, of ES signal ![$$\varvec{u} \in \mathscr {L}^m_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq2.gif)
represents the electrical stimulation applied to the ith muscle. Since each array element does not necessarily correspond to a single muscle, we now introduce signal ![$$\varvec{z} \in \mathscr {L}^n_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq3.gif)
containing the stimulation applied to each of the n elements of the array over ![$$t \in [0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq4.gif)
. The stimulation provided to the ith muscle is assumed to be a linear combination of those array elements within spatial range, and hence can be modeled using the relationship
![$$\begin{aligned} u_i(t) = \sum ^{n}_{j = 1} a_{i,j} z_j (t), \quad i = 1, \ldots , m, \quad t \in [0,T]. \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ1.gif)
(8.1)
![$$\begin{aligned} M : \varvec{\varPhi } = H_{RB} F_m (\varvec{\varPhi },\dot{\varvec{\varPhi }}) H_{LAD} h_{IRC}(A \varvec{z}), \quad A :\varvec{z} \mapsto \varvec{u}: \varvec{u}(t) = \left[ \begin{array}{ccc} a_{1,1} &{}\cdots &{} a_{1,n} \\ \vdots &{}\ddots &{}\vdots \\ a_{m,1} &{}\cdots &{}a_{m,n} \end{array} \right] \varvec{z}(t) \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ2.gif)
(8.2)
and 
(i.e. each ES channel is associated with a single muscle).It is possible to identify parameters in (8.2) by extending the approach developed in Chap. 2. However, in the case of the wrist and hand there are at least 41 musculo-tendon units actuating16 joints with 23 degrees of freedom [6]. This makes measurement of all joint angles and application and sensing of force/moments about each axis highly challenging. In addition, since the muscles are small and closely packed, the model system is sensitive to small changes in array position as well as to physiological changes such as fatigue and spasticity, and environmental conditions such as temperature and humidity. This makes the previous approach impractical, and therefore it will be exchanged for the identification of local linear models (as incorporated as an option within the design procedures of Chaps. 3, 4 and 6).
8.2 General Array Control Framework
Application of control design Procedure 4 of Sect. 6.4 to the array control problem first requires designing feedback controller K to stabilise the linearised model description 
. The impractically of obtaining a global non-linear model means that we must identify 
around each operating point 
. The structure of 
is given by (8.2) and has n inputs and p outputs, making it potentially high dimension. To recover a tractable identification problem we therefore embed a restricted plant stimulation space with dimension 
. From Definition 6.1, this necessitates feedback structure (6.11), and hence the control scheme takes the form shown in Fig. 8.1, where operator X is given by (6.6) with full rank 
. We can expand the previous design procedure as follows:
. The impractically of obtaining a global non-linear model means that we must identify around each operating point . The structure of is given by (8.2) and has n inputs and p outputs, making it potentially high dimension. To recover a tractable identification problem we therefore embed a restricted plant stimulation space with dimension . From Definition 6.1, this necessitates feedback structure (6.11), and hence the control scheme takes the form shown in Fig. 8.1, where operator X is given by (6.6) with full rank . We can expand the previous design procedure as follows:Procedure 5
(Design for Robust stability: specification to electrode array)
Define task: Choose P via (6.1) and Lemma 6.3 to capture the required functional task (or for simplicity choose 
to track reference 
over ![$$t \in [0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq15.gif)
).
to track reference over ).Model identification: Apply input sequence 
to the ES electrode array, and use resulting input-output data set 
to identify a linear approximation to the dynamics 
about 
. Here
![$$\begin{aligned}&M |_{\varvec{z}(k)} : \mathscr {L}^n_2[0,T] \rightarrow \mathscr {L}^p_2[0,T] : \varvec{z} \mapsto \varvec{\varPhi } \nonumber \\&\,\;\;\;\;\;\qquad \qquad \qquad : \varvec{\varPhi } = \big ( H_{RB} |_{\varvec{w}} F_m (\varvec{\varPhi },\dot{\varvec{\varPhi }}) H_{LAD} \varvec{h}_{IRC} (A \varvec{z}) \big ) |_{\varvec{z}_k} \varvec{z} \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ3.gif)
Feedback controller design: Design ![$$K_X : \mathscr {L}^p_2[0,T] \rightarrow \mathscr {L}^q_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq20.gif)
to stabilize 
. This is equivalent to designing K to stabilize 
given structure (6.11).
to the ES electrode array, and use resulting input-output data set to identify a linear approximation to the dynamics about . Here(8.3)
to stabilize . This is equivalent to designing K to stabilize given structure (6.11).ILC design: Design L to satisfy condition (6.14) or (6.17) of Theorem 6.1 for the resulting closed-loop system
![$$\begin{aligned}&G |_{\hat{\varvec{\varPhi }} + \varvec{v}_k} : \mathscr {L}^p_2[0,T] \rightarrow \mathscr {L}^p_2[0,T] : \varvec{v} \mapsto \varvec{\varPhi } : \varvec{\varPhi } = (I + M |_{\varvec{z}(k)} K)^{-1} M |_{\varvec{z}(k)} K \varvec{v}. \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ4.gif)
The first condition guarantees nominal convergence to zero error, but requires 
. In both cases, implement ILC update using (6.26).
(8.4)
. In both cases, implement ILC update using (6.26).
Fig. 8.1
Feedback and ILC control scheme specified to electrode array structure
We next present examples of explicit forms of 
and L for application in the above procedure. These assume that the form of linearized system (8.3) is stipulated by the designer as a static 
mapping, i.e. with form
![$$\begin{aligned} F : \mathscr {L}^n[0,T] \rightarrow \mathscr {L}^p[0,T] : \varvec{z} \mapsto \varvec{\varPhi } : \varvec{\varPhi }(t) = \bar{F} \varvec{z}(t), \;\; \bar{F} \in \mathbb {R}^{p \times n}, \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ5.gif)
followed by SISO linear dynamics. In practice this assumption is supported by similar muscle and rigid body dynamics in the wrist and hand, together with the presence of spasticity, inherent stiffness, and the low bandwidth of required movements.
and L for application in the above procedure. These assume that the form of linearized system (8.3) is stipulated by the designer as a static mapping, i.e. with form(8.5)
Proposition 8.1
Let linearized dynamics (8.3) have form 
where F is given by (8.5) and 
is a SISO system. The feedback control action 
, where 
is a SISO system, applied to system 
realizes stimulation input
where 
is the unique minimizer of a weighted norm of the tracking error, 
, and the SISO system
The resulting closed-loop system dynamics are
where the orthogonal projection onto the range of FX is 
,
![$$\begin{aligned} (FX)^\perp : \mathscr {L}^p[0,T] \rightarrow \mathscr {L}^p[0,T] : \hat{\varvec{\varPhi }} \mapsto \varvec{x} : \varvec{x}(t) = \bar{F} \bar{X}(\bar{F} \bar{X})^\dagger \hat{\varvec{\varPhi }}(t). \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ9.gif)
where F is given by (8.5) and is a SISO system. The feedback control action , where is a SISO system, applied to system realizes stimulation input(8.6)
is the unique minimizer of a weighted norm of the tracking error, , and the SISO system(8.7)
(8.8)
,(8.9)
Proof
Consider the weighted tracking error 
where weight 
with 
the adjoint operator. This has solution
The proposed control action 
realizes (when 
)
The corresponding closed-loop dynamics are
where weight with the adjoint operator. This has solution realizes (when )(8.10)
The feedback control action 
of Proposition 8.1 therefore ensures that 
tracks the demand input 
(or 
if ILC is also applied) as closely as possible, subject to dynamics 
specified by the designer. It requires only that SISO feedback controller 
be selected to stabilize dynamics (8.7), but has the property of stabilizing all p joints. By enforcing dynamics 
, it also facilitates the following simplified ILC update:
of Proposition 8.1 therefore ensures that tracks the demand input (or if ILC is also applied) as closely as possible, subject to dynamics specified by the designer. It requires only that SISO feedback controller be selected to stabilize dynamics (8.7), but has the property of stabilizing all p joints. By enforcing dynamics , it also facilitates the following simplified ILC update:Proposition 8.2
The system of Fig. 8.1 with 
, the feedback action of Proposition 8.1, and ILC update
![$$\begin{aligned} \varvec{v}_{k+1} = \varvec{v}_{k} + L P \varvec{e}_{k}, \quad L = l (P N_w (F X)^\perp )^\dagger , \quad l \in (0,1] \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ11.gif)
satisfies (6.17) and enforces convergence to the minimum extended error norm, i.e.
Furthermore, if 
is tuned so that 
approximates a pure delay of 
seconds then (8.11) corresponds to the phaselead update
![$$\begin{aligned} \varvec{v}_{k+1}(t) = \varvec{v}_{k}(t) + l (P_j (\bar{F} \bar{X})^\perp )^\dagger \varvec{e}_k^e (t + \lambda ), \quad \quad t \in [t_{j-1},t_j], \; j = 1, \ldots S. \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ13.gif)
, the feedback action of Proposition 8.1, and ILC update(8.11)
(8.12)
is tuned so that approximates a pure delay of seconds then (8.11) corresponds to the phaselead update(8.13)
Proof
We set 
in Theorem 6.1 and assume 
. Since Proposition 8.1 yields closed-loop dynamics 
, we then substitute 
and 
into 
to give 
which satisfies (6.17). The corresponding limiting error solution (6.22), 
, is given by 
which is the orthogonal projection onto the kernel of 
. This is the minimum achievable error and hence solves (8.12). If 
then 
with time-based implementation (8.13). 
in Theorem 6.1 and assume . Since Proposition 8.1 yields closed-loop dynamics , we then substitute and into to give which satisfies (6.17). The corresponding limiting error solution (6.22), , is given by which is the orthogonal projection onto the kernel of . This is the minimum achievable error and hence solves (8.12). If then with time-based implementation (8.13). Note that update (8.13) takes the ‘phase-lead ILC’ form, which has received significant research attention due to its simple structure (with only two parameters, l, 
) that enables heuristic tuning [7–9].
) that enables heuristic tuning [7–9].We next examine how stimulation subspace X can be chosen to balance tracking accuracy with practicality of identification.
8.3 Subspace Identification
The identification of subspace 
is addressed in this section, with subsequent identification of the q input, p output system 
following in Sect. 8.4. The purpose of 
is to reduce the dimension of the latter problem so that it can feasibly be performed within the limited time available in practice. The use of 
is motivated by the observation that only a subset of muscles are required for a given posture, and the underlying muscle locations can be assumed to not change. It is hence possible to select an input subspace that covers those muscles needed to perform a required set of tasks, together with the possible variation in array placement. In practice a suitable input subspace can be constructed using:
is addressed in this section, with subsequent identification of the q input, p output system following in Sect. 8.4. The purpose of is to reduce the dimension of the latter problem so that it can feasibly be performed within the limited time available in practice. The use of is motivated by the observation that only a subset of muscles are required for a given posture, and the underlying muscle locations can be assumed to not change. It is hence possible to select an input subspace that covers those muscles needed to perform a required set of tasks, together with the possible variation in array placement. In practice a suitable input subspace can be constructed using:
previous experimental input and output data, and/or
structural information based on prior system knowledge.
