Helmet attempt to protect the user’s head by absorbing mechanical energy and protecting against penetration. Every year many people or riders are killed or seriously injured in the accidents as a result of head injuries. Wearing an appropriate safety helmet significantly reduces the risk of injury or even death. Protective headwear could save your life.In this research work a modified helmet will be designed with different compositions in CATIA for modelling. After modelling the helmet the critical positions of impact was identified.To use test with different material combination like 3mm Acrylonitrile Butadiene Styrene (ABS) as outer surface and 1mmExpanded Poly-Styrene (EPS) as inner surface, as well as 3mm Poly Carbon (PC) as outer surface and 1mm Expanded Poly-Styrene (EPS) as inner surface are considered. The main objective of the research is to enhance the safety of the rider and find out the best suitable material of the helmet

1. http://www.iaeme.com/IJMET/index.asp 198 editor@iaeme.com
International Journal of Mechanical Engineering and Technology (IJMET)
Volume 9, Issue 11, November 2018, pp. 198–210, Article ID: IJMET_09_11_022
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=11
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
A NOVEL APPROACH OVER INVERSE
KINEMATIC ANALYSIS OF 5-AXIS HYBRID
PARALLEL MANIPULATOR FOR CONTOUR
TRAJECTORY
Suryam L.V
Department of mechanical engineering, Vignan’s institute of engineering for women,
Visakhapatnam, AP, India,
A.V.Pradeep
Department of mechanical engineering, Vignan’s institute of engineering for women,
Visakhapatnam, AP, India,
S.V.Satya Prasad
Department of mechanical engineering, Vignan’s institute of engineering for women,
Visakhapatnam, AP, India,
K.Vahini
Department of mechanical engineering, Vignan’s institute of engineering for women,
Visakhapatnam, AP, India,
Y.Sai ratnakar
Department of mechanical engineering, Welfare Institute of science,technology and
Management, Visakhapatnam, AP, India,
ABSTRACT
In the recent past, the PKM technology was brought to the next level by invention of
Hybrid Parallel Kinematic Machines (HPKMs) because both serial and Parallel
Kinematic Machine features are incorporated in these machines.Kinematics and
dynamics play a crucial role in functioning of any parallel kinematic machine. The
Inverse kinematic analysis is tedious for finding the velocity and acceleration of legs in
PKMs having more than three Degree of freedom (DOF). In this paper, 5-axis Hybrid
parallel kinematic machine with hemisphere workspace has been modeled and assembled
in CATIA. The inverse kinematic analysis of PKM was carried out in digital mockup unit
(DMU). The variations in velocities and accelerations of all the three legs and joint angles
were found along work plane axes at desired feed rate. On the other hand, the regression
equations were generated for velocity and acceleration of each leg, joint angles with
respect to position and time, while the tool travels along the v-shape contour trajectory.

2. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 199 editor@iaeme.com
Key words: 5-Axis HPKM; Inverse Kinematics; DMU; contour trajectory.
Cite this Article Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai
ratnakar, A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel
Manipulator for Contour Trajectory, International Journal of Mechanical Engineering and
Technology, 9(11), 2018, pp. 198–210.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=11
1. INTRODUCTION
Parallel Kinematic machines are based on Parallel mechanism machine tools. These mechanisms
offer better stiffness, due to the existence of multiple legs. The Stewart Gough platform illustrated
this benefit by hexapods [1-2]. The researchers in robotics community insist that parallel
kinematic machines (PKMs) have considerable merits over conventional serial mechanisms in
terms of operation speed, rigidity, accuracy and dynamic load.[3]. The design of PKMs usually
contains two steps namely dimension synthesis and topology synthesis. A good performance
obtained by providing good topology with optimized geometrical parameters. Based on existing
methods, the dimension synthesis is categorized into two types, i.e., objective function based
optimal design [4-9] and performance-chart based design [10-11]. Defining of suitable
performance indices, employing proper optimization algorithms as well as reducing the number
of design variables are still challenging issues [12-13].
