CHAPTER 3 MODELLING, SIMULATION AND KINEMATIC STUDY …shodhganga.inflibnet.ac.in/bitstream/10603/16739/8/08_chapter 3.pdf · chapter 3 modelling, simulation and kinematic study of

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CHAPTER 3

MODELLING, SIMULATION AND KINEMATIC STUDY

OF 3 - DOF PARALLEL MANIPULATORS

3.1 INTRODUCTION

Most existing PKM can be classified into two main families. The

PKM of the first family generally called as “hexapods”. They have a Stewart-

Gought parallel kinematic architecture. Many prototypes and commercial

hexapod PKM already exist like the CMW300, the TORNADO 2000, the

MIKROMAT 6X, the hexapod OKUMA, the hexapod G500. In this first

family, a hybrid architecture with a 2-axis wrist mounted in series to a 3-DOF

tripod positioning structure (the TRICEPT from Neos Robotics) also found.

The second family of PKM has been more recently investigated. In this

category the HEXAGLIDE which features six parallel and coplanar linear

joints is found. The HexaM is another example with non coplanar linear

joints. A 3-axis translational version of the hexaglide is the TRIGLIDE

(Mikron), which has three coplanar and parallel linear joints. Another 3-axis

translational PKM is proposed by the ISW Uni Stuttgart with the LINAPOD.

This PKM has three vertical (non coplanar) linear joints. The URANE SX and

the QUICK-STEP are 3-axis PKM with three non coplanar horizontal linear

joints. The SPRINT Z3 is a 3-axis PKM with one degree of translation and

two degrees of rotations. A hybrid parallel/serial PKM with three parallel

inclined linear joints and a two-axis wrist is the GEORGE V (Philippe

Wenger and Damien Chablat 2002). Dan Zhang et al (2006) addressed that

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since machining operation requires five axes at most, new configurations with

less than six parallel axes would be more appropriate. Development on new

configurations is mainly on three axes PKMs. Examples include Tri-Glide

and Tripod.

From the literatures, the existing PKM structures Tripod and Tri-Glide

are considered for the study. In conventional drilling machine, drilling an

angular hole requires special fixtures for each specific angle. In this present

work, the Tri-Glide and Tripod parallel manipulators mechanisms have been

modified to an angular drilling machine for performing angular drilling

operation. The mobile platform of the Tri-Glide and Tripod were used as

worktable to attain the required angular tilt. This chapter deals with two types

of 3-DOF PMs for the kinematic study. In the first type, the lead screws are

kept or placed horizontally (Glide type), and in the second type the lead

screws are kept vertically (POD type).

This study is carried out to find the parallel configuration, which will

give a better MP tilt for the smaller linear displacement of the nut, a larger

work volume and better singular positions. At first, the architectural

description and mobility of the PMs are briefly given. Secondly, the

kinematic analyses with single link movement of the PMs are illustrated.

Later, the work volume by the Pappus-Guldinus theorem is determined, and

the singular positions are simulated. Finally, a study is carried out based on

the experimental, analytical and simulation results.

3.2 ARCHITECTURAL DESCRIPTION

Figures 3.1 and 3.2 show the 3-PRS Tripod and Tri-Glide PMs,

which depict the various names of the linkage assembly. These mechanisms

typically consist of a circular plate, referred to as the MP. This MP is

connected to a base platform through links. The link is connected to a revolute

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joint at the bottom end, and a spherical joint at the other end. The revolute

joint is attached to the nut, which is mounted on the guide way. The guide

way consists of a lead screw and two guide rods. The mobility or motion of

the MP is accomplished by the screw and nut pairs on the guide ways.

Figure 3.1 Tripod PM

Figure 3.2 Tri-Glide PM

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3.3 MOBILITY EQUATION

The degrees of freedom of the PM are mainly dependent on the

number of links which connect the MP and the base platform. In this work,

the links are connected by spherical joints to the MP at one end, and the other

end is connected by pin joints to the nuts. Nuts are mounted on the lead

screws which are actuated by the stepper motors. The mobility of the

mechanism is calculated by Equation (1.2).

DOF = 6(8-9-1) + 15 = 3 (3.1)

= 6 for the spatial PM, Fi = 1 for the Revolute joints and Prismatic

joints and Fi = 3 for the Spherical joints. For the proposed mechanisms, N=8

(3 links, 3 nuts, 1 MP and 1 BP), J=9 and Fi = 15. Therefore, the given

mechanisms have 3-DOF. The 3-DOF of the PMs are 1) the rotation about the

x axis, 2) rotation about the y axis and 3) translation along the z axis.

3.4 KINEMATICS OF SINGLE LINK MOVEMENT

The fundamental problem of robot kinematics deals with mapping

between vectors in two spaces, viz., joint space ( ) and Cartesian space (X),

where ‘ ’ represents the position and orientation of the manipulator in the

kinematic analysis.

a) MP b) BP

Figure 3.3 Geometrical representation of the MP and BP of the two PMs

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Two coordinate systems (global and local) are used to describe the

position of the mobile platform of the manipulator, as shown in Figures 3.3.

The kinematics also considers the motion conversion in the spherical joints

(S1, S2 and S3) and the pin joints (P1, P2 and P3). The ball joints are used to

rotate the mobile platform in any specified direction, and the pin joint

connects the lower end of the link. The forward kinematics refers to the

computation of the position or motion of each link as a function of the joint

variables. The kinematic equation for finding the angle of the mobile platform

is to be found, in terms of the link length, joint angles and the radius of the

mobile platform.

3.4.1 Kinematics of Tripod

Since the ball joints are placed at the vertices of an equilateral

triangle, the Cartesian position or the origin of the X, Y and Z frame is

essentially the centroid of the triangle. The spherical joints 1 and 3 are

considered as fixed when the link connected with spherical joint 2 is actuated.

Figure 3.4 Single link movement of Tripod

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Equations (3.2) to (3.9) are formulated based on the arrangement of

the single link movement of the Tripod PM, as shown in Figure 3.4.

L = 1.5r (3.2)

X = L + L Cos (3.3)

D = p × n (3.4)

H = Z D (3.5)

From the triangle AOE in Figure 3.4,

Cos =(( )( ) )

(3.6)

From the triangle AEC in Figure 3.4,

Cos =(( )

(3.7)

= = tan (3.8)

= + (3.9)

The input parameters of the single link movement are link length

(L), nut displacement (D), mobile platform radius (r), initial angle between

link and base platform ( ). The output parameter is mobile platform tilt ( ).

3.4.2 Kinematics of Tri-Glide

From Figure 3.5, the kinematics of the single link movement of the

Tri-Glide PM is formulated for the nut movement of towards centre (Tri-

Glide-A) and the nut movement of away from the centre (Tri-Glide-B) for the

MP tilts are calculated from Equations (3.10) to (3.14).

Nut movement towards centre,

From the triangle AEC in Figure 3.5a,

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Cos = ( )(( ( ) )

(3.10)

From the triangle AOF in Figure 3.5a,

Cos =( ) (( ( ) )

( ) (( ( ) )(3.11)

(a) Towards the centre

(b) Away from the centre

Figure 3.5 Single link movement of the Tri-Glide PM

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Nut movement away from centre,

From the triangle AOG in Figure 3.5b,

Cos = ( )(( ( ) )

(3.12)

From the triangle AOF in Figure 3.5b,

Cos =( ) (( )

( ) )(3.13)

= (3.14)

3.4.3 Kinematic Synthesis of the PM

The synthesis of the mechanism is the design or criteria of the mechanism to

produce a desired output motion for a given input motion. Merlet (2005)

stated that dimensional synthesis is to determine the length of the links, the

axis and location of the joints, etc. The word dimension will have the broad

sense of any parameter that will influence the robot behavior and is needed

for the manufacturing of the robot.

In this thesis, the dimensional synthesis is taken into account, for

the determination of the suitable dimensions of the mechanism, by a logical

approach. The greater mobile platform tilt for smaller displacement of nut is

considered as an important factor for the proposed study. The geometrical

parameters are taken by logical approach. The link lengths are considered to

be 200, 300,400 and 500 mm. Similarly, the dimensions of the radius of the

MP are considered to be 60, 70, 80 90 and 100 mm. The initial angle between

MP and link are considered as 65º, 70º, 75º and 80º. The simulations of the

PMs are carried out using the ADAMS package. Based on the simulation

results, the synthesis of the mechanisms is carried out.

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3.4.4 Kinematic Modelling and Simulation of the PM

The various parts of two PMs are modeled. The parts are the lead

screws, links, spherical joints (ball & socket joint), base plate etc. The ball

and socket joint is selected to withstand the load and the compactness for light

weight. Similarly, the lead screw is also designed to withstand the force acting

on it. Proper bearings are provided for support by considering the friction and

the force acting on them. The pay load is calculated from the thrust force of

the drill by the Equation (3.15),

Thrust force (T) = k d f 0.7 (3.15)

Where,

k = 84.7 for steel (Rao 2011), d = Diameter of the drill in mm, f = Feed rate in

mm / rev.

T = 84.7 x 4 x (0.1) 0.7 = 67.6 N

The calculated thrust force is considered for pay load calculations.

The PMs are modeled by considering a payload of 150N with a factor of

safety more than 2. Considering these specifications as the target parameters,

the PMs are modeled and simulated.

3.5 WORK VOLUME ANALYSIS

Work space analysis is a specified problem in Direct Kinematics,

and it can be conveniently solved by formulating input – output equations in a

suitable form for easy repetitive calculations, which are needed to compute all

the reachable positions and the orientations of the MP. The work volume can

be considered as the union of the unit volumes, which have the same manifold

geometry. For a PM, the work volume is limited only because of the bounded

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range of the linear actuators, the mechanical limits on the passive joints, and

the interference between the links.

The workspace of the mechanism was studied, using different

methods, e.g., geometric and numerical approaches. But most of them are

related to the position workspace, which is a part of the workspace. In fact,

the workspace can be divided into the position workspace and the orientation

workspace for a manipulator with rotational capability.

The problem of the determination of the workspace, in terms of the

volume of the PMs, is dealt here. The spherical joints connected to the MP

and the circumferential points of the MP are considered here, for finding the

work volume.

3.5.1 Analytical Approach

The analytical approach is carried out to determine the work

volume of the PMs by considering the MP geometry. Knowing the MP radius

(r) and its angle of tilt ( ), the work volume is determined using the Pappus-

Guldinus theorem. The theorem states that “the volume of the body of

revolution is equal to the generating area times the distance travelled by the

centroid area while the body is being generated”.

Figure 3.6 Schematic representation of MP for work volume analysis

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a) MP b) I region c) II region

Figure 3.7 Regions of surface area of the MP

The circle ABCD of Figure 3.6 represents the MP of the PMs. In

this approach, the total surface is divided into two separate regions to

calculate their area and the centroid as shown in Figure 3.7. The work volume

profiles of different parts of the MP are shown in Figures 3.8 and 3.9.

Figure 3.8 Work volume of I region Figure 3.9 Work volume of II region

Using the Pappus-Guldinus theorem, the work volume is calculated

from the following Equations (3.16) to (3.22).

