On the dynamic model and kinematic analysis of a class of Stewart platforms

Robotics and Autonomous Systems 9 (1992) 237-254 Elsevier

237

On the dynamic model and kinematic analysis of a class of Stewart platforms Zheng Geng and Leonard S. Haynes Intelligent Automation, Inc., 1370 Piccard Drive, Rockville, MD 29850, USA

James D. Lee and Robert L, Carroll School of Engineering and Applied Science, George Washington University, Washington, DC 200052, USA

Abstract Geng, Z., Haynes, L.S., Lee, J.D. and Carroll, R.L., On the dynamic model and kinematic analysis of a class of Stewart platforms, Robotics and Autonomous Systems, 9 (1992) 237-254. In this paper, a dynamic model for a class of Stewart platform (six degrees of freedom parallel link robotic manipulators) is derived by using tensor representation. A set of six Lagrange's equations are obtained. The kinematics analysis for a class of Stewart platform is conducted and a sixteenth order polynomial equation corresponding to the forward kinematic solution of the Stewart platform is obtained, which gives all possible global solutions of a manipulator configuration for a given set of six leg lengths.

Keywords: Robot dynamics; Robot kinematics; Parallel link manipulator; Stewart platform.

1. Introduction The Stewart platform based parallel link robotic manipulators [7] have recently received considerable research interests due to their successful applications and potential advantages over the conventional serial link manipulators. A general Stewart platform is shown in Fig. 1, which has six adjustable actuators connecting an mobile platform and a base platform. The six actuators are six pistons, as shown in Fig. 2, which are controlled independently by a hydraulic or electrical system. As the lengths of the pistons Zheng Geng received his Doctor of Science degree from the Department of Electrical Engineering, George Washington University. Before joining Intelligent Automation, Inc., he previously held various positions with industrials and universities. His accomplishments at the CAD Lab for Systems/Robotics, University of New Mexico, Albuquerque, NM, included the development of an extensive design and analysis technique and software CAD package for flight control system design based on large-scale system theory (Decentralized control and Hierarchical control systems). He was also involved in a 'hardened robot' research project which is used for enhancing the performance of robots in hazardous environments of outer space and nuclear reactor. At Project Design System, Inc., Arlington, VA, he worked on the Engineering Change Proposal (ECP) database management system project for U.S. Naval Sea Command. As a senior distinguished teaching and research assistant at the George Washington University, Washington, D.C., his research activities were in the areas of self learning control system design and analysis, including iterative learning control systems, neural networks and real-time expert system applications. In his current position as Director of Robotics Research in Intelligent Automation, Inc., Dr. Geng has involved in projects for the research and development of advanced technology for intelligent control and robotic systems. Dr. Geng is an active member of the IEEE Control System Society, IEEE Robotics and Automation Society, and IEEE Communication Society. He has served as a member of the Program Committee of the International Conference on Robotics and Manufacturing. Dr. Geng has authored over 40 journal and conference papers in the fields of flight control system design, robot control, intelligent control systems and neural network development and applications.

0921-8890/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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Z. (;czzg, I,.S. ttaynes, ,I,l). Lee, R.L. Carroll

change, an end-effector attached to the mobile platform can be moved in six degree-of-freedom space to obtain desired configuration (position and orientation) and motion. The mechanical design of the Stewart platform based parallel link manipulator is relatively simpler. They have higher structural strength and higher stiffness since the load is proportionally distributed by all six actuators. They can achieve higher accuracy in positioning and motion because the positioning error on each actuator is averaged out instead of being accumulated at the end-effector. Some assumptions we made for this Stewart platform are as follows: (1) The mobile and base platform are rigid bodies. The moving part and the stationary part of the pistons are also rigid; (2) The origin of the coordinate embedded in the mobile platform need not to be assigned to the mass center of the mobile platform. It can be assigned at the tip of the tool held by the mobile platform; (3) The six pistons are not necessarily identical in their mechanical designs and electrical characteristics, such as their masses, strokes, control gains, etc.; (4) The attaching points of these pistons at the mobile platform, denoted as a i, i = 1, 2 . . . . . 6, are not necessarily coplanar. The same applies to the attaching points at the base platform b i, i = 1, 2 . . . . . 6,. One reason we consider this non-coplanar assumption is due to a practical consideration of singularity of a Stewart platform. Some singularity will be avoided by employing the non-coplanar a i a n d / o r b i, i = 1, 2 . . . . . 6. The dynamic model of a mechanical system like a Stewart platform defines the relationship between the force exerted on the platform and the position, velocity, and the acceleration of the mobile platform. Do

Leonard S. Haynes is the founder and President of Intelligent Automation, Incorporated. IAI specializes in Robotics and Artificial Intelligence, and has successfully completed many contracts in these areas. Before founding IAI, Dr. Haynes worked for five years at the National Institute of Standards and Technology (formerly the National Bureau of Standards) where he was Leader of the Robotic Assembly Group and Acting Leader of the Real-time Control Systems Group within the Robot Systems Division. Before joining the National Institute of Standards and Technology, Dr. Haynes was program manager of the Federal Aviation Administration's Automated En-route Air Traffic Control System - the FAA's attempt to automate the En-route Air Traffic Control System. Dr. Haynes was the founding Chairman of the Robotic Industries Association Standards Committee R15.04 and is the founding and current Chairman of the IEEE Robotics and Automation Society's Standards Committee. It was largely his initiative which began the effort to develop a companion standard to MMS (Manufacturing Message Service) for robots, and that work has resulted in an international standard. Dr. Haynes is the author of over 25 referred papers, including eight papers in international archival journals, and is the holder of seven patents. James D. Lee received his M.S. from Rice University in Civil Engineering in 1967 and his Ph.D. from Princeton University in mechanical and aerospace engineering in 1971. He has held various teaching and research position in universities, industrials and government organizations. He was research associate in Purdue University from 1971 to 1972 and associate research professor/coordinator of fracture mechanics group of George Washington University from 1972 to 1981. Then he served as associate professor in west Virginia University, research scientist in General Tire and Rubber Company, and associate professor in University of Minnesota, respectively, from 1982 to 1985. From 1985 to 1989, he was a supervisory mechanical engineer/group leader in Robot Systems Division, National Institute of Standards and Technology. He worked as an aerospace engineer in robotics branch, NASA/Goddard Space Flight Center from 1989 to 1990. Presently, he is a professor with the department of ~ivil, mechanical, and environmental engineering, George Washington University. He is a member of ASME, Society for Industrial and Applied Mathematics, Society of Engineering Science, and The American Academy of Mechanics. He: has published over 60 journal and conference papers in the fields of robotics, control systems, finite element analysis, continuum mechanics, fracture mechanics, composite materials. Robert L. Carroll received the Ph.D. degree from the University of Connecticut in 1973. He was employed at the Honeywell Systems and Research Center, Minneapolis, and was Associate Professor at the University of South Carolina before joining the Department of Electrical Engineering and Computer Science at the George Washington University in 1979, where he now holds the rank of Professor. He is a Senior Member of IEEE. His research interests are in control of multidimensional systems and in target tracking.

