Lower-Limb Exoskeleton With Variable-Structure Series ...

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IEEE TRANSACTIONS ON ROBOTICS 1

Lower-Limb Exoskeleton With Variable-StructureSeries Elastic Actuators: Phase-Synchronized Force

Control for Gait Asymmetry CorrectionGabriel Aguirre-Ollinger and Haoyong Yu

Abstract—Series elastic actuators (SEAs) can provide accurateforce control and backdrivability in physical human–robot interac-tion. Control of SEA-generated forces or torques makes allowancefor the user’s own volitional control and allows implementing awide variety of assistive strategies. A novel force control methodfor a SEA-driven lower-limb assistive exoskeleton is presented.The device features variable-structure SEAs coupled via Bowdencables. The actuator alternates between two discrete levels of stiff-ness depending on the amplitude of the commanded force. Thealgorithm features a switching force-tracking control based onthe forward-propagating Riccati equation. A disturbance-rejectioncomponent increases the device’s transparency in zero assistancemode. The force control was used to implement an assistive strategythat aims to correct the asymmetric gait typical of stroke survivors.Assistive joint torques synchronize with the user’s gait by meansof an adaptive frequency oscillator, which extracts the continuousphase and frequency of the patient’s gait using data from both theparetic and the healthy sides. The control was tested with healthysubjects wearing the exoskeleton while subject to a simulated kneeflexion impairment. The control proved effective in restoring spa-tial and temporal knee flexion symmetry to levels comparable tounobstructed gait.

Index Terms—Exoskeleton, force control, legged locomotion,medical robotics, rehabilitation robotics.

I. INTRODUCTION

IN RECENT years, different types of lower-limb exoskele-tons and robotic orthoses have been developed to provide

assistance or rehabilitation to people suffering from neurologicalgait impairments. Generally, the robotic device features a controllayer that receives sensory information and uses it to providemechanical power to the user. This high-level control defines theessential task performed by the robot and, thus, can be referredto as the assistive strategy [1]. A key issue in assistive robotics

Manuscript received June 18, 2020; accepted September 4, 2020. This workwas supported in part by the Agency for Science, Technology and Research,Singapore, under the National Robotics Program, with A*Star SERC Grant 19225 00045, and in part by the FRC Tier 1 under Grant R397-000-302-114 andGrant R-397-000-328-114, National University of Singapore. This article wasrecommended for publication by Associate Editor F. Ficuciello and Editor E.Yoshida upon evaluation of the reviewers’ comments. (Corresponding author:Gabriel Aguirre-Ollinger.)

The authors are with the Department of Biomedical Engineering, NationalUniversity of Singapore, 117583, Singapore (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this article are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2020.3034017

is that the robot’s hardware determines, to a great extent, whatkind of assistive strategies can be successfully implemented.For example, systems, such as the ReWalk [2], eLEGS [3], andthe Vanderbilt lower-limb exoskeleton [4], feature high-ratiotransmissions that make them especially suitable for assistivestrategies based on position control. These devices have foundsuccess in providing therapy to patients with spinal cord injury,which involves making the device follow a normative gait tra-jectory with little or no input from the user. By contrast, thiskind of strategy is less suitable for individuals that retain somedegree of volitional control, such as stroke patients. As part ofthe motor learning process, stroke patients need to adjust theirlower-limb trajectories freely, rather than follow a prescribedtrajectory. Furthermore, a high gearing ratio typically produceslarge mechanical impedance in the exoskeleton’s joints. The re-sulting resistive torques make it difficult for the user to backdrivethe device without continuous intervention from the control [5].Position control typically produces high interaction torques thatinterfere with the user’s desired movements and may present anobstacle to motor recovery [6]–[8]. By contrast, control of theinteraction forces can deliver assistance while allowing the userto choose their lower-limb trajectory [5], [9], [10].

Series elastic actuators (SEAs) are a type of compliant actua-tor conceived for superior force tracking performance with rela-tively simple control. End-effector force is regulated basically bycontrolling the deflection of the elastic element. Backdrivabilityis obtained by zeroing the deflection. The actuator’s passive com-pliance improves user safety in human–robot interaction and re-duces shock loads [11]. Assistive devices featuring SEAs includethe multijoint lower-limb exoskeletons [12], [13], ankle footorthoses [14], and specialized knee–ankle–foot orthoses [15].

Applications of SEAs involving physical human–robot inter-action often employ impedance control to modify the apparentdynamics of the robot, particularly the stiffness of the joints.The impedance controller typically consists of an impedancemodel followed by an inner-loop force controller. While simplePID control on the inner loop can achieve stable behavior [16],performance requirements may demand more complex schemes.For applications requiring passivity in addition to stability, ithas been proposed to use a torque controller cascaded with aninner velocity loop [17]. To compensate for environment uncer-tainties, a combination of disturbance observer and feedforwardcontroller was proposed in [18]. Time delay control provides analternative approach to cancelling uncertainties, by modifying

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2 IEEE TRANSACTIONS ON ROBOTICS

the control actions based on past observations of system responseand control inputs [19].

In this article, a novel assistive control for a lower-limb ex-oskeleton, driven by a variable-structure SEA [20], is presented.Its main objective is to deliver precisely timed joint torques tocorrect gait asymmetries in patients suffering from lower-limbimpairments. This research builds upon a previous study orgroup [21]. The present contribution implements the followingimprovements.

1) A force feedback control for the variable-structure SEAand proof of its asymptotic stability.

2) A disturbance rejection control that minimizes interactionforces between the exoskeleton and the user during phasesof zero assistance.

3) An assistive control method for correcting asymmetry inspecific gait events.

4) An evaluation method for gait asymmetry. A novel set ofmeasures was developed for both discrete and continuousgait variables.

A crucial aspect in the design of a SEA is selecting the stiffnessof the elastic element. A very stiff spring increases the actuator’sopen-loop bandwidth, which may improve performance in appli-cations that require high force bandwidth requirements. On theother hand, for a given force, a soft spring will store more energythan stiff springs [22]; it will also be subject to larger deflections,thereby increasing the resolution of the force control [20]. Thevariable-structure SEA combines the advantages of each case byproviding two discrete levels of stiffness: low stiffness for tasksthat involve low-magnitude assistive forces with high resolu-tion, and high stiffness for generating large-magnitude forcesat lower resolution but with increased bandwidth. Switchingbetween conditions involves enforcing a contact constraint inthe SEA structure. Force tracking is performed by a switch-ing control based on the forward-propagating Riccati equation(FPRE) [23], which responds stably to changes in the SEAstructure.

The use of adaptive oscillators as a means of synchronizing awearable robots with a human’s cyclic movements is currently anactive area of research. Adaptive oscillators can be employed toestimate the gait phase of human walking by means of wearablesensors [24], or to estimate the velocity and acceleration esti-mates of near-periodic movements [25]. A controller based ona phase oscillator has been shown to deliver energy into the gaitcycle, thereby achieving a reduction of the effort provided by theuser [26]; oscillator-based control has also produced improvedphysiological response during periodic squatting [27].

Gait phase estimates produced by an adaptive oscillator can beused to generate the reference assistive torque, for example, byindexing the phase to a normative database of joint torques [28].A single assistive controller has been adapted to assist differentlocomotion modes by combining adaptive oscillators with motorprimitives [29]. In another study, oscillator-based control wascombined with a support vector machine in order to detect thewalking environment (level, slope up/down, etc.) and modulatethe assistive action accordingly [30]. In the exoskeleton controlpresented here, an adaptive frequency oscillator (AFO) extractsthe continuous phase and frequency of the patient’s gait usingkinematic data from both the paretic and the healthy sides.

Assistive torques are synchronized with the extracted phase andcan adapt to changes in gait pace.

The exoskeleton’s assistive action aims to correct the asym-metric gait patterns typical of stroke survivors. By generatingless propulsive force with the paretic leg, stroke patients tend towalk asymmetrically, with longer paretic steps than nonpareticsteps [31]. Asymmetry is caused, in turn, by abnormal coac-tivation in pairs of agonist and antagonist muscles or musclegroups [32]. Gait asymmetry is important from a clinical pointof view because it leads to problems, such as increased energyexpenditure, lack of balance control, risk of musculoskeletalinjury to the nonparetic limb, and loss of bone mass density inthe paretic limb. These deficits can restrict functional mobility inpatients and have a negative impact on their quality of life [33],[34].

This study tested the control’s ability to treat one specificanomaly, namely, spatial and temporal asymmetry in peak kneeflexion between the impaired and the healthy side. Validationexperiments were conducted with healthy subjects wearing theexoskeleton together with a customized brace that simulatedthe knee flexion impairment. The control proved effective atrestoring flexion symmetry to levels comparable to those ofunobstructed gait.

II. EXOSKELETON DESIGN: GENERAL FEATURES

A. Semiautonomous Lower-Limb Exoskeleton

A lightweight unilateral exoskeleton was developed to aid theimpaired limb of stroke patients during gait training. The deviceprovides torque to assist the angular motion of the knee jointand/or the ankle joint while walking. The exoskeleton has twomodalities of use: walking on a treadmill, with the actuatorsmounted on a fixed support frame [see Fig. 1(a)], and walkingover ground, with the actuators mounted on a wearable harness(see [35] for an earlier embodiment of the concept). The presentstudy covers only walking on a treadmill.

Torque is provided by a SEA that connects to the joint via apair of Bowden cables. Transmitting torque via Bowden cableshas the advantage of eliminating the loading of the patient’slimbs that would otherwise be caused by the actuators’ weightand inertia. Fig. 1(a) shows a detail of the exoskeleton in itsconfiguration for assisting only the knee joint. For the studypresented here, subjects walked on a treadmill while wearing theexoskeleton [see Fig. 1(b)]. The treadmill’s velocity control hasthe capability to adapt automatically to the user’s gait, therebyallowing the user to adjust their gait speed by consciouslyaccelerating or decelerating their pace.

