Robotics Fundamentals
Degrees of Freedom
Degrees of freedom count the independent ways a robot can move. Learn the six DOF of a rigid body, how joints supply them and why six is the magic number for an arm.
beginner · 12 min read
Lesson content
"How many axes does it have?" is the first question anyone asks about a robot arm. The answer is its degrees of freedom. The concept is simple, the counting is easy and the consequences reach into every other lesson in this curriculum.
Definition
Degree of freedom (DOF)
One independent way something can move. One number you would have to state to describe where it is.
Formal definition
The number of independent position variables that must be specified to locate all parts of a mechanism. For an open kinematic chain with single-variable joints, DOF equals the number of joints.
A rigid body floating in space has exactly six degrees of freedom: three ways to slide and three ways to turn. There is no seventh. Anything else you can do to it is some combination of those six.
| Kind | Axis | Aircraft name | Marine name |
|---|---|---|---|
| Translation | along X | not named | surge |
| Translation | along Y | not named | sway |
| Translation | along Z | not named | heave |
| Rotation | about X | roll | roll |
| Rotation | about Y | pitch | pitch |
| Rotation | about Z | yaw | yaw |
The same six appear as position and orientation, as the six numbers in a pose and as the six rows of a Jacobian. It is one idea wearing several hats.
Why do industrial arms so often have exactly six joints?
Because six is the smallest number that can hit any position and any orientation in space.
The job splits neatly in two. The first three joints carry the wrist to a point, which covers the three translations. The last three, all meeting at the wrist, aim the tool, which covers the three rotations.
With five joints there are poses you can reach the place of, but not the angle. With six you can, in principle, put the tool anywhere in the workspace at any angle you like.
So why does anything have fewer? Because fewer joints is cheaper, stiffer and faster. A job that never needs full orientation control uses four (a SCARA) or three (a delta) and is better for it.
The math
Counting for an open chain is arithmetic, not geometry: add up what each joint contributes. Only closed loops make it harder.
DOF (open chain) = Σ fᵢ over all joints
revolute → f = 1 · prismatic → f = 1 · cylindrical → f = 2 · spherical → f = 3
- fᵢ
- degrees of freedom the i-th joint permits
With real numbers
A 6-axis industrial arm: six revolute joints × 1 = 6 DOF.
A SCARA: two revolute + one prismatic + one revolute = 4 DOF.
A differential-drive mobile robot: two wheel speeds → 2 controllable DOF, in a plane that has 3, which is exactly why it cannot slide sideways.
A drone: four rotor speeds → 4 controllable DOF, in a space with 6.
Controllable DOF is not the same as total DOF
A quadcopter lives in a world with six degrees of freedom. It has four controls, one per rotor.
So it cannot simply slide sideways. It has to tilt first, then let the tilt push it across. A machine with fewer controls than degrees of freedom is called underactuated.
Cars are underactuated too. You cannot drive sideways into a parking space. So are differential-drive robots and walking bipeds.
Keep the two numbers apart in your head: total DOF is the space the robot lives in, controllable DOF is what it can command directly. The gap between them is most of what makes motion planning hard.
| Robot | Total DOF in its space | Controllable DOF | Consequence |
|---|---|---|---|
| Cartesian gantry | 3 | 3 | Fully actuated: command x, y, z directly |
| 6-axis arm | 6 | 6 | Any pose in reach |
| 7-axis arm | 6 | 7 | Redundant: can dodge obstacles while holding the tool still |
| Differential drive | 3 (x, y, θ) | 2 | Cannot move sideways; must manoeuvre |
| Quadcopter | 6 | 4 | Must tilt to translate |
| Mecanum base | 3 | 3 | Holonomic: slides in any direction |
Cartesian gantry
- Total DOF in its space
- 3
- Controllable DOF
- 3
- Consequence
- Fully actuated: command x, y, z directly
6-axis arm
- Total DOF in its space
- 6
- Controllable DOF
- 6
- Consequence
- Any pose in reach
7-axis arm
- Total DOF in its space
- 6
- Controllable DOF
- 7
- Consequence
- Redundant: can dodge obstacles while holding the tool still
Differential drive
- Total DOF in its space
- 3 (x, y, θ)
- Controllable DOF
- 2
- Consequence
- Cannot move sideways; must manoeuvre
Quadcopter
- Total DOF in its space
- 6
- Controllable DOF
- 4
- Consequence
- Must tilt to translate
Mecanum base
- Total DOF in its space
- 3
- Controllable DOF
- 3
- Consequence
- Holonomic: slides in any direction
Example
What a seventh joint actually buys
Try this now. Put your hand flat on the table and keep it exactly where it is. Then move your elbow up and down.
Your hand has not moved. Your arm has.
That is redundancy. Your arm has seven degrees of freedom (three at the shoulder, one at the elbow, three at the wrist) but placing a hand only needs six. The extra one means a whole family of arm shapes puts your hand in the same spot.
Robots with seven joints get the same trick. They use it constantly: reaching around an obstacle, staying clear of a joint limit, dodging a singularity, all while the tool holds perfectly still.
Degrees of freedom are the questions you must answer
"Where is the pen on my desk?"
If you only care about the tip, three numbers do it: left-right, forward-back, up-down.
But a pen also points somewhere and it can be rolled about its own axis. That is three more numbers.
Six questions, six answers, six degrees of freedom. Use any fewer and someone could put the pen back differently and still match your description perfectly.
Counting the gripper's fingers as arm DOF.
Gripper freedoms are usually quoted separately. A "6-axis arm with a 1-DOF gripper" has six axes for reaching and one for holding.
Assuming more DOF is always better.
Each joint adds cost, mass, backlash, a failure mode and a term in every equation. Beyond six you buy dexterity, not reach.
Confusing DOF with the number of motors.
A delta robot has three motors and three DOF but a dozen joints, because the closed loops absorb the rest. Count freedoms, not parts.
Key takeaway
- A degree of freedom is one independent way to move, or one number you must state.
- A free rigid body in 3D has six: three slides and three turns.
- In an open chain, DOF is just the sum of the joints' freedoms.
- Six DOF lets an arm reach any position and orientation within its workspace.
- Controllable DOF can be fewer than total DOF. Underactuation is why cars park in several moves.
- A seventh joint gives redundancy: the same tool pose with a family of arm postures.
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