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To cite this version:

Benoussaad, Mourad and Rotella, Frédéric and Chaibi, Imen Flatness of musculoskeletal systems under functional electrical stimulation. (2020) Medical & Biological Engineering & Computing, 58 (5). 1113-1126. ISSN 0140-0118

Official URL:

https://doi.org/10.1007/s11517-020-02139-3

Flatness of Musculoskeletal Systems Under Functional

Electrical Stimulation

Mourad BENOUSSAAD∗,∗∗∗·

Frederic ROTELLA∗

· Imen CHAIBI∗∗

Abstract Functional Electrical Stimulation (FES) is an effective technique formovement rehabilitation after spinal cord injury (SCI). One of the main issuesof this technique is the choice of appropriate FES patterns for movement con-trolling which ensure efficiency and less muscular fatigue. To reach those objec-tives, the knowledge of the muscle behavior under FES is mandatory. Therefore,a physiologically-based model is commonly used. Existing musculoskeletal modelsare often complex and highly non-linear, thus the model-based FES patterns syn-thesis and control still remains complex and challenging process. On other hand,the system flatness has been proved to be an efficient method for nonlinear systemcontrol, however it never has been explored for musculoskeletal systems.

The aim of this work is to explore the flatness property of musculoskeletalsystems under FES in dynamic condition for movement control purposes. For thisstudy, the knee joint controlled by electrically stimulated quadriceps muscle wasused.

Results highlights that the two-inputs musculoskeletal system is flat, where twoflat outputs were found. It also shows that the single-input musculoskeletal systemis not flat. These results are crucial for flatness-based control of musculoskeletalsystems since the most used musculoskeletal models in literature deal with only asingle input.

Keywords Movement rehabilitation · Functional Electrical Stimulation (FES) ·

Musculoskeletal modeling · Flatness of nonlinear system

∗ LGP, INP-ENIT, 47 av. d’Azereix, Tarbes, France∗∗University of Sousse, BP 264 Sousse Erriadh 4023, Tunisia∗∗∗Corresponding Author:Tel.: +33 (0)5 67 45 01 11; Fax.: +33 (0)5 62 44 27 08E-mail: mourad.benoussaad@enit.fr

Graphical abstract

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Author biographies

Mourad Benoussaad is an associate professor atEcole Nationale d’Ingenieurs de Tarbes (ENIT; Na-tional Engineering School of Tarbes) since 2014. He re-ceived a Ph.D in Robotics and Control Theory from theUniversity of Montpellier 2, at LIRMM-CNRS-INRIA,in 2009. He was a postdoc researcher at The Univer-sity of Tokyo (JSPS-Japan) in 2010-2011 and then apostdoc researcher at Heidelberg University (Germany)in 2012-2013. In 2013-2014, he joint LIRMM-CNRS-INRIA as a researcher in Robotics Lab. His researchfocus mainly on Physical Human-Robot Interaction(pHRI) in context of collaborative robotics (cobotics)and active exoskeleton. He also works on human motionand control of human musculoskeletal system.

Frederic Rotella received his Ph.D. in 1983 and hisDoctorat d’Etat in 1987 from the University of Sci-ence and Technology of Lille-Flandres-Artois, France.During this period, he served at the Ecole Centrale deLille, France, as assistant professor in automatic con-trol. In 1994, he joined the Ecole Nationale d’Ingenieursde Tarbes, France, as professor of automatic controland mathematics where he is in charge of the Depart-ment of Electrical Engineering. His research fields ofinterest are about control of nonlinear systems andlinear time-varying systems with a particular focuson observation. Professor Rotella has coauthored eightFrench textbooks in automatic control (e.g. Automa-tique elementaire, Hermes-Lavoisier, 2008) or in ap-plied mathematics (e.g. Theorie et pratique du calculmatriciel, Technip, 1995, and, Traitement des systemeslineaires, Ellipses, 2015).

Imen Chaibi is currently a Ph.D candidate in col-laboration between LATIS Lab. (University of Sousse,Tunisia) and LGP Lab. (INP-ENI of Tarbes, France).In 2016, she accomplished an internship of master inLGP. Lab (INP-ENI of Tarbes, France) about musculo-skeletal system modeling and nonlinear control. In2015, she received a master of science in applied com-puter engineering (2I) at ENISo (University of Sousse,Tunisia) and an engineering diplomat in industrial com-puting from the same university.

1 INTRODUCTION

Under spinal cord injury, the natural control of limbs declines and becomes impos-sible. It leads to total or partial paralysis, represented by a paraplegia or a tetraple-gia. Functional Electrical Stimulation (FES) is proving to be a very promisingsolution to ensure movement rehabilitation of the paralyzed limbs by contract-ing skeletal muscles [6,13]. In addition, FES improves paralyzed limbs health bymaintaining a muscular activity. Also it reduces side effects of limbs paralysis suchas atrophy, eschar and cardiovascular problems [9,13]. However, many problemslimits the success of this artificial activation such as the rapid change of musclefibers properties and the lack of muscle behavior knowledge. In these conditions,the choice of FES pattern is often chosen empirically without using any musculo-skeletal model, which leads to a early muscular fatigue. Therefore, an accuratemodeling of the muscle-limb dynamics is needed. Furthermore, it gives insightinto the natural control of musculoskeletal system. A wide variety of muscle mod-els were proposed and used in literature which differ in the intended applicationsuch as the mathematical complexity and the level of physiological structures bytaking into account the biological behavior. However, two main muscle models werecommonly used, highlighting either a macroscopic representation of muscle and itsbehavior [8] or a microscopic muscle behavior, which is based on Huxley’s slidingfilament theory of muscle fibers [10]. In [4], authors worked on modeling of theskeletal muscle system under FES by proposing an original multi-scale model thatcombines the macroscopic [8] and the microscopic [10] dynamic behavior of themuscle. In this work, the authors identified experimentally the model parametersin isometric mode on animals with parameters estimation techniques such as theLevenberg-Marquardt and the Extended Kalman Filter. Based on the identifiedmodel, the next step consists in searching the strategy to generate the adequateFES patterns that restore the movement of the paralyzed limb.Based on this original model [4], several works used it and focused on generat-ing an appropriate FES patterns which allows to restore some basic functions.In [19], authors generated stimulation patterns that synthesized movement byusing both static and dynamic approaches of antagonist muscles. They appliedan input-output linearization and a predictive control strategy on this nonlinearmusculoskeletal model. To achieve the same goal, a high order sliding mode wasalso applied in [18].