Several PKMs have been proposed over last three decades, but only some of them have been
effectively used in production [14-18]. The main reason is that, PKM usually needs lot of passive
joints, and these joints arise new problems like geometric deviations and deformations under a
force or thermal load [19].To decrease the number of passive joints, Neumann has patented a
novel HPKM(Hybrid Parallel Kinematic machine) called Exechon machine tool[20]. Exechon
machines can be applied to fill the space between the industrial robots and traditional machine
tools[21]. The HPKM architecture consists of a 3-degreeof-freedom (dof) parallel mechanism
(PM) connected in series with a t three degree of freedom spherical wrist.[22] . The Exechon 5-
Axis Parallel Kinematic Machine (PKM) is a successful design created in Sweden and adopted
by many producers of machine tools around the world. A new version of the manipulator is being
developed as a component of a mobile self-reconfigurable fixture system within an inter-
European project [23]. Kinematics and Dynamics form basics in functioning of a parallel
kinematic machine. However, due to the highly nonlinear connection between joint variables and
position/orientation of the end effectors, it has been a challenging work to establish a kinematic
model on Hybrid–parallel kinematic machine in closed form [21,24].Z.M.Bi solved the forward
and inverse kinematics of 5-axis HPKM [23].
In the present work a regression analysis approach is proposed to solve the inverse kinematic
analysis of 5 –axis HPKM. For that various parts of it are modeled and assembled in CATIA.
Further, inverse kinematic analysis of the modeled pkm is carried out in the milling of a v-shape
pocket. Moreover, joint angles, variations in velocity and accelerations of each leg have been
studied, while the tool moves along the desired trajectory at 800mm/min feed rate. Finally, the
regression equations are generated for velocity and acceleration of each leg and joint angles with
respect to position and time, along the tool travel trajectory. Mini tab.17 was employed for
generating regression equations.
2. STRUCTURE OF 5-AXIS HPKM
The endeffector platform is attached by three Legs namely, Leg_1, Leg_2, and Leg_3,
respectively which is shown in Figure 1. At home position, the Leg 1 and Leg 3 are symmetrical
to Leg 2. The universal joint is used to connect the leg1 and the base with the rotational axes of
£11 and £12 and it is followed by a prismatic joint with the axis of £13. Leg1 and the end-effector

3. A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel Manipulator for
Contour Trajectory
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platform are connected by a revolute joint with the axis of £14. Similarly, the Leg1 and Leg3 are
connected to the base and the endeffector platform by means of revolute joint, prismatic joints.
So, £11 and £13 function with each other. Leg 2 is different from Leg1 and Leg3.The universal
joint is used to connect the Leg2 and base with the rotational axes of £21 and £22 and it is followed
by a Prismatic joint with the axis of £23.With respect to the axis £23, the Leg 2is allowed to spin
by itself. The other end of Leg 2 and Leg 3 is connected to the endeffector by a revolute joint
with the axes of £24, £34. Three translation motions are actuated along £13, £23, and £33, and the
rest of the motions are passive. The centers of universal joints on the base are denoted as P1, P2,
and P3. The connected location of a leg and the platform are represented by Q1, Q2, and Q3
respectively. Consider the global coordinate system is , , , , where Ok is the
central point of P1P3. Xk is a unit vector from P1to P3. Yk is at right angles P1P3 and in the direction
of P2, Zk is toward down to the end-effector platform. The coordinate system is assigned
as , , , , where Og is the middle point of Q1Q3. Xg is a unit vector from Q1 to Q3.
Yg is perpendicular to Q1Q3 and in the direction of Q2, Zg is down to the end-effector platform. A
serial 2R manipulator is connected to the endeffector platform to implement manipulating tasks.
Figure 1 Axes representations of 5-Axis HPKM
3. MODELING AND ASSEMBLY OF 5-AXIS HPKM
The design parameters of HPKM are taken from existing 5-axis Parallel kinematic machine. The parts
classification of 5-axis HPKM is shown in Figure 2. To perform assembly, the parts of HPKM were
designed using 3D modeling tool, CATIA. The DMU Kinematics module was employed to assemble all
the modeled parts. The insert command was used to export the components. All the components were
arranged in proper positions by using move command. For assembling all parts, the Part.1 is fixed using
anchor command. To generate kinematic motion, the revolute joint command was used. The revolute
joints were formed between part.1 and part .2, part.1 and part .3, part.2 and part .4, part.3 and part .5,
part.3 and part .6, part.7 and part .10, part.7 and part .8 and part.8 and part .9. Similarly, the prismatic
joint command was used to create prismatic joints between part.4 and part.10, part.5 and part.10.The

4. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 201 editor@iaeme.com
final assembly of 5-axis HPKM is illustrated in Figure 3.
