The authors describe and implement a gait for sidewinding snake robots to enhance directional stability by improving static stability with vertical wave modulation. Gaits are tested over a variety of spatial frequency ratios and temporal frequencies, in two different robots, and the results evaluated in terms of open-loop directional stability. The gaits are simulated, and the model is used to evaluate stability, with predictions compared to the experimental results. Trial counts should be included in figures or their captions (figs 5-7). It would also be helpful to have a visual indication of which robot (or simulation) is being used plotted in each of the figures. How certain are the authors that the static stability accounts for all of the locomotion discrepancies? It would be helpful to have a statistical measure of the correlation (e.g. between the two fig 4 panels), rather than a qualitative inference. It would also improve confidence to address/eliminate other possible experimental/simulation fidelity effects, for example: - How well does the actual gait of the robot follow the prescribed gait in terms of joint angles at different speeds/wavelengths (especially important given the use of a series elastic actuator in one system)? - The model omits inertia but the authors acknowledge its importance, can a sense of the size of the inertial forces vs. temporal frequency be given? - How well is friction predicted? The robophysical model looks to have a variety of contact surfaces. These sorts of things would be useful to have information on when evaluating the data. The only performance measure the authors consider is directional stability, I think expanding the evaluation could be beneficial. Things such as joint work / cost of transport? The authors could also look at robustness more in the context of external perturbations - what does the robot performance look like on uneven hard ground, or over a small obstacle? The definition of robustness as directional stability is a little narrow to me. "...careful manipulation and protection of motor modules are required." Some elaboration on this point would be good, i.e. what is the specific deficiency of the dynamixel robot? Why could the more sophisticated robot not be used for all trials? The authors should show a pair of equivalent trials with both robots side by side so how comparable they are is clear This is a semantic point, but the distinction the authors are making between robotic and robophysical systems is not clear to me. To my mind the difference is mainly to due with the intended purpose (i.e. applied robotics vs. physical modelling). Here the robots are used in the same way and to the same purpose. The paper is well written and easy to follow. This is very minor but the paper would read more comfortably if the figures were better positioned relative to their references in the text, particularly the results plots.

This paper does a good job to present the problem being solved, the current state of work in the field, as well as the contributions made with this work. Limbless, snake-like robots currently seem to suffer from undesireable motions. The authors use geometric mechanics to stabilize the robot for controlled, desirable lateral and rotational movements, which they validate on 2 separate robotic systems. The paper is well-written and easy to follow. Section 2 presents a succinct description of sidewinder locomotion and the geometric mechanics that are used throughout the rest of the paper to analyze and control the system, which is important for the clarity of the paper and in particular for readers that may not be as familiar with the background work. Figures clearly convey the methods used and their effects on the system. However, while stated in the Results section and shown in the accompanying video, the final results showing the dependency of the frequency on the gait performance could be shown through a series of time-lapse images in the paper directly. There is plenty of space to do this and would be a good addition to make the paper more self contained and not rely on the video to demonstrate the success of the robot. A study into the frequency of contact sequences to take advantage of both static and dynamic stability is significant in that it can be extended to other systems that use similar principles for control. For example, the same principles are present in legged systems such as bipeds and quadrupeds where gaits are rarely statically stable and require a minimum contact switching frequency for underactuated control modes in order to prevent it from falling over. Work looking at the concept of "dynamic stability" is important and methods to analyze systems that can move in a controlled manner even during temporarily underactuated systems is necessary to create robots capable of being controlled while executing dynamic maneuvers. This work addresses both high speed movements, as well as stable low speed movements.