Gruebler’s Equation

Degrees of freedom for planar linkages joined with common joints can be calculated through Gruebler’s equation.

Gruebler’s equation is given by the formula:

where,
n = total number of links in the mechanism
j_p = total number of primary joints (pins or sliding joints)
j_h = total number of higher-order joints (cam or gear joints)

Most linkages used in machines have a single degree of freedom. An example of single degree-of-freedom linkage is shown in figure (a).

Linkages with zero or negative degrees of freedom are termed locked mechanisms. Locked mechanisms are unable to move and form a structure. A truss is a structure composed of simple links and connected with pin joints and zero degrees of freedom. An example of locked mechanism is shown in figure (b).

Linkages with multiple degrees of freedom need more than one driver to precisely operate them. Generally multi-degree of freedom mechanisms are open-loop kinematic chains used for reaching and positioning, such as robotic arms and backhoes. In general, multi-degree of freedom linkages offer greater ability to precisely position a link. An example of multi-degree of freedom mechanism is shown in figure (c).

Exception to Gruebler’s Equation:

The Gruebler’s equation does not account for link geometry, in rare instances, it can lead to misleading results.
One such example is shown in the below figure (d).

Notice that above figure (d) linkage contains five links and six pin joints. Using Gruebler’s equation, this linkage has zero degrees of freedom. Of course, this suggests that the mechanism is locked. However, if all pivoted links were the same size and the distance between the joints on the frame and coupler were identical, this mechanism is capable of motion, with a single degree of freedom. The center link is redundant and because it is identical in length to the other two links attached to the frame, it does not alter the action of the linkage.

There are other examples of mechanisms that violate the Gruebler’s equation because of unique geometry. A designer must be aware that the mobility equation can, at times, lead to inconsistencies.