Decomposing the Performance of Admittance and Series Elastic Haptic Rendering Architectures

Transcription

1 Decomposing the Performance of Admittance and Series Elastic Haptic Rendering Architectures Emma Treadway 1, Yi Yang 1, and R. Brent Gillespie 1 Abstract In this paper, we explore certain tradeoffs in haptic rendering performance with different device architectures, specifically examining admittance and series elastic actuator (SEA) architectures. We apply performance analysis techniques that decompose the driving point impedance frequency response into effective stiffness, inertia, and damping and discuss device behaviors as rendering reverts to hardware dynamics with increasing frequency. SEA has previously been discussed in terms of stiffness limitations due to stability; here we discuss a frequency limit for haptic rendering with both architectures, and discuss differing high-frequency behaviors to consider when selecting a device architecture. Theoretical conclusions are accompanied by experimental frequency response data and a perceptual experiment using a reconfigurable haptic device. I. INTRODUCTION Virtual environments (VEs) often consist of unilateral constraints known as virtual walls. Outside the wall, free motion is allowed. At the position of the wall, a stiff virtual spring is rendered, pushing back as the user attempts to penetrate the wall with the haptic device. Thus, stiffness and free space rendering are both of concern in the display of VEs. The choice of a device architecture for haptic rendering necessarily involves a tradeoff; a haptic interface optimized for display of very stiff walls will be sub-optimal for rendering free space. The most popular device architectures are admittance and impedance displays, which, respectively, source actuator position in response to measured force and source actuator force in response to measured position. In a sense, the admittance and impedance architectures lie on opposite ends of a spectrum. Admittance-type devices (Fig. 1a) are non-backdrivable, and naturally (in their unpowered state) admit large operator forces with little motion response, allowing very stiff virtual walls to be rendered. Impedance-type devices, on the other hand, are designed to be as transparent and backdrivable as possible, featuring direct-drive motors that allow quite free unpowered movement. These design choices have drawbacks as well as benefits, of course. Admittance-type devices can close a loop to render free space, but this ability breaks down at higher frequencies, where reversion to unpowered device dynamics causes free space to begin to feel like the motor inertia This work was supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health, Grant No. R1EB1983, and by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE1566. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or NIH. 1 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 819, USA etreadwa, yeeyoung and brentg@umich.edu reflected through the gear ratio. Impedance-type devices, on the other hand, can easily saturate the torque limits of the direct-drive actuators when rendering virtual walls (even at low frequencies), resulting in mushy constraints. (a) Admittance Architecture (b) SEA Architecture Fig. 1: Device architecture schematics. If admittance- and impedance-type haptic devices lie at opposite ends of a spectrum, then there exist a number of intermediate device architectures that represent compromises between these extremes. This spectrum also includes devices such as variable impedance actuators (VIAs) [1]. In [], we presented a comparison of haptic rendering with admittance and series elastic actuator (SEA) devices (SEAs are essentially VIAs with fixed stiffness elements between geared motors and the point of user interaction [3] see Fig. 1b). SEA for haptic rendering had been previously suggested in other papers [], [5], but has not been widely adopted in the haptics community due to its inability to passively render stiffnesses greater than the physical stiffness of the series elastic element [6], [7]. Arguably, SEA is closer to the admittance end of the haptic rendering spectrum, featuring a highly geared motor with the ability to present high forces; however, the elastic element introduces some more impedance-like characteristics at high frequencies, putting SEA somewhere between the two extremes. In our previous work [], we presented a theoretical analysis accompanied by data collected with a 1 degree of freedom (DOF) haptic device that could be configured either as an admittance or SEA device. We noted that at high frequencies, the elastic element of an SEA device masks the inertia reflected through the gear ratio, yielding a free-er freespace than admittance. While we were able to make comparisons between the architectures using a physical device that allowed an apples-to-apples comparison through reconfiguration of the same hardware, our experimental data collection methods were limited by the use of human excitation as an input signal, and our work lacked a perceptual experiment to validate whether trends shown in the model would indeed be felt by users. Colonnese et al. [8] introduced a method for analyzing the rendered dynamics of impedance-type haptic devices, which

