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12thIFToMM World Congress, Besanon (France), June18-21, 2007

Synthesis of a 6-RUS Parallel Manipulator Using Its Stationary Configurations

I. Zabalza* J. RosPublic University of Navarre Public University of Navarre

Pamplona, Spain Pamplona, Spain

AbstractIn this paper a synthesis method for the 6-RUS1parallel manipulator is studied. With this method, several

moving platform positions are prefixed to be reached in totalstationary configuration of the manipulator. A numericalexample is presented to obtain the manipulator dimensions so

that it can reach two prefixed positions of the moving platform intotal stationary configuration.

The 6-RUS parallel manipulator, with a suitablegeometry, is able to reach a maximum of 64 totalstationary configurations (TSC) [9], [10].

At these configurations:

- The moving platform has a great accuracy in position,because small perturbations in the position of the inputelements have nearly no effect on the platform position.

- The velocity of the moving platform vanishes for anyvalue of the velocities of the input elements.

Keywords: synthesis, robotics, parallel manipulator,

singularities

- The torques exerted by the input elements are almost

zero because the loads applied to the moving platformare supported by the reaction forces acting on thekinematics joints.

I. Introduction

In order to design a mechanism, first a mechanical

architecture has to be chosen. This may be obtained eitherby synthesis starting from the constraints on the task, or

by using an a priori solution. Secondly, the chosenmechanical architecture must be modelled, and the modelthen used to make a geometric synthesis, i.e. to determinethe physical and geometrical characteristics of the

mechanism that are the best for the task.

Taking into account the above-mentioned advantages,the subject of this work is the design of a 6-RUS parallelmanipulator so that some of its moving platform positions,

being the manipulator in TSC, are the prefixed positions.These prefixed moving platform positions can be useful

in different industrial applications, for example inmachine tools.

The same process is carried out in the design of parallelmanipulators. II. Manipulator topology and notation

A lot of mechanical architectures of parallelmanipulators exist. Some of these are shown in [5].Several examples and references can also be found at the

following URLs:

The 6-RUS (6-RKS) parallel manipulator wasintroduced by Hunt in 1983 [4] and is composed of twotriangular platforms, one of them fixed to the ground.

On the fixed platform there are six rotating actuators (R)located on the edges of the triangle. These actuators arethe input elements on which the motors are located.

- [http://www-sop.inria.fr/coprin/equipe/merlet/merlet.html].

- [http://www.parallemic.org/].

Between these a priori solutions, the 6-RUS parallel

manipulator is chosen in this work, because it is able toreach stationary configurations.

Each actuator is linked to the moving platform through arod. In each rod, one tip is linked to an actuator through auniversal joint (U) and the other tip is connected to themoving platform by means of a spherical joint (S), asdepicted in Fig. 1.

In order to obtain the best solution for the task, a designcriterion must be defined.

For a given parallel manipulator many design criterionshave been used by different authors. For example:

- Geometrical criterions: workspace [3], isotropy [6].- Static criterions: stiffness [2], [8], balancing [7]

The design criterion in this work is to let the 6-RUSparallel manipulator to be able to reach some prefixedmoving platform positions with a great accuracy using itsstationary configurations.

According to the knowledge of the authors, this criterionhas not been used before.

*E-mail: [email protected]. 1. The 6-RUS parallel manipulator introduced by Hunt

E-mail: [email protected] Also 6-RKS


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The notation used to describe the topology of thisparallel manipulator is summarized in the following pointsand shown in Fig. 2.

Fig. 2. Notation used to describe the parallel manipulator

- 0i: center of the rotation axis of the i-th actuator.- 1i: center of the universal joint between the i-th

actuator and the i-th rod. The segment defined bypoints 0i and 1i is perpendicular to the actuatoraxis.

- Ri: length of the crank between centers 0i and 1i.

- Li: length of the i-th rod.- 1ik: center of the spherical joint between rods i-th and

k-th. These centers coincide with the vertices of themoving platform.- Apq: length of the edge of the moving platform that

links the p-th and q-th rods.

III. Total stationary configurations (TSC)

A. Stationary configurations condition

The actuators crank and the rod compose eachkinematics chain, or leg, between moving and fixedplatforms.

One leg of the manipulator is in stationaryconfiguration (SC) when the actuators axis, theactuators crank and the rod are in the same plane,

Zabalza and others [9], [10].Being the moving platform over the fixed one as in

Fig. 1, each leg can reach two SC: One when the actuatorcrank and the rod are nearly aligned and other when theactuator crank and the rod are nearly superimposed.

