**6.2 Performance comparison**

To compare the kinematic performance of the proposed 4-legged mechanism with the 3 legged mechanism and Steward platform, abovementioned performance indices were used. The results obtained are shown in Figs. 7 to 10. These figures show how performance indices vary on a plate (*Z=0.75*) within the workspace.

Fig. 7 depicts the Stewart 3-3 has the higher isotropy index for manipulability comparing with Stewart 3-6, 3-legged and 4-legged mechanisms. After Stewart 3-3, the 4-legged mechanism presents the better performance. This figure illustrates that adding a leg to a 3 legged mechanism can significantly improve the manipulability of the mechanism. Furthermore, a 4-legged mechanism can have a performance comparable with the Stewart platforms or even better.

Fig. 8 depicts the Stewart 3-3 compared with the other mechanisms under study has the higher dexterity index. After Stewart 3-3, again, the 4-legged mechanism presents the better performance. This figure also illustrates significant enhancement of the dexterity of the mechanism due to the additional leg. Also it shows that, in terms of dexterity, the 4-legged mechanism can have a comparable performance against the Stewart platforms or even better.

In the next step of performance comparison of manipulators, displacement and rotation sensitivities for mechanisms of our interest are compared. Fig. 9 shows the amount of displacement sensitivity indices of the abovementioned mechanisms on the selected plane

Table 1 shows that adding one leg to the 3-legged mechanism reduces the *R.P.* (Reachable points Percentage) by 5.03%. However, it should be noted that, although the non-redundant mechanism has a wider workspace, it has much more singular points than the redundant mechanism, and actuator forces and torques are also less in the redundant mechanism. As it can be seen from Table 1, *RP*s in the Stewart platforms are smaller than 3-legged and 4 legged mechanisms. In the 6-legged Stewart-like UPS mechanisms (Stewart, 1965), the workspace is constructed by intersecting of 6 spheres. On the other hand, in the proposed 4 legged UPS mechanism, the workspace is constructed by intersecting of only 4 spheres.

> Mechanism % RP Stewart (6-6) 76.35 Stewart (3-6) 74.21 Stewart (6-3) 73.70 Stewart (3-3) 63.46 3-legged mechanism 84.75 4-legged mechanism 79.72

Assuming similar dimensions for the two mechanisms result in a larger workspace for the 4-

To compare the workspace of the proposed 4-legged mechanism with the 3-legged mechanism and Steward platforms, the reachable points for them were calculated and obtained results are shown in Fig. 6 It is appear that the 3-legged mechanism has the largest

To compare the kinematic performance of the proposed 4-legged mechanism with the 3 legged mechanism and Steward platform, abovementioned performance indices were used. The results obtained are shown in Figs. 7 to 10. These figures show how performance indices

Fig. 7 depicts the Stewart 3-3 has the higher isotropy index for manipulability comparing with Stewart 3-6, 3-legged and 4-legged mechanisms. After Stewart 3-3, the 4-legged mechanism presents the better performance. This figure illustrates that adding a leg to a 3 legged mechanism can significantly improve the manipulability of the mechanism. Furthermore, a 4-legged mechanism can have a performance comparable with the Stewart

Fig. 8 depicts the Stewart 3-3 compared with the other mechanisms under study has the higher dexterity index. After Stewart 3-3, again, the 4-legged mechanism presents the better performance. This figure also illustrates significant enhancement of the dexterity of the mechanism due to the additional leg. Also it shows that, in terms of dexterity, the 4-legged mechanism can have a comparable performance against the Stewart platforms or even

In the next step of performance comparison of manipulators, displacement and rotation sensitivities for mechanisms of our interest are compared. Fig. 9 shows the amount of displacement sensitivity indices of the abovementioned mechanisms on the selected plane

workspace followed by 4-legged mechanism, 3-6 and 3-3 Steward Platforms.

Table 1. Reachable points

**6.2 Performance comparison** 

platforms or even better.

better.

vary on a plate (*Z=0.75*) within the workspace.

legged mechanism.

Fig. 6. Workspaces of Stewart (3-6), Stewart (3-3), 3-legged and 4-legged mechanisms.

