SELECTING A MANIPULATOR FOR THE TASK OF SORTING …€¦ · The orientation and position and of the end-effector can be presented as transformation matrix: =[] (1) where: R is 3 х

http://www.iaeme.com/IJMET/index.asp 147 [email protected]

International Journal of Mechanical Engineering and Technology (IJMET)

Volume 9, Issue 6, June 2018, pp. 147–167, Article ID: IJMET_09_06_018

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=6

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

SELECTING A MANIPULATOR FOR THE TASK

OF SORTING OBJECTS IN THE CONTINUOUS

FLOW: SCARA MANIPULATOR STUDY

Ivan Krechetov

Office of Scientific Research and Development,

Moscow Polytechnic University, Moscow, Russia

Arkady Skvortsov



Ivan Poselsky



Vladislavs Korotkovs

RU.Robotics, Moscow, Russia

Pavel Lavrikov

RU.Robotics, Moscow, Russia

ABSTRACT

The objective of this work is to study the kinematic diagrams of robot

manipulators for determination of the optimal kinematic scheme for use as a waste

sorting complex to perform relocations with objects on a conveyor belt. The paper

presents a description and a complete analytical solution to inverse kinematic of 4-

DOF SCARA robot. The kinematics and dynamics of the SCARA robot have been

analyzed.

Key words: SCARA, Delta robot, Cartesian robot, joints, end-effector, kinematic

diagram, forward kinematics, Denavit-Hartenberg (DH), coordinate system, degrees of

freedom (DOF), inverse kinematics, robotics sorting unit, robotic waste sorting.

Cite this Article: Ivan Krechetov, Arkady Skvortsov, Ivan Poselsky, Vladislavs

Korotkovs and Pavel Lavrikov, Selecting a Manipulator for the Task of Sorting

Objects in the Continuous Flow: SCARA Manipulator Study, International Journal of

Mechanical Engineering and Technology 9(6), 2018, pp. 147–167.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=6

Ivan Krechetov, Arkady Skvortsov, Ivan Poselsky, Vladislavs Korotkovs and Pavel Lavrikov


1. INTRODUCTION

Nowadays, there are two basic types of complexes for sorting of solid waste at the market:

with the use of manual sorting and automated ones. One type of automated sorting is optical

sorting. Optical sorting allows to sort different types of materials (glass, plastic, paper, wood,

cardboard, metal and textile) in automatic mode.

Optical sorting complexes have high quality of material recognition and allow sorting

with a low error probability. The most famous companies dealing with optical sorting

complexes are TOMRA, Green Machine, MEYER, Envac, CPG Group and Paprec Group.

Optical sorting line has complicated design and high price (several millions of euro).

Robotic waste sorting complexes have smaller price. The company ZenRobotics from

Finland made a system for waste recycling for automatic sorting the demolition and

construction waste. System can sort up to 4000 objects per hour.

Usage of robots will allow to sort waste without crushing. Robot has an advantage over

traditional air nozzles, has higher payload and almost the same performance.

1.1. Introduce the Problem

Comparison shows that automated complexes more efficiently select useful types of waste (in

general, by 2-3 times), they do not require a lot of personnel to work (about 3-4 times less

than for lines with manual sorting). However, the cost of automated lines is much higher than

that of manual ones - approximately by 14-30 times. Frequent equipment failures are another

major drawback of automated lines, and since all of them are manufactured abroad, in

practice it is necessary to wait for half a year to replace a broken part or assembly. The above

mentioned features make it practically unprofitable to use such complexes in our country in

the context of commercial use.

As a summary, at the moment complexes with manual sorting are in demand in our

country. Automated complexes are sold individually, and usually with the financial

participation of the state, since the payback period is several times higher (up to 10 times)

than for complexes with manual sorting, despite all the advantages.

