Positioning between systems analysis, optimization and simulation.

Teodorescu, Adriana ; Dolga, Valer

1. INTRODUCTION

The mechatronic hyper systems frequently include positioning systems, which perform a specific task, consistent with the role of the hyper system. Positioning refers to the movement in general and usually involves the motion speed and precision. Speed may be related with productivity, while precision with the operating performance of the system.

Mechatronic positioning systems use various typical- or special actuators. They introduce specific mechatronic solutions due to the operating principles and by reason of the control and actuating procedures (Balekics & Dolga, 1980), (Dolga et al., 1980), (Yoo et al., 2003), (Scott & Tesor, 1999).

In the authors' point of view, achieving optimal positioning systems requires an overall analysis and the application of design principles specific to mechatronic design. The paper aims at presenting the problem structuring and at highlighting particular applications. The research materializes the optimal design approach for the transmission gears of the system by a set of recurrence relations and the systemic structural analysis mode of the remaining mechanical components.

Including nonlinear issues within the models of the mechatronic positioning systems and the validation of the accepted assumptions are future aims of the research.

2. ANALYSIS OF SPECIFIC SOLUTIONS

2.1 Possible solutions

An analysis of principles for achieving positioning systems highlights the possibilities offered by pneumatic actuators (with a number of disadvantages related to multi-point positioning and the flexibility offered during the amendment of the position cycle) and the electrical actuators.

Available versions, with electric motor and integrated brake or electromagnetic coupling, are noticed by facilities for constructive simplicity and good flexibility in changing positioning cycle (Dolga et al., 2008).

The structure of a positioning system is shown in Figure 1 (TP-transducer position, B-electromagnetic brake, M-electric engine, T-transmission, C-clutch, MA-mobile element).

[FIGURE 1 OMITTED]

The analysis of workable actuators emphasizes the availability of the step-by-step motors, DC servomotors, AC servomotors and synchronous servomotors with permanent magnets, but a growing trend to use the latter solution, based on synchronous motors, is evident. The selection of the optimal actuator involves a multi criteria decision making process, and an assessment of the performance parameters for the available types of actuators. Next performance parameters are requisites:

* The peek value of the torque cmax- defined by the maximum value of the current intensity;

* The transient power available for a moment of inertia J :

[P.sub.S] = [c.sup.2.sub.max]/J (1)

* The maximum acceleration: [[epsilon].sub.max] = [C.sub.max]/J

* The launch time--defined as the time required to achieve the nominal speed for the nominal value of the torque:

[T.sub.0] = J[[OMEGA].sub.n]/[C.sub.n] (2)

The motion resolution is a compulsory requirement for the positioning system and depends mainly on the qualitative parameters of the position transducers that are used. The precision adjustment requires appropriate safety measures to remove or decrease non-linearity within the system: eliminate backlash, reduce friction and increase the rigidity.

The system dynamic behaviour varies with the components, the displacement law and the control system.

2.2 Analysis of the optimum transmission gears

The total transmission ratio of the gear may also be subject to the optimization problem providing as goal functions: maximum start-up acceleration, maximum transmitted power, correlation of the mechanical impedances, minimum duration of the positioning cycle (Dolga, et al., 1980).

The step ratios within the gear influence the gear's overall weight. To choose step ratios, various optimization criteria may be applied, like the minimum distance between the axes.

For the gear in Figure 2, consisting of n geometric axes, recurrence relations depending on the gear tooth load are proposed (Balekics & Dolga, 1980). If considering the gear contact load, one determines the recurrence relation that defines the link between the gear ratios on two consecutive gear steps:

2 x [i.sub.j]--1/[cube root of [i.sup.2.sub.j]] = k x [i.sub.j+1] - 2/[cube root of [i.sub.j+1]] (3)

[i.sub.j] is the transmission ratio on the gear step j (j=1, 2 ... n); k is a factor that considers the material and the geometry of the gear.

k = [cube root of [([[sigma].sub.ka,j]]/ [[sigma].sub.ka,j+1]]).sup.2] [[psi].sub.aj]/ [[psi].sub.aj+1]] x [[sigma].sub.ka,j] is the permissible contact load on the step j and [[psi].sub.aj] is the width coefficient on this step j.

