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Dynamics Simulation

Toolbox for IndustrialRobot Manipulators

METIN TOZ, SERDAR KUCUK

Department of Electronics and Computer Education, University of Kocaeli, Umuttepe Campus, Kocaeli 41380, Turkey

Received 18 March 2008; accepted 23 June 2008

ABSTRACT: A new robot toolbox for dynamics simulation based on MATLAB GraphicalUser Interface (GUI) is developed for educational purposes. It is built on the previous version

named as ROBOLAB performing only the kinematics analysis of robot manipulators. The

toolbox presented in this paper provides interactive real-time simulation and visualization of

the industrial robot manipulator’s dynamics based on LangrangeEuler and NewtonEulerformulations. Since most of the industrial robot manipulators are produced with six degrees of

freedom (DOF) such as the PUMA 560, the Fanuc ArcMate 120iB and Stanford Arm, the library

of the toolbox includes sixteen fundamental 6-DOF robot manipulators with Euler wrist. The

software can be used to interactively perform robotics analysis and off-line programming of

robot dynamics such as forward and inverse dynamics as well as to interactively teach andsimulate the basic principles of robot dynamics in a very realistic way. In order to demonstrate

the user-friendly features of the toolbox much better, simulation of the NS robot manipulator

(Stanford Arm) is provided as an example. 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18:

319330, 2010; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20262

Keywords: GUI; off-line programming; simulation; visualization; robot dynamics; industrial

robots

INTRODUCTION

The subject of manipulator dynamics has beenfocused on by the researchers for more then three

decades. In particular, dynamics equations are impor-

tant for real-time control applications in determining

appropriate forces/torques that will produce desired

trajectories (position, velocity and acceleration). In

the literature, several methods such as LangrangeE-

Euler [1], recursive Lagrange [2], recursive New-

tonEuler [3], generalized D’Alembert [4] and Kane

[5] have been proposed for deriving the manipulatordynamics equations. Featherstone [68] presentedvery in-depth work on robot dynamics and especially

the minimization of dynamic computational coast

which is one of the most important issues in

developing new formulations and algorithms for

control and simulation. He gave main algorithms

and associated equations with a common concise

notation. He also interested in new factorization

algorithms and new formulae that show how the costCorrespondence to S. Kucuk ([email protected]).

2009 Wiley Periodicals Inc.

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of calculating and factorizing the joint-space inertia

matrix vary with the topology of a kinematics tree.

These formulae can be used for comparing the cost of

calculating forward dynamics for a branched tree and

an unbranched tree of the same size [9]. It should be

noticed that although the efficiency and speed of the

dynamics computations are very important, the mainissue of this article is to improve the understanding of

the fundamentals of robot dynamics based on

LangrangeEuler and NewtonEuler algorithmsthrough an interactive real-time dynamic simulation

toolbox. Since LangrangeEuler and NewtonEuleramong the various formulations have been accepted to

be most suitable as computational algorithms, is

selected for computing the joint actuator torques/

forces in this toolbox [10].

Computer simulation is a powerful visualization,

and a very helpful tool in different fields of science

and technology. A researcher can examine, design,

and test a mechanical system using computersimulation, even it does not exist. Since deriving

and analyzing the robot dynamic equations without

using any computerized tool is a very tedious and

error-prone process, the robotics research community

has paid considerable attention to the computer

simulation in recent years. It plays a very important

role in dynamic modeling, possibly more significant

then any field in robotics. Supporting the computer

simulation with visualization tools and GUIs; a

researcher can simulate the operation of the robotics

systems in a realistic way near to nature.

A GUI is a kind of user interface which allows

users to interact with a computer. GUIs have been

developed against to the tiresome usage of command

line interfaces, which require commands to be typed

on the keyboard. The numerous commands available

in command line interface require grater efficiency

and take more time in order to be learned. A well-

designed GUI enables users to eliminate learning

complex command languages. Users also can take

advantages of the computer graphics capabilities to

make the program easier to use. Windows, sliders,

menus and other visual components, on the other

hand, allow users to move data from one application

to another in an easy manner.In this article, a new simulation toolbox based on

powerful MATLAB GUI is presented for interactive

real-time simulation and visualization of industrial

robot manipulator’s dynamics using LangrangeEu-Euler and NewtonEuler formulations. MATLAB isone of the most powerful environments for the

modeling, simulation, and visualization of robotics

systems. Matrix based problem solutions, dynamic

simulation with graphical interfaces and easy pro-

gramming are some powerful features of MATLAB.

The proposed toolbox successfully performs forward/

inverse dynamics and eigenvalues analyses. The

forward dynamics computes the resulting end-effector

acceleration, given the joint positions, velocities,

joint forces/torques and mass distribution of each

link. It is required for simulation and model-basedcontrol of robotics manipulators [11]. Inverse dynam-

ics computes joint forces/torques, given the joint

positions, velocities, accelerations and mass distribu-

tion of each link. It is important for the computed

torque control of robotic manipulators. Any control

system uses the inverse dynamics calculation to

linearize the plant. The eigenvalues of a manipulator

mass matrix are extremely important for design,

analysis and control of serial robot manipulators.

They provide information about the controller

parameters.

The proposed toolbox has four unique aspects for

educational purposes. First of all, it has a graphicaluser interface with very simple menus allowing

students to study the basic principles of robot

dynamics easily. Secondly, it provides an inter-

active animation of forward and inverse dynamics of

sixteen 6-DOF fundamental robot manipulators

with Euler wrist. Its library also includes the

dynamics models of well-known robot manipulators

like Puma, Stanford arm and Scara, which enable

students to learn the subjects in a realistic way.

Thirdly, it illustrates the joint forces/torques, accel-

erations, mass matrices, Coriolis/centrifugal force

vectors, gravity vectors, kinetic and potential energy,

eigenvalues either as tables, or as graphs. Students

can also use these outputs for numerical analysis.

Finally, it allows students to make dynamics analysis

using both NewtonEuler and Lagrange Eulerformulations.

Nearly one decade of our robotics teaching

experience has showed that the students are often

frustrated when they encounter complex theoretical

equations of robot dynamics. Having used the

dynamic simulation toolbox in robotic classes, we

have observed that the students well overcome this

learning problem smoothly and deal with much more

complicated robot dynamics problems themselves.We have also noticed following remarkable differ-

ences in student teaching process:

(I) The toolbox has provided students a wider

insight of the theoretical aspects in robot

dynamics and given straightforward individual

experience in LangrangeEuler and New-tonEuler methods that were just taught anddiscussed in the classroom.

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(II) The interactive environment of the toolbox has

increased the interaction between the students

and the instructor while reducing the learn-

ing time of the students as they establish the

relationship between the theoretical principles

and their physical robot applications readily.

(III) The toolbox has enhanced the students’motivation to robotics courses and encourages

them to adapt greater responsibilities for their

own learning.

PREVIOUS RESEARCH

In literature, several computer programs about robot

dynamics have been proposed with different capa-

bilities and properties. In general, these computer

programs have been developed for either symbolic

calculation or numeric calculation utilizing a simu-lation tool. The computer programs based on

symbolic calculation were mostly studied in early

2000. They use such tools as algebraic system with

concise notations, advanced algorithms minimizing

the computational costs and clever reorganization of

computations for generation of efficient manipulator

dynamics equations in symbolic form [6,12]. Some

popular computer programs based on symbolic

calculation can be given as follows.

Dynamic Models of Industrial Robot (DYMIR)

[13] was developed for automatic generation of

dynamic equations in a symbolic form using some

powerful features of a commercially available sym-

bolic manipulation Lisp-based system REDUCE [14].

Given the spatial geometry and structural parameters,

it computes the nx1 generalized torque vector for rigid

case or the corresponding quantities for elastic case.

The elastic case was developed for the revolute joints

only. It consists of n-times iteration of five steps for

computing the nxn inertia matrix ‘‘M(q),’’ nx1 vector

of centrifugal and Coriolis forces ‘‘C ðq; _qÞ’’ and nx1vector of gravity term ‘‘G(q).’’ This structure saves

considerable amount of memory and CPU time. The

open code tool DYMIR allows the symbolic calcu-

lation of dynamics models as many as twelve DOFusing Lagrangian formulation.

The computer program Algebraic Robot Modeler

(ARM) written in C and Lisp languages was

implemented to generate symbolically the forward

solution and complete Lagrangian dynamic robot

model for control engineering applications [15]. The

inertial, centrifugal, and Coriolis, and gravitational

parts of the Lagrangian dynamic robot model are

computed symbolically using a special Q matrix

formulation. The computational requirements are

enumerated as functions of the number of degrees-

of-freedom of the manipulator.

Another Lisp-based program Efficient Manipu-

lator Dynamic Equation Generator (EMDEG) auto-

matically generates forward and inverse dynamics

based on simplified LagrangeChristoffel formula-tions [16]. It uses some simplification rules during the

process of equation generation in order to reduce the

amount of real time computation required to compute

the complete set of configuration dependent dynamic

parameters (i.e., M (q), C ðq; _qÞ and G(q)). Thesesimplification rules are based on the structure of the

manipulator dynamics, common manipulator config-

urations, and algebraic system, used for deriving the

dynamic equations.

