Dynamic analysis and control of a Stewart Platform kinematics, dynamics

7/23/2019 Dynamic analysis and control of a Stewart Platform kinematics, dynamics

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Dynamic Analysis and Control of a

Stewart Platform Manipu lator

G. Lebret,

K.

Liu,' and

F.

L. Lewis

Automat ion

&

Robot ics Research Inst i tute

The University

of

Texas at A rl ington

7300

Jack Newel1 Blvd., S.

Fort Worth,

TX

76118

Received May

3,

1992; accepted October

20,

1992

The Stewart platform is a six-axis parallel robot manipulator with a force-to-weight ratio

and positioning accuracy

far

exceeding that of a conventional serial-link arm. Its stiff-

ness and accuracy approach that of a machine tool yet its workspace dexterity ap-

proaches that of a conventional manipulator. In this article, we study the dynamic

equations

of

the Stewart platform manipulator. Our derivation is closed to that of

Nguyen and Pooran because the dynamics are not explicitly given but are in a step-by-

step algorithm. However, we give some insight into the structure and properties of these

equations:

We

obtain compact expressions of some coefficients. These expressions

should be interesting from a control point of view. A stiffness control scheme is de-

signed

for

milling application. Some path-planning notions are discussed that take into

account singularity positions and the required task. The objective is to make the milling

station into a semiautonomous robotic tool needing some operator interaction but hav-

ing some intelligence of its own. It should interface naturally with part delivery and

other higher-level tasks.

0

1993

John Wiley Sons,

Inc.

*Addressee

for

all correspondence.

Journal

of

Robotic

Systems 10(5), 629-655 (1993).

0

1993

by John Wiley & Sons, Inc.

CCC

0741 2223/93/050629-027


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Robot ic

Systems-1

993

1. INTRODUCTION

The Stewart platform manipulator is a parallel kinematic linkage system that

has major mechanical differences from typical serial link robots. In this re-

search, interest focuses on configuration offers the following significant advan-

tages when applied as a manufacturing manipulator:

(1)

Major components of

actuation forces are usually additive; (2) actuator position errors are not addi-

tive.

The Stewart platform manipulator appears simple and refined to the point of

elegance. However, the same closed kinematics that provide mechanical stiff-

ness also present an extremely difficult theoretical problem for forward kine-

matics analysis. This problem has blocked the development of a practical con-

trol algorithm capable of real-time trajectory generation, a necessity for

application of the manipulator.

Much effort has been devoted to finding an efficient algorithm for giving an

accurate kinematic solution since this kind of platform was proposed by

Stewart' in 1965. In Lee and Shah,2 the kinematic behavior of a three-link,

three-degrees-of-freedom (do0 platform was investigated. Though this re-

search gave the reader a clear geometric background, the 3-dof nature of the

device (two for orientation and one for position) limits its application. In Fich-

ter,3 he kinematic behavior of a 6-dof Stewart platform was studied. To solve

for the position of the upper platform in terms of the given link lengths, 30

nonlinear algebraic equations must be solved simultaneously. Due to the time-

consuming nature of this procedure, it is impossible to compute the kinematic

solutions on-line. In Waldron et aL4 and Nanua et aI.,s similar efforts have been

made. Instead of solving 30 equations, a 24th-order polynomial4or a 16th-order

polynomial5 in a single variable must be solved. Not to mention the computa-

tional complexity involved in solving such high-order polynomials, the multiple

solutions alone (there may exist up to 64 solutions) make this a difficult ap-

proach to use practically.

Liu et aL6 proposed a simple algorithm that involves solving only three

nonlinear algebraic equations. This algorithm provides not only a simpler kine-

matics analysis approach but also a relatively simpler approach to deal with the

Jacobian matrix and singularities.

In Liu et aL6 and this article interest focuses on the Stewart platform config-

uration, which consists of

a

semiregular hexagonal lower platform, an equilat-

eral triangular upper platform, and six identical linear actuators as shown in

Figure

1.

The detailed forward and inverse kinematics are in Liu et aL6 In this

article, we concentrate our effort upon the dynamics of the Stewart platform

based on our previous work on the kinematics and Jacobian. Then, we present

some potential applications of the Stewart platform manipulator in manufactur-

ing systems.