Consider the region ABCD,

Area 1 (OEAFO) = (3.16)

Area 2 (OEGBO) = (3.17)

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Area 3 (OBDO) = 1 2 × 0.866 × 2 (3.18)

Area 4 (BCD) = Area 1- Area 3 (3.19)

Work volume = Surface area generated × Distance traveled by the centroid

Work volume 1(BAD) = ( + 2 + ) 2360 (3.20)

Work volume 2 (BCD) = ( ) 2360 (3.21)

Total work volume = Work volume 1 + Work volume 2 (3.22)

3.5.1.1 Normalization of Work volume

Normalization is the process to transform different scales and units

among various criteria into common measurable units to allow comparisons

across the work volume. Assume fij to be of the evaluation value R of link

length j under work volume i, then an element rij of the normalized evaluation

value R can be calculated from the Equation (3.23).

J

jij

ijij

f

fr

1

2

j = 1,2,3,….,J, i = 1,2,3,….,n. (3.23)

3.5.2 Methodology

ADAMS is used to simulate the PMs. Once the simulation is done,

the positions of the joints on the MP are noted, with respect to the simulation

time. The simulations of the PMs are done and shown in Figures 3.10 a to f.

The positions of the spherical joints connected to the MP and the

circumferential points of the MP are noted, with respect to the simulation

time. The noted points are given as the input to the MATLAB software, for

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getting the work envelope, and later to the AUTOCAD package for

determining the work volume.

a b c

d e f

Figure 3.10 Simulation of the MP circumferential points a) Tripod Link

1 Movement b) Tripod Link 2 Movement c) Tripod Link 3

Movement d) Tri-Glide Link 1 Movement e) Tri-Glide Link

2 Movement f) Tri-Glide Link 3 Movement

3.6 SINGULARITY ANALYSIS

Singularity is a significant problem in parallel mechanisms,

because it leads to loss of controllability and degradation of the stiffness of

the system. The Figures 3.11 and 3.12 shows the schematic diagram of Type I

and Type II Singularity positions.

Type I Singularity position shows the posture of the link 1 and the

mobile platform being in the same plane. Similarly, the posture of link 2, link

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3 and the mobile platform being in the same plane in the Type II Singularity

position (Xiang Cheng et al 2004).

Figure 3.11 Singular Poses - Type I Singularity

Figure 3.12 Singular Poses - Type II Singularity

The proposed mechanisms are simulated to find the various

singularities, and the positions of the PMs are as shown in Figures 3.13a to f.

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Figures 3.13a and 3.13d show the first kind (Type I) of singularity of the PRS

PMs, when the MP has 180º with any one of the links. Similarly, the MP is in

line with the other two links (L2, L3) while link (L1) is kept constant.

a b c

d e f

Figure 3.13 Three kinds of singularity positions a) Tripod Type I

singularity b) Tripod Type II singularity c) Tripod Type III

singularity d) Tri-Glide Type I singularity e) Tri-Glide Type

II singularity f) Tri-Glide Type III singularity

The second kind (Type II) of singularity of the mechanism is

obtained as shown in Figures 3.13b and 3.13e. The third kind (Type III) of

singularity is obtained when all the links are actuated in different positions to

reach the first or second type of singularity position as shown in Figure 3.13c

and 3.13f. The above mentioned three kinds of singularities are taken in to

consideration for the comparative study of the two PMs.

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3.7 EXPERIMENTAL MODELS

The experimental models of the Tripod and the Tri-Glide PMs are

built, and are shown in Figures 3.14 and 3.17. The models are made of the

same geometrical size of the simulation models. Each link is actuated at

various positions to get various MP tilts.

Figure 3.14 One Link arrangement of Tri-Glide PM

Figure 3.15 Prototype model of Tri-Glide PM

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Figure 3.16 One Link arrangement of Tripod PM

Figure 3.17 Prototype model of Tripod PM

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3.8 METHODS USED IN THE POSITION ANALYSIS

3.8.1 Experiment

An experiment was carried out for the position analysis by

actuating one of the links at a time. A laser torch is mounted on the center ‘C’

of the MP, and the laser light is projected on the vertical screen at a point ‘A’,

which is at a predetermined distance ‘AC’ from the point of the laser source.

When the MP is tilted by the actuation of the link, the laser beam gets

deflected to some other point ‘B’ on the vertical screen. The angle of tilt of

the MP is measured from the orientation of the source. From Figure 3.18, the

MP tilt angle ( ) is measured from the Equation (3.24).

= tan-1(AB / AC) (3.24)

Figure 3.18 Calculation of angle of tilt of the MP

Figure 3.19 and 3.20 show the experimental setup of laser torch on

PMs. The Laser torch is placed on the MP in such a way that the laser torch

axis is aligned with one of the joint axis from the MP. The experimental

procedures of actual experiments are shown in Figure 3.21 to 3.26. In initial

position all the links are placed in same distance in such a way that the nut

positions are same in the lead screws. Similarly, at the final position any one

or two links are actuated and the other link or links are kept constant. Due to

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the various positions of nut, the MP is tilted. The tilt is measured from the

positions of laser source.

Figure 3.19 Experimental setup with laser torch – Tripod PM

Figure 3.20 Experimental setup with laser torch - Tri-Glide PM

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Figure 3.21 Positioning of laser torch

Figure 3.22 Initial position of PM with laser torch

In Figure 3.24, the link1 is kept constant and the links 2, 3 are

actuated. In initial position all the links are placed in same distance in such a

way that the nut positions are same in the lead screws.

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Figure 3.23 Laser source on screen at initial position

Figure 3.24 Final position of PM with laser torch

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Figure 3.25 Laser source on screen at final position

Figure 3.26 Laser source positions on screen

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3.8.2 Analytical

Software programs were written in C and JAVA languages for

finding the displacement of the nut for the PMs. The programs were written

based on the kinematic equations formulated. The tilt of the MP obtained by

actuating the links was obtained from the programs. In the analytical method,

in order to verify the displacement of the nut, the angle of tilt of the MP is

given as the input. Once the input is given, the program calculates the linear

displacement of the nut and the number of rotations of the screw, which will

be useful for giving the pulses to the stepper motor.

3.8.3 Measuring the MP tilt using the Accelerometer

To measure the angle of tilt of the MP, an accelerometer which

works on 2.6 V to 5 V power supply is used, and it is directly interfaced to the

ADC of a microcontroller. This module is used to sense the motion or the tilt

in 3 axes.

The accelerometer senses tilt angle which is manifestation of

acceleration. The reference frame of the accelerometer and the coordinate

frame of the mobile platform are aligned with each other and hence the tilt

angle obtained from the accelerometer corresponds to the orientation of the

mobile platform. For the present study the ADXL 335 and ADXL 203

accelerometers are used and its specifications are given in Appendix 1.

The only drawback of the accelerometer is that it gives outputs only

in the form of voltages. The angle at which the MP is directly tilted is not

measured directly. This is why the calibration of the accelerometer module is

very much essential.

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3.8.3.1 Calibration of accelerometer

The calibration of the accelerometer is carried out by considering

the output voltages, that the accelerometer module gives for certain specific

standard angles. By knowing these voltages, the necessary angle of tilt of the

MP can be found. The following steps are used for the calibration of the

accelerometer,

1) A sine bar is first placed on a flat surface plate. The sine bar is

used to get the different angles with the help of the gauge

blocks and the corresponding voltage values of the

accelerometer are obtained.

2) At first, the sine bar is tilted at zero degrees when placed over

the surface plate. Now the accelerometer is kept over the sine

bar as shown in Figure 3.27, and the readings are noted when

the sine bar is tilted to various angles.

Figure 3.27 Calibration of the accelerometer

3) With the help of the gauge blocks, different angles are taken,

the readings are tabulated and the corresponding voltages are

tabulated.

4) The circuit connections are made as per the diagram shown in

Figure 3.28. Two multimeters are used to find the MP tilt in

the X direction and in the Y direction.

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Figure 3.28 Circuit diagram of the accelerometer

3.8.4 ADAMS

The manipulator models are constructed in ADAMS by building

the physical attributes of the elements, or the parts in the mechanical systems

that have rigid bodies, point masses, flexible bodies and constraints. The

working models are created using ADAMS, as shown in Figure 3.29. The

models are simulated for the nut displacement of 50 mm and the angles of tilt

of the MP about the X and Y axes are obtained from the simulation graphs.

a b

Figure 3.29 Various constraints of ADAMS models

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The detailed description of Adams model is shown in Figures 3.30

and 3.31. The Figure 3.30 shows the various constrains associated with base

and lead screw, the description of joints are as follows,

Fixed Joint 1 – Between base platform and Channel section

Fixed joint 2 – Between Plate 1 and Channel section

Rotational motion – Between Lead screw and stepper motor

Fixed joint 3 – Between Plate 2 and Guide rod 1

Fixed joint 4 – Between Plate 2 and Guide rod 2

Cylindrical joint 1 – Between Nut and Guide rod 1

Cylindrical joint 2 – Between Nut and Guide rod 2

Figure 3.30 Various constraints associated with base and lead screw

Similarly, the Figure 3.31 shows the various constrains associated

with base and lead screw, the description of joints are as follows,

Fixed Joint 5 – Between Plate1and Channel section

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Pin joint – Between the Nut and Link

Screw joint – Between Nut and Lead Screw

Spherical joint – Between Link and Mobile Platform

Fixed Joint 6 – Between Plate 3 and Guide rod 1

Fixed Joint 7 – Between Plate 3 and Guide rod 2

Figure 3.31 Various constraints associated with link and MP

Similarly, the other link associated joints are made in Tripod

and Tri-Glide PMs. The necessary nut movement is actuated by rotational

joint in terms of angle of rotation with respect to time. The model is simulated

based on the nut displacement.

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3.9 SIMULATION STUDY OF 3- PRS, 3-PRR AND 3-PUS PMs

In the present work, simulation study on three Tripod PMs (3- PRS,

3-PRR and 3-PUS PMs) is carried out to find which Tripod configuration is

better to achieve better singular positions and the MP tilt (by considering the

Transmission angle between MP and the link).

The PUS PM is considered with PRR and PRS PMs here because

the base location of actuators of PUS PM lead to: (i) reduction of the (motor)

weight carried by the legs; (ii) elimination of the actuation transmission

requirement (through intermediary joints as in the case of the Stewart-Gough

platform); and (iii) most importantly absorption of reaction-forces by the

ground. By selecting the base actuated joint to be prismatic, the proximal

links are not subjected to the bending moments and the corresponding stresses

(Madhusudanan Sathia Narayanan 2010).

In this study, first, the Transmission angle and singularity positions

are briefly described and next, the simulation results are compared based on

the geometrical parameters, tilt, singularity and driving torque for the input

motion of the mechanism.

3.9.1 Transmission Angle

Transmission angle is the angle between the coupler link and the

output link of the four bar mechanism. The Transmission angle is an

important index evaluating the quality of motion / force transmission. It helps

to decide the best among a family of possible mechanisms for the most

effective force transmission. Though a good Transmission angle is not a cure

to all for every design problem, for many mechanical applications it can

guarantee the performance of a linkage at high speed without unfavorable

vibrations. The study of link mechanisms shows that Transmission angle is

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significant not only as an indicator of good force and motion transmission but

also a prime factor in the linkage sensitivity to small design parameter errors.

For the purpose of high speed, high accuracy and high quality of

motion transmission, the most widely accepted design limits for the

Transmission angle are (45º , 135º) or (40º , 140º). Additionally, the

Transmission angle does not consider the dynamic forces due to velocity and

accelerations. For this reason, it is widely used in the kinematic synthesis.

