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A symmetry-preserving, reduced-order state observer is presented for the unmeasured part of a system's state, where the nonlinear system dynamics exhibit symmetry under the action of a Lie group. The proposed observer takes advantage of this symmetry through the use of a moving frame that constructs invariant mappings of the measurements. Sufficient conditions for the observer to be asymptotically stable are developed by studying the stability of an invariant error system. As an illustrative example, the observer is applied to the problem of rigid-body velocity estimation, which demonstrate how exploiting the symmetry of the system can simplify the stabilization of the estimation error dynamics., Comment: 6 pages, 6 figures, Submission to the 2025 American Control Conference

This paper extends algorithms that remove the fixed point bias of decentralized gradient descent to solve the more general problem of distributed optimization over subspace constraints. Leveraging the integral quadratic constraint framework, we analyze the performance of these generalized algorithms in terms of worst-case robustness and convergence rate. The utility of our framework is demonstrated by showing how one of the extended algorithms, originally designed for consensus, is now able to solve a multitask inference problem.

In this paper, we present two Hermite polynomial based approaches to derive one-step numerical integrators for mechanical systems. These methods are based on discretizing the configuration using Hermite polynomials which leads to numerical trajectories continuous in both configuration and velocity. First, we incorporate Hermite polynomials for time-discretization and derive one-step variational methods by discretizing the Lagrange-d'Alembert principle over a single time step. Second, we present the Galerkin approach to derive one-step numerical integrators by setting the weighted average of the residual of the equations of motion over a time step to zero. We consider three numerical examples to understand the numerical performance of the one-step variational and Galerkin methods. We first study a particle in a double-well potential and compare the variational approach results with the corresponding results for the Galerkin approach. We then study the Duffing oscillator to understand the numerical behavior in presence of dissipative forces. Finally, we apply the proposed methods to a nonlinear aeroelastic system with two degrees of freedom. Both variational and Galerkin one-step methods capture conservative and nonconservative dynamics accurately with excellent energy behavior. The one-step Galerkin methods exhibit better trajectory and energy performance than the one-step variational methods and the variational integrators.

A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive time-step variational integrators that conserve the energy in addition to being symplectic and momentum-preserving. Their utility, however, is still an open question due to the numerical difficulties associated with solving the discrete governing equations. In this work, we investigate the numerical performance of energy-preserving, adaptive time-step variational integrators. First, we compare the time adaptation and energy performance of the energy-preserving adaptive algorithm with the adaptive variational integrator for Kepler's two-body problem. Second, we apply tools from Lagrangian backward error analysis to investigate numerical stability of the energy-preserving adaptive algorithm. Finally, we consider a simple mechanical system example to illustrate the backward stability of this energy-preserving, adaptive time-step variational integrator.

We present a model-based approach to wind velocity profiling using motion perturbations of a multirotor unmanned aircraft system (UAS) in both hovering and steady ascending flight. A state estimation framework was adapted to a set of closed-loop rigid body models identified for an off-the-shelf quadrotor. The quadrotor models used for wind estimation were characterized for hovering and steady ascending flight conditions ranging between 0 and 2 m/s. The closed-loop models were obtained using system identification algorithms to determine model structures and estimate model parameters. The wind measurement method was validated experimentally above the Virginia Tech Kentland Experimental Aircraft Systems Laboratory by comparing quadrotor and independent sensor measurements from a sonic anemometer and two SoDARs. Comparison results demonstrated quadrotor wind estimation in close agreement with the independent wind velocity measurements. Wind velocity profiles were difficult to validate using time-synchronized SoDAR measurements, however. Analysis of the noise intensity and signal-to-noise ratio of the SoDARs proved that close-proximity quadrotor operations can corrupt wind measurement from SoDARs., Comment: 30 pages, 19 figures

The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d'Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider two numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. We show a trade-off between energy-preserving performance and accurate discrete trajectories while choosing an initial time step. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step increases through a condition number analysis. As a second example, we numerically simulate a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.