Identification of Gait-Cycle Phases for Prosthesis Control

biomimetics

Article Identiﬁcation of Gait-Cycle Phases for Prosthesis Control

Raffaele Di Gregorio * and Lucas Vocenas

LaMaViP, Department of Engineering, University of Ferrara, 44122 Ferrara, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-0532974828

Abstract: The major problem with transfemoral prostheses is their capacity to compensate for the loss of the knee joint. The identification of gait-cycle phases plays an important role in the control of these prostheses. Such control is completely up to the patient in passive prostheses or partly facilitated by the prosthesis in semiactive prostheses. In both cases, the patient recovers his/her walking ability through a suitable rehabilitation procedure that aims at recreating proprioception in the patient. Understanding proprioception passes through the identification of conditions and parameters that make the patient aware of lower-limb body segments’ postures, and the recognition of the current gait-cycle phase/period is the first step of this awareness. Here, a proposal is presented for the identification of the gait-cycle phases/periods under different walking conditions together with a control logic for a possible active/semiactive prosthesis. The proposal is based on the detection of different gait-cycle events as well as on different walking conditions through a load sensor, which is implemented by analyzing the variations in some gait parameters. The validation of the proposed method is done by using gait-cycle data present in the literature. The proposal assumes the prosthesis is equipped with an energy-storing foot without mobility.

Keywords: above-knee amputee; prosthesis; gait cycle; proprioception; lower-limb control

Citation: Di Gregorio, R.; Vocenas, L. Identification of Gait-Cycle Phases for 1. Introduction Prosthesis Control. Biomimetics 2021, Human locomotion is an efficient biomechanical process. A healthy individual can 6, 22. https://doi.org/10.3390/ travel long distances with low energy consumption. Despite the progress in prosthetic biomimetics6020022 design, the replacement of lower-limb segments with a prosthesis affects the efficiency of this locomotion. The purpose of a lower-limb prosthesis is to minimize the impact of the Academic Editor: Josep Samitier amputation and make the patient somehow autonomous again. That is why prosthesis technology mainly tries to mimic the joint behavior of human lower limbs during walking. Received: 7 March 2021 The study of asymptomatic walking, therefore, presents the basis for thinking about Accepted: 22 March 2021 Published: 26 March 2021 the development of prosthetic components [1]. It seems indeed judicious to have maximum data on the movement, which one seeks to reproduce. Unfortunately, although walking

Publisher’s Note: MDPI stays neutral seems relatively simple to healthy people, since it does not require any concentration, it is with regard to jurisdictional claims in an extremely complex process to model, involving many mechanisms. The complexity of published maps and institutional affil- this modeling explains the large number of works dedicated to it (see [2,3] for References). iations. Numerous studies carried out over the years have laid the foundations for the var- ious techniques of the current analysis. There are many different walking models, but none of them are completely satisfactory, and the research on optimal modeling is still in progress [2–15]. Each observation technique of human walking has limits, and many are the external factors that influence the measurements, which explains the disparity of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. results present in the literature [16]. This article is an open access article Each person has a specific gait. However, a pattern common to all individuals is distributed under the terms and identifiable: the walking (or gait) cycle (Figure1). Indeed, walking is substantially a conditions of the Creative Commons repetitive activity; thus, its analysis refers to the motion cycle repeated during walking. Attribution (CC BY) license (https:// Two successive impacts on the ground of the same heel delimit one cycle. The analysis of creativecommons.org/licenses/by/ the gait cycle can consider only one (named “ipsilateral limb”) out of the two lower limbs, 4.0/). which is usually the right leg. These aspects are independent of individual characteristics,

Biomimetics 2021, 6, 22. https://doi.org/10.3390/biomimetics6020022 https://www.mdpi.com/journal/biomimetics Biomimetics 2021, 6, 22 2 of 15

successive impacts on the ground of the same heel delimit one cycle. The analysis of the gait cycle can consider only one (named “ipsilateral limb”) out of the two lower limbs, which is usually the right leg. These aspects are independent of individual characteristics, whereas stride, step length, step width, step angle, and cadence (see [17,18] for definitions) vary according to the physical characteristics of individuals and, for the same individual, according to his/her physiological conditions. There are two different phases during the gait cycle [17,18]: the stance phase and the swing phase (Figure 1). The stance phase, which corresponds to 60% of the cycle in the “normal” gait1, occurs when the foot of the ipsilateral limb is in contact with the ground. It begins with the “initial contact (IC)” event, which is when the heel touches the ground, and ends with the “toe-off (TO)” event, which is

Biomimetics 2021, 6, 22 when the toe is lifted. During this phase, the body weight is transferred2 of 16from the rear leg to the front leg. The swing phase, which corresponds to 40% of the cycle in the “normal” gait, occurs when the foot does not touch the ground and the leg oscillates. It begins with whereasthe TO stride, event step and length, ends step with width, the stepnext angle, IC event. and cadence (see [17,18] for deﬁnitions) vary accordingIf both tothe the limbs physical are characteristics considered of [18], individuals the stance and, for phase the same can individual, be further divided into accordingthree periods: to his/her two physiological “double-support” conditions. periods There are (i.e two., periods different in phases which during both the the feet touch the gait cycle [17,18]: the stance phase and the swing phase (Figure1). The stance phase, which correspondsground), one to 60% at ofthe the beginning cycle in the “normal”(initial double gait1, occurs support) when the and foot the of the other ipsilateral at the end (terminal limbdouble is in contactsupport), with and the ground. a third, It begins“single-limb with the “initialsupport” contact period (IC)” event,for the which remaining is part. The whenextension the heel in touches the percentage the ground, andof the ends cycle with theof “toe-offthese three (TO)” event,periods which depends is when on the walking thespeed. toe is lifted.In addition, During thisit is phase, worth the mentioning body weight isa transferredfiner repartition from the rear[19], leg which to the refers to seven front leg. The swing phase, which corresponds to 40% of the cycle in the “normal” gait, occursevents when that the make foot doesit possible not touch to theidentify ground four and theperiods leg oscillates. in the stance It begins phase with the and three periods TOin eventthe swing and ends phase with (Figure the next IC1). event.

FigureFigure 1. Gait-cycle1. Gait-cycle phases phases (reproduced (reproduced with permission with permission from [19]). from [19]).

