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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 46, NO. 7, JULY 2016 1679

Identification-Based Closed-Loop NMES LimbTracking With Amplitude-Modulated

Control InputTeng-Hu Cheng, Qiang Wang, Rushikesh Kamalapurkar, Huyen T. Dinh, Matthew Bellman, and Warren E. Dixon

Abstract—An upper motor neuron lesion (UMNL) can becaused by various neurological disorders or trauma and leadsto disabilities. Neuromuscular electrical stimulation (NMES) is atechnique that is widely used for rehabilitation and restorationof motor function for people suffering from UMNL. Typically,stability analysis for closed-loop NMES ignores the modulatedimplementation of NMES. However, electrical stimulation mustbe applied to muscle as a modulated series of pulses. In thispaper, a muscle activation model with an amplitude modu-lated control input is developed to capture the discontinuousnature of muscle activation, and an identification-based closed-loop NMES controller is designed and analyzed for the uncertainamplitude modulated muscle activation model. Semi-global uni-formly ultimately bounded tracking is guaranteed. The stabilityof the closed-loop system is analyzed with Lyapunov-based meth-ods, and a pulse frequency related gain condition is obtained.Experiments are performed with five able-bodied subjects todemonstrate the interplay between the control gains and the pulsefrequency, and results are provided which indicate that controlgains should be increased to maintain stability if the stimulationpulse frequency is decreased to mitigate muscle fatigue. For thefirst time, this paper brings together an analysis of the controllerand modulation scheme.

Index Terms—Artificial neural networks, dynamics, fatigue,neuromuscular stimulation, stability analysis.

I. INTRODUCTION

UPPER motor neuron lesions (UMNL) cause disability andparalysis in millions of people. UMNL is usually caused

by neural disorders such as stroke or cerebrovascular accident,spinal cord injury, multiple sclerosis, cerebral palsy, or trau-matic brain injury. The overall reported prevalence is 37 000people/million/year for an UMNL [1]. Since the lower motor

Manuscript received February 27, 2015; revised May 13, 2015; acceptedJune 26, 2015. Date of publication July 28, 2015; date of current ver-sion June 14, 2016. This work was supported by the National ScienceFoundation (NSF) under Award 1161260 and Award 1217908. This paperwas recommended by Associate Editor J. Q. Gan.

T.-H. Cheng, R. Kamalapurkar, H. T. Dinh, and M. Bellman are withthe Department of Mechanical and Aerospace Engineering, Universityof Florida, Gainesville, FL 32611-6250 USA (e-mail: [email protected];[email protected]; [email protected]; [email protected]).

Q. Wang is with the Department of Electrical and ComputerEngineering, University of Florida, Gainesville, FL 32611-6250 USA (e-mail:[email protected]).

W. E. Dixon is with the Department of Mechanical and AerospaceEngineering, University of Florida, Gainesville, FL 32611-6250 USA, andalso with the Department of Electrical and Computer Engineering, Universityof Florida, Gainesville, FL 32611-6250 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCYB.2015.2453402

neuron system and muscles are intact in those with UMNL,muscle contractions can be evoked by directly applying elec-trical stimulus to the muscles; this technique is widely usedfor rehabilitation [2] and restoration of motor function and isreferred to as neuromuscular electrical stimulation (NMES) orfunctional electrical stimulation (FES) when applied to pro-duce a functional outcome. The development of an NMESmethod to provide a desired outcome is challenging dueto the nonlinear response from muscle to electrical input,load changes during functional movement, unexpected mus-cle spasticity, time lag between muscle activation and muscleforce output, and muscle fatigue. Closed-loop NMES con-trol is promising with regard to its ability to achieve preciselimb movement and disturbance rejection, both of which areessential for functional rehabilitation.

Several proportional–integral–derivative-based linearNMES controllers have been developed [3]–[6], but thesemethods typically rely on a linear muscle model and lacka stability analysis. Neural network (NN)-based NMEScontrollers [7]–[20] have been developed which utilize theuniversal approximation property of NNs to approximate thenonlinear (unstructured) dynamics. Robust NMES methodshave been developed in [21] and [22] that achieve guaranteedasymptotic limb tracking. In [23] and [24], inverse and directoptimal NMES controllers were developed with guaranteedstability, and the authors addressed the problem of musclefatigue from overstimulation by balancing the performanceand the control effort. In [6] and [25], muscle contractiondynamics were modeled with known parameters or with bestestimates of the parameters, and nonlinear controllers weredeveloped which yielded asymptotically stable closed-looperror systems. In [26], the control method used model-free,adaptive pattern generator/pattern shapers, but stability analy-sis of the closed-loop error system was not analyzed. In [18],an adaptive, neuro-sliding-mode controller was developedbased on uncertain human knee-joint dynamics, and it wasdemonstrated in able-bodied and paralyzed individuals that thedeveloped controller was able to successfully track a desiredknee-joint trajectory. In [27], an adaptive robust controlsystem for FES-induced ankle dorsiflexion and plantarflexionwas developed based on fuzzy logic and sliding modecontrol methods that yielded accurate tracking performanceof an ankle trajectory in able-bodied and paraplegic subjectswithout requiring a priori model knowledge. In [28], thecontrol system developed in [27] was extended to the task of

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walker-assisted FES-walking with FES controllers for the hip,knee, and ankle joints. Despite the success of these previousapproaches, controllers were developed without accountingfor how the control signal is modulated during application tothe muscle. Previous approaches assumed that the stimulationapplied across the muscle groups is continuous. However,in practice, stimulation is typically applied in the formof discrete pulses, and the pulse amplitude, duration, andfrequency is modulated to control the muscle force outputduring FES. Since modulation strategies are not consideredin any previous system models, the developed controllers andsubsequent stability analyses, if performed, yield no insightinto the interplay between the controller and the modulationparameters.

NMES is delivered in the form of electrical pulses whichcreate a localized electric field to elicit action potentials inthe nearby neurons. Muscle force output is determined by thepulse amplitude, duration, frequency, and the muscle fatiguestate. Pulse duration and amplitude determine the activationregion, i.e., how many motor units are recruited, and are equiv-alent with regard to the total applied electric charge. Thiseffect is often referred to as spatial summation. Each electricpulse causes a twitch in the muscle fibers. If a second pulse isapplied before the first twitch finishes, the two twitches sumand a higher force output from the muscle is achieved. Thiseffect is often referred to as temporal summation. When thepulse frequency is higher than a threshold called the fusion fre-quency, continuous muscle force output is observed, and largerforces can be achieved with higher frequencies. However,higher stimulation frequencies cause the muscle to fatiguefaster. In practice, muscle force is controlled by modulatingthe pulse amplitude or pulse duration, and the frequency isset to a constant value that is as low as possible to main-tain fused force output while avoiding fatiguing the muscleprematurely [29]. Recent results demonstrated that frequency-modulation can yield better performance for both peak forcesand force-time integrals than pulse-duration-modulation, whileproducing similar levels of muscle fatigue [30]. However,it is still unclear how different amplitude-modulation strate-gies affect the performance of FES-induced activities andwhat modulation strategy should be used to maximizeperformance.

In [31], an identification-based controller was developed forthe muscle-limb model which includes an uncertain first-orderdynamic system that models muscle contraction dynamics.The parameters of the limb dynamics and the muscle con-traction model are unknown. Since the NMES control inputis implemented as a series of pulses and the modulationstrategy has significant impact on the muscle performanceand fatigue, the ability to examine the impact of the con-trol signal and modulation strategy analytically may yieldnew insights into the development of NMES controllers. Inthis paper, a muscle activation model with an amplitude-modulated control input is developed to capture the discon-tinuous nature of muscle activation. Measurement of angularacceleration is required to guarantee stability, but sensor noiseconstraints typically make acceleration measurements unre-liable, so an observer is designed to provide an estimate

of angular acceleration. An identification-based, closed-loopNMES controller is designed and analyzed for the uncer-tain pulse muscle activation model. Semi-global uniformlyultimately bounded (SUUB) tracking is guaranteed throughLyapunov-based methods. A sufficient condition for stabil-ity that relates the pulse frequency and the control gains isobtained. Experiments are performed with five able-bodiedsubjects to demonstrate the interplay between the control gainsand the pulse frequency, and results are provided which indi-cate that control gains should be increased to maintain trackingperformance if the stimulation pulse frequency is decreased tomitigate muscle fatigue.

II. DESCRIPTION OF THE DYNAMICS

A freely swinging lower limb with respect to the knee-jointcan be segregated into body segmental dynamics and mus-cle activation and contraction dynamics. The body segmentaldynamics can be expressed as

MI(q) + Me(q) + Mg(q) + Mv(q) + τd = τm (1)

where q, q, and q ∈ R denote the angular position, veloc-ity, and acceleration of the lower limb about the knee-joint,respectively; τm : [0, ∞) → R denotes the active torque atthe knee-joint produced by muscle through electrical stimu-lation; MI : R → R denotes the inertial effects of the lowerlimb-foot complex about the knee-joint; Me : R → R denotesthe elastic effects due to joint stiffness; Mg : R → R denotesthe gravitational component; Mv : R → R denotes the viscouseffects due to damping in the musculotendon complex [32];and τd : [0, ∞) → R includes all other unmodeled effectsand disturbances such as external loads. In (1) and in the sub-sequent development, the dependence on time is suppressed.The inertial component MI in (1) is defined as

MI(q) � JIq (2)

where JI ∈ R is an unknown constant, denoted as the iner-tia of the lower limb-foot complex about the knee-joint. Thetotal muscle torque τm generated at the knee-joint is consid-ered as an unknown nonlinear function ζ : R → R (which isthe moment arm) multiplied by the muscle contraction forcexf : R × R → R generated by electric stimulation as

τm � ζ (q)xf (q, q). (3)

The elastic effects are modeled on the empirical findings byFerrarin and Pedotti [32] as

Me(q) = −K1(e−K2q)(q − K3)

where K1, K2, and K3 ∈ R are unknown positive constants.As in [6], the viscous moment Mv can be modeled as

Mv(q) = B1 tanh(−B2q) − B3q

where B1, B2, and B3 ∈ R are unknown positive constants.For complete details of the dynamics in (1), see [22].

The following assumptions and properties are used tofacilitate the subsequent control development and stabilityanalysis.

CHENG et al.: IDENTIFICATION-BASED CLOSED-LOOP NMES LIMB TRACKING WITH AMPLITUDE-MODULATED CONTROL INPUT 1681

Property 1: The moment arm ζ is a continuously dif-ferentiable, nonzero, positive, monotonic, and boundedfunction [33].

Assumption 1: The disturbance term τd and its time deriva-tives τd, τd are bounded.

The model developed in (1)−(3) is used to examine thestability of the subsequently developed controller, but thecontroller does not explicitly depend on the model. Aftersubstituting (2) and (3), and dividing both sides by ζ , theexpression in (1) can be written as

ζ−1(q)JIq + ζ−1(q)(Me(q) + Me(q) + Mv(q))

+ ζ−1(q)τd = xf (q, q). (4)

Muscle activation and contraction dynamics can be mod-eled as in [6], [25], [34], and [35], which can be generalizedas (see the Appendix for further details)

xf (q, q, q) + Af (q, q)xf (q, q) + f1(q, q, q) + τ1 = b(q, q)u

(5)

where Af , f1 : R × R → R and b : R × R × R → R

are uncertain continuous functions, τ1 : [0, ∞) → R rep-resents disturbances in the muscle (e.g., muscle spasticity,fatigue, and volitional muscle activation), and u : [0, ∞) → R

is the applied electric stimulation voltage. The first-order,nonlinear differential equation in (5) represents a general-ization of the relationship between the excitation input, u,and muscle fiber force output, xf , which combines the mus-cle activation and contraction dynamics based on the modelsin [25], [26], and [34]. The introduction of the unknown non-linear functions Af and f1 enables the muscle contraction to beconsidered under general conditions in the subsequent controldevelopment.

Property 2: Based on the empirical data in [36] and [37],the muscle gain (recruiting) function b is a bounded, positivefunction with bounded, first-order time derivatives.

