## 1 Introduction  
Generally, the robotic mechanisms can be categorized into three main groups: the open-loop mechanism, the closed-loop mechanism and the hybrid mechanism. A robot of the first group is usually designed with an end-effector connected to a fixed base by means of a single serial kinematic chain. Meanwhile, each robot of the second group is a mechanism which is usually composed of a moving end-effector connected to a fixed base by at least two kinematic chains, such as the parallel robots. A hybrid manipulator is a mechanism which is usually composed of some open-loop kinematic chains combined with some closed-loop linkages [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. The main advantages of the serial robots are the large workspace, the high dexterity of the end-effector and the large non-singular range of the joint variables. However, the robots of this group suffer from several drawbacks, such as relatively low stiffness and accuracy, low nominal load/weight ratio and heavy structure [21]. Therefore, in the last decades, several types of robots having closed-loop kinematic chains have been investigated in order to obtain better kinematic performances. The main advantages of these robots are low weight, compact structure, better accuracy and stiffness. Nevertheless, this robot group also have some disadvantages, such as a small workspace, a complex mechanical design and a complex procedure for dynamic modelling and control. For these reasons, in recent years, great attention has been paid to the development of hybrid robots which have combined advantages of the serial robots and the closed-loop architectures. More and more hybrid robots are designed and developed to meet the increasing demand for a wide spectrum of hybrid robotic applications [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. In practice, the hybrid robots have been designed in a large variety of kinematic configurations and structures, and they can be classified into three main classes as follows:  
The first class (Class I) includes all the hybrid robots whose structure has several parallel modules connected in series [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] (Figure 1a). Class II consists of the hybrid robots which have a basic parallel module combined with a serial manipulator [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24] (Figure 1b). Class III is a family of the hybrid robots which are composed of some local closed-loop chains appended to a main serial mechanism [25; 26; 27; 28; 29; 30; 31; 32; 33] (Figure 1c). It is clearly seen that the main architecture of a hybrid robot Class I or II is a parallel mechanism. Other parallel modules or serial links are connected to such the main structure to complete a hybrid serial-parallel robot. Different from the robots Class I and II, each hybrid robot Class III is usually designed with a serial module as the main structure of a robot. To provide with desirable mechanical advantages for a hybrid robot, some local closed-loop mechanisms are added to the main serial module. For example, when designing manipulators which suffer a heavy payload and operate in a large workspace, parallelograms were added and hydraulic cylinders were used for driving some revolute joints of the manipulators [26; 28; 29; 30; 31]. The addition of parallelograms to the main arm is to increase the rigidity of the robot, and also to restrict the orientation of the end- effector as desired. The use of the hydraulic  
Figure 1: Three main classes of the hybrid robots: (**a**) Class I; (**b**) Class II and (**c**) Class III  
cylinders is to increase the structural stiffness and to avoid the counterweights for the robot structure. The parallelograms and cylinders and other links of the serial arm naturally compose local closed loops of the entire robot mechanism.