Tracking and stabilization of pneumatically actuated cart-inverted pendulum.

Zilic, Tihomir ; Essert, Mario ; Situm, Zeljko 等

1. INTRODUCTION

Model of the cart-IP is usually used for testing controllers because it represents simple version of the class of underactuated mechanical systems with nonholonomic constraints of second order. 'Underactuated' means that mechanical system has some of degrees of freedom unactuated. More complex examples in applications are underwater vehicles, space crafts, underactuated manipulators, surface vessels etc. Because IP uses gravitational forces as 'additional' actuator, then there exist even linear controllers that will stabilize the system around equilibrium point (Oriolo & Nakamura, 1991). Dynamics between control voltage and air force on the cart is described by second order differential equation. This dynamics of pneumatic cylinder is used for horizontal moving of the cart and parameters of differential equation were found through process of identification. Linear motor of first order actuator dynamics is used by (Riachy et al., 2007). For control purposes in this work is used linear quadratic (LQ) controller. Stabilization of pneumatically actuated cart-IP with LQ controller is described by (Zilic et al., 2006). Positioning of cart using nonlinear control such as sliding mode controller is given by (Situm et al., 2003). The main idea of this work is to show how the cart can follow desired trajectory and simultaneously stabilize the IP around unstable equilibrium using mentioned linear controller. Nonlinear model of the cart- IP plus actuator is linearized around unstable equilibrium of IP and whole system is presented in state space of sixth order. To use this controller, system must be controllable, and in this case it is controllable. Viscous friction model is used for controller synthesis, but real friction model is more like Karnopp friction model. The problem in direct implementing Karnopp and more similar nonlinear friction models in synthesis of linear controller makes linearization process more difficult. Because static friction is not used in synthesis of the controller, experimental results will show the permanent tracking error. Experiment will also show robustness property of used controller when the system is under influence of external disturbances. Simulation results will not be presented due to lack of writing space. Further research will be directed toward modelling of stick-slip friction, and also change linear controller with nonlinear one.

2. EXPERIMENTAL SETUP

The pneumatically-actuated inverted pendulum (Fig. 1.) is based on the linear pneumatic motor experimental setup, which has been developed at the University of Zagreb recently.

[FIGURE 1 OMITTED]

The pendulum is hinged to the rod-less pneumatic actuator via a pivot. The pendulum deflection angle (angular position) is measured by a rotary servo-potentiometer. Linear motion of the piston is controlled by a voltage-controlled proportional directional control valve connected to both cylinder chambers. The proportional valve bandwidth is approximately 100 Hz. The cylinder position is measured by the horizontal linear potentiometer with 0.5 m travel length, attached directly to the actuator. Pressure transducers are added to the setup in order to measure air pressures in both cylinder chambers. Identification of second order dynamic between pneumatic force and control voltage is given in Fig.2. Maximal pressure in system is 5 bars. Data acquisition and control of the experimental setup are implemented by utilizing a portable (notebook) PC computer equipped with a PCMCIA acquisition and control card NI- 6036E. The card comprises 16 analog inputs and two analog outputs with 16-bit resolution and [+ or -] 10 V range. The data acquisition and control routines are implemented under Matlab Real-Time Workshop.

[FIGURE 2 OMITTED]

3. MATHEMATICAL MODELLING

3.1 Modeling of pneumatically actuated cart-IP

Nonlinear model of cart-IP plus dynamics of the actuator is linearized around zero state x = [0 0 0 0 0 0], which means around vertical position of IP. Used approximations are: sin [x.sub.3] [approximately equal to] [x.sub.3], cos [x.sub.3] [approximately equal to] 1, [x.sub.4] [approximately equal to] 0. System is given in state space form (1).