IfThe both observation the limbs are consideredof the three-dimensional [18], the stance phase kinematics can be further of the divided gait cycle into can relate the three periods: two “double-support” periods (i.e., periods in which both the feet touch the ground),joint angles one at of the the beginning knee and (initial ankle double to a support) particular and thepercentage other at the of end gait-cycle (terminal completion [20]. doubleConsequently, support), and an aattempt third, “single-limb of obtaining support” the post periodures for of the lower-limb remaining part.body The segments by iden- extensiontifying the in thegait-cycle percentage phases/periods of the cycle of thesemay threesucceed. periods The depends same observation on the walking reveals that, dur- speed.ing the In gait addition, cycle, it isthe worth position mentioning of the a center ﬁner repartition of mass[ 19oscillates], which refersboth tovertically seven and laterally events that make it possible to identify four periods in the stance phase and three periods in[19]. the swing phase (Figure1). TheThe observation dynamic of analysis the three-dimensional of the gait cycle kinematics characterizes of the gait the cycle force can relatesystems the both externally jointand angles internally of the knee(i.e., and in the ankle joints) to a particular acting percentageon the lower of gait-cycle limbs. completionThe dynamic [20]. equilibrium of Consequently,the pedestrian an attemptcan be stated of obtaining by saying the postures that the of lower-limbground reaction body segments forces (GRFs), by which the identifyingground applies the gait-cycle to the phases/periodsfeet, must equilibrate may succeed. the force The same system observation consisting reveals of the body weight that, during the gait cycle, the position of the center of mass oscillates both vertically and laterallyand the [19 inertia]. forces. Therefore, measuring the three GRF components (i.e., vertical force, anterior–posteriorThe dynamic analysis shear, of the and gait cyclemedial–lateral characterizes theshear) force systemsprovides both relevant externally pieces of infor- andmation internally on the (i.e., gait in the cycle joints) phase/period. acting on the lowerThe GR limbs.F is The non-null dynamic only equilibrium during ofthe stance phase. theDuring pedestrian this canphase be stated [19,21,22], by saying in the that normal the ground gait: reaction forces (GRFs), which the ground applies to the feet, must equilibrate the force system consisting of the body weight and- theThe inertia vertical forces. force Therefore, has measuringtwo peaks the of three about GRF 120% components of the (i.e., body vertical weight force, (BW) that occur anterior–posteriorapproximately shear, andat the medial–lateral end of the shear) initial provides double-support relevant pieces period of information (i.e., 10% of gait-cycle on the gait cycle phase/period. The GRF is non-null only during the stance phase. During this phase [19,21,22], in the normal gait: 1 According to ([19], p. 632), the reference data for “normal” walking are 1.37 m/s for the walking speed, 1.87 steps/s (110 steps/min) - The vertical force has two peaks of about 120% of the body weight (BW) that occur of step rate (cadence), and 0.72 m of stepapproximately length. at the end of the initial double-support period (i.e., 10% of gait-cycle completion (Figure1)) and earlier than the beginning of the terminal double-support

1 According to ([19], p. 632), the reference data for “normal” walking are 1.37 m/s for the walking speed, 1.87 steps/s (110 steps/min) of step rate (cadence), and 0.72 m of step length. Biomimetics 2021, 6, 22 3 of 16

period (i.e., 45–50% of gait-cycle completion (Figure1)) with a minimum of about 80% BW in the middle (i.e., 30% of gait-cycle completion (Figure1)); - The anterior–posterior shear has two peaks with opposite signs of about 20% BW, which approximately occur when the vertical force has two peaks and vanish in the middle of the stance phase (i.e., 30% of gait-cycle completion (Figure1)); - The medial–lateral shear is much smaller than the other two components. It has no sharp peaks, and it is comprised in the range [−5, +5]%BW These results and GRF diagrams reported in the literature [1,17–19,21,22] make it possible to relate a signal obtained by measuring the GRF components to the kinematic data of the ankle and knee joints to recognize the current gait-cycle phase/period. Active/semiactive lower-limb prostheses have been extensively studied in the last two decades [23–33], and up to 21 [32] or 26 [31] types of active prostheses, according to the adopted classification criterion, have been counted in recent reviews, among which 12 are for above-knee amputees. Three different types of control systems have been adopted for them: echo control, gait-mode control, and, recently, direct myoelectric control. Echo control aims at mimicking the motion of the healthy limb. It is applicable only to unilateral amputees and needs the introduction of sensors both on the healthy limb and on the prosthesis; it is not able to reproduce asymmetric walking. Gait-mode control tries to recognize the current gait-cycle phase/period by means of a number of sensors inserted into the prosthesis and adapts the prosthesis behavior to the current gait-cycle phase/period by using software based on artificial-intelligence algorithms. Direct myoelectric control interprets the signals coming from the contraction of the stump muscles of the patient to control the prosthesis behavior. It is a novel approach aiming at making the patient able to generate prosthesis commands with his/her brain. At the moment, the gait-mode control guarantees better performances [32], and it is implemented into commercial active/semiactive knee prostheses [33]. This paper presents a method for the detection of the gait-cycle events on different gait conditions through a load sensor and its use for controlling an active/semiactive prosthetic knee. The method is based on the analysis of the variations of a number of gait parameters. The experimental data on the asymptomatic gait reported in the literature are exploited to establish the control model. Only the behavior of a prosthetic knee equipped with an energy-storing foot without mobility is considered in this work. Differently from other gait-mode prosthesis control, the proposed method allows devising a control logic based only on real-time GRF measurements and on deterministic computations. This approach reduces the prosthesis hardware and simplifies the control software, thus moving toward low-cost prescriptions and increase of potential users in developing countries. The proposal is applicable to any type of prosthetic-knee architecture. The paper is organized as follows. Section2 illustrates the proposed method. Section3 reports the results, and Section4 discusses them. Finally, Section5 draws the conclusions.

2. Materials and Methods The values reported in [18] for the GRF vertical component (Figure2) and those reported in [34] for the knee flexion–extension (Figure3) as functions of the gait-cycle percentage have been used to simulate datasets measured on a specific patient during normal walking. Two continuous curves, representable through algebraic equations, which give the gait parameters as a function of the gait-cycle percentage, have been generated by means of polynomial regressions on these datasets. In particular, these data have been imported into MATLAB, and a polynomial regression with the least-square method has been implemented on each monotonic part of the two data sets, thus obtaining two piecewise polynomials (splines). For the knee-flexion angle, four monotonic parts are BiomimeticsBiomimetics 20212021, ,66, ,22 22 4 4of of 15 16 Biomimetics 2021, 6, 22 4 of 15

