performance and stability limitations of admittance-based ......electromechanical haptic interface...

Abstract—In this paper, we describe an analytical model

developed to investigate the performance limitations of

admittance-based haptic interfaces. The model is used to

investigate the effects that position control bandwidth and outer

loop delay have on the stability and rendering range of an

admittance-based interface. We show that the performance, as

defined by both the minimum renderable mass and the

rendering frequency range, is directly related to the closed-loop

bandwidth of the inner-position loop and the amount of

additional delay in the outer rendering loop. In addition, we

show that the minimum renderable mass is directly proportional

to the damping provided by the user which implies a stronger

grip, with a higher damping, decreases the stable rendering

region of the admittance-based haptics device as opposed to an

impedance-based device where increased damping enhances

stability. Our results are validated using a one degree-of-

freedom admittance-based device.

I. INTRODUCTION

In general, haptic devices can be categorized as impedance-

type or admittance-type devices, based on the device

characteristics and the control schemes used. Impedance-type

devices are generally back drivable, operated open-loop, can

easily render low inertia and low stiffness and generally have

less output force. Impedance-based devices typically have

difficulty rendering high-impedances, such as a stiff wall or

large mass or inertia. Contrary to this, admittance-type

devices are generally non-back drivable, operated closed-

loop, used to render high inertia and high stiffness and can

generally render high output forces. Admittance-based

devices typically have difficulty rendering low-impedances,

such as small mass or inertias Performance of an impedance-based device is generally

evaluated based on its transparency or output impedance and its capabilities to render stiff virtual objects. Several researchers have studied the relationship between the physical characteristics of impedance-based devices, including device friction, stiffness, and encoder quantization, and the device’s rendering performance and stability limitations [1-8]. For example stability conditions for the simulation of a virtual wall using passivity is presented in [2] and factors such as sample-and-hold, inherent interface dynamics, displacement sensor quantization, and velocity filtering affecting the dynamic range of achievable impedances- Z-Width is presented in [3].

More recent work [9, 10] has focused on using energy-based approaches to account for phenomena affecting energy generation and dissipation during rendering and to improve control performance. Performance of impedance-based systems have also been measured by comparing desired vs the rendered closed-loop impedance which is related to the accuracy of the haptic display [11, 12].

Contrary to impedance-based devices, the performance of admittance-based devices is generally evaluated based on the device’s capability to render small mass or inertias. While considerable effort has gone into the design and control of high performance admittance-based devices [13-16], few have investigated the factors that affect performance. In this paper, we seek to identify and study the elements that affect the stability and performance of admittance-based haptic devices.

II. APPROACH

A. Dynamic System Model

Toward this end we have developed a dynamic model of an

admittance-based device and human user with sufficient

detail to allow for the analytical investigation of the various

characteristics that affect device performance and stability.

An overview of the one-degree-of-freedom model is given in

Fig. 1. The main elements of the model include the

electromechanical haptic interface dynamics, a coupled

human impedance model [17], and the rendered virtual

admittance.

Figure 1. Lumped-parameter model of admittance-type haptic device,

including coupling to human impedance model [17]. Im: motor inertia, md:

drive mass, N: gear ratio, fh: interaction force between user and device, xo:

motion of the device, xh: motion of the human, xm: desired user input, xd:

desired device input, mv, cv, kv: rendered virtual mass, damping and

stiffness respectively.

The device parameters were chosen to emulate a non-back

drivable haptic interface with the high gear reduction, typical

of an admittance-based device. The specific device

parameters used are listed in Table I. A block diagram

representation of system dynamic model, including the

Performance and Stability Limitations of Admittance-Based

Haptic Interfaces*

Chembian Parthiban, Michael Zinn, Member, IEEE

*Research supported in part by a grant from the National Science

Foundation (NSF Award No. IIS-1316271).

Chembian Parthiban is a Graduate Student at the University of

Wisconsin-Madison, Madison, WI – 53706 USA (e-mail:

[email protected]).

