analysis of the direct kinematic problem

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Analysis of the Direct Kinematic Problemin 3-DOF Parallel Manipulators

M. Urizar, V. Petuya, O. Altuzarra, E. Macho, and A. Hernandez

Abstract In this paper, the authors will show a methodology for computing the

configuration space with one constant input, basing on the principles of the dis-

cretization methods. Taking chance of an entity called the reduced configuration

space, the Direct Kinematic Problem will be solved. Moreover, this entity allows

the transition between different solutions to be performed, with the purpose of

enlarging the range of motion of the manipulator.

Keywords Dkp solutions Parallel manipulator Path planning Reduced

configuration space

1 Introduction

Parallel manipulators are an interesting alternative to serial manipulators given the

important mechanical and kinematics advantages offered, i.e., better stiffness, high

accuracy, higher velocity and acceleration of the end-effector, etc. Nevertheless, in

counterpart they present limited and complex workspaces with internal singulari-

ties. Thus, the workspace size and shape, as well as the singularity loci are

considered the main design criteria of these robots. The representation of the

workspace normally stands for the reachable volume of the end-effector.

Concerning the workspace obtaining methods, there exist several viewpoints;

discretization methods, geometric methods and analytical methods can be distin-

guished. Discretization methods [1, 7], consist in establishing a mesh of nodes with

end-effectors positions and orientations. Each node is checked to find whether it

M. Urizar (*)

Faculty of Engineering in Bilbao, Department of Mechanical Engineering, University of the

Basque Country, Basque Country, Spain

I. Visa (ed.), SYROM 2009,

DOI 10.1007/978-90-481-3522-6_36,# Springer ScienceBusiness Media, B.V. 2010445


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Fig. 1 Positional

discretization algorithm

Analysis of the Direct Kinematic Problem in 3-DOF Parallel Manipulators 447


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Subsequently, the verification of which discrete pose constitutes a solution is

carried out analytically, by solving the IKP of the remaining kinematic chains

separately. One pose will belong to the workspace when the three kinematic chains

can be assembled. The discrete poses belonging to the workspace are joined in an

interpolation process, hence, obtaining a continuous surface.The kinematic analysis is carried out by evaluating the DKP and IKP Jacobian

matrices determinants for each pose. One position equation for each limb is

set up, called the characteristic equation (fi for i = 1, 2, 3). Mathematical

assembly is performed expressing in the characteristic equations, fi, each limb

joining-node coordinates as a function of the output variables. Then, the system

governing the velocity problem is obtained and can be expressed in matrix form

as follows:

@f1@x1@f1@x2

@f1@x3@f2@x1

@f2@x2

@f2@x3

@f3@x1

@f3@x2

@f3@x3

264

375

_x1_x2_x3

8>:(2)

The dimensional values are: a = b = R2 = 2.5 and R3 = 2a = 5. Fixing input

angle a3 constrains the moving platforms motion to a constant value of the output

variable y. Thus, both the reduced configuration spaces representing output

variable or x can be depicted, obtaining different surfaces for each case, and

having each one of them the same information related to the motion capacity

analysis.

Selecting constant input a3 = 170 and representing, i.e., output variable yields

the reduced configuration space shown in Fig. 3. Note that it is not a closed surface

due to the angular character of input a2 and output .

Fig. 2 (a) Parallel

manipulator RPR-2PRR; (b)

Software representation



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In Fig. 3, the different regions associated with the positive and negative sign of

the |JDKP| are shown, separated between them by the DKP singularity curves

depicted in red colour. Besides, the two regions associated with the two possible

configurations of limb two are separated by the IKP singularity curves, plotted in

green.

The two possible configurations of limb two establish the two working modes of

this limb. The manipulator is in its WM1 when 90 a2 90, WM2 beingassociated with the range 90 a2 270.

3.1 Finding All DKP Solutions

The DKP consists in obtaining the output variables for specific input values. For a

given input, the reduced configuration space can be computed, and the two remain-

ing inputs represent a vertical line in the reduced configuration space that intersects

the surface at several points. Each of these intersection points will belong to

triangles made up by the three closer exact solutions. Hence, it is feasible to

make an interpolation among those point values to obtain the approximate solution

(x,y,f) of the intersection point, as it is shown in Fig. 4. The solutions of all the

intersection points will bring out all the DKP solutions.