Fig. 7. Isotropy index for manipulability of Stewart 6-6, Stewart 3-6, 3 Legged mechanism and 4 legged mechanism on the plane z= 0.75 m

Exploiting Higher Kinematic Performance –

compared with other mechanisms.

results were listed in Table 2.

appropriate candidate.

kinematic performances

Stewart 6-6, and Stewart 3-6

effector.

performance Index (GPI) can be evalueated as:

mechanism have the better kinematics accuracy.

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 59

(*Z=0.75*). It clearly shows that the 4-legged mechanism has less displacement sensitivity index by far. Fig. 10 depicts the 4-legged mechanism has also less rotation sensitivity

So far from Figs. 7 to 10, the amounts of performance indices are shown on the planes. To compare the kinetics performance of manipulators over the entire workspace, the global

*W*

*GPI*

which is the average value of local performance index (*PI*) over the workspace (*W*).

*W*

The amounts of GPI for Isotropy Index for Manipulability (*IIM*), Dexterity (*DX*), Displacement Sensitivity (*DS*) and Rotation Sensitivity (*RS*) were calculated and obtained

Table 2 shows that the 4-legged mechanism has a better *IIM* within the selected workspace, which explicitly indicates a better ability for transmitting a certain velocity to its end-

As it is seen from Table 2, the Stewart 3-3 platform has the biggest global *DX* compared with other mechanisms and the 4-legged mechanism has the second (i.e. difference in *DX* is only 5.71% less). It represents that the Stewart 3-3 platform and the proposed 4-legged

Having the lower sensitivity is a demand for industrial mechanisms. By comparing the values of *DS* and *RS*, which are listed in Table 2, it is obvious the 4-legged mechanism is an

Based on the results shown in Table 2, the 4-legged mechanism is recommended for better

Mechaniasm *IIM DX DS RS* Stewart (6-6) 0.0429 0.1589 1.3249 3.5791 Stewart (3-6) 0.1296 0.3969 1.2070 2.5725 Stewart (6-3) 0.1020 0.3449 1.2113 2.8991 Stewart (3-3) 0.1978 **0.5284** 1.0485 2.6077 3-legged mechanism 0.1136 0.3423 0.8538 2.3861 4-legged mechanism **0.2140** 0.4982 **0.7279 1.8441** 

Table 2. Performance comparison between 3-legged mechanism, 4-legged mechanism,

Several types of workspace can be considered. For example, the 3D constant orientation workspace, which describes all possible locations of an arbitrary point P in the moving system with a constant orientation of the moving platform, the reachable workspace (all the

**6.3 Singularity analysis of 3-legged and 4-legged mechanisms** 

The higher, the better The lower, the better

*dW* 

*PI dW*

, (40)

Fig. 8. Dexterity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and 4-legged mechanism on the plane z= 0.75 m.

Fig. 9. Displacement Sensitivity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and 4-legged mechanism on the plane z= 0.75 m.

Fig. 10. Rotation Sensitivity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and 4 legged mechanism on the plane z= 0.75 m.

Fig. 8. Dexterity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and 4-legged

Fig. 9. Displacement Sensitivity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and

Fig. 10. Rotation Sensitivity index of Stewart 6-6, Stewart 3-6, 3-Legged mechanism and 4-

mechanism on the plane z= 0.75 m.

4-legged mechanism on the plane z= 0.75 m.

legged mechanism on the plane z= 0.75 m.

(*Z=0.75*). It clearly shows that the 4-legged mechanism has less displacement sensitivity index by far. Fig. 10 depicts the 4-legged mechanism has also less rotation sensitivity compared with other mechanisms.

So far from Figs. 7 to 10, the amounts of performance indices are shown on the planes. To compare the kinetics performance of manipulators over the entire workspace, the global performance Index (GPI) can be evalueated as:

$$GPI = \frac{\int PI \cdot d\mathcal{W}}{\int\_{\mathcal{W}} d\mathcal{W}} \, , \tag{40}$$

which is the average value of local performance index (*PI*) over the workspace (*W*).

The amounts of GPI for Isotropy Index for Manipulability (*IIM*), Dexterity (*DX*), Displacement Sensitivity (*DS*) and Rotation Sensitivity (*RS*) were calculated and obtained results were listed in Table 2.

Table 2 shows that the 4-legged mechanism has a better *IIM* within the selected workspace, which explicitly indicates a better ability for transmitting a certain velocity to its endeffector.