By substituting xf and xf into (5), the dynamics in (5) canbe expressed as

J(q, q, q)...q = −f2(q, q, q) − τ2 + u (6)

where J, f2 : R × R × R → R and τ2 : [0, ∞) → R aredefined as

J(q, q, q) � b−1(q, q)ζ−1(q)JI

f2(q, q, q) � b−1(q, q)(−JIζ

−2(q)ζ (q, q)

+ Af (q, q)JIζ−1(q)

)q + b−1(q, q)

× f1(q, q, q) + b−1(q, q)ζ−1(q)Af (q, q)

× (Me(q) + Mg(q) + Mv(q)

)

+ b−1(q, q)ζ−1(q)(Me(q, q)

+ Mg(q, q) + Mv(q, q))

− b−1(q, q)ζ−2(q)ζ (q, q)

× (Me(q) + Mg(q) + Mv(q)

)

τ2 � b−1(q, q)(τdζ

−1(q) − τdζ−2(q)ζ (q, q)

+ τ1 + Af (q, q)τdζ−1(q)

). (7)

Based on Properties 1 and 2, the following inequalities canbe developed:

ξ0 ≤ J(q, q, q) ≤ ξ1, |τ2| ≤ ξ2 (8)

where ξi ∈ R, i = 0, 1, 2 are known positive constants.The electrical pulse input u can be modeled as

u ={

v, nT ≤ t < nT + d−vb, otherwise

n = 0, 1, 2, . . . (9)

where v : [0, ∞) → R and d, T ∈ R denote pulse ampli-tude, width, and period,1 respectively, and vb : [0, ∞) → R≥0denotes a charge-balancing pulse, which has a known boundvb ∈ R≥0, defined such that vb ≤ vb. In other words, the con-troller is modulated in the sense that u = v when t satisfiesnT ≤ t ≤ nT + d, whereas u = −vb at nT + d < t < (n + 1)T .The pulse frequency is defined as f � (1/T). Based on (9) thesystem in (6) can be expressed as

J...q =

{−f2(q, q, q) − τ2 + v, nT ≤ t < nT + d−f2(q, q, q) − τ2 − vb, otherwise

n = 0, 1, 2, 3, . . . . (10)

III. CONTROLLER DEVELOPMENT

The control objective is to ensure the knee angle q tracksa desired trajectory, denoted by qd : [0, ∞) → R, which isan essential task in many rehabilitative exercises and func-tion restoration tasks. To quantify the tracking objective,a lower limb angular position tracking error, denoted bye : [0, ∞) → R, is defined as

e � qd − q (11)

where qd is a known trajectory, designed such that qd andqi

d ∈ L∞, where qid denotes the ith derivative of qd for

i = 1, 2, 3. To facilitate the subsequent control design andstability analysis, filtered tracking errors denoted by e1 ande2 : [0, ∞) → R, are also defined as

e1 � e + α1e (12)

e2 � e1 + α2e1 (13)

where α1, α2 ∈ R are positive constant control gains.Using (11)–(13), e2 can be expressed as

e2 = qd − q + (α1 + α2)(qd − q) + α1α2e. (14)

The subsequent development is based on the assumption thatq and q are measurable. The error dynamics in (14) dependon the unmeasurable limb acceleration. To compensate forthe acceleration dependency, an acceleration estimation errore2 : [0, ∞) → R is designed based on (14) as

e2 � qd − ¨q + (α1 + α2)(

qd − ˙q)

+ α1α2e (15)

where ˙q, ¨q : [0, ∞) → R denote the subsequently designedobserver outputs.

1In this paper, fixed-period pulses were considered for simplicity, but theproposed method could be extended to n-let pulse trains.


To facilitate the subsequent analysis, let f2d : R × R ×R → R be defined as

f2d(qd, qd, qd) � f2(qd, qd, qd) (16)

where f 2d is the same function as f2, except that its argu-ments q, q, and q are replaced by the desired argumentsqd, qd, and qd. Using the bounded desired arguments ensuresthat the domain of f 2d is a compact set, which is a require-ment for NN approximation. Based on the universal functionapproximation property [38], the unknown function in (16) canbe approximated by a multilayer NN as

f2d(qd, qd, qd) = WTσ(VTXd

) + ε(qd, qd, qd) (17)

where Xd ∈ R4 is defined as

Xd �[1, qd, qd, qd

]T

σ : Rn0 → R

n0+1 denotes the activation function; W ∈ Rn0+1

and V ∈ R4×n0 denote the bounded constant ideal weights for

the hidden layer neurons and the input layer neurons, respec-tively, where the number of hidden layer neurons is selected asn0; and ε : R × R×R → R denotes the reconstruction error.

Assumption 2: The activation function σ and its first orderderivative with respect to its arguments σ ′ are bounded byknown constants [39].

Property 3: The reconstruction error ε and its first orderpartial derivative ε′ are bounded by known constants [39].

For notational brevity the dependence of all the functionson the states is suppressed hereafter.

After multiplying the time derivative of (14) by J, andusing (6), (11)–(13), (16), and (17), the open-loop error systemfor e2 is

Je2 = −1

2Je2 + f2 − f2d + WTσ

(VTXd

)

+ 1

2Je2 − Jα2

1 e + J(α1 + α2)e1

+ J...q d + τ2 + ε − u. (18)

Let W : [0, ∞) → Rn0+1 and V : [0, ∞) → R

4×n0 denotethe estimated weights for W, V , and let σ, σ , σ ′, σ , W ∈ R

n0 ,and V ∈ R

4×n0 be defined as

σ � σ(VTXd

)(19)

σ � σ(

VTXd

)(20)

σ ′ �∂(σ(

VTX))

∂(

VTX)

∣∣∣∣∣∣VT X=VT Xd

(21)

σ � σ − σ (22)

W � W − W (23)

V � V − V. (24)

By using a Taylor series approximation, σ can be expressed asσ = σ ′VTXd + o(VTXd)

2, where o(·)2 ∈ Rn0 denotes higher

order terms. By using (19)−(24), WTσ can be expressed as

WTσ = WT σ ′VTXd + WT σ

+ WT σ ′VTXd + WTo(VTXd

)2. (25)

The update laws ˙W ∈ Rn0+1 and ˙V ∈ R

4×n0 can be arbitrarilyselected as

˙W � proj(·), ˙V � proj(·) (26)

where proj(·) is a smooth projection operator(see [40, Sec. 4.3]). Gradient-based update laws withpositive definite gain matrices 1 ∈ R and 2 ∈ R

4×4 wereused in the following experiments. Since proj(·) guaranteesW, V are bounded

∣∣∣WT σ

∣∣∣ ≤ a1 (27)

where a1 ∈ R is a known positive constant.The error system in (18) can be expressed as

Je2 = −1

2Je2 + f3 + WT σ − u (28)

where the auxiliary function f3 ∈ R is defined as

f3 � f2 − f2d + WT σ

+ WT σ ′VTXd + WT σ ′VTXd

+ WTo(VTXd

)2

− Jα21 e + J(α1 + α2)e1

+ 1

2Je2 + J

...q d + ε + τ2.