Michael Zinn is an Associate Professor in the Mechanical Engineering

Department at the University of Wisconsin-Madison, Madison, WI –

53706 USA (e-mail: mzinn@ wisc.edu).

© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

control structure, is shown in Fig. 2.

While several control structures have been suggested for

admittance-based devices, the most common approach is to

implement a high-bandwidth inner position loop nested inside

an outer rendering loop. Specifically, a force sensor is used

to measure the interaction forces between the device and the

user. The reference position command for the inner position

loop is derived from the desired virtual admittance and the

measured interaction force.

Here a lead-type compensator was implemented for the

inner position loop controller. Contrary to a simple PD

controller, the lead compensator can provide similar stability

margins while limiting high frequency amplification of the

force-sensor based signals. In this case, the lead compensator

was tuned to provide a phase and gain margin of 60 degrees

and 20 dB, respectively. These margins were chosen to

achieve a balance between position control speed and overall

stability robustness.

Figure 2. Block diagram representation of admittance-type haptic device.

Fh: measured force, Th,o: human impedances, Ie: Effective inertia given by

Im + Id/N2, Td: Time delay.

For our analysis, we have chosen a five-parameter human

impedance model specifically developed for the study of

haptic interface-human interactions [17]. Human-arm

impedance values vary greatly between subjects, tasks,

experimental apparatuses, and perturbation patterns and is

strongly dependent on the arm configuration [18, 19]. The

five parameter human impedance model developed by Speich

et al. [17] is a more general model than earlier models

developed and can be applied for both rigid and soft contact

grips. and better approximates the dynamics within the

specific frequency ranges for which compensators are

designed.

An overview of the model is shown in Fig. 1. The human

impedance model’s parameter values are listed in Table I. The

transfer function of the human impedance model is given as

( )

( )

( ) ( )

( ) ( )

3 2

1 1 1 2 1 2 1 2 1 2

2

1 2 1 2

h h h

o h

F s m b s m k b b s k b b k s k k

X s m s b b s k k

+ + + + +=

+ + + + (1)

1 The Nyquist plot has two distinct regions when the inner position loop’s

closed-loop bandwidth is greater than the resonant frequency of the human

TABLE I. MODEL PARAMETERS

Rotor Inertia (Im) 6.96 x 10-6 [Kgm2]

Gear Ratio (N) 260 [rad/m]

Drive Inertia (Id) 0.4181 [ Kgm2]

Mass (Mh) 4.5 [Kg]

Stiffness (k1) 48.8 [N/m]

Stiffness (k2) 375 [N/m]

Damping (b1) 4.5 [Ns/m]

Damping (b2) 7.9 [Ns/m]

B. Stability Assessment

Noting from Fig. 2 that the control structure is in the form of a simple feedback loop, the stability of the system can be directly analyzed using the Nyquist stability criterion. In this case, the open-loop transfer function used in the Nyquist stability analysis is a cascade of the virtual admittance with the coupled inner position loop controller and the human impedance model. The Nyquist plot for the modeled system is shown in Fig. 3. In this case, the virtual admittance has been specified as a unit mass and the closed-loop bandwidth has been set to 30 Hz.

Figure 3. Nyquist plot of the open loop transfer function, GOL(s), showing

stable, unstable and the conditionally stable regions (Note: The virtual

admittance is a unit mass).

Figure 4. Frequency response of the open loop transfer function, GOL(s),

showing the gain margin, GM, and the minimum renderable mass, mmin.

From the Nyquist plot we see that, as a function of the loop

gain, two distinct regions are present1. Specifically, for values of loop gain less than the threshold value, identified as Kcond in Fig. 3, the system is stable while for values of gain greater than Kcond the system is unstable. In the case where the virtual

impedance model (1 Hz – 5 Hz). For high performance systems under

consideration, this is generally the case.


admittance is a pure mass, these gain values correspond directly to the inverse of the rendered mass and the threshold value, Kcond , corresponds to the inverse of the minimum mass value that can be stably rendered by the system. In this case we can also assess the system stability directly from the frequency response plot (Fig. 4), which offers a more convenient method to visualize the system’s stability as a function of parameter variations. The minimum renderable mass can be found from the frequency response plot by evaluating the gain margin of the system with a virtual admittance set equal to 1. Here, the minimum mass is equal to the inverse of the gain margin.

