Autors Guide for Preparation of the Camera-Ready Article for the 4Th International IMEKO

Tehran International Congress on Manufacturing Engineering (TICME2007)

December 10-13, 2007, Tehran, Iran

Linearization and Robust Control of Scara Robot Used in Manufacturing

M. Moavenian [1], M.R. Gharib[2], A.A. Akbari [3] and A.H. Samiee [4]

Solid Mechanic Engineering Group, Mechanical Engineering Department,

Ferdowsi University, Mashhad, Iran

Corresponding Author E-mail:

Abstract

This paper presents a new linearization technique to be used for robust control of an SCARA robot manipulator. Some nonlinear terms in the model of robot manipulator are compensated by linear terms which are introduced in the paper. Linearization process is carried out for four arms but QFT is only applied to design feedback controller for the first arm. The research indicates that applying the proposed technique successfully overcomes obstacles for robust control of nonlinear SCARA robot.

Keywords: SCARA, Robot, Robust Control, Linearization

1 Introduction

SCARA robots are widely used in assembly manufacturing (‘Figure’ 1). The robot is a horizontally articulated manipulator with a vertical joint at the wrist end. The arm is very stiff in the vertical direction but is relative compliant laterally. This feature is convenient for a variety of assembly tasks. The potential problem arises mainly due to the positioning errors in assembly. Adaptive and model-based controls are two of the popular control strategies used to control robotic systems. These control schemes cannot overcome the structure uncertainties of a robotic system. [1]

Dynamic model of robot manipulator consist of highly nonlinear coupled second order differential equations. Linear time invariant control laws which utilize linear models of robot manipulator dynamics are often used for industrial robot manipulators because of the simplicity of the control algorithms. However nonlinearity and parameter variation in real systems prevents, ordinary linear time invariant control schemes to achieve satisfactory control performance.

Linearization tecniques for robust control of robot manipulators with uncertainty has been the subject of many research studies.For example Kawabata, et al. (1993) and Takayanagi et al. (1993) studied robust position controller for 2 link manipulator. [2] T.J. Tern, A.K. Bejeay, A. Lotdorl, and Y. Chen have presented dynamic modeling and linearization technique for a SCARA robot. [3]

2 Scara robot

2.1 Arm matrix

Arm matrix shows the position and the direction of the tool, with respect to base coordinate system.

(1)

Where p is a vector of position and R is a vector of direction for the tool.

The shaping space of the tool has 6 dimensions because every form of tool placing can be found by its position and its direction characteristics yaw, pitch, roll. Generally reversed kinematic problem is as follows:

For each P position and R direction for the tool, find the space variation value to satisfy the above equation.

If shows the roll angel of the tool, then tool forming vector is w that in is defined as follows: (2)

For SCARA robot, tool forming vector is as follows.

(3)

Figure 1: Selective compliant assembly robot arm (SCARA). [4]

2.2 Inverse kinematic for SCARA robot

‘Figure’ 2 shows the inverse kinematic chart for SCARA robot.[4]

Figure 2: Inverse kinematic chart for SCARA robot.

Where si and ci are sin () and cos () respectively.

2.3 Continuous Movement

If the speed of each joint can be controlled individually, robot can pass continuous directions. For problems which their inverse kinematic is available analytically, we can have the speed of each joint calculating the derivative of it with respect to time. The results as obtained are given respectively as follows: [4]

Figure 3: Link coordinate diagram of the scara robot [4]

(4)

(5)

(6) (7)

(8)

2.4 Dynamics of Robot

When the Newton-Euler equations are evaluated symbolically for any manipulator, they yield a dynamic equation which can be written in the form (9) Whereis the nn mass matrix of the manipulator is an n1 vector of centrifugal and Coriolis terms, and G (q) is an n 1 vector of gravity terms.

2.5 System Linearization

In order to reduce the nonlinear robust controller, design to a much easy and simpler problem. A linearization technique is presented here, which covers all features of the real nonlinear system in working space. Each arm is considered as a load system which is connected to the motor. Then the following simple governing equation can be considered for each arm.

Ignoring all non linear terms in equation (1), we have (10)

Angular velocity (), acceleration() and required torque () for different trajectories can be calculated, by simulating the nonlinear model using "Matlab Robotic Toolbox" based on numerical values found for them, an equivalent linear plant will be derived for each arm as follows:

(11)

In equation (9), matrix and vector can be obtained carrying out identifying tests [3] or simulating robot in a defined trajectory. and are identifiable terms with respect to defined points in every trajectory. Therefore, for n points of a trajectory we have:

Suppose , B= (12)

We have. Where (13)

For example equation (9) of SCARA robot used in [5] has the following given numerical values.

(14)

By running the simulation for multiple trajectories of all arms, and are found respectively.

Arm 1 (15)

Arm 2 (16)

Arm 3 (17)

Arm 4 (18)

The above founded values were chosen, selecting certain known operating points which sufficiently represent the range of variation in joint dynamics.

As a result linearized, transfer function for each arm can be represented as follows:

i=1,2,3,4 (19)

3 QFT controller design

Using QFT method ([6], [7], [8], [9]) for controlling the selected SCARA. The nonlinear plant needs to be converted to family of linear and uncertain processes implementing the new technique mentioned in part 2.5.

A suitable QFT controller and prefilter ‘Figure’4 were then designed for the four joints to satisfy the closed loop specifications ((=20%) and (=0.08 s) where and are the overshoot and the settling time respectively).

Figure 4: Structure of a two degrees of freedom system

Template generation (reveals frequency domains ‘Figure’ 5); robust margins for five selected trajectories of the first arm based on frequencies found in template generation are shown in ‘Figure’ 6, robust tracking and bounds intersection of the same arm are presented in ‘Figures’ 7 and 8 respectively. [6]

Tehran International Congress on Manufacturing Engineering (TICME2007)

December 10-13, 2007, Tehran, Iran

Figure 5: Template generation

Figure 7: Robust tracking for arm 1

Figure 6: Robust margin for arm 1

Figure 8: Intersection of bounds for arm1

Tehran International Congress on Manufacturing Engineering (TICME2007)

December 10-13, 2007, Tehran, Iran

The loop shaping and prefilter functions are found running QFT toolbox. The results are presented in ‘Figures’ 9 and 10 respectively. [6], [9]

Figure 9: Loop-shaping in Nichols chart for arm 1

The respected controller is found as:

(20)

Figure 10: Pre-filter shaping in Nichols chart for arm 1

The respected prefilter is found as:

(21)

The time domain closed loop response before and after adding the designed controller and prefilter are shown respectively in ‘Figures’ 11 and 12 which indicate the strength and practibility of the proposed design method and supports the frequency domain results.[6]

Figure 11: Time domain simulation without controller and prefilter for arm 1

Figure 12: Time domain simulation for arm 1

4 Conclusion

This paper proposes new linearization technique implemented for controlling an SCARRA using QFT method. The technique is not restricted to QFT and can be tested on other methods as well. Our research indicates that applying the proposed technique successfully overcomes obstacles for robust control of nonlinear SCARA robot. A comparative verification of the technique will show the strength of our findings.

References

[1] E. Meng, and K.Liew, “Control of adept one scara robot using neural networks“. IEEE Trans on Industrial Electronics, Vol. 44, (1997) pp.762-768.

[2] Z. B. Kong, T. Y. Chai, K. Oshima, J. M. Yang, and S. Fujii, “Robust Vibration Control of Scara-Type Robot Manipulators“. http://www.paper.edu.cn.

[3] T.J. Tern, A.K. Bejeay, A. Lotdorl, and Y. Chen “Nonlinear Feedback in Robust Arm Control“, Proce, IEEE control, NY, Dec. (1984) pp. 38-51.

[4] M.S. Asnaashari,“Design and product of a software for simulating behavior of robot manipulators in inverse kinamatics and control problems by classic and neuro-fuzzy methods “ Master of Science Thesis, Sept. (2005).

[5] H. Taghirad, and Sh. Afshar, “Linearization and PID controller for Scara Robot“, ICEE Conference, Isfahan, (2000) pp. 161-168.

[6] M. Gharib, A.A. Amiri Moghadam, and M. Moavenian, presented to International Symposium on Automation and Robotics in Construction (ISARC), Kochi, Kerala, India 19-21, Sept. (2007).

[7] I. M. Horowitz, Survey of Quantitative Feedback Theory, Int. J. Control Journal, Vol. 33, (1991) pp. 255-261.

[8] A.C. Zoloas, and G.D. Halikias, “Optimal Design of PID Controllers Using the QFT Method“, IEE Proc-Control Theory Appl, Vol. 146, (1999)

[9] P.S.V. Nataraj, “Computation of QFT Bounds for Robust Tracking Specifications“, Automatica 38, (2002) pp. 327-334.

[1] - Assistant Professor, Mechanic Engineering

[2] - MSc Student, Mechanic Engineering

[3] - Assistant Professor, Mechanic Engineering

[4] - MSc Student, Mechanic Engineering