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This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent $${\mathscr {L}}_1$$ adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.

SUMMARYThis paper reports on laboratory and field experimental results for controlled robotic manipulators operating on moving platforms with unmodeled dynamics. The aim is to validate theoretical predictions for the dependence on control parameters of an adaptive control strategy. In addition, the results provide insight into different discretizations of the continuous-time formulation, suggesting the most suitable discretization scheme for hardware implementation. The second set of experimental results, obtained from an implementation of the control framework for synchronization and consensus in networks of robotic manipulators, similarly validate theoretical predictions on the sensitivity to network communication delays.

This article proposes a methodology for integrating adaptive control with the control-based continuation paradigm for a class of uncertain, linear, discrete-time systems. The proposed adaptive control strategies aim to stabilize the closed-loop dynamics with convergence toward a known reference input, such that the dynamics approach the open-loop fixed point if the reference input is chosen to make the steady-state control input equal 0. This enables the tracking of a parameterized branch of open-loop fixed points using methods of numerical continuation without specific knowledge about the system. We implement two different adaptive control strategies: model-reference adaptive control and pole-placement adaptive control. Both implementations achieve the desired objectives for the closed-loop dynamics and support parameter continuation. These properties, as well as the boundedness of system states and control inputs, are guaranteed provided that certain stability conditions are satisfied. Besides, the tuning effort is significantly reduced in the adaptive control schemes compared with traditional proportional–derivative controllers and linear state-space feedback controllers.

This paper studies the delay robustness of an $\mathcal {L}_1$ adaptive controller designed for systems with unknown matched nonlinearities and unknown input-gain matrices. The analysis establishes rigorously the existence of a positive lower bound for the closed-loop system's time-delay margin (TDM), provided that a filter bandwidth and an adaptive gain are chosen sufficiently large. In this case, if the input delay is below a critical value, then the state and control input of the control system follow those of a nonadaptive, robust reference system closely. The analysis also suggests a way to estimate this lower bound for the delay robustness using Pade approximants. Results from forward simulation are consistent with the Pade estimate and with an explicit upper bound on the TDM which decays to 0 as the filter bandwidth grows without bound.

In rigid-body mechanics, models that capture collisional contact as an instantaneous exchange of momentum may exhibit dynamics that include infinite sequences of impacts accumulating in finite time to a state of persistent contact, often referred to as chatter. In this paper, we review theoretical tools for the analysis of transient and steady-state behavior in the vicinity of critical periodic orbits for which chatter terminates at a point corresponding to the imminent release from persistent contact, and illustrate the application of this theory to a simplified model of a mechanical pressure relief valve. A general theory for single-degree-of-freedom impact oscillators, previously described in an unpublished manuscript by Nordmark and Kisitu1, is shown to yield both qualitative and quantitative agreement with model simulation results. The predicted bifurcation structure shows that the border orbit unfolds supercritically into a universal cascade of local attractors with nontrivial scaling relationships.

This paper aims to analyze the emergence of task differentiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coordinated behavior and collective function. The analysis focuses on linear network filters, i.e., networks of coupled linear oscillators with a differentiated steady-state response to exogenous harmonic excitation. It demonstrates how an optimal allocation of excitation sensitivities across the network nodes in a condition of resonance may be constructed either using global information about the network topology and spectral properties or through the iterated dynamics of a nonlinear, nonsmooth learning paradigm that only relies on local information within the network. Explicit conditions on the topology and desired resonant mode shape are derived to guarantee local asymptotic stability of fixed points of the learning dynamics. The analysis demonstrates the possibly semistable nature of the fixed point with all zero excitation sensitivities, ...

Stability analysis of an L1 adaptive control design decomposes into i) the derivation of an inequality condition imposed on a filter in the control channel that guarantees transient performance bounds on the state and control input for a theoretical nonadaptive reference system, and ii) a Lyapunov-type proof of the decay of a predictor error as the inverse square root of the adaptive gain, which in turn can be made arbitrarily large without sacrificing robustness to time delay in the control input. This paper reviews recent results on the robustness to input and communication time delays for L1 control systems with uncertain nonlinearities and in nontrivial network configurations, as well as a possible scheme for compensating for large, constant input delays.

