Workspace generation of 3-DOF parallel manipulator · workspace analysis of 3-PSP parallel robot is carried out and three different types of singularities are analyzed recently by

102

Chapter 4

Workspace generation of 3-DOF parallel manipulator

4.1 Introduction

The workspace is one of an important step in design phase to determine

feasibility of 3- PRS spatial parallel manipulator with 3-DOF as a machine tool

structure. Closed loop nature of the parallel mechanism limits the motion of the

platform and creates complex kinematic singularities inside the workspace. There are

other design criteria based on kinematic point of view such as kinematic performance

indices, singularity avoidance, task development etc. Workspace development is a

prime focus in this chapter as one of the major criteria for appraising the kinematic

design of parallel manipulator. The workspace of a robot is defined as set of all end-

effector configurations which can be reached by some choice of joints coordinates.

The reachable workspace of the mechanism can be determined if values of all

kinematic constants are given. Parallel mechanism has been designed to obtain an

appropriate maneuverability and normally categorized as planar, spherical and spatial

in accordance with the number of the workspace dimensions. Hence, it is always

necessary to identify the workspace boundary for a new mechanism. Size, shape and

quality of workspace are very important design aspects for parallel manipulator under

investigation.

In the literature, it is stated that the computation of the workspace of parallel

manipulator is far more complex and highly non-linear in relation between joint

coordinates and Cartesian coordinates than serial manipulator as its translational

ability dependent upon the orientation of end-effector [72]. Various methods to

determine workspace of a parallel robot have been proposed using geometric or

numerical approaches. Algorithm to trace boundary of workspace of mechanical

manipulator is presented by Kumar and Waldron [73]. Algorithm for workspace of

parallel manipulator is investigated and reported by Gosselin [74]. The reachable pose

of configuration possible through forward and inverse kinematic solutions can be

considered as operational workspace [75]. Numerical integration method to find void

inside the workspace is determined for three legged parallel manipulator. It is very

much useful information before trajectory planning of the manipulator [76]. There is

103

an analogous symmetrical theorem of workspace for spatial parallel manipulators with

identical kinematic chain. Three different cases of any symmetric spatial parallel

manipulator: a) The identical kinematic chains of a spatial parallel manipulator are

symmetric about a certain plane b) The identical kinematic chains of a spatial parallel

manipulator are rotational symmetric about a certain axis c) The identical kinematic

chains of a spatial parallel manipulator are centro-symmetric about a certain point are

considered and presented in [77]. However, their method cannot be utilized for

mechanisms with non-symmetric and non-identical kinematic chains. Systematic

methodology to identify maximal regular-shaped dexterous workspace (MRsDW) of

parallel manipulators (PMs) is proposed and the concept of the utilizable ratio of

dexterous workspace (URDW) is introduced, which is a new measure for the

rationality of design parameters of a PMs by Z. Wang et al. [78]. Generally, a closed-

form solution for workspace boundary of a spatial parallel mechanism is challenging

task due to a complex surface boundary. The author believes that among the existing

algorithms for the workspace of parallel mechanisms, the algorithm provided by

Gosselin [79] is a good solution. Gosselin's methodology is general and usable for

different configurations of the robot with any sort of structural parameters.

The knowledge of the overall size and shape of workspace and boundary of

SPM is of a great importance to locate the work-piece properly in order to avoid

collisions between the work-piece and the cutting tool. A new geometrical

methodology is introduced for determining the reachable workspace of 6-3 stewart

platform mechanism [80]. Unconstraint motion due to passive joint clearance is

analyzed for large class of parallel manipulator by Philip Voglewede et al. [81]. The

workspace is investigated for six-degrees of freedom three prismatic-prismatic-

spherical-revolute parallel manipulator and the effects of joint limit and limb

interference on the workspace shape and size are numerically studied by M. Z. A.

Majid, Z. Huang and Y. L. Yao [82]. Concept of constant orientation workspace, total

orientation workspace and inclusive orientation workspace is presented and

algorithms for Gough type parallel manipulator are developed. It is deduced that

robots of similar dimensions the joints layout has a large influence on the workspace

volume by J P Merlet [83]. Procedure for determining the maximal singularity-free

orientation workspace for Gough–Stewart platform is developed by Qimi Jiang,

Clément M. Gosselin [84]. One of the method commonly adopted for estimating the

workspace of 6-DOF parallel manipulators have involved the discretization of

104

workspace for the computation of the orientation workspace is presented as it is quite

simple concept. The space around the manipulators is filled with uniformly distributed

points. The considerations of joint limits, actuators' stroke, link interference,

compatibility constraint are incorporated in the study of the robot workspaces.

Discretized algorithm for axi-symmetric parallel manipulator’s orientation workspace

is developed using modified Euler angles by Ilian A. Bonev, Jeha Ryu [85]. Methods

for predicting possible constraint equations for the boundary curves of neighboring

sections are proposed to facilitate the evaluation of orientation workspace of Stewart–

Gough parallel manipulator by K.Y. Tsai , J.C. Lin [86]. Z. Affi , L. Romdhane, A.

