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## Publications

Publications (212)

Stabilization of the inverted pendulum by fractional-order proportional-derivative (PD) feedback with two delays is investigated. This feedback law is obtained as a combination of PD feedback with two delays and fractional-order PD feedback with a single delay. Different types of stabilizability boundaries and the corresponding geometric and multip...

A planar stick balancing task was investigated using stabilometry parameters (SP); a concept initially developed to assess the stability of human postural sway. Two subject groups were investigated: 6 subjects (MD) with many days of balancing a 90 cm stick on a linear track and 25 subjects (OD) with only one day of balancing experience. The underly...

In this paper, we apply the semi-discretization method to controllable single-input linear systems with input delay and fractional-order feedback. This requires the fractional derivative to be discretized in a way that the resulting discrete map is linear and time-invariant. To this end, three different techniques are investigated, namely, the shor...

This paper considers the stabilization of a rolling balance board by means of the multiplicity-induced-dominancy property. A two degree-of-freedom mechanical model of a human balancing on a rolling balance board is analyzed in the sagittal plane. The human body is modeled by an inverted pendulum which connects to the balance board through the ankle...

Human stick balancing is investigated in terms of reaction time delay and sensory dead zones for position and velocity perception using a special combination of delayed state feedback and mismatched predictor feedback as a control model. The corresponding mathematical model is a delay-differential equation with event-driven switching in the control...

The relation between balancing performance on rolling balance board and reaction time is investigated. Ten young healthy adults performed balancing trials on a rolling balance board with different wheel radius R and stance widths d in the frontal plane. 2- and a 3-degree-of-freedom models subject to delayed state feedback with a single lumped react...

Human reaction delay significantly limits manual control of unstable systems. It is more difficult to balance a short stick on a fingertip than a long one, because a shorter stick falls faster and therefore requires faster reactions. In this study, a virtual stick balancing environment was developed where the reaction delay can be artificially modu...

In this paper, we consider the stabilization of φ̈(t)−a0φ(t)=−kpφ(t−τ−δp)−kdφ̇(t−τ−δd)−kaφ̈(t−τ−δa),which describes the control of an inverted pendulum by detuned proportional-derivative-acceleration (PDA) feedback. We show that the system can be stabilized using an appropriate choice of the control parameters kp, kd, ka and δp, δd, δa ≥0 if τ is s...

Postural sway refers to the fluctuations in the horizontal movements of the center of mass (COM) while a human stands quietly. The fluctuations in the center of pressure (COP) measured with a force platform provide a measure of the neuromuscular response to the motions of the COM.

The stability radius provides a scalar measure of the distance from instability in the space of the system and control parameters. It is not difficult to imagine that the risk of falling might be inversely proportional to the stability radius, i.e the smaller the stability radius the higher the risk of falling due to sensory uncertainty.

Human balancing tasks typically involve the stabilization of an inverted pendulum. The desired position is the upright one, which, without control, is an unstable position. Newtonian dynamics and reaction time delay implies that the control is described by second-order delay differential equations.

Microchaos is a low amplitude type of chaos that arises in time-delayed dynamical systems from the interplay between a sensory dead zone and time-sampled feedback. In the previous chapter we suggested that the Weibull-type survival curves observed for pole balancing at the fingertip were a manifestation of transient microchaos.

Experimental paradigms that involve the stabilization of an inverted pendulum with time delayed feedback are important for developing and understanding the neuroscience of balance control and skill acquisition.

A pendulum-cart model is developed for human pole balancing at the fingertip in which the mass of the “cart” is much greater than the mass of the pole. It is shown that the main control problem for this model is the stabilization of the angular displacement of the pole.

An expert pole balancer cannot balance a 0.3 m stick at their fingertip for longer than 5 s if one eye is patched! This is a consequence of the limited ability of human depth perception in the sagittal (anterior-posterior) plane.

In this article we consider the ‐stabilization of nth‐order linear time‐invariant dynamical systems using multiplicity‐induced‐dominancy‐based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least and at least n using an integral f...

The critical length that limits stabilizability for delayed proportional-derivative-acceleration (PDA) feedback and for predictor feedback (PF) is analyzed for the inverted pendulum paradigm. The aim of this work is to improve the understanding of human balancing tasks such as stick balancing on the fingertip, which can be modeled as a pendulum car...

Postural sway is a result of a complex action–reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the respo...

The relation between balancing performance and reaction time is investigated for human subjects balancing on rolling balance board of adjustable physical parameters: adjustable rolling radius R and adjustable board elevation h . A well-defined measure of balancing performance is whether a subject can or cannot balance on balance board with a given...

Stability analysis has been always a key issue in nonlinear dynamics and engineering applications, and it is still a challenging task for time-delay control systems when tuning some parameters like feedback gains and delays. In this paper, firstly we propose a parametric continuation algorithm for calculating the rightmost characteristic root(s), b...