B. Exoskeleton SEA

The performance of a SEA is highly dependent on the stiffnessof the springs employed. A soft spring allows high force controlfidelity and low output impedance, but also limits the force rangeand the control bandwidth. A stiff spring increases the forcerange and bandwidth at the expense of force control fidelity.The exoskeleton features a variable-structure SEA designed bythe authors’ research group to overcome these limitations [20].Its underlying principle is the use of two springs with vastly



AGUIRRE-OLLINGER AND YU: LOWER-LIMB EXOSKELETON WITH VARIABLE-STRUCTURE SERIES ELASTIC ACTUATORS 3

Fig. 1. Lower-limb exoskeleton and its associated systems. (a) Exoskeleton assembly, configured for assisting only the knee joint. A second Bowden cablemechanism can be added to assist the ankle joint. (b) Experimental station for gait training: Exoskeleton actuator system and variable velocity treadmill. The SEApackage is mounted on a fixed structure behind the user. The routing of the Bowden cables is highlighted for clarity. For walking over ground, the subject canwear the SEAs on the back by means of a custom-designed harness. (c) Detail view of the SEA package. (d) Mechanism for coupling the Bowden cables to theexoskeleton joint: the mechanism allows quick release of the cables, for example to switch assistance to the other leg.

Fig. 2. Series elastic actuator design. Depending on the amount of force transmitted, the system will operate at one of two stiffness levels: low stiffness, providedmainly by the linear spring, and high stiffness, provided by the torsional spring when the linear spring is fully compressed.

different stiffness values, connected in series to handle differentforce ranges. A soft translational spring handles the lower forcerange, allowing high force tracking fidelity. A torsional springwith high stiffness handles the higher force range when the softspring is fully compressed, thereby achieving increased controlbandwidth.

The SEA is powered by a dc brushless motor, which transmitsforce to the Bowden cables via the springs (see Fig. 2). Force iscontrolled by regulating the deflection of the springs. On theirpart, the Bowden cables exert torque on the exoskeleton joint bymeans of a pulley and a fixed-axis rotary coupling. The springs

ensure that the coupling between the user and the motor becompliant, thereby protecting the users body from impact loadsand other undesirable interactions.

The stiffness and the deflection range of the linear springare chosen such that the spring can transmit up to one-thirdthe actuator’s peak force before becoming fully compressed.The torsional spring’s stiffness is chosen such that its equivalentspring constant is about 100 times that of the linear spring. TheSEA has two modes of operation, which shall be referred toas “low force” and “high force.” In the low force mode, thelinear spring is only partially compressed; hence, the actuator’s




Fig. 3. Linearized model of the SEA and exoskeleton system.

dynamics are dominated by the stiffness of the linear spring.When in low force mode, the actuator can track a referenceforce with high fidelity and low output impedance. In the highforce range, the linear spring is fully compressed, and the forcecontrol depends on the stiffer torsional spring. This increasesthe control bandwidth of the actuation system, at the cost of acertain reduction in tracking fidelity due to the smaller deflec-tions involved. The design specifications of the SEA prototypeemployed in this study are provided in Section VI-A.

III. EXOSKELETON CONTROL: REFERENCE FORCE TRACKING

A. Modeling the Actuator/Exoskeleton System

The objective of the force controller is to track an arbitraryreference forceFref(t). The control design employs an equivalentlinear motion model of the SEA and the exoskeleton componentsthat assist a single joint (see Fig. 3). The soft spring k23 handlesthe low force range with high tracking fidelity; the high-stiffnessspring k12 handles the high force range when k23 is fullycompressed, thereby increasing the control bandwidth. Massm3 consists mainly of the reflected moment of inertia of theexoskeleton; b3 is a damping coefficient representing friction onthe cable and k3 is a stiffness coefficient representing the actionof gravity on the exoskeleton. Mass position x1 is equal to themotor-side encoder angle multiplied by p/(2π), where p is thepitch of the ball screw (see Fig. 2); position x2 is obtained in thesame manner from the second encoder. Position x3 is obtainedfrom the linear potentiometer reading x3 − x2, which is equalto the linear spring deflection. The corresponding velocities v1,v2, and v3 are obtained by means of a Kalman filter estimatorfeaturing a triple integrator model [36].

The SEA exoskeleton model needs to include the constraintimposed by the full compression of k23. Given the systempositions and velocity vectors

x = [x1 x2 x3]T (1)

v = [v1 v2 v3]T (2)

the unconstrained system is given by the state equations

v = Fxx+ Fvv +Gu (3)

where Fx, Fv , and G are matrices that define the system’s dy-namics. Full compression of k23 is represented by the constraint

Av = 0

A = [0 1 −1]. (4)

The constrained SEA exoskeleton is modeled using theUdwadia–Kalaba formulation for constrained multibody sys-tems [37], [38]. Given the system’s mass matrix M, the con-straint term is defined as

D(hc(t)) = I− hc(t)C (5)

where hc(t) is a switching variable equal to 1 when the com-pression constraint is active and to 0 when it is inactive, and

C = M−1AT (AM−1AT )−1. (6)

The constrained dynamical system is then

w = F(hc(t))w +G(hc(t))u (7)

where w = [xT vT ]T and

F(hc(t)) =

[0 I

D(hc(t))Fx D(hc(t))Fv

]G(hc(t)) =

[0

D(hc(t))G

]. (8)

Equation (7) is the basis for the control design. To this end,Fout(t) is defined as the system’s output, namely, the forceexerted by the actuator on mass m3, which is given by

Fout(t) = H(hc(t))w(t) (9)

where

H(hc(t)) =

{k23 · [0 1 −1 0 0 0] if hc(t) = 0

k12 · [1 −1 0 0 0 0] if hc(t) = 1. (10)

Note that (10) assumes that the damping coefficients b1 and b2in Fig. 3 are negligible. Computation of hc(t) for the purposesof control is described in Appendix A.

Tracking of Fref(t) by Fout(t) is performed by a combinationof feedback and feedforward control. The feedforward com-ponent is a straightforward application of inverse dynamics.In addition, a disturbance observer is employed to increasethe backdrivability of the exoskeleton when no assistive forceis commanded. The complete force tracking control systemis shown in block diagram form in Fig. 4. The individualcomponents of the control are discussed in the sections thatfollow.

B. Feedback Control

As previously discussed, during operation, the actuator mayswitch from a low force condition (spring k23 partially com-pressed) to a high force condition (spring k23 fully compressed).This switching, signaled by the term hc(t) in (7) and (8),constitutes a change in the physical structure of the system.Therefore, the SEA exoskeleton must be defined as a lineartime-varying system. In principle, it is possible to implementa linear quadratic control for a linear time-varying system byintegrating the Riccati equation backward from a final time con-dition. However, this requires knowledge of the future dynamicsof the system, which in the case of (7) is not possible as it wouldrequire knowing the behavior of hc(t) in advance. Instead, analternative approach will be used, based on forward integrationof the Riccati equation. While this method does not achieveoptimality in the sense of a linear quadratic regulator, it has the




Fig. 4. Force tracking control of the exoskeleton: block diagram.

crucial advantage of only requiring knowledge of the system’sdynamics at the present time.

Different control schemes based on the FPRE have describedin the literature [23], [39], [40]. A feedback force control basedon a discrete-time FPRE is derived here. The result is a controlcapable of guaranteeing asymptotic stability, despite the changein the SEA structure that occurs when the full compressionconstraint is active.

The discrete-time equivalents of the original constrained sys-tem (7) and the output equation (9) are

wk+1 = F(hc,k)wk +G(hc,k)uk (11)

Fout,k = H(hc,k)wk. (12)

Each of the matrices F(hc,k), G(hc,k), and H(hc,k) can switchbetween two possible values depending on whether the con-straint is active (hc,k = 1) or inactive (hc,k = 0).

A stabilizing feedback control is given by

uk = −Kk wk (13)

where

Kk =[G(hc,k)

T Pk G(hc,k) +R]−1

G(hc,k)T Pk F(hc,k).

(14)

Pk is computed recursively using the discrete-time FPRE

Pk+1 = F(hc,k)T Pk F(hc,k)

− F(hc,k)T Pk G(hc,k)

[G(hc,k)

T Pk G(hc,k) +R]−1

×G(hc,k)T Pk F(hc,k) + Q(hc,k) (15)

where Q(hc,k) = H(hc,k)T QF H(hc,k). Both QF and R are

positive definite matrices (Note: QF and R are not consideredcost matrices since the concepts of state cost and control costonly apply in the context of the linear quadratic regulator, whichemploys backward integration of the Riccati equation).

As will be shown, the feedback control (13) can stabilizethe system if the Riccati equation (15) is subject to the ap-propriate initial conditions. To this end, (15) is denoted byPk+1 = f(Pk), with the initial condition P0 to be reset ev-ery time hc changes value. Noting that Pk is symmetric andnonnegative, the following Lyapunov function is defined:

V (wk) = wTk Pk wk. (16)

Using Lyapunov analysis, initial conditions are derived as toguarantee that V (wk) be monotonically nonincreasing, whichin turn guarantees the stability of the closed-loop system.

