Co-design is a high-dimensional search problem in the robot morphology and control design space. Efficient search requires exploiting the structure shaped by their interaction. To understand this structure, we analyze the landscapes of soft locomotion and manipulation tasks. We identify three patterns consistent across regions of their co-design spaces: 1) Within a region, quality varies along a low-dimensional manifold, with minimal variation orthogonal to it, reducing the effective search space dimensionality. 2) In higher-quality regions, the variance in quality is spread across more dimensions, necessitating search to expand dimensionality as quality improves. 3) In higher-quality regions, quality varies along joint morphology-control dimensions, requiring search along them. Using these insights, we devise an efficient co-design algorithm that yields 36% better co-designs than state-of-the-art baselines. We examine their exploration patterns and show that these baselines required an order of magnitude more function evaluations to find co-designs of comparable quality. Finally, we ablate our algorithm to verify that exploiting the identified structure was the key to efficient co-design.

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