A planar four bar mechanism is a single closed loop system. A

parallel robot is a multi closed loop mechanism. Usually, a fully parallel robot

has more or less the characteristic of a planar four bar mechanism. Wang

Jinsong et al (2009) suggested that the design concept of the four bar

mechanism using Transmission angle should be used in the design of parallel

robot. They also proposed the local and global transmission indices in the

design of a special 3-DOF parallel robot, which is kinematically considered as

the combination of two planar mechanisms.

Chang (1988) explained that for a general RSSR-linkage, the axes

of the two R-pairs do not intersect. When these two axes do intersect, the

RSSR-linkage becomes a spherical four-bar.

From the Figure 3.32, the PM is considered as 4 bar linkage when

single link is actuated. In this study, the parallel manipulator is selected for

angular machining application and the term transmission angle is related to

quality motion of the system is considered. Hence, the transmission angle is

introduced in the four bar mechanism and to evaluate the performance of the

system.

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a) Front view

b) Top view

Figure 3.32 Single link movement of PM

a b

c d

Figure 3.33 Synthesis of mechanism using transmission angle ( )

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The transmission angle is an important parameter to evaluate the

quality of motion/force transmission, for this reason, Local Transmission

Index (LTI) is used in the design of parallel manipulator. The presented

mechanisms are synthesized for the transmission angles of 135º to obtain high

quality of motion/force transmission as per LTI. Figures 3.33a and d shows

the simulation of 3-PRR and 3-PUS parallel manipulator single link

movement, similarly Figures 3.33b and c shows two link movement at equal

vertical displacement of the nut.

3.9.2 Simulation of Singularity Positions of PMs

The proposed mechanisms are simulated to find various singularity

positions as explained in section 3.6. Figure 3.34a shows the Type I

singularity of the PUS and PRS PMs. Similarly, Type II singularity of the

mechanism is shown in Figure 3.34b for 3-PRR mechanism. The above

mentioned two kind of singularity is taken in to account for the study.

a) 3- PUS and 3-PRS b) 3-PRR

Figure 3.34 Singularity positions of tripod PMs

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3.10 STRUCTURAL ANALYSIS

Neugebauer et al (2006) summarized that a FEM tool was required

for to optimize the PM structure. The main aim of this structural analysis is to

analyze the structure of the Tripod, Tri-Glide and 3-PRR parallel

manipulators for angular drilling applications. The Parallel manipulator

structures are analyzed based on characteristics like stress, deformation and

moment reactions in PMs joints. The geometrical parameters of the link

structure is taken as 10 mm,12 mm,14 mm,16 mm and18mm diameter for

solid structure. For hollow structure, 18 mm is taken as outer diameter and

inner diameters are varied as 11mm, 12 mm, 13 mm, 14 mm and 15 mm.

Similarly, taking 18 mm as inner diameter the outer diameters are varied as

21mm, 22 mm, 23 mm, 24 mm, and 25 mm.

Figure 3.35 ANSYS model of Tripod PM

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Figure 3.36 ANSYS model of Tri-Glide PM

Figure 3.37 ANSYS model of 3-PRR PM

To reduce the weight of the mild steel (MS) structure, the link is

replaced by the aluminum (Al) alloy. Figures 3.35 to 3.37 show the ANSYS

simulated models of the Tripod, Tri-Glide and 3-PRR PMs using BEAM 189

element. The mechanism is modeled by considering the geometrical

parameters of link length 200 mm, mobile platform radius 90 mm and initial

angle between the link and base platform is 70º. A load of 150N is applied at

the centre of the mobile platform.

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The Materials used in this structural analysis are mild steel and aluminum alloy. Table 3.1 shows the properties of the two materials.

Table 3.1 Material property of mild steel and aluminium alloy

Properties Mild Steel Aluminium Alloy

Young's Modulus 2.1×105 N/mm2 0.7×105 N/mm2

Poisson's Ratio 0.3 0.33

Density 7.85 e-006 kg/mm³ 2.7e-006 kg/mm³

3.11 RESULTS AND DISCUSSION

In the kinematic study of the two PMs, the dimensional synthesis

and the work volume were carried out, to evaluate the influence of the

geometrical parameters (MP radius, Link length and the initial angle between

link and the base platform). Similarly, the simulation of the singularity

positions and the experiments on the position analysis were conducted, to

evaluate the singular configurations and the tilt of the MP of the two PMs.

3.11.1 Dimensional Synthesis

The link lengths (L), the initial angle between the link and the base

), and the radius of the MP (r) are considered to be the important

parameters, which play a vital role in increasing or decreasing the tilt of the

MP of the two PMs, and are compared here. The same geometrical parameters

are used for the purpose of comparing these two configurations. The

geometrical parameters for the two models are as follows, the MP radius is

varied from 60 to 100 mm (60, 70, 80, 90 and100), the link length is varied

from 200 to 500 mm (200, 300, 400 and 500), and is varied from 65° to 80°

(65°, 70°, 75° and 80°).

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Table 3.2 shows the theoretical results of the MP tilt for the Tri-

Glide and Tripod PMs. From the results presented, it is observed that when

the MP radius is 60 mm, the Tripod PM has a maximum MP tilt of 31.80° (for

= 80°and L = 500 mm), and a minimum of 29.37° (for = 65° and L = 200

mm) for the nut displacement of 50 mm. The Tri-Glide (B) PM has a

maximum MP tilt of 30.39° (for = 65° and L = 200 mm), and a minimum of

7.47° (for = 80° and L = 500 mm). Similarly, the Tri-Glide (A) PM has a

maximum MP tilt of 12.44° (for = 65° and L = 500 mm) and a minimum of

1.56° (for = 80° and L = 200 mm).

Table 3.2 MP tilt for MP radius of 60 mm and nut displacement of 50 mm

Initial angle ( )In Deg

Link Length (L)

in mm

Angle of Tilt of MP ( ) in Deg Tri-Glide

(Towards center (A))

Tri-Glide (Away from center (B))

Tripod (C)

65 200 9.91 30.39 29.3765 300 11.31 22.73 29.4665 400 12.01 20.60 29.5165 500 12.44 19.53 29.5370 200 7.12 20.10 30.1070 300 8.48 16.73 30.1970 400 9.16 15.40 30.2370 500 9.58 14.68 30.2675 200 4.34 14.66 30.8475 300 5.68 12.37 30.9375 400 6.35 11.37 30.9875 500 6.76 10.81 31.0180 200 1.56 10.58 31.6180 300 2.90 8.78 31.7180 400 3.56 7.95 31.7780 500 3.96 7.47 31.80

84

From Table 3.3, it is observed that, when the MP radius is 70 mm,

the Tripod PM has a maximum MP tilt of 27.10° (for = 80° and L = 500

mm) and a minimum of 25.34° (for = 65° and L = 200 mm). The Tri-Glide

(B) PM has a maximum MP tilt of 23.29° (for = 65° and L = 200 mm) and a

minimum of 6.38° (for = 80°, L = 500 mm). Similarly, the Tri-Glide (A) PM

has a maximum MP tilt of 10.69° (for = 65° and L = 200 mm) and a

minimum of 1.34° (for = 80° and L = 200 mm).


Initial angle ( )In

Deg

Link Length (L)

in mm




Tripod (C)

65 200 8.51 23.29 25.3465 300 9.71 18.69 25.4065 400 10.32 17.13 25.4365 500 10.69 16.31 25.4470 200 6.10 16.65 25.8870 300 7.27 14.06 25.9470 400 7.87 13.00 25.9770 500 8.22 12.41 25.9875 200 3.72 12.36 26.4275 300 4.87 10.49 26.4875 400 5.45 9.67 26.5175 500 5.80 9.20 26.5380 200 1.34 8.99 26.9880 300 2.48 7.48 27.0580 400 3.06 6.79 27.0880 500 3.40 6.38 27.10

85






has a maximum MP tilt of 9.38° (for = 65° and L = 200 mm) and a minimum

of 1.17° (for = 80° and L = 200 mm).



Link Length (L)

in mm




Tripod (C)

65 200 7.45 19.32 22.3065 300 8.51 15.94 22.3465 400 9.05 14.69 22.3665 500 9.38 14.03 22.3770 200 5.34 14.25 22.7270 300 6.37 12.13 22.7670 400 6.89 11.25 22.7870 500 7.21 10.75 22.7975 200 3.25 10.69 23.1475 300 4.26 9.11 23.1875 400 4.77 8.41 23.2075 500 5.07 8.00 23.2180 200 1.17 7.82 23.5680 300 2.17 6.52 23.6180 400 2.67 5.92 23.6380 500 2.97 5.57 23.64

86







of 1.04° (for = 80° and L = 200 mm).



Link Length (L)

in mm




Tripod (C)

65 200 6.63 16.61 19.9365 300 7.58 13.91 19.9565 400 8.06 12.87 19.9765 500 8.36 12.31 19.9870 200 4.75 12.47 20.2670 300 5.66 10.68 20.2970 400 6.13 9.92 20.3070 500 6.41 9.49 20.3175 200 2.89 9.42 20.5975 300 3.79 8.06 20.6275 400 4.24 7.44 20.6375 500 4.51 7.09 20.6480 200 1.04 6.92 20.9380 300 1.93 5.78 20.9680 400 2.38 5.25 20.9780 500 2.64 4.94 20.98

87







of 0.94° (for = 80° and L = 200 mm).


Initial

angle

) in

Deg

Link Length

(L)

in mm

Angle of Tilt of MP ( ) in Deg

Tri-Glide

(Towards center

(A))

Tri-Glide

(Away from

center (B))

Tripod (C)

65 200 5.97 14.60 18.01

65 300 6.83 12.35 18.04

65 400 7.27 11.46 18.05

65 500 7.53 10.98 18.05

70 200 4.27 11.09 18.29

70 300 5.10 9.54 18.31

70 400 5.52 8.87 18.32

70 500 5.78 8.49 18.33

75 200 2.60 8.43 18.56

75 300 3.41 7.22 18.58

75 400 3.82 6.67 18.59

75 500 4.06 6.36 18.60

80 200 0.94 6.20 18.83

80 300 1.74 5.19 18.85

80 400 2.14 4.72 18.86

80 500 2.38 4.44 18.87

88

3.11.1.1 Influence of the link length on the MP tilt

The dimensional synthesis was carried out by the analytical method, and the results are shown in Figures 3.38 to 3.41.

For the Tripod PM, the dimensional synthesis results show that by increasing the link length from 200 mm to 500 mm, the angle of tilt of the MP was found to increase by 0.54 % (for r = 60 mm and = 65°). Similarly, 0.53% for = 70°, 0.55% for = 75° and 0.59% for = 80°. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to increase by 7.08 % (from = 65° to = 80°) and 7.14 % for the MP radius of 60 mm.

Similarly, by increasing the link length from 200 mm to 500 mm for the Tripod PM, the angle of tilt of the MP was found to increase by 0.22 % (for r = 100 mm and = 65°). Similarly, 0.22% for = 70°, 0.21% for = 75° and 0.21% for = 80°. For the constant link lengths of 200 mm and

500 mm, the MP tilt was found to increase by 4.35 % (from = 65° to = 80°) and 4.34 % for the MP radius of 100 mm.

For the Tri-Glide PM (A), by increasing the link length from 200 mm to 500 mm, the angle of tilt of the MP was found to increase by 20.34 % (for r = 60 mm and = 65°). Similarly, 25.68% for = 70°, 35.80% for = 75° and 60.60% for = 80°. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by 84.26 % (from = 65° to = 80°) and 68.16 % for the MP radius of 60 mm.