The stability proof is presented in Appendix B. Its results aresummarized as follows. Given

1) (F(hc)T , G(hc)

T ) is detectable for both hc = 0 andhc = 1;

2) (F(hc)T , Q(hc)) is stabilizable for both hc = 0 and

hc = 1;3) the initial condition is reset to P0 such that P0 ≥ 0 and

P0 − f(P0,Q(hc)) +Q(hc) ≥ 0 every timehc switchesvalue,

the system (11) with control law (13) is asymptotically stable.The feedback control will be used to track a reference force

trajectory Fref,k. The reference state is given by

wref,k = H(hc,k)+ Fref,k (17)




Fig. 5. Step response of the feedback and feedforward force control combina-tion for a simple exoskeleton and human limb model. Top: step response of theoutput force. Bottom: evolution of the eigenvalues of the solution to the Riccatiequation, Pk . Decreasing values indicate the decreasing behavior of Pk per(53).

where (+) denotes the matrix pseudoinverse. The feedbackcontrol is then

ufb,k = Kk (wref,k −west,k) (18)

where west,k is the current state estimate.Fig. 5 shows the theoretical step response of the combined

feedback and feedforward control for the linearized model ofthe SEA and exoskeleton combination coupled to a human limb.The model uses the stiffness values provided in Section VI-Awith QF = 200 and R = 1. A change in behavior occurs whenhc switches from 0 to 1, indicating compression of spring k23.However, the system’s response remains stable throughout.

C. Force Disturbance Rejection

The objective of the feedback and feedforward controls isto make the spring force Fout track the desired assistive forceFref. However, the force acting on the human limb will differfrom Fout due to additional forces introduced by the Bowdencable friction, as well as the action of inertia and gravity onthe shank brace [see Fig. 1(a)]. While these forces are typi-cally small in comparison with Fref, they can disrupt the user’sgait when no assistance is commanded. In view of this, anadditional component in the control minimizes the interactionforces between the exoskeleton and the human limb whenFref =0. This component can be referred to as a “force disturbancerejection” control. It should be noted that, in robotics literature,disturbance rejection is frequently understood as rejection of po-sition disturbances caused by environmental forces. The controlcomponent described here has the opposite effect: it increasesthe exoskeleton’s responsiveness to the user’s movements.

The force disturbance rejection control (see Fig. 4) consists ofa disturbance observer [41], [42], a smooth switching function,and a feedback transfer function. These elements are connectedin series to form a positive feedback loop. The input to theobserver is the theoretical value of the net force acting on massm3, given by

F3,net = Fout − Fc (19)

Fig. 6. Disturbance observer: block diagram.

where Fc is a dissipation force caused by the Bowden cable andapproximated as

Fc = b′3 v3 (20)

where b′3 is a nominal damping value smaller than b3 (In theory,dissipation on the Bowden cable could be fully compensatedsimply by setting b′3 = 0. However, this may lead to instability).

Any forces acting on m3 in addition to F3,net are lumpedinto the disturbance Fd; these include any unmodeled cabledynamics as well as the reflected interaction forces between theexoskeleton and the human limb. Fd is estimated by means of adisturbance observer, shown as a block diagram in Fig. 6:Ydob(z)represents the admittance of the physical mass m3 in discretetime and Ydob(z) is the corresponding model in the observer,given by

Ydob(z) =T

2m3

1 + z−1

1− z−1(21)

where T is the sampling period. Cdob(z) is a first-order low-passfilter that controls the bandwidth of the observer and is given by

Cdob(z) = Π(z, ωdob) (22)

where

Π(z, ω) =ωT2 (1 + z−1)(

1 + ωT2

)− (1− ωT

2

)z−1

(23)

and ωdob is the observer’s bandwidth. The disturbance estimateFd generated by the observer is used as the feedback signal.

Disturbance rejection is meant to be active only when Fref iszero. However, in order to avoid discontinuous behavior in thecontrol, the disturbance rejection component is deactivated ina gradual manner. To that end, Fd is multiplied by a switchingterm hd given by

hd = Λ(Fref, 0, Flim). (24)




Fig. 7. Smooth switching function for the disturbance rejection control.

Λ(·) is a smooth function defined for some input variable y as

Λ(y, yo, yf ) =1

2

{1 + tanh

[μ

(1

2− y − yo

yf − yo

)]}(25)

where μ is a positive constant. An example plot of Λ(y, yo, yf )is shown in Fig. 7; from there it can be seen that, in (24), hd =0 for Fref � Flim (In the current implementation, Flim = 20 Nand μ = 6). Finally, the modified disturbance estimate hdFd isrun through a transfer function Tu

d (z) that reflects a disturbanceinput onto the control input port.

IV. ASSISTIVE CONTROL FOR IMPROVING GAIT SYMMETRY

The authors previously designed an exoskeleton control tocorrect a typical kinematic deficit poststroke, namely, reducedknee flexion on the paretic side during swing [21]. It wasassumed that the target value of peak knee flexion and its timingwithin the gait cycle were known a priori. The present studypursues a more general objective: achieving symmetry in theknee flexion trajectories of the normal and the impaired sides. Anassistive control was developed to minimize gait asymmetriesby delivering joint torques synchronized with the phase of thepatients gait. Phase in this context is a variableσ(t) that measuresthe progress of the gait cycle; σ(t) increases monotonically overtime and wraps around modulo 1 every time the user completesone gait cycle.

In the discussion that follows, the following nomenclature isused.

1) γ is the knee flexion angle, defined per Fig. 1(a).2) γp is the peak value of the knee flexion angle.3) σp is the value of σ(t) at which peak flexion occurs; σp is

referred to as the “associated phase” of γp.4) A superscript nrm refers to the patient’s normal (healthy)

side; the superscript imp refers to the impaired (paretic)side.

The first specific objective is to make the peak flexion angle ofthe impaired-side knee γimp

p match the peak flexion angle of thenormal knee γnrm

p . A key consideration is that the normal-sideflexion angle changes from one stride to the next, due both tothe normal variability of human gait and to voluntary changesin the subject’s gait (for example, changes in gait speed). Asa consequence, there is not a fixed target for γimp

p . Instead, thecontrol must constantly monitor the knee flexion trajectory onthe normal side to determine, for the qth stride, what the targetvalue of γimp

p should be.Symmetry also requires that the timing of gait events on the

impaired side be congruent with the timing of those on the

Fig. 8. Detail of the thigh-mounted IMUs.

normal side. For the impaired side, the peak knee flexion eventis denoted by t | γimp(t) = γimp

p ; this event happens at a cer-tain phase value σimp(t) = σimp

p . Likewise, for the correspond-ing normal-side event t | γnrm(t) = γnrm

p the associated phaseis σnrm(t) = σnrm

p . Temporal symmetry requires the associatedphases of these two events to be half a cycle apart. By linkingthe peak angle value to a specific phase value, it is assumed thatthe associated phase of the “peak knee flexion” event is invariantwith step frequency.

A. Continuous Gait Phase: Extraction

The angle of the thigh on the sagittal plane θ(t) is defined perFig. 1(a). The exoskeleton control uses the instantaneous thighangle difference

Δθ(t) = θnrm(t)− θimp(t) (26)

to extract the wrap-around phase of the gait cycle σ(t). Assistivetorque for the impaired leg will be computed as a functionof σ(t). The thigh angles are measured by means of inertialmeasurement units (IMUs) mounted on the back of the thighswith their roll axis aligned perpendicularly to the sagittal plane(see Fig. 8). Each IMU tracks orientation as a quaternion, thevalue of which is relayed wirelessly to the exoskeleton control inreal time. The thigh angle is then calculated using the standardconversion of quaternion to Euler angles, with θ(t) representingrotation about the roll axis.

A virtual dynamical system driven by an AFO extracts theinstantaneous phase and frequency of the measured thigh angledifference [43]. The AFO input is the error signal

e(t) = Δθm(t)−Δθrec(t) (27)

where Δθm(t) is the measured angle difference and Δθrec(t) isthe reconstructed angle difference, given by a finite-term Fourierdecomposition performed in real time. The dynamical system isgiven by

ς = f − 1

2πε e(t) sin(2πς) (28)

f = − 1

2πε e(t) sin(2πς) (29)

αk = η cos(2πkς) e(t) (k = 0, . . . , Nf ) (30)




Fig. 9. Reference profile for the assistive torque as a function of the contin-uous gait cycle phase. ς is the continuous phase generated by (28); it growsmonotonically without bounds. σ is the phase with wrap-around (modulo 1).The midpoint phase σc is measured with respect to the most recent σ = 0 pointin the cycle.

βk = η sin(2πkς) e(t) (31)

Δθrec =

Nf∑k=0

αk cos(2πkς) + βk sin(2πkς) (32)

σ = ς (mod 1). (33)

Here, ς is the continuous oscillator phase, which grows mono-tonically without bounds, and σ is the phase with modulo 1wrap-around. f is the oscillator’s intrinsic frequency in cyclesper second, ε is the coupling strength of the AFO, and η isa learning constant. The Fourier coefficients αk and βk areupdated online using (30) and (31).

B. Assistive Force Profile as a Function of Phase

The reference assistive torque is based on a bell-shaped pulsefunction hF (σ, σc), shown in Fig. 9. The timing of the pulse iscontrolled by the midpoint phase value σc, its duration by Δσrise

hF (σ, σc) =1

2{tanh [κ (Ψ(σ, σc) + Δσrise)]

+ tanh [κ (Ψ(σ, σc)−Δσrise)]} (34)

where

Ψ(σ, σc) = 2(σ − σc + 1)

κ = − tanh−1(2χ− 1)

Δσrise

and χ � 1.Given a target peak force Fmax, the required assistive force

profile to be delivered by the SEA is

Fref(t) = Fmax hF (σ(t), σc) (35)

which corresponds to a reference joint torque

τref(t) = r · Fref(t) (36)

where the radius r is given by the distance of the attachmentpoint of the Bowden cable to the center of the exoskeleton’srevolute joint [see Fig. 1(a)].

In order to assist the exoskeleton user toward achieving sym-metric knee flexion, the exoskeleton control computes, once perstride, the required values of peak forceFmax and midpoint phaseσc in (35). This computation process is described below.