The algebraic dynamic robot modeling program

PIODYM-I employs an algorithm for both manual and

automatic derivation of dynamic models of robot

manipulators using Piogram symbolic representationmethod [17]. It was developed for using New-

tonEuler formalism. It is applicable to the manip-ulators of any degrees freedom.

Automatic Robot Dynamic Equation Generator

(ARDEG) based on the modified LagrangeChris-Christoffel formulation automatically generates the

equation of motion for open chain manipulators for

the purpose of robot system analysis and control [18].

It was developed in Turbo Basic on an IBM PC/AT.

and programmed with an internal algebraic represen-

tation which was designed to assist the implementa-

tion of the symbolic mathematical operations.

Yin and Yuh [18] compared ARDEG with

DYMR, ARM, EMDEG, and PIODYM-I in terms of

the computing time and compactness of the resulting

equations. The comparisons showed that ARDEG

requires much less computing time to symbolically

derive the equation of motion than the others.

The software package SYMORO þ (A Systemfor the Symbolic Modeling of Robots) [19] developed

under Mathematica and C languages was intended for

the automatic symbolic modeling of robots. It permits

generating the direct/inverse geometric, the direct/

inverse kinematics, the dynamic and the inertial

parameters identification models. The computationof the dynamic model of robots with a reduced

number of operations allows SYMOROþ to be usedin real time simulation and control. Since the codes of

the generated models can be directly used by Matlab,

Mathematica, C, and Maple, this software package is

more advanced than those are mentioned above.

All of the software packages explained in detail

above have been developed for automatic generation

of dynamic equations in a symbolic form only. They

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are not intended for educational purposes and do not

have a GUI.

In addition to above computer programs based on

symbolic calculation, following computer programs

based on numeric calculation have been studied in the

literature.

Planar Manipulator Toolbox implemented inMATLAB enables modeling and simulation of n-

DOF planar robot manipulator’s kinematics and

dynamics with revolute joints only [20]. The dynamic

equations are derived based on Lagrangian formula-

tion. The toolbox allows the users to calculate the

system vectors/matrices (such as vector of Coriolis

and Centrifugal forces, gravity forces, inertia and

Jacobian matrix), joint accelerations and external

forces acting on the end-effector. It is activated via

command line and does not provide an interactive

environment to the users. It has a very limited

educational functionality. The library of the toolbox

includes only planar manipulators.Robotics Toolbox for MATLAB provides a set of

functions that can be configured by the user to model

and analyze the manipulators [21]. It is capable of

performing many essential implementations required

for robotics such as kinematics, dynamics and

trajectory generation of serial-link robot manipula-

tors. It provides inverse dynamics calculations using

recursive NewtonEuler formulation. It also per-forms simulation and enables analyzing the exper-

imental results obtained from real robotic

applications. Although it contains very powerful tools

for student education, it uses command line interface

for robotic applications and its library includes a few

examples (i.e., PUMA and Stanford). It has poor

visualization and does not provide interactive GUI

facilities for the users. The library of Robotics

Toolbox includes only PUMA and Stanford industrial

robot manipulator as examples.

ROBOTECT [22] is a software package for

modeling, visualization and performance analysis of

serial link robot manipulators. It provides simple

commands for configuring three dimensional models

and dynamic force torque analysis. This analysis

incorporates the additional forces corresponding to

inertial and Coriolis effects in the motion usingNewtonEuler formulation. Although Robotect pro-vides a graphical user interface, it is originally not

developed as an educational tool and its library does

not include fundamental serial robot manipulators.

ROBOTİCA [23] is a Mathematica-based user

friendly computer-aided analysis and design package.

It requires over 30 functions for performing symbolic

and numeric calculation of kinematics and dynamics

equations for multi-degree-of-freedom robot manipu-

lators. The computation of the robot dynamics is

performed using LangrangeEuler formulation only.Although it is an educational tool, it has very limited

GUI facilities. It does not provide interactive environ-

ment to the students and not include any industrial

robot manipulators in its library.

SD/FAST [24] based on the advance Kane’sformulation performs dynamics analysis of the robot

manipulators as well as a wide range of mechanical

systems such as ground vehicles, gear trains, articu-

lated spacecraft, and high speed electromechanical

devices. It is not intended for educational purposes

and does not provide interactive environments to the

users and GUI facilities. The library of this toolbox

does not include fundamental industrial robot manip-

ulator.