In general, there exists a duality between parallel robots and serial linkage

robots. For

a

6-dof Stewart platform, the complexity of the forward kinematics

has to be compared with that of the direct kinematics of the serial robot and, on

the contrary, inverse kinematics are as easy as the forward kinematics of a


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Lebret,

Liu,

and Lewis: Stewart Platform Dynamics Analysis

631

Figure

1. Stewart platform geometry.

serial robot arm. The same remark can be made about the study of the direct

and inverse Ja~obian.~,

As far as the dynamics are concerned, some articles*.9 how the complexity

of this study. In the first, a Newton-Euler approach is used. In the second, the

Lagrange formulation is used. In this research, we also use Lagrange formula-

tion. However, we are more interested in the structure and properties of the

dynamics. Indeed, our aim here is to give more insight into the different coeffi-

cient of the dynamic equations and more precisely into the inertia matrix [M(q)

in

(4)j. We will show how it is related to the Jacobians defined

in

Liu et al.OThe

reason for this development is that a good knowledge of these terms is useful

for the study of adequate control schemes. For example, only a particular

factorization of the Coriolis and centrifugal terms

[V(q,$

in (4)] eads to the

well-known skew-symmetric relation used in the proof of the convergence of

several robust and adaptive In the same way, we hope that the de-

scription of the matrix M(q) will allow us to use some concepts like imaginary

robot developed by Gu and Loh.I4 This theory is based on a factorization of

the inertia matrix using Jacobians.

As a manufacturing manipulator, the Stewart platform has great potential to

be applied for the automation of many light machining applications such as

surface finishing, edge finishing, routing, and profile milling. New manipulator

applications to manufacturing processes requiring high-force and -power out-

put such as combined assembly pressing are also possible. Therefore, in section

4 the potential applicationsof the Stewart platform in the manufacturing indus-

try are discussed. As a special application, we will study the stiffness control

in

the milling process.

The milling machine is one of the most important machining tools used in

todays manufacturing industry. It can be used to process either the tools or the

parts. Due to its ability to quickly cut material and accurately follow contours,

the milling machine is used to machine small pieces such as helical gear or


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632 Journal of Robot ic Systems-1993

large-scale parts such as cylinder heads for an oceangoing freighter. However,

because

in

general the cutter has only

2 or 3

dof to process a complex work

several configuration adjustments of the cutter, table, and fixture must be made

to let the combination of the movements meet the desired contour. In some

cases, a special fixture must even be designed to meet a specific need.

Another problem in milling is the shock that may occur when teeth on a

cutter make contact with the work. To minimize the effects of shock and give a

smoother cutting action, the cutter holder the spindle) must be very cumber-

some and powerful.

The Stewart-platform-based manipulator (SPBM) as some innate merits

that make it suitable for milling actions. Its link ends are simply supported,

making it far more rigid in proportion to size and weight than any serial link

robot. Furthermore, its links are arranged

so

that major force components of all

six prismatic joints may add together, yielding a ratio of force-output-to-manip-

ulator-weight more than one order of magnitude greater than most industrial

robots. The closed kinematic chain and parallel linkage structure not only give

the

SPBM

great rigidity and a high force-to-weight ratio but also provide the

capability of improved accuracy over serial link robots because the parallel link

positioning errors are distributed, not cumulative.

2.

NOTATIONS

To introduce the notations, we first briefly summarize the approach used in

Section

3

to derive the dynamic equations, Then, we describe the different

frames and coordinates needed to modelize the robot.

Procedures using the Lagrange formulation are

1.

Calculate the kinetic energy K)nd potential energy

P)

s a function of

the generalized coordinate q :

(1)

P = P q)

where P(q) E

53

(2)

1

K

=

K q,Q) =

2 QTM(q)Q

where

q E

3

and M

E R n x n

2.

Develop the Lagrange equations using

where 7 is the torque

or

force to apply to the body.

This finally yields the dynamic equations in the standard form:


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Lebret, Liu, and Lewis: Stewart Platform Dynamics Analysis

633

The inertia matrix

M(q)

is directly given by the expression of the kinetic

energy K(q,$. The gravity term is obtained from the potential energy P(q) by

G(q) =

aP(q)/aq. However, the Coriolis and centrifugal term (V(q,$ E R n ) s

not

so

easy to express.

2.1.

Corio l is

and Centrifugal Term V(q,a)

Using the Christoffel symbols

[Cu&(q);

ee ref.

121, V(q,q)

can be obtained by

v(q,qIk=

e (eC,qi)C

fork = 1,

. . . ,

6

j i

with

This is the most practical expression to compute the Coriolis and centrifugal

term

V(q,q)

for a numerical application. But, for a theoretical study the de-

scription of Koditschek" is also interesting:

where

V,,,(q,q)

is an

n x n )

matrix that can be expressed by

Here,

U

s a matrix obtained from M(q) and 4 by the following relation, where

@ represents the Kronecker product:

with


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634

Journal

of

Robotic Systems-1 993

Remark

1

The equivalence of the two descriptions can be understood from the rela-

tions

The skew-symmetric property commonly used (see refs.

12

and

13) in

the

Rroof of the convergence of several control laws is easy to derive from

(8):

M - 2V,,

=

UM -

UL

is a skew-symmetric matrix.