Maalej have analyzed workspace of a 3-translational-DOF in-parallel manipulator (3-

T-P-M) having 3 linear actuators. The concept of ‘active workspace’ and ‘passive

workspace’ is introduced. It is highlighted that workspace of platform is reduced

significantly due to presence of passive kinematic chain [87]. The parallel robots

TRIGLIDE and 3-RPS realize a wide workspace and simulated using virtual reality

tool of MATLAB/Simulink module by Dan Verdeș, Sergiu-Dan Stan [88]. A concept

of joint workspace is introduced. It means workspace definition in terms of joint

coordinates system for the joint motions. An approach is developed to determine joint

workspace based on the structural constraints of a PKM. It is observed that the

trajectory planning in the joint coordinate system is not reliable without taking into

considerations of cavities or holes in the joint workspace by Z M Bi, S Y T Lang [89].

Geometric and non-geometric constraints and numerical algorithm of work space

generation is presented. Significance of developing workspace using cutter point

while machining is highlighted by Zhe Wang, Zhixing Wang, Wentao Liu, Yucheng

Lei [90]. Direct position analysis, inverse kinematics analysis, jacobian analysis and

workspace analysis of 3-PSP parallel robot is carried out and three different types of

singularities are analyzed recently by Amir Rezaei, Alireza Akbarzadeh, Payam

Mahmoodi Nia, Mohammad-R.Akbarzadeh-T [91]. A general methodology for

obtaining the maximal operational workspace for parallel manipulator is described. A

case study on Delta like translational manipulator is presented for different working

modes by E. Macho, O. Altuzarra and A. Hernandez [92]. Geometric approach to

evaluate reachable workspace for 6-3 stewart platform as machine tool is analyzed by

Serdar Ay, O.Erguven Vatandas and Abdurrahman Hacioglu [93].

Many researchers, in the past years, proposed methodologies and optimization

criteria to be used as design criteria particularly in order to obtain large and suitably

105

shaped workspaces, to avoid functional singularities and to reach satisfactory parallel

manipulator behaviors. In next section, workspace definition, workspace generation

using forward and inverse kinematic, workspace boundary generation and workspace

analysis using simulation software is presented for 3-PRS configuration.

4.2 Definitions and types of workspace

The workspace 𝑊(𝐻) is defined as a region of points that can be reached by a

reference point 𝐻 of the manipulator that is moved to reach all possible positions

within the scope of mobility ranges of the joints. This is the position workspace.

Fundamental characteristics of the manipulator position workspace are recognized as:

i) shape and volume of workspace ii) the voids within workspace.

In other words, the workspace means surface or volume regions generated by

a reference point 𝐻 of the end-effector, which may be bounded within two (2D) or

three dimensional (3D) space due to manipulator joints extremity. Workspace is normally divided into two categories:

a) Constant orientation workspace: Set of all the possible locations of the center

of the mobile platform that can be reached with a constant orientation of

platform [83, 84, 85].

b) Orientation workspace: The set of orientations that can be reached by the

manipulator extremity [84-86]. It can be classified as,

Figure 4.1 Types of orientation workspace

Reachable Workspace (RW) determination is very important for all kind of

parallel robots because it surrounds all other types of workspace and provides

information on the size of a space in which the robot end-effector can manipulate

safely. In present work, the workspace determination for 3-PRS configuration using

direct kinematic formulation is the reachable workspace. The total orientation

workspace means all possible locations of the center of platform that can be reached

with any orientation in a set defined by three ranges for orientation angles [83]. The

Reachable Workspace

Total Orientation

Dexterous Workspace

Orientation workspace

106

dexterous workspace means volume of space where at each point the end-effector can

be arbitrarily oriented. It is a subspace of a reachable workspace having an adequate

performance of a dexterity measure, e.g. the condition number of the jacobian matrix

by Carretero J.A., Nahon M., Podhorodeski R.P. [94]. The dexterous workspace is

often null for parallel manipulators.

Numerous studies are available in literature to evaluate workspace for parallel

manipulators: a) Geometric approach b) Analytical (numerical) approach c) Graphical

approach. The geometrical approach employed for solution of workspace generation

is simple, easy to deduce and can be simulated with a commercial software package.

Analytical approach is normally utilized to determine boundary of workspace.

Graphical approach has been proposed to determine workspace using the position and

mechanism constraints of a manipulator. Various graphical approaches are utilized to

develop workspace plots of parallel manipulators using discrete-points generated by

kinematics equations [90, 94], computer aided design (CAD) tool on calculating

physical model of the workspace [95]. Various indices may be used to characterize

the workspace of parallel robots. A volume index is one of such parameters and is

defined as ratio of workspace volume to volume of a robot by J P Merlet [96].

The generated workspace of any parallel manipulator is always constrained

due to,

1) Limited bounded range for linear actuation (Recirculating ball screws-Prismatic

joint range limitation)

2) Limits of passive joints (Revolute joints)

3) Links interference in any limb plane

In next section, the workspace generation for 3-PRS parallel manipulator

configuration with assumed structural parameters is carried out using Sylvester

method as well as Bezout’s approach as a direct kinematics problem. Normally,

structural parameters in terms of link lengths and joint locations are chosen to more

than adequately satisfy the workspace requirements of the specification. The inverse

kinematics formulation of mechanism is also used to determine the workspace of the

manipulator. The process of workspace generation using both approaches is depicted

in figure 4.2. The workspace is also developed using Pro/Mechanism simulation

software. Moreover, the workspace boundary are defined for maximum velocity and

joint motion ranges for single and multiple actuations and analyzed further with and

107

without time lags for multiple actuations. Volume index is also determined for the

configuration under consideration.