In [2] authors achieved a parameter identification of this musculoskeletal modelon real spinal cord injured patients and in [3], they used the identified modelto generate optimal FES patterns which synthesizes a desired movement of theparalyzed leg.

In all these previous works, FES patterns were generated offline or the controlwere applied in simulation. For online FES generation and closed loop control,authors in [16] used a simpler model for a real-time FES control in only isometriccondition (fixed knee-joint angle) based on an electromyography (EMG) feedback[15]. The authors applied the model-based predictive control strategy and theresults showed a promising control performances in a very specific conditions. Inall these previous works, none explored the musculoskeletal system by keeping itsnonlinear property and its physiological complexity in dynamic condition for anonline control context.

Furthermore, flatness-based control strategy proved to be an efficient solutionfor a nonlinear systems control [5]. However, as far as we know, this method has notbe explored in musculoskeletal system. Note that the main obstacle to overcomeis to prove the flatness, or not, of the considered system.

In the present work, we used the musculoskeletal model introduced in [4] in dy-namic condition, toward an online FES patterns generation and closed loop controlof paralyzed limbs. Therefore, the main objective of this study is to explore theflatness property of the musculoskeletal system for a purpose of flatness-based con-trol application. For that, we focus in the current work on the model of knee jointactuated by quadriceps muscle, which model is a combination of both macroscopicand microscopic properties [4].

Thus, in the next section, the whole musculoskeletal model of knee-quadricepswill be introduced and then some physiologically-based assumptions, to reducethe model complexity, are presented and discussed. in Section (2.2), an overviewof flatness is presented, with a short reminder of systems flatness property andits advantage in nonlinear system control. In Section (3.1), we explore the flatnessproperties of musculoskeletal systems by considering both models, with one andwith two inputs. Once flatness is proved, in Section (3.2), we introduce the appli-cation of a reference trajectory motion planning (open-loop control) using flatnessproperties and current work results are summarized and discussed in Section (4).Finally, conclusions and perspectives of this work are presented in Section (5).

2 METHODS

2.1 Musculoskeletal system modeling

In the musculoskeletal model of FES-controlled knee joint, we can distinguish twoparts: the FES-activated muscle model which apply a force (and then a torque) onthe second part of the model which is the knee joint biomechanical model. In thissection, the two parts of the model are detailed separately and the whole musculo-skeletal system, with some simplifying assumptions on the model, are presentedat the end of the section.

2.1.1 Model of muscle under FES

The muscle model under FES used here [4] is based on the integration of the Hux-ley’s microscopic model [10] in the Hill’s macroscopic model [8]. The first model isbased on the physiology of the muscle and the sliding filament theory, which de-scribes the relative sliding of two sets of filaments (actin and myosin), whereas thesecond model is based on the mechanical properties of the muscle and its differentfunctional descriptions.

The model of electrically stimulated muscle acting on skeletal is illustrated byFigure 1. It is well known that electrical stimulation is a pulse train characterizedby a pulse width (PW ), an amplitude (I) and a frequency (f). In this musclemodel, Activation model generates two inputs α and uch, considered as the inputof our system (Fig.1-Top), where α is the fibers recruitment rate that depends onboth PW and I, while uch is the chemical control that depends on f . The muscle

dynamical model is represented with the well-known Hill three elements model(Fig.1-Bottom), where inputs (α, uch) control the contractile element (CE) thatgenerates the muscle force output (F ). This force applied on the joint, makeschange on muscular length and then on CE length (Lc).

[Fig. 1 about here.]

At the level of contractile element, the dynamics is described by the followingset of equations [4]:

KC =

(

s0αK0 − suKC + svqs0αF0KC − suFCKC

1 + pKC − svqFC

)

uch −svaKC

1 + pKC − svqFC

εc

FC =s0αF0 − suFC

1 + pKC − svqFC

uch +bKC − svaFC

1 + pKC − svqFC

εc

(1)Where the state vector is XT =

[KC FC

], with Kc and Fc represent, respectively,

the stiffness and the force of the CE. The control input is UT =[α uch

]. In this

study, we considered these control inputs, since relationships between them andthe FES patterns are static and can be solved furthermore. su and sv are twodiscontinuous Sign function depending on uch and εc, such as: su = sign(uch) andsv = sign(εc), where εc = (Lc − Lc0)/Lc0 is the contractile element deformationand Lc0 is theCE length at the rest condition. Based on the mechanical descriptionof the muscle (Fig.1-Bottom), this CE deformation can be defined as follows [20]:

εc =L0

Lc0ε−

Fc

KsLc0(2)

Where ε represents the deformation at the whole muscle level, such as:

ε =L− L0

L0(3)

With L is the muscle length and L0 its rest condition length.F0 and K0 are maximal force and stiffness of contractile element at its current

length. These variables are governed by a so-called force-length relationship (Flc)as follows [7,22]: F0(εc) = Fm Flc(εc) and K0(εc) = Km Flc(εc), where constantsFm and Km are maximal force and stiffness at optimal muscle length, and theforce-length relationship (Flc) is the one introduced in [7,22]:

Flc(εc) = exp

[

−(εcb

)2]

(4)

Where b is called shape parameter, which describes the filaments overlapping levelin sarcomeres.

In order to have a compact presentation of Eq.(1), an intermediate parametersa, b, p, q and s0 were defined as follows:

a =L0

Lc0; b = L0; p =

1

Ks; q =

1

Lc0Ks; s0 =

1 + su2

Main muscle model parameters with their physiological signification are summa-rized in Table (1).

2.1.2 Knee joint biomechanical model

The knee joint biomechanical model, considered in this study, consists of twosegments representing respectively the shank and the leg (thigh+foot) connectedto each other by one degree of freedom joint (knee). The thigh is supposed fixed,which represents a patient in sitting position, while the shank is free to movearound the knee joint (Fig.2). We consider here the FES-based activation of onlyquadriceps muscle, while other muscles around the knee are considered as havinga passive effects [2]. Therefore, the quadriceps muscle achieves the knee extensionwhile the flexion is made by the leg gravity.

[Fig. 2 about here.]

Based on the pulley model of knee joint, active muscular torques can be eas-ily obtained from the quadriceps forces (Fq) through a constant moment arm r1(Fig.2). θ is the knee joint angle around the rotation center o (θ = 0◦ corre-sponds to full extension of the knee and θ = 90◦ represents the rest position).Tg = m g d cos(θ) is the gravity torque of the leg around the knee. Based on thismodel, the quadriceps muscle length can be expressed as a function of knee jointangle θ:

L(θ) = Lext + r1θ (5)

where Lext is the muscle length at the leg full extension (i.e., θ = 0◦).The leg motion is governed by a nonlinear second order equation, where the

highly nonlinear elastic torque is neglected in the movement range of the kneeextension explored here [26]. This equation is then given by:

Jθ = mgd cos θ − r1Fq − Fv θ (6)

Parameters of this model are summarized (with muscle parameters) in Table 1.

[Table 1 about here.]

2.1.3 Assumptions and model reduction complexity

The musculoskeletal system dynamic (1) is highly complex and nonlinear. Forthis first study of flatness on the musculoskeletal system, we made here somephysiologically based assumptions to reduce the muscle model complexity andexplore the flatness in some specific conditions. These assumptions were based onones used in [19], as summarized and detailed bellow:

1. The stiffness of the series element (SE), which represents the tendon, is higherthan the stiffness of the contractile element as long as the musculoskeletalsystem work in dynamic condition (non isometric). Thus, in Eq.(2), the term

1Lc0Ks

Fc is neglected and the deformation of the whole muscle is almost due tothe deformation of the contractile element. This assumption is physiologicallyjustified by the fact that at peak active muscular force, the relative deformationof tendon (called tendon slack length) is about 3.3% w.r.t its initial length,when the muscle length can have a deformation of about 50% [27].

2. Muscle contractions are concentric, i.e. the active muscle is shortening. Con-sequently, we have:

sv = sign(εc) = −1

3. According to the model improvement introduced in [4], the chemical controlfunction uch is positive in all conditions (activation and relaxation phases ofcontractile element). Therefore, we obtain:

su = 1, s0 = 1

2.1.4 Quadriceps-leg Muscloskeletal Model

In the current work, we are in the context of non-isolated muscle, i.e. the muscle isa part of a whole musculoskeletal system. In this case, the parallel element (PE)effect (Fig.1-Bottom) is considered at the knee joint level, as all passive musclesaround the joint [2]. Therefore, the quadriceps muscle force (Fq), applied to movethe leg, becomes equal to the force generated by the contractile element Fc:

Fq = Fc

Similarly, based on previous assumptions about the high level of the series elementstiffness (Ks) compared to the contractile element stiffness (Kc), we can alsoconclude that the quadriceps muscle stiffness (Kq) is mainly due to CE stiffness:

Kq = Kc

By combining models at muscle and knee joint levels, the relative deformation ofthe quadriceps muscle and its derivative can be obtained from Eq.(3) and Eq.(5)as:

ε(θ) =r1L0

θ, ε(θ) =r1L0

θ (7)

From Eq.(2), (7) and assumptions above-mentioned (§2.1.3), we can obtain therelative deformation of muscle contractile element and its derivative as a functionof the joint states:

εc(θ) =r1Lc0

θ, εc(θ) =r1Lc0

θ (8)

On other hand, since F0 and K0 are dependent of muscle length, thorough aforce-length relationship, it is a function of knee joint angle θ. We can then write:F0(θ) and K0(θ). The quadriceps-leg musculoskeletal model is a combination ofthe previously detailed muscle model (§2.1.1) and the current biomechanical kneejoint model (§2.1.2). Thus, by defining the state vector XT =

[

Kq Fq θ θ]=

[x1 x2 x3 x4

]and the control input UT =

[α uch

], we obtain the following state

space system equations from Eqs. (1), (6) and (8):

x1 =

(

αK0(x3)− x1 − qαF0(x3)x1 − x2x1

1 + px1 + qx2

)

uch +ax1

1 + px1 + qx2

r1Lc0

x4

x2 =αF0(x3)− x2

1 + px1 + qx2uch +

bx1 + ax2

1 + px1 + qx2

r1Lc0

x4

x3 = x4

x4 =1

J(mgd cos(x3)− r1x2 − Fvx4)

(9)

From this set of equations, we can notice that the term 1 + px1 + qx2 does notvanish since parameters p, q and states x1, x2 are all positive based on theirphysical meanings.

