Proceedings of RAAD 03, 12Th International Workshop on Robotics in Alpe-Adria-Danube Region

Proceedings of RAAD’03, 12th International Workshop on Robotics in Alpe-Adria-Danube Region

Cassino May 7-10, 2003

AN EXPERIMENTAL CHARACTERIZATION OF SINGULARITIES

IN A 6-WIRE PARALLEL ARCHITECTURE

Erika Ottaviano1, Marco Ceccarelli1, Federico Thomas2

1 Laboratory of Robotics and Mechatronics 2 Institut de Robotica i Informatica Industrial

DiMSAT - University of Cassino CSIC - UPC

Via Di Biasio 43, 03043 Cassino (Fr), Italy Llorens Artigas 4-6, 2 planta, 08028 Barcelona, Spain

email:ottaviano/ email:

ABSTRACT- This paper deals with the experimental characterization of singularities of a 6-wire parallel architecture. In particular, a formulation for the singularity characterization has been formulated as based to the geometry of polyhedra. Singular configuration have been classified as a function of polyhedron volume. Then, experimental characterization has been carried out by considering a wire parallel architecture CATRASYS, which is a measuring system designed and built at LARM, Laboratory of Robotics and Mechatronics in Cassino.

Keywords: Parallel Architecture, Wire Measuring Systems, Singularities, Experimental Determination.

INTRODUCTION

Several measuring systems have been developed to determine the position and the orientation of a moving object, such as cameras, theodolites, laser tracking systems and wire systems.

Most of those measuring system have a trilateration or triangulation based measure.

In this paper a wire-based parallel architecture is considered to determine the position and orientation of moving objects. The architecture can be modeled as a Gough-Stewart platform [Merlet, ]. Indeed, the problem of determining position and orientation can be seen as to solve the Direct Kinematics of the architecture.

The parallel architecture measuring system can be modeled as a special type of Gough-Stewart platform.

A 6-WIRE PARALLEL ARCHITECTURE

CATRASYS (Cassino Tracking System) has been conceived and designed to perform the determination of a position and orientation of a rigid body by using trilateration formulation for a 6-wire parallel mechanism, [Ceccarelli et al. 1999; Ceccarelli et al. 2000; Ottaviano et al. 2002a]. CATRASYS system has been designed as composed of a mechanical part, an electronic/informatic interface unit and a software package.

The core of the mechanical part can be identified in an end-effector for CATRASYS and the 6-wire parallel architecture mechanism, that is given by the 6 wires that connect the end-effector for CATRASYS to the wire transducers installed on so-called Trilateral Sensing Platform. The end-effector for CATRASYS is a coupling device: it connects the wires of the six transducers to the extremity of the movable system. It allows the wires to track the system while it moves.

Signals from wire transducers are fed though an amplified connector to the electronic interface unit, which consists of a Personal Computer for data analysis.

In particular, b1 is the reference point of the moving platform. Points b2 and b3 have been used to determine the orientation of the moving platform. The base reference frame is O-XYZ.

Kinematics of the measuring system CATRASYS has been solved in closed-form due to the special arrangement of the system by using the trilateration technique [Ceccarelli et al.1999].

The prototype of the measuring system is shown in Fig. 2. It has been used for the determination of the workspace and Kinematic parameters of complex mechanical systems, as reported in [Ottaviano et al. 2002; Ottaviano et al. 2002].

Fig 1. A scheme of 6-wire parallel architecture measuring system.

Fig 2. A prototype layout of CATRASYS applied to a PUMA robot.

Fig 3. The end-effector for CATRASYS.

THE PROBLEM FOR SINGULARITIES DETERMINATION

A singularity of the parallel architecture measuring system can be defined as the configuration in which the measure is not univocally determined.

The analysis of singularities is an important issue since they set the limits of the kinematic motion and static load equilibrium. Singularities of parallel architectures have been extensively studied by several authors. Merlet has used Plucker line coordinates, Grassmann line geometry and Screw Theory in [Merlet 1989] in order to represent a parallel robot geometrically.