[??](t) = Ax(t) + Bu(t), y(t) = Cx(t) (1)

with matrices and vectors defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Parameters of model matrices are:

M = [m.sub.s] + [m.sub.p], T = [I.sub.s]/[m.sub.p]L, [I.sub.s] = [I.sub.cm] + [m.sub.p][L.sup.2], [I.sub.cm] = [m.sub.p]/3 [L.sup.2]

where: g -acceleration of gravity, b -coefficient of viscous friction, [A.sub.p] -- piston cross-section, [[OMEGA].sub.0] -- applied force model natural frequency, [K.sub.1] -- applied force model gain parameter, [zeta] -- applied force model damping ratio.

Numerical values of parameters are: g = 9.81 m/[s.sup.2], [m.sub.S] = 1.5 kg, [m.sub.p] = 0.06 kg, L = 0.2 m, [A.sub.p] = 1.8[10.sup.-4] [m.sup.2], [K.sub.1] = 3.45[10.sup.5] Pa/V, [[OMEGA].sub.0] = 100 rad/s, [zeta] = 0.65, b = 200 Ns/m, u = 0 / 10 V.

3.2 Modeling of LQ tracking controller

Purpose of control u(t) of system (1) is used for tracking with respect to desired reference trajectory x1d(t). Tracking means that cart will follow the trajectory and IP will be simultaneously stabilized in vertical position. Controller design will consist of LQR and reference trajectory design as follows (Podsialdo):

[??](t) = Fv(t), [micro](.t) = Hv(t), v(0) = [v.sub.0] (2)

The problem is to find the linear feedback control law u(t) = Kx(t) that minimizes performance index

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Transforming the tracking problem into optimal regulator problem using equations (1) and (2) we get new augmented system in the state space as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Now the new performance index is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

and the control law is u(t) = K[xi] = -[K.sub.1]x(t) - [K.sub.2] v(t).

System state weighting matrix is chosen as [??] = 100, R = 0.5 . Trajectory design (2) is based on solution of differential equation of m-th order. If desired trajectory is sinusoidal then it is represented as solution of differ. eq. of m = 2 with matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Desired reference of the form [x.sub.1d](t) = 0.1cos(0.5t) is the solution of (2) with [zeta] = 0, [omega] = 0.5 rad/s and [v.sub.0]=(0.1,0). Using Matlab function K = lqr(W,V,Q,R) we will find controller parameter K = [-14.1421, -12.1269, -20.3418, - 3.3525, 0.0037, 0.0000, 13.4721, 7.9859]

4. EXPERIMENTAL RESULTS

Experimental results are shown at Fig.3. Reference [x.sub.1d](t) that cart follows has form [x.sub.1d](t) = 0.1cos(0.5t).

[FIGURE 3 OMITTED]

5. CONCLUSION

This work shows that uncompensated friction will result in permanent error in tracking, but robustness of the controller will keep system stable. LQ controller has ability for tracking and stabilization of high nonlinear underactuated systems, but with gravity help. Future work will be focused on implementing nonlinear controllers for better trajectory tracking.

6. REFERENCES

Oriolo, G. & Nakamura, Y. (1991). Control of Mechanical Systems with Second-Order Nonholonomic Constraints: Underactuated Manipulators, 30th IEEE Conference on Decision and Control, Brighton, UK, December 11-13, 1991

Podsialdo, P. Advanced Control Engineering 630.428, Optimal control part, Available from: http://www.mech.uwa.edu.au/ undergrads/sem2.html

Riachy, S.; Floquet, T. & Richard J.-P. (2007). Nonlinear control for linear motors with friction -- Application to an inverted pendulum system, LDIA'07, 6th Internal Symposium on Linear Drives for Industry Applications, Lille, France, 2007

Situm, Z.; Petric, J. & Crnekovic, M. (2003). Sliding Mode Control Applied to Pneumatic Servo Drive, Proceedings of the 11th Mediterianean Conference on Control and Automation MED'03, June 18-20, 2003, Rodos Palace Hotel, Rhodes, Greece

Zilic, T; Pavkovic, D & Zorc, D. (2009). Modeling and Control of a Pneumatically Actuated Inverted Pendulum, ISA Transactions, Vol. 48, No. 3, (July 2009) 327-335, ISSN 00190578