Degrees of Freedom description

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This is a description of the term   “Degrees of Freedom”   (DoF)   as applied to robotics.

How do we keep track of a robot’s position in the physical, task, or work space?

In robotics, the term  “Degrees of Freedom”  (DoF)  represents the set of independent translations and rotations that specify the displaced position of the rigid robot part, body or system   (its coordinates in three-dimensional space).  A simple robot or other rigid body would have at most six DoF.  These include  three axis rotations  and  three translationsTranslation  is the ability to move on an axis without rotating, while  Rotation  is the revolving about an axis without any translations.

Degrees of Freedom image.

X  axis  rotation is named the roll axis and is represented by   ά alpha,  or theta-x   θx.
X  Axis  translation forward / back, is also called,   surge.
Y  axis  rotation is named the pitch axis (Tilt) axis and is represented by   β beta  or theta-y   θy.
Y  Axis  translation is left / right, is also called,   sway.
Z  axis  rotation is named the yaw axis (pan) and is represented by   γ gamma  or theta-z   θz.
Z  axis  translation up / down, is also called,   heave.

Examining a one - dimensional work space only allows for one DoF. 

A two - dimensional workspace allows three DoF; and a three dimensional workspace allows for   six DoF.  A robot or robot part like an arm or leg with several movable parts or segments would have the  combined DoF summed  from the individual DoF of the parts or segments, minus any internal constraints occurring when in motion.

Holonomicity  refers to robotics sum of its Degrees of Freedom and the relationship to the controllable Degrees of Freedom. 

A robot that controls all of its  work space Degrees of Freedom   (DoF)  is said to be  Holonomic.  A robot or robotic part with fewer controllable DoF than total workspace DoF is said to be non-holonomic, and a robot with more controllable DoF than total workspace DoF is said to be redundant.  The term Holonomic is usually applied to robotic arms.  Holonomic Redundancy  usually has more DoF controllable motions than DoF available in its task or work space. 

Red Train picture.

For example, a train can only move on the X axis work space, one DoF   (the train tracks),   and can control its position on that one axis, thus it is holonomic.  Most automobiles can be orientated and move to any position in two - Dimensional   (2D)   work space.  The automobile requires three DoF to describe its position  (X, Y and gamma),  but at any point, it can move only along the vehicles centerline and turn with a steering angle input.  (ignoring skidding or “drifting”)  Thus, it has only two control DoF and three positional DoF;  so, the automobile is non-holonomic.

Ackermann Steering picture.

The DoF term can be used to describe a particular movement, as well as a unique representation of movement possibilities. 

When used by actuators for position and manipulator configuration, DoF control definitions can use  forward and inverse kinematic relationships  for movement computations.

Movements are calculated using mathematics, ranging from a simple  one-X axis Algebraic move from point A to point B,  such as Xb – Xa = move distance,   to a  vector Quaternion positional rotation  in three dimensions using  Linear Algebra matrix theory.

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