I Introduction
The field of prosthesis design has been growing considerably over the past years, addressing the needs of both transtibial and transfemoral amputees [1, 2, 3]
. Upon understanding the limitations of passive prostheses, researchers have made strides to develop powered prostheses
[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Said prostheses implement control strategies that fall into two major groups: impedance controllers that attempt to mimic human joint impedance [7, 14] and trajectory-tracking controllers that follow optimized joint trajectories [15, 16, 17, 10]. Of the two classes, the former has displayed greater promise in mimicking human-like gait kinetics. As stated in [7], an impedance controller enables the user to interact with the device much like in the case of healthy walking. An impedance controller consists of parameters pertaining to stiffness, damping and the equilibrium angle of the joints. By modulating these parameters, the joint torque required for support and propulsion of the human body can be generated. According to [7], researchers sectioned the gait cycle into 4-6 phases based on kinematic changes observed in a healthy human gait cycle (refer Fig. 1).Each phase has a set of three constant values–stiffness, damping, and the equilibrium angle. These values were initially estimated using a least squares optimizer that reduced the error between the torque of the impedance controller and human torque data [18]. During testing, these estimates were tuned. Though successful, this approach mandated the manual tuning of several parameters. The study [19] implemented a series elastic actuator to modulate the impedance of the ankle joint in a transtibial prosthesis. Assuming the ankle to be a spring-damper system, the study estimated the stiffness parameters using a least squares optimization approach. The study, [20], proposed an ankle-foot exoskeleton that aided stroke patients in combating foot-drops. The system was manipulated using an impedance controller that was automatically tuned using a simple algorithm. Other notable attempts at estimating joint impedance during walking are [21] and [22]. The former proposes quasi-stiffness, which estimates joint stiffness by calculating the slope of the torque vs. angle curve during stance. The latter solved a constrained optimization problem where the joint impedance was the decision variable and the dynamics of a humanoid bipedal served as constraints. A common attribute of these studies is that the estimated impedance does not vary smoothly and continuously throughout the gait cycle. This paper will refer to the above group of estimation methods as theoretical approaches.
In the field of science, the most accepted method of identifying a system’s parameters is by experimentally inducing perturbations. This paper will refer to such a methods as empirical approaches. With the objective of empirically determining the ankle impedance while walking, researchers conducted experimental studies on the ankle [23, 24, 25]. These studies perturbed the ankle at various instances of the gait cycle. The ankle’s response to the perturbation was gathered and analyzed to evaluate empirical values for stiffness and damping. It was reported that the ankle stiffness increases upon heel-strike until terminal-stance, following which the stiffness reduces until toe-off and maintains an almost constant value during swing phase. The damping parameter was observed to be high during heel-strike and toe-off. Unlike the results of the prior theoretical approaches, the empirical impedance parameters varied continuously and smoothly throughout the gait cycle. The first objective of this paper is to bridge the gap between the theoretical and empirical approaches by extending the least squares method proposed by [7] to produce results that conform to [23, 24, 25]. Specifically, the stiffness and damping parameters will be allowed to vary continuously throughout the gait cycle.
In [14], researchers successfully implemented the empirical impedance parameters reported in [23] to control the ankle of a transfemoral prosthesis. Though the study enforced impedance control on the knee, it utilized quasi-stiffness as the impedance parameter. Unfortunately, to the authors’ knowledge, there have been no published attempts at empirically estimating knee or hip impedance during the gait cycle via perturbation studies. Any insight gained in this matter is limited to the swing phase [26]. Possible reasons for this gap in knowledge are (i) the huge investment required to conduct perturbation studies, and (ii) the grand challenge of isolating of the effects induced by the perturbation to the joint being studied. Thus, the research community would highly benefit from theoretical approaches to estimating impedance. It is hoped that the least squares optimization method proposed in this study can be used to estimate knee and hip impedance. The concluding section of this paper presents a preliminary estimate of knee impedance. The resulting impedance will be compared with other literary works.