For a easier use of the model further, we define new parameters as:

c1 =r1Lc0

; c2 = −r1J; c3 =

mgd

J; c4 = −

Fv

Jand two new inputs as:

{v = α uch

u = uch(10)

Then, the model can be written as:

x1 =

(

K0(x3)− qF0(x3)x1

1 + px1 + qx2

)

v −

(

x1 − qx2x1

1 + px1 + qx2

)

u+ac1x1x4

1 + px1 + qx2

x2 =F0(x3)

1 + px1 + qx2v −

x2

1 + px1 + qx2u+

(bx1 + ax2)c1x4

1 + px1 + qx2

x3 = x4

x4 = c2 x2 + c3 cos(x3) + c4 x4

(11)

2.2 Brief overview on Flatness

2.2.1 Definition

Since the introduction of differential flat systems theory and the flatness-basedcontrol, proposed by Fliess and al.[5], several efficient solutions for advanced con-trol and state estimation problems were developed and provided. In this section,a brief description of flatness theory is presented.

Definition 1 Let’s consider a nonlinear system of the form

x = f(x, u), x ∈ Rn, u ∈ R

m (12)

It is differentially flat if and only if there exists a set of variables z(t) ∈ Rm of the

form:z = h(x, u, u, · · · , u(α)) (13)

such that there exist:x = A(z, z, · · · , z(β)) (14)

u = B(z, z, · · · , z(β+1)) (15)

where α and β are integers. The set of variables z are the flat outputs of the system,called also the endogenous variables. It makes possible to parameterize any variableof the system (components of the system, states, inputs and the outputs). We cannotice that the components of z must be differentially independent [1,5].

The real output of the process can be written:

y = g(x, u) (16)

Thus, from Eqs.(14) and (15), the real output y is written as a function of the flatoutput z and its derivatives as:

y = C(z, z, · · · , z(β+1)) (17)

We point out a few important details:

– The dimension of the flat outputs is equal to the system input dimension.– There is an infinity of flat outputs. In other words, the set of found flat outputs

is not unique.– We can often find flat outputs that have a physical meaning.

2.2.2 Flatness-based control

The objective of the flatness-based control is to obtain the asymptotic tracking ofa reference trajectory and this is ensured through the following steps [23] :

– Motion planning (open loop): by imposing desired trajectories on flat out-puts, we can obtain exactly and explicitly the necessary control to generatethem, without any integration of differential equations. These desired trajec-tories zd must be (β + 1)-times continuously differentiable.

– Motion tracking (closed loop): For a desired trajectories tracking, thecontrol v is defined as follows:

v = z(β+1)d −

β∑

i=0

ki(z(i)

− z(i)d ) (18)

with a good choice of the ki, namely the polynomial k(p) = pl+1 +∑β

i=0 kipi,

where kipi is Hurwitz; The trajectory can be stabilized and errors go asymp-

totically to zero. The complete control is then written as follows :

u = B(z, · · · , z(β), v) (19)

This relation leads to the asymptotic tracking of desired trajectories.Notice that the information needed by this control can be obtained throughobservers [24] with a major advantage, compared to other nonlinear controlstrategies, related to the fact that it overcomes problems of a non stable ze-ros dynamics [11], [21]. Moreover, the relationship of Eq.(17) leads to get theoutput trajectory from the desired flat output trajectory zd. Notice that ifthis output trajectory is not admissible, the key is then to design a piecewisetrajectory where some conditions for smoothness are verified on the cuttingpoints. However, this relationship (Eq.(17)) is still remain valid between thesepoints.

The power of flatness is that it does not transform a nonlinear system into alinear one. On one hand, when system is flat, it is an indication that its nonlin-ear structure is well characterized and it can be exploitable in designing controlalgorithms for trajectory generation and stabilization. On the other hand, flatnessrepresents, in a way, a proper geometric notion of linearity even when the systemis nonlinear in any chosen coordinates [5]. Finally, the principle of flatness can beextended to distributed parameters systems with boundary control and is usefuleven for controlling linear systems.

As mentioned before, the application of flatness-based control requires to findthe flat output of the system. However, when no flat output can be found, itdoesn’t mean that the system is not flat. Furthermore, to demonstrate that agiven system is not flat, the ruled-manifold criterion can be applied, as detailedin the next section.

2.2.3 The ruled manifold criterion

This criterion represents only a necessary condition of flatness. It means when thesystem is flat, it does satisfy the criterion. Therefore, the negation of the criterion

highlights the not flatness of the system [25,17]. To introduce the ruled manifoldcriterion, let’s consider the nonlinear system of the form:

x = f(x,u); x ∈ Rn, u ∈ R

m (20)

for which the elimination of the control input leads to a system of (n−m) relations:

F (x, x) = 0 (21)

If this system is flat, then we can show that [17]:

∀(λ, µ) ∈ R2n, such that F (λ, µ) = 0 (22)

there exists v ∈ Rn, v 6= 0, such that:

∀e ∈ R, F (λ, µ+ ev) = 0 (23)

This result shows the fact that the subvariety F (λ, µ) = 0 is regulated withrespect to the second parameter. By taking the negation of this mathematicalstatement, we obtain that if this subvariety is not regulated then the system is notflat, which is stated in the form of the following criterion.

Theoreme 1 If:

(∀(λ, µ) ∈ R2n, F (λ, µ) = 0, ∀e ∈ R, F (λ, µ+ ev) = 0) =⇒ v = 0 (24)

Then the system is not flat.

Therefore, this criterion can not be used to prove the flatness of a given system,but its no flatness.

