I Introduction
While the concept of a personal augmentation device or exoskeleton is an old idea [1, 2, 3], a system which delivers on the dream of transparent interaction, of “feeling like the system is not there,” through augmentation of sensed human interaction forces is still an ambitious goal of force control technology today [4, 5, 6, 7]. Unlike assistive exoskeletons which help complete predictable behaviors [8, 9] or rehabilitation exoskeletons [10, 11] which simulate rehabilitation therapy, human augmentation exoskeletons [4, 12] use nonpassive feedback control to amplify the user’s strength. But this type of feedback control brings the system closer to instability. And since the exoskeleton is in a feedback interconnection with the human, a model of the human’s dynamic behavior plays a critical role in determining the stability of an augmentation exoskeleton [13, 14].
Among all different kinds of dynamic model of an individual human joint, perhaps the most popular one is the massspringdamper model—with the additional nonlinearity that the spring stiffness of the human joint can be modified by both voluntary muscle contractions or external torques exerted on the joint [15]. Several studies demonstrated a linear relationship between the stiffness of the human (found by fitting a linear massspringdamper model for a single joint) and an external torque [16, 17, 18]. For modeling the human joint damping, some other studies explored the fact that not only the stiffness but also the damping increases with muscle contractions [19] and external torques [20]. A linear relationship between the damping and the external torque has also been identified for the human ankle joint, but it is statistically weaker than the strong linear relationship between the stiffness of the ankle and the external torques [16, 18]. However, it is not clear from the literature that a linear relationship between the damping and the stiffness of a human joint can be expected in more general cases.
Another way to model the damping in the linear massspringdamper model is through the empirical observation that a relatively consistent damping ratio is maintained by the human elbow across different joint stiffnesses [14]. Frequency domain identification of the ankle joint impedance [16, 21] also showed a consistent damping ratio within the range from 0.22 to 0.49. This damping ratio consistency on the ankle is also supported by the fact that the ankle damping ratio does not have significant change with large variations of mean external torques exerted on the subjects [20]. For upper limbs, a multijoint impedance study on human arms [22]
showed that the damping ratio of the minimally damped mode for the 2D endpoint impedance is distributed with a mean of 0.26 and a standard deviation of 0.08. Although this could be explained as the effect of humans adapting their damping to stabilize movement
[23], a more detailed explanation of how humans achieve this consistency remains unclear.Hysteretic damping models have seen success in biomechanical modelling before. In [16], experimental results showed a hysteretic relationship between the applied torque and the ankle angle at very low frequencies. Hysteretic damping is shown indirectly in [17] (see Fig. 6 of that paper), where the human elbow stiffness has a phase shift around 25 degrees in a wide range of low frequencies—contradicting the viscous damping hypothesis. This type of phase behavior is explained (in the field of structural mechanics) by defining a hysteretic damping whose damping coefficient is proportional to the inverse of frequency [24]. Models with hysteretic damping have also been adapted to describe the dynamic properties of the whole body of a seated human [25] as well as cockroach legs [26].
In this paper we study the human stiffness and damping behavior when coupled to an exoskeleton inertia, and test the effectiveness of a hysteretic damping term in the system model. More specifically we compare three models 1) a linear mass, spring, and viscous damper model, 2) a nonlinear complexstiffnessspring and mass model (that is, a spring, mass, and hysteretic damper model), and 3) a combination model with mass, spring, and both viscous and hysteretic damping. Our results show that there is a statistically significant benefit of the hysteretic damping term (comparing model 1 to model 3 with an Ftest), and a less significant benefit for the viscous damping term (comparing model 2 to model 3). This hysteretic damping explains the consistent dampingratio of the human–exoskeleton resonant peak even as the stiffness and exoskeleton inertia change—which is not well explained by the linear model. And it also explains the low frequency phase lag in human stiffness (previously observed in [17]). Our elbow joint experiments vary parameters which would result in a differing damping ratio if the linear model were true: we change the inertia of the exoskeleton, and (indirectly, using an adjustable exercise hand grip and a bias torque) the stiffness of the human joint. We also test different exoskeleton strength amplification factors, and it does not appear to elicit a different human behavior than when the inertia is simply reduced. One further contribution of the paper is the theorizing of an amplification controller which uses fractional order filtering to exploit the hysteretic damping of the human, offering improved performance over previous strategies.
