Variable Stiffness Mechanism for suppressing unintended

Variable Stiffness Mechanism for suppressingunintended forces in physical Human Robot

Interaction.

Sri Sadhan JujjavarapuHuman In the Loop System Lab.

University at Buffalo, SUNYBuffalo, New York 14260

Email: [email protected]

Amirhossein H. MemarHuman In the Loop System Lab.



M. Amin KaramiIDEAS Laboratory,



Ehsan T. Esfahani ∗Human In the Loop System Lab.



ABSTRACTThis paper presents the design of a two-degree-of-freedom variable stiffness mechanism and demonstrates how

its adjustable compliance can enhance the robustness of physical human-robot interaction. Compliance on the grasphandle is achieved by suspending it in between magnets in pre-loaded repelling configuration to act as nonlinearsprings. By adjusting the air gaps between the outer magnets, the stiffness of the mechanism in each direction canbe adjusted independently. Moreover, the capability of the proposed design in suppressing unintended interactionforces is evaluated in two different experiments. In the first experiment, improper admittance controller gain leadsto unstable interaction, whereas in the second case, high-frequency involuntary forces are caused by the tremor.

1 INTRODUCTIONIn recent years, adjustable compliance actuators or variable stiffness actuators (VSA) have received much attention in

wearable robotics, rehabilitation robotics, and home-based interactive robots [1, 2]. In comparison with stiff actuators, theycan improve stability during physical Human-Robot Interaction (pHRI). The interaction safety can also be improved byminimizing large external forces due to shocks [1, 2]. Placing an elastic element at the port of interaction provides two mainbenefits: first, it can store and release energy when required and second, it can act as a low-pass filter for attenuating high-frequency involuntary forces. Moreover, compliant actuators reduce the reflected inertia during human-robot interaction[3, 4].

Complaint actuators can be classified into two major categories, active and passive depending upon their adaptabilityand energy storing capability [1]. Passive compliant actuators consist of an elastic element to store energy, whereas activecompliant actuators use control strategies to mimic the spring-like behavior [5]. Therefore, the adjustable behavior is aninherent hardware property [6] of passive actuators, whereas it’s a control strategy in the active actuators.

During a pHRI task, the robot senses the input force or motion from a human and produces corresponding reactions.Several control strategies are proposed for effective pHRI, out of which two popular control strategies are the impedancecontrol [7] and the admittance control [8,9]. They both deal with the modulation of inertia, stiffness, and damping parametersof the apparent dynamics simulated by the robot for efficient interaction. Hogan [7] suggested the use of impedance controlfor manipulators by describing the environment as admittance and the manipulator as impedance. However, the bandwidth ofachievable forces is highly limited due to the possible non-collocation of the force sensor and the actuator [10]. On the otherhand, the admittance control strategy can achieve a wider bandwidth of apparent dynamics [11] and thus are more effectivein human-robot cooperation [12]. However, designing the admittance controller for a low-effort task by selecting the desired

∗Address all correspondence to this author.

ESFAHANI JMR-18-1384 1

inertia below the end-effector’s physical inertia can lead to instability [12]. The interaction stability in such situations isgreatly affected by the human limb stiffness [13] Attaching a passive compliant element to the end-effector and controllingthe endpoint stiffness can improve the interaction stability in these cases.

Passive compliant elements are also used as low-pass filters in some health-care robotic applications for suppressingtremor [14, 15]. Tremor is a rhythmic and involuntary movement caused due to the oscillatory innervation of antagonisticmuscle fibers [16]. In particular, physiological tremor has small amplitudes in the bandwidth of 8-16 Hz, and affects thelimbs and causes significant difference in the precision of hand movements. Some of the tremor suppression strategiesinclude biomechanical loading of the limb [17, 18], passive devices such as dynamic vibration absorbers [15] and activecontrol strategies based on jerk minimization [19].