3 RESULTS

3.1 Flatness of musculoskeletal systems

As mentioned before, the system flatness proof requires to find the system flatoutputs. One intuitive way consists on exploring flat output candidates and prove(or not) that they are flat outputs. However, if no flat output is found, it doesn’tmean that the system is not flat. In this case, the ruled manifold criterion, definedin 2.2.3, is another approach that explore the system flatness property by using anecessary condition of flatness.

In order to study the flatness of the musculoskeletal system, we consider firstour model (Eqs.11) with a single input and then with both controlled inputs.

3.1.1 Model with a single input

Since in many existing FES-based rehabilitation strategies, model with a singleinput is commonly used, let’s explore in this section the flatness of the currentmodel with only one output. From the previously detailed model (Eqs.11) andbased on previous assumptions (§2.1.3), we consider the model with only onecontrol input U = [v], while the second input u is regarded as a constant. Thismeans that in the FES input, the frequency is fixed and not modulated duringmovement control. The system state vector is still the same defined before, i.e.XT=

[x1 x2 x3 x4

]=[

Kq Fq θ θ].

After exploring few intuitive flat output candidates, we couldn’t prove theexistence of flat output. However, it doesn’t prove that the system is not flat.Therefore, instead of proving the flatness of this one-input model, we will exploreits no-flatness through the ruled manifold criterion. The ruled manifold criterionwas applied on one-input model as detailed in Appendix (A). Therefore, the state-ment of Eq.(24) is demonstrated, which highlights the no flatness of the one-inputmusculoskeletal model.

3.1.2 Model with two inputs

Since the one-input musculoskeletal model is proved to be not flat, let’s considerthe original musculoskeletal two-inputs model (Eqs.(11))

For a easier further manipulation, two first equations can be presented as:

[x1

x2

]

=

[ac1x1x4

(1+px1+qx2)(bx1+ax2)c1x4

1+px1+qx2

]

︸ ︷︷ ︸

f(x)

+

[(

K0(x3)− q F0(x3)x1

1+px1+qx2

)

−(

x1 − q x2x1

1+px1+qx2

)

F0(x3)1+px1+qx2

− x2

1+px1+qx2

]

︸ ︷︷ ︸

G(x)

[vu

]

(25)

and two last equations as:

{x3 = x4

x4 = c2 x2 + c3 cos(x3) + c4 x4

(26a)

(26b)

Since in this system dim(U) = 2, therefore, the model admits two flat output

candidates

(z1z2

)

(i.e., dim(z) = 2). Let’s consider following flat output candi-

dates:

(z1, z2) = (x1, x3) = (Kq, θ) (27)

Notice immediately that these outputs have a physical interpretation: z1 is themuscle stiffness and z2 is the knee joint angle.

Using the flatness property proof described above (§2.2.1), let’s explore thesechosen candidates are flat outputs. For that, we try to express each state andcontrol input as a function of flat outputs and their derivatives.

– Equation (26a) leads to write:

x4 = z2 (28)

– From equation (26b) and by replacing x3 and x4 by chosen flat output candi-dates we get:

z2 = c2x2 + c3 cos(z2) + c4z2

which leads to :

x2 =1

c2(−c3 cos(z2)− c4z2 + z2) (29)

We can see that each state was expressed as a function of flat outputs candidates.Afterwards, we have to do the same for control inputs, by expressing each inputas a function of flat outputs, by solving Eq.(25). The condition to do it, is that thematrix G(x) is invertible. Then, let’s study this condition through the determinantof G(x):

detG(x) =

∣∣∣∣∣

(

K0(x3)− q F0(x3)x1

1+px1+qx2

)

−(

x1 − q x2x1

1+px1+qx2

)

F0(x3)1+px1+qx2

− x2

1+px1+qx2

∣∣∣∣∣

=F0(x3)x1 −K0(x3)x2

1 + px1 + qx2(30)

If by controlling the trajectories of x1 and x2, we ensure that the conditionF0(x3)x1 6= K0(x3)x2 is satisfied, thus the matrix G(x) is invertible and thesolution of Eq.(25) leads to express input controls as follows:

[vu

]

= G−1(x)

{[x1

x2

]

− f(x)

}

= 1+px1+qx2

F0(x3)x1−K0(x3)x2

[

− x2

1+px1+qx2

x1 − q x2x1

1+px1+qx2

−F0(x3)

1+px1+qx2

K0(x3)− q F0(x3)x1

1+px1+qx2

][

x1 −ac1x1x4

1+px1+qx2

x2 −(bx1+ax2)c1x4

1+px1+qx2

] (31)

Where, based on Eqs.(27), (29), we have:

x1 = z1

x2 =1

c2(c3z2 sin(z2)− c4z2 +

...z 2)

(32)

we can see in equation (31) that all inputs of musculoskeletal system are expressedas function of states and their derivatives x1, x2. On other hands, all these statesand their derivatives are expressed as function of chosen flat outputs and theirderivatives. Therefore, inputs are function of chosen flat outputs and their deriva-tives as well. Therefore, we proved that the two-inputs system is flat since weexpressed all its variables (states and inputs) as a function of both flat output (z1,z2) and their derivatives, and we obtained thus functions h, A and B of equations(13), (14), (15).

As mentioned before (§2.2.2), the flatness properties of system serves to applya flatness-based control, which can be done in open-loop (motion planning) orclosed-loop (motion tracking). The purpose of this article is mainly to explorethe flatness properties of musculoskeletal systems, however the motion planingprinciple and its application are explored and detailed in next section, while theclosed loop control is planned in further works.