The manipulator is in TSC when the six legs are in SC.Consequently, the manipulator can reach 26 = 64 TSCs.

B. Constraint equations for the manipulator in TSC

To model the 6-RUS parallel manipulator, a set ofnatural coordinates introduced by Garca de Jaln andBayo [1], is used. Natural coordinates is a set of

coordinates that define the position of all the elements ofthe mechanical system with respect to an inertial referenceframe using the Cartesian coordinates of some points and

the Cartesian components of some unit vectors, usually

located at the kinematics joints.An inertial reference frame is considered on the fixed

platform, as shown in Fig. 2, being Z-axis normal to theground and facing upwards.

For the synthesis methodology to be developed, the

prefixed positions of the moving platform (points 123,145 and 161) are known, so the following constraintequations can be written:- Constant length condition for the actuators cranks, a set

of six quadratic equations like (only the first one iswritten):

0R)zz()yy()xx( 212

0111

2

0111

2

0111 =++ (1-6)

- The points 11 to 16 are located on the tips of theactuators crank and consequently their trajectories mustbe circumferences in planes perpendicular to the rotationaxis of the corresponding actuator. These conditions arealso defined by means of a set of six quadraticconstraints like (only the first one is written):

0)zz)(zz(

)yy)(yy()xx)(xx(

01020111

0102011101020111

=+

++

(7-12)

- To define the constant length condition for the rods, aset of six quadratic equations like (only the first one iswritten):

0L)zz()yy()xx( 212

11161

2

11161

2

11161 =++

(13-18)

- Stationary configuration is expressed using cubicconstraint in the natural coordinates. This conditionfulfils if the actuators axis, the actuators crank and therod are in the same plane, and can be imposed byequating to zero the volume of the pyramid defined by

two natural points that define the actuators axis and thetwo natural points that define the rod. To be themanipulator in TSC this condition gives six equationslike (only the first one is written):

0)xx)(yy)(zz(

)yy)(xx)(zz(

)zz)(xx)(yy(

)xx)(zz)(yy(

)yy)(zz)(xx(

)zz)(yy)(xx(

1116101110102

1116101110102

1116101110102

1116101110102

1116101110102

1116101110102

=

+

+

(19-24)


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The set of equations (1) to (24) is a system of non-linearalgebraic equations that must be fulfilled for all TSCs.

C. Two prefixed positions of the moving platform

For two prefixed positions of the moving platform in

TSCs, Fig. 4, 48 constraint equations must be fulfilled,

and only there are 36 unknowns, for this reason must beintroduced 12 variables, for example, the lengths of the

cranks and the rods.

IV. Prefixed positions of the moving platform

A. Possible prefixed positions

As the 6-RUS parallel manipulator can reach 64 TSCs,the manipulator geometry can be theoretically set so that

64 prefixed positions of the moving platform can bereached in TCS. However, this objective is very difficultto reach because in each TSC a set of 24 constraintequations must be fulfilled and the unknowns are only6x3 = 18 natural coordinates of the actuators cranks tips.

This means for the 64 prefixed positions must be

fulfilled 64 x 24 = 1536 constraint equations and onlythere are 64 x 18 = 1152 actuators cranks tips coordinates

as unknowns. Therefore, to make the system compatible,

1536 - 1152 = 384 new unknowns must be introduced.Since the manipulator has six legs, 64 new variables(actuators crank and rod lengths, actuators

position,) must be associated with each leg.Definitively, too much variables.

Fig. 4. Manipulator 6-RUS in two TSCs

Having in mind that in a fabrication process, normallyfew positions of the piece or tool are needed, and thedesign difficulty of introducing new variables, themanipulator synthesis will be made for a few prefixed

positions of the moving platform in TSC of themanipulator.

In theory, these two prefixed positions of the moving

platform in TSC can be obtained in 64 x 63 = 4032

different manners, number of possibilities of taking 2

TSCs between the 64 TSCs that the manipulator can

reach.

B. One prefixed position of the moving platform D. N prefixed positions of the moving platform

For one prefixed position of the moving platform inTSC, Fig. 3, 24 constraint equations must be fulfilled, andonly there are 18 coordinates of the crank tips asunknowns. Therefore, six variables must be introduced. Inthis case, for example, the lengths of the cranks can be

taken as variables.