Since W, V, σ, and ε are bounded, using the mean valuetheorem, (26), and the assumption that

...q d is bounded, f3 can

be bounded as

| f3| ≤ a2 + ρ1(∥∥zf

∥∥)∥∥zf∥∥ (29)

where a2 ∈ R is a positive constant, zf ∈ R3 is defined as

zf � (e, e1, e2)T , and ρ1 : [0, ∞) → [0, ∞) is a positive,

strictly increasing, and radially unbounded function.2 Basedon (26) and (28), and the subsequent stability analysis, thecontrol input is designed as

v = kf e2 + WT σ (30)

where kf ∈ R is a positive control gain.After substituting (9) and (30) into (28), the closed-loop

error system can be obtained as

Je2 ={

− 12 Je2 + f3 − kf e2, nT ≤ t < nT + d

− 12 Je2 + f3 + WT σ − vb, otherwise

n = 0, 1, 2, 3, . . . . (31)

IV. OBSERVER DESIGN

The objective of this section is to design anobserver/identifier to generate estimations of ¨q and ˙q,which are used to generate e2 in (15), so that the controllerin (30) can be implemented with only measurements ofq and q.

2For some classes of systems, the bounding function ρ1 could be selected asa constant, and a global uniformly ultimately bounded result can be obtained.


To facilitate the observer design, let x, x, x, r ∈ R2, and

z ∈ R4 be defined as

x �[q, q

]T (32)

x �[q, ˙q

]T(33)

x � x − x (34)

r � [r1, r2]T (35)

= ˙x + αx (36)

z �[xT , rT]T

(37)

where α � α1 + α2. By using (13) and (15), the differencebetween e2 and e2 is

r2 = e2 − e2. (38)

After substituting (2) and (3), the dynamics in (1) can beexpressed as

JIq + Me + Mg + Mv + τd = ζxf (39)

which can be rewritten as

x = −αx + g1(q, q) + h (40)

where x is defined in (32), and g1 : R × R → R2 and

h : [0, ∞) → R2 are defined as

g1(q, q) � αx +(

q−J−1

I

(Me + Mg + Mv − ζxf

))

h �(

0−J−1

I τd

).

Let g1d : R×R → R2 be defined as g1d(qd, qd) � g1(qd, qd).

The unknown function g1d can be approximated by a mul-tilayer NN with a reconstruction error as g1d(qd, qd) =WT

1 σ1(VT1 xd) + ε1(qd, qd), where xd ∈ R

2 is defined asxd � [qd, qd]T , and W1 ∈ R

(n1+1)×2, V1 ∈ R2×n1 denote the

ideal constant weights for the hidden layer neurons and theinput layer neurons, respectively, where the number of hiddenlayer neurons is selected as n1, and ε1 : R × R → R denotesthe reconstruction error.

Assumption 3: The activation function σ1 : Rn1 → R

n1+1

and its first order derivative with respect to its arguments σ ′1

are bounded by known constants [39].Property 4: The reconstruction error ε1 and its first order

derivative ε′1 are bounded by known constants [39].

The dynamics in (40) can be rewritten as

x = −αx + g1 − g1d + WT1 σ1d + ε1 + h (41)

where σ1d � σ1(VT

1 xd). Based on (41), a multilayer dynamic

NN observer is designed as˙x = −αx + WT

1 σ1 + μ (42)

where σ1 � σ1(VT1 x), and W1 : [0, ∞) → R

(n1+1)×2,V1 : [0, ∞) → R

2×n1 denote the estimated weights for W1,

V1, and μ ∈ R2 is defined as

μ(x) � kx − kx(0) +t∫

0

kαxdτ (43)

where k ∈ R is a positive control gain.

Based on (41) and (42), the observer error dynamics can bewritten as

˙x = −αx + ε1 + ε2 + g1 − g1d + h − μ (44)

where ε2 ∈ R2 is defined as

ε2 � WT1 σ1d − WT

1 σ1. (45)

After some algebraic manipulation, the time derivative of (45)can be written as

ε2 = WT1 σ

′1dVT

1 xd − ˙WT1 σ1 − WT

1 σ′1˙VT

1 x − WT1 σ

′1VT

1˙x

+ WT1 σ

′1VT

1˙x + WT

1 σ′1VT

1˙x + WT

1 σ′1VT

1˙x (46)

where W1 � W1−W1, V1 � V1−V1 denote the mismatches forthe ideal weight estimates, σ ′

1d � (∂(σ1(VT1 xd)))/(∂(VT

1 xd)),

and σ ′1 � (∂(σ1(VT

1 x)))/(∂(V1T

x)). The update laws ˙W1 ∈R

(n1+1)×2,˙V1 ∈ R

2×n1 are designed as

˙W1 � proj(w1σ

′1VT

1˙xxT

)(47)

˙V1 � proj(v1

˙xxT WT1 σ

′1

)(48)

where w1 ∈ R(n1+1)×(n1+1) and v1 ∈ R

2×2 are positivedefinite gain matrices. The update laws in (47) and (48)ensure that

αxT(

WT1 σ

′1VT

1˙x + WT

1 σ′1VT

1˙x)

+ G = 0 (49)

where G : [0, ∞) → R is defined as

G � α

2tr(

W1T−1

w1 W1

)+ α

2tr(

VT1 −1

v1 V1

).

By using the mean value theorem, Assumptions 3 and 4,(47), and (48), the following inequalities can be obtained:

N1 ≤ ρ2(‖ϕ‖)‖ϕ‖ + a3 (50)

N2 ≤ a4‖z‖ + a5∥∥zf

∥∥ + a6 (51)

where N1, N2 ∈ R are defined as

N1(ϕ) � WT1 σ

′1VT

1 xd − ˙WT1 σ1

(VT

1 x)

+ ε1 + h

− WT1 σ

′1˙VT

1 x − WT1 σ

′1VT

1˙x + WT

1 σ′1VT

1˙x

+ g1(q, q, q) − g1(qd, qd, qd) (52)

N2(ϕ) � WT1 σ

′1VT

1˙x + WT

1 σ′1VT

1˙x (53)

a3, a4, a5, a6 ∈ R are positive constants; ρ2 : [0, ∞) →[0, ∞) is a positive, strictly increasing, and radiallyunbounded function3; and ϕ ∈ R

7 is defined as ϕ � [zT , zTf ]T .

3For some classes of systems, the bounding function ρ2 could be selected asa constant, and a global uniformly ultimately bounded result can be obtained.


Fig. 1. System block diagram.