Ii Methods
Iia Apparatus
For this study we employed the P0 series elastic elbowjoint exoskeleton from Apptronik Systems, as shown in Fig. 1
. This exoskeleton has a moment of inertia of 0.1
with no load on it, but allows for attaching additional weights to it. A load, attached 0.45 m from the exoskeleton joint, is pictured in Fig. 1.b. The contact force between the human and the exoskeleton is measured by a sixaxis force/torque sensor situated below the white 3D printed “cuff” (which includes the adjustable strap which clamps the forearm). This force torque signal is cast as a torque () using the motion Jacobian of the sensor frame (). Rubber pads are adhered to the inside surfaces of the cuff and the cuff strap to improve user comfort. Joint position
is directly measured by a dedicated encoder at the exoskeleton joint. The series elastic actuator (SEA) has a spring force control bandwidth of 10 Hz and provides high fidelity actuator torque tracking using the force control disturbance observer of [27].In parallel with an excitation chirp command (which essentially performs system identification of the human subject), a gravity compensation controller, a human augmentation controller, and a bias torque comprise the desired actuator torque signal. The gravity compensation controller takes the measurement of to calculate and compensate the gravity torque acting on the exoskeleton system. The human augmentation controller takes the measurement of and multiplies by negative . With the assistance of actuator torques produced from the augmentation command, the human’s interaction forces with the exoskeleton are amplified by a factor of . This exoskeleton augmentation strategy differs from the one we applied in [14] in the directness of the augmentation feedback.
IiB Experimental Protocol
The experimental protocol was approved by the Institutional Review Board (IRB) at the University of Texas at Austin. It consists of fifteen perturbation experiments with a 28year old male subject. The experiments are separated into three groups (Exp. IIII) of five experiments. The first three experiments in each group are conducted with loads of 1.25 lbs, 5 lbs and 10 lbs and an value of 1 (corresponding to a nonaugmentation controller) while the last two experiment in each group are conducted with a load of 10 lbs and values of 2 and 4. The mass of the loads and the mass of the exoskeleton have their gravitational bias torque fully compensated through gravity compensation control, while their inertia is attenuated by a factor of due to the cuff torque feedback.
The stiffness of the human elbow is influenced by muscle cocontraction as well as by contraction to resist the bias torque. In order to obtain different values of elbow stiffness for the three experiment groups, both the bias torque component of the controller and the cocontraction are varied. The three groups have, respectively, Nm, Nm, and Nm of bias torque. Cocontraction is controlled by having the subject squeeze an adjustable force hand grip. The three groups have a 22lb, a 30lb, and a 60lb gripping force respectively. The amplitude of the perturbation chirp signal is set to be Nm.
To avoid fatigue of the subject, the duration of each perturbation experiment is set to be 100 seconds. The perturbation is set to be an exponential chirp signal, and the results are typically analyzed in the frequency domain. To sufficiently capture the natural frequency for damping feature identification, we set different ranges of frequency for the chirp signal according to the stiffness values the subject achieved from the bias torque and the gripping force. Frequency ranges of 220 rad/s, 330 rad/s and 440 rad/s are set for the chirp signals for the three experiment groups.
After the chirp perturbation experiments, we transfer the time domain data into the frequency domain and identify the dynamic stiffness model of the subject by linear regression. The parameters of the three experiment groups are summarized in Tab.
I.IiC Models
In our models, we define as the human elbowjoint real stiffness, as the human elbowjoint hysteretic damping, as the human elbowjoint viscous damping, as the moment of inertia of the human and as the moment of inertia of the exoskeleton. See list of symbols in Tab. II.
A passive linear model of human dynamic stiffness with viscous damping can be expressed as
(1) 
If we consider a human model with hysteretic damping (complex stiffness) we have a nonlinear model
(2) 
And to generalize the two, we also consider a nonlinear model with both viscous and hysteretic damping
(3) 
However, these models are difficult to identify from the experimental and values because the natural frequency of the human dynamic stiffness can easily go beyond the range of the frequency for the experiments. With the augmentation controller, the operator feels an attenuated inertia from the exoskeleton. Therefore, we added a nominal attenuated inertia of to the frequency domain data of for the model identification. In essence, we desensitize our identification to errors far above the natural frequency of the human spring and the exoskeleton inertia. Combining this additional term with (1), (2) and (3), the three models of humanexoskeleton interaction can be expressed as
(M1)  
(M2)  
(M3) 
where is the perceived inertia at the human joint.
