Mech. Mach. Theorr Vol. 25, No. 6, pp. 61 I--621, 1990 0094-114X/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ~ 1990 Pergamon Press plc

DIRECT POSITION ANALYSIS OF THE STEWART PLATFORM MECHANISM

CARLO INNOCENTI and VINCENZO PARENTI-CASTELLI Dipartimento di Ingegneria delle costruzioni meccaniche, nucleari, aeronautiche e di Metallurgia,

Universit~i di Bologna, Viale Risorgimento, 2-40136 Bologna, Italia

(Received 9 August 1989; received for publication 16 January 1990)

Abstract--The direct position analysis of the Stewart platform mechanism (SPM), that is to find position and orientation of the platform when a set of actuator displacements is given, has been performed in closed-form. The analysis has been carded out by referring to a general kinematic model which has been derived on the basis of the characterization presented by Stewart. All the different SPM arrangements can be analysed in an unified way by this model. The analysis leads to a 16th order polynomial equation in one unknown from which 16 different positions and orientations of the platform can be derived. This new theoretical result is confirmed by numerical examples.

1. INTRODUCTION

In recent years an increasing number of papers have focused their attention on the study of parallel mechanisms, mainly for the attractive performances they can offer in robotic applications [1-16]. Parallel mechanisms are kinematic chains containing one or more closed loops, and only a certain number of pairs of the mechanism are actuated. They can be used in several cases as a promising alternative to serial manipulators [1-3] which, on the contrary, are basically open loop chains where all pairs are actuated. Parallel manipulators have some advantages over serial manipulators. Among these are a stiffer mechanical structure with consequent higher natural frequencies, less sensitivity to variation of external load on position accuracy, and a superior position accuracy of the output-controlled link (platform). Parallel mechanisms can also be used as force-torque sensors. The main disadvantages of parallel manipulators are the limited working space and reduced dextrous maneuverability of the platform. Moreover, the direct position analysis for parallel mechanisms, that is, to find the location (position and orientation) of the output link when a set of inputs (actuator displacements) is given, is a difficult analytical problem. Indeed, it involves non-linear equations, and many solutions exist. On the contrary, the reverse position analysis--to determine the input values when the location of the platform is given--is generally an easier problem. The opposite happens to be true when dealing with serial manipulators.

Recently, particular attention has been attracted by the parallel mechanism proposed by Stewart in 1965 [17]. Originally designed for flight simulation, it can today represent a basic component for parallel manipulators. Basically the mechanism (see Fig. 1) has a movable link W (platform) connected to the fixed base by three kinematic chains (legs). The platform is connected to the legs by three spherical pairs, centered at points Pj, P2 and P3. A proper number of pairs is actuated to provide the platform with six degrees of freedom. Stewart suggested four different leg arrangements two of which are shown in Figs 1 and 2, where P and R are for prismatic and revolute pairs respectively and P* and R* represent linear and rotary actuators.

The Stewart platform mechanism (SPM) has been widely studied by many au- thors [3, 5, 7-10, 14-16] who proposed a large variety of arrangements. A careful inspection of these mechanisms shows that they represent a class that, as regards the direct position analysis, can be studied with reference to the same kinematic model, to be called the Stewart platform model, and will be described in detail in the next section.

Another attractive parallel mechanism, frequently considered as a generalization of the SPM, is the one represented in Fig. 3. It has six linearly actuated legs A~B~, i = 1, 6, connecting the platform W to the base through spherical pairs at A~ (or Bj) and universal joints at B~ (or A~). This mechanism has also been extensively studied [4, 8, 12, 13].

611

CARLO INNOCENTI and VINCENZO PARENTI-CASTELLI 612

Fig. 1. One of the Stewart platform arrangements with linear and rotary actuators.

This paper will focus the attention on the direct position analysis of the SPM, which has been addressed already by several authors [3, 7, 9, 10, 14-16]. Many of them solved it numerically by iterative methods [9, 14] or, in practice, by measurements with sensors [7, 10]. Only a few faced the task of solving it in closed form [15, 16].

Solving the direct position analysis in closed form would give more insight on the kinematic behavior of the mechanism. Besides, together with the exact number of locations of the platform, it would allow: (1) evaluation of the effect of the input errors on the location of the platform: (2) precise platform location, with respect to the base through accurate control of the input values. Moreover, devices based on SPM could be used as position-orientation sensors of a rigid body; (3) development of a new type of force-torque sensors with high compliance and, consequently, with improved sensitivity, while maintaining the position-orientation control of the platform [15].

