about zp to superimpose triangle t on the half plane (xp, g,), xp positive. The values of 0~ and 02 defining the location of the platform can be determined by imposing that the distances A2 B2 and Aj Bj equal the leg lengths L2 and L3, respectively. To this respect, let us consider the 3 x 3 rotation matrix R for the coordinate transformation from system Wp to system Wo:

ct " c2 - u " st " s2 - c t " s 2 - u " su " c2 v " s , [ 1

R = s j . c 2 + u . c l . s 2 - s ~ . s 2 + u . c ~ . c 2 - v ' c t ] , (I)

t ' • S 2 V " C 2 U

which has been written according to the Denavit-Hartenberg notation with:

c j = c o s 0 / s j=sin0j ( j = l , 2 ) ; u = c o s a ; v = s i n a , (2)

where a is the angle between the axes zb and zp. With reference to Fig. 4 it can be written:

+ -

cos cc = " sin a = (I - cos 2 ~)1/2 (3) 2 " I A i I ' I B I I '

where all quantities can be determined, and A I and B, are position vectors of points A~ and B~ in Wb and in Wp, respectively.

The closure equations for the spherical wrist structure are:

( R ' B : - A : ) 2=L~, ( R ' B s - A 3 ) 2=L~, (4)

which can be written as follows:

A t - R . B , = ( A ~ + B ~ - L ~ ) / 2 , A t . R - B , = ( A ~ + B ] - L ~ ) / 2 , (5)

where Aj is the position vector of the point Aj in W b and Bj is the position vector of the point Bj in Wp.

General fully.parallel spherical wrist 557

Holding the positions Aj = (x o, x2s, x3s)-and Bs = 0%, Y2s, Y3s), where x~s, x2j, x~j and Yo, Y2~, Y3s are the Cartesian components o f posit ion vectors A s and B s in W~ and W~, respectively, expansion o f equat ions (5) yields:

DI~" ct • c2 + D ~ • ct • s2 + D3~ • st • cz + D ~ • st • s2 + D ~ • ct

where

+ D s . s t + D T ~ . c 2 + D ~ ' h + D g , = O ( i = ! , 2 ) , (6)

Dl~ = x, /" Yls + u " xzS" yzj , (7.1)

D2~ = u • x2s" Yts - xts" Yzs, (7.2)

D3, = xzs" Yu - - U • X t j " Y 2 / , (7.3)

D4, = - u • xt/" Ytj - x2s " Y2/, (7.4)

D~ = - v • xzs" Y3/, (7.5)

D6, = v • x t / " Y3/, (7.6)

DT, = v • x~j" Yzj, (7.7)

D,, = v • x,s" Ytj, (7.8)

D~, u " x~j )'3/ (x~s + x~/ + 2 2 2 2 L~)/2, (7.9) = • _ x 3 j + y u + y 2 j + y 3 1 -

with i -- i, 2 and j = 1 + i. Each equat ion in system (6) is cosine--sine linear in 01 and 02; i.e. in each equat ion any term, if dependent on 0 s q = I, 2), contains either cos 0 s or sin Oj.

By substituting in system (6) the well-known relations:

i - t~ 2 . t , c ,= I +t--'-'~; s , = I +t2," (8)

where t, = tan(0,/2), and by rationalizing, it follows:

a a , ' t l ' t ~ = O ( i = 1 , 2 ) , (9) h.k -0.2

with the constant quantities a ~ (h, k = 0, 2; i = I, 2) defined as follows:

a0o, = Dr1 + Ds~ + DT~ + Dg,, (10.1)

aot, ffi 2 . ( O ~ + D.,), (10.2)

ao2, ---- - -Dl~ + Ds, - DT~ + Dg,, (10.3)

al0 ~ = 2" (D3~ + D~), (10.4)

art ~ ffi 4" D ~ , ( 1 0 . 5 )

a12 t - - 2 ' ( - D j i + Ds) , (10.6)

azo, = - Dr1 - Ds, + DT~ + D 9 i , ( 1 0 . 7 )

a2t , = 2 " ( - D ~ + Ds,), (10.8)

a . ~ = D u - D51 - D7~ + D ~ . ( 1 0 . 9 )

2.2. S y s t e m solution and back subst i tut ion

In order to solve system (9) one o f the two unknowns, for instance h , must be eliminated. By evidencing the dependence o f the two equations on t 2 it can be written:

