In one line
Degree of freedom (DOF) is the number of independent variables needed to fully describe a robot's configuration; for a serial robot it equals the number of one-DOF joints.
Explained simply
A light switch can only go up or down: one DOF. A door rotates about its hinges: one DOF. Your arm (shoulder, elbow and wrist together) has about seven, which is why you can hold a cup still while moving your elbow.
Technically, a free rigid body in space has 6 DOF (3 translations, 3 rotations); in a plane it has 3 (2 translations, 1 rotation). Joints remove freedoms. Each revolute or prismatic joint allows 1 DOF.
How it works
- For a serial (open-chain) robot with only R and P joints: DOF = number of joints.
- To place a tool anywhere in 3D space with any orientation, at least 6 DOF are needed.
- To reach any point in a plane with a fixed orientation not required, 2 DOF are enough; with orientation, 3.
- For general planar mechanisms, use Grubler's criterion (formula below), counting the base as a link.
- Robots with more DOF than needed are called redundant.
Key formula
Planar Grubler: \( F = 3(n - 1) - 2j_1 - j_2 \), where n = number of links including the fixed base, \( j_1 \) = number of one-DOF joints (R or P), \( j_2 \) = number of two-DOF joints. Spatial: \( F = 6(n-1) - \sum (6 - f_i) \), \( f_i \) = DOF of joint i.
Quick example
Planar 2R arm: n = 3 (base + 2 links), \( j_1 = 2 \): \( F = 3(2) - 4 = 2 \) DOF, matching the two joints. A typical 6-axis industrial robot has 6 revolute joints, so 6 DOF.
Common mistakes
- Forgetting to include the base when counting n in Grubler's formula.
- Counting the gripper's open/close as a DOF of the arm.
- Assuming more joints always means more useful DOF; joints in a closed loop reduce mobility.
How MU asks it
A 3-mark short part: 'Define degree of freedom; find the DOF of the given manipulator'. Write the definition, the 6 (space) and 3 (plane) DOF facts, and show the count. Syllabus numericals are limited to 2-DOF serial robots.