Dynamic model for a class of Stewart platform

239

a a a a I~atlorm

Fig. 1. Generalstructureof Stewartplatform. and Yang [8] studied the inverse dynamics problem for Stewart platform using Newtonian mechanics. A set of thirty-six equations were derived assuming frictionless joints and idealizing the linear actuators as slender rods. A path tracking control algorithm was given. Nguyen and Pooran [9] employed the Lagrangian approach to derive a set of equations of motion for Stewart platform where the generalized coordinates were selected to be the Cartesian coordinates. Both Do and Yang's model and Nguyen and Pooran's equations assume the origin of the mobile platform coordinate to be the center of mass of the mobile platform. Furthermore, the mass moment of inertia matrix is assumed to be diagonal. For the type of Stewart platform mentioned in this paper, a complete dynamic model has not appeared in the literature. The kinematics analysis establishes the relationship between the lengths of six actuators and the position and configuration of the mobile platform or end-effector. Unlike its counterpart in conventional serial link robotic manipulators, the forward kinematic problem for parallel link mechanism is difficult

a

I, Fig. 2. Schematicdrawingof the structureof a piston.

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Z. Geng, L.S. Haynes, J.D. Lee, R.L. Carroll

while the inverse problem is comparatively simple. In general, the solution for the forward kinematic problem is not unique. The approaches to forward kinematic problem proposed in the literature so far can be catalogued into two categories: iteratit,e methods and direct solutions. Among iterative approaches, Dieudonne et al. [1] proposed a treatment based on Newton-Raphson's method. This method consists of first defining a function of the unknowns and find its Jacobian matrix, then iteratively applying the recursive algorithm until a convergence criterion is met. Landsberger [5] suggested a search method based on geometrical interpretation. The inverse kinematic problem has to be solved for a series of candidate configurations, then find one which satisfies the given leg length constraints. The common drawbacks shared by iterative approaches are: the performance and the result of the algorithms heavily depend on the selection of the initial condition, because they can only find a local solution. There is no systematic way in this category which is able to give all possible kinematic solutions (pose of mobile plate) for a given set of leg lengths. There are relatively few methods available in the category of direct solution. Nanua and Waldron [6] presented a solution for a special case of Stewart platform shown in Fig. 1, in which the attaching points a/, i = 1, 2 . . . . . 6, are connected in pair-wise fashion at three points on the mobile platform. In addition to this restriction, all points b i, i = 1, 2 . . . . . 6, have to be coplanar. They formulated the problem as a 24-th order polynomial equation. Griffis and Duffy [4] recently gave a solution for an ever deteriorated case of Stewart platform shown in Fig. 1 in which bi, i = 1, 2 , . . . , 6, are coplanar and meet in pair-wise fashion at three points on the base. They derived a 16-th order polynomial equation for the problem based on the results for spherical four-bar linkage. In this paper, we first establish a dynamic model for the general Stewart platform of Fig. 1 by using tensor expression. A set of six Lagrange equations are derived. Then we present a solution for the forward kinematic problem for a class of Stewart platform. The forward kinematic problem is converted through a series of symbolic calculations into the problem of solving a 16-th order polynomial equation. All possible solutions for the pose of a mobile platform can be found for a given set of leg lengths.

2. Dynamic model of the Stewart platform Let the (x = x 1, y = x 2, z = x 3) coordinate system be fixed in the base platform and the (2 =21, ~ = 2 2, 2 = 2 3) coordinate system be embeded in the mobile platform. The a t h piston ( a = 1, 2 . . . . . 6) connects point b, on the base platform and the point a , on the mobile platform. Throughout this section, the standard tensor summation convention is adopted for repeated Latin indices over the range of the coordinate indices (from one to three); a superposed dot indicates the time derivatives; a variable with a Greek index as the subscript refers to the quantity associated with the a t h piston; and the Greek index may be omitted if no ambiguity arises. Also, for the sake of simplicity, the following notations are employed: ( A l Az...An),~=--A1, A 2 , . . . A , , ~ 6 Z ( A 1 A x . . . A n ) c ~ =- Y'~ A l e ~ A 2 a . . . A n , . a=l

(2.1) (2.2)

2.1. Motion of the mobile platform The mobile and the base platforms are assumed to be rigid bodies. The position and the motion of the mobile platform with respect to the base platform can be determined by three translations and three successive rotations performed in a specific sequence. In this work the Eulerian angles are adopted to define the three successive rotations. The sequence is started by rotating the mobile platform by an angle 0 3 about the 2 3 axis. Then the mobile platform is rotated by an angle 01 about the 21 axis and, finally, by an angle 0 2 about the 2 2 axis. Let u 1, u 2, u 3 be the displacement components of the origin of the mobile

Dynamic modelfor a class of Stewartplatform

241

platform along Xl, x2, x 3 axes, respectively. Then the base coordinates of a generic point in the mobile platform with coordinates 2 may be calculated as:

X i =Rim.~m q- Ui,

(2.3)

where Rim denotes the ith raw and mth column element of the rotation matrix R. The rotation matrix R is specified by the rotation angle 0i, i = 1, 2, 3, and the rotating sequence. In our case, R can be expressed in terms of the three Eulerian angles as:

[C2C3--S1S2S 3

R=lC2S3+S,SzC3 L

--CIS2

--CaS3 $2C3 +S1C2S3]

clc3 s2s3-s,c2c3l, SI

ClC2

(2.4)

]

where c i =- cos Oi, si - sin Oi. The six pairs of connecting points are denoted as: a,~: ( x ~ , x~, x ~ ) ,

ba:(Xl, X2, X3). For the a t h piston (a = 1, 2 . . . . . 6), the vector connecting point a~ on the mobile platform to point b~ on the base platform is obtained as:

( qi)~ = Rim( X* ) , + ui - ( xi)~,

(2.5)

qi is the ith element of vector q. The length of a t h piston can be calculated as: l~ = ~'( qi)c~( qi)c~ .

(2.6)

Eqs. (2.5) and (2.6) are actually the inverse kinematics formulae for the Stewart platform. The velocity of point ~ can be obtained as:

Xi = RimjOjXm "q-(di,

(2.7)

where

ORim Rim J= t)Oj Let p be the mass density (per unit volume) of the mobile platform, then the kinetic energy of the mobile platform T 1 is the integration of pYciYci/2 over the volume, i.e.:

= i f p[ RimjOj~m ~ ~i][RinkOk~n dF~i] du = }[M~i~li-JF2RimjOjl~iMXm-}-RimjginkOj[~kJrnn],

(2.8)

where

M =

fp dv,

(2.9)

1

Xm = "M f P£m dr,

(2.10)

Jm~ = f P~m~,, dr.