Figure 4.2 Work space generation using two kinematic formulations

4.3 Constraints

4.3.1 Geometric Constraints

Link length Limitations:

The prismatic joint movement from predefined reference (prismatic joint

actuation length) constraint is expressed as,

𝑇𝑖𝑚𝑖𝑛≤ 𝑇𝑖 ≤ 𝑇𝑖𝑚𝑎𝑥

(4.1)

The links lengths up to spherical joints with respect to fixed reference frame

𝑆1, 𝑆2, 𝑆3 are expressed using equation (3.32) by,

𝑆𝑖 = 𝑂𝑂1

+ 𝑂1𝑆1 = 𝑑 + 𝑋𝑖

(4.2)

Joint angle constraints:

The motion cone precisely determines the motion range permitted in a

spherical joint. The angular capacity of ball joint is physically constrained motion and

can be defined as angle between Z-axis of base frame and vector along the connecting

link to spherical joint.

Joint coordinates

ranges

Loop closure equations of

parallel manipulators

Method of solving non-linear

loop closure equations

Manipulator’s Workspace

Graphical Representation

3-D

OF

3-P

RS

Parallel

Man

ipu

lator

Co

nfig

uratio

ns

Inverse Kinematic problem (IKP)

formulation of the parallel manipulator’s

architecture

108

The rotation angle of ball joint and its constraint can be computed by,

𝜃𝑠𝑖 = cos−1𝑙𝑖 ∙ 𝑇 𝑛𝑝𝑖

𝑙𝑖 ≤ 𝜃𝑠𝑚𝑎𝑥 (4.3)

Figure 4.3 Angular capacity of a spherical joint, the socket opening and motion cone [55]

The relative motion in a spherical joint through the function 𝜎 = 𝑓 𝜏 relying

on the azimuth angle (𝜎) and tilt angles (𝜏) as shown in figure 4.3.

Link interference constraints:

Links have physical dimensions, interference might have occurred. Many

softwares determine volumetric interference between links using different algorithms.

In virtual environment, collision detection is one of the most computationally

demanding in field of robotics, path planning and rigid as well as deformable body

simulations. Any volumetric interference among any moving or stationary component

of the assembly leads to end of the mechanism simulation automatically using global

collision detection feature in Pro/Engineer.

4.3.2 Non-geometric constraints

Controllable motion condition:

The relationship between moving platform velocity and velocity of actuations

is expressed using jacobian matrix as per equation (3.68),

𝑣1, 𝑣2 , 𝑣3 𝑇 = 𝐽 𝑉, 𝜔 𝑇 (4.4)

Where,

𝑣1, 𝑣2 , 𝑣3 𝑇 = Velocity of actuations for limb1, limb2 and limb3

𝐽 = Jacobian matrix (It should be full rank matrix for controllable motion)

𝑉, 𝜔 𝑇 = Velocity of moving platform

109

4.4 3D Workspace development of 3-PRS manipulator using direct

kinematics

Depending on the linkage and a configuration of the joints, workspace helps to

determine working area/volume and applications of the manipulator for various

purposes. Before developing a physical prototype, one has to estimate maximum

limits on workspace. The process for obtaining a workspace is divided into two basic

stages.

In a first step, set of all possible poses the manipulator can reach is obtained.

Therefore, 3D workspace of 3-PRS configuration is developed through direct

kinematics and presented in this section. In the second stage, a singularity analysis of

these positions is performed, based on Jacobian matrices computation. Hence,

singularity analysis is carried out and presented in Chapter 5. Three non-linear

simultaneous loop closures equations are derived analytically for 3-PRS

configurations as discussed in Chapter 2 for forward kinematics. Forward or direct

kinematics is always complicated with loop closure equations due to non linear

equations in case of parallel manipulators. The procedure to determine solution of

passive joints variables 𝜃𝑖 for known values of active joints actuations is programmed

in MATLAB using Bezout’s method and Sylvester method. An appropriate solution

set for three passive joints variables is identified out of available solutions sets. Then,

centre point coordinates of spherical joints and tip coordinates are determined as

discussed in Chapter 3.

The tool tip coordinates are used as point clouds to define workspace

boundaries through direct kinematics. The efficient usage of 3-DOF 3-PRS

configuration parallel manipulator can be realized for machine tool applications

through clarity on workspace generation. Three prismatic joints are 120° apart above

the fixed base for configuration under consideration. Manipulator’s symmetry is

exploited and can be proved as shown in table 4.1 and 4.2. The boundary of

workspace is attained whenever at least one of the actuators reaches one of its limits.

Hence, reachable workspace of mobile platform for any prismatic joint actuation 𝑇𝑖

adhere to following limitations,

𝑇𝑚𝑖𝑛 < 𝑇𝑖 < 𝑇𝑚𝑎𝑥

The outer workspace boundary is developed for single actuation within its

joint range limit analytically and inner workspace boundary is identified for

110

simultaneous double actuations from same reference positions for both actuators

without time lag using mechanism simulation through software. It is also possible to

carry out workspace analysis further by considering all possible cases by double and

triple actuation with time lag between various actuations. It becomes difficult to find

out all voids inside the workspace in forward kinematics.

