Degrees of Freedom

Grubler’s Formula

Consider a mechanism consisting of $N$ links, where ground is also regarded as a link. Let $J$ be the number of joints, $m$ be the number of degrees of freedom of a rigid body ($m=3$ for planar mechanisms and $m=6$ for spatial mechanisms), $f_i$ be the number of freedoms provided by joint $i$, and $c_i$ be the number of constraints provided by joint $i$, where $f_i+c_i=m$ for all $i$. $\text{dof}=m(N-1-J)+{\sum }_{i=1}^J{f}_i$ #to-review This formula holds only if all joint constraints are independent. If they are not independent then the formula provides a lower bound on the number of degrees of freedom.