Griffis and Duffy[15] solved a particular case of SPM in closed form. They considered the case (see Fig. 4) when six legs, all linearly actuated, are concurrent at three spherical pairs both at the base and at the platform. They derived a polynomial equation which proves the existence of eight reflected solution pairs. This remarkable result can also be applied to the particular arrangement where the legs are connected to the base at six different points belonging to the same plane [5, 8, 15].

RR

R

R,

R

~R

p,,

,W

)R p o

R

R

,RR

Fig. 2. One of the Stewart platform arrangements with linear actuators.

Direct position analysis of the SPM 613

Fig. 3. A generalization of the SPM.

Recently Nanua and Waldron [16] presented a closed form solution that does not guarantee against extraneous roots and resulted in a final 24th order polynomial equation from which 24 different configurations of the mechanism would be expected.

The aim of this paper is to present a novel direct position analysis for the SPM in closed form. Based on the SPM model mentioned above, it will be shown that the problem can be reduced to the solution of a system of three non-linear equations which can be solved in closed form.The analysis results in a 16th order polynomial equation in one unknown. Hence, for a given set of input values, 16 different configurations of the mechanism are possible. As far as the authors are aware, this is a new result. Finally numerical verifications are reported.

2. MODELING AND SOLUTION OF THE DIRECT POSITION ANALYSIS

2.1. Modeling of the SPM According to the characterization given by Stewart [17] and based on the following consider-

ations, a general kinematic model for the direct position analysis of the SPM with any leg arrangements is derived.

The basic geometric structure of the SPM consists of a movable platform W connected to the adjacent links at three distinct points P,, r -- 1, 3, by spherical kinematic pairs. Three distinct legs with various arrangements of links and pairs (see Figs 1, 2 and 4 for instance) six of which are suitably actuated, can be devised to connect the platform to the base, providing the platform with six degrees of freedom. Each leg has two actuated pairs and, considering the platform momentarily disconnected at P,, it allows point P, to move in the three-dimensional space. When a set of inputs is given point P, of each leg describes a circle F, (see Fig. 5) with definite center and radius, and lying on a plane perpendicular to a definite direction w,. It is worth noting that whenever the inputs are modified the circles F, change. Specifically, the axes w, can change when particular arrange- ments are adopted. The three points P, also belong to the platform and this provides the condition for defining the locations of the points on the corresponding F, circles. Several locations are possible. Points P, are determined by the position vectors Q,P, that can be defined by the three angles 0, measured with respect to arbitrary directions n, perpendicular to the axes w,. Because there is a one-to-one correspondence between the three 0, angles and the location of the platform, the direct analysis problem reduces to determination of all the possible triplet of 0, when a set of inputs is given. A mechanism that can be analysed by this model, regardless of its leg arrangements, will be called an SPM.

614 CARLO INNOCENTI and VINCENZO PARENTI-CAS'gELLI

%

R2

Fig. 4. The octahedron arrangement of the SPM.

Noting the fact that when the actuator displacements are given any mechanism is essentially a structure, the above-said model can be adopted also in those cases where the number of inputs is different from six. Moreover, the model can also be adapted with slight modifications to SPM type mechanisms where one or more points P, of the platform describe straight lines instead of circles. Finally, it is worth mentioning that for some particular leg arrangements, once a set of inputs is given, several configurations of the leg links may be possible, but the SPM model can nevertheless be applied to all possible configurations each providing a definite F, circle as in the case of one of the four arrangements proposed by Stewart [17] which is shown in Fig. 2.

2.2. Closure equations

Henceforth whenever r and s appear simultaneously in the same equation it is r = 1, 3 and s = mod(r, 3 )+ 1, where mod(xl, x2) is a function that returns the remainder when the first argument is divided by the second.

With reference to Fig. 5 for each loop P , P , Q , Qr the following vector equation can be written:

P, - P, = (Ps - Q,) + (Q, - Q,) - (P, - Q,). (l)

Here

[P, - P , [ = (2 )

P, - Q, = H, (u , cos 0, + v, sin 0,), (3)

P, - Qs = H,(us cos 0, + v, sin 0,), (4)

P2

/r2

Fig. 5. The kinematic model of the SPM.

Direct position analysis of the SPM 615

where u, and v, are unit mutual orthogonal vectors parallel to the plane of the F, circles, (u, x v, = w,). and with direction chosen arbitrarily; L,, represents the distance between P, and P~; Hr is the radius of the F, circle centered at Q,.

