Application of a Pid+fuzzy controller on the motion control system in machine tools

Application of a Pid+fuzzy Controller on the Motion Control System in Machine Tools

H. B. Lacerda, Dr. Eng.

Fabrication and Machine Design Department

Federal University of Santa Maria

Santa Maria, RS Brazil

[e-mail]

E. M. Belo, Ph.D.

Mechanical Engineering Department

College of Engineering at São Carlos

University of São Paulo

São Carlos, SP Brazil

[e-mail]

Abstract

This work deals with an hybrid PID+fuzzy logic controller applied to control the machine tool biaxial table motions. The non-linear model includes backlash and the axis elasticity. Two PID controllers do the primary table control. A third PID+fuzzy controller has a cross coupled structure whose function is to minimise the trajectory contour errors. Once with the three PID controllers tuned, the system is simulated with and without the third controller. The responses results are plotted and compared to analyse the effectiveness of this hybrid controller over the system. They show that the proposed methodology reduces the contour error in a proportion of 70:1.

Keywords: Machine Tools, Machine Control System, PID

Introduction

One of the most important elements in machine tools is their motion controller. The complex design of modern products and the pressure for higher efficiency demand much improvement in the motion controller dynamic behaviour. The complex surfaces created by CAD systems must be transformed, as precisely as possible, in paths to be followed by the tool. The greater the path discontinuities and the required velocity are, more difficult it will be to achieve the necessary precision. Thus, new motion controller design methods have been developed. There are three main types of motion controllers (Koren, and Lo, 1992) feedback controllers, like PID; feedforward controllers (Tung, and Tomizuka, (1993) and contour error controllers (Koren, and Lo, (1992), Srinivasan, and Kulkarni. (1990) and Kulkarni, P.K. and Srinivasan (1989)). Here, an hybrid PID+fuzzy controller with a cross coupled structure is applied to reduce the table path contour error e (see Fig. 6). A pure fuzzy logic controller is not a good solution, as is shown in Lacerda and Belo (1996). If the table has no motions, the same system can be used to control the tool.

Modelling the Axis

A non-linear dynamic model of a numerically controlled biaxial machine tool table is used to simulate the system. Generally, each axis consists of a feed-drive powered by a DC motor, gear box, power screw and table. This set contains many elements, each one having mechanical characteristics such as inertia, axial and torsional elasticity, friction and backlash. There is thermal deformation in all of these elements. To reduce the model complexity, the following hypotheses were used: the elasticity of the gears and their axis are small when compared to the global system combined elasticity. Thermal deformations were not considered, because the axis temperature variation is small. Guideways and ball screw pitch errors were not considered.

DC motor model

This type of motor is widely used in control systems, because it is easy to control, has a low time constant and high torque-inertia ratio. Its characteristics allow linearisation, simplifying the analytical work. A DC motor model can be found in many control textbooks, as in (Kuo, 1985):

(1)

where T_L(t) is the load torque for the motor and is given by Eq. (3). The output equation is:

(2)

Gear box model

In most machines, it is necessary to use a gear box to reduce the motor speed. There is viscous friction in the bearings and dry friction between the gears. The gear box input torque is:

(3)

where T_s(t) is the power screw torque, given by Eq. (21) and z is the gear box ratio.

Power screw model

The equation relating the power screw applied torque and resulting axial force was obtained from a power screw maker (Rexroth Mannesmann, 1990):

T_s(t) = a . F(t) (4)

where:

(5)

Here, l is the lead, in m/rad and h is the mechanical efficiency.

In the gear box output axis, the rotational displacement is q _m/z, where q _m is the motor angular displacement and z is the transmission ratio. This is not the effective angular displacement at the nut position due to the power screw, coupling and bearing elasticity. Screw length changes because of the axial force F(t):

d (t) = l .q _s,a (t) (6)

The screw is a circular section rod with variable length L(t). From the strength of materials theory, we have:

(7)

Substitution of Eq. (4) and Eq. (6) in Eq. (7) yields:

(8)

The screw torsional elasticity is:

(9)

The coupling angular displacement under the action of torque T_s(t) is:

(10)

where K_c is the coupling torsional stiffness coefficient.

The bearings axial displacement under axial force F(t) is:

(11)

K_b,a is the screw bearings axial stiffness coefficient. The equivalent angular displacement is:

(12)

Therefore, the elastic angular displacement is:

q _el(t) = q _s,a(t) + q _s,t(t) + q _c,t(t) + q _b,a(t) (13)

or

(14)

where K_EQ is the equivalent system elasticity coefficient:

(15)

The effective angular displacement at the nut position is:

(16)

Equation of motion

The forces involved in the axis linear motion are represented in Fig. 1. The system is in equilibrium under the action of these forces, so we can write:

(17)

This linear motion is converted to an equivalent rotational motion, using the kinematics equation:

x(t) = l . q (t) (18)

where l is constant. Substituting Eq. (4) and Eq. (18) in Eq. (17), yields:

(19)

where T_s,f(t) is the torque supplied by the gear box reduced as a consequence of screw inertia and viscous friction at the nut and screw bearings.

(20)

Substituting Eq. (20) in Eq. (19), we obtain:

(21)

where:

(22)

With Equations Eq. (3) and Eq. (21), it is possible to obtain the motor load torque and substituting it in Eq. (1), also obtain q _m(t) from Eq. (2). The effective angular displacement can be calculated using Eq. (14) and Eq. (16). Finally, x(t) is obtained from Eq. (18).