Figure 2 Parts of 5-axis PKM
4. KINEMATIC ANALYSIS OF 5-AXIS PKM USING DMU IN CATIA
It is necessary to simulate the machine after the assembly. The point curve joint command is used
to make a joint between the tool and v-shape work path to simulate the pkm. In order to simulate
the machine, the simulation command was chosen from DMU generic animation tool bar. The
collision between the parts were analysed through simulation. So as to perform inverse kinematic
analysis for the complete mechanism, a kinematic law should be established. The simulation time
has been manually calculated for desired feed rate to create the law. The obtained value is 6.342
s for 800 mm/min respectively. A law ((84.5399mm)/ (6.342s)*(Mechanism.1/KINTime)) was
established for the whole mechanism to identify the velocity and accelerations of each joint. This
was achieved using formula icon from the knowledge tool bar.
A sensor point was created on each leg for finding the positions, velocities and acceleration.
The correlation was established between base of the machine and the sensor on the leg by using
speed acceleration command in DMU kinematic tool bar. The sensors were activated by using
simulation with laws command from simulation tool bar. The desired graphs (linear velocity,
linear acceleration, angular velocity and angular acceleration with respect to time) were obtained
by simulating the tool along the trajectory. The corresponding values in the form of .xls file were
generated very prompt after the simulation.
Figure 3 Assembly of HPKM

5. A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel Manipulator for
Contour Trajectory
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5. RESULTS AND DISCUSSIONS:
The assembly of 5-axis PKM has been modeled in CATIA.DMU kinematic analysis has been
performed for PKM considering the tool motion along the v-shape trajectory, with 800 mm/min
feed rate. The path traced by the tool along with work plane axes is indicated in Figure 4.The
perimeter of the v-shape pocket is 84.5399 mm. The machining time obtained for desired feed
rate during the v-shape motion is 6.342 s. . To find the variation in length of the legs, the motion
is constrained between tool post to tool and tool post to endeffector (θ11 =0 and θ12=0). In the
DMU kinematic analysis, 41 points were taken on the perimeter of v-shape trajectory. For those
points, the instantaneous time and the position of the tool are obtained for required feed rate,
which are shown in the Table.1.
Figure 4.Tool path and work plane axes
Table.1.Positions and instantaneous time of tool at 800mm/min
S.N
O
Positio
n
P (mm)
800
mm/min
S.N
O
Positio
n
P (mm)
800mm
/min S.N
O
Positio
n
P (mm)
800
mm/min
S.N
O
Positio
ns
P (mm)
800
mm/mi
n
T(s) T(s) T(s) T(s)
1 0 0 12
23.248
5
1.7440
7
23
46.496
9
3.48814 34
69.745
4
5.2322
1
2 2.1135
0.15855
2
13 25.362
1.9026
2
24
48.610
4
3.64669 35
71.858
9
5.3907
7
3
4.2269
9
0.31710
4
14
27.475
5
2.0611
8
25
50.723
9
3.80525 36
73.972
4
5.5493
2
4
6.3404
9
0.47565
6
15 29.589
2.2197
3
26
52.837
4
3.9638 37
76.085
9
5.7078
7
5
8.4539
9
0.63420
8
16
31.702
5
2.3782
8
27
54.950
9
4.12235 38
78.199
4
5.8664
2
6
10.567
5
0.79276 17 33.816
2.5368
3
28
57.064
4
4.2809 39
80.312
9
6.0249
7
7 12.681
0.95131
2
18
35.929
5
2.6953
8
29
59.177
9
4.43945 40
82.426
4
6.1835
3
8
14.794
5
1.10986 19 38.043
2.8539
3
30
61.291
4
4.59801 41
84.539
9
6.3420
8
9 16.908 1.26842 20
40.156
5
3.0124
9
31
63.404
9
4.75656
10
19.021
5
1.42697 21
42.269
9
3.1710
4
32
65.518
4
4.91511
11 21.135 1.58552 22
44.383
4
3.3295
9
33
67.631
9
5.07366

6. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 203 editor@iaeme.com
During the machining process of v-shape pocket, it is observed that slight variations in joint
angles were obtained at given feed rate and the corresponding values are shown in
Appendix.Table.1. To complete the machining, the maximum variation of length in Leg_1 is
15.3mm, Leg_2 is 14.1mm and Leg_3 is 15.4mm with respect to initial positions. As the
workpiece existed in XY plane, there exists different velocity for each leg in both X and Y
directions. The corresponding values of linear velocity, linear acceleration, angular velocity and
angular acceleration for each leg have been obtained, which are presented in Appendix Table (2-
4).