Table 4.1 Analytical solution of tool tip coordinates as a proof of axis symmetry for single

actuation

Actuation of limb-1 ∇𝑇1 in step of 5 mm

𝜃𝑧 = 0°

Actuation of limb-2 ∇𝑇2 in step of 5 mm

𝜃𝑧 = −120°

Tool Tip coordinates Tool Tip coordinates

∇𝑇1

(mm)

𝑥

(𝑚𝑚)

𝑦

(𝑚𝑚)

𝑧

(𝑚𝑚)

∇𝑇2

(mm)

𝑥

(𝑚𝑚)

𝑦

(𝑚𝑚)

𝑧

(𝑚𝑚)

5 -2.9171 -1.6842 762.7655 5 -2.9171 -1.6842 762.7655

10 -5.8350 -3.3689 760.9742 10 -5.8350 -3.3689 760.9742

15 -8.7540 -5.0541 759.0964 15 -8.7540 -5.0541 759.0964

20 -11.6741 -6.7400 757.1290 20 -11.6741 -6.7400 757.1290

25 -14.5955 -8.4267 755.0687 25 -14.5955 -8.4267 755.0686

30 -17.5187 -10.1144 752.9114 30 -17.5187 -10.1144 752.9114

35 -20.4440 -11.8033 750.6530 35 -20.4440 -11.8033 750.6530

40 -23.3719 -13.4938 748.2885 40 -23.3719 -13.4938 748.2885

45 -26.3030 -15.1861 745.8124 45 -26.3030 -15.1861 745.8124

50 -29.2382 -16.8807 743.2182 50 -29.2382 -16.8807 743.2182

55 -32.1784 -18.5782 740.4988 55 -32.1784 -18.5782 740.4988

60 -35.1247 -20.2793 737.6456 60 -35.1247 -20.2793 737.6456

65 -38.0787 -21.9847 734.6488 65 -38.0787 -21.9847 734.6488

70 -41.0420 -23.6956 731.4967 70 -41.0420 -23.6956 731.4967

75 -44.0168 -25.4131 728.1755 75 -44.0168 -25.4131 728.1755

80 -47.0059 -27.1389 724.6684 80 -47.0059 -27.1389 724.6684

85 -50.0127 -28.8748 720.9549 85 -50.0127 -28.8748 720.9549

90 -53.0415 -30.6235 717.0095 90 -53.0415 -30.6235 717.0095

95 -56.0981 -32.3882 712.7996 95 -56.0980 -32.3882 712.7996

100 -59.1900 -34.1734 708.2828 100 -59.1900 -34.1734 708.2828

111

Table 4.2 Analytical solution of moving platform coordinates as a proof of axis symmetry for

single actuation

Actuation of limb-1 ∇𝑇1 in step of 5 mm

𝜃𝑧 = 0°

Actuation of limb-1 ∇𝑇2 in step of 5 mm

𝜃𝑧 = 120°

Moving centre point coordinates Moving centre point coordinates

∇𝑇1

(mm)

𝑥

(𝑚𝑚)

𝑦

(𝑚𝑚)

𝑧

(𝑚𝑚)

∇𝑇2

(mm)

𝑥

(𝑚𝑚)

𝑦

(𝑚𝑚)

𝑧

(𝑚𝑚)

5 0.0140 0.0082 587.7983 5 0.0140 0.0082 587.7983

10 0.0567 0.0327 586.1065 10 0.0567 0.0327 586.1065

15 0.1289 0.0744 584.3972 15 0.1289 0.0744 584.3972

20 0.2318 0.1338 582.6698 20 0.2318 0.1338 582.6698

25 0.3664 0.2115 580.9235 25 0.3664 0.2116 580.9235

30 0.5340 0.3083 579.1574 30 0.5340 0.3083 579.1574

35 0.7360 0.4249 577.3704 35 0.7360 0.4249 577.3704

40 0.9740 0.5624 575.5613 40 0.9740 0.5624 575.5612

45 1.2499 0.7216 573.7287 45 1.2499 0.7216 573.7287

50 1.5655 0.9038 571.8711 50 1.5655 0.9038 571.8711

55 1.9233 1.1104 569.9865 55 1.9233 1.1105 569.9865

60 2.3260 1.3429 568.0728 60 2.3260 1.3429 568.0728

65 2.7765 1.6030 566.1274 65 2.7765 1.6030 566.1274

70 3.2787 1.8930 564.1471 70 3.2787 1.8930 564.1471

75 3.8369 2.2152 562.1282 75 3.8369 2.2152 562.1282

80 4.4562 2.5728 560.0662 80 4.4562 2.5728 560.0662

85 5.1432 2.9694 557.9556 85 5.1432 2.9694 557.9556

90 5.9056 3.4096 555.7892 90 5.9056 3.4096 555.7891

95 6.7536 3.8992 553.5580 95 6.7536 3.8992 553.5580

100 7.7003 4.4458 551.2502 100 7.7003 4.4458 551.2502

112

4.4.1 Workspace development using Sylvester’s method

Case-1: Workspace development using individual limb linear actuation at constant

velocity from a reference:

Initially, kinematic analysis of this configuration is carried out using Sylvester

method and solved using symbolic mathematics in MATLAB software. The solutions

of the three non-linear higher order equations are also worked out. The coordinates of

tool tip of the end effector are determined. The obtained tool tip coordinates are

exported to excel program. The coordinate’s data are captured for the rotary base

positions of (ϕ0): 0˚, 30˚, 60˚, 90˚, 120˚. A 3-PRS configuration repeats the same

coordinates for any angular increment after 120˚ due to its axi-symmetry nature. The

workspace of parallel manipulator is developed using such tool tip point clouds

process sequentially as represented in figure 4.4.