Q,, w, and H, are uniquely defined when the actuator displacements are given. By squaring equation (1) the following scalar equation is obtained:

'qIC, C , + ' q , C , S , + ' q 3 S , C , + ' q 4 S , S, + ' q s C , + ' q s S , + ' q T C , + ' q s S , + ' q g = O , (5)

where

and

Ck ---- COS 0k, Sk -- sin 0h,

'q, = 2H, H,u,u~, (5.1)

'q~ = 2H, H,u,v,, (5.2)

'g3 = 2H, H,v,u,, (5.3)

'q4 = 2H, H,v,v,, (5.4)

"qs = 2H, (Q , - Q,)u,, (5.5)

'q6 -~ 2 H , ( Q , - Q,)v,, (5.6)

"q, = 2H, (Q , - Q,)u,, (5.7)

"q8 = 2H, (Q, - Q,)v , , (5.8)

"q9 L~, 2 2 - - - H , - H , - ( Q , - Q , )2 . ( 5 . 9 )

Substituting for sine and cosine the well known expressions:

Sk = 2tk/(1 + t~) and Ck = (1 -- t~)/(1 + t~),

where h-- tan(0J2) , equation (5) can be written as follows:

~, "aut~t ~-~ 0, / - 0 , 2

where i - 0.2

'a0o = 'ql + 'qs + 'q7 + 'qg,

'ao. = 2('q2 + 'qs),

'ao~ = - 'q l + 'q5 - 'q~ + 'qg,

'alo = 2('q3 + 'qt),

"all = 4'q4 ,

' a p = - 2 ( ' q 3 - ' q t ) .

'azo -- --'ql - 'qs + 'q7 + 'qg, 'a21 = - 2 ( ' q , - ' q s ) ,

(6)

"a , ffi "q] - 'qs - 'q7 + 'q9

and "qj, i = !, 9, are given by equations (5.1)-(5.9). The three equations (6), when simultaneously satisfied, represent the closure of the SPM. When

a set of inputs is given they represent a system of three second-order algebraic equations in the unknown t~, h and h. The solutions of this system define the locations of points P, [see equations (3) and (4)] and, consequently, the locations of the platform are determined.

Inspection of the system of equations (6) reveals that each equation contains only two unknowns (t, and t , , respectively) and this makes it feasible to solve the system without introducing extraneous roots. The following procedure may be used to eliminate two unknowns from equations (6) in two steps. First h, for instance, can be eliminated from the second and the third equation of system (6). The result will be an equation in the unknowns tm and h. Then, from this equation and the

(6.1)

(6.2) (6.3) (6.4) (6.5) (6.6) (6.7)

( 6 . 8 )

(6.9)

616 CARLO INNOCENTI a n d VINCENZO PAI~NTI-CASTELLI

first equation of system (6), the variable t2 can be eliminated, thus obtaining the final polynomial equation in the unknown tl.

Elimination o f tj. Equation system (6), for r = 3 and r = 2 provides two equations in the unknowns tl and t3, and t2 and t3, respectively.

The two equations can be written as follows:

At~ + Bh + C = 0, (7)

Rt] + S t 3 + T --- 0. (8) where

A= Z Ait~, i=0,2

B= Z s,t , i= 0.2

(9.1)

(9.2)

C = ~ C~t~, (9.3) i=0.2

R-- Z Rit~2 ' i - 0 , 2

(9.4)

and

The eliminant of equations (7) and (8) is

S---- ~ Sit~, i - 0,2

T = ~ T,t~,

(9.5)

(9.6) i=0,2

A i = 3a2i, (10.1)

Bi ---- 3ali, (10.2)

c, = % . (10.3)

R~ = 2a a, (10.4)

S i = 2ai l , ( 1 0 . 5 )

T~ = 2aio, (10.6)

the following:

"0 A B C"

A B C 0 --o. (11)

0 R S T

R S T 0

Equation (! l) represents the condition under which equations (7) and (8) have the same solutions for t3.

Developing equation (I 1) it results:

(AT - CR) 2 + (AS - BR) (CS - BT) = 0. (12)

Equation (12) is a fourth degree equation in the two unknowns fi and t2 and, taking into account the relations (9), it can be written as follows:

~., but~t~=O. (13) i--0.4 j--0.4

The full analytical expressions of the coefficients b U are reported in Table 1. Elimination of t.,. The first equation of system (6) (r -- 1), is:

'aot~t{=O. (14) i=0.2 j = 0.2

Direct position analysis of the SPM

Table 1. Analytic expression for the bq coefficients of equation (13)

