Robotics
Rigid-body transformation
Definition
A rigid-body transformation changes a body's position and orientation without changing its shape or size. In three-dimensional robotics it consists of a proper rotation and a translation.
Also known as: Rigid transformation, Rigid-body transform
Updated
Moving a body without deforming it
Imagine moving a rigid gripper from one pose to another. Every point moves consistently with the same rotation and translation, and distances between points remain unchanged. Scaling, shearing, and bending are outside this model.
Modern Robotics describes the set of spatial rigid transforms as SE(3), the special Euclidean group. A transform can represent a body's pose, a displacement, or a relationship between coordinate frames.
A transform and its representation
A homogeneous transformation matrix is one way to store and compose the operation. Another stores a translation together with an orientation representation such as a unit quaternion.
The underlying geometric relationship is independent of how it is stored. MIT's spatial-algebra discussion emphasizes keeping track of the frames when composing poses and converting coordinates.
Composition requires matching frames
A camera-to-torso relationship can be combined with a torso-to-world relationship to locate observations in the world. Reversing a relationship requires its inverse. Swapping multiplication order generally describes a different operation.
A rigid transform describes pose, not velocity, force, or deformation. Twists represent instantaneous rigid motion, while wrenches represent force and moment about a specified reference point.
Sources
Related terms
Homogeneous transformation
In rigid-body robotics, a homogeneous transformation is a 4-by-4 matrix that combines a three-dimensional rotation and translation. It represents a pose or changes coordinates between reference frames.
Rotation matrix
A rotation matrix represents an orientation or rotation while preserving lengths and angles. In three dimensions it is a 3-by-3 orthonormal matrix with determinant positive one.
Screw theory
Screw theory is a geometric framework for describing rigid-body motion and forces using axes, rotation, translation, and pitch. In robotics it provides the basis for twist and wrench representations and screw-axis formulations of kinematics.