Virtual Stiffness Research Articles

Experimental study on the stability of an impedance-type force-feedback architecture based on an augmented-state observer for a haptic system under time delay using a LMI approach

One of the most important performance measures of a haptic interface may be characterized by its behaviour in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness that still leaves the device stable. This value depends on many factors including time delays and the virtual world implementation. In this paper we propose a new architecture for this latter factor based on the use of an augmented-state observer which contributes to improving, without additional costs, the performance of a haptic device with respect to the standard implementation based on the use of the backward finite-difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, some conditions expressed in terms of LMI (linear matrix inequality) for the stability of discrete-time haptic systems under constant or variable time delay are given. An implementation of a force-feedback haptic test bench is presented. The obtained experimental results confirm the efficiency of the proposed force-feedback architecture with respect to the traditional backward finite-difference method in enlarging the choice for the parameters of the virtual environment for which the haptic system is stable.

Stability and quantization-error analysis of haptic rendering of virtual stiffness and damping

The stable, quantization-error noise-free, rendering of high-stiffness dynamics can be challenging using impedance-type haptic displays. In this paper we examine a canonical, one degree of freedom, haptic display rendering a virtual spring and damper, including the effects of the device and human dynamics, sampling, position quantization, time delay, and the low-pass filter operating on the device velocity estimate. We construct various stability and quantization-error regions as a function of the system parameters and show the necessary trade-offs that occur between them. Although we apply the quantization-error analysis to virtual spring and damper rendering, it applies to a general virtual environment. We present sufficiency for quantization-error passivity, necessity for no malicious-touch limit cycles, and necessity for no uncoupled-touch limit cycles. Using these results, aided by the presented supplementary code, we find control parameters to render the largest, renderable, virtual stiffness for a given haptic display. The analytical results are experimentally verified using a Phantom Premium 1.5 haptic device.

A constraint and algorithm for stress-based evolutionary structural optimization of the tie-beam problem

Purpose – The purpose of this paper is to present a constraint and corresponding algorithm enhancing the evolutionary structural optimization (ESO) method, aiming to circumvent its structure break down problem in some special cases, such as the tie-beam problem. Design/methodology/approach – A virtual soft material introduced to an element will change the stiffness of the element and may consequently change the stress distribution of that element and its neighbors. With this property, the virtual stiffness of the selected element is calculated and the threshold of the stress changes is derived. The stress threshold is used to evaluate the role of an element on the load path and therefore decide the contribution of the element to the structure. Adding this checking operation into the original ESO iterations enables validation of element removal. Findings – The reason for structure break down with the ESO method is that the element removal criterion of ESO only works for certain optimal objectives. It cannot guarantee that the structure does not fail. The proposed operation offers a stronger and stricter constraint condition for ESO’s element removal process, preventing the structure from breaking down in some special cases. Originality/value – The tests on several examples reported in the literature show that the proposed method has the same ability to achieve an optimum solution as the original ESO methods do, while avoiding incorrect deletion of structurally important elements. The benchmark tie-beam problem is solved successfully with this algorithm. The method can be used in other situations as well.

Rendered and Characterized Closed-Loop Accuracy of Impedance-Type Haptic Displays.

Impedance-type kinesthetic haptic displays aim to render arbitrary desired dynamics to a human operator using force feedback. To render realistic virtual environments, the difference between desired and rendered dynamics must be small. In this paper, we analyze the closed-loop dynamics of haptic displays for three common virtual environments: a spring, a damper, and a spring-damper, including the effects of time delay and low-pass filtering. Using a linear model, we identify important parameters for the rendered dynamics in terms of effective impedances, a conceptual tool that decomposes the displays closed-loop impedance into components with physical analogs. Our results establish bandwidth limits for rendering effective stiffness and damping. The effective stiffness bandwidth is limited by the virtual stiffness and device mass, and the effective damping bandwidth is limited by the cut-off frequency of the low-pass filter which filters the device velocity estimate. We show that a general system impedance can be characterized by a mass, damper, and spring optimally by the solution to a convex optimization problem, and we present a quantitative metric, the Average Distortion Error (ADE), to describe the fidelity of this model. Time delay has no significant effect on characterized stiffness, and reduces characterized damping by the product of virtual stiffness and total time delay. Reducing the low-pass filter cut-off frequency reduces the characterized damping. Experimental data gathered with a Phantom Premium 1.5 validates the theoretical analysis. We also conducted human user experiments to investigate the effects of time delay and low-pass filtering on perceived stiffness and damping. Similar to the characterized dynamics results, we observed no significant effect of time delay on perceived stiffness, and increasing time delay resulted in reduced perceived damping. Lower filter cut-off frequencies resulted in lower perceived damping. This work informs haptic display design by presenting how closed-loop behavior changes with key parameters.