1) Extracting the Peak Knee Flexion Angle and its Phase:For both the normal and the impaired sides, the swing portionof the leg’s motion is detected by polling the thigh angle mea-surement θ(t) and detecting the extreme values of θ(t) overone stride. Although less accurate than detecting toe-off andheel-strike events, this method has the advantage of requiringonly an angle reading from a single IMU for each leg.

During leg swing, an extreme detection algorithm recordsevery time the knee joint angle γ(t) reaches a local maximum,and stores the corresponding maximum angle and phase pair〈γi, σi〉 (i = 1, 2, . . .). Then, the peak angle γp,m and its asso-ciated phase σp,m for the entire swing phase are easily obtainedas

〈γp,m, σp,m〉 = 〈γj , σj〉 | γj = maxi

γi. (37)

2) Computing the Peak Assistive Force: The primary objec-tive of the assistive control is to minimize the difference betweenthe normal-side peak flexion angle and its impaired-side counter-part. In keeping with the “assist as needed” principle, the controlshould attempt to apply as little torque as possible, therebyallowing the subject to learn to overcome the impairment, atleast partially. One way to satisfy these competing requisites isby making the assistive control minimize the cost function [44]

J =1

2

(γnrmp,m − γimp

p,m

)2+

R

2F 2

max (38)

where R is a constant that weighs the relative cost of the peakangle difference and the actuator force.

A simple control that minimizes the preceding cost functionfor the qth stride has the form

Fmax(q) = λFFmax(q − 1) +Kp (γnrmp,m(q − 1)− γimp

p,m(q − 1))(39)

where λF < 1 is a forgetting factor and Kp acts as a feedbackgain for the peak angle difference [44].

3) Computing the Midpoint Phase: For the qth stride, thepeak knee joint angle on the normal side has an associated phaseσnrmp (q). To achieve temporal symmetry, the assistive control

should in principle make the impaired-side peak angle occurhalf a cycle apart fromσnrm

p (q). However, the normal fluctuationsthat occur in σnrm

p during walking can make it difficult for thesubject to adapt to the exoskeleton’s assistance. In order to filterout these fluctuations, the assistive control maintains a runningaverage of associated phase values for the last Ns strides. Thisallows computing the best estimate of σnrm

p for the qth stride as

σnrmp,est(q) =

1

Ns

Ns−1∑n=0

σnrmp,m(q − n). (40)

The desired value for the associated phase on the impairedside is then equal to σnrm

p,est(q) shifted by half a cycle. This is




Fig. 10. Computation of the reference assistive force for one gait cycle. Thehorizontal axis is the unbounded phase ς . Each point where σ = 0 represents thestart of a new gait cycle. The unity amplitude force pulse hF (σ, σc) (34) hasa midpoint phase σc equal to the desired value for the associated phase σnrm

p+ ,which is computed using data from the previous cycles [(40) and (41)]. The peakamplitude of the reference force Fmax is proportional to the difference in peakflexion angles from the preceding stride γnrm

p,m and γimpp,m. The reference assistive

force for the current cycle is then Fref = Fmax hF (σ, σc).

computed as

σnrmp+ (q) ≡

(σnrmp,est(q) +

1

2

)(mod 1). (41)

Temporal symmetry requires ensuring that σimpp = σnrm

p+ for theqth stride. In turn, this requires making the midpoint phase σc

in (35) a function of σnrmp+ . For the experiments presented here,

σc was simply set as σc = σnrmp+ . In the results section, the actual

behavior of the impaired-side phase with respect to σnrmp+ (q) is

analyzed. The complete process for computing the referenceforce (35) is illustrated in Fig. 10.

V. ASYMMETRY MEASURES

In order to assess the user’s gait asymmetry and the potentialimpact of the exoskeleton on it, a novel set of measures wasdeveloped for both discrete and continuous gait variables.

A. Asymmetry Measure for Discrete Variables

With the artificial impairment in place, the key objective isfor the exoskeleton control to reduce the difference in peakknee flexion among sides. Because the impairment would likelyaffect the duration of the swing phase, differences in swingphase duration among sides should be tested as well. Bothvariables are discrete, in the sense that their values are com-puted once per stride of each leg. Gait studies often employthe “symmetry angle” (SA) to quantify asymmetry betweendiscrete variables [45]. However, this metric has two importantdrawbacks: it is nonlinear with respect to the deviation fromthe symmetry axis, and it requires a test to choose among twopossible formulas.

A novel measure, referred to as the “asymmetry index” (AI),shall be used to quantify asymmetry among discrete variableswithout the drawbacks of the SA. Given a “left-side” value xL

and a “right-side” value xR, the AI has the following properties.1) AI = 0 for xL = xR (perfect symmetry).2) |AI| = 1 for xL = −xR (maximum asymmetry).3) |AI| = 1

2 for either xL = 0 or xR = 0.

Fig. 11. AI for different cases of the mapping 〈xL, xR〉 → [uL uR]T . Thecircumference represents the locus of valid [uL uR]T vectors. Points on thecircumference are labeled with the AI value associated to the corresponding[uL uR]T . Thick vectors represent the two possible cases of perfect symmetry(uL = uR, yielding AI = 0). Maximum asymmetry occurs when uL = −uR

(with AI = 1 for positive uL and AI = −1 for positive uR).

Computation of AI begins with the mapping of the pair〈xL, xR〉 to a unit vector as follows:

〈xL, xR〉 →[uL

uR

]≡

⎡⎢⎣xL√

x2L + x2

RxR√x2L + x2

R

⎤⎥⎦ . (42)

AI is then obtained as

AI =2

πarcsin

(uL − uR√

2

). (43)

AI values for different combinations of uL and uR are shownschematically in Fig. 11 .

B. Special Case: Associated Phase of Peak Knee Flexion

Temporal symmetry requires the impaired-side peak flexionangle to occur half a cycle apart from the healthy-side peak angle.From Section IV-B3, this is equivalent to requiring σimp

p to equalσnrmp+ (Recall thatσnrm

p+ is the healthy-side associated phase shiftedhalf a cycle). However, due to the wrap-around property of σ(t),quantifying the asymmetry of these two variables requires aspecial treatment. If, for example, σimp

p = 0.95 and σnrmp+ = 0.05,

one might erroneously conclude that their difference in value is0.9 (indicating large asymmetry), whereas the correct differenceis−0.1. To account for wrap-around correctly, an operatorDσ(·)is used to compute the difference between phase variables.Dσ(·) is defined as follows. Given two phase values σL and

σR, and the auxiliary variables

D1 = σL − σR (44)

and

D2 =

{D1 − 1 if σL ≥ σR

D1 + 1 otherwise(45)

the difference between phase values is then

Dσ(σL, σR) =

{D1 if |D1| ≤ |D2|D2 otherwise

. (46)




A key property is that |Dσ| <= 0.5, indicating that the max-imum difference between phase values can be half a cycle atmost. Therefore, the AI for the associated phases is defined as

AIσ(σimpp , σnrm

p+ ) = 2Dσ(σimpp , σnrm

p+ ). (47)

Thus, maximum asymmetry corresponds to AIσ = ± 1, whichis consistent with the AI defined in Section V-A.

C. Symmetry of Continuous Variables

Although the exoskeleton control is designed primarily toreduce the asymmetry of discrete variables, it is instructive toanalyze the symmetry of continuous variables as well. Specif-ically, the symmetry of the time trajectories of the knee jointangle (γ(t)) shall be examined. In order to compare betweenthe healthy-side and the impaired-side trajectories, it must betaken into account that these are approximately half a cycle outof phase. Therefore, their corresponding time series must beshifted in phase first. Noting that both time series are indexedto the oscillator’s continuous phase (28), i.e., γimp = γimp(ς(t))and γnrm = γnrm(ς(t)), the following phase-shifted time seriesare defined:

γimp− (ς) = γimp

(ς − 1

4

)(48)

γnrm+ (ς) = γnrm

(ς +

1

4

)(49)

where γimp− (ς) is the impaired-side time series shifted backward

in time by one quarter of a cycle, and γnrm+ (ς) is the healthy-side

time series shifted forward by the same amount. Thus, for aperfectly symmetric gait, γimp

− (ς) = γnrm+ (ς). Taking a cue from

the VAF definition (51), the following symmetry measure fortime series is proposed:

St(γimp− , γnrm

+ ) = 1− var(γimp− − γnrm

+ )

var

(1

2(γimp

− + γnrm+ )

) . (50)

For perfect symmetry, St = 1. Using the average of γimp− (ς) and

γnrm+ (ς) in the denominator (51) indicates that neither trajectory

constitutes the reference. Instead, both trajectories should be asclose to the mean as possible at every point in time.

VI. EXPERIMENTS

A. Experimental Setup

Gait trials were conducted on an experimental station consist-ing of the exoskeleton SEA system and a treadmill adapted forvariable velocity control [see Fig. 1(b)]. With all subjects, theleft side was treated as the “impaired” one. Therefore, only theleft exoskeleton brace was actuated; the right-side brace wasdisconnected from the SEA altogether. However, the systemlogged IMU and knee-joint encoder data from both sides in orderto perform the necessary control computations.

The SEA (see Fig. 2) uses a dc brushless motor (EC 4-pole200 W 48 V, Maxon, Switzerland). The actuator structure ismade of carbon fiber plate. With a total weight of about 800

TABLE IEXOSKELETON CONTROL PARAMETER VALUES

grams, the SEA is suitable also for wearing on a harness whenwalking over ground.

The angular positions of the motor shaft before and afterthe torsional spring are measured by rotary encoders with aresolution of 2048 counts per revolution after quadrature. Rotarymotion is converted to linear by a ball screw with a pitch of0.002 m per revolution. Deflection of the linear spring (k23)is measured by a linear potentiometer. Per the linearized SEAmodel (see Fig. 3), the spring constants are k12 = 2.86 ×106 N/m and k23 = 2.40 × 104 N/m. The joint radius r (36)is 0.03 m; this yields an equivalent torsional stiffness at the kneeof 21.6 N·m/rad in the low force range (spring k23 only partiallycompressed) and 2574 N·m/rad in the high force range.