617

boo: AoZToZ-AoBoSoTo+AoCoSoZ-2AoCoRoTo +BoZRoTo-BoCoRoSo+ColRo z

bo~: 2AoZToT~-AoBo(SoT~+S~To)+2AoCoSoS~ -2AoCo(RoT~+R~To)+BoZ(RoT~+R~To) -BoCo(RoSI+R~So)+2ColRoR~

be2= AoZ(T~2+2ToT2)-AoBo(SoTz+S~T~+S2To) +AoCo(S~+2SoS=)-2AoCo(RoTz+R~T~+R2To) +Boe(RoTz+R~TI+R2To)-BoCo(RoSz+R~S~+R2So) +Co=(R~=+2RoR2)

bo3: 2Ao2TITI-AoBo(S~T2+SzT~ )+2AoCoS~S2 -2AoCo(R~T2+R2T~ )+Bo2(R~T2+RzT~ ) -BoCo(R~S2+R2S1)+2Co~R~R2

bo4: AoWTz~-AoBoSzT=+AoCoS=~-2AoCoR=T= +BoZRzTz-BoCoR=Sz+CoZR~ ~

b~o: 2AoA~To=-(AoB~+A~Bo)SoTo+(AoC~+A~Co)So * -2(AoC~+A~ Co)RoTo+2BoB~RoTo -(BoC~+B*Co)RoSo+2CoC~Ro z

b~ : 4AoA~ToT~-(AoB~+A~Bo)(SoT*+S~To) +2(AoC~+A~Co)SoS~-2(AoC~+A~Co)(RoT~+R~To) +2BoB~(RoT~+R~To)-(BoC~+B*Co)(RoS~+R*So) +4CoC~RoR~

b~z: 2AoA*(T~=+2ToTz)-(AoB~+A~Bo)(SoT*+S*T~ +SzTo)+(AoCl+A~Co)(S~=+2SoSz)-2(AoC* +A1Co)(RoTe+R~T~+R=To)+2BoB~(RoT=+R~T~ +RzTo)-(BoC~+B~Co)(RoS~+R~S~+R~So) +2CoC~(R~Z+2RoRz)

b~: 4AoA~T~Tz-(AoB~+A~Bo)(S~Te+S~T1 )+2(AoC~ +AICo)S~S=-2(AoC~+A~Co)(R~T~+R=T~) +2BoB~(R~Tz+R=T~)-(BoC*+B~Co)(R~S=+R~S~) +4CoC~R~Rz

b~4:2AoA~T=~-(AoB~+A1Bo)S~Tz+(AoC~+A~Co)S2 z -2(AoC~+A~Co)R~Tz+2BoB~R~T~ -(BoC~+B~ Co)RzSz+2CoC~Rz =

bzo: (A~e+2AoA~)To=-(AoB~+A~B~+A=Bo)SoTo +(AoCz+A~C~+A~Co)SoZ-2(AoC=+A~C1 +AaCo)RoTo+(B~+2BoB~)RoTo-(BoCz +B~C~+BzCo)RoSo+(C~+2CoC~)Ro z

b2~: 2(AIZ+2AoA=)ToTI-(AoB=+A~B~+A~Bo)(SoT~ +S~To)+2(AoC~+A~C~+A=Co)SoS~-2(AoCz+A~C~ +A~Co)(RoT~+R~To)+(BI=+2BoBz)(RoT~+R~To) -(BoC=+B~C~+B=Co)(RoS~+R~So) +2(C~z+2CoCz)RoR~

bz=: (A~+2AoA=)(TI~+2ToT=)-(AoB=+A1B~ +A2Bo)(SoT=+S~T~+S~To)+(AoC=+A~C~ +A=Co)(S~+2SoS~ )-2(AoCz+A~C~+A=Co)(RoT= +R~T~+R~To)+(B~+2BoB=)(RoT~+R~T~+RzTo) -(BoCa+B~C~+B=Co)(RoS~+R~S~+R=So)÷(C~ = +2CoC=)(R~Z+2RoR=)

bzs: 2(A~Z+2AoA~)T~T:-(AoB:+A~B~+AzBo)(S~Tz +SeT~ )+2(AoCe+A~C~+A~Co)S~S~-2(AoCe+A~C~ +AeCo)(R~TI+ReT~)+(B~|+2BoBz)(R~Tz+RzT~) -(BoCw+B~C~+BeCo)(R~S=+R~S~) +2(C~z+2CoC=)R~R=

b=4:(A~+2AoAz)T|I-(AoB=+A1BI+AzBo)SIT= +(AoCz+A~C~+AwCo)Sz~-2(AoCz+A~C~+AzCo)RzTz +(B~=+2BoB~)R=T~-(BoC=+B~C~+B~Co)RzS~ +(C~=+2CoC=)R= w