2 is a decomposition of the frequency response into effective impedances. They focus on 3 mechanical primitives available in a set of many effective impedances: virtual stiffness, damping, and mass. For an impedance architecture device rendering a virtual spring of stiffness K V E, they show that the effective stiffness is near the desired stiffness up to the critical frequency ωs,imp = K V E /m, where m is the mass of the device. Above that frequency, the effective mass reverts to the physical mass of the haptic device. In this paper, we expand our previous work [] by applying the method of analysis in [8] to admittance and SEA devices, decomposing the rendered impedance into effective stiffness, damping, and mass. We use this decomposition to characterize the performance of the devices in the frequency domain. Our theoretical conclusions are supported by two sets of experimental data: first, experimental frequency response data that improves upon the data collection and analysis methods we previously used, and second, a perceptual experiment in which participants classify VEs by stiffness at different motion frequencies for the two device architectures. A. Admittance Architecture II. MODEL An admittance device for haptic rendering features a highly geared motor (Fig. 1a) made more transparent through feedback control based on measured interaction forces (Fig. a). The closed-loop driving point impedance Z F u /Ẋ is Z ADM (s) = nc(s)z V E + Z h, (1) nc(s) + 1 where the inherent impedance of the unpowered stage reflected through the ballscrew is abbreviated as Z h = Ms+B. (a) Admittance Architecture These models for the admittance and SEA architectures were given in []; see there for further detail. For the analysis presented here, we use proportional feedback, C(s) = C p. III. THEORETICAL PERFORMANCE We present the theoretical performance of the haptic devices in two ways: as Bode plots, commonly used to present the frequency-dependent rendering of a VE, and also as effective impedances, applied in [8] for impedance devices. In particular, we present the effective mass (EM), stiffness (ES), and damping (ED) components of the decomposition. The decomposition makes use of the phase-angle separation of Z for different mechanical primitives. The response Z(jω) is separated by phase angle, projected onto the complex axis corresponding to the particular mechanical primitive s response, and normalized by frequency. This captures the dominance of certain primitives at different frequencies (for example, a spring-mass Z = ms + k/s excited at low frequency presents force in phase with position yielding an ES of k, but an EM of due to low acceleration; at high frequency, force and position are out of phase, with EM of m and an ES of ). A. Virtual Stiffness Both admittance and SEA devices can be used to render virtual stiffnesses (Z V E = K V E /s). Using an admittance device (1) to render a virtual spring yields the driving point impedance frequency response shown as the red line in Fig. 3a. An SEA device () yields the frequency response given by the blue line. The device responses at low frequency overlap, and stay true to the desired response (dotted black line); as frequency increases, both devices ability to render the VE breaks down, and the responses diverge from the ideal response (and from each other). Using the definitions of effective impedances [8], the responses in Fig. 3a can be decomposed into the ES, ED, and EM shown in Fig. 3b. For our admittance display of a virtual spring, it can be seen that the ES bandwidth is low, breaking down well within the human motion generation frequency range (DC to no more than 1 Hz [9]). We found that the critical frequency (above which effective stiffness is zero) occurs at ω ncp K V E s,adm = M. (3) (b) SEA Architecture Above this critical frequency, the effective mass becomes Fig. : Device control schemes. quite high, approaching M/(1 + nc p ), which is consistent with the high frequency behavior derived in []. B. Series Elastic Architecture For SEA rendering a virtual spring, low frequency effective impedances are quite similar to those created by To form an SEA device, an elastic element is introduced in series with the stage (see Fig. 1b). The addition of a small the admittance display. However, as frequency increases, a end effector mass (Z e = ms), along with physical stiffness k nonzero ES is present at a wider range of frequencies than the and damping b between the end effector and the stage alters admittance display allows, including most frequencies within the performance. With k = k/s + b, the resulting driving the range of human motion generation. However, it should point impedance Z F u /Ẏ for Fig. b is: be noted that in the high end of frequencies with nonzero Z SEA (s) = Z e(z h + knc(s)/s + k ) + k (Z h + nc(s)z V E ) ES, the rendered stiffness no longer matches the desired VE, Z h + knc(s)/s + k. and this stiffness is accompanied by a spike in ED and EM () caused by the resonance of the elastic element. However,

3 (a) (b) Effective Impedances (Freespace Rendering) Z (Freespace Rendering) Magnitude [db] Admittance SEA Ideal VE 1 Effective Impedances (Stiffness Rendering) Z (Stiffness Rendering) Magnitude [db] Frequency [Hz} (c) (d) Fig. 3: Theoretical (a) Bode plot and (b) decomposition into effective impedances for rendering a virtual stiffness (ZV E = KV E /s, KV E = N/m) with SEA and admittance architectures. Theoretical (c) Bode plot and (d) decomposition into effective impedances for rendering freespace (ZV E = ). It should be noted that the ideal magnitude response for freespace is, and therefore is not visible in the top pane of (c). All responses use Cp = 3 and Table I parameters. increasing frequency even further, the ED and EM become quite low, with the EM approaching simply the mass m of the end effector. This is similar to the tendency of an impedance device s high-frequency EM to approach the device mass [8]. B. Free Space To complement the virtual stiffness required to render a wall, the theoretical response when rendering free space (ZV E =, or equivalently, Fdes = ) is given in Fig. 3c. As with the virtual stiffness, this response is decomposed into the effective impedances shown in Fig. 3d. Comparing the responses of the admittance