Adelstein [17] modeled the human musculoskeletal system as a second order dynamical system whose frequency re-sponse is characterized by a low-pass filter with cut-off frequency subject to the stiffness, inertia and damping parameters ofthe limb. This concept can be used to design an elastic element that can act as a low-pass filter to suppress the effect of invol-untary forces. By placing this elastic element at the end-effector of a lightweight robotic arm, the high-frequency impedanceof the actuator is limited to the low-frequency stiffness of the elastic element [20]. This can prevent the transmission ofhigh-frequency forces and displacements to the system being operated. In this context, a VSA can be used in suppressing aparticular bandwidth of frequency by adjusting the stiffness of the end-effector without affecting the inherent dynamics ofthe robot.

In this regard, this paper presents a two degree of freedom (DoF) variable stiffness mechanism (VSM) that can be used toimprove the robustness of interaction in pHRI. The proposed design is based on permanent magnets in a pre-loaded repellingconfiguration. A data-driven model is used to estimate the stiffness of the mechanism based on the air gap between actuatedmagnets. Moreover, frequency responses are used to study the behavior of the system around the characteristic frequency. Fi-nally, the effectiveness of this mechanism in suppressing high-frequency forces caused due to tremor or admittance controlleris demonstrated and discussed.

2 MODELING AND DESIGN OF VSMThe design concept of the proposed variable stiffness mechanism is illustrated in Fig.1. The handle is attached to a

passive magnet that is suspended by springs in between two actuated magnets in an antagonistic configuration. The nonlinearsprings between the blocks provide the capability of adjusting the stiffness of the system by controlling the distance betweenthe actuated blocks [1]. We previously used permanent magnets in a pre-loaded repelling configuration as nonlinear springsto develop a variable stiffness gripper [21] that can be used for safe object handling during dynamic manipulation [22].Motivated from this design, permanent magnets in repelling antagonistic configuration are used in two orthogonal directionsto allow stiffness adjustment in planar motions. Magnets are selected in the design of the VSM as the interaction forcesbetween them are non-contact and thus can allow zero separation, unlike springs that impose minimum and maximumseparation constraints. For convenience, the external magnets are referred to as ‘active magnets’ and the suspended magnetas the ‘passive magnet’ in this paper. A pulley-belt assembly attached to a servo motor with position control is used tocontrol the position of active magnets. The active magnets can be displaced in an equal and opposite direction as shown bysimilar color arrows in Fig.1C. Cylindrical permanent magnets of diameter 15 mm, height 5 mm, and uniform magnetization6.05x105 A/m in the pre-loaded repelling configuration are chosen to serve as nonlinear springs. The selection of thesemagnets is based on the size (compact design) and the required bandwidth of repulsive forces (maximum of 25 N for alow-effort HRI) in an antagonistic configuration.

2.1 MECHANICAL DESIGNFig.2 displays the two DoF variable stiffness mechanism and its components. The movement of all magnet blocks are

linearly constrained with the help of low friction linear guides and the displacement of passive magnets from the equilibriumposition are measured using sliding potentiometers as shown in Fig.2B.

Active magnets are controlled by a pulley belt sub-assembly mounted on the shaft of a servo motor with positionencoder. The interaction between each magnet pair is non-contact as long as the applied force is within permissible limits.If the applied force exceeds the opposing magnetic forces, the passive block makes a contact with the active block in thedirection of applied force and the stiffness can no longer play a role.

2.2 MAGNET FORCE MODELA data-driven model has shown to best represent the behavior of the magnets within the feasible range of motion [22].

For this purpose, we conduct a series of experiments in which a force sensor (PCB Piezotronics 208C03) is attached to thebase of VSM to measure the interaction forces between each magnet pair. The handle attached to the passive magnet is thendisplaced using a low-frequency sine sweep (0.5 Hz) and the force data is recorded for five different air-gaps. An expo-nential model, Fm(x) = C1e−C2(x), is used to fit the experimental data for different magnet air-gaps. The force-displacement


Fig. 1: A) Single DoF spring model B) Equivalent magnet model C) Interaction forces in magnet model.

relationship of the antagonistic configuration can be described by Eq.1.

