Iterative learning vector field for FES-supported cyclic upper limbmovements in combination with robotic weight compensation*

Arne Passon1, Thomas Seel1, Jonas Massmann1, Chris Freeman2 and Thomas Schauer1

Abstract— Robotics and Functional Electrical Stimulation(FES) are well-established technologies for the rehabilitationof stroke and spinal cord injured (SCI) patients. We proposea hybrid solution that combines feedback-controlled FES ofbiceps and triceps as well as posterior and anterior deltoid witha cable-driven robotic system to support repetitive arm move-ments, like “breaststroke swimming” exercises. The roboticsystem partially compensates the arm weight by controllingthe cable tension forces, and the FES promotes motion in thetransversal plane. To adjust the FES support to the needsof the individual patients we use an iterative learning vectorfield (ILVF) which encodes the stimulation intensities that areapplied to guide the patient along a pre-specified referencetrajectory in the joint angle space. In contrast to previousiterative learning control approaches, the ILVF allows thepatient to perform the motion at self-selected cadence. Theproposed learning algorithm explicitly takes the dynamics of theartificially activated muscles into account and assures smoothstimulation intensity profiles. The control algorithm is testedin simulations using a complex neuro-musculoskeletal model.For “breaststroke” motions, the initial RMS error of purelyvolitional movements is reduced from 38◦ to 10◦ within 21cycles by the adaptive FES support. After 50 iterations of theILVF, the algorithm converges to a steady state RMS errorof 4◦. Changes in the patient’s muscle activity and cadencewere well tolerated by the control system and did not cause anoticable increase in the steady state RMS error.

I. INTRODUCTIONRobotics and functional electrical stimulation (FES) are

commonly used in the rehabilitation of stroke patients andincomplete spinal cord injured patients [1]–[3]. Robotics of-fer a persistent movement support with high precision whileFES actively involves the paretic muscles in the movementgeneration causing an increased proprioceptive feedback.Recent findings in motor learning advocate the synchronousapplication of FES with volitional muscle activity for achiev-ing the same motor function. A major limitation of FES isthe rapid muscular fatigue of the artificially activated motorunits. Lifting an arm over a longer period of time by FESis especially problematic. Therefore, it is promising to use ahybrid solution in which a robotic system compensates thearm weight persistently and FES supports arm motion in thehorizontal plane. A primary objective of FES-based motionsupport is to support the patient only as much as needed to

*The work was partially conducted within the research project BeMobil,which is supported by the German Federal Ministry of Education andResearch (BMBF) (FKZ16SV7069K).

1Arne Passon, Thomas Seel, Jonas Massmann and Thomas Schauerare with the Control Systems Group at TU Berlin, Einsteinufer17 EN-11, 10587 Berlin{passon,seel,schauer}@control.tu-berlin.de

2Chris Freeman is with Electronics and Computer Science, University ofSouthampton, United Kingdom

Fig. 1. “Breaststroke swimming” exercise with the cable-driven roboticsystem Diego. The robotic system compensates the arm weight and func-tional electrical stimulation supports the horizontal arm movements.

assure completion of the task without accelerating muscularfatigue or making the patient become passive. Since residualmotor function and abilities vary largely from one patientto the next, feedback control is often used to automaticallyadjust the FES support to an individual patient [3].

Iterative Learning Control (ILC) has been used in severalprevious contributions to achieve the aforementioned adjust-ment of stimulation parameters to the individual needs of thepatient [4], and experimental proofs of concept have beenachieved both in lower limbs [5]–[7] and in the upper limb[8]–[10]. The fact that cyclic motions such as strides may notalways have the same length has been addressed by severalauthors [11]–[14]. However, in all of these contributions, thereference trajectory has a fixed time scale, i.e. the controllerdictates the speed at which the motion must be performed.

