Robust Adaptive Control Barrier Functions: An Adaptive & Data-Driven Approach to Safety (Extended Version)

A new framework is developed for control of constrained nonlinear systems with structured parametric uncertainties. Forward invariance of a safe set is achieved through online parameter adaptation and data-driven model estimation. The new adaptive data-driven safety paradigm is merged with a recent adaptive control algorithm for systems nominally contracting in closed-loop. This unification is more general than other safety controllers as closed-loop contraction does not require the system be invertible or in a particular form. Additionally, the approach is less expensive than nonlinear model predictive control as it does not require a full desired trajectory, but rather only a desired terminal state. The approach is illustrated on the pitch dynamics of an aircraft with uncertain nonlinear aerodynamics.Comment: Added aCBF non-Lipschitz example and discussion on approach implementatio

Money and Goldstone modes

Why is ``worthless'' fiat money generally accepted as payment for goods and services? In equilibrium theory, the value of money is generally not determined: the number of equations is one less than the number of unknowns, so only relative prices are determined. In the language of mathematics, the equations are ``homogeneous of order one''. Using the language of physics, this represents a continuous ``Goldstone'' symmetry. However, the continuous symmetry is often broken by the dynamics of the system, thus fixing the value of the otherwise undetermined variable. In economics, the value of money is a strategic variable which each agent must determine at each transaction by estimating the effect of future interactions with other agents. This idea is illustrated by a simple network model of monopolistic vendors and buyers, with bounded rationality. We submit that dynamical, spontaneous symmetry breaking is the fundamental principle for fixing the value of money. Perhaps the continuous symmetry representing the lack of restoring force is also the fundamental reason for large fluctuations in stock markets.Comment: 7 pages, 3 figure

Ladder operators and endomorphisms in combinatorial Physics

Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we first show how the ordering of the non-commuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but row-finite, matrices, which may also be considered as endomorphisms of C[x]. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladder-operators familiar in physics

Robust incremental SLAM with consistency-checking

Incorrect landmark and loop closure measurements can cause standard SLAM algorithms to fail catastrophically. Recently, several SLAM algorithms have been proposed that are robust to loop closure errors, but it is shown in this paper that they cannot provide robust solutions when landmark measurement errors occur. The root cause of this problem is that the robust SLAM algorithms only focus on generating solutions that are locally consistent (i.e. each measurement agrees with its corresponding estimates) rather than globally consistent (i.e. all of the measurements in the solution agree with each other). Moreover, these algorithms do not attempt to maximize the number of correct measurements included in the solution, meaning that often correct measurements are ignored and the solution quality suffers as a result. This paper proposes a new formulation of the robust SLAM problem that seeks a globally consistent map that also maximizes the number of measurements included in the solution. In addition, a novel incremental SLAM algorithm, called incremental SLAM with consistency-checking, is developed to solve the new robust SLAM problem. Finally, simulated and experimental results show that the new algorithm significantly outperforms state-of-the-art robust SLAM methods for datasets with incorrect landmark measurements and can match their performance for datasets with incorrect loop closures.Charles Stark Draper Laboratory. Internal Research and Development Progra

S-matrix approach to quantum gases in the unitary limit II: the three-dimensional case

A new analytic treatment of three-dimensional homogeneous Bose and Fermi gases in the unitary limit of negative infinite scattering length is presented, based on the S-matrix approach to statistical mechanics we recently developed. The unitary limit occurs at a fixed point of the renormalization group with dynamical exponent z=2 where the S-matrix equals -1. For fermions we find T_c /T_F is approximately 0.1. For bosons we present evidence that the gas does not collapse, but rather has a critical point that is a strongly interacting form of Bose-Einstein condensation. This bosonic critical point occurs at n lambda^3 approximately 1.3 where n is the density and lambda the thermal wavelength, which is lower than the ideal gas value of 2.61.Comment: 26 pages, 16 figure

On the growth of Boletus elegans Schum in pure culture

The thesis consists of two parts; the first is concerned with the morphological characters of Boletus elegans in pure culture and contains a description of the mycelium of the fungus which is compared with the descriptions of the mycelia of other Boletus spp. given by previous workers. The second and larger part deals with certain aspects of the physiology of B elegans. Chief consideration is given to the effect of the pH value of the medium on the growth of the fungus. The pH range is found to differ significantly on silica gels and gelatin media and an attempt is made to elucidate the phenomena shown in the gelatin media. In connection with the investigation into the relation "between pH value and growth, attention is drawn to the influence of staling products present in the inocula on the rate of growth of the cultures derived from them and to the effect upon growth of the molecular concentration of the medium. A brief enquiry into the nitrogen diet of B. elegans is also included.<p

Cooperation in an Uncertain and Dynamic World

We investigate how reputational uncertainty and the rate of change of the social environment interact to influence cooperation in social networks. Reputational uncertainty significantly decreases cooperation and welfare, induces more forgiveness toward defectors, and promotes opportunistic play. Compared to reputational uncertainty, a fast-changing social environment only causes a second-order qualitative increase in cooperation by making individuals more lenient in imposing a network-punishment (link removal). The interaction between reputational uncertainty and a fast-changing social environment induces more lenient strategies by reducing the frequency of action-punishment (retaliatory defection). Although neither of them affects the aggregate network metrics, their interaction decreases homophily among cooperators

Cooperation in an Uncertain and Dynamic World

We investigate how reputational uncertainty and the rate of change of the social environment interact to influence cooperation in social networks. Reputational uncertainty significantly decreases cooperation and welfare, induces more forgiveness toward defectors, and promotes opportunistic play. Compared to reputational uncertainty, a fast-changing social environment only causes a second-order qualitative increase in cooperation by making individuals more lenient in imposing a network-punishment (link removal). The interaction between reputational uncertainty and a fast-changing social environment induces more lenient strategies by reducing the frequency of action-punishment (retaliatory defection). Although neither of them affects the aggregate network metrics, their interaction decreases homophily among cooperators