Proceedings in Manufacturing Systems, Volume 10, Issue 4, 2015, 157−164 ISSN 2067-9238

FORWARD AND INVERSE KINEMATICS STUDY OF INDUSTRIAL ROBOTS TAKING

INTO ACCOUNT CONSTRUCTIVE AND FUNCTIONAL PARAMETER'S MODELING

Adrian-FlorinNICOLESCU1,*, Florentin-Marian ILIE2, Tudor-George ALEXANDRU3

1)
Prof. PhD., Department of Machines and Manufacturing Systems, University "Politehnica" of Bucharest, Romania
2)
Eng., Student, Department of Machines and Manufacturing Systems, University "Politehnica" of Bucharest, Romania
3)
Eng., Student, Department of Machines and Manufacturing Systems, University "Politehnica" of Bucharest, Romania

Abstract: Forward and inverse kinematic studies of industrial robots (IR) have been developed and
presented in a large number of papers. However, even general mathematic formalization is usually
almost correct, (basically following up general Hartenberg - Denavit (H-D) conventions and associated
homogenous transformation matrix), only few papers presents kinematic models ready to be directly
implemented on a real scale industrial robot or as well able to evaluate kinematics behavior of a real
scale IR specific model. That is usually due to some inconsistencies in modeling, the most frequently of
these referring on: the incomplete formalization of the full set of constructive and functional parameters
(that mandatory need to be considered in case of a specific real IR's model), avoidance of considering
IR's specific design features, (as joint dimensions and links dimensions are) leading to wrongly locating
the reference frames used for expressing homogenous coordinate transformations, as well as missing of
the validation procedures able to check the correctitude of the mathematical models, previously to its
implementing in a real scale IR's controller. That is why present paper shows first a completely new
approach for IR's forward an inverse kinematics, in terms of IR's analytical modeling by taking into
account the full set of IR's constructive and functional parameters of two different IR's models. Then, for
both direct and inverse mathematical models complete symbolic formalization and full set of solutions for
forward and inverse kinematics are presented for both IR types. In order to study mathematical models
applicability on the real scale IR, two specific IR models were studied: an ABB serial-link open chain
kinematics IR and a Fanuc serial-link closed chain kinematics IR. Numerical results were verified by
cross validation using both analytically calculations results and by mean of a constrained 3D CAD model
used to geometrically verify the results. The parametric form of the model elaborated in PTC Mathcad 14
allows a quick reconfiguration for other robot's models having similar configurations. Results can be also
used for solving dynamics, path planning and control problems in case of real scale IR.

Key words: industrial robot, extended parametric modeling, homogenous transformation matrix, forward
kinematics, inverse kinematics.

1. INTRODUCTION1 the two) [2, 9]. The kinematics of the robot is ussually
Industrial robots have long been used to replace the represented using a symbolic structure describing each
necessity of the human operator in repetitive tasks or joint and link and the relationship between the two in the
dangerous environments. From the early design stages, path of the motion. Modeling the chains of bodies
with respect to the specificity of the application each connected by joints is done using the Denavit-Hartenberg
specific IR model may be efficiently integrated into a (D-H) conventions [2, 3, 7, 8, and 9]. The representation
flexible manufacturing cells by primary taking into of D-H convention results in a 4 × 4 matrix called the
account its maximum reach ability and payload. Both of homogenous transformation matrix. For an n degrees of
these major functional features are basically depending freedom robot, the resulting number of homogenous
by the length of each IR's link, the available limits for transformation matrix is n + 1. The products of the
each IR's joint orientation as well as some specific coordinates frames transformations matrices for each link
constructive features of each IR's subassemblies [1]. are used to determine the forward kinematics. By this
From this point of view the IR's work space can be mean the position and orientation of the tool attached to
defined as a relationship between the length of the arms, the robot can be computed using a given set of joint
the number of degrees of freedom and the type of the angles. Also, the form and limit of the robot’s working
joint (rotational, translational or combinations between envelope can be analyzed using the position vector from
the transformation matrix describing the forward
*
Corresponding author: Splaiul Independentei, sector 6, Bucharest, kinematics [3]. The inverse kinematics problem is solved
Romania to achieve a desired position and orientation of the tool
Tel.: +40744923533
Fax: +40212691332 relative to the workstation. The inverse kinematics
E-mail addresses:afnicolescu@yahoo.com (A.Nicolescu) problem r equires solving at first the inverse kinematics
158 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164

equations for the position fallowed by the equations for [5] and the second one a closed-chain kinematics (Fanuc
the orientation of the tool relatively to the robot’s base m2000iA 900L) [6]. For both IR models the full set of
frame. The number of solutions for the joint angles is constructive and functional parameters have been take
affected by the number of IR specific configurations and account into modeling [4]. In both models the IR's work
number of degrees of freedom. While the problem of space limits are determined and results for extremity
inverse kinematics may have 8 solutions for the most 6 points are validated using by mean of a constrained 3D
DOF robots, the problem for 7 DOF robots is more CAD geometric model. Thus, the presented parametric
complex because an infinity of solutions may be model can be further used for dynamic analysis, path
identified. Therefore, sometimes one DOF in suppressed planning and control of presented IR's real scale models,
when solving the kinematics equations. To analytically as well as for other IR type's (with similar configuration)
validate the inverse kinematics results, the value of joint modeling due to facilities offered for their quick
angles determined for a specified pose of the IR's end reconfiguration.
effector should correspond with the value of joint angles
from the inverse kinematics [15]. In order for this 2. TECHNICAL SPECIFICATIONS AND
purpose a geometric validation may be used too by mean FUNCTIONAL PARAMETERS OF THE
of constrained 3D models of the studied real scale IR STUDIED ROBOTS
achieved by taking into account their complete set of The studied robots are ABB IRB6620 (Fig. 1) and
constructive parameters and design features [4, 5, and 6]. Fanuc m2000iA (Fig. 2).
After defining each value corresponding to each joint • ABB IRB6620 is a medium payload serial link robot
angle, the final tool frame should correspond with the with open-chain kinematics and 6 DOF, [5];
desired pose frame. Many kinematic studies for industrial • Fanuc m2000iA 900L is a heavy payload serial link
robots behavior evaluation were recently published. robot with closed-chain kinematics and 6 DOF [6].
Several analytical and numerical approaches are The technical specifications and functional
available for solving forward and inverse kinematics parameters of the ABB IRB6620 and Fanuc m2000iA
problem for industrial robots. Forward and inverse robots are detailed in Table 1.
kinematics was analytically computed in [7] for a serial- Table 1
link opened kinematics KUKA KR60 6 DOF IR, the Technical specifications of the ABB IRB6620 Industrial
accuracy of the results being verified using a simulation robot [5, 6]
program which uses the Unified System for Automation
Manufacturer and model ABB IRB6620 Fanuc
and Robot Simulation (USARSim). The forward m2000iA 900L
kinematics, inverse kinematics, workspace and joint Maximum payload (kg) 150 900
accelerations and velocities were determined and results Number of NC axis 6 6
were also verified using Robo-analyzer software [8]. Repeatability (mm) 0.03 0.5
Similar work was done for a closed kinematics 4 DOF Reach (mm) 2200 4638
palletizing robot type including specific of its specific Maximum speed 1 100 °/s 45 °/s
design features and constraints regarding mechanical 2 90 °/s 30 °/s
behavior when solving the forward kinematics problem, 3 90 °/s 30 °/s
the validation of the results being made in MATLAB [9]. 4 150 °/s 50 °/s
Also, complete direct kinematic models were presented 5 120 °/s 50 °/s
6 90 °/s 70 °/s
for the real case of Kawasaki FS10E industrial robot, by
Working range 1 +170° / ˗170° +165° / ˗165°
using an extended set of constructive and functional
2 +140° / ˗65° +100° / ˗60°
parameter's modeling [4] and the conventional D-H 3 +70° / ˗180° +35° / -130°
formalization algorithm [10 and 12], as well as the 4 +300° / ˗300° +360° / ˗360°
quaternion mathematical formalization algorithm [11]. 5 +130° / ˗130° +120° / ˗120°
For both previously approaches, the validation of 6 +300° / ˗300° +360° / ˗360°
analytic calculus results was performed using Kawasaki Weight (kg) 900 9600
PC Roset Offline programming software and Dynalog
experimental measuring system [10], and respectively
Microsoft Robotics Developer Studio software [11]. A 3. IR'S CONSTRUCTIVE PARAMETERS AND D-
wide range of numerical approaches have been H MODIFIED CONVENTION
elaborated, with focus on fuzzy logic inverse kinematics The D-H convention is used for the modeling of
mapping model for redundant manipulators [13], chains of bodies connected by joints. Originally the D-H
adaptive neuro-fuzzy interference system for inverse convention only was applied to single-loop chains, but
kinematics modeling [14] and approximation method for now is almost universally applicable to most serial chains
inverse kinematics modeling using MLP training [15]. structures.
Also, combined analytical and numeric approaches are For representing the frames of the studied industrial
available for modeling the kinematics of redundant robots, the modified convention of the D-H parameters
manipulators having more than 6 DOF [16]. will be used herein [17]. To attach the link frames the
However the presented approach use PTC Mathcad fallowing procedure has been fallowed (Figs. 3 and 4)
14 for parametric modeling of forward and inverse [4]:
kinematics for two serial-link industrial robots, the first 1) Identification of the joint axes and drawing of
one having an open-chain kinematics (ABB IRB 6620) infinite lines along them. For steps 2 to 5 below, two of
 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164 159