E ' t ] + F ' t 2 + G = O , L ' t ~ + M ' t 2 + N = O , (11)

558 CARLO ty.,~oct.,~rn and VINCENZO PARENT1-CA.VrELLI

where E, F, G, L, M and N are second-order polynomials in fi, which can be written as:

where

E = Z es't~, 1" 0.2

F = ~ f s ' t ~ , .i - 0.2"

1- 0.2

L= Z j - 0.2

M = ~ m,. t~, j - 0.2

N = Z ,,,-ti, j=0.2

eo = az2t fo = azll go = a2ol 1o = a22z

el = a121 f l ~ d l l l gl = alOl [I : al22

e2 = ao21 f,- = aoll g: = aool 12=ao2,,

The elimination of the unknown t2 from system (8)

0 E F

E F G

0 L M

L M N

(12.1)

(12.2)

(12.3)

(12.4)

(12.5)

(12.6)

mo -- a212 n 0 ~ a202

ml = all2 /'/i = al02

m2 = aol., n2 = a0o2. (I 3)

can be worked out by imposing the condition:

G

0 = 0 , (14)

N

0

where only the unknown t, appears. The left-hand side of equation (14) is the eliminant of equations (I I), and equation (14) itself is the condition under which the closure equations (9) have the same solutions for t2.

By developing equation (14) it follows that:

(E . N - G • L) ~ + ( E . M - F . L ) . ( G • M - F . N) = 0, (15)

represents an 8th order algebraic equation in ft. Hence it can be written in the form:

"3" t~ = 0, (16) 1-0,~

where the coefficients % ( j = 0, 8) depend, through equations (13), only on the link geometry of the spherical wrist structure. Their analytical expression is reported in Table 1.

Table 1. Analytic expression for the % ¢oemcients of equation (16)

w I . . e['n[ +e2.[g z . ( m ] - 2" Iz'n 2 ) - f2 .mz ' n,] +f~" 6"n2-f2 'g2"6"m: +g~'l~

w , = e t ' [ 2 ' e z ' n J + g 2 . ( m ~ - 2 ' / z ' n : ) - f z ' m 2 ' n z ] + g , ' [ e : ' ( m ~ - 2 ' l z ' n 2 ) - f 2 . / 2 ' m 2 + 2"g2"l~l

+ m l ' [ e 2 ' ( 2 ' g 2 " m 2 - f 2 " n z ) - f 2 " g 2 ' l : ] + f , ' ( - e 2 'm 2"n2 + 2 " f , ' / 2 " n : - g : ' / 2 " m : ) + l l ' ( - 2 " e : ' g : ' n ,

+f['n2-] '2'g2"m2+ 2"g~'l 2)+n, '[2"e~'n 2 + e 2 " ( - f 2 ' m z - 2 .g2 ' l z )+ f~ . l z ]

~,o - Co. [2. e2" ,,~ + g2" ("~ - 2 . 6 'n2) - A ' "2 "n2] + d ' n~ + e," {g,-(, , ,~ - 2 . 6 " n:)

+ m ~ . ( 2 ' g , . m 2 - f 2 ' n 2 ) +n~ . ( 4 . e : . n : - f : ' m 2 - 2 ' g : ' 1 2 ) - f l ' m z ' n 2 - 2.g:-I~ .n2]

+go ' [e2"(mJ- 2 ' 12 - n t ) - f 2 " l z ' m z + 2 ' g z ' l ] ] + f l ' [ l i ' ( 2 ' f 2 " n : - g 2 " m z ) +m." ( - e 2. n : - g : . I,)

+ ( 2 " f z ' l : ~ - e z.m2)'n, - g l "lz' m2] + too' [e2' (2 'g2-m2-f2"n2) - f2"g2 ' lz ] +gt ' [11 " ( - 2 " e 2 'n 2

- f 2 " m z + 4 " g 2 " 1 2 ) - 2 ' e 2" l : 'n , + m I ' ( 2 ' e 2.m 2 - f 2 . / 2 ) ] + f o ' ( - e 2 - m : ' n 2 + 2 . f + ' l : ' n : - g z - 1 2 ' m 2 )

+1 o ' ( - 2 . e 2 . g : 'n: + f ~ ' n z - f z ' g : ' m 2 + 2"g~'l 2 )+no '12"e i 'n2+e2 . ( - f , . 'mz - 2"gz ' l : )+f[ ' t2]

