Feedback control of oxygen uptake during robotics-assisted end-effector-based stair climbing

ABSTRACT A heart rate (HR) feedback control system for end-effector gait rehabilitation robots was previously developed and successfully tested, but oxygen uptake () is thought to better characterize physiological exercise intensity. The aim of the present study was to identify and compare and HR dynamics, and to develop and test a controller for an end-effector robot operated in stair climbing mode. Six able-bodied subjects were recruited for controller testing. Command response, disturbance rejection and robustness were assessed by means of three quantitative outcome measures: root-mean-square (RMS) error of (), average control signal power () and RMS error of volitionally controlled power (). The nominal first-order linear model for had time constant  s and steady-state gain k=0.0174 (l/min)/W. The mean time constant  s for HR was significantly higher than for , where (p=0.048). Command responses for a target profile gave consistent and accurate tracking with  l/min,  W2 and  W (). Disturbance rejection performance was also found to be satisfactory. The results of the controller tests confirm the feasibility of the proposed feedback control strategy. Robustness was verified as the single LTI controller was specific to only one of the subjects and no difference in outcome values was apparent across all subjects. Subject-specific variability in breath-by-breath respiratory noise is the main challenge in feedback control of .


Introduction
Stroke is the second-most-common cause of death and long-term disability worldwide (Donnan, Fisher, Macleod, & Davis, 2008). Every second stroke leads to total or partial loss of walking ability (Mozaffarian et al., 2014). Furthermore, aerobic exercise capacity, which is defined using peak oxygen uptake, is 50% lower 30 days post stroke compared to standard values of age-matched healthy adults (Kelly, Kilbreath, Davis, Zeman, & Raymond, 2003;MacKay-Lyons & Makrides, 2002). Evidence suggests that intensive task-specific rehabilitation in all phases after stroke has a positive influence on the rehabilitation outcome (Billinger et al., 2014;Veerbeek et al., 2014). Not only are physical improvements possible, there is also evidence that cognitive function can be enhanced by early rehabilitation, depending on the amount of training (El-Tamawy, Abd-Allah, Ahmed, Darwish, & Khalifa, 2014).
It has been shown that a large volume of task-specific training is important (French et al., 2007;Langhorne, CONTACT Jan Riedo jan.riedo@bfh.ch Coupar, & Pollock, 2009). Since conventional therapy is limited by fatigue of the physiotherapists, roboticsassisted devices can improve the quality of rehabilitation. Several studies investigated the effects of robotics-assisted gait rehabilitation, concluding that it is more effective when conventional therapy is combined with roboticsassisted training due to the higher volume and intensity of training that can be applied (Chang & Kim, 2013;Hesse, Tomelleri, Bardeleben, Werner, & Waldner, 2012;Mehrholz & Pohl, 2012). Intensive task-specific training not only enhances physical and coordination skills, but also enhances plastic changes in neural circuits (Hornby et al., 2011). Even in non-impaired individuals who are trying to improve in a certain field of physical activity, the volume and intensity of training are the most important factors for success (Schmidt & Lee, 2011). Due to the benefits of, and demand for, repetitive task-specific training, there is a steadily growing market for rehabilitation robots. Two main kinds of gait rehabilitation robots can be identified: end-effector robots (Hesse, Waldner, & Tomelleri, 2010;Stoller, Schindelholz, Bichsel, & Hunt, 2014) and exoskeletons (Westlake & Patten, 2009). End-effector gait rehabilitation robots have an advantage concerning task-specific training because of the possibility to implement stair climbing activity (Hesse et al., 2010).
The studies reviewed above have focused on potential improvements in the ability to walk which can come about due to intensive, task-specific therapy and associated neurological adaptations. As a complement to this line of research, recent work has examined the applicability of gait-assistance robots for cardiovascular training and assessment. In this regard, several studies investigated the cardiopulmonary and cardiovascular responses to end-effector-based walking and stair climbing (Stoller et al., 2014;Stoller, Schindelholz, & Hunt, 2016), which confirmed that substantial increases in heart rate and respiration can be obtained from this form of exercise.