To overcome this limitation, we have proposed a methodusing an iterative learning vector field (ILVF) in a previouspublication [15]. This ILVF is used for adjusting the FESsupport when performing cycling motion tasks like “breast-stroke swimming” with the cable-driven arm robotic systemDiego (Tyromotion GmbH, Austria) (see Figure 1). Themajor advantage of the applied approach, which had beeninspired by previous work in rehabilitation robotics [16],[17], is that only the desired spatial movement trajectory

2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)Madrid, Spain, October 1-5, 2018

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needs to be specified without any timing restriction on themovement. The user is able to freely determine the cadence.

The approach was preliminarily evaluated in experimentaltrials with healthy subjects. The subjects performed repetitivemotions that deviated largely from a given desired movementpath. The controller automatically adapted the support byadjusting the stimulation and thereby reduced the deviationto less than a third of the initial deviation within lessthan ten strokes [15], i.e. the controller learned to assistthe subject to better follow the desired path. Recently, westudied the convergence of the learning algorithm in moredetail based on a complex neuro-musculoskeletal model [18].These studies revealed several limitations of the initiallyproposed method: 1st) The algorithm only converged whenthe purely volitional cyclic motion was very close to thedesired reference movement. 2nd) Initial movement trajec-tories close to the reference converged within the first 30to 50 cycles to the reference but then diverged again. 3rd)The generated stimulation intensity profiles became more andmore aggressive (high-frequency) and therefore unpleasantfor a potential user.

In the current contribution a new learning law is proposedthat aims at solving the above outlined problems by 1st)defining a new error for the learning law, 2nd) taking thedelay in FES-based torque generation into account and3rd) adding a spatial low-pass filter to the learning lawto avoid undesired high-frequency signal components. Theeffectiveness of the method is demonstrated in computersimulations using the previously mentioned complex neuro-musculoskeletal model. The structure of the paper is asfollows: Section II presents an overview on the proposedsimulation model and the new learning control methods. Theobtained simulation results are presented in Section III fol-lowed by the final discussion and conclusions in Section IV.

II. METHODS

A. Simulation Model of Upper Limb Motions

The control task is to support bilateral cyclic arm move-ments in the horizontal plane while assuming almost perfectcompensation of the gravitational forces by a robotic system.The movement is described by the joint angles ϕu and ϕf as

ϕf

ϕu T

B

aD

pD

Fig. 2. Definition of the joint angles for the weight-compensated arm inthe transversal plane. The shoulder joint is actuated by the anterior (aD) andposterior deltoid (pD). Biceps (B) and triceps (T) act at the elbow joint. Thecyclic movement path of the wrist is indicated in red.

shown in Fig. 2. For simplicity only the right arm is shown.The shoulder torque Tu and elbow torque Tf are given by

Ti(u, ϕi, ϕi, t) = T FESi (u, ϕi, ϕi, t) + T vol

i (t) +

T passivei (ϕi, ϕi) + T limit

i (ϕi), i ∈ {u, f},

where T FESi represents the joint torque generated by FES at

the joint i, u = [uB, uT, uaD, upD]T is the vector of positive

control signals (stimulation intensities (pulse widths)) of theartificially activated antagonistic muscle pairs (biceps (B)–triceps (T), anterior deltoid (aD)–posterior deltoid (pD)), andT passivei is the passive joint torque caused by viscous-elastic

properties of muscles and tendons [18]. The term

T limiti (ϕi) = αi

[2ϕi − ϕi,max − ϕi,min

ϕi,max − ϕi,min

]pi

i ∈ {u, f}.

guarantees that the physiological angular limits[ϕi,min, ϕi,max] of the corresponding joint i are not violated[19]. We have chosen pf = 21, pu = 15, αf = 20Nm,αu = −1Nm for our simulation model. The used joint-anglelimits are reported in [19]. The volitionally induced jointtorque T vol

i at joint i is simply modeled as

T voli (t) = Ai sin(2πft) +Bi, i ∈ {u, f}.

