Degrees of Freedom
Independent motions of a mechanism from bodies and constraints.
Formula
Variables
Example
A typical industrial arm has 6 DoF — enough to reach any position and orientation.
How it works
In a 3D mechanism, each moving rigid body can in principle translate in three directions and rotate about three axes, giving 6 possible motions per body. The formula starts with that maximum and subtracts the restrictions imposed by joints, links, or other constraints, because every constraint removes one or more independent motions. The result is the number of independent input motions you must control to determine the mechanism’s configuration.
When to use it
Use this when analysing the mobility of robots, linkages, machine assemblies, or any mechanism made of rigid bodies connected by joints. It is especially useful for checking whether a design is underconstrained, properly constrained, or overconstrained before building it.
Worked example, step by step
A robotic inspection platform has 2 moving rigid bodies. Each body could have 6 possible motions, so start with 6n = 6 × 2 = 12. The joints and stops together impose 7 independent constraints, so Σc = 7. Apply the formula: DoF = 6n - Σc = 12 - 7 = 5. Final answer: the mechanism has 5 degrees of freedom.
Common mistakes to avoid
- Counting the fixed base as a moving body in n
- Subtracting the number of joints instead of the number of independent constraints
- Using this 3D formula for a purely planar mechanism without adjusting the motion count
Did You Know?
Six degrees of freedom — three to move and three to rotate — let a robot arm place a tool anywhere in its workspace.
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