5.1. Velocities and accelerations of Legs
Leg 1: At the feed rate of 800mm/min, the maximum linear velocity obtained in X and Y
directions are 0.0041099m/s and 0.007455 m/s at 4.439s and 4.756 s respectively. The angular
velocity in X and Y directions are 0.039423 rev/min at 6.024s and 0.04644 rev/min at 0.317 s.
Similarly, the maximum linear acceleration in X and Y directions are 0.0070796 m/s2
at 4.122s
and 0.00200188 m/s2
at 4.439 s, and angular acceleration in X and Y directions are0.0128737
rad/s2
at 4.439s and 0.005316 rad/s2
at 6.024 s.
Leg 2: linear velocity in X and Y directions are 0.0042127m/s at 4.439s and 0.0047007 m/s
at 4.756 s.angular velocity in X and Y directions are0.039423 rev/min at 6.024 s and 0.0457
rev/min at 1.902 s. linear acceleration in X and Y directions are 0.006623 m/s2
at 4.122s
and0.0021075 m/s2
at 4.439 s. Angular acceleration in X and Y directions are 0.01287 rad/s2
at
4.439s and 0.006325 rad/s2
at 6.024 s.
Leg 3: Linear velocity in X and Y direction are 0.004751m/s at 4.439s and 0.00863 m/s at
4.756 s.Angular velocity in X and Y direction are 0.0394721rev/min at 6.024 s and 0.0481092
rev/min at 1.902 s. Linear acceleration in X and Y direction are 0.007874 rev/min at 4.122 s and
0.026456 rev/min at 4.439 s.Angular acceleration in X and Y direction are 0.004485 rad/s2
at
4.439 s and 0.0073014 rad/s2
at 4.756 s.
The combined variations of all the parameters with respect to time in both X and Y directions
are plotted using 3D surface graphs. The plots are shown in Figure.5.

7. A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel Manipulator for
Contour Trajectory
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Figure 5 Variations in Linear velocity(V), Linear acceleration(ω), Angular velocity(a), Angular
acceleration (α) of Leg 1, Leg 2 and Leg 3 at 800mm/min feed rate
While the tool moves along the v-shape path at given feed rate, the corresponding values of
V, ω, a and α of each Leg varies. Figure.5 shows the surface plots for variation in linear velocity,
linear acceleration, angular velocity, angular acceleration of three legs in X-Y directions. From
the plots it is evident that, V, ω, a and α with respect to time are higher y-direction than x-
direction.
5.2. Regression analysis:
The values obtained in DMU kinematic analysis have been exported to Minitab to generate the
regression equations. The equations were generated for each variation in joint angle in relation
with position and time. The regression equations for joint angles at given feed rate are as follows:

8. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 205 editor@iaeme.com
θ1= -46.8623 + 0.00766 P- 0.0548 T
θ2= 23.780 + 0.0073 P-0.133 T
θ3= -46.8623 + 0.00766 P- 0.0548 T
θ4= 11.660 + 0.0016 P- 0.057 T
θ5 = -11.0352 - 0.00499 P+ 0.0355 T
θ6= 7.368 + 0.0074 P- 0.129 T
θ7= 0.35029 + 0.000775 P- 0.00842 T
θ8= 0.83432 - 0.000117 P- 0.00349 T
θ9= -1.5050 + 0.00498 P- 0.0355 T
Spin_ θ10 = 23.780 + 0.0073 P- 0.133T
d1 = -160.29 + 0.260 P- 2.55 T
d2 = 937.43 + 0.135 P- 0.26 T
d3 = 530.33 + 0.279 P- 1.99 T
Similarly, the regression equation is generated for both in x and y direction for vx , vy, ax, ay,
ωx ωy αx αy respectively. This is done in relation to the position and time for all positions at
800mm/min feed rate.
