Feedback Control Design

Fig. 3.1

Feedback control scheme with $K: \varvec{e} \mapsto \varvec{u}: \mathscr {L}_2^p \rightarrow \mathscr {L}_2^m$

This chapter addresses the design of feedback controllers to stabilize all joints in the upper limb system. Moreover, robust stability margins are derived to ensure that changes in the dynamics do not destabilize the system. To maximize practical utility, these are employed to derive bounds on the most significant sources of model inaccuracy, with explicit focus on muscle fatigue. In the next chapter a feedforward control action will be combined with the feedback controllers developed here in order to further improve tracking accuracy and hence ensure successful task completion.

3.1 General Feedback Control Framework

The combined mechanical and anthropomorphic system was shown to take the form

$\begin{aligned} M : \left\{ \begin{array}{rl} \dot{ \varvec{x} }_{s}(t) &{}= \varvec{f}_s ( \varvec{x}_{s}(t) ) + \varvec{g}_s ( \varvec{u}(t) ) \\ \varvec{\varPhi }(t) &{}= \varvec{h}_s ( \varvec{x}_{s}(t) ), \quad \quad \quad \quad \quad t \ge 0 \end{array} \right. \end{aligned}$

(3.1)

with components given by (2.7). Since $\varvec{f}_s (\cdot ), \varvec{g}_s (\cdot ), \varvec{h}_s (\cdot )$ are continuously differentiable, M has the properties of uniqueness and continuity [11]. In this chapter we consider a general feedback control structure given by

$\begin{aligned} K : \left\{ \begin{array}{rl} \dot{ \varvec{x} }_{c}(t) &{}= \varvec{f}_c ( \varvec{x}_{c}(t), \varvec{e}(t) ) \\ \varvec{u}(t) &{}= \varvec{h}_c ( \varvec{x}_{c}(t), \varvec{e}(t) ), \quad \quad t \ge 0 \end{array} \right. \end{aligned}$

(3.2)

where functions $\varvec{f}_c (\cdot ), \varvec{h}_c (\cdot )$ are continuously differentiable, so that K also has the properties of uniqueness and continuity. Figure 3.1 shows the combined structure, where the reference vector is denoted $\hat{\varvec{\varPhi }} \in \mathscr {L}^p_2$ and the tracking error is $\varvec{e} = \hat{\varvec{\varPhi }} - \varvec{\varPhi }$ . This must be designed to embed robustness to model uncertainty and disturbance, together with baseline tracking performance of the resulting closed-loop system

$\begin{aligned} \underbrace{ \left[ \begin{array}{c} \dot{\varvec{x}}_{s}(t) \\ \dot{\varvec{x}}_{c}(t) \end{array} \right] }_{\displaystyle { \dot{\varvec{x}}(t) }}&= \underbrace{ \left[ \begin{array}{c} \varvec{f}_s ( \varvec{x}_{s}(t) ) + \varvec{g}_s ( \varvec{h}_c ( \varvec{x}_{c}(t), \hat{\varvec{\varPhi }}(t) - \varvec{h}_s ( \varvec{x}_{s}(t) ) ) ) \\ \varvec{f}_c ( \varvec{x}_{c}(t), \hat{\varvec{\varPhi }}(t) - \varvec{h}_s ( \varvec{x}_{s}(t) ) ) \end{array} \right] }_{\displaystyle { \varvec{f} ( \varvec{x}(t), \hat{\varvec{\varPhi }}(t) )}} \nonumber \\ \varvec{\varPhi }(t)&= \underbrace{ \varvec{h}_s ( \varvec{x}_{s}(t) ) }_{\displaystyle { \varvec{h} ( \varvec{x}(t) ) }}, \quad \quad t \ge 0. \end{aligned}$

(3.3)

The functional movements used in rehabilitation may not involve all joint axes. Equally movement about certain joints may need to be actively avoided, due to the presence of subluxations, stiffness, or limited angular range of movement for example. To embed this flexibility in controlled joint selection, we define the set $\mathscr {P}$ containing the controlled joint indices, with elements $\mathscr {P} = \big \{ p_1, \ldots , p_{n_p} \big \}$ , $n_p \le p$ . We denote the complement of $\mathscr {P}$ by $\bar{\mathscr {P}} = \{ 1, \ldots p \} \setminus \mathscr {P}$ . For signal $\varvec{x}$ and set of distinct indices $\mathscr {S}$ we use notation $\varvec{x}_\mathscr {S}(t) = [\varvec{x}_{\mathscr {S}(1)}(t), \ldots \varvec{x}_{\mathscr {S}_{|\mathscr {S}|}}(t)]^\top$ where $\mathscr {S}(i)$ is the ith smallest element of $\mathscr {S}$ . With this notation, the controlled and uncontrolled joint angle signals are $\varvec{\varPhi }_{ \mathscr {P}} \in \mathscr {L}_2^{n_p}$ and $\varvec{\varPhi }_{ \bar{\mathscr {P}}} \in \mathscr {L}_2^{p - n_p}$ respectively. Armed with this notation, we can now introduce a definition of stability for use in control design:

