Kalman filter implementation for a driving simulation in a final project
$begingroup$
Currently I am designing a Kalman filter-based steering for my final paper in a driving simulator. I'm actually new to the Kalman filtering method but I've studied a couple of journals I can find, though I do have problems when designing my system. Apologies if there are mistakes in the following question since this is basically my first time into the system.
The proposed system has a purpose to predict the steering based on the target position.
From here, the state vector I'll be using is defined as
$$x = begin{bmatrix}
alpha \
v \
end{bmatrix}
$$
where $alpha$ is the steering angle and $v$ is the car's velocity. I'm also using the basic kinematics for the $F$ vector with
$$F = begin{bmatrix}
1 & Delta T \
0 & 1 \
end{bmatrix}
$$
where $Delta T$ will be the time step for the system. As for the measurement vector $z$, it'll comprise of
$$z = begin{bmatrix}
alpha_x \
alpha_y \
v_x \
v_y \
end{bmatrix}
$$
which are positions and current speeds in regards to x and y axes, all in Cartesian. The positions for x and y are defined as
$$
x = rho times cos theta
$$
$$
y = rho times sin theta
$$
Where $rho$ is the distance calculated based on the sensor readings and $theta$ is the angle.
Given the measurements, I would also have to map the $h(x)$ vector (making this an Extended Kalman Filter) so that it'll form the eventual steering and velocity estimates using the following mapping.
$$h(hat x) = begin{bmatrix}
arctan(frac {2y}{x}) \
frac {v}{3.6} \
end{bmatrix}
$$
With this, I have a couple of questions regarding to the difficulty I have:
1) According to the formula, the error $y$ is calculated by $z - h(hat x)$, but given the difference of the matrices, are both the $z$ and $hat x$ matrices being subtracted first before transformed into $h$ or not? I might perceive this the wrong way and this is one of the difficulties I have in understanding it.
2) How do I express $arctan(frac {2y}{x})$ in matrix form such that it could be along the lines of the following normal filter $H$ matrix?
$$H = begin{bmatrix}
1 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
end{bmatrix}
$$
Given that this would eventually form the Jacobian matrices for $F_j$ and $H_j$?
Thanks in advance!
kalman-filter
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
$endgroup$
add a comment |
$begingroup$
Currently I am designing a Kalman filter-based steering for my final paper in a driving simulator. I'm actually new to the Kalman filtering method but I've studied a couple of journals I can find, though I do have problems when designing my system. Apologies if there are mistakes in the following question since this is basically my first time into the system.
The proposed system has a purpose to predict the steering based on the target position.
From here, the state vector I'll be using is defined as
$$x = begin{bmatrix}
alpha \
v \
end{bmatrix}
$$
where $alpha$ is the steering angle and $v$ is the car's velocity. I'm also using the basic kinematics for the $F$ vector with
$$F = begin{bmatrix}
1 & Delta T \
0 & 1 \
end{bmatrix}
$$
where $Delta T$ will be the time step for the system. As for the measurement vector $z$, it'll comprise of
$$z = begin{bmatrix}
alpha_x \
alpha_y \
v_x \
v_y \
end{bmatrix}
$$
which are positions and current speeds in regards to x and y axes, all in Cartesian. The positions for x and y are defined as
$$
x = rho times cos theta
$$
$$
y = rho times sin theta
$$
Where $rho$ is the distance calculated based on the sensor readings and $theta$ is the angle.
Given the measurements, I would also have to map the $h(x)$ vector (making this an Extended Kalman Filter) so that it'll form the eventual steering and velocity estimates using the following mapping.
$$h(hat x) = begin{bmatrix}
arctan(frac {2y}{x}) \
frac {v}{3.6} \
end{bmatrix}
$$
With this, I have a couple of questions regarding to the difficulty I have:
1) According to the formula, the error $y$ is calculated by $z - h(hat x)$, but given the difference of the matrices, are both the $z$ and $hat x$ matrices being subtracted first before transformed into $h$ or not? I might perceive this the wrong way and this is one of the difficulties I have in understanding it.
2) How do I express $arctan(frac {2y}{x})$ in matrix form such that it could be along the lines of the following normal filter $H$ matrix?
$$H = begin{bmatrix}
1 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
end{bmatrix}
$$
Given that this would eventually form the Jacobian matrices for $F_j$ and $H_j$?
