A stochastic Hammerstein model for control of oxygen uptake during robotics-assisted gait

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2009; 23:472–484Published online 15 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.1060

A stochastic Hammerstein model for control of oxygen uptakeduring robotics-assisted gait

K. J. Hunt1,2,∗,† and D. B. Allan2,3

1Centre for Rehabilitation Engineering, Department of Mechanical Engineering, University of Glasgow,

Glasgow G12 8QQ, U.K.2The Scottish Centre for Innovation in Spinal Cord Injury, U.K.

3Queen Elizabeth National Spinal Injuries Unit, Southern General Hospital, 1345 Govan Road,Glasgow G51 4TF, U.K.

SUMMARY

Robotics-assisted gait devices have been introduced as part of rehabilitation therapy for people withneurological impairment. The scope of application has recently been expanded to encompass cardiopul-monary training and assessment, but the response of oxygen uptake to progressively increasing workrate while walking with robotics assistance can be strongly nonlinear. This hampers efforts to estimatecardiopulmonary performance parameters.

We hypothesized that a linear increase in oxygen uptake can be achieved by employing feedbackcontrol. We aimed to develop a nonlinear stochastic model of oxygen uptake response to work rate duringrobotics-assisted gait, and to employ this model to design and test feedback strategies for control ofoxygen uptake during incremental exercise testing.

A new model structure was developed consisting of a Hammerstein component, i.e. a static nonlineargain combined with a linear time-invariant transfer function, and a stochastic approximation of regularizedbreath-by-breath fluctuations in oxygen uptake.

Simulation results using the model and a range of control design parameters confirmed the ability offeedback to achieve a linear increase in work rate in the face of strong plant nonlinearity.

This new approach could be applied during incremental exercise tests and may lead to improvedunderstanding and estimates of the sub-maximal gas exchange threshold and peak cardiorespiratoryperformance parameters. Copyright q 2008 John Wiley & Sons, Ltd.

Received 31 March 2008; Revised 13 June 2008; Accepted 16 June 2008

KEY WORDS: rehabilitation engineering; rehabilitation robotics; spinal cord injury; feedback control

∗Correspondence to: K. J. Hunt, Centre for Rehabilitation Engineering, Department of Mechanical Engineering,University of Glasgow, Glasgow G12 8QQ, U.K.

†E-mail: [email protected], URL: http://www.SCISCI.ac.uk

Copyright q 2008 John Wiley & Sons, Ltd.

A STOCHASTIC HAMMERSTEIN MODEL 473

1. INTRODUCTION

Rehabilitation robotics have been introduced in clinical practice to support therapeutic interventionsin a wide range of neurological conditions [1]. Several devices have been developed to providerobotics-assisted gait training for people who have suffered a spinal cord injury or a stroke [2–4].These include driven-gait orthoses (DGOs) of the type illustrated in Figure 1 [6]. The researchfocus and clinical application of DGOs has been to promote positive neurological adaptations,plasticity and recovery of function, which may result in improvements in the ability to walk.

Recent work has investigated cardiopulmonary responses to robotics-assisted walking exer-cise [7, 8]. The potential application of these gait-support devices for cardiopulmonary fitnesstraining and assessment has been explored [9]. Drawing on approaches from exercise scienceand physiology, exercise intensity can be specified using measures of heart rate, work rate orratings of perceived exertion, but the most reliable and direct indicator appears to be the rate ofoxygen consumption in the working muscles, which is usually estimated using real-time respiratorymeasurement of oxygen uptake, denoted by VO2 [10].

From a control engineering perspective, the exercise driver, i.e. the demanded work rate, canbe considered as the plant input. The plant output, oxygen uptake, then responds dynamically ina fashion that depends on the physiology of the exercising subject (e.g. Figure 2). For ergometersin which there is a systematic and linear relationship between work rate and oxygen uptake,the use of work rate to set exercise intensity is possible, thereby obviating the need for real-time, breath-by-breath VO2 measurement. This approach can be applied during training and duringtesting for assessment of the main cardiopulmonary performance parameters. An important testin this regard is the incremental exercise test (IET) in which imposed work rate increases as

Figure 1. Robotics-assisted gait. A neurologically impaired subject is assisted in walking on a treadmillby a pair of driven-gait leg orthoses [4]. These use linear drives to power the hip and knee joints. Part of

the subject’s body weight is relieved by a dynamic overhead unloading system [5].