For N prefixed positions of the moving platform in TSC,

24 x N constraint equations must be fulfilled, and only

there are 18 x N unknowns. Therefore, 6 x N variables

must be introduced.

In this case, the positions of the actuators can be

introduced as variables, in addition to the length of the

cranks and the rods.

In theory, these N prefixed positions of the moving

platform in TSC can be obtained in 64 x 63 xx

(64N) different manners.

V. Manipulator synthesis

Given the N prefixed positions of the moving platform,

(coordinates of points 123, 145 and 161 for the N

positions), a system of 24 x N constraint equations can be

written. In this system, in addition to the

18 x N coordinates of the crank tips, 6 x N new variables

must be introduced, for example, lengths of the crank and

the rods and actuators positions coordinates.

Fig. 3. Manipulator 6-RUS in one TSC

Since the manipulator can reach 64 TCSs, one prefixed

position of the moving platform in TSC can be obtained in

64 different ways, one for each TSC of the manipulator.

The synthesis of the manipulator is achieved by solving

the system presented in the above paragraph for the

18 x N + 6 x N unknowns.


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If the set of unknowns is grouped in a vector q, the

system of non-linear equations can be rewritten in the

form:

0(q) = (25)

The system of equations (25) can be solved using the

Newton-Raphson iterative method. The iterative process

can be written as:

)(q)q)(q(q ii1iiq =+ (26)

If the Newton-Raphson method converges, one solution

is obtained, but theoretically 64 x 63 x x (64-N)

different solutions can be found.

The solution obtained with the Newton-Raphson method

depends of the initial values of the vector q. For this

reason the iterative Newton-Raphson method is applied 64

x 63 x x (64-N) times, taking as initial values of the

vector q the lengths of the crank and the rods, the

actuators positions coordinates and the 64 x 63 x x

(64-N) combinations of the crank tips coordinates being a

manipulator of a known dimensions in all the 64 TSCs.

This set of initial configurations for the solution of the

non-linear problem using the Newton-Raphson iteration,

has been proved useful to obtain all or a set of the

different solutions in a number of design problems.The non-linear nature of the problem cannot guarantee

that any possible design solution will be obtained. For

example two different initial conditions can converge to

the same solution, or convergence for a given initial

condition cannot be reached.

Nevertheless the number of solutions attained using the

proposed set of initial values uses to be big enough so that

a suitable solution can be chosen between them.

Geometrical analyses are known to have problems with

convergence when singularities are present. This is not the

case for the synthesis problem here proposed. In this

synthesis problem, the N desired platform positions are

the data, so the 4xN equations that determine the position

and geometry of each leg for the N desired positions are

mutually independent (any variable appearing in the

equations related to one leg will not appear in any other

leg). This implies that even solutions that correspond to

singular positions of the manipulator will not be singular

for the mechanism represented by a single leg and then

convergence can be reached without singularity related

problems.

So the only possible reason for non-convergence of the

Newton-Raphson iterative process for a given set of initial

values is mathematical in nature, meaning that the initial

values are not within the attractor of any solution. Simply

stated, it is far away from any possible solution.

VI. Numerical example

It is desired to mechanize two faces of the icosahedra in

one sphere of diameter 100 millimeters located over the

moving platform barycenter, Fig. 5, using two TSCs of the

manipulator.

Fig. 5. Two positions of moving platform in TSCs

The faces to mechanize must be horizontal; with its

barycenter coinciding with the Z-axis and the edge

between faces parallel to the Y-axis. In the first position

the moving platform must be horizontal.

Fig. 6. Dimensions of the manipulator

Taking a manipulator with the platforms dimensions and

actuators positions shown in Fig. 6, and initially with the

cranks lengths 0.1 and the rods lengths 0.6 meters, and

knowing that the angle between the faces of the

icosahedra, 41.81, the coordinates of the moving platform


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vertex (points 123, 145 and 161) for the two prefixed

positions will be:

[x123A

, y123A

, z123A

, x145A

, y145A

, z145A

, x161A

, y161A

, z161A

]=

=[-0.1441, 0.25, 0.6498, 0.2887, 0, 0.6498, -0.1441, -0.25, 0.6498]

[x123B, y123B, z123B, x145B, y145B, z145B, x161B, y161B, z161B]=

=[-0.1409, 0.25, 0.7587, 01819, 0, 0.4701, -0.1409, -0.25, 0.7587]

To realize the manipulator synthesis, the method

presented in the section V is applied: A system of 48 non-

linear equations is obtained. As unknowns the 36

coordinates of the cranks tips in the two positions, and

the 12 lengths of the cranks and rods are taken. And the

system is solved using the iterative Newton-Raphson

method.To obtain the optimum solution, the iterative process is

repeated 64 x 63 = 4032 times, taking as initial unknowns

the combinations of cranks tips coordinates in the 64

TSCs of the parallel manipulator with the initial

dimensions shown in Fig. 6.