and SEA architectures reveals that at low frequencies, the two devices perform comparably at rendering free space. In the band of frequencies around the resonant frequency of the elastic element and end effector, the SEA device presents high ES, ED, and EM. As the frequency increases, however, the two responses diverge. At high frequencies (which the user can certainly feel), the SEA architecture has a much lower EM than the admittance architecture. It is interesting to note that by increasing the proportional control gain Cp, the resonant peak for SEA freespace rendering can be pushed to higher frequencies (and can easily be pushed above 1 Hz, where humans will not generate input motions). However, since humans have tactile perception up to about 1 Hz [9], these non-ideal behaviors may still be noticed (for example if higher frequencies are excited by bouncing off of a virtual or physical barrier). IV. E XPERIMENTAL VALIDATION The apparatus we used to experimentally validate the theoretical performance of admittance and SEA rendering is a single degree of freedom device, consisting of a motorized stage actuated through a ballscrew, with a spring-mounted end effector and load cell mounted atop the stage, as pictured in Fig.. Complete details of the experimental apparatus are described in []. Two minor changes have been made Fig. : Experimental apparatus, capable of functioning as an admittance device through the load cell or as an SEA device through the end effector between the springs. to the device, however: the linear encoder measuring the spring displacement has been replaced with an encoder of higher resolution (US Digital EM---N), and the load cell has been mounted vertically closer to the ballscrew to reduce unmodeled compliance in the 3D printed stage. The parameter values to fit to the theoretical model, found through system identification, are presented in Table I1. TABLE I: Physical device parameter values from system ID. stage and motor inertia stage damping end effector mass end effector damping physical spring stiffness gain (motor, amp, and gear) Symbol M B m b k n Value Units [kg] [kg/s] [kg] [kg/s] [N/m] [-] 1 Value of m given in table is for mass of the plastic end effector shown being grasped in Fig. only. Mass used in results figures is actually m = kg, which also includes the mass of a small motor (not pictured) affixed to the end effector to improve high-frequency frequency response estimation. During normal use, however, this motor is removed.

4 Magnitude [db] Coherence 1 5 Z (Stiffness Rendering) Adm (Sim) (a) Experimentally measured and theoretical frequency responses, with coherence showing quality of measurement. Magnitude [db] Coherence Z (Freespace Rendering) Adm (Sim) (a) Experimentally measured and theoretical frequency responses, with coherence showing quality of measurement. 3 1 Effective Impedances (Stiffness Rendering) Adm (Sim) (b) Experimental and theoretical effective impedance decomposition. Fig. 5: Experimental results for stiffness rendering. 3 1 Effective Impedances (Freespace Rendering) Adm (Sim) (b) Experimental and theoretical effective impedance decomposition. Fig. 6: Experimental results for freespace rendering. A. Data Collection and Analysis Methods Excitation of the device was performed by hand, with additional excitation provided by a small 6mm diameter Maxon DC motor mounted on the load cell or end effector with an affixed eccentric mass. Since humans cannot generate frequencies above 1 Hz (and even reaching this point can be quite difficult), the rotation of the eccentric mass was used to expand the input frequency spectrum as well as provide a more complete frequency sweep. Displacement and user force (measured through the load cell for admittance or through spring displacement for SEA) were recorded. Each frequency response plot was generated from at least 1 minutes of recorded data. Both the frequency of the motions provided by the human and the supply voltage powering rotation of the eccentric mass were swept to ensure coverage of a wide spectrum of frequencies. High frequency noise is removed by a low-pass filtering with a cutoff frequency of ω = 5 Hz. The filtered data are used to estimate the driving point impedance frequency response (using the Matlab function tfestimate). To assess the quality of our frequency response data, we also calculated the coherence (using mscohere). Coherence quantifies the strength of the relationship between the input and output in the frequency domain, with values close to 1 indicating a good linear relationship with high input strength and low noise. B. Results For rendering a virtual spring, the experimental results are shown in Fig. 5a. As with the theoretical response, the experimental magnitude and phase response of the devices are decomposed into EM, ED, and ES, as shown in Fig. 5b. High coherence for both device architectures was achieved from low frequency up to about 17 Hz. Results from rendering freespace are shown in Fig. 6a. The decomposition into effective impedances is shown in Fig. 6b. Frequencies were excited up to about 15 Hz, although low frequencies do not exhibit coherence quite as high as in the stiffness rendering case. C. Discussion The coherence is a useful metric to add to the experimental analysis, as it gives us a sense of the frequencies at which we can trust the measured response. We can see in Fig. 5a and Fig. 6a that when the coherence is close to 1, there is a good match between the theoretical and experimental curves, barring perhaps some parameter mismatch between the physical device and the theoretical response. As predicted by the models, experimental results show comparable performance at low frequencies for the two architectures. As frequency increases, the responses diverge.

