Kinematics Modeling of a 4-DOF Robotic Arm

Amin A. Mohammed and M. SunarMechanical Engineering Department

King Fahd University of Petroleum & MineralsDhahran, Saudi Arabia

e-mail: mehmets@kfupm.edu.sa

Abstract—This work presents the kinematics model of an RA-02 (a 4 DOF) robotic arm. The direct kinematic problem isaddressed using both the Denavit-Hartenberg (DH) conventionand the product of exponential formula, which is based on thescrew theory. By comparing the results of both approaches, itturns out that they provide identical solutions for themanipulator kinematics. Furthermore, an algebraic solution ofthe inverse kinematics problem based on trigonometricformulas is also provided. Finally, simulation results for thekinematics model using the Matlab program based on the DHconvention are presented. Since the two approaches areidentical, the product of exponential formula is supposed toproduce same simulation results on the robotic arm studied.

Keywords-Robotics; DH convention; product of exponentials;kinematics; simulations

I. INTRODUCTION

Nowadays, robotics is a rich area of research, in terms oftheir kinematics, dynamics and control. Kinematics, inparticular, plays a significant role in robotics and especiallyfor the study of industrial manipulators' behavior. Therefore,a decisive step in any robotics system is the analysis andmodeling of the manipulator kinematics [1]. It can be dividedgenerally into forward and inverse kinematics. The formerrefers to direct finding of the end effector (EE) position andposture for the given joint coordinates, on the contrary, thelatter is the determination of joint variables in terms of theEE position and posture [2]. In other words, it is the processof obtaining a configuration space that corresponds to agiven work space. It has several disciplines of applicationsincluding the computer graphics [3].

While the forward kinematics is straightforward, theinverse kinematics is not an easy task and it usually hasmultiple solutions. Numerous techniques and theircombinations could be used for solving and analyzing theinverse kinematics problem. However each method beside itsadvantages possesses some handicaps. Therefore, it isadvantageous to apply more than one method and tointegrate them with additional procedures [3]. Severalapproaches have been investigated and compared byAristides et al [4]. Jasjit Kaur et al [5] have analyzed andsimulated a robotic arm having three links-manipulator,where the inverse kinematics results in two solutions, out ofwhich the best one has been chosen with the usage of thegenetic algorithm. They have concluded that the robotic armmovement optimization can be achieved by the genetic

algorithms in a practical and effective way. A geometricapproach to solve the unspecified joint angles needed for theautonomous positioning of a robot arm is presented in [1].Based on few assumptions concerning the workingenvironment of the manipulator and the kind of manipulationrequired, in addition to the basic trigonometry, an easiersolution has been proposed. The introduced methodology hasbeen tested using five degrees of freedom robotic armcompounded to an i-Robot. By carrying out the mechanicaldesign, all components have been assembled together inaddition to two infrared sensors to detect the object and tofind its position. Finally, the electrical design was done, andthe proposed method was tested and justified. However, thisapproach lacks its generality due to the abundantassumptions considered, and what is more is that the last linkis restricted to just two orientations. In what we introducehere is that the last link could have any arbitrary orientation,which makes the analysis more general to differentconfigurations.

Ge and Jin [6] proposed an algorithm of a redundantrobot kinematics based on product of exponentials (PE),where the forward and inverse kinematics problems wereaddressed. The inverse kinematics problem was decomposedinto several sub problems and by studying them separately, asolution of the inverse kinematic problem was proposed.Kinematics and interactive simulation system modeling forrobot manipulators is given in [7], where the forward andinverse kinematics of a robotic arm called Katana450 arestudied by the exponential product method, then aninteractive simulation system is built based on three-meritvirtual scene modeling techniques proposed for robotmanipulators. These techniques are OpenGL, VRML andanother one based on Visual C++, Lab View and Matlabplatform. A robot simulator, which is introduced by Lodes [8]and is based on Matlab, enables the modeling of any roboticarm provided that the corresponding DH parameters areknown. Links visualization in terms of shape and size wasmade after some considerations. The program was shown toadequately model several example robots.

This paper focuses on modeling and simulation of theRA-02 robotic arm, a small special robot with 4 revolutejoints [9]. An excellent survey of robotics kits for post-secondary education is given in [10]. The rest of this paper isorganized as follows: Section II below presents thekinematics analysis based on the DH convention andexponentials formulation. The inverse kinematics solution

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2015 International Conference on Control, Automation and Robotics

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based on the algebraic method is given in section III. Insection IV, the manipulator simulation procedure and resultsare presented. Section V concludes this work.

II. MANIPULATOR KINEMATICS

The DH convention and product of exponentials oftenused in kinematics analysis of robot manipulators and othermechanical structures. The below section utilizes bothmethods to drive the kinematics model of the robot. Tosimplify the analysis, the effect of the gripper is ignoredmaking the robotic arm a four degree of freedom, a four-jointspatial manipulator. The treatment of this manipulator isanalogue to that of PUMA 560 after removing the wrist andadding one more revolute joint.