By using (32), (35), (46), (52), and (53), the observer errorsystem in (44) can be rewritten as

r = −kr + N1 + N2. (54)

V. STABILITY ANALYSIS

The stability of the overall system, depicted in Fig. 1, is sub-sequently analyzed based on Lyapunov methods for switchedsystems.

Theorem 1: The controller defined by (9) and (30),along with the estimates in (15) and (20), the update lawsin (26), (47) and (48), and the observer in (42) ensure thatall closed-loop signals are bounded, and the tracking error isSUUB in the sense that ‖ϕ‖ uniformly converges to a ball witha constant radius, provided the control gains are selected suf-ficiently large based on the initial conditions of the states (seethe subsequent stability analysis) and the following sufficientconditions are satisfied:

α, α1 >1

2(55)

α2 > 1 (56)

kf > 4 (57)

min

(α1 − 1

2, α2 − 1,

1

8kf − 1

2

)>

ε

2a5 (58)

min

(α − 1

2,

k

4− 1 + kf

2− a5

2ε

)>

a24

k(59)

γ3(T − d) − γ1d < ln

(β1

β2

)(60)

where ε ∈ R is an arbitrary positive constant; a4 and a5 areintroduced in (51); T and d are introduced in (9); γ3 ∈ R

is a known, positive bounding constant; γ1 ∈ R is a con-stant gain that can be made arbitrarily large by selectingα, α1, α2, kf , and k in (12), (13), (30), and (43) arbitrar-ily large; and β1, β2 ∈ R are positive constants defined asβ1 � (1/2) min(1, ξ0), β2 � (1/2) max(1, ξ1).

Proof: Consider the Lyapunov candidate functionV : R

7→ R, which is a continuously differentiable, positivedefinite function defined as

V(ϕ, t) � 1

2e2 + 1

2e2

1 + 1

2Je2

2 + 1

2xT x + 1

2rTr (61)

which satisfies the following inequalities:

β1‖ϕ‖2 ≤ V ≤ β2‖ϕ‖2. (62)

Taking the time derivative of (61), substituting the dynamicsin (28) and (54), and using (35) and (38) yields

V(ϕ, t) = ee1 − α1e2 + e1e2 − α2e21 + e2f3 + e2WT σ

− e2u + rTN1 + rTN2 − krTr + xT r − xTαx.

The function V can be expressed in segments Vn(ϕ, τ ), whereVn(ϕ, τ ) ∈ R is defined as

Vn(ϕ, τ ) � V(ϕ(nT + τ)) (63)

where τ � t − nT, n � t/T�. Using (29), (50), and (51) onthe interval 0 ≤ τ < d (i.e., u = v) yields

Vn(ϕ, τ ) = ee1 − α1e2 + e1e2 − α2e21

+ e2ρ1(∥∥zf

∥∥)∥∥zf∥∥ + a2e2 + e2WT σ

+ rT(ρ2(‖ϕ‖)‖ϕ‖ + a3)

− e2

(kf e2 + WT σ

)− krTr + xT r

+ rT(a4‖z‖ + a5

∥∥zf∥∥ + a6

) − xTαx.

Applying Young’s inequality yields

Vn(ϕ, τ ) ≤ −(

α1 − 1

2

)e2 − (α2 − 1)e2

1

−(

kf

8− 1

2

)e2

2 −(

k

4− 1 + kf

2

)‖r‖2

−(

α − 1

2

)‖x‖2 − kf

4e2

2

+ e2ρ1(∥∥zf

∥∥)∥∥zf

∥∥ + εa5

2

∥∥zf

∥∥2

+ a5

2ε‖r‖2 − 3k

4‖r‖2 + rTρ2(‖ϕ‖)‖ϕ‖

+ rTa4‖z‖ + rT(a3 + a6) − kf

8e2

2 + a2e2.

Completing the squares and upper-bounding the resultyields

Vn(ϕ, τ ) ≤ −(

λ1 − ρ21

kf

)∥∥zf

∥∥2 − λ2‖z‖2

+ ρ22

k‖ϕ‖2 + (a3 + a6)

2

k+ 2a2

2

kf(64)


where λ1, λ2 ∈ R are positive constants, provided the sufficientconditions in (55)–(59) are satisfied, defined as

λ1 � min

{α1 − 1

2, α2 − 1,

kf

8− 1

2

}− εa5

2

λ2 � min

{k

4− 1 + kf

2− a5

2ε, α − 1

2

}− a2

4

k.

Let two sets D and SD be defined as

D �{ϕ⊂R

7∣∣ ‖ϕ‖ ≤ min

(inf

{ρ−1([λ1kkf , ∞))}

,

inf{ρ−1

2

([√λ2k, ∞

))})}(65)

SD �{

ϕ⊂D ∣∣ ‖ϕ‖ ≤√

β1

β2min

(inf

{ρ−1([λ1kkf , ∞))}

,

inf{ρ−1

2

([√λ2k, ∞

))})}

(66)

where for a set A ⊂ R, the inverse image ρ−1(A) ⊂ R isdefined as ρ−1(A) � {a ∈ R | ρ(a) ∈ A}, and the functionρ(·) is defined as ρ(·) � kρ2

1(·) + kf ρ22(·). Using (62), (64)

can be rewritten as

Vn(ϕ, τ ) ≤ −γ1Vn + γ2, ∀ϕ ∈ D (67)

where γ1, γ2 ∈ R are defined as

γ1 � 1

β2min

{

λ1 − ρ21(‖ϕ(0)‖)

kf− ρ2

2(‖ϕ(0)‖)k

,

λ2 − ρ22(‖ϕ(0)‖)

k

}

γ2 � (a3 + a6)2

k+ 2a2

2

kf.

The region of attraction D in (65) can be made arbitrarily largeto include any initial condition by increasing the control gainsα, α1, α2, k, and kf (i.e., a semi-global result). Likewise, onthe interval d ≤ τ < T (i.e., u = −vb)

Vn(ϕ, τ ) ≤ γ3Vn(ϕ, τ ) + γ4 (68)

where γ3, γ4 ∈ R are constants defined as

γ3 � max{λ3, λ4}, γ4 � (a3 + a6)2

k

where λ3, λ4 ∈ R are constants defined as

λ3 � max

{−

(α1 − 1

2

),−(α2 − 1),

(3

4+ a2 + a1 + vb

)}

+ a25

k+ ρ2

1(‖ϕ(0)‖)4

+ ρ22(‖ϕ(0)‖)

4

λ4 � max

{−

(α − 1

2

),−

(k

8− 1

2

)}+ 2a2

4

k+ ρ2

2(‖ϕ(0)‖)4

.

