# Analyses of hydrostatic transmission system using the modelling and simulation method.

Abstract: The modeling of any hydrostatic system begins by writing the differential equations that define its behavior. The differential equations contain some parameters that can be directly measured by various standard procedures. Apart from the measured parameters, for a well defining of the system, there are some unknown parameters that can be identified and included in equations. The correct estimation of the values of the time varying parameters can be proved by a comparison of measured signals and simulation results.

Key words: hydrostatic system, differential equation, unknown parameter

1. INTRODUCTION

The hydraulic system is the important and kernel part of the control and power transmission in many great mechanical equipments and systems and has been used widely in astronautics, military affairs, metallurgy, robots construction, etc. No other system is so compact and allows such high levels of force and power.

Many researchers (Isermann et al., 1992; Vasiliu, 1999) studied and proposed different models of hydrostatic transmission. The present paper offers a way to do and to verify the correctness of the simulation made.

2. HYDROSTATIC SYSTEM MODEL

2.1 Elements of a hydrostatic system

Basically a hydraulic system consists in a pump P, driven by an electrical motor [M.sub.E], and a hydraulic motor [M.sub.M]; the hydraulic motor is driven by the potential energy of the oil, pressurized by the pump. Figure 1 shows this simplified hydraulic circuit diagram of the hydrostatic transmission under consideration.

2.2 The equations of a hydrostatic system

The dynamic behavior of hydrostatic systems can be modeled by ordinary deferential equations. Theses equations contain some parameters that can directly be measured (masses, the sections of cylinder, the length of piston course, etc.) and some unknown parameters. Some authors use the identification method for condition monitoring in hydrostatic systems, which is based directly on the system deferential equations. In our case the unknown model parameters of a hydrostatic transmission will be identified from measured data.

[FIGURE 1 OMITTED]

We will note the oil flow with Q. The leakage is modeled by the product of a constant leakage coefficient [K.sub.D] and the pressure p in the pressure pipe. The oil flow passing through the hydraulic motor is equal to ([V.sub.M]/2[pi]) x [omega], where [omega] denotes the angular speed of the motor shaft and [V.sub.M] is the displacement engine. With the hydraulic capacity [C.sub.H] of the pressure pipe, the continuity equation is

[C.sub.H] dp/dt = Q--[K.sub.D]P--[V.sub.M]/2[pi] [omega] (1)

The equation of motion has the form

J[??] + [k.sub.d] [omega] = [V.sub.M]/2[pi] p--[M.sub.M] (2)

In this equation we note with J the mass moment of inertia of the machine the motor, with [k.sub.d] the damping coefficient, ([V.sub.M]/2[pi]) x p is the torque transmitted through the motor shaft, and [M.sub.L] is the torque exerted on the machine by external loads. The measurement of the angular speed [omega] can be approximated by a first-order delay, such that the between [omega] and the measured value [bar.[omega]] is described by

T[??] + [bar.[omega]] = [omega] (3)

Equations (1), (2), and (3) contain two well-known coefficients, J and ([V.sub.M] / 2[pi]). The coefficients [C.sub.H], [K.sub.D] and T are constant with unknown values; the coefficient [k.sub.d] and the torque [M.sub.M] shall be denoted as time-varying load parameters since they depend on time-varying external loads.

In the practical application considered, the quantities Q, p, and angular speed [bar.[omega]] have been measured and for solving the equations we used the MATLAB/SIMULINK program.

3. THE MODEL OF UNKNOWN PARAMETERS

The unknown parameters must be identified and for this, we try to obtain a simplified form of the system represented by the equations (1) (2) (3). For the identification of the parameters [C.sub.H], [K.sub.D], and T, e insert equation (3) into equation (1). Q, p, [??], [bar.[omega]] are substituted by discrete-time values. This Yields the difference equation

[x.sup.T] (k) j = y(k) (4)

Where

[x.sup.T (k) = [[P.sub.k]--[P.sub.k-1]/[T.sub.S] [P.sub.k] + [P.sub.k-1]/2 [V.sub.M]/2[pi] [bar.[[omega].sub.k] [bar.[[omega].sub.k-1]/[T.sub.S]] (5)

[T.sub.S] is the time step and [P.sub.k] = p(k[T.sub.S], the parameter vector

J = [[[C.sub.H] [K.sub.D] T].sup.T], (6)

and

y(k) = [Q.sub.k] + [Q.sub.k-1]/2--[V.sub.M]/2[pi] x [bar.[[omega].sub.k]] + [[bar.[[omega].sub.k-1]]/2 (7)

Equation (4) can be applied for k = 1, 2, ... n, which can be written in matrix notation as

for k = 1, 2,..., n, which can be

X(n)j = y(n) (8)

Using the well known least squares method the estimate of [??] of parameter vector is given by

[??](n) = ([X.sup.T](n)X[(n)).sup.-1] [X.sup.T] (n)y(n) (9)

The validation of the estimation is made by checking the Convergence of the results for n = N; N + 1; .... N + M. If used, which is given by the following equations: [??](n) was calculated from equation (9) each time. M+ matrix inversions would have to be performed.

