Frames (Robotics)

A frame consists of an origin O and an orthogonal x,y,z axis, and is used to represent rigid body configurations. We assume all frames to be right-handed and stationary.

To represent the position and orientation of a rigid body in space, we first first a body frame {b} and a space/world frame {s}. The configuration of the body is then given by:

Where both of the above are expressed in the space frame coordinates.

The position and orientation of an object's coordinate frame is referred to as its pose.

The relative pose of a frame {A} relative to {B}, denoted AξB. Suppose we have a point p, and represented relative to frame {A},{B} denoted Ap,Bp respectively. They are related by Ap=AξBBp. More formally, AξB is any mathematical object that can represent rigid body transformations.

In practical applications, one would have a frame for various entities for e.g. a frame for each joint of a robot, a frame for the camera of the robot and so on. This can be visualized as a collection of coordinate axes, with the world coordinate frame O fixed as the origin. The pose ξR would denote the pose of frame B relative to the world frame. This can also be visualized as a Directed Acyclic Graph, with each coordinate frame as the nodes, and the directed edges of the form OR is labelled with ξR and edges of the form AB labelled as AξB.

Below are the key aspects we need to represent with respect to frames:

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