This paper considers an opportunity for robotic automation in the seeding of field crops. It reports on progress in designing a partially automated seed refilling system that transfers seeds to a centralized tank on a tractor-pulled planter during travel, while the tractor continues to follow a trajectory determined by a desired seeding strategy The paper describes challenges associated with accommodating variability among commercial planters, as well as with formulating a control strategy for a robotic manipulator that compensates for time-dependent uncertainty and unknown disturbance loads due to travel of a mounting platform across uneven terrain. To address such challenges, the paper reviews theoretical predictions for a recently proposed adaptive control framework regarding the existence of transient performance bounds in the presence of fast uncertainty, as well as robustness to unmodeled dynamics in the form of time delays in the control signals. In particular, the theoretical treatment guarantees the existence of a positive time-delay margin, below which stable operation can be expected. Several experimental designs involving one or multiple robotic manipulators are considered in order to validate the theoretical results and to explore possible seed transfer strategies.

This paper develops an adaptive controller for underactuated robotic systems with unmodeled dynamics. The control scheme is motivated by the applications of manipulators operating on dynamic platforms. The design decouples the system's adaptation and control loops to allow for fast estimation rates, while guaranteeing bounded deviation from a nonadaptive reference system. The proposed formulation is independent of detailed information about the system model. The control scheme is tested in different trajectory-tracking scenarios: (i) a manipulator installed on a ship operating in a high-sea state with uncertain environmental disturbances and (ii) a mobile manipulator moving across a rough terrain of unknown geometry. The simulation results illustrate the tracking performance of the proposed control algorithm, its ability to deal with unmodeled dynamics, and its robustness to measurement noise and time delay, while maintaining smooth control signals. An adaptive control formulation for underactuated robotic systems with unmodeled dynamics is proposed.The design allows for fast estimation while guaranteeing bounded deviation from a nonadaptive reference system.The proposed formulation is independent of detailed information about the system model.The proof of stability is established by the analysis based on input-output maps.The system's robustness to measurement noise and time delay is demonstrated.

A modification to the ${\cal L}_1$ control framework for uncertain systems with actuator delay is presented. Specifically, a time delay is introduced in the control input of the state predictor to compensate for the destabilizing effect of input delay in the plant. For this modified framework, the analysis shows that the output of the adaptive system closely follows the behavior of a suitably defined, nonadaptive, stable reference system provided that a delay-dependent stability condition is satisfied and the adaptive gain is chosen sufficiently large. The set of combinations of input delay and compensation delay for which the stability condition is satisfied contains an open set of pairs of positive values provided that a filter bandwidth, characteristic of ${\cal L}_1$ adaptive control is chosen sufficiently large. The efficacy of the delay compensation is illustrated by a simple example. A numerical continuation is also performed to explore the stability region for a case where this can be approximated a priori., 18 pages, 6 figures

Summary This paper presents an analysis of the synchronization and consensus problems of networked manipulators operating on an underactuated dynamic platform in the presence of communication delays. The proposed formulation does not require detailed information about the system model. A theoretical formulation based on input–output maps of functional differential equations shows that the control system's behavior matches closely that of a non-adaptive reference system. The tracking synchronization objective is achieved despite the effects of the communication delay and the unknown dynamics of the platform. When there is no common desired trajectory, the modified controller drives all robots to average consensus for an unsigned graph and to bipartite consensus for a structurally balanced signed digraph. In addition, a leader–follower scheme is proposed that allows for the control of the constant and time-varying consensus values. Simulation results illustrate the performance of the proposed control algorithms. Copyright © 2016 John Wiley & Sons, Ltd.

This paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near-grazing dynamics. In particular, it is shown that for critical choices of parameter values, commonly used physical models of rigid or compliant mechanical contact, capillary adhesion, and cell division satisfy the conditions required by the general framework. The paper relies on the well-known discontinuity-mapping formalism. In contrast to previous treatments, the proposed abstract framework more clearly establishes the origin of the large state-space stretching in the initial (and possibly only) step of the construction of a discontinuity mapping. It further highlights the nonuniqueness in the formulation of the discontinuity mapping and its connection to the choice of a locally smooth map with which the discontinuity mapping is composed to describe the near-grazing dynamics. The analysis is illustrated with examples from tapping-mode atomic force microscopy in the presence of thin fluid layers on the sample and the probe tip and from the study of protein activity during a eukaryotic cell cycle.