Leg_1 Leg_2 Leg_3
vx = -0.00369 + 0.00076 P - 0.0089 T
vx = 0.00335 + 0.190 P
- 2.54 T
vx = 0.00426 + 0.221 P
- 2.95 T
vy = -0.00887 - 0.00234 P + 0.0338 T
vy = -0.02448 - 0.916 P
+ 12.22 T
vy = -0.02685 + 0.515 P
- 6.9 T
ax= 0.00197 - 0.000208 P + 0.00276 T
ax = -0.00238 + 1.420 P
- 18.94 T
ax = 0.00277 + 0.127 P- 1.70 T
ay = 0.0090 - 0.000673 P + 0.00874 T ay = 0.0025 + 1.76 P - 23.5 T ay = 0.0207 + 3.08 P - 41.1 T
ωx = -0.1636 + 0.00173 P + 0.0152 T ωx = -0.1464 - 4.83 P + 64.4 T
ωx = 0.0001+0.00021 P-
0.0041 T
ωy = -0.0195 + 0.00133 P - 0.0172 T ωy = -0.0028 - 5.80 P + 77.3 T ωy = -0.0358 - 5.37 P + 71.6 T
αx = 0.00548 - 0.000372 P
+ 0.00480 T
αx= 0.00064 + 1.79 P - 23.8 T
αx = 0.00346 + 0.517 P
- 6.90 T
αy = -0.00227 + 0.000227 P
- 0.00302 T
αy = 0.00226 - 1.707 P + 22.75
T
αy = -0.00381 - 0.349 P
+ 4.66 T
6. CONCLUSIONS
The following conclusions have been drawn in this paper as follows:
• The 5-axis PKM have been modeled in CATIA, which combine a 2-dof wrist and a
3-dof tripod. The clashes between parts were studied. The linear velocities and
accelerations, Angular velocities and accelerations of each leg along work plane axes
were identified.
• The maximum linear velocity is obtained for leg_1 at a position of 59.1779mm in X-
direction and 63.4049mm in Y-direction. Similarly the angular velocities are obtained
at 80.3129mm in x-direction and 4.22699mm in Y-direction.
• The maximum linear acceleration is obtained for leg_1 at a position of 54.9509mm in
X-direction and 59.1779mm in Y-direction. Similarly the angular accelerations are
obtained at 59.1779mm in x-direction and 80.3129mm in Y-direction.
• The maximum linear velocity is obtained for leg_2 at a position of 59.1779mm in X-
direction and 63.4049mm in Y-direction. Similarly the angular velocities are obtained
at 80.3129mm in x-direction and 25.362mm in Y-direction.
• The maximum linear acceleration is obtained for leg_2 at a position of 54.9509mm in
X-direction and 59.1779mm in Y-direction. Similarly the angular accelerations are
obtained at 59.1779mm in x-direction and 80.3129mm in Y-direction.

9. A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel Manipulator for
Contour Trajectory
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• The maximum linear velocity is obtained for leg_3 at a position of 59.1779mm in X-
direction and 80.3129mm in Y-direction. Similarly the angular velocitiesare obtained
at 80.3129mm in x-direction and 25.362mm in Y-direction.
• The maximum linear acceleration is obtained for leg_3 at a position of 54.9509mm in
X-direction and 59.1779mm in Y-direction. Similarly the angular accelerations are
obtained at 59.1779mm in x-direction and 63.409mm in Y-direction.