bso: 2A1A~To=-(A~B=+AzB~ )6oTo+(A1Cz+A=C~ )So = -2(A~Cz+A~C~)RoTo+2B~B=RoTo-(B~Cz+BzC~)RoSo +2C~C=Ro =

bs~: 4A~A=ToT~-(A~B=+A~B~)(SoT~+S~To) +2(A~C~+AzC~)SoS~-2(A~Cz+AzC~)(RoT~+R~To) +2B~Bz(RoT~+R~To)-(B~C=+BzC~)(RoS~+R~So) +4C~C=RoR~

bsz: 2A~Az(T~+2ToTz)-(A~Bw+A~B~)(SoTz+S~T~ +S=To)+(A,C=+A=C~ )(S~+2SoS~ )-2(A~C=

• +AzC~ )(RoT~+R~TI+RzTo)+2B~B~(RoT=+R~T~ +RzTo)-(B~Ce+B~C~)(RoSz+R~S~+R=So) +2C~C=(R~Z+2RoR=)

b:s: 4A~A=T~Tz-(A~B:+A:B~ )(S~Tz+S:TI )+2(A~Cz +A:C~)S~Sz-2(A~C:+A:C~)(R~T:+RzT~) +2B~Bz(R~Tz+R:T~)-(B~C:+B:C~)(R~Sz+R:S~) +4C~C:R~Rz

ha4: 2A~A=T::-(A~Bz+A~B~)SzTa+(A~C2+AzC:)Sz z -2(A~C~+AzC~)RzTz+2B~B~RzTz-(B~Cz+BzC~)RzS~ +2C~C:Rz z

b4o: A22To2-A2BzSoTo+A2C2So:-2AzC:RoTo+B:~RoTo -B:C:RoSo+C::Ro ~

b41= 2Az2ToT1-/~Bz(SoTI+S1To)+2A2C2SoSI -2AzC2(RoTI+RtTo)+BzZ(RoTI+R1To) -BzC2(RoSI+R1So)+2Cz2RoRI

b42= A2Z(T12+2ToT2)-A282(SoT2+S1Tl+S2To) +AzCz(SI2+2SOSZ)-2A2C2(RoTz+R1Tl+R2To) +Bz2(RoTz+R1TI+R2To)-BeC*(RoS2+R1SI+R2$o) +Cz2(R12+2RoR2)

bds= 2Az:T1T2-A2B2(S1T2+S2T1)+2A2CzSIS2 -2A2Cz(R1Te+R2TI)+B22(R1T*+RzTI) -BzC2(R1S2+R2S1)+2C2ZR1P,2

b44:A2|T22-A2B2S2T2+A:C2S22-2A2C2R:T2 +B2ZR2T2-B2C2ReS2+C22R22

Equations (13) and (14) represent a system of two equations in the two unknowns tl and t.,. Then t z, for instance, can be eliminated between them obtaining one equation in the only unknown ft.

Equations (13) and (14) can be rearranged as follows:

Gt~ + Mt~ + Nt~ + Ut2 + V = O, (15)

Dt~ + Et 2 + F = 0, (16)

618 CARL.O INNOCENTI and VINCENZO PARENTI-CASTELLI

where

and

G = Z bi4t~l , i=0,4

M = ~ bi3tit, i~0,4

N = Z ba t l , i~0,4

U = ~. bM~, i=0.4

v = b t'i, i~0,4

D =

E =

F =

The eliminant of equations (15) and (16)

0 G M G M N 0 0 0 0 0 D 0 D E D E F

Z la,2l~, i= 0.2

y,

i-0.2

is the following:

N U V U V 0 D E F

= 0 E F 0 F 0 0 0 0 0

(17.1)

(17.2)

(17.3)

(17.4)

(17.5)

(18.1)

(18.2)

(18.3)

(19)

Equation (19), after developing the 6 x 6 determinant, can be written as:

G V ( D 2 F 2 + E 4 - 3 D E 2 F ) - G U E F ( E 2 - 2 D F ) - G F 2 ( M E F + N D F - N E 2 - G F 2)

+ V D 2 ( 2 M E F + V D 2 - N D F ) + V D E ( N D E - U D 2 - M E 2) - D F 3 ( G N - M 2)

+ D E F 2 ( G U - M N ) + D F ( G V - M U ) ( D F - E 2) - D 2 F 2 ( M U - N 2)

+ D 2 E F ( M V - N U ) - D J F ( N V - U 2) = 0. (20)

It is recognized that equation (20) results in a 16th order polynomial equation in the variable ft. Therefore 16 real and complex solutions for fi are possible.

Determina t ion o f t . , a n d t3. It can be shown that, for every solution t t --- t I of equation (20), unique values for t 2 and t3 do exist. Let them be t2 and t 3, respectively.

For t = t~ equations (15) and (16) are indeed algebraic equations in the only unknown t2, and these have a common root, t2, whose value can be found by equating to zero the first-degree greatest common divisor (G.C.D.) of the polynomials at the left-hand side of equations (15) and (16). Similarly, once tm and t2 are known, the corresponding value t3 can be found by equating to zero the first-degree G.C.D. of the polynomials at the left-hand side of equations (7) and (8).

Hence a unique solution (t~, t2, t3) of system (6) is derived for every solution tj of equation (20) and, consequently, through equations (3) and (4) a unique location of the platform is obtained. Therefore the direct position analysis of the SPM provides 16 solutions in the field of complex numbers.

This confirms the outlook advanced by Hunt [3].

3. NUMERICAL EXAMPLE The direct position analysis of the Stewart platform arrangement shown in Fig. 6 is considered.

Here the extremities of six linearly actuated variable length legs are connected at three points Pr,

Direct position analysis of the SPM 619

r = 1, 3, on the platform by three double spherical pairs and at six points R,, S,, r = 1, 3, on the base by universal joints. Points R, and S, are arbitrarily located in the base. For the purpose of the present analysis all non-actuated pairs can be considered as spherical. When practical arrangements are considered the internal mobility can be eliminated by a suitable design.

This mechanism has been proposed by Stewart [17] and can be solved by the kinematic model presented in Section 2. Indeed when the leg lengths IR, = I P, -- R, I and Is, = ]P, - S,[ are given and, considering a momentary disconnection of the platform, point P, describes a circle of radius H, = P,Q,, where Q, is the projection of P, on the line R,S, .

Determination o f Q, and H,. With reference to Fig. 6 it can be written:

(P, - R,) ~ = l~,,

( P , - S,) 2 _- 12,

(21)

(22)

(23) (P , - R , ) = o , ( S , - R , ) + V , k , x (S , - R , ) ,

where k, is a unit vector orthogonal to the plane containing the points P,, R, and S,; a, and #, are scalar quantities to be determined, and

o,(S, - - R,) = (Q, -- R,), (24)

/*,IS,- R, I = H,. (25)

Equations (21), (22) and vector equation (23) represent a system of five linear equations in the five unknowns a,,/a, and the three coordinates of point P,. The system, solved for o, and/~,, leads to:

o', = 1 +, (S , - R , )2 , ]

and

(26)

(27)

and from equations (24) and (25) point Q, and the radius H, can be obtained.

P.

, !

z I W1 Wr= u r x X r

Or

Fig. 6. The numerically solved Stewart platform arrangement.

MMT 25/6--C

620 CARLO I~NOCEbrn and V~NCEr~ZO PAILENTI-CASTELLI

The negative value for/~, could bc taken in equation (27) and the final results would not be affected by this choice. Moreover, if in equation (27) /~ < 0 it means that the triangle P,R,S, is not a real triangle.

Table 2. PL. P,. and P~ x. y, : coordinates (with real and imaginary parts) in reference system So, for all solutions

Solution point P1 point P2 point P3 No.

10

11

12

13

14

15

16

( 79.535380, 0.000000) ( -26.094293, 0.000000) ( -70.922245, 0.000000) ( -45.880935, 0.000000) ( -68.945779, 0.000000) ( 54.310495, 0.000000) ( 152.901806, 0.000000) ( 62.395535, 0.000000) ( 94.385310, 0.000000)

( 68.867648, 0.000000) ( -47.021189, 0.000000) ( 21.249326, 0.000000) ( -33.006202, 0.000000) ( 21.088681, 0.000000) ( 137.061240, 0.000000) ( 165.807314, 0.000000) ( 106.439635, 0.000000) ( 95.739055, 0.000000)

( 82.538913, 0.000000) ( -48.826171, 0.000000) ( -6.782282, 0.000000) ( 51.078311, 0.000000) ( 24.727683, 0.000000) (-100.517259, 0.000000) ( 145.915415, 0.000000) ( 101.985254, 0.000000) ( 74.218082, 0.000000)

( 90.901681, 0.000000) ( -40.776390, 0.000000) ( 14.067563, 0.000000) ( 53.394494, 0.000000) ( 6.285160, 0.000000) (-I08.359226, 0.000000) ( 135.384749, 0.000000) ( 117.423785, 0.000000) ( 71.884731, 0.000000)