(2.11)

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Z. Geng, L.S. Haynes, J.D. Lee, R.L. Carroll

It is noticed that M is the total mass of the mobile platform; ( ~ , x 2, x 3) is the location of the center of mass in the mobile platform coordinate system (2 I, 22, 23); and the matrix J relates to the mass moment of inertia I of the mobile platform as: lmn = Jkk(~mn - J . . . .

(2.12)

where (~mn is the Kronecker delta. The potential energy of the mobile platform due to gravitational field is simply: V~ = - g i M [

R i m 2 m + ui] ,

(2.13)

where g is the gravitational constant multiplied by the unit vector in the direction of the gravity. 2.2. Motion o f the pistons The piston considered in this paper, as shown in Fig. 2, consists of two parts: (1) a stationary part which is a hollow cylinder with length 2d and mass ~ uniformly distributed along its length, and (2) a moving part with a fixed length r / a n d mass m uniformly distributed along its length, q is the vector from point b to point a with length l. Let s be the one-dimerlsional coordinate of a generic point on the stationary part or moving part. The stationary part and the moving part extend from s = 0 to s = 2 d and from s = 6 to s = l, respectively. Notice that for a rigid stationary and moving parts, l - (5 = r/. An axial force f, called piston force, is acting on the moving part in the direction of q. The axial force on each piston fi, i = 1, 2 . . . . . 6, are considered as the control inputs for the Stewart platform. The kinetic energy of the piston may be derived as follows. For the moving part, the base coordinate of a generic point is x i = s l - l q i.

(2.14)

Based on the expression of the coordinate of a generic point, it is easy to find the velocity: 2 i = sl-~dli - s[l-2qi .

(2.15)

Define a dimensionless variable e as: s-3+e(l-6),

(2.16)

so that e goes from 0 to 1 as s goes from (~ to I. Also the mass density per unit length p* is obtained as: m

p*-

(2.17)

l-6' because the mass of the moving part is uniformly distributed along its length. The kinetic energy of the moving part of the piston may now be obtained as: T2

= ½fP*2i2 i ds 1

f

1.

.

= 5rnJo x i x ~ d e ~ 1 2 + --gm [3/ 2 - 3lrt + r/2lK,

(2.18)

where K - l-4b,,,,(1,,4,,

(2.19)

b,,, - 12(~mn -- q m q n "

(2.20)

The kinetic energy of the stationary part of the piston can be easily obtained as:

1"3=

~md2K.

(2.21)

Dynamicmodelfor a classof Stewartplatform

243

The potential energy of the piston due to gravitational field is: V2= - g i [ ½( l + ~)m +

md]l-~qi.

(2.22)

2.3. Lagrange equations of motion The general Lagrange equations, which can be derived from either Hamiliton's principle or D'Alembert's principle, can be written as: at

~

-

=P/,

i = 1 , 2 . . . . . 6,

(2.23)

where L denotes the kinetic energy T minus the potential energy V of the system, Pi denote the generalized coordinates, and Pi are the generalized forces:

p = (Ul, U2, U3, 01, 02, 03) T P = (F1, /72, /73, M1, Mz, M3) T where F~ (i = 1, 2, 3) is the applied force acting at the origin of the mobile platform along the x i axis, and M 1 = M x cos 03 + My sin 03, M 2 = ( - M x sin 03 + My cos 03) cos 01 + My sin 01,

(2.24)

M3=Mz, where, M i (i =x, y, z) are applied moments acting on the mobile platform about the x, y, and z axis, respectively. The total kinetic energy is: m

Z~-l[Mui~li]-2RimjOj~liMXm-t-RimjRinkOjOkJmnl "t- E ( 2 [2+ ['-6-( 3 / 2 - 3l~ "t-~2)'~ 2md2]g)a. (2.25) The total potential energy is:

V = - g i M [ Rim2 m + ui] - Y'~ {gi[ ½( l + 6 ) m + ~d]l-'qi}, .

(2.26)

Recall that qi, l, and K can be written as

qi =Rimx*m + ui --~i, Ii = (qiqi) 1/2, g - l-4(12~mn - qmqn){ln(lm. The time derivative of qi and l can be obtained as:

(1i = QikO,, + fti,

(2.27)

[ = I- lqi~li,

(2.28)

where

Qik ~ Rimk X'm"

(2.29)

Z. Geng, L.S. Haynes, J.D. Lee, R.L, Carroll

244

Then it is straightforward to obtain: Oqi

Ouj - 6ij,

aq i

~Oj - Qij,

o/

Ogli 3 7 = SijkOk,

Ogli

o~,

-- ~ij,

o[ = l-3bifli,

dt

Oui

d (~K)

OK

0{1~ --=Qij,

Ol

o4

out

~1 --=l-lq~Qij,

~oj

o/

o/

- - = l-3bikQijglk + l-lqiSijkO k

- l - l q i,

- l - l q i,

- - = l lqiQii,

(2.30) (2.31)

QjiHj

(2.32)

d5 where Sij k -- RimjkX

Hi-

(2.33)

m .

(2.34)

2 l - 6 b i m ( 1 2 O m - 2glmglnqn).

Now write T* a s m T * " __[2 -t- ½IK, 2 where

(2.35)

1 - _m3_ [ ( 3/2 _ 31~7+ ~72) + g4~d211"

(2.36)

It

is

to obtain:

straightforward

aK 2 [l_3binq n + ~m(3l - r l ) l - t q i K + ½I -Oui -"

OT* Oui

m

0T* &ii

m . OK ll-lq ' + 1 I - - . 2 , Z 0U i

dt

'l

m

l

(2.37) (2.38)

m3

t

a t t &~i ] = --fll- qi + - ~ l l - bingln + l m ( 2 1 - "q)l-lqngIn + 5I~-;, , ~ / •

--dt ~

Ofti ] + Oui - ½ I

dt

~ 01~ i

--

~u i

+ ~ m r l l - ' q i K +ml-2qiq"{iin + QnkO'k + SnikOiOk}/2

OT * O0i

+ m ( 2 l -- rt)l-abi, gl,. m m OK 2 /(l-3bjkQjiglk + l-lqjSjikOk) + --ff( 3 / - r t ) l - l Q j i q j K + !21 -00- i .

OT* 00 i

m OK 2 l- lq, Q, i + ½I 00 i

d(0T* t

at

m.. 1

(2:39)

(2.40) (2.41) (2.42)

m.

o0i ] = - ~ l l - qnQni + ~ l { - l - 3 q m g l m q n Q m + l-l(l~Qni +

m +-~-(21-~7)g+~I~--;

~i

"

l-'q.S.ijOj} (2.43)

Dynamic model for a class of Stewart platform

dt ~ OOi

O0i

lI

245

( d (OK) OK} m m 2 --~ ~ -- ~ -- --~'rll-lqnani g -t- - ~ l - qnaniqm[iim + amjO; + SmjkOjOk]

+ m(21 - rl)l-4bm,Qmi(l,.