Here, Ai, f and Bi describe the amplitude, frequency andbias of the volitional torque at joint i. These values have beenempirically tuned to simulate a paretic patient with weakmuscle control assuming linear superposition of the FES andvolitionally induced torques. The FES-induced joint torquesat the joint i ∈ {u, f} are given by

T FESf (uB, uT, ϕf, ϕf, t) = T FES

f,B (uB, ϕf, ϕf, t) +

+T FESf,T (uT, ϕf, ϕf, t),

T FESu (uaD, upD, ϕu, ϕu, t) = T FES

u,aD(uaD, ϕu, ϕu, t) +

+T FESu,pD(upD, ϕu, ϕu, t)

where T FESf,B , T FES

f,T , T FESu,aD and T FES

u,pD are the joint torquesgenerated by biceps, triceps, anterior deltoid and posteriordeltoid due to FES, respectively. These torques are simu-lated by nonlinear muscle models that have been identifiedexperimentally [18]:

T FESi,j = hj(uj , t)F

MAj,i (ϕi, ϕi),

i ∈ {u, f}, j ∈ {B,T, aD, pD}.

A Hammerstein model is used to describe the activation dy-namics hj(uj , t) that consist of a nonlinear static recruitmentcurve with the stimulation intensity (pulse width) as inputfollowed by a linear second-order activation dynamics. Thelower and upper limits of the recruitment curve are denotedby uthr

j and usatj , respectively. The output of the Hammerstein

model is scaled with a nonlinear function FMAj,i (ϕi, ϕi) which

captures the dependency of the generated joint torque onthe corresponding joint angle and velocity. Parameters of thefunctions FMA

j,i (ϕi, ϕi) and T passivei (ϕi, ϕi) were taken from

literature [18] (patient D).The skeletal dynamics has been simulated by using Sim-

scape Multibody (The MathWorks Inc., USA). Segment

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PrefPpredict,ref

Pfw,ref

P

Ppredict

verr

Ocycle

Oref

vfw

vfw

vdeform

copy

reference

last cycle

new cycleϕf

ϕu

vector field

β

β

Fig. 3. Illustration of the iterative learning vector field control. Shownis the last complete cycle (in blue), the current ongoing cycle (in black)and the reference path (in red) together with positions and vectors used todetermine the control actions.

lengths and inertia have been estimated from body heightand weight as outlined in [20]. In our simulation model, thebody height and body weight were set to 180 cm and 80 kg,respectively.

B. Vector Field Control

Each arm position represents a point in a two-dimensionalspace that is spanned by the angles ϕu and ϕf. To drive thearm closer to a pre-specified reference trajectory we mustactivate the two antagonistic muscle pairs in an adequatemanner. For each antagonistic muscle pair we first choose aunique control signal, uf ∈ [−1, 1] for the biceps–triceps anduu ∈ [−1, 1] for the posterior–anterior deltoid. Then thesetwo new control signals are mapped to the four pulse widthsas follows:

uB = uthrB + uf

(usat

B − uthrB

)for uf > 0,

uT = uthrT + |uf|

(usat

T − uthrT

)for uf ≤ 0,

uaD = uthraD + uu

(usat

aD − uthraD

)for uu > 0,

upD = uthrpD + |uu|

(usat

pD − uthrpD

)for uu ≤ 0,

where uthrj and usat

j are the aforementioned limits of therecruitment curve. Only one muscle of each antagonisticmuscle pair will be stimulated at any given time. In thefollowing we further take for granted the following:

• The patient can voluntarily generate cyclic arm motionsthat correspond to some closed path around some geo-metric center in the joint-angle space, cf. Fig. 3.

• Cadence and cycle path do not differ much from cyclel to the next cycle l+ 1, where l = 1, 2, ..., i.e. neitherthe volitional nor the FES-induced torques vary stronglyfrom cycle to cycle.

• The control actions uf and uu generate torques withan approximate delay Td of 500 ms (by analyzing theactivation dynamics given in [18]) and with the samesign as the applied control signals.