The effect of reaction delay, temporal sampling, sensory quantization, and control torque saturation is investigated numerically for a single-degree-of-freedom model of postural sway with respect to stability, stabilizability, and control effort. It is known that reaction delay has a destabilizing effect on the balancing process: the later one reac...

Postural sway is a result of a complex action-reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the respo...

This book demonstrates how delay differential equations (DDEs) can be used to compliment the laboratory investigation of human balancing tasks. This approach is made accessible to non-specialists by comparing mathematical predictions and experimental observations. For example, the observation that a longer pole is easier to balance on a fingertip t...

Dynamic balance conditions were realized by asking eight volunteers to stand on uniaxial balance board with adjustable geometry and to carry out 60 s long balancing trials. Four different balance board geometry were used, each associated with different difficulty level. Balancing trials were repeated five times weekly (learning period) in order to...

Human balancing on rolling balance board in the sagittal plane is analyzed such that the geometry of the balance board can be adjusted: the radius R of the wheels and the elevation h between the top of the wheels and the board can be changed. These two parameters have a significant influence on the stability of standing on the board as shown by pre...

In today’s aging society special attention is devoted to the research of human balance control. Our research group focuses on understanding the control mechanism of the brain during balancing from an engineering point of view. The model of stick balancing is derived and the dynamical behaviour of the mechanical system is investigated numerically. T...

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher mu...

A virtual stick balancing environment is developed using a computer mouse as input device. The development process is presented both on the hardware and software level. Two possible concepts are suggested to obtain the acceleration of the input device: discrete differentiation of the cursor position measured in pixels on the screen and by direct me...

The paper presents a pseudospectral approach to assess the stability robustness of linear time‐periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real‐valued time‐periodic uncertainties affecting the coefficient matrices...

The control mechanism of the human nervous system during stick balancing is analyzed using signal processing techniques. It is assumed that during stick balancing the inclination angle, the angular velocity and the angular acceleration of the stick are all perceived and used for feedback. Taking into account the human reaction time delay, the balan...

Understanding how dexterity improves with practice is a fundamental challenge of motor control and neurorehabilitation. Here we investigate a ball and beam implementation of a dexterity puzzle in which subjects stabilize a ball at the mid-point of a beam by manipulating the angular position of the beam. Stabilizability analysis of different biomech...

In this study we consider the stabilization of nth-order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input/output channels. A sufficient condition is given for the dominancy of a real root with multiplicity n + 1 using an integral representation of...

The prediction of chatter-free machining parameters is not reliable for industrial applications without guaranteed robustness against modeling uncertainties. Measurement inaccuracies, fitting uncertainties, simplifications and modeling errors typically lead to a mismatch between the mathematical model and the real physical system. This paper presen...

Sensory uncertainties and imperfections in motor control play important roles in neural control and Bayesian approaches to neural encoding. However, it is difficult to estimate these uncertainties experimentally. Here, we show that magnitude of the uncertainties during the generation of motor control force can be measured for a virtual stick balanc...

Understanding the human balancing is a fundamental question. Investigation of simple tasks can help in this challenging problem. In order to describe the nature of the underlying control mechanism, first of all, the balancing force has to be determined. As a second step one can identify the behaviour of the controller. There are two main problems i...

Human and robotic legged locomotion can be described with complex multi-degree-of-freedom dynamic models, whose bifurcation or parameter analysis may explain some features of typical patterns during motion. In this paper, we focus on the effect of kinematic parameters and foot placement techniques on the ground-foot impact intensity. The work is ba...

The causes of falling in the elderly are multi-factorial. Three factors that influence balance stability are the time delay, a sensory dead zone and the maximum ankle torque that can be generated by muscular contraction. Here, the effects of these contributions are evaluated in the context of a model of an inverted pendulum stabilized by time-delay...

Connected cruise controllers that utilize motion information of human-driven vehicles must guarantee stability against uncertainties in driver behavior. Here, we present the robustness analysis of such connected vehicle systems using structured singular value analysis. In particular, we consider uncertainties arising from the feedback gains and rea...

This chapter deals with predictor feedback controllers to compensate time delays in feedback loops. The concept and the governing equations of the Smith predictor, the modified Smith predictor and the finite spectrum assignment are discussed in detail. The relationship between the three control strategies is established both in frequency and time d...

The dynamic analysis of legged locomotion typically involves issues related to multibody dynamics, underactuation, motion planning and stability. In addition to biomechanics of humans and animals, the dynamic analysis of legged locomotion is also an important issue in the control development of pedal robots. For these robots, stable internal dynami...

Regenerative chatter vibrations arising during machining operations are a severe problem affecting the manufacturing production of high-precision mechanical components. A relatively deep understanding of this phenomenon when the cutting tool has equally spaced cutting teeth without runout has been achieved by now. However, real cutting tools are al...

