Introduction

Assume we have two coordinate frames: world frame W and robot frame R. Let $T_{WR}$ denote the transformation from world frame to robot frame. Assume $P^W$ and $P^R$ are the corresponding points in world frame and robot frame respectively. They satisfy:

\[P^W = T_{WR} * P^R\]

If $P^R$ is the origin of the robot frame, that is $P^R=[0,0,0,1]^T$, then $P^W=[\vec{t}, 1]^T$ where $\vec{t}$ is the translation part of $T_{WR}$. Obviously, $\vec{t}$ in $T_{WR}$ is a vector starting from the origin of frame W and ending at the origin of frame R expressed in frame W.

The effect of shifting frames on rigid body transformation.

Assume there is a point cloud, which is described in robot frame R with resect to the world frame R. The point cloud pose is denoted as $T_{WR}$. Now we want to shift the robot frame R using a transformation, for example, shift robot frame R in +y axis by 2 meters w.r.t the original robot frame R. Notice only the coordinate frame instead of the points is shifted. Assume the shifted robot frame is R’. Since points are not shifted, any point in the point cloud still has the same cooridnate as before in the world frame W after this shifting process. For example, point A has a coordinate [0, 2, 0] in R, now its coordinate is [0, 0, 0] in R’, but its world frame coordinate is the same.

We can then construct the transformation matrix among frame R and R’ : $T_{RR’} = \{I, \vec{t}\}$ where $\vec{t} = [0, 2, 0]^T$. Afterwards, the point cloud in frame R’ can be obtained by: $T_{R’R}* P^R$. The point cloud pose (the pose of frame R’) is: $T_{WR}*T_{RR’}$.

The effect of shifting points on rigid body transformation.

Assume there is a point cloud, which is described in robot frame R with resect to the world frame R. The point cloud pose is denoted as $T_{WR}$. Now we want to shift the point cloud in robot frame R using a transformation, for example, shift point cloud in -y axis by 2 meters in the robot frame R. For example, point A has a coordinate $[0, 2, 0]^T$ in R, after shifting, its corridnate becames $[0, 0, 0]^T$. We can construct a transformation for this process:

\[Cloud^{Transformed} = T * Cloud^R\]

where $T=\{ I, [0,-2,0]^T \}$.

So the transformed point cloud is $Cloud^{Transformed}$ and its pose is $T_{WR}$.