(2.44)

Based on above derivations, a set of six Lagrange equations can be obtained as:

Miii + MOiib~ + MSi~OjOg + E ( N i - g j h i j ) ~ - M g i = F i

(2.45)

+ E(fl-lqi)~,

mOjii~ j -J¢-JmnRkmigknjO; + JmnRlniRlmjkOjOk "Jr-E ( ( ] Q j - - g k h j k ) e j i ) a - - m g j O

j`

(2.46)

=Mi + ]~_.(fl-lqjQji)~, where

m lVi = ( II-4b~ + ml-2qiqj)iij + [m( l + ~) - 211-1]l-Sbimqngtnttm + -~rll-lqi K, hij=-m'ij + (md-

m

-~rl)l

-3

(2.47) (2.48)

biy,

(2.49)

Qji = RjmiX'm"

2.4. Discussions It is well known that in general the equations of motion for a serial link manipulator may be expressed in a canonical form:

(2.50)

Mij( P)i~j + Cijk( P)lbklSj + Gi( P) + Bi( P, P) = ei,

where M is the generalized mass matrix; the C matrix is due to the contributions of centrifugal and Coriolis forces; the G vector is due to the contribution of potential energy; the B vector specifies dissipative effects such as friction and/or damping; p is the vector of generalized coordinates called joint variables; and P is the vector of generalized forces, i.e, the forces and torques at joints. It can be shown that such canonical form of equations of motion can also be obtained for the rigid parallel link robotic Table 1 Tabular expression of the dynamic equations

Mij iii

ui

Oi

M6ij + E{ll-4bij + ml-2qiqj}~ MxmRim j + ~2{[II-4bin + ml- 2qiqn]Qnj}a

MfmRy. . + 7~{[ll-4bjm + ml- 2qjqm]Q,,J,~ JmnRkmjRkni + Y2{[II-4bmn + ml- 2qmqn]QmiQmj},~

Cijk ~2{½mrll- SqmQmibjk + I r a ( 2 / - "q)1-5 - 21l-6]qkbOqli}a fijUk Y~1½mrll-Sqibyk + [ m ( 2 1 - r / ) 1 - 5 -2ll-6]qkbij}a E{mrll- SqibmjQmk + [ m ( 2 1 - r l ) 1 - 5 - 211-6](bimqj + bijqm)Qmk} a Y~{mrll-Sq~QtibjmQm,~ + [ m ( 2 l - rl)1- 5 - 211-6 ](btmqj + btjqm)QtiQ m o~ok MxmRimjk + E{[ ll-4bin + ml- 2qiqn]Snjk JmnRlmjk Rln i + •{[ ll-4bmn + ml 2qmqn]QmiSnjk 1 --5 + ½mrll-SqtbmnQtiQmjQnk + ~mrll qibmnQmyQnk + [ m ( 2 1 - r l ) 1 - 5 - 211-6]q.,btnQtiQmjQnk},, + [ m ( 2 1 - r / ) 1 - 5 - 21tl-6]qnbimQmjQnk}~ i_ F.{mgi +( - 7mr 1 I + Nd)l-3biygy},.

ai

_

e~

E{ + fl- Iqi},, + Fi

Mg

1 -3 -- MRjmi.~mgj - gjE{mQj i + ( -m- d - 7m'q)l bjnQni},~

E{fl-lqnQni}~ + M i

Z. Geng, L.S. ttaynes, J.D. Lee, R.L, Carroll

246

manipulator such as the Stewart platform. The elements of the matrices M, C, and G, as well as the elements of vector P, are listed in Table 1.

3. Direct forward kinematic solution for a class of Stewart platforms

In this section we consider the forward kinematics problem for a special class of Stewart platform as shown in Fig. 3. The platform in Fig. 3 differs from that in Fig. 1 in the connecting pattern of the attaching points on the mobile platform. The points ai, i = 1, 2 , . . . , 6 , in Fig. 3 are connected in the fashion that three pairs of actuators meet at three points, R, S, and T, in the mobile plate and at the other ends of the actuators b i, i = 1, 2 . . . . . 6, are located arbitrarily at six points, Pg, i = 1, 2 . . . . . 6, on the base platform, which are not necessarily coplanar. The forward kinematics problem of the general Stewart Platform is addressed as following: Let R, S and T be fixed on the moving mobile platform, and P~, i = 1, 2 . . . . . 6, be fixed on the static base platform. Determine the position, x, y, z, and orientation, 01, 02, 03, of the moving mobile plate relative to the base platform, given a set of six leg lengths, namely Li, i = 1, 2 . . . . ,6. Notice that in general for a set of leg lengths, the forward kinematics problem can have no solution, or multiple solutions. 3.1. Geometrical interpretation o f the forward kinematic problem

If we take off the mobile platform R S T from the Stewart Platform shown in Fig. 3, the mechanism then becomes three independent triangle link pairs ( L l, L 2) ~ P 1 R P 2 , ( L 2, L 4 ) ~ P 3 S P 4 , ( L s , L 6) PsTP6. It can be noticed from Fig. 4 that for a given set of length L i, i = 1, 2 . . . . . 6, the shapes of these three triangles are fixed. We can calculate the positions of Or, 0 s, 0 t and the values of m r, m s, m t. Since P~ and P2 are fixed on the base of the Stewart Platform and R has one degree of freedom to move, the locus of point R must be a circle with center located on O r and radius m,.. This circle is in a plane perpendicular to line P1/'2. Similarly, point S has one degree of freedom to move on a circle with a center located on O s and a radius ms. Point T has one degree of freedom to move on a circle with a center located on 0 t and a radius m t. Therefore, given leg length, three independent variables 0,, 0 s,

R

z

T

1"2 P5

p4 P3

Fig. 3. A class of Stewart platform in which the attaching points on the mobile platform are of pair-wise fashion.

