For the design and analysis of the manipulators, the basic thing to keep track of and to plan is the position as well as the orientation of objects. The objects for which we are concerned are the components of the manipulators, like their links, joints, and tools, and also the objects in the space around the manipulator as well as all the objects with which the manipulator interacts.
For the effective design of the manipulator, the position and orientation of the objects concerned need to be represented suitably such that it can be mathematically processed to make the manipulator move in the desired manner.