5 Rendering at the high end of human motion generation (near 1Hz) with SEA is dominated by the physical stiffness. In Fig. 5b and 6b, it can be clearly seen that the large effective mass that characterizes the admittance architecture s response at high frequencies begins to dominate well within the range of frequency that human motion can generate. Unfortunately, the upper limit of frequency input achieved by the rotating eccentric mass is just below the frequency at which we would expect to see the small end effector mass begin to dominate the SEA response, so we were not able to verify this behavior. V. PERCEPTUAL EXPERIMENT To evaluate the theoretical performance differences between the SEA and admittance architectures, we performed a perceptual classification experiment to test subjects ability to discern between stiffnesses and freespace on the two devices. A. Procedure The experiment was designed to test subjects VE discrimination capability using the two architectures (SEA and admittance), when interacting with VEs with slow (.5 Hz) or fast (8 Hz) motions. For each of the four conditions (SEA/slow, SEA/fast, admittance/slow, and admittance/fast), the subject was asked to classify 5 randomly generated environments as being freespace (environment A), low stiffness (environment B, 15 N/m), or high stiffness (environment C, 8 N/m). The order of the four sets was randomized, as were the 5 environments presented in each set. All environments were rendered with C p = to ensure passivity []. Ten able-bodied participants (mean age 6.9 years, nine right-handed, two female) were recruited from the community of University of Michigan graduate students and acquaintances. Before starting the study, each participant was consented according to a protocol approved by the Institutional Review Board of the University of Michigan. Participants were not compensated. Testing lasted about.5 hours for each participant. During testing, subjects sat or stood at a comfortable height in front of a computer and the haptic device. During each trial, a blind was placed over the haptic device to occlude visual feedback. Movement frequency was enforced by coaching the participants to move along with a metronome. For each trial, the participant placed their right hand on the interface, was allowed /8 back-and-forth wiggles for slow/fast trials, and was then asked to remove their hand from the device before classifying the presented environment as either A, B, or C via the keyboard. Before each set of 5 trials began, the subject was presented with the three environments in the architecture for the upcoming trial, and allowed to explore and switch between them at will, using the A, B, and C keys on the keyboard. At a minimum, the subject was instructed to feel all 3 environments and to confirm that they could feel the differences, and was given information about how quickly they would be moving during the upcoming set of trials. Participants were asked to practice the slow and fast movements before trials began to ensure that the correct tempo was reached. If subjects stopped moving with the metronome, they were reminded by the experimenter of what a correct motion should look like between trials (e.g. It s actually twice that fast ). To verify that differences in perception were due to device and controller dynamics rather than limitations of perception, we also performed a pilot test with physical springs, matched as closely as possible to the VEs (, 157.6, and 85.6 N/m), mounted on the SEA end effector apparatus. 3 subjects (mean age 5. years, all right-handed, 1 female) followed the same procedure as in the VE discrimination test, with sets (physical/slow, physical/fast) of 5 trials, order randomized. Verbal answers were recorded by the experimenter. B. Results Results from the perceptual experiment are shown in Fig. 7, in the form of a confusion matrix. Each plot contains three Fig. 7: VE (left/middle columns) and physical environment (right column) classification results, averaged across subjects. rows corresponding to the true (presented) VE, and three columns corresponding to the environments that subjects identified. The darkness of each block corresponds to the frequency with which subjects classified the environment (row) as being a given environment (column). Perfect classification would be presented as three black blocks diagonally (classification frequency 1) from top left to bottom right, with the other cells perfectly white (classification frequency ). Random guessing would result in a classification frequency of.33 for all conditions. These matrices were produced first for each subject s 5 trials in each condition. The classification frequencies were then averaged across all 1 subjects to produce the results shown. At low frequency, subjects were able to discern between freespace, low stiffness, and high stiffness quite well with both the SEA and admittance architectures. At high frequency, much more confusion arose. For physical springs, however, subjects performed comparably at both frequencies. One subject s SEA/fast condition contains only 1 data points, since a device malfunction resulted in only the physical spring being felt.

