Corrigendum: Optimizing trajectories for highway driving with offline reinforcement learning

A Corrigendum on
Optimizing trajectories for highway driving with offline reinforcement learning

by Mirchevska B, Werling M and Boedecker J (2023). Front. Future Transp. 4:1076439. doi: 10.3389/ffutr.2023.1076439

In the published article, there was an error. Algorithm 2: a_lo should be $a_{la{t}_p}$.

A correction has been made to 3 Approach, 3.2 Decision making. This sentence previously stated:

“${\pi }_{\theta }(s)=(a_{tv},a_{la{t}_d},a_{lo{n}_d},a_{lo})$.”

The corrected sentence appears below:

“${\pi }_{\theta }(s)=(a_{tv},a_{la{t}_d},a_{lo{n}_d},a_{la{t}_p})$.”

In the published article, there was an error. Algorithm 2: a_lo should be $a_{la{t}_p}$.

A correction has been made to 3 Approach, 3.2 Decision making. This sentence previously stated:

“$t=generate\text{\_}traj(s,a_{tv},a_{la{t}_d},a_{lo{n}_d},a_{lo})$.”

The corrected sentence appears below:

“$t=generate\text{\_}traj(s,a_{tv},a_{la{t}_d},a_{lo{n}_d},a_{la{t}_p})$.”

A correction has been made to 4 MDP Formalization, 4.3 Reward. This sentence previously stated:

“For the first objective, not causing collisions and remaining within the road boundaries, we define an indicator ind_f signaling when the agent has failed in the following way:”

The corrected sentence appears below:

“For the first objective, not causing collisions and remaining within the road boundaries, we define an indicator f signaling when the agent has failed in the following way:”

A correction has been made to 4 MDP Formalization, 4.3 Reward. This equation previously stated:

$in{d}_f=\left\{\begin{array}{cc}1,\hfill & \text{if\hspace{0.17em}the\hspace{0.17em}agent\hspace{0.17em}has\hspace{0.17em}failed}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.(1)$

The corrected equation appears below:

$f=\left\{\begin{array}{cc}1,\hfill & \text{if\hspace{0.17em}the\hspace{0.17em}agent\hspace{0.17em}has\hspace{0.17em}failed}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.(1)$

A correction has been made to 4 MDP Formalization, 4.3 Reward. This equation previously stated:

$in{d}_v=\left\{\begin{array}{cc}1,\hfill & v_{\mathit{lon}}<v_{\mathit{des}}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.(3)$

The corrected equation appears below:

$v_s=\left\{\begin{array}{cc}1,\hfill & v_{\mathit{lon}}<v_{\mathit{des}}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.(3)$

A correction has been made to 4 MDP formalization, 4.3 Reward. This equation previously stated:

$\begin{array}{l}\hfill r\left(s,a\right)=in{d}_f\left(-0.5\right)+\left(1-in{d}_f\right)\left[in{d}_v\left(1-{\delta }_v/v_{\mathit{des}}\right)+\left(1-in{d}_v\right)\right.\\ \hfill +in{d}_{\mathit{jlon}}\left(p{j}_{\mathit{lon}}\left(sq{j}_{\mathit{lon}}\left(a\right)/j_{\mathit{lon}}^{\mathit{max}}\right)\right)+\left(1-in{d}_{\mathit{jlon}}\right)\left(p{j}_{\mathit{lon}}\right)\\ \hfill \left.+in{d}_{\mathit{jlat}}\left(p{j}_{\mathit{lat}}\left(sq{j}_{\mathit{lat}}\left(a\right)/j_{\mathit{lat}}^{\mathit{max}}\right)\right)+\left(1-in{d}_{\mathit{jlat}}\right)\left(p{j}_{\mathit{lat}}\right)\right]\end{array}(7)$

The corrected equation appears below:

$\begin{array}{l}\hfill r\left(s,a\right)=f\left(-0.5\right)+\left(1-f\right)\left[v_s\left(1-{\delta }_v/v_{\mathit{des}}\right)+\left(1-v_s\right)\right.\\ \hfill +in{d}_{\mathit{jlon}}\left(p{j}_{\mathit{lon}}\left(sq{j}_{\mathit{lon}}\left(a\right)/j_{\mathit{lon}}^{\mathit{max}}\right)\right)+\left(1-in{d}_{\mathit{jlon}}\right)\left(p{j}_{\mathit{lon}}\right)\\ \hfill \left.+in{d}_{\mathit{jlat}}\left(p{j}_{\mathit{lat}}\left(sq{j}_{\mathit{lat}}\left(a\right)/j_{\mathit{lat}}^{\mathit{max}}\right)\right)+\left(1-in{d}_{\mathit{jlat}}\right)\left(p{j}_{\mathit{lat}}\right)\right]\end{array}(7)$