Fig. 4.4 Sequential processing of captured tip coordinates with single leg actuation

for workspace generation

Using MATLAB program, the workspace is developed after sequential

processing of tool tip coordinates data captured earlier. The workspace graph as

shown in figure 4.5 is developed in MATLAB by surf command to generate the

pattern of tool tip for single actuation of individual link at maximum velocity to

develop outer workspace boundary.

Leg1_30°

Leg1_60°

Leg1_120°

Leg2_30°

Leg3_120°

Leg2_60°

Leg3_90°

Leg3_60°

Leg3_60°

Leg2_120°

Leg2_90°

Leg1_0°

Leg1_90°

113

Figure 4.5 Work space generation using individual link actuation

with constant velocity & without time lag

Case-2: Workspace development using double limb linear actuation at constant

velocity from same reference:

In this case, linear actuation of pair of any two legs simultaneously from a

same reference level. It means actuation in limb plane either of limb-1 (𝑇1) and limb-2

(𝑇2), limb-2 (𝑇2) or limb-3 (𝑇3) and limb-1 (𝑇1) and limb-3 (𝑇3) plane actuations

simultaneously for same range and with a specified step size is applied without time

lag for determination of tip coordinates. The tool tip coordinates are captured in excel

file. The generated workspace boundary workspace of the manipulator can be seen in

figure 4.6 and 4.7 respectively. The workspace graph is made in MATLAB by surf

command to generate the pattern of tool tip coordinates. The inner workspace

boundary is also identified for double actuation simultaneously within its joint range

limits without time lag and same position with respect to base.

Figure 4.7 shows combination of above two figures 4.5 and 4.6. The outer

work space boundary is described by individual legs actuation, whereas inner one is

described by combination of any two legs actuation. The parallel manipulator has 3-

DOF with rotary base and can be used for parts assembly and light machining tasks

that require large workspace, high dexterity, high accuracy, high loading capacity

with considerable stiffness.

114

Figure 4.6 Work space development using any pair of links actuation

with constant velocity & without time lag

Figure 4.7 Work space boundaries generation using all possible combination of link

actuation with constant velocity & without lag.

115

4.4.2 Workspace development using Bezout’s approach

A fixed reference frame OXYZ is attached to the base platform. Workspace

theoretically extends to infinity in the Z-direction, a realistic limit on the stroke of the

prismatic actuators must be imposed in order to obtain meaningful results. Generally a

trajectory in the workspace of a Spatial Parallel Manipulator (SPM) is specified by a

set of points.

Figure 4.8 Linear actuation of 𝑇1 by 20 mm only and rotary actuation in step of 10°

Figure 4.9 Linear actuation of 𝑇1 by 10 mm only and rotary actuation in step of 1°

-100-50

050

100

-100

-50

0

50

100700

710

720

730

740

750

760

X- Co-ordinates(mm)

Y- Co-ordinates(mm)

Z-

Co

-ord

ina

tes(m

m)

-100

-50

050

100

-100

-50

0

50

100700

710

720

730

740

750

760

770

X- Co-ordinates(mm)

Y- Co-ordinates(mm)

Z-

Co

-ord

ina

tes(m

m)

116

Figure 4.10 Outer workspace boundary generation with linear actuation of T1 by 1 mm and

rotary actuation of base by 1°

In this case of SPM, workspace is estimated for discrete value of linear

actuation of T1 by 20 mm only for entire joint range from initial assumed position,

which results in discrete points of planar curve. Similarly, rotary actuation in step of

10° of rotary base and linear actuation T1 by 20 mm results in planar curve in another

plane. In this way, discrete points are obtained in 36 different planes to identify outer

workspace boundary as shown in figure 4.8 using MATLAB program for Bezout’s

approach. The more refined workspace boundary is presented in figure 4.9 for linear

actuation of T1 by 10 mm and rotary actuation in step of 1°. The more realistic outer

workspace boundary of 3-PRS parallel manipulator is depicted in figure 4.10, which

consists of 36000 tool-tip coordinates. The combined inner and outer work space

boundaries are shown in figure 4.11 using Bezout’s approach. The result of the

workspace development is quite satisfactory by comparing the result with available

result using Sylvester’s method. The processing time for workspace development is

comparatively less in case of Bezout’s approach compared to Sylvester’s method due

to variation in resultant matrix size as mentioned in Chapter 3.

-80-60

-40-20

020

4060

80

-100

-50

0

50

100700

710

720

730

740

750

760

770

X-Coordinates (mm)Y-Coordinates (mm)

Z-c

oord

inate

s (

mm

)

117

Figure 4.11 Outer and inner workspace boundaries development using Bezout’s approach

-80-60

-40-20

020

4060

80

-80-60

-40-20

020

4060

80680

690

700

710

720

730

740

750

760

770

X- Co-ordinates(mm)

Y- Co-ordinates(mm)

Z-

Co

-ord

ina

tes(m

m)

Outer workspace

boundary Inner workspace

boundary

118

4.5 3D Workspace development and analysis using Pro/Engineer

simulations

3D solid modeling assembly using mechanism constraints and virtual

simulation are carried out to minimize the problems encountered during its real time

performance. Graphical simulation of workspace generation is investigated using

Pro/Mechanism. The main emphasis is given to the use of CAD tools used to develop

and analyze workspace and understanding behavior of spatial parallel robot

manipulators. The three servo motor actuations sequences are shown in table 4.3 for

tip coordinates determination.