wisewiseidentifiable polynomials polynomials2, and (splines). the(splines). best fittingFor For the the was knee-flexion knee-flexion obtained withangle, angle, a four 3-3-4-3 four monotonic monotonic spline3 (Figure parts parts are 4area). identifi- identifi- For the 2 3 ableableGRF,2 ,and verticaland the the component,best best fitting fitting was fourwas obtained monotonicobtained with with parts a a 3-3-4-3 are3-3-4-3 identifiable spline spline 3 (Figure4(Figure, and the 4a). 4a). best For For fitting the the GRF GRF was 4 verticalverticalobtained component, component, with a 3-3-4-5 four four splinemonotonic monotonic (Figure parts parts4b). are Inare theidentifiable identifiable stance phase,,4 ,and and thesethe the best best two fitting fitting fitting was was curves, ob- ob- tainedtainedwhich with with both a a have3-3-4-5 3-3-4-5 the spline spline gait-cycle (Figure (Figure percentage 4b). 4b). In In the the on stance stance the abscissa, phase, phase, these canthese be two two combined fitting fitting curves, curves, into a which unique which bothbothspatial have have curve the the gait-cycle (Figuregait-cycle5) percentage topercentage state a one-to-one on on the the abscissa, abscissa, correspondence can can be be combined combined between into into any a a unique twounique variables spatial spatial curvecurveamong (Figure (Figure the knee-flexion 5) 5) to to state state a angle,a one-to-one one-to-one GRF vertical correspondence correspondence component, between between and gait-cycle any any two two percentage. variables variables among among Such a thethecurve knee-flexion knee-flexion makes a possibleangle, angle, GRF GRF control vertical vertical system componen componen able tot, roughlyt, and and gait-cycle gait-cycle determine percentage. percentage. the knee-flexion Such Such a a curve curve angle makesmakesand the a a possible gait-cyclepossible control control percentage system system by able able measuring to to roughly roughly only determine determine the GRF the verticalthe knee-flexion knee-flexion component. angle angle The and and same the the gait-cyclegait-cycleGRF measure, percentage percentage if timed, by by gives measuring measuring the gait only cadence.only the the ThisGRFGRF procedurevertical vertical component. component. can be directly The The implemented same same GRF GRF measure,measure,on datasets if if timed, timed, measured gives gives onthe the agait gait specific cadence. cadence. patient; Th Thisis procedure then, procedure the results can can be be candirectly directly be used implemented implemented to adjust on theon datasetsdatasetsparameters measured measured of the on prosthesis on a a specific specific control patient; patient; system. then then, ,the the results results can can be be used used to to adjust adjust the the param- param- eterseters of of the the prosthesis prosthesis control control system. system.

FigureFigure 2 2. 2. .Ground GroundGround reaction reaction reaction force force force (GRF) (GRF) (GRF) vertical vertical vertical component component component in in in percentage percentage percentage of ofof body body body weight weight weight (BW) (BW) (BW) as asas a a a functionfunctionfunction of of of the the gait-cycle gait-cycle percentage percentage obta obta obtainedinedined from from the the dataset dataset reported reported in inin [18]. [[18].18].

Figure 3. Diagram of the knee-flexion angle as a function of the gait-cycle percentage obtained Figure 3 3.. DiagramDiagram of the knee-flexion knee-ﬂexion angle as a functionfunction of thethe gait-cyclegait-cycle percentagepercentage obtainedobtained from from the dataset reported in [34]. fromthe dataset the dataset reported reported in [34 in]. [34].

2 2 The The first first between between the the IC IC event event and and the the 1s 1st tpeak peak occurrence, occurrence, the the second second between between the the 1st 1st peak peak occurrence occurrence and and the the central central minimum minimum, , thethe third third between between the the central central minimum minimum and and th the e2nd 2nd peak peak occurrence, occurrence, and and the the fourth fourth betw betweeneen the the 2nd 2nd peak peak occurrence occurrence and and the the next next 2 ICIC event. event.The ﬁrst between the IC event and the 1st peak occurrence, the second between the 1st peak occurrence and the central minimum, the third between 3 the central minimum and the 2nd peak occurrence, and the fourth between the 2nd peak occurrence and the next IC event. 3 The numbers separated by the hyphens indicate the degrees of the polynomials that compose the spline. 3 The numbers separated by the hyphens indicate the degrees of the polynomials that compose the spline. 4 The numbers separated by the hyphens indicate the degrees of the polynomials that compose the spline. 4 The first between the IC event and the 1st peak occurrence, the second between the 1st peak occurrence and the central minimum, 4 The first between the IC event and the 1st peak occurrence, the second between the 1st peak occurrence and the central minimum, the thirdThe ﬁrst between between the the central IC event minimum and the 1st and peak the occurrence, 2nd peak the occurrenc second betweene, and the the 1st fourth peak occurrence between andthe the2nd central peak minimum, occurrence the and third the between TO thethe third central between minimum the andcentral the 2ndminimum peak occurrence, and the and2nd thepeak fourth occurrenc betweene, theand 2nd the peak fourth occurrence between and the the 2nd TO event.peak occurrence and the TO event.event.

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Figure 4. Curve fitting: (a) knee-flexion angle and (b) GRF vertical component. Figure 4. Curve fitting: ﬁtting: ( a)) knee-flexion knee-ﬂexion angle and ( b) GRF vertical component.

Figure 5. Spatial curve relating knee-flexion angle, GRF vertical component, and gait-cycle percentage. Figure 5. Spatial curve relating knee-flexion angle, GRF vertical component, and gait-cycle per- Figure 5. Spatial curve relating knee-flexion angle, GRF vertical component, and gait-cycle percentage. centage.2.1. Detection of Gait-Cycle Events The superposition of the two above-deduced fitting curves (Figure6) reveals that: 2.1. Detection of Gait-Cycle Events 2.1.- DetectionA first flexion of Gait-Cycle of the knee Events starts at the IC event (1st event), which is easily identified by The superposition of the two above-deduced fitting curves (Figure 6) reveals that: Thethe transitionsuperposition of the of GRFthe two vertical above-deduced component fitting from zero curves to a (Figure positive 6) value, reveals increases that: - A first flexion of the knee starts at the IC event (1st event), which is easily identified - Awith first the flexion GRF of increase the knee and starts reaches at the its IC maximum event (1st aevent), bit earlier which than is easily the 1st identified peak of by the transition of the GRF vertical component from zero to a positive value, in- bythe the GRF transition vertical componentof the GRF (2ndvertical event). comp Thisonent is becausefrom zero the to knee a positive flexion somehowvalue, increases with the GRF increase and reaches its maximum a bit earlier than the 1st peak creasescompensates with the for GRF the shockincrease of and the suddenreaches appearanceits maximum of a a bit non-null earlier than GRF the and 1st gives peak a of the GRF vertical component (2nd event). This is because the knee flexion somehow ofsmooth the GRF transition vertical fromcomponent the swing (2nd phase event). to This the stance is because phase. the knee flexion somehow - compensatesThe first knee for flexion the shock is followed of the sudden by a complete appearance knee of extension a non-null that GRF has and its gives middle a compensates for the shock of the sudden appearance of a non-null GRF and gives a smoothconfiguration transition at the from minimum the swing of thephase GRF to vertical the stance component phase. (3rd event) and approx- smooth transition from the swing phase to the stance phase. - Theimately first terminatesknee flexion when is followed the GRF by vertical a complete component knee reachesextension its that 2nd has GRF its peak middle (4th - The first knee flexion is followed by a complete knee extension that has its middle configurationevent), which at is atthe the minimum beginning of ofthe the GRF second vertical double-support component period.(3rd event) and ap- configuration at the minimum of the GRF vertical component (3rd event) and ap- - proximatelyA second knee terminates flexion then when starts, the GRF which vertical accompanies component the decrease reaches ofits the 2nd GRF GRF vertical peak proximately terminates when the GRF vertical component reaches its 2nd GRF peak (4thcomponent event), which until theis at TO the event beginning (5th event), of the second easilyidentified double-support by the period. transition of the (4th event), which is at the beginning of the second double-support period. - AGRF second vertical knee component flexion then from starts, a positive which value accompanies to zero, and the continuesdecrease of during the GRF the swingverti- - A second knee flexion then starts, which accompanies the decrease of the GRF verti- calphase component until the until reaching the TO of aevent maximum (5th event), flexion easily angle identified that occurs by nearlythe transition at the middle of the cal component until the TO event (5th event), easily identified by the transition of the GRFof the vertical swing phasecomponent (6th event). from a positive value to zero, and continues during the GRF vertical component from a positive value to zero, and continues during the - swingThe second phase kneeuntil flexion the reaching is followed of a maximum by a second flexion knee extensionangle that that occurs terminates nearly at at the the swing phase until the reaching of a maximum flexion angle that occurs nearly at the middlenext IC of event the swing (1st event phase of (6th the nextevent). cycle). middle of the swing phase (6th event). - The second knee flexion is followed by a second knee extension that terminates at the - The second knee flexion is followed by a second knee extension that terminates at the next IC event (1st event of the next cycle). next IC event (1st event of the next cycle).

