**2. Mechanisms description**

The schematics of the 6-DOF non-redundant,3-legged and redundant 4-legged mechanisms are shown in Figs. 1 and 2, respectively.

The non-redundant 3-legged mechanism was first introduced by Beji & Pascal (1999). The redundant 4-legged mechanism has the similar structure, with a single leg added to the 3 legged system, which keeps symmetry. Each leg is composed of three joints; universal,

Gao et al. (2010) proposed a novel 3DOF parallel manipulator and they increased the stiffness of their system, using an optimization technique. Lopes (2010) introduced a new 6- DOF moving base platform, which is capable of being used in micro robotic applications after processing serial combination with another industrial manipulator. It is in fact a non-

Deidda et al. (2010) presented a 3-DOF 3-leeged spherical robotic wrist. They analyzed mobility and singularity. Tale Masouleh et al. (2011) investigated the kinematic problem of a 5-DOF 5-RPUR mechanism with two different approaches, which differ by their concepts of eliminating passive variables. Zhao and Gao (2010) investigated the kinematic and dynamic properties of a 6-DOF 8-PSS redundant manipulator. They presented a series of new jointcapability indices, which are general and can be used for other types of parallel

Li et al. (2007) worked on the singularity-free workspace analysis of the general Gough– Stewart platform. In a similar line of work, Jiang and Gosselin (Jiang & Gosselin, 2009a;b;c) determined the maximal singularity-free orientation workspace at a prescribed position of the Gough–Stewart platform. Alp and Ozkol (2008) described how to extend the workspace of the 6-3 and 6-4 Stewart platforms in a chosen direction by finding the optimal combination of leg lengths and joint angles. They showed that the workspace of the 6-3

Mayer and Gosselin (2000) developed a mathematical technique to analytically derive the singularity loci of the Gough-Stewart platform. Their method is based on deriving an

To demonstrate the redundancy effects, Wu et al. (2010) compared a planar 2-DOF redundant mechanism with its non-redundant counterpart. Arata et al. (2011) proposed a new 3-DOF redundant parallel mechanism entitled as Delta-R, based on its famous non-

Sadjadian and Taghirad (2006) compared a 3-DOF redundant mechanism, hydraulic shoulder, to its non-redundant counterpart. They concluded that the actuator redundancy in the mechanism is the major element to improve the Cartesian stiffness and hence the dexterity of the hydraulic shoulder. They also found that losing one limb reduces the

The rest of the chapter is organized as follows. In Section 2, in addition to introduction and comparison of non-redundant 3-legged and redundant 4-legged UPS PMs, four different architectures of the Gough-Stewart platforms are depicted. The kinematics of the abovementioned mechanisms are reviewed in Section 3. The jacobian matrices using the screw theory is derived in Section 4. In Sections 5 and 6, the performances of the redundant and non-redundant mechanisms are studied and compared with four well-known

The schematics of the 6-DOF non-redundant,3-legged and redundant 4-legged mechanisms

The non-redundant 3-legged mechanism was first introduced by Beji & Pascal (1999). The redundant 4-legged mechanism has the similar structure, with a single leg added to the 3 legged system, which keeps symmetry. Each leg is composed of three joints; universal,

explicit expression for the determinant of the jacobian matrix of the manipulator.

redundant counterpart, Delta, which was developed by Vischer & Clavel (1998).

redundant parallel mechanism with 6 linear actuators.

Stewart platform is smaller than that of the 6-4 one.

stiffness of the manipulator significantly.

**2. Mechanisms description** 

are shown in Figs. 1 and 2, respectively.

architectures of hexapods. Finally we conclude in Section 7.

manipulators.

Fig. 1. Schematic of the non-redundant mechanism.

Fig. 2. Schematic of the redundant mechanism.

prismatic, and spherical (Fig. 3). A rotary actuator and a linear actuator are used to actuate each leg. The rotary actuators, whose shafts are attached to the lower parts of the linear actuators through the universal joints, are placed on the corners of the fixed platform (Abedinnasab & Vossoughi, 2009; Aghababai, 2005). The spherical joints connect the upper parts of the linear actuators to the moving platform.

Rotary actuators are situated on the corners *Ai* (for *i=1, 2, 3, 4*) of the base platform and each shaft is connected to the lower part of linear actuators through a universal joint (Figs. 1 and 2). The upper parts of linear actuators are connected to the corners of the moving platform, *Bi* points, through spherical joints (Fig. 3).

Exploiting Higher Kinematic Performance –

devices, etc.

*i a*

and

**3. Kinematic analysis** 

represents the vector *OAi*

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 47

change makes the mechanism to be lighter, since the rotary actuators are resting on the fixed platform, which causes higher accelerations to be available due to smaller inertial effects. The applications of this type of robots can be found in flight simulators, high precision surgical tools, positioning devices, motion generators, ultra-fast pick and place robots, haptic devices, entertainment, multi-axis machine tools, micro manipulators, rehabilitation

Advantages of high rigidity and low inertia make these PMs ideal for force feedback control, assembly, manufacturing, underground projects, space technologies, and biology projects.

One of the most important issues in the study of parallel mechanisms is the kinematic analysis, where the generated results form the base for the application of the mechanism.

. e . e . 12 3 0 , 120 , <sup>120</sup> *Non Red Non R d Non R d*

*and* ,

*and* ,

23 321 31 321 31 23 321 31 321 31 2 2 1 2 1

s s c c c c

 *<sup>A</sup> <sup>i</sup> B i <sup>B</sup>*

*i i r p b*

*ii ii r a p b a*

s s s c

 

 

ss s s ss

1 , <sup>2</sup> , and

denote the position vectors for *P* and *Bi* in the reference frame*A* ,

s ss

 , in which,

> 

 

. We can represent *<sup>A</sup>*

   

 

*i b* 

*b R PB* . (2)

. (3)

. (4)

*<sup>i</sup> <sup>i</sup> <sup>B</sup> b PB* . *Bbi*

3 are three Euler angles

would be expressed in

*B ij R r* , the rotation

, (1)

represents

 is

 (Fig. 1). [cos sin 0]*<sup>T</sup> i ii a g* 

Re . Re . Re . Re . 1 23 4 45 , 45 , 135 , <sup>135</sup> *d dd d*

where *g* and *r* are the radius of the fixed and moving platforms, respectively. *Bbi*

*<sup>i</sup> i i b h* 

c c c

from both sides of (3) one obtains

 

 

defined according to the *zyx* convention. Thus, the vector *<sup>B</sup>*

the position of the ith joint on the platform in the moving frame {B}, *<sup>B</sup>*

 

cc c cc c

s

, and so on.

 

constant and is equal to [cos sin 0] *B T*

matrix from *B* to*A* , using Euler angles as

 

 

respectively. From the geometry, it is obvious that

 and 1 1 c cos 

*A BR*

where 1 1 s sin 

and *ir*

Subtracting vector *<sup>i</sup> a*

Let *p*

the fixed frame {A} as

Fig. 3. Schematic of the universal joint, and the joints variables.

Cartesian coordinates *A* (*O, x ,y, z*) and *B* (*P, u, v, w*) represented by {A} and {B} are connected to the base and moving platforms, respectively. In Fig.3, *is* represents the unit vector along the axes of ith rotary actuator and *<sup>i</sup> d* is the vector along *A Bi i* with the length of *<sup>i</sup> d* . Assuming that each limb is connected to the fixed base by a universal joint, the orientation of ith limb with respect to the fixed base can be described by two Euler angles, rotation *<sup>i</sup>* around the axis *is* , followed by rotation *<sup>i</sup>* around *ni* , which is perpendicular to *<sup>i</sup> d* and *is* (Fig. 3). It is to be noted that *<sup>i</sup>* and *<sup>i</sup> d* are active joints actuated by the rotary and linear actuators, respectively. However, *<sup>i</sup>* is inactive.

By replacing the passive universal joints in the Stewart mechanism with active joints in the above mentioned mechanisms, the number of legs could be reduced from 6 to 3 or 4. This

Fig. 4. Schematics of well-known Stewart platforms.

change makes the mechanism to be lighter, since the rotary actuators are resting on the fixed platform, which causes higher accelerations to be available due to smaller inertial effects. The applications of this type of robots can be found in flight simulators, high precision surgical tools, positioning devices, motion generators, ultra-fast pick and place robots, haptic devices, entertainment, multi-axis machine tools, micro manipulators, rehabilitation devices, etc.

Advantages of high rigidity and low inertia make these PMs ideal for force feedback control, assembly, manufacturing, underground projects, space technologies, and biology projects.