Augmented observer approach for high-impedance haptic system with time delay

Among others measures of performance, a haptic device may be characterized by its behavior in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness leaving stable the device. This value depends on many factors including time delays and the virtual world implementation. The paper proposes a new architecture for this latter one based on the use of an augmented state observer which contributes to improve without additional costs the performance of a haptic device with respect to the standard implementation based on the use of the backward finite difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, a new LMI criterion for the stability of discrete-time systems with time-varying delay is given. Some numerical simulation results are included to prove the effectiveness of the proposed approach.

On the stability and accuracy of high stiffness rendering in non-backdrivable actuators through series elasticity

This paper addresses the problem of accuracy and coupled stability of stiffness-controlled series elastic actuators, where the motor is modeled as a non-backdrivable velocity source, and the desired value of virtual stiffness is above the physical stiffness of the compliant element. We first demonstrate that, in the mentioned conditions, no linear outer-loop force control action can be applied on the velocity-sourced motor to passify the system. Relaxing the constraint of passivity, we exhaustively search the control design space defined by parametric force and stiffness controllers, expressed in a general lead-lag form, and define a lead-type stiffness compensator that results in acceptable conditions for both coupled stability and accuracy. We also address the effect of a non-ideality in the velocity control loop, such as limited-bandwidth velocity control, and derive relationships between the value of the inner velocity loop time constant and parameters of the stiffness compensator that provide the best performance in terms of both stability and accuracy of haptic display.We show that the parameters of a simple outer-loop stiffness compensator can be optimized to result in a stable and accurate display of virtual environments with stiffness values in a large range, that also comprises values of virtual stiffness higher than the physical stiffness of the compliant element. A requirement for coupled stability is that the actuator is designed such that the minimum value of inertia connected to the compliant actuator load is higher than a control-defined threshold. Finally, we extensively analyze how the minimum value of interaction mass for coupled stability can be minimized through modulation of the stiffness compensator zeros and poles, considering realistic limitations in the velocity control bandwidth of non-backdrivable motors. Our analysis, validated through both numerical simulations and experiments, opens the possibility for alternative approaches to the design of compliant actuators, whereby rendering of high stiffness is possible if the load mass is always higher than a determined threshold.

Passive reach and grasp with functional electrical stimulation and robotic arm support.

Rehabilitation of arm and hand function is crucial to increase functional independence of stroke subjects. Here, we investigate the technical feasibility of an integrated training system combining robotics and functional electrical stimulation (FES) to support reach and grasp during functional manipulation of objects. To support grasp and release, FES controlled the thumb and fingers using Model Predictive Control (MPC), while a novel 3D robotic manipulator provided reach support. The system's performance was assessed in both stroke and blindfolded healthy subjects, where the subject's passive arm and hand made functional reach, grasp, move and release movements while manipulating objects. The success rate of complete grasp, move and release tasks with different objects ranged from 33% to 87% in healthy subjects. In severe chronic stroke subjects especially the hand opening had a low success rate (<25%) and no complete movements could be made. We demonstrated that our developed integrated training system can move the passive arm and hand for functional pick and place movements. In the current setup, the positioning accuracy of the robot with respect to the object position was critical for the overall performance. The use of a higher virtual stiffness and including feedback of object position in the robot control would likely improve the relative position accuracy. The system has potential for post-stroke rehabilitation, where support could be reduced based on patient performance which is needed to aid motor relearning of reach, grasp and release.

Stable Discrete-Time Impedances for Haptic Systems With Vibration Modes and Delay

The stability of haptic rendering is affected by many factors that limit the range of impedances that can be applied to virtual objects. This paper addresses the effect of the vibration modes of the haptic interface on the stability of the impedance control loop. It is well known that experimental stability boundaries present complex shapes, making it difficult to predict the final Z-width of the haptic system. This paper shows how the vibration modes of the mechanical interface highly affect the size of the Z-width, causing a sudden reduction in the critical virtual stiffness K if the virtual damping B is increased beyond a certain value. The inclusion of the most significant vibration modes in the theoretical model of the haptic system-together with the viscous damping, time delay and sampling rate-makes it possible to obtain the stable impedances associated with the haptic device. A PHANToM Premium 1.0 haptic interface was used as a test bed to validate this paper. Although results have been tested only on this device, this paper proposes a methodology for obtaining the Z-width that can be generalized for any other haptic system.

일차 홀드 방식의 반력 구현 시스템에 대한 안정성 해석

This paper presents the effect of a reflective force computed from a first-order-hold method on the stability of a haptic system. A haptic system is composed of a haptic device with a mass and a damper, a virtual spring, a sampler and a sample-and-hold. The boundary condition of the maximum virtual stiffness is analytically derived by using the Routh-Hurwitz criterion and the condition shows that the maximum virtual stiffness is proportional to the square root of the mass and the damper of a haptic device and also is inversely proportional to the sampling time to the power of three over two. The effectiveness of the derived condition is evaluated by the simulation. When the reflective forces are computed by using the first-order-hold method, the maximum available stiffness to guarantee the stability is increased several hundred times as large as when the zero-order-hold method is applied.