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Figure 6. Identification of the gait-cycle events: superposition of the two fitting curves (i.e., knee- Figure 6. Identification of the gait-cycle events: superposition of the two fitting curves (i.e., knee- flexion angle and GRF vertical component as functions of the gait-cycle percentage). flexion angle and GRF vertical component as functions of the gait-cycle percentage). Figure 6. Identification of the gait-cycle events: superposition of the two fitting curves (i.e., knee- TheThe above above analysis analysis highlights highlights that that six six events events must must be be detected detected to to monitor monitor and and control control flexion angle and GRF vertical component as functions of the gait-cycle percentage). thethe configuration configuration of of a a possible possible prosthesis prosthesis and and that that only only the the 6th 6th cannot cannot be be detected detected through through the GRF measure. The 6th event, which is the reaching of the maximum knee flexion dur- the GRF measure.The above 6th event, analysis which highlights is the that reaching six events of must the be maximum detected to knee monitor flexion and control duringing the the swing swing thephase, phase, configuration could could be be ofdetected/controlled a detected/controlled possible prosthesis throughand through that onlyan anadjustable the adjustable 6th cannot limit limitbe switchdetected switch in- through insertedserted in in the the prosthesis’ prosthesis’the GRF measure. knee knee joint joint The that6th event, limits which the maximummaximum is the reaching kneeknee of flexionflexion the maximum according according knee to toflexion the the dur- cadencecadence [ 34[34]] in ining order order the to swingto mimic mimic phase, the the asymptomaticcould asymptomatic be detected/controlled walking. walking. through an adjustable limit switch in- TableTable1 summarizes1 summarizesserted in the the the prosthesis’ reference reference knee events events joint to thatto use use limits in thein the prosthesis maximumprosthesis control, knee control, flexion and and Figureaccording Figure7 to the shows7 shows the the control controlcadence scheme scheme [34] forin fororder the the prosthesis to prosthesis mimic the with asymptomaticwith a finite-statea finite-state walking. machine machine and and a a sequential sequential functionalfunctional chart. chart. Table 1 summarizes the reference events to use in the prosthesis control, and Figure 7 shows the control scheme for the prosthesis with a finite-state machine and a sequential TableTable 1. 1.Detection Detectionfunctional of of gait-cycle gait-cycle chart. events events based based on on GRF-vertical-component GRF-vertical-component measurements measurements and and the the knee-flexionknee-flexion limit limit switch. switch. Table 1. Detection of gait-cycle events based on GRF-vertical-component measurements and the knee-flexion limit switch. DetectedDetected Condition EventEvent Gait-CycleGait-Cycle Percentage Percentage (t(t = = Time Time Instant)Instant)Detected Condition (Number/Name)(Number/Name)Event Gait-Cycle(%)(%) Percentage GRF(t − dt) = 0 & GRF(t) > 0 1/IC 0 GRF(t − dt) = 0 & GRF(t)(t = Time > 0 Instant) 1/IC(Number/Name) 0 (%) GRFGRF (t (t − dt)dt) < < GRF(t) GRF(t)GRF(t >> GRF(t−GRF(t dt) =+ 0+ dt) &dt) GRF(t) > 0 2/1st2/1st GRF GRF peak 1/IC 15 15 0 GRF(tGRF(t − dt)dt) > > GRF(t) GRF(t)GRF (t −<< dt) GRF(tGRF(t < GRF(t) + + dt) dt) > GRF(t + dt) 3/heel 3/heel rise 2/1st GRF peak 32 32 15 GRF(tGRF(t − dt)dt) < < GRF(t) GRF(t)GRF(t − >> dt) GRF(tGRF(t > GRF(t) + + dt) dt) < GRF(t + 4/2nd4/2nd dt) GRF GRF peak 3/heel rise 47 47 32 GRF(t) > 0 & GRF(t + dt) = 0 5/TO 64 GRF(t) > 0 &GRF(t GRF(t − dt) + dt)< GRF(t) = 0 > GRF(t + dt) 5/TO 4/2nd GRF peak 64 47 Limit SwitchGRF(t) Reached > 0 & GRF(t + dt) = 6/max 0 knee flexion 5/TO 78 64 Limit Switch Reached 6/max knee flexion 78 Limit Switch Reached 6/max knee flexion 78

Figure 7.Figure Modeling 7. ofModeling the prosthesis of the control prosthesis scheme: control (a) finite-state scheme: machine (a) ﬁnite-state and (b) sequential machine functional and (b chart) sequential (St: stance; Figure 7. ModelingSw: of theswing; prosthesisfunctional Fl: flexure; control chart Ex: extension). (St: scheme: stance; (a Sw:) finite-state swing; Fl: machine ﬂexure; and Ex: ( extension).b) sequential functional chart (St: stance; Sw: swing; Fl: flexure; Ex: extension).

