Isotropic Design of a Parallel Machine-Tool Mechanism

7/28/2019 Isotropic Design of a Parallel Machine-Tool Mechanism

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D. CHABLAT, PH. WENGER, J. ANGELES

ISOTROPIC DESIGN OF

A PARALLEL MACHINE-TOOL MECHANISM

Abstract: The subject of this paper is the isotropic design of a hybrid mechanism intended for three-axis

machining applications. Parallel mechanisms are interesting alternative designs in this context. We

compare machine-tool mechanism, with a hybrid serial parallel structure, of which we optimize the first

two-axis subsystem, i.e. the table of the machine tool. The comparative study is conducted on the basis of

a prescribed workspace and given kinetostatic performance. The two-degree-of-freedom mechanisms

analyzed in this paper can be extended to three-axis machines by adding a third axis in series with the

first two.

1. PRELIMINARIES

1.1 Serial Mechanism with Three Degrees of Freedom

Most industrial machine tools use a simple PPP serial kinematic chain with threeorthogonal prismatic joint axes (Figure 1).

In this case, the motion in the XY plane andthe motion along the Z-axis , where the axis of rotation of the tool is located.

The problem of the PPP mechanism is thatthe actuator controlling the Y -axis supports at

the same time the workpiece and the actuator controlling the displacement of the X -axis,which affects the dynamic performance.

To solve this problem, it is possible to usemore suitable architectures like parallel or hybrid mechanisms. For a PPP mechanism,the kinematic model takes the form

Figure 1: Traditional mechanism of three-axis machine tool

J .ρ =

.p, with J = 13 × 3

where.p= [

.x

.y

.z]T is the velocity-vector of the tool center P and

.ρ = [

. ρ1

. ρ2

. ρ3]

T isthe velocity-vector of the prismatic joints.

The Jacobian matrix J being the identity matrix, the manipulability and force[Yoshikawa 85] ellipsoids are both a unit sphere for all configurations in theCartesian workspace. The aim of this paper is to replace this structure by a hybridserial-parallel mechanism.

1.2 Planar Parallel Mechanisms with Two Degrees of Freedom

T.H.E. Editor(s) (ed.), Book title, 1—6.

© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

We focus on a two-DOF parallel mechanism, Figure 2, for the motion of the table of



D. CHABLAT, PH. WENGER , J. A NGELES

the machine tool depicted in Figure 1.

Figure 2: Parallel mechanism with two

degrees of freedom

Figure 3: Parallel mechanism with two degrees

of freedom with control of the orientation

The joint variables are ρ1 and ρ2 associated with the two ismatic join and theoutput variables are the Cartesian coordinates of the tool enter P =[ x T. Themechanism can be parameterized by the lengths L1 and L2, the angles α1 an α2, the position of points A and B, and the actuated joint ranges ∆ρ1 d ∆ρ2. To reduce thenumber of design variables, we set L1 = L2 and ∆ρ1 = ∆ρ2. is simplification also provides symmetry and, in turn, reduces the manufacturing c s.

pr tsc y]

dan

Thostto nte

sidin eir dit to

is one can be loc

ta

To control the orientation of the reference frame attached the tool ce r point P , two parallelograms can be used, which also increases the rigidity of the structure,Figure 3. The dimensioning of the parallelogram links is out e of the scope of the paper. However, the rigidity of the structure as well as the jo t limits of th passive joints depends on the technological solutions adopted for the esign.

To produce the third motion of the machine tool, is possible place

orthogonally a third prismatic joint on top of the first two. Th ated asin the case of Figure 1.

1.3 Kinematics of Planar Parallel Mechanisms

The velocity.p of point P can be expressed in two different ways. By traversing the

closed loop ( ACP-BDP ) in the two possible directions, we ob in

where E is a matrix of rotation through 90°, c and d representing the position vectorsof the points C and D, respectively. Moreover, the velocities

.c and

.d of the points C

and D are given by

where the angles α1 and α2are the orientations of the prismatic actuators from line

AB of Figs. 2 and 3. We would like to eliminate the two idle joint rates.θ 1 and

.θ 2

from equations (1), which we do upon dot-multiply the former by (p - c)T and thelatter by (p - d)T, thus obtaining

.c =

c - a

||c - a||

. ρ1 =

cos( α1)

sin( α1)

. ρ1 ,

.d =

d - b

||d - b||

. ρ2 =

cos( α2)

sin( α2)

. ρ2



ISOTROPIC DESIGN OF A PARALLEL MECHANISM: APPLICATION TO MACHINING 3

(p - c)T .p = (p - c)

T

c - a

||c - a||

. ρ1 and (p - d)

T .p = (p - d)

T

d - b

||d - b||

. ρ2 (2)

Equations (2) can now be cast in vector form, namely, A.p = B

.ρ, with A and B

denoted, respectively, as the parallel and serial Jacobian matrices,.