Motor adaptation with passive machines: A first study on the effect of real and virtual stiffness

Motor adaptation to novel force fields is considered as a key mechanism not only for the understanding of skills learning in healthy subjects but also for rehabilitation of neurological subjects. Several studies conducted over the last two decades used active robotic manipulanda to generate force fields capable of perturbing the baseline trajectories of both healthy and impaired subjects.Recent studies showed how motor adaptation to novel force fields can be induced also via virtual environments, whereas the effects of the force are projected onto a virtual hand, while the real hand remains constrained within a channel. This has great potentials of being translated into passive devices, rather than robotic ones, with clear benefits in terms of costs and availability of the devices. However, passive devices and virtual environments have received much less attention at least with regard to motor adaptation.This paper investigates the effects of both the real and virtual stiffness on motor adaptation. In particular, we tested 20 healthy subjects under two different real stiffness conditions (Stiff Channel vs Compliant Channel) and two different virtual conditions (Viscous vs Springy). Our main finding is that compliance of the channel favours a better adaptation featured with less lateral errors and longer retention of the after-effect. We posit that the physical compliance of the channel induces a proprioceptive feedback which is otherwise absent in a stiff condition.

Learning Collaborative Impedance-Based Robot Behaviors

Research in learning from demonstration has focused on transferring movements from humans to robots. However, a need is arising for robots that do not just replicate the task on their own, but that also interact with humans in a safe and natural way to accomplish tasks cooperatively. Robots with variable impedance capabilities opens the door to new challenging applications, where the learning algorithms must be extended by encapsulating force and vision information. In this paper we propose a framework to transfer impedance-based behaviors to a torque-controlled robot by kinesthetic teaching. The proposed model encodes the examples as a task-parameterized statistical dynamical system, where the robot impedance is shaped by estimating virtual stiffness matrices from the set of demonstrations. A collaborative assembly task is used as testbed. The results show that the model can be used to modify the robot impedance along task execution to facilitate the collaboration, by triggering stiff and compliant behaviors in an on-line manner to adapt to the user's actions.

Passive Multirate Wave Communications for Haptic Interaction in Slow Virtual Environments

Haptic interaction in slow virtual environments (VEs) can become unstable due to the phase lag introduced in the control loop by the slow update rate of the VE. Increasing the physical damping and/or limiting the contact stiffness rendered to users can mitigate the destabilizing effect of the low VE update rate. However, large physical damping and compliant virtual contacts decrease the sense of presence in VEs, especially during interaction with rigid virtual objects. To increase the maximum virtual contact stiffness that can be rendered to users without increasing the interface damping, this paper proposes a control strategy based on multirate wave communications between a haptic interface and a VE updated at a slow and fixed rate. The multirate wave communications are shown to be guaranteed passive only if the decrease of the wave sampling rate at the connection between the haptic interface and the VE does not cause aliasing. Therefore, an antialiasing low-pass filter is placed before the wave rate drop in the communications. The passivity condition is verified analytically and numerically for multirate haptic interaction in VEs with various contact stiffnesses and update rates. The transparency of haptic interaction in slow VEs to which users connect via passive multirate wave communications is investigated analytically in the frequency domain. Experiments validate that passive multirate wave communications can render stiffer contact in slow VEs than conventional direct coupling, and illustrate the destabilizing effect of the aliasing caused by the sampling rate drop in the communications.

Control of Haptic Systems with Time-varying Delay: a Novel Approach

This paper addresses the stable implementation of high stiffness interactions of objects in the virtual environment. The virtual wall is modeled as a mechanical system including a linear spring characterized by a virtual stiffness and a damper with a virtual damping in parallel. A novel approach for improving the stability of discrete-time haptic systems in the case of time-varying delay is proposed by using an augmented state observer instead of the traditional backward finite differentiator. Based on Lyapunov's stability theory, the delay-dependent asymptotic stability condition expressed in terms of LMI for choosing the virtual-wall parameters is presented. The numerical simulation results in the case of the PHANTOM 1.0 haptic devices are included in order to prove the effectiveness of the proposed method.

The bipenalty method for arbitrary multipoint constraints

SUMMARY In finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single-point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one-dimensional contact–impact formulation and a two-dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility. Copyright © 2012 John Wiley & Sons, Ltd.