Definition 3.1

Feedback controller (3.2) is said to stabilize the closed-loop system [M, K] about operating-point $\bar{\varvec{\varPhi }} \in \mathscr {L}_2^{n_p}$ if it achieves global asymptotic stability of the controlled joints, $\varvec{\varPhi }_{ \mathscr {P}}$ , about $\bar{\varvec{\varPhi }}$ .

Satisfying the condition of Definition 3.1 stabilizes joints with indices in set $\mathscr {P}$ , but musculo-tendon interaction and dynamic rigid body coupling cause movement in the remaining joints. We therefore next derive conditions to ensure stability of the uncontrolled joints, $\phi _i$ , $i \in \bar{\mathscr {P}}$ .

3.1.1 Stability of Unactuated Joints

To examine stability of uncontrolled joints, $\phi _i$ , $i \in \bar{\mathscr {P}}$ , first express components of $\varvec{C} ( \varvec{\varPhi }, \dot{\varvec{\varPhi }})$ in standard form as

$\begin{aligned} c_{i,j} = \sum ^{p}_{k = 1} c_{i,j,k} \dot{\phi }_k, \quad c_{i,j,k} = \frac{1}{2} \left( \frac{\partial b_{i,j}}{\partial \phi _k} + \frac{\partial b_{i,k}}{\partial \phi _j} - \frac{\partial b_{j,k}}{\partial \phi _i} \right) , \quad i,j = 1, \ldots , p \end{aligned}$

(3.4)

where $b_{i,j}$ are components of $\varvec{B}(\varvec{\varPhi })$ . Then using $\varvec{\eta }_1 = \varvec{\varPhi }_{ \bar{\mathscr {P}}}$ , $\varvec{\eta } = [\varvec{\eta }_1^\top , \; \varvec{\eta }_2^\top ]^\top$ , $\varvec{\xi } = [ \varvec{\xi }_1^\top , \; \varvec{\xi }_2^\top , \; \varvec{\xi }_3^\top ]^\top$ , the system (2.7) and controller (3.2) can be represented as

$\begin{aligned} \dot{\varvec{x}}= & {} \varvec{f} (\varvec{x}), \end{aligned}$

(3.5)

$\begin{aligned} \dot{\varvec{\eta }}= & {} \varvec{\omega } \left( \varvec{\xi } , \varvec{\eta } , t \right) , \end{aligned}$

(3.6)

$\begin{aligned} \varvec{\xi } (\varvec{x})= & {} \left[ \begin{array}{c} \bar{\varvec{\varPhi }} - \varvec{\varPhi }_{ \mathscr {P}} \\ \bar{\varvec{\varPhi }}^{(1)} - \varvec{\varPhi }_{ \mathscr {P}}^{(1)} \\ \bar{\varvec{\varPhi }}^{(2)} - \varvec{\varPhi }_{ \mathscr {P}}^{(2)} \end{array} \right] = \left[ \begin{array}{c} \bar{\varvec{\varPhi }} - \varvec{h}_{ \mathscr {P}} (\varvec{x}(t)) \\ \bar{\varvec{\varPhi }}^{(1)} - \varvec{h}_{ \mathscr {P}} ( \varvec{f} ( \varvec{x}(t) )) \\ \bar{\varvec{\varPhi }}^{(2)} - \varvec{h}_{ \mathscr {P}} ( \varvec{f}^\prime ( \varvec{x}(t) ) \varvec{f} ( \varvec{x}(t) )) \end{array} \right] , \end{aligned}$

(3.7)

where $\varvec{\varPhi } = [ \varvec{\eta }_1^\top , \bar{\varvec{\varPhi }}^\top - \varvec{\xi }_1^\top ]^\top$ , $\dot{\varvec{\varPhi }} = [ \varvec{\eta }_2^\top , (\bar{\varvec{\varPhi }}^{(1)})^\top - \varvec{\xi }_2^\top ]^\top$ , and the uncontrolled joint dynamics are

$\begin{aligned}&\varvec{\omega } \left( \varvec{\xi } , \varvec{\eta }, t \right) \\&\quad =\left( \begin{array}{c} \varvec{\eta }_2 \\ \varvec{B}_{ \bar{\mathscr {P}}}^{-1}(\varvec{\varPhi }) \Big ( \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{\varvec{\varPhi }} + \varvec{\xi }_1 ), \bar{\varvec{\varPhi }} - \varvec{\xi }_1, \bar{\varvec{\varPhi }}^{(1)} - \varvec{\xi }_2 \big ) - \varvec{C}_{ \bar{\mathscr {P}}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}) \varvec{\eta }_2 - \varvec{G}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi } ) \\ - \varvec{C}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }}) \big ( \bar{\varvec{\varPhi }}^{(1)} - \varvec{\xi }_2 \big ) - \varvec{F}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi }, \dot{\varvec{\varPhi }} ) - \varvec{K}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi } ) - \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi }) \big ( \bar{\varvec{\varPhi }}^{(2)} - \varvec{\xi }_3 \big ) \Big ) \end{array} \right) . \end{aligned}$

Terms $\varvec{C}_{ \bar{\mathscr {P}}}(\varvec{\varPhi },\dot{\varvec{\varPhi }})$ and $\varvec{C}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }})$ respectively have elements

$\begin{aligned} C_{{ \bar{\mathscr {P}}},i,j} = \sum ^{n}_{k = 1} c_{{ \bar{\mathscr {P}}}(i),{ \bar{\mathscr {P}}}(j),k} \dot{\phi }_k, \quad C_{{ \bar{\mathscr {P}}} \mathscr {P},i,j} = \sum ^{n}_{k = 1} c_{{ \bar{\mathscr {P}}}(i),\mathscr {P}(j),k} \dot{\phi }_k \end{aligned}$

and likewise $\varvec{B}_{ \bar{\mathscr {P}}}(\varvec{\varPhi })$ and $\varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi })$ have elements

$\begin{aligned} B_{{ \bar{\mathscr {P}}},i,j} = b_{{ \bar{\mathscr {P}}}(i),{ \bar{\mathscr {P}}}(j)}, \quad B_{{ \bar{\mathscr {P}}} \mathscr {P},i,j} = b_{{ \bar{\mathscr {P}}}(i),\mathscr {P}(j)}. \end{aligned}$

(3.8)

Assuming the passive parameter form (2.12), $\varvec{F}_{ \bar{\mathscr {P}}}(\varvec{\varPhi },\dot{\varvec{\varPhi }})$ has elements

$\begin{aligned} F_{ \bar{\mathscr {P}},i}( \varvec{\varPhi }, \dot{\varvec{\varPhi }} )&= F_{e,{{ \bar{\mathscr {P}}}(i)}} (\phi _{{ \bar{\mathscr {P}}}(i)}) + F_{v,{{ \bar{\mathscr {P}}}(i)}} (\dot{\phi }_{{ \bar{\mathscr {P}}}(i)}) \\&:= F_{e,i} (\varvec{\varPhi }_{ \bar{\mathscr {P}}} ) + F_{v,i} (\dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}} ), \quad \quad \quad \quad i = 1, \cdots p - n_p \end{aligned}$

From (3.5)–(3.7) the surface $\varvec{\xi } = \varvec{0}$ defines an integral manifold for the system

$\begin{aligned} \dot{\varvec{\eta }} = \varvec{\omega } \left( \varvec{0} , \varvec{\eta }, t \right) . \end{aligned}$

(3.9)

Since the controlled joints are assumed to be stable about this surface via Definition 3.1, system (3.9) is globally attractive and defines the zero dynamics relative to the controlled output $\varvec{\varPhi }_{ \mathscr {P}} = \bar{\varvec{\varPhi }}$ . We next state the Center Manifold Theorem, see [12].

Theorem 3.1

Suppose that $\varvec{\omega } \left( \varvec{0} , \varvec{\eta }_1^*, t \right) = \varvec{0}$ for $t \ge 0$ , i.e. $(\varvec{0}, \varvec{\eta }_1^*)$ is an equilibrium of the full system (3.5)–(3.7), and $\varvec{\eta }_1^*$ is an equilibrium of the zero dynamics (3.9), and that Definition 3.1 is satisfied. Then $(\varvec{0}, \varvec{\eta }_1^*)$ of the full system (3.5)–(3.7) is locally stable if $\varvec{\eta }_1^*$ is locally stable for dynamics (3.9).

Stability of all joints, $\varvec{\varPhi }$ , is hence assured if both the controlled and uncontrolled joints are independently stable. The former is guaranteed via Definition 3.1, and the following theorem gives conditions for the latter.