To understand how each element in the model affects the stability of the system, an expanded frequency response plot, showing the response of the individual system components, is shown in Fig. 5. Features which are relevant to the performance assessment described in the next section include the following: � The human impedance can be approximated using

simplified low and high frequency models, partitioned by the resonant frequency of the human-impedance model. The low and high frequency impedance approximations are

a spring, � ��

��, and damper, � � ��,respectively.

� The closed-loop position controller is 2nd order, where the phase below and above the corner frequency approaches 0 and -180 degrees, respectively. The phase at the corner frequency is -90 degrees.

� The virtual admittance terms has a constant phase of -180 degrees

Figure 5. Overlaid frequency response plot of the closed loop position

controller tuned at 30 Hz bandwidth (Blue), the human impedance model

(Red), and the rendered virtual inertia of 1 kilogram (orange).

III. PERFORMANCE ASSESSMENT

In this section, we investigate the elements that play an important role in the performance of an admittance-based device. In general, there are several linear and non-linear characteristics present in a haptic interface which arises from the mechanical device itself and the computer interface such as device friction, backlash, actuator saturation, encoder

2 The corner frequency is located at the intersection of the low and high

frequency asymptotes of the closed-loop system’s frequency response

quantization and delays from sampling, to name a few. As mentioned earlier these characteristics have been studied extensively for impedance-based devices [10].

In the case of admittance-based systems, the device characteristics can be broadly categorized as affecting either the achievable closed loop bandwidth or impacting the signal sampling portion of the loop - manifested as loop delays. Characteristics such as backlash, friction and actuator saturation limit the achievable closed loop bandwidth while characteristics such as the ZOH, anti-alias or noise reduction filtering, contribute to an overall delay in the loop. As such, we can investigate the effect of a variety of device characteristics on system performance by focusing on the effects of (1) position loop bandwidth and (2) overall loop delay.

A. Closed Loop Position Controller Bandwidth

To analyze the effect of the closed loop bandwidth on the stability more precisely, the frequency response of the open loop transfer function for a range of position controller bandwidths was evaluated (see Fig. 6). As seen from Fig. 6, as the closed-loop bandwidth is increased, the minimum renderable mass decreases, as evidenced by the increase in gain margin. The relationship between minimum renderable mass and closed-loop position controller’s corner-frequency2,

ωc, has been plotted for a range of values (see Fig. 7).

Figure 6. Open-loop transfer function of the system model shown in

Figure 2, with varying bandwidth of the closed loop position controller.

Upon closer inspection of the open-loop system’s transfer function shown in Fig. 6, we see that when the controller’s corner-frequency is greater than the resonant frequency of the human-impedance model, the stability of the system can be evaluated using the high-frequency approximation of the human-impedance model discussed earlier. The open loop transfer function, GOL(s), using the high frequency approximation, is given in (2), where Xo(s) is the output position, Xd(s) is the desired position, b1 is the damping from the human impedance model and mv is the virtual mass being rendered.

( )( )

( )( )1 2

1oOL

d v

X sG s b s

X s m s

=

(2)

magnitude plot. The closed-loop bandwidth, defined at the -3 dB point of the

magnitude plot, is typically 1 to 2 times higher than the corner frequency.

Mag

nitud

e P

has

e (d

eg)

Frequency (Hz)

Human

impedance

Position

controller

( )

( )h

o

F s

X s 21

vm s

Rendered

mass

( )

( )o

d

X s

X s


The frequency at which the gain margin (and thus

minimum renderable mass) is evaluated, ωGM, is located at the point where the phase of the open-loop transfer function equals minus 180 degrees.

( )( )

( ) ( )2

11180

o GM

d GM v GM

X j

OL GM GMX j m jG j b j

ω

ω ωω ω∠ = − = ∠ + ∠ + ∠�

Noting that the phase of the human impedance model and rendered inertia are constant and equal to +90 and -180 degrees, respectively, it becomes apparent that the gain margin is evaluated at the frequency where the phase of the position controller equals -90 degrees, which occurs precisely at the

corner frequency, ωc, of the closed-loop inner position controller.

( )

( )90

o GM

GM C

d GM

X j

X j

ωω ω

ω∠ = − → =

� (3)

The minimum renderable mass, mmin, is determined by evaluating the magnitude of the open-loop transfer function,

GOL(s), at ωGM = ωc

( )( )

( ) ( )

( )

( )

1 2

1

1o COL C C

d C v C

o C

d C v C

X jG j b j

X j m j

X j b

X j m

ωω ω

ω ω

ω

ω ω

= ⋅ ⋅

= ⋅

Finally, noting that the magnitude of the Xo(s)/Xd(s) is approximately equal to 1 at the corner frequency and setting

( ) 1OL CG jω = , we can evaluate the minimum renderable

mass, mmin, as

1 1min

min

1C C

b bm

m ω ω= → = (4)

From (4) we see that the minimum renderable mass is inversely proportional to the inner position controller’s corner frequency or, equivalently, inversely proportional with the closed-loop bandwidth. A comparison of (4) with the numerical results obtained using the full human impedance model (see Fig 7) shows a close correlation between the complete and approximate model used to develop (4). The accuracy of (4) is good in cases where the inner position controller’s corner frequency is greater than the resonant frequency of the human impedance model. For typical values of human resonance [1-5 Hz], this assumption is quite reasonable.

Figure 7. Minimum renderable mass as a function of the closed-loop

position controller’s corner-frequency, ωc.

It is interesting to note that mmin is directly proportional to the damping provided by the user, which implies a stronger grip, with a higher damping, decreases the stable rendering region of the admittance-based haptics device as opposed to an impedance-based device where increased damping enhances stability [6].

B. Delay

The delay we consider here is in the outer loop, specifically

arising from the force sensor measurement that is dependent

on the sample rate and the delays associated with filtering of

the force signal. To analyze the effect of delay on the stability more

precisely, the frequency response of the open loop transfer function was evaluated over a range of delay values (see Fig. 8). To separate the effects of delay and bandwidth, we have assumed here that the closed loop position bandwidth is

infinite, such that ( ) ( ) 1o dX s X s = . As seen from Fig. 8, as the delay is increased, the minimum

renderable mass increases, as evidenced by the decrease in gain margin. The relationship between minimum renderable mass and loop delay, Td, has been plotted for a range of values (see Fig. 9).

Figure 8. Open-loop transfer function of the system model shown in

Figure 2, with varying loop delays.

Figure 9. Plot shows the minimum renderable inertia in kilograms as a

function of the delay Td in seconds.

Upon closer inspection of the open-loop system’s transfer

function shown in Fig. 8, we see that the frequency at which the gain margin (and thus minimum renderable mass) is


evaluated, ωGM, is located above the resonant frequency of the human impedance model for values of delay less than 30 ms. In these cases, the stability of the system can be evaluated using the high-frequency approximation of the human-impedance model discussed earlier. The open loop transfer function, GOL(s), using the high frequency approximation and

assuming that ( ) ( ) 1o dX s X s = , is given in (4), where Td is the delay in seconds.

( ) ( )( )1 21

dT s

OL

v

G s e b sm s

− =

(5)

The frequency at which the gain margin is evaluated, ωGM, is located at the point where the phase of the open-loop transfer function equals minus 180 degrees. Recalling that the phase of the human impedance model and rendered mass are constant and equal to +90 and -180 degrees, respectively, it becomes apparent that the gain margin is evaluated at the frequency where the phase of the delay term equals -90 degrees.