2.2. Frames and Coo rdinates Asso ciated with the Robot

In our study, we will use the following notations to modelize the parallel

robot. Referring to Figure 1, fix an inertial frame X, ,Z) at the center

0)

f

the lower platform with the Z-axis pointing vertically upward. Fix another

moving coordinate system (x,y,z) at the center of gravity of the upper platform

with the z-axis normal to the platform, pointing outward. In the sequel, these

two coordinate systems are called the

BASE

or fixed frame (ff)and the TOP or

moving from (mf). The physical dimensions of the lower and upper platforms

and the coordinates of their vertices

in

terms of the

BASE

frame or the TOP

frame are shown in Figure 2.

To

specify the configuration of the 6-dof Stewart platform, six independent

position-orientation variables are needed. Denote the location of the origin of

-4

+

1

X

Figure2. Physical

dimensions

Y

of

base

and top.


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Lebret,

Liu, and

Lewis: Stewart

Platform Dynam ics Analysis

635

1

unity

approach vector

Y

Figure

3.

Rotation

angles.

the TOP frame with respect to the BASE frame

[ p x , p y , p z l T .

The orientation is

not defined by the standard Euler angles but by rotating the

TOP

fram e first

about the fixed X-axis by a degrees, then about the fixed Y-axis by

p

degrees,

and finally about the z-axis by y degrees (Fig. 3).

All

angles are m easured in the

right-handed sense. We will denote R x ( a ) , R y ( p ) , and R , ( y ) the three matrices

that represent these basic rotations.

1

R , ( y ) =

P 0 SP

-sp

0 cp

c y

- s y

0

0 0 1

(13)

This definition of orientation not only gives

us

a clear physical meaning but

also avoids violating the o ne-to-one relationship between the system configura-

tion

and the value of Xpp0,hich may cause the Jacobian matrix to lose its rank

even if the system is not in a singular position (see ref.

10).

Thu s, the position and orientation of the upp er platform is specified by the

Cartesian coordinates

XP- ,

[ p x , p ~ , p z , a , p , y I ~ .

Other notations will be introduced in the text. Just note that to make clearer

t h e reading we will differentiate a vector from

a

scalar by adding an arrow and

will denote the scalar product of two vec tors u and v either by u *

v'

o r iTv u

will be the module of 6).

3.

DYNA MIC EQUATIONS OF THE STEWART PLATFORM

As noted in Section 1, there exists a duality between serial and parallel

robots. It app ears here in the fact that the dynam ic equations are less difficult to

obtain in the Cartesian spa ce than in the joint space. Thus, o ne see ks the


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Jou rnai of

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equations

M(X,.,-,)X,,-, +

Vm(Xp-,,X,,-,)Xp-,G X,-,) =

J*(X,,-,)T

(14)

The notations are the same than in eq.

(4);

we just replace the joint coordi-

nates q by the Cartesian coordinates X,,-()nd as a consequence replace the

torque

T

by the force

J

* X,,-,)T. T = V; fiA

,f4,fs ,fs] ,

where is the force

applied in the actuator of the leg Legi. Note that the Jacobian ( J ) that

is

introduced here, and more generally for parallel robots,

is

the classical

in-

verse Jacobian (of serial robot). This explains why we have here J * and not

J-*, as is the case for serial robots.

In the following, we divide the robot into two subsystems: the upper platform

and the six legs. We compute the kinetic energy and the potential energy for

both of these subsystems and then derive the global dynamic equations.

3.1. Kin etic and Poten tial Energies

3.1.1. Kinetic and Potential Energies

of

the Upper Platform

The characteristics of the upper platform (up) are

mass m u .

tensor of inertia

in

the moving frame l(d).

angular velocity in the fixed frame

f l u p ( ~ )

Kinetic energy

of

the upper platform. The kinetic energy (Kup) an be di-

vided into two terms:

Let

Translational energy. Energy of the center of mass when one considers

that the entire mass of the platform is concentrated there:

Rotational energy. Energy of the body due to the rotation around its center

of mass:

us develop the last expression.

The tensor of inertia O(m has a simple form in the moving frame

O(,,,f)

= 0 Iy 0

.

Given the definition of the angles a, p and y , the

[ I


9/27


angular velocity fiUp fi)of the upper platform is

or

CP 0

S P

- s p

0

cp

fiup(ff,

= ([

0 1 0

:

, j +

0 0 0

, :

:]

0 0 0

637

(17)

+ ['

C a

-saI[:

3

:

(18)

Sa

ca

0 0 1

In the moving frame,

it

is defined by

f iup mf) = R, y)TRx a)rRy P)r

iup fi).

Thus,

we obtain

1

Kupbt ) = 5

ci

IxC2y

+ f y S 2 y 1,

- y ) C a C y S y

0

0

-1zS2a

I Z

( f x - f y ) C a C y S y C 2 a ( l x S 2 y + ZyC2y) + fzS*a

- f z s 2 a ]

[i]

[

[

(19)

Finally, we can write the total kinetic energy o f the upper platform in a compact

form as follows:

Result 1

or

K,, =

2

m u

0

0

0 m ,

0

0 0

m ,

0 0 0

0 0 0

0 0 0

Potential energy of the upper platform


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638 Journal of

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Systems-1993

Result

2

Pup

= mu

p z

= [0 0

mug 0

O]Xp-o

(21)

where

g

is the gravity

3.1.2.