Fig. 1. Dimensional specifications and joint numbering of the ABB IRB6620 Industrial robot [5]
J1 to J3 - Corresponding joints for position and J4 to J6 corresponding joints for orientation.

Fig. 2. Dimensional specifications and joint numbering of the Fanuc m2000iA Industrial robot [6]
J1 to J3 = Corresponding joints for position and J4 to J6 corresponding joints for orientation.

these neighboring lines are considered (at axes i and i + 6) Assignment of {0} to match {1} when the first
1); joint variable is zero.
2) Identification of the common perpendicular For {N} an origin location is chosen. The Xn direction
between them, or point of intersection. The link frame is determined freely, but generally so as to cause as many
origin is assigned at the point of intersection, or at the linkage parameters as possible to become zero;
point where the common perpendicular meets the ith axis;
3) Assignment of the Zi axis pointing along the ith 4. LINK PARAMETERS
joint axis;
The link parameters describe the dimensions of the
4) Assignment of the Xi axis pointing along the robot arm, the joint offsets (resulting from D-H
common perpendicular, or if the axes intersect, the Xi is convention) and the position and orientation of each joint
assigned to be normal to the plane containing the two referencing the previous joint. The link parameters for
axes; ABB IRB6620 and Fanuc m2000iA 900L robot are
5) Assignment of the Yi axis to complete a right hand detailed in Tables 2 and 3. The sets of parameters will
coordinate system; define the Homogenous transformation matrix.
160 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164

Fig. 3. Assignment of link frames for ABB IRB6620 Robot [4]:

ai = the distance from Zi to Zi-1 measured along Xi; αi = the angle between Zi and Zi+1 measured along Xi; di = distance from Xi-1 to Xi
measured along Zi-1 and θi = angle between Xi-1 and Xi measured about Zi; n, m, p and o represent robot arm joint offset parameters.
parameters

Fig. 4. Assignment of link frames for Fanuc m2000iA 900L Robot [4];
all the notations are the same as in Fig. 2, the only difference being the g robot arm joint offset parameter.
parameter

Table 2 Table 3
Link parameters derived for ABB IRB6620 Robot Link parameters derived for Fanuc m2000iA Robot
i ai-1 αi-1 (mm) di θi i ai-1 αi-1 (mm) di θi
1 0° 0 0 θ1 1 0° 0 0 θ1
2 -90° 320 0 θ2 2 -90° 500 0 θ2
3 0° 975 0 θ3 3 0° 1700 0 θ3
4 -90° 280 887 θ4 4 -90° 180 2850 θ4
5 90° 0 0 θ5 5 90° 0 0 θ5
6 -90 0 0 θ6 6 -90 0 0 θ6
 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164 161