( )2 2

d GMT j

d GM GM

d

e TT

ω π πω ω

−∠ = − = − → = (6)

The minimum renderable mass, mmin, is determined by evaluating the magnitude of the open-loop transfer function,

GOL(s), at ωGM.

( )( )

1 2

1d GMT j

OL GM GM

v GM

G j e b jm j

ωω ωω

−= ⋅ ⋅

( ) 1OL GMv GM

bG j

mω

ω=

Recalling that 2GM dTω π= and rearranging, we see that

the minimum renderable mass, mmin, as a function of delay is evaluated as:

( )11 1

min

min 2

21 1

d

d

v GM T

b Tb bm

m m πω π= → = → = (7)

From (7) we see that the minimum renderable mass is

proportional to the loop delay. A comparison of (7) with the

numerical results obtained using the full human impedance

model (see Fig 9) shows a close correlation between the

complete and approximate model used to develop (7). The

accuracy of (7) is good in cases where ωGM in (6) is greater

than the resonance frequency of the human impedance model.

For values of delay less than 30 ms this assumption is quite

reasonable. C. Combined Effects of Delay and

Bandwidth

By examining the open-loop frequency response we can

analyze the combined effects of loop delay and inner-position

loop corner frequency on the minimum renderable mass (see

Fig. 10). For small values of loop delay, the minimum

renderable mass is inversely proportional to the inner position

loops corner frequency while for large values of inner position

loop corner frequency the minimum renderable mass is

proportional to loop delay. While the minimum renderable

mass is a function of both the delay and the inner-position

loop corner frequency, the contour plot in Fig. 10 can be

partitioned according to the dominant factor (either delay or

corner frequency) determining the minimum renderable mass.

The boundary between the two regions is determined by

equating the frequency, ωGM, at which the gain margin (and

thus minimum renderable mass) is evaluated.

Postion loop:

Loopdelay: 22

GM C

C

GM d

d

TT

ω ωπ

ωπω

=

==

Figure 10. Contour plot of the minimum renderable mass as a function of

the closed loop bandwidth and delay in the outer loop.

IV. OUTPUT IMPEDANCE

In the performance evaluation of impedance-type and admittance-type haptic devices, transparency, or output impedance more generally, is an important performance characteristic, as it directly measures the ability of the device to render zero forces in the presence of user-imposed device motion. Here we define the output impedance, Z(s) as the ratio of the device rendering force, F(s), to the device velocity, V(s):

( ) ( ) ( )Z s F s V s= .

In evaluating the device impedance, we use a modified version of the model shown in Fig. 2, where the coupled human impedance model is removed. The modified model is shown in Fig. 11 where

( )2

1

e

G sI s

= and ( )s z

D s Ks p

+=

+

The transfer function relating device velocity, V(s), to force, F(s), is derived from the modified model in Fig 11.

( )

( )

( ) ( )( )( ) ( )( )( )

2

21

dT s

v

v

G s m s ND s e sV s

F s G s D s Nm s

−+

=+

Figure 11. Block diagram showing the device and the virtual admittance

term and the user reflected force being fed back as a force disturbance.

The output impedance in general is defined as force output to the velocity

input Z(s) = F(S)/V(s).


Substituting in for G(s) and D(s), and forming the output impedance, Z(s)

( )( )( )

( )

2 3 2

3 2 2 2d d

v e e

T s T s

v v

m N I s I ps Ks Kz sZ s

m s m ps N Ke s N Ke z− −

+ + +=

+ + + (8)

In the case where there is no delay, the output impedance is given as

( )( ) ( )

( ) ( )

2 3 2

2 2

v e e

v

m N I s I ps Ks Kz sZ s

s p m s N K

+ + +=

+ + (9)

Using (8), we can investigate the output impedance and define a renderable frequency range over which the device can create the desired impedance. We present how (1) position loop corner frequency and (2) delay affect the desired output impedance of the device and the renderable frequency range.