Under these assumptions, we determine control signalsui, i ∈ {u, f}, as follows: Based on the currently measuredarm position P (t) in the joint space we predict the expectedarm position Ppredict at time t+Td. Using the found positionwe extract the intensities udeform

f and udeformu by interpolation

from a given vector field Ψl. In Fig. 3 nine exemplaryvectors of the field are shown. The purpose of the stimulationintensities udeform

f (t) and udeformu (t) applied at time t is to

generate the correction torques at time t + Td that arenecessary to bring the arm position P (t + Td) closer tothe given reference trajectory. Additionally, stimulation in-tensities upropel

f and upropelu are determined that shall generate

supportive joint torques for maintaining a cyclic clockwisemovement of the right arm. The sum of both intensitiesui = udeform

i + upropeli , i ∈ {u, f} is then applied to the

biceps–triceps and anterior–posterior deltoid muscle pair asdescribed above.

In detail, the following steps need to be performed toobtain the control signals:

1) Ahead prediction: Figure 3 illustrates the desired ref-erence trajectory together with the current arm position Pand the last completed cycle of the arm movement (inblue). The movement direction is indicated by arrows onthe trajectories. The geometric centers Oref and Ocycle of thereference trajectory and the last completed arm cycle arecalculated. At time t, we determine the position Pref on thereference trajectory for which the angle with respect to thecenter Oref matches the angle β of the current arm positionP with respect to the center Ocycle of the last completedcycle. We assume that, for the time period [t, t + Td], thearm will continue its movement in a similar way as it wouldmove on the corresponding part of the reference trajectory(green paths in Fig. 3), where the path velocity is estimatedfrom the cadence of the last completed cycle. Applying thisdisplacement to the point P in the current cycle yields thepredicted position Ppredict.

2) Control actions for bringing the patient’s cycle closerto the reference: By interpolation of the vector field Ψl atthe predicted position Ppredict we obtain the intensities udeform

fand udeform

u [udeform

f (t)udeform

u (t)

]= vdeform(t)

where vdeform is the interpolated vector of the field Ψl at thepoint Ppredict. The vector field stores FES support information(the two intensities) with a spatial resolution of 2.5◦ forboth joint angles to save memory and computational effort.Additionally, we limit the joint angles to values between−90◦ and 180◦. This results in a vector field with n × ngrid points, where n = 109. The vector field Ψl can then bedescribed by two matrices Ψl,f ∈ Rn×n and Ψl,u ∈ Rn×n

that describe the intensities udeformf and udeform

u at the n2 gridpoints, respectively.

3) Control actions for propelling support: The requiredpropelling support at the predicted position Ppredict is calcu-lated by taking the future path of the reference trajectoryinto account as shown in Figure 3. Starting from Pref, we

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determine the points Ppredict, ref and Pfw,ref that are Td andTd + Tforward seconds ahead on the reference, respectively.As above, the path velocity is estimated from the cadence ofthe last completed cycle. Let vfw be the vector that pointsfrom Ppredict,ref to Pfw,ref, and choose a scalar gain kP > 0 toobtain the following stimulation intensities for the propellingsupport: [

upropelf (t)

upropelu (t)

]= kPvfw(t)

Here we exploit the aforementioned assumption that a posi-tive/negative upropel

u will always promote an increase/decreaseof ϕu, while a positive/negative upropel

f will always promotean increase/decrease of ϕf.

4) Cycle detection: A simple cycle detection is requiredto determine the above outlined quantities and to estimatethe cadence. A new cycle starts when the phase angle ofthe arm position in joint space with respect to the geometriccenter of the previous cycle exceeds a predefined value.