+f~" I 2"n z + e2z .n I + I," [OC~-- 2" e 2.g,)" n, - f2"g: .m,] - e , "f. 'm, .n, + e:'g2"m] +g~. I~ +g~" I~ continued on next page

G e n e r a l f u l l y -pa r a l l e l s p h e r i c a l wr i s t

Table i . - - c o n t i n ~ d

559

ws,,.eo.[2.e , .n~+g i .(m~-2"l z • nz)+ m , "(2 "gz'm:-]'z'n2)+n, .(4 .e z .nz-J'z.mz-2 'tz "Iz)

- - f l " m z ' n z - 2 . e z . l l ' n 2 ] + e ," [ f ~ - ( - m I "n z - m : ' n I ) + g , - ( - - 2 . 1 1 ' n : - - 2 . i 2 . n I + 2" m I " m z)

4"2 "e z • OS~ - - f2" r e l " nl -- 2 ' g 2 ' 11" nl "4" gZ" re~] 4. gO' [ e l ' (m~ -- 2" ! : . nz) + I , . ( - 2 . e , . n 2 - / 2 " m ,

4 . 4 - 1 2 " iz) - 2 - e z • i 2 • n, 4- m , . (2 - • 2 • m z - f z ' Iz) - f l " iz" mz "4" 2 ' g , " 1~] + f o " i l l " (2 "f2" nz - g : " mz )

+re," (-ez" nz - gz"/2) - el" re," n, + 2 "f,' I~- nz + (2 "fz" I, - ez" m2)" nl - g,' I,. m,] + g. ~. (2 'f2" nz

- - g z ' m z ) 4 " & l ' ( - - 2 " e z ' n z - - f z ' m z + 4 ' g : ' / 2 ) - - 2 " e I . g z . n z + ( f [ - 2" e 2 - g z ) " n, - f : ' g 2 " m , 4- 2 - g [ . i , ]

4.reo" [e l" (2 ' g z ' mz --.fz" nz) 4 . f i " ( - -e2 'n2 - e z !z) - e , f z " nl 4" gl • (2 . e 2 • m 2 - f 2 " Iz) 4. 2 . e z • e l " nil

- f 2 "g2" I l l + no" [e l" ( 4 . e2 ' nz - f 2 ' m : - 2 " . '2" /2) + 2" e~ "nl -4-]', ' (2 " f2 ' / 2 - e : . m : )

- e z " ~ " re, - 2 . e2" g," 12 + ( f~ - Z. e: "g2)" It I + f ~ " ( I , - n: + /2" n, ) + 2 ' eT" n, . n2 + A ' 1/," (2 ".fz .n, - e , - re,)

- e , . m, . n, + el" ( - I t ' m: - 12" m,)] + g , . [1~. ( - 2- ez" n, - ~ ' m, ) + e, ' m~ + 2. g," I~] + 2- g~" I , . /2

w 4 ~ e 2 o ' n ~ + e o ' [ ~ ' ( - r e l " n z - m z ' n l ) + g o ' ( m ~ - 2 " 1 2 " n 2 ) + g J " ( - 2 . 1 , ' n 2 - 2 " 1 2 " n , + 2 - m , . re: )

4 -m o • (2 "g2 "mz - - f z " n2) 4- n o • ( 4 . • z • n z - f z ' m2 - 2 . gz" Iz) 4" 4 . e I • n I • n 2 - f o " m 2 ' n, -- 2 " g , ' I o • n 2

+ 2 ' e z ' n ~ - f z . m , . n , - 2 ' g 2 " l , ' n I + g 2 " r e ~ ] + f o ' l e , ' ( - r e , ' n 2 - r e : ' n , ) + ~ ' ( 2 " l , ' n 2 + 2.12 "n,)

4.10 • (2 'fz .nz - g z " mz) 4. too' ( - e z " nz - g z ' Iz) + II - (2 ' fz" at - g 2 " m, ) - e z • me" nl 4- (2 "fz" Iz - e2" mz) "no

+ el" ( - I," re, - 12" eel) - go" 12' re,] + go' [e,. ( - 2 ' ll" n2 - 2.12' at + 2 . re,. re:) + Io' ( - 2 . e," n, - ~ . m,

+ 4 " g z . l , ) + l l ' ( - - 2 . e z ' n I - ~ . m l ) - 2 . e , . I z ' n o + ~ . ( - I I . m : - I z . m , ) + m o . ( 2 . ¢ z . r e z - f z ' l z ) 4 . e 2 . m ~

