The tensometric force sensor. Design, evaluation and optimal selection.

Dolga, Lia ; Dolga, Valer

1. INTRODUCTION

Effective monitoring of a manufacturing process is essential to guarantee the quality of the final product. Numerous industrial systems require data acquisition regarding a torque vector; in addition, the value of other external forces is often required. The measuring devices are specific; they are included in the multicomponent (multi-axes) force sensors. The existing references comprehensively present constructive solutions, criteria and design methods for this kind of sensors (Ahili, 1997), (Dickert, 1999), (Kang, 2005). A large range of sensorial elements for force measuring (in a generalized meaning) is available. The requirement of a stable, accredited and responsive method for evaluation and selection of appropriate force sensors is evident (Dolga, 2002), (Wang et al., 2003).

The paper outlines the authors' opinion concerning a unitary methodology about the analysis, the comparison, the assessment and the selection of tensometric sensorial elements. This concept was implemented through a software application, "PROTRA", developed in Visual Basic, which can be later upgraded for other types of sensors.

2. THE STRUCTURE OF "PROTRA"

Starting from the main request of the force sensors design, the authors modularly organized "PROTRA" in four sections; the user can retrieve each one either separately or sequentially.

The first design strategy, "Selecting the optimal sensor's variant", defines the "PROTRA" first module. This selection is applied to a set of "n" sensor variants [V.sub.i] (i=1, 2 ... n), compared by "m" evaluation criteria, [C.sub.j] = 1, 2 ... m). The "TOPSIS" method (Technical Ordered Preference by Similarity to Ideal Solution) ranks the variants (Dolga, 2002).

The module "Expert system" allows accessing a specific knowledge base, including heuristic knowledge of experienced designers. The module recommends potential decisions on the design alternative, proposes the sensor type and structure, etc.

The module "Sensors databases" comprises information about ready-made force sensors (commercial) or sensor components (Figure 1). These databases are relational, including "BLOB" fields (binary large objects). The user may consult either constructive databases or databases referring to the functional parameters and the potential applications.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The module "Calculus" ensures the conditions to apply electrical and mechanical analytical determinations for different components of the force sensors. Figure 2 shows a page in the module with recommended elastic elements for a single component force sensor. This module can call databases detailing the materials for the elastic element/ structure or strain gauges databases. The proof and confirmation of the designed solution should be completed using finite element programs.

The structure of the sensorial element often requires an analysis starting from the mathematical model of this element. An evaluation of the agreed solution follows, based on a set of reference criteria, availed by a proper mathematical support.

3. THE MATHEMATICAL MODEL OF THE SENSOR

3.1 Introduction

The input-output quantities dependence, specified in the relationship (1), gives the static characteristic of a force sensor:

[Y] = f ([X], [P]) = [[PSI]] * [X] (1)

where [X] = [[[F][M]].sup.T.sub.6x1] is the vector of the input signals, including the components of the measured force-torque vector, [P] comprises the perturbation signals and perturbation effects, [Y] is the output quantity, represented by the output signals [S.sub.i] (i=1,2 ... n) and [PSI] signifies a transfer function that is called "the measuring matrix of the sensor" (Bicchi, 1992).

Once correlating these considerations with the real input-and output quantities and with the applied notations, the equation system (1) becomes:

[[S].sub.nx1] = [[C].sub.nx6] x [[F][[M].sup.T.sub.6x1] (2)

The "coupling matrix" [C] outlines the influence of each force-torque vector component on the measuring channel [S.sub.i].

The equation system (2) relieves studying the system stability and the errors propagation during the design stage and the analysis stage. An important algebraic instrument in evaluating the perturbation sensitivity derived from the coupling matrix is the condition number (Bicchi, 1992):

[K.sub.(C)] = [parallel] C [parallel] x [parallel] [C.sup.-1] [parallel]] (3)

[parallel] * [parallel] signifies the norm of the matrix.

A minimum value for the condition number--the unit value being the reference rate--involves a higher stability of the sensor against gaussian perturbations, because a minimum ratio noise/signal is guaranteed.