Dynamic model for a class of Stewart platform

247

R

T

r]

P2

1 P5

S T

R

PI

Or

i>2

Fig. 4. Geometrical interpretation of the forward kinematics calculation.

and 0 t, as shown in Fig. 4, determine the positions of points R, S, and T. In the following derivation, a capital letter such as R, S, T, P, etc. may be interpreted as a point in space or a vector in base platform coordinate if no ambiguity arises. It can be easily verified that I PIP2

= rl + r2,

[ P3P4 1 = Sl + s2,

I Pse6 [tl + t2,

ms=~(L23-s2),

mt:~(L2-s

(3.1)

and m r = ~ ( L l - 2- r ( ) ,

2)

where [ P1P2 I 2 + L~ - L~ rl=

2lele2l

[ PaP4 [2 + L23 _ L24 '

sl=

21Pae4[

I PsP6 12 + L 2 - L 2 '

tl=

2[ese61

The coordinates of points O r, Os, and 0 t are calculated as follows: Or =P~ + (P2 - P~)r~/] P2 - P1 I, Os =P3 + (P4 - Pa)s~/] P4 - P3 [, Ot =1"5 + (P6 - Ps)t~/[ P6 - t ' 5 ]. (3.2)

Z. Geng, L.S. Haynes,J.D. Lee, R.L. Carroll

248

Let L'r, Vs, a n d t, t d e n o t e t h e v e c t o r s R - Or, S - - O s , a n d T - O t, respectively. T h e s e v e c t o r s c a n be o b t a i n e d t h r o u g h t h r e e successive r o t a t i o n t r a n s f o r m a t i o n s (/3, a , 0) f r o m t h e b a s e p l a t f o r m c o o r d i n a t e axis. G i v e n t h e p o s i t i o n of p o i n t Pi, i = 1, 2 . . . . . 6, t h e r o t a t i o n a n g l e s / 3 a n d a c a n be c a l c u l a t e d as:

'( e2 - P , ) x + ( e2 - P, )-~,

( P2 - PI )x /3r = COS-I

i ( P2 - p , ) 2 + (P2

-

pI)2y

1

i(p4_P3)2x+(e4_P3)!v

/3,=cos,[¢( (P.-P~)~ ------2 . . . .

1

"

I P 2 - PI [

]

(P4 - P3)x /3s = COS-

a r = COS-

O~s = COS

~(P4-P3)2x+(P4-P3)~ , ]

-1

iY44-Ki

'

]

O/t = C O S

2

1

IP6-PsI

P6 - es ) x + ( P6 - es ) y

(3.3)

T h e r e f o r e t h e p o s i t i o n v e c t o r s Vr, Vs, a n d v t c a n b e r e p r e s e n t e d as follows: Vr = m r ( C ° S /3r sin a r sin Or - s i n

/3r COS O r ) i + m r ( S i n /3r sin a r sin 0 r + C O S /3 r COS Or)j

+ mr(COS a r sin Or)k,

Vs = ms(COS fls sin a s sin 0 s - s i n /3s cos Os)i + m s ( s i n /3s sin a s sin 0 s + cos /3s cos Os)j + ms(COS a s sin Os)k,

v t = mr(COS /3 t sin a t s i n 0 t - s i n /3 t cos Ot)i + m t ( s i n /3 t sin a t sin 0 t + cos /3 t cos Ot)J + r o t ( c o s a t sin Ot)k.

(3.4)

T h e p o s i t i o n v e c t o r s o f p o i n t s R, S, a n d T b e c o m e

R=Or+V

r,

S=Os+v

s,

T=Ot+u

t.

(3.5)

D e f i n e G1, G2, a n d G 3 as t h e l e n g t h o f v e r t i c e RS, S T a n d R T , respectively. G1, G2, a n d G 3 a r e c o n s t a n t s for a g i v e n p l a t f o r m . T h u s t h e a d d i t i o n a l physical c o n s t r a i n t s to t h e p o s i t i o n s o f p o i n t s R, S, a n d T a r e t h e d i s t a n c e b e t w e e n R a n d S s h o u l d e q u a l to t h e l e n g t h o f t h e v e r t i c e G l, t h a t o f S a n d T e q u a l to G 2, a n d t h a t o f R a n d T e q u a l to G 3. T h e s e c o n d i t i o n s c a n b e e x p r e s s e d as:

=Gl,

IR-SI

IR-TI

= G 1,

IS-TI

=a,,

or

(Xn-Xs)

2 + ( Y R - Y s ) 2 + ( Z R - - Z s ) 2 = G~.

(3.6)

( XR-- XT) 2 + ( Y R - - Y T ) 2 + ( ZR-- ZT) 2 = G~.

(3.7)

( Xs - X r ) 2 + ( Ys - Y r ) 2 + ( Zs - Zr) e = G~.

(3.8)

3.2. Equations for direct forward kinematics S u b s t i t u t e Eq. (3.4) i n t o Eq. (3.6) gives:

[( Or -

Os) x + mr cos /3 r sin a r sin Or - m r sin /3r COS Or

+[(Or-Os);.+m + m r cos /3r COS Or

-

-

m s cos /3 s sin a s sin 0 s + m s sin /3s cos Os] 2

r sin /3 r sin o~r sin Or -

-

+ m r cos a r sin Or - m

m s sin /3~ sin a s sin 0s - m~ cos /3s cos Os] 2 + [ ( O r - Os) ~ s cos a s s i n 0~] 2 = G 2.

(3.9)

249

Dynamic model for a class of Stewart platform

T h r o u g h some m a n i p u l a t i o n , we have COS f r sin a r sin 0 r - 2 ( 0

2(0 r --Os)xmr

r -Q)xmr

sin f r COS 0 r - 2 ( O r - Q ) x m s

cos f s sin a s sin 0 s

+ 2 ( 0 r - O s ) , m s sin f s cos 0 s 2mjn

-

s c o s / 3 r sin a r sin 0r c o s / 3 s sin a s sin 0 s + 2 m r m s cos f r sin a r sin Or sin f s COS 0~

+ 2 m r m s sin f r COS Or COS f s sin a~ sin 0 s - 2 m j n

s sin f r COS Or sin f~ cos 0 s

+ 2 ( 0 r -- O s ) y m r s i n / 3 r sin a r sin 0 r + 2 ( O r - O s ) y m r c o s f r COS Or

- 2(0 r - Q)ym

s sin f s sin a s sin 0s - 2 ( 0 r - Q ) y m

s c o s / 3 s cos'0s

2 m r m s sin /3r sin a r sin 0r sin f r s i n a s sin Os -- 2 m r m s sin f r sin a r sin 0 r cos f~ cos 0~

-

- 2 m r m s cos f r COS Or sin f~ sin a s sin 0 s 2 m r m s cos fir cos Or cos f , cos 0 s + 2(Or - O ~ ) ~ m r cos a r sin Or - 2 ( O r -- O , ) z m s . 2 m r m s cos .

a r

sin .Or cos a s sin . 0s

G 2.

(O r-

Os)2x .