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2009; 23:472–484DOI: 10.1002/acs

474 K. J. HUNT AND D. B. ALLAN

a ramp until the exercising subject’s peak aerobic capacity is reached, marked by a peak inthe VO2 response [11, 12]. This test delivers an estimate of peak oxygen uptake, VO2peak

, andan estimate of the sub-maximal gas exchange threshold (GET) marking the onset of anaerobicmetabolism: this is observed via respiratory measurements during the IET by an increase in therate of carbon dioxide output, VCO2, indicated by a deflection point in the VCO2–VO2 relationshipand supported by additional respiratory phenomena [13, 14]. To reliably obtain an estimate ofthe GET, it is important that the VO2 response linearly follows the ramping work rate during thetest.

It has been observed in robotics-assisted gait that the VO2 response during an IET, where workrate increases linearly, can be strongly nonlinear and as a consequence the ability to detect a GET isimpaired [9]. This is in part due to the work rate measure available giving only a partial indicationof external work done: it is derived from forces measured from subject–orthosis interactions, butit neglects work done on the treadmill surface and by the subject’s arms. Neurologically impairedsubjects tend to employ increasing upper-body effort as the IET progresses and they approachmaximum volitional performance. This additional work is not accounted for and the associatedoxygen requirement therefore distorts linearity of the VO2 response.

We hypothesized that in robotics-assisted gait a feedback approach for control of oxygen uptakecan be used to achieve a linear increase in VO2 with respect to time during the IET. The samework rate variable would be used as the exercise driver for the subject, but it would be determinedautomatically in real time by the dynamic feedback, and the nonlinear characteristics of therelationship between work rate and VO2 would be neutralized to an extent determined by theclosed-loop bandwidth.

To investigate this hypothesis, our aim here was to develop a nonlinear stochastic model ofoxygen uptake response to work rate during robotics-assisted gait, and to employ this modelto design and test feedback strategies for control of oxygen uptake during incremental exercisetesting.

Figure 2. Structure for feedback control of oxygen uptake during robotics-assisted gait. In the work ratecalculation, Mi are the joint torques and �i the corresponding angular velocities with i=1 . . .4 ranging

over the actuated hip and knee joints.



2. MODEL DEVELOPMENT

Oxygen uptake during exercise depends dynamically on the work rate of the exercising subject.For robotics-assisted gait, we previously developed a feedback structure that allows the subjectto control work rate to desired values by means of real-time visual feedback of target and actualwork rates [9, 15] (see also Figure 2). This is a manual feedback approach relying on adjustmentby the subject of volitional effort. The oxygen uptake response then follows work rate, but withsubstantially slower dynamics. Our aim here is to develop an appropriate model structure describingthe relationship between target work rate (input) and oxygen uptake (output), i.e. the open-looprelationship linking u(t) and y(t) in the block labelled ‘plant’ in Figure 2. The plant is a complexsystem consisting of biomechanical subject–machine interactions, volitional feedback control ofwork rate and the physiological responses. Since oxygen uptake responds much more slowlythan the work rate dynamics, however, the overall input–output relationship is dominated by thephysiological components of the system.

The proposed model for the dynamic relationship between target work rate u(t) and oxygenuptake y(t) has at its core a linear time-invariant function G(q−1) of the form

G(q−1)= q−k B(q−1)

A(q−1)(1)

Here, q−1 is the delay operator and q−k is a discrete delay of k sample periods. A and B arepolynomials in q−1:

A(q−1)=1+a1q−1+·· ·+anaq

−na (2)

B(q−1)=b0+b1q−1+·· ·+bnbq

−nb (3)

This linear component can be taken as valid for small-signal variations around some operatingpoint. For large-signal changes in work rate and oxygen uptake, a strongly nonlinear relationshiphas previously been observed during robotics-assisted gait [9]. This relationship exhibits relativelylow gain at low work rates, but the gain is seen to increase smoothly by a factor of as much as 5as work rate increases. Since this nonlinear behaviour is dependent on the plant input (i.e. workrate), it is deemed appropriate to augment the small-signal linear dynamics G(q−1) with a staticnonlinear gain, f (.), at the plant input: f (.) is defined as a smoothly increasing function dependenton the current value of work rate. This leads to the Hammerstein model structure shown in Figure 3(the Hammerstein element comprises the static nonlinearity and the linear transfer function).

Development of a model for disturbances influencing the oxygen uptake signal proceeds fromthe observation that individual breaths are irregularly spaced in time and that the volume of oxygentaken up at the lungs with each breath can vary widely, depending on the volume of inspired air.An appropriate model for this situation is a compound Poisson process, a non-stationary stochasticprocess consisting of changes of random magnitude occurring at random times (this type of modelhas previously been utilized within optimal control theory [16]). Although each breath occursirregularly over time, discrete-time control usually operates at regularly spaced sample instants. Inorder to synchronize oxygen uptake measurements with the discrete sample instants, the followingdata regularization approach has been adopted [17, 18]: at times corresponding to each sampleinstant, all breath-by-breath values during the preceding sample interval are averaged and this istaken as the VO2 value at that time. Averaging and resampling a compound Poisson process in this



Figure 3. Stochastic Hammerstein model.

way leads naturally to a random-walk process e(t), which we model here as the outcome of a zero-mean white noise signal d(t) driving a discrete integrator. The complete stochastic Hammersteinmodel developed here is shown in Figure 3.

The open-loop results presented below were obtained using the stochastic Hammerstein model(Figure 3). The model was then embedded within the proposed feedback structure to investigateclosed-loop performance (Figure 4). The nonlinear function f (.) was implemented on the range[0,24]W with the gain smoothly increasing from 1 to 6 over that range. This approximatesexperimental observations seen previously [9], where a DGO device was employed (Lokomat,Hocoma AG, Volketswil, Switzerland). The linear transfer function used in control design wasobtained empirically in previous work [19] and was

G(q−1)= 0.01127q−1

1−0.8602q−1(4)

with a sample time of 20 s. d(t) was a zero-mean white noise signal with power as specified below.

3. FEEDBACK DESIGN

The overall structure for feedback control of oxygen uptake during robotics-assisted gait is shownin Figure 2. The oxygen uptake controller CVO2

is a dynamical system that operates on the target

and measured oxygen uptake signals, V ∗O2

and VO2, and produces a target work rate P∗ for theinner loop within the plant.

Our linear feedback design approach is to define the nominal plant model as the linear componentof the open-loop model of Figure 3 scaled by the gain of the nonlinear function f (.) at a fixedoperating point, together with the stochastic disturbance signal e(t). The controller is computed onthe basis of this nominal model, but the full stochastic Hammerstein structure can then be used forclosed-loop robustness analysis and simulation can be carried out using the closed-loop simulationstructure shown in Figure 4.



Figure 4. Closed-loop simulation framework. This corresponds exactly with the real system structureshown in Figure 2. The plant here is the nonlinear stochastic model of Figure 3 and comprises all of thecomponents within the block labelled ‘plant’ in Figure 2. The dynamic controller CVO2

is governed by

Equation (5), u(t)=(T (q−1)r(t)−S(q−1)y(t))/R(q−1).

The oxygen uptake controller CVO2is governed by the equation

u(t)= 1

R(q−1)(T (q−1)r(t)−S(q−1)y(t)) (5)

where R, S and T are polynomials in q−1. With reference to Figures 2 and 4, r(t) is the oxygenuptake target signal V ∗

O2(t), u(t) is the target work rate P∗(t) and y(t) is the oxygen uptake VO2(t).