5. HOMOGENOUS TRANSFORMATION 6. FORWARD KINEMATICS

MATRIX
For an n-axis rigid-link manipulator, solving the
The homogenous transformation matrix represents forward kinematics problem gives the position and
each coordinate frame of the link with respect to the orientation of the robot’s end effector relative to its base.
coordinate system of the previous link. The resulting The solution is obtained by repeated application of
matrix is a 4 × 4 matrix: equation (4):

cosθi − sin θi cosαi sin θi sin αi ai cosθi 

0
Tn =0A1 0 A2 ...n −1An = K ( q) . (4)
 sin θi cosθi cosαi − cosθi sin αi ai sin θi 
i −1
Ai =   , (1) which represent the multiplication of the frame
 0 sin αi cosαi di 
  transformation matrices for each link.
 0 0 0 1  The matrix of transformation between the final frame
and the base frame can be defined by equation (5):
0
Ti −1 i −1 Ai , (2)
0 T = T 01 ⋅ T 12 ⋅ ... ⋅ T 56 ⋅ TCPF ⋅ TB ⋅ TEF =
where Ti is the homogenous transformation describing
the pose of coordinate frame i with respect to the world  R11 R12 R13 Px 
  . (5)
coordinate system 0.  R 21 R 22 R 23 Py 
The resulting homogenous transformation matrix are  R31 R32 R33 Pz 
 
detailed in Eq. (3):  0 0 0 1 


 cos(θ1) − sin( θ1) 0 0 Forward kinematics is used to determine the form and
 
 sin( θ1) cos(θ1) 0 0 limit of the working envelope.
T 01 =  ; (3)
0 0 1 0 The functional limits for θ1, θ2 and θ3 are defined
  base on IR's working ranges that may be identified from
 0 0 0 1  the technical specifications Table 1.
 cos( θ2) − sin( θ2)0 a1 A constant step value is defined for all joint limits.
  Using the position vector from the homogenous
 0 0 1 0
T 12 =  ; transformation matrix corresponding to the forward
− sin( θ2) − cos(θ2) 0 0 
  kinematics, a vector describing the position vectors for
 0 0 0 1  Px and Pz can be generated. Each row of the vector
 cos(θ3) − sin(θ3) 0 a 2  represents the coordinates for the given position in the
  Cartesian space. From the points describing the shape of
 sin(θ3) cos(θ3) 0 0 
T 23 =  ; the working envelope corresponding to the XOZ plane, a
0 0 1 0 3D model can be obtained revolving the resulting spline
 
 0 0 0 1 around the rotation axis of θ1. For each of the studied
robots, the resulting plots are presented in Fig. 5,a for
 cos(θ4) − sin( θ4) 0 a 3  ABB IRB6620 and Fig. 5,b for Fanuc m2000iA 900L.
 
 0 0 1 d 4 Comparing the volume of the working envelope for the
T 34 =  ;
− sin(θ4) − cos(θ4) 1 0  two robots, the first robot uses all the combinations of
  joint limits defined because of its serial-link open
 0 0 0 1 
kinematics. On the other hand, the second robot can only
 cos(θ5) − sin( θ5) 0 0  use a limited combination of joint limits. This is due to
 
 0 0 − 1 0 the existence of a link corresponding to the lever that
T 45 =  ;
0 0
requires a special geometrical constraint (6):
sin(θ5) cos(θ5)
 
 0 0 0 1 θ2 + θ3 + 90
≥ −θ3 max . (6)
 cos(θ6) − sin(θ6) 0 0  This is why the working limit of the lever’s joint
 
 0 0 − 1 0 should not exceed the inferior limit of θ3 (Fig. 6).
T 56 =  ;
− sin( θ6) − cos(θ6) 1 0 
  7. INVERSE KINEMATICS
 0 0 0 1 
Inverse kinematics is used to compute the joint angles
1 0 0 0 1 0 0 0 
    which will achieve a desired position and orientation of
0 1 0 0 0 1 0 0  the end-effector relative to the base frame.
TCPF =  ; TB =  0 0 1 a + a0 ;
0 0 1 d Both studied robots have two joint offsets (on X and Z
   
1  1 
axis).
0 0 0 0 0 0 The problem of inverse kinematics results in sets of
1 0 0 0  joint angles, depending on the configuration of the robot.
 
