In feedback controlled FES, often Hammerstein muscle models are employed that mainly consist of a non-linear static input function followed by linear transfer function. The input function, also called recruitment function, yields the muscle recruitment (amount of active motor units) depending on the normalized stimulation intensity *v* (range [0,1]). The dynamic model part (the transfer function) describes the muscle activation dynamics and the joint motion driven by the muscle recruitment. To consider hybrid muscle activations, we extend the static input function by adding a second input – the voluntary activity estimate . As shown in Fig. 2, this extended recruitment function is described by an Artificial Neural Network (ANN) that uses Local Linear Models (LLM) weighted by Radial Basis (RB) functions (c.f. Fig. 3). Four normalized radial basis functions

Figure 2 The used model consists of a static input function (artifi-cial neural network), that describe the motor unit recruitment, and linear transfer function model (AutoRegressive model with eXogenous input (ARX)), that captures the muscle activation dynamics and the joint motion.

$${\Phi}_{i}(v,y)=\frac{{\mu}_{i}(v,y)}{{\displaystyle \sum _{n=1}^{4}{\mu}_{n}}(v,y)},i=\{1,2,3,4\}$$(1)based on standard radial basis functions

$${\mu}_{i}(v,y)=\mathrm{exp}\left(-\frac{1}{2}(\frac{{(v-{c}_{i,v})}^{2}}{{\sigma}_{i,v}^{2}}+\frac{{(y-{c}_{i,y})}^{2}}{{\sigma}_{i,y}^{2}})\right)$$(2)are used. The normalized RB functions are combined with four local linear models to yield the neurons whose outputs are superposed yielding the output of the ANN:

$$\widehat{y}={\displaystyle \sum _{i=1}^{4}\underset{{\widehat{y}}_{i}}{\underbrace{({w}_{i,0}+{w}_{i,v}+{w}_{i,y}y)}}\cdot {\Phi}_{i}(v,y).}$$(3)Figure 3 The used structure for the artificial neural network using four neurons.

The parameters of the radial basis functions are chosen with respect to [5] and summarized in .

Table 1 Parameters of the ANN.

To describe the combined muscle activation dynamics and joint motion, an AutoRegressive model with eXogenous input (ARX)) [5] (a linear dynamic transfer function model) is used:

$$\widehat{\vartheta}[k]=\frac{{q}^{-m}}{1+{a}_{1}{q}^{-1}+{a}_{2}{q}^{-2}}\widehat{y}[k],$$(4)where $\widehat{\vartheta}[k]$ is the one-step backwards shift operator (*q*^{−1}*s*(*k*) = *s* (*k* − 1)). The time delay of *m* = 1 sampling instants matches the typically observed delay in recorded I/O-data. The tunable parameters are combined in the parameter vector

$$\Theta =[{w}_{1,0},{w}_{1,y},{w}_{1,y},\cdots ,{w}_{4,0},{w}_{4,v},{w}_{4,y},{a}_{1},{a}_{2}].$$(5)In order to adapt them to an individual subject and muscle condition, I/O data are recorded during an identification experiment and a successive linear least squares optimization is performed yielding the optimal parameter set *Θ*^{*} that minimizes the cost function

$$J(\Theta )={\displaystyle \sum _{k=0}^{N}{(\widehat{\vartheta}[k](\Theta ,y[k],v[k])-\vartheta [k])}^{2}},$$(6)where *ϑ*[*k*] is the recorded joint angle.

To obtain I/O data, an experimental procedure is proposed in which the stimulation intensity is increased stepwise (five levels, linear increase of the intensity) to the upper well tolerated intensity. During the time periods in which the stimulation intensity remains constant (lasting always 6 s), the subject is instructed to voluntarily elevate his arm to a given joint angle of approximately 50° for 2 s.

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