Using (61) and (63), Vn(ϕ, τ ) can be upper-bounded as

Vn(ϕ, τ ) ≤{−γ1Vn(ϕ, τ ) + γ2, 0 ≤ τ < d

γ3Vn(ϕ, τ ) + γ4, d ≤ τ < T

which can be solved to obtain upper bounds for Vn(ϕ, d) andVn(ϕ, T) as

Vn(ϕ, d) ≤(

Vn(ϕ, 0) − γ2

γ1

)e−γ1d + γ2

γ1(69)

Vn(ϕ, T) ≤(

Vn(ϕ, d) + γ4

γ3

)eγ3(T−d) − γ4

γ3. (70)

By using (69) and (70), and the fact that Vn+1(z, 0) =Vn(z, T), the change in V across a pulse can be defined andupper-bounded as

Vn � Vn+1(ϕ, 0) − Vn(ϕ, 0)

≤ Vn(ϕ, d)eγ3(T−d) − Vn(ϕ, 0)

+ γ4

γ3(eγ3(T−d) − 1)

≤ Vn(ϕ, 0)(

e−γ1deγ3(T−d) − 1)

+ γ2

γ1

(1 − e−γ1d

)eγ3(T−d)

+ γ4

γ3

(eγ3(T−d) − 1

). (71)

To ensure that ‖ϕ(nT)‖ > ‖ϕ((n + 1)T)‖, Vn must satisfy thefollowing condition:

Vn < Vn(ϕ, 0)β1 − β2

β2. (72)

To satisfy (72), it is sufficient to demonstrate that (71) satisfies

Vn ≤ Vn(ϕ, 0)(

e−γ1deγ3(T−d) − 1)

+ γ2

γ1

(1 − e−γ1d

)eγ3(T−d) + γ4

γ3

(eγ3(T−d) − 1

)

< Vn(ϕ, 0)β1 − β2

β2. (73)

If the condition in (60) is satisfied and V(ϕ(nT)) > β1d2,where d ∈ R is defined as

d �

√√√√√

γ4γ3

(1 − e−γ3(T−d)

) + γ2γ1

(1 − e−γ1d

)

β1

(β1β2

e−γ3(T−d) − e−γ1d) (74)

then V(ϕ(nT)) < V(ϕ((n + 1)T)), that is

V(ϕ(0)) > V(ϕ(T)) > V(ϕ(2T)) > . . . . (75)

The size of d is based on the period, pulse width, and controlgains.

Given (61), (62), (65), and (75), e is SUUB [41, Th. 4.18]in the sense that

|e| ≤ ‖ϕ‖ < d, ∀t ≥ T(d, ‖ϕ(0)‖),∀‖ϕ(0)‖ ∈ SD

where T(d, ‖ϕ(0)‖) ∈ R is a positive constant that denotesthe ultimate time to reach the ball.

Remark 1: Based on (60) and (74), the interplay betweenthe modulation strategy and the controller can be determined.To minimize muscle fatigue, one is motivated to decreasethe stimulation frequency (i.e., increase T; see [42]–[44]).From (60) and (74), decreasing the stimulation frequencyindicates that the control gains should be selected larger (mak-ing γ1 larger) to maintain a similar level of steady state


tracking performance. If the frequency is increased (leadingto accelerated muscle fatigue), then the control gains maybe selected lower to maintain a similar level of trackingperformance.

VI. EXPERIMENTS

The controller defined by (9) and (30), along with the esti-mates in (15) and (20), the update laws in (26), (47), and (48),and the observer in (42) was implemented on five able-bodiedvolunteers with written informed consent approved by theUniversity of Florida Institutional Review Board. The purposeof the experiments was to evaluate the control performance andto investigate the interplay between stimulation frequency andcontrol gains described in Remark 1. In the experiments, eachsubject was instructed to sit on a customized leg extensionmachine, described in [22], with an additional free swingingrigid arm which was attached to the subject’s shank. The arm’scenter of rotation was aligned with the subject’s lateral femoralcondyle for each trial, allowing the angular position of the armto coincide with the knee angle. An optical encoder was usedto measure the angular position q of the rigid arm at a sam-pling rate of 1 kHz. The angular velocity q was obtained usingbackward differencing methods without filtering. Bipolar, self-adhesive 3"× 5" PALS oval electrodes4 were used to deliverthe electrical stimulus. One electrode was placed over thedistal-medial portion of the quadriceps femoris muscle groupand the other was placed over the proximal-lateral portion.The electrical stimulation was delivered through a custom builtstimulator as monophasic, rectangular pulses with a constantpulse width of d = 400 μs and pulse frequency ( f = 1/T)of 25 and 60 Hz. The controller was implemented with a NN(i.e., five sigmoid neurons) along with two sets of parame-ters: 1) low frequency with high gains (LFHGs) and 2) highfrequency with low gains (HFLGs). First, the control gainswere tuned for each subject at a frequency of 60 Hz (highfrequency) until a set of gains were found that yielded satis-factory performance (high gains). The gain tuning adopted wasbased on a trial-and-error approach, where γ1 was increasedby simultaneously increasing k, kf , α1, and α2. Then, LFHGwas defined as using the controller with the high gains imple-mented at a frequency of 25 Hz (low frequency). Finally,HFLG was defined as using the controller with low gains,defined as 20% less than the high gains, implemented at highfrequency (60 Hz). The amplitude of the electrical pulses wasmodulated by the output of the controller.

For each trial, the subjects were instructed to relax as muchas possible to reduce voluntary participation in the trackingtask, and they were given no indication of their tracking per-formance during the experiment. Each session was 20 s longand between sessions the subjects were given at least five min-utes of rest to minimize the effect of muscle fatigue on theresults.

Remark 2: Able-bodied subjects and subjects with UMNLcan both fit the developed model. However, specific diseaseor injury conditions can lead to differences in performance

4Surface electrodes for the study were provided compliments of AxelgaardManufacturing Company Ltd.

(a)

(b)

(c)

(d)

Fig. 2. (a) Tracking errors from the two protocols with the representativesubject. Desired versus measured joint angles with the representative subjectusing (b) LFHG strategy and (c) HFLG strategy. (d) Comparison of controlinputs for the two protocols with the representative subject.