3.2 motion planning

3.2.1 Flatness-based open-loop control

To show the main features of system flatness property, we focus first on the motionplanning of of system flat outputs. Since the musculoskeletal system with two in-puts is proved to be flat, by defining a desired trajectory of flat output, we obtainexactly and explicitly the necessary control for it, as mentioned in §(2.2.2). Thechoice of these trajectories can be a problem if the flat outputs have no physicalmeaning. In our case, both flat outputs have physical meaning, however, the def-inition of some trajectories (such as a stiffness) can be an issue if we don’t haveany measurement of it in practice. To explore the flatness advantage in control-ling the musculoskeletal system, we have to define desired trajectories (and theirderivatives) of both flat outputs: zd(t) = (zd1(t), z

d2(t)) and then, the corresponding

inputs are obtained as follows (Eq.31):

zd(t) → zd(t) → zd(t) →...z d(t)

︸ ︷︷ ︸

vd(t), (ud(t))

Based on equations which relates flat outputs with states, their derivatives andinputs (Eqs.(31), (27), (28), (29), (32)), the choice of desired trajectories shouldsatisfy following conditions:

– zd1 : 1-time continuously derivable– zd2 : 3-times continuously derivable

3.2.2 Trajectory generation

To satisfy previous derivability conditions, we propose here a polynomial functionsas desired trajectories of flat outputs. For simplicity reason, we considered the twopolynomial trajectories with the same order, which connect an initial to a finalconditions of flat outputs. On other hand, as it is commonly used in robotics andmotion planing, we defined a polynomial trajectory of 5th order as follows [12].

{

zd = zid + (zfd − zid)r(t)

r(t) = a3(

tT

)3+ a4

(tT

)4+ a5

(tT

)5 (33)

With t is the time variable (0 ≤ t ≤ T ) and T is the whole movement duration.

zd, zid and zfd are respectively the flat desired trajectory, its initial and its final

values.The parameters (a3, a4, a5) are determined by considering the boundary condi-tions. These conditions are initial and final states of desired flat trajectory, whichare defined for each of both flat outputs. To set parameters (a3, a4, a5) for eachpolynomial desired trajectory, we defined first and second derivatives of flat de-sired outputs, at the start and end of movement, as follows:

{

zid = zfd = [0, 0]T

zid = zfd = [0, 0]T

We can thus obtain the function r of Eq.(33) as follows [12]:

r(t) = 10

(t

T

)3

− 15

(t

T

)4

+ 6

(t

T

)5

(34)

3.2.3 Open-loop control application

The open-loop flatness based control is described by the following scheme:

[Fig. 3 about here.]

As we can see in figure 3, from chosen initial and final flat outputs and accord-ing to Eq.(33), trajectories (zd1 , z

d2) and their obtained derivative are generated

for the whole movement. Then, an open-loop controller, based on Eq.(31), gener-ates inputs (vd, ud) throughout the movement, which are needed to track desiredtrajectories. These inputs are then applied directly on musculoskeletal system. Insimulation, the obtained output trajectories are (zs1, z

s2).

In ideal case of system (model in simulation) the obtained (zs1, zs2) fit per-

fectly with the desired flat outputs trajectories (zd1 , zd2). However, in a real system

with modeling errors and external perturbation, a deviation should exist betweendesired and real flat outputs. In this case, a closed loop flatness-based control isneeded (§2.2.2).

[Fig. 4 about here.]

[Fig. 5 about here.]

4 DISCUSSION

Since the flatness of the two-input musculoskeletal depends on the inversion ofG(x) matrix (Eq.(31)) and the condition F0(x3)x1 6= K0(x3)x2, the choice ofdesired flat output trajectories should satisfy that condition throughout the wholemovement. Since we proved the flatness and established a relationship betweenflat outputs and system states (Eq.(15), (14)), we can thus control any state andmake it flowing a predefined trajectory.

Control inputs, obtained thanks to system flatness property and applied inopen loop control strategy, generates a flat outputs that matches perfectly with adesired flat trajectories in ideal case, with a perfect model (simulation). However,in a real case, there should be a deviation, related to model and parameters errorsand perturbation, which define limits and drawbacks of open-loop control. In thiscase, the motion tracking which corresponds to a flatness-based closed-loop controlis required (§2.2.2). Test and comparison in simulation of open-loop and closed-loop control will be explored in further works.

In the current work, chosen flat outputs corresponds to a meaningful states,such as knee joint angle and muscle stiffness. In this case, defining initial and finalvalues of these flat outputs were intuitive and presents an obvious link with systemstates and outputs. However, in general case, it is initial and final values of states(X) and inputs (U) which should be given, and then initial and final values of flatoutputs can be obtained using Eq.(13). Therefore, a trajectory of flat output canbe generated and used to control the system.

Since the mostly used musculoskeletal models have only one input, or thesecond input is ignored and fixed to a constant value, results of our study arecrucial and establish that for these type of model, the flatness-based control can

not be applied. In addition, these results are very interesting since the flatness-based control is comparable with the natural muscular control which, deal withtwo inputs: recruitment rate (spatial summation) and chemical control (temporalsummation) [2,4].

As specified in overview of flatness (§2.2), the flatness of a system and partic-ularly a nonlinear system offers several advantage for controlling it. Indeed, one ofthe most important issues in FES-based rehabilitation is to synthesize and controlFES patterns to generate a desired movement. In most of previous works, non-linear optimization have been applied to inverse numerically the musculoskeletalmodel [3] with some drawback related to calculation time consuming and localminimum issues of optimization. Therefore, the application of flatness propertyleads to an analytic inversion of model without linearizing it.

In the current work, we focus on proving flatness of musculoskeletal systemto use for control through inputs α and uch (Eq.(10)). However, these inputsare not the ones used in practice since we applied a stimulation patterns, whichfrequency influences uch and amplitude/pulse width influences α. As mentionedbefore (§2.1.1), the amplitude/pulse width can be deduced from recruitment rate αthanks to a recruitment function [2], however it seems more complex to get directlythe stimulation frequency from the chemical control uch since the estimated one,using the flatness-based control, have not been constrained in terms of signalform. This relationship and impact of uch signal form constraints will be exploredin future works.