Similarly, by increasing the link length from 200 mm to 500 mm for the Tri-Glide PM (A), the angle of tilt of the MP was found to increase by 20.17 % (for r = 100 mm and = 65°). Similarly, 26.12% for = 70°, 35.96% for = 75° and 60.50% for = 80°. For the constant link lengths of 200 mm and 500 mm, the MP tilt was found to decrease by 84.25 % (from = 65° to = 80°) and 68.39 % for the MP radius of 100 mm.

89

For the Tri-Glide PM (B), by increasing the link length from 200

mm to 500 mm, the angle of tilt of the MP was found to decrease by 35.73 %

(for r = 60 mm and = 65°). Similarly, 26.96% for = 70°, 26.26% for =

75° and 29.39% for = 80°. For the constant link lengths of 200 mm and 500

mm, the MP tilt was found to decrease by 65.18 % (from = 65° to = 80°)

and 61.75 % for the MP radius of 60 mm.

Similarly, by increasing the link length from 200 mm to 500 mm

for the Tri-Glide PM (B), the angle of tilt of the MP was found to decrease by

24.79 % (for r = 100 mm and = 65°). Similarly, 23.44% for = 70°, 24.55%

for = 75° and 28.38% for = 80°. For the constant link lengths of 200 mm

and 500 mm, the MP tilt was found to decrease by 57.53 % (from = 65° to

= 80°) and 59.56 % for the MP radius of 100 mm.

3.11.1.2 Influence of the MP radius on the MP tilt

For the Tripod PM, by increasing the radius of MP from 60 mm to

100 mm, the angle of tilt of the MP was found to decrease by 38.67% (for L =

200 mm and = 65°), 39.23% for = 70°, 39.82% for = 75°, 40.43% for

= 80°. Similarly, by increasing the radius of the MP from 50 mm to 100 mm,

the angle of tilt of the MP was found to decrease by 38.87% (for L = 500 mm

and = 65°). Similarly, 39.42% for = 70°, 40% for = 75° and 40.66% for =

80°.

For the Tri-Glide PM (A), by increasing the radius of the MP from

60 mm to 100 mm, the angle of tilt of the MP was found to decrease by

39.76% (for L = 200 mm and = 65°), 40.02% for = 70°, 40.09% for =

75°, 39.74% for = 80°. Similarly, by increasing the radius of the MP from

60 mm to 100 mm, the angle of tilt of the MP was found to decrease by

39.47% (for L = 500 mm and = 65°). Similarly, 39.66% for = 70°, 39.94%

for = 75° and 39.89% for = 80°.

90

For the Tri-Glide PM (B), by increasing the radius of the MP from 60 mm to 100 mm, the angle of tilt of the MP was found to decrease by 51.96% (for L = 200 mm and = 65°), 44.82% for = 70°, 42.49% for = 75°, 41.39% for = 80°. Similarly, by increasing the radius of the MP from 50 mm to 100 mm, the angle of tilt of the MP was found to decrease by 43.78% (for L = 500 mm and = 65°). Similarly, 42.16% for = 70°, 41.16% for = 75° and 40.56% for = 80°.

3.11.1.3 Influence of the initial angle on the MP tilt

For the Tripod PM, by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to increase by 7.08% (for L = 200 mm and r = 60 mm), 6.07% for r = 70 mm, 5.34% for r = 80 mm, 4.77% for r = 90 mm and 4.35% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to increase by 7.14% (for L = 500 mm and r = 60 mm), 6.12% for r = 70 mm, 5.37% for r = 80 mm, 4.76% for r = 90 mm and 4.34% for r = 100 mm.

For the Tri-Glide PM (A), by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to decrease by 84.25% (for L = 200 mm and r = 60 mm), 84.25% for r = 70 mm, 84.29% for r = 80 mm, 84.31% for r = 90 mm and 84.25% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to decrease by 68.17% (for L = 500 mm and r = 60 mm), 68.19% for r = 70 mm, 68.33% for r = 80 mm, 68.42% for r = 90 mm and 68.39% for r = 100 mm.

For the Tri-Glide PM (B), by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to decrease by 65.18% (for L = 200 mm and r = 60 mm), 61.39% for r = 70 mm, 59.52% for r = 80 mm, 58.33% for r = 90 mm and 57.53% for r = 100 mm. Similarly, by increasing the initial angle between the link and the base platform from 65° to 80°, the angle of tilt of the MP was found to decrease by 61.75% (for L = 500 mm and r = 60 mm), 60.88% for r = 70 mm, 60.29% for r = 80 mm, 59.87% for r = 90 mm and 59.56% for r = 100 mm.

91

a) MP = 60 mm b) MP = 70 mm

c) MP = 80 mm d) MP = 90 mm

e) MP = 100 mm

Figure 3.38 Influence of = 65° over MP tilt for various MP radii and

link lengths

92

a) MP = 60 mm b) MP = 70 mm

c) MP = 80 mm d) MP = 90 mm

e) MP = 100 mm


link lengths

93

a) MP = 60 mm b) MP = 70 mm

c) MP = 80 mm d) MP = 90 mm

e) MP = 100 mm


link lengths

94

a) MP = 60 mm b) MP = 70 mm

c) MP = 80 mm d) MP = 90 mm

d) MP = 100 mm


link lengths

95

3.11.1.4 Comparison on maximum and minimum tilt of the MP

The kinematic study on the single link movement of the two PMs

structure is based on the MP tilt, by considering the geometrical parameters,

like the link length, MP radius and initial angle between the link and the base

platform.

Table 3.7 shows the results of the maximum and minimum MP tilt

of the Tri-Glide (A), Tri-Glide (B) and Tripod PMs. By considering the nut

displacement of 50 mm, for the Tripod PM, it is observed that the maximum

angle of tilt of the MP is found to be 31.80°, when the link length is 500 mm

with = 80°, and a minimum of 18.01° is obtained when the link length is

200 mm with = 65°.

Table 3.7 MP tilt comparison for various geometrical parameters for

the nut displacement of 50 mm

MPradius in mm

The angle of the MP tilt in ° Tri-Glide (A) Tri-Glide (B) Tripod (C)

Max(L=500

mm, =65°)

Min(L=200

mm, =80°)

Max(L=200

mm, =65°)

Min(L=500

mm, =80°)

Max(L=500

mm, =80°)

Min(L=200

mm, =65°)

60 12.44 1.56 30.39 7.47 31.80 29.3770 10.69 1.34 23.29 6.38 27.10 25.3480 9.38 1.17 19.32 5.57 22.30 23.6490 8.36 1.04 16.61 4.94 20.98 19.93

100 7.53 0.94 14.60 4.44 18.87 18.01

Similarly, for the Tri-Glide (A) PM, it is observed that a maximum

angle of tilt of the MP of 12.44° is obtained when the link length is 500 mm

with = 65° and a minimum of 0.94° is obtained when the link length is 200

mm with = 80°. For the Tri-Glide (B) PM, it is observed that a maximum

angle of tilt of the MP 30.39° is obtained, when the link length is 200 mm

96

with = 65° and a minimum of 4.44° is obtained, when the link length is 500

mm with = 80°.

From the study, when the variation of the initial angle is 15° ( =

65° to 80°) and the MP radius is 100 mm, the maximum variation of the angle

of tilt of the MP was 0.86° for the Tripod PM. Similarly, under the same

conditions, the angle of tilt of the MP was 10.16° for the Tri-Glide (B) PM

and 6.53° for the Tri-Glide (A) PM respectively. From the above results, it is

observed that the MP tilt of the Tripod PM is more precise than that of the

Tri-Glide (B) PM and Tri-Glide (A) PM, when the link length is increased for

a constant MP radius.

Similarly, when the MP radius is varied for 40 mm (from r = 60

mm to 100 mm) for the constant initial angle = 80°, the maximum variation

of the angle of tilt of the MP was 0.62° for the Tri-Glide (A) PM. Similarly,

under the same condition, the angle of tilt of the MP was 4.38° for the Tri-

Glide (B) PM and 12.78° for the Tripod PM.

From the analytical results, it is observed that the Tri-Glide (B) PM

MP tilt was 30.39° (For r = 60 mm, L = 200 mm and = 65°). Similarly,

under the same condition, the angle of tilt of the MP was 9.91° for the Tri-

Glide (A) and 29.37° for the Tripod PMs.

3.11.2 Work Volume Analysis

In the work volume analysis, the geometrical parameters are taken as

important parameters. To analyze the Tripod and Tri-Glide parallel

manipulators, the same geometrical parameters are used. The geometrical

parameters include the link length, MP radius and initial angle between the link

and the base platform.

97

Table 3.8 shows the results of the Tri-Glide and Tripod PMs for

MP radius 60 mm. From the results presented, it is observed that when the

MP radius is 60 mm, the Tripod PM work volume is found to be a maximum

of 218317.10 mm3 (for = 80°and L = 500 mm) and a minimum of 201642.42

mm3 (for = 65° and L = 200 mm) for the nut displacement of 50 mm. The

Tri-Glide (B) PM work volume is found to be a maximum of 208623.13 mm3

(for = 65° and L = 200 mm) and a minimum of 51312.90 mm3 (for = 80°

and L = 500 mm). Similarly, the Tri-Glide (A) work volume is found to be a

maximum of 85395.20 mm3 (for = 65° and L = 500 mm) and a minimum of

10740.82 mm3 (for = 80° and L = 200 mm).

Table 3.8 Work volume for MP radius of 60 mm for nut displacement 50 mm

Initial angle In Deg

Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))

Tri-Glide (Away from center(B))

Tripod (C)

65 200 68065.89 208623.13 201642.4265 300 77628.40 156019.22 202255.3465 400 82468.43 141449.92 202562.2365 500 85395.20 134086.99 202746.5470 200 48847.49 137983.07 206613.6470 300 58197.29 114878.38 207241.8970 400 62910.46 105747.56 207557.5470 500 65754.25 100760.68 207747.4575 200 29768.78 100639.40 211703.7875 300 38982.05 84925.67 212365.8475 400 43607.42 78074.40 212699.6975 500 46392.12 74198.46 212900.9580 200 10740.82 72610.66 217015.8780 300 19892.75 60254.81 217733.9080 400 24468.37 54587.55 218097.4580 500 27217.06 51312.90 218317.10

98

From Table 3.9 normalized work volume results, it is observed that

the Tripod PM work volume is found to be a maximum of 0.259864 (for =

80°and L = 500 mm) and a minimum of 0.240016 (for = 65° and L = 200

mm). Similarly, Tri-Glide (B) and Tri-Glide (A) PM work volume is found to

be a maximum of 0.4631702 (for = 65° and L = 200 mm), 0.3934092 (for

= 65° and L = 500 mm) and a minimum of 0.1139212 (for = 80° and L = 500

mm), 0.0494821 (for = 80° and L = 200 mm).

Table 3.9 Normalized work volume for MP radius of 60 mm for nut displacement 50 mm


Link Length in mm

Work volume

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 0.3135744 0.4631702 0.24001665 300 0.3576281 0.3463827 0.24074665 400 0.3799258 0.314037 0.24111165 500 0.3934092 0.2976904 0.24133170 200 0.2250367 0.3063402 0.24593470 300 0.2681105 0.2550448 0.24668170 400 0.2898237 0.2347732 0.24705770 500 0.3029248 0.2237017 0.24728375 200 0.1371425 0.2234324 0.25199375 300 0.1795873 0.1885459 0.25278175 400 0.2008961 0.1733352 0.25317875 500 0.213725 0.1647301 0.25341880 200 0.0494821 0.161205 0.25831680 300 0.0916444 0.1337734 0.2591780 400 0.1127239 0.1211914 0.25960380 500 0.1253869 0.1139212 0.259864

99

Table 3.10 shows the results of the Tri-Glide and Tripod PMs for

MP radius 60 mm. From the results presented, it is observed that when the

MP radius is 60 mm, the Tripod, Tri-Glide (B) and Tri-Glide (A) PMs work

volume is found to be a maximum of 295414.12 mm3 (for = 80°and L = 500

mm), 208623.13 mm3 (for = 65° and L = 200 mm), 85395.20 mm3 (for =

65° and L = 500 mm) and a minimum of 201642.42 mm3 (for = 65° and L

= 200 mm), 51312.90 mm3 (for = 80° and L = 500 mm) and 10740.82 mm3

(for = 80° and L = 200 mm).



Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 92728.41 253935.79 276243.75

65 300 105872.81 203798.41 276877.14

65 400 112547.12 186751.63 277193.61

65 500 116590.78 177826.54 277383.45

70 200 66497.78 181494.22 282112.28

70 300 79286.42 153269.39 282749.59

70 400 85747.99 141676.14 283068.92

70 500 89652.24 135265.02 283260.76

75 200 40510.53 134724.43 288045.25

75 300 53071.03 114384.56 288703.57

75 400 59385.98 105378.99 289034.37

75 500 63191.47 100254.49 289233.42

80 200 14616.79 97992.19 294151.43

80 300 27073.90 81577.39 294849.86

80 400 33305.74 73991.69 295201.92

80 500 37051.16 69595.16 295414.12

100

From Table 3.11 normalized work volume results, it is observed

that when the MP radius is 70 mm, the work volume of the Tripod PM, Tri-

Glide (B) and Tri-Glide (A) PMs is found to be a maximum of 0.258297757,

(for = 80°and L = 500 mm), 0.4315281(for = 65°and L = 200 mm) and

0.3939853 (for = 65°and L = 500 mm) and a minimum of 0.241535987 (for

= 65° and L = 200 mm), 0.1182672 (for = 80°and L = 500 mm) and

0.0493933 (for = 80°and L = 200 mm).



Link Length in mm

Work volume

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 0.3133493 0.4315281 0.24153598765 300 0.357767 0.3463267 0.24208979765 400 0.3803209 0.3173581 0.24236650565 500 0.3939853 0.3021911 0.24253249370 200 0.2247103 0.3084238 0.24666718570 300 0.2679259 0.2604597 0.24722442270 400 0.2897609 0.2407586 0.24750363170 500 0.3029542 0.2298638 0.24767136775 200 0.1368938 0.2289452 0.25185472575 300 0.1793384 0.1943804 0.25243033375 400 0.200678 0.1790767 0.25271957175 500 0.2135376 0.1703684 0.25289361280 200 0.0493933 0.1665239 0.25719371480 300 0.0914885 0.1386293 0.25780439280 400 0.1125473 0.1257384 0.25811221980 500 0.1252038 0.1182672 0.258297757

101

From Table 3.12, it is observed that when the MP radius is 80 mm,

the work volume of the Tripod PM is found to be a maximum of 384696.98

mm3 (for = 80°and L = 500 mm) and a minimum of 362918.59 mm3 (for =

65° and L = 200 mm). Similarly, Tri-Glide (B) and Tri-Glide (A) PM work

volume is found to be a maximum of 314391.09 mm3 (for = 65° and L = 200

mm) and 152666.76 mm3 (for = 65° and L = 500 mm) and a minimum of

90667.76 mm3 (for = 80° and L = 500 mm), 19088.77 mm3 (for = 80° and

L = 200 mm).



Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 121211.47 314391.09 362918.5965 300 138514.39 259322.82 363571.4065 400 147322.37 239070.13 363897.1965 500 152666.76 228259.42 364092.4870 200 86870.65 231836.71 369712.3270 300 103639.63 197473.96 370360.2870 400 112127.63 183031.43 370684.4370 500 117261.99 174980.93 370879.0075 200 52905.13 173954.53 376517.6575 300 69332.12 148296.76 377177.3775 400 77600.36 136820.18 377508.2175 500 82586.61 130263.51 377707.0580 200 19088.77 127215.64 383452.8580 300 35359.93 106143.12 384141.9880 400 43503.65 96353.75 384488.4680 500 48399.99 90667.76 384696.98

102



(for = 80°and L = 500 mm) and a minimum of 0.24263046 (for = 65° and

L = 200 mm) for the nut displacement of 50 mm. The Tri-Glide (B) PM work

volume is found to be a maximum 0.418220696 (for = 65° and L = 200

mm) and a minimum of 0.12061135 (for = 80° and L = 500 mm). Similarly,

the) work volume is found to be a maximum of 0.39444926 (for = 65° and L

= 500 mm) and a minimum of 0.049320174 (for = 80° and L = 200 mm).

Table 3.13 Normalized Work volume for MP radius of 80 mm for nut displacement 50 mm


Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 0.313177372 0.418220696 0.24263046

65 300 0.357883397 0.344965789 0.2430669

65 400 0.380640814 0.318024522 0.24328471

65 500 0.39444926 0.303643508 0.24341527

70 200 0.224450061 0.308402221 0.24717243

70 300 0.267776531 0.262690959 0.24760563

70 400 0.289707207 0.243478694 0.24782234

70 500 0.302972993 0.232769467 0.24795242

75 200 0.136692423 0.231404092 0.25172216

75 300 0.179135284 0.197272684 0.25216322

75 400 0.200498161 0.182005893 0.2523844

75 500 0.21338127 0.173283842 0.25251734

80 200 0.049320174 0.169229394 0.25635871

80 300 0.091360413 0.141197543 0.25681943

80 400 0.112401564 0.128175173 0.25705107

80 500 0.125052371 0.12061135 0.25719048

103



mm3 (for = 80°and L = 500 mm) and a minimum of 461681.47 mm3 (for

= 65° and L = 200 mm). The Tri-Glide (B) PM work volume is found to be a

maximum of 384810.15 mm3 (for = 65° and L = 200 mm) and a minimum

of 114530.45 mm3 (for = 80° and L = 500 mm). Similarly, the Tri-Glide (A)

work volume is found to be a maximum of 193623.90 mm3 (for = 65° and L

= 500 mm) and a minimum of 24156.77 mm3 (for = 80° and L = 200 mm).



Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 153515.18 384810.15 461681.4765 300 175553.46 322287.26 462352.0765 400 186794.74 298279.69 462686.5165 500 193623.90 285306.69 462886.9170 200 109966.13 288842.06 469420.0070 300 131256.99 247457.65 470078.6470 400 142049.50 229795.81 470407.8170 500 148583.68 219896.33 470605.2975 200 66952.57 218311.08 477117.2875 300 87765.32 186657.18 477780.5475 400 98250.57 172395.19 478112.7375 500 104577.56 164223.57 478312.2480 200 24156.77 160278.50 484903.3380 300 44750.85 133951.30 485588.1280 400 55062.10 121673.36 485931.8480 500 61263.55 114530.45 486138.52

104

From Table 3.15, it is observed that when the MP radius is 90

mm, the work volume of the Tripod PM is found to be a maximum of

0.256359 (for = 80°and L = 500 mm) and a minimum of 0.243462 (for

= 65° and L = 200 mm). The Tri-Glide (B) PM work volume is found to be a

maximum of 0.4104684 (for = 65° and L = 200 mm) and a minimum of

0.1221671 (for = 80° and L = 500 mm). Similarly, the Tri-Glide (A) work

volume is found to be a maximum of 0.3948296 (for = 65° and L = 500 mm)

and a minimum of 0.0492595 (for = 80° and L = 200 mm).


Initial angle in Deg

Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 0.3130416 0.4104684 0.243462

65 300 0.3579811 0.3437766 0.243816

65 400 0.3809039 0.3181683 0.243992

65 500 0.3948296 0.3043303 0.244098

70 200 0.2242382 0.3081014 0.247543

70 300 0.2676537 0.2639576 0.24789

70 400 0.2896613 0.2451181 0.248064

70 500 0.3029855 0.2345585 0.248168

75 200 0.1365268 0.2328676 0.251602

75 300 0.1789673 0.1991031 0.251952

75 400 0.2003484 0.1838901 0.252127

75 500 0.2132501 0.1751736 0.252232

80 200 0.0492595 0.1709655 0.255708

80 300 0.091254 0.1428829 0.256069

80 400 0.1122803 0.1297863 0.25625

80 500 0.124926 0.1221671 0.256359

105



599727.41 mm3 (for = 80°and L = 500 mm) and a minimum of 572543.37

mm3 (for = 65° and L = 200 mm). The Tri-Glide (B) and Tri-Glide (A) PM

work volume is found to be a maximum of 464132.64 mm3 (for = 65° and L

= 200 mm) and 239462.70 mm3 (for = 65° and L = 500 mm) and a minimum

of 141183.09 mm3 (for = 80° and L = 500 mm) and 29820.78 mm3 (for =

80° and L = 200 mm).



Link Length in mm

Work volume in mm^3

Tri-Glide (Towards

center (A))


Tripod (C)

65 200 189639.60 464132.64 572543.3765 300 216990.27 392570.38 573230.0965 400 230964.62 364325.01 573572.4165 500 239462.70 348932.19 573777.4970 200 135784.20 352439.11 581241.2170 300 162138.55 303203.79 581910.0170 400 175513.69 281960.39 582244.0570 500 183617.44 270005.03 582444.3875 200 82652.84 267784.81 589844.4075 300 108370.64 229463.13 590512.0475 400 121336.65 212102.54 590846.1475 500 129164.34 202133.62 591046.7080 200 29820.78 197179.43 598496.1180 300 55246.64 165001.53 599179.1580 400 67981.10 149950.32 599521.6180 500 75641.82 141183.09 599727.41

106



0.255708741 (for = 80°and L = 500 mm) and a minimum of 0.244118148

(for = 65° and L = 200 mm). The Tri-Glide (B) PM work volume is found to

be a maximum of 0.4053139 (for = 65° and L = 200 mm) and a minimum of

0.1232912 (for = 80° and L = 500 mm). Similarly, the Tri-Glide (A) work

volume is found to be a maximum of 0.3951465 (for = 65° and L = 500 mm)

and a minimum of 0.0492084 (for = 80° and L = 200 mm).

Table 3.17 Normalized Work volume for MP radius of 100 mm for nut displacement 50 mm


Link Length in mm

Work volume in mm^3

Tri-Glide(Towards

center (A))


Tripod (C)

65 200 0.3129315 0.4053139 0.24411814865 300 0.3580639 0.3428206 0.24441094865 400 0.3811235 0.3181547 0.24455690565 500 0.3951465 0.3047126 0.24464434670 200 0.2240627 0.3077751 0.24782668970 300 0.267551 0.2647793 0.24811184870 400 0.2896218 0.246228 0.24825427570 500 0.3029941 0.2357877 0.2483396975 200 0.1363886 0.2338489 0.25149487375 300 0.1788265 0.2003836 0.25177953875 400 0.2002222 0.1852231 0.2519219975 500 0.213139 0.1765176 0.25200750480 200 0.0492084 0.1721912 0.25518374680 300 0.0911646 0.1440912 0.25547497780 400 0.1121782 0.1309474 0.25562099380 500 0.1248195 0.1232912 0.255708741

107

3.11.2.1 Influence of the link length on the work volume

The work volume is calculated by the Pappus-Guldinus theorem.

