## 1 Introduction

A common side effect of stroke and some other neurological disorders is the drop foot syndrome. It impedes the lifting of the inner or the outer edge, or both, of the patient’s paretic foot by voluntary muscle activation. This leads to a pathological gait with an increased risk of falling. As illustrated in Figure 1, drop foot neuroprostheses can be used to activate the affected shank muscles during the swing phase via Functional Electrical Stimulation (FES) [1]. Therein, a main challenge is the fact that the ankle joint has two degrees of freedom and is actuated by multiple muscles in a non-trivial way. The underlying anatomy is depicted in Figure 1 along with commonly applied electrode positions. Experiments show that FES applied via the frontal electrode lifts the foot and turns it by raising its inner edge. On the contrary, FES applied to the lateral electrode causes foot roll in the opposite direction by lifting the outer edge and, at higher intensities, also the inner edge to some (patient-individual) extent. The issue of coordinating those movements can be solved by means of automatic feedback control if a sensor is employed that measures the foot motion. Various control techniques have been implemented previously, see [1] for an overview. However, most approaches as well as commercially available systems only aim at providing sufficient pitch, while few work exists that takes both degrees of freedom into account. With respect to the latter, valuable results were obtained by Veltink et al. [5] for the tuning of an implanted two-channel stimulator based on foot orientation measurements. However, to the best of our knowledge, there exists no thorough description of the interrelations and couplings between multichannel stimulation intensities applied via surface electrodes and the resulting foot motion. Moreover, experiments show that the maximum stimulation intensity tolerated by the subject depends on the combined sensation caused by all channels. This impedes the implementation of decentralized control schemes, since it implies that the control channel’s saturation limits are interdependent. The present contribution addresses both issues by proposing suitable parameterizations of the domain of tolerated stimulation intensities and by an in-depth experimental analysis of the couplings between FES parameters and foot motion.

## 2 Hardware setup and inertial assessment of foot motion

We employ the hardware setup of the Adaptive Peroneal Stimulator described in Figure 1 and in [3]. Two self-adhesive electrodes are attached to the shank of a drop foot patient. FES is applied by means of a novel stimulator that generates tri-phasic stimulation pulses at a frequency of *f*_{FES} = 50 Hz. As explained in [4], this setup facilitates the independent manipulation of two effective stimulation pulses: one that mainly stimulates the nervous tissue below the frontal electrode, and one that mainly stimulates the nervous tissue below the lateral electrode. Both pulse charges *q*_{L} and *q*_{F}, respectively, are manipulated in realtime via a laptop that is connected to the stimulator and receives the measurements of the inertial sensor attached to the patient’s paretic foot. By means of the algorithm described in [2], the measured angular rates and accelerations are used to detect the instant *t*_{to} of *toe-off* and the instant*t*_{ic} of*initial contact* for each step in realtime. Furthermore, we use the method presented in [3] to obtain real-time measurements of the current foot orientation angles (*ψ*) and *φ*(*t*) in pitch and roll direction at a sampling rate of *f*_{IMU} = 100 Hz. Both angles are defined with respect to the horizontal plane. While *ψ* > 0 refers to the toes pointing upwards, *φ* > 0 implies that the outer edge of the foot is above the inner edge, cf. Figure 1.

When analyzing the influence of stimulation parameters on the foot motion, it is cumbersome to consider the entire roll and pitch angle trajectories of each step. In order to capture each stride’s foot motion in a scalar measure, we define the following pitch and roll indicators:

where *ψ _{b}* = −40° and

*φ*

_{b}= 0° are chosen to approximate typical toe-off angles in slow physiological gait. Please note that the cubic average operation maintains the sign and gives more weight to large deviations from the base values

*φ*and

_{b}*ψ*. As demonstrated in [3], such indicators correlate well with the gait rating of experienced clinicians. Therefore, they are considered satisfactory scalar measures of the clinically relevant foot motion in a step.

_{b}## 3 Choosing suitable stimulation intensity parameters

Since applying large electrical pulses via skin electrodes causes discomfort and pain, it is advisable to limit the FES intensity to the maximum values tolerated by the subject who receives the stimulation. Figure 2 shows typical maximum values *q*_{L} and *q*_{F}, as well as for equal pulse charge on both electrodes (*q*_{L} = *q*_{F}). By linear interpolation, we obtain the depicted quadrangle that defines the domain*Q* of admissible stimulation intensities. As discussed in Section 1, current research aims at employing a decentralized control scheme (i.e. two independent controllers) for the foot pitch and roll in feedback controlled drop foot neuroprostheses. If the stimulation intensities*q*_{L},*q*_{F} are used as manipulated variables, then the saturation limit of each depends on the current value of the other. This problem can be avoided by suitable parameterization of the domain*Q*. In the following, we propose two such parameterizations. The first approach uses two parameters*u*_{L}*2* [0, 1] and*u*_{F}*2* [0, 1] to describe the domain*Q* in Cartesian-like coordinates:

Alternatively, one may use the polar-type coordinates *u*_{Σ} ∈ [0, 1] and *ρ* ∈ [−1, 1] to parameterize*Q* as follows:

As illustrated in Figure 2, *u*_{L} = 0 implies *q*_{L} = 0, and *u*_{L} = 1 refers to points on the interpolated line between *u*_{F}, *q*_{F}, and *ρ* = −1 refers to *q*_{L} = 0 and *ρ* = +1 refers to *q*_{F} = 0, while *u*_{Σ} scales both *q*_{L} and*q*_{F} between zero and their *ρ*-dependent maximum values. Both of the proposed parameterizations allow the implementation of two independent single-input single-output controllers with properly defined saturation limits. In the following, we investigate the influence of both pairs of stimulation intensity coordinates on the foot pitch and roll during swing phase.