Theorem 3.2

Let feedback controller K satisfy Definition 3.1 and uncontrolled joints, $\varvec{\varPhi }_{ \bar{\mathscr {P}}}$ , be passive with respect to $(\varvec{\tau }^*_{ \bar{\mathscr {P}} }(\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }),\varvec{\varPhi }^*_{ \bar{\mathscr {P}} })$ , i.e.

$\begin{aligned}&( \varvec{\varPhi }_{ \bar{\mathscr {P}}} - \varvec{\varPhi }^*_{ \bar{\mathscr {P}}} )^\top \big ( \varvec{F}_{e} ( \varvec{\varPhi }_{ \bar{\mathscr {P}}}) + \varvec{G}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi }_{ \bar{\mathscr {P}}} ) + \varvec{K}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi }_{ \bar{\mathscr {P}}} ) - \varvec{\tau }^*_{ \bar{\mathscr {P}} } ( \varvec{\varPhi }_{ \bar{\mathscr {P}}} ) \big ) \ge 0 \end{aligned}$

(3.10)

where $\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }$ satisfies $\varvec{F}_e ( \varvec{\varPhi }^*_{ \bar{\mathscr {P}} } ) + \varvec{G}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi }_{ \bar{\mathscr {P}}}^*) + \varvec{K}_{ \bar{\mathscr {P}}} ( \varvec{\varPhi }_{ \bar{\mathscr {P}}}^*) = \varvec{\tau }^*_{ \bar{\mathscr {P}} } ( \varvec{\varPhi }_{ \bar{\mathscr {P}}}^*)$ , with $\varvec{\tau }^*_{ \bar{\mathscr {P}} } ( \varvec{\varPhi }^*_{ \bar{\mathscr {P}}} ) = \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K (\hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }}, \dot{\bar{\varvec{\varPhi }}} \big ) - \bar{\varvec{C}}_{ \bar{\mathscr {P}} \mathscr {P}} (\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }) \dot{\bar{\varvec{\varPhi }}} - \varvec{B}_{ \bar{\mathscr {P}} \mathscr {P}} (\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }) \ddot{\bar{\varvec{\varPhi }}}$ the moment transferred from controlled to uncontrolled joints, and let the uncontrolled joint damping function satisfy the sector bounds

$\begin{aligned} F_{v,i}( \dot{\phi } ) {\left\{ \begin{array}{ll} {>}\bar{F}_{v,i} \dot{\phi }_i &{} \quad \text {if} \quad \dot{\phi } > 0,\\ {<}\bar{F}_{v,i} \dot{\phi }_i &{} \quad \text {otherwise.} \end{array}\right. } \quad i \in \bar{\mathscr {P}} \quad \text {where} \quad \bar{F}_{v,i} = \sum _{i, k \notin \mathscr {P}, \; i \ne k} \Big | \sum _{j \in \mathscr {P}} c_{i,j,k} \dot{\bar{\phi }}_{j} \Big | \end{aligned}$

(3.11)

Then the uncontrolled joints are locally stable about $(\varvec{\tau }^*_{ \bar{\mathscr {P}} }(\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }),\varvec{\varPhi }^*_{ \bar{\mathscr {P}} })$ .

Proof

From (2.7) the uncontrolled system dynamics are given by

$\begin{aligned}&\varvec{B}_{ \bar{\mathscr {P}}}(\varvec{\varPhi }) \ddot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}} + \varvec{C}_{ \bar{\mathscr {P}}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}) \dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}} + \varvec{C}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }}) \dot{\varvec{\varPhi }}_{\mathscr {P}} + \varvec{F}_{ \bar{\mathscr {P}}}(\varvec{\varPhi },\dot{\varvec{\varPhi }}) + \varvec{G}_{ \bar{\mathscr {P}}}(\varvec{\varPhi }) + \varvec{K}_{ \bar{\mathscr {P}}}(\varvec{\varPhi }) \nonumber \\&\quad + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\varPhi }) \ddot{\varvec{\varPhi }}_{\mathscr {P}} = \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \varvec{\xi }_1 + \hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }} - \varvec{\xi }_1, \bar{\varvec{\varPhi }}^{(1)} - \varvec{\xi }_2 \big ). \end{aligned}$

(3.12)

The term $\varvec{C}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }})$ can be partitioned as $\overline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }}_{\mathscr {P}}) + \underline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}})$ , where