Fmag = Fm(s− x)−Fm(s+ x) = 2C1e−C2s sinh (C2x) (1)

Where s is the air-gap between external magnets and x is the displacement of the passive magnet from the equilibrium posi-tion. A nonlinear least squares approach with Levenberg-Marquardt algorithm is used to calculate the modeling parametersas C1=28.41 and C2=206.35.

The experimental data and the corresponding fitted model are shown in Fig.3. Negative displacement represents move-ments towards the left and positive towards the right. The resultant stiffness can be obtained by differentiating Eq.1 withrespect to x and thus the slope of the force-displacement curve represents stiffness which also decreases as the air-gapbetween active magnets increases.

The forces between each magnet pair diminish significantly when the air-gap is greater than 0.02 m. Therefore, themodel is only effective if the air-gap on each side of the passive magnets is within a 0.02 m limit. It is also observed thatthe resultant force and stiffness at the equilibrium position is zero and exponentially increases towards both the ends. Anyexternal force Fe is compensated by the repelling forces due to magnets as shown in Fig.1C. More details on the bandwidthof stiffness and its dependency on air-gap are discussed in [23].

2.3 MODEL EVALUATION - SIMULATIONSUsing the fitted model for magnetic repulsive forces, one can study the change in frequency response of the system

with respect to its resultant stiffness. Two main factors affecting the frequency response of the system are the air-gap andmagnitude of the external force. For the purpose of verification, a single DoF sub-assembly is considered for simulation. Inthe presence of an external force Fe(t) applied on the passive magnet, the equation of motion for the single DoF-VSM aboutthe equilibrium position is described by Eq.2.

m Üx+ c Ûx+Fm(s, x) = Fe(t) (2)

Fig. 2: Mechanical Design of two DoF variable stiffness in A) Isometric view and B) Top View


Where m is the mass attached to the passive magnet, F(s− x) and F(s+ x) are the repelling forces between magnet pairs asa function of the air-gap (s− x) and (s+ x) respectively, and c Ûx represents the viscous damping term (to compensate for thedynamic friction due to the linear guide). In our simulation study, the external force is considered to be a sinusoidal functionof Fe(t) = Aesin(2π f t). Thus, Eq.2 can be simplified as Eq.3.

m Üx+ c Ûx+2C1e−C2s sinhC2x = Aesin(2π f t) (3)

The frequency response of the system is characterized by the resultant stiffness which in turn depends on the air-gapbetween magnets and the displacement of the handle (Fig.3). To understand the influence of air-gap, magnitude of force, andfrequency on the dynamic behavior of the single DoF system, the differential equation (Eq.2) is evaluated using an explicit4th order Runge-Kutta method with variable time step for the force range of 0.4–1.2 N, frequency of 2–20 Hz and the air-gapbetween active magnets 0.04–0.01 m. In all the simulation cases, a damping constant of 0.1 N-s/m and attached mass of 0.18kg is considered.

A bandwidth of force and frequency are considered to evaluate the mechanism for tremor suppression, because tremoris known to exist at frequencies greater than that of voluntary movement (i.e. in the range of 4-16 Hz) and the amplitudeof tremor is significantly lower than voluntary movement. However, the intensity of tremor can vary from task to task andindividual to individual. Therefore, the response of the handle for different force and frequency is studied in the first set ofsimulations. Force is varied (0.4-1.2 N) for a constant air-gap (10 mm) and the results for this case are presented in Fig.4A.

It is observed that the peak shifts towards the right with an increase in the magnitude of applied force. This is obviousfrom the fact that large force amplitudes cause larger deviations (x) of the handle from its equilibrium position and thusexperiences higher stiffness (displacement from equilibrium position increases slope). Therefore, the magnetic air gapshould be properly selected to suppress a particular bandwidth of frequency.

In the second set of simulations, the frequency response is studied for varying air-gaps and constant input force (0.45N). Fig.4B illustrates the shift in peak corresponding to the natural frequency towards a lower frequency by increasing theair-gap, this shows the significance of the air-gap in decreasing the stiffness of the system.

2.4 MODEL EVALUATION - EXPERIMENTSThe single DoF mechanism is attached to the base of a shaker table in the horizontal direction to avoid the effect of

gravity on the magnets (Fig.5). The linear actuator is connected to the handle by aligning it’s axis to the allowable motion ofthe passive magnet. A force sensor is attached at the interface connecting the handle and the actuator to monitor and ensureconstant input force throughout the experiment. The position of active magnets is controlled using a servo motor constantlypowered at 11 Volts.

A constant amplitude sine wave is used for actuating the shaker with the oscillation frequency increasing at the rate of1 octave/min in the 2-20 Hz range. A high precision laser vibrometer is used to record the displacement and velocity of thehandle.

Fig. 3: Force-Displacement characteristics of the antagonistic configuration measured for different air-gaps between themagnets. The solid lines are the average of the loading and unloading cycle (dotted lines) for each condition. The dashedline corresponds to the equilibrium point.


Fig. 4: Frequency response of simulation data A) Constant air-gap (10 mm) and varying force input, B) Constant force input(0.45 N) and varying air-gap

The first set of experiments are performed by keeping the air-gap constant (10 mm) and changing the force (0.4-1.2N). The displacement transfer function for each amplitude is then plotted against frequency in Fig.6A. It can be seen thatwith the increase in the amplitude of input force, the peak shifts towards the right (similar to simulation results in Fig.4A).When the air-gap is decreased, higher forces are encountered by the passive magnet on each side which in turn increases thestiffness and natural frequency.

In the second set of experiments, response of the system for a constant input force (0.45 N) and varied air-gap from 3mm (high stiffness) to 19 mm (low stiffness) are demonstrated in Fig.6B. Experimental results are similar to the simulationresults (shown in Fig.4B) i.e. the air gap affects the natural frequency of the system. The frequency bandwidth of the VSMobserved in these experiments is in the range of 3-20 Hz due to the minimum and maximum separation constraints. Theslight difference between the natural frequency of the system in experimental and simulation results is due to inadequateinformation to model friction during simulations.

3 EXPERIMENTAL VALIDATION3.1 EXPERIMENT SETUP

An experiment is designed to demonstrate the effectiveness of VSM for stabilizing and suppressing high-frequencyforces during physical human-robot interaction. The two DoF VSM is mounted on top of a 6-axis force/torque sensorattached to the end effector of a light-weight 6-DoF robot arm (SCHUNK PowerBall-LWA). Furthermore, a cartesian space-based admittance controller is implemented on the robot to mimic the behavior of a second order dynamic system with null

Fig. 5: Test rig for experimentation.


Fig. 6: Frequency response of experimental data A) Constant air-gap (10 mm) and varying force input, B) Constant forceinput (0.45 N) and varying air-gap

stiffness. This controller consists of two feedback loops, the first one to convert the input force from human to a desiredend-effector position (Eq.4) and the second to track the desired joint velocities of the robot (Eq.5).

Eq.4 represents a virtual second order dynamical system of inertia, M ε R6X6 and damping, C ε R6X6 that converts thehuman input force (Fh ε R

6) to the end-effector velocity (vd ε R6).

M Ûvd +Cvd = Fh (4)

The end-effector velocity is mapped to the joint space velocity using the inverse of Jacobian (J) as represented in Eq.5.The input force is sampled at 256 Hz and the desired joint space velocity is attained with the help of a closed loop feedbackcontrol.

Ûqd = J(q)−1vd (5)

Input force to the admittance controller (Fi) is modulated by the VSM that is used to couple the human and robot at theinteraction port, the equivalent block diagram representation of the complete setup is shown in Fig.7.

3.2 HUMAN SUBJECT STUDYA single subject study with eight different stiffness settings (randomized during each session) is conducted after ob-

taining an approval from the Institutional Review Board (IRB no.030-801361). The experiment comprises of four differentinteraction configurations to assess the performance of the variable stiffness mechanism in suppressing unintended forces inpHRI. These cases are shown in Fig.8 and are as follow:

Case-1: Stiff handle which is used as the control case.Case-2: 2 Dof variable stiffness mechanism.Case-3: Stiff handle in presence of simulated tremorCase-4: VSM in presence of simulated tremor

To simulate the tremor in the last two cases, the subject is asked to wear a glove on which a DC motor with an offsetmass mounted (Fig.8C). The frequency of oscillation is tuned to be around ∼ 12 Hz by changing the input to the DC motor.

In all cases, the subject is asked to perform a visuomotor task which includes moving the handle inside the boundaries ofa maze (Fig.7B) consisting of a fine and a gross motor region. For the purpose of this experiment, the end-effector movementis restricted to planar motion (X-Y plane) and the corresponding velocities are transformed to position and displayed on acomputer screen as visual feedback to the subject. A total of 18 levels were selected, 1 each in case-1 and case-3, and 8 each


in case-2 and case-4 (8 different magnet separations). The magnet separation can be varied continuously from 0-30 mm oneach side of the passive magnet but for the subject study, 8 separations equally spaced between 3-17 mm were considered.Prior to performing the experiment, the subject has practiced the levels several times and is accustomed to the settings of theVSM and the robot. Each level requires the subject to perform two trials of the visuomotor task during which the input force,the end-effector’s velocity and the displacement in X and Y directions are recorded.

4 RESULTS AND DISCUSSION4.1 STABILITY DURING INTERACTION

The selected inertia and damping values for the admittance controller of the robot are 3 kg and 25 Ns/m respectively.Here, the inertia of the apparent dynamics simulated by the admittance controller is selected below the physical inertia ofthe end-effector. These admittance parameters are chosen such that the task requires low-effort (low inertia and damping)during an interaction, but at the expense of interaction stability caused due to the stiffness variation in human limb [12].

Comparing the outcome of case-1 and case-2, one can examine if adjusting the stiffness in VSM would, in fact, improvethe stability of interaction. Furthermore, the comparison of case-3 and case-4 can demonstrate the effectiveness of VSM insuppressing the high-frequency unintended forces during pHRI.

The mechanism proposed in this work creates an interface to stabilize the cooperative task by varying the stiffness of themechanism and thus affecting the resultant stiffness of the combined setup (human and mechanism) at the interaction port.Note that the VSM modulates the components of the human input force (Fh) to the admittance controller Fi by varying theend-point stiffness.

To better understand the components of force (Fi) causing instability in an admittance controller, a frequency domainanalysis is performed using force and velocity signals recorded during the experiment [12]. The Power Spectrum Density(PSD) of the force and velocity signals from each trial are obtained using the Welch method with a two second Hanningwindow and 50% overlap. Post experiment completion, the resultant PSD’s of force and velocity are averaged across trialsand is shown in Fig.9A.