Krenn and Schafer [25] developed a software tool

for dynamics modeling and simulation environment

for rapid manipulator design. The assembly and

simulation environment permit engineers to rapidlydesign a fully operating concept of a robotics system

in an easy manner. Its GUI is not satisfactory because

of limited number of facilities provided. Its main

purpose is design and analysis rather then uses it in

education. In addition to these, there are some other

simulation tools for robotics systems [2630] thathave not been mainly intended for examining

dynamics of the serial robot manipulators.

A new robot Dynamics Simulation Toolbox for

Industrial Robot Manipulators built on the previous

version ROBOLAB [26] has been developed for

merely educational purposes. The novelty and utility

of this toolbox in contrast to its counterparts

mentioned above can be stated as that it provides

interactive real-time simulation and visualization of

the industrial robot manipulator’s dynamics based on

both LangrangeEuler and NewtonEuler formula-tions. It has a robot library of the sixteen different 6-

DOF robot manipulators with Euler wrist including

well known robot manipulators such as Prismatic,

Scara, PUMA, and Stanford. Educational users can

change the related joint variables (revolute or

prismatic) using both sliders and edit boxes, and

interactively animate the three dimensional solid

models of robot manipulators through a user friendlyGUI written in MATLAB. After setting the new

variables in the edit boxes, the animations are

performed in a few milliseconds. It is important to

emphasize that the program routines are written in a

straightforward manner for educational purposes

rather than for obtaining maximum computational

efficiency. The toolbox encapsulates over 50 functions

for manipulating data types such as vectors and

homogeneous transformation, which are necessary for

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dynamics computations, and robot animations. These

functions provide a modular structure; hence, the

students can easily include new functions to exploit

additional features (extension of the toolbox to other

kinematics configurations besides the 16 listed,

simulations of manipulators with multiple branches

or kinematics loops and additional joint frictionmodels). However the dynamics simulation toolbox

in this paper is mainly intended for animation of the

only 16 kinematics configurations based on the two-

letter-code classification which includes all possible

‘‘useful’’ and ‘‘distinct’’ robots configurations [31].

Useful means that the linkage is able to produce gross

motion in all three spatial dimension and distinct

denote that each linkage is kinematically unique

among the different categories.

In order to make useful and meaningful compar-

isons, some major properties of the current software

packages about robot dynamics already mentioned in

this article are provided in Table 1.

ROBOT CONFIGURATIONS

Huang and Milenkovic [31] used a two-letter code to

classify 3 DOF robot configurations. The first letter

characterizes the first joint and the first joint’s

relationship to the second joint. The second letter

identifies the third joint and third joint’s association to

the second joint. The code letters and their meanings

are: S is slider, C is rotary parallel to slider, N is rotary

perpendicular to rotary and R is rotary perpendicular

to rotary or rotary parallel to rotary. The combination

of these rotary and prismatic joints compose the

sixteen robot configurations which are named as SS,

SC, SN, CS, CC, CR, NS, NN, NR, RC, RN, RR, RS,

SR, CN, and NC. In order to have 6-DOF robot

manipulators, each robot structure is equipped with

Euler wrist.

THEORETICAL BACKGROUND

A spatial transformation between two consecutivelinks of a serial robot manipulator can be described by

a set of parameters, namely ai1, ai1, yi and d i [32].

The definitions of these parameters are given as

follows:ai1 is the angle from the zi-1 axis to the zi axis

measured about xi1 axis, ai1 the distance between

from the zi1 axis to the zi axis measured along xi1axis, yi the angle from the xi1 axis to the xi axis

