Coordinate Systems

Robotic systems consist of many parts (links) connected with revolute, prismatic, fixed, floating, or planar joints. The configuration of each link can be represented with a set of 6 coordinates, three coordinates for translation and three coordinates for orientation. It is common practice to associate each link with a coordinate frame rigidly attached to a point on the body.

• Transformation Matrix
• Naming Convention
• Common Frame Names

Transformation Matrix

Often we want to transform the coordinates of a point between coordinate systems. Let A and B be two arbitrary coordinate systems. Let p_B be the coordinates of a point p with respect to coordinate system B.

The coordinates of p with respect to the coordinate system A are then given by

Naming Convention

Symbol	Code	Meaning
TAB	T_A_B	Transformation matrix that transforms points from frame B to frame A.
RAB	R_A_B	Rotation matrix that rotates points from coordinate system B to coordinate system A.
tAB	t_A_B	Position of frame B with respect to coordinate system A.
pA	p_A	Position of point p with respect to coordinate system A.

Common Frame Names

Name	Description
world	Fixed world reference frame, mostly used in simulation.
task	Dynamically defined reference frame for a manipulation task, e.g. corner of a table.
base	Base coordinate system located at the base of the robot.
tool0	Robot tool mounting point.
TCP	Center point of the tool.