( 440.465978, 374.285925) ( -19.859203, 31.816581) ( 251.634723, -25.530623) ( -613.153096, 538.924920) ( -23.907915, -109.034298) ( 47.574791, 158.691077) ( -279.352370, -485.821570) ( 192.999348, -11.250822) ( 69.674410, 17.673445)

( 440.465978, -374.285925) ( -19.859203, -31.816581) ( 251.634723, 25.530623) ( -613.153096, -538.924920) ( -23.907915, 109.034298) ( 47.574791, -158.691077) ( -279.352370, 485.821570) ( 192.999348, 11.260822) ( 69.674410, -17.673445)

( 134.783717, -13.768332) ( -23.399877, 16.930693) (-I05.225277, -56.666947) ( 30.649023, 47.624021) ( -33.184715, -62.880575) ( 94.752932, -106.252866) ( 81.290387, 15.622948) ( 151.431368, -15.711648) ( 100.961182, -6.335668)

( 134.783717, 13.768332) ( -23.399877, -16.930693) (-I05.225277, 56.666947) ( 30.649023, -47.624021) ( -33.184715, 62.880575) ( 94.752932, 106.252866) ( 81.290387, -15.622948) ( 151.431368, 15.711648) ( 100.961182, 6.335668)

( 62.678338, 61.636535) ( -16.585413, -8.876874) ( 132.340119, -44.373474) ( 111.960982, 41.145860) ( -64.775609, 48.627638) ( -96.171441, -33.662626) ( 168.711711, -78.417197) ( 132.543596, 39.555595) ( 64.280593, -0.038014)

( 62.678338, -61.636535) ( -16.585413, 8.876874) ( 132.340119, 44.373474) ( 111.960982, -41.145860) ( -64.775609, -48.627638) ( -96.171441, 33.662626) ( 168.711711, 78.417197) ( 132.543596, -39.555595) ( 64.280593, 0.038014)

( 161.766239, -20.267102) ( -62.228315, 23.910237) ( -89.049127, 43.765497) ( 34.617424, 98.900324) ( -0.258491, -87.297976) ( -61.675481, -80.609270) ( 47.429954, 22.037201) ( -35.101032, -19.179409) ( 84.209825, -11.272884)

( 161.766239, 20.267102) ( -62.228315, -23.910237) ( -89.049127, -43.765497) ( 34.617424, -98.900324) ( -0.258491, 87.297976) ( -61.675481, 80.609270) ( 47.429954, -22.037201) ( -35.101032, 19.179409) ( 84.209825, 11.272884)

( 1116.688788, 1245.910259) ( -15.529244, (-2034.513400, 1584.965505) ( -35.238232, 4-1077.252205,-1610.220008) ( 198.483448,

( 1116.688788,-1245.910259) ( -15.529244, (-2034.513400,-1584.965505) ( -35.238232, (-1077.252205, 1610.220008) ( 198.483448,

32.143142) (-101.012698, 20.521219) -115.534194) ( -23.452268, -84.944858) -22.337967) ( 88.900100, -9.971257)

-32.143142) (-101.012698, -20.521219) 115.634194) ( -23.452268, 84.944858) 22.337967) ( 88.900100, 9.971257)

( 161.002959, 21.615369) ( -63.280792, -23.503310) (-376.311773, 210.670859) ( 37.096968, -97.789807) ( 3.211428, 87.548720) ( 228.585289, 410.706778) ( 48.301403, -23.759551) ( -35.002291, 22.325926) ( 134.466496, 25.113405)

( 161.002959, -21.615369) ( -63.280792, 23.503310) (-376.311773, -210.670859) ( 37.096968, 97.789807) ( 3.211428, -87.548720) ( 228.585289, -410.706778) ( 48.301403, 23.759551) ( -35.002291, -22.325926) ( 134.466496, -25.113405)

Direct position analysis of the SPM 621

Definition of the angles 0,. A reference coordinate system So =-(x,y, z) fixed to the base can be chosen arbitrarily. The directions of vectors u,, r = 1, 3, are chosen to be parallel to the x, y plane of system So, while vectors w, are directed from points R, to points S,, and vectors v, = w, x u,. The angles 0, are given by rotation of the position vector (P, - Q,) about the axis w, with respect to the unit vectors u, and measured according to the right-hand screw rule. Points Q, change their positions on the lines ( S , - R,) according to the actuated leg lengths.

Case study. The coordinates of points R, and S, in the So reference system are Rj (50.0, 0.0, 100.0), S j ( - 2 5 . 0 , - 2 . 0 , 4 0 . 0 ) , R2(80.0,20.0,50.0), $ 2 ( - 5 0 . 0 , - 2 0 . 0 , 7 0 . 0 ) and R3(48.0,15.0,68.0), S.~(36.0, 31.0 , -93.0) . The distance between the P, points are L n = 141.0, L23 = 135.0 and L31 = 190.0.