Kinet ic and Potent ial Energies of the Six Legs

A precise study of the dynamics of the legs should require a decomposition

into two parts: the fixed part (to the base) and the moving part (see Fig. 1). And,

we should consider them as two bodies with their own inertia. However, this

will lead to a complex form ulation.

As

a simplified yet still accurate model, we

will consider that each leg can be represen ted by a m oving point (its cen ter of

mass

Gi)

and assume that the mass is concentrated there. This takes into

account the motion of the center of gravity due

to

the change in leg actuator

length.

No te the following specification of a leg (Legi

i = 1, . . . ,

6) The ce nter of

mass

GI;

of the fixed part is not in the middle of the body due to the great mass

of the DC motor (see Fig.

4).

Let us denote by lI and

rnl

the length and the

mass, respectively, of this part and 6 he distance B,Gli.On the contrary, the

moving part of each ac tuato r is uniform. Its center

of

mass Gzi is in its middle.

Let us denote 12 and

m2

the length and m ass, respectively, of this second part.

Denote by

i

the unitary vector that gives the direction of a leg (I;i

=

m,/

Bi7j)-wherej

=

i

+

1)/2

if

i

is odd a n d j

=

i / 2

if

i

is even ). Finally,

Li

is the

total length of the leg Legi (note that this is a function of the time; its time

derivative will be noted

i i ) .

Figure

4. Leg

of

the

Stewart platform.


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639

To obtain the kinetic and potential energies of a leg (Legi

=

1 , .

. . ,

6) we

need to determine the position and velocity of its center of mass

Gi :

The velocity vG, f the center of mass G ; is then

1

i

+

1)/2

if

i

is odd

where

= i / 2

if

i

is even

Note that

Kinetic energy

of

the legs. Because a leg is just modeled by one point, one

can write

Note that v, *

Ci)CJT VC

- (vq GJC;)

=

0; thus, after some calculations one

obtains


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640

Let

us

denote

hi =

(-

+

L;

mI

+

m2

Then:

Result 3

Journal

of Robotic Systems-1

993

1)/2 if i is odd

(29)

if i

is even

where

Remark 2. One can also write

Remember that

is

the vector that gives the direction of the leg

i.

Thus, if

qT

s the velocity of the end of the leg,

* ; t i j

corresponds to the speed of the

elongation of the leg, that

is =

dL;/dt. As a consequence, K L , can be

divided into two parts with a physical meaning as follows:

The kinetic energy due to the rotation around the f ix point B; of the leg (if

one supposes that the length of the leg is constant), given by

The kinetic energy due to the elongation (or the translation motion) of the

leg (if one supposes that the direction of the leg is constant), given by

Let us now give a compact expression for the kinetic energy of the six legs.

For

this, note that one can write

r

hi 0

0

0 hi 0

0

0

hi

(33)


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Lebret, Liu, and Lewis: Stewart Platform Dynamic s Analysis

w

Hence

-

hi

+

h,

0

0 hj + hz

0 0

-

where

and

K =

k l O O O

O k 2 O O

O O k 3 O

0

0

hi + h2

h3 +

h4

0 0

0 h3 + h,

0

0 0 h3

+

h,

T

K

h5

+ h6

0

0

with

1

0

0

h5 +

h,

64 1

(34)

(35)

[ k i = hi- (

m 2

) 2

ml + m2

Let us now introduce the Jacobians J I and J (see Appendix A

or

ref.

10.

for

details). One can write


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642

- -

h i 0 0 0 0 0

O h 2 0 0 0 0

O O h J O O O

and K2

= ( m2 2

0 0

0

hq

0

0 mi + m2

0 0

0

0

hs

0

-

0 0 0 0 h 6

Jou rnal of Robotic

Systems-1 993

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

I

0 0 0 1 0

Thus, one finally obtains:

Result

4

561 =

where

~ o o o o o

]

(40)

h . =

(-+

'

Li ml + m2

Potential energy of the legs

j

=

i

t

1)/2

j

= i / 2

if

i

is odd

if

i

is

even

Z7;

=

pz + 2

mj.fi)

nd

One can finally write:


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Lebret, Liu, and Lewis: Stewart Platform Dyn amics Analysis

643

Result 5

with

Remark

4.

mj(,f,

s given by the geometry of the platform.

3.2. Dynamic Equations

of eq.