$\begin{aligned}&\overline{C}_{{ \bar{\mathscr {P}}} \mathscr {P},i,j} = \sum ^{n_p}_{k = 1} c_{{ \bar{\mathscr {P}}}(i),\mathscr {P}(j),\mathscr {P}(k)} \dot{\phi }_{\mathscr {P}(k)} \quad \text{ and } \quad \underline{C}_{{ \bar{\mathscr {P}}} \mathscr {P},i,j} = \sum ^{p - n_p}_{k = 1} c_{{ \bar{\mathscr {P}}}(i),\mathscr {P}(j),{ \bar{\mathscr {P}}}(k)} \dot{\phi }_{{ \bar{\mathscr {P}}}(k)}. \end{aligned}$

Furthermore $\underline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}}) \dot{\varvec{\varPhi }}_{\mathscr {P}}$ can be written as $\underline{\varvec{C}}_{ \bar{\mathscr {P}}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}_{\mathscr {P}}) \dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}}$ with

$\begin{aligned} \underline{C}_{{ \bar{\mathscr {P}}},i,j} =&\sum ^{n_p}_{k = 1} c_{{ \bar{\mathscr {P}}}(i),\mathscr {P}(k),{ \bar{\mathscr {P}}}(j)} \dot{\phi }_{\mathscr {P}(k)}. \end{aligned}$

This enables (3.12) to be rewritten using substitutions $\varvec{C}_{{ \bar{\mathscr {P}}} \mathscr {P}} \Leftrightarrow \overline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}$ and $\varvec{C}_{{ \bar{\mathscr {P}}}} \Leftrightarrow \overline{\varvec{C}}_{ \bar{\mathscr {P}}}$ , where $\overline{\varvec{C}}_{ \bar{\mathscr {P}}} = \varvec{C}_{ \bar{\mathscr {P}}} + \underline{\varvec{C}}_{ \bar{\mathscr {P}}}$ , to give

$\begin{aligned}&\varvec{B}_{ \bar{\mathscr {P}}} (\varvec{\varPhi }) \ddot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}} + \overline{\varvec{C}}_{ \bar{\mathscr {P}}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}) \dot{\varvec{\varPhi }}_{ \bar{\mathscr {P}}} + \overline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\varPhi },\dot{\varvec{\varPhi }}_{\mathscr {P}}) \dot{\varvec{\varPhi }}_{\mathscr {P}} + \varvec{F}_{ \bar{\mathscr {P}}} (\varvec{\varPhi },\dot{\varvec{\varPhi }}) + \varvec{G}_{ \bar{\mathscr {P}}} (\varvec{\varPhi }) \nonumber \\&\quad + \varvec{K}_{ \bar{\mathscr {P}}} (\varvec{\varPhi }) + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\varPhi }) \ddot{\varvec{\varPhi }}_{\mathscr {P}} = \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \varvec{\xi }_1 + \hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }} - \varvec{\xi }_1, \bar{\varvec{\varPhi }}^{(1)} - \varvec{\xi }_2 \big ). \end{aligned}$

(3.13)

When $\varvec{\xi } = \varvec{0}$ the zero dynamics correspond to the system

$\begin{aligned}&\varvec{B}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1) \dot{\varvec{\eta }}_2 + \overline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) \varvec{\eta }_2 + \overline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\eta }_1) \bar{\varvec{\varPhi }}^{(1)} + \varvec{F}_{ \bar{\mathscr {P}}} ( \varvec{\eta }_1, \varvec{\eta }_2 ) + \varvec{G}_{ \bar{\mathscr {P}}} ( \varvec{\eta }_1 ) + \varvec{K}_{ \bar{\mathscr {P}}} ( \varvec{\eta }_1 ) \nonumber \\&\quad + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\eta }_1) \bar{\varvec{\varPhi }}^{(2)} = \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \hat{\varvec{\varPhi }}_{\mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }}, \bar{\varvec{\varPhi }}^{(1)} \big ) \end{aligned}$

(3.14)

where the muscle dynamic forms (2.2)–(2.4) mean $\varvec{\tau }_{ \bar{\mathscr {P}}} ( \cdot )$ is bounded input, bounded output stable, and functional dependence on $\bar{\varvec{\varPhi }}$ , $\bar{\varvec{\varPhi }}^{(1)}$ has been omitted. System (3.14) equates to $\dot{\varvec{\eta }}_2 = - \varvec{h}(\varvec{\eta }_1,\varvec{\eta }_2) - \varvec{g}(\varvec{\eta }_1)$ where