NOMENCLATURE:
θ1= Angle of rotation for Axis £11
θ2= Angle of rotation for Axis £21
θ3 = Angle of rotation for Axis £31
θ4= Angle of rotation for Axis £12
θ5 = Angle of rotation for Axis £22
θ6= Angle of rotation for Axis £32
d1= linear displacement of £13
d2= linear displacement of Axis £23
d3 = linear displacement of Axis £33
θ7 = Angle of rotation for Axis £14
θ8= Angle of rotation for Axis £24
θ9 = Angle of rotation for Axis £34
θ10= Spin of Leg 2.
θ11= Angle of rotation between End effector to
tool post
θ12= Angle of rotation between tool post to tool
vx =linear velocity in X-direction
vy=linear velocity in y-direction
ax=linear acceleration in X-direction
ay=linear acceleration in y-direction
ωx= Angular velocity in X-direction
ωy= Angular velocity in Y-direction
αx= Angular acceleration in X-direction
αy= Angular acceleration in Y-direction
s= Time in seconds
Appendix.
S.No
Time
(s)
θ1
(rad)
θ3
(rad)
θ2
(rad)
θ4
(rad)
θ5
(rad)
θ6
(rad)
θ7
(rad)
θ8
(rad)
θ9
(rad)
d1
(mm)
d2 (mm) d3 (mm)
1 0
-
0.817
0.406
-
0.817
0.195 0.12
-
0.193
0.014 0.01
-
0.0258
-
163.799
942.842 531.677
2 0.317
-
0.817
0.408
-
0.817
0.197 0.122
-
0.193
0.014 0.01
-
0.0259
-
162.743
941.735 531.65
3 0.634
-
0.817
0.41
-
0.817
0.199 0.124
-
0.193
0.014 0.01
-
0.0259
-
161.678
940.637 531.631
4 0.951
-
0.817
0.413
-
0.817
0.201 0.126
-
0.193
0.014 0.01
-
0.0259
-
160.604
939.547 531.62
5 1.268
-
0.817
0.415
-
0.817
0.203 0.128
-
0.193
0.015 0.01
-
0.0259
-
159.522
938.466 531.618
6 1.585
-
0.817
0.417
-
0.817
0.205 0.13
-
0.193
0.015 0.01
-
0.0259
-158.43 937.394 531.624
7 1.902
-
0.817
0.419
-
0.817
0.207 0.133
-
0.193
0.015 0.01
-
0.0259
-157.33 936.331 531.638
8 2.219
-
0.817
0.421
-
0.817
0.209 0.135
-
0.193
0.015 0.01
-
0.0259
-
156.221
935.277 531.661
9 2.536
-
0.817
0.422
-
0.817
0.21 0.135
-
0.193
0.015 0.01
-
0.0255
-
155.018
935.758 532.844
10 2.853
-
0.815
0.421
-
0.815
0.209 0.134
-
0.194
0.015 0.01
-
0.0246
-
153.695
938.271 535.563

10. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 207 editor@iaeme.com
11 3.171
-
0.814
0.42
-
0.814
0.207 0.133
-
0.195
0.015 0.01
-
0.0238
-
152.362
940.791 538.286
12 3.488
-
0.813
0.418
-
0.813
0.206 0.132
-
0.196
0.014 0.01
-
0.0229
-
151.021
943.316 541.015
13 3.805
-
0.811
0.417
-
0.811
0.205 0.131
-
0.197
0.014 0.01
-
0.0221
-
149.672
945.847 543.749
14 4.122 -0.81 0.416 -0.81 0.204 0.13
-
0.198
0.014 0.01
-
0.0213
-
148.455
948.343 546.36
15 4.439 -0.81 0.414 -0.81 0.202 0.128
-
0.198
0.014 0.01 -0.021
-
148.925
949.892 547.122
16 4.756
-
0.811
0.412
-
0.811
0.2 0.126
-
0.197
0.014 0.01
-
0.0216
-
151.083
949.464 545.309
17 5.073