(14),

i.e.,

We now have to calculate the terms

M(Xp-,,),Vfn(Xp-o,

p - J , and

G(Xp-,,)

M(Xp-,,)Xp-,, V f n ( X P - ,p - JX P - , )+ G(X,-,) = JT(Xp-,,)F

More precisely

From the expressions of the kinetic energies of the platform and the legs, we

directly have an expression of the coefficient M(XP-J. Vm (Xp-,,

Xp-J

will be

obtained from relations

5 )

and (6) [or from (8)]. The gravity term

G(X,-,)

has

to be calculated from the potential energy.

Although it is not difficult to obtain an explicit expression for

Mu,,

mbp,nd

G,,

(see Appendix B), it is impossible to obtain explicitly

MLegs, VmLlsr

nd

G L ~ ~ ~ .ven with a symbolic package like Mathematica, the expressions are too

complicated to obtain In this case, a step-by-step formulation must be used.

3.2.7. Terms

Mu,,

Vmup, nd

G,,

See Appendix B.

3.2.2. Terms MLegs,mLwa,nd GLegs

diagonal matrix. Thus

From (381, we know that M L e g s

=

ml

+

m2)[JT[kU - J ~ K J I I J 2 l , here is a

9 9

6 i = l , .

. .

, 6

m = l n = l

p = l

j = 1 , . . . , 6

MLegsij =

2 C [ J2mi M m n

- 2

J l p r n K p p J l p n ) J z n j ]

{

(43)


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Journal of Robot ic Systems 1

993

Now,

from relation (6) of Section 2, we have

So, we have

to

calculate a mLegsij)laXp-o k))

or

i, j , k =

1,

. .

. ,

:

d J ~ ~ i / a X ~ - ~ k ) )r aJ2nj/aXp-o k,) re not difficult to obtain from the expres-

W m n

f

0 only i f m = n. In this case , if one takes i = 1 for

1

5 , i = 3 for

sion of J2 given in Appendix A.

4 5 6,

and

i

=

5 f o r 7

n

5

9

aJlpn/aXp-dk,) r aJlpmlaXp-dk))an be calculated from the expression

of

J I

given in Appendix A:


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Lebret, Liu, and Lewis: Stewart Platform Dyn amics Analysis

645

Note that to get

(46),

(47), and

(48)

we need the expression of (aLnlaXp-o(k))or

n k = I , . ,

.

,

6. These terms can be obtained from

J I

and JZ by

The other terms

( m l , m2, q. ,,,f)...)

re either information available from the

geometry of the robot or ( a ( p ) r Ln.J information that has to be computed.

Note that these last terms are intermediate or final results of the computations

of the inverse kinematic that might be included

in

the algorithm described

hereafter.

3.2.3.

Term GLegs

It

is necessary to include the computation of G L ~ ~ ~n the algorithm. Indeed,

one can see from the following expression of GLegs(Xp- ) hat we need

the expressions of

(dL2;- tlaXp-,,(k)), (aLzJaX,-,,(k)), 1

/Lit-

I), 1 lLij)y

and

( a z T p x p - o ( k ) ) :

where


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Journal of Robotic

Systems-1

993

3.3.

Summary

of

the Calculat ion

of

the Dynamics and Algor i thm

To obtain the dynamic equations, we use a combination of direct and step-

by-step Lagrangian formulations: direct formulation for the platform and step-

by-step formulation for the legs.

The expressions for Mu,, V, ,, and G,, are explicitly given in Appendix B.

The algorithm for computing MLegs, VmLCEvnd G L ~ ~ ~re summarized below.

Algorithm

for

the computation of the terms M L ~ ~ ~ ,mkp,,nd GLegs

1. Compute the inverse kinematics (see Appendix A for the theoretical ex-

pression)-This gives the length of the legs: L ; , i

=

1 , . . . , 6 . Note that

it also includes (see Appendix

A

the calculation of the termsm.(p)p = 1 ,

.

. . 3), and

ZT, ,

j = 1 ,

. .

. 3).

2.

Compute the

Jacobian

J I

(see Appendix

A

for the theoretical expres-

sion)-This needs the results of the previous step ( L i , i = 1 , .

. .

6 and

B i T , , j

=

1, .

. . 3).

3. Compute the Jacobian J2 (see Appendix A for the theoretical expression).

4. Compute the matrices and

06

[see relations (35) and (36)l-This needs

the values of

Li i

= 1,

. .

.

,

6).

5 . Compute the JacobianJ = J I 2to obtain dLn/aXp-o k))=

1,

. . . ,6, k =

1,

. . .

, 6 [see relation (49)l.

6. Compute the components of the tensors

(dJ2mi/dXp-o(k)) ,

dW,,,,/dXp-o(k)),

(dK,,,,/dXp-o~~ ,nd

( d J ~ , , ~ / d X ~ - ~ ~ , k ~ )

o calculate the other tensor defined by

(d(mLegs; j) /dXp-o(k))relations (461, (47), (48), and (4511.

-

7. Compute the Chrisroeffeelsymbols Cijk with relation (6).

8. Compute the coefficients MLegs, V,, and GLegs [see relations 43),

9,

and

SO)].

Remark and Conclusion.

Note that steps

1

to 4 are sufficient to obtain

MLegs.

Step 5 has to be added to obtain GLegs.But most of the computer time will be

dedicated to 6 and 7, that is, for the computations of VmLIEI.n fact, these last

steps will probably not be necessary.

As

pointed out by Reboulet and Bertho-

mieu* or Nguyen and P ~ o r a n , ~or most applications the contribution of the

Coriolis and centrifugal term is small and may usually be neglected.

As a conclusion of this section, we have to say that our study is close to that

of Nguyen and P ~ o r a n . ~owever, our contribution here is result 4, which

gives a compact expression of the inertia matrix MLegs n terms of the Jacobians

JI and J2. The interest of this result is essentially theoretical. Indeed, we think

that such an expression will allow us to develop some special control approach,

similar to the one described in Gu and L0h,14 where the matrix of inertia (M)

needs to be written in terms of some Jacobians.

Work in this direction as well as simulation of the dynamic equations will

have to be done to make possible the application in the milling process pro-

posed in the next section.


19/27

Lebret, Liu, and Lewis: Stewart Platform Dynam ics Analysis

647

4. STIFFNESS CONTROL OF A STEWART

PLATFORM-BA SED MANIPULA TOR

In this section, we

will

discuss the applications of a Stewart-platform-based

manipulator (SPBM)

in

manufacturing industry.

As

an example, we will con-

sider a special case-the stiffness control of SPBM in milling process.

As mentioned in Section 1, the milling machine is a popular machine tool that

can be found

in

any modern machine shop.

A

typical milling machine consists

of a swivelling (or nonswivelling) table, which is used to hold the parts to be

machined, and a horizontal- or vertical-spindle, which is used to hold the

cutter. Usually, the table (holding the part) is fixed during the milling process.

The spindle (with the cutter) can make 2-D movement. To machine a complex

part, several adjustments of the configuration of the table and spindle must be

made manually. In some cases, a specially designed fixture

or

attachment may

even be used.

In

milling actions, the cutter is required to be in continuous contact with the

part being machined to exert certain cutting forces along certain directions. It is

known that

when

the teeth on a cutter make contact

with

the part, shock or

vibration may be experienced.

The deficiencies of the regular milling process may be remedied by introduc-

ing an SPBM into the milling system. Its

6

dof including 3 dof in position and

3

dof in orientation and its precise positioning capacity make it fit for coordinat-

ing motion control

with

the cutter to machine a complex part. More important,

its far greater stiffness and high force-to-weight ratio make it suitable for reduc-

ing

the shock

or

vibration during the milling actions, yielding a smoother

mill-

ing path.

In this research, we propose to mount an SPBM on the machine table. The

SPBM base is fixed to the table and its upper platform holds the part being

machined.

From the discussion of Section 3,

it

can be seen

that

the dynamics of an

SPBM is complicated. Further, some parts of the dynamics are not understood

well. This makes controller design difficult. Standard control schemes such as

linearization techniques cannot be easily implemented because of a lack of the

complete knowledge of the dynamics.

However, in the milling process we may take advantage of this particular

application (e.g., the motion of an SPBM is not

so

fast) to design a simplified

controller. In this controller, a simplified model of an SPBM will be used in

which the Coriolis-centrifugal terms are ignored due to the assumption of low

velocities. A possible computed-torque controller for this situation was pro-

posed in Liu et al.

In

this research, a good deal of effort has been put into reducing the shock to

a minimum by using stiffness control.

Let (x, y ,

z)

be an orthogonal coordinate system associated with the fixed

task space. Its origin is at the location of

( X , ,

Y,) ,

2,

in

terms of the base frame

X - Y - Z

and its rotation angles with respect to the base frame are a,, o ,y o ) .


20/27

648

Journal of

Robotic

Systems-1993

(xo~~@zo* so~7d

Figure 5.

Base frame and task frame.

Let

R (a , , P o ,yo) E

R3')

be the constant rotation matrix from the task space to

the base space (Fig.

5 .

Then,

it

is easy to express the transformation from task

space to base space by

Let

e l ,

e2 ,

03)

be the yaw-pitch-roll angles about the x - ,

y-,

and z-axes. Then,

the yaw-pitch-roll matrix

in

task space can be written as

cos

O2

cos

O 3

sin 8, sin O2

cos e3-

cos

sin

f13

cos

82

sin

O3

sin

el

sin 62 sin

O3

+ cos

8,

cos O 3

-sin

02

sin 0, cos

tI2

cos

e l

sin

e2

sin

e3 -

sin

61

cos

83 =

[ne

Og ael

cos

e l

sin z cos O3

+

sin 8, sin

e3

cos

e l

cos

e2

where no, 0 8 , and

a6

represent the normal vector, sliding vector, and approach

vector, respectively. It can be shown that

any

rotation movement in the task


21/27

Lebret, Liu, and Lewis: Stewart Platform Dynamics Analysis 649

space

e l , 02,63)

can be transformed to the rotation movement

in

the base frame

a ,@, y ) by using the "equivalent transformation matrix"

Comparing both sides of (52) term by term gives the rotation angles

in

base

frame expressed in terms of the rotation movement in task space

by

where

R j

is the ith column of the rotation maxtrix

R

and

(a . b)

represents the

dot product of two vectors a and

b.