The linear spring force at full compression is 240 N; thus, theSEA can deliver joint torque up to 7.2 N·m in the low force range.Based on the motor’s torque capacity, the maximum continuousjoint torque in the high force condition is about 14.0 N·m.Control of the SEA exoskeleton system is implemented as aSimulink model running on Simulink Desktop Real-Time (TheMathworks, Natick, MA, USA) with an update rate of 200 Hz.Thigh angle (θ(t)) measurements are obtained from IMU data(see Fig. 8) sampled at 50 Hz and relayed via user datagramprotocol (UDP). The joint flexion angle (γ(t)) is measured witha noncontact rotary encoder with analog output (RLS MerilnaTehnika, Slovenia), sampled at 200 Hz .

Velocity commands for the treadmill are sent via UDP at anupdate rate of 5 Hz. The treadmill’s velocity control is designedto adapt automatically to the user’s gait as detected by the AFO,thereby allowing the user to adjust their gait speed voluntarily(see Section II-A). Further details can be found in [21].

Table I provides a summary of the exoskeleton control param-eters. The values of QF and R were determined via simulation;the rest were determined statistically from preliminary trialswith healthy subjects that did not participate in the experimentalstudy.

B. Simulated Impairment

The experiments involved simulating a typical poststroke gaitdeficit in the healthy subjects. Knee flexion deficit was chosenbecause it is a frequent outcome of stroke that contributes todeveloping an asymmetric gait pattern. In a common scenario,reduced knee flexion compromises foot clearance during the




Fig. 12. Lower brace for trials in the IMPAIR and ASSIST conditions. Thefixed ankle joint impedes ankle push-off in the stance phase, which results inreduced knee flexion during swing.

swing phase [46], [47]. To facilitate foot clearance, patientsresort to compensatory movements, such as hip hiking andcircumduction, which results in asymmetric gait [47], [48].

Multiple factors contribute to knee flexion deficit, knownin biomechanics literature as stiff-knee gait [49], [50]. Oneis overactivity of the rectus femoris muscle, which producesexcessive knee extension moment (spasticity). Another one isinsufficient push-off power at the ankle, which results in reducedknee flexion velocity at toe-off, with a consequent lack ofpassive knee flexion. The latter is considered to be the prevalentcause of stiff-knee gait [49]. To simulate reduced push-off, theexoskeleton’s standard lower brace [see Fig. 1(b)] was replacedwith one featuring a rigid ankle joint (see Fig. 12). The rigidjoint impeded the rotation of the ankle joint, thereby creating apush-off deficit; this was expected to reduce the usual range ofknee flexion while walking.

C. Experimental Protocol

An experimental study was conducted to test the effectivenessof the exoskeleton’s assistive force (35) in correcting asymmetryin the flexion angle of the knee joints during walking. The experi-ments consisted of walking in the gait training system composedof the exoskeleton and the treadmill. The objective was solelyto detect changes in gait pattern while the subjects walked inthe exoskeleton; no attempt was made to study the retention ofthese effects after gait training. Eight healthy male subjects (age25–34 years, body mass 60–82 kg, height 1.67–1.81 m), all ofthem members of the authors’ research group, volunteered totake part in the study. All of the participants gave their informedconsent prior to taking part in the experiments.

Kinematic data were logged from the actuator sensors (en-coders and linear potentiometer), the knee joint encoder and the

thigh-mounted IMUs. Before the experiment, the participantswere allowed to wear the exoskeleton and walk on the treadmillfor 3 to 5 min. This allowed them to practice adapting their gaitspeed while walking.

During the actual experiment, subjects walked on the tread-mill for three successive trials, each of 4 min duration. Theinstruction for them was to find a comfortable pace and to main-tain it until the end of the trial. Trials involved the experimentalconditions described below in succession.

1) Free Walking (FREE): The subjects walked with the ex-oskeleton control set to baseline mode, namely zero assistiveforce and maximum force disturbance rejection (see Section III-C). The standard lower brace was employed [see Fig. 1(b)]. Theobjective was to obtain, from the exoskeleton data, represen-tative values of peak knee flexion angle γp, and its associatedphase σp during normal walking.

2) Artificial Impairment (IMPAIR): With the exoskeleton’sassistive force still set to zero, the subjects walked with theartificial knee flexion impairment described in Section VI-B.

3) Artificial Impairment With Exoskeleton Assistance (AS-SIST): Participants walked with the artificial impairment inplace while receiving assistive knee torque from the exoskeletonper (36) and (39). The main expected effect was a reduction ofthe differences in peak knee flexion angle and associated phasebetween the “impaired” and the “healthy” side.

D. Force Tracking and Disturbance Rejection Quality

The effectiveness of the exoskeleton’s assistance dependson the quality the force tracking measured as the fit betweenthe reference Fref and the output force Fout. For the studypresented here, the peak output force was limited to 300 N, whichyields about 50% the typical knee peak flexion moment duringswing [51]. Prior to the gait experiments, the force control’sperformance in the absence of disturbances was tested. Thetest procedure involved tracking reference forces of increasingfrequency, first with a peak value (200 N) that kept the SEA inlow force condition, and then with the maximum allowable peakvalue (300 N), which required the control to switch between thelow force and the high force conditions. To asses the overallquality of the fit among signals, a “variance accounted for”(VAF) metric was used [52], defined as

VAF(xref, xout) = 1− var(xref − xout)

var(xref)(51)

where xref is the reference signal and xout is the output signal.Tracking behavior from both tests is illustrated in Fig. 13. Asexpected, there is a certain loss of tracking accuracy in thehigh force condition, due mainly to the increased stiffness ofthe actuator. Table II lists, for different input frequencies, themaximum and minimum values of VAF, controller gain, andcontroller phase delay in both the low force and the high forceconditions. These results confirm that the control can track areference within the chosen force range with high fidelity andminimal phase delay, at frequencies within the typical range oflower-limb movements (≤ 2 Hz).




Fig. 13. Tracking sinusoidal reference forces: outputs for the minimum (0.5 Hz) and maximum (2 Hz) test frequencies. Top row: operation in the low force range(< 240 N). Bottom row: the peak value of the reference force (300 N) causes switching between low force and high force conditions. For reference, the switchingvariable hc(t) is shown scaled by the peak force value.

TABLE IITRACKING QUALITY FOR SINUSOIDAL REFERENCE FORCE

In the absence of an assistive torque to apply (FREE andIMPAIR conditions), the important quality of the control istransparency; this is the objective of the disturbance rejectioncomponent. Although the exoskeleton does not feature sensorsto measure disturbance torques directly, a simple quality metricis provided by the SEA output force Fout. When Fref is zero,Fout should be nearly zero as well, indicating that the subjectexperiences almost no reactive forces from the SEA. In theFREE condition, the root mean square (rms) value of Fout(t)for all subjects was 9.35± 1.047 N [mean ± standard errorof the mean (s.e.m.)]. This is equivalent to an rms torque of0.281± 0.0314 N·m on the knee joint, which indicates a largedegree of transparency. For illustration purposes, Fig. 14 showsthe SEA output force and the knee flexion angle for one subjectwalking at a uniform speed in the FREE condition.

E. Statistics

One-way ANOVA was used to compare changes in gait amongthe different experimental conditions. Comparisons were madeamong the averaged values of the following variables.

Fig. 14. SEA output force for one subject in the FREE condition. The flexionangle of the artificially impaired knee (in degrees) is shown for reference.

1) AIs of peak knee flexion angle (γp), the associated phase(σp), and swing phase duration (%Tsw).

2) Self-selected gait speed and step frequency.3) Symmetry measure (St) of the time trajectories of the knee

flexion angle (γ(t)).In order to ensure that statistics were computed from steady-

state walking data, the averages were taken from the last 2 minof the trial. Where statistical significance was detected (definedas p ≤ 0.05), post-hoc Tukey’s honest significant difference(HSD) tests were performed to determine which differenceswere significant among conditions.

F. Experimental Results

Fig. 15 shows, for one subject walking in the ASSIST con-dition, the initial synchronization of the AFO with the subject’sgait and the consequent assistive torque computation. Therewere no instances in any trial of the AFO losing synchronizationafter the initial adaptation. Per Section VI-E, the primary interestin this study was to determine how the artificial impairment andthe exoskeleton’s subsequent intervention affected the symmetryof the following variables: peak knee flexion angle (γp), associ-ated phase (σp), and swing phase duration (%Tsw).




Fig. 15. Gait trial in the ASSIST condition for one subject. These plots show the initial adaptation of the AFO to the subject’s gait, and the consequent generationof the exoskeleton’s assistive force. Top plot: thigh angle difference Δθm(t) measured from IMU data. Superimposed are the wrap-around instantaneous phaseσ(t) and the gait cycle frequency f(t) as computed by the AFO-based dynamical system. Middle plot: knee flexion angles and peak flexion angle for the impairedand the normal sides. Bottom plot: reference assistive torque τref(t) and actual assistive torque τout(t) acting on the “impaired” knee joint; τlow,max is the maximumtorque delivered by the SEA in the “low force” condition (Note: individual variables’ units are indicated on the plots’ legends, but the numerical values on thevertical axes are valid for all the variables plotted).

Fig. 16. Experimental results. Square brackets with asterisks indicate significant differences (p < 0.05). (a) AI for the peak knee flexion angles under the differentexperimental conditions (mean ± s.e.m.). (b) AI for the associated phase for the peak angles (mean ± s.e.m.). (c) AI for the duration of the swing phase.