As it is seen from Table 2, the Stewart 3-3 platform has the biggest global *DX* compared with other mechanisms and the 4-legged mechanism has the second (i.e. difference in *DX* is only 5.71% less). It represents that the Stewart 3-3 platform and the proposed 4-legged mechanism have the better kinematics accuracy.

Having the lower sensitivity is a demand for industrial mechanisms. By comparing the values of *DS* and *RS*, which are listed in Table 2, it is obvious the 4-legged mechanism is an appropriate candidate.

Based on the results shown in Table 2, the 4-legged mechanism is recommended for better kinematic performances


Table 2. Performance comparison between 3-legged mechanism, 4-legged mechanism, Stewart 6-6, and Stewart 3-6

#### **6.3 Singularity analysis of 3-legged and 4-legged mechanisms**

Several types of workspace can be considered. For example, the 3D constant orientation workspace, which describes all possible locations of an arbitrary point P in the moving system with a constant orientation of the moving platform, the reachable workspace (all the

Exploiting Higher Kinematic Performance –

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 61

Fig. 12. Singularity analysis in planes 1, 2 and 3 for both 3-legged (non-redundant) and 4-

legged (redundant) mechanisms.

locations that can be reached by P), the orientation workspace (all possible orientations of the end-effector around P for a given position) or the inclusive orientation workspace (all the locations that can be reached by the origin of the end-effector with every orientation in a given set) (Abedinnasab & Vossoughi, 2009).

Out of those types, we have used the inclusive orientation workspace, where for every position in a fixed surface, the moving platform is rotated in every possible orientation to determine if that configuration is singular or not. After trials and errors, we figured out that for a better determination of the singular configurations, the roll-pitch-yaw rotation about the global coordinate is the most critical set of rotations compared to the other rotations such as the reduced Euler rotations.

To illustrate the positive effects of redundancy on eliminating singular configurations, we have done jacobian analysis in planes in different orientations of the workspace as shown in Fig. 11. The results are shown in Figs. 12 and 13. In Figs. 12 and 13, the jacobian determinant of center of the moving platforms of 3-legged and 4-legged mechanisms has been calculated. The platform is rotated simultaneously in three different directions according to the rollpitch-yaw Euler angles discussed above. Each angle is free to rotate up to ±20º. After the rotations in each position, if the mechanism did not encounter any singular configuration, the color of that position is represented by light gray. If there was any singular configuration inside ±20º region and beyond ±10º region, the color is dark gray. At last, if the singular configuration was encountered in ±10º rotations, the color is black.

As seen from Fig. 12, in the 3-legged mechanism, there exist singular configurations in the most of the X-Y, X-Z and Y-Z planes (black and dark gray regions). However, in the 4 legged mechanism, the singular points do not exist at the most of the plane. Figure 13 shows the same patterns as in the other planes. Figures 12 and 13 simply illustrate the great effect of a simple redundancy; namely, the addition of a leg to the 3-legged mechanism can remove vast singular configurations.

Fig. 11. (a) Schematic view of planes 1, 2, and 3. (b) Schematic view of planes 4, 5, and 6.

locations that can be reached by P), the orientation workspace (all possible orientations of the end-effector around P for a given position) or the inclusive orientation workspace (all the locations that can be reached by the origin of the end-effector with every orientation in a

Out of those types, we have used the inclusive orientation workspace, where for every position in a fixed surface, the moving platform is rotated in every possible orientation to determine if that configuration is singular or not. After trials and errors, we figured out that for a better determination of the singular configurations, the roll-pitch-yaw rotation about the global coordinate is the most critical set of rotations compared to the other rotations such

To illustrate the positive effects of redundancy on eliminating singular configurations, we have done jacobian analysis in planes in different orientations of the workspace as shown in Fig. 11. The results are shown in Figs. 12 and 13. In Figs. 12 and 13, the jacobian determinant of center of the moving platforms of 3-legged and 4-legged mechanisms has been calculated. The platform is rotated simultaneously in three different directions according to the rollpitch-yaw Euler angles discussed above. Each angle is free to rotate up to ±20º. After the rotations in each position, if the mechanism did not encounter any singular configuration, the color of that position is represented by light gray. If there was any singular configuration inside ±20º region and beyond ±10º region, the color is dark gray. At last, if the singular