From the comments made following Theorem 6.3, any basis can be used to define the stimulation subspace without affecting subsequent performance (i.e. X may be exchanged for 
with any full rank 
).
with any full rank ).8.3.1 Selection Using Experimental Data
Assume that previous experiments (with any choice of input subspace) have yielded input and output signal pairs 
for plant M given by (8.2) (where 
is the stimulation applied to the n elements of the array). From these select those with outputs close to the required movement(s), and denote as 
. These can be used to produce a basis for the input set by directly inserting in X as
![$$\begin{aligned} \bar{X}(t) = [\varvec{z}^1(t), \varvec{z}^2(t), \dots , \varvec{z}^c(t)] \in \mathbb {R}^{n \times c} \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ14.gif)
and setting 
. If X is time-invariant, then a finite set of 
can instead be employed. In the case of linear M, any reference in the set spanned by a linear combination of 
(corresponding to the range of MX, 
), will be tracked with zero error using the ILC algorithms of Theorem 6.1. If 
does not belong to this set, then the subsequent error is the orthogonal projection of 
onto 
, as shown in Fig. 8.2. If 
, the dimension of MX is far smaller than that of M. The direct use of previous inputs is not ideal as q cannot be independently prescribed (since 
). Inputs may also be linearly dependent, and thereby provide no new information. This is now addressed.
for plant M given by (8.2) (where is the stimulation applied to the n elements of the array). From these select those with outputs close to the required movement(s), and denote as . These can be used to produce a basis for the input set by directly inserting in X as(8.14)
. If X is time-invariant, then a finite set of can instead be employed. In the case of linear M, any reference in the set spanned by a linear combination of (corresponding to the range of MX, ), will be tracked with zero error using the ILC algorithms of Theorem 6.1. If does not belong to this set, then the subsequent error is the orthogonal projection of onto , as shown in Fig. 8.2. If , the dimension of MX is far smaller than that of M. The direct use of previous inputs is not ideal as q cannot be independently prescribed (since ). Inputs may also be linearly dependent, and thereby provide no new information. This is now addressed.Fig. 8.2
Output space showing error as an orthogonal projection onto achievable plant dynamics
8.3.1.1 Previous Input Data
Let subspace dimension q be prescribed and data sets 
available. The next proposition provides optimizations to yield ![$$X : \mathscr {L}^q_2[0,T] \rightarrow \mathscr {L}^n_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq85.gif)
:
available. The next proposition provides optimizations to yield :Proposition 8.3
The distance between each ![$$\varvec{z}^i \in \mathscr {L}^n_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq86.gif)
and the closest element in the image of X is minimized by solving
where 
is the Hilbert-Schmidt norm and operators
![$$\begin{aligned} H&: \mathbb {R}^c \rightarrow \mathscr {L}^q_2[0,T] : \varvec{a} \mapsto \varvec{b} : \varvec{b}_i = \sum ^c_{i = 1} H_i \varvec{a}_i, \nonumber \\ Z&: \mathbb {R}^c \rightarrow \mathscr {L}^n_2[0,T] : \varvec{a} \mapsto \varvec{b} : \varvec{b}_i = \sum ^c_{i = 1} \varvec{z}^i \varvec{a}_i. \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ16.gif)
Using the Frobenius norm 
, this can also be expressed as
and the closest element in the image of X is minimized by solving(8.15)
is the Hilbert-Schmidt norm and operators(8.16)
, this can also be expressed as(8.17)
Proof
The total distance between each ![$$\varvec{z}^i \in \mathscr {L}^n_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq89.gif)
and the closest element in the image of
![$$\begin{aligned} X : \mathscr {L}^q_2[0,T] \rightarrow \mathscr {L}^n_2[0,T] : \varvec{r} \mapsto \varvec{z} : \varvec{z}(t) = \bar{X}(t) \varvec{r}(t). \end{aligned}$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_Equ18.gif)
is minimized by solving
where ![$$H_i \in \mathscr {L}^q_2[0,T]$$](/wp-content/uploads/2016/09/A352940_1_En_8_Chapter_IEq90.gif)
. This is simplified by using
since 
is an orthogonal basis of 
, it follows that
Note, from (8.20), that this can also be expressed as
and the closest element in the image of(8.18)
(8.19)
(8.20)
. This is simplified by using is an orthogonal basis of , it follows that(8.21)