This paper presents the robustness analysis for the head-to-tail string stability of connected cruise controllers that utilize motion information of human-driven vehicles ahead. In particular, we consider uncertainties arising from the feedback gains and reaction time delays of the human drivers. We utilize the linear fractional transformation and...

Understanding the mechanism of human balancing is a scientifically challenging task. In order to describe the nature of the underlying control mechanism, the control force has to be determined experimentally. A main feature of balancing tasks is that the open-loop system is unstable. Therefore, reconstruction of the trajectories using the measured...

Our research aims the study of balancing on a rolling balance board with respect to dynamic properties such as stability and stabilizability. The goal is to identify the parameter regions where human subjects are able to keep themselves stable in the upright position for at least 60 s. The radius of the balance board and the height of the foot plat...

Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machi...

A two-degree-of-freedom mechanical model was developed to analyze human balancing on rolling balance board in the frontal plane. The human nervous system is modeled as a proportionalderivative controller with constant feedback delay. The radius R of the wheels and the board distance h measured from the center of the wheel are adjustable parameters....

The aim of this study is to investigate the dynamics of an accelerating skater–board system modeling downhill motion. The governing mathematical model is a system of time-varying neutral delay-differential equations. Stability analysis is performed based on the frozen-time method, and the results are verified via numerical simulations. It is shown...

Models for the stabilization of an inverted pendulum figure prominently in studies of human balance control. Surprisingly, fluctuations in measures related to the vertical displacement angle for quietly standing adults with eyes closed exhibit chaos. Here we show that small-amplitude chaotic fluctuations (“microchaos”) can be generated by the inter...

Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the...

Bifurcations of the periodic stationary solutions of nonlinear time-periodic time-delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay-differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map th...

In this paper, we investigate a vehicular string traveling on a single lane, where vehicles use connected cruise control to regulate their longitudinal motion based on data received from other vehicles via wireless vehicle-to-vehicle communication. Assuming digital controllers, the sample-and-hold units introduce time-periodic time delays in the co...

A nonlinear model for human balancing subjected to a saturated delayed proportional-derivative-acceleration (PDA) feedback is analysed. Compared to the proportional-derivative (PD) controller, it is confirmed that the PDA controller improves local stability even for large feedback delays. However, it is shown that the saturated PDA controller typic...

Virtual stick balancing by human subjects is analyzed with special attention to the reaction time delay and the sensory dead zones. A pendulum-cart system displayed on a computer screen is controlled via an optical computer mouse. Realization of the balancing task involves additional computational delay in the feedback loop. The measurement results...

Human balancing on rolling balance board in the frontal plane is analyzed using a two DoF mechanical model, where the human body is modeled by a four-bar linkage mechanism and the geometry of the balance board can be adjusted. Human nervous system is assumed to employ a proportional-derivative controller with constant feedback delay identical to th...

In the last few decades the advantages of fractional-order control was demonstrated with several examples in comparison with integer-order control. In this paper, stabilizability of a second-order unstable system subject to a delayed PDµ and PDµDρ controller is investigated in terms of the critical delay. Stabilizability diagrams are determined as...

Mechanical models of balancing a ball rolling on a see-saw (“ball and beam” system) and balancing an inverted pendulum attached to a cart rolling on a see-saw (“pendulum-cart and beam” system) are analyzed. A delayed proportional-derivative controller is modeled with four different actuation schemes. The angular position, the angular velocity, the...

Reliable stability predictions of machine tool chatter have high potential in the optimization of machining processes. Industrial applications can benefit from the corresponding stability lobe diagrams, which can guarantee improvement in the performance. However, predictions and experiments often do not match during cutting tests due to uncertain d...

In order to increase the material removal rate in milling, advanced cutting tools with complex geometry are typically applied under extreme cutting conditions which may trigger undesired chatter vibrations of the machining system. Recently some dynamic milling models were proposed in the literature which take into account the higher geometrical com...

A possible explanation for a transient behavior of digitally controlled machines is presented. The mechanical system under study is an inverted pendulum controlled by a reaction wheel. A proportional-derivative feedback is used to control the unstable equilibrium of the pendulum. Modeling the quantization and the saturation of the applied control t...

The dynamic analysis of legged locomotion is challenging because of many reasons, such as the possibly high degrees of freedom of the model, the alternating topology in certain phases of walking or running, the presence of under-actuation and over-actuation, the geometric nonlinearities and ground-foot impact induced non-smoothness. Control issues...

A two-wheeled vehicle balancing a passive inverted pendulum is analyzed based on an experimental device. The corresponding mechanical model is a wheeled double pendulum, where only the position of the lower pendulum is measured. The sampling effect of the digital control is modeled as a zero-order hold. It is shown that the stabilization of the upr...