A recent study by [27] proposed a continuum of equilibria in contrast to the discrete set of equilibria implemented in [7]. The study also estimated the impedance of the knee joint using a theoretical approach. This study has raised questions regarding the effect of multiple equilibria on the performance of impedance controllers. The second objective of this paper is to fill this gap in knowledge by investigating the role of multiple equilibria on the proposed least squares estimation method. The validity of the resulting impedance will be assessed via implementation on an existing prosthesis AMPRO II. Additionally, attempts will be made to reduce the required tuning process during implementation.
Ii Least Squares Estimation of Impedance Parameters
The optimization problem solved in this paper is fundamentally similar to the one used by [7]. The lower limb joints are modeled as a spring-damper system. Let and represent the stiffness and damping of the joint, respectively. The generated torque can be calculated as follows.
(1) |
In (1), and signify the position and velocity of the joint, while is the equilibrium angle of the joint. It is desired that the generated torque be similar to that found in healthy human walking [18], say . Thus, the optimization problem minimizes the error between the torque and . Per [23, 24, 25], the stiffness and damping parameters continuously vary throughout the gait cycle in a smooth manner. Most of the variation in the impedance parameters are observed during the stance phase, while the parameters adopt an almost constant value during the swing phase. To permit the continuous variation of stiffness and damping, while maintaining minimal decision variables, the stiffness and damping parameters were represented by polynomials during the stance phase. The orders of the polynomials were adjusted to get a better fit (i.e. reduce difference between and ). During the swing phase the impedance parameters were assigned constant values: and . Supposing and represent the order of the stiffness and damping polynomials, the impedance parameters at any instant during the gait cycle are determined as follows.
(2) | |||
(3) |
Per the requirement for continuity in the impedance parameters, and . Much like [7], the gait is sectioned based on kinematic changes; making a set of angles. The optimization problem can be summarized as follows.
(4) | ||||
Subject to: | (5) | |||
(6) | ||||
(7) |
The constraints listed in (5) force the positivity of the impedance parameters. Further, (6) ensures that the parameters maintain continuity between gait cycles. The last constraint, (7), forces the resulting to be continuous using a Lipschitz constant, . Additional bounds were added, as needed, to restrict the value of the equilibrium angles.
Ii-a Multiple Equilibria
Set label | Sections of the gait cycle | |||
---|---|---|---|---|
Set A | 0% - 13% | 13% - 40% | 40% - 63% | 63% - 100% |
Set B | 0% - 40% | 40% - 63% | 63% - 100% | |
Set C | 0% - 60% | 63% - 100% | ||
Set D | 0% - 100% |
To study the impact of multiple equilibria on the impedance controller, four sets of equilibria were established. The first set echoes the one found in [7]. The gait is sectioned in accordance to the foot contact sequence during stance phase. The first phase initiates at heel-strike (0%) and continues until foot-drop (13%), followed by the second phase that terminates at heel-off (40%). The third phase exists between heel-off and toe-off (63%). Unlike [7], the swing phase of the gait cycle is not sectioned. This set of equilibria has been labeled as Set A. One of the objectives of this paper is to reduce the number of impedance control parameters that require tuning. The number of equilibria contributes heavily towards the number of tuning parameters. Thus, this paper wishes to determine the minimum number of equilibria required to generate natural human-like walking (both kinematically and kinetically). The remaining three sets of equilibria implement fewer sections of the gait cycle. Table I lists said sets. The sectioning proposed in Set B is similar to the one proposed by [23].
Ii-B Results of the optimization
The minimum order of the stiffness and damping polynomials required to lower the optimal cost was determined to be 4. This study fixed the order of the stiffness and damping polynomial to be the same. It was observed that the trend of the stiffness and damping parameters was not sensitive to the equilibria set. Fig. 2 depicts the stiffness and damping parameters. The corresponding torques, , has been presented in the Appendix. It is evident that all sets of parameters attained a suitable cost to the optimization problem. While the trend of the stiffness parameter, during stance, resembled that reported by [23, 24, 25], the values are greatly lower. The trend of the damping parameters, on the other hand, did not entirely conform to the results presented in [23, 24, 25]. Though it portrayed high damping post heel-strike, there is little to no damping during terminal-stance. Better results could be attained by increasing the order of the damping parameter. Table II presents the equilibrium angles that resulted from the optimization. Sets A to C showed similarities by having the ankle plantar-flexed during terminal-stance and dorsi-flexed during swing. The equilibrium angle for Set D resembled a foot-drop condition–the state a human foot would conform to when physically unconstrained. It was anticipated that the foot-drop condition would pose a challenge during swing phase. It is likely that certain compensatory actions will be needed to assure sufficient foot clearance during swing.