1.2. Explore Importance of the Problem

A feature of the optical sorting line is that the image of objects on the conveyor belt is formed

by synchronizing the exposure time and the conveyor speed. The system of technical vision

carries out vertical scanning perpendicular to the conveyor belt, and horizontal scanning is

provided by means of the displacement of the belt itself. Thus, the optical system forms a

static image containing information about the position and materials of objects in the flow. A

disadvantage is that the mutual orientation and position of objects moving on the conveyor

belt can be violated, both due to the high speed of movement, and under the influence of

compressed debris on a dense stream of objects. As a result of the combined effect of all these

factors, in order to maintain the efficiency of the automatic sorting line, it is required either to

reduce overall capacity adequately with the composition and configuration of the incoming

flow, or to use additional automation equipment such as continuous video surveillance,

selecting and tracking of objects on a conveyor belt, the use of high-speed robotic

manipulators, for example, Delta robots capable of performing up to 2 pick and place

operations per second for moving objects.

Selecting a Manipulator for the Task of Sorting Objects in the Continuous Flow:

SCARA Manipulator Study


Factors affecting the efficiency of automatic sorting:

Weight of objects;

The non-determinism of the weight of individual fractions causes situations in which, due to

insufficient pressure at the outlet of the pneumatic valve, the "shot" object does not reach

acceptance, and when the required pressure is exceeded, collisions and ricochets occur during

the interaction of the particle flow. The speed of the conveyor along with the errors in the

identification of the dimensions of the fractions can also lead to incorrect control of the

opening time (the time of the air jet impact on the object) of the pneumatic valve.

Density of placement of objects on the conveyor belt;

Density determines the degree of overlap between the objects. High density leads to errors in

the optical system when analyzing and classifying the composition of the flow, and also it

hampers the process of distribution by containers at the stage of exposure to compressed air

by pneumatic valves. In automatic sorting lines, various special separators are used to

maintain high productivity, performing preliminary separation of objects before feeding to the

conveyor.

The displacement of the center of mass of the object relative to the geometric center;

The air jet, as a rule, impacts the geometric center, which, in the case of the displacement of

the center of mass, leads to the occurrence of undesirable rotational movements. These

movements can disrupt the flight path and reduce range.

Orientation of objects on the conveyor belt;

Along with the dimensions and weight of the object, the orientation of the object is an

important factor for successful exposure to compressed air. Thus, for example, the impact on

elongated objects is more effective when oriented perpendicular to the conveyor belt. This is

determined by the position of the pneumatic valves, which are placed at the end of the

conveyor belt perpendicular to the direction of motion

Let us consider the robot SCARA [1] as a mechanism for waste sorting complex.

Figure 1 shows SCARA robot. It is designed as a series of four movable links. SCARA

can move and rotate objects in the planes that are parallel to the base link. SCARA has high

speed but it is less than that of the Delta robot.

Figure 1 SCARA robot

Figure 2 shows a kinematic diagram of the SCARA robot.



Figure 2 Kinematic diagram of the SCARA robot

For moving small objects with weight up to 3 kg the SCARA robot DRS40L (Delta

Group) is one of the optimal options [3].

1.3. Related work

Main criterion for choosing the most applicable robot is the velocity.

The Cartesian robot is not suitable for this task due to the robust design and low velocity.

The 6 DOF industrial robots doesn’t have such high speed as the Delta or SCARA robots.

Delta robot has low inertia and able to accelerate up to high speed.

The SCARA robot has smaller speed than Delta robot but it is more reliable and has

simple construction.

For making a choice in favor of one or another robot should be done complete analysis of

kinematic and dynamic parameters by solving equations:

The forward kinematics of position refers to use of the kinematic equations for calculation the

orientation and position of the end-effector from joint angles

The inverse kinematics of position refers to use of the kinematic equations for calculating

joints angles from orientation and position of the end-effector;

The forward kinematics of velocities and accelerations evaluates linear velocities and

accelerations for the end-effector from joint velocities and accelerations;

The inverse kinematics of velocities and accelerations evaluates joint velocities and

accelerations from linear velocities and accelerations of the end;

The forward dynamics calculates required forces and moments for moving along the

trajectory;

The inverse dynamics calculates the trajectory of the end-effector from forces and moments

acting on links.

Solving the forward kinematics for sorting objects is not necessary, because we know the

position and orientations of object on the conveyor belt by using machine vision system. First

of all is required to solve inverse kinematics.

2. METHODS




2.1. Solution of the forward Kinematics

The Denavit-Hartenberg (also called DH) method is used for solving the forward kinematics

[4]. DH method is computational effective and simple for understanding.

Let's analyze the principle of the algorithm [5]. The state of the robot is defined by the

values of joints (angles of links). The forward kinematics is used for calculating orientation

and position of the robot end-effector:: by the given vector of joints of the robot 𝑞 =(𝑞1, 𝑞2, … , 𝑞𝑁)

𝑇 is required to find the position and orientation of the end-effector 𝑠 = 𝑓(𝑞).

The orientation and position and of the end-effector can be presented as transformation

matrix:

= [

] (1)

where:

R is 3 х 3 rotation matrix defining the end-effector orientation relative to the OXYZ initial

coordinate system;

p is 3 х 1 vector defining the X, Y, Z end-effector position.

Let 𝐴𝑖(𝑖 = , 2, … ,𝑁) be homogeneous matrices defining the transform from the

coordinate system of the i-th link to the coordinate system of the (i-1)-th link. Then, the

matrix

𝑁 = 𝐴1𝐴2 𝐴𝑁 (2)

is the solution of forward kinematics. By introducing the matrix:

𝑖 = 𝐴1𝐴2 𝐴𝑖 (3)

the following recurrent relation is obtained:

𝑖 = 𝑖−1𝐴𝑖 , 𝑖 = , 2, … ,𝑁 (4)

According to DH method in order to align the (i-1)-th coordinate system

𝑂𝑖−1𝑋𝑖−1𝑌𝑖−1𝑍𝑖−1 with the i-th coordinate system 𝑂𝑖𝑋𝑖𝑌𝑖𝑍𝑖 it is necessary to perform the

operations with coordinate systems:

Rotation around the axis 𝑍𝑖−1 by 𝑞𝑖. Axes 𝑋𝑖−1 and 𝑋𝑖 are parallel;

Shift along the axis 𝑍𝑖−1 by 𝑖. Axes 𝑋𝑖−1 and 𝑋𝑖 are coincident;

Shift along the axis 𝑋𝑖−1 by 𝑎𝑖. Origins of 𝑂𝑖−1 and 𝑂𝑖 are coincident;

Rotation around the axis 𝑋𝑖−1 by 𝑖. Coordinate systems 𝑂𝑖𝑋𝑖𝑌𝑖𝑍𝑖 and 𝑂𝑖−1𝑋𝑖−1𝑌𝑖−1𝑍𝑖−1 are coincident.

In this way it is possible to find the position of the end-effector.

2.2. Solution of the Inverse Kinematics

Let us consider the inverse kinematics [6]. It is a method for determination joints of links

from position and orientation of the end-effector in the Cartesian coordinates. The values of

joints are required for the robot motion driver to move the end-effector to the desired point

with desired orientation.



Inverse kinematics can be solved using numerical or analytical approaches.

The analytical method of inverse kinematics calculation can be done by two approaches

[7]:

trigonometric approach;

inverse transformation method.

There is a program (IKFast [8,9]), which allow automatically compute the equations for

the analytical solution, however it generates lengthy mathematical equations, are limited by

the number of DOF and do not work with all kinematic diagram.

Most of all modern inverse kinematics solvers are numerical, and computational speed

depends on specified accuracy. Sometimes numerical solution is absent or can’t converge

within the permissible limits. One of these methods [10] is based on the least squares method

[11].

Let us consider kinematic chain with n elements. Each Cartesian coordinate is a function

of each joint. Let x1, x2, …, x6 are Cartesian coordinates, and j1, j2, …, j6 are angles in n joints.

Then:

{

1 = 1( 1, 2, … , )

2 = 1( 1, 2, … , )

= 1( 1, 2, … , )

(5)

In matrix form this system looks like X = F (J), where J is a vector of joints, and X is a

vector of coordinates (X, Y, Z, roll, pitch, yaw). If we take the partial derivatives of each

coordinate relative to each joint:

{

1 =

1

2

1 =

1

2

=

1

2

(6)

In the matrix form can be written as = ( ) . The solution is to minimize the

equation ‖ – ( ) ‖.