measured about zi axis, and d i the distance from the

xi1 axis to the xi axis measured along z i axis. T a b l e

1

S o m e M a j o r P r o p

e r t i e s o f t h e C u r r e n t S o f t w a r e P a c k a g e s

A b o u t R o b o t D y n a m i c s

P r o g r a m

C

a l c u l a t i o n t y p e

P r o g r a m m i n g

l a n g u a g e

M e t h o d s

V i s u a l i z i n g

P r o g r a m i n p u t

I n t e r a c t i v e

G U I

R o b o t s i n l i b r a r y

E d u c a t i o n a l

1

D Y M I R

S y

m b o l i c

L i s p

L a g r a n g i a n E u l e r

—

C o m m a n d l i n e

N o

N o

—

N o

2

A R M

S y

m b o l i c

C a n d L i s p


—


N o

N o

—

N o

3

E M D E G

S y

m b o l i c

L i s p

L a g r a n g e

C h r i s t o f f e l

—


N o

N o

—

N o

4

P I O D I M - I

S y

m b o l i c

A P C b a s e d

N e w t o n E

u l e r

—


N o

N o

—

N o

5

A R D E G

S y

m b o l i c

T u r b o b a s i c

L a g r a n g e

C h r i s t o f f e l

—


N o

N o

—

N o

6

S Y M O R O þ

S y

m b o l i c

M a t h e m a t i c a a n d C

N e w t o n E

u l e r

—


N o

N o

—

N o

7

P l a n a r m a n i p u l a t o r

t o o l b o x

N u

m e r i c

M A T L A B


W i r e


N o

N o

O n l y p l a n a r m a n i p u l a t o r s

N o

8

A r o b o t i c s t o o l b o x

f o r M A T L A B

N u

m e r i c

M A T L A B

N e w t o n E

u l e r

W i r e


N o

N o

P u m a a n d S t a n f o r d

Y e s

9

R O B O T E C T

N u

m e r i c

—

N e w t o n E

u l e r

S o l i d

M o u s e

Y e s

Y e s

—

Y e s

1 0

R O B O T I C A

N u

m e r i c

M a t h e m a t i c a

N e w t o n E

u l e r

W i r e / S o l i d

M o u s e

N o

L i m i t e d

—

Y e s

1 1

S D / F A S T

S y

m b o l i c / n u m e r i c

M a t l a b

K a n e ’ s f o r m u l a t i o n

S o l i d


N o

N o

—

N o

1 2

K r e n n a n d S c h a f e r ’ s t o o l

N u

m e r i c

M a t l a b S i m u l i n k


S o l i d

M o u s e

N o

L i m i t e d

—

N o

1 3

D y n a m i c s s i m u l a t i o n

t o o l b o x f o r i n d u s t r i a l

r o b o t m a n i p u l a t o r s

N u

m e r i c

M a t l a b

L a g r a n g i a n E u l e r /

N e w t o n E

u l e r

S o l i d

M o u s e

Y e s

Y e s

P u m a , s t a n f o r d , s c a r a

, p r i s m a t i c

a n d o t h e r 1 2 r o b o t s

Y e s


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Using these parameters, the general form of the

transformation matrix for adjacent coordinate frames,

i1 and i is obtained as

The transformation of the link n coordinate frame

into the base coordinate frame of the robot manipu-

lator is given by

i1n T ¼

i1i T

iiþ1T

iþ1iþ2T

n1n T ð2Þ

where i ¼ 1,2,. . .,n. The manipulator dynamic equa-tions can be derived systamatically using the two

major methods, namely LagrangeEuler and New-tonEuler dynamic formulations. The dynamicequations are particularly crucial for controlling and

simulating the robot manipulators.

LagrangeEuler Formulation

The difference between kinetic and potential energy

produces Lagrangian function given by

L ðy; _yÞ ¼ K ðy; _yÞ PðyÞ ð3Þ

where y and _y represent joint position and velocities,

respectively. While the potential energy (P) depends

on position only, the kinetic energy (K ) depends onboth position and velocity of the manipulator. The

total kinetic energy of the manipulator is defined as

K ðy; _yÞ ¼ 1

2

Xni¼1

ðviÞT

mivi þ ðoiÞT I ioi ð4Þ

where mi is the mass of link i and I i denotes the 3 3inertia tensor for the center of the link i with respect to

the base coordinate frame. I i can be expressed as

I i ¼ 0i RI m

0i R

T ð5Þ

where 0i R represents the rotation matrix and I m stands

for the inertia tensor of a rigid object about its centerof mass. The terms vi and oi refer to the linear and

angular velocities of the center of mass of link i with

respect to base coordinate frame, respectively.

vi ¼ AiðyÞ _y ð6Þ

and

oi ¼ BiðyÞ _y ð7Þ

where Ai(y) and Bi(y) are obtained from the Jacobian

matrix, J i(y). If the Equations (6) and (7) are

substituted in Equation (4), the total kinetic energy

is obtained as follows.

K ðy; _yÞ ¼ 1

2_yT

Xni¼1

½ AiðyÞT

mi AiðyÞ þ BiðyÞT I i BiðyÞ _y

ð8Þ

The Equation (8) can be written in terms of

manipulator mass matrix and joint velocities.

K ðy; _yÞ ¼

1

2 _qT

M ðyÞ_y ð9Þ

where M (y) denotes mass matrix given by

M ðyÞ ¼Xni¼1

AiðyÞT

mi AiðyÞ þ BiðyÞT I i BiðyÞ ð10Þ

The total potential energy is determined in order to

complete the Lagrangian function.

PðyÞ ¼ Xni¼1

migT hiðyÞ ð11Þ

where g and hi(q) denotes gravitational acce-

leration vector and the center of mass of link irelative to the base coordinate frame, respectively.

As a result, the Lagrangian function can be ob-

tained by combining the Equations (9) and (11) as

follows.