ρ defined as thevector of actuated joint rates and

.p is the velocity of point P , i.e.,

A≡ (p-c)T

(p-d)T , B ≡

(p-c)T

c-a

||c-a||,0,0, (p-d)T

d-b

||d-b||,

. ρ=[ ].

ρ1 ,. ρ2 ,

.p= [ ].

x ,.

y

When A and B are not singular, we obtain the relations,

.p = J

.ρ with J = A-1 B and

.ρ = K

.p with K = B-1 A (3)

We base our design on the Jacobian matrix K .

1.4 Parallel Singularities

Parallel singularities occur when the determinant of A vanishes [Chablat 98], i.e.when det(A) = 0. In the presence of such a singularity, the tool center P can movewith the actuators locked.

Figure 4: Parallel singularity Figure 5: Structural singularity

These singularities are to be avoided, because the structure cannot resist anyforce, and control is lost. To avoid any performance deterioration, it is necessary tolimit the range of the actuated joints. For the mechanism under study, parallelsingularities occur whenever the points C , D, and P are aligned (Figure 4), i.e. when

θ1 - θ2 = k π, for k = 1,2,..... These singularities are located inside the Cartesianworkspace and form the boundaries of the joint workspace. Moreover, structuralsingularities can occur when L1 = L2 (Figure 5). In these configurations, the controlof point P is lost.

1.5 Serial Singularities

Serial singularities occur when det(B) = 0. In the presence of theses singularities,there is a direction along which no Cartesian velocity can be produced. Serial




singularities define the boundary of the Cartesian workspace [Merlet 97]. For themechanism under study, serial singularities occur whenever θ1 - α1 = π / 2 + k π, or θ2 - α2 = π / 2 + k π, for k = 1, 2,..., i.e. whenever AC is orthogonal to CP or BD isorthogonal to D P .

1.6 Application to Machine-Tool Design

For a three-axis machine tool, as that in Figure 1, the table moves along two perpendicular axes. The joint limits of each actuator determine the dimension of theCartesian workspace. For the parallel mechanisms under study, this transformationis not direct. The resulting Cartesian workspace is more complex and its size smaller

for equal strokes of the actuators. We want to have a Cartesian workspace whichwill be close to the Cartesian workspace of an industrial serial machine tool. For our two-DOF mechanism, we prescribe a rectangular Cartesian workspace. In addition,the workspace must be reduced to a t-connected region, i.e. a region free of serialand parallel singularities. Finally, we want to prescribe relatively stable kinetostatic properties in the workspace.

2. ISOTROPIC DESIGN

2.1 Matrix Condition Number

During our process of design, we will define the loci of equal condition number of

the Jacobian matrices. To do this, we first recall the definition of the conditionnumber κ(M) of an m × n, matrix M with m ≤ n. This number can be defined invarious ways; for our purposes, we define κ(M) as the ratio of the largest, σ l, to thesmallest, σ s singular values of M [Golub 89]. The singular values {σ k }

m

1 of matrixM are defined as the square roots of the nonnegative eigenvalues of the positivesemi-definite m × m matrix M MT.

2.2 Conditioning of the Parallel Jacobian Matrix

To calculate the condition number of A, we need the product AAT, which wecalculate below:

AA

T

= L

2

1

1 cos(θ1 - θ2)

cos(θ1 - θ2) 1

The eigenvalues η1 and η2 of the above product are given by η1 = L

2

1 (1 + cos(θ1 - θ2)) and η2 = L2

1 (1 - cos(θ1 - θ2)). Upon simplification, thecondition number of matrix A is:

κ(A) = 1 / |tan((θ 2 - θ 1) / 2)|

In light of the above equation, it is apparent that κ(A) attains its minimum of 1




when |θ1 - θ2| = π / 2 + k π for k =1, 2, .... At the other end of the spectrum,κ(A) →∝ when |θ1 - θ2| = k π, for k =1, 2, .... The configurations for which κ(A) = 1are called isotropic (Figure 7), whereas configurations for which κ(A) →∝ are the parallel singularities of the manipulator (Figure 4). The study of the conditioning of A enables us to find a set of configurations for which A is isotropic. Theseconditions, however, do not give to the orientations α1 and α2 of the actuated joints.