+4 .e l ./," 12 + 2. e, . I~l + g. [el' ( -2" e, "n2--2"ez'n,)+ff,'n:+f," (2 "f2' n,--g, "m2--e,'re,) + e, " ( - 2 . e~ . nl - ] ' 2 " re, + 4 . e2 " I , ) + ( f ~ - 2" e, " e , ) " no - ] ' 2 " e, " ,,Io 4. 2" e~ 121 + too" (e, . ( - / , " n, -.t"2. n,

+ 2 ' g , ' re2 + 2 "gz" ml ) + / , ' ( - e2 "hi - g, "I, - g , " I , ) - ez "f2" no + g , " (2" e , ' m I - f : ' It )l + no" [2 ' e~ "n 2

+ e l " ( 4 . e2" nl - f ~ ' re2 - f , ' re, - 2 . ~ . / 2 - 2" e , ' I t) + f ~ - (2 ' / 2 ' I, - e , ' m~) + / ~ . I, - 2 . e : ' e , I , I +fo" 12.,,, Z. , gz m o + g o " l~ z z + d ' , , ~ 4 . e , ' t ~ ' , ' ( r e Z - 2 ' l l ' , , , ) - A ' r e l ' , , , l + f f ' , ' , ' , , , + e 2 ,,o - A ' e, ' l, ' re, + ,', ' ' ~ ~ + e l l ,

+ e l . Io ~

w~ = eo " [ fo " ( - m , " n , - m z " n, ) + eo " ( - 2 " ll " n , - 2 "12 " n, 4 " 2 ' m l ' m , ) + / o ' ( - - 2 " g l ' n , - - 2 " , e , ' n , )

4"too' ( - - / i ' nz - - f z ' nl + 2. ,¢1" re, + 2 ' e~" ml ) + no' (4 . el • nz -6 4 . e 2 • at - f , • m, - f z ' ml - 2 ' e, • 12 - 2. ez' I, )

+ 2 . e , ' n~ + e~" (m~ - Z. I , . hi) - / , " m, . n,] +/o" [Io" (2 . f , ' n z + 2 .f~. n, - g,- re2 - .¢:' m, ) + ,no. ( - e l . n,

- e , ' n l - e , ' l : - e 2 " l , ) - e , ' r e , . n , + 2 ' f ~ . I~ "nl + ( - e I . m z - e z . m ~ + 2 . f ~ . I , + 2 . f 2 . 1 1 ) . n o + g o . ( - - l l . m z

- - I z . m ~ ) - - & e . l l . m , l + e o . [ I o . ( - - 2 . ¢ ~ . n ~ - - 2 . e z . n l - - f l . m 2 - - f , . r e , + 4 . g , "l , + 4 " e 2 " l ~ ) + e l ' ( m ~ - 2 " l ~ . n , )

+ ( - 2 "e,'/2 - 2 "e2' I,)' no + reo' (2 .e~. re2 + 2. ez- re, - / i • I 2 - f~ . I,) -f~ • I, "re2 + 2 "el" I~] +]'o ~' (I,. n z

+ 1 2 " n l ) + 2"e~o'n, "n~ + m o ' [ e , " (2"g , "ml -- f~ ' n i ) + ( - - e l "f~ - - e : ' . f ~ ) ' n o - - f~ "g, "l~] + l o ' [ - - 2 " e , "e, "nl + f ~ . n ,

+ ( - - 2 . e l ' g : - - 2 . e ~ . g ~ + 2 . ~ . ~ ) ' n o - - f ~ . g , . r e , + ( - - ~ . & z - - ~ . g , ) . r e o + 2 . g ~ . l , l + n o . [ 2 . e ~ ' n l

4 . e I " ( - - f~ " re, -- 2" e l ' l , ) + f ~ . l , I + 2 . e I . e2 . nZo + (e, . g,2 4- e: " e , ) " mZo 4. 2 " eZo " 1 , ' 1 z + 2 " g , " e: " Io

wz - eo . [ fo . ( - m o . nz - m , . n~ - mz " no) + go " ( - 2 " lo " n: - 2 " l~ " nl - 2 " l~ " no ÷ 2 " reo " m : + m~ ) + lo " ( - 2 " e~ . n~

- 2 . g 2 . n o ) + m o - ( - f ~ . n I - f z . n o + 2 . g ~ . m l ) + n o . ( 4 . e ~ .n~ - f ~ " r e 1 - 2 . g , . l l ) + 2 . e z . n 2 o + g , . m o ]

+eo~" (2 "no'n~+n~)+fg'(Io'n2+l, .n, + I, ' no)+fo" [g" (2 'f~ .n, +2" /2 ' n o - s , - m , - e , 'reo)

4" ,'n o " ( -- e, • n I -- e' z • n o -- g~" !, ) + (2 .f~ • I t -- e , " m I ) ' n o 4" go" ( -- g ' re , -- I, • rel - - / z ' too)] "4" .¢o" [/o" ( - 2 . e, . n,

- - 2 ' e~ " n o - - f , "re, - - / 2 " m o + 4 . g , • I , ) -- 2 ' e , . I t • n o + m o • ( 2 - e~. m, - - ~ • I , ) + e : " re~ + 2 - g : " I~,J + e~ . n o

2 2 4./o" { ( /~ - 2" • I - e l ) ' no - - A "g," tool - e, 'f~" too" no 4" e, "e," reo + eo" (2 . /o" /2 + 1~) -4- .¢~. Io ~

w~ s e o . [ f o . ( - m o . n ~ - m ~ . n o ) + e o . ( - 2 . 1 o . n ~ -- 2 . 1 ~ . n o 4. 2 . m o . m l ) + 2 . e l . n o - - f ~ . m o . n o - - 2 . g l . l o . n o

+ e l " reo~l + f g " (g ' n, + I, 'no) + 2 ' eg" no- n, +fo" I/o" (2 . / , " no - e," 'no) - el" too' no +.co' ( - / o " re,

-12" reo)] + go" llo" ( - 2 . e , . no - / , "reo) + e , . reo ~ + 2 "el" Io ~ ] + 2 . go ~. Io' I,

Wo , , e~o . nZo + ¢o . l eo . (re~o - 2 .1o . no ) - f o . mo . no l +.l'Zo . lo . no - / o . eo . lo . mo 4. g~o . I o

Equat ion (16) has eight solut ions in the complex field. Let t~k (/c -- I, 8) be a generic root o f equat ion (16); then the po lynomia l s E, F, G, L, M, N assume definite values, and system (i 1) can be linearly solved in the u n k n o w n s ( t l ) and (h) . In particular the so lut ion h , o f 12, together with

560 CARLO INNOCENTI and VIr~L'F_'SZO PARENTI-CASTELLI

Table 2. B~. B: and B~ (x. yr:) coordinates in reference system W b for all closures

Closure No. Point x y :

B~ -0.62359300 -0.72988477 0.72000000 I B: 0.24738360 -0.63107461 1.24118741

B, - I. 13909607 0.85731825 1.54918222

B I - 0.45610510 - 0.84472962 0.72000000 2 B, 0.54258048 -0.94695361 0.89938050

B~ - 0.40795784 0.50643197 2.00239783

B I - 0.60484099 0.74549808 0.72000000 3 B, 0.31209619 0.73650559 I. 16625704

B 3 0.10044801 2.06669265 -0.38883377

B I 0.45853364 -0.84341384 0.72000000 4 B 2 1.24165277 -0.67097786 0.08992837

B~ 1.09592040 0.61500551 1.68914970

B I 0.95980694 -0.01925177 0.72000000 5 B: 0.94037525 0.96081736 0.43877603

B~ - 0.26563732 0.47620150 2.03351148

BI - 0.28335509 0.91722947 0.72000000 6 B, 0.12342221 0.25944732 1.38472165

B~ 1.46992178 1.45576604 0.39060876

B I 0.92689519 0.24993061 0.72000000 7 B: 1.40528417 - 0.12356488 - 0.09953956

B~ 0.93450327 1.87719529 -0.18799327

B I 0.67969178 0.67795213 0.72000000 8 B: 1.33982820 0.45199160 -0.02374844

B~ 0.11721967 2.06575756 -0.38910830

the value t~k for tt, is a solution of system (9). Moreover it is straightforward, by means of relations (8), to find matrix R [see equa t ion (I)], and the coordinates Bj, = R • Bj of points Bj ( j = !, 3) in reference system W h. it can be concluded that, in the complex field, each of the eight roots of equation (16) leads to a closure of the spherical wrist structure.

3. CASE STUDY

The DPA of a three-degree-of-freedom fully-parallel spherical wrist schematically represented in Fig. 2 has been performed.