The results are shown in Figures 3.42 to 3.45. For the Tripod PM, the results

show that by increasing the link length from 200 mm to 500 mm, the work

volume of the MP was found to increase by 0.54 % (for r = 60 mm and

= 65°). Similarly, 0.54% for = 70°, 0.56% for = 75° and 0.59% for

= 80°. For a constant link length of 200 mm, the work volume of the MP

was found to increase by 7.08 % (from = 65° to = 80°) for the MP radius

of 60 mm. Similarly, 7.13 % of the work volume of the MP was found to

increase for the constant link length of 500 mm. Similarly, by increasing the

link length from 200 mm to 500 mm for the Tripod PM, the work volume of

the MP was found to increase by 0.21% (for r = 100 mm and = 65°).

Similarly, 0.20% for = 70°, 0.20% for = 75° and 0.20% for = 80°. For a

constant link length of 200 mm, the work volume of the MP was found to

increase by 4.33 % (from = 65° to = 80°) for the MP radius of 100 mm.

Similarly, 4.33 % of the work volume of the MP was found to increase for the

constant link length of 500 mm.

For the Tri-Glide PM (A), the results show that by increasing the

link length from 200 mm to 500 mm, the work volume of the MP was found

to increase by 20.29% (for r = 60 mm and = 65°). Similarly, 25.71% for =

70°, 35.83% for = 75° and 60.53% for = 80°. For a constant link length of

200 mm, the work volume of the MP was found to decrease by 84.23 % (from

= 65° to = 80°) for the MP radius of 60 mm. Similarly, 68.13 % of the

work volume of the MP was found to decrease for the constant link length of

500 mm. Similarly, by increasing the link length from 200 mm to 500 mm for

the Tri-Glide PM (A), the work volume of the MP was found to increase by

20.80% (for r = 100 mm and = 65°). Similarly, 26.05% for = 70°, 36% for

= 75° and 60.57% for = 80°. For a constant link length of 200 mm, the

108

work volume of the MP was found to decrease by 84.27 % (from = 65° to

= 80°) for the MP radius of 100 mm. Similarly, 68.41 % of the work volume

of the MP was found to decrease for the constant link length of 500 mm.

For the Tri-Glide PM (B), the results show that by increasing the

link length from 200 mm to 500 mm, the work volume of the MP was found

to decrease by 35.72% (for r = 60 mm and = 65°). Similarly, 26.97% for =

70°, 26.27% for = 75° and 29.33% for = 80°. For a constant link length of

200 mm, the work volume of the MP was found to decrease by 65.19 % (from

= 65° to = 80°) for the MP radius of 60 mm.

Similarly, 61.73 % of the work volume of the MP was found to

decrease for the constant link length of 500 mm. Similarly, by increasing the

link length from 200 mm to 500 mm for the Tri-Glide PM (B), the work

volume of the MP was found to decrease by 24.82% (for r = 100 mm and =

65°). Similarly, 23.38% for = 70°, 24.52% for = 75° and 28.39% for =

80°. For a constant link length of 200 mm, the work volume of the MP was

found to decrease by 57.52 % (from = 65° to = 80°) for the MP radius of

100 mm. Similarly, 59.54 % of the work volume of the MP was found to

decrease for the constant link length of 500 mm.

3.11.2.2 Influence of the MP radius on the work volume

For the Tripod PM, by increasing the radius of the MP from 60 mm

to 100 mm, the work volume of the MP was found to increase by 64.78% (for

L = 200 mm and = 65°), 64.45% for = 70°, 64.12% for = 75°, 63.74%

for = 80°. Similarly, by increasing the radius of the MP from 60 mm to 100

mm, the work volume of the MP was found to increase by 64.66% (for L =

500 mm and = 65°). Similarly, 62.61% for = 70°, 63.98% for = 75° and

63.60% for = 80°.

109

For the Tri-Glide PM (A), by increasing the radius of the MP from

60 mm to 100 mm, the work volume of the MP was found to increase by

64.10% (for L = 200 mm and = 65°), 64.02% for = 70°, 63.98% for =

75°, 63.98% for = 80°. Similarly, by increasing the radius of the MP from

60 mm to 100 mm, the work volume of the MP was found to increase by

64.33% (for L = 500 mm and = 65°).

Similarly, 64.19% for = 70°, 64.08% for = 75° and 64.02% for

= 80°. For the Tri-Glide PM (B), by increasing the radius of the MP from 60

mm to 100 mm, the work volume of the MP was found to increase by 55.05%

(for L = 200 mm and = 65°), 60.85% for = 70°, 62.41% for = 75°,

63.17% for = 80°. Similarly, by increasing the radius of the MP from 60

mm to 100 mm, the work volume of the MP was found to increase by 61.57%

(for L = 500 mm and = 65°). Similarly, 62.68% for = 70°, 63.29% for =

75° and 63.65% for = 80°.

3.11.2.3 Influence of the initial angle on the work volume

For the Tripod PM, by increasing the initial angle between the link

and the base platform from 65° to 80°, the work volume of the MP was found

to increase by 7.08% (for L = 200 mm and r = 60 mm), 6.08% for r = 70 mm,

5.35% for r = 80 mm, 4.79% for r = 90 mm and 4.33% for r = 100 mm.

Similarly, by increasing the initial angle between the link and the base

platform from 65° to 80°, the work volume of the MP was found to increase

by 7.13% (for L = 500 mm and r = 60 mm), 6.10% for r = 70 mm, 5.36% for r

= 80 mm, 4.78% for r = 90 mm and 4.32% for r = 100 mm.

For the Tri-Glide PM (A), by increasing the initial angle between

the link and the base platform from 65° to 80°, the work volume of the MP

was found to decrease by 84.22% (for L = 200 mm and r = 60 mm), 84.23%

for r = 70 mm, 84.25% for r = 80 mm, 84.26% for r = 90 mm and 84.27% for

110

r = 100 mm. Similarly, by increasing the initial angle between the link and the

base platform from 65° to 80°, the work volume of the MP was found to

decrease by 68.12% (for L = 500 mm and r = 60 mm), 68.22% for r = 70 mm,


For the Tri-Glide PM (B), by increasing the initial angle between

the link and the base platform from 65° to 80°, the work volume of the MP

was found to decrease by 65.19% (for L = 200 mm and r = 60 mm), 61.41%

for r = 70 mm, 59.53% for r = 80 mm, 58.35% for r = 90 mm and 57.52% for

r = 100 mm. Similarly, by increasing the initial angle between the link and the

base platform from 65° to 80°, the work volume of the MP was found to

decrease by 61.73% (for L = 500 mm and r = 60 mm), 60.86% for r = 70 mm,


a) L = 200 mm b) L = 300 mm

c) L = 400 mm d) L = 500 mm

Figure 3.42 Influence of = 65° over the work volume for various MP

radii and link lengths

111

a) L = 200 mm b) L = 300 mm

c) L = 400 mm d) L = 500 mm


radii and link Lengths

a) L = 200 mm b) L = 300 mm

c) L = 400 mm d) L = 500 mm



112

a) L = 200 mm b) L = 300 mm

c) L = 400 mm d) L = 500 mm



3.11.2.4 Comparison of maximum and minimum work volume of the

PMs

The kinematic study of the single link movement of the two PMs

structure is based on the work volume, by considering the geometrical

parameters, such as the link length, the MP radius and the initial angle

between the link and the base platform.

Table 3.18 shows the results of the maximum and minimum work

volume of the PMs. By considering the nut displacement of 50 mm, for the

Tripod PM, it is observed that a maximum work volume is found to be

599727.41mm3 when the link length is 500 mm with = 80°, and a minimum

work volume is found to be 201642.42mm3, when the link length is 200 mm

with = 65°.

113

Table 3.18 Work volume comparison for various geometrical

parameters for the nut displacement of 50 mm by analytical

method

MPradius in mm

The work volume in mm^3 Tri-Glide (A) Tri-Glide (B) Tripod (C)

Max(L=500

mm, =65°)

Min(L=200

mm, =80°)

Max(L=200

mm, =65°)

Min(L=500

mm, =80°)

Max(L=500

mm, =80°)

Min(L=200

mm, =65°)

60 85395.20 10740.82 208623.13 51312.90 218317.10 201642.4270 116590.78 14616.79 253935.79 69595.16 295414.12 276243.7580 152666.76 19088.77 314391.09 90667.76 384696.98 362918.5990 193623.90 24156.77 384810.15 114530.45 486138.52 461681.47100 239462.70 29820.78 464132.64 141183.09 599727.41 572543.37

Similarly, for the Tri-Glide (A) PM, it is observed that a maximum

work volume is found to be 239462.70 mm3, when the link length is 500 mm

with = 65° and a minimum of 10740.82 mm3 is obtained, when the link

length is 200 mm with = 80°. For the Tri-Glide (B) PM, it is observed that a

maximum work volume is found to be 464132.64 mm3 when the link length is

200 mm with = 65°, and a minimum of 51312.90 mm3 when the link length

is 500 mm with = 80°.

The work volume analysis has been dealt with in detail, to find the

better work volume among the PMs. From Tables 3.11 to 3.18, it was found

that the geometrical parameters have a scaling effect over the work volume.

As the radius of the MP is increased the Tripod PM work volume is found to

be a maximum of 572543.37 mm3 for a radius of 100 mm to a minimum of

201642.42 mm3 for a MP radius of 60 mm at a constant link length of 200

114

mm and = 65°. Similarly, under the same conditions the Tri-Glide (A) PM

work volume is found to be a maximum of 189639.60 mm3 for a MP radius of

100 mm to a minimum of 68065.89 mm3 for a MP radius of 60 mm and for

Tri-Glide (B) PM the work volume is found to be a maximum of 464132.64

mm3 for a radius of 100 mm to a minimum of 208623.13 mm3 for a MP

radius of 60 mm.

Similarly, as the link length is increased the Tripod PM work

volume is found to be a maximum of 202746.54 mm3 for a link length of 500

mm to a minimum of 201642.42 mm3 for a link length of 200 mm at a

constant MP radius of 60 mm and = 65°. Similarly, under the same

conditions, the Tri-Glide (A) PM work volume is found to be a maximum of

85395.20 mm3 for a link length of 500 mm to a minimum of 68065.89 mm3

for a link length of 200 mm and for Tri-Glide (B) PM the work volume is

found to be a maximum of 208623.13 mm3 for a link length of 200 mm to a

minimum of 134086.99 mm3 for a link length of 500 mm.

As the initial angle ( ) is increased, the Tripod PM work volume is

found to be a maximum of 217015.87 mm3 for a = 80° to a minimum of

201642.42 mm3 for a = 65° at a constant MP radius of 60 mm and link

length 200 mm. Similarly, under the same condition the Tri-Glide (A) PM

work volume is found to be a maximum of 68065.89 mm3 for a = 65° to a

minimum of 10740.82 mm3 for a = 80° at a constant MP radius of 60 mm

and link length 200 mm and for Tri-Glide (B) PM the work volume is found

to be a maximum of 208623.13 mm3 for a = 65° to a minimum of 72610.66

mm3 for a = 80° at a constant MP radius of 60 mm and link length 200 mm.

3.11.3 Simulation Study on the MP tilt and Work Volume

For this simulation study, the Tripod PM and Tri-Glide PM (B) are

considered, because of their dimensional synthesis result is comparatively

115

better than that of the Tri-Glide PM (A). The geometrical parameters

considered for this study are, the link length varied from 200 to 500 mm , the

MP radius is 90 mm, initial angle = 750, and the nut displacement of 50 mm.