A correction has been made to 6 Experiments and results, 6.3 Smoothness analysis. This equation previously stated:

$\begin{array}{l}\hfill r\left(s,a\right)=f\left(-0.5\right)+\left(1-f\right)\left[v_s\left(1-{\delta }_{\text{vel}}/v_{\mathit{des}}\right)+\left(1-v_s\right)\right.\\ \hfill \left.+j_s\left(j_{rw}\left(-j_{\mathit{cost}}\left(a\right)/j_{\mathit{cost}}^{ub}\right)\right)+\left(1-j_s\right)\left(-j_{rw}\right)\right]\end{array}(8)$

The corrected equation appears below:

$\begin{array}{l}\hfill r\left(s,a\right)=f\left(-0.5\right)+\left(1-f\right)\left[v_s\left(1-{\delta }_v/v_{\mathit{des}}\right)+\left(1-v_s\right)\right.\\ \hfill \left.+j_s\left(j_{rw}\left(-j_{\mathit{cost}}\left(a\right)/j_{\mathit{cost}}^{ub}\right)\right)+\left(1-j_s\right)\left(-j_{rw}\right)\right]\end{array}(8)$

A correction has been made to 6 Experiments and results, 6.3 Smoothness analysis. This sentence previously stated:

“The results indicate that the best performance in terms of jerk is yielded when the reward function from Eq. 8 is used and when j_w is assigned a value around 2. However, is important to note that the performance is not very sensitive to the value chosen for j_w and performs similarly well in a range of values. It is interesting to note that when the value for j_w is too low, e.g., 0.5, the agent deems the jerk-related reward component less significant which results in higher jerk values.”

The corrected sentence appears below:

“The results indicate that the best performance in terms of jerk is yielded when the reward function from Eq. 8 is used and when j_rw is assigned a value around 2. However, is important to note that the performance is not very sensitive to the value chosen for j_rw and performs similarly well in a range of values. It is interesting to note that when the value for j_rw is too low, e.g., 0.5, the agent deems the jerk-related reward component less significant which results in higher jerk values.”

A correction has been made to Appendix, Trajectory generation details. This equation previously stated:

$\begin{array}{ll}\hfill & tra{j}_{\mathit{lonp}}=b_0+b_{1}t+b_2{t}^2+b_3{t}^3+b_4{t}^4\hfill \\ \hfill & \text{where}t=\left\{0.0,dt,2dt,\dots ,a_{lo{n}_p}dt\right\}\hfill \end{array}(\mathrm{A1})$

The corrected equation appears below:

$\begin{array}{ll}\hfill & tra{j}_{\mathit{lonp}}=b_0+b_{1}t+b_2{t}^2+b_3{t}^3+b_4{t}^4\hfill \\ \hfill & \text{where}t=\left\{0.0,dt,2dt,\dots ,a_{lo{n}_d}\right\}\hfill \end{array}(\mathrm{A1})$

A correction has been made to Appendix, Trajectory generation details. This equation previously stated:

$\begin{array}{ll}\hfill & tra{j}_{\mathit{latp}}=c_0+c_{1}t+c_2{t}^2+c_3{t}^3+c_4{t}^4+c_5{t}^5\hfill \\ \hfill & \text{where}t=\left\{0.0,dt,2dt,\dots ,a_{la{t}_p}dt\right\}\hfill \end{array}(\mathrm{A2})$

The corrected equation appears below:

$\begin{array}{ll}\hfill & tra{j}_{\mathit{latp}}=c_0+c_{1}t+c_2{t}^2+c_3{t}^3+c_4{t}^4+c_5{t}^5\hfill \\ \hfill & \text{where}t=\left\{0.0,dt,2dt,\dots ,a_{la{t}_d}\right\}\hfill \end{array}(\mathrm{A2})$

The authors apologize for these errors and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.

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