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2.2. Detection of the Longitudinal Slope The datasets used in the above analysis refer to the normal gait on a flat surface. When the surface has a positive (uphill walking)/negative (downhill walking) longitudinal slope, the patterns both of the GRF vertical component and of the knee-flexion angle vary with the slope [1,35]. Consequently, the prosthesis control has to detect the slope change and, accordingly, has to change the prosthesis behavior. The literature [1,35] shows that: - In uphill walking, the GRF vertical component still has two peaks, but the 2nd peak is higher than the 1st, and the difference increases with the slope, whereas the knee- flexion angle still has two flexions and two extensions, but the knee flexion at the IC increases with the slope; - In downhill walking, the GRF vertical component still has two peaks, but the 1st peak is higher than the 2nd, and the difference increases with the slope, whereas the knee-flexion angle keeps the same IC value, but all the intermediate values are amplified with an amplification factor that increases with the slope; - Both in uphill and in downhill walking the minimum value between the two peaks does not change appreciably. These observations bring to choice the difference, ∆F, between the maximum GRF value at the 2nd peak and the maximum GRF value at the 1st peak as a reference variable for detecting the slope. Indeed, ∆F will be positive with a positive slope (uphill walking) and negative with a negative slope (downhill walking). Therefore, the two additional events reported in Table2 can be added, and the finite-state machine of the prosthesis control system becomes that reported in Figure8. With reference to Figure8, once either of Events 7 or 8 is detected, the reference curves (knee-flexion angle, ankle-joint torque, etc.) the prosthesis control system uses to control the prosthesis behavior are changed by taking the data from a memorized database, where these curves are recorded through the coefficients of the polynomial regressions computed during the prosthesis calibration. The modified reference curves are then used after the next event 1 (IC event). Of course, all the calibration values that modify the knee-flexion fitting curves according to the slope must be adjusted on the patient during his/her rehabilitation.

Table 2. Detection of the longitudinal slope based on GRF-vertical-component measurements.

Detected Condition Event Slope Variation (i = Cycle Index) (Number) (Sign)

∆Fi > ∆Fi+1 7 Negative ∆Fi+1 > ∆Fi 8 Positive

2.3. Effects of Cadence The measure of the GRF, if timed, provides the gait cadence. Indeed, since the time interval, ∆t1, between two successive IC events corresponds to two steps (i.e., one stride), the cadence is 2/∆t1 steps/s, and the average value of the cadence on a number, n, of strides is 2n/∆tn, where ∆tn is the time interval between the ﬁrst and the (n + 1)th IC events. All the digital control systems have a clock that synchronizes its actions on the controlled system; consequently, the assumption that the cadence measurement is included in the GRF measure is plausible and does not imply the addition of further sensors. Biomimetics 2021, 6, 22 8 of 16 Biomimetics 2021, 6, 22 8 of 15

Figure 8. Finite-state-machine modeling of the pr prosthesisosthesis control scheme that takes into account the longitudinal slope of the walking.

Consequently,Cadence influences by introducing all the kinetic/kinematic the following calibration parameters procedure: of the gait cycle [34,36–38]. In particular,(i) The “normal” in the GRF walking vertical cadence components, of the the pa 1sttient peak is determined increases, and, as the simultaneously, one at which the minimum between the two peaks decreases as the cadence increases, while the 2nd the two GRF peaks are about equal; peak substantially does not change; additionally, in the knee-flexion angle, the maximum (ii) The reference peak difference ΔF0, the maximum knee-flexion angle, etc. at each flexion angle increases with the cadence increase. Eventually, the gait-cycle percentage cadence are determined (e.g., through a regression on a finite number of covered by the stance phase decreases with the increase of the cadence. measures on the patient); Consequently, by introducing the following calibration procedure: (iii) The parameter ΔF that appears in Table 2 for identifying a possible slope is rede- (i) Thefined “normal” as follows walking cadence of the patient is determined as the one at which the two GRF peaks are about equal; (ii) The reference peak difference ∆FΔ0F, the= ΔF maximum0 − ΔFm knee-flexion angle, etc. at each(1) cadence are determined (e.g., through a regression on a finite number of measures on where ΔFm is the peak difference measured in real time during walking; the prosthesis the patient); control system can have the measured cadence, as a primary reference parameter, and the (iii) The parameter ∆F that appears in Table2 for identifying a possible slope is redefined longitudinal slope, as a secondary reference parameter, for identifying which patterns for as follows the knee-flexion angle must be used in the prosthesis control without changing the finite- state machine of Figure 8. ∆F = ∆F0 − ∆Fm (1)

2.4.where Control∆Fm Algorithmis the peak difference measured in real time during walking; the prosthesis controlThe system above procedure can have the (Figures measured 7 and cadence, 8) for ma asking a primary the prosthesis reference control parameter, system able and tothe identify longitudinal the current slope, asconfiguration a secondary of reference the lower parameter, limb during for identifying walking is whichbased patternson real- timefor the GRF knee-flexion measures and angle on must a database be used (memoriz in the prosthesised in the controlcontroller) without containing changing a set the of referencefinite-state data/curves machine of obtained Figure8. by calibrating some prosthesis parameters through direct measurements on the patient during his/her rehabilitation training. These data bring to 2.4. Control Algorithm delineate the following control algorithm StepThe above1: The procedurecadence is (Figures measured7 and and,8) forfrom making the database, the prosthesis the values control of Δ systemF0 to use able in Equationto identify (1) the and current of all the configuration other reference of the parameters lower limb (e.g., during the maximum walking is knee based flexion on real- to usetime for GRF adjusting measures knee’s and limit on a switch) database depending (memorized on the in thecadence controller) are determined; containing a set of referenceStep 2: data/curves The output obtained data of Step by calibrating 1 and the Δ someF obtained prosthesis from parameters Equation (1) through are used direct for selecting,measurements from the on thedatabase, patient the during reference his/her knee-flexion-angle rehabilitation training. curve to These use for data identifying bring to thedelineate current the knee following configuration control as algorithm. a function of the real-time GRF measure (Figures 7 and 8); Step 1: The cadence is measured and, from the database, the values of ∆F0 to use in EquationStep (1)3: The and control of all the system other referenceuses the meas parametersured GRF (e.g., and the the maximum selected knee knee-flection- flexion to use for adjusting knee’s limit switch) depending on the cadence are determined; angle curve as input data of a simple program. This program solves the inverse dynamics of the lower limb with a prosthesis for determining the torque that the active or semiactive actuation system of the prosthesis has to apply to the prosthetic knee for making the limb mimic the selected knee-flection-angle curve.