A. DH Convention.Denavit-Hartenberg (DH) convention is commonly used

in the kinematics analysis of the robotic manipulator [2]. It isbased on attaching a coordinate frame at each joint andspecifying four parameters known as DH parameters foreach link, and utilizing these parameters to construct a DHtable. Finally, a transformation matrix between differentcoordinate frames is obtained. The major objective is tocontrol both the position and orientation of the EE or thegripper in its work space. We will first derive therelationship between the joint variables and the position, andorientation of the gripper, using the DH method [2]. Therobotic arm under study is shown below in Fig. 1 (RA-02Aof Images SI Inc [9] ) and DH parameters in Fig. 2.

Figure 1. The robot arm.

Figure 2. DH parameters.

Various rotary and linear motions of the robotic arm areshown below in Fig 3. Each motion is made possible by aservomotor (or just a servo) at the corresponding location.

Figure 3. Robotic arm with 5 independent motions [9].

B. Forward KinematicsAs Fig. 2 depicts a coordinate frame attached to every

link in order to find its configuration in the neighboringframes using the rigid motion formula. To do so a DH tableis needed as follows:

TABLE I. DH PARAMETERS.

By applying the Denavit-Hartenberg (DH) notations [2]for the joints coordinates, the DH-table can be constructed aslisted above in Table II. The link lengths as shown in Fig. 2are

1 1 1 .5 ,l cm= 2 1 2 ,l cm= and3 4 9 .l l c m= =

C. Forward Transformation MatricesOnce the DH table is ready, the transformation matrices

are easy to find. Generally, the matrix of transformation fromthe frame

iB to the frame1iB −

for the standard DH methodis given by [2]:

1

cos sin cos sin sin cossin cos cos cos sin sin

0 sin cos0 0 0 1

i i i i i i i

i i i i i i iii

i i i

aa

Td

−

− − =

(1)

Then the individual transformation matrices for1 , 2 , 3 , 4i a n d= can be easily obtained. Thereafter, the

complete transformation 04T is found from:

0 0 1 2 34 1 2 3 4T T T T T= (2)

Upon the determination of, 04T we can find the global

coordinates of the end effector. The tip point of the arm is atthe origin of frame

4B (Fig. 2), i.e. it is at [ ]0 0 0 1 T .So, its position in the global frame becomes:

Frame ( )iia ia id i

1 0 90 1l 12

2l 0 0 23

3l 0 0 34

4l 0 0 4

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1 4

2 40 0 4 04 4

3 4

0001 1 1

p p

r d xr d y

r T r Tr d z

= = = =

(3)

Obviously, this is the last column of the transformationmatrix 0

4T . After simplifications using trigonometricformulas, the previous equation becomes:

( ) ( )1 2 2 3 2 3 4 1 2 3 4dx c l c l c l c c = + + + + + (4)

( ) ( )1 2 2 3 2 3 4 1 2 3 4dy s l c l c l s c = + + + + + (5)

( ) ( )1 2 2 3 2 3 4 2 3 4dz l l s l s l s = + + + + + + (6)

where s stands for s in and c stands for c o s . In addition,dx , dy and dz are the global end effector coordinates.

Moreover, the end effector orientation is:

2 3 4 = + + (7)

The DH parameters are akin to the configuration of therobot. For different manipulator structures, the kinematicsequations are not unique. Moreover, kinematics equations ofthe manipulator based on the DH convention provide somesingularity making the equations difficult to solve orunsolvable in some cases. In addition, in the DH convention,the common normal is not defined properly when axes oftwo joints are parallel. In this case, the DH method has asingularity, where a little change in the spatial coordinates ofthe parallel joint axes can create a huge misconfiguration inrepresentation of the DH coordinates of their relativeposition. In the following section, an alternative to the DHconvention, the PE is presented.

D. Product of ExponentialsAside from the DH convention, another method is the so-

called product of exponentials (PE). Using the reality that themotion of each joint is generated by a twist accompaniedwith the joint axis, a more geometric representation of thekinematics can be acquired. Remember that if ξ is a twist,according to reference [11], the forward kinematics is givenby:

( ) ( )1 1 2 2ˆˆ ˆ ... 0n n

st stg e e e g = (8)

The above equation is called the product of exponentialformula for the robot forward kinematics, where ( )0stg is

the final configuration of the robot and n̂ ne is a matrixexponential given by [7]:

( ) ( )ˆ

0 1

n n n n

n n

Tn n n n n ne I e v

e

− × +=

(9)

For a prismatic joint the twist i is given by:0

ii

v

=

,

and for a revolute joint by: i ii

i

q

− ×

=

,

where 3i R ∈ is a unit vector in the direction of the axis

of the twist, 3iq R∈ is any point on the axis, and 3

iv R∈ isa unit vector directing in the translation direction. In thiscase, the twist ξ’s for different links of the robot are givenby:

1

000001

=

1

2

0001

0

L

= −

1

23

0

01

0

L

L

−

= −

( )