In this paper we detail the design, construction, experimental investigation, theoretical modeling, and numerical exploration of a laboratory-scale electromechanical device, which includes the possibility of recurrent mechanical contact and exploitable multidomain signal transduction. In particular, we illustrate the use of inductive coupling between electrical and mechanical degrees of freedom to actuate and interrogate changes in the mechanical response due to the onset of contact through the coupled behavior of the device current. It is shown that nonsmooth bifurcations associated with a transition from nonimpacting to impacting dynamics, induced by changes in critical parameter values, may result in detectable changes in the current amplitude and, significantly, frequency content. The device design and the observed results are argued to support a similar functionality in microscale devices, where multidomain coupling is common.

The focus of this paper is on the possibility of formulating a consistent and unambiguous forward simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painleve paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-stiffness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward simulation model is afforded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter—a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-stiffness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model.

This paper develops a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. In particular, it is shown that a discrete-in-time linear feedback correction to a parameter governing a state-space discontinuity surface can suppress discontinuity-induced fold bifurcations of limit cycles that achieve near-tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise-linear spring stiffness as well as with a piecewise-nonlinear model of a capacitively excited mechanical oscillator. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, nonsmooth fold bifurcations associated with the onset of low-relative-velocity (near-grazing) impacts in an oscillatory mechanical system are proposed as a potential operating principle for high-speed limit switches. Specifically, analytical, numerical, and experimental methods are employed to investigate the near-grazing transient behavior in a representative system. It is shown that the rate of growth of successive impact velocities increases beyond all bounds as the threshold parameter value is approached. A limit switch based on the proposed nonsmooth fold scenario would thus be expected to outperform one that relies on a smooth bifurcation, such as the cyclic-fold bifurcation, in terms of switching speed and sensitivity.

This paper presents an event-driven, discrete-in-time feedback strategy for tracking and stabilizing naturally occurring periodic oscillations in the probe-tip dynamics of atomic force microscope (AFM) cantilevers in tapping-mode operation. Specifically, robust dynamic tracking and stabilization is achieved by the imposition of discrete changes in the vertical offset between the cantilever support and the sample surface based on an estimated linearization of the system dynamics about a dynamically generated reference trajectory. Here, use is made not only of the oscillation amplitude, as is typical in commercial control implementations for AFMs, but also of the instantaneous phase information. It is shown that stabilization and desirable performance during surface scanning is possible, even in the presence of uncertainty and limited state access. In particular, the methodology enables robust tracking and use of low-contact-velocity periodic system responses that are unstable in the absence of control.

This paper describes the underlying formulation and functionality of the newly developed software program $\widehat{\text{{\sc tc}}}$ (“tc-hat”), to perform bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, often referred to as hybrid dynamical systems. Boundary-value-problem formulations corresponding to single- and two-parameter continuations of periodic trajectories and selected associated codimension-one bifurcations in such systems are presented. Finally, the capabilities of the program are illustrated by performing bifurcation analysis of a few example hybrid dynamical systems.

This paper investigates the dynamic response of an initially stationary part of a mechanism in the presence of a restoring force and dry friction to low-velocity collisions with a relatively more massive oscillating element. Of particular interest is the persistence of a local attractor in the motion of the less massive part as the path of the oscillating element grows to encompass the entire set of possible equilibrium positions in the absence of contact. It is argued that loss of a local attractor and the associated large-amplitude oscillations of the less massive part affords a means for energy transfer through the mechanism and a means for energy damping. The paper contains a rigorous derivation of conditions that appear sufficient for the persistence of a local attractor in the case where the massive oscillating element is replaced by an oscillating rigid unilateral constraint corresponding to an infinite mass ratio. Numerical simulations are subsequently used to investigate the response in the case where the mass ratio is assumed finite.

This paper argues for the possibility of purposely approximating smooth vector fields with highly localized variability in terms of piecewise smooth vector fields for the purpose of analyzing the bifurcation characteristics of the corresponding dynamical systems. Here, emphasis is placed on the changes in system response that result as a periodic trajectory begins to incorporate a brief flow segment in the region of high variability under variations in some system parameter. In particular, it is shown that tools from the theory of grazing bifurcations in piecewise-smooth systems may be employed to qualitatively predict the bifurcation scenario associated with such a transition both in terms of the shape of the branch of periodic trajectories and in terms of the persistence of a local attractor in the vicinity of the original periodic trajectory.