The Fuzzy Logic Controller

The usual fuzzy logic controller design involves three steps:

1. The membership functions for input and output controller signals are determined;

2. The rules are written, relating inputs and desired outputs;

3. The defuzzification procedure is chosen.

The fuzzy inference system was created using helpful suggestions from Ross, (1995) and the Matlab Fuzzy Logic Toolbox from Jang and Gulley (1995). There are two inputs, sixteen rules and two outputs. The inputs are the X and Y axis positions. The outputs are auxiliary signals which drive each DC motor accordingly to reduce the contour error. Trapezoidal and triangular membership functions were used to fuzzify the input and output variables, which can be seen in Figs. 2 and 3.

After fuzzifying the input signals, the rules are applied. Depending on the inputs, each rule contributes more or less to the output. The process is known as aggregation. Table 1 shows the rules.

There are some methods to obtain a numerical value for the output signal from the figure generated by aggregation. Here, it was used the centre of gravity method (Hussu, 1995).

The contour error controller (CEC)

The contour error controller consists of two main parts: The first is the contour error model, which is used to calculate the table path deviation in real time and the second is the control law, which send appropriate correction signals to the individual axes. In this paper, the control law is performed by the fuzzy logic rules. The segment PS in Fig. 4 is the contour error.

The contour error mathematical model is obtained making geometric considerations and is given by the following equation (Lacerda , 1998):

e=E_x.sina - E_y.cosa + r.(secg - 1)(23)

Where a is the angle between the tangent of the desired trajectory and the X-axis. The angle g is a function of the delay between the reference point and the tool position. It can be observed in Fig. 4.

The simulations were performed on a non-linear biaxial table model created with Simulink software (Hicklin, et. al., 1992). The blocks FD_x and FD_y in the simplified block diagram shown in Fig. 5 represent the linear motion axes. Two PID controllers utilise the axial positions errors signals to do the primary control of the X-Y table, reducing the axial tracking errors E_x and E_y. They were tuned by simulation of the classical method of Ziegler and Nichols, (1942). A final manual tuning was required. These signals and the command signals from the interpolator feed the "ErC" block, which calculates the contour error and the trajectory curvature radius

The contour error feeds the "PIDce" controller, resulting in a signal which is a function of the contour error. This controller was tuned by optimisation techniques (Lacerda, 1998). The "Fuzzy" block outputs drive the DC motors accordingly to reduce the contour error. The function of the "Uxy" block is to compensate the different dynamics of the axes. The PID+fuzzy controller objective is to minimise the error component orthogonal to the desired trajectory, correcting the table deviation from the desired path, as can be seen in Fig. 6.

Simulation results

A circular contour was simulated to analyse the controller capacity to reject different error sources. Initially, only the axial controllers are working. The system behaves like a common PID controlled XY-table. The simulation results are shown in Fig. 7, where we have the contour error in each circle angular position, from 0 to 2p radians.

The results show considerable errors, mainly where occurs an inversion of the axis motion direction. This effect is known as "quadrant glitch" and is due to friction and inertia.

The cross-coupling controller is switched on and the simulation is performed again. The PID gains were obtained by optimisation. The other parameters had no alterations. The results can be seen in Figs. 7 and 8.

It can be noted that the action of the cross-coupling controller with the PID+Fuzzy control law resulted in contour error reduction of 70:1 when compared with the common axial PID controllers.

Conclusion

Previous work showed that a contour error controller with a pure fuzzy logic control law is difficult to design, because it is necessary to choose many membership functions, many rules and they must be tuned by exhaustive trial and error. The performance is poor. The hybrid PID+fuzzy logic controller shown here is simple and can achieve considerable reduction in the contour error when the table performs a linear or non-linear trajectory, even in presence of friction, backlash and cutting forces. The controller adjustment is simple. It is only necessary to tune the PID driven by the contour error (PIDce in Fig. 5). The X-Y table simulator described here could reproduce well the machine expected behaviour.

Nomenclature

A = area of section [m²]

B = viscous damping [N.m.s / rad]

E = elasticity module [N/m²]

F = force [N]

G = stiffness module [N / m²]

I = inertia momentum [Kg . m²]

J = area momentum of inertia [m⁴]

H = electrical inductance

[Henry, H]

L = variable length [m]

M = mass [Kg]

R = electrical resistance [Ohm, W ]

T = torque [N.m]

d = diameter [m]

¦ = friction coefficient [N / N]

i = electrical current [Ampère, A]

k = constant

l = lead [m / rad]

r = instantaneous trajectory curvature radius [m]

v = voltage [Volt, V]

z = transmission ratio [rad / rad]

Subscripts

a = axial

e = elasticity

f = dry friction coefficient

g = gear

m = motor

r = resistant

s = screw

t = relative to torque or torsional.

ar = armature

gw = guideways

Greek letters

d = axial deformation [m]

e = contour error [m]

w = angular velocity [rad / s]

q = angular position [rad]

Presented at DINAME 97 - 7th International Conference on Dynamic Problems in Mechanics, 3 - 7 March 1997, Angra dos Reis, RJ, Brazil. Technical Editor: Agenor de Toledo Fleury.

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Publication Dates

• Publication in this collection
  20 Nov 2002
• Date of issue
  Sept 1999