Set label | 0% - 13% | 13% - 40% | 40% - 63% | 63% - 100% |
---|---|---|---|---|
Set A | 0.0294 | -0.3428 | -0.3491 | 0.3029 |
Set B | -0.4258 | -0.4363 | 0.0000 | |
Set C | -0.4363 | 0.1453 | ||
Set D | -0.4655 |
Iii Testing Methodology
The proposed sets of impedance parameters were tested on a custom-built powered transfemoral prosthesis shown in Fig. 3. AMPRO II (Fig. 3) has an actuated ankle and knee joint, and a passive spring-loaded toe joint. While the proposed impedance controller was implemented at the ankle, a previously published controller–a hybrid of impedance and trajectory tracking–was used to manipulate the knee. The latter has been discussed in [28]. The prosthesis was operated under a time-based scheme which utilizes a parameter that linearly increases from 0 to 1 as the gait progresses from 0% to 100%. This parameter is used to identify the progress in the gait cycle. A force sensor placed under the heel was used to initialize the parameter. To the authors’ knowledge, current state-based control schemes have limitations that are yet to be overcome [29, 28]. For instance, the study [29] investigated the usage of thigh angle as an indicator (or phase variable) of gait progression. It was demonstrated that the resulting phase variable does not display the ideal linear behavior during mid-stance, making state identification difficult. Since the focus of this study was to evaluate the performance of continuously varying impedance parameters, it was preferred the performance be unaffected by the possible inaccuracies of state-based control. Further, it was unclear whether the stiffness at AMPRO II’s toe joint would impact the ankle’s performance. To study the effect of each impedance controller, in an isolated manner, the toe joint was restrained using a rigid element. Future studies will investigate the impact of toe stiffness on the generated gait.
Iii-a Experimental protocol
To validate the proposed idea, an indoor experiment was designed using the aforementioned powered prosthesis in Fig. 3. A healthy young subject (male, 5’7” height, 150 lb weight) participated in the experiment using an L-shape simulator that helped emulate prosthetic walking. The subject was asked to walk on a treadmill at his preferred walking speed (0.7 m/s). The subject’s safety was assured by handrails located on either side of the treadmill. The experiment protocol has been reviewed and approved by the Institutional Review Board (IRB) at Texas A&M University (IRB2015-0607F).
Iii-B Tuning
The subject recruited showed a considerable height difference between his limbs while wearing the prosthesis. In an attempt to solve this issue, the subject was asked to wear boots during the experiments; unfortunately, the difference in height still persisted. This height difference significantly limited the amount ankle dorsi-flexion observed in the prosthetic during mid-stance. In compensation, the equilibrium angles were tuned to reduce the magnitude of the plantar-flexed angles. The tuned equilibrium angles have been documented in Table III. In addition, the stiffness and damping curves were scaled down by factors and , respectively. A major drawback of scaling was that the stiffness during swing phase was no longer sufficient to transition from the plantar-flexed equilibrium angle during terminal-stance to the swing dorsi-flexion angle. Thus, a constant stiffness term () was uniformly added to the stiffness curve. The following equations describe the tuning process.
(8) | ||||
(9) |
The scaling factors were reduced until certain dorsi-flexion was observed during the mid-stance phase. The term was increased until the ankle displayed dorsi-flexion during the swing phase. Note that was not required for Set D since the equilibrium angle remained constant throughout the gait cycle. The stiffness curve for Set A was scaled by a factor of , while for the remaining sets. Further, while for Set A, for the remaining sets. The constant term was equal to 20 for all sets. Though Set D did not necessitate a constant term, the term was implemented in the interest of conducting a systematic study where the stiffness curves of each set were approximately of same magnitude.
