In GO-SAIL Compact, *u*_{a} = [*u*_{1}, *u*_{2}]^{⊤} is the stimulation applied to the shoulder and elbow, and *u*_{w} = [*u*_{3}, ⋯, *u*_{26}]^{⊤} is the stimulation applied to the forearm muscles via the electrode array. In addition, Φ_{a} = [*ϕ*_{1}, ⋯, *ϕ*_{5}]^{⊤} and Φ_{w} = [*ϕ*_{6}, ⋯, *ϕ*_{17}]^{⊤} contain the joint angles of the upper arm and wrist respectively. The control scheme is shown in Figure 5, in which the feedback controller is partitioned as ${C}_{c}=\text{diag}\{{C}_{c,a},{C}_{c,w}\}$ and is designed to establish stability and baseline tracking during each trial. The requirement to repeatedly perform a set of finite duration tasks with a fixed initial arm position enables ILC to be utilised to improve tracking performance. ILC uses the performance error from each trial to update the input ${v}_{k}={[{v}_{a,k}^{\top},{v}_{w,k}^{\top}]}^{\top}$ in an attempt to increase the accuracy of the subsequent attempt. On trial *k*, Φ_{k}(*t*) denotes the joint angles and the associated error is given by ${e}_{k}=\widehat{\mathrm{\Phi}}-{\mathrm{\Phi}}_{k}$. Feedback controller component, *C*_{c, a}, is chosen to be an input-output linearising controller, in series with feedback controller, *K*_{a}(*s*) = [0, *K*_{a, 2}(*s*), 0, *K*_{a, 4}(*s*), 0]^{⊤} is then selected to stabilise the resultant closed-loop dynamics

$$\begin{array}{c}{G}_{a,i}:\left({\widehat{\varphi}}_{i}+{v}_{k,i}\right)\mapsto {\varphi}_{k,i}:{\varphi}_{k,i}\left(s\right)=\hfill \\ \hfill {\left(I+{H}_{a}\left(s\right){K}_{a,i}\left(s\right)\right)}^{-1}{H}_{a}\left(s\right){K}_{a,i}\left(s\right)\left({\widehat{\varphi}}_{i}\left(s\right)+{v}_{k,i}\left(s\right)\right),\\ \hfill i=2,4.\end{array}$$(2)

Note that *ϕ*_{2} and *ϕ*_{4} denote the anterior deltoid and elbow joint angles respectively.

A linearised model of the wrist *H*_{w}(*s*) : *u*_{w, k} ↦ Φ_{w,k} is identified about a suitable operating point. Feedback controller *C*_{c, w} = *K*_{w}(*s*) is then chosen to stabilise system *H*_{w}(*s*) to yield the resulting closed-loop dynamics

$$\begin{array}{c}{G}_{w}:\left({\widehat{\mathrm{\Phi}}}_{w}+{v}_{w,k}\right)\mapsto {\mathrm{\Phi}}_{w}:{\mathrm{\Phi}}_{w}\left(s\right)=\hfill \\ \hfill {\left(I+{H}_{w}\left(s\right){K}_{w}\left(s\right)\right)}^{-1}{H}_{w}\left(s\right){K}_{w}\left(s\right)\left({\widehat{\mathrm{\Phi}}}_{w}\left(s\right)+{v}_{w,k}\left(s\right)\right).\end{array}$$(3)

An ILC scheme is then implemented in order to provide input *v*_{k} such that the error is minimised, i.e. $\underset{k\to \mathrm{\infty}}{lim}{v}_{k}={v}_{k}^{\star}$ with ${v}_{k}^{\star}:={\mathrm{min}}_{{v}_{k}}{\parallel \widehat{\mathrm{\Phi}}-{\mathrm{\Phi}}_{k}\parallel}^{2}$. This is achieved through the update structure

$${v}_{k+1}={v}_{k}+L{e}_{k},{v}_{0}=0,k=0,1,\mathrm{\cdots}$$(4)

where $L=\text{diag}\{{L}_{a},{L}_{w}\}$.

For the arm dynamics, design of *L*_{a,i} to satisfy $\parallel I-{G}_{a,i}{L}_{a,i}\parallel <1$, *i* = 2, 4, guarantees convergence of *ϕ*_{i} to zero error, and many suitable schemes are available, see [13] and examples therein. Similarity, $\parallel I-{G}_{w}{L}_{w}\parallel <1$ guarantees convergence of the wrist and hand joints to zero error. If this is not possible, the alternative form of $\parallel I-{G}_{w}{L}_{w}\parallel <1$ may be employed to guarantee convergence of error *e*_{w, k} to the limiting solution (*I* − *G*_{w} (*L*_{w}G_{w})^{−1} *L*_{w})${\widehat{\mathrm{\Phi}}}_{w}$.

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