37 CHAPTER 3 KINEMATIC MODELING OF ROBOTIC MANIPULATORS This Chapter addresses the kinematic analysis performed on two different multi-DOF robotic manipulators under study. Here, the kinematic modeling has been performed using Denavit-Hartenberg convention for forward kinematics. Two different approaches viz. analytical approach and geometrical approach has been investigated to obtain solutions for inverse kinematics. The detailed discussion is given below. 3.1 3-DOF Omni-Bundle Robotic Manipulator In the present research work, a Quanser make Omni Bundle robotic manipulator (as shown in Figure 3.1) consisting of 3-DOF in joints viz. θ 1 , θ 2 , θ 3, has been used. The digital encoders are used to measure for proper positioning of the end- effector along x, y and z axes. The frame convention and Denavit-Hartenberg convention is as shown in Figure 3.2. Figure 3.1 Representation of 3-DOF Omni-Bundle robotic manipulator
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37
CHAPTER 3
KINEMATIC MODELING OF ROBOTIC
MANIPULATORS
This Chapter addresses the kinematic analysis performed on two different multi-DOF
robotic manipulators under study. Here, the kinematic modeling has been performed
using Denavit-Hartenberg convention for forward kinematics. Two different
approaches viz. analytical approach and geometrical approach has been investigated
to obtain solutions for inverse kinematics. The detailed discussion is given below.
3.1 3-DOF Omni-Bundle Robotic Manipulator
In the present research work, a Quanser make Omni Bundle robotic
manipulator (as shown in Figure 3.1) consisting of 3-DOF in joints viz. θ1, θ2, θ3, has
been used. The digital encoders are used to measure for proper positioning of the end-
effector along x, y and z axes. The frame convention and Denavit-Hartenberg
convention is as shown in Figure 3.2.
Figure 3.1 Representation of 3-DOF Omni-Bundle robotic manipulator
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Figure 3.2 Schematic representation of Denavit-Hartenberg convention for 3-DOF Omni-
Bundle robotic manipulator
3.1.1 Analytical analysis
The kinematic analysis of any robotic system is performed in two ways i.e.
forward kinematics and inverse kinematics. The forward kinematics problem is to find
the position and orientation as a function of joint variables, achieved by end-effector
of robotic manipulator, as given in equation 3.1. The forward kinematics of multi-
DOF robotic manipulators is an easy task due to the availability of Denavit-
Hartenberg convention.
𝑥(𝑡) = 𝑓(𝜃(𝑡)) (3.1)
The calculation of joint variables to bring the end-effector of robotic
manipulator to the required position and orientation is defined by inverse kinematics
problem, as given in equation 3.2.
𝜃(𝑡) = 𝑓′(𝑥(𝑡)) (3.2)
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As compared to forward kinematics, calculation of inverse kinematic solutions
is a complex task since there is no possible unique solution due to non-linear and
time-varying nature of its governing equation. The inverse kinematics of multi-DOF
robotic manipulator can be obtained using three different techniques, viz. algebraic
approach, geometric approach and iterative approach.
By substituting the Denavit-Hartenberg parameters (θi, di, ai, αi) in the general
matrix given in equation 3.3; the transformation matrices A1 to A3 are obtained as
below:
𝐴𝑛+1 = [
𝐶𝜃𝑛+1 𝑆𝜃𝑛+1𝐶𝛼𝑛+1𝑆𝜃𝑛+1 𝐶𝜃𝑛+1𝐶𝛼𝑛+1
𝑆𝜃𝑛+1𝑆𝛼𝑛+1 𝑎𝑛+1𝐶𝜃𝑛+1−𝐶𝜃𝑛+1𝑆𝛼𝑛+1 𝑎𝑛+1𝑆𝜃𝑛+1
0 𝑆𝛼𝑛+10 0
𝐶𝛼𝑛+1 𝑑𝑛+10 1
] (3.3)
𝐴1 = [
𝐶1 0𝑆1 0
−𝑆1 0𝐶1 0
0 −10 0
0 00 1
] 𝐴2 = [
𝐶2 −𝑆2𝑆2 𝐶2
0 𝐿1𝐶20 𝐿1𝑆2
0 00 0
1 00 1
]
𝐴3 =
[ 𝐶(𝜃3−
𝜋
2) −𝑆(𝜃3−
𝜋
2)
𝑆(𝜃3−𝜋
2) 𝐶(𝜃3−
𝜋
2)
0 𝐿2𝐶(𝜃3−𝜋
2)
0 𝐿2𝑆(𝜃3−𝜋
2)
0 00 0
1 00 1 ]
Mathematically, the forward kinematics equations can be obtained by
multiplying A1 to A3 matrices as given in equation 3.4:
𝐴30 = 𝐴1…… . 𝐴3 (3.4)
which results to, 𝐴30 = [
𝑅3×3 𝑝1×30 0 0 1
]
After applying the above steps, the forward kinematic equations for 3-DOF
Omni-Bundle robotic manipulator under study is obtained as given in equation 3.5 to
equation 3.7:
𝑝𝑥 = 𝐿1 cos 𝜃1 cos 𝜃2 +𝐿2 cos 𝜃1 cos 𝜃2 cos(𝜃3 −𝜋
2) − 𝐿2 cos 𝜃1 sin 𝜃2 sin(𝜃3 −
𝜋
2)
(3.5)
𝑝𝑦 = 𝐿1 sin 𝜃1 cos 𝜃2 +𝐿2 sin 𝜃1 cos 𝜃2 cos(𝜃3 −𝜋
2) − 𝐿2 sin 𝜃1 sin 𝜃2 sin(𝜃3 −
𝜋
2)
(3.6)
𝑝𝑧 = 𝐿1 sin 𝜃2 −𝐿2 sin 𝜃2 cos(𝜃3 −𝜋
2) − 𝐿2 cos 𝜃2 sin(𝜃3 −
𝜋
2) (3.7)
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The movement of each link and variation of joint angles of robotic
manipulator is as given in Table 3.1. The forward kinematic equations have been
formulated using the Denavit-Hartenberg convention, as given in Table 3.2.
Table 3.1 Description of movement of 3-DOF Omni-Bundle robotic manipulator
S. No. Link Link movement Variation of joint angles (rad)
1 Link 1 Clockwise/Anti-clockwise -π/3.24 < Ө1 < π/3.24
2 Link 2 Front-Back 0 < Ө2 < -π/10.98
3 Link 3 Up-Down π/2.97 < Ө3 < π/1.4
Table 3.2 Denavit-Hartenberg parameters of 3-DOF Omni-Bundle robotic manipulator
Links Ө [rad] d [mm] a [mm] α [rad]
Link 1 Ө1 0 0 -π/2
Link 2 Ө2 0 132 0
Link 3 Ө3 – π/2 0 132 0
The values of link lengths are L1 = L2 = 132 mm, (θ1, θ2, θ3) are respective
joint angles and (px, py, pz) are coordinates at any position of end effector.
Here, analytical inverse kinematic analysis for 3-DOF robotic manipulator has
been performed using geometrical approach. The inverse kinematic equations are
obtained geometrically (as shown in Figure 3.3) as given below:
𝜃1 = tan−1(𝑦 𝑥⁄ ) (3.8)
𝜃3 = 3𝜋
2− cos−1 (
𝐿12+ 𝐿2
2− 𝑘2
2𝐿1𝐿2) (3.9)
𝜃2 = ∅ − 𝛾 (3.10)
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where, 𝑘 = √𝑥2 + 𝑦2 + 𝑧2 , ∅ = cos−1 (𝑑
𝑘) 𝑓𝑜𝑟 𝑧 ≤ 0 or
∅ = −cos−1 (𝑑
𝑘) 𝑓𝑜𝑟 𝑧 > 0, 𝑑 = √𝑥2 + 𝑦2 and 𝛾 = sin−1 (
𝐿2 sin(3𝜋
2− 𝜃3)
𝑘),
respectively.
Figure 3.3 Geometrical representation to derive inverse kinematics of 3-DOF Omni-
Bundle robotic manipulator
Due to physical limitations of robotic manipulator, the joint angle θ3 always
remain positive. In this case, the robotic manipulator has been designed in such a way
that it executes unique inverse kinematic solution for any possible movement. Both
the forward and inverse kinematic equations as given from equation 3.5 to equation
3.10 are used to implement ANFIS on the robotic manipulator, discussed later in
Chapter 4.
3.2 5-DOF Pravak Robotic Manipulator
In this section of work, a 5-DOF Pravak robotic manipulator comprising of 3-
DOF at joints and 2-DOF at wrist has been considered. The available degree of
freedom in links is sufficient to bring the end effector to the required position;
however, the wrist movement provides additional flexibility to reach a particular
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position by the end effector. The extra degrees of freedom made available at the wrist
provide greater flexibility and applicability to the complete robotic system. It also
enhances the accuracy of experiments performed, discussed later in Chapter 5.
The robotic manipulator has been plotted using Peter Corke Robotics Toolbox
[84] for MATLAB (release 9.8), as shown in Figure 3.4 (a) and (b). Table 3.3 gives
the complete description of movement of each link and wrist of 5-DOF Pravak robotic
manipulator. The frame convention and Denavit-Hartenberg convention is as shown
in Figure 3.5.
(a) (b)
Figure 3.4 Representation of 5-DOF Pravak robotic manipulator
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Figure 3.5 Schematic representation of Denavit-Hartenberg convention for 5-DOF Pravak
robotic manipulator
Table 3.3 Description of movement of 5-DOF Pravak robotic manipulator
S. No. Type Part of Manipulator Movement Rotation
1 Link 1 Waist Left/Right -90o – 90
o
2 Link 2 Shoulder Forward/Backward 0o – 180
o
3 Link 3 Elbow Up/Down 0o – 180
o
4 Wrist Wrist pitch Sky-turn/Earth-turn 0o – 180
o
5 Wrist Wrist Roll Clock-wise/Anti-clock-wise 0o – 360
o
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Table 3.4 Denavit-Hartenberg parameters of 5-DOF Pravak robotic manipulator
Joint θi (o) αi (
o) ai di
1 θ1 -90
0 L0
2 θ2 0 L1 0
3 θ3 0 L2 0
4 θ4 - 90 -90
0 0
5 θ5 0 0 L3
3.2.1 Analytical analysis
Here, analytical kinematic analysis for 5-DOF pick and place type robotic
manipulator has been performed using algebraic approach. As given in Table 3.4, the
Denavit-Hartenberg convention has been used to obtain the forward kinematic
equations.
By substituting the Denavit-Hartenberg parameters (θi, di, ai, αi) in the general
matrix given in equation 3.3; the transformation matrices A1 to A5 are obtained as
below:
𝐴1 = [
𝐶1 0𝑆1 0
−𝑆1 0𝐶1 0
0 −10 0
0 𝐿00 1
] 𝐴2 = [
𝐶2 −𝑆2𝑆2 𝐶2
0 𝐿1𝐶20 𝐿1𝑆2
0 00 0
1 00 1
]
𝐴3 = [
𝐶3 −𝑆3𝑆3 𝐶3
0 𝐿2𝐶30 𝐿2𝑆3
0 00 0
1 00 1
] 𝐴4 = [
𝐶4 0𝑆4 0
−𝑆4 0𝐶4 0
0 −10 0
0 00 1
]
𝐴5 = [
𝐶5 −𝑆5𝑆5 𝐶5
0 00 0
0 00 0
1 𝐿30 1
]
Mathematically, the forward kinematics equations can be obtained by
multiplying A1 to A5 matrices as given in equation 3.11:
𝐴50 = 𝐴1…… . 𝐴5 (3.11)
which results to, 𝐴50 = [
𝑅3×3 𝑝1×30 0 0 1
]
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After applying the above steps, the forward kinematic equations for 5-degree
of freedom robotic manipulator under study is obtained as given in equation 3.12 to