0 1 0 f  A total of 8 sets of joint angles can be achieved for a
TEF = .
0 0 1 e
given position for the ABB IRB6620 robot while 6
  solutions can be achieved for the Fanuc m2000iA 900L
0 0 0 1  robot.
162 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164

a b

Fig. 5. Working envelope: a – ABB IRB6620; b – Fanuc m2000iA.

Fig. 6. Geometrical constraint applied to the Fanuc m2000iA Robot.

Analytically solving the inverse kinematics problem

t 3 = a tan 2(d 4, a3) − a tan 2[( a32 − d 4 2 − K 2 ), K ]
can be done by two steps:
1. Computation of θ1, θ2 and θ3 using the elements of (9)
the position vector from the T matrix (5).
For θ2:
Px = c1( a 3 ⋅ c 32 − d 4 ⋅ s 23 + a1 + a 2 ⋅ c 2 );
a = − d 4 ⋅ sin( t 3) + a 2 + a 3 ⋅ cos(t 3);
Py = s1( a 3 ⋅ c 32 − d 4 ⋅ s 23 + a1 + a 2 ⋅ c 2 ); (7)
b = a 3 ⋅ sin( t 3) + d 4 ⋅ cos( t 3);
Pz = − a 3 ⋅ s 23 − d 4 ⋅ c 23 − a 2 ⋅ s 2.
Px
c= − a1;
To choose the optimal equations for each joint, a cos( t1)
study of the motion plane is done. To compute the d = − Pz;
solutions for each joint, the form of the equations can be
simplified using the inverse kinematics identities
t 2 = a tan 2[( a ⋅ c + b ⋅ d ), ( a ⋅ d − b ⋅ c )] . (10)
proposed by John Craig. The resulting equations are
presented below: 2. For θ4, θ5 and θ6, 3 equations are used (7):

For θ1: T 03−1 ⋅ T 06 = T 36;

T 04 −1 ⋅ T 06 = T 46; (11)
C1 = Px , −1
T 05 ⋅ T 06 = T 56.
S1 = Py , (8)
π For T 0 n −1 the values are known. Each solution uses
t1 = − a tan 2( Py , Px ) . the a tan2 geometrical identity that transforms a specific
2
position from the Cartesian space to the joint space using
For θ3: (8):

θ = a tan 2( x, y ) [rad]. (12)

[ Px + Py + Pz − 2a1( Px ⋅ cos( t1)]
2 2 2
K= +
2a 2 To convert from rad to deg, Eq. (13) is used:
[ Py ⋅ sin( t1)] + a12 − ( a 32 + d 4 2 + a 2 2 )] 180
; θdeg = θ ⋅ . (13)
2a 2 π
 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164 163

For θ4:  R11 R12 R13   0 0 1 

   
S 4 = [ − R13 ⋅ sin( t1) + R 23 ⋅ cos( t1)]; R =  R21 R22 R23 =  0 − 1 0  . (17)
 R31 R32 R33  1 0 0 
C 4 = [− R13 ⋅ cos(t1) ⋅ cos(t 2 + t 3)]    
(14)
− [ R 23 ⋅ sin(t1) ⋅ cos(t 2 + t 3)] + [ R33 ⋅ sin(t 2 + t 3)];
After defining this set of joint angles, the numeric
π
t 4 = − a tan 2( S 4, C 4 ) . result for the homogenous transformation matrix can be
2 achieved. These results are used as input data for the
solutions of the inverse kinematics for the orientation
For θ5: angles.
• The homogenous transformation matrix for the base
s 5 = −{R13 ⋅ [cos( t1) ⋅ cos( t 2 + t 3) ⋅ cos( t 4 ) ⋅ sin( t1) ⋅ of the robot, tool center point frame and end-
sin( t 4 )] + R 23 ⋅ [sin( t 2 + t 3) ⋅ cos( t 4 ) ⋅ cos( t 2 + t 3) ⋅ effector.
cos(t 4) − cos(t1) ⋅ sin(t 4)] The homogenous transformation matrixes have the
same rotation matrix (the unit matrix). Different values
− R33 ⋅ [sin(t 2 + t 3) ⋅ cos(t 4)]}; are defined for the components of the position vectors.
c5 = {R13 ⋅ [cos(t1) ⋅ cos(t 2 + t 3) ⋅ cos(t 4) + sin(t1) ⋅ The parameters for the position vectors for the base,
sin(t 4)] + R 23 ⋅ [sin(t 2 + t 3) ⋅ cos(t 4) ⋅ cos(t 2 + t 3) ⋅ TCPF and end-effector are presented in Table 4.
After defining these parameters the solutions for the
cos(t 4) − cos(t1) ⋅ sin(t 4)] − R33 ⋅ [ − cos(t 2 + t 3)];
inverse kinematics are solved with 8 and 6 solutions
π being achieved. In this case, only one solution identical
t 5 = − a tan 2( s 5, c5). (15)
2 to the solutions computed by the direct kinematics is
chosen (Table 5 for ABB IR and Table 6 for Fanuc IR).

