Homogeneous Transformation Matrix

A 4×4 matrix combining rotation and translation into one, describing one frame's pose relative to another.

The homogeneous transformation matrix T is the standard way robotics represents pose: the upper-left 3×3 block is the rotation matrix R (orientation), the upper-right 3×1 block is the translation vector p (position), and the bottom row is fixed at [0 0 0 1]. The set of all such matrices forms the special Euclidean group SE(3). Adding that bottom row lets a point, written as [x, y, z, 1], have rotation and translation applied together in a single matrix multiplication. It's used three ways: to represent frame {b}'s pose relative to {s} as T_sb; to change reference frames, chaining transforms by canceling matching subscripts, as in T_sc = T_sb·T_bc; and to apply a translation and rotation to a frame or an object. Forward kinematics is exactly the product of each joint's transform, giving the end effector's pose relative to the base; ROS's TF tree also stores poses this way. Note that matrix multiplication doesn't commute, so left-multiplying and right-multiplying mean different things.

ExampleA camera mounted on an arm's wrist: given the base-to-end-effector transform T_base_ee and the end-effector-to-camera transform T_ee_cam (from hand-eye calibration), multiplying them gives T_base_cam, which converts whatever point the camera sees into the base frame so the arm can grasp it.

Also called

SE(3) Matrix, Rigid Transformation

Related

Rotation Matrix · Coordinate Transformation · Pose · Forward Kinematics (FK) · Hand-Eye Calibration · TF / tf2 Transform Tree

Sources

Modern Robotics 3.3.1: Homogeneous Transformation Matrices (Northwestern)
Modern Robotics (Lynch & Park) preprint PDF