A key observation was that response magnitudes are within the range required for positive training adaptations to take place within the cardiovascular system. A further breakthrough was made when the feasibility of implementing formal cardiopulmonary exercise testing (commonly referred to as CPET) in robotics-assisted end-effector-based stair climbing was confirmed (Stoller et al., 2016). From the perspective of control engineering applications, a novel and major advance was made when for the first-time automatic feedback principles were applied and successfully tested for control of heart rate and oxygen uptake on an exoskeleton system (Lokomat, Hocoma AG, Switzerland), (Schindelholz & Hunt, 2012. A previous study, conducted using an end-effector gait rehabilitation robot (G-EO Evolution system, Reha Technology AG, Switzerland), implemented and tested a feedback system for control of heart rate (HR) (Riedo & Hunt, 2016). However, a more comprehensive physiological assessment consists of measurement of both heart rate and the rate of oxygen uptake (denotedVO 2 and referred to in the sequel simply as 'oxygen uptake').
This observation provides the motivation for the problem investigated in the present work, namely the development of controllers for oxygen uptake. Oxygen uptake provides certain advantages over heart rate, as detailed below, but also presents additional difficulties. The principal difficulty lies in the more sophisticated sensor technology required to obtain oxygen uptake in real time, and also in the substantial breath-by-breath noise which affects this signal and which must be considered in the feedback design.
The principal advantage of using oxygen uptake rather than heart rate is that a more accurate determination of the physiological exercise intensity is possible. Exercise intensity is directly correlated with oxygen uptake and with cardiac output (rate of blood flow). Cardiac output is given by the product of heart rate and stroke volume (SV) but, for the sake of convenience, usually only the heart rate is measured. Since the autonomic nervous system continuously adjusts HR and SV to achieve a required cardiac output, the HR variable displays substantial lability. The problem of using HR to characterize exercise intensity is compounded by the fact that it is susceptible to external sensory influences such as noise, disturbances appearing in the visual field of the subject, or by other psychological factors.
Oxygen uptake, on the other hand, is given by the product of the volume of oxygen taken up per breath (cf. heart: stroke volume) and breathing frequency (cf. heart: heart rate). Direct measurement ofVO 2 therefore implicitly includes both of the subsidiary variables which are continuously adapted by the autonomic nervous system (namely uptake per breath and breathing frequency), and thus gives a stable and direct characterization of exercise intensity. This explains the observation that oxygen uptake is constant when training at a constant work rate but heart rate tends to continually increase over time: despite the increase in heart rate, cardiac output stays the same, presumably due to a downward autonomous regulation of stroke volume. It is therefore recommended to prescribe training intensity using oxygen uptake (Garber et al., 2011;Kenney, Wilmore, & Costill, 2015;Pescatello, Arena, Riebe, & Thompson, 2014).
The feedback structure employed here for control of oxygen uptake is based on a method previously proposed for HR control (Riedo & Hunt, 2016). The principal challenge for HR control was to deal appropriately with broad-spectrum heart rate variability, whereas for control of oxygen uptake it is the breath-by-breath variability ofVO 2 which is important. The purposely low-pass characteristics of the control approach developed in Riedo and Hunt (2016) therefore promised to be suitable for this new application also. A further contribution of the present work is a comparison of HR andVO 2 dynamics, because HR can conveniently be measured simultaneously withVO 2 : based on previous studies using other exercise modalities (Bearden & Moffatt, 2001;Zhang et al., 2014), it was hypothesized that the time constant of the heart rate response would be higher than that for oxygen uptake.
The aim of this study was to identify and compare oxygen uptake and heart rate dynamics, and to develop and test an oxygen uptake controller for an end-effector gait rehabilitation robot using the stair climbing mode of operation.

Materials, overall identification and control structures
An end-effector gait rehabilitation robot was employed (G-EO Evolution system, Reha Technology AG, Switzerland; Figure 1). This provides three different trajectories: walking, stair climbing and stair descent. Stepping cadence and stepping height can be adjusted in the ranges 1-70 steps/min and 5-20 cm, respectively. The G-EO was augmented with a biofeedback-screen (Figure 1(B)), enabling the subject to perform volitional control of exercise work rate P (Figure 2(a)); the goal for the subject was to keep the actual value of P as close as possible to a target work rate P target . P was calculated using the speed of the end-effectors (Figure 1(A)) and the forces applied on the footplates by the subject. The volitional control structure with the human-in-theloop work-rate controller was used for identification oḟ VO 2 and HR dynamics. For system identification, as far as HR andVO 2 are concerned, this represents an open-loop structure wherein target work rate P target was predefined as a series of steps (Figure 3(a)). For feedback control ofVO 2 , the identification structure was then embedded within an outer automatic feedback control loop with the effective feedback controller transfer function C(z −1 ) = S(z −1 )/R(z −1 ) (Figure 2(b)). The automatic controller continuously updated target work rate P target on the basis of the measured actual oxygen uptakeVO 2 and a target/reference oxygen uptake profileVO 2 target .
A breath-by-breath cardiorespiratory monitoring system (Metamax 3B, Cortex Biophysik GmbH, Germany) was used to measureVO 2 in real time. Heart rate was measured with a chest belt (model T34, Polar Electro Oy, Finland) and a receiver (HRMI, Sparkfun Electronics, USA). With this system, individual breaths are analysed by means of gas sensors connected via a sampling line to the mask worn by the subject and a volume sensor embedded in the mask (Figure 1(C)). Real-time identification and feedback control systems used Labview (National Instruments Inc., USA), implemented directly on the computer embedded in the G-EO (Figure 1(D)). Offline data analysis and simulations were carried out using Matlab (Math-Works Inc., USA).

Plant model and system identification
For both oxygen uptake and heart rate dynamics identification, a first-order linear time-invariant (LTI) system was used, with parameters k, the steady-state gain, and τ , the time constant. Other studies have shown that this simple model can be sufficient for accurate feedback control ofVO 2 (Schindelholz & Hunt, 2015) as well as HR Hunt, Fankhauser, & Saengsuwan, 2015;Riedo & Hunt, 2016). The model is expressed in continuous (P c ) and discrete (P d ) time as ( 1 ) Here, the double arrow denotes transformation between the continuous-and discrete-time domains with sample period T s . Overall structure for closed-loop control of oxygen uptake. The controlled variable of the outer loop is the actual oxygen uptakeVO 2 . The transfer function u → y from the target work rate P target to actual oxygen uptakeVO 2 corresponds to Equation (1): P d (z −1 ), the plant model. Figure 2(a), this model structure can be employed for the transfer functions from the target work rate P target toVO 2 and HR. These transfer functions thus include the inner volitional feedback control loop involving the subject as well as the physiological reaction of the subject.

As shown in
Heart rate and oxygen uptake dynamics were both identified with a linear least-squares method on the basis of the first-order plant model in Equation (1). All six subjects (see Table 1) participated in the identification tests. Heart rate and oxygen uptake dynamics were identified using a single test for each subject. According to Figure 3(a), an individualized target work rate was defined with four changes, thus providing four step responses in HR andVO 2 . Identification was conducted within the time interval from 590 s to 1790 s in order to include all changes in target work rate but excluding start and end transients. Outcomes of the identification were the absolute root-mean-square model error (RMSE) and the normalized RMSE, which indicates the goodness of fit in percent (fit).