For θ6: 8.2. Geometrical validation

The geometrical validation requires the 3D CAD
s6 = R11⋅ [cos(t 4) ⋅ sin(t1) − cos(t1) ⋅ model for each robot. The joints are constrained
cos(t 2 + t 3) ⋅ sin(t1) ⋅ sin(t 4)] + R31⋅ sin(t 4) ⋅ according to the D-H convention. Values corresponding
to each joint angle are inserted and the model is updated.
sin(t 2 + t 3);
The desired position is achieved (Fig. 8) the TCPF
c6 = R12 ⋅ [cos(t 4) ⋅ sin(t1) − cos(t1) ⋅ cos(t 2 + t 3) ⋅ coincides with the defined point.
sin(t 4)] − R22 ⋅ [cos(t1) ⋅ cos(t 4) + cos(t 2 + t 3) ⋅
sin(t1) ⋅ sin(t 4)] + R31⋅ sin(t 4) ⋅ sin(t 2 + t 3);
Table 4
π Position vector parameters for ABB IRB6620 Robot
t 6 = − a tan 2( s 6, c6). (16)
2 ABB IRB6620 Fanuc m2000iA
a + a0 680 1300
D 200 445
8. RESULTS VALIDATION E 200 100
F 100 200
To verify the results of the inverse kinematics, an
arbitrary point within the working limit of the robot is
defined. The inverse kinematics will compute the set of Table 5
joint angles. An optimal set of joints is defined when the ABB IRB6620 (Position of effector Px = 1975, Py = 0,
first arm is positioned perpendicular to the ground. Pz = 1100)
Results can be validated analytically and geometrically. Radians Degrees
θ1 0 0°
8.1. Analytical validation θ2 ˗1.089302 ˗62.412413°
For the studied industrial robots the fallowing are θ3 0.381833 21.877473°
θ4 3.141592 180°
defined:
θ5 0.863328 49.465059°
• A position in the Cartesian space of the robot (the θ6 3.141592 180°
point of extremity from the robot’s working range)
• The rotation matrix for the positioning of the End-
effector given by θ 4, 5 and 6. Table 6
The rotation matrix is denoted using the direct Fanuc m2000iA (Position of effector Px = 2875, Py = 0 and
kinematics by defining a set of joint angles that will Pz = 4909)
bring the robot in an optimal position (the first arm in
Radians Degrees
vertical position and the second arm parallel to the θ1 0 0
ground). Values are assigned for θ 4, 5 and 6 to complete θ2 1.57132945 ˗90.030548°
the rotation matrix (for example θ4 = 0, θ5 = 0 and θ3 ˗0.610337 ˗34.964052°
θ6 = 0 so that the axis of the robot’s output flange to θ4 0 0°
coincide with the motion axis of the second wrist). The θ5 0.6107707 ˗35.9946°
components of the resulting matrix are described in (17): θ6 0 0
164 A.F. Nicolescu, F.M. Ilie and T.G. Alexandru / Proceedings in Manufacturing Systems, Vol. 10, Iss. 4, 2015 / 157−164

b
Fig. 8. Geometrical validation for: a – ABB IRB6620; b – Fanuc m2000iA 900L.