Figure 3.46 Link length Vs MP Tilt for the MP radius 90 mm and nut

displacement of 50 mm

Figure 3.46 shows the comparative results of the MP tilt of two

PMs. From the results obtained, it can be observed that for the link length of

200 mm and the MP radius 90 mm, the MP tilt of the Tripod and the Tri-

Glide PMs is found to be 20.59° and 9.42° by the analytical method.

Similarly, by the simulation packages, the MP tilt is found to be 21.11° and

9.16°.

For the link length of 300 mm, the Tripod and the Tri-Glide (B)

PMs MP tilt is found to be 20.62° and 8.06° by the analytical method (refer

Appendix 2 for analytical calculation using C / JAVA programs). Similarly,

by the simulation packages, the MP tilt is found to be 21.23° and 7.74°. The

Tripod and the Tri-Glide (B) PMs MP tilt for the link length of 400 mm are

116

found to be 20.63° and 7.44° by the analytical method. Similarly, by the

simulation packages, the MP tilt is found to be 21.36° and 7.12°.

For the link length of 500 mm, the Tripod and the Tri-Glide (B)

PMs the MP tilt is found to be 20.64° and 7.09° by the analytical method.

Similarly, by the simulation packages, the MP tilt is found to be 21.50° and

6.68°. From the results, it is observed that the analytical and simulation results

are found to be close, and having a minimum deviation of 2.53% and a

maximum of 5.78%.

Figure 3.47 shows the work volume results of the two PMs. From

the results presented, it can be observed that for the link length of 200 mm

and the MP radius of 100 mm, the Tripod and the Tri-Glide (B) PMs work

volume is found to be 477117.28 mm3 and 218311.08 mm3 by the analytical

method. Similarly, by the simulation packages (refer Appendix 3), the work

volume is found to be 489545.85 mm3 and 212422.56 mm3.

For the link length of 300 mm, the Tripod and the Tri-Glide PMs

work volume is found to be 477780.54 mm3 and 186657.18 mm3 by the

analytical method. Similarly, by the simulation packages, the work volume is

found to be 492328.68 mm3 and 179492.43 mm3. The Tripod and the Tri-

Glide PMs work volumes for the link length of 400 mm are found to be

478112.73mm3 and 172395.19 mm3 by the analytical method. Similarly by

the simulation packages, the work volume is found to be 495343.41 mm3 and

165114.48 mm3.

For the link length of 500 mm, the Tripod and the Tri-Glide PMs

work volume is found to be 478312.24 mm3 and 164223.57 mm3 by the

analytical method. Similarly, by the simulation packages, the work volume is

found to be 498590.05 mm3 and 154910.78 mm3. From the results, it is

117

observed that the analytical and simulation results are found to be close, and

having a minimum deviation of 2.60% and a maximum of 5.67%.

Figure 3.47 Link length Vs work volume for the MP radius of 90 mm

and Nut displacement of 50 mm

3.11.4 Singularity Analysis

Based on the dimensional synthesis and the work volume analysis,

the Tripod and Tri-Glide (B) PMs are considered for the singularity analysis.

For this simulation study, the same geometrical parameters are considered by

logical approach. The geometrical parameters are, MP radius = 90 mm, Link

length =200 mm and the initial angle between the link and the base platform =

77°.

Table 3.19 shows the simulation results for the two 3-DOF PMs

with their MP tilt for the singular positions. From the results, it can be

observed that in Type I singularity, the Tripod and the Tri-Glide (B) MP tilts

are found to be 78.97° and 34.58° about the X-axis ( ). Similarly, in Type II

singularity, the Tripod and the Tri-Glide (B) MP tilts are found to be 53.41°

and 75.73° about the X-axis. In Type III singularity, the Tripod and the Tri-

Glide (B) MP tilts are found to be 8.24° and 10.28° about the X-axis and

59.1° and 30.74° about the Y-axis ( ).

118

Table 3.19 Results of the MP tilt and nut displacement at singular positions

ParametersTripod Tri-Glide (B)

I II III I II III

in ° 78.97 53.41 8.24 34.58 75.73 10.28

in ° 0 0 59.1 0 0 30.74

D1 (mm) 200 0 60 94 0 50

D2 (mm) 0 88 120 0 156 100

D3 (mm) 0 88 0 0 156 0

The linear displacement of the nut 1(D1) for the Type I singularity

is found to be 200 mm and 94 mm respectively for the Tripod and Tri-Glide

(B) PMs. Similarly, the linear displacement of the nuts 2 and 3 for the Type II

singularity is found to be 88 mm for the Tripod PM and 156 mm for the Tri-

Glide (B) PM. In Type III singularity, the linear displacement of the nuts 1

and 2 is found to be 60 mm and 120 mm for the Tripod PM and 50 mm and

100 mm for the Tri-Glide (B) PM.

Figures 3.48 and 3.59 show the result of the pin joints (PJs) force

and torque values of the two PMs. From the results, it can be observed that

the Tri-Glide (B) PM and the Tripod PM have the joint force of 117 N and

8.84 N for Type I singularity. Similarly, for Type II singularity the Tri-Glide

(B) and Tripod PMs have 288.99 N and 24 N. For the Type III singularity Tri-

Glide (B) and Tripod PMs have 54.32 N and 35 N.

Similarly, the pin joints torque values of the Tri-Glide (B) and

Tripod PMs are found to be 12056.90 and 80.84 N mm for Type I singularity.

For the Type II singularity, the values are found to be 28531.65 and 1336.29

N mm. The pin joints torque values of the Tri-Glide (B) and Tripod PMs, for

Type III singularity are found to be 6879.09 and 2459.13 N mm.

119

Figure 3.48 Type I singularity pin joints force (Tripod)

Figure 3.49 Type I singularity pin joints force (Tri-Glide)

Figure 3.50 Type II singularity pin joints force (Tripod)

120

Figure 3.51 Type II Singularity pin joints force (Tri-Glide)

Figure 3.52 Type III singularity pin joints force (Tripod)

Figure 3.53 Type III singularity pin joints force (Tri-Glide)

121

Figure 3.54 Type I singularity pin joints Torque (Tripod)

Figure 3.55 Type I singularity pin joints torque (Tri-Glide)

Figure 3.56 Type II singularity pin joints torque (Tripod)

122

Figure 3.57 Type II singularity pin joints torque (Tri-Glide)

Figure 3.58 Type III singularity pin joints torque (Tripod)

Figure 3.59 Type III singularity pin joints torque (Tri-Glide)

123

3.11.5 Position Analysis

The kinematic analysis of the 3-DOF PMs has been carried out and the

results are verified by experiments, analyses and the software package ADAMS.

The Experimental models of the Tripod and Tri-Glide (B) PMs are fabricated with

a Link length of 300 mm, MP radius of 90 mm and the initial angle of =70°.

3.11.5.1 Tripod PM

Table 3.20 shows the predetermined distance (AC) from the point

of the laser source to the vertical screen, and various vertical heights of the

laser beam on the vertical screen (AB), to calculate the Tripod MP tilt by the

experimental method.

Table 3.20 Tripod results of the angle of tilt of the MP for the nut displacement of 90 mm by experimental method

Actuation Link

Number

Predetermined distance (AC) from the point

of the laser source to the

vertical screen in mm

Vertical height of the laser

beam on the vertical

screen(AB) in mm (Average of

5 Trials)

Angle of tilt of the MP (degrees)

About the X axis

About the Y axis

About the X axis

About the Y axis

1 195 135 0 34.69 02 195 57 114 16.29 30.313 195 56 114 16.02 30.31

124

Table 3.21 Results of the angle of tilt of the MP for the nut displacement of 90 mm by experimental and ADAMS methods

Actuation Link

Number

Angle of tilt of MP (degrees)Experiment ADAMS

About the X axis

About the Y axis

About the X axis

About the Y axis

1 34.69 0 35.25 02 16.29 30.31 16.58 30.513 16.02 30.31 16.58 30.51

From Table 3.21, the Tripod PM result shows, that for the 90 mm

linear displacement of the nut, the MP has tilted to an angle of 34.69° about

the x axis by the actuation of link 1 in the experimental method. Similarly, by

the derived single link movement kinematic equations (analytical method), it

is found that the angle of tilt about the x axis is 35.95°. The angle of tilt of the

MP about the x axis from the ADAMS simulation is found to be 35.25°.

From the above results, the experimental method shows a deviation

of 3.5% (for the x axis MP tilt) when compared with the analytical method.

The ADAMS results are compared with the experimental results. It is found

that it has a maximum deviation of 3.5%. Since the values obtained from the

experimental and ADAMS methods have minimum deviation from the

analytical results, the values are considered to be closer.

3.11.5.2 Tri-Glide (B) PM

Table 3.22 shows the predetermined distance (AC) from the point

of the laser source to the vertical screen, and the various vertical heights of

125

the laser beam on the vertical screen (AB) to calculate the Tri-Glide (B) MP

tilt by the experimental method. The Tri-Glide (B) PM result shows, that for a

90 mm linear displacement of the nut, the MP has tilted to an angle of 30.72°

about the x axis by the actuation of link 1 in experimental method. Similarly,

by the derived single link movement kinematic equations (analytical method),

it is found that the angle of tilt about the x axis is 31°. The angle of tilt of the

MP about the x axis from the ADAMS simulation is found to be 30.98°.

From Table 3.23, the experimental method shows a deviation of

0.9% (for the x axis MP tilt), when compared with the analytical method. The

ADAMS results are compared with experimental results. It is found that it has

a maximum deviation of 3.54%. Since the values obtained from the

experimental method and ADAMS have a minimum deviation from analytical

results the values are considered to be closer. The ADAMS simulation results

of the Tripod and Tri-Glide PMs for the Link actuation 2 are shown in Figures

3.60 to 3.63. Similarly, the other simulation results are obtained.

Table 3.22 Tri-Glide (B) results of the angle of tilt of the MP for the nut displacement of 90 mm by the experimental method

Actuation Link

Number

Predetermined distance ‘AC’

from the point of the laser source to the vertical screen in mm

Vertical height of the laser

beam on the vertical screen

in mm (Average of 5 Trials)

Angle of tilt of the MP (degrees)

About X axis

About Y axis

About X axis

About Y axis

1 249 148 0 30.72 0

2 249 64 129 14.41 27.38

3 249 64 128 14.41 27.20

126

Table 3.23 Results of the angle of tilt of the MP for the nut

displacement of 90 mm by experimental and ADAMS

methods

ActuationLink No.

Angular Tilt of MP (degrees)

Experiment ADAMS

About X axis About Y axis About X axis

About Y axis

1 30.72 0 30.98 0

2 14.41 27.38 14.92 27.48

3 14.41 27.20 14.92 27.48

Figure 3.60 MP tilt about the x axis for tripod PM for link actuation 2

Figure 3.61 MP tilt about the y axis for tripod PM for link actuation 2

127

Figure 3.62 MP tilt about the x axis for tri-glide PM for link actuation 2

Figure 3.63 MP tilt about the y axis for tri-glide PM for link actuation 2

3.11.6 A Performance Study of the 3- PRS, 3-PRR and 3-PUS PMs

The performance study of the three PMs is carried out by fixing the

MP tilt up to 45º in order to achieve µ = 135 º . The revolute / universal joint

forces and torque values of the PMs are considered in this study, to get a

better design of PMs. The geometrical parameters considered for this

simulation study are link length of 200 mm, MP radius of 90mm, and an

initial angle between the base and link of 75º.