Controller design and analysis
For automatic feedback control ofVO 2 , the structure in Figure 2(a) was supplemented with the outer closed loop shown in Figure 2(b). The measured plant outputVO 2 is fed back as a controller input, enabling the controller to continuously calculate P target based on the control error betweenVO 2 target andVO 2 . The control design method developed in Riedo & Hunt (2016) for HR control was also applied here for control ofVO 2 . This control approach applies three important constraints for the effective feedback controller transfer function C(z −1 ) = S(z −1 )/R(z −1 ): Figure 3. The step functions for identification, command response and disturbance rejection tests. (a) Time course of individually set target power P target during identification sessions. Difference between P low and P high was 40 W for all subjects. The specific values can be found in Tables 2 and 3. (b) Time course ofVO 2 target during command response tests.VO 2 high andVO 2 low were set individually for each subject. However, the difference between both levels was the same for all:VO 2 high −VO 2 low = 0.7 l/min. Numerical values are summarized in Table 4. (c) Time course of cadence (CD) during the disturbance rejection test. CD = 35 steps/min, CD high = 70 steps/min and CD low = 35 steps/min. integrator (inclusion of (1 − z −1 ) in R(z −1 )), low-pass characteristics (inclusion of z −1 in S(z −1 ), i.e. strictly causal) and a conservative pole cancellation strategy (inclusion of plant polynomial A(z −1 ) in S(z −1 )). Full details of this feedback design method can be found in Riedo and Hunt (2016), and a brief summary is given here. Under the above constraints, the controller function C takes the form With the plant pole cancellation strategy, the characteristic polynomial of the feedback system is denoted = A . The reduced polynomial is used in a pole assignment approach to define the behaviour of the controller. The coefficients φ 1 and φ 2 of are defined by specifying the nominal closed-loop rise time t r and the damping factor ζ . Employing the above constraints, can be expressed in terms of the plant and controller parameters as The algebraic solution for the controller parameters is then readily obtained by equating coefficients of like powers, giving A deterministic reference pre-filter C pf was also employed in order to decouple the command signal from the feedback loop, as detailed in Riedo and Hunt (2016). This allows the definition of a separate rise time t r pf and damping ζ pf for the reference tracking response. C pf is obtained from the overall pole polynomial cl for the reference response, which is in turn defined by the chosen t r pf and ζ pf values. This results in In order to perform a frequency-domain analysis of the influence of the disturbance d and the measurement noise n on the control signal u (see Figure 2(b)), the input sensitivity function U o is employed, because U o is the transfer function from the disturbance input to the control signal (d → u) as well as from the noise input to the control signal (n → u). The input sensitivity function U o is defined as Because of the constraint that the controller transfer function C has low-pass characteristics, it follows from the above equations that the input sensitivity function must be low-pass as well. Further reasoning for this can be found in Riedo and Hunt (2016). This feature is important as it ensures the control signal will not be excited by high-frequency components of the disturbance and measurement noise signals.

Data processing, outcome measures and analysis
Due to substantial breath-by-breath noise on the mea-suredVO 2 signal, which is typical for respiratory data, real-time data processing is essential. Three aspects were considered when designing a filtering strategy: • Due to the real-time feedback control ofVO 2 , the use of a moving average filter with zero phase shift was not possible. Thus, the more values that are averaged, the greater is the phase lag, resulting in a delayed reaction of the controller. • There is no universal standard for data processing concerning breath-by-breath respiratory data. The most common procedure is either time averaging over a period of 15-30 s or a moving average of a certain number of breaths (e.g. 11 breaths as in Robergs, Dwyer, & Astorino, 2010). There is no recommendation specifically for real-time application as in feedback control of oxygen uptake. • A factor of about 10 is recommended between averaging time and the system time constant (Åström & Wittenmark, 2011). The identification ofVO 2 in Section 3.1 revealed a mean time constant of τ = 48.3 s thus the averaging time should be around 5 s.