9. CONCLUSIONS KukaKR5 Robot for Welding Application, International

Journalof Engineering Research and Applications
In the present work, a forward an inverse kinematics (IJERA), Vol. 3, Issue 2, 2013, pp. 820−827.
parametric analytical model was discussed. Symbolic [9] T. Yong, F. Chen, X. Hegen, Kinematics and Workspace
solutions for the inverse kinematics are presented. Two of a 4-DOF Hybrid Palletizing Robot, Advances in
industrial robots were studied, ABB IRB6620 serial-link Mechanical Engineering, Vol. 2014, 2014.
open chain kinematics and a Fanuc m2000iA 900L. [10] A.M. Ivan, Cercetari privind optimizarea exploatarii
robotilor industriali pentru prelucrari prin aschiere (PhD
Thesis), University "Politehnica" of Bucharest, 2011.
REFERENCES [11] S. Popa, A. Dorin, A.F. Nicolescu, A.M. Ivan,
Quaternion-Based Algorithm for Direct Kinematic Model
[1] A.F. Nicolescu, Roboți Industriali (Industrial Robots),
of a Kawasaki FS10E Articulated Arm Robot, Applied
EDP Publishing House, Bucharest, 2005.
Mechanics and Materials, Vol. 762, 2015, pp. 249−254.
[2] G. Khushdeep, D. Sethi, An analytical method to find
[12] A.F. Nicolescu, A.M. Ivan, Advanced Off-line
workspace of a robotic manipulator, Journal of
programming and simulation of a flexible robotic
Mechanical Engineering, Vol. 41, No. 1, 2010, pp. 25‒30.
manufacturing cell using Kawasaki PC-ROSET
[3] A. Srikanth, M. Sravanth, V. Sreechand, K.K. Kumar,
Proceedings in Manufacturing Systems, Vol. 6, Iss. 4,
Kinematic Analysis of 3 dof of serial robot for industrial
2011, pp. 239−244.
applications, International Journal of Engineering Trends
[13] Y. Xu, M.C. Nechyba, Fuzzy inverse kinematic mapping:
and Technology (IJETT), Vol. 4, Iss. 4, 2013, pp.
Rule generation, efficiency, and implementation,
1000‒1004.
Proceedings of the 1993 IEEE/RSJ International
[4] A.F. Nicolescu, Proiectarea și operarea roboților
Conference, Vol. 2, 1993, pp. 911−918.
industriali (Industrial robots design and operation), course
[14] S. Gurjeet, K.B. Vijay, ANFIS Implementation for Robotic
notes, University "Politehnica" of Bucharest, available at:
http://imst.curs.pub.ro Arm Manipulator, International Journal of Engineering
/2015/login/index.php Research & Technology (IJERT), Vol. 1, Iss. 4, 2012.
[5] *** ABB Robotics, Product specification IRB 6620, [15] V.C. Moulianitis, E.M. Kokkinopoulos, N.A. Aspragathos,
Sweden, 2004. A Method for the Approximation of the Multiple IK
[6] *** Fanuc Robotics, M-2000iA Series, available at: Solutions of Regular Manipulators Based on the
http://www.fanucrobotics.com/cmsmedia/ Uniqueness Domains and Using MLP, Robotics and
datasheets/M-2000iA%20Series_23.pdf, accessed: Mechatronics, Vol. 37, 2016, pp. 273−281.
2015-09-04. [16] H. Ananthanarayanan, R. Ordóñez, Real-time Inverse
[7] A. Khatamian, Solving Kinematics Problems of a 6-DOF Kinematics of (2n+ 1) DOF hyper-redundant manipulator
Robot Manipulator, Proceedings of the International arm via a combined numerical and analytical approach,
Conference on Scientific Computing (CSC), pp. 228−233, Mechanism and Machine Theory, Vol. 91, 2015,
Las Vegas – Nevada, 07.2015. pp. 209−226.
[8] K.K. Kumar, A. Srinath, G. Jugalanvesh, P. Premsai, M. [17] J.C. Craig, Introduction to robotics: mechanics and
Suresh, Kinematic Analysis and Simulation of a 6 DOF control, Addison-Wesley, 1990.