What are degrees of freedom (DoF) and how many are there?

To identify the position of an object in three-dimensional space, we use a coordinate system that defines three axes: X, Y, and Z. If the object is a point mass, we only need three coordinates (X, Y, and Z) to locate its position. But a rigid body can both move, or translate, along these three axes and rotate about them, so we need three translational (X, Y, and Z) and three rotational coordinates (rotation about X, Y, and Z) to locate its position.

The classic example of a rigid body in three-dimensional space is an aircraft in flight. It can make translational movements forward and back, left and right, and up and down in the X, Y, and Z axes. But it can also rotate around the X, Y, and Z axes, creating rotational motions referred to as roll, pitch, and yaw, respectively.

These three translational and three rotational movements define the six degrees of freedom (DoF) of a rigid body in 3D space.

An example of degrees of freedom in linear motion is a bearing block mounted to a profiled linear guide. The bearing has only one degree of freedom, since it can only move along one axis, typically referred to as the X axis. Motions in the other five degrees of freedom — translation along the Y and Z axes and all three rotational motions — are constrained by the guide being mounting to the rail.

However, just because motion is constrained in the other five degrees of freedom doesn’t mean that there is zero movement in those axes. This is because deflection of the bearing block can introduce small motions in the constrained degrees of freedom. For example, loads placed on the bearing in the downward (Z) or lateral (Y) direction can cause the bearing to deflect in those directions. And offset, or moment, loads applied to the bearing can cause it to rotate slightly around any of the three axes. These motions due to deflection in the constrained degrees of freedom are planar and angular errors.

motion errors — Motions in the constrained degrees of freedom represent planar and angular errors.
*Image credit: Dover Motion*

Can robots have more than six degrees of freedom?

We established earlier that only six degrees of freedom (three translational and three rotational) exist in three-dimensional space, but it’s not uncommon to hear of a robot with seven or more “degrees of freedom.” So how can a robot have more than six degrees of freedom?

In robot lexicon, “degrees of freedom” often refers to the number of robot joints or axes of motion. And although some robot designs can have seven or more axes of motion, it’s important to note that a robot with more than six axes of motion is kinematically redundant — meaning it can reach a given position from multiple joint states.

A good example of a kinematically redundant system is the human arm. If you place your hand on a table, you can change the position of your wrist and shoulder without changing the position of your hand. This means there are an infinite number of ways in which your arm can move to place your hand at a specific location on the table.

Fortunately, our brains are designed to determine the “best” solution when we need to do something like pick up an object. But kinematic redundancy makes robots with more than six axes, or degrees of motion, difficult to program and control, requiring the introduction of additional constraints or dependencies in order to arrive at a single set of joint motions for a given target position.