L ðy; _yÞ ¼ 1

2_yT M ðyÞ _y þ

Xni¼1

migT hiðyÞ ð12Þ

When the Lagrangian function is evaluated symboli-

cally, the dynamic model of a robot manipulator can

be written in the form of

Xn j¼1

M ijðyÞ€y j þXnk ¼1

Xn j¼1

C ikjðyÞ _yk _y j þ GiðyÞ

¼ ti 1 i; j; k n

ð13Þ

where, t is the nx1 generalized torque vector applied

at joints. y; _y and €y are the nx1 joint position, velocity

and acceleration vectors, respectively. M (y) is the nxn

mass matrix, C ðy; _yÞ is an nx1 vector of centrifugaland Coriolis terms given by

i1i T ¼

cos yi sin yi 0 ai1sin y

i cos a

i1 cos y

icosa

i1 sina

i1 sina

i1d

isin yi sin ai1 cos yisinai1 cosai1 cosai1d i0 0 0 1

2664

3775 ð1Þ

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C ikjðyÞ ¼ @

@ qk M ijðyÞ

1

2

@

@ qi M kjðyÞ ð14Þ

G(y) i s a n nx1 vector of gravity terms of actual

mechanism expressed as

GiðyÞ ¼ X3k ¼1

Xn j¼1

gk m j A jkiðyÞ ð15Þ

The friction term is omitted in Equation (13). The

detailed information about Lagrangian dynamic for-

mulation can be found in the text [33].

NewtonEuler Formulation

The NewtonEuler algorithm consists of two sets of iterations: the outward iterations that transform the

kinematics parameters from the base to the end-

effector, and the inward iterations that transform the

dynamics variables and compute the joint torques.The outward iterations can be described as follows:

The angular velocity (o) from one link to the next is

given by

iþ1oiþ1 ¼ iþ1i R

ioi þ _yiþ1iþ1 ^ Z iþ1 ði ¼ 0; 1; 2; . . . ; nÞ

ð16Þ

where iþ1 Riþ1 and the vector iþ1 ^ Z iþ1 denote the

rotation matrix relating coordinate frame i þ 1 toframe i and the rotation axis relating the coordinate

frame i þ 1 to frame i þ 1, respectively. If the joint isprismatic, this equation reduces to

iþ1oiþ1 ¼ iþ1i R

ioi ð17Þ

The angular acceleration ( _o) for a revolute joint is

iþ1_oiþ1 ¼

iþ1i R

i_oi þ €yiþ1

iþ1 ^ Z iþ1 þiþ1i R

ioi

_yiþ1iþ1 ^ Z iþ1

ð18Þ

If the joint is prismatic, this equation becomes as

iþ1_oiþ1 ¼

iþ1i R

i_oi ð19Þ

The linear acceleration ( _v) for a revolute joint is

denoted as

iþ1

_viþ1 ¼ iþ1i R

i

_oi i

Piþ1 þi

oi i

oi i

Piþ1

þi

_vi

ð20Þ

Consequently the equation becomes as follows for a

prismatic joint.

iþ1_viþ1 ¼

iþ1i R½

i_oi

iPiþ1 þioi ð

ioi iPiþ1Þ

þ i _vi þ 2iþ1oiþ1 _d iþ1

iþ1 ^ Z iþ1 þ €d iþ1iþ1 ^ Z iþ1

ð21Þ

where _d and €d denote the joint velocities and

accelerations for prismatic joints. The linear accel-

eration ( _vC) for the center of mass of a link is

iþ1_vCiþ1 ¼

i þ1_oiþ1

iþ1PCiþ1 þiþ1oiþ1 ð

iþ1oiþ1

iþ1PCiþ1 Þ þiþ1

_viþ1 ð22Þ

If the linear and angular accelerations of the mass

center of each link are obtained, the inertial force and

torque acting on the center of mass of each link can be

computed as follows

iþ1F iþ1 ¼ miþ1iþ1

_vCiþ1 ð23Þ

and

iþ1 N iþ1 ¼ C iþ1 I iþ1

iþ1_oiþ1 þ

iþ1oiþ1

ðCiþ1 I iþ1iþ1oiþ1Þ

ð24Þ

where C iþ1 I iþ1 is the inertia tensor. Having computed

the velocities and accelerations of each link, the forceand moment equations acting on each link can then be

found using the inward iterations. These equations are

evaluated link by link starting from the end-effector

and working inward the base of the robot as follows.

i f i ¼ iiþ1 R

iþ1 f iþ1 þiF i ð25Þ

and

ini ¼ i N i þ

iiþ1 R

iþ1niþ1 þiPC i

iF i þiPiþ1

ðiiþ1 Riþ1 f iþ1Þ

ð26Þ

Finally, the required joint torques are determined as

ti ¼ i nT ii ^ Z i ðfor prismatic jointsÞ ð27Þ

and

ti ¼ i f T i

i ^ Z i ðfor prismatic jointsÞ ð28Þ

The detailed information about NewtonEulerdynamic formulation can be found in the text [11].