Determinant of the Jacobian varies with time in dynamical systems. The number of DOF

in the Cartesian space is equal to the number of rows of the Jacobian and the number of joints

is equal to the number of columns. Thus, robot with k DOF will have the Jacobian with 6 x k

dimension. If the angular positions of the joints are represented as θ and the velocity as ν, we

obtain equation:

= ( )−1 (7)

where

is a vector of joint velocities n x 1;

is a vector of translation 3 x 1 and angular 3 x 1 velocities.




Therefore, for determination values of joints at a particular time, it is required to evaluate

the inverse matrix of the Jacobian. A direct inversion is possible only when Jacobian is a

square matrix, that is (the number of DOF in the Cartesian space is equal to the number of

joints). It is valid for the robot with 6 DOF. In case of more than 6 DOF, it is necessary to use

the Moore–Penrose inverse (pseudoinverse) method to approximate the inverse

transformation of the Jacobian. There can be used singular value decomposition method [9].

2.3. Analysis of the SCARA robot

The SCARA robot design is shown in Figure. 3.

Figure 3 An example of the SCARA robot design

2.3.1. Forward kinematics of position

Using the rules of the DH homogeneous transformations, it is possible to define the matrices

of a homogeneous transformation and find the position of the end-effector. The kinematic

diagram of the SCARA manipulator and coordinate systems selected in such a manner are

shown in Figure 4.

Figure 4 Kinematic diagram of the SCARA robot

Assume the following system of designations:

1 1sinS q

1 1cosC q

12 1 2sin( )S q q



12 1 2cos( )C q q

1000

0100

0

0

1111

1111

11

SLCS

CLSC

AT

1000

0100

0

0

2222

2222

2

SLCS

CLSC

A

1000

0100

0

0

122111212

122111212

212

SLSLCS

CLCLSC

AAT

1000

100

0010

0001

3h

A

1000

100

0

0

122111212

122111212

3213h

SLSLCS

CLCLSC

AAAT

Define the position vector of the gripper p

:

h

SLSL

CLCL

P 12211

12211

3,0

Now the forward kinematics can be solved -

hZ

qqLqLY

qqLqLX

)sin(sin

)cos(cos

21211

21211

Thus, the formulas obtained make it possible to find the Cartesian coordinates of the

manipulator gripper by the known generalized coordinates.

2.3.2. Inverse kinematics

1000

0100

0

0

1111

1111

1

SLCS

CLSC

A




On the operation stage, the manipulator movement is defined by the start and end points in the

Cartesian coordinate system. For manipulator operation, it is necessary to determine

generalized coordinates T

Nqqqq ),...,,(**

2

*

1

* .

If designated as:

)(qfs

then the required angles *q will be defined by the relation

)( *1* sfq

Let us designate the coordinates of each point

dSSLSLY

dCCLCLX

12211

12211

Then

22

12121

2

12

2

2

2

1

2

1

2

22

12121

2

12

2

2

2

1

2

1

2

2

2

SdSSLLSLSLY

CdCCLLCLCLX

Therefore,

2

221

2

2

2

1 2 dCLLLL

Hence it is found:

21

2

2

2

1

2

2

21

2

2

2

1

2

2

2

2

LL

LLdq

LL

LLdq

arccos

cos

The value of 1q can be determined similarly:

)cos(

1221

11122

2

111

11122

2

111

qdCLL

SdSSSLSLSY

CdCCCLCLCX

dL

LLdq

qdL

LLdL

qdLL

LLdLL

1

2

1

2

2

2

1

1

1

2

2

2

1

22

1

1

21

2

2

2

1

2

21

2

2

2

2

)cos(

)cos(

)cos(

dL

LLdq

1

2

1

2

2

2

12

arccos)(

Therefore,

dL

LLdq

1

2

1

2

2

2

12

arccos



For vertical motion

zh

Thus, the formulas obtained make it possible to find the Cartesian coordinates of the

manipulator gripper if the generalized coordinates are known.