A simplified model of stick balancing on the fingertip subjected to predictor feedback is investigated, which accounts for three important modeling issues: (1) feedback delay; (2) the sensory dead zone; and (3) limitation of the control force corresponding to the maximum acceleration and the maximum jerk of human hand movement. Eight different case...

This paper presents a study on the limits of stabilizability of unstable second-order dynamical systems by means of digital proportional-integral-derivative-acceleration (PIDA) feedback. Four different models are considered, which are all governed by the same dimensionless second-order differential equation. The mathematical model under analysis is...

Bifurcation analysis of the orthogonal cutting model with cutting force nonlinearity is presented with special attention to double Hopf bifurcations. The normal form of the system in the vicinity of the double Hopf point is derived analytically by means of center manifold reduction. The dynamics is restricted to a four-dimensional center manifold,...

Reliable prediction of machine tool chatter is an essential problem in efficiency-oriented machine tool centers, since it requires the precise characterization of the dynamics of the machine-tool-workpiece system and the cutting force characteristics. Due to imperfect measurements, noise, uncertain and varying operational conditions, the mathematic...

This paper extends two numerical methods for the stability analysis of a class of time-periodic hybrid time-delay systems. In particular, the pseudospectral tau and spectral element methods are extended to hybrid systems. The analyzed delay-differential equation involves delayed terms with both continuous and piecewise constant arguments, in other...

The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is d...

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possib...

One of the most significant limitation of the productivity of machining operations is the regenerative machine tool vibration, also called machine tool chatter, which is a self-excited vibration between the tool and the workpiece induced by the chip formation mechanism. Extension of the chatter-free parameters is possible by active chatter control...

This paper presents an application of the spectral element method for the stability analysis of regenerative machine tool chatter models in milling operations. An extension of the spectral element method is introduced in order to handle the discontinuities in the cutting force in an efficient way. The efficiency of the method is demonstrated on som...

This paper investigates the stabilizing effect of process damping at low cutting speeds for regenerative machine tool vibrations of milling processes. The process damping is induced by a velocity-dependent cutting force model, which takes into account that the actual cutting velocity is different from the nominal one during machine tool vibrations....

The dynamics of connected vehicle systems are investigated where vehicles exchange information via wireless vehicle-to-vehicle (V2V) communication. In particular, connected cruise control (CCC) strategies are considered when using different delay configurations. Disturbance attenuation (string stability) along open chains is compared to the linear...

Prediction of machine tool chatter requires the characterization of dynamic of the machine-tool-workpiece system by means of frequency response functions (FRFs). Uncertainties of the measured FRFs result in uncertainties of the calculated stability diagrams, therefore robustness of stability prediction against parameter perturbations is of high imp...

Human balancing on a balance board is modelled as a delayed proportional-derivative control mechanism with unknown feedback delay. The mechanical model implies that there exists a critical delay, for which no control gain parameters can stabilize the system. This theoretical critical delay is determined by numerical analysis for different geometrie...

Experimental cutting tests often show different dynamic behavior from the one predicted by means of theoretical mechanical models. One reason behind this observation is that stability lobe diagrams determined by standard linear stability analysis often under- or overestimate the region of chatter-free parameters. Therefore, reliable selection of te...

Prediction of machine tool chatter requires a dynamic characterization of the machine-tool-workpiece system by means of frequency response functions (FRFs). Stability lobe diagrams are sensitive to the uncertainties of the measured FRF, which reduces the reliability of their industrial application. In this paper, a frequency-domain method is presen...

For implementation in real traffic, connected vehicle systems should be designed to be robust against uncertainties arising from human-driven vehicles. Assuming that the bounds of uncertainties are known, we propose a frequency domain approach to guarantee robust string stability and to select optimal control parameters. The method is demonstrated...

A new mechanical model is presented for regenerative machine tool vibrations in milling processes, which accounts with not only the regenerative effect but also with the sampling and actuation scheme of the control loop of the feed drive of the workpiece holder. It is shown that depending on the choice of control parameters, the stability lobe diag...

This paper presents a method which applies pseudospectral tau approximation for retarded functional differential equations (RFDEs). The goal is to construct a system of ordinary differential equations (ODEs), which provides a finite dimensional approximation of the original RFDE. The method can be used to determine approximate stability diagrams fo...

The ability of humans to maintain balance about an unstable position in a continuously changing environment attests to the robustness of their balance control mechanisms to perturbations. A mathematical tool to analyze robust stabilization of unstable equilibria is the stability radius. Based on the pseudospectra, the stability radius gives a measu...

This paper presents a study on the stability of linear one-degree-of-freedom mechanical systems subjected to digital PIDA control. The mathematical models under analysis are hybrid systems involving terms with piecewise constant arguments due to the discrete sampling and actuation of the controller. Closed form formulas are determined for the domai...

The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root funct...