The analysis has been performed for the following leg lengths IR~ = 76.0, ls~ = 160.0, IR2 = 139.0, Is2 = 55.0, IR3 = 128.0, Is3 = 217.0.

The results of the direct position analysis are shown in Table 2, where the coordinates x, y, z of points P, are reported for all 16 solutions. Four solutions are real, the remaining ones complex. All solutions have been numerically verified by substitution of the corresponding 0, in equation system (6).

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

This p a p e r presents the c losed form so lu t ion o f the direct pos i t ion analysis o f the SPM. The analysis has been deve loped with reference to an S P M kinemat ic m o d e l der ived f rom the

cha rac te r i za t ion given by S tewar t [17]. The closed form so lu t ion results in a 16th o rde r p o l y n o m i a l equa t ion in one unknown. Con-

sequently, 16 loca t ions o f the p l a t fo rm are poss ible when a set o f a c t u a t o r d i sp lacements is assigned.

This result has been conf i rmed by numer ica l examples .

Acknowledgements--The financial support of the Research National Council (CNR) and the Ministry of Education (MPI) is gratefully acknowledged.

R E F E R E N C E S 1. K. J. Hunt, Kinematic Geometry of Mechanisms, p. 426. Clarendon Press, Oxford (1978). 2. C. F. Earl and J. Rooney, Trans. ASME J! Mech. Trans. Automn Des. 105, 15 (1983). 3. K. J. Hunt, Trans. ASME JI Mech. Trans. Automn Des. 105, 705 (1983). 4. D. C. H. Yang and T. W. Lee, Trans. ASME J! Mech. Trans. Automn Des. 106, 191 (1984). 5. E. F. Fichter, Des. Eng. Tech. Conf., Cambridge, Mass., ASME Paper 84-DET-45 (1984). 6. M. G. Mobamed and J. Duffy, Trans. ASME JI Mech. Trans. Auwmn Des. 107, 226 (1985). 7. H Inoue, Y Tsusaka and T. Fukuizumi, 3rd ISRR, Gouvieux, France, p. 321 (1985). 8. E. F. Fichter, Int. J. Robot. Res. 5, 157 (1986). 9. K. Sugimoto, Trans. ASME JI Mech. Trans. Automn Des. 109, 3 (1987).

10. K. J. Waldron, M. Raghavan and B. Roth, Proc. Model. Control Robot. Manip. Manufact. (Ed. R. Shoureshi et aL) ASME Book, No. DSC-Vol. 6, p. 127 (1987).

I I. M Sklar and D. Tesar, Trans. ASME JI Mech. Trans. Automn Des. 110, 109 (1988). 12. W. Q. D. Do and D. C. H. Yang, J. Robot. Systems 5, 209 (1988). 13. J. P. Metier, Int. Syrup. Theory Pract. Robots Manip. Ro.man.sy 88, Udine, Italy (1988). 14. F. lkhi, IEEE JI Robot. Automn. 4, 561 (1988). 15. M. Griffis and J. Duffy, Private communication, Center for Intelligent Machines and Robotics, University of Florida,

Gainesville, Fla, Jan. (1989). 16. P. Nanua and K. J. Waldron, IEEE Proc. Conf. Robot. Auwmn. p. 431 (1989). 17. D. Stewart Proc. lnstn Mech. Engrs, Vol. 180, Part 1, No. 15, p. 371 (1965/66).

ANALYSE DIRECTE DE POSITION DU MECHANISME DE LA PLATE-FORME DE STEWART

R~sumL~--Les auteurs pr~entent la solution analytique du probl~me directe de I'analyse de position du m~canisme de la plate-forme de Stewart (SPM), c'est d dire de trouver la position et l'orientation de la plate-forme, Iorsque les valeurs des d@placements des moteurs sont donn~.

L'analyse a ~t~ execut~ en rffer~nce ~ un mod&le cin&matique g~n~ral, sur la base de la caract~risation donn~e par Stewart. Avec ce module tousles diff~rents arrangements du m~canisme peuvent ¢:tre analys~s d'une faqon unitize.

L'analyse aboutit ~i une ~quation polynrmiale de seizi~me ordre avec une seule inconnue, de laquelle on obtient seize positions et orientations diff~rentes de la plate-forme.

Les nouveaux r@sultats obtenus par cette analyse sont confirm~s par des exemples num~riques.