In order to generalize the study and to remove the inconsistence of the relationships due to different unit measures for the force components and for the moment components, the normalized coupling matrix was introduced (Svinin, 1995):

[[bar.C]] = [[N.sub.[epsilon]].sup.-1] x [C] x [[N.sub.n]] (4)

[[N.sub.e] = diag [[epsilon].sub.1m] [[epsilon].sub.2m] ... [[epsilon].sub.nm]] is the diagonal matrix of the maximal stresses on the "n" measuring channels of the sensor; [[N.sub.e]] = diag [??][F.sub.xm] [F.sub.ym] [F.sub.zm] [M.sub.xm] [M.sub.ym] [M.sub.zm][??] is the diagonal matrix containing the nominal values of each component in the force-torque vector.

Based on the normalized coupling matrix, the condition number is defined (Svinin, 1995) as below:

[K.sub.([bar.C])] = [[lambda].sub.max]/[[lambda].sub.min] [greater than or equal to] 1 (5)

The numbers [[lambda].sub.max] and [[lambda].sub.min] are the largest singular value and the smallest singular value of the normalized coupling matrix. The ideal value [L.sub.([bar.C])] = 1 points out a perfect isotropy of the measuring structure.

3.2 The dimensional optimisation

One can group the design parameters of the force sensors in three categories: geometric parameters (X), material parameters (Y) and loading parameters (Z). A set of design parameters, a set of constraints [G.sub.i] and a set of possible optimisation criteria characterize the sensor during the design stage. The optimisation problem, expressed in (6), requires finding the design parameters U(X, Y, Z).

[f.sub.(U)] [right arrow] {min, max} (6)

3.3 Application

Figure 3 presents a basis construction of a force sensor for two components whose measuring matrix is given in (7):

[FIGURE 3 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If the clause of minimizing the condition number is applied, one obtains the solution (8) (Dolga, 2002):

[x.sub.2] x [f.sub.ymax]/B = [M.sub.ymax]/H (8)

Besides the relationship (8), the geometrical constraints relationships and resistance relationships must be also satisfied for a comprehensive optimization of the force sensor.

4. CONCLUSION

The optimal design of a force sensor is an essential requirement in many types of mechatronic systems. The tensometric principle defines the most used force sensor type. A unitary methodology to design and evaluate these devices became necessary. To answer this requirement, the authors grouped appropriate design-, evaluation--and optimization procedures in a dedicated software application, "PROTRA".

In designing the elastic structure of the force sensors, helpful mathematical instruments are the pseudo-inverse matrix and the condition number. Appropriate selection--and optimization procedures were added to the shape calculation of the elastic element. Suitable databases make up the modular application. "PROTRA" evolved from a didactic tool to a design--and research assistant. New force sensor types stay in mind, but further progress focuses too on creating a simulator to select the sensorial elements for reconfigurable systems.

5. REFERENCES

Ahili, F. (1997). Dynamics and Control of direct-drive robots with positive joint torque feedback, Proc. IEEE Int. Conference on Robotics and Automation, pp. 2865-2870

Bicchi, A. (1992). A criterion for optimal design of multi-axis force sensors. Journal of Robotics and Autonomous Systems, Vol.10, No.4, pp. 269-286

Dolga, V. (2002). The optimal design of force-torque sensors (in Rom.)--moment, Proc. of 6th Conf. on Fine Mechanic and Mechatronic, pp.245-252, ISSN 1220-6830, Brasov, October, "Transilvania" University Publishing House

Kang, CG. (2005). Performance Improvement of a 6-axis force -torque sensor via novel electronics and cross-shaped double-hole structure, International Journal of Control, Automation and Systems, Vol.3, No.3, pp.469-475

Svinin, M.; Uchiyama, M. (1995). Optimal geometric structures of force / torque sensors. Journal of Robotics Research, Vol. 14, No.6, pp.560-573

Wang, L.; Kannatey-Asibu, E.; Mehrabi, M. (2003). A method for sensor selection in reconfigurable process monitoring. Journal of Manufacturing Science and Engineering, Vol.125, February 2003, p.95-99