Exp  Load (lb)  Grip (lb)  Bias (Nm)  Amplitude (Nm)  Frequency (rad/s)  

I.1  
I.2  
I.3  
I.4  
I.5  
II.1  
II.2  
II.3  
II.4  
II.5  
III.1  
III.2  
III.3  
III.4  
III.5 
Symbol  Meaning 

Actuator desired torque  
Actuator actual torque  
Transfer function from to  
Humanexoskeleton interaction torque  
Exoskeleton gravity torque  
Elbowjoint angular position  
Human elbowjoint real stiffness parameter  
Human elbowjoint hysteretic damping parameter  
Human elbowjoint viscous damping parameter  
Moment of inertia of human forearm  
Moment of inertia of exoskeleton  
Moment of inertia of human with exoskeleton  
Moment of inertia of human with attenuated exoskeleton  
Elbowjoint dynamic stiffness of human  
Elbowjoint dynamic stiffness of human with exoskeleton  
Elbowjoint dynamic stiffness of human with attenuated exoskeleton  
Transfer functions of augmentation plant and augmentation controller  
Natural frequency of  
Natural frequency and damping ratio of  
Natural frequency and damping ratio of  
Natural frequency and damping ratio of 
We also calculate the damping ratio of , as a measure of the degree of oscillation at the resonant zeropair. Because M2 and M3 have the term which provides a damping effect in addition to , we define the damping ratio of each model using the imaginary part of the transfer function evaluated at the resonance:
(4)  
(5)  
(6) 
where is the natural frequency of .
IiD Statistical Analysis
In order to compare the significance of and in the humanexoskeleton interaction model, we calculate the residual square sum (RSS) for all three models, denoted , and respectively. For each experiment, we conduct Ftests for each of the two threeparameter models (M1 and M2) against the generalizing fourparameter model (M3). Our Fstatistic accounts for the complex number data,
(7) 
where is the number of complex value samples at the frequency domain and the real and imaginary parts of each sample are statistically independent. The significance of and
then will be evaluated by comparing this F statistic against a critical F statistic threshold based on a 0.05 falserejection probability.
We split the 100 seconds of time domain data for each experiment into 10 sequences. For each of the 10 second sequences, only the data from the first 5.78 seconds is used for calculating the frequency domain sample. The remainder period of 4.22 seconds is greater than the 2% settling time for all the 2nd order dynamics of
identified in the experiments. By this method we can safely assume statistical independence between the 10 singlefrequency data points comprising our estimate of the frequency response function for the purposes of statistical testing.
Iii Results
Iiia Phase Shift
In the frequency domain results of (Fig. 3), the phase starts (at low frequencies) from a value between to instead of zero and changes very little across all the frequencies before it reaches the second order zero at for each experiment. This type of phase shift is very different from the phase shift usually experienced by a linear system with a constant time delay or a constant damping property in which the phase shift approaches zero in the limit as . As shown in Fig. 4, this phase shift is clearly visible even in time domain comparisons of and . The data show that the human joint motion is not perfectly sinusoidal—it stops following the trend of the torque after they both reach their peak values and “waits” before following the torque in its descent. At low frequencies, these peaks seem especially flat.
IiiB Model Comparisons
The results of the identified parameters (Tab. III) show that the three models give the same values of , and consequently to two decimal places for each experiment. This is because the difference between the three models is restricted to the imaginary part of while and are the coefficients of the real part of . Although the identified values of and are quite different between the three models, the values of are still very close for each experiment. This means that the three models give very similar values for the slope of the phase at the resonant frequency .
From M1 to M3, the values of have been reduced considerably. This means that M3 uses the term to replace part of the term in M1 while maintaining a similar phase behavior at the frequency . From M2 to M3, the values of have been reduced except for Exp. I.3, III.3 and III.5 in which M3 gives a negative value for . These negative value of is because there is no lower bound constraint on the value of during the frequency domain regression for M3. Although a negative value of brings nonpassivity to a linear massspringdamper system in the common sense, the term in M3 enforces the dynamics of to remain passive across the range of frequencies in our experiments.