6 3 1 Effective Impedances - Stiffness Rendering Adm A SEA A 1 ADM B SEA B Adm C SEA 1 C Fig. 8: Theoretical effective impedances of the SEA and Admittance devices rendering environments A (free space), B (15 N/m spring), and C (8 N/m spring). Vertical blue lines denote the slow (.5 Hz) and fast (8 Hz) frequencies of the metronome. C. Discussion The results from low frequency conditions confirm what is shown by the model: at low frequencies, both SEA and Admittance-type haptic devices are capable of rendering identifiable VEs. The results at high frequency were different from our initial expectation. As discussed in Section III, SEA features a free-er freespace rendering than admittance, in the sense that there is not a large EM obscuring the VE at high frequency. Thus, we expected participants to be able to correctly identify environment A more easily with the SEA device than with the admittance device. As can be seen in the results, performance was quite poor for both the SEA and admittance devices in the high frequency conditions. The reason for the resulting performance can be seen in a comparison of the three VEs rendered with the two devices at the enforced motion frequencies. The effective impedances for all cases are shown in Fig. 8. At the low frequency (.5 Hz, marked in the plot by the leftmost vertical line), the three environments have distinct ES (equal to the desired stiffness), and could be expected to feel quite distinct. However, at the 8 Hz mark (the right vertical line) the VEs rendered with both the SEA and admittance devices have broken down. At this frequency, all three environments rendered with SEA feel primarily like three quite similar stiff springs (high ES); the three environments rendered through the admittance device feel primarily like three large masses (high EM). Support for this theoretical behavior can be seen in the fact that in the SEA Fast condition (bottom left, Fig. 7), subjects were much more inclined to answer B or C, regardless of the actual presented VE; even when free space was being rendered, subjects tended to perceive a spring. VI. CONCLUSIONS While we had previously undertaken a comparison of admittance- and SEA-type haptic devices [], the additions of the analysis by decomposition into effective impedances and of a perceptual experiment have made some of the similarities and differences much clearer. Essentially, both architectures ability to faithfully render stiffness breaks down at about the same frequency (dictated by the device inertia, gear ratio, rendered stiffness, and controller gain). The main difference is in the behavior above this breakdown in rendering capability: the SEA device initially feels quite springy, resolving into a small mass at very high frequencies, while the admittance device s response is dominated by a large effective mass. The choice between an admittance and SEA device architecture comes down to a tradeoff. For SEA device architectures, a stiffness limit is imposed by stability constraints. For admittance device architectures, the freespace response at high frequencies breaks down to feel like a large mass, which is apparent to the user after a collision with an obstacle (physical or virtual) that elicits high frequencies. Our analysis here was based on simple proportional control for both the admittance and SEA device. A wide variety of control schemes that would lead to different frequency responses for admittance control have been proposed in the literature, and different frequency responses for stiffness rendering have also been achieved for SEA with different control schemes [6]. Another limitation of this work is that we have only explored the rendering of freespace and virtual stiffness; our future plans include the application of the effective impedance decomposition to additional types of virtual environments for SEA and admittance device architectures. While we are able to make some comparisons with the work done on impedance-type devices in [8], we have further plans to make use of a 1 DOF haptic device capable of acting not only as admittance or SEA device, but also as an impedance device. REFERENCES [1] B. Vanderborght, A. Albu-Schaeffer et al., Variable impedance actuators: A review, Robotics and Autonomous Systems, vol. 61, no. 1, pp , 13. [] T. Horibe, E. Treadway, and R. B. 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