2.3. Conditioning of the Serial Jacobian Matrix

By virtue of the diagonal form of B, its singular values, β1 and β2, are simply theabsolute values of it diagonal entries. The condition number κ(B) of B is thus

κ(B) =β max

β min

with B ≡ L1

cos(θ1 - α1) 0

0 cos(θ2 - α2)

where, if |cos(θ1 - α1)| < |cos(θ2 - α2)| then βmin = |cos(θ1 - α 1)| andβmax = |cos(θ2 - α2)|, else, βmin = |cos(θ2 - α2)| and βmax = |cos(θ1 - α1)|. It is apparentthat κ(B) attains its minimum of 1 when |cos(θ1 - α1)| = |cos(θ2 - α2)| ≠ 0. At theother end of the spectrum, κ(B) →∝ when |cos(θ1 - α1)| = 0 or |cos(θ2 - α2)| = 0.The configurations for which κ(B)=1 are the isotropic configurations and theconfigurations for which κ(B) →∝ are the serial singularities. As for the study of the conditioning of matrix B, the isotropy conditions of the matrix B do not give theorientations α1 and α2 of the actuators.

2.4. Conditioning of the Kinematic Jacobian Matrix

To define the orientation of the prismatic joints, we study the conditioning of theJacobian matrix K given in the equation (3b). In this case, matrices B-1 and K arewritten simply,

B-1=1

L1

(1/c1) 0

0 (1/c2), K =

(1/c1) (p - c)T

(1/c2) (p - d)T with ci = cos(θi - αi), i = 1, 2

The isotropy conditions of the matrix K are,

(1/c1) ||p - c|| =(1/c

2) ||p - d|| and (p - c)T (p - d) = 0

Now, we set α1 = 0 and α2 = π / 2 in order to have K = 12×2 in the isotropicconfiguration when θ1 = 0 and θ2 = π / 2. In this configuration, a motion of the prismatic joint ρ1 produces a motion along X (Figure 7), and a motion of the prismatic joint ρ2 produces a motion along Y as a machine tool with a PP architecture.




Figure 6: Motion of the point P generated bythe motion of the joint ρ1

Figure 7: Isotropic configuration

Moreover, if we place point A at the origin and B at ( M M ) (Figure 7), the valuesof ρ1 and ρ2 are equal when the manipulator reaches its isotropic configuration. Thisremark enables us to study the problem of velocity amplification due to the parallelmechanism. Indeed, in the case of a serial mechanism with three linear axes of motion, a motion of an actuated joint yields the same motion of the tool (or of theworkpiece). For parallel mechanisms, these motions are not equivalent. When themanipulator is close to a parallel singularity, there is an amplification of motion (or velocity), i.e. a unit motion of one actuated joint can produce a motion of the toolwhich is several times larger. Thus, to move the tool with a constant accuracy, the precision of the actuated joints must be very high.

2.5. Study of the Joint and Cartesian Workspaces

In this study, we will see that isotropy conditions give very interesting properties for the manipulator under study. To this end, we study two mechanisms, one withcoaxial actuators, of the biglide type (Figure 8), i.e. the mechanism most usuallyfound, and the other with actuator axes at right angles, as shown in Figure 9.

Figure 8: Biglide mechanism Figure 9: Isotropic mechanism

To define the joint limits, we will study the manipulability ellipsoids of matrix K [Yoshikawa 85]. By using equation (3), we can write a relation between the velocity.p of point P and the joint velocity

.ρ vector. With ||

. ρ|| ≤ 1, we have

.pT KK T

.p ≤ 1.