(O r

Os)y__(Or2

Q)

cos a s sin Os 2

2

(3.1o) or we can express Eq. (3.10) as A~ sin Or + A 2

COS Or

+ A 3 sin 0s + A 4 c o s Os + A 5 sin 0 r sin 0 s + A 6 sin 0 r cos 0 s

+ A 7 cos Or sin 0s + A 8 cos 0r cos 0s = A 0,

(3.11)

where Z 1 = 2 ( 0 r - O s ) x m r c o s f r sin a r + 2 ( O r - O s ) y m r sin f r sin a r + 2 ( 0 r -- O s ) z m r COS a r , r sin/3 r + 2(0 r - Os)ym r cos f r ,

A 2 = -2(0

r -Os)xm

A 3 = -2(0

r - O s ) x m s COS f s

A4=

r -- O s ) x m s

+2(O

sin a s - 2(Or - Q ) r m s

sin f~ - 2 ( 0 r - Q ) y m

s sin a r sin a s cos ( f s - f r )

A 5 = -2rnjn

sin f s sin a s - 2(Or - O s ) z m ~ cos a s,

s cos f ~ ,

-2mrms

cos a r cos a s,

A 6 = 2 m r r G sin a r s i n ( f s - f r ) , A 7 = 2 m r m s sin a s s i n ( f ~ - - f r ) , As = - 2mrms A o . G 1 z.

c o s ( f l s - fir),

( O r. _ O s ) 2.x

(O r

O s ) y2- - ( O r - - O s ) z - - m2r - - m s .2

2

Eq. (3.11) c o n t a i n s b o t h sin a n d cos of 0s a n d Or. T o make the expression concise, the t a n - h a l f - a n g l e r e l a t i o n s h i p is used; u r=tan

(or) ~- ,

us=tan

(0s) ~-

,u,=tan

(0) -~

.

(3.12)

Eq. (3.11) t h e n b e c o m e s 2Amur(1 + u 2) + A 2 ( l - u r ) ( 12

+ u 2) + 2 A 3 ( 1 "bU r2)/,~s + A 4 ( 1 + u 2 ) ( 1 _ Us2)

+ 4 A s U r U s + 2 A 6 U r ( 1 - u 2) + 2 A 7 ( 1

- Ur2 ) u , + A 8 ( 1 - u 2)(1 - u 2) = A o ( 1 + Ur2)(1 + U~),

or B l U r2U 2s 4- B 2 u 2 U s + B3UrU2 4- B4UrU s 4- B s u 2 + B6 u2 4- BTU r -t- B s u s 4- B 9 = O,

(3.13)

251)

Z. Geng, L.S. Haynes, J.D. Lee, R.L. Carroll

where B l = - A 2 - A 4 + A 8 - A 0,

B 2 = +2A 3 - 2A7,

B3 = 2 A I - 2 A 6 ,

B5 = -A 2 +A 4 -As -A0,

B¢,=A2-A4-As-A

o,

B 4 = 4As,

B7=2A 1+2A6,

Bs= +2A3+2A7,

B 9 = - A 2 q.-A 4 q.-A 8 - A 0. In a similar fashion, substitute Eq. (3.4) into Eq. (3.7) gives:

[(Or-Ot)x+m

r cos fir sin a r sin O r - m r sin /3r cos O , - m t cos /3t sin a, sin O , + m t sin /3t cos Ot] z

+[(Or-Ot)~+mr

sin /3r sin a r sin O r - m r cos /3r cos O r - m , sin /3, sin a t sin 0 t

-m, cos/3tcos0t]2+[(Or-Ot)~+m

rcosflr sin0r-m

,cos/3,sinO,]2=G2

.

(3.14)

U p o n simplification and variable transformation, the following equations are obtained: C 1 sin0r+C2cos0

r+C 3sinO tC4cos0,C 5sin0 r sin0,+C 6sin0 r cos0,

+ C 7 cos Or sin 0t + C 8 cos 0 r cos 0, = C 0.

(3.15)

2 2 + Dzu2U, + D3UrU2 + D4uru, + Dsu 2 + D6U2 + D7Ur + D s u , + Do = O. DlUrUt

(3.16)

Also from Eq. (3.5) into Eq. (3.8):

[(0 s -0,)

x + m , cos /3, sin % sin 0 , - m , sin /3s cos 0 s - m t

+ [(Os-Ot)y+m

, sin /33 sin % sin O , - m ,

cos /3t sin a, sin 0t + m,sin /3t cos 0,] 2

cos /3, cos O , - m t sin /3, sin a, sin 0,

- r e , cos /3, cos 0 , ] e + [ ( O , - O , ) z + m

, c o s /3, sin O , - m ,

cos /3, sin o , ] 2 = G , 2,

(3.17)

E l sin 0, + E 2 cos 0s + E 3 sin Ot + E 4 cos O, + E 5 sin 0, sin O, + E 6 sin 0, cos O, + E 7 cos Os sin 0, + E8 cos 0, cos 0 t = E 0,

(3.18)

FlU,U,2e + Fzu2U, + F3usU2 + Fau,U, + Fsu;~ + F6u 2 + F7us + Fsu, + F9 = 0,

(3.19)

where C~, D~, E~, and F i are all constants. N o w the forward kinematics problem has b e e n converted into the algebraic p r o b l e m of solving simultaneously three equations (3.13), (3.16), and (3.19).

3.3. Solution o f the forward kinematics problem Express Eq. (3.16) and (3.19) as the quadratic equations in terms of u, and u,: al( Ur)U2s -- a2( u r ) u , - a 3 ( u r ) = 0,

(3.20)

where 01 = ( B I u 2 + B3u r + B6) ,

a 2 = - ( n 2 u 2 + B4u r + B s ) ,

a 3 = - ( n 5 u2 + n T u r + n 9 ) ,

bl( Ur)U 2 - b2( ur)U t - b 3 ( u r ) = 0,

(3.21)

(3.22)

where bl=(OlU2+O3Ur+O6),

b2=-(O2u2+O4Ur+Os),

b3=-(Osu2WOTUr+O9),

(3.23)

ai(u ,) and bi(ur) , (i = 1, 2, 3), are second o r d e r polynomials of u r. Substitute (3.20) and (3.22) into (3.19) gives: Fl( a2u s + + a 3 ) ( b2u t + b3) + b l F z U , ( a2u s + a3) + alF3Us( b2u t + b3) + alblF4UsU ,

+ b l F s ( a 2 u s + a3) + a i F 6 ( b z u , + b3) + a l b l F T u , + a l b l F s u t + alblF9 = O, or ClU,U t + CzU s + c3u , + c 4 = 0,

(3.24)

Dynamic model for a class of Stewart platform

251

where c I = F l a z b 2 + F z a 2 b 1 + F 3 a l b 2 + F 4 a l b l, c 2 = F i a 2 b 3 + F 3 a l b 3 + F s a 2 b 1 + F 7 a l b 1,

(3.25)-

c3 = F t a 3 b 2 + F z a 3 b l + F 6 a l b 2 + F s a l b l , . c 4 = F l a 3 b 3 + F s a 3 b I + F 6 a l b 3 + F 9 a l b I.

Notice that c i, i = 1, 2, 3, 4, are the fourth o r d e r polynomials of u r. Define: a4

~

,,1/2

--1-(02 + q.ala3)

,

b4

+ ( b 2 + 4 b i b 3 ) 1/2,

(3.26)

we then have the relationships (solution for a quadratic equation): 2 a l u s = a z + an,

2 b l u ~ = b 2 + b 4.