Set label | 0% - 13% | 13% - 40% | 40% - 63% | 63% - 100% |
---|---|---|---|---|
Set A | 0.0100 | -0.0875 | -0.3490 | 0.0873 |
Set B | -0.1745 | -0.2617 | 0.0000 | |
Set C | -0.2617 | 0.1452 | ||
Set D | -0.2617 |
Iv Results and Discussion
Figure 4
presents the average angular trajectory, torque, and power of the ankle for all four sets of impedance parameters. The averaged values represent 15-20 consecutive gait cycles. The standard deviation was well bounded, indicating the consistency of the observed results. Barring
Set D, the trend of the kinematic and dynamic curves are similar across the impedance sets. The trend also bears resemblance to healthy human data reported in [18]. On the other hand, the magnitude of the results was greatly lower in comparison. It is strongly believed that the height difference explained in Section III-B is the key reason behind these discrepancies. The following subsections compare the performance of each impedance set in terms of kinematics and dynamics of the generated gait. Following which the limitations of this study have been discussed.Iv-a Comparison of kinematics
It should be emphasized that the stiffness and damping curves portrayed similar trends across all sets of impedance. Thus, the kinematics of the generated gait was dictated by the equilibrium angles. The following observations form the basis of this claim: (i) Set A displayed lesser plantar-flexion proceeding heel-strike in comparison to the other sets owing to the dorsi-flexed equilibrium angle between heel-strike and foot-drop (ii) Lesser dorsi-flexion was observed during terminal-stance in Set C and D. Unlike these sets, Set A and B
increase the plantar-flexed equilibrium angle in increments. It is likely that such an incremental ascension assisted the subject in achieving higher dorsi-flexion during mid and terminal-stance (iii) The variance in ankle angles, among sets, at the beginning and end of the gait cycle is due to varying swing equilibrium angles. It would be beneficial to implement a higher swing equilibrium angle since it ensures sufficient foot clearance during swing (iv) Plantar-flexion at push off was greater in
Set A due to the higher equilibrium angle. Additionally, Set A showed an earlier descent from dorsi-flexion to plantar-flexion at heel-off (40%). A plausible explanation is that the combined effect of heightened stiffness and higher plantar-flexion forced an earlier push-off (v) Evidently, Set D showed an absence of dorsi-flexion during swing phase due to the plantar-flexed equilibrium angle. As anticipated, the foot-drop equilibrium angle of Set D resulted in few stumbles during the swing phase [30].A kinematic abnormality that cannot be overlooked is the absence of push-off in Set D. As stated earlier, the impedance control strategy was implemented using a time parameter that linearly increased as the gait progressed. With that being said, the success of time-based control heavily depends on the subject’s ability to synchronize his/her gait with said time parameter. This synchronization task proved to be a mighty challenge while testing Set D. Specifically, the constant foot-dropped equilibrium angle introduced gait abnormalities such as exaggerated hip extension at toe-off. In preparation for the over-extended hip angle, the subject forcibly maintained a dorsi-flexed ankle beyond peak stiffness (which occurs at 50% of the gait cycle). When toe-off eventually occurred, the stiffness was thus insufficient to quickly restore the ankle to the plantar-flexed equilibrium.
Iv-B Comparison of dynamics
Set A and B resulted in higher torque during terminal-stance in comparison to the other sets. This is likely due to higher dorsi-flexion in mid and terminal-stance (as discussed in Section IV-A). As a consequence, the corresponding power was higher in Set A and B. Most interesting to note was the abrupt change in the torque corresponding to Set A at foot-drop (13%). Such a change was not observed in the results of the other sets. This is undoubtedly a consequence of the change in Set A’s equilibrium angle at foot-drop. A similar behavior was observed at heel-off (40%) in the results of both Set A and B. Thus, fewer changes in equilibrium angles ensure a smoother torque output. Further, the re-positioning of the ankle joint to the swing dorsi-flexed angle resulted in positive torque at the beginning of the swing phase for Sets A to C. In regards to the power curves, Set A’s power output displays an aberrant increase at heel-off. The backing rationale is the high velocity arising from the hastened push-off detailed in Section IV-A. Finally, the push-off power associated with Set D was significantly lower due to the previously discussed delayed push-off.
