Coordinate System Translation, Rotation, and Position Offset: Basic Principles and Application Examples

Basic Principles

Basic Definitions

Coordinate System: A reference frame defined by an origin and a set of mutually orthogonal, oriented basis axes (usually X, Y, Z), used to uniquely represent the position and orientation of a point in space under specific units.

• 2D contains two axes (X, Y); 3D contains three axes (X, Y, Z);
• It is necessary to agree on chirality (usually the right-hand system) and units (mm, m, etc.);
• Vectors and orientations given relative to the same coordinate system are comparable and composable.

Common Types of Coordinate Transformations

Translation: poseAdd()

• Meaning: Move the position of the same point within the same coordinate system, with its orientation unchanged.
• Formula (with translation when a point expressed in frame B is transformed to frame A):

$${\mathbf{p}}_{\mathrm{A}}={\mathbf{R}}_{\mathrm{A}}^{\mathrm{B}}{\mathbf{p}}_{\mathrm{B}}+{\mathbf{t}}_{\mathrm{A}}^{\mathrm{B}}$$

Here ${\mathbf{t}}_A^B=[t_x,t_y,t_z{]}^T$ is the translation vector expressed in coordinate system A.

• $\mathbf{R}$ is responsible for "changing direction", and $\mathbf{t}$ is responsible for "adding position". For translation only, ${\mathbf{p}}_A={\mathbf{p}}_B+\mathbf{t}$.

Rotation: poseRotation()

• Meaning: Rotate the orientation within the same coordinate system; the position changes in a circular path relative to the origin.

• Formula: ${\mathbf{p}}_A^{\prime }=\mathbf{R}{\mathbf{p}}_A$, where $\mathbf{R}\in SO(3)$, satisfying ${\mathbf{R}}^T\mathbf{R}=\mathbf{I}$ and $det\mathbf{R}=+1$.

• Common representations (3D):

  • Euler angles ZYX (yaw-pitch-roll): $\mathbf{R}={\mathbf{R}}_z(\psi ){\mathbf{R}}_y(\theta ){\mathbf{R}}_x(\varphi )$
  • Axis-angle (rotation by angle $\theta$ about the unit axis $\hat{\omega }$): Rodrigues' formula $\mathbf{R}=\mathbf{I}+\sin \theta [\hat{\omega }{]}_{\times }+(1-\cos \theta )([\hat{\omega }{]}_{\times }{)}^2$

Translation + Rotation Combined (Pose / Homogeneous Transformation): poseTrans()

Pack rotation and translation into a 4×4 matrix:

$$\mathbf{T}=\left[\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ {\mathbf{0}}^T& 1\end{array}\right],\left[\begin{array}{c}{\mathbf{p}}_A\\ 1\end{array}\right]={\mathbf{T}}_A^B\left[\begin{array}{c}{\mathbf{p}}_B\\ 1\end{array}\right]$$

• Composition order: Right-multiply by the transformation that is closer to the point. For example, first offset/rotate within the TCP's own frame ($\mathrm{\Delta }{\mathbf{T}}_{tcp}$), then place it back into the base:

$${\mathbf{T}}_{\mathrm{base}}^{\mathrm{tcp},\mathrm{new}}={\mathbf{T}}_{\mathrm{base}}^{\mathrm{tcp}}\mathrm{\Delta }{\mathbf{T}}_{\mathrm{tcp}}$$

• Inverse transformation poseInverse(): Convert the "pose from B to A" into the "pose from A to B",

$${\mathbf{T}}^{-1}=\left(\begin{array}{cc}{\mathbf{R}}^{\mathsf{T}}& -{\mathbf{R}}^{\mathsf{T}}\mathbf{t}\\ \mathbf{0}& 1\end{array}\right)$$

Application Examples

1. Point Offset

Example program package

1. Point offset via the Direction-Until node

   

2. Feature in the 0.31 branch: point offset relative to a variable or waypoint

   

2. Coordinate System Offset

1. Implement offset via a custom coordinate system

   

3. Coordinate System + Point Offset