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Information theory is used to design robots with guaranteed arrival over noisy terrain. 

Contact planning is crucial to the locomotion performance of robots: to properly self-propel forward, it is not only important to determine the sequence of internal shape changes (e.g., body bending and limb shoulder joint oscillation) but also the sequence by which contact is made and broken between the mechanism and its environment. Prior work observed that properly coupling contact patterns and shape changes allows for computationally tractable gait design and efficient gait performance. The state of the art, however, made assumptions, albeit motivated by biological observation, as to how contact and shape changes can be coupled. In this paper, we extend the geometric mechanics (GM) framework to design contact patterns. Specifically, we introduce the concept of “contact space” to the GM framework. By establishing the connection between velocities in shape and position spaces, we can estimate the benefits of each contact pattern change and therefore optimize the sequence of contact patterns. In doing so, we can also analyze how a contact pattern sequence will respond to perturbations. We apply our framework to sidewinding robots and enable (1) effective locomotion direction control and (2) robust locomotion performance as the spatial resolution decreases. We also apply our framework to a hexapod robot with two back-bending joints and show that we can simplify existing hexapod gaits by properly reducing the number of contact state switches (during a gait cycle) without significant loss of locomotion speed. We test our designed gaits with robophysical experiments, and we obtain good agreement between theory and experiments. 

Abstract A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment. 

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its “combinatorial shadow”. We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities.