Our responses to the main concerns are given as follows. Section 4.4 for a related discussion and generalizations to non-unit norms. We would be happy to move some of the less central corollaries (e.g., Sections 4.2 and 4.5) to the We will also correct the typo in Line 202.

Legged robots offer several advantages when navigating unstructured environments, but they often fall short of the efficiency achieved by wheeled robots. One promising strategy to improve their energy economy is to leverage their natural (unactuated) dynamics using elastic elements. This work explores that concept by designing energy-optimal control inputs through a unified, multi-stage framework. It starts with a novel energy injection technique to identify passive motion patterns by harnessing the system's natural dynamics. This enables the discovery of passive solutions even in systems with energy dissipation caused by factors such as friction or plastic collisions. Building on these passive solutions, we then employ a continuation approach to derive energy-optimal control inputs for the fully actuated, dissipative robotic system. The method is tested on simulated models to demonstrate its applicability in both single- and multi-legged robotic systems. This analysis provides valuable insights into the design and operation of elastic legged robots, offering pathways to improve their efficiency and adaptability by exploiting the natural system dynamics.

We thank the reviewers for their valuable comments. We reply to each outstanding point below. The scaling properties of our method depend on the specific setting considered. Thanks also for the additional reference, which we will add to the paper. I expected more meta-learning insights.

We will incorporate this feedback.

Assume the conditions in Proposition 1 hold.

We slightly improve the statement regarding the cluster-wise augmentations according to the reviewers' responses.

Regarding the minor clarity issues, we will adjust Figure 1 according to these suggestions and fix the typos stated. If we understand correctly, the reviewer's main concerns are that the numerical results are not comprehensive. We compared COMP/DD/SSS/LP experimentally because these all use the same test matrix (i.i.d. Reviewer 3's suggestions, we believe it would belong in the supplementary material and not the main body.