Iv-C Limitations of this study
The aforementioned height difference between the subject’s limbs forced him to adopt a limp; i.e. a shorter step length and longer stance phase on the limb without the prosthetic. A major drawback of these gait asymmetries was insufficient loading of the prosthetic ankle during mid-stance; resulting in the reduced dorsi-flexion. Further, the swing equilibrium angle could have been tuned in a more systematic manner; i.e. the enforced swing dorsi-flexed angle could have been uniform across all sets. Finally, the usage of time-based control enforced the stringent requirement of gait synchronization on the subject.
V Conclusion
This study proposed a least squares approach to estimating ankle impedance based on the work of [7]. The resulting impedance values followed a trend consistent with perturbation studies [23, 24, 25]. The estimated impedance parameters were not sensitive to the number of equilibria enforced. On studying the effect of multiple equilibria on the impedance controller’s performance, the following results were revealed: (i) Multiple equilibria during mid and terminal-stance phase that increase the plantar-flexed equilibrium angle (in increments) can assure more ankle dorsi-flexion during mid-stance. Consequently, the generated torque and power at push-off are higher (ii) Abrupt changes in torque can be expected at the instances when the equilibrium angle switches. Such changes impact the robustness of the system to perturbations. This ideology is the motivation behind studies in continuum of equilibria [27] (iii) While overly plantar-flexed equilibrium angle during terminal-stance results in higher push-off torque, it can give rise to premature push-off (iv) Most importantly, a single equilibrium angle during stance phase is sufficient to generate human-like kinematics and dynamics.
Vi Future Work
To overcome the prior listed limitations, a state-based control scheme will be implemented to enable flexibility in gait speed. The basis of such a control scheme can be found in [28]. Current efforts are targeted at overcoming the limitations of the control scheme by using sensor-fusion. This paper assumed the order of the stiffness and damping polynomials to be the same during the stance phase. Future attempts will investigate the validity of this assumption. Presently, a height adjustable prosthesis is under development. This prosthesis will be used in all future studies of the impedance controller. Based on this study, an automated tuning algorithm will be developed and consecutively implemented on a transfemoral prosthesis. The dorsi-flexion observed during mid-stance will serve as the primary indication of well-tuned stiffness and damping parameters. Further, the impedance parameters proposed in this study will be used in a hybrid control strategy that would implement impedance control during stance phase followed by trajectory tracking during swing [28].
In regards to the knee, a preliminary estimation of knee impedance during stance has been presented in Fig. 5. The estimation utilized the sectioning proposed in Set B. The optimized equilibrium angles were 0.1413 rad of knee flexion during initial stance, followed by 0.1968 rad until toe-off. The result is consistent with the values implemented by [7] and [27].
Considering the extensive knee movement during swing, it may be more desirable to use trajectory tracking during swing. Another path worthy of investigation is the concept of a continuum of equilibria proposed by [27].
Appendix
The following figure depicts the fit attained using the least squares optimization. Also included is human torque data from [18]. Table IV presents the optimized coefficients of the stiffness and damping polynomials.
Stiffness | |||||
---|---|---|---|---|---|
Set label | |||||
Set A | -29870.57 | 28322.46 | -7061.82 | 586.04 | 2.21 |
Set B | -19977.92 | 17340.71 | -3424.51 | 199.97 | 0.32 |
Set C | -19822.71 | 17146.19 | -3333.05 | 181.16 | 0.75 |
Set D | -16520.32 | 14144.17 | -2596.67 | 136.56 | 0.00 |
Damping | |||||
Set label | |||||
Set A | -22.45 | 88.08 | -76.20 | 18.76 | 0.12 |
Set B | -140.21 | 261.35 | -158.46 | 31.21 | 0.12 |
Set C | -164.32 | 303.05 | -181.22 | 35.04 | 0.18 |
Set D | -171.23 | 311.36 | -182.97 | 34.53 | 0.26 |
Acknowledgment
The authors thank Kenny Chour for his assistance during experimentation.
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