Root Shifting

• Root Shifting

 

 

- Which is about changing the reference coordinate frame (or "root") used to describe 3D positions and orientations.

- You have data (like a 3D map or trajectory) that’s expressed in the global frame {g}, but now you want to express that data in a local or vehicle frame {v} instead.

- Because sometimes it’s easier or more useful to:

View motion relative to the robot

Align local maps

Do sensor fusion

- This transformation from {g} to {v} is called root shifting.

- Root shifting = expressing a point in a new coordinate frame.

- Your point cloud or position is in {g}, and you want to re-express it in {v}.

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- {g} : global coordinate frame in a point cloud (blue) in 3D

- {v} : the vehicle pose at different points

- The coordinate (X, Y, Z) is in {g} → you want to convert this to {v}

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- \( P_g \) : is known in global frame.

- You want to compute \( P_v \), the same point expressed in the {v} frame.

- To do this, you apply a coordinate transformation: \( P_v = T_{vg} \cdot P_g \), where \( T_{vg} \) is the transformation from global {g} to vehicle {v}.

- This matrix includes rotation and translation.

- So You’re shifting the “root” coordinate system from {g} to {v}.

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- This is the same idea used to compute the transform between frames using the SE(3) composition rule :

To go from {c} to {a}, you can go from {c} to {b} to {a}.

\[ T_{ac} = T_{ab} \cdot T_{bc} \Rightarrow \begin{bmatrix} R_{ac} & \mathbf{w}_a \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} R_{ab} & \mathbf{u}_a \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} R_{bc} & \mathbf{v}_b \\ 0 & 1 \end{bmatrix} \]

You want to express a point from {g} (global) in {v} (vehicle).

So you compute the transformation \( T_{vg} = T_{gv}^{-1} \)