Since we demonstrated the flatness of a given system, there is an infinity of flatoutputs. The choice of flat outputs for the two-input musculoskeletal system wasmade intuitively, however we demonstrated that they satisfy the output flatnesscriteria. More generally, the construction procedure of flat outputs is divided intotwo steps. The first step consist on the construction of an implicit model. The sec-ond step is to express certain variables algebraically in function of other variablesand their derivatives. To go further, readers can see [5,14], which we will explorein future works for a general flat output construction of musculoskeletal systems.In the current work, we made some assumptions based on a specific case, wherequadriceps is always shortening during contraction without any co-contractionphenomena. A more general cases should be considered in future works.

5 CONCLUSION

In this study, the flatness of electrically stimulated musculoskeletal system wereexplored for the first time, as far as we know. This flatness property is a firststep for flatness-based control of such a nonlinear system, which is an importantissue in FES-based rehabilitation. As it is known in previous works about flatness,this property is very crucial to control a nonlinear system without linearizing itnor using an optimization procedure which presents some known drawback. In-deed, this property allows to generate required inputs simply by defining a de-sired flat outputs trajectories. For the current study, a multi-scale muscular modelwith two inputs (recruitment rate and chemical control), based on combinationof macroscopic Hill’s model and microscopic Huxley’s muscle fibers model. Themuscloskeletal model use for the study is a knee joint actuated by a quadricepsmuscle in sitting position. In the current work, we demonstrated that the musculo-

skeletal model with only one input is not flat, while we proved that the two-inputsmusculoskeletal model is flat since we found one set of two flat outputs. This isan important results since most of musculoskeletal models under FES consideronly one controlled input while the natural muscular control deal with two inputs,which represents spatial and temporal summation of muscle fibers. The flatness-based open-loop control principle is presented, which can be enough for a modelas close as possible with real system. Simulation test of flatness-based open-loopand closed loop control is planned in future works

A Ruled Manifold criterion of one-input musculoskeletal model

Considering only one input (α) and uch as a constant, we can define the input (Eq.(10)):

v = αuch

and parameters,

uc = uch; c1 =r1

Lc0; c2 = −

r1

J; c3 =

mgd

J; c4 = −

Fv

J

Therefore, the one-input musculoskeletal model can described by equations (Eqs.(11)), byconsidering uc as a constant:

x1 =

(

K0(x3)− qF0(x3)x1

1 + px1 + qx2

)

v −

(

x1 − qx2x1

1 + px1 + qx2

)

uc

+ac1x1x4

(1 + px1 + qx2)

x2 =F0(x3)

1 + px1 + qx2v −

x2

1 + px1 + qx2uc +

(bx1 + ax2)c1x4

1 + px1 + qx2

x3 = x4

x4 = c2 x2 + c3 cos(x3) + c4 x4

(35a)

(35b)

(35c)

(35d)

(35e)

The first stage of ruled manifold criterion consists in expressing the implicit form (Eq. 21)of the model by eliminating the input control v from Eqs. (35b) and (35c). Let’s extract theinput v from Eq.(35c) as follows:

(1 + px1 + qx2) x2 = F0(x3) v − x2 uc + (bx1 + ax2) c1 x4 (36)

⇔ v =(1+px1+qx2) x2+x2 uc−(bx1+ax2) c1 x4

F0(x3)(37)

From Eq.(35b) we get:

(1 + px1 + qx2) x1 = ac1x1x4 − (x1 (1 + px1 + qx2) + qx2x1)uc

+ (K0(x3)(1 + px1 + qx2)− qF0(x3)x1) v (38)

Then, by replacing v in this equation we obtain an equation without control input:

(1 + px1 + qx2) F0(x3)x1 = aF0(x3)c1x1x4 − (x1 (1 + px1 + qx2) + qx2x1)F0(x3)uc

+ (K0(x3)(1 + px1 + qx2)− qF0(x3)x1) ((1 + px1 + qx2) x2 + x2 uc − (bx1 + ax2)c1x4)

Based on this input elimination and Eqs.(35d), (35e), we define the implicit form of the one-input model as F (x, x) = 0:

(1 + px1 + qx2) F0(x3)x1 + (x1 (1 + px1 + qx2) + qx2x1)F0(x3)uc

− (K0(x3)(1 + px1 + qx2)− qF0(x3)x1) ((1 + px1 + qx2) x2 + x2 uc − (bx1 + ax2) c1 x4)

−aF0(x3)c1x1x4 = 0

x3 − x4 = 0

x4 + c2 x2 + c3 cos(x3) + c4 x4 = 0

(39)

Let’s then explore a necessary condition of flatness by using the negation of ruled manifold

criterion (Eq.(24)). Then we have:∀(λ, µ) ∈ R

2n

where λ = (λ1, λ2, λ3, λ4) and µ = (µ1, µ2, µ3, µ4).Considering, from the implicit form of the model (Eqs.(39)), the following set of equation,

which corresponds to F (λ, µ) = 0:

(1 + pλ1 + qλ2) F0(λ3)µ1 + (λ1 (1 + pλ1 + qλ2) + qλ2λ1)F0(λ3)uc

− (K0(λ3)(1 + pλ1 + qλ2)− qF0(λ3)λ1) ((1 + pλ1 + qλ2) µ2 + λ2 uc − (bλ1 + aλ2) c1 λ4)

−aF0(λ3)c1λ1λ4 = 0

µ3 − λ4 = 0

µ4 + c2 λ2 + c3 cos(λ3) + c4 λ4 = 0

(40)Then, ∀e ∈ R, let’s consider the statement F (λ, µ+ ew) = 0, where w = (w1, w2, w3, w4).