The black solid line represents the PSD corresponding to the experiment performed without the mechanism (case-1), andthe blue and red dashed lines represents the PSD corresponding to maximum and minimum stiffness respectively (case-2).From the PSD values of force, it is clear that the oscillations due to an unstable motion for the designed admittance controllerare more predominant around the ∼ 5 Hz region (represented by a rectangular patch in the figure). It is also observed duringthe experiments that the power of the peak decreased significantly (from blue to red) by decreasing the stiffness of themechanism. The bounded grey band represents the bandwidth of PSD’s achievable for different levels of stiffness (from high

Fig. 7: A) Human-robot coupling at the interaction port using the 2 DoF VSM. Fh and Vh represents the input force andoutput velocity respectively at the interface. Fi and Vo represents the input-output pair of robot’s admittance control. B)Visuomotor task (Fine and gross region)


Fig. 8: Representation of three different configuration: A) Stiff handle, B) Mechanism attached, C) Motor with offset massattached to human limb to simulate tremor

Fig. 9: PSD of force and velocity for A) case-1 and case-2, B) case-3 and case-4. Grey band corresponds to regions ofinterest, 3–6.5 Hz in A) and both 3–6.5 Hz and 9–16 Hz bands in B).

and low).Based on these observations, the contribution of unstable force components can be decreased by decreasing the stiffness

of the mechanism during the interaction.

4.2 HIGH FREQUENCY FORCE SUPPRESSIONTo assess the effectiveness of VSM in suppressing unintended high-frequency forces, we present and discuss the results

of case-3 and case-4 in the presence of tremor. Haptic stability observer (Eq.6) [24] is often used to quantify the level ofstability in pHRI with an admittance control robot [12, 25].

I =∑ωs/2

ω=ωcPf (ω)∑ωs/2

ω=ωoPf (ω)

(6)

A modified version of this index (Eq.7) is used to differentiate tremor from the controller instability.

IR =

∑ωuω=ωl

Pf (ω)∑ωs/2ω=ω0 Pf (ω)

(7)

Where ωs is the sampling rate (ωs/2 using Nyquist criteria), ω0 is the lowest frequency corresponding to the PSD (Pf ).ωl and ωu are the lower and upper bounds of the frequency band of interest. In the present experiment, two frequencybands are considered, the first band corresponds to controller instability (3–6.5 Hz) as mentioned in the previous sub-sectionand the second band corresponds to the frequency caused due to tremor (9–16 Hz) as shown in Fig.9B. The effect of thesehigh-frequency forces on the end-effector displacement for four different settings of the VSM is shown in Fig.10.


Fig. 10: Displacement of the end-effector in the fine motor region A) Case-3, B–D) Case-4 with maximum, optimum andminimum stiffness, respectively.

Table 1: Instability indices corresponding to the two frequency bands for different levels of stiffness.

Air-gap Stiffness IR (3-6.5Hz) IR (9-16Hz)

X (Handle) High .356 0.072

3 mm 6937 N/m .230 0.019

5 mm 5392 N/m .097 0.019

7 mm 4477 N/m .028 0.019

9 mm 3918 N/m .012 0.013

11 mm 3568 N/m .011 0.012

13 mm 3346 N/m .005 0.016

15 mm 3202 N/m .004 0.026

17 mm 3109 N/m .007 0.020

The modified instability index is calculated for these two frequency bands using the PSD of force signal without (case-3)and with (case-4) the mechanism and the results are tabulated in Table 1. The range of IR lies in-between 0 and 1, wherevalues close to 0 correspond to higher attenuation of forces and values close to 1 correspond to lower attenuation of forcesin that frequency band.

The instability index values for eight different magnet separations are represented alongside the index for the stiff handlein the two frequency bands using a bar plot (Fig.11). It is observed that decreasing the stiffness leads to higher stability inthe first frequency band whereas the attenuation increases and then decreases with the continuous decrease in stiffness in thesecond frequency band. Such kind of behavior is observed when the tremor frequency is close to the natural frequency of thesystem for that particular level of stiffness. Therefore in such situations, a choice of optimum stiffness level has to be madebased on the instability index values to suppress most part of the involuntary forces in both the frequency bands. It is alsoobserved that when the mechanism is attached (case-4), index values in both the frequency bands (for all stiffness levels) arelower compared to the results in case-3. This proves the effectiveness of the mechanism in stabilizing the controller duringstiff interaction and also in the case of the tremor.

The index values presented in this paper are obtained from the experimental data post-trial completion. These indexvalues act as a reference to obtain an optimum level of stiffness for a subject.

5 CONCLUSIONThis paper presents the design and validation of a two-DoF passive variable stiffness mechanism using permanent

magnets. The interaction forces between magnets in an antagonistic configuration are modeled as a function of air-gapbetween them and the displacement of the center magnet from the equilibrium position. It is shown that the mechanismendpoint stiffness can be varied by changing the air-gap between magnets. The design concept is validated using simulationsand experimental results by characterizing the stiffness variation as a change in the behavior of its frequency response. Theseresults show that the stiffness at the interaction port is effected by air-gap between active magnets as well as the magnitude


Fig. 11: Instability index of handle (black) and VSM with 8 different stiffness (green) in two frequency bands: A) 3-6.5Hzband (instability due to controller), B) 9-16Hz band (tremor)

of the input force.Moreover, a human subject study is conducted with the mechanism attached to the robot with an admittance controller

and validated for a motor-task. It is observed that by changing the stiffness of the mechanism at the port of interaction,the instability caused due to the stiff environment could be significantly reduced. The above experiment is further extendedto verify the effect of stiffness in suppressing the high-frequency involuntary forces caused due to tremor. Choosing aproper level of stiffness can suppress the high-frequency forces to some extent indicating the effectiveness of the proposedmechanism in reducing the backlash of involuntary components during pHRI.

Future work should address a real-time control strategy to extract the involuntary movements and reduce its effectby controlling the position of magnets. Moreover, because of the individual differences in muscle stiffness, the optimumstiffness of the proposed mechanism in pHRI is likely to be subject dependent which should be investigated further in futureresearch.

AcknowledgementsThis work was supported by the National Science Foundation under Grant No.1502287.

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List of Tables1 Instability indices corresponding to the two frequency bands for different levels of stiffness. . . . . . . . . . 9

List of Figures1 A) Single DoF spring model B) Equivalent magnet model C) Interaction forces in magnet model. . . . . . . 32 Mechanical Design of two DoF variable stiffness in A) Isometric view and B) Top View . . . . . . . . . . . 33 Force-Displacement characteristics of the antagonistic configuration measured for different air-gaps between

the magnets. The solid lines are the average of the loading and unloading cycle (dotted lines) for eachcondition. The dashed line corresponds to the equilibrium point. . . . . . . . . . . . . . . . . . . . . . . . 4

4 Frequency response of simulation data A) Constant air-gap (10 mm) and varying force input, B) Constantforce input (0.45 N) and varying air-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Test rig for experimentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Frequency response of experimental data A) Constant air-gap (10 mm) and varying force input, B) Constant

force input (0.45 N) and varying air-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A) Human-robot coupling at the interaction port using the 2 DoF VSM. Fh and Vh represents the input

force and output velocity respectively at the interface. Fi and Vo represents the input-output pair of robot’sadmittance control. B) Visuomotor task (Fine and gross region) . . . . . . . . . . . . . . . . . . . . . . . 7

8 Representation of three different configuration: A) Stiff handle, B) Mechanism attached, C) Motor withoffset mass attached to human limb to simulate tremor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9 PSD of force and velocity for A) case-1 and case-2, B) case-3 and case-4. Grey band corresponds to regionsof interest, 3–6.5 Hz in A) and both 3–6.5 Hz and 9–16 Hz bands in B). . . . . . . . . . . . . . . . . . . . 8

10 Displacement of the end-effector in the fine motor region A) Case-3, B–D) Case-4 with maximum, optimumand minimum stiffness, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

11 Instability index of handle (black) and VSM with 8 different stiffness (green) in two frequency bands: A)3-6.5Hz band (instability due to controller), B) 9-16Hz band (tremor) . . . . . . . . . . . . . . . . . . . . 10


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Variable Stiffness Mechanism for suppressing unintended