What Is the Best Way to Optimally Parameterize the MPC Cost Function for Vehicle Guidance?

Round 1

Reviewer 1 Report

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

I cannot understand Fig.1 because: (1) Which is the simulation environment and which is the parameter optimizer? (2) It seems that Fig. 1 only shows the coordinate used in this study.

The simulation environment is depicted in blue and the parameter optimizer is depicted in orange. We have additionally described in more detail the information flow between the simulation environment and the parameter optimizer in Fig 1, and given additional explanations in the text and the caption of Fig. 1.

More detail information should be provided about the vehicle model.

The vehicle model is a combination of a simple drive train model, a steering actuator model and a bicycle model. It is described in detail in Ref. [25]. We have added a short description and referred to [25] for more details.

It is suggested to explain which function is used to describe the path, otherwise the reader will be hard to understand the path parameter. The details about variable u, x and y are also necessary.

The revised manuscript adds explanations about the description of the path with the new Eq. (6). The paths are modelled as cubic splines. In addition, we also added the description of u, x and y following Eq. (7).

Please explain that how to calculate e(t), especially the determination of p(s(t))?

By e(t), we denote the deviation of the system output from the path. The path p(s(t)) is now defined by the new Eq. (6), see also 3.  In the text, we have added an explanation following Eq. (8).

Please give out the equation to show how to approximate the lateral acceleration.

The lateral acceleration is approximated by a_lat(t) = v(t), \dot{\psi}(t). We have added the formula following Eq. (8). 

Please explain the relationship between Vs.des, Vs(t) and v(t).

The desired path velocity v_s,des is determined prior to MPC optimization based on current and future speed limits. The path velocity v_s(t) describes how fast the path parameter changes and thus also the dynamics of the geometric reference point (x_des, y_des) which the vehicle should follow. If the vehicle follows the path exactly, then v(t) = v_s(t). Otherwise, the ego velocity v(t) can deviate from v_s(t), for example when cutting a curve.We have added these explanations.

First paragraph in section 3.3, the assumptions are unreasonable, because: (1) The assumptions doesn't agree with the practices; (2) The assumptions are made only for conveniences and the author should discuss these assumptions from the engineering practice.

The simplifications regarding the weights in the weighting matrices Q, P and R discussed in Sec. 3.3 are done in order to bound the calculation time of the simulation and thus be able to obtain statistically significant benchmarks of different optimizers. However, these simplifications do not limit the results of the MPC as they appear reasonable in the application context. For instance, tracking in x- and y-direction is equally important and therefore we weight the deviation from the path in the x- and y-directions equally. Similarly, we do not expect a substantial benefit from differentiating between terminal and stage cost. Finally, we set the weighting factor q_\psi to a fixed value since the absolute value of the weighting factors is not important for the cost function, but only the relation of the factors to each other. We have added some remarks about our motivation for the simplifications in Sec. 3.3.

How to determine the range in (9)?

These boundaries result from heuristics and the experience gained from the numerous driving with the MPC system over the past few years. They thus describe the range of values to be expected. We have added an explanation in the text.

How to determine Vdes in (10)?

The desired velocity v_des is determined during the simulation based on the speed limit v_lim(s) of the used track. We have added this remark and a new lower figure in Fig. 2 as an explanation.

How to calculate e1a, in (11)?

The lateral deviation e_lat is the shortest distance from the reference point of the vehicle (middle of the front axle) to the target line shown in Fig. 2 for the sampling time t_k. It is calculated based on the coordinates of the vehicle in the world coordinate system returned by the vehicle model of the simulation. We have added this remark.

Fig. 2 should be re-arranged following its context. Algorithm 1 also should be re-arranged for readability.

We have rearranged Fig. 2 and the algorithms.

What does different color mean in Fig. 6. For example, which one denotes the best Jl, or worst Jl.

The colors in Fig. 6 have the following meaning. They show the Pareto fronts for different parameterizations, which correspond to the left image and are marked in the equivalent colors: minimizing acceleration (green dots), lateral error (blue dots) and velocity (red dots).    All other parameterizations on the Pareto front (black dots) represent alternative non-dominated compromises. We have added these explanations in the caption of Fig. 6.

Some latest studies (Accurate Pseudospectral Optimization of Nonlinear Model Predictive Control for High­ performance Motion Planning. IEEE Transactions on Intelligent Vehicles, 2022, online, doi: 10.1109/TIV.2022.3153633) about NMPC based motion planning are suggested to be summarized in the introduction.

We have added the study and discussed it in the introduction.

Reviewer 2 Report

2022/12/13 (Mathematics)

Review Comments for Manuscript Number: mathematics-2121900-peer-review-v1

Multi-objective particle swarm optimization (MOPSO), a genetic algorithm (NSGA-II), and multiple versions of Bayesian optimization (BO) were studied in this research for autonomous vehicles' MPC. The authors target the prior algorithms considering the budget of manufacturing. This research is beneficial in case of a crash. I recommend this article after the following:

1. Please replace the word contribution in the abstract with other words such as research, paper, article, .…
2. It would be great if the authors add a figure that explains the MPC system and how it works. Add all of the illustrations to the figure.
3. With the currently available technology of AI and Machine Learning, cyber-attacks may happen easily. How will the current proposed system work in such a case?
4. The unit should be added between brackets in the figures.
5. Minor English revision is required. I don’t suggest repeating the same words in the manuscript such as numerous.
6. The manuscript is well-written and clear. The methodology is very clear as well. I have no questions about the mathematical models (algorithms). The references are up-to-date. Please consider the previous points when submitting your revised manuscript.

Author Response

Please replace the word contribution in the abstract with other words such as research, paper, article, .…

We have revised the abstract and replaced the word contribution.

It would be great if the authors add a figure that explains the MPC system and how it works. Add all of the illustrations to the figure.

We have added a figure explaining the MPC system used in the paper, see the new Fig. 2 of the revised manuscript.

With the currently available technology of AI and Machine Learning, cyber-attacks may happen easily. How will the current proposed system work in such a case?

Although we acknowledge that cyber-attacks are potentially a relevant topic for automated and autonomous driving, we do not see the need to deal with this issue in our current paper. The system and the control described in the paper work without information flow via information channels publicly accessible and therefore there is no outside interference. However, if the proposed system is implemented in vehicles with publicly accessible information channels, the same safety measure as for other relevant IT infrastructures are needed.  

The unit should be added between brackets in the figures.

We have added units with brackets.

Minor English revision is required. I don’t suggest repeating the same words in the manuscript such as numerous.

We have carefully checked the text and removed repeated wordings.

The manuscript is well-written and clear. The methodology is very clear as well. I have no questions about the mathematical models (algorithms). The references are up-to-date. Please consider the previous points when submitting your revised manuscript.

Thanks for your comments. We have duly taken into account all the questions you raised.

Round 2

Reviewer 1 Report

Though the author has addressed most of my concerns, I still have the following one:

1. Fig. 1 should be re-drawn to show the detail about the tuning framework. I cannot find any useful information from the current version.

Author Response

Fig. 1 should be re-drawn to show the detail about the tuning framework. I cannot find any useful information from the current version.

Actually, we are not quite sure what the Reviewer actually wants. The tuning framework has many details and we do not know that the Reviewer means with "the detail about the tuning framework."

Anyway, we have added that the MPC depicted in Fig. 1 is described in more detail in Fig. 2. In addition, we have described the elements of the Parameter optimizer (preprocessing, database and optimization) in more detail in Sec. 3.1. The additions are marked in blue in the manuscript.