Modelling and simulation of robot system using Matlab and Simulink.

Damic, Vjekoslav ; Cohodar, Maida ; Kulenovic, Malik 等

1. INTRODUCTION

Robotic systems has already become standard and very useful devices in modern industrial manufacturing, which characterized processes as metal cutting, welding, part handling, etc. Robots are often integral parts of flexible manufacturing cells and thus, integral parts of the complete production processes. To improve dynamic performances of robots and industrial manufacturing processes, increasing attention has been paid to modelling and simulation of such systems (Siciliano et al., 2009; Damic & Cohodar, 2006; Wehage et al., 1992).

This paper presents model-based simulation of a robot system using Matlab and Simulink. Model of the anthropomorphic arm was developed and the simulation results are presented. This example was chosen because one of the most commonly used robot's configuration is anthropomorphic arm (e.g. PUMA, RV-M1, etc.). In (Damic & Cohodar, 2006) model of an anthropomorphic arm was developed by Bond Graphs using program BonSim(R), (Damic & Mongtomery, 2003). The cases in which all three links are rigid, and the last two links taken as flexible, are analysed.

2. MODELLING APPROACH

For the industrial purposes, the end-effector trajectory is usually defined in the operational space as a time-depended trajectory:

r = [[X Y Z].sup.T] = r(t). (1)

It is necessary to map this trajectory to the joint space by inverse kinematics:

[theta] = [F.sup.-1] (r), (2)

where [theta] is vector of the joint angles.

The joint velocities are calculated via Jacobian matrix as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

The Simulink supports the model-based simulation, which enables development of a complex system model as an aggregation of already defined elementary components stored in the library. To develop model of a robot system, the components from SimMechanics library, as well as the other common models, are used as basic building blocks. For example, Body model can be used to model rigid links; different kind of the joints (revolute, prismatic, spherical, universal), taken from the library Joints, provide models of the joint, etc.

3. NUMERICAL EXAMPLE

An anthropomorphic arm, shown in Fig. 1, is analysed in this paper.

[FIGURE 1 OMITTED]

The manipulator is composed of three links with revolute joints and is controlled by a controller that implements PD joint space control law (Damic & Cohodar, 2006; Siciliano et al., 2009):

[tau] = [K.sub.P] ([theta].sub.r]--[theta]) + [K.sub.D] ([??].sub.r] - [??]), (4)

where index r denotes the reference values of the joint variables defined by Eqs. (2) and (3), [tau] is vector of generalized forces and [K.sub.P] and [K.sub.D] are matrices of the proportional and derivative gains.

The Simulink model of the anthropomorphic arm is depicted in Fig.2. Rigid links are represented by Body models, denoted as Link1, Link2 and Link3. The inertia frame (coordinate frame [X.sub.0][Y.sub.0][Z.sub.0] in Fig.1) is presented using Absolute frame model. Three revolute joints, modelled by components Joint1, Joint2 and Joint3, are driven by the joint actuators.

[FIGURE 2 OMITTED]

The generalized forces are generated in the framework of Subsystem2 according to the control law in Eq.(4). The joint sensors collect the necessary information; e.g. the actual joint angles and velocities, but could be other values as well, e.g. the reaction forces, accelerations, etc. The initial values of the joint angles and velocities are defined in components IC.

The simulation results could be represented in different ways--e.g. using Scope, XY graph, Display, Floating Scope. In addition, they could be exported as arrays to Matlab workspace. Thus, after performing new simulation under different conditions, it is possible to export results into the same workspace to make comparison and further analyses of the system dynamics.

4. SIMULATION RESULTS

The geometric and material properties of the robot are taken from (Wehage,1982). First two links are made of aluminium with [[rho].subA1]=2750 kg/[m.sup.3] (mass density) and [E.sup.A1]=0,699e11 Pa (modulus elasticity), while the third one is of graphite/epoxy with [[rho].sub.gr]=1300 kg/[m.sup.3] and [E.sub.gr]=1,727e11 Pa. The lengths of the links are [L.sub.1]=0,3 m, [L.sub.2]=[L.sub.3]=0,5 m. Moment (product) of inertia and dimensions of cross-section area of the links are given in Table 1.

The end-effector follows the trajectory (from point A to point B in Fig.3) defined in the operational space (in the coordinate frame [X.sub.0][Y.sub.0][Z.sub.0], Fig. 1) by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

The values of proportional and derivative gains of Eq.(4) are taken as [K.sub.P] = 1,5e5, [K.sub.D] = 1,5e5 for the all axes. The simulations were undertaken for 1 s, with minimum time step of 0,001 s. The standard absolute and relative error tolerances of 1e-6 were used. The results are shown in Fig. 4a-f.

The accuracy of joint angles is order of 1e-4 rad, Fig.4c. The time histories of achieved end-effector trajectory's coordinates are shown in Fig. 4e. Their deviations with respect to the exact values are given in Fig.4f.

PD control provides a good of accuracy end-effector trajectory and is of order of 1e-4 m.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. CONCLUSION

This paper presents general and effective algorithm for modelling and simulation of an anthropomorphic arm using Simulink. Using the component model approach makes these processes relatively comfortable. It is easy to change the model in order to analyze dynamic system behaviour under different conditions. The simulation results achieved show a good accuracy and confirm that proposed approach could be used for solving such class of problems. In the future research work, authors plan to compare different approaches to the modelling and simulation of mechatronics systems, e.g. the approach presented in this paper and modelling and simulation using bond graphs.

6. REFERENCES

Cohodar, M. (2005). Modelling and Simulation of Robot with Flexible Links using Bond Graphs, PhD thesis, Faculty of Mechanical Engineering, University of Sarajevo

Damic, V. & Cohodar, M. (2006). Bond Graph Based Modelling and Simulation of Flexible Robotic Manipulators, 20th European Conference on Modelling and Simulation ECMS, Bonn

Damic, V. & Montgomery, V. (2003). Mechatronic by Bond Graphs, Springer--Verlag, ISBN 3-540-42375-3, Berlin Heidelberg

Siciliano, B.; Sciavicco, L.; Villani, L. & Oriolo, G. (2009). Robotics, Modelling, Planning and Control, Springer-Verlag, ISBN 978-1-84628-641-4, London

Wehage, R. A.; Shabana, A. A. & Hwang, Y.L. (1992). Projection methods in Flexible Multibody Dynamics. Part II: Dynamics and Recursive Projection Methods. Int. Journal for Numerical methods in Engineering, Vol.35, No.35, (1941-1966)

*** (2009) http://www.mathworks.com--Mathworks products: Matlab--The language for technical computing and Simulink--Simulation and model based design, Accesed on: July, 2009

Tab. 1. Inertia properties and dimensions of the links

 Moment (product) inertia
Link (kg[m.sup.2])

 [I.sub.xx] [I.sub.xx] [I.sub.xx]

1 2.488e-1 1.244e-1 2.488e-1

2 6.875e-3 1.389e-1 1.430e-1

3 3.385e-3 6.839e-2 7.042e-2

 Moment (product) inertia
Link (kg[m.sup.2]) Cross-
 section
 [I.sub.xy]=[I.sub.yz]=[I.sub.xz] (m)

1 0 r=0,15
 (circular)

2 0 0.1x0,05
 (rectangular)

3 0 0.1x0,05
 (rectangular)