Point tracking from a mobile robot

How can I track a fixed point $P=(x_P, y_P)$ from a moving robot?

Coordinates of $P$ are relative to the state/pose of the robot (x axis looks forward the robot and y axis is positive on the right of the robot). Suppose that the initial robot state/pose is at $S_{R}=(x_R, y_R, \theta_R)$. The next frame (namely after $\Delta t$) with the applied control $(v, \omega)$ the robot is at state $S_{R'}=(x_{R'}, y_{R'}, \theta_{R'})$.

Where (I set the axes as OpenCV):

$x_{R'} = x_R + v cos(\theta_R) \Delta t$

$y_{R'} = y_R + v sin(\theta_R) \Delta t$

$\theta_{R'} = \theta_{R} + \omega\Delta t$

The question is: which are the coordinates $(x_P', y_P')$ of the same point $P$ relative to $S_{R'}$? As visible in the picture, I know the transformation from the initial state to the next state of the robot and the coordinate of P in reference to the initial state

$$t = \begin{pmatrix} cos(\theta_{R'}) & -sin(\theta_{R'}) & x_{R'}\\ sin(\theta_{R'}) & cos(\theta_{R'}) & y_{R'}\\ 0 & 0 & 1\\ \end{pmatrix}$$

Please correct me if I made some mistakes!

Thank you, any help is appreciated.

• Please try to rephrase your question. You are talking about some point P = (xp,yp), then talking about the state Sr (using some (xr, yr) point) and then asking for the relative position of the point P which you never told us what it is. If you cannot define it mathematically, at least try to explain it with words, which points are your talking about. Thanks. – Damjan Dakic Apr 16 '14 at 16:25
• Thankyou for the feedback, I edited the question, is it a bit clearer now? – Michele mpp Marostica Apr 17 '14 at 8:08