Non-singular slip (NSS) method for longitudinal tire force calculations in a sudden braking simulation

In a dynamic vehicle simulation, longitudinal tire force is primarily based on the longitudinal slip (ratio). In the longitudinal slip formula, state variables are used in the denominator. This causes a divergence problem for numerical simulations of vehicle dynamics. To avoid this numerical singularity, a differential slip calculation method was developed for use in dynamic vehicle simulations. However, this method also causes a singularity when the wheel velocity approaches zero in a pure slip state, such as during sudden braking. In this paper, a new longitudinal slip calculation method, which can overcome singularities in all velocity conditions, is proposed. For this purpose, the Taylor series is adapted to the slip formula and the idea of virtual wheel rotation stiffness is introduced for the development of the slip equation. The physical phenomenon at the zero slip state is analyzed. Finally, the proposed slip formula is used to solve the numerical singularity problem, and the non-singular slip (NSS) calculation method is proposed. The proposed NSS method is applied to tire model performance test (TMPT) simulations to validate its performance.

Influence of Vibration Modes and Human Operator on the Stability of Haptic Rendering

Developing stable controllers, which are able to display virtual objects with high stiffness, is a persistent challenge in the field of haptics. This paper addresses the effect of internal vibration modes and the human operator on the maximum achievable virtual stiffness. An 11-parameter mechanical model is used to adequately characterize the overall system dynamics. Experiments that are carried out on LHIfAM and PHANToM haptic interfaces demonstrate the importance of vibration modes to determine the critical stiffness when the user grasps the device.

Modeling of Configuration-Dependent Flexible Joints for a Parallel Robot

This paper provides a method to determine the variable flexible joint parameters which are dependent on configurations for a PRS Parallel Robot. Based on the continuous force approach, virtual springs were used between the joint components to simulate the joint flexibility. The stiffness matrix of the joint virtual springs was derived. The method uses system dynamic characteristics in different configurations to set the virtual spring stiffness for all the joints in the system. Modal testing was conducted on a set of selected robot configurations to determine the system natural frequencies and mode shapes along with their variation. To obtain the virtual spring stiffness, the system was condensed at the joint nodal coordinates. Then eigen-sensitivity analysis was conducted on the condensed system matrices with respect to the stiffness parameters of the joint virtual springs. Thus, the virtual spring parameters in the model can be set to match the variation of the system dynamic responses with the robot configuration changes. The virtual spring parameters between the selected robot configurations were obtained by interpolation. The research indicates that the method is effective and relatively easy to conduct, compared to other methods. The variable flexible joint model is applicable to flexible multibody systems with variable configurations.

Minimisation of the sound power radiated by a submarine through optimisation of its resonance changer

An important cause of sound radiation from a submarine in the low frequency range is fluctuating forces at the propeller. The forces are transmitted to the hull via the shaft and the fluid. Sound radiation occurs due to hull and propeller vibrations as well as dipole sound radiation caused by the operation of the propeller in a non-uniform wake. In order to minimise sound radiation caused by propeller forces, a hydraulic vibration attenuation device known as a resonance changer can be implemented in the propeller/shafting system. In this work, cost functions that represent the overall radiated sound power are investigated, where the virtual stiffness, damping and mass of the resonance changer were chosen as design parameters. The minima of the cost functions are found by applying gradient based optimisation techniques. The finite element and boundary element methods are used to model the structure and the fluid, respectively. The adjoint operator is employed to calculate the sensitivity of the cost function to the design parameters. The influence of sound radiation due to propeller vibration on the optimisation of the resonance changer as well as the influence of the reduction in amplitude for higher harmonics of the blade-passing frequency on the control performance is investigated.

Stability Boundary for Haptic Rendering: Influence of Damping and Delay

The influence of viscous damping and delay on the stability of haptic systems is studied in this paper. The stability boundaries have been found by means of different approaches. Although the shape of these stability boundaries is quite complex, a new linear condition, which summarizes the relation between virtual stiffness, viscous damping, and delay, is proposed under certain assumptions. These assumptions include a linear system, short delays, fast sampling frequency, and relatively low physical and virtual damping. The theoretical results presented in this paper are supported by simulations and experimental data using the DLR light-weight robot and the large haptic interface for aeronautic maintainability (LHIfAM).

Adaptive vibration control using a virtual-vibration-absorber controller

A control algorithm emulating a dynamic vibration absorber (DVA) is developed for a flexible structure subject to harmonic disturbances of uncertain frequency. The virtual vibration absorber is mathematically equivalent to a passive DVA, but its stiffness, inertia and damping coefficient are adjustable by software. Stiffness of the virtual spring is tuned according to the phase difference between the acceleration of the primary body and the displacement of the virtual mass. The adaptation algorithm consists of a phase detector with a low-pass filter, similar to that found in a phase-locked loop. Both undamped and damped vibration absorbers are developed; the former has the advantage of cleaner vibration neutralization while the latter has a smoother stiffness adaptation. Adaptation rate of the virtual stiffness is analyzed in detail. The effectiveness of the proposed method is confirmed by simulations and real-time experiments.