(e.g., rapid fatigue, spasticity, etc.). The developed controlleris analyzed in the presence of general disturbance terms thatcan be used to model such effects. While the experiments onlydemonstrate the controller’s performance in able-bodied indi-viduals, and hence demonstrate efficacy of the controller andrelationships between the controller and modulation parame-ters, results will potentially vary when the controller is appliedto specific disease or injury populations.

Without loss of generality, the desired trajectory wasdesigned to be sinusoidal5 with a period of 2.5 s, rangingfrom 5◦ to 60◦ from the subject’s rest position, where the anglebetween the shank and the vertical line is 0◦. Each subject per-formed eight repetitions over the course of the 20 s trial. Tovalidate the interplay between control gains and stimulationfrequencies stated in Remark 1, two different control strate-gies were employed: 1) LFHG and 2) HFLG. The NN used isconstructed based on five sigmoid neurons (i.e., n0 = 5) withone hidden layer, and the control gain tuning adopted is basedon a trial-and-error approach. A pulse frequency of 25 Hz wasused for LFHG, and 60 Hz was used for HFLG. For LFHG,γ1 was increased by simultaneously increasing k, kf , α1, andα2 until satisfactory performance was achieved (i.e., until thesteady-state tracking error response was approximately fivedegrees root mean square (RMS) or less, similar to the resultssuch as [45] and [46]). For HFLG, low gains were defined as15%–20% less than the gains used for LFHG.

5The tracking performance of the closed-loop system with different desiredtrajectories, including multiple frequencies, remains the same provided thatAssumption 1 is satisfied.


TABLE ICOMPARISON OF AVERAGE RMS TRACKING ERRORS FOR ALL FIVE

SUBJECTS WITH MEAN VALUES, MEDIAN VALUES, AND STANDARD

DEVIATIONS, AND NRMS TRACKING ERRORS

TABLE IICOMPARISON OF TRANSIENT RMS TRACKING ERRORS FOR ALL FIVE


DEVIATIONS, AND TRANSIENT NRMS TRACKING ERRORS

Representative results for one individual are presented inFig. 2, which illustrates the tracking errors, desired and mea-sured angles from the representative subject’s trial using HFLGand LFHG approaches, and the control inputs. The RMSand normalized root mean square (NRMS) tracking errorsof all five subjects are summarized in Tables I–III. Therange of voltages applied to all five subjects is 15–34 V.Fig. 2(b) and (c) depicts the tracking errors and desired versusmeasured angles from a representative subject’s trial. Duringthe transient period (i.e., 0–10 s), the tracking errors arebounded, oscillating trajectories, implying that the system isstable. However, the oscillations make comparison of HFLGand LFHG approaches difficult so that the RMS tracking errorswere calculated over each 2.5 s period of the desired trajec-tory. For all subjects, the HFLG strategy consistently resultedin higher transient error when compared to LFHG strategy, butthe steady state RMS tracking errors of both strategies behavedsimilarly, reflecting the predicted interplay between the stim-ulation frequencies and control gains derived in (74). Controlinputs from the two approaches for the representative subjectis given in Fig. 2(d). For all subjects, the LFHG approachyielded a higher control input voltage, but the tracking errorsof the two approaches were similar. One interpretation of theresults is that the high-frequency input generates more muscleforce output than the lower frequency, thereby requiring lesscontrol input voltage to achieve similar tracking performance.

TABLE IIICOMPARISON OF STEADY STATE RMS TRACKING ERRORS FOR ALL FIVE


DEVIATIONS, AND STEADY STATE NRMS TRACKING ERRORS

TABLE IVCONTROL GAINS USED FOR THE TWO STRATEGIES

WITH THE FIVE SUBJECTS

TABLE VWEIGHTS FOR THE UPDATE LAWS WITH THE FIVE SUBJECTS, WHERE

1n, 2n, w1n, and v1n ∈ R>0 ARE POSITIVE CONSTANTS

FOR THE MATRICES 1 = 1nI , 2 = 2nI4, w1 = w1nIn1+1,AND v1 = v1nI2, WHERE I IS AN IDENTITY MATRIX

WITH DENOTED DIMENSION

Tables I–III summarize the experimental results for theLFHG and HFLG protocols by the RMS and NRMS trackingerrors calculated over the entire trials (0–20 s), transient period(0–10 s), and during steady state (10–20 s), respectively. WhileHFLG appears to have a greater RMS error over the entire trial(see Tables I–III), a Wilcoxon signed-rank test on the paireddata resulted in a P-value of 0.138. Thus, one cannot concludeat a significance level of 0.05 that there is a significant differ-ence between the HFLG and LFHG protocols in terms of themedian RMS tracking errors over the entire duration of the


TABLE VIPEAK, MEAN, AND MEDIAN VALUE OF THE CONTROL INPUTS OF THE

TWO STRATEGIES WITH THE FIVE SUBJECTS

trials (see Table I). In Table II, the transient performances ofboth protocols are compared with a Wilcoxon signed-rank teston the paired data resulted in a P-value of 0.10, which onecannot conclude at a significance level of 0.05 that there is asignificant difference between the HFLG and LFHG protocols.Similarly, both protocols resulted in similar steady state RMStracking errors (see Table III). A Wilcoxon signed-rank test onthe paired data during steady state resulted in a P-value of 1.0,further indicating that the LFHG and HFLG protocols result instatistically similar steady state performance, as concluded inRemark 1. The control gains implemented in the experimentsare listed in Tables IV and V, and the peak, mean, and medianvalues of the control efforts are summarized in Table VI.

VII. CONCLUSION

An identification-based closed-loop NMES controller wasdesigned based on an uncertain muscle activation model withan amplitude-modulated control input. Based on Lyapunovstability analysis methods for switched systems, the con-troller is proven to ensure SUUB tracking, provided sufficientconditions on the control gains and stimulation modulationparameters are satisfied. To support the main result of thispaper, experiments on five able-bodied volunteers are imple-mented. The theoretical link between frequency and the controlgains was demonstrated in the sense that, as discussed inRemark 1, higher control gains paired with a low-frequencymodulation strategy yielded similar tracking performance to ahigh-frequency modulation scheme with lower control gains.Future work will seek to extend these results to functionalactivities involving multiple muscle groups and several degreesof freedom (e.g., cycling and walking).