Repeating the iterative process, in all of them the

process converge, and 64 different solutions are obtained.

These 64 solutions obtained correspond to two solutions

for each leg: One with short crank and long rod and

other with long crank and short rod.

In each leg, the solution long crank and short rod isunacceptable because it can reach uncertainty singular

configurations.

Taking into account the above comments, only one of

the 64 solutions is acceptable, the solution with the six

short cranks and six long rods.

This solution can be defined as the solution that

minimizes the sum of crank lengths, and in this case, is

the following:

Cranks lengths, in meters:

[R1, R2, R3, R4, R5, R6] = [0.0537, 0.0537, 0.0528, 0.0899,

0.0899, 0.0528]

Rods lengths, in meters:

[L1, L2, L3, L4, L5, L6] = [0.7466, 0.7466, 0.7457, 0.6095,

0.6095, 0.7457]

This solution is acceptable because applying the process

shown by Zabalza and others [9-10], the parallel

manipulator with these dimensions is able to reach the 64

TSCs without reach any uncertainty singular

configuration.

In this solution: For the first position the cranks 1, 2, 3

and 6 are under and the cranks 4 and 5 over the fixed

platform. For the second position the cranks 1, 2, 3 and 6

are over and the cranks 4 and 5 under the fixed platform.

VII. Future work

More cases with two or more prefixed positions of the

moving platform have to be studied in order to draw more

general conclusions.

In these new cases, the variable values of the vector q

for the iterative Newton-Raphson process can be a

combination of crank and rod lengths and actuators

position and orientation coordinates.

A new algorithm can be proposed to solve the equations

of this synthesis problem that exploits the fact that the

equations for each leg of the manipulator are mutually

independent.

Also other aspects like workspace, isotropy, stiffness,

uncertainty singular configurations,.... of the designed

parallel manipulators can be analysed.

VIII. Conclusions

In this paper a synthesis method that allows to the 6-

RUS parallel manipulator to reach several positions of the

moving platform in total stationary configuration (TSC) is

presented. A numerical example for a manipulator that

reaches two prefixed positions of the moving platform in

TSC is presented.

References

[1] Garca de Jaln, J., Bayo, E.Kinematic and dynamic simulation of

multibody system. Springer-Verlag, 1994.

[2] Gosselin C. Stiffness mapping for parallel manipulators. IEEETrans. on Robotics and Automation,6(3):377-382, June 1990.

[3] Gosselin C. Determination of the workspace of a 6-dof parallel

manipulator. ASME J. of Mechanical Design, 112(3):331-336,

September 1990.

[4] Hunt, K.H. Structural kinematics of in-parallel-actuated robot-arms,J. of Mechanisms, Transmissions, and Automation in Design,

105(4):705-712, December 1983.[5] Merlet J.P.Parallel robots. Kluwer Academic Publishers, 2000.

[6] Pashkevich A., Wenger P., Chablat D. Design strategies for the

geometric synthesis of Orthoglide-type mechanisms, Mechanism

and Machine Theory, 40(8):907-930, August 2005

[7] Russo A., Sinatra R., Xi F. Static balancig of parallel robots,Mechanism and Machine Theory, 40(2):191-202, February 2005

[8] Takeda Y.. Funabashi H. Kinematic synthesis of in-parallel

actuated mechanisms based on the global isotropy index, J. ofRobotics and Mechatronics, 11(5):404-410, October 1999.

[9] Zabalza I. and others. Evaluation of the 64 Insensitivity Positions

for a 6-RKS Hunt-Type Parallel Manipulator. In Tenth World

Congress on the Theory of Machine and Mechanisms, pages 1152-1157, Oulu, June, 20-24, 1999.

[10] Zabalza I. and others. Total and Partial Stationary Configurations

for a 6-RUS Hunt-Type Parallel Manipulator. In Eleventh worldCongress in Mechanism and Machine Science, pages 1987-1991,

Tianjin, April 1-4, 2004.

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