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KAAP686
Neuromusculoskeletal Modeling
W. Rose
2017-02-22
Notes on Buchanan & Manal (2004), Neuromusculoskeletal Modeling, J. Appl. Biomech.
30: 367-395.
EMG-based Forward Dynamics vs. Inverse Dynamics
EMG-based Forward Dynamics
Predict forces & movements from the EMGs. Requires a model of relation between
EMG & muscle force, and a model of relation between muscle force & movement. Each
of these two stages has many uncertainties. First step (EMG to force) is based on the idea
that both EMG and force are results of the neural commands to muscle. It is possible or
likely that the transfer function from neural command to EMG varies with time, position,
coactivation, etc., and that the transfer function from neural command to force also varies
with those factors, but that they don’t vary the exact same way, and therefore the transfer
function between EMG and force is state-dependent, i.e. not constant. The second step
(force to movement) requires detailed knowledge of muscle lines of action, moment
arms, moments of inertia, etc. Furthermore, because position is doubly-integrated
version of force (or torque), small error in force can accumulate to make large error in
position. See integration_error.xls.
Effect of Force Offset
2.5
6
F (b=0.00)
F (b=0.02)
2.0
5
F (b=0.05)
X (b=0.00)
1.5
4
1.0
3
X (b=0.05)
0.5
2
0.0
1
-0.5
0
-1.0
-1
0
2
4
6
8
Time
Inverse Dynamics
10
12
14
Position
Force
X (b=0.02)
Measure displacements and external forces, estimate muscle forces. Done by
differentiating positions and angles twice to get accelerations and angular accelerations.
From those, and from external forces and from limb inertias and moments of inertia,
estimate forces and torques across joints. From those forces and torques one might try to
estimate muscle forces and neural commands.
Problems with this approach: Accuracy of the estimated forces is limited by accuracy of
the estimates of limb inertias & moments of inertia. Differentiation and double
differentiation amplifies noise. Joint reaction forces and torