Fuzzy logic controller design for a planar parallel robot.

Lapusan, Ciprian ; Maties, Vistrian ; Balan, Radu 等

1. INTRODUCTION

Parallel robots are mechanisms that have the end-effector connected to the fixed frame by more than one kinematic chain (Merlet, 2006). This construction offer them advantages like: high speed, higher accuracy, bigger loads. Applications of this type of robots can be found in the positioning devices for high precision surgical tools, multi-axis machining tools, and precision assembly tools or in training simulators for pilots.

Different approaches when it comes to the control system for this type of robots can be developed. But mainly the control of such robots is done using PID algorithms. Because of the highly non-linear dynamics of the robot, this type of control doesn't offer always the best results. Here is introduced a novel approach as an alternative to the classical approaches by using the fuzzy logic controller.

2. THE PARALLEL ROBOT MODEL

2.1 Robot description

The robot structure analyzed in the paper is a three degree of freedom parallel robot also known as the 3-RRR robot. The CAD model of the developed robot is presented in figure 1 a). As can be seen in the picture the end-effector is connected to the fixed frame by three kinematic chains. The driving of the robot is made by three DC motors with gears. More detailed information regarding the mechanical structure and the kinematics can be found in (Lapusan et al., 2005, 2008).

In figure 1 b) is present the experimental stand used for control implementing. The parallel robot (5) is connected to the dSPACE control board through the interface panel (3). The parameters of the system are supervised and modified using the ControlDesk (4), the interface allow to enter the position of the end effector and to display the data from the sensors. The dSPACE DAC port are interfaced with the DC motor using a power amplifier (2), the energy needed for the actuators is provided by the power supply (1).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.2 Mechanical structure model

In order to simulate the behaviour of the robot, the dynamic model of the structure has to be built. This is a complex subject and different methods were developed in order to solve it. A classical method for calculating the dynamic models of closed-chains is to be considered first as an equivalent tree-structure, and then to consider the system constraints by the usage of Lagrange multipliers or d'Alembert's principle (Duffy, 1996). Other approaches include the use of virtual work, Lagrange formalism, Hamilton's principle, and Newton-Euler equations.

In this article the dynamic model of the robot is built using the SimMechanics toolbox from Simulink. The toolbox uses the standard Newtonian dynamic of forces and torques in order to solve booth direct and inverse kinematics problem. The model is built using Simulink blocks; blocks that represents the kinematic elements and the joints of the robot. The blocks from the toolbox allow to model mechanical systems consisting of any number of rigid bodies, connected by joints representing translational and rotational degrees of freedom.

The model of the mechanical structure is presented in fig. 2. All three kinematic chains are defined by bodies and joint blocks. The inertia properties and the coordinates of the joints for each body were imported from the CAD model of the parallel robot.

The connection of the dynamic model of the mechanical parts to the rest of the robot model is made using actuators and joints block. As input in the model, using actuators, can be chose between the generalized force and position/speed/acceleration of the motor joints. In this chase the inputs are the speed of all three motor joints of the robot. As outputs is chosen the angles for each motor element. Also in simulation sensors are used for determine the position of the end-effector.

2.3 Actuator model

The actuation of the robot is made using three DC motors with gears. The equations that define the dynamic behavior of one actuator are described in the followings.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 3 OMITTED]

The model of Equation 1, based on the Simulink blocks, is shown in Figure 3. The inputs of the model are the voltage and the resistance torque and the output represents the angular velocity of the shaft and current.

3. FUZZY CONTROLLER

The fuzzy controller in this case will have two inputs, the error e between the angle reference for each actuator and the actual value in the system and the variation speed of the error [DELTA]e. The output of the controller will be the voltage for the actuator, the range of the voltage is -9 ... 9 [V].

In designing a fuzzy logic controller three steps must be perform: fuzzification, inference and defuzzification. Fuzzification is an interface that produces a fuzzy subset from the measurement. The defuzzification is an interface that produces a crisp output from the results of the interface (Kandel, 1993). Figures 4 and 5 present the membership functions for inputs and output.

The results from the input are processed using a set of rules, this process is called inference. The total number of rules is twenty five in this study; the rules are presented in table 1.

There are three DC motor for three arm of the robot. So, three fuzzy controller units are used for each motor. The model of the robot is presented.

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4. SIMULATION RESULTS

Figure 7 present the simulation results for a step signal, representing a change in position from angle 20 to angle 50. The figure presents the response of the system for two controllers, a Fuzzy controller (thin line) and a PID controlled (dash line).

The designed fuzzy controller moves the actuator at the desired angle faster and more accurate than the PID controller.

[FIGURE 7 OMITTED]

5. EXPERIMENTAL RESULTS

In order to load the model in the dSpace real time hardware the model of the robot is modified and the dynamic models of the actuators and mechanical structure are replaced with DAC and ADC ports. Matlab generate automatically the code for the control board.

Figure 8 presents the experimental results, the response of the system is the same as the response obtained in simulation.

6. CONCLUSION

The paper presented a control approach using fuzzy logic for a planar parallel robot. The control system was developed and tested using MATLAB/Simulink. The results of the fuzzy controller were compared with a PID controller response, the fuzzy controller responded faster and more accurate than the PID one. The simulated results were tested using real time hardware from dSPACE control board and the real robot.

7. REFERENCES

Duffy J, (1996). Statics and Kinematics with Applications to Robotics. Cambridge University Press, New-York, USA

Kandel, A., Langholz, G. (1993), Fuzzy Control Systems, ISBN 0849344964, CRC Press

Lapusan, P., (2005), Diploma work, Design and realization of control system for a parallel robot with 3 DOF, Technical University of Cluj-Napoca, Romania

Lapusan, C., Maties, V., Balan, R., Hancu, O., Stan, S., Lates, R. (2008). Rapid control prototyping using Matlab and dSpace. Application for a planar parallel robot, ISBN9784244-25761, 2008 The 16th edition IEEE International Conference on Automation, Quality and Testing, Robotics AQTR 2008, Cluj-Napoca, Romania

Merlet, J-P.(2006). The parallel robots, Second edition, ISBN 1402041322, Springer

Tab. 1. Fuzzy logic rule base

 derivative error [DELTA]e

Error e n MN Z MP P

 N N N N Z MP
 MN N MN MN Z MP
 Z N MN Z MP P
 MP MN Z MP MP P
 P MN Z P P P