[FIGURE 2 OMITTED]

The total gear ratio i is added to the equations system (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Similar relationships can be determined for the gear bending load and for the criterion that minimizes the volume of the material within the gear.

3. CLASSES OF POSITIONING SYSTEMS

The performance of a system depends on a series of factors, but the mathematical model of the system may discard a number of them, by neglecting their contribution. Some experts divide the servomechanisms according to their performance--own pulsation, accelerating capacity, accuracy--in three classes: of high performance, of medium performance and of low performance (Yoo et al., 2003). Others apply the predominant elasticity within the system:

* elasticity due to the mechanism / transmission parts;

* elasticity of the elastic assemblage actuator--frame;

* elasticity of the actuator's affix;

* elasticity of the guiding part.

Each definite case has to be investigated in a particular manner, regardless the classification mode.

Figure 3 shows an example of a system with the elasticity caused by the mechanism / transmission parts. Taking into account the elasticity of the flexible elements, the system can be assimilated to a single deformable element and an inertial mass, as shown in Figure. 4.

The system dynamics is expressed by the state model (5), with the usual symbols for the input, output and state quantities.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

dx/dt = A x x + B x u

y = C x x + D x u (5)

The matrices A and B in the state equation and the input u are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

B = [[0 0 1 0].sup.T] (7)

u = [0 0 F 0] (8)

The state model is used in simulations and early performance estimation; the results provide desired information to find a proper positioning system for a given request.

4. CONCLUSION

The paper summarized the authors' research on the design and optimization of the positioning systems from within mechatronic hyper systems. The successful design and execution of a positioning system may be critical because the component essentially contributes to the performance parameters, engineering-oriented parameters, and cost parameters of the product. A single error might increase the total system errors or costs. Actually, positioning applications are suitably to the mechatronics design approach, as all the core technologies are there and interact. The authors propose analyses and optimization solutions specific to the mechatronic philosophy, outlining the diversity of the studied objects.

Concrete applications were studied, which highlighted that any constructive optimization must be correlated with the control methods. The design and interaction of both the control and mechanical system define the product working frame. As they are risky, resonant modes and frequencies of the mechanical system must be avoided. Authors recommend mathematical state models for dynamic analyses and simulations in order to verify expected performance and provide advantages in an advanced control system.

Further studies will be focused on the nonlinear aspects and on the micro-positioning applications, in the micron field.

5. REFERENCES

Balekics, M. & Dolga, V. (1980). Choosing the best gear ratios to intermediate cylindrical reducers to obtain the total distance between the axles, Symposium of Mechanisms and Mechanical Transmissions p.111, Timisoara (Romania)

Dolga, V.; Dolga, L.; Teodorescu, A. et al. (2008). Features of movement control in mechatronics, Contract 112 CEEX-II 03/2006, phase 4-4-2/2008, Timisoara (Romania)

Dolga, V.; Vacarescu, I. & Radulescu, I. (1980). Principles of calculation of the engine step by step motors and screws for Ball industrial robots, Symposium of Mechanisms and Mechanical Transmissions, pp.165, Timisoara (Romania)

Scott, E.L. & Tesor, D.(1999). Criteria Based Actuator Control, Ph.D. Dissertation, Department of Mechanical Engineering, University of Texas, Austin (U.S.A.)

Yoo, J.; Ashok, P.; Kapoor, C. & Tesar, D. (2003). Operational Performance Criteria of Intelligent Actuators, Robotics Research Gr. Report to the Dep. of Energy, grant DEFG04 -94 EW37966, AMD A007, Univ. of Texas, Austin (USA)