Simulation Parameters:

Base Servo motors: 10 deg/s

Simulation time for base servo motor: As per desired angular position

Link Servomotor: 1 mm/s

Simulation Time for actuation of prismatic joint: 100 sec

Frame Rate: 10

Number of frames: 1000

Linear Actuation: 100 mm

No. of frames: 100

Table 4.3 Servo motor actuations for Tip Coordinates determination

Rotary

Base

Angle

Servo motors

(1 ,2 or 3)

Actuation T1

(mm)

Actuation T2

(mm)

Actuation T3

(mm)

Body

Lock

Links No.

0°

30°

60°

90°

120°

Only T1 100 *** *** 2 & 3

Only T2 *** 100 *** 1 & 3

Only T3 *** *** 100 1 & 2

Only T1 & T2 100 100 *** 3

Only T2 & T3 *** 100 100 1

Only T3 & T1 100 *** 100 2

T1 & T2 & T3 100 100 100 ***

Placement of coordinate system at top and centre of fixed base plate.

BOTTOM_PLATE_FIXED:YDP

Parallel manipulators have smaller workspace compared to serial manipulator

as found from various literatures [11, 17]. The workspace means the set of all spatial

119

coordinates of the centre of the moving platform positions for entire working range of

active joints actuation. The workspace analysis is always imperative to avoid singular

configuration. Moreover, many facts can be observed to enhance the parallel

manipulator configuration further. It is always desirable to analyze the shape and

volume of the workspace for the particular application requirements point of view. It

is difficult to express complete workspace as it does not reveal the actual tool tip

orientation information of the machine tool, which is essential for user at time of

machining in many cases for physical constraints avoidance. The position coordinates

(x, y, z) are exported to excel program. In workspace of parallel manipulator one has

to actuate individual legs of the parallel manipulator. E.g. there are three legs, Leg 1

or Leg 2 or Leg 3. The workspace is developed after actuating individual screw pair

shown in figure 4.12.

The workspace of the proposed manipulator is developed using MATLAB coding.

Surface is generated using surf command.

Table 4.4 Consideration of different cases to trace curves for workspace analysis

Single Actuation Case Double Actuation Triple actuation

1

2

3

Prismatic joint

actuation

individually with

maximum

velocity

A

1,2 , 2,3 , 1,3 limbs actuation

simultaneously with same maximum

velocity without time lag

1,2,3 limbs actuation

with same velocity

without time lag

B

1,2 , 2,3 , 1,3 limbs actuation

simultaneously with same maximum

velocity with time lag

1,2,3 limbs actuation

with same velocity

with time lag

C

1,2 , 2,3 , 1,3 limbs actuation

simultaneously with different velocity

without time lag

1,2,3 limbs actuation

with different velocity

without time lag

D

1,2 , 2,3 , 1,3 limbs actuation

simultaneously with different velocity

with time lag

1,2,3 limbs actuation

with different velocity

with time lag

Single Actuation: For 3-PRS configuration, the position coordinates are captured for

the limiting range of the spherical joints movement as well as prismatic joints to avoid

the interference between link and mobile platform. The coordinate’s data are captured

for different angular positions (θZ): 0˚, 10˚, 20˚………..110˚, 120˚ for each prismatic

joint actuation of 100 mm. A Tripod configuration repeats the same coordinates for

the angles incremented by 10˚ after 120˚ of the next limb as mentioned in table 4.4.

120

The outer work space boundary is developed using curve tracing for angular step size

of 10° about z-axis with single prismatic joint actuation of 100 mm as shown in figure

4.12, 4.13 and 4.14.

Figure 4.12 Traces of planar curves for workspace boundary representation using

Pro/Engineer

Figure 4.13 Workspace boundary with angular step size of 1° for initial 30º and in step of 30°

afterward using single actuation

Workspace

boundary with

angular step size of

1° for initial 30º

Workspace

boundary with

angular step of 30º

121

Double Actuation:

Case-A: 1,2 , 2,3 , 1,3 limbs actuation simultaneously with same maximum

velocity without time lag

The inner workspace boundary as highlighted using red color is obtained due

to simultaneous actuations of any two limbs with same maximum velocity without

time lag as shown in figure 4.13 for a step size of 10°. The symmetry is observed after

120° for any limb pair prismatic joints actuations with above conditions.

Figure 4.14 Workspace boundary with angular step size of 10° using double actuation with

maximum velocity without time lag

Inner workspace boundary points are determined to separate points obtained

through actuation of various combinations of all actuators. Using Pro/Mechanism, 36

curves are traced with step size of 10° as shown in figure 4.14 for double actuation

with maximum velocity without time lag. These results confirm the results obtained

analytically

Case-B: 1,2 , 2,3 , 1,3 limbs actuation simultaneously with same maximum

velocity with time lag

As time lag increases the traced curve is more offset from the curve traced

without time lag and which is little short compared to curve traced out for actuation

with maximum velocity without time lag. Time lag between servo motors 1 and 2 is

positive the curve is offset towards limb-1, otherwise it is offset towards limb-2 for

Workspace

boundary for

double actuation

Workspace

boundary for

single actuation

122

negative time lag between servo motors 1 and 2. Moreover, these curves are lying on

the same surface of case-A.

