**1. Introduction**

42 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

Wang , Y. (2007). A direct numerical solution to forward kinematics of general Stewart–

It is believed that Gough and Whitehall (1962) first introduced parallel robots with an application in tire-testing equipments, followed by Stewart (1965) , who designed a parallel mechanism to be used in a flight simulator. With ever-increasing demand on the robot's rigidity, redundant mechanisms, which are stiffer than their non-redundant counterparts, are attracting more attention.

Actuation redundancy eliminates singularity, and greatly improves dexterity and manipulability. Redundant actuation increases the dynamical capability of a PM by increasing the load-carrying capacity and acceleration of motion, optimizing the load distribution among the actuators and reducing the energy consumption of the drivers. Moreover, it enhances the transmission characteristics by increasing the homogeneity of the force transmission and the manipulator stiffness (Yi et al., 1989). From a kinematic viewpoint, redundant actuation eliminates singularities (Ropponen & Nakamura, 1990) which enlarge the usable workspace, as well. The kinematic analysis on general redundantly actuated parallel mechanisms was investigated by Müller (2005).

A number of redundantly full-actuated mechanisms have been proposed over the years and some of them which are more significant are listed in this section. The spatial octopod, which is a hexapod with 2 additional struts, is one of them (Tsai, 1999). A 5-DOF 3-legged mechanism was proposed by Lu et al. (2008), who studied its kinematics, statics, and workspace. Staicu (2009) introduced a new 3-DOF symmetric spherical 3-UPS/S parallel mechanism having three prismatic actuators. As another work of Lu et al. (2009), they introduced and used 2(SP+SPR+SPU) serial–parallel manipulators. Wang and Gosselin (2004) addressed an issue of singularity and designed three new types of kinematically redundant parallel mechanisms, including a new redundant 7-DOF Stewart platform. They concluded that such manipulators can be used to avoid singularities inside the workspace of non-redundant manipulators.

Choi et al. (2010) developed a new 4-DOF parallel mechanism with three translational and one rotational movements and found this mechanism to be ideal for high-speed machining.

Exploiting Higher Kinematic Performance –

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

Fig. 2. Schematic of the redundant mechanism.

parts of the linear actuators to the moving platform.

*Bi* points, through spherical joints (Fig. 3).

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 45

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

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,

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 nonredundant parallel mechanism with 6 linear actuators.

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 manipulators.

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

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

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

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 stiffness of the manipulator significantly.

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 architectures of hexapods. Finally we conclude in Section 7.