Biomimetics 2021, 6, 22 9 of 16

Step 2: The output data of Step 1 and the ∆F obtained from Equation (1) are used for selecting, from the database, the reference knee-flexion-angle curve to use for identifying the current knee configuration as a function of the real-time GRF measure (Figures7 and8); Step 3: The control system uses the measured GRF and the selected knee-flection-angle Biomimetics 2021, 6, 22 9 of 15 curve as input data of a simple program. This program solves the inverse dynamics of the lower limb with a prosthesis for determining the torque that the active or semiactive actuation system of the prosthesis has to apply to the prosthetic knee for making the limb mimicIt the is worth selected noting knee-flection-angle that all the GRF curve. components can be measured through load cells insertedIt is worthin the noting prosthetic that allankle the of GRF the components prosthesis, canwhich be measuredalso provides through the loadankle-joint cells insertedtorque. in the prosthetic ankle of the prosthesis, which also provides the ankle-joint torque.

3.3. Results Results ThisThis section section presents presents the the validation validation results results of the of above-deﬁnedthe above-defined control control logic obtainedlogic ob- throughtained through an ad-hoc an simulation ad-hoc simulation program developedprogram developed in MATLAB, in MATLAB, which resorts which to the resorts planar to modelthe planar of the model lower of limb the lower with alimb prosthesis with a shownprosthesis inFigure shown9 .in In Figure these 9. simulations, In these simula- the databasetions, the measured database measured on the patient on the is patient replaced is byreplaced some publishedby some published datasets ofdatasets the APSIC of the projectAPSIC [1project,23,39]. [1,23,39]. Figures 10Figures–12 show 10–12 the show continuous the continuous curves, reconstructed curves, reconstructed by means ofby polynomialmeans of polynomial regressions regressions from the APSIC from datasets,the APSIC for datasets, level ground, for level slope ground, +12%, slope and slope+12%, −and12% slope of the –12% GRF of anterior–posterior the GRF anterior–posterior and vertical and components, vertical components, of the prosthetic of the ankle prosthetic [23], adoptedankle [23], in theadopted model in of the Figure model9, andof Figure of the 9, knee and ﬂexion, of the knee respectively. flexion, respectively.

FigureFigure 9. 9.Planar Planar model model of of the the lower lower limb limb with with a a prosthesis prosthesis (a (a = = b b = =0.5 0.5m). m).

TheThe chosen chosen datasets datasets refer refer to to walking walking speeds speeds that that are are common common in in daily-life daily-life activities. activities. TheseThese walking walking speeds speeds are are not not high; high; consequently, consequently, the the inertia inertia forces forces are are much much lower lower than than thethe other other loads loads and, and, in in the the solution solution of of the the inverse inverse dynamics dynamics problem, problem, will will be be neglected. neglected. AssumingAssuming the the “absence” “absence” of inertiaof inertia forces forces in the in prosthetic the prosthetic foot equilibrium foot equilibrium equations equations makes themakes force the system force that system the footthat appliesthe foot to applies the shin to through the shin the through ankle equivalentthe ankle equivalent to GRF; that to is,GRF; the resultantthat is, the force resultant of this force system of isthis equal system to the is GRF, equal and to itsthe resultant GRF, and moment, its resultant which mo- is equalment, to which the ankle is equal torque, to the NA ankle, arises torque, from theNA, misalignmentarises from the between misalignment these two between forces. these So, intwo the forces. model So, of in Figure the model9, point of AFigure is indeed 9, point the A rotation is indeed center the rotation of the ankle, center the of shapethe ankle, of thethe foot shape does of notthe affectfoot does the computation,not affect the computation, and the components and the Y componentsA and XA are Y theA and vertical XA are andthe thevertical anterior–posterior and the anterior–posterior components ofcomp theonents GRF, respectively. of the GRF, Therespectively. force equilibrium The force ofequilibrium the whole limbof the without whole limb inertia without forces inertia also reveals forces thatalso thereveals components that the components YH and XH ofYH theand force XH of applied the force to applied the hip to must the behip equal must andbe equal opposite and toopposite YA and to X YAA (i.e.,and X YAH (i.e.,= − YHA = −YA and XH = −XA) and the internal torque of the hip, NH, is computable from the moment equilibrium equation about H of the whole limb; that is:

NH + NA − YA[b cos(α1 + α2) + a cos(α1)] + XA[b sin(α1 + α2) + a sin(α1)] = 0 (2) which gives

NH = YA[b cos(α1 + α2) + a cos(α1)] − NA − XA[b sin(α1 + α2) + a sin(α1)] (3)

BiomimeticsBiomimetics 20212021,, 66,, 2222 1010 ofof 1515

Biomimetics 2021, 6, 22 10 of 16

H TheThe torquetorque NNH isis appliedapplied byby thethe patientpatient duringduring wawalking.lking. FigureFigure 1313 showsshows thethe valuesvalues H A A A 1 2 ofof NNH computedcomputed throughthrough FormulaFormula (3),(3), wherewhere thethe valuesvalues ofof XXA,, YYA,, NNA,, αα1 andand αα2 areare thosethose and X = −X ) and the internal torque of the hip, N , is computable from the moment reportedreportedH inin thetheA datasetsdatasets displayeddisplayed inin FiguresFigures 10–1210–12 (Suplementary(SuplementaryH Data).Data). equilibrium equation about H of the wholeK limb; that is: Eventually,Eventually, thethe internalinternal torque,torque, MMK,, ofof thethe knee,knee, whichwhich thethe prosthesisprosthesis actuationactuation sys-sys- temtem mustmust provide,provide, isis computablecomputable fromfrom thethe momentmoment equilibriumequilibrium eqequationuation aboutabout pointpoint KK NH + NA − YA[b cos(α1 + α2) + a cos(α1)] + XA[b sin(α1 + α2) + a sin(α1)] = 0 (2) ofof thethe prostheticprosthetic shank,shank, thatthat is:is:

which gives K A A 1 A 1 MMK ++ NNA −− YYA aa cos(cos(αα1)) ++ XXA aa sin(sin(αα1)) == 00 (4) (4)

whichwhich givesgivesNH = YA[b cos(α1 + α2) + a cos(α1)] − NA − XA[b sin(α1 + α2) + a sin(α1)] (3)

MK = YA a cos(α1) − NA − XA a sin(α1) (5) The torque NH is appliedMK by = Y theA a patientcos(α1) during− NA − X walking.A a sin(α Figure1) 13 shows the values(5) of N computed through Formula (3),K where the values of X ,Y ,N , α and α are those HFigureFigure 1414 showsshows thethe valuesvalues ofof MMK computedcomputed throughthrough FormulaAFormulaA A (5),(5), 1 wherewhere the2the valuesvalues reportedA A in theA datasets 1 displayed in Figures 10–12 (Supplementary Data). ofof XXA,, YYA,, NNA,, andand αα 1 areare thosethose reportedreported inin thethe datasetsdatasets displayeddisplayed inin FiguresFigures 1010 andand 11.11.

Figure 10. a FigureFigure 10.10. GRFGRF componentscomponents as as a a function function of of the the gait-cycle gait-cycle pe percentagepercentagercentage for for three three different different longitudinal longitudinal slopes: slopes: ( (aa)) anterior- anterior- posteriorposterior component,component, andand ((bb)) verticalvertical component.component.component.

FigureFigureFigure 11.11. ProstheticProsthetic 11. Prosthetic ankleankle ankle data:data: data: ((aa) ()a dorsi-plantardorsi-plantar) dorsi-plantar flexion ﬂexionflexion angle angleangle and andand (b ()(bb ankle-joint)) ankle-jointankle-joint torque. torque.torque.

BiomimeticsBiomimeticsBiomimetics 20212021 2021, ,6,6 ,6 ,22, 22 22 111111 of ofof 15 1615

FigureFigureFigure 12 12.12. Knee-flexion. Knee-ﬂexionKnee-flexion angle angleangle for forfor three threethree different differentdifferent longitudinal longitudinallongitudinal slopes slopesslopes of ofof the thethe walkway. walkway.walkway.

Figure 13. Internal torque, NH, of the hip that the patient must apply during the stance phase to FigureFigure 13.13. InternalInternal torque,torque, N NH,, ofof thethe hiphip that that the the patient patient must must apply apply during during the the stance stance phase phase to to walk walk with the prosthetic kneeH on walkways with three different longitudinal slopes. withwalk thewith prosthetic the prosthetic knee onknee walkways on walkways with three with differentthree different longitudinal longitudinal slopes. slopes.

Eventually, the internal torque, MK, of the knee, which the prosthesis actuation system must provide, is computable from the moment equilibrium equation about point K of the prosthetic shank, that is:

MK + NA − YA a cos(α1) + XA a sin(α1) = 0 (4)

which gives MK = YA a cos(α1) − NA − XA a sin(α1) (5)

Figure 14 shows the values of MK computed through Formula (5), where the values of XA,YA,NA, and α 1 are those reported in the datasets displayed in Figures 10 and 11.

Figure 14. Internal torque, MK, of the knee that the prosthesis actuation system must provide for Figure 14. Internal torque, MK, of the knee that the prosthesis actuation system must provide for makingmaking the the prosthetic prosthetic knee knee mimic mimic the the asymptomatic asymptomatic motion motion on on walkways walkways with with three three different different lon- lon- gitudinalgitudinal slopes. slopes.

4.4. Discussion Discussion TheThe above-reported above-reported results results prove prove that, that, for for wa walkinglking speeds speeds of of daily-life daily-life activities, activities, it it is is possiblepossible to to extract extract the the pieces pieces of of information information ne necessarycessary to to control control an an active active or or semiactive semiactive kneeknee prosthesis prosthesis by by measuring measuring only only the the GRF, GRF, for for instance, instance, by by means means of of load load cells cells inserted inserted

Biomimetics 2021, 6, 22 11 of 15

Figure 12. Knee-flexion angle for three different longitudinal slopes of the walkway.

Biomimetics 2021, 6, 22 12 of 16 Figure 13. Internal torque, NH, of the hip that the patient must apply during the stance phase to walk with the prosthetic knee on walkways with three different longitudinal slopes.

Figure 14. Internal torque, MK, of the knee that the prosthesis actuation system must provide for Figure 14. Internal torque, MK, of the knee that the prosthesis actuation system must provide for making the prosthetic knee mimic the asymptomatic motion on walkways with three different lon- making the prosthetic knee mimic the asymptomatic motion on walkways with three different gitudinal slopes. longitudinal slopes.

4.4. Discussion Discussion TheThe above-reported above-reported results results prove prove that, that, for for wa walkinglking speeds speeds of of daily-life daily-life activities, activities, it it is is possiblepossible to to extract extract the the pieces pieces of of information ne necessarycessary to to control control an active or semiactive kneeknee prosthesis prosthesis by measuring onlyonly thethe GRF,GRF, for for instance, instance, by by means means of of load load cells cells inserted inserted in the prosthetic ankle. The comparison of the internal torque, NH, of the hip that the patient has to apply in the case of a limb with a prosthesis (Figure 13), and in the case of a healthy limb ([19], p. 659), also reveals that they are similar. This observation allows concluding

that, over mimicking the motion of a healthy limb, the proposed control procedure does not overload/stress the patient. The proposal does not consider a particular actuation system of the active/semiactive knee prosthesis. Therefore, it can be implemented on any actuation system that is able to generate the knee-torque values, MK, as shown in Figure 14. The hypothesis that the inertia forces are negligible at daily-life walking speed, which greatly simpliﬁes the control software, needs a deeper discussion. If the inertia forces are considered, the following equation replaces Equation (5)

d2α M = Y a cos(α ) − N − X a sin(α ) + J 1 (6) K A 1 A A 1 dt2 where J is the inertia moment of the prosthetic shank. Smith et al. [40] report realistic values of J. According to [40], J = 0.274 kg m2 has been chosen, and Equation (6) has been used to compute MK for 2 s and 1 s of gait-cycle duration, which are durations that cover daily-life walking speeds [34]. The results of these computations are reported in Figure 15. The comparison of the diagrams reported in Figures 14 and 15a,b shows that the variations are really small. Further simulations with gait-cycle durations under 1 s show that distortions are negligible until 0.75 s, which roughly corresponds to 1.92 m/s (=6.91 km/h) of walking speed and is a limit value outside of daily-life activities’ walking speeds. Consequently, the fact that inertia forces are negligible at daily-life walking speed is confirmed. In order to evaluate if the proposed control logic is implementable in real time, the computation burden has to be determined. The only computations the controller has to perform in real time are those necessary to compute the torque MK by means of Equation (5). Such computations involve 30 FLOP for computing the two trigonometric functions plus 6 FLOP for the remaining sums/subtractions/multiplications. Over the 36 FLOP for evaluating Equation (5), the preliminary computation of the ankle-flexion angle from the memorized polynomial coefficients requires a computation burden that depends on the polynomial degree and, for polynomial degrees lower than 13 (which is definitely much more than the actual degrees used in the regressions), it requires 90 more FLOP. Biomimetics 2021, 6, 22 12 of 15

