**6. Comparison between 4-legged mechanism and other mechanisms**

In order to investigate the kinematic performance of 4-legged mechanism, the response of the mechanisms are compared in several different aspects; reachable points, performance indices, and singularity analysis.

#### **6.1 Reachable points and workspace comparison**

Consider the 3-legged and the 4-legged mechanisms with g=1 m and h=0.5 m, respectively; g and h are the radii of the fixed and moving platforms, respectively.

By assuming a cubic with 1m length, 1 m width and 1 m height located 0.25 m above the base platform, we are interested in determining the reachability percentage in which each mechanism can successfully reach to the locations within this cubic space.

Exploiting Higher Kinematic Performance –

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 57

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

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.


Table 1. Reachable points

Assuming similar dimensions for the two mechanisms result in a larger workspace for the 4 legged mechanism.

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