To clarify the procedure, the example under study will be used. Being the input

values: L1 = 4, a2 = 21 and a3 = 170, it yields the four DKP solutions represented

in Fig. 5. At the left side of Fig. 5, both the reduced configuration space at the

top and its projection onto the joint space (L1,a2), underneath, are depicted. The

Fig. 3 Reduced Configuration

Space fora3 = 170

450 M. Urizar et al.


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selected inputs correspond to one point in the joint space inside the region with four

DKP solutions, meanwhile this point represents a vertical line in the 3D plot.

The four solutions represented on the right hand side of Fig. 5 have a constant

value of output variable y, due to the constraint of fixing input a3. Hence, the

moving frames origin lies in a horizontal line plotted in red colour.

By varying the point in the joint space, the DKP solutions for any input values

can be obtained.

Fig. 5 Four DKP solutions for inputs: L1 = 4, a2 = 21 and a3 = 170

Fig. 4 Interpolation process for obtaining the DKP solutions



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Cusp points correspond to points in the DKP singularity curves where three DKP

solutions coalesce. It has been researched in previous papers [5], that encircling

cusp points permits making non-singular solution changes between different solu-

tions or assembly modes.

Let us demonstrate these affirmations in the example under study. Focusing onthe DKP solutions location for constant a3 = 170 (Fig. 5), it can be observed

that there are four regions associated to |JDKP| < 0 (plotted in blue, two regions) or

|JDKP| > 0 (plotted in green, two regions) which are separated between them by

the DKP singularity curves. Besides, as explained previously, the IKP singularity

curves divide the reduced configuration space into two regions associated to the two

working modes of the manipulator.

In this case (fora3 = 170), the DKP solutions lie one in each singularity-free

region and the impossibility exists for finding a path or trajectory to move from one

solution to another without crossing any singularity.Nevertheless, focusing on Fig. 6, which corresponds to a3 = 160, the reduced

configuration space has acquired a shape such that there are only two singularity-

free regions, separated between them by the DKP singularity curves. These two

regions are called sheets of the reduced configuration space, and define the

maximum region free of any singularity.

Yet again, similarly to the previous case for a 3 = 170, the IKP singularity

curves divide the reduced configuration space into two regions according to each

working mode. Thus, for each working mode, there are two sheets free of internal

singularities.Due to the existence of only two singularity-free sheets, in each of them

lie two different DKP solutions. Solutions (13) are located on the region with

|JDKP| < 0 (sheet1), meanwhile solutions (24) lie on the region with |JDKP| > 0

(sheet2). Hence, it is feasible to search for paths inside the singularity-free sheets

so as to move from one solution to another.

In fact, the existence of the aforementioned cusp points is the key factor for

planning transitions between solutions. In Fig. 7, the paths joining the two pairs of

solutions in each sheet, by encircling cusp points c1 and c2, are shown. Each one of

these paths remains completely inside each singularity-free sheet associated withWM1. Thus, the transitions between solutions can be performed in a completely

safety way.

Working now with the first of the sheets, i.e., the one associated to |JDKP| < 0

(sheet1), the singularity-free path joining the two DKP solutions 1 and 3 is shown in

Fig. 8, which does not cross at any time neither the DKP nor the IKP singularity

curves.

This path encircles cusp point c1, standing for a circular trajectory in the joint

space. Poses correspondent to solutions 1 and 3, and two intermediate poses pi,

along the path, have been represented in Fig. 8.

Therefore, visualizing the reduced configuration space for different input values,

allows searching for the optimal input for non-singular transitions to be possible,

thus, getting a wider operational workspace.



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It must be emphasized that the reduced configuration space has all the informa-tion of the configuration space. However, as it has been explained, one of the inputs

must remain constant and depending on the selected value of the input it is feasible

or not to perform non-singular transitions.

5 Conclusions

In this paper, a useful computational tool for obtaining the reduced configuration

space with constant input has been presented. It allows the Direct Kinematic

Problem to be solved, and the locations of the solutions across the different surfaces

visualized. In addition, the reduced configuration space permits searching for

Fig. 7 Singularity-free paths encircling cusp points



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trajectories joining different solutions so as to enlarge the manipulators range

of motion.

Acknowledgments The authors of this paper wish to acknowledge the financial support received

from the Spanish Government via the Ministerio de Educacion y Ciencia (Project DPI2008-00159)

and the University of the Basque Country (Project GIC07/78).

References

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Mech. Mach. Theory 28(1), 3142 (1993)

Fig. 8 Path planning between solutions 1 and 3



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