Case-C: 1,2 , 2,3 , 1,3 limbs actuation simultaneously with different velocity

without time lag

One limb with maximum velocity and other with less velocity will trace a

curve in between outer and inner workspace boundary. Limb-1 with maximum

velocity and limb-2 with half of the maximum velocity of limb-1 leads to this

situation as shown in figure 4.14. Two limbs prismatic joints are actuated with

different velocity, but both velocities are less than maximum velocity. In this case

also traced curve is lying in between outer and inner workspace boundary.

Case-D: 1,2 , 2,3 , 1,3 limbs actuation simultaneously with different velocity with

time lag

Figure 4.15 Traced curve for double actuation with different velocity with time lag

Traced 3D curve trajectory is lying in between inner and outer workspace

boundary as represented by dotted lines. It is observed that any of the actuator is near

to maximum velocity the traced curve is near to outer workspace boundary. It means

it is more away from inner workspace boundary and larger curve length for smaller

time lag as shown in figure 4.15.

Triple Actuation:

Case-A: 1,2,3 limbs actuation with same velocity without time lag

Straight line path is generated along the z-axis of fixed reference frame. The

length of traced straight line is equal to velocity of actuator multiply by simulation

time in such case.

Limb-1: 0.7 mm/s, Limb-3: 0.4

mm/s, Time Lag: 5 sec

Limb-1: 0.9 mm/s, Limb-3: 0.3

mm/s, Time Lag: 3 sec

123

Case-B: 1,2,3 limbs actuation with same velocity with time lag

3D curve is generated and lying inside the inner workspace boundary.

Case-C: 1,2,3 limbs actuation with different velocity without time lag

Planar curve is generated in 3D workspace within inner workspace boundary.

It is observed that generated planer curve is lying in between the two highest velocity

actuators limb planes. Moreover, angular position of the planar curve in space can be

determined with respect to XZ-plane of fixed reference frame using following

procedure,

Equation of plane passing through any three points of generated planar curve

XZ-Plane of fixed reference frame located on fixed base

Determination of normal 𝑛 1 and 𝑛 2 of these two planes

The angular position of planar curve can be determined by

cos 𝛼1 =𝑛 1 ∙ 𝑛 2

𝑛 1 𝑛 2 =

𝑛 1 ∙ 𝑗

𝑛 1 (4.5)

Case-D: 1,2,3 limbs actuation with different velocity with time lag

3D space curve is generated using any three combinations of actuator with

different velocity and with time lag in this case.

Sylvester’s Theory is adopted to validate the results of Pro/Engineer software

as well as Bezout’s elimination approach. The outer and inner workspace boundaries

are created using Bezout’s elimination theory is utilized further for singularity

evaluation. After using all the above three methods, it is concluded that the workspace

boundaries developed by each method is almost similar.

Normally, near the singular configurations either of serial or parallel

manipulator’s experiences poor performance as found from various literature.

Singularity can be a kind of situation where the manipulator has additional

uncontrollable DOF or loss of any existing DOF. Algebraically, singularities represent

rank deficiency of Jacobian matrix. Force transmission is relatively very poor near

singular region. Singular configurations can be determined using various approaches

like screw theory, instantaneous center of motion for planar configurations and using

coordinate transformations. The singularity analysis is explained for configuration

under consideration in next chapter.

124

Volume Index: It is one of the workspace performance index. It reflects the ratio of

workspace size to physical size of manipulator.

Workspace volume can be computed through following procedure approximately,

Run mechanism kinematic analysis using double actuation for limb-1 and

limb-2

Measure results of X, Y and Z coordinates of tool tip and export file in excel

format separately for each coordinates.

Combine X, Y and Z- coordinates in a single excel file. Save As –and change

the file type to Text (tab delimited) (*.txt). Manually change the file name

with a .pts extension.

Create a new ProE part file. To add the point array, use INSERT >> Model

Datum >> Point >> Offset Coordinate System.

In the Offset CSys Datum Point Dialog box, select the default coordinate

system to be the reference. Import .pts file

Figure 4.16 Offset coordinate system datum points of tool tip array

Create a Sketch. Select the Front datum plane as the Sketch plane. Generate

planar curve using spline. Close figure and use revolve feature to develop

workspace volume for single and double actuation.

Use ANALYSIS<<Measure<<Volume

125

Figure 4.17 Volume index of 3-PRS configuration for assumed structure parameters

Similarly, manipulator’s volume can be developed using revolve feature after

drawing line diagram for any limb plane for assumed structure parameters.

Manipulator’s Volume: 207414651 𝑚𝑚2

Workspace Volume: 604490 𝑚𝑚2

Workspace volume index: 342757/207414651 =0.00291

It is observed that workspace volume index is very low, which is one of the

characteristic of parallel manipulator. They exhibit less but very useful workspace

compared to serial manipulators for performing precise operations.

4.6 3D Workspace development using IKP solutions

It is necessary to change orientation of tool while manipulating some complex

surfaces. Machining on hemispherical surfaces or performing inclined internal

operations on prismatic surfaces requires a complex orientation of the end-effector

normal to the operating surface. Therefore, it is imperative to have inverse kinematic

formulation of workspace that visualizes the viability of required task

accomplishment.

