# Observed position update after motion of the robot

The equation (14) in this paper, $$P_{Cj}=RP_{Pj}+T+e_j$$ where $$P_{Cj}$$ and $$P_{Pj}$$ are the observed positions of feature $$j$$ prior to and after the current robot motion, $$R$$(otation) and $$T$$(ranslation)

If ignoring $$e_j$$ and assuming there is no translation,

I don't understand why applying $$R$$ and not $$R^{-1}$$.

For example,

the previous robot orientation is GREEN and the blue dot in GREEN coord is $$(0,3)$$.

Then the robot rotates clockwise $$\frac{\pi}{2}$$ degree to RED coord and the blue dot in RED coord is (-3,0)

If clockwise rotation is negative,

$$R$$ in this case is:

$$\begin{bmatrix}\cos(\frac{-\pi}{2})&-sin(\frac{-\pi}{2})\\sin(\frac{-\pi}{2})&cos(\frac{-\pi}{2})\end{bmatrix}$$

equal to

$$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

So to get $$(-3,0)$$, $$R^{-1}$$ seems to be the correct one.

Do I misunderstand anything?

Thank you for precious time on my question.

So it's the rotation of the coordinate system GREEN into RED that uses the inverse $$\color{red}{C} = R^{-1}\color{lime}{C}$$ and determining the location of a fixed point under the new coordinates uses $$R$$ itself.