The results from the Ftests (Fig. 5) relate to the significance of and in M3. Based on the 20 statistically independent data values for each experiment, a critical Fstatistic value of 4.49 is calculated for 0.05 falserejection probability. The results show that values of for all the experiments are much higher than the critical Fstatistic value, with the values of in Exp. II.3 and II.5 exceeding 100 (c.f. the critical value of 4.49). This proves that the existence of the term in M3 significantly improves modeling accuracy of . The values of are mostly below the critical Fstatistic value except for Exp. I.5, II.1, III.1 and III.2. The other observation is that the value of is always much lower than the value of for all experiments. Although the effect of the term cannot be completely ignored based on the results of these Ftests, we can claim that the term is still much more significant than the term in M3.
IiiC Linear Regression between and
Because the term is created to describe the phase shift effect from the complex human stiffness in M2 and M3, we suspect that the identified value of has a linear relation with the value of . Therefore, we apply linear regression between the values of and identified from M2 and M3 (Fig. 6). Compared with M3, the linear regression result with M2 shows a stronger linear relationship with a much higher coefficient of determination (). The regression equation identified from the M2 parameters also has a smaller value of bias from the origin of the plane compared with the regression equation identified from the M3 parameters. Intuition leads us to expect low bias in the regression equation, since a nonzero value of when the value of is zero could not be explained as hysteretic spring behavior.
Exp  Model  

I.1  M1  
M2  
M3  
5pt. I.2  M1  
M2  
M3  
5pt. I.3  M1  
M2  
M3  
5pt. I.4  M1  
M2  
M3  
5pt. I.5  M1  
M2  
M3  
II.1  M1  
M2  
M3  
5pt. II.2  M1  
M2  
M3  
5pt. II.3  M1  
M2  
M3  
5pt. II.4  M1  
M2  
M3  
5pt. II.5  M1  
M2  
M3  
III.1  M1  
M2  
M3  
5pt. III.2  M1  
M2  
M3  
5pt. III.3  M1  
M2  
M3  
5pt. III.4  M1  
M2  
M3  
5pt. III.5  M1  
M2  
M3 
Based on linear regression equations, we can express the phase shift at the low frequencies as
(8)  
(9) 
where is the regression equation identified from the values of and in M2 and M3 with and being the slope and the bias of the regression equation. By substituting into (5) and (6), the value of for M2 and M3 can be expressed as
(10)  
(11) 
Because the values of of the regression equations for M2 and M3 and the values of for M3 are relatively small, the phase shift at the low frequencies is dominated by the value of and the value of is dominated by the constant term. This explains the fact that the phase shift is nonzero at low frequencies and the fact that the value of changes very little compared to the changes of and across all our experiments.
Iv Implications for Control of Performance Augmentation Exoskeletons
Iva 1Parameter Complex Stiffness Model
One of the challenges of augmentation control is to design an augmentation controller to stabilize the exoskeleton with all possible human impedances. This requires a robust human impedance model with bounded parameter uncertainties for the augmentation controller design.
Similar to [13], a robust model version of M1 can be defined with bounded uncertainties for and , which could be obtained from multiple measurements in advance. (We assume does not change for the elbowjoint.) Because both and vary in large ranges, the 2D uncertain parameter space of becomes very huge, and the augmentation controller can easily end up as an extremely lowbandwidth conservative controller. Since such an uncertain model includes all combinations of possible and , the damping ratio can be a very limiting design constraint. This is not realistic, given that is relatively consistent in our experiment results. Therefore, we propose a 1parameter model simplification to reduce the uncertain parameter space for augmentation controller design.
One strategy is to model as a linear function of which allows us to create a robust model of M1 with bounded uncertainty for only . Based on a linear relationship between and , M1 can be expressed as
(12) 
By substituting to (4), can be expressed as
(13) 
which is proportional to . However, we do not observe this proportional relationship between and from our experimental results for M1 in Tab. III. On the other hand, because (12) is a simplification from M1, it also fails to explain the nonzero phase shift at low frequencies.
If we assume , a simplified complex stiffness model of M2 can be expressed as
(14) 
Based on (8) and (10), (14) is able to explain both the nonzero phase shift at low frequencies and the near constant value of across all the experiments. This, in turn, supports the use of (14) as a 1parameter model of for augmentation controller design.
Adopting this 1parameter model allows simplification of (2),
(15) 
and the dynamic stiffness of the human coupled with the exoskeleton ,
(16) 
where is the combined inertia between the human and the exoskeleton. We consider to be the natural frequency of , despite the term.
