__ Summary and Contributions__: This paper proposes the extension of safety barrier certificates to consider uncertainty in multi-robot systems, in particular they provide a system that satisfies collision-avoidance chance constraints.

__ Strengths__: The topic of the paper is quite relevant as safety barrier certificates have attracted quite a lot of attention from the control community and extending them the inherent uncertainty of robot systems is a worthwhile endeavour. Furthermore the formalisation looks elegant to me, even though I’m not an expert in the field of control.

__ Weaknesses__: The chance constraints are imposed on a per timestep basis, but there’s no notion of probability of collision for a full trajectory of the robots, nor is that discussed. I’d assume that with chance constraints lower than 1 one will be building systems that are surely brittle in the long run, as the probability of collision will tend to 1 as the number of conflicts between robots goes to infinity. Furthermore, the experimental section doesn't explicitly consider different levels of conservativeness, which could allow the reader to have some extra insight on the former point.
The paper is quite relevant to the control community, but I have some doubts on its relevance to the NeurIPS community. In particular, the proposed approach has no learning component at all and is very much pure control. I’d expect the control theory papers in NeurIPS to tackle problems like partially known dynamics for example, and this is nothing of that sort.

__ Correctness__: As far as I understand, the paper is correct.

__ Clarity__: Yes, the paper is well organised and written.

__ Relation to Prior Work__: As far as I am concerned the related work section looks good, but I’m completely unaware of the research on safety barrier certificates.

__ Reproducibility__: Yes

__ Additional Feedback__: Equations 2 and 3 are pretty much the same, so it’s a bit redundant writing both down.
After the rebuttal phase, I'm happy with the general response the authors gave regarding relevance to NeurIPS. However, given that most in the NeurIPS community will not be familiar with SBCs, I think that 1) the discussion given regarding integration with learnt models should be provided in the paper itself; and 2) more care should be given to the presentation of SBCs to ensure the paper is sufficiently self contained and there are no assumptions on prior knowledge on the topic. Given this, I keep my opinion that this is a very borderline paper.

__ Summary and Contributions__: This paper proposed PrSBC, which extended the safety barrier certification in probabilistic settings, and used it to guarantee safety (collision avoidance) for robotic applications. This method formulates the safety controller as a constrained QP, which minimally changes the nominal control input, and subject to linear constraints derived from PrSBC. The paper compared PrSBC with SBC in two simple simulated environments and showed no safety violations when the noises are present.

__ Strengths__: The paper has the two following strengths:
1) Since the real world is inherently stochastic, it is extremely important to reason about safety by taking into consideration noise and uncertainty. For this reason, extending a popular safety algorithm (SBC) to probabilistic settings is a significant contribution.
2) The proposed approach can be generalized to different forms of uncertainty. In theory, its derivation can work with any noise models with finite support.

__ Weaknesses__: In my personal opinion, the weaknesses of this paper are:
1) The algorithm is evaluated in two relatively simple scenarios. It is not clear to me the generality of the method. For example, can the proposed algorithm be applied to safety control problems in a higher dimensional space (e.g. self collision avoidance for a 7-dof robot arm) or with more realistic dynamics (e.g. heavy unicycle with low ground friction).
2) Parts of the paper is not crystal clear to me. Please see "Clarity" for more details.

__ Correctness__: The high-level derivations of the method seem reasonable. I did not carefully check the details or the proofs in the Appendix, though.

__ Clarity__: The paper is mostly well written. The accompanying video is helpful to visualize the results.
I have one major confusion though, which probably is due to my ignorance of the background of Safety Barrier Certificates. Since many readers of NeurIPS papers may also not be experts in this field, it would be great to give more detailed explanations that may eliminate this potential confusion:
I believe that the optimization (15) is to find a safe u for the current time step. This optimal u is not the entire trajectory over time. In the next time step, the QP is formed and solved again.
If this is the case, how long are the PrSBC constraints valid? In other words, if u belongs to S_u^\sigma, does it guarantee that collision will not happen in the next time step, or for the next few time steps, or forever? The discussion about "forward-invariant" in Section 4 suggests that the certificate is a subspace of all possible controls in the current time step that can guarantee no collision forever in the future (all t > 0). Does it mean that if I choose u \in S_u^\sigma, collision cannot happen in the future? This would be too good to be true. Otherwise, does it mean that if I choose u \in S_u^\sigma, collision cannot happen in the near future (e.g. next 10 time steps)? If so, what is the time horizon that safety is guaranteed? Which parameter controls this time horizon?

__ Relation to Prior Work__: The relation to prior wok is clearly and sufficiently discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: ----------------Post rebuttal comments-----------------
Thanks for the response. It addressed my questions.

__ Summary and Contributions__: The main contribution of this paper is to apply the Barrier Certificate formulation of safety constraint to the problem of multi-robot collision-free path planning via optimization methods. This is demonstrated using the example of multiple UAVs performing coordinated motion planning in a simulation environment.