In line with these considerations, a moving average of 4 breaths resulting in an averaging time of roughly 5 s (this is dependent on the changing breathing frequency) was implemented in real time for the variableVO 2 . Three outcome measures serve as the basis for quantitative evaluation of controller performance, allowing comparison between the subjects and with other studies. RMS error for oxygen uptake tracking RMSEV O 2 , average power of changes in target work rate P P , and RMS tracking error of power RMSE P were calculated as follows: Here,VO 2 sim (i) is obtained from simulation of the nominal closed-loop system, using the nominal plant model from Equation (1) and the calculated controller parameters.
The average power of the control signal P P gives a quantitative indication of the intensity of the control signal.
This value indicates how accurately a subject was able to follow the target work rate.
All three outcome values were calculated over a time period of 20 min, starting at 5 min to exclude the startup transient. This time interval thus includes 5 min before the first change in target work rate as well as 5 min after the last change in target work rate.

Test protocol and subjects
All test profiles were individually tailored for each subject in order to apply an appropriate mean level of exercise intensity in each case, but the amplitude of steps in P target (open-loop identification) andVO 2 target (closed-loop control) were kept the same for all subjects. All subjects took part in an identification and a control (command response) session. These tests were all conducted with a stepping cadence of 70 steps/min and a stepping height of 18 cm. Additionally, one subject participated in a disturbance rejection test (cadence was repeatedly changed as noted below), and another in a command response test with a non-low-pass controller. Each test was carried out on a different day.
Using data recorded in a previous study (Riedo & Hunt, 2016), appropriate work rate levels for each subject were determined for the identification tests. These measurements were conducted at moderate to vigorous intensity as predicted by heart rate and each subject's age. The mean value of P target was subjected to a squarewave signal with an amplitude of 20 W (see Figure 3(a)).
To determine appropriate individual meanVO 2 levels for the feedback control tests, the mean value ofVO 2 during identification sessions within the identification interval (590-1790 s) was calculated for each subject. This value was then employed as the mean value ofVO 2 target for the command response tests. The amplitude of thė VO 2 target square wave was calculated based on the intention to have approximately the same 20 W amplitude of target power during the command response tests as in the identification tests.
This was calculated based on the steady-state gain of the nominal model in Section 3.2, i.e. k = 0.0174 (l/min)/W. Thus, In order to test the disturbance rejection properties of the controller, a separate test with constantVO 2 target at the mean identification oxygen uptake level was conducted. A major disturbance was induced changing cadence stepwise between 70 and 35 steps/min. The downward change initially causes a reduction in velocity, therefore, the subject needs to increase the forces applied to the footplates in order to maintain the target power. Due to this increased effort, a higher oxygen uptake follows. The expectation is then that the controller will reduce target power with the result thatVO 2 should return to the target value. For changes from 35 back up to 70 steps/min, the converse response would be expected.
Six healthy males participated in this study (Table 1). No subject had any history of cardiovascular, pulmonary or musculoskeletal problems or current limitations in their physical ability. The study was reviewed and approved by the ethics committee of the Canton of Bern in Switzerland and all subjects provided written informed consent.

Identification
The identified values for oxygen uptake dynamics were: steady-state gain k = 0.0186 ± 0.0029 (l/min)/W (mean ± SD) and time constant τ = 53.4 ± 4.1 s. The overall fit was 62.5 ± 12.6% and RMSE was 0.12 ± 0.05 (l/min)/W. For the heart rate dynamics, the following values were identified: steady-state gain k = 0.532 ± 0.132 bpm/W and time constant τ = 67.3 ± 15.2 s with overall fit 62.7 ± 3.6% and RMSE 3.27 ± 0.74 bpm/W. Individual values for steady-state gain k and time constant τ are summarized in Tables 2 and 3. Subject S02 had to be excluded from this analysis due to measurement error leading to a high level of breath-by-breath noise inVO 2 .
Exemplary plots of raw identification data are shown in Figure 4(a, c), and the corresponding model validation results in Figure 4(b, d). The identification interval (range of the x-axis in Figure 4(b, d)) is displayed using red bars in Figure 4(a, c) (590-1790 s ). This includes 10 s before the first change in P target and lasts for 20 min, including all changes in P target .
