# 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

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

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. Apr 16 '14 at 16:25
• Thankyou for the feedback, I edited the question, is it a bit clearer now? Apr 17 '14 at 8:08