in the prosthetic ankle. The comparison of the internal torque, NH, of the hip that the patient has to apply in the case of a limb with a prosthesis (Figure 13), and in the case of a healthy limb ([19], p. 659), also reveals that they are similar. This observation allows concluding that, over mimicking the motion of a healthy limb, the proposed control procedure does not overload/stress the patient. The proposal does not consider a particular actuation system of the active/semiactive knee prosthesis. Therefore, it can be implemented on any actuation system that is able to generate the knee-torque values, MK, as shown in Figure 14. The hypothesis that the inertia forces are negligible at daily-life walking speed, which greatly simplifies the control software, needs a deeper discussion. If the inertia forces are considered, the following equation replaces Equation (5) Biomimetics 2021, 6, 22 13 of 16 2 𝑑 𝛼1 MK = YA a cos(α1) − NA − XA a sin(α1) + J (6) 𝑑𝑡2 where J is the inertia moment of the prosthetic shank. Smith et al. [40] report realistic val- Consequently, 126 FLOP (=90 + 36) are necessary for each MK evaluation. This result allowsues of J. concluding According that, to [40], in aJ = gait 0.274 cycle kg m with2 has a durationbeen chosen, of 0.75 and s Equation (i.e., for a(6) walking has been speed used fasterto compute than those MK for of 2 daily-life s and 1 s activities), of gait-cycle 1000 duration, computations which are per durations gait cycle that would cover require daily- alife computation walking speeds power [34]. of only The 168,000results of FLOP/s these computations to implement are the reported control algorithm in Figure in15. real The time.comparison An old of microprocessor the diagrams suchreported as the in IntelFigures 80,486 14 and (dated 15a,b back shows on 1989), that the with variations a clock ofare 16 really MHz, small. 0.128 Further FLOP/clock-cycle, simulations andwith 32gait bit-cycle of precision, durations provides under 1 2.048s show MFPLOP/s that distor- oftions computation are negligible power until at 320.75 bits, s, which which roughly is deﬁnitely corresponds much more to 1.92 than m/s 168,000 (=6.91 FLOP/s.km/h) of Therefore,walking speed the simplicity and is a limit of the value algorithm outside guarantees of daily-life its real-timeactivities’ implementation walking speeds. and Conse- the possibilityquently, the of fact using that cheap inertia hardware forces are components. negligible at daily-life walking speed is confirmed.

2 FigureFigure 15. 15.Internal Internal torque, torque, M MKK,, of of the the knee knee prosthesis prosthesis computed computed with with Equation Equation (6) (6) for for J J = = 0.274 0.274 kg kg m m2and and two two gait-cycle gait-cycle durations:durations: ( a(a)) 2 2 s s and and ( b(b)) 1 1 s. s.

Eventually,In order to theevaluate adopted if the validation proposed technique control deserveslogic is implementable a final discussion. in real Indeed, time, one the couldcomputation object that, burden differently has to frombe determined. the simulations The only presented computations above, thethe noisecontroller in the has real- to timeperform measurements in real time could are those compromise necessary the to realcompute possibility the torque of using MK theby means proposed of Equation control logic.(5). Such On thiscomputations point, it is worthinvolve stressing 30 FLOP that, for in computing the proposed the control two trigonometric logic, only the functions GRF is measuredplus 6 FLOP in realfor time.the remaining All the other sums/subtracti curves areons/multiplications. fitting curves that come Over from the 36 a databaseFLOP for (memorizedevaluating Equation in the controller), (5), the preliminary where these comp curvesutation are recorded of the ankle-flexion through the angle coefficients from the of thememorized polynomial polynomial regressions coeffici computedents requires during the a computation prosthesis calibration; burden that consequently, depends on they the arepolynomial practically degree the same and, as for those polynomial used in thedegrees above lower validation than 13 technique. (which is definitely much moreThe than noise the inactual the measured degrees used GRF, in which the re mainlygressions), comes it requires from the 90 shock more accompanying FLOP. Conse- the IC event, can also be managed through either analog or digital filters. In particular, it introduces signal distortions, the spectral components of which have frequencies much higher than those of the GRF clean signal, the main spectral components of which have frequencies lower than 25 Hz. Actually, a 0.75 s gait-cycle duration corresponds to 1.33 Hz;

consequently, cutting the signal at 25 Hz corresponds to keeping the ﬁrst 18 spectral components in the worst case, which is much more than necessary. For instance, the C-Leg Ottobock [33] uses a sampling rate of 50 Hz for acquiring signals, which, for the Shannon theorem, means keeping frequency components not higher than 25 Hz. One simple digital ﬁlter could be the use of the average of 10 sequentially acquired GRF values for replacing the GRF value at the end of the acquisition time. This criterion would require the acquisition of ten times more values than those that are really processed and 10 FLOP more for computing the average value. In the case of 1000 computations per gait cycle and 0.75 s of gait-cycle duration, this yields 13,333 FLOP/s to add to the above-computed 168,000 FLOP/s, which gives a total computation burden of 181,333 FLOP/s, still much lower than the computation power of an old microprocessor. Therefore, the assumption that Biomimetics 2021, 6, 22 14 of 16

the real-time-measured GRF signal is smooth is realistic, and its usage in the above-reported validation technique, based on simulated input data, is correct.

5. Conclusions By reviewing the data reported in the literature for the gait-cycle kinematics/kinetics, a control logic has been devised for extracting the pieces of information necessary to control an active or semiactive knee prosthesis from real-time GRF measurements. Since real-time GRF measurements can be done by means of load cells inserted in the prosthetic ankle, the proposed control logic simpliﬁes the prosthesis hardware. In particular, the proposed control logic is able to identify the current limb conﬁgura- tion for different longitudinal slopes of the walkway and for different cadences, provided that the walking speed falls in the range usually spanned in daily-life activities. It then uses this information to compute the knee torque that the actuation system of the prosthesis has to apply. The implementation of this proposal relies on datasets measured directly on the patient during his/her rehabilitation training and on general motion patterns, and it does not require a particular actuation system of the prosthesis. The validation of the proposal has been done through simulations on a planar model. The results of these simulations showed that its implementation yields hip-joint internal torques that are comparable with those of a healthy limb, thus avoiding overload/stress for the patient.

Supplementary Materials: The files containing the datasets used in this paper and the MATLAB program used to generate Figures 10–15 are in the zipped file MatLabProgram+Data.zip available online at https://www.mdpi.com/article/10.3390/biomimetics6020022/s1. Author Contributions: Conceptualization, R.D.G. and L.V.; methodology, R.D.G. and L.V.; software, L.V.; validation, R.D.G. and L.V.; formal analysis, R.D.G. and L.V.; project administration, R.D.G.; funding acquisition, R.D.G. All authors have read and agreed to the published version of the manuscript. Funding: This research was developed at the Laboratory of Mechatronics and Virtual Prototyping (LaMaViP) of Ferrara Technopole and was funded by the University of Ferrara (UNIFE), grant number FAR2019. Data Availability Statement: The data sets used in this work have been uploaded as “Supplementary Materials” accompanying the paper. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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