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

38

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)

39

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)

40

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)

41

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

42

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

43

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

44

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

]

45

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

equation 3.14:

𝑝𝑥 = −𝐿3 × 𝐶1 × 𝑆234 + 𝐿2 × 𝐶1 × 𝐶23 + 𝐿1 × 𝐶1 × 𝐶2 (3.12)

𝑝𝑦 = −𝐿3 × 𝑆1 × 𝑆234 + 𝐿2 × 𝑆1 × 𝐶23 + 𝐿1 × 𝑆1 × 𝐶2 (3.13)

𝑝𝑧 = −𝐿3 × 𝐶234 − 𝐿2 × 𝑆23 − 𝐿1 × 𝑆2 + 𝐿0 (3.14)

where, 32323223 sin SCCSS and 32323223 cos SSCCC and

(px, py and pz) are end-effector coordinates at any position.

The orientation of end-effector of 5-DOF robotic manipulator is given below

in equation 3.15 to equation 3.23:

𝑛𝑥 = 𝐶1𝑆234𝐶5 + 𝑆1𝑆5 (3.15)

𝑛𝑦 = 𝑆1𝐶234𝐶5 − 𝐶1𝑆5 (3.16)

𝑛𝑧 = −𝐶234𝐶5 (3.17)

𝑜𝑥 = −𝐶1𝑆234𝑆5 + 𝑆1𝐶5 (3.18)

𝑜𝑦 = −𝑆1𝑆234𝑆5 − 𝐶1𝐶5 (3.19)

𝑜𝑧 = 𝐶234𝑆5 (3.20)

𝑎𝑥 = 𝐶1𝐶234 (3.21)

𝑎𝑦 = 𝑆1𝐶234 (3.22)

𝑎𝑧 = −𝑆234 (3.23)

where, {n, o, a} are normal, orientation and approach vectors represented by the three

components along x, y and z directions.

The 5-DOF Pravak robotic manipulator used in this study comprises of a 2-

DOF wrist motion. The sufficient condition to solve inverse kinematics is that it has

two intersecting axes. As discussed in section 1.5.2, for these types of manipulators it

is possible to separate inverse kinematic problem into two sub-problems: position and

orientation. To put it in another way, the 5-DOF robotic manipulator has 3-DOF

46

available at links to find the end position of wrist and 2-DOF available at wrist to find

the orientation of the wrist. It implicates that the robotic manipulator under study has

closed form solutions. Thus, the wrist position pW can be calculated as given in

equation 3.24:

𝑝𝑊 = 𝑝𝑒 − 𝐿3𝑎𝑒 (3.24)

where, pe denotes the end-effector position and orientation is specified in terms of

𝑅𝑒 = [𝑛𝑒 𝑜𝑒 𝑎𝑒], respectively. Here, {ne, oe, ae} are normal, orientation and

approach vectors for end-effector represented by the three components along x, y and

z directions.

Finally, the position of the wrist is obtained by:

[

𝑝𝑤𝑥𝑝𝑤𝑦𝑝𝑤𝑧

] = [

𝑝𝑒𝑥 − 𝐿3𝑎𝑒𝑥𝑝𝑒𝑦 − 𝐿3𝑎𝑒𝑦𝑝𝑒𝑧 − 𝐿3𝑎𝑒𝑧

] (3.25)

The inverse kinematics solution for the complete 5-DOF robotic manipulator

is obtained by a closed solution of the above equation. Thus, the general solutions for

the joint angles are given as:

𝜃1 = tan−1 (𝑝𝑦

𝑝𝑥) (3.26)

𝜃2 = tan−1 (

𝐿0−𝑝𝑧

𝑝𝑥𝐶1+𝑝𝑦𝑆1) − tan−1(

𝐿2𝑆3

√(𝑝𝑥𝐶1+𝑝𝑦𝑆1)2+(𝐿0−𝑝𝑧)2−(𝐿2𝑆3)2

) (3.27)

where, 𝑆1 = ±√1 − 𝐶12

𝜃3 = tan−1 (

𝑆3

𝐶3) (3.28)

where,

𝐶3 = (𝑝𝑥𝐶1+𝑝𝑦𝑆1)

2+(𝐿0−𝑝𝑧)