-
0.812
0.411
-
0.812
0.199 0.125
-
0.196
0.014 0.01
-
0.0224
-
153.638
948.123 542.573
18 5.39
-
0.813
0.41
-
0.813
0.198 0.124
-
0.196
0.014 0.01
-
0.0233
-
156.187
946.79 539.841
19 5.707
-
0.815
0.408
-
0.815
0.197 0.123
-
0.195
0.014 0.01
-
0.0241
-158.73 945.465 537.114
20 6.024
-
0.816
0.407
-
0.816
0.196 0.121
-
0.194
0.014 0.01 -0.025
-
161.267
944.149 534.393
21 6.342
-
0.817
0.406
-
0.817
0.195 0.12
-
0.193
0.014 0.01
-
0.0258
-
163.798
942.842 531.677
Table 1 Variation in joint angles with respect to the tool position
s.no
Time Leg 1
(s)
V(m/s) a(m/s2
) ω (rad/s) α (rad/s2
)
X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis
1 0 0 0 0 0 0 0 0 0
2 0.317 -0.00407 -0.002893 -3.98E-05 -3.96E-05 0 0.0464425 0 0
3 0.634 -0.0040827 -0.0029055 -4.02E-05 -3.94E-05 0 0.0463689 0 0
4 0.951 -0.0040955 -0.0029179 -4.05E-05 -3.92E-05 0 0.0463194 0 0
5 1.268 -0.0041084 -0.0029303 -4.08E-05 -3.91E-05 0 0.0462697 0 0
6 1.585 -0.0041214 -0.0029427 -4.11E-05 -3.89E-05 0 0.0462198 0 0
7 1.902 -0.0041345 -0.002955 -4.15E-05 -3.87E-05 0 0.0461696 0 0
8 2.219 -0.0041543 -0.0029673 -4.18E-05 -3.86E-05 0 0.0461192 0 0.00513
9 2.536 0.0021889 -0.0041241 2.38E-05 -5.70E-05 -0.0394733 0 0 0
10 2.853 0.0021964 -0.0041422 2.39E-05 -5.70E-05 -0.0393856 0 0 0
11 3.171 0.0022041 -0.0041603 2.41E-05 -5.70E-05 -0.0392979 0 0 0
12 3.488 0.0022117 -0.0041784 2.42E-05 -5.70E-05 -0.0392104 0 0 0
13 3.805 0.0022194 -0.0042055 2.43E-05 -5.71E-05 -0.039123 0 0 0
14 4.122 0.0032225 -0.0020096 0.0070796 0.0195276 -0.0288952 -0.0372544 0.0103321 -0.00745
15 4.439 0.0041099 0.0041502 -0.0018202 0.0200188 0 -0.0449383 0.0128737 0
16 4.756 0.0023654 0.0074553 -4.85E-05 -6.13E-05 0.0390732 0 0 0
17 5.073 0.00235 0.0074358 -4.86E-05 -6.16E-05 0.0391605 0 0 0
18 5.39 0.0023345 0.0074162 -4.86E-05 -6.19E-05 0.039248 0 0 0
19 5.707 0.0023191 0.0073966 -4.87E-05 -6.21E-05 0.0393356 0 0 0
20 6.024 0.0023036 0.0073768 -4.88E-05 -6.24E-05 0.0394233 0 0 0.005316
21 6.342 -0.0009362 0.002155 -1.24851 -2.01419 0 0 -0.814038 0

11. A Novel Approach over Inverse Kinematic Analysis of 5-Axis Hybrid Parallel Manipulator for
Contour Trajectory
http://www.iaeme.com/IJMET/index.asp 208 editor@iaeme.com
Table 2 Variation of linear velocity (V), linear acceleration (A), Angular velocity (ω), Angular
acceleration (α) of Leg 1
Leg 2
V(m/s) a(m/s2
) ω (rad/s) α (rad/s2
)
X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis
0 0 0 0 0 0 0 0
-0.0043149 0.0028353 4.14E-05 -3.69E-05 0 0.0454338 0 0
-0.0042953 0.0028236 4.10E-05 -3.71E-05 0 0.0454865 0 0
-0.0042823 0.0028118 4.07E-05 -3.72E-05 0 0.045539 0 0
