International Journal of Mechanical Engineering and Technology (IJMET)
Volume 8, Issue 2, February 2017, pp. 08–15, Article ID: IJMET_08_02_002
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=8&IType=2
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
FORWARD KINEMATIC ANALYSIS OF A ROBOTIC
MANIPULATOR WITH TRIANGULAR PRISM
STRUCTURED LINKS
Nalin Raut, Abhilasha Rathod, Vipul Ruiwale
Department of Mechanical Engineering, MIT-College of Engineering, Pune, India
ABSTRACT
To control robot manipulators as per the requirement, it is important to consider its kinematic
model. In robotics, we use the kinematic relations of manipulators to set up the fundamental
equations for dynamics and control. The objective of this paper is to introduce triangular prism
structured manipulator and derive the forward kinematic model using Denavit-Hartenberg
representation.
Key words: Forward kinematics, Robotic Manipulators, Triangular prism structure, DenavitHartenberg convention.
Cite this Article: Nalin Raut, Abhilasha Rathod and Vipul Ruiwale. Forward Kinematic Analysis
of a Robotic Manipulator with Triangular Prism Structured Links. International Journal of
Mechanical Engineering and Technology, 8(2), 2017, pp. 08–15.
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1. INTRODUCTION
A robot manipulator is composed of a set of links connected together by various joints. The joints can be
very simple, such as a revolute joint or a prismatic joint, or else they can be more complex, such as a ball
and socket joint. Kinematics is the relationships between the positions, velocities, and accelerations of the
links of a manipulator.
In the kinematic analysis of manipulator position, there are two separate problems to solve: direct or
forward kinematics, and inverse kinematics: Forward kinematics refers to the use of the kinematic
equations of a robot to compute the position of the end-effector from specified values for the joint
parameters Inverse kinematics refers to the use of the kinematics equations of a robot to determine the joint
parameters that provide a desired position of the end-effector. In robotics, we use the kinematic relations of
manipulators to set up the fundamental equations for dynamics and control.
The Denavit and Hartenberg representation [1], gives us a standard methodology to list the kinematic
equations of a manipulator. This is especially useful for serial manipulators where a matrix is used to
represent the position and the orientation of one body with respect to another. The purpose of this paper is
to present a manipulator with triangular prism structured links and develop the forward kinematics using
Denavit-Hartenberg convention. MATLAB has been used to calculate and plot the varying positions of end
frame with respect to varying joint angles.
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Forward Kinematic Analysis of a Robotic Manipulator with Triangular Prism Structured Links
2. PRISM STRUCTURED LINK
As stated earlier the shape of each link
link is a triangular prism with revolute joints at the centre of slant
surfaces as shown in Fig.1. The triangle considered for the prism in this paper is a right isosceles triangle.
Figure 1 Triangular prism structured link.
Fig.2
g.2 below shows some of the combinations that can be achieved using five such links. As the number
of links increase the degree of freedom of the manipulator is known to increase and the combinations that
can be achieved also increase.
Figure 2 Combinations achieved using triangular prism links.
3. DENAVIT HARTENBERG REPRESENTATION
R
Forward kinematics is concerned with the relationship between the individual joints of the robot
manipulator which governs the position and orientation
orientation of the tool or end effector. A serial-link
serial
manipulator comprises a set of bodies, called links, in a chain connected by joints. A link is considered a
rigid body that defines the spatial relationship between two neighboring joint axes. The objective of
forward kinematic analysis is to determine the cumulative effect of the entire set of joint variables on the
end effector.
The Denavit and Hartenberg convention or D-H
D H convention geometry is the most commonly used
fundamental tool for selecting frames
frame of reference and describing serial--link mechanism in robotic
applications. In this, the homogeneous transformation matrix Ai for each link is represented as a product of
four basic transformations. [2]
(1)
Ai = Rot z ,θ × Trans z ,d × Trans x,a × Rot x,α
i
i
i
i
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c (θ i ) − s (θ i )
 s (θ ) c (θ )
i
Ai =  i
0
0
 0
0

c (θ i )
 s (θ )
= i
0
0
− s (θ i ) c (α i )
c (θ i ) c (α i )
s (α i )
0
0
0
1
0
0
0

0

1
1
0
× 0
0
0
1
0
0
s (θ i ) s (α i )
− c (θ i ) s (α i )
c (α i )
0
0 0  1
0 0  0
1 d i  × 0
0 1  0
0
1
0
0
0
0
0 a i  1
0 c (α i ) − s (α i )


0 0 ×
1 0  0 s (α i ) c (α i )
0 1  0
0
0
0
0
0

1
ai c (θ i ) 
ai s (θ i ) 

di

1

where the four quantities θi, ai, di, αi are parameters associated with link i and joint i. In the above
equation ‘c’ represents cosine and ‘s’ represents sine.
The four parameters θi, ai, di, αi in equation (1) are generally known as joint angle, length of the
common normal, link offset and link twist respectively. These names derive from specific aspects of the
geometric relationship between two coordinate frames. Matrix Ai is a function of a single variable while
three of the above four quantities remain constant for a given link. The fourth parameter, in our case, θi for
a revolute joint, is variable.
To perform a forward kinematic analysis of a serial-link robot, based on Denavit-Hartenberg (D-H)
convention it is necessary to follow an algorithm [1,3,4],
i.
Numbering the joints and links
A serial-link robot with n joints will have n +1 links. Numbering of links starts from 0 for the fixed
grounded base link and increases sequentially up to ‘n’ for the end-effector link. Numbering of joints starts
from 1, for the joint connecting the first movable link to the base link, and increases sequentially up to n.
Therefore, the link i is connected to its lower link i-1 at its proximal end by joint i and is connected to its
upper link i+1 at its distal end by joint i+1.
ii.
Attaching a local coordinate reference frame for each link i and joint i+1.
The coordinate systems are attached to each link as per the rules stated below,
•
The origin of coordinate system i is located at the point of intersection of the axis of joint i+1 and
common normal between the axes of joints i and i+1.
•
The zi - axis is aligned with the axis of (i + 1)th joint. The positive direction of this axis can be chosen
arbitrarily.
•
The xi and yi axes can be chosen in any convenient manner so long as the resulting frame is right handed.
•
zi-1 - axis should always intersect xi+1 - axis.
Figure 3 Denavit-Hartenberg Frame assignment and parameters.
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Forward Kinematic Analysis of a Robotic Manipulator with Triangular Prism Structured Links
iii.
Establish the D-H
H parameters for each link.
Using the attached frames (fig.1), the four parameters that locate one frame relative to another are defined
as:
•
ai = distance along xi from
om oi to the intersection of the xi and zi-1 axes.
•
di = distance along zi-1 from oi-1 to the intersection of the xi and zi-1 axes.
•
αi = the angle between zi-1 and zi measured about xi.
•
θi = the angle between xi--1 and xi measured about zi-1.
iv Calculate the matrix of homogeneous transformation for each link and compute the overall
transformation matrix.
The homogeneous transformation matrix Ai for each link is calculated using equation (1). The overall
transformation matrix T is given by,
T = A1 A2 A3 ......... Ai
(2)
4. APPLICATION OF D-H
H CONVENTION TO TRIANGULAR
TRIANGULAR PRISM
STRUCTURED MANIPULATOR (ANALYTICAL
ANALYTICAL SOLUTION)
Consider the manipulator modell (Fig.4) for applying the D-H
D H convention. Link 1 is fixed and is the ground
link. Each link has slant length of 5 cm with each slant surface having a revolute joint at the center as
shown earlier.
Fig
Figure
4 Model for applying D-H convention
We establish
sh the coordinate system for each link (Fig.5) following the algorithm mentioned above,
Figure 5 Assignment of frames and D-H
D H parameters to triangular prism structured links.
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Nalin Raut, Abhilasha Rathod and Vipul Ruiwale
In this case the only variable quantity is θi and ai is zero since both zi-1 and zi axes are co-planar and
intersect each other. xi is chosen normal to the plane formed by zi and zi-1. The positive direction of xi is
arbitrary. The most natural choice for origin oi in this case is at the point of intersection of zi and zi-1.
However, any point along zi as per convenience suffices. The D-H parameters for the given manipulator
(Table1):
Table 1 D-H Parameters
Link
ai
αi
θi
di
1.
0
π/2
θ1*
5.0cm
2.
0
-π/2
θ2*
5.0cm
3.
0
0
θ3=0
2.5cm
The Ai -matrices for the manipulator are given by equation (1).
c (θ1 )
 s (θ )
A = 1
1
0
0
0 s (θ )
1
0 − c (θ )
1
1 0
0 0
c (θ 2 )
 s (θ )
A = 2
2
0
0
0