All data sets for bothVO 2 and HR were used for the comparative analysis (Section 3.2). Furthermore, the identification results forVO 2 dynamics were used to select a nominal model forVO 2 controller calculation (Section 3.2).

Comparison of time constants and nominal model selection
The hypothesis that the time constant of heart rate dynamics is significantly higher than that of oxygen uptake was confirmed. Individual identified values of τ forVO 2 and HR are plotted in Figure 5 (see also Tables 2  and 3). The mean values of τ forVO 2 and HR were 53.4 ± 4.1 s and 67.3 ± 15.2 s, respectively. A single-sided t-test showed this difference to be significant with a p-value of 0.048 (the data were tested for a normal distribution with  the Lilliefors test). The mean difference was 14.0 s with a 95% confidence interval from 0.24 to ∞; the fact that the value 0 lies outwith this interval reflects the observed difference at the significance level of 5%. All values listed in Tables 2 and 3 were used for comparison, Subject S02 having been excluded as noted above. In order to investigate the robustness of the overall VO 2 control strategy, a single LTI model was selected as the nominal model for controller calculation. For this purpose, one of the individually identified models was selected as nominal, rather than using an average over several models/subjects. With this strategy, controller robustness can be better verified because potential differences between the identification subject and the other subjects can be analysed. The model selection procedure is visualized in Figure 5. Subject S05 was excluded from further consideration for the nominal model because of a distinctly higher value for steadystate gain k. It can be seen that the model for Subject S06 was closest to the average model of Subjects S01, S03, S04 and S06. Thus, the identified model for Subject S06 was selected as the nominal model and used for controller calculation. The nominal steady-state gain was therefore k = 0.0174 (l/min)/W and the nominal time constant was τ = 52.4 s. A sampling period of T s = 5 s was employed according to recommendations in Åström & Wittenmark (2011 and . These choices give the nominal discrete-time model, Equation (1),

Controller calculation
A single LTI controller was calculated according to Section 2.3 using the nominal model. The specifications used were: rise time t r = 195 s, critical damping ζ = 1 and sample period T s = 5 s. This gave the controller Equation (2) as The rise time of the reference pre-filter was set to t r pf = 150 s with the aim of having a relatively dynamicVO 2 command response, when compared to the slower specification for the feedback loop. With ζ pf = 1, Equation (5) was used to obtain With feedback controller Equation (12), the input sensitivity function U o , Equation (6), has the desired low-pass characteristics ( Figure 6, red line). The input sensitivity function for a controller with non-low-pass characteristics (used for comparative tests) is also plotted (Figure 6, blue line).

Controller tests
The quantitative outcome measures for all command response tests are summarized in Table 4 and the corresponding plots are shown in Figure 7. As with the identification tests, Subject S02 was excluded from the analysis due to excessive noise on theVO 2 signal, which appears    Notes: ID: subject number; SD: standard deviation;VO 2 mid : mid-level oxygen uptake (see Section 2.5);VO 2 low : low-level oxygen uptake;VO 2 high : high-level oxygen uptake; RMSEV O2 : RMS error of oxygen uptake; P P : average power of the control signal; RMSE P : root-mean-square error of volitionally controlled power.
to be a peculiarity of this particular subject's respiration rather than a property of the controller. The results otherwise demonstrate accurate and robust performance of the single LTI controller across all subjects. Target oxygen uptake was tracked well in all cases with low RMSEV O 2 and the dynamic response ofVO 2 to changes in the target value was close to the nominal/simulated response. The control signal (P target ) was smooth in all cases, enabling the subjects to accurately follow P target with low RMSE P . Specifically, taken across all subjects, the performance outcomes were RMSEV O 2 = 0.198 ± 0.070 l/min, P P = 2.15 ± 0.70 W 2 and RMSE P = 39.2 ± 15.3 W (mean ± SD; Table 4).
