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If G is a finite connected graph, then an arithmetical structure on $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries such that $(\diag(\mathbf{d}) - A)\cdot \mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and the entries of $\mathbf{r}$ have no common factor other than $1$. In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, Garc\'ia Puente, Glass, and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs $\CT{p}{2}$) to all coconut tree graphs $\CT{p}{s}$ which consists of a path on $p>0$ vertices to which we append $s>0$ leaves to the right most vertex on the path. We also give a characterization of smooth arithmetical structures on coconut trees when given number assignments to the leaf nodes., Comment: 18 pages, 9 figures, comments are welcomed

For dynamical systems involving decision making, the success of the system greatly depends on its ability to make good decisions with incomplete and uncertain information. By leveraging the Koopman operator and its adjoint property, we introduce the Koopman Expectation, an efficient method for computing expectations as propagated through a dynamical system. Unlike other Koopman operator-based approaches in the literature, this is possible without an explicit representation of the Koopman operator. Furthermore, the efficiencies enabled by the Koopman Expectation are leveraged for optimization under uncertainty when expected losses and constraints are considered. We show how the Koopman Expectation is applicable to discrete, continuous, and hybrid non-linear systems driven by process noise with non-Gaussian initial condition and parametric uncertainties. We finish by demonstrating a 1700x acceleration for calculating probabilistic quantities of a hybrid dynamical system over the naive Monte Carlo approach with many orders of magnitudes improvement in accuracy., Comment: 7 figures, 2 tables

In this paper, we present an online adaptive robust control framework for underactuated brachiating robots traversing flexible cables. Since the dynamic model of a flexible body is unknown in practice, we propose an indirect adaptive estimation scheme to approximate the unknown dynamic effects of the flexible cable as an external force with parametric uncertainties. A boundary layer-based sliding mode control is then designed to compensate for the residual unmodeled dynamics and time-varying disturbances, in which the control gain is updated by an auxiliary direct adaptive control mechanism. Stability analysis and derivation of adaptation laws are carried out through a Lyapunov approach, which formally guarantees the stability and tracking performance of the robot-cable system. Simulation experiments and comparison with a baseline controller show that the combined direct-indirect adaptive robust control framework achieves reliable tracking performance and adaptive system identification, enabling the robot to traverse flexible cables in the presence of unmodeled dynamics, parametric uncertainties and unstructured disturbances., Comment: 8 pages, 8 figures, 2020 IEEE Conference on Decision and Control (CDC)

Control of wire-borne underactuated brachiating robots requires a robust feedback control design that can deal with dynamic uncertainties, actuator constraints and unmeasurable states. In this paper, we develop a robust feedback control for brachiating on flexible cables, building on previous work on optimal trajectory generation and time-varying LQR controller design. We propose a novel simplified model for approximation of the flexible cable dynamics, which enables inclusion of parametric model uncertainties in the system. We then use semidefinite programming (SDP) and sum-of-squares (SOS) optimization to synthesize a time-varying feedback control with formal robustness guarantees to account for model uncertainties and unmeasurable states in the system. Through simulation, hardware experiments and comparison with a time-varying LQR controller, it is shown that the proposed robust controller results in relatively large robust backward reachable sets and is able to reliably track a pre-generated optimal trajectory and achieve the desired brachiating motion in the presence of parametric model uncertainties, actuator limits, and unobservable states., Comment: 8 pages, 12 figures, 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

Motivated towards achieving multi-modal locomotion, in this paper, we develop a framework for a bipedal robot to dynamically ride a pair of Hovershoes over various terrain. Our developed control strategy enables the Cassie bipedal robot to interact with the Hovershoes to balance, regulate forward and rotational velocities, achieve fast turns, and move over flat terrain, slopes, stairs, and rough outdoor terrain. Our sensor suite comprising of tracking and depth cameras for visual SLAM as well as our Dijkstra-based global planner and timed elastic band-based local planning framework enables us to achieve autonomous riding on the Hovershoes while navigating an obstacle course. We present numerical and experimental validations of our work.

The active galactic nuclei (AGN) lobe radio luminosities modelled in hydrodynamical simulations and most analytical models do not address the redistribution of the electron energies due to adiabatic expansion, synchrotron radiation and inverse-Compton scattering of CMB photons. We present a synchrotron emissivity model for resolved sources which includes a full treatment of the loss mechanisms spatially across the lobe, and apply it to a dynamical radio source model with known pressure and volume expansion rates. The bulk flow and dispersion of discrete electron packets is represented by tracer fields in hydrodynamical simulations; we show that the mixing of different aged electrons strongly effects the spectrum at each point of the radio map in high-powered FR-II sources. The inclusion of this mixing leads to a factor of a few discrepancy between the spectral age measured using impulsive injection models (e.g. JP model) and the dynamical age. The observable properties of radio sources are predicted to be strongly frequency dependent: FR-II lobes are expected to appear more elongated at higher frequencies, while jetted FR-I sources appear less extended. The emerging FR0 class of radio sources, comprising gigahertz peaked and compact steep spectrum sources, can potentially be explained by a population of low-powered FR-Is. The extended emission from such sources is shown to be undetectable for objects within a few orders of magnitude of the survey detection limit and to not contribute to the curvature of the radio SED., Comment: 20 pages, 17 figures, 1 table; accepted in MNRAS

We examine the relationship between source position stability and astrophysical properties of radio-loud quasars making up the International Celestial Reference Frame. Understanding this relationship is important for improving quasar selection and analysis strategies, and therefore reference frame stability. We construct light curves for 95 of the most frequently observed ICRF2 quasars at both the 2.3 and 8.4 GHz geodetic VLBI observing bands. Because the appearance of new quasar components corresponds to an increase in quasar flux density, these light curves alert us to potential changes in source structure before they appear in VLBI images. We test how source position stability depends on three astrophysical parameters: (1) Flux density variability at X-band; (2) Time lag between flares in S and X-bands; (3) Spectral index rms, defined as the variability in the ratio between S and X-band flux densities. We find that small time lags between S and X-band light curves, and low spectral index variability, are good indicators of position stability. On the other hand, there is no strong dependence of source position stability on flux density variability in a single frequency band. These findings can be understood by interpreting the time lag between S and X-band light curves as a measure of the size of the source structure. Monitoring of source flux density at multiple frequencies therefore appears to provide a useful probe of quasar structure on scales important to geodesy. We show how multi-frequency flux density monitoring may allow the dependence on frequency of the relative core positions along the jet to be elucidated. Knowledge of the position-frequency relation has important implications for current and future geodetic VLBI programs, as well as the alignment between the radio and optical celestial reference frames. (Abridged), Comment: 13 pages, 7 figures, 1 table. Changed to match version accepted by Journal of Geodesy