5
1 
0
− s (θ )
2
0
c (θ )
2
−1 0
0
0
0
0


5
1 
0
1 0 0 0 
0 1 0 0 
A =
3 0 0 1 2.5
0 0 0 1 
The overall transpose matrix, T is given by equation (2),
T = A1 A2 A3
 r11 r12 r13 p x 
r
r
r
py 
=  21 22 23

 r31 r32 r33 p z 
0
0
1 
 0
Where,
r11 = cos(θ1 ) cos(θ 2 )
r12 = − sin(θ 2 )
r13 = − cos(θ1 ) sin(θ 2 )
r21 = sin(θ1 ) cos(θ 2 )
r22 = cos(θ1 )
r23 = − sin(θ1 ) sin(θ 2 )
r31 = sin(θ 2 )
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Forward Kinematic Analysis of a Robotic Manipulator with Triangular Prism Structured Links
r32 = 0
r33 = cos(θ 2 )
p x = −2.5 cos(θ1 ) sin(θ 2 ) + 5 sin(θ1 )
p y = −2.5 sin(θ1 ) sin(θ 2 ) − 5 cos(θ1 )
p z = 2.5 cos(θ 2 ) + 5
Position of end frame with respect to the base frame along the x-axis.
axis. This is represented by the
Red colored curve in Figures 6 to 9.
px =
p y = Position of end frame with respect to the base frame
fr
along the y -axis. This is represented by the Green colored curve in Figures 6 to 9.
p z = Position of end frame with respect to the base frame
along the z -axis. This is represented by the Blue colored curve in Figures 6 to 9.
Following
llowing are figures showing the position of frame 4 at different joint angles.
(1) At joint angle 1, θ1 = 0 o
Figure 6 Position of end frame vs. Joint angle 2, θ 2 (at
θ1 = 0
o
)
(2) At joint angle 1, θ = 45 o
1
Figure 7 Position of end frame vs. Joint angle 2, θ 2 (at
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θ 1 = 45
o
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Nalin Raut, Abhilasha Rathod and Vipul Ruiwale
(3) At joint angle 1, θ1 = 90o
Figure 8 Position of end frame vs. Joint angle 2, θ 2 (at
θ 1 = 90
o
)
o
)
(4) At joint angle 1, θ1 = 180o
Figure 9 Position of end frame vs. Joint angle 2, θ 2 (at
θ 1 = 180
5. CONCLUSION
In this paper, we studied forward kinematics for triangular prism structured
structured links analytically using
Denavit-Hartenberg
Hartenberg convention. Furthermore, we used MATLAB to calculate and plot different positions
of the end frame with respect to base frame. In our future research, we intend to study the inverse
kinematics, dynamics and
nd control of triangular prism structured links.
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Denavit, Jacques; Hartenberg, Richard Scheunemann (1955), "A kinematic notation for lower-pair
lower
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[2]
M. Spong, S. Hutchinson, and M. Vidyasagar, Robot modeling and control. Wiley, 2006.
[3]
L-W.
W. Tsai. "Robot Analysis: The Mechanics of Serial and Parallel Manipulators". NY, 1999, John
Wiley & Sons, Inc.
[4]
W. W. Melek. "ME 547: Robot Manipulators: Kinematics, Dynamics, and Control". Waterloo,
Wate
ON,
2010, University of Waterloo.
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Forward Kinematic Analysis of a Robotic Manipulator with Triangular Prism Structured Links
[5]
Aldoomshareef, Ji-Ping Zhou, Hong Miao and Hui Shen, Inverse Kinematics Analysis and Simulation
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[6]
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International Journal of Mechanical Engineering and Technology, 4(3), 2013, pp. 125–129.
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