This paper presents the first application of the discontinuity-mapping approach to the study of near-grazing bifurcations of originally quasiperiodic, co-dimension-two system attractors. The paper establishes an exact formulation for the discontinuity-mapping methodology under the assumption that a Poincare section can be found that is everywhere transversal to the grazing attractor. In particular, it is shown that, while a reduced formulation may be employed successfully in the case of co-dimension-one attractors, it fails to capture dynamics in directions transversal to the original quasiperiodic attractor. This shortcoming necessitates the full machinery presented here. The generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four.

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

International audience; A method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities. In particular, it is shown that a discrete, linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory, such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact. The implementation relies on a local analysis of the near-grazing dynamics using the concept of discontinuity mappings. Numerical results are presented for a linear and a nonlinear oscillator.

Small, discrete, corrective adjustments to foot geometry in a class of bipedal passive walkers are employed to affect the local stability properties of a periodic reference gait. It is demonstrated that successful stabilization can be accomplished for otherwise strongly unstable motions of vertically constrained as well as entirely unconstrained model mechanisms. In particular, recent results by the authors on stabilization of repetitive motion in hybrid dynamical systems are implemented to formulate a rigorous analytical methodology for predicting the stability characteristics of the controlled system. It is demonstrated that these predictions can be explicitly obtained based entirely on knowledge of the local stability properties of the reference motion in the absence of control.

A combined control scheme relying on feedback-based local control in the vicinity of periodic system responses and global control based on a coarse-grained approximation to the nonlinear dynamics is developed to achieve a desirable dynamical behavior of a Braille printer impact hammer. The proposed control methodology introduces discrete changes in the position of a system discontinuity at opportune moments during the hammer motion while the hammer is away from the discontinuity, thereby exploiting the recurrent contacts with the discontinuity to achieve the desired changes in the transient dynamics. It is argued that, as the changes in the position of the discontinuity affect the motion only indirectly through changes in the timing and state at the subsequent contact, the control actuation can be applied over an interval of time during the free-flight motion as long as it is completed prior to contact. A forced, piecewise smooth, single-degree-of-freedom model of a Braille impact hammer is used to illustrate the methodology and to yield representative numerical results.

Electrically driven impact microactuators generate nanoscale displacements without large driving distances and high voltages. These systems exhibit complex dynamics because of inherent nonlinearities due to impacts, friction, and electric forces. As a result, dramatic changes in system behavior, associated with so-called grazing bifurcations, may occur during the transition between impacting and nonimpacting dynamics, including the presence of robust chaos. For successful open-loop operating conditions, the system design is limited to certain parameter regions, where desired system responses reside. The objective of this paper is to overcome this limitation to allow for a more precise displacement manipulation using impact microactuators. This is achieved through a closed-loop feedback scheme that successfully controls the system dynamics in the near-grazing region.

Impact microactuators rely on repeated collisions to generate gross displacements of a microelectromechanical machine element without the need for large applied forces. Their design and control rely on an understanding of the critical transition between non-impacting and impacting long-term system dynamics and the associated changes in system behavior. In this paper, we present three co-dimension-one, characteristically distinct transition scenarios associated with grazing conditions for a periodic response of an impact microactuator: a discontinuous jump to an impacting periodic response (associated with parameter hysteresis), a continuous transition to an impacting chaotic attractor, and a discontinuous jump to an impacting chaotic attractor. Using the concept of discontinuity mappings, a theoretical analysis is presented that predicts the character of each transition from a set of quantities that are computable in terms of system properties at grazing. Specifically, we show how this analysis can be applied to predict the bifurcation behavior on neighborhoods of two co-dimension-two bifurcation points that separate the co-dimension-one bifurcation scenarios. The predictions are validated against results from numerical simulations of a model impact microactuator.

A rigorous mathematical technique is presented for exploiting the presence of discontinuities in non-smooth dynamical systems in order to control the local stability of periodic or other recurrent motions. In particular, the formalism allows one to predict the effects of the control strategy based entirely on information about the uncontrolled system. The methodology is illustrated with examples from impacting systems, namely a model hopping robot and a Braille printer. It is shown how initially unstable motions can be successfully stabilized at negligible cost and without active energy injection.