$\begin{aligned} \varvec{h}(\varvec{\eta }_1,\varvec{\eta }_2) &= \varvec{B}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1)^{-1} \Big ( \overline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) \varvec{\eta }_2 + \varvec{F}_{ e} (\varvec{\eta }_1 ) + \varvec{F}_{ v} (\varvec{\eta }_2 ) + \varvec{G}_{ \bar{\mathscr {P}}} ( \varvec{\eta }_1 ) + \varvec{K}_{ \bar{\mathscr {P}}} ( \varvec{\eta }_1 ) \Big ), \\ \varvec{g}(\varvec{\eta }_1) &= \varvec{B}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1)^{-1} \Big ( \overline{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\eta }_1) \bar{\varvec{\varPhi }}^{(1)} + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} ( \varvec{\eta }_1) \bar{\varvec{\varPhi }}^{(2)} - \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \hat{\varvec{\varPhi }}_{ \mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }}, \bar{\varvec{\varPhi }}^{(1)} \big ) \Big ). \end{aligned}$

The equilibrium point of the uncontrolled joints satisfies $\varvec{h}(\varvec{\eta }_1^*,\varvec{0}) + \varvec{g}(\varvec{\eta }_1^*) = \varvec{0}$ , and, following [13], the system can be interpreted as conservative system $\dot{\varvec{\eta }}_2 + \varvec{g}( \tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) = \varvec{0}$ acted on by external force $- \varvec{h}(\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*,\varvec{\eta }_2)$ where $\tilde{\varvec{\eta }}_1 = \varvec{\eta }_1 - \varvec{\eta }_1^*$ . Accordingly, introduce energy function

$\begin{aligned} V(\tilde{\varvec{\eta }}_1,\varvec{\eta }_2)&= \varvec{\eta }_2^\top \frac{\varvec{B}_{ \bar{\mathscr {P}}}(\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*)}{2} \varvec{\eta }_2 + \int ^{\tilde{\varvec{\eta }}_1}_0 \big ( \varvec{F}_{e} (\varvec{\sigma }) + \varvec{G}_{ \bar{\mathscr {P}}} (\varvec{\sigma }) + \varvec{K}_{ \bar{\mathscr {P}}} (\varvec{\sigma }) \big ) \delta \varvec{\sigma } \\&\quad + \int ^{\tilde{\varvec{\eta }}_1}_0 \Big ( \bar{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\varvec{\sigma }) \bar{\varvec{\varPhi }}^{(1)} + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\varvec{\sigma }) \bar{\varvec{\varPhi }}^{(2)} - \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \hat{\varvec{\varPhi }}_{ \mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }}, \bar{\varvec{\varPhi }}^{(1)} \big ) \Big ) \delta \varvec{\sigma }. \end{aligned}$

The first and second terms respectively correspond to the kinetic and potential energy in the uncontrolled joint system, and the third to the potential energy transferred from the controlled joints. The rate of energy satisfies

$\begin{aligned} \dot{V}(\tilde{\varvec{\eta }}_1,\varvec{\eta }_2)&= \varvec{\eta }_2^\top \varvec{B}_{ \bar{\mathscr {P}}}(\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) \dot{\varvec{\eta }}_2 + \varvec{\eta }_2^\top \frac{\dot{\varvec{B}}_{ \bar{\mathscr {P}}}(\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*)}{2} \varvec{\eta }_2 \\&\quad + \varvec{\eta }_2^\top \big ( \varvec{F}_{e} (\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) +\varvec{G}_{ \bar{\mathscr {P}}} (\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) + \varvec{K}_{ \bar{\mathscr {P}}} (\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) \big ) \\&\quad + \varvec{\eta }_2^\top \Big ( \bar{\varvec{C}}_{{ \bar{\mathscr {P}}} \mathscr {P}}(\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) \bar{\varvec{\varPhi }}^{(1)} + \varvec{B}_{{ \bar{\mathscr {P}}} \mathscr {P}} (\tilde{\varvec{\eta }}_1 + \varvec{\eta }_1^*) \bar{\varvec{\varPhi }}^{(2)} \\&\qquad \qquad \;\;- \varvec{\tau }_{ \bar{\mathscr {P}}} \big ( K ( \bar{\varvec{\varPhi }}_{ \mathscr {P}} - \bar{\varvec{\varPhi }} ), \bar{\varvec{\varPhi }}, \bar{\varvec{\varPhi }}^{(1)} \big ) \Big ) \\&= \varvec{\eta }_2^\top \left( \frac{\dot{\varvec{B}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1)}{2} - \overline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) \right) \varvec{\eta }_2 - \varvec{\eta }_2^T \varvec{F}_v ( \varvec{\eta }_2 ) \\&\le \; \varvec{\eta }_2^\top \left( \frac{1}{2} \dot{\varvec{B}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1) - \overline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) - \bar{\varvec{F}}_v \right) \varvec{\eta }_2. \end{aligned}$

Hence the system converges to $\varvec{\eta }_1 = \varvec{\eta }_1^*$ , $\varvec{\eta }_2 = \dot{\varvec{\eta }}_1 = \varvec{0}$ if

$\begin{aligned}&\min _i \mathfrak {R}\left( \lambda _i \left( \frac{\dot{\varvec{B}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1)}{2} - \varvec{C}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) - \underline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) - \bar{\varvec{F}}_v \right) \right) < 0. \end{aligned}$

As $\frac{1}{2} \dot{\varvec{B}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1) - \varvec{C}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2)$ is skew-symmetric, a sufficient condition is that the term $\underline{\varvec{C}}_{ \bar{\mathscr {P}}}(\varvec{\eta }_1,\varvec{\eta }_2) + \bar{\varvec{F}}_v$ is diagonally dominant with positive diagonal entries, which is satisfied by (3.11). $\quad \square$

The conditions of Theorem 3.2 motivate the following intuitive guidelines for ensuring stability of the uncontrolled joints:

Procedure 1

(Design guidelines for stabilizing uncontrolled joints)

Add damping: The condition on $\varvec{F}_v( \dot{\phi } )$ , given by (3.11), can always be met by adding viscous damping to the uncontrolled joints.

Feedback controller tuning: Bounds on $\varvec{F}_v$ scale with $| \bar{\varPhi }^{(1)} |$ , and hence (3.10) and (3.11) are easy to satisfy if the controlled joint equilibrium trajectory is smooth. This motivates (de)-tuning of feedback controller (3.2).

Reference selection: The controlled joint equilibrium trajectory can also be made smoother through selection of the reference trajectory $\hat{\varvec{\varPhi }}_{ \mathscr {P}}$ .

Arm structure selection: The amount of damping required for stability is dictated by the degree of axis coupling which is reflected in the magnitude of elements $\underline{\varvec{C}}_{ \bar{\mathscr {P}}} (\cdot )$ . The components of $\underline{\varvec{C}}_{ \bar{\mathscr {P}}}(\cdot )$ are related to the elements of $\varvec{B}(\varvec{\varPhi })$ via (3.4). Note that they do not involve components on the principal diagonal of $\varvec{B}(\varvec{\varPhi })$ and hence the bound is solely dependent on the amount of interaction between the system joints. With no interaction $\overline{F}_{v,i} = 0$ , reducing to the requirement that $\varvec{F}_v (\cdot )$ is passive.

Stimulated muscle selection: Musculo-tendon coupling produces moments $\varvec{\tau }_{ \bar{\mathscr {P}}}( \cdot )$ about uncontrolled joints due to applied ES. This solely has the effect of displacing the equilibrium point $\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }$ .

Mechanical support: This also displaces the equilibrium point $\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }$ , but can also be used to satisfy passivity condition (3.10). Note that the mechanical support must provide sufficient support such that an equilibrium point, $\varvec{\varPhi }^*_{ \bar{\mathscr {P}} }$ , exists for the uncontrolled joints.

3.2 Case Study: Input-Output Linearizing Controller

The feedback control design approach is next illustrated by applying it to the clinically relevant system that was introduced in Sect. 2.2.5. Here ES is applied to the anterior deltoid and triceps muscles using inputs

and

respectively. The kinematics are shown in Fig. 2.3, and the clinical objective is to ensure $\phi _2(t)$ and $\phi _5(t)$ track reference signals $\hat{\phi }_2(t)$ and $\hat{\phi }_5(t)$ respectively, with the remaining joint angles stable [14, 15]. Hence we set

and $\mathscr {P} = \{ 2, 5 \}$ .

The linear actuation dynamics $h_{\textit{LAD},i}(t)$ ,

appearing in dynamic model (2.7) can be assumed to be second order [16], so that without loss of generality $\mathscr {L} \left\{ h_{{\textit{LAD}},i}(t) \right\} = \frac{n_{i,1} s + n_{i,2}}{ s^2 + d_{i,1} s + d_{i,2}}$ . This gives rise to the Hammerstein structures