Thus,

a n y

position/orientation changes in

task space can be transformed to the base space using

X l

where

Jh E R h x 6

s the Jacobian between the task space and the reference

(world) frame. We assume that

Jh

is nonsingular.

To model the situation of contact between the cutter and the environment

(e.g., the part), another term

F,.

must be added into the dynamic equations

(9)

to

yield

MX,-,, + VJX, -,,, xp-,,)xp-,, G(x,-,J = F

- J ~ F , .

(57)

where

F,.E

R h x l epresents a vector of the reaction forces/torques between the

cutter and the environment in task space.

Intuitively.

if

a mechanism can automatically generate a force that will tend

to return any deflected states to the original, nominal states then that mecha-

nism can be used

to

solve our problem.

A n

ideal, linear spring will generate a

force that is proportional to the deflection from its nominal position. In this

research, we assume that there are no "twisting" actions occurring in the

cutter and the part; thus, a

3-D

linear spring can be used to model the contact


22/27

650

Journal of

Robotic

Systems-1 993

between the cutter and the part. Controlling the variation rate of the forces is

equivalent to controlling the stiffness of the spring. Let F, be

F, =

k , o o o o d

0

kt 0 0 0 0

O O k O O O

0 0 0

k, 0

0

0

0

0

0

k, 0

0 0

0

0

0

k ,

-

x - o

Y - Y u

z

-

u

0

0

:

=

K, 1 58)

X (x,

, o ,

zo)

is a vector of the commanded, nominal contact position in task

space,

x T

=

(x,

y ,

z)

is the tip position of the cutter in task space, and

k,

represents the stiffness constant. Figure

6

shows a

1-D

spring

in

the z-direction.

Before the cutter makes contact with the part, only position control is needed

to command the mobile platform (holding the part) to follow the desired posi-

tiodorientation trajectories Xp-ond Xp-o.s long as the cutter

is

in contact

with the part, stiffness control must be included

to

force the velocity of the

mobile platform in Cartesian space to go to zero while maintaining the desired

cutting forces

Fcd

in task space.

Because the contact force is modeled as a 3-D spring as shown in 58). we

may express the desired force in task space as

Figure 6. One-dimensional spring model.


23/27

Lebret,

Liu,

and Lewis: Stewart Platform Dyn amics Analysis 651

where

x:

= (xd, Y d , z d ) represents a vector of the "virtual" desired tip position

of the cutter

in

task space. We call it virtual because

it

only plays a role in

deriving the controller formulation. It

will

never be reached.

Define a Lyapunov function candidate"

where P,.

is

the force error defined by F,.d -

F,.

.Taking into account models

(58)

and 59) , (60) can be written as

u X,,-,, ,

F,) =

2

. pT-oMXp-,,

+ ["i

I T K ,

[ ,1

(61)

Differentiating (61) with respect time along with the dynamic eq.

57)

yieldsi6

[ (I ]} [:] , [ ' 1

(62)

F

-

G(X,-,)

- Jh'K,

Note that in

(62)

we already use the skew-symmetric property of the dynamics

(see Sections 2 and 3). If the generalized control force F is selected as

where K E R6x6 s a diagnonal, positive, constant matrix, then

The result of 64) an only guarantee that the Cartesian velocity of the mobile

platform goes to zero and hence the Cartesian acceleration goes to 2er0.I~

However, checking with the closed-loop system may find that at steady state

Because we assumed that the Jacobian Jh is nonsingular; it is easy to infer a

zero solution of

F,

from

65).

Note that expression (63) only gives us the "generalized" control input

in

Cartesian space. To manipulate the linear actuators, this control input must be

transformed to link space. Another problem is that in (63) measurements of

Xp-uand XP-


24/27

652

Journal of Robotic Systems-1 993

synthesized by

There is no need to measure positiodvelocity in task space. That is why we call

x d a

virtual desired position vector.

5.

CONCLUSION

In this article, we study the dynamics of a Stewart platform manipulator and

its potential applications in manufacturing industry. Although the explicit dy-

namic formulations cannot be obtained due to the high nonlinearities existing in

the kinematics and the strong coupling among the legs, a feasible dynamic

algorithm is proposed. As

a

special application

of

the Stewart platform, the

SPBM

milling station is discussed. Applying a simple stiffness control, the

mobile platform the part) not only follows the preplanned position/orientation

trajectories but also maintains desired forces in certain directions. Two proto-

type Stewart platforms have been built by the Advanced Controls and Sensors

Group at Automation Robotics Research Institute ARRI), the University of

Texas at Arlington. A TMS320C30-based real-time control system

is

being

developed and integrated at

ARRI

for SPBM control.