The controller polynomials R, S and T are determined in a pole assignment design algorithm[16, 20] as follows. The closed-loop equation resulting from the nominal dynamics (1) and thecontroller (5) is

y(t)= q−k B(q−1)T (q−1)

�(q−1)r(t)+ A(q−1)R(q−1)

�(q−1)e(t) (6)

Here, the closed-loop characteristic polynomial is denoted as � and is given by �= AR+q−k BS.(For simplicity of exposition we assume here that the gain of f (.) at the nominal operating pointis unity, or that any non-unity gain is subsumed into B.) Desired closed-loop poles are specifiedas the roots of a model polynomial Am(q−1) and an observer polynomial Ao(q−1) (i.e. we set�= AmAo). Integral action can be included in the controller by factoring the controller denominatoras R= R1R2. For integral action, R1 is pre-specified as R1=(1−q−1); to place a double integratorin the controller (used here for tracking of ramp reference signals) R1 is chosen as R1=(1−q−1)2.The design equation to be solved to obtain R2 and S is then

AR1R2+q−k BS= AmAo (7)

T is chosen as a scalar multiple of Ao to cancel observer dynamics from the reference trackingresponse and to ensure unity steady-state gain from r to y, i.e. T (q−1)=(Am(1)/B(1))Ao(q−1)

(see Equation (6)). Closed-loop poles were chosen here by setting Am and Ao as the denominatorof discretized second-order transfer functions with specified step-response damping and rise time.



Several closed-loop controllers were designed following the procedures detailed above; detailedresults are presented using two of these controllers. The first controller was designed with adouble integrator, R1=(1−q−1)2, a model rise time of 200 s, observer rise time of 50 s anda damping factor of 0.9 for both model and observer. The second controller was designed tohave a lower bandwidth: it had a model rise time of 450 s (all other design variables wereunchanged). Summary closed-loop performance measures are provided for these controllers, andfor two additional controllers with intermediate rise times (300 and 400 s).

4. RESULTS

The noise-free, open-loop response of the plant model to a ramp work rate input demonstratesstrongly nonlinear VO2 dynamics (Figure 5). This approximates quite well the experimental resultsseen previously in tests with paraplegic subjects [9].

The noise-free closed-loop response with the first controller shows that, following an initialtransient at the start of the VO2 ramp phase, the VO2 response exactly follows the ramp VO2 targetin a linear fashion (Figure 6). This result supports our hypothesis that a linear increase in oxygenuptake during incremental exercise testing can be achieved by employing feedback control, withthe target work rate being determined in real time by the dynamic feedback.

When noise is introduced into the system, the first controller is still able to achieve close trackingof the linear VO2 target signal (Figure 7, noise power for d(t) of 0.005).

When the lower-bandwidth, second controller is employed, variation in the control signal (targetwork rate) is reduced (Figure 8(b)). Tracking of the VO2 target signal remains linear, but thetracking is less tight (Figure 8(a)). This result confirms our hypothesis that the chosen closed-loopbandwidth determines the extent to which the nonlinear characteristics of the relationship betweenwork rate and VO2 are neutralized by the feedback.

Summary closed-loop performance measures for four controllers in total confirm that the effectof reduced bandwidth (i.e. increased model rise time) is to increase the variance of the trackingerror and to decrease the variance of the control signal (Table I).

5. DISCUSSION

It was recently observed that oxygen uptake responds nonlinearly to changes in work rate duringrobotics-assisted gait and that this presents difficulties in estimation of important cardiopulmonaryperformance parameters [9]. We set out to investigate the hypothesis that a linear increase in oxygenuptake can be achieved during incremental exercise testing by employing feedback control.

We developed a novel nonlinear stochastic model of oxygen uptake response to work rate duringrobotics-assisted gait. The static nonlinear function f (.) was selected to give qualitatively similargain characteristics seen experimentally in a small cohort (n=3) of subjects with incomplete spinalcord injury [9]. This gives an adequate representation for testing of feedback control strategies,but scope remains for application of a systematic approach to estimation and validation of thisnonlinear function for specific subjects, or for classes of subjects (such an approach has beenapplied to identify a Hammerstein model of oxygen uptake for able-bodied subjects walking on atreadmill [21]). It should be noted that f is required only for simulation and evaluation of feedback



400 600 800 1000 1200 1400 1600 1800 20000

0.5

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oxyg

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400 600 800 1000 1200 1400 1600 1800 2000

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Figure 5. Open-loop response (noise free): (a) VO2 output y(t) and (b) work rate target u(t).

control performance and is not needed for the feedback design itself. Uncertainty in this functiondoes not therefore limit practical application of the feedback approach.