Given a pose (position and orientation) of the manipulator, reference point of

the moving platform determines an allowable point within the workspace, if the

inverse kinematics of the given pose exists under all the kinematic constraints. By

giving a series of poses and obtaining a series of allowable points of the upper

platform, the workspace becomes as an assembly of all the allowable points. The

workspace is discretized as a uniform grid of nodes in Cartesian coordinate system.

Each node is then examined in order to determine whether it belongs to the workspace

Manipulator’s volume

Outer workspace boundary

Workspace volume

126

or not. For a given angular orientation of the end-effector, each candidate workspace

point is then considered and inverse kinematics problem is solved. The leg lengths

obtained as solutions are then compared to the minimum and maximum allowed leg

lengths to ascertain if the candidate point is part of workspace. All such points can

then be marked to obtain a graphical representation of the workspace.

Leg1 actuation by 100 mm:

Leg2 actuation by 100 mm:

Leg1-2 actuation by 100 mm:

Leg3 actuation by 100 mm:

Leg1-3 actuation by 100 mm:

127

Leg2-3 actuation by 100 mm:

Figure 4.18 Snap shot of position coordinates for range determination of inverse kinematics

Table 4.5 Variation of tool tip coordinates for maximum actuation of 100mm for any

combination

Initially assumed Configuration X-coordinate: 0 Y-coordinate: 0 Z-coordinate: 764.473

Direction Minimum Maximum

𝑥 -59.19 59.19

𝑦 -66.1998 68.3467

𝑧 683.735 708.283/764.473

Table 4.6 Variation of moving platform centre point coordinates for maximum actuation of

100mm for any combination

Initially assumed Configuration X-coordinate: 0 Y-coordinate: 0 Z-coordinate: 589.473

Direction Minimum Maximum

𝑥 -7.003 7.003

𝑦 -8.89154 4.44577

𝑧 519.864 551.25/589.473

Generally, the workspace of parallel mechanism is obtained through inverse

kinematics simulation offline. The extremities of tool tip coordinates and moving

platform centre point coordinates are decided from obtained direct kinematics

solutions as shown in figure 4.18. The following ranges are considered based on

assumed structure parameters for tool pose of proposed configuration within

workspace:

X-coordinates of tool tip: Range [-60, 60] in step size of 0.1

Y-coordinates of tool tip: Range [-67, 69] in step size of 0.1

Z-coordinates of tool tip: Range [683,765] in step size of 20

128

X-coordinates of moving platform centre point: Range [-7, 7] in step size of 0.1

Y-coordinates of moving platform centre point: Range [-9, 5] in step size of 0.1

Z-coordinates of moving platform centre point: Range [519,590] in step size of 20

One can judge the position point belong the workspace or not.

Figure 4.19 Workspace generation using inverse kinematics for course step size

Numerical algorithm for workspace generation using inverse kinematics:

1. Give constant structural parameters of the 3-PRS configuration 𝑇, 𝑝, 𝑈, 𝑏1, 𝑞

2. Input tool tip and moving platform coordinates or tool tip and orientation angle of

moving platform

3. Compute orientation angle of moving platform using tool tip and moving platform

coordinates

4. Determine intersection of plane and sphere defined by spherical joint coordinates

and centre of moving platform along with constraints of manipulator

𝑆3𝑥 = 0

𝑆2𝑥 = − 3𝑆2𝑦

𝑆1𝑥 = 3𝑆1𝑦

5. Determination of spherical joint coordinates 𝑠𝑖 𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 , 𝑖𝜖 1, 2, 3 and

𝜃1 , 𝜃2𝑎𝑛𝑑 𝜃3 using expressions (3.45)

𝑐𝑜𝑠𝜃1 =1

𝑈 2𝑦1 +

𝑞

3− 𝑏 , 𝑐𝑜𝑠𝜃2 =

1

𝑈 2𝑦2 +

𝑞

3− 𝑏

𝑐𝑜𝑠𝜃3 =1

𝑈 −𝑦3 +

𝑞

3− 𝑏

129

6. Compute linear translational actuations using equation (3.46),

𝑇 − 𝑇𝑖 = 𝑧𝑖 − 𝑈𝑠𝑖𝑛𝜃𝑖

7. Check after required actuations,

If determined final actuators positions are within specified joint ranges for all

prismatic joint actuators then

Point is within workspace

else

Point outside workspace

8. The workspace of 3-PRS parallel manipulator is set of all points satisfying the

conditions stated in step-7.

9. Workspace generation as shown in figure 4.19

4.7 Concluding remarks

The concept of reachable, orientation and dexterous workspace is expressed.

Geometric and non-geometric constraints are explained in this work. The advanced

computer tools are utilized to develop the workspace completely. Tool tip coordinates

obtained analytically are plotted for workspace development. Simulation software

(Pro/Mechanism) is used for workspace generation for 3-PRS manipulator with

assumed structural parameters. The shape and size of generated workspace boundaries

using analytical approach and through simulation are same. Inner and outer

workspace boundaries are developed and workspace analysis is carried out for four

different cases as shown table 4.4. Workspace volume index for the given structural

parameters is identified for the proposed mechanism. Algorithm for 3D workspace

development using inverse kinematics is also reported.