The wrist geometric data, in arbitrary length unit, are: A, = ( 0 , 0 , I), A2=(I .1 ,0 ,0 .95 ) and A3 = (0.1, 1.5, I) in reference system W b, and B; = (0, 0, 1.2), B2 = (1, 0, I) and B3 = (0.1, 1.85, 1) in reference system Wp. The leg lengths are: Lt = 1, L2 = i.! and L3 = !.5.

Eight closures of the wrist have been determined that are reported in Table 2 in terms of the Cartesian components of points Bj, j = I, 3, of the platform measured in reference system Wb. All solutions resulted to be real. It has also been verified that for each closure the distance between all pairs of points connected by a leg equals the corresponding leg length.

4. C O N C L U S I O N S

In this paper the echelon form direct position analysis of a class of fully-parallel mechanisms for the orientation of a rigid body with a fixed point has been presented. The analysis provides a final 8th order polynomial equation in one unknown. That is, when the leg lengths are given, eight closures of the mechanisms in the complex field are possible.

The class of mechanisms considered here collects the most general arrangements for manipulator spherical wrists with three degrees of freedom and fully-parallel architecture. The architecture has no concentric spherical pairs, that makes it attractive from a design standpoint, and can represent an alternative scheme to serial-chain-based spherical wrist designs.

An example has been reported which confirms the new theoretical result.

Acknowledgement--Th¢ financial support of the Ministry of Education (MPI-40% and MPI-60%) is gratefully acknowl- edged.

Genera] fully-parallel spherical wrist 561

REFERENCES 1. K. J. Hunt, Trans. ASME JI Mech. Autonm Des. 105, 705--712 (1983). 2. M. Gri~s and J. Duff'y, J. Robot. Systems 6, 703-720 (1989). 3. L. Wei, J. Duffy and M. Griffts, 21st ASME Mech. Conf, DE-Voi. 25. pp. 263-269, Chicago, I11., Sept. (1990). 4. C. lnnocenti and V. Parenti Castelli, Trans. ASME JI Mech. Des. (in press). 5. J. P. Merlet, Les Robots Parail~les. Hermes, Paris (1990). 6. V. Parenti CasteUi and C. lnnocenti, 21st ASME Mech. Conf., DE-Vol. 25, pp. 111-116, Chicago, 111., Sept. (1990).

Also Trans. ASME JI Mech. Des. 114, 68-73 (1992). 7. H. Asada and J. A. Cro Granito, Proc. IEEE Int. Conf. Robot. Autor, m, pp. 244-250, St Louis, Mo. (1985). 8. C. Gosselin and J. Angeles, Trans. ASME JI Mech. Trans. Automn Des. ! i i , 202-207 (1989). 9. F. Pierrot and E. Bombre, ~ Int. Workshop Adcances in Robot Kinematics, Linz, Austria, Sept. (1990).

10. P. Nanua and K. J. Waldron, Proc. 8th int. Syrup. Theory and Practice of Robots and Manipulators, Ro.man.sy "90, pp. 131-139, Cracow, Poland, Jul. (1990).

il . C. D. Zhan 8 and S. M. Song, 21st ASME Mech. Conf. DE-Vol. 25, pp. 271-278. Chicago, i11, Sept. (1990). 12. S. Li and G. K. Matthew, 7th IFToMM WId Congr. Theory of Machines and Mechanisms, pp. 141-145. Sevilla, Spain,

Sept. (1987).

E C H E L O N - F O R M I G E P O S I T I O N S A N A L Y S E D E S A L L G E M E I N E N V O L L I G P A R A L L E L A N G E T R I E B E N E R K U G E L H A N D G E L E N K S

Zusammenfassung--Es wird die direkte echelon-formige Positionsanalyse einer Klasse vollig paralle- langetriebener Getriebe f~r die Orientierung eines starren K6rpers um einen festen Punkt beschrieben. Die Analyse f'tihrt zu einer polynomialen Gleichun8 achten Grades yon einer einzigen Unbekannte, die zeigt, daB. bei jeder beliebi8 gegebenen Kombination yon Stellantrieb, acht Gestaltungen des Getriebes m6glich sind. Die Struktur der Getriebe, die hier analysiert werden, entspricht der allgemeinsten Anordnung, die man im Bereich parallelgeschalteter Manipulatorskugelhandgelenke yon drei Freiheitsgraden finden kann. Die erfolgreiche Anwendung der Theorie auf einen Ikzugsfall wird schlieBlich dargestellt.