For [??](N + 1)....(N + M), the recursive least squares algorithm Is used, which is given by the following equations:

[gamma](n) = 1/1 + [X.sup.T] (n + 1)P(n)X(n + 1) P(n)X(n + 1) (10)

[??](n + 1) = [??](n) + [gamma](n)[y(n + 1)--[x.sup.T] (n + 1) [??] (n)] (11)

P(n + 1) = [I--[gamma](n)[x.sup.T] (n + 1)]P(n) (12)

the matrix P(N) is given by

P(N) = ([X.sup.T] (N)X[(N)).sup.-1] (13)

Accomplishing these equations various sets of measured data from different experiments are available. One of these data sets has to be selected for the calculation of the final estimation result. This can be done by the convergence check as mentioned above; another indication for suitable data is the presence of significant vibration amplitudes, i.e. there is no problem to identify the coefficients of the signals' derivatives. To avoid the influence of the noise, the signals must be filtered using a first-order low pass filter with a known time constant. Figure 2 shows the convergence of the estimated values of [C.sub.H] and T from the most suitable set of measured data.

Using equation (2) and (3) and the values of [C.sub.H], [K.sub.D], and T as identified in the chosen section, the load parameters [K.sub.d] and [M.sub.M] may be identified by a similar procedure. To estimate the functions [K.sub.d](t) and [M.sub.M](t), a factor [lambda] is introduced into the recursive least squares equations (Isermann et al., 1992).

4. SIMULATION OF THE CLOSED-LOOP SYSTEM

In the different cases of robots, especially mobile robots, the most important parts of hydraulic transmissions are closed-loop systems. Theses types of system offer a good mobility; typically they have 3-9 degrees of freedom. The nature of mobile hydraulics robots makes the simulation approach exceptionally well justified. Considering mobile robots production series are relatively small and unit prices are high, due to the fact that these physical prototypes are rarely available for designers and researchers. There are also a lot of variables to be measured from the prototype. Comprehensive study of the fluid power system of a mobile machine may require 50 flow and pressure measurements. Therefore the measurement equipment with the same amount of sensors must be installed. In the same time the sensors and wiring connections also tend to be sensitive to environmental circumstances. Changes in mechanical structure require a lot of manufacturing and installation work and a very limited number of variations can be tested in practice (Zetu, D. et al. 1997). The basic construction of the mobile machines can be assumed to be quite constant over a long period of time. In this carefully made case and verified simulation model can be used as a design tool for years. All design modifications can be examined on the model. Based on these facts, simulation studies can be very beneficial by both technical and financial means in the area of different types of robots.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Having determined the unknown constant model parameters and the time-varying load parameters, the closed-loop system can be simulated (Ellman & al 1996; Topliceanu 1998). The simulation results for Q[rho], and [omega] should match the measured signals; this fact has been used for model validation and for the choice of the factor[lambda]. In Figure 3 measured signals and simulation results are compared for one of our different experiments.

5. CONCLUSIONS

In the conditions of the great progress of computational technique is important to use the simulation method to study the characteristics of different types of mechanical and hydraulic systems, by technical and financial reasons. In the specific area of our activity we can confirm that there is a good agreement between the simulation results and the measured signals, even though the identification was based on rather noisy data. This model can be used for the design of a state controller and for the simulation of the closed-loop system under various operating conditions of static and mobile hydraulic robots.

6. REFERENCES

Ellman, A.U., Kappi,T.J.;Vilenius, M.J. (1996). Simulation and Analysis of Hydraulically Driven Boom Mechanism. 9th Bath International Fluid Power Workshop, September 9-11, Bath, England, University of Bath, Published in book Fluid Power Systems, edited by C. Burrows and K. Edge Research Studies Press, Somerset, England. pp. 413-429.

Isermann, R., Lachmann, K.-H., Matko, D., (1992) Adaptiv Control Systems. Prentice Hall International Ltd. England

Topliceanu, L., (1998). A Numerical Method for the Study of the Adiabatic Flow of the Gas in the Radial Damper; The 13th Conference on Machine Tools University of Miskolci Egyetem, oct. 1998, pag. 27-29, ISSN 0016-6572, Miskolci.

Topliceanu, L. (2004). The Modeling and Analyses of a Electrohydraulic Servovalve, The 10th International Conference of Fracture Mechanics, pag iulie, 2004, ISBN 973-8392-25-X, Bacau.

Vasiliu, N., Vasiliu, D., Seteanu, I., Radulescu, V. (1999) Fluid Mechanics and Hydraulic Systems Tec ISBN 978-973-718-710-9, Bucharest.

Zetu, D., Gojinetchi, N., Dulhariu, V. (1997). Industrial Robots Publishing House Satya, Iasi, ISBN 973-97945-3-XIasi.