Topology of Vibro-Impact Systems in the Neighborhood of Grazing

The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors may be invisible or, at least, very sensitive to changes of parameters of the system. On the other hand, they are observed in experiments and numerical simulations. We offer (Theorem 2) an approach which allows to explain this contradiction and give a new robust mathematical model of the non-hyperbolic dynamics in the neighborhood of grazing.Comment: Submitted to Physica

Prediction of stable walking for a toy that cannot stand

Previous experiments [M. J. Coleman and A. Ruina, Phys. Rev. Lett. 80, 3658 (1998)] showed that a gravity-powered toy with no control and which has no statically stable near-standing configurations can walk stably. We show here that a simple rigid-body statically-unstable mathematical model based loosely on the physical toy can predict stable limit-cycle walking motions. These calculations add to the repertoire of rigid-body mechanism behaviors as well as further implicating passive-dynamics as a possible contributor to stability of animal motions.Comment: Note: only corrections so far have been fixing typo's in these comments. 3 pages, 2 eps figures, uses epsf.tex, revtex.sty, amsfonts.sty, aps.sty, aps10.sty, prabib.sty; Accepted for publication in Phys. Rev. E. 4/9/2001 ; information about Andy Ruina's lab (including Coleman's, Garcia's and Ruina's other publications and associated video clips) can be found at: http://www.tam.cornell.edu/~ruina/hplab/index.html and more about Georg Bock's Simulation Group with whom Katja Mombaur is affiliated can be found at http://www.iwr.uni-heidelberg.de/~agboc

Homoclinic crossing in open systems: Chaos in periodically perturbed monopole plus quadrupolelike potentials

The Melnikov method is applied to periodically perturbed open systems modeled by an inverse--square--law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced. The (modified) Smale-Birkhoff homoclinic theorem is used to study transversal homoclinic intersections. A larger class of open systems with degenerated (nonhyperbolic) unstable periodic orbits after regularization is also briefly considered.Comment: 19 pages, 15 figures, Revtex

Sensitivity Analysis for Periodic Orbits and Quasiperiodic Invariant Tori Using the Adjoint Method

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation.Comment: revised manuscript, source code for demonstrations available at doi:10.6084/m9.figshare.19252055 and github.com/jansieber/adjoint-sensitivity2022-sup

Model-free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design

Switching adaptive control of a bioassistive exoskeleton

The effectiveness of existing control designs for bioassistive, exoskeletal devices, especially in highly uncertain working environments, depends on the degree of certainty associated with the overall system model. Of particular concern is the robustness of a control design to large-bandwidth exogenous disturbances, time delays in the sensor and actuator loops, and kinematic and inertial variability across the population of likely users. In this study, we propose an adaptive control framework for robotic exoskeletons that uses a low-pass filter structure in the feedback channel to decouple the estimation loop from the control loop. The design facilitates a significant increase in the rate of estimation and adaptation, without a corresponding loss of robustness. In particular, the control implementation is tolerant of time delays in the control loop and maintains clean control channels even in the presence of measurement noise. Tuning of the filter also allows for shaping the nominal response and enhancing the time-delay margin. Importantly, the proposed formulation is independent of detailed model information. The performance of the proposed architecture is demonstrated in simulation for two basic control scenarios, namely, (i) static positioning, for which the predefined desired joint motions are constant; and (ii) command following, where the desired motions are not known a priori and instead inferred using interaction measurements. We consider, in addition, an operating modality in which the control scheme switches between static positioning and command following to facilitate flexible integration of a human operator in the loop. Here, the transition from static positioning to command following is triggered when either the human–machine interaction force at the wrist or the end-effector velocity exceeds the corresponding critical value. The controller switches from command following back to static positioning when both the interaction force and the velocity fall below the corresponding thresholds. This strategy allows for smooth transition between two phases of operation and provides an alternative to an implementation relying on wearable electromyographic sensors