The model is based on previous research on the linearization of a multiplicative Hill-type model (Stroeve
1998), in which the muscle force is described by the multiplication of the activation signal
a, the force–length relationship
f
l
(
L
m
), the force–velocity relationship
f
v
(
V
m
) and the maximal force
F
max:
$$ F_{s} = a \, \cdot \, f_{{l}} \left( {L_{m} } \right) \cdot \, f_{v} \left( {V_{m} } \right) \cdot F_{ \max } $$
(11)
The first-order linearized activation term is a partial derivative of
F
s
to
a, and is approximated by:
$$ \partial F_{s} /\partial a = f_{{l}} \left( {L_{m} } \right) \cdot f_{v} \left( {V_{m} } \right) \cdot F_{ \max } $$
(12)
The variations of muscle force δ
F
s
caused by changing muscle activation δ
a around an operating point can now be described as:
$$ \begin{aligned} \delta F_{s} & = \left[ { \, f_{{l}} \left( {L_{m} } \right) \cdot f_{v} \left( {V_{m} } \right) \cdot F_{ \max } \, } \right] \cdot \delta a \\ &\quad + \left[ {a \cdot f_{v} \left( {V_{m} } \right) \cdot F_{ \max } \cdot \partial f_{{l}} \left( {L_{m} } \right)/\partial L_{m} } \right] \cdot \delta L_{m} \\ &\quad + \left[ {a \cdot f_{{l}} \left( {L_{m} } \right) \cdot F_{ \max } \cdot \partial f_{v} \left( {V_{m} } \right)/\partial V_{m} } \right] \cdot \delta V_{m} , \\ \end{aligned} $$
(13)
which can be rewritten in the nomenclature of the current study as:
$$ T_{{muscle}} = \left[ { \, f_{l} \left( {L_{m} } \right) \cdot f_{v} \left( {V_{m} } \right) \cdot F_{ \max } } \right] \cdot {\rm A} + [k_{a} ]\theta_{\text{muscle}} + \left[ { \, b_{a} } \right]\dot{\theta }_{\text{muscle}} $$
(14)
with
T
muscle now denoting deviations in muscle force (δ
F
s
),
A deviations in muscle activation (δ
a) due to reflexive activity, θ
muscle deviations in muscle stretch (δ
L
m
), and
\( \dot{\theta }_{\text{muscle}} \) the deviations in muscle stretch velocity (δ
V
m
). Note that the intrinsic stiffness and damping due to already contracted muscles is described by
k
a
and
b
a
, respectively. The term [
f
l
(
L
m
)
. f
v
(
V
m
)
. F
max] is for this model implicitly incorporated in the loop gain of each of the reflexive feedback loops (
k
p
, k
v
, k
f
).