Dynamics Equations in Cartesian Space

The dynamic equations for a robot manipulator can be

obtained applying LangrangeEuler or NewtonEu-

Euler formulations in joint space with a generalform as

t ¼ M ðyÞ€y þ C ðy; _yÞ þ GðyÞ ð29Þ

where M (y) is the nxn mass matrix, C ðy; _yÞ is an nx1vector of centrifugal and Coriolis force and G(y) is an

nx1 vector of gravity term. The actuators work in joint

space, whereas tasks to be performed by a robot

manipulator are in Cartesian space. The manipulator


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dynamics in Cartesian space is described as

F ¼ J ðyÞT M ðyÞ€y þ J ðyÞT C ðy; _yÞ þ J ðyÞT GðyÞ

ð30Þ

where F is the force-torque vector acting on the end-

effector of the manipulator and J (y)T inverse of the

Jacobian transpose in the tool frame as F . The

relationship between joint space and Cartesian space

can be defined as

t ¼ J ðyÞT F ð31Þ

The Cartesian acceleration (end-effector acceleration)

of the manipulator can be found as

€w ¼ _ J _y þ J €y ð32Þ

The Jacobian is defined in the same frame as

Cartesian force and Cartesian acceleration €w.

Eigenvalues

Eigenvalues are a special set of scalars associated with

a linear system of equations (i.e., a matrix equation)

that are sometimes also known as characteristic roots

or characteristic values [34]. Let M and x be a square

matrix and a vector, respectively.

Mx ¼ l x ð33Þ

where M 2C nxn, x2C n, and l is a scalar and called theeigenvalues of matrix M with corresponding eigen-

vector x. Equation (33) can be expressed in the form

ð M l I Þ x ¼ 0 ð34Þ

where I is the identity matrix of order n. The Equation

(34) presents a system of homogenous linear algebraic

equations that has a non-zero solution x, so the

solutions of Equation (34) are given by

detð M l I Þ ¼ 0 ð35Þ

This equation is known as the characteristic equation

of matrix M , and the left-hand side is known as the

characteristic polynomial. The eigenvalues of thematrix M are the roots of the polynomial given by

Equation (35). M denotes the mass matrix of the

manipulator.

EXAMPLE

The NS robot manipulator (Stanford arm) is consid-

ered as an example for illustrating the features of the

proposed toolbox. The overview of the toolbox is

illustrated in Figure 1 before explaining it in detail.

The symbols ‘‘M.M.,’’ ‘‘C. C.,’’ ‘‘J. A.,’’ ‘‘C. A.,’’ ‘‘C.F.,’’ and ‘‘Nm. Val.’’ in Figure 1 denote the Mass

Matrix, Coriolis and Centrifugal forces, Joint Accel-

eration, Cartesian Acceleration (End-effector accel-

eration), Cartesian Forces (End-effector forces), and

Numerical Values, respectively.

Once the toolbox is activated, user can select the

NS robot manipulator from the library that includes

sixteen 6-DOF robot manipulators with Euler wrist.

The dynamics applications of the NS robot manipu-

lator can be performed based on LagrangeEuler andNewtonEuler methods by selecting the ‘‘Lagrange-Euler’’ and ‘‘NewtonEuler’’ buttons, respectively.The way to perform NewtonEuler and Lagrange-Euler applications is similar to each other except ina few differences.

Figure 1 The overview of the toolbox.

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The toolbox allows students to handle the

following specifications by choosing ‘‘LagrangeEuler’’ button: (1) each element of the mass matrix,

centrifugal/Coriolis and gravity vectors, (2) torques

acting on each joint and Cartesian forces acting on the

end-effector, (3) joint and end-effector (Cartesian)

accelerations, (4) total kinetic and potential energiesof the manipulator, (5) eigenvalues of the mass matrix.

They can be monitored as either tables with numerical

values or graphs. In order to show some features

of toolbox, let revolute joints move with the angles,

for example, y1 ¼ 150o, y2 ¼ 45

o, y4 ¼ 85o,

y5 ¼ 25o and y6 ¼ 120, prismatic joint moves with

certain unit length value of d 3 ¼ 10. The force-torque vector acting on each joint appears on the

screen as ‘‘Tourqe’’ by simply clicking the ‘‘Lagran-

geEuler’’ button. This implementation is illustratedin Figure 2 where the symbol ‘‘t’’ denotes the joint

torques.

