analysis of dynamics of scora-er14 robot in...

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163 Volume 1 Issue 4 (May 2014) http://ijirae.com

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Analysis of Dynamics of SCORA-ER14 Robot in MATLAB

Ashok Kumar Jha Ajoy Kumar Dutta Jyotirmoy Saha NIT Jamshedpur Jadavpur University Jadavpur University

[email protected] [email protected] [email protected] Abstract— SCORA-ER14 robot is a SCARA based robot used in industries for assembly and many other purposes. In this paper the dynamics of this robot is discussed. The Euler-Lagrange method of dynamics analysis is followed here to deduce the equations relating the joint torques to the joint positions, velocities and acceleration. The simulations are done in MATLAB on the basis of these equations to show the variation of joint torques, while the end-effector of the robot is following a certain trajectory. This work has a significant interest for the users, as it provides them valuable information about the characteristics and behaviour of this robot while designing a specific application for it.

Keywords— SCARA Robot, SCORA-ER14, Dynamics, MATLAB, Euler-Lagrange.

I. INTRODUCTION At first a brief description about the mechanical configuration of the SCORA-ER14 robot is given before entering into

the details of dynamic analysis. There are several methods of analysing the dynamics of a robot manipulator, like Newton-Euler formulation, Lagrange –Euler formulation etc.[1][2] But here the Euler-Lagrange formulation is chosen to analyse the dynamics of SCORA-ER 14 robot because this formulation is consistent with the algorithmic steps essential for implementing it through computer simulation[3]. The discussion begins with the derivation of the kinetic energy equations and then the potential energy equations for this robot. Later the Lagrangian is formed from the kinetic and potential energy expressions. Thus we finally get the torque equations for different joints of SCORA-ER14 robot. Next, the results of the MATLAB simulation on the basis of these torque equations are shown for a deeper insight and the trajectory of the joints is shown in graphical form followed by the representation of the variation of toque at different joint throughout the followed trajectory. The significance of these results is discussed in the concluding section which highlights the key features of this manipulator.

II. BRIEF ABOUT SCORA-ER14 ROBOT

The SCORA-ER 14 robot is a SCARA type manipulator. It has four degrees of freedom. The first two joints of it are revolute joints for the movement of the robot arm in a horizontal x-y plane [4]. The third joint is a prismatic joint for the movement of the arm in vertical z direction. The fourth joint is again a revolute joint which is basically for changing the orientation of robot’s end-effector [5][6]. Often it is a key issue in grasping an object that the robot needs orient its gripper, attached with the end-effector, according to the orientation of the object. So, the fourth joint handles this purpose. The schematic of SCORA-ER14 Robot with proper dimensions is shown in Fig.1.

Fig.1 Schematic of SCORA-ER14 robot.

III. EULER-LAGRANGE FORMULATION There are two ways to formulate the dynamics of a manipulator. The first one is the force balance approach where the

forces and moments are calculated for the constraints acting between the links. This is the famous Newton-Euler


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approach. Another approach is the energy based approach which is called the Lagrangian dynamics formulation. Both the approaches give us obviously the same equations of motion. The Lagrange’s approach is described here briefly for serial link manipulators with rigid links. The expression for the kinetic energy of i-th link of the manipulator is given by ki as:

The total kinetic energy of the manipulator can be written as the sum of all the kinetic energies of the links of the

manipulator. That is,

In general the kinetic energy of a manipulator can be given by the following formula,

The potential energy ui for the i-th link of the manipulator is expressed in the form,

Where, g0= 3x1 matrix representing the gravity vector. Pci= Vector locating the centre of mass of the i th link urefi= This is a constant chosen such as the minimum value of ui is zero. The total potential energy can be expressed as the sum of all the potential energy of the links of the manipulator, as:

The Lagrangian dynamics formulation gives the means to express the equations of motion from a scalar function

called Lagrangian. This Lagrangian is basically the difference between the kinetic energy and the potential energy of the manipulator,

Finally the equation of motion of the manipulator can be given as,

Here τ is the nx1 vector of actuation torques.