From the simulation results, it is found that, the displacement of the

nut is found to be 75mm and 45mm respectively for the 3-PRR mechanism

128

for the single link and two link simultaneous movements to achieve a

Transmission angle (µ) = 135 º. Similarly, for the 3-PUS and 3-PRS PMs, the

nut displacement is found to be 110 mm and 82 mm respectively. From the

displacement of the nut to achieve µ = 135°, the 3-PRR mechanism is better

compared to the other PMs to get the required MP tilt, with a smaller

displacement of the nut.

Figures 3.64 to 3.67 show the simulation results of various forces

and torques along x (1), y (2) and z (3) directions of the revolute joint (R) in

PRR PM, universal joint (U) in PUS PM and revolute joint (R) in PRS PM, to

achieve µ = 135°. Figure 3.64 shows the forces at the revolute joints of

3-PRR (Joint 1) and 3-PRS (Joint 3) PMs, and the universal joint of the

3-PUS (Joint 2) PM. From the results, it is found that, the 3-PRS revolute

joint and 3-PUS universal joint has a maximum force of 45.3N for the single

link movement for the required MP tilt than 3-PRR revolute joint force value

of 42.3N. Similarly, in Figure 3.65, the 3-PRS revolute joint and 3-PUS

universal joint have the maximum joint force of 52.4N when two links move

simultaneously than 3-PRR revolute joint force value of 51N. Hence, the

revolute joint force acting on the 3-PRR PM is comparatively lesser than that

of the other Tripod PMs.

In Figures 3.66 and 3.67, the torque values of the three PMs are

compared; from the results it is found that the 3-PRR revolute joint has the

maximum torque of 810 Nmm compared to the 3-PRS revolute joint and

3-PUS universal joint maximum torque values of 230 Nmm and 215 Nmm

respectively, when one link is moved, but when two links are moved

simultaneously the torque values of the 3-PRS revolute joint and 3-PUS

universal joint is found to be 3233 Nmm and 3349 Nmm respectively, and

they are higher than that of the 3-PRR maximum torque value of 1841Nmm.

129

Figure 3.64 R-U-R force values for one link movement

Figure 3.65 R-U-R force values for two link movements simultaneously

Figure 3.66 R-U-R joint torque values for one link movement

-55

1525354555

0 1 2 3 4

Forc

e(i

nN

ewto

n)

Single Link Movement Fx, Fy and Fz values

PRRPUSPRS

0102030405060

0 1 2 3 4

Forc

e(i

nN

ewto

n)

Two Link Movement Fx, Fy and Fz values

PRRPUSPRS

0200400600800

1000

0 1 2 3 4

Torq

ue(i

nN

mm

)

Single Link Movement Tx, Ty and Tz values

PRRPUSPRS

130

Figure 3.67 R-U-R joint torque values for two link movements

simultaneously

From this comparison, it is clearly found that the 3-PRS and 3-PUS

PMs have closer force and torque values than the 3-PRR. Similarly, a

comparative study is carried out, for the singularity positions of the proposed

mechanisms; from the simulation results it is found that for any kind of link

movement the singularity positions of 3-PRS is better than the 3-PUS and 3-

PRR PMs, because of the spherical and revolute joint combination in the

linear movement.

The simulation and performance studies of the 3-PRR, 3-PUS and

3-PRS PMs were carried out. The following inferences were drawn based on

the simulation results; 1.The PRR has a better MP tilt for shorter displacement

of the nut when one link and two links (simultaneously) movement. 2. The

PRR developed a low torque for two links (simultaneously) movement. 3.

Each link of the PRS and PUS PMs can be moved up to singularity position to

get the maximum tilt. 4. In the singularity analysis, the PRS mechanism is

better than that of PUS, and that of PUS is better than that of PRR.

0

1000

2000

3000

4000

0 1 2 3 4

Torq

ue(i

nN

mm

)

Two Link Movement Tx, Ty and Tz values

PRRPUSPRS

131

3.11.7 Structural analysis of the Tripod, Tri-Glide and 3-PRR PMs

The structural analysis is carried out, the values of deformations,

stresses and moments are found from the ANSYS, and the results are shown

in Table 3.24 to 3.27 for solid structure. From the Figure 3.68 to 3.70, for

mild steel structure, the Tripod, Tri-Glide and 3 PRR PMs have the maximum

deformation value of 0.0084 mm, 0.0082 mm and 0.0081 mm for 10 mm link

diameter and a minimum of 0.0013 mm for 18 mm link diameter. Similarly,

for aluminium alloy, the PMs have the maximum deformation value of 0.0251

mm, 0.0248 mm and 0.0245 mm for 10 mm link diameter and a minimum of

0.004 mm, 0.0038 and 0.0038 for 18 mm link diameter.

Table 3.24 Deformation values of PMs for solid link

Link Diameter in mm

Tripod Tri-Glide PRRDeformation in mm

Mild Steel 10 0.0084 0.0082 0.008112 0.0044 0.0043 0.004214 0.0027 0.0026 0.002516 0.0018 0.0017 0.001718 0.0013 0.0013 0.0013

Aluminium Alloy 10 0.0251 0.0248 0.024512 0.0131 0.0129 0.012714 0.0079 0.0077 0.007616 0.0054 0.0051 0.005118 0.0040 0.0038 0.0038

Table 3.25 shows the moment values of PMs, from the results it is

observed that the Tripod, Tri-Glide and 3-PRR PMs have the maximum

moment value of 3795 N mm, 3184 N mm and 3540 N mm for 18 mm link

132

diameter and a minimum of 3544 N mm, 2969 N mm and 3408 N mm for 10

mm link diameter. Figure 3.71 shows the solid link moment values of PMs.

Table 3.25 Moment values of PMs for solid link

Link Diameter in mm

Tripod Tri-Glide PRRMoment in N mm

10 3544 2969 340812 3612 3015 345214 3692 3074 350116 3761 3134 353118 3795 3184 3540

Table 3.26 Axial stress values of PMs for solid link

Link Diameter in mm

Tripod Tri-Glide PRR

Axial Stress in N/mm2

10 0.3435 0.3430 0.312812 0.2390 0.2391 0.217814 0.1770 0.1765 0.160816 0.1360 0.1360 0.123818 0.1080 0.1080 0.0980

Table 3.27 Bending stress values of PMs for solid link

Link Diameter in mm

Tripod Tri-Glide PRRBending Stress in N/mm2

10 3.6590 3.6610 3.5350

12 2.2290 2.2300 2.134914 1.5030 1.5040 1.423016 1.0880 1.0890 1.108518 0.8300 0.8260 0.7659

133

The solid MS and Al structures of the PMs have axial stress values

between 0.098 N/mm2 and 0.03435 N/mm2. Similarly, the bending stress

values between 0.7659 N/mm2 and 3.661 N/mm2.

Figure 3.68 Deformation values for Tripod PM

Figure 3.69 Deformation values for Tri-Glide PM

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

9 10 11 12 13 14 15 16 17 18

Def

orm

atio

nin

mm

Link Diameter in mm

Tripod (MS)Tripod (AL)

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

9 10 11 12 13 14 15 16 17 18

Def

orm

atio

nin

mm

Link Diameter in mm

Tri-Glide (MS)

Tri-Glide (AL)

134

Figure 3.70 Deformation values for 3-PRR

Figure 3.71 Moment values for solid link

Figures 3.72 to 3.74 shows the deformation results of the hollow

structures of Tripod, Tri-Glide and 3-PRR PMs and the maximum value is

found to be 0.0023 mm, 0.00226 and 0.00255 mm for the link 18 mm outer

diameter (OD) and 15 mm inner diameter (ID), whereas in the link 18 mm ID

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

9 10 11 12 13 14 15 16 17 18

Def

orm

atio

nin

mm

Link Diameter in mm

3-PRR(MS)3-PRR(AL)

240026002800300032003400360038004000

9 10 11 12 13 14 15 16 17 18

Mom

ent

inN

mm

Link Diameter in mm

TripodTri-Glide3-PRR

135

and 21 mm OD, it is found to be 0.00174 mm, 0.00169 and 0.00167 mm.

Similarly, the minimum value is found to be 0.00155 mm, 0.00149 and

0.001488 mm for the link 18 mm OD and 11 mm ID, whereas in the link 18

mm ID and 25mm OD, it is found to be 0.0009 mm, 0.0008 and 0.00087 mm.

Figures 3.75 and 3.76 shows the moment values of PMs, from the

results it is observed that the Tripod, Tri-Glide and 3-PRR PMs have the

maximum moment value of 3736 N mm, 3135 N mm and 3496 N mm for the

link OD 18 mm and ID 11 mm and a minimum of 3652 N mm, 3059 N mm

and 3448 N mm for the link OD 18 mm and ID 15 mm. Similarly, for the

constant link ID 18 mm, the maximum moment value of Tripod and 3-PRR

PMs is found to be 3677 N mm, 3433 N mm for the link OD 21 mm. For Tri-

Glide it is found to be 3133 N mm for the link OD 25 mm. The minimum

moment value for Tripod and 3-PRR PMs are found to be 3573 N mm and

3275 N mm for the link OD 25 mm. For the Tri-Glide PM the minimum value

is found to be 3089 N mm for the OD 21 mm.

Figure 3.72 Tripod PM link deformation values for hollow structures

0.0005

0.001

0.0015

0.002

0.0025

10 12 14 16 18 20 22 24 26

Def

orm

atio

nin

mm

Link Diameter in mm

Tripod (18 ID)

Tripod (18 OD)

136

Figure 3.73 Tri-Glide link deformation values for hollow structure

Figure 3.74 3-PRR link deformation values for hollow structure

0

0.0005

0.001

0.0015

0.002

0.0025

10 12 14 16 18 20 22 24 26

Def

orm

atio

nin

mm

Link Diameter in mm

Tri-Glide (18 ID)

Tri-Glide (18 OD)

0.0005

0.001

0.0015

0.002

0.0025

0.003

10 12 14 16 18 20 22 24 26

Def

orm

atio

nin

mm

Link Diameter in mm

3-PRR (18 ID)

3-PRR (18 OD)

137

Figure 3.75 Moment values for hollow link OD 18 mm

Figure 3.76 Moment values for hollow link ID 18 mm

The ANSYS output results are given in Appendix 4.

240026002800300032003400360038004000

10 11 12 13 14 15 16

Mom

ent

inN

mm

Link Diameter in mm


24002600280030003200340036003800

20 21 22 23 24 25 26

Mom

ent

inN

mm

Link Diameter in mm


138

3.12 SUMMARY

The kinematic equations for the single link movement, work

volume and singularity analysis of Tri-Glide and Tripod PMs are studied in

this chapter. Based on the study, it is found that the Tripod PM has a better

angle of tilt of the MP, work volume and singular positions. A performance

study on the Tripod PMs was carried out. From the study, it is found that the

3-PRR PM has a better angle of tilt about one axis (x axis) of the MP, for a

smaller linear displacement of the nut than the other PMs. Similarly, the 3-

PRS PM has better singular positions than the other PMs. Hence, the 3-PRR

and 3-PRS PMs are considered for further study and discussed in the chapter

4 and 5. In structural analysis, for the same geometrical parameters Tripod,

Tri-Glide and 3-PRR PMs were analyzed. From the study, it is found that the

PMs have minimum variation in displacement, stress and moment values.

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