Significant differences were found in the asymmetry ofthe peak knee flexion angle, AI(γimp

p , γnrmp ) among the dif-

ferent experimental conditions [p = 2.38× 10−7; HSD: IM-PAIR < FREE, ASSIST > IMPAIR, ASSIST < FREE;Fig. 16(a)]. While γp was essentially symmetric in the FREEcondition, the artificial impairment generated an AI value of−0.1114± 0.00704 (mean ± s.e.m.) in the IMPAIR condition,equaling a deficit in knee flexion of −15.8◦ ± 0.9◦ on theimpaired side with respect to healthy one. On its part, the ex-oskeleton’s assistance reduced the magnitude of the asymmetrysignificantly, yielding an AI of −0.0386± 0.00589, or a flexiondeficit of about −6.2◦ ± 0.9◦.

The phase associated with peak knee flexion σp exhibited anegative mean AI in all conditions [see Fig. 16(b)]. This meansthat peak knee flexion during swing tended to occur earlier on theimpaired side than on the healthy one. However, no significantdifferences were found in AI(σimp

p , σnrmp ) among conditions, due

in part to the large variations in asymmetry among subjects [seeFig. 16(b)].

To test the asymmetry of the duration of the swing phase,%Tsw is defined as the “swing phase percentage,” i.e., thefraction of the gait cycle percentage that correspondsto the swing phase. Significant differences were foundin AI(%T imp

sw , %T nrmsw ) among experimental conditions




Fig. 17. Phase-shifted trajectories for the impaired flexion angle (γimp− ) and

the normal one (γnrm+ ) under the different experimental conditions, for one

subject. Horizontal coordinate is the continuous phase ς . Perfect symmetrySt(γ

imp− , γnrm

+ ) = 1 would show plots that coincide perfectly.

[p = 0.000915, HSD: IMPAIR > FREE, ASSIST < IMPAIR,ASSIST > FREE; Fig. 16(c)]. In the FREE condition the swingduration was slightly lower on the “impaired” side, hence thenegative AI. While this was an unplanned asymmetry, it wasnot altogether unexpected since the “healthy” side brace wasdetached from its own SEA. Although the control’s baselinemode attempts to make the SEA transparent to the user, theload on the “impaired” side still differed from the load on the“healthy” side.

The IMPAIR condition caused a change in the direction of theswing-phase asymmetry. The difference in swing phase dura-tion among sides changed from −0.0566± 0.01334 s in FREEcondition to 0.0596± 0.01632 s in IMPAIR. In other words,the impaired-side swing phase became longer in the impairedcondition. This most likely reflects the change in loading causedby the artificial impairment. The ASSIST condition yieldedthe smallest swing-phase asymmetry, bringing the differenceamong sides to 0.01209± 0.0203 s. No significant differenceswere found among conditions for the step frequency f or theself-selected gait-speed vs.

For the flexion angles’ time series symmetry (St(γimp− , γnrm

+ )),the following values (mean ± s.e.m.) were obtained: 0.846±0.0231FREE, 0.748± 0.0322 IMPAIR, and 0.695± 0.044AS-SIST. By way of example, Fig. 17 shows the phase-shifted flex-ion angle trajectories for one subject. In general, the angle flex-iontrajectories showed asymmetry even in the FREE condition,even though the symmetry of the peak flexion angle events wasquite strong in the FREE case [see Fig. 16(a)]. A decrease in timeseries symmetry was observed in the IMPAIR condition, andfurther decrease in the ASSIST condition, although significanceonly occurred between the FREE and the ASSIST conditions(p = 0.0258; HSD: ASSIST < FREE). Thus, although the AS-SIST condition managed to recover symmetry in the peak flexion

angle events, it yielded the lowest symmetry in overall flexionangle trajectories.

VII. CONCLUSION

FPRE-based feedback control was chosen for the SEA on thebasis of the actuator’s variable structure and the peculiarities ofthe exoskeleton’s task, namely the correction of gait asymmetry.Switching between the low force (unconstrained) and the highforce (constrained) modes depends on the magnitude of theforce being transmitted. But the intrinsic variability of the taskmakes it impossible to predict the behavior of this force, letalone the timing of the mode transitions. Therefore, the controlmust not rely on any forecast of the system matrices (8). Thecontrol method guarantees the stability of the closed-loop systemdespite the discontinuity in the system matrices and the fact thatthe high force mode introduces a singularity. Stationary testsinvolving sinusoidal reference forces showed good quality offorce tracking with the FPRE, for both the low force and the highforce conditions (see Fig. 13). Equally important, no evidencewas found of instability during mode switching.

The FPRE-based force control was employed to implementa novel assistive strategy aimed at counteracting the joint flex-ion deficits that give rise to gait asymmetry. Synchronizationbetween the user’s gait and the controller’s action was affectedby a dynamical system featuring an AFO. Experimental datarevealed no instances of the AFO losing synchronization. Thissuggests that the dynamical system (28)–(32) may prove robustenough for treadmill-based training for stroke patients.

The artificial knee flexion deficit emulated one typicalanomaly of poststroke gait and served to test the ability ofthe control to correct it. However, the assistive strategy can beextended to other anomalies, such as deficit in ankle propul-sion before toe-off, or deficit in ankle dorsiflexion during theswing phase. Correction of the spatial asymmetry was satis-factory; in the ASSIST condition, the peak flexion angles onthe “impaired”-side became much closer to the “healthy”-sidevalues. The control was also effective at improving symmetryin the duration of the swing phase. Interestingly, swing phasesymmetry in the ASSIST condition was better even than in theFREE condition. This is probably because, in the latter, mechan-ical differences between the “impaired”-side and “healthy”-sidebraces had introduced an unplanned asymmetry.

The associated phase σp showed little sensitivity to the exper-imental condition. At best, this shows that the straightforwardcontrol of the midpoint phase did not cause any significantdisruption in the timing of the peak knee flexion. However,it must be kept in mind that this experiment only attemptedto reproduce some of the abnormal kinematics of poststrokegait; it did not reproduce the abnormal muscle activations thatcause the kinematic anomalies. Therefore, it is quite likely thatthe subjects maintained control over the timing of their muscleactivations throughout the experiment. This probably explainsalso why other gait parameters, such as step frequency andself-selected velocity, showed little change in the IMPAIR andASSIST conditions.




In general, the objective of recovering symmetry for onediscrete gait event (peak knee flexion) was achieved satisfac-torily. However, excluding the peak angle landmark, the timetrajectories of knee flexion angle still displayed considerableasymmetry in the ASSIST condition (see Fig. 17). This maybe due in part to the aforementioned mechanical differencesbetween the braces. On the other hand, some of the parameters ofthe bell-shaped torque function (σrise,χ) were chosen arbitrarily.Thus, better symmetry of the flexion angle time trajectoriesmay be achieved by employing an adaptive control scheme tooptimize those parameters.

The experimental study presented here can be seen as atemplate for future gait training experiments involving neuro-logically impaired patients. In order to assist the recovery ofgait symmetry in impaired people, the robotic hardware and thecontrol will have to act on other gait cycle events besides peakknee flexion, such as peak ankle dorsiflexion angle during swingand peak plantarflexion during ground contact. Gait symmetryalso requires temporal symmetry among the gait cycles fromboth sides; in theory, the same gait event should occur on bothsides with a half-cycle phase difference. The experiment did notattempt to quantify error in the phase of the assistive torquesfor the purposes of temporal symmetry, as that would requirefeedback from the unimpaired side. In general, experiments withimpaired patients will require the robotic device and its controlto assist multiple joints simultaneously, and at different stagesof the gait cycle.

The control strategy presented here is aimed principally atrehabilitation in a controlled environment. It presupposes thatthe subject is walking under conditions that normally require asymmetric gait, such as walking on a flat terrain with a relativelyuniform speed. However, oscillator-based control can in princi-ple be adapted to walking on other environments, such as stairsor ramps. Transitions between environments can be detected,for example, by means of a support vector machine [30]. Inthis way, a different set of symmetry objectives could be set foreach environment. Alternatively, continuous assistance could beprovided by means of dynamic movement primitives; smoothtransitions between motor tasks could be achieved by gradualswitching of the primitive weights [29]. Adaptive oscillators alsohave the ability to maintain synchronization with nonuniformgaits, which in principle would allow to use the exoskeletoncontrol when turning or walking on uneven terrain. An openquestion is how the assistive action should be modified in termsof phase or torque amplitude in such cases.

APPENDIX A

Constraint Detection: Switching Variable ComputationThe deflection forces of the linear motion spring equivalents

are defined as F12 = k12 (x1 − x2) and F23 = k23 (x2 − x3)(see Fig. 3). Full compression of the low-stiffness spring occurswhen |F23| exceeds a threshold value Flow,max. In order tocompute hc, following two auxiliary variables are used.

1) The resultant force acting on the right-hand side of massm2 (see Fig. 3), defined as Fr = F12 −m2 v2 (Note thatthis definition is independent of the state of spring k23).

2) The force difference ΔF = F23 − Fr. If the constraint isactive (k23 fully compressed), the condition signaling thatit is about to become inactive is sign(F23 ·ΔF ) ≥ 0.

For discrete-time control, given the value of hc at time stepk − 1, the value of hc at time step k is given by the followinghysteresis conditions.

1) For hc,k−1 = 0: if F23 ≥ Flow,max, then hc,k = 1.2) For hc,k−1 = 1: if sign(F23 ·ΔF ) ≥ 0, then hc,k = 0.

