# Line based Extended Kalman Filter for localization : Measurement Jacobian

I have to estimate location w.r.t a given map of the environment. The state is expressed as

$X = \begin{bmatrix} x \\ y \\ \theta \end{bmatrix}$

Given ,

$X_{t-1} = \begin{bmatrix} x_{t-1} \\ y_{t-1} \\ \theta_{t-1} \end{bmatrix}$ and it's covariance $P_{t-1},$ a 3X3 matrix .

I have a measurement model which detects lines representing obstacles , so , in one measurement there can be any number of lines , these lines are represented as $$Z = \begin{bmatrix} \alpha \\ r \end{bmatrix}$$ normal form of line.

I have successfully completed all the steps until the calculation of innovation covariance i.e.

$S = H_t\hat{P_t}H_t^T + R_t$ ,

where $H_t$ is the measurement Jacobian , $\hat{P_t}$ is the estimated motion covariance and $R_t$ is the measurement noise matrix.

When I have single line measurement , the measurement Jacobian, $H_t$ is a 2-Dimensional $2\times 3$ matrix and the measurement noise matrix , $R_t$ is $2\times 2$. But , when there are multiple line measurements , for example 8 lines , $H_t$ becomes $2\times 3 \times 8$ three-dimensional matrix and $R_t$ a $2\times 2 \times 8$ three-dimensional matrix.

Because of this change in dimension I cannot carry out the calculation of innovation covariance $S$ as $\hat{P_t}$ is a $3 \times 3$ matrix.

How can I incorporate multiple line readings and suppress the three-dimensional matrix to match innovation covariance equation ?

• Welcome to DSP.SE! What you haven't told us is what your measurement / signal model is? How are you modeling the state transitions? How are you modeling the output equation i.e. How does the state get to the measurements? Without that information, we can't tell you much about how to get your Jacobian right. Usually, with multiple measurements, you just stack the state / observations so things go from $2\times 3$ to $2\times 24$ in the case of 8 measurements. – Peter K. Jan 5 '18 at 17:34
• Thanks @PeterK. Sir. Your suggestion to stack the observation worked perfectly. Actually , for 8 measurements the Jacobian was 16X3 , I only had to stack one above another. – shubham sharma Jan 5 '18 at 19:21
• OK! I've added that comment as an answer. – Peter K. Jan 5 '18 at 20:23

Usually, with multiple measurements, you just stack the state / observations so things go from $2 \times 3$ to $2\times 24$ in the case of 8 measurements, instead of augmenting in another dimension.