$\begin{aligned} \dot{\varvec{x}}_{i}(t)&= \underbrace{\left[ \begin{array}{cc} -d_{i,1} &{} -d_{i,2} \\ 1 &{} 0 \end{array} \right] }_{\varvec{M}_{A,i}} \varvec{x}_{i}(t) + \underbrace{\left[ \begin{array}{c} 1 \\ 0 \end{array} \right] }_{\varvec{M}_{B,i}} h_{{\textit{IRC}},i} ( u_{i}(t) ), \nonumber \\ h_i(u_i,t)&= \underbrace{[ \begin{array}{cc} n_{i,1}&n_{i,2} \end{array} ]}_{\varvec{M}_{C,i}} \varvec{x}_{i}(t), \quad \qquad \qquad \quad \quad \; i \in \{ 1, 2 \}. \end{aligned}$

(3.15)

We employ musclo-tendon mapping (2.4) and since muscles are aligned with joints

$\begin{aligned}&\varvec{\tau }_1 = 0, \; \varvec{\tau }_2(\varvec{u},\varvec{\varPhi },\dot{\varvec{\varPhi }}) = F_{M,2,1}( \phi _2(t), \dot{\phi }_2(t) ) h_1(u_1,t), \; \varvec{\tau }_3 = 0, \; \varvec{\tau }_4 = 0, \\&\qquad \qquad \quad \varvec{\tau }_5(\varvec{u},\varvec{\varPhi },\dot{\varvec{\varPhi }}) = F_{M,5,2}( \phi _5(t), \dot{\phi }_5(t) ) h_2(u_2,t). \end{aligned}$

Using $\varvec{x}_s = [\varvec{\varPhi }^\top , \dot{\varvec{\varPhi }}^\top , \varvec{x}_2^\top , \varvec{x}_5^\top ]^\top$ the controlled dynamics, M, of system (3.1) are hence

$\begin{aligned} \dot{\varvec{x}}_s&= \underbrace{ \left[ \begin{array}{c} \dot{\varvec{ \varPhi }} \\ \varvec{B} ( \varvec{ \varPhi })^{-1} \left( \left[ \begin{array}{c} 0 \\ F_{M,2,1}( \phi _2, \dot{\phi }_2 ) \varvec{M}_{C,1} \varvec{x}_1 \\ 0 \\ 0 \\ F_{M,5,2}( \phi _5, \dot{\phi }_5 ) \varvec{M}_{C,2} \varvec{x}_2 \end{array} \right] - \varvec{X}(\varvec{\varPhi },\dot{\varvec{\varPhi }}) \right) \\ \varvec{M}_{A,1} \varvec{x}_1 \\ \varvec{M}_{A,2} \varvec{x}_2 \end{array} \right] }_{\varvec{f}_s(\varvec{x}_s)} + \underbrace{ \overbrace{ \left[ \begin{array}{cc} 0 &{}0 \\ 0 &{}0 \\ 0 &{}0 \\ 0 &{}0 \\ 0 &{}0 \\ \varvec{M}_{B,1} &{}0 \\ 0 &{}\varvec{M}_{B,2} \end{array} \right] }^{\displaystyle {[g_1(\varvec{x}_s), \; g_2(\varvec{x}_s)]}} \left[ \begin{array}{c} h_{\textit{IRC},1}(u_1) \\ h_{\textit{IRC},2}(u_2) \end{array} \right] }_{\varvec{g}_s(\varvec{u})} \nonumber \\ \varvec{\varPhi }_{ \mathscr {P}}&= \left[ \begin{array}{c} \phi _2 \\ \phi _5 \end{array} \right] = \left[ \begin{array}{c} h_1(\varvec{x}_s) \\ h_2(\varvec{x}_s) \end{array} \right] \end{aligned}$

(3.16)

where

$\begin{aligned} \varvec{X}(\varvec{\varPhi },\dot{\varvec{\varPhi }}) = \varvec{C} (\varvec{ \varPhi },\dot{\varvec{ \varPhi }}) \dot{\varvec{ \varPhi }} + \varvec{F} (\varvec{ \varPhi },\dot{\varvec{ \varPhi }}) + \varvec{G} (\varvec{ \varPhi }) + \varvec{K} (\varvec{ \varPhi }). \end{aligned}$

To satisfy Definition 3.1, we next design K using input-output linearization in order to control $\varvec{\varPhi }_{ \mathscr {P}}$ using $\varvec{u} = [u_1, u_2]^\top$ . As described in [12], for an $m \times m$ system the control action is

$\begin{aligned} \Big [ \begin{array}{c} h_{\textit{IRC},1}(u_1) \\ h_{\textit{IRC},2}(u_2) \end{array} \Big ] = \varvec{\chi }(\varvec{x}_s)^{-1} \big ( \varvec{v} - \varvec{\mu }(\varvec{x}_s) \big ) \end{aligned}$