The red bars in Figures 7-9 indicate the formal outcome evaluation interval from 5 to 30 min. In addition to the variables which were already defined above, a variable denotedVO 2 smooth is shown in Figures 7-9: this is a moving average of 25 values of rawVO 2 , and was computed in order to better allow qualitative/visual assessment of the control results. These plots also show the measured HR data in order to demonstrate the phenomenon of a trend in the HR signal as described in Section 1.
The disturbance test with Subject S03 showed that the controller was able to rapidly reject the effects of changes in stepping cadence (Figure 8). It can be seen that, when the cadence is reduced, an initial rise in oxygen uptake is counterbalanced by the controller in the form of a reduction in P target , resulting in rapid recovery ofVO 2 towardsVO 2 target with RMSEV O 2 = 0.279 l/min. The disturbance here was smaller than in a former study of HR control (Riedo & Hunt, 2016). In comparison to that study, the smaller reduction in cadence enabled the subject to follow P target more accurately, resulting in the relatively low value of RMSE P = 45.5 W. Despite the disturbance, the control signal showed smooth and calm behaviour, which is reflected in the low average control signal power of P P = 1.11 W 2 .
A further test with Subject S05 investigated the behaviour of a non-low-pass controller. This test underlines the importance of low-pass characteristics in the controller and, consequently, in the input sensitivity function (cf. Figure 6). The low-pass and non-low-pass tests with Subject S05 are shown for comparison in Figure 9. The low-pass controller shows better results in all performance outcomes. The values for the low-pass vs. non-low-pass controllers were: RMSEV O 2 = 0.148 vs. 0.152 l/min, P P = 1.66 vs. 106.63 W 2 and RMSE P = 24.6 vs. 29.3 W. The most striking difference here is the average control signal power P P , which was two orders of magnitude higher with the non-low-pass controller, reflecting this controller's sensitivity to high-frequency measurement noise and disturbances.

Discussion
The aim of this work was to identify and compare oxygen uptake and heart rate dynamics, and to develop and test an oxygen uptake controller for an end-effector gait rehabilitation robot using the stair climbing mode of operation. As stated in Section 1,VO 2 offers a more direct way to characterize the physiological level of exercise intensity. While the main challenge in feedback control of HR was physiological heart rate variability (Riedo & Hunt, 2016), in the present study the substantial breath-by-breath noise onVO 2 was of primary concern.
The identification ofVO 2 and HR dynamics was successful for all subjects except for Subject S02 due to excessive noise causing fluctuation of theVO 2 signal. In a previous study, Schindelholz and Hunt (2015), Subject S05 completed aVO 2 identification using an exoskeletontype rehabilitation robot (Lokomat, Hocoma AG, Switzerland). The findings there of k = 0.0238 (l/min)/W and τ = 48.3 s are very close to the values obtained here on the G-EO with k = 0.0227 (l/min)/W and τ = 47.1 s. A comparison of the meanVO 2 time constant here of τ = 53.4 s ( Table 2) with the mean time constant of six subjects running normally on a treadmill found in Hunt, Ajayi, Gollee, & Jamieson (2008) of τ = 47.1 s, indicates thatVO 2 dynamics are similar across very different exercise modalities. Furthermore, the time constant for oxygen uptake dynamics according to Wasserman, Hansen, Casaburi, & Whipp (1999) is generally on the range τ = 40 − 60 s at medium intensity. The mean HR time constant τ = 67.3 s can be compared with a study  conducted with 24 subjects running on a treadmill, which revealed a time constant of 57.6 s. Thus allVO 2 and HR identification results from the present study are in broad agreement with previous reports using differing exercise devices.
