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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
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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.