Grubler’s Formula

Formula for determining the number of Degrees of Freedom for planar and spatial robots.

Definition

Consider a mechanism consisting of $N$ links, where ground is also considered 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 mechanism, $m=6$ for spatial mechanisms). Let $f_i$ be the number of freedoms provided by joint $i$, and $c_i$ be the number of constraints provided by $i$ .where $f_i+c_i=m$ for all $i$. Then, Grubler’s formula for the number of degrees of freedom of the robot is:

$$\begin{array}{rl}\text{dof}& =\underset{\text{rigid body freedoms}}{\underbrace{m(N-1)}}-\underset{\text{constraints}}{\underbrace{\sum _{i=1}^J{c}_i}}\\ & =m(N-1)-\sum _{i=1}^J(m-f_i)\\ & \boxed{=m(N-1-J)+\sum _{i=1}^J{f}_i}\\ & \\ & =(\text{number of DOF})(\text{links}-1-\text{joints})+\text{freedoms}\end{array}$$

This only holds if all joint constraints are independent. If they are not independent, the formula provides a lower bound on the number of DOF.