(3.27)

Substitute Eq. (3.27) into (3.24), we have d l a 4 b 4 + d2a 4 + d3b 4 + d 4 = 0 d 1 =c~,

d 2 = b 2 c 1 + 2 b l c 2,

d3=a2c I +2alc 3

d 4 = a2b2c I + 2a2blC 2 + 2alb2c 3 + 4alblC4,

( E i g h t h o r d e r polynomial of Ur).

(3.28)

Squaring Eq. (3.28), we have 2(dld4_d2d3)a4b

4

=

- a-l2a 4 02 4- 2 + d z 2a 42 + d 32b 42 - d 2.

(3.29)

Based on Eq. (3.25), the left h a n d side of Eq. (3.29) is 2 ( d i d 4 - d z d 3 ) a 4 b 4 = 8 a l b l ( c l c 4 - c 2 c 3 ) a 4 b 4. F u r t h e r simplification generates 2 ( d l d 4 - d z d 3 ) a 4 b 4 = 8 a 2 b 2 [ ( F4F9 - F 7 F s ) a , b I + (F3F9 - F 6 F 7 ) a l b 2 + ( F 4 F 6 - V 3 F s ) a l b 3 + ( F z F 9 - F s F 8 ) a z b 1 + ( F , F9 - F s F 6 ) a z b 2 + ( F z F 6 - F, F s ) a z b 3 + ( F 4 F 5 - F z F 7 ) a 3 b I + ( F 3 F 5 - F 1 F T ) a 3 b 2 + ( F I F 4 - F z F 3 ) a 3 b 3 ] a 4 b 4.

(3.30)

Canceling the c o m m o n factor 8, a~b 2 2 1 m a k e Eq. (3.30) an 8th o r d e r polynomial of Ur: [(F,F 9 -F7F8)alb , + (F3F 9 -F6Fv)alb 2 + (F4F6 -F3F8)alb 3 + (FzF 9 -FsF8)azb 1 + (FIF9 - FsF6)azb 2 + (FzF 6 - VlFs)a2b 3 + (F4F 5 - FzFv)a3b I + (F3F s - F1F7)a3b 2 + ( F I F 4 - FzF3) a3b3] a4b4 = (F3F 5 + F1F7)aza3b ~ + (F4F 5 + F2Fv)aza3b,b 2 + 2(F3F 7 - F,F 9 - FsF6)ala3b ~

+ 2 ( F 2 - 2 F 1 F 6 ) a l a 3 b2 + 2 ( F 2 - 2FsFg)ala3b21 + 2 ( F 1 F 3 + F, F 7 ) a 2 a 3 b2 + 4 ( F 3 F 7 + F 2 F 8 - 0.5F4z - F s F 6 - F 1 F 9 ) a l a 3 b l b 3 + 2 F s F 7 a z a 3 b ~ + 2 ( F 2 - 2 F 1 F s ) a Z b , b 3 + 2(F~ - 2F6Fg)a~blb 3 + 2F1F2aZb2b3 + 2 ( F , F 8 + FzF 6 - F3F4)ala3b2b 3 + 2 F 6 F 8 a Z b z b 3 _ -rg2 2a,- 203 + ( F z F 6 + F 1 F s ) a Z b z b 3 + ( F 4 F 6 + F3F8) a , a 2 b 2 b 3

+ 2 ( F 4 F 8 - F 3 F6 - F 3 F 9 - F 6 F 7 ) a x a 2 b ~ b 3 + 2 ( F 4 F 7 - F s F s ) a l a 3 b l b 2 - 2 F 2 F 4 a 2 a 3 b l b 3 - ( F , F 4 + F 2 F 3 ) a 2 a 3 b 2 b 3 - ( F I F 9 + F s F 6 + F z F 9 + F s F s ) a Z b ~ - ( F 3 F 9 + F 6 F T ) a l a 2 b2

- ( F 4 F 9 + F 7 F 8 ) a a a z b l b 2 - 2F~F6a22b~ - 2 ( F , F 9 + F s F 6 - F 2 F 8 ) a ~ b l b 3 - 2 F s F g a Z b z - 2Fga,azb 2 - 2FiFsaZb~ - 2F2FsaZb,b2 - 2FzF9a,a3blb 2 - 2F6F9a~b ~ - 2F8Fga~blb 2 - 2FlZaZb 2 - 2FsZa~b 2 - 2FgZa~b~.

(3.31)

252

Z. (;eng, L.S. Haynes, J.D. Lee, R.L. Carroll

Eq. (3.31) still contains the square root term a4b 4. Squaring both sides of (3.31), a sixteenth order polynomial of u r is obtained. Solving the resulted equation, we are able to find out all possible solutions of the forward kinematic problem. If the points Pt, P2, P3, /94, Ps, and P6 are arranged to be coplanar and pair-wise, it can be shown that the coefficients of the odd-order terms in the 16th polynomial equation are all zeros. Therefore the equation can be reduced to an 8th order polynomial equation by a simple variable transform w = u~. The derivation of the 16th order polynomial is constructive, therefore the following algorithm can be summarized as follows: Step Step Step Step Step

1: 2: 3: 4: 5:

Calculate the coefficients B i, Oi, and F i, i = 1, 2 . . . . . 9 in Eq. (13), (16), and (19); Calculate the coefficients of polynomial a i and b~, i = 1, 2, 3; Construct the 16th order polynomial equation according to Eq. (3.31); Solve the 16th order polynomial equation to get all possible solutions: Select a unique solution or a set of solutions for a particular application.

In Step 2 and Step 3 of above algorithm, the polynomial multiplication can easily be implemented by using convolution algorithm. Suppose that we have two polynomials whose coefficients can be expressed by two vectors a and b. The coefficient vector c of the multiplication of these two polynomials are obtained as:

N-1 Cn+ t = ~

ak+lbn_k,

(3.32)

k=0 where N is the maximum vector length of a and b. There are many algorithms for finding the roots of a 16th order polynomial equation encountered in Step 4. For example, we can solve for the roots of a polynomial by finding the eigenvalues of an associated companion matrix using EISPACK eigenvalue subroutines which are based on similarity transformations. We can also adopt Laguerre's method of polynomial root finding as described in [10]. As we mentioned before, in general there is more than one possible solution for the forward kinematics problem, given a set of actuator lengths. In a particular application, one should develop an appropriate procedure to select a unique solution or a set of solutions for the specific application. For example, among all possible solutions for points R, S and T, half of them are reciprocal to the other half by z = 0 plane if all points Pi, i = 1, 2 . . . . . 6 are embeded on z = 0 plane. Due to this 'mirror effect', the only difference between a pair of reciprocal solutions are the sign of their z components. There are also some solutions which are theoretically correct but are not implementable because of the interference among actuators and between actuators and base or mobile platform. The forward kinematics algorithm presented in this section can be used in the motion control and f o r c e / t o r q u e sensing of the Stewart platform. In general, the forward kinematics is used in a position feedback control problem where the lengths are measured by sensors mounted on the actuators and are then converted to the position and orientation of the moving platform with respect to the base platform using the forward kinematic transformation. In each position control cycle, the initial point (position and orientation) of the moving platform is known and the maximum moving distance of the platform in one control cycle can be estimated by the current velocity. Denote this distance as a. An algorithm can be designed to pick up an unique forward kinematics solution among all possible solutions by judging the distance between solution points and initial point. The solution point must locate inside the /t-neighborhood of the initial point. The selected solution point then serves as the initial point for next control cycle. 3.4. Numerical example Consider a Stewart platform shown in Fig. 3, where the coordinates of points PI, P2, e3, P4, /95, and P6 are as following (units for all lengths are cm):