This research was supported by Texas Advanced Technology Program under Grant

003656-008

nd Le Ministitre Franais des M air es Etrangtres Francaises, Programme

Lavoisier.

APPENDIX A: INVERSE K INEMATICS AND JA COBIANS6*10

Inverse Klnematlcs

Li

=

F(X,-,)

F o r i =

1 , .

. .

, 6

Li = BiTj =

OTj -

OBi

where

OBi

the norm of m) s given by the geometry of the base and

where

j

=

i

+

1)/2 if

i

is odd

if i

is even

i t2

and

G T J ~

s given by the geometry of the platform.


25/27

Lebret, Liu, and Lewis: Stewart Platform Dynam ics Analysis

653

Jacoblans

E

L

j

with

where

where

1)/2 if i is odd

if i is even

and

platform and if

u =

v1 -v3 V2

-u2] then S u) =

[

i2

.v3

And,

13

is the identity

matrix of order 3.

Explicit expressions for

J1

and

Jz

are given in Liu et

d.Io


26/27

654

Journal of Robotic

Systems-1 993

APPENDIX

B:

UPPER PLATFORM DYNA MICS TERMS

M

=

with

Gup

= [O

0 mup 0 0 OI

References

1 . D. Stewart, A platform with six degrees of freedom, Proc. of the Institute for

Mechanical Engineering, London,

1%5,

Vol. 180, pp.

371-386.


27/27

Lebret, Liu, and Lewis: Stewart Platfor m Dynam ics Analysis 655

2 .

3 .

4 .

5 .

6 .

7 .

8 .

9 .

10.

I I .

12.

13.

14.

15.

16.

17.

K. M. Lee, and D. K. Shah, Kinematics analysis of a three-degrees-of-freedom

in-parallel actuated manipulator, IEEE

J

of Robotics and Automation,

4,

354-

360 (1988).

E. F. Fitch er, A S tewa rt platform-based manipulator: G enera l theory an d practi-

cal construction, Int. J of Robotics Research, 157-182 (1986)

K. J . Waldron, M. Raghavan, and

B.

Roth, Kinematics of a hybrid series-parallel

manipulation systems, J of Dynamic S ystem s, Measurement, and Con trol,

111,

P. Nanua, K . J. Waldron, and V. M urthy, D irect kinematics solution of a Stewart

platform,

IEEE Trans. on Robotics and Automation, 6, 438-444 (1990).

K. Liu, M. Fitzgerald, and

F.

L. Lewis, (1991 a) Some Issue s about Modeling of

the Stewart Platform Manipulator, Proc. Second Int. Symposium on Implicit and

Robust Systems, Warsaw, Poland, pp.

131-135.

J. P. M erlet, Les Robots Paralleles, Herm es, 1990.

C. Reboulet and T. Berthomieu, Dynamic models of a six degree of freedom

parallel manipulators,

Proc.

of

the Con5 K A R 91,

Pise, Italy, 1153-1 157 (1991).

C. C.

Nguyen and

F.

J.

Poo ran, Dynam ic analysis of a

6

DOF

CKCM robot end-

effector for dual-arm telerobot systems,

Robotics and Autonomous Systems,

K. Liu, F. L . Lewis, G. Lebr et, and D. T aylor, The singularities and dynam ics of

a Ste war t platform m anipulator,

J of Intelligent Robotic Syste ms.

D .

Koditschek, Natural motion for robot arms,

Proc. of the 23rd Conf,

on

Decision and Control, Las Vegas, NV, 733-735 (1984).

21 1-221 (1989).

377-394 (1989).

M. Spong, and M. Vidyasagar; Robot Dynamics and Control, Wiley, New York,

1989.

J. J. Slotine and W. Li, Adaptive manipulator control: a case study, IEEE

Trans. on Automatic Control 33,

995-1003 (1988).

Y. L. Gu and N. K. Loh, A dynamic modeling and control by utilizing an

imaginary robot,

IEEE J . of

Robotics and Automation,

532-540 (1988).

K. Liu, M. Fitzgerald, D. Dawson, and F. L. Lewis, Modeling and control of a

Stewart platform manipulator,

Proc.

of

the Symp. of Control of Systems with

Inexact Dynamic Models, Atlanta, GA, 1991, pp. 83-89.

F. L. L ewis , C. T. Abdallah, and D. M. D aws on, Control of Robot Manipulators,

New York: Macmillan, 1993.

J .

J. Craig, Introduction to Robotics, Addison-Wesley, Reading, MA, 1989.

Documents

Dynamic analysis and control of a Stewart Platform kinematics, dynamics