2.3.3. Forward kinematics of velocities

Now the velocity problem should be formulated similarly, as it plays an important role in the

construction of the manipulator motion control algorithms.

Forward kinematics is to determine the gripper speed vector

qqJs )(

by speeds in joints q .

The position of the manipulator gripper is characterized by six numbers: three coordinates

and three angles. Let is designate these parameters via s

)(qfs

Differentiating with respect to time, we obtain

qqJs )(

where s - 6 х 1 vector of the generalized speed of grasping:

TTT vs ),(

)(qJ - 6 х N Jacobian matrix for the transformation f

q

fqJ

)(

Let the rate of change of the generalized coordinates q be specified. It is necessary to find

the angular and linear speeds of grasping. The Jacobian matrix can be written as follows:

)...,,()( 21 NjjjqJ

The 6-dimensional vector kj will be determined by expression:

Nkk

k

k pz

zj

11

1

- for rotational joint

or

1

0

k

kz

j

- for telescopic joint,

Thus, the ratio for the speeds has the form:




qqJ )(

For a manipulator of the SCARA type,

23,113,.00

10 0

zPzPz

zzJ

1

0

0

0z

1

0

0

1z

1

0

0

2z

T

T

T

T

hSLCLP

hSLSLCLCLP

SLSLCLCLP

SLCLP

1221223,1

12211122113,0

12211122112,0

11111,0

0

0

0

100

0

100

122

122

122122

3,.00

12211

12211

1221112211

3,.00

CL

SL

hSLCL

kji

Pz

CLCL

SLSL

hSLSLCLCL

kji

Pz

Let us generate the Jacobian matrix:

100

0

0

011

000

000

12212211

12212211

CLCLCL

SLSLSLJ

h

q

q

CLCLCL

SLSLSL

qqJs

2

1

12212211

12212211

100

0

0

011

000

000

)(

It indicates that the angular speed vector is equal to

21

0

0

qq



Linear speed vector:

Z

Y

X

V

V

V

v

,

where

hV

qCLqCLCLV

qSLqSLSLV

Z

Y

X

2122112211

2122112211

)(

)(

The formulas obtained allow finding the speed of robotic working element motion if the

speeds of the generalized coordinates are known.

2.3.4. Inverse kinematics of velocities

Let us consider only linear speeds of grasping. In this case the Jacobian matrix is equal to:

100

0

0

12212211

12212211

CLCLCL

SLSLSL

J

It is not difficult to obtain the following ratios:

221det SLLJ

221

1221112211

122122

221

1

00

0

01

SLL

SLSLCLCL

SLCL

SLLJ

Z

Y

X

V

V

V

SLL

SLSLCLCL

SLCL

SLLh

q

q

sqJq

221

1221112211

122122

221

2

1

1

00

0

01

)(

Z

YX

YX

Vh

VSLL

SLSLV

SLL

CLCLq

VSL

SV

SL

Cq

221

12211

221

122112

21

12

21

121

The formulas obtained allow finding the speeds of generalized coordinates if the speed of

robotic working element motion is known. Forward kinematics of position has the following

form:

hZ

qqLqLY

qqLqLX

)sin(sin

)cos(cos

21211

21211

Inverse kinematics of position has the following form:




zh

LL

LLdq

dL

LLdq

21

2

2

2

1

2

2

1

2

1

2

2

2

1

2

2

arccos

arccos

Forward kinematics of velocities is written as follows:

hV

qCLqCLCLV

qSLqSLSLV

Z

Y

X

2122112211

2122112211

)(

)(

Inverse kinematics of velocities is written as follows:

Z

YX

YX

Vh

VSLL

SLSLV

SLL

CLCLq

VSL

SV

SL

Cq

221

12211

221

122112

21

12

21

121

2.3.5. Dynamics for the SCARA manipulator

Let us write down the equation of the manipulator motion:

)()(),()( qGqFqqqCqqMQ

where:

q is a vector of generalized coordinates;

q is a vector of generalized speeds;

q is a vector of generalized accelerations;

M moment of inertia matrix;

C are the Coriolis and centripetal forces (the centripetal force is proportional to 2

iq and

the Coriolis force is proportional to ji qq );

F is dry friction;

G is the force of gravity;

Q is the generalized external acting force (force for the translational link or torque for the

rotational link).