__ Strengths__: The main feature of this paper is the application of the barrier certificates methodology in the context of multiple UAV path planning, taking into account uncertainty in perception and actions. So, the authors adopt a probabilistic version of barrier certificates, defining it in terms of level sets of confidence. So, the authors derive a chance constraint over pairs of controllers, in order to enforce minimum distance limits etc. Given a desired level of probability, this is then turned into a linear constraint in space which can then be enforced in control computation. This then facilitates optimization based trajectory generation, which is a contribution, and also decentralized schemes based on defining a common protocol for how different robots interact with each other.

__ Weaknesses__: While the methodology is overall quite sound, I am unsure about two main points:
(1) The overall approach is to gradually reduce the various forms of variability in the problem. So, for instance, we start with a chance constraint formulation which point-wise and pair-wise delineates the concerns of collision-avoidance. Then, we have a specific protocol for turning the problem into a decentralised form, etc. All this limits the expressivity of the overall framework. So, for instance, if the overall constraint were not just collision avoidance - say, we had a max restriction on numbers of agents allowed within a volume (timely, as I write this review!) - then it is not clear there is an easy adaptation. So, in this sense, the paper seems quite closely tied to the specific case of collision-avoidance rather than more general forms of safety. To give a different example, how would we compile the constraints similarly if the safety constraint were a temporal statement such as needing to visit a charging station at some point within every 5 min window? I would have appreciated a more detailed discussion of such issues around modelling.
(2) The literature on distributed control of robots (e.g., http://coordinationbook.info/) includes several methods for achieving spatial configurations that also admit dynamical systems analysis for convergence and correctness akin to the formalism here. Those authors would not have phrased their claims in the same way but I am sure the present authors can see that the bounds being used to define SBCs here are also the same as those known in dynamical systems. Indeed, even some of these concerns about decentralisation have been considered before, and the tasks shown in fig 2 are familiar in that setting. If so, how do the methods compare and to the sceptic how much of what is shown here is new? I am not necessarily claiming there isn't novelty but the paper would be stronger by clearly discussing the comparisons.

__ Correctness__: The arguments in this paper seem to me to be correct.

__ Clarity__: The paper is well written. I have offered comments above to suggest improvements, but I was able to follow the arguments reasonably well.

__ Relation to Prior Work__: I believe the authors have appropriately cited related work. I have made a few suggestions above for the authors to consider.

__ Reproducibility__: Yes

__ Additional Feedback__: [Post-author-response comments] Thanks for addressing some of my questions, and I hope this will be expanded upon within the main text of the paper.

__ Summary and Contributions__: The paper contributes by presenting an approach for the collision avoidance problem in multi-robot systems, which considers the uncertainty in sensors’ measurements and the inaccuracy of the robots’ model to provide guarantees on safety when computing motion controls for the robots.
The problem is well-defined and formally presented. The Related Work section shows relevant and recent works on collision avoidance for multi-robot systems.
Safety is defined as a function of distance between pairs of robots. Then, the probability that the robots maintain this safety distance is defined to be above a certain level σ (this is a parameter given by the end user). By following the control barrier functions formulation, in a previous work the authors presented the Safety Barrier Certificates, defined as the set of admissible controls for the robots, computed at each time step, which are collision-free at all times. From this formulation, the authors present the Probabilistic Safety Barrier Certificates, to compute the set of admissible controls that ensure the probability of the robots to avoid collision is above or equal to σ. This formulation translates into defining, for each robot, the intersection of half-spaces between pairs of robots and robot-obstacles in the joint control space. The approach is theoretically proven. From this resultant set of admissible controls, an optimization problem is solved to compute, for each robot, the nearest control from a nominal control (it is assumed the robots are moving towards their goal by following a task-related control). Both a centralized and decentralized versions are presented. In the latter, the effort that each robot makes to avoid the collision has to be considered, which translates into restricting the set of admissible controls. The evaluation of the system demonstrates the suitability of the approach.

__ Strengths__: Theoretical validation of the approach.
Thorough evaluation through simulation of the solution proposed.

__ Weaknesses__: The authors could elaborate more on the probabilistic model used for evaluation.

__ Correctness__: Correctness is observed in the claims, methodology and evaluation.

__ Clarity__: The paper is very well written, and easy to understand. Though there are few grammatical errors to amend.

__ Relation to Prior Work__: The paper extends on previous work, though the authors clearly differentiate the contribution with respect to the previous work.

__ Reproducibility__: Yes

__ Additional Feedback__: The rebuttal has been taking into account. We are happy in general with the information that was provided. We agree with other reviewers that a more extensive simulation would further strengthen the work.
Minor issues to be amended:
- Sometimes the space of control is indicated as R^d (in the main paper) and R^m (in the appendix and paper), which is the correct one?
- Missing “/” in formula 16, the second part: “eij / γ · uj ≤ pji(pij + pji) · bσij ”, should be “eij / γ · uj ≤ pji / (pij + pji) · bσij ”
- In line 248, “∀ j > i” should be “∀ i > j”, for consistency with the rest of formulation in the paper.
- In the video, in the example “2 unicycle robots swap positions”, the size of the box for robot 1 is bigger than the box for robot 2. Is this because the error is greater? How you select the error distribution for each of the robots?