Coordinate frames: the language behind robot motion

Foundation lesson · Study time: about 20 minutes. Prerequisite: Cartesian coordinates and a right-angle rotation. Aim: name the reference frame of a point and verify a simple frame transformation.

Coordinates need a reference

A point can have different coordinates in the robot base frame and the tool frame while remaining the same physical point. A frame supplies an origin and oriented axes. A rigid-body pose describes both position and orientation relative to a named frame.

Use the convention pA = RAB pB + tAB: pB is a point expressed in frame B; RAB rotates its coordinate components into frame A; tAB is the B-origin position expressed in A. In three dimensions, a 4 × 4 homogeneous transformation combines rotation and translation.

An original planar example

Let frame A have x pointing right and y pointing up. Place the origin of B at (0.4, 0.2) metres in A. Rotate the B axes 90° counterclockwise relative to A. Then +xB points along +yA, and +yB points along −xA. For this arrangement:

xA = −yB + 0.4
yA = xB + 0.2

Take a point at (0.1, 0) metres in B. Substitution gives (0.4, 0.3) metres in A. Check the geometry: the point lies 0.1 metre along +xB, which points upward in A, starting from the B origin. The B origin itself, (0, 0) in B, maps to (0.4, 0.2) in A.

Order and labels prevent mistakes

For column-vector conventions, transformations compose as TAB × TBC = TAC: the rightmost transform acts first. Reversing the order generally changes the result. The inverse converts in the opposite direction. State the convention and units alongside any matrix.

Check your understanding

Where is (0, 0.2) metres in B? (0.2, 0.2) metres in A. It moves left from the B origin because +yB points along −xA. Recover the B coordinates of (0.4, 0.3) in A using xB = yA − 0.2 and yB = 0.4 − xA. The result should be (0.1, 0).

Editorial note

Written independently for IIMEEE with AI assistance. Technical statements and the worked example were checked against the references listed below. No external prose, code, diagrams, or tables have been reproduced. These introductory notes use simplified assumptions; they are not instructions for operating machinery or live electrical equipment.