C. Iterative Learning of the Vector Field

The vector field Ψl is initialized with zero support andis then iteratively updated after each completed cycle byanalyzing the deviations between observed movement andreference. To this end, the cycle trajectory is quantized alongthe grid points. For each passed grid point, an error vectoris determined that describes the deviation between that pointand the corresponding point on the reference trajectory, i.e.the point that has the same phase angle β with respect to thecorresponding geometric center. Figure 3 illustrates this foran exemplary pair of points (P and Pref ). The correspondingerror vector verr is shown. The error at all points that are notpassed by the trajectory is set to zero. The first and secondentries of all errors vectors of the cycle l are stored in theerror matrices Ef,l ∈ Rn×n and Eu,l ∈ Rn×n, respectively.After the (l)-th cycle has finished, we apply an integratingupdate law to adjust the vector field for the next cycle l+1:

Ψl+1,i = LP(Ψl,i + kIEi,l), i ∈ {u, f}. (1)

Here, kI represents the learning gain and LP is a spatiallow-pass filter (Butterworth, 2nd order, cut-off frequency fcof 5◦, width twice the cut-off frequency). This spatial filteris introduced for three reasons:

1) to adapt the vector field also in the vicinity of the pointspassed during the last completed cycle,

2) to avoid that high-frequency components in the er-ror, e.g. due to noise and quantization effects, causeunpleasant high-frequency components in the controlaction, and

3) to realize a slow forgetting of the previously learnedvector field in order to realize an assist-as-neededsupport, e.g. the vector field should slowly decay tozero if the reference trajectory is tracked perfectly.

The latter is achieved as the repeated application of the filterwill continuously reduce the energy of the vector field. Pleasealso note the following: As the tracking is improved by

TABLE IVOLITIONAL SUPPORT MODEL PARAMETERS.

Time periodParameter 0. . . 500 s 500. . . 850 s 850. . . 1200 s

l=1. . . 124 l=125. . . 212 l=213. . . 277Af [Nm] 5.2 3.9 3.9Bf [Nm] 0 0 0ff [rpm] 15 15 11.25Au [Nm] 4.8 3.6 3.6Bu [Nm] 0 0 0fu [rpm] 15 15 11.25

TABLE IICONTROLLER PARAMETERS.

Parameter ValuekI [1/◦] 0.01 0.03 0.04kP [1/◦] 0.02Td [ms] 500Tfw [ms] 150

learning from trial to trial, the entries of the error matricesget smaller and the corrections kIEi,l in (1) are absorbedby the filter. Thus, the learning vector field does not lead tozero tracking error. Instead, it only helps the patient to getclose to the desired path. This should motivate the patient toincrease his effort for making the remaining errors small.

III. RESULTS

The proposed methods are evaluated in computer simu-lations using the model from Section II-A. Table I reportsthe parameter values used for the voluntary torques. After25 cycles without FES support, the ILVF is activated at the26-th cycle. After 500 seconds (124 cycles), the amount

50 100 150 200 2500

10

20

30

40

Cycle l

e l[◦

]

kI = 0.01kI = 0.03kI = 0.04

Fig. 4. Progression of the RMS error el between the observed motionand the reference for different learning gains kI . The ILVF was activatedat the 26-th cycle. No FES support was active during the period shown ingrey. In the volitional torque generation, the amplitude and the frequencyare changed at cycle 125 and at cycle 213, respectively.

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0 0.5 1 1.5 2 2.5 3 3.5 4

40

60

80

100

120

140

160

Time [s]

Puls

ew

idth

s[µ

s]

BicepsTriceps

0 0.5 1 1.5 2 2.5 3 3.5 4

40

60

80

100

120

140

160

Time [s]

Puls

ew

idth

s[µ

s]

Ant. DeltoidPost. Deltoid

Fig. 5. Applied stimulation intensities during the 100-th cycle (solid lines) and after changing the amplitude in the voluntarily torque (164-th cycle,dashed). Since the reduced voluntary activity requires considerably different support, both stimulation patterns differ largely. Nevertheless, both of thesepatterns are determined automatically by the ILVF.

of the voluntarily torque is reduced by 25% percent. At850 seconds (213-th cycle), we change the cadence of thepatient’s volitional support from 15 to 11.25 rpm.

Reference20-th cycle: no FES46-th cycle / 21-st iteration of ILVF100-th cycle / 75-th iteration of ILVF

222-nd cycle (after changing freq. in vol. torque)164-th cycle (after changing ampl. in vol. torque)

100

90

80

70

60

50

40

30

20

10

0806040200-20

ϕf[◦

]

ϕu [◦]

Fig. 6. Observed cycles in the joint-angle space together with the referencetrajectory (in red) and the learned vector field at the 100-th cycle.

Fig. 4 shows the observed root-mean-square (RMS) error

el =

√√√√ 1

nl

nl∑k=1

||verr(k)||2

for all cycles using the empirically tuned controller parame-ters given in Table II. nl is the number of sampling instantsin the cycle l. Fig. 4 reveals that smaller values of kIreduce the convergence speed as expected and lead to largersteady state RMS errors. Too large values of kI may leadto divergence in the sense that the patient’s movement doesnot converge to the reference. This is illustrated in Fig. 4.The other parameters kP , Td and Tfw were set to empiricallydetermined optimal values. Large changes of any of theseparameters into any direction were found to lead to instabilityof the learning process. In the following, results are reportedfor the best setting of kI = 0.03.

In Fig. 6, the twentieth cycle (without FES) is plottedtogether with the reference trajectory (in red with a toleranceband of 10◦) and several cycles with FES support. Thecycles for l = 164 and l = 222 show the convergedtrajectories after the change in amplitude and frequency bythe user, respectively. In the same figure, we plotted thelearned vector field Ψl after 74 learning steps (100-th cycle).For the very same cycle, Fig. 5 presents the pattern of thestimulation intensities over time along with the stimulationpattern to which the ILVF converged after changing theamplitude of the voluntarily torque (164-th cycle). Bothpatterns are not just scaled versions of each other. Instead,when voluntary activitiy is reduced, a considerably differentpattern is required to achieve motion along the desired path.The ILVF has determined these four-dimensional patternsautomatically by learning from cycle to cycle.

The observed cadence over all cycles is shown in Fig. 7together with the cadence of the patient’s voluntary torquegeneration.

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50 100 150 200 25011

12

13

14

15

16

17

Cycle l

Cad

ence

[rpm

]Cadence of the volitional joint torque T vol

i

Observed cadence of cyclic motion

Fig. 7. Cadence of the joint torque T voli volitionally induced by the patient

and observed cadence of the FES-supported cyclic motion. The patient candictate the movement speed.

IV. DISCUSSION AND CONCLUSIONS

The algorithm supported the weak voluntary movementand converged to a steady state in about 50 iterations ofthe ILFV. The resulting stimulation intensities were smoothenough to be tolerated by real patients. After only 21 itera-tions the error had been reduced to values below 10◦, lessthan a third of the initial value. Changes in patient’s effort(voluntary torque) were automatically compensated. Thesmaller voluntary torque lead to a larger residual trackingerror, i.e. the ILVF yields a motion support that encouragesthe patient to maximize his contribution. Beyond this, itis particularly notable that the resulting cycle cadence isdictated by the user’s torque generation frequency, i.e. thesystem follows the patient.

There are several limitations of this study. First, we didnot provide a mathematically firm proof of convergenceat the moment. Second, we have to study the selectionof the controller parameters in more detail in order tofacilitate a fast and patient-individual tuning. Third, we haveto validate the methods in experiments with patients andreal sensors, since the used model is time-invariant and doesnot capture effects like muscular fatigue, changing spasticityand sensor noise. Fourth, the necessary assumption of cyclicunidirectional volitional movements by the patient may limitthe suitable group of neurological impaired patients thatmay benefit from this approach. Fifth, the approach doesnot directly promote a continues movement of the patient tothe desired movement – a biofeedback of the patients effort(derived from the observed FES-support) could mitigate thisproblem.

In summary, the iterative learning vector field is found tobe an appropriate method for adjusting the FES support incyclic movements to the needs of the patient. It learns toprovide the support that is needed to help the individual

patient to get close to the desired path. Nevertheless, itassures a steady-state error that motivates the patient toincrease his/her efforts. Finally, and in contrast to standarditerative learning control, the ILVC allows the patient toperform the motion at his/her own comfortable speed.

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