1

2 34

0

01

0

L

L L

− +

= −

Moreover, ( )0s tg is the initial configuration of the robotgiven by

( )2 3 4

1

1 0 00 1 0 0

00 0 10 0 0 1

s t

l l l

gl

+ + =

The forward kinematics map of the manipulator has theform:

( ) ( ) ( ) ( )3 31 1 2 2 4 4

ˆˆ ˆ ˆ 00 1st st

R pg e e e e g

= =

(10)

By expanding terms in the product of exponentialsformula, Eq. 10 yields

( )( ) ( )( ) ( )

( ) ( )

2 3 4 1 1 2 3 4 1

2 3 4 1 1 2 3 4 1

2 3 4 2 3 4

cos cos sin sin coscos sin cos sin sin

sin 0 cosR

+ + − − + + = + + − + + + + + +

(11)

( )( ) ( )( )( ) ( )( )( ) ( )

1 3 2 3 2 2 4 2 3 4

1 3 2 3 2 2 4 2 3 4

1 3 2 3 2 2 4 2 3 4

cos L cos L cos cos

sin L cos L cos cos

L sin L sin sin

L

p L

L L

+ + + + +

= + + + + + + + + + + +

(12)

It is clear that the above equation (12) is identical tothose in equations (4) through (6). Thus, the kinematicsmodeling using the DH convention and product ofexponentials agrees with each other.

III. INVERSE KINEMATICS

We end up with a set of four nonlinear equations withfour unknowns. Solving these equations algebraically,known as the inverse kinematics, requires that we need toknow the joint variables

1 2 3 4, , , a n d for a given EEposition [ ], dy, dzdx and orientation . We get fromequations (4) to (7), by dividing, squaring, adding and usingsome trigonometric formulas:

89

11 tan d y

d x − =

(13)

( ) ( )1 2 2 12 tan , tan ,c r c a b − −= ± − − (14)

2 2 2 2 21 2 3

32 3

c o s2

A B C l ll l

− + + − −=

(15)

where2 2

3 3 2 3 3 1 4sin , cos , sin , .a l b l l c dz l l andr a b = = + = − − = +

In addition( ) ( ) ( )4 1 4 1 1 4, ,A dx l c c B dy l s c andC dz l l s = − = − = − −

Having determined1 2 3, , a n d , we can then find

4 from the EE orientation of as follows:

4 2 3 = − − (16)

IV. SIMULATION OF THE ROBOTIC ARM

Since robotics is a multidisciplinary subject, its studyrequires skills form different fields of knowledge, not easilyavailable. Thus, simulation has been recognized as a suitabletool that combines all these features together enabling theuser of a direct visualization of different kinds of motion thata robot may perform, making the role of simulation veryimportant in robotics [12]. Using the robotics toolboxtogether with the Matlab software [13]-[15], the kinematicsof a robotic arm can be simulated and analyzed based on theDH convention described before. The toolbox takes aconventional approach to represent the kinematics anddynamics of serial-link robotic arms. Besides, it providesseveral functions and routines, which are handy for thesimulation and scrutiny of robotic manipulators, likekinematics, dynamics and trajectory creation.

Based on the aforementioned toolbox, manyconfigurations of the robot manipulator are easily visualized.In order to simulate the robotic manipulator, first we erect avector of Link objects and insert our four groups of DHparameters as in Table I. Then, these are passed to theconstructor SerialLink, which is the key step to utilize thetoolbox. A detailed procedure can be found in [15]. In orderto generate the plot four joint variables are needed. In thefirst case all joint variables are entered in the form of zeros(row vector), such as [ ] [ ]1 2 3 4 0 0 0 0 = . Fig.4 shows rest or home position of the robot, where all jointsvariables are zeros. In the toolbox, each revolute joint isresembled by a small cylinder. Since we have four revolutejoints, four cylinders can be seen in Figs. 4-7.

A second interesting configuration is that when all linksare standing upright as Fig. 5 shows. The correspondinginput in this case is given by: [ ]0 / 2 0 0 , where all theangles are defined in radian by default. To test thefunctionality of all the joints, other two configurations areselected as given by the vectors of joint variables

[ ]/ 2 0 /2 0 − − and [ ]/ 4 / 2 / 4 − , which aredepicted in Figs. 6 and 7, respectively.

Figure 4. Home position.

Figure 5. Upright position.

Figure 6. Left-down position.

Figure 7. All joints are given angles.

V. CONCLUSION

Kinematics model of a 4 degree-of-freedom robotic armis presented using both the DH method and product ofexponentials formula. It is proven that both approachesprovide the same solution for the robot manipulator understudy. In addition, the simulation of the robot manipulator iscarried out using the Matlab software via the roboticstoolbox, through which several positions of the manipulatorare realized based on the DH convention. Although results ofthe product of exponentials formula are not given, they areexpected to be same as those of the DH convention.

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ACKNOWLEDGMENT

Authors gratefully acknowledge the support provided byKing Fahd University of Petroleum & Minerals through theNSTIP project 09-ELE786-04.

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