APPENDIX B

Stability of the SEA Control Based on the FPREThe present analysis focuses on the period between resets

of the initial condition P0. During this period, hc remainsconstant and so do the matrices F(hc), G(hc), and Q(hc).Proof of the asymptotic stability of the system starts by definingΦ = F(hc)

T and Γ = G(hc)T and substituting into (15) to

yield

Pk+1 = ΦPk ΦT

− ΦPk ΓT(ΓPk Γ

T +R)−1

ΓPk ΦT + Q. (52)

Equation (52) has the structure of a linear quadratic estimator orKalman filter. This allows to use certain results about Kalmanfilters to analyze the stability of the control FPRE. The followingtheorem establishes well-known sufficient conditions for thestability of the Kalman filter [53].

Given the difference Riccati equation (52) subject to1) (Φ, Γ) is detectable,2) (Φ, Q) is stabilizable,3) the initial condition P0 ≥ 0 (positive semidefinite),the solution Pk converges asymptotically to the unique, sta-

bilizing solution P∗ of the algebraic Riccati equation (ARE)that results from substituting Pk+1 = Pk = P∗. However, thetheorem assumes Φ to be nonsingular. This poses a problembecause, while (52) meets the enumerated conditions in the caseof the SEA, Φ becomes singular when the full-compressionconstraint (4) is active (hc = 1).

In order to prove the asymptotic stability of (52), it is neces-sary to prove thatV (wk) (16) can only decrease or, equivalently,that Pk is monotonically nonincreasing. This is expressed as

Pk+1 −Pk ≤ 0. (53)

The first step is to prove that if Pk is monotonically nonin-creasing at some time k, then it remains monotonically nonin-creasing at all subsequent times. In other words

Pk+1 ≤ Pk ⇒ Pk+1+j ≤ Pk+j ∀ j > 0. (54)

To this end, the following lemma by De Souza [54] shall be used.Let P1,k and P2,k be the solutions of two Riccati difference

equations with the same matrices Φ, Γ, and R but possiblydifferent matrices Q1 and Q2, and different initial conditionsP1,0 and P2,0. Define

Pk = P2,k −P1,k. (55)

Pk satisfies




Pk+1 = Φk Pk ΦT

k

− Φk Pk ΓT(Γ Pk Γ

T + Rk

)−1

Γ Pk ΦT

k + Q (56)

where

Φk = Φ−ΦP1,k ΓT(ΓP1,k Γ

T +R)−1

ΓT

Rk = ΓP1,k ΓT +R

Q = Q2 −Q1.

The proof proceeds by induction. The first step is to makeP1,k = Pk, P2,k = Pk+1 and Q1 = Q2. Substituting in (56)yields, after some algebra

Pk+1 = Φk Pk ΦT

k − Φk Pk ΓT Σ−1

k+1 Γ Pk ΦT

k (57)

where Σ−1k+1 = (ΓPk+1 Γ

T +R)−1. It should be noted thatboth summands on the right-hand side of (57) are nonpositivesince Pk ≤ 0 [from the assumption in (54)] and Σ−1

k+1 > 0.Therefore

Pk+1 = Pk+2 −Pk+1 ≤ 0 (58)

which can be generalized to (54) by induction.The next step is to find an initial conditionP0 for (52) such that

the assumption in (54) is met from the start of the closed-loopsystem’s evolution (k = 0). The solution Pk of (52) is equal tothe solution of the following ARE:

Pk = ΦPk ΦT

− ΦPk ΓT(ΓPk Γ

T +R)−1

ΓPk ΦT + Qk (59)

where

Qk = Pk −Pk+1 +Q. (60)

The monotonicity condition (53) requires Pk to be nonneg-ative. On the basis of a theorem for AREs [53], for the AREdefined at timek to have a unique nonnegative solution, sufficientconditions are that (Φ, Γ) is detectable, (Φ, Q) is stabilizable,and Qk ≥ 0. The last condition is the crucial one as it was notestablished previously. It implies that the initial condition mustsatisfy Q0 = P0 −P1 +Q ≥ 0 or

P0 − f(P0,Q) +Q ≥ 0 (61)

where f(·) represents the right-hand side of (52).Thus, in order to guarantee the stability of the control Riccati

equation (15), a set of sufficient conditions consists of thefollowing.

1) The stability conditions of the Kalman filter theorem [53],expressed in terms of F(hc)

T and G(hc)T .

2) The initial condition requisite (61).The key outcome is that, for stability to be guaranteed, the sin-

gularityF(hc) imposes an additional restriction on the allowableinitial condition for (15).

REFERENCES

[1] T. Yan, M. Cempini, C. M. Oddo, and N. Vitiello, “Review of assistivestrategies in powered lower-limb orthoses and exoskeletons,” Robot. Au-ton. Syst., vol. 64, pp. 120–136, Feb. 2015.

[2] A. Esquenazi, M. Talaty, A. Packel, and M. Saulino, “The ReWalk poweredexoskeleton to restore ambulatory function to individuals with thoracic-level motor-complete spinal cord injury,” Amer. J. Phys. Med. Rehabil.,vol. 91, no. 11, pp. 911–921, Nov. 2012.

[3] K. A. Strausser and H. Kazerooni, “The development and testing of ahuman machine interface for a mobile medical exoskeleton,” in Proc.IEEE/RSJ Int. Conf. Intell. Robots Syst., 2011, pp. 4911–4916.

[4] R. J. Farris, H. A. Quintero, and M. Goldfarb, “Preliminary evaluation of apowered lower limb orthosis to aid walking in paraplegic individuals,”IEEE Trans. Neural Syst. Rehabil. Eng., vol. 19, no. 6, pp. 652–659,Dec. 2011.

[5] G. Lv, H. Zhu, and R. D. Gregg, “On the design and control of highlybackdrivable lower-limb exoskeletons: A discussion of past and ongoingwork,” IEEE Control Syst., vol. 38, no. 6, pp. 88–113, Dec. 2018.

[6] J. Hidler and A. Wall, “Alterations in muscle activation patterns duringrobotic assisted walking,” Clin. Biomech., vol. 20, pp. 184–193, 2005.

[7] A. Duschau-Wicke, T. Brunsch, L. Lunenburger, and R. Riener, “Adaptivesupport for patient-cooperative gait rehabilitation with the Lokomat,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Sep. 2008, pp. 2357–2361.

[8] L. L. Cai et al., “Implications of assist-as-needed robotic step training aftera complete spinal cord injury on intrinsic strategies of motor learning,” J.Neurosci., vol. 26, no. 41, pp. 10564–10568, Oct. 2006.

[9] J. Ghan, R. Steger, and H. Kazerooni, “Control and system identificationfor the Berkeley lower extremity exoskeleton (BLEEX),” Adv. Robot.,vol. 20, no. 9, pp. 989–1014, Jan. 2006.

[10] S. A. Murray, K. H. Ha, C. Hartigan, and M. Goldfarb, “An assistivecontrol approach for a lower-limb exoskeleton to facilitate recovery ofwalking following stroke,” IEEE Trans. Neural Syst. Rehabil. Eng., vol.23, no. 3, pp. 441–449, May 2015.

[11] R. Ham, T. Sugar, B. Vanderborght, K. Hollander, and D. Lefeber,“Compliant actuator designs,” IEEE Robot. Autom. Mag., vol. 16, no. 3,pp. 81–94, Sep. 2009.

[12] J. Veneman, R. Ekkelenkamp, R. Kruidhof, F. Van der Helm, and H. Vander Kooij, “A series-elastic and Bowden-cable-based actuation system foruse as torque-actuator in exoskeleton-type training,” Int. J. Robot. Res.,vol. 25, no. 3, pp. 261–281, 2006.

[13] T. Yan et al., “A novel adaptive oscillators-based control for a poweredmulti-joint lower-limb orthosis,” in Proc. IEEE Int. Conf. Rehabil. Robot.,Aug. 2015, pp. 386–391.

[14] J. Zhang and S. H. Collins, “The passive series stiffness that optimizestorque tracking for a lower-limb exoskeleton in human walking,” Front.Neurorobot., vol. 11, Dec. 2017, Art. no. 68.

[15] C. B. Sanz-Morere et al., “A knee–ankle–foot orthosis to assist the soundlimb of transfemoral amputees,” IEEE Trans. Med. Robot. Bionics, vol. 1,no. 1, pp. 38–48, Feb. 2019.

[16] G. Pratt, P. Willisson, C. Bolton, and A. Hofman, “Late motor processingin low-impedance robots: Impedance control of series-elastic actuators,”in Proc. Amer. Control Conf., 2004, pp. 3245–3251.

[17] H. Vallery, J. Veneman, E. van Asseldonk, R. Ekkelenkamp, M. Buss, andH. van Der Kooij, “Compliant actuation of rehabilitation robots,” Robot.Autom. Mag., vol. 15, no. 3, pp. 60–69, Sep. 2008.

[18] K. Kong, J. Bae, and M. Tomizuka, “Control of rotary series elasticactuator for ideal force-mode actuation in human–robot interaction appli-cations,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 1, pp. 105–118,Feb. 2009.

[19] S. Kim and J. Bae, “Force-mode control of rotary series elastic actuatorsin a lower extremity exoskeleton using model-inverse time delay con-trol,” IEEE/ASME Trans. Mechatronics, vol. 22, no. 3, pp. 1392–1400,Jun. 2017.

[20] G. Chen, P. Qi, Z. Guo, and H. Yu, “Mechanical design and evaluationof a compact portable knee– ankle– foot robot for gait rehabilitation,”Mech. Mach. Theory, vol. 103, pp. 51–64, 2016. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0094114X16300465

[21] G. Aguirre-Ollinger, A. Narayan, and H. Yu, “Phase-synchronized as-sistive torque control for the correction of kinematic anomalies in thegait cycle,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 27, no. 11,pp. 2305–2314, Nov. 2019.