IvB FractionalOrder Augmentation Controller
As in [14], the augmentation control we discuss here is designed to eliminate the augmentation error signal by feeding it back to the actuator command with an augmentation controller. Different from the direct augmentation feedback shown in Fig. 2 in which the augmentation command is multiplied by , this strategy allows us to design an augmentation controller completely separated from the augmentation factor .
By substituting (15) and (16), the transfer function from to can be expressed as
(17) 
Based on (17), the augmentation plant transfer function from to can then be expressed as
(18) 
where the SEA transfer function acts as a 2nd order lowpass filter. Because of the high bandwidth of the SEA force controller, the natural frequency of is much greater than both and .
Looking at the bode plot from low to high frequencies, has a pair of conjugate poles at , then a pair of conjugate zeros at and then another pair of conjugate poles at (Fig. 7). Before , both and are dominated by the complex stiffness. Therefore, has magnitude and phase. Between and , is dominated by its inertia effect and the magnitude of decreases while the phase leaves . On the other hand, is still dominated by the complex stiffness and prevents the phase moving below . At the frequency between and , the inertia effects in and completely dominate their frequency behaviors. The magnitude of stays at which is in the range from 1 to .
However, the gain crossover of falls beyond without an augmentation controller. The phase margin with such crossover is very close to zero because of the 2nd order SEA dynamics. Also, the closed loop behavior amplifies the high frequency sensor noise from the actual signal of . ( is usually denoised by a lowpass filter beyond the frequency of which makes the closed loop even more unstable.) Therefore, the augmentation controller must lower the crossover frequency in order to achieve a minimum phase margin.
Similar to [14], the new crossover cannot be placed at the frequency between and because multiple other crossovers can be easily triggered. Instead, a new crossover can be safely placed at the frequency between and with a fractionalorder augmentation controller expressed as
(19) 
where is the fractional order (that is, a noninteger power of s) of and is a gain which allows tuning the magnitude of in the frequency domain. The fractionalorder controller in (19) has its magnitude decreasing dB per decade and its phase staying at degrees at all frequencies. Because of the nonzero phase shift from the complex stiffness, a positive phase margin can be guaranteed if
(20) 
where is chosen in the range of . The value of can then be tuned to achieve a crossover in the frequency range from to .
As a fractionalorder controller, (19) cannot be implemented directly into the control system. However, we can approximate it as the product of many 1st order lag filters,
(21)  
(22)  
(23) 
where is the number of lag filters and the zero and the pole for each lag filter are and . We define such that all the lag filters have an equal distance between the zero and the pole, and we define such that there is a constant distance between adjacent lag filters (in log frequency space). The augmentation controller in (21) functions as a fractionalorder filter in the frequency range of rad/s. The fractional order can be approximated as .
In Fig. 4 the flat peaks of could be explained by a hysteretic coulombfrictionlike nonlinearity of the human. If this was the case, then our frequency domain measurements would be measurements of the describing function of the nonlinearity. It is not yet clear how this hypothesis would hold up to testing at different amplitudes of the force input, since we did not include such tests in our experimental plan. It would be a complex task to measure the relationship between the amplitude and the hysteretic damping, since increasing the amplitude would potentially increase the stiffness as well. Regardless of the cause, such hysteresis can not be modeled by M1, and we find that the additional term in M2 and M3 helps to model the hysteresis by creating a non zero phase shift for the human stiffness.
As for the proposed fractionalorder controller, if our frequency domain model was due to hysteretic coulombfrictionlike nonlinearity of the human, then we would expect that the value would be a function of the interaction force signal amplitude. Fitting the model to tests performed at some maximal force amplitude, our controller would also be stable for lower amplitudes. The result would be that our controller would perform well below a force threshold, at which point we would need to switch to another controller—perhaps saturating the desired force or employing a backup safety controller [28]—to avoid having a nonrobust controller in situations with high force amplitude.
V Conclusion
Augmentation exoskeletons rely on a human model to determine stability. While ideal force feedback maps passive environments to passive human experiences, force feedback with finite bandwidth will add energy due to the inevitable phase lag. Human damping directly helps system stability by removing this energy. So the more we know about the damping, the more augmentation can safely be achieved. And in this paper we have presented compelling evidence that this damping is better modeled as hysteretic damping than as viscous damping. With this higher quality model of the human, it should be possible to design augmentation controllers with less conservatism and more performance. We have additionally theorized a fractionalorder controller to take maximal advantage of this model.
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