-0.0042695 0.0028 4.04E-05 -3.74E-05 0 0.0455913 0 0
-0.0042567 0.0027881 4.01E-05 -3.75E-05 0 0.0456433 0 0
-0.004244 0.0027762 3.98E-05 -3.77E-05 0 0.0457725 0 0
-0.0042315 0.0027642 3.95E-05 -3.79E-05 0 0.0457468 0 0
0.0025281 -0.0077245 4.70E-05 -5.98E-05 -0.0394733 0 0 0
0.002543 -0.0077434 4.69E-05 -5.95E-05 -0.0393856 0 0 0
0.0025578 -0.0077622 4.69E-05 -5.93E-05 -0.0392979 0 0 0
0.0025727 -0.007781 4.68E-05 -5.90E-05 -0.0392104 0 0 0
0.0025875 -0.0077996 4.68E-05 -5.87E-05 -0.039123 0 0 0
0.0035557 -0.0068419 0.0066223 0.0111716 -0.0288952 -0.0339201 0.0103321 -0.0083477
0.0042127 -0.0013999 -0.0027519 0.0210753 0 -0.0448884 0.0128737 0
0.0021764 0.0047007 -2.32E-05 -5.47E-05 0.0390732 0 0 0
0.0021691 0.0046833 -2.31E-05 -5.47E-05 0.0391605 0 0 0
0.0021618 0.004666 -2.29E-05 -5.47E-05 0.039248 0 0 0
0.0021545 0.0046486 -2.28E-05 -5.47E-05 0.0393356 0 0 0
0.0021473 0.0046313 -2.27E-05 -5.46E-05 0.0394233 0 0 0.006325
0 0 0 0 0 0 0 0
Table 3 Variation of linear velocity (V), linear acceleration (A), Angular velocity (ω), Angular
acceleration (α) of Leg 2
Leg3
V(m/s) a(m/s2
) ω (rad/s) α (rad/s2
)
X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis X-Axis Y-Axis
0 0 0 0 0 0 0 0
-0.00477 -0.000353 0 -4.10E-05 0 0.0480995 0 0
-0.00478 -0.000366 0 -4.09E-05 0 0.0481029 0 0
-0.004781 -0.000379 -3.17648 -4.09E-05 0 0.0481055 0 0
-0.004782 -0.000392 -3.48971 -4.09E-05 0 0.0481074 0 0
-0.004783 -0.000405 -3.80281 -4.09E-05 0 0.0481087 0 0
-0.004785 -0.000418 -4.11574 -4.09E-05 0 0.0481092 0 0
-0.004786 -0.000431 -4.4285 -4.09E-05 0 0.048109 0 0
0.002671 -0.008148 3.62393 -4.15E-05 -0.03863 0 0 0
0.002682 -0.008161 3.62201 -4.14E-05 -0.03886 0 0 0
0.002694 -0.008174 3.62006 -4.13E-05 -0.03879 0 0 0
0.002705 -0.00818 3.61801 -4.11E-05 -0.03824 0 0 0
0.002717 -0.008207 3.61587 -4.10E-05 -0.03723 0 0 0
0.003846 -0.00592 0.007874 0.0218991 -0.027852 -0.0369874 0.0035854 -0.0082268
0.004751 0.002212 -0.002523 0.026456 0 -0.0466971 0.0044865 0
0.002617 0.00864 -3.87382 -4.10E-05 0.0390032 0 0 0.0073014
0.002604 0.008633 -3.87772 -4.12E-05 0.0391605 0 0 0
0.002592 0.008614 -3.88152 -4.13E-05 0.0391948 0 0 0
0.00258 0.0086009 -3.88524 -4.15E-05 0.0392356 0 0 0
0.002568 0.00858 -3.88887 -4.16E-05 0.0394721 0 0 0
-0.00117 0.004044 -1.4431 -1.75402 0 0 -0.291355 0
Table 4: Variation of linear velocity (V), linear acceleration (A), Angular velocity (ω), Angular acceleration (α) of Leg 3

12. Suryam L.V, A.V.Pradeep, S.V.Satya Prasad, K.Vahini and Y.Sai ratnakar
http://www.iaeme.com/IJMET/index.asp 209 editor@iaeme.com
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