Before we can analyze the motion of a robot, we must first quantify its capacity for motion. This is done using the concept of Degrees of Freedom (DOF).

The Degrees of Freedom (DOF) of a body is the number of independent parameters required to completely specify its configuration (position and orientation) in space.

The DOF of a robot's end-effector determines its ability to reach different points and achieve different orientations. The number of joints in a manipulator is often referred to as its degrees of freedom, as each joint typically provides one independent parameter of motion.

Let's build this concept from a simple point to a full rigid body.

DOF of a Point:

A point is a geometric entity with no size or orientation. We only need to specify its position.

In 1D space (a line), a point can only move along that line. Its position can be described by a single variable (e.g.,\((x,y)\)). Therefore, it has 1 DOF.

In 2D space (a plane), a point's position is described by two coordinates (e.g., \((x,y)\)). Therefore, it has 2 DOF.

In 3D space, a point's position is described by three coordinates (e.g., \((x,y)\)). Therefore, it has 3 DOF.

DOF of a Rigid Body:

A rigid body is an object of a fixed size and shape. Unlike a point, we must specify not only its position but also its orientation.

In 2D Space (Planar): Consider a flat object moving on a tabletop. To specify its configuration, we need to know the position of a reference point on the object (2 parameters: \((x,y)\) and its angle of rotation about an axis perpendicular to the plane (1 parameter: \( \theta \)).

• Total DOF = 2 (for position) + 1 (for orientation) = 3 DOF.

In 3D Space: This is the most general case for robotics. To specify the configuration of a rigid body in 3D space, we need:

• 3 parameters for position: The \( (x,y,z) \) coordinates of a reference point on the body.

• 3 parameters for orientation: Describing the body's rotation in space. This is more complex than the 2D case. While there are many ways to represent 3D orientation (e.g., Euler angles, quaternions), it can be proven that three independent parameters are required. Common conventions include describing rotations about three principal axes (e.g., roll, pitch, and yaw).

• Total DOF = 3 (for position) + 3 (for orientation) = 6 DOF.

This value is fundamental: an unconstrained rigid body in 3D space has 6 degrees of freedom. This means that for a general-purpose robotic manipulator to be able to place its end-effector in any arbitrary position and orientation within its workspace, it must have at least 6 degrees of freedom.