Equation (4) defines the range of variation of .p. The transformation of a unit circle

of the whole joint workspace by matrix KK T gives an ellipse in the Cartesianworkspace [Lallemand 94]. The square roots γ1 and γ2 of the eigenvalues of matrix




KK T are the values of the semi-axes of the ellipse which define the two velocity-amplification factors, λ = 1 / γ and λ = 1 / γ , according to these principal axes.1 1 2 2

To limit the variations of this factor in the Cartesian workspace, we set theconstraints

1/3 < λi < 3. (4)

This means that for a given joint velocity, the output velocity is at most threetimes larger or, at least, three times smaller than the velocity of P . This constraintalso permits us to limit the loss of rigidity (velocity amplification lowers rigidity)and of accuracy (velocity amplification also amplifies the encoder resolution). The

values in equation (4) were chosen as an example and should be defined preciselydepending on the type of machining tasks.

To be able to compare the two mechanisms studied here, the distance between points A and B is the same in two cases, as well as the lengths L1 and L2. Likewise,the scale factors of the joint and the Cartesian workspaces are also equal.

In our design, we do not want any singular configuration in the Cartesianworkspace. Also, the velocity-amplification factors are bounded in the Cartesainworkspace when using the constraints introduced in equation (4). Thus, a useful joint workspace of a square shape is defined in the joint workspace as well as its image inthe Cartesian workspace, which is the useful Cartesian workspace. In the case of theisotropic mechanism, the definition of a zone without singularities in the jointworkspace, in the presence of constraints (4), leads to smaller joint limits than in thecase of the biglide mechanism (Figures 10a and 11a). Moreover, the useful Cartesian

workspace of the isotropic mechanism is better for machining because the registeredsquare is seven times larger than that of the biglide mechanism (Figures 10b and11b).

(a)

(b)

(a)(b)

Figure 10: Joint and Cartesian workspaces

for the biglide mechanism

Figure 11: Joint and Cartesian workspaces for

the isotropic mechanism

For the isotropic mechanism, we show in Figure 12 the curves of iso-values of

the velocity-amplification factors. It is noted that velocity amplification λi varieswithin [0.4 , 1.6]. Moreover, if one limits a square zone in the useful Cartesianworkspace, one observes that the constraints are reached only in one small zone(noted I), placed on the boundary.




(a) (b)

Figure 12: Curves of iso-values of the factor of velocity amplification factor (a) λ1 and (b) λ2, in the useful Cartesian workspace of the isotropic mechanism

3. CONCLUSIONS

Introduced in this paper is the isotropic design of a parallel mechanism with twodegrees of freedom. The optimum design, based on the conditioning of the Jacobianmatrices, led simultaneously to, good kinematic performance and a simple, regular Cartesian workspace free of singular configurations. The velocity amplificationfactor is used to define joint ranges. Its values are bounded by reasonably values inthe workspace. Future work will aim at generalizing this mechanism to the spatial

case, to obtain a parallel mechanism with three degrees of freedom.

12. REFERENCES

[Chablat 98]: Chablat D., " Domaines d'unicité et parcourabilité pour les manipulateurs pleinement

parallèles ", Thèse de Doctorat, Nantes, Novembre, 1998.

[Lallemand 94]: Lallemand, J.P. et S. Zeghoul, Robotique: Aspects fondamentaux, Masson, Paris, 1994.

[Merlet 97]: Merlet J.P., Les robots parallèles, 2ème édition, Hermes, Paris, 1997.

[Yoshikawa 85]: Yoshikawa T., " Manipulability and redundant control of mechanisms", Proc. IEEE, Int.

Conf. Rob. And Aut., pp. 1004-1009, 1985.

[Daniali 95]: Daniali M., " Contributions of the Kinematics Synthesis of Parallel Manipulators ", Thèse

de doctorat, McGill, 1995.

[Gosselin 88]: Gosselin C., " Kinematic Analysis, Optimisation and Programming of Parallel robotic

Manipulators ", Thèse de doctorat, McGill, 1988.

[Golub 89]: Golub G.H. et Van Loan C.F., Matrix Computations, The John Hopkins University Press,

Baltimore, 1989.

13. AFFILIATIONS

D. Chablat, Ph. Wenger, Institut de Recherche en Communication et Cybernétiquede Mantes, 1 rue de la Noë, 44321 Nantes, France.J. Angeles, McGill Centre for Intelligent Machines, McGill University, 817Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6.

Isotropic Design of a Parallel Machine-Tool Mechanism

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