[22] N. Paine, S. Oh, and L. Sentis, “Design and control considerations for high-performance series elastic actuators,” IEEE/ASME Trans. Mechatronics,vol. 19, no. 3, pp. 1080–1091, Jun. 2014.


http://www.sciencedirect.com/science/article/pii/S0094114X16300465



[23] A. Prach, O. Tekinalp, and D. S. Bernstein, “Output-feedback control oflinear time-varying and nonlinear systems using the forward propagatingRiccati equation,” J. Vib. Control, vol. 24, no. 7, pp. 1239–1263, Jun. 2016.

[24] T. Yan, A. Parri, V. R. Garate, M. Cempini, R. Ronsse, and N. Vitiello,“An oscillator-based smooth real-time estimate of gait phase for wearablerobotics,” Auton. Robots, vol. 41, no. 3, pp. 759–774, May 2016.

[25] R. Ronsse, S. M. M. D. Rossi, N. Vitiello, T. Lenzi, M. C. Carrozza, andA. J. Ijspeert, “Real-time estimate of velocity and acceleration of quasi-periodic signals using adaptive oscillators,” IEEE Trans. Robot., vol. 29,no. 3, pp. 783–791, Jun. 2013.

[26] T. Sugar et al., “Limit cycles to enhance human performance based onphase oscillators,” J. Mech. Robot., vol. 7, no. 1, 2015, Art. no. 011001.

[27] A. Gams, T. Petric, T. Debevec, and J. Babic, “Effects of robotic kneeexoskeleton on human energy expenditure,” IEEE Trans. Biomed. Eng.,vol. 60, no. 6, pp. 1636–1644, Jun. 2013.

[28] T. Xue, Z. Wang, T. Zhang, and M. Zhang, “Adaptive oscillator-basedrobust control for flexible hip assistive exoskeleton,” IEEE Robot. Autom.Lett., vol. 4, no. 4, pp. 3318–3323, Oct. 2019.

[29] V. R. Garate et al., “Experimental validation of motor primitive-basedcontrol for leg exoskeletons during continuous multi-locomotion tasks,”Front. Neurorobot., vol. 11, Mar. 2017, Art. no. 15.

[30] K. Seo et al., “Adaptive oscillator-based control for active lower-limbexoskeleton and its metabolic impact,” in Proc. IEEE Int. Conf. Robot.Autom., May 2018, pp. 6752–6758.

[31] C. K. Balasubramanian, M. G. Bowden, R. R. Neptune, and S. A. Kautz,“Relationship between step length asymmetry and walking performancein subjects with chronic hemiparesis,” Arch. Phys. Med. Rehabil., vol. 88,no. 1, pp. 43–49, Jan. 2007.

[32] D. J. Clark, L. H. Ting, F. E. Zajac, R. R. Neptune, and S. A. Kautz, “Merg-ing of healthy motor modules predicts reduced locomotor performance andmuscle coordination complexity post-stroke,” J. Neurophysiol., vol. 103,no. 2, pp. 844–857, Feb. 2010.

[33] K. K. Patterson, W. H. Gage, D. Brooks, S. E. Black, and W. E. McIl-roy, “Evaluation of gait symmetry after stroke: A comparison of currentmethods and recommendations for standardization,” Gait Posture, vol. 31,no. 2, pp. 241–246, Feb. 2010.

[34] S.-C. Yen, B. D. Schmit, and M. Wu, “Using swing resistance and as-sistance to improve gait symmetry in individuals post-stroke,” HumanMovement Sci., vol. 42, pp. 212–224, Aug. 2015.

[35] G. Aguirre-Ollinger, A. Narayan, H.-J. Cheng, and H. Yu, “Exoskeletoncontrol for post-stroke gait training of a paretic limb based on extraction ofthe contralateral gait phase,” in Biosyst. & Biorobotics. Berlin, Germany:Springer, Oct. 2018, pp. 294–298.

[36] P. Belanger, P. Dobrovolny, A. Helmy, and X. Zhang, “Estimation ofangular velocity and acceleration from shaft-encoder measurements,” Int.J. Robot. Res., vol. 17, no. 11, pp. 1225–1233, 1998.

[37] R. E. Kalaba and F. E. Udwadia, “Equations of motion for nonholonomic,constrained dynamical systems via Gauss’s principle,” J. Appl. Mech.,vol. 60, no. 3, pp. 662–668, 1993.

[38] F. E. Udwadia and P. Phohomsiri, “Explicit equations of motion for con-strained mechanical systems with singular mass matrices and applicationsto multi-body dynamics,” Proc. Roy. Soc. A: Math., Phys., Eng. Sci.,vol. 462, no. 2071, pp. 2097–2117, Jul. 2006.

[39] A. Prach, O. Tekinalp, and D. S. Bernstein, “Infinite-horizon linear-quadratic control by forward propagation of the differential Riccati equa-tion,” IEEE Control Syst., vol. 35, no. 2, pp. 78–93, Apr. 2015.

[40] B. Basu and A. Staino, “Control of a linear time-varying system witha forward Riccati formulation in wavelet domain,” J. Dyn. Syst., Meas.,Control, vol. 138, no. 10, Jun. 2016, Art. no. 104502.

[41] W.-H. Chen, J. Yang, L. Guo, and S. Li, “Disturbance-observer-basedcontrol and related methods—An overview,” IEEE Trans. Ind. Electron.,vol. 63, no. 2, pp. 1083–1095, Feb. 2016.

[42] S. Li, J. Yang, W.-H. Chen, and X. Chen, Disturbance Observer-BasedControl: Methods and Applications, 1st ed. Boca Raton, FL, USA: CRCPress, 2014.

[43] T. Petric, A. Gams, A. Ijspeert, and L. Zlajpah, “On-line frequencyadaptation and movement imitation for rhythmic robotic tasks,” Int. J.Robot. Res., vol. 30, no. 14, pp. 1775–1788, 2011.

[44] J. Emken, R. Benitez, and D. Reinkensmeyer, “Human-robot cooperativemovement training: Learning a novel sensory motor transformation duringwalking with robotic assistance-as-needed,” J. Neuroeng. Rehabil., vol. 4,no. 1, 2007, Art. no. 8.

[45] R. A. Zifchock, I. Davis, J. Higginson, and T. Royer, “The symmetry angle:A novel, robust method of quantifying asymmetry,” Gait Posture, vol. 27,no. 4, pp. 622–627, May 2008.

[46] S. Moore, K. Schurr, A. Wales, A. Moseley, and R. Herbert, “Observationand analysis of hemiplegic gait: Swing phase,” Aust. J. Physiotherapy,vol. 39, no. 4, pp. 271–278, 1993.

[47] D. Mazzoli et al., “Electromyographic activity of the vastus intermediusmuscle in patients with stiff-knee gait after stroke. A retrospective observa-tional study,” Gait Posture, vol. 60, pp. 273–278, 2018. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0966636217307063

[48] D. C. Kerrigan, E. P. Frates, S. Rogan, and P. O. Riley, “Hip hiking andcircumduction,” Amer. J. Phys. Med. Rehabil., vol. 79, no. 3, pp. 247–252,May 2000.

[49] I. Campanini, A. Merlo, and B. Damiano, “A method to differentiate thecauses of stiff-knee gait in stroke patients,” Gait Posture, vol. 38, no. 2,pp. 165–169, Jun. 2013.

[50] S. R. Goldberg, S. Õunpuu, and S. L. Delp, “The importance of swing-phase initial conditions in stiff-knee gait,” J. Biomech., vol. 36, no. 8,pp. 1111–1116, Aug. 2003.

[51] D. Winter, Biomechanics and Control of Human Movement, 3rd ed.Hoboken, NJ, USA: Wiley, 2005.

[52] H. van der Kooij, S. Donker, M. de Vrijer, and F. van der Helm, “Identi-fication of human balance control in standing,” in Proc. IEEE Int. Conf.Syst., Man, Cybern., 2004, pp. 2535–2541.

[53] S. Chan, G. Goodwin, and K. Sin, “Convergence properties of the Riccatidifference equation in optimal filtering of nonstabilizable systems,” IEEETrans. Autom. Control, vol. AC-29, no. 2, pp. 110–118, Feb. 1984.

[54] C. De Souza, “Monotonicity and stabilizability results for the solutions ofthe Riccati difference equation,” in Proc. Workshop Riccati Equ. Control,Syst., Signals, Como, Italy, 1989, pp. 38–41.

Gabriel Aguirre-Ollinger received the M.S. degreein mechanical engineering from Carnegie MellonUniversity, Pittsburgh, PA, USA, in 2001, and thePh.D. degree in mechanical engineering from North-western University, Evanston, IL, USA, in 2009.

He was a Lecturer with the School of Electrical,Mechanical and Mechatronic Systems, University ofTechnology Sydney, Ultimo, NSW, Australia, and aResearch Fellow with the Department of BiomedicalEngineering, National University of Singapore, Sin-gapore. He is currently the Manager for Research and

Development with Articares Pte Ltd., Singapore. His research interests includerehabilitation robotics, physical human–robot interaction, bipedal walking, andmanipulator programming by demonstration.

Haoyong Yu received the B.S. and M.S. degrees fromShanghai Jiao Tong University, Shanghai, China,in 1988 and 1991, respectively, and the Ph.D. de-gree from the Massachusetts Institute of Technology,Cambridge, MA, USA, in 2002, all in mechanicalengineering.

He was with the DSO National Laboratories, Sin-gapore, as a Principal Member of Technical Staffbefore he joined the Faculty of the Department ofBiomedical Engineering, National University of Sin-gapore (NUS), Singapore, in 2010. His research in-

terests include DSO-included exoskeleton and humanoid robots as well asintelligent ground and aerial robots. His current research at NUS is focusedon robotics for neurorehabilitation, especially on using exoskeleton systemsand smart mobility aids for patients with stroke and Parkinson’s disease.


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