As seen from Fig. 12, in the 3-legged mechanism, there exist singular configurations in the most of the X-Y, X-Z and Y-Z planes (black and dark gray regions). However, in the 4 legged mechanism, the singular points do not exist at the most of the plane. Figure 13 shows the same patterns as in the other planes. Figures 12 and 13 simply illustrate the great effect of a simple redundancy; namely, the addition of a leg to the 3-legged mechanism can

Fig. 11. (a) Schematic view of planes 1, 2, and 3. (b) Schematic view of planes 4, 5, and 6.

configuration was encountered in ±10º rotations, the color is black.

given set) (Abedinnasab & Vossoughi, 2009).

as the reduced Euler rotations.

remove vast singular configurations.

Fig. 12. Singularity analysis in planes 1, 2 and 3 for both 3-legged (non-redundant) and 4 legged (redundant) mechanisms.

Exploiting Higher Kinematic Performance –

determined using the concept of reciprocal screws.

**7. Conclusion** 

smaller inertial effects.

parallel mechanisms.

**8. References** 

173.

rehabilitation devices, etc.

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 63

The effects of redundant actuation are studied. The redundant 4-legged and non-redundant 3-legged parallel mechanisms are compared with 4 well-known architectures of Gough-Stewart platforms. It is shown that the inverse kinematics of the proposed 3-legged and 4 legged mechanisms have a closed-form solution. Also the Jacobian matrix has been

From the design point of view, by replacing the passive universal joints in the Stewart platforms with active joints in the above mentioned mechanisms, the number of legs could be reduced from 6 to 3 or 4. It makes the mechanism to be lighter, since the rotary actuators are resting on the fixed platform, which allows higher accelerations to be available due to

It is illustrated that redundancy improves the ability and performance of the non-redundant parallel manipulator. The redundancy brings some advantages for parallel manipulators such as avoiding kinematic singularities, increasing workspace, improving performance indices, such as dexterity, manipulability, and sensitivity. Finally, we conclude that the redundancy is a key choice to remove singular points, which are common in nearly all

It is worthy to state that the applications of these robots can be found in flight simulators, high precision surgical tools, positioning devices, motion generators, ultra-fast pick-andplace robots, haptic devices, entertainment, multi-axis machine tools, micro manipulators,

Abedinnasab, M. H. & Vossoughi, G. R. (2009). Analysis of a 6-dof redundantly actuated 4-

Aghababai, O. (2005). Design, Kinematic and Dynamic Analysis and Optimization of a 6

Alp, H. & Özkol, I. (2008). Extending the workspace of parallel working mechanisms, Proc.

Angeles, J. & Lopez-Cajun, C. S. (1992). Kinematic isotropy and the conditioning index of

Angeles, J. & Rojas, A. A. (1987). Manipulator inverse kinematics via condition-number

Arata, J., Kondo, J., Ikedo, N. & Fujimoto, H. (2011). Haptic device using a newly developed

Bai, S. (2010). Optimum design of spherical parallel manipulators for a prescribed

Beji, L. & Pascal, M. (1999). Kinematics and the full minimal dynamic model of a 6-dof

Cardou, P., Bouchard, S. & Gosselin, C. (2010). Kinematic-sensitivity indices for

Choi, H. B., Konno, A. & Uchiyama, M. (2010). Design, implementation, and performance

dimensionally nonhomogeneous jacobian matrices, IEEE Trans. Rob. 26(1): 166–

DOF Redundantly Actuated Parallel Mechanism for Use in Haptic Systems, MSc

legged parallel mechanism, Nonlinear Dyn. 58(4): 611–622.

Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 222(7): 1305–1313.

serial robotic manipulators, Int. J. Rob. Res. 11(6): 560–571.

minimization and continuation, Int. J. Rob. Autom. 2: 61–69.

workspace, Mech. Mach. Theory 45(2): 200–211.

parallel robot manipulator, Nonlinear Dyn. 18: 339–356.

evaluation of a 4-DOF parallel robot, Robotica 28(1): 107–118.

redundant parallel mechanism, IEEE Trans. Rob. 27(2): 201–214.

Thesis, Sharif University of Technology, Tehran, Iran.

Fig. 13. Singularity analysis in planes 4, 5 and 6 for both 3-legged (non-redundant) and 4 legged (redundant) mechanisms.