Line geometry, wrench singularity analyses for parallel architectures have been presented in [Hao and McCarthy 1998; Nothash 1998] and others. Most of the work is based on the analysis performed by Dandurand, [Dandurand 1994].

Parenti castelli!!!!!!

Based on their nature, singularities of the measuring system can be classified into three categories: configuration, architecture, and formulation singularities. The above-mentioned classification has been developed by [Ma and Angeles, 1992].

The first type of singularity is an inherent property of the device that occurs at some points within its workspace. Architecture singularities are due to the architecture and can prevail over the entire workspace. Formulation singularities are caused due to the adopted analysis and can be avoided easily by changing the formulation method.

For the proposed singularity analysis it has been introduced characteristic tetrahedra. By observing the singularity of those three tetrahedra one can deduce the singularity of the parallel architecture measuring system.

A characteristic tetrahedron is introduced to identify the singularities of the parallel architecture measuring system.

For the following analysis the base of the tetrahedron is formed by three points, whose position is known or determined, the position of the apex is unknown. Indeed, by considering Fig.1 tetraheron T1 is defined by its base formed by a1, a2 and a3, three edges d1, d2 and d3, and its apex b1. The tetrahedron T2 is identified by its base

The characteristic tetrahedra dor the parallel architecture measuring system are shown in Fig. 4.

A tetrahedron is non singular if and only if its four faces do not have a common point.

By considering this definition singular tetrahedra have a great variety of forms. We do not consider tetrahedra with infinite elements, since

In particular, the position of the apex is determined if and only if the tetrahedron is non-singular. Indeed, one of the basic requirements to obtain a measure is that the parallel architecture

Fig. 4. Non singular characteristic tetrahedra T1, T2 and T3 for the parallel architecture measuring system.

A PROCEDURE FOR AN EXPERIMENTAL CHARACTERIZATION OF SINGULARITIES

Applying the above mentioned considerations to the measuring system, an architecture singularity arises for particular arrangements of the geometry of the parallel architecture measuring system. Indeed, a study for the positioning of the wire transducers on the trilateral sensing platform must be carried out in order to avoid singularities of the tetrahedra.

A singularity analysis have been performed by considering the geometry of tetrahedra.

The determinant expressed by Eq. (??) is equal to zero if p1, p2, p3 and p4 lie on a plane, this is the case of the edge is zero; or if p1,p2, p3 are aligned, this is the case of base area equal to zero.

Indeed, all singular configurations for the measuring system can be determined by observing the three polyhedra P1, P2 and P3 defined by points (a1, a2, a3, b1); (a4, a5, b1, b2) and (a6, b1, b2, b3).

In particular, singular conditions can be expressed in Eq. (??)

1) singularity of polyhedron P1:

1a) a1, a2, a3 aligned: architecture singularity

1b) a1, a2, a3, b1 lie on a plane: configuration singularity.

2) singularity of polyhedron P2

2a) a4, a5, b2 aligned: configuration singularity

2b) a4, a5, b1, b2 lie on a plane: configuration singularity

3) singularity of polyhedron P3:

3a) a6, b1, b2 aligned: configuration singularity

3b) a6, b1, b2, b3 lie on a plane: configuration singularity

4) Combined singularity:

Note that special case can arise if P2 and P3 are contemporaneously singular. This condition can be named as combined singularity 23.

Regarding to condition 1a) it can be avoided by considering an arrangements of the wire transducers on the trilateral sensing platform to avoid the alignment of points a1, a2, a3.

Conditions 1b) and 2a) can be avoided by choosing the working area of the measuring system outside the trilateral sensing platform. This means that none of points b1 and b2 can be on the plane of the trilateral sensing platform.

Conditions 2b); 3a); 3b) and combined singularity 23 occur in the working area of CATRASYS and they should be avoided by properly choosing the working area of the measuring system.

Fig 1. Singular configuration of the polyhedron P1:

a) zero area;

a) b)

Fig 1. Singular configuration of the polyhedron P1:

a) zero area; b) zero area.

a) b)

Fig 2. Singular configuration of the polyhedron P3:

a) zero high; b) zero high.

a) b)

Fig 3. Singular configuration of the polyhedron:

a) combined singularity of P1 and P3;

b) singularity of P2.

a) b)

Fig 4. Singular configuration of the polyhedron:

a) combined singularity of P2 and P3; b) singularity of P1.

singular configuration of T3.

singular configuration of T2 and T3.

THE EXPERIMENTAL DETERMINATION FOR A 6-WIRE PARALLEL ARCHITECTURE PROTOTYPE

Experimental activity has been carried out with CATRASYS at the LARM, Laboratory of Robotics and Mechatronics in Cassino by using a PUMA 562 robot, [Unimation 1990]. The lay-out with PUMA robot is shown in Fig. and a built prototype for CATRASYS end-effector is shown in Fig. .

In particular, by considering the above-mentioned formulation and singularity classification have been useful to determine singular configurations of CATRASYS system.

A singular configuration for CATRASYS, which has been sketched in Fig is shown in Fig. . Experimental results for this measuring configuration are represented in Fig., in which coordinates of points H, F and Q of Fig. are measured for a trajectory including singular configuration for CATRASYS. It is shown that a motion passing through the singular configuration gives no reliable measures for point Q, and a discontinuity can be detected for x and y coordinates of Q, too.

Some of the above-mentioned configurations expressed by conditions 2b) 3a); 3b) and 23 have been experimentally detected by using CATRASYS and results are shown in Figs.???.

In particular, Fig. 8 shows the experimental layout for the condition sketched in Fig. 3a). For this case, one obtain the singular configuration for the polyhedra P3, since b3 belongs to the plane of b1, b2 and b6.

Figure 9 shows the experimental layout for the singularity of P3.

Figure 10 shows the experimental layout for the combined singularity of P2 and P3. In such case, b3 belongs to the plane of b1, b2 and O6 (zero high) and b2 belongs to the plane of O4, O5 and b1 (zero high).

Fig 8. Singular configuration of the polyhedron P3.

Fig 9. Singular configuration of polyhedron P3.

Fig 10. Singular configuration of P2 and P3.

CONCLUSION

REFERENCES

Dandurand A., “The rigidity of Compound Spatial Grid”, Structural Topology, Vol.10, pp.41-44, (1984).

Ottaviano E. Ceccarelli M., Sbardella F., Thomas F., “Experimental Determination of Kinematic Parameters and Workspace of Human Arms”, 11th International Workshop on Robotics in Alpe-Adria-Danube Region RAAD 2002, Balatonfured, 2002, pp.271-276.

Thomas F., Ottaviano E., Ros L., Ceccarelli M., “Uncertainty Model and Singularities of 3-2-1 Wire-Based Tracking Systems”, Advances in Robot Kinematics, Ed. Kluwer Academic Publishers, 2002, pp.107-116.

Ottaviano E., Ceccarelli M., Toti M., Avila Carrasco C., “CaTraSys (Cassino Traking System): A Wire System for Experimental Evaluation of Robot Workspace”, Journal of Robotics and Mechatronics, Vol. 14, No.1, 2002, pp.78-87.

Ceccarelli M., Toti M.E., Ottaviano E., "CATRASYS (Cassino Tracking System): A New Measuring System for Workspace Evaluation of Robots”, 8th International Workshop on Robotics in Alpe-Adria-Danube Region RAAD'99, Munich, 1999, pp. 19-24.

Ceccarelli M., Avila Carrasco C., Ottaviano E., “ Error Analysis and Experimental Tests of CATRASYS (Cassino Tracking System)”, 2000 International Conference on Industrial Electronics, Control and Instrumentation IECON 2000, Nagoya, 2000, paper SPC11-SP2-4.