This paper studies repetitive gaits found in a 3D passive walking mechanism descending an inclined plane. By using direct numerical integration and implementing a semi-analytical scheme for stability analysis and root finding, we follow the corresponding limit cycles under parameter variations. The 3D walking model, which is fully described in the paper, contains both force discontinuities and impact-like instantaneous changes of state variables. As a result, the standard use of the variational equations is suitably modified. The problem of finding initial conditions for the 3D walker is solved by starting in an almost planar configuration, making it possible to use parameters and initial conditions found for planar walkers. The walker is gradually transformed into a 3D walker having dynamics in all spatial directions. We present such a parameter variation showing the stability and the amplitude of the hip sway motion. We also show the dependence of gait cycle measurements, such as stride time, stride length, average velocity, and power consumption, on the plane inclination. The paper concludes with a discussion of some ideas on how to extend the present 3D walker using the tools derived in this paper.

The inherent dynamics of bipedal, kneed mechanisms are studied with emphasis on the existence and stability of repetitive gait in a three-dimensional environment, in the absence of external, active control. The investigation is motivated by observations that sustained anthropomorphic locomotion is largely a consequence of geometric and inertial properties of the mechanism. While the modeling excludes active control, the energy dissipated in ground and knee collisions is continuously re-injected by considering gait down slight inclines. The paper describes the dependence of the resulting passive gait in vertically constrained and unconstrained mechanisms on model parameters, such as ground compliance and ground slope. We also show the possibility of achieving statically unstable gait with appropriate parameter choices.

We combine a detailed understanding of the dynamics of low-dimensional models of burstingin the turbulent boundary layer with the method of Genetic Programming to obtain appropriate control strategies for the suppressionof such bursting in these models. The study is applicable toO(2) symmetric systems for which structurally stable heteroclinic cycles are the dominant dynamical features. We argue that such a combined approach can prove a useful tool in achieving control in higher-dimensional models where actual analysis is prohibitively complicated. The results of the present study are compared to near-optimal controllers derived in previous studies.

In this paper, we aim to model a functional task affected by injury, along with the corresponding neuromuscular compensation strategy, in order to understand differences in task performance during recovery from the injury. This study is motivated by differing rates of functional task improvements during recovery from anterior cruciate ligament repair. In particular, clinical studies have shown faster recovery times for single-limb forward hopping versus single-limb crossover hopping (hopping back and forth laterally while moving forward). Modeling this hopping task will help us understand whether the main factor of the differing functional results is from the physical restrictions of the injury, the compensation strategies used to overcome these restrictions, or a combination of both. Our hypothesis is that the discrepancies in clinical functional results will be reproduced by employing a feedforward compensation strategy, where the compensation is learned and adjusted over time.Copyright © 2013 by ASME

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom, and with desirable symmetry properties, and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may coexist. We further formulate a candidate optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol may exhibit discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.

This paper presents a preliminary analysis of the robustness of a recently-proposed adaptive controller for robot manipulators based on the ℒ1 control paradigm. Here, the use of a low-pass filter in the control input decouples the estimation loop from the control loop, thereby facilitating an arbitrary increase of estimation rates (limited only by hardware) without sacrificing robustness. Tuning of the filter also allows for shaping the nominal response and enhancing the system’s robustness. The paper further demonstrates improvements in the critical time delay associated with a static reference input achieved through the introduction of time delay in the state-predictor formulation. Guided by results from the theory of single-input-single-output ℒ1 control systems, a linear, time-invariant system is proposed in order to derive a conservative lower bound on the system’s actuator critical time delay for static reference input in the limit of large estimation gains, as an indicator of robustness. Finally, a numerical method is proposed for quantifying the robustness against time delay of the system’s response to a given static reference input based on techniques of parameter continuation. This method computes the critical time delay at which local stability is lost in a Hopf bifurcation. The dependence of this critical time delay on control parameters, such as adaptive gains and filter bandwidth, is here obtained using advanced algorithms for computing approximate covers of implicitly defined manifolds.