Equations can be found using some methods, including by the Lagrange method, the

Newton-Euler method, and so on. Here the Lagrange method is used as it is based only on the

kinetic and potential energy of the system.



According to Lagrange's equation of the second kind, any mechanical system can be

represented in the form:

i

ii

Qq

L

q

L

dt

d

where:

L is the Lagrangian operator;

Q - the generalized external acting force.

Let us spell out L:

)(),(),( qVqqTqqL

where:

),( qqT - is the kinetic energy

)(qV - is the potential energy

For a manipulator of the SCARA 3 DOF type, the equations of motion can be found in

this way (Figure 5):

Figure 5 Three-linked SCARA robot

weight of link 1 - 1m



length of link1 - 1L

length of link2 - 2L

length of link3 - d




distance from base to center of mass 1 - 1r

distance from base to center of mass 2 - 2r

distance from base to center of mass 3 - d

The center-of-gravity position:

111 cosqrX

111 sinqrY

01 Z

)cos(cos 212112 qqrqLX

)sin(sin 212112 qqrqLY

02 Z

)cos(cos 212113 qqLqLX

)sin(sin 212113 qqLqLY

dZ 3

Speeds:

1111 sin qqrX

1111 cos qqrY

01 Z

))sin(())sin(sin( 21222121112 qqrqqqrqLqX

))cos(())cos(cos( 21222121112 qqrqqqrqLqY

02 Z

))sin(())sin(sin( 21222121113 qqLqqqLqLqX

))cos(())cos(cos( 21222121113 qqLqqqLqLqY

dZ 3

Kinetic energy:

332211),( RTRTRTqqT

where:

1T is translational energy of link 1;



1R - rotational energy of link 1;

2T - translational energy of link 2;

2R - rotational energy of link 2;

3T - translational energy of link 3;

3R - rotational energy of link 3.

Let us express the above values as follows:

)(2

1 2

1

2

111 YXmT )cossin(2

1 2

11

22

1

2

11

22

11 qqrqqrm 2

1

2

112

1qrm

2

1112

1qIR Z 2

1

2

11 )12

(2

1q

Lm

2

2122 )(2

1qqIR Z

21

2

222

2

2

222

1

2

22 )12

(2

1)

12(

2

1)

12(

2

1qq

Lmq

Lmq

Lm

Assuming the following system of designations:

1S = 1sinq

1C = 1cosq

12S = )sin( 21 qq

12C = )cos( 21 qq ,

the following is obtained:

))((2 1221221121

2

122

2

2

2

12211

2

1

2

2 SrSrSLqqSrqSrSLqX

2

12

2

21212121

2

12

2

2

2

212121

2

12

2

2

2

1

2

1

2

1 22 SrSSrLqqSrqSSrLSrSLq

))((2 1221221121

2

122

2

2

2

12211

2

1

2

2 CrCrCLqqCrqCrCLqY

2

12

2

21212121

2

12

2

2

2

212121

2

12

2

2

2

1

2

1

2

1 22 CrCCrLqqCrqCCrLCrCLq

2

21211212121

2

2

2

212112121

2

2

2

1

2

1

2

2

2

2 )(2)(2 rCCSSrLqqrqCCSSrLrLqYX

2

222121

2

2

2

2221

2

2

2

1

2

1 2)2 rCrLqqrqCrLrLq

)(2

1 2

2

2

222 YXmT

2

2221212

2

2

2

22221

2

2

2

1

2

122

1)2

2

1rCrLqqmqrmCrLrLqm

))((2 1221221121

2

122

2

2

2

12211

2

1

2

3 SLSLSLqqSLqSLSLqX

2

12

2

21212121

2

12

2

2

2

212121

2

12

2

2

2

1

2

1

2

1 22 SLSSLLqqSLqSSLLSLSLq




22

3 dZ

22

21211212121

2

2

2

212112121

2

2

2

1

2

1

2

3

2

3

2

3

)(2

)(2

dLCCSSLLqq

LqCCSSLLLLqZYX

2

222121

2

2

2

2221

2

2

2

1

2

1 2)2 LCLLqqLqCLLLLq

)(2

1 2

3

2

3

2

333 ZYXmT

2

3

2

2221213

2

2

2

23221

2

2

2

1

2

132

1

2

1)2

2

1dmLCLLqqmqLmCLLLLqm

Link 3 is translational and independent of the first two links. There is no centrifugal force

here and therefore no need to take into account rotational kinetic energy.

The sum of kinetic energies

2

3212213

2

12213212212

2

12212

2

222

23

2

2221

2

222

23

2

22

2

2

2

22

2

112

23

2

13

2

12

2

22

2

11

2

1

2222

)12

(2)12

(

)12

()12

(2

1

dmqqCLLmqCLLmqqCrLmqCrLm

LmLmrmqq

LmLmrmq

LmLmLmLmLmrmrmqT

By substituting 2

11

Lr

and 2

22

Lr

, we shall obtain:

2

3212213

2

12213212212

2

12212

2

222

23

2

2221

2

222

23

2

222

2

2

22

2

112

23

2

13

2

12

2

22

2

112

1

22

)12

(4

2)12

(4

)12

()12

(442

1

dmqqCLLmqCLLmqqCLLmqCLLm

LmLm

Lmqq

LmLm

Lmq

LmLmLmLmLm

LmLmq

2

3322122132

2

1221

322

221322

2

2

2322

23212

1

2

1

)2(2

)3

(2)3

()3

()3

(2

1

dmmmqqCLLmmqCLL

mm

Lqqmm

Lqmm

Lmmm

Lq

Lagrangian L:

)(),(),( qVqqTqqL

For horizontally moving links 1 and 2, the potential energy is 0. For translational link 3

the potential energy makes:

dgmqV 3)(

Thus, Lagrangian L is obtained:

dgmdm

mmqqCLLmmqCLLmm

Lqq

mm

Lqmm

Lmmm

LqqqL

3

2

3

322122132

2

1221322

221

322

2

2

2322

23212

1

2

1

)2(2)3

(2

)3

()3

()3

(2

1),(

Let us designate:



)3

()3

( 322

23212

1 mm

Lmmm

L

as 1A

)3

( 322

2 mm

L as 2A

)2( 3221 mmLL as K2

Then:

dgm

dmKqqqKqqAqqAqAqqqL

3

2

3212

2

122212

2

21

2

1 )2()cos()2()cos()2(2

1),(

For any mechanical system:

i

ii

Qq

L

q

L

dt

d

for i=1:

01

q

L

22122211

1

)cos(2)cos(22222

1qqKqqKqAqA

q

L

KqqKqqAqAq 22122211 )cos()2()cos(

2

2222212122211

1

)sin()cos()sin()cos(2 qqqqKqqqqqKAqAqq

L

dt

d

2

22212222211 )sin()sin(2)cos()cos(2 qqKqqqKqKAqqKAq

for i=2:

212

2

12

212

2

12

2

)sin()sin(

)sin(2)sin(22

1

qqqKqqK

qqqKqqKq

L

122122

2

)cos(2222

1qqKAqAq

q

L

122122 )cos( qqKAqAq

212122122

2

)sin()cos( qqqKqqKAqAqq

L

dt

d

21222221 )sin()cos( qqqKAqqKAq

for i=3:




gmd

L3

dmd

L 3

dmd

L

dt

d 3

Let us write down the dynamic equation of the manipulator:

3

222

2221

00

0)cos(

0)cos()cos(2

m

AqKA

qKAqKA

d

q

q

2

1

+

+

3

2

1

3

2

1

22

2222

0

0

000

00

02

gmd

q

q

qqK

qqKqqK

)sin(

)sin()sin(

where:

1A =

)3

()3

( 322

23212

1 mm

Lmmm

L

2A =)