2−𝐿12−𝐿2

2

2𝐿1𝐿2 , 𝑆3 = ±√1 − 𝐶3

2

Also, the general solutions for wrist rotations are obtained as:

𝜃4 = tan−1((𝑎𝑥𝐶1𝑆23+𝑎𝑦𝑆1𝑆23+𝑎𝑧𝐶23))

(𝑎𝑧𝑆23−𝑎𝑥𝐶1𝐶23−𝑎𝑦𝑆1𝐶23) (3.29)

47

𝜃5 = tan−1

((𝑛𝑦𝐶1−𝑛𝑥𝑆1))

(𝑜𝑦𝐶1−𝑜𝑥𝑆1) (3.30)

where, L0 = 226 mm, L1=179 mm, L2 = 177 mm are the respective link lengths and L3

= 80 mm, is the distance between wrist center and end-effector center.

The above inverse kinematic solution gives one value of θ1 and two values of

θ2 and θ3 each as per the rotations of link 1, link 2 and link 3 given in Table 3.3. Two

solutions are possible for θ4 and one solution exist for θ5 as per wrist rotations. It is

clear from the obtained solutions that a number of multiple solutions are possible. It

can be seen from equation 3.26 to equation 3.30, that a total of 08 multiple solutions

exist for all possible combinations of joint angles with wrist rotation for the 5-DOF

robotic manipulator.

3.3 Minimum, Maximum Reach and Workspace

Reach is a function of robotic manipulator’s link lengths and its configuration.

The minimum distance a robotic manipulator can attain within the work-volume is

called as minimum reach. The maximum distance attained by robotic manipulator is

said to be maximum reach. To put it another way, it leads to boundary singularities. It

is possible only when the links are close enough to the body of the manipulator or

stretched out completely. Figure 3.6 shows the two possible extreme conditions of

boundary conditions for 5-DOF Pravak robotic manipulator. The link lengths for 5-

DOF Pravak robotic manipulator are 226 mm, 179 mm and 177 mm, respectively. It

can be clearly seen from Figure 3.6 that the maximum reach of the used robotic

manipulator are points on the circle of radius (l1 + l2 + l3) and the closest it can reach

from the origin are points on the circle of radius (l1 – l2 – l3), respectively.

As discussed in section 1.5.2, a workspace of any robotic manipulator is

defined as the space where a manipulator can move and operate its wrist end, or it is

the set of values of position and orientation of end position of wrist for which the

inverse kinematics solution exists. Due to the physical limitations of robotic

manipulator, given in terms of range of each joint angle and wrist motion, the work-

volume includes boundary singularities. Since, both the robotic manipulators under

study move in a three dimensional space, thus, the workspace obtained is as shown in

Figure 3.7 and Figure 3.8, respectively.

48

(a) Links close to the body

(b) Links stretched out completely

Figure 3.6 Boundary singularities of robotic manipulator

49

Figure 3.7 Workspace of 3-DOF Omni Bundle robotic manipulator

Figure 3.8 Workspace of 5-DOF Pravak robotic manipulator

150200

250

-200

-100

0

100

200

-50

0

50

X

X-Y-Z co-ordinates generated for all theta1, theta2 and theta3 combinations using forward kinematics formula

Y

Z

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1.25

1.3

1.35

1.4

1.45

1.5

1.55

TH

ET

A1D

- T

HE

TA

1P

Deduced theta1 - Predicted theta1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100008

8.5

9

9.5

10

10.5

11

11.5

12

12.5

13

TH

ET

A2D

- T

HE

TA

2P

Deduced theta2 - Predicted theta2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-2.91

-2.905

-2.9

-2.895

-2.89

-2.885

-2.88

-2.875

-2.87

-2.865

-2.86

TH

ET

A3D

- T

HE

TA

3P

Deduced theta3 - Predicted theta3

-500

0

500

-1

-0.5

0

0.5

1-400

-200

0

200

400

600

X axisY axis

Z a

xis

50

3.4 Conclusions

This Chapter introduced two different multi-DOF robotic manipulators of

industrial use. The forward and inverse kinematic analysis was performed for both 3-

DOF and 5-DOF Pravak robotic manipulators. Also, the reach of robotic manipulators

along with their workspace was demonstrated.