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom and with desirable symmetry properties and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which open-loop fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may co-exist. We further formulate a candidate open-loop optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol exhibits discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.

This paper discusses the development of a method for on-the-fly, self-calibration of a mass-flow sensing system on a harvest combine in order to enable accurate mass-flow sensing and, consequently, yield mapping, under varying input conditions, for example due to changes in the moisture content of harvested grains or the aging of elevator paddles. The proposed method relies on a known or estimated physics-based relationship between the grain dynamics through the grain elevator and a measured force imparted on an impact plate through collisions with the grains. The paper provides a summary description of the modeled relationship between the rate of mass flow and impact-plate sensor readout and conceptualizes the self-calibration scheme through the introduction of an additional, controllable, degree of freedom of the plate. The validity of the self-calibration technique is illustrated through off-line application to bench-top and full-scale experimental test data.Copyright © 2012 by ASME

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc., in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with a distinctly different amplitude and frequency content. The results reported here include a review of an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively driven circuit protection device.

The onset of low-velocity collisions in vibro-impacting systems induces instabilities in the system dynamics that, when not checked, may result in sudden, and unanticipated discontinuous transitions between distinct steady-state responses. This paper illustrates this phenomenology in an example system that includes dry friction. Here, the instability is associated with the zero-velocity contact of an oscillatory unilateral constraint and a stationary mass suspended through a preloaded spring. The analysis summarizes observations on the passive response of the mass under variations in the oscillation amplitude of the constraint. A control strategy is subsequently shown to successfully suppress the instability. The paper concludes with suggestions for applications of this phenomenology as well as a description of similar observation in mechanical systems with or without friction and with rigid as well as compliant contact.

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc. in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with distinctly different amplitude and frequency content. The results reported here include an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively-driven circuit protection device.Copyright © 2009 by ASME

Vehicle manufacturers are constantly pushed to reduce the aerodynamic drag of vehicles, for example by constructing lower vehicles with less road clearance. This, however, reduces the available margin for oscillations within the suspension. If the oscillation amplitude exceeds a critical value, the suspension will impact a bumpstop. Under periodic excitation, the onset of low-velocity impacts is associated with a strong instability in favor of high-velocity impacts. Such impacts reduce comfort and could be damaging to the vehicle. Efforts should therefore be made to limit impact velocities with the bumpstop, for example by suppressing the instability associated with low-velocity impacts. This paper proposes a low-cost feedback-control strategy, based on making small adjustments to the position of the bumpstop, that serve to suppress the transition to high-velocity impacts with the bumpstop in the case of periodic excitation. The control law is derived from the theory of discontinuity maps. The results demonstrate that the feedback strategy works even when wheel-hop is present.

In this paper, dynamical systems analysis and optimization tools are used to investigate the local dynamic stability of periodic task-related motions of simple models of the lower-body musculoskeletal apparatus and to seek parameter values guaranteeing their stability. In particular, the dynamics of a two-link model of a leg undergoing periodic excitation through one or several contractile muscle elements corresponding to a simple knee-bending motion is studied. Several muscle models incorporating various active and passive elements are included and the notion of self-stabilization of the rigid-body dynamics through the imposition of muscle-like actuation is investigated. It is found that self-stabilization depends both on muscle architecture and configuration as well as the properties of the reference motion. Additionally, antagonistic muscles (flexor-extensor muscle couples) are shown to enable stable motions over larger ranges in parameter space and that even the simplest neuronal feedback mechanism can stabilize the repetitive motions. The work provides a review of the necessary concepts of stability and a commentary on existing incorrect results that have appeared in the literature on muscle self-stabilization.Copyright © 2007 by ASME

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.Copyright © 2007 by ASME

Biological tissues are plastic with respect to the mechanical environment to which they are exposed. This makes them able to modify their architecture and inner structure in order to respond to different loading conditions with the smallest biological effort (functional adaptation). As a result, tissues can optimally adapt their structures to the task they have to perform. Based on these concepts, a kinetic model of the ankle joint has been recently developed. The tibio-talar relative motion was obtained by imposing the congruence maximization as a biological optimum throughout the entire flexion range. The aim of this work is to investigate the applicability of the proposed approach to the knee and to evaluate the weight of the meniscal contribution to the global femoro-tibial congruence.