In general I believe this is a nice paper that is well-written. I worry that there is not a lot of novelty, and the parts that are new to me (using a function approximation over different bounds on the disturbance) are not rigorous enough to guarantee safety. I think this paper would be better suited for a conference like ICRA or IROS. Notes on related work: Contrary to the authors’ assertion, I believe that guaranteed following of a path is studied quite a bit, often by using an adversarial game formulation similar to the one proposed in this paper. Some examples: Majumdar, Anirudha, and Russ Tedrake. "Funnel libraries for real-time robust feedback motion planning." The International Journal of Robotics Research 36.8 (2017): 947-982. Herbert, Sylvia L., et al. "FaSTrack: A modular framework for fast and guaranteed safe motion planning." 2017 IEEE 56th Annual Conference on Decision and Control (CDC). IEEE, 2017. Singh, Sumeet, et al. "Robust online motion planning via contraction theory and convex optimization." 2017 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2017. Smith, Stanley W., He Yin, and Murat Arcak. "Continuous abstraction of nonlinear systems using sum-of-squares programming." 2019 IEEE International Conference on Decision and Control (CDC). IEEE, 2019. Questions - I am unclear what the general assumptions on the dynamics might be. In this paper a discrete bicycle model is used with steering and acceleration inputs Why is the information pattern giving an advantage to the control player, rather than the disturbance? - In (7), are assuming we have a discrete set of control and disturbance actions? If so, how finely discretized are they? How does this affect the computation of the set? Later you say that they are discretized. Can you comment more on how the discretization affects the computation? - How did you choose the discretization of the state space? What impact does that have on the computation? - Why did the computation times vary so much? - Do you have any theory on if the sets will be guaranteed to be continuous across a continuous range of k_max? This is an assumption that seems to be made - I don’t see how the neural net implementation is guaranteed to not have false positives (where I define false positives as assuming a state is safe when it is not). In fact, in the tests each method did in fact have false positives. Given that this paper’s emphasis is on safety and recursive feasibility, I think this should be addressed.

This paper is well-written and introduces a novel approach to address the design of the terminal constraint for nonlinear MPC. The application considered in this paper is path following. While the method and result presented in this paper are all convincing, there are some suggestion to improve the paper. #1 Presentation of the results. I would suggest to make the subplots in Fig. 5 and 6 landscape instead of portrait. Currently, it is very hard to see the details in a printed version. #2 What is the difference between the mixed dynamic-kinematic model and the model in (1)? Is the purpose here to test the robustness of the method on a mismatched model? #3 Need to add unit to Table VI. #4 Extendability of the method to other dynamic models and under collision avoidance constraint?

The most original part is the problem formulation of safe motion planing in autonomous driving as a game between the motion planner, which aims to follow the road unknown ahead, and the road as adversarial player with bounded curvature, which aims to get the car off road. Under this formulation, the paper employs viability theory and game theory to design a motion planner to follow a path with safety guarantee. Both analysis and MPC simulations are provided to validate the proposed algorithm. Overall, it is a nice contribution to safe planning in autonomous driving.