The linear transfer function G(q−1) was obtained previously in identification and validationexperiments with one able-bodied subject in a DGO device [19]. This single model was usedto design a linear feedback controller that was tested in five able-bodied subjects and found togive satisfactory performance and stability robustness [19]. This suggests that, although the linearcomponent of the overall model developed here is theoretically a small-signal model valid arounda given operating point, repeated identification experiments to obtain an estimate of G for specificsubjects can often be avoided.



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Figure 6. Closed-loop response (noise free, model rise time 200 s). The open-loop responsesare repeated here for ease of comparison (thin dashed lines): (a) VO2 target r(t) and outputs

y(t) and (b) work rate target u(t).

The stochastic signal e(t) was developed here as a model of random-walk-like fluctuations inoxygen uptake, derived as a regularly sampled and integrated compound Poisson process. Thisgives a plausible representation of the physiological characteristics of breathing and encompassesthe requirement for resampling of the data to the control sample instants. There is a need to carryout systematic identification experiments to validate this component of the model and to evaluatethe extent to which the statistical properties of the model match the observed phenomena.

The linear component of the model, G(q−1), was used to design feedback compensators forcontrol of oxygen uptake. A similar approach for control of oxygen uptake was previously applied



400 600 800 1000 1200 1400 1600 1800 20000

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Figure 7. Closed-loop response (noise power 0.005, model rise time 200 s): (a) VO2 target r(t) and outputy(t) and (b) work rate target u(t).

in the context of treadmill running with able-bodied subjects [18], stimulated cycling exercise insubjects with complete paralysis resulting from spinal cord injury [17] and in able-bodied subjectsduring robotics-assisted gait [19]. In all of these cases the basic principle of VO2 control wasconfirmed, but evaluation was limited to regulation around a constant setpoint or to tracking ofsquare wave reference signals. Here, the approach was extended to tracking of ramp VO2 targetsignals similar to those applied during peak cardiorespiratory performance testing. Appealingto the internal model principle of control theory [22], a model of the unstable reference signalwas incorporated in the feedback loop by including a double integrator in the compensator (aramp can be modelled as a double integrator driven by a unit pulse function). This ensured zero



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Figure 8. Closed-loop response (noise power 0.005, model rise time 450 s): (a) VO2 target r(t) and outputy(t) and (b) work rate target u(t).

steady-state tracking error with the ramp reference. Controllers performed robustly across the fullrange of operation of the stochastic Hammerstein model and our basic hypothesis was confirmed.Experimental evaluation of the approach within the target population of subjects with incompletespinal cord injury performing robotics-assisted gait is now warranted.

The results presented here provide further support for the general concept of direct regulation ofexercise intensity using feedback control of oxygen uptake. Within the context of robotics-assistedgait, our hypothesis that feedback can in principle achieve a linear increase in oxygen uptakeduring incremental exercise testing, despite strong plant nonlinearity, was confirmed. This new



Table I. Summary closed-loop performance measures for various model risetimes and noise power of 0.005.

Rise time (s) Error variance (×103) Control variance

200 0.53 2.84300 0.54 2.31400 0.66 2.28450 0.68 2.29

‘Error variance’ is the variance of the closed-loop tracking error signal r− y. ‘Controlvariance’ is the variance of the signal given by the difference between the noise-free controlsignal (Figure 6(b)) and the actual control signal u for the various conditions.

approach could be applied during IETs and may lead to improved estimates of the sub-maximalGET and peak cardiorespiratory performance parameters.

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A stochastic Hammerstein model for control of oxygen uptake during robotics-assisted gait