## 4 Experiments and discussion

A chronic drop foot patient walks on a treadmill at constant, self selected speed of about *u*_{L} and *u*_{F} within ranges that provide sufficient support to enable the patients to walk properly: At first we decrease *u*_{L} in small steps from 1 to 0 while keeping *u*_{F} constant. This is done for various values of *u*_{F}. Since hysteresis effects are common in FES, we repeat the process vice versa, i.e. keeping *u*_{L} constant while decreasing*u*_{F} from 1 to 0. Subplot (c) of Figure 2 shows pitch and roll indicators plotted against the Cartesian-like stimulation parameters *u*_{L} and *u*_{F}, while subplot (d) presents the result of the entire experimental procedure carried out for the polar-like stimulation parameters *u*_{Σ} and *ρ*.

For each constant *u*_{Σ}, the roll indicator increases almost (i.e. we simplify the discussion by ignoring very short non-monotonous sections) monotonically with *ρ*. The foot pitch, however, is almost constant if *ρ* ≥ 0. For the Cartesian-like coordinates, foot roll depends on *u*_{L} only if *u*_{F} ≤ 0.75 and foot pitch is influenced by *u*_{F} only if *u*_{L} ≥ 0.5. Recall that our goal is to generate and maintain physiological foot pitch and roll by manipulation of *u*_{L},*u*_{F} or *u*_{Σ}, *ρ*. Based on the observations from Section 1, the most intuitive approach would be to simply raise/lower *u*_{F} (or *u*_{Σ}) whenever the pitch is too low/high, respectively, and to raise/lower *u*_{L} (or *ρ*) whenever the roll is too low/high. The results in Figure 2 indicate that such a decentralized approach would be successful. For example, consider any parameter combination *u*_{Σ}, *ρ* in subplot (d) that yields too little roll. Then an increase of *ρ* eventually leads to roll values within the physiological ranges found in healthy foot motion. Likewise, a decrease of *ρ* brings too high roll indicators back to the physiological range. Analogous properties are found for the *u*_{Σ}-pitch-relation. However, at least in the considered patient, it is not possible to avoid slightly exaggerated foot roll if *u*_{F} ≥ 0.9, and the *u*_{F}-pitch-relation shows that it is not possible to avoid slightly exaggerated foot pitch if *u*_{L} ≥ 0.9.

Beyond the sole reachability of physiological foot motion, Figure 2 also allows discussion of the cross couplings, i.e. the influence of *u*_{L} and*ρ* on pitch and the influence of *u*_{F} and *u*_{Σ} on roll. If they were zero, the lines in each plot would lie on top of each other. This is approximately the case for several ranges of both coordinates. However, modifying *ρ* has a significant effect on pitch when *u*_{Σ} ≥ 0.8, and *u*_{F} as well as *u*_{Σ} influence roll for some values of *u*_{L} and *Σ*, respectively. Therefore, we find that neither of the two proposed parameterizations eliminates these cross couplings completely. Similar results are obtained in another chronic drop foot patients.

## 5 Conclusion

We considered multichannel FES of the shank muscles that control foot motion during the swing phase of gait. We highlighted the fact that the maximum stimulation intensity tolerated by the subject depends on the combined sensation caused by both channels. Therefore, the domain of admissible stimulation intensities needs to be parameterized properly to enable feedback control of foot pitch and roll in a decentralized manner. Two suitable parameterizations were suggested and compared in terms of the cross-couplings between each of the new stimulation intensity coordinates and the foot motion caused during swing phase in drop foot patients walking on a treadmill. Based on the discussed results, both parameterizations are found to yield monotonous input-output behavior and therefore facilitate full control over the two-dimensional foot motion. On a marginal scale, the polar-like parameterization of the domain of admissible stimulation intensities is found to yield slightly better control of foot roll in the considered patients. Beyond drop foot treatment, the proposed parameterization methods can be extended to more than two channels and represent a useful tool for other multichannel FES applications.

In future research, the closed-loop dynamics of a decentralized iterative learning control scheme will be investigated in combination with both proposed parameterizations of the domain of admissible stimulation intensities, and additional decoupling networks will be employed to further reduce cross couplings.

We would like to express our deep gratitude to the patients who participated in the experimental trials. We are also indebted to Boris Henckell for his skillful support in conducting and evaluating the experimental trials as well as to Cordula Werner and her colleagues from Charité Univesitätsmedizin Berlin for their valuable cooperation and medical guidance.

**Funding**: Being conducted in the research project APeroStim, this work is funded by the German Federal Ministry of Research and Education (FKZ 01EZ1204B).

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Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

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