Degrees of Freedom (DOF) refers to the number of independent variables needed to completely specify the position and orientation of a system. In robotics, DOF determines what movements a robot can make.
Rigid Body in Space
A rigid body floating freely in 3D space has 6 DOF:
DOF
Type
Description
3
Translation
Movement along X, Y, Z axes
3
Rotation
Rotation about X (roll), Y (pitch), Z (yaw)
This is why robot end-effectors often need 6+ joints to reach any position and orientation.
Common Robot Configurations
Robot Type
Typical DOF
Notes
2D mobile base
3
x, y, θ (heading)
Quadcopter
6
Full 3D position + orientation
Industrial arm
6
Minimum for arbitrary pose
Collaborative arm
7
Redundant (extra flexibility)
Humanoid arm
7+
Mimics human arm
Human hand
27
Extremely dexterous
Redundancy
A robot is kinematically redundant when it has more DOF than needed for a task:
6-DOF arm reaching a 3D point (needs only 3 DOF) → Redundant
7-DOF arm reaching a 6D pose → 1 DOF redundancy
Redundancy enables:
Obstacle avoidance while maintaining end-effector pose
Optimizing for secondary objectives (minimize energy, avoid singularities)
More natural, human-like motion
Constraints Reduce DOF
Real robots have constraints that reduce effective DOF:
Joint limits: Can’t rotate beyond certain angles
Obstacles: Some configurations cause collisions
Singularities: Configurations where DOF is temporarily lost
Calculating DOF
For a mechanism with joints:
DOF = Σ(joint freedoms) - constraints
Grübler’s Formula (planar mechanisms):
DOF = 3(n-1) - 2j₁ - j₂
Where:
n = number of links (including ground)
j₁ = number of 1-DOF joints
j₂ = number of 2-DOF joints
For spatial (3D) mechanisms, replace the coefficient 3 with 6.
Workspace vs Configuration Space
Workspace: The set of positions/orientations the end-effector can reach (in Cartesian space)
Configuration Space (C-space): The set of all possible joint configurations (dimension = DOF)
A 6-DOF robot arm has a 6-dimensional configuration space, even though it operates in 3D workspace.