If the users want to see each torque graph indetail, they simply click on the zoom button depicted

in Figure 2. In addition, the instantaneous numerical

values of the joint torques can be monitored by

selecting ‘‘Num. Val.’’ button. It can be noticed that

since the time beetween initial and goal position is

devided in to 30 time intervals, all of the numerical

values for an application are achived with 30 slider

steps. For instance, the user can move the indicator bar

on the slider to compute and see the force/torque

vector of the 24th time step as in Figure 3.

Since the NS robot manipulator has 6-DOF, its

mass matrix is composed of 36 elements. Any element

of the mass matrix can be viewed by selecting the

corresponding row in ‘‘M. M.’’ pop-up menu on

Figure 2. Users can monitor the all mass matrix

elements in a table for a specific time. Figure 4

illustrates the mass matrix elements at the 16th time

interval. It can be easily noticed from Figure 4 that the

mass matrix is symmetric.

The eigenvalues of the mass matrix can be

displayed by using the ‘‘M. M.’’ pop-up menu

illustrated in Figure 5. If the graphs of the eigenvalues

are closely examined, it can be seen that there is not

any value equal or less then zero. This can also be

proofed by displaying each time step values of the eigenvalues using ‘‘Num. Val.’’ button. These

implementations verify that the mass matrix is

positive definite as well as symmetric. The symbols

eig1, eig2,. . .,eig6 illustrate the first, second,. . .,sixth

eigenvalue of the mass matrix.

The ‘‘update’’ button allows users to change the

current configuration of the robot manipulator and to

enter the data for the next example. Although some

features of the toolbox are presented in this section

with an example, others can be better noticed and

utilized by hands-on experiences.

CONCLUSION

In this paper, a new industrial robot manipulator

toolbox for dynamics simulation based on MATLAB

Figure 2 The force-torque vector acting on each joint in

LagrangeEuler method. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.

com.]


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GUI is developed to help users (e.g., instructors,students) to interactively learn the theoretical basic

concepts in robot dynamics much better in less time.

The graphical user interface with very simple menus

provides students to study the basic principles of robot

dynamics such as joint forces/torques, accelerations

both in Cartesian space & joint space, mass matrices,

Coriolis/centrifugal force vectors, gravity vectors,

kinetic and potential energy, eigenvalues both as

tables and graphs in an easy manner. The solid

dynamics models of some well-known robot manip-

ulators like Puma, Stanford arm and Scara available in

the toolbox library enable students to learn the

subjects in a realistic way.

A completely flexible educational real-time

simulation toolbox based on Matlab GUI including

wide range of robot implementations (such as building

any kind of robot configuration, changing the

DenavitHartenberg parameters, link masses andinertias, implementation of other dynamics formula-

tions, realization of the quaternion and exponential

algebras, accommodation of variable end-effector

loads, contacts, collisions etc.) is considered as the

further work.

Figure 3 The numerical values of the force/torque vector

for the 24th step in LagrangeEuler method. [Color figurecan be viewed in the online issue, which is available at

www.interscience.wiley.com.]

Figure 4 The numerical values of the mass matrix

elements at the 16th time interval. [Color figure can be

viewed in the online issue, which is available at www.

interscience.wiley.com.]

Figure 5 The eigenvalues of the mass matrix. [Color

figure can be viewed in the online issue, which is available

at www.interscience.wiley.com.]

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ACKNOWLEDGMENTS

The authors gratefully acknowledge the support

of The Scientific and Technological Research Council

of Turkey (TUBITAK).

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BIOGRAPHIES

Metin Toz received the degree in electronics

and computer education from the University

of Kocaeli, Turkey, in 2006 and the MSc

degree in electronics and computer education

from the University of Kocaeli, Turkey, in

2008. Currently, he is working as an operator

at the information technologies department

of University of Kocaeli, Turkey. His mas-

ter’s thesis was focused on interactive real-

time simulation and visualization of the industrial robot manipu-

lator’s dynamics based on LangrangeEuler and NewtonEulerformulations.

Serdar Kucuk received the degree in

electronics and computer education from

the University of Marmara, Istanbul, Turkey,

in 1995, the MSc degree in electronics and

computer education from the University of

Marmara, Istanbul, Turkey, in 1998, and the

PhD degree in electrical education from the

University of Kocaeli, Kocaeli, Turkey, in

2004. Currently, he is working as an Assistant

Professor at the University of Kocaeli, in Kocaeli, Turkey. His

research interest includes kinematics and dynamics analysis of

serial and parallel manipulators.

330 TOZ AND KUCUK

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