Thanks in advance!
kalman-filter
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
$endgroup$
add a comment |
$begingroup$
Currently I am designing a Kalman filter-based steering for my final paper in a driving simulator. I'm actually new to the Kalman filtering method but I've studied a couple of journals I can find, though I do have problems when designing my system. Apologies if there are mistakes in the following question since this is basically my first time into the system.
The proposed system has a purpose to predict the steering based on the target position.
From here, the state vector I'll be using is defined as
$$x = begin{bmatrix}
alpha \
v \
end{bmatrix}
$$
where $alpha$ is the steering angle and $v$ is the car's velocity. I'm also using the basic kinematics for the $F$ vector with
$$F = begin{bmatrix}
1 & Delta T \
0 & 1 \
end{bmatrix}
$$
where $Delta T$ will be the time step for the system. As for the measurement vector $z$, it'll comprise of
$$z = begin{bmatrix}
alpha_x \
alpha_y \
v_x \
v_y \
end{bmatrix}
$$
which are positions and current speeds in regards to x and y axes, all in Cartesian. The positions for x and y are defined as
$$
x = rho times cos theta
$$
$$
y = rho times sin theta
$$
Where $rho$ is the distance calculated based on the sensor readings and $theta$ is the angle.
Given the measurements, I would also have to map the $h(x)$ vector (making this an Extended Kalman Filter) so that it'll form the eventual steering and velocity estimates using the following mapping.
$$h(hat x) = begin{bmatrix}
arctan(frac {2y}{x}) \
frac {v}{3.6} \
end{bmatrix}
$$
With this, I have a couple of questions regarding to the difficulty I have:
1) According to the formula, the error $y$ is calculated by $z - h(hat x)$, but given the difference of the matrices, are both the $z$ and $hat x$ matrices being subtracted first before transformed into $h$ or not? I might perceive this the wrong way and this is one of the difficulties I have in understanding it.
2) How do I express $arctan(frac {2y}{x})$ in matrix form such that it could be along the lines of the following normal filter $H$ matrix?
$$H = begin{bmatrix}
1 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
end{bmatrix}
$$
Given that this would eventually form the Jacobian matrices for $F_j$ and $H_j$?
Thanks in advance!
kalman-filter
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
$endgroup$
Currently I am designing a Kalman filter-based steering for my final paper in a driving simulator. I'm actually new to the Kalman filtering method but I've studied a couple of journals I can find, though I do have problems when designing my system. Apologies if there are mistakes in the following question since this is basically my first time into the system.
The proposed system has a purpose to predict the steering based on the target position.
From here, the state vector I'll be using is defined as
$$x = begin{bmatrix}
alpha \
v \
end{bmatrix}
$$
where $alpha$ is the steering angle and $v$ is the car's velocity. I'm also using the basic kinematics for the $F$ vector with
$$F = begin{bmatrix}
1 & Delta T \
0 & 1 \
end{bmatrix}
$$
where $Delta T$ will be the time step for the system. As for the measurement vector $z$, it'll comprise of
$$z = begin{bmatrix}
alpha_x \
alpha_y \
v_x \
v_y \
end{bmatrix}
$$
which are positions and current speeds in regards to x and y axes, all in Cartesian. The positions for x and y are defined as
$$
x = rho times cos theta
$$
$$
y = rho times sin theta
$$
Where $rho$ is the distance calculated based on the sensor readings and $theta$ is the angle.
Given the measurements, I would also have to map the $h(x)$ vector (making this an Extended Kalman Filter) so that it'll form the eventual steering and velocity estimates using the following mapping.
$$h(hat x) = begin{bmatrix}
arctan(frac {2y}{x}) \
frac {v}{3.6} \
end{bmatrix}
$$
With this, I have a couple of questions regarding to the difficulty I have:
1) According to the formula, the error $y$ is calculated by $z - h(hat x)$, but given the difference of the matrices, are both the $z$ and $hat x$ matrices being subtracted first before transformed into $h$ or not? I might perceive this the wrong way and this is one of the difficulties I have in understanding it.
2) How do I express $arctan(frac {2y}{x})$ in matrix form such that it could be along the lines of the following normal filter $H$ matrix?
$$H = begin{bmatrix}
1 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
end{bmatrix}
$$
Given that this would eventually form the Jacobian matrices for $F_j$ and $H_j$?
Thanks in advance!
kalman-filter
kalman-filter
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
asked Dec 5 '18 at 9:23
Setsuna R SeieiSetsuna R Seiei
11
11
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