The identification ofVO 2 and HR dynamics using a single test for each subject provided a good basis for direct comparison of the respective time constants. Previous studies suggest that τ for HR is generally higher than for  VO 2 (Bearden & Moffatt, 2001;Zhang et al., 2014). The statistical analysis of the five data sets obtained here confirm this hypothesis for the end-effector robot form of exercise, with a p-value for the observed difference of 0.048.
The command response tests showed highly accurate and fast tracking performance for all subjects evaluated, with low RMSE outcomes. The control signal, which in this application is the target work rate to be achieved by volitional control of the subject, displayed calm and smooth behaviour throughout, with a low value of average control signal power. This property is important because it eases the subject's volitional control task.
Robustness was tested by using a controller which was specific to only one subject, i.e. the nominal model for controller calculation was taken as the identified model for Subject S06. The results showed good robustness properties because there was no substantial difference in any of the controller performance outcome measures between the identification subject and the others.
The results demonstrate a very strong positive correlation between RMS tracking error and average control signal power. Ranking these variables according to Table 4, it is apparent that RMSEV O 2 and P P have the same order when ranked, i.e. the subject with the highest RMSEV O 2 also has the highest P P , and so on. The correlation coefficient for these two variables was computed as r = 0.9920 (p = 0.0009). This result suggest that, for feedback control ofVO 2 , the individual level of breath-by-breath noise is the principal determinant of objective controller perfor-mance, whereas the controller tuning parameter t r likely plays a secondary role.
The single disturbance test with Subject S03 showed a good disturbance rejection performance. The controller showed good behaviour and was able to bringVO 2 back to the target in a short time after the perturbation caused when stepping cadence was changed.
The essential difficulty encountered in this work was the substantial breath-by-breath noise observed with some subjects. Moreover, this phenomenon appeared to vary considerably between subjects. This difficulty was overcome by purposely shaping the frequency response of the input sensitivity function to be low pass, such that the control signal remained calm for even worst-case subjects displaying the strongest noise effects.
The importance of this design strategy can be seen in the single test with a non-low-pass controller, which clearly illustrates the importance of suppression of highfrequency noise and disturbances in this application. With the non-low-pass controller, the average control signal power P P was very high, which substantially increased the cognitive load on the subject and made it very difficult for him to follow the rapidly changing control signal P target . Despite the highly dynamic control signal, where the controller reacted to even small deviations inVO 2 , RMSEV O 2 was still higher than for the low-pass controller. Thus, low-pass characteristics of the controller are essential in this case.
A limitation of the present work is that only healthy, able-bodied subjects were included. Further study is required for application to patients who might have cardiopulmonary or cardiovascular disease. Such patients should be subjected to specific identification experiments in case the plant model gains and time constants are substantially different to the ones employed here, which might in turn warrant recalculation of the controller parameters.
A further limitation is that only a small number of subjects was studied, within the limited age range of 23-26 (with the exception of Subject S05, aged 52). Although S05 was not used for determination of the nominal model parameters, he was included in the feedback control experiments; the stable and accurate results achieved with this subject, and with all other subjects, point empirically to an inherent robustness of the feedback system. Nevertheless, future work should include both male and female subjects from target populations of patients undergoing rehabilitation due to neurological deficits, and who will likely be in an older age range than the subjects used in the present work. These investigations should focus first on identification of model parameters and analysis of any differences to the values obtained here.

Conclusions
The proposed feedback control strategy for oxygen uptake,VO 2 , was found to be feasible during roboticsassisted stair climbing in healthy subjects. Using quantitative outcome measures, highly accurate and stable control performance was observed in the command response and disturbance rejection tests. Robustness was demonstrated on an empirical basis because a single LTI controller was used which was specific to just one of the subjects, yet the outcome measures were similar across all subjects. The hypothesis that the time constant τ of HR is significantly higher than that forVO 2 was confirmed. Subject-specific differences in breath-by-breath respiratory noise is the main challenge in the design of feedback control systems forVO 2 . Further clinical testing of this approach with target groups of neurologically impaired subjects is warranted.