⇒

(1 + pλ1 + qλ2) F0(λ3)(µ1 + ew1) + (λ1 (1 + pλ1 + qλ2) + qλ2λ1)F0(λ3)uc

− (K0(λ3)(1 + pλ1 + qλ2)− qF0(λ3)λ1) ((1 + pλ1 + qλ2) (µ2 + ew2) + λ2 uc − (bλ1 + aλ2) c1 λ4)

−aF0(λ3)c1λ1λ4 = 0

µ3 + ew3 − λ4 = 0

µ4 + ew4 + c2 λ2 + c3 cos(λ3) + c4 λ4 = 0

(41)By taking into account that F (λ, µ) = 0 (Eq.40), we obtain:

(1 + pλ1 + qλ2) F0(λ3)ew1 − (K0(λ3)(1 + pλ1 + qλ2)− qF0(λ3)λ1)(1 + pλ1 + qλ2) ew2 = 0

ew3 = 0

ew4 = 0

(42)

⇒

eF0(λ3)w1 = ew2(K0(λ3)(1+pλ1+qλ2)−qF0(λ3)λ1)(1+pλ1+qλ2)

1+pλ1+qλ2= ew2K(λ1, λ2)

ew3 = 0

ew4 = 0

(43)

Thus, for all λ1, λ2, λ3, λ4, the only possible values of w that satisfy this last statement are(w1, w2, w3, w4) = (0, 0, 0, 0).

Since w = (w1, w2, w3, w4) are proved to be equal to zero in order to respect the ruledmanifold criterion negation (Eq.(24)), the musculoskeletal system with a single input is thusproved to be not flat.

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List of Figures

1 Physiological model of electrically stimulated musculoskeletal system 212 Biomechanical pulley model (2D) of the knee joint . . . . . . . . . 223 Flatness-based open-loop control scheme. From a predefined initial

and final flat outputs (z...), their desired trajectories were generatedzd.... The required inputs were calculated using the flatness prop-erties, and applied on the system to get the simulated flat outputzs1... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Desired and simulated flat outputs in open-loop control (Fig.3)where the simulation system is considered ideal w.r.t the model.Flat outputs are the muscle stiffness (left) and the knee joint angle(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Desired and simulated flat outputs in open-loop control (Fig.3),where the simulation system is different from the model (variationof 10% in ks parameter value). Flat outputs are the muscle stiffness(left) and the knee joint angle (right). . . . . . . . . . . . . . . . . 25

FIGURES

SE

PE

L

cL s

L

uch

,α

CE

Activation model

Fibre Recruitment Muscle

dynamical

model

Dynamical

skeletal

model

Muscular force

Muscular length

Muscle model

FES patterns

PW, I

f

PW

I

T=1/f

Calcium dynamics

α

uch

Joint angle

,α

SESS

F

CL

Fig. 1 Electrically stimulated muscle Model. Top: Overview of muscle model. The stimulatedmuscle applies a force (F ) on the skeletal system. The joint position has a feedback on themuscle length and then on the contractile element length Lc; Bottom: Focus on muscle dy-namical model. It is based on Hill three elements model and includes the contractile elementCE, controlled by the recruitment rate α and the chemical control uch, the serial element SE(with a stiffness Ks) and the parallel element PE;

FIGURES

Thigh

Quadriceps

Passives muscles θO

r1

Fq

FES d

mg

Fig. 2 Biomechanical pulley model (2D) of the knee joint. Only quadriceps muscle is activatedwith FES, whereas the other inactivated muscles around knee joint are considered with apassive effects.

FIGURES

!"#$%&'(")*

+%,%"#'-(,**

./01.2233*

45%,67((5*

&(,'"(7**

./01.289*2:33*

*

;<=&<7(=>%7%'#7*

=)='%?**

./0=1*.8833*

Fig. 3 Flatness-based open-loop control scheme. From a predefined initial and final flat out-puts (z...), their desired trajectories were generated zd.... The required inputs were calculatedusing the flatness properties, and applied on the system to get the simulated flat output zs1...

FIGURES

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [sec]

0

200

400

600

800

1000

Sti

ffn

es

s [

N/m

]

Desired

Simulated

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [sec]

40

50

60

70

80

90

An

gle

s [

de

g]

Desired

Simulated

Fig. 4 Desired and simulated flat outputs in open-loop control (Fig.3) where the simulationsystem is considered ideal w.r.t the model. Flat outputs are the muscle stiffness (left) and theknee joint angle (right).

FIGURES

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time [sec]

0

200

400

600

800

1000

Sti

ffn

es

s [

N/m

]

Desired

Simulated

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [sec]

40

45

50

55

60

65

70

75

80

85

90

An

gle

s [

de

g]

Desired

Simulated

Fig. 5 Desired and simulated flat outputs in open-loop control (Fig.3), where the simulationsystem is different from the model (variation of 10% in ks parameter value). Flat outputs arethe muscle stiffness (left) and the knee joint angle (right).

FIGURES

List of Tables

1 Musculoskeletal model parameters . . . . . . . . . . . . . . . . . . 27

TABLES

Parameters Physiological and biome-

chanical definition

Km Maximum stiffness of the mus-cle

Fm Maximum force of the muscleKs Stiffness of the serial element

(SE)Lc0 The contractile element length

at the rest conditionFv Viscosity parameter at knee

joint leveld Distance between the knee cen-

ter and leg’s center of massm Mass of the leg (shank+foot)J Moment inertia of the system

shank+foot around knee jointg Gravity acceleration

Table 1 Musculoskeletal model parameters