APPENDIX

From [34, eq. (35)], the active musculotendon forcexf (q, q) ∈ R can be modeled as

xf (q, q) = Foη1(q)η2(q)a(t)cos(φ(q))

where adapting some of the notation to that of [25], Fo ∈ R isthe constant, maximum isometric muscle fiber force, η1(q) ∈R denotes the muscle’s force-length relationship, η2(q) ∈R denotes the muscle’s velocity-length relationship, and

a(t) ∈ [0, 1] denotes the muscle activation. In [25], the mus-culotendon force is multiplied by the moment arm ζ(q) ∈ R toyield the active moment. Similarly, in [6], the active momentis defined as Ma = Fm(q, q)a(t), where Fm(q, q) ∈ R

denotes the nonlinear contraction dynamics. Comparing mod-els, Fm(q, q), the contraction dynamics, can be defined as

Fm(q, q) = Foη1(q)η2(q)cos(φ(q))

and multiplied by a(t), the activation function, to yieldxf (q, q). In this paper, the model in (5) was developed bytaking the time derivative of xf (q, q) and substituting the acti-vation dynamics from [35]. Performing these operations andrearranging terms yields the model in (5) where

Af (q, q) � φ(q, q)tan(φ(q)) + k2

f1(q, q, q) � −[η1(q, q)η2(q) + η1(q)η2(q, q)

]

× Foa(t)cos(φ(q))

b(q, q) � [k2 + k1(1 − a(t))]Foη1(q)η2(q) · cos(φ(q)).

ACKNOWLEDGMENT

Any opinions, findings, and conclusions or recommenda-tions expressed in this paper are those of the author(s) and donot necessarily reflect the views of the sponsoring agency.

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Teng-Hu Cheng received the B.Sc. and M.Sc.degrees in mechanical engineering from NationalTaiwan University, Taipei, Taiwan, in 2007 and2009, respectively. He is currently pursuing thedoctoral degree in mechanical engineering withthe Nonlinear Controls and Robotics Laboratory,University of Florida, Gainesville, FL, USA, underthe supervision of Dr. W. E. Dixon.

In 2013 and 2014, he was a Research Internwith the Air Force Research Laboratory, Eglin AFB,Shalimar, FL, USA, and the University of Florida

REEF, Shalimar, FL, aiding in designing cooperative control for networkedsystems with time-varying network topologies and developing event-drivencontrol for networked systems. His current research interests include net-worked system control, switched control, event-driven control, and nonlinearcontrol.

Qiang Wang was born in Hubei, China. He receivedthe bachelor’s degree from the Huazhong Universityof Science and Technology, Wuhan, China, the mas-ter’s and Ph.D. degrees from Peking Union MedicalCollege, Beijing, China, all in biomedical engineer-ing, and the Ph.D. degree in electronic engineeringfrom the University of Florida, Gainesville, FL,USA, in 2012. He is currently pursuing the seconddoctoral degree in electrical engineering with theNonlinear Controls and Robotics Research Group,under the supervision of Dr. W. E. Dixon.

He is currently the Founder and the CEO of Peaceful Mind, LLC,Gainesville, dedicated to develop wearable biomedical devices.

http://ncr.mae.ufl.edu/papers/muscle07.pdf


Rushikesh Kamalapurkar received the B.Tech.degree in mechanical engineering from theVisvesvaraya National Institute of Technology,Nagpur, India, and the M.Sc. and Doctor ofPhilosophy degrees from the Department ofMechanical and Aerospace Engineering, Universityof Florida, Gainesville, FL, USA, in 2014, underthe supervision of Dr. W. E. Dixon.

He was a Design Engineer with Larsen andToubro Ltd., Mumbai, India, for two years. Heis currently a Post-Doctoral Researcher with the

Nonlinear Controls and Robotics Laboratory, University of Florida. Hiscurrent research interests include dynamic programming, optimal control,reinforcement learning, and data-driven adaptive control for uncertainnonlinear dynamical systems.

Huyen T. Dinh received the B.S. degree in mecha-tronics from the Hanoi University of Science andTechnology, Hanoi, Vietnam, in 2006, and theM.Eng. and Ph.D. degrees in mechanical engineer-ing from the University of Florida, Gainesville, FL,USA, in 2010 and 2012, respectively.

She is currently an Assistant Professor with theDepartment of Mechanical Engineering, Universityof Transport and Communications, Hanoi. Hercurrent research interests include development ofLyapunov-based control and applications for uncer-

tain nonlinear systems, learning-based control, and adaptive control foruncertain nonlinear systems.

Matthew Bellman received the bachelor’s andmaster’s (magna cum laude) degrees in mechanicalengineering with a minor in biomechanics from theUniversity of Florida, Gainesville, FL, USA. He iscurrently pursuing the doctoral degree in mechanicalengineering with a continued focus on the theoreticaldevelopment of robust, adaptive control systems forapplications involving functional electrical stimula-tion, specifically those involving rehabilitation andmobility of the lower extremities.

Mr. Bellman has been a member of the NonlinearControls and Robotics Group, University of Florida since 2009. He is aNational Defense Science and Engineering Fellow.

Warren E. Dixon received the Ph.D. degreefrom the Department of Electrical and ComputerEngineering, Clemson University, Clemson, SC,USA, in 2000.

In 2004, he joined the Department of Mechanicaland Aerospace Engineering, University of Florida.He was selected as an Eugene P. WignerFellow at Oak Ridge National Laboratory (ORNL),Oak Ridge, TN, USA. His current research interestsinclude development and application of Lyapunov-based control techniques for uncertain nonlinear

systems. He has published three books, over ten book chapters, and over100 journal and 200 conference papers.

Dr. Dixon was a recipient of the 2015 and 2009 American AutomaticControl Council O. Hugo Schuck (Best Paper) Award, the 2013 Fred EllersickAward for Best Overall MILCOM Paper, the 2012–2013 University of FloridaCollege of Engineering Doctoral Dissertation Mentoring Award, the 2011American Society of Mechanical Engineers Dynamics Systems and ControlDivision Outstanding Young Investigator Award, the 2006 IEEE Roboticsand Automation Society Early Academic Career Award, the NSF CAREERAward from 2006 to 2011, the 2004 U. S. Department of Energy OutstandingMentor Award, and the 2001 ORNL Early Career Award for EngineeringAchievement. He serves (or has served) as an Associate Editor for numerousjournals focused on control systems. He is an IEEE Control Systems Society(CSS) Distinguished Lecturer. He currently serves as a member of the U.S.Air Force Science Advisory Board and the Director of Operations for theExecutive Committee of the IEEE CSS Board of Governors.

ieee transactions on cybernetics, vol. 46, no. 7, july ...ncr.mae.ufl.edu/papers/cyber16.pdfcheng et...

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