3( 3

22

2 mm

L

1 2 2 32 ( 2 )K L L m m

Let us write it down in another form:

33

2221

1211

00

0

0

M

MM

MM

d

q

q

2

1

+

3

2

1

3

2

1

21

1211

0

0

000

00

0

Hd

q

q

N

NN

where:

)cos()2()3

()3

( 22132

2

2322

1321

11 qLLmmLmm

Lmmm

M

)cos()2

()3

( 221322

232

2112 qLLmm

Lmm

MM

2

232

22 )3

( Lmm

M

333 mM

)sin()2( 22213211 qqLLmmN



)sin()2

( 222132

12 qqLLmm

N

)sin()2

( 222132

21 qqLLmm

N

gmH 33

It is apparent that the mathematical model of the SCARA robot is much easier to

implement than the mathematical model of the Delta robot.

3. RESULTS

Thus, a large working area is the main advantage of the SCARA robot. When developing the

kinematic diagram of the manipulator as part of the robotized sorting unit, it should be taken

into account that the size of the working area, sufficient for manipulating objects, is

determined not only by the width of the conveyor belt, but also by the geometric arrangement

of the receiving containers.

4. DISCUSSION

The use of scalable automated sections based on the developed prototype of a robotic sorting

unit will increase the number of useful fractions by 40-60% at the manual sorting stage as

compared to human labor with a decrease in operating expenses associated with a reduction in

the wage fund and increased insurance premiums for compulsory social insurance for

workers. The increased profitability of solid waste sorting lines may become a positive

economic effect. At the country level, the implementation of the developed sorting unit will

increase the share of waste that is involved in recycling and simultaneously solve many

problems in the field of ecology and development of the MSW sorting and processing

industry.

5. CONCLUSIONS

The obtained results will allow us to use of backlog in solving similar scientific problems

when sorting and moving various objects, including in adjacent areas, for example, sorting

vegetables and fruits, classifying objects according to their physical properties. The

expressions for the inverse kinematic solutions for SCARA manipulator tasks will allow

developing own manipulator in accordance with the requirements of the work area size.

ACKNOWLEDGMENTS

This research was financially supported by the Ministry of Education and Science of the

Russian Federation under the Grant agreement #14.586.21.0029 as of “28” July 2016.(Unique

identifier of the agreement: RFMEFI58616X0029 ); the grant is provided to perform the

applied research on the topic: “Research and development of scientific and technical solutions

in the field of conducting sorting operations in real time, with objects that have complex

characteristics, using highly efficient robotic automation equipment”.

REFERENCES

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of a SCARA robot. Simulation Modelling Practice and Theory, 2005, 13(3), 257-271.




[2] Gonzales, C. What’s the Difference between Industrial Robots? 2016. Available

at:http://www.newequipment.com/industry-trends/what-s-difference-between-industrial-

robots.

[3] Delta SCARA Robot. Automation for a Changing World. Available at:

http://robotforum.ru/assets/files/Delta/drs40l.pdf

[4] Hartenberg R.S., Denavit J. A kinematic notation for lower pair mechanisms based on

matrices. Journal of Applied Mechanics, 1955, 77(2), 215-221.

[5] Tarabarin V.B. Industrial robots and manipulators. 2003. Available at: http://tmm-

umk.bmstu.ru/lectures/lect_20.htm

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Moscow: NE Bauman Moscow State Technical University Press, 2004.

[8] Creating a custom IKFast Plugin. 2013. Available at:

http://docs.ros.org/hydro/api/moveit_ikfast/html/doc/ikfast_tutorial.html

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http://openrave.org/docs/0.8.2/openravepy/ikfast/

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on the Atlas Humanoid Robot. Procedia Computer Science, 2015, 46, 1441-1448. DOI:

10.1016/j.procs.2015.02.063

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Applied Mathematics, 1995.

Documents

SELECTING A MANIPULATOR FOR THE TASK OF SORTING …€¦ · The orientation and position and of the end-effector can be presented as transformation matrix: =[] (1) where: R is 3 х