Dynamic model for a class of Stewart platform P1 = [29.5442

5.2094

10.0000],

P2 = [ - 1 0 . 2 6 0 6

28.1908

10.0000],

P3 = [ -

22.9813

0],

19.2836

P4 = [ - 1 9 . 2 8 3 6

-22.9813

0],

/'5 = [ - 1 0 . 2 6 0 6

-28.1908

0],

P6=[29.5442

253

-5.2094

0].

T h e mobile p l a t f o r m is an e q u i l a t e r a l triangle with the vertice G 1= G z

= G 3 = 34.6410.

T h e six leg l e n g t h are given as follows: L 1 =31.3048,

L 2=36.2831,

L 3=33.9809,

L 4 = 45.5407,

L 5 = 47.4476,

L 6 = 31.5523.

Based o n the a l g o r i t h m given previously, a 16th o r d e r p o l y n o m i a l e q u a t i o n is o b t a i n e d : 0.0285Ulr 6 -- 0.1231ulr 5 + 0.2708Ulr4 -- 1.4759Ul 3 + 2.6136U1~2 -- 4.2820U n + 4.7004Ulr0 + 0.8267U 9 + 0.1644U 8 -- 4.0600UTr -- 1.3452U 6 -- 1.9692U~ -- 0.2900U 4 -- 0.3019U 3 -- 0.0119U 2 -- 0.015Ur + 0.0008 = 0. T h e sixteen roots of this e q u a t i o n are: + 2.5585i

-0.1922 -

0.1922 - 2.5585i 0.1963 + 2.4241i 0.1963 - 2.4241i 2.0227 1.9526 1.7207 + 0.4664i

-0.6104 -

0.6104 - 0.4664i

Table 2

~o. 2 3 5 5 7 9 10 11 12 13 14 15 16

Or 2.2233 2.2233 2.2233 2.2233 2.1950 2.1950 2.1950 2.1950 2.0877 2.0877 2.0877 2.0877 0.0963 0.0963 0.0963 0.0963

Os 3.1092 3.1092 1.8905 1.8905 3.1079 3.1079 3.1079 3.1079 3.1052 3.1052 1.8018 1.8018 -2.5495 -2.5495 1.4346 1.4346

O, 1.4534 0.0009 1.4534 0.0009 1.4724 1.4724 1.4724 1.4724 1.5421 0.0144 1.5421 0.0144 2.0264 -0.5429 2.0264 -0.5429

R 20.2895 20.2895 20.2895 20.2895 20.0093 20.0093 20.0093 20.0093 18.9087 18.9087 18.9087 18.9087 0.5532 0.5532 0.5532 0.5532

27.8219 27.8219 27.8219 27.8219 27.3367 27.3367 27.3367 27.3367 25.4304 25.4304 25.4304 25.4304 -6.3623 -6.3623 -6.3623 -6.3623

S 29.5709 29.5709 29.5709 29.5709 29.9866 29.9866 29.9866 29.9866 31.4008 31.4008 31.4008 31.4008 12.3674 12.3674 12.3674 12.3674

12.1035 12.1035 -9.4145 9.4145 12.1022 12.1022 -9.9813 -9.9813 12.0992 12.0992 - 12.0947 - 12.0947 6.7751 6.7751 23.5485 -23.5485 -

-

10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

T 1.0162 1.0162 29.8125 29.8125 1.0571 1.0571 29.9942 29.9942 1.1430 1.1430 30.5697 30.5697 - 17.5251 - 17.5251 31.1126 31.1126

19.7066 6.3999 19.7066 6.3999 19.9907 6.4000 19.9907 6.4000 21.0399 6.4015 21.0399 6.4015 28.1037 8.5670 28.1037 8.5670

-6.8123 16.2356 -6.8123 16.2356 - 7.3044 16.2353 - 7.3044 16.2353 -9.1217 16.2329 - 9.1217 16.2329 - 21.3565 12.4820 -21.3565 12.4820

29.9366 0.0270 29.9366 0.0270 29.9982 0.1278 29.9982 0.1278 30.1317 0.4334 30.1317 0.4334 27.0693 - 15.5730 27.0693 - 15.5730

254 -0.1067 -0.1067 0.0539 0.0539 -0.0564 - 0.0564 0.0482

Z. Geng, L.S. Haynes, J.D. Lee. R.L. Carroll

+ + + -

0.4331i 0.4331 i 0.3732i 0.3732i 0.3600i 0.3600i

T h e r e a r e four real r o o t s for t h e e q u a t i o n : u ~ = 2.0227,

u r = 1.9526,

u r = 1.7207,

and

0.0482.

D u e to t h e sign o f a 4 a n d b 4, a t o t a l o f sixteen solutions a r e o b t a i n e d . Table 2 shows the possible p o s i t i o n s o f p o i n t s R, S, a n d T c o o r r e s p o n d i n g to t h e s e sixteen solutions.

4. Conclusion In this p a p e r , a rigid b o d y d y n a m i c m o d e l for a class o f S t e w a r t p l a t f o r m b a s e d on s i x - d e g r e e - o f - f r e e d o m p a r a l l e l link r o b o t i c m a n i p u l a t o r s is d e r i v e d by using t e n s o r r e p r e s e n t a t i o n . A set o f six L a g r a n g e e q u a t i o n s a r e o b t a i n e d . T h e k i n e m a t i c s analysis for a class o f S t e w a r t p l a t f o r m s a r e c o n d u c t e d a n d a 16th o r d e r p o l y n o m i a l e q u a t i o n c o r r e s p o n d i n g to t h e f o r w a r d k i n e m a t i c s o l u t i o n o f t h e S t e w a r t p l a t f o r m is o b t a i n e d which gives all p o s s i b l e global s o l u t i o n s o f a m a n i p u l a t o r c o n f i g u r a t i o n for a given set o f six leg length.

Acknowledgement T h e a u t h o r s (Z. G e n g a n d L.S. H a y n e s ) wish to a c k n o w l e d g e the s u p p o r t for this w o r k t h r o u g h t h e N a t i o n a l Science F o u n d a t i o n G r a n t # I S I 8821640.

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