IV. FORMULATION OF KINETIC ENERGY The formulation of the kinetic energy of the manipulator consists of the translational part and rotational part. In our

case the expression composed of both of these parts is given by the following equation.



Where the elements of D(q) is given as,

So,

V. FORMULATION OF THE POTENTIAL ENERGY The potential energy of the manipulator is the sum of all the potential energies of all the links. For each link, the

potential energy is just the mass multiplied by the gravitational acceleration and the height of its center of mass, Thus,

Here, d1, d2, d4 are constants, and d3 = q3

VI. DEVELOPMENT OF THE FINAL EXPRESSIONS So, the Lagrangian can be written as,

Where,

The partial derivative of the Lagrangian with respect to the kth joint velocity ( ) is given by,

For, partial derivative, with respect to , we have to consider as the variable and all other qi as constant.



And,

Similarly, the partial derivative of the Lagrangian with respect to kth joint position (qk) is given by,

Thus for each k= 1,2,3,4 the Euler-Lagrange equation can be written as,

Where,

= The generalized force associated with qk.

Where, K=1,2,3,4

, these are known as Christottel symbols (of first kind)

And

Here,

Finally we can write the equations for joint torques as,

VII. MATLAB SIMULATION RESULTS Using the above equations the simulation of dynamics analysis is done in MATLAB platform. The first three joints are

given definite trajectory to follow for 10s, mentioning the position, velocity and acceleration of each joint at every instant of time, which is shown in Fig.2. In Fig.3 the torques produced at joint1 and joint2 are plotted.



Fig.2 Change in position, velocity and acceleration of different joints against time

Fig.3 Torque produced at joint1 (T1) and joint2 (T2).

VIII. DISCUSSION OF RESULTS

For analysing the dynamics of the robot first we have chosen three different functions to vary joint variables θ1, θ2 and d3 and differentiating those functions we get the corresponding velocity and acceleration of the respective joints. Then these values of position, velocity and acceleration are put in the equation of the dynamics to find out the torques produced at joint1 and joint2. The functions for varying the joint variables are chosen arbitrarily.

θ1=3*(1-cos(π*x/10); θ2=3*(1-sin(π*x/10);

d3=0.0019*x



The torque values we have got are dependent on the mass and moment of inertia of every link. The values of these parameters are taken as follows (indexes 1,2,3,4 are for link1, link2, link3, link4 respectively):

m1=23.687; m2=17.15;

m3=0.4179; m4=2.0125;

I1=0.099379; I2=0.050526; I3=0.000405; I4=0.000702;

IX. CONCLUSION The dynamics analysis clearly indicates that the amount of torque need in the base joint, i.e. the first revolute joint is

much higher than the torque required at the second revolute joint, which is sensible as the coupling load is much less in the successive joints rather than the joints which are preceding. Also we notice the increment of the torque value with the increment of the velocity, which can easily be predicted from the structure of the dynamics equations.

REFERENCES

[1] “SCORA-ER 14Pro User manual”, Catalog # 200035 Rev.B., Intelitek. [2] “Kinematic Modeling and Simulation of a SCARA Robot by Using Solid Dynamics and Verification by

MATLAB/Simulink,” Mahdi Salman Alshamasin, Florin Ionescu, Riad Taha Al-Kasasbeh, European journal of Scientific Research,, ISSN 1450-216X, Vol.37 No.3, pp.388-405.

[3] Faglia R., Legnani G. Harmonic Drive transmissions: the effects of their elasticity, clearance and irregularity on the dynamic behavior of an actual SCARA robot. Robotica, vol.10, pp.369-375 (1992).

[4] Corke P. I., Matlab Robotics Toolbox (release 5), CSIRO, Australia, 1999. [5] Fu K. S., Gonzalez R. C., Lee C. S. G., Robotics, conrrol, sensing, vision, and inteligence, McGraw-

Hill Book Company, 1987. [6] Kurek J. E., Calculation ofRohot Manipulator Model Using Neural Ner, European Control Conference ECC '99, Karlsruhe, Germany 1999.

analysis of dynamics of scora-er14 robot in...

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