## Fundamental concepts, methods and classification

The number of degrees of freedom of a mechanism is normally called the mobility, which is the number of inputs required to determine the position

of all the links, also known as outputs, with respect to a fixed reference frame, sometimes referred to as ground. The mechanism becomes a locked chain, or a conventional structure, if no mobility remains. In three dimensional space, the number of degrees of freedom of a rigid link is six: three directional displacements and three directional rotations. Thus n free links will have 6n degrees of freedom. By fixing one link as ground, the remaining degrees of freedom are 6(n – 1). A joint with f degrees of freedom connecting two links reduces the total degrees of

freedom by 6 – f. For a mechanism composed of n links that are connected with a total of j joints, each of which has fi (i = 1, 2, . . ., j) degrees of freedom, the mobility of the linkage, m, is

,

or

(2.1)

This is called the Grübler-Kutzbach mobility criterion (Hunt, 1978) or Kutzbach criterion. Figure 2.1(a) shows a spatial linkage composed of a chain of four links and four joints, including one revolute joint (R), one spherical joint (S), one cylindrical joint (C) and one prismatic joint (P), which is commonly referred to as an RSCP linkage. According to Eq. (2.1), its mobility is

.