**1. Introduction and basic definitions**

In the following sections, this chapter will deal with direct and inverse kinematics of open-chain multibody systems consisting of rigid bodies. The whole problematics is analyzed from the view of robotics. Each manipulator or mechanism investigated in this chapter will be of serial kinematic structure (open chain).

Open-chain multibody systems are mechanically constructed by connecting a set of bodies, called links, by means of various types of joints. In general, the joints can be passive or active. The joints, which are moved by actuators, are active joints.

In general, from the view of robotics, there are two tasks in kinematics:


Before these terms are explained and demonstrated by some study cases, we have to mention the basic definitions, necessary for the further analyses.

**Degrees of freedom** (DOF): is the smallest number of coordinates needed to represent the robot configuration. Thus, the number of DOF equals to the dimension of configuration space.

**Joint space**: Let us define all the joint variables in a vector **q** ¼ *q*1, *q*2, … , *qn <sup>T</sup>* ∈ ⊂ *<sup>N</sup>*. The set we call the so-called joint space and it contains all the possible values, which joint variables may acquire.

**Workspace**: Workspace is a subset of the Euclidean space , in which the robot executes its tasks. From the view of robotics, workspace is the set of all the points that mechanism may reach in Euclidean space by end-effector. The workspace can be categorized as follows [1, 2]:

*Maximal workspace*—it is defined as locations that can be reached by endeffector at least with one orientation.

*Inclusive-orientation workspace*—it is defined as locations that can be reached by end-effector with at least one orientation among a range of orientations (maximal workspace is particular case).

*Constant-orientation workspace*—it is defined as location that can be reached by the end-effector with fixed orientation of joints.

*Total-orientation workspace*—it is defined as location that can be reached by the end-effector with any orientation.

*Dexterity workspace*—it is defined as location that can be reached by the endeffector with any orientation and without kinematic singularities.

**Task space**–space of positions and orientations of the end-effector frame. The workspace is a subset of task space that the end-effector frame can reach [3].
