Adjoint Representation

The 6×6 matrix that transforms a twist or a wrench from one coordinate frame into another.

The adjoint representation is a basic tool in screw theory, and Lynch and Park's textbook Modern Robotics builds it into the core of its chapter on rigid-body motion. Given a pose T = (R, p) (R is the rotation matrix, p is the translation vector), the adjoint representation [Ad_T] is a 6×6 matrix: the upper-left and lower-right blocks are both R, the lower-left block is [p]R (where [p] is the skew-symmetric matrix of p), and the upper-right block is zero. Its job is to change coordinate frames: the same twist (angular velocity plus linear velocity), expressed in two frames {a} and {b}, satisfies V_a = [Ad_Tab]·V_b; a wrench (torque plus force) transforms using its transpose instead. It shows up constantly in the product of exponentials formula, in deriving Jacobians, and in recursive dynamics algorithms.

ExampleA wrist-mounted six-axis force-torque sensor reads a wrench F_s expressed in the sensor's own frame; to get the force in the tool center point frame instead, convert it in one step with F_tcp = [Ad_T]ᵀ·F_s, where T is the pose of the TCP frame relative to the sensor frame.

Also called

Ad, Adjoint Map, Adjoint Transformation

Related

Screw Theory · Twist · Wrench · Homogeneous Transformation Matrix · Lie Group · Product of Exponentials Formula

Sources

Modern Robotics（Lynch & Park）预印本，3.3.2 节 Definition 3.20 与 Proposition 3.27 (Chinese)