A. Effect of Position Controller Bandwidth

Using (9), the output impedance was evaluated for various position controller corner frequencies (see Fig. 12).

Figure 12. Device output impedance plot for different closed loop

bandwidths (5, 30 & 100 Hz) rendering their respective minimum

rederable inertias.

The rendered virtual admittance for each plot is set equal to the minimum mass corresponding to the bandwidth calculated previously from the complete model (Fig. 7) – 65 grams, 9.5 grams and 2.8 grams respectively. It can be observed that within the corner frequency of the position controller, the output impedance matches closely to that of the desired rendered impedance, in this case a pure mass, and above the corner frequency of the closed loop controller, it is equal to that of the device’s uncontrolled reflected mass. The resonant peak is due to the effect of user reflected forces on the

device and, per (9), is located at 2vN K mω = .

Here we define the rendering range as the frequency, ωr, below which the output impedance has a phase of approximately 90 degrees, corresponding to that of a pure mass. We define this boundary at 135 degrees, which is equidistant between a pure mass, with 90 degrees of phase, and its derivative (jerk), with a phase of 180 degrees. Using this

definition, the rendering range can be shown to be directly

proportional to the corner frequency, ωC, of the position controller:

34r C

ω ω≅

B. Effect of Delay

Using (8) and assuming that the closed loop position

bandwidth is infinite, such that ( ) ( ) 1o dX s X s = , the output impedance was evaluated for various values of loop delay (see Fig. 13).

Fig. 13 shows the device output impedance as the loop delay is varied – 1 ms (blue), 5 ms (red), and 10 ms (yellow). The rendered virtual admittance for each plot is set equal to the minimum mass corresponding to the respective loop delay values calculated previously from the complete model (Fig. 9) – 2 grams, 29 grams and 46 grams respectively.

Figure 13. Device output impedance plot for 1,5 and 10 ms delay, with

the closed loop position controller tuned to a very high bandwidth.

Evaluating the rendering range, ωr, in the same manner as

described above, we see from Fig. 13 that ωr occurs at the frequency where the pure delay term has a phase of -45 degrees. The resulting expression shows that the rendering range is inversely proportional to the loop delay, Td.

4 4d rjT

r

d

eT

ω π πω

−∠ = − → =

V. EXPERIMENTAL VALIDATION

Experimental validation of the approach is demonstrated with an admittance-type haptic device shown in Fig. 14. The system under consideration was originally built as an MR compatible haptic device testbed [20] and was repurposed as an admittance device testbed. The testbed employs high output impedance piezoelectric actuators and a high gain position controller that can be tuned to achieve a bandwidth of 50 Hz [20].

A. Human Impedance Model Parameter Estimation

To evaluate the results obtained in Section III, we experimentally determined the human impedance model parameters, allowing us to reconstruct the experimental open loop frequency response of the system which in turn can be


used to predict the minimum renderable mass using the results described in in Section III.

Figure 14. Admittance-type haptic device used to demonstrate proposed

approach.

To evaluate the parameters of the human impedance model (see Fig. 1), the device was sinusoidally driven (in position) while a user grasped the input knob. Both device position and input force were measured. The user was instructed to maintain a consistent grip while trying not to resist the motion of the input knob. The frequencies and amplitude of the command were restricted to 30 Hz or less and 5 mm, respectively, to prevent damage to the hardware. The frequency response of two separate users was experimentally determined. The human impedance model given in (1) was fit to these data sets. The human impedance parameters were determined by fitting the magnitude and phase data as described [17]. The experimental and fitted frequency response plot for user 1 is shown in Fig. 15 and the fitted parameters for both users are summarized in Table II. The differences in model parameters measured here and those presented in [17] can be attributed to differences in grasp posture and device force range.

Figure 15. Overlaid frequency response plot of the experimental and the

fitted model depicting the open loop transfer function for a user 1.

TABLE II. EXPERIMENTAL MODEL PARAMETERS

Parameter User 1 User 2

Mass (Mh) 4.5 [Kg] 4.9 [Kg]

Stiffness (k1) 975[N/m] 985 [N/m]

Stiffness (k2) 1486 [N/m] 1475 [N/m]

Damping (b1) 198 [Ns/m] 192 [Ns/m]

Damping (b2) 48[Ns/m] 38.5 [Ns/m]

B. Stability Model Validation

To verify the results obtained in Section II, we experimentally determined the minimum renderable mass using the device described above and compared these results with our model predictions. In the experiments, the minimum renderable mass was evaluated by applying a continuous motion to the device with the identical grasp and posture used by the users in the human impedance model parameter evaluations. The rendered mass was progressively decreased until instability was observed in the form of oscillations. The inner-position loop corner-frequency and loop delay (implemented via software) results are shown in Fig. 16.

The predicted and experimentally measured minimum renderable mass values match reasonably well for both users, with the predicted mass values lying somewhat below the measured experimental values in most cases. The difference between theoretical and experimental values is likely due to unmodeled nonlinear characteristics of the experimental system, the most significant of which comes from backlash inherent to the planetary gear-set of the experimental system. Importantly, the experimental values reflect both the predicted inverse relationship between the minimum renderable mass and inner position loop’s corner frequency given in (4) and the predicted linear relationship between minimum renderable mass and delay given in (7).

Figure 16. Bar graphs showing the experimental (Light gray) and

predicated (Dark gray) minimum renderable mass values for different

corner frequencies of the inner loop position controller and the

programmed delay in the outer loop for the two users.

To verify the relationship established with the human impedance damping in (4) a set of experiments to evaluate the minimum mass with varying grasp and posture was conducted using a similar approach described above. The human impedance model parameter b1 was evaluated for each of these grasps before measuring the minimum mass. The model predicted and the experimentally evaluated minimum mass values for various human impedance damping values are shown in Fig. 17 with the closed loop bandwidth set to 30 Hz and no delay introduced in the outer-loop.

Ultrasonic

actuators

Differential

mechanism

Linear slide with

flexure constraints

actuatorencoders

output

encoder

rack andpinion

input

knob

force

sensor

Planetary

gears


Figure 17. Plot shows the predicted and experimental renderable

minimum mass values for various human impedance damping ranging

from a weak to a strong grasp.

The linear relationship between the human damping and the minimum renderable mass can be see in Fig. 17 where the experimental values match the predicted results within ten percent. The differences likely arise from variations in the human impedance evaluation, unmodeled nonlinear characteristics described earlier.

VI. CONCLUSION

Through use of an experimentally validated analytical model we have shown that the performance limitations of admittance-based haptic interfaces can be understood by examining the effects of inner position loop bandwidth (or corner frequency) and outer loop delay on stability and rendering range. Specifically, our results demonstrate that the minimum renderable mass, a common performance index for admittance-based devices, is inversely proportional to the position loop’s corner frequency, suggesting that device performance is directly related to the performance of the position controller. Characteristics that would adversely affect the bandwidth of the position controller, such as backlash and drive-train compliance, will have a direct and negative impact on rendering performance. Similarly, characteristics that introduce delay, either directly or through necessity (as would be the case when using a noisy sensor requiring pre-filtering), will adversely affect the performance. We have shown that the mass rendering range, defined as the frequency below which the output impedance approximates a pure mass, is proportional to the position loop’s corner frequency (and thus bandwidth) and inversely proportional to loop delay, further underlining the performance ramifications of both. Finally, we have show that the minimum renderable mass is directly proportional to the damping provided by the user which implies a stronger grip, with a higher damping, decreases the stable rendering region of the admittance-based haptics device as opposed to an impedance-based device where increased damping enhances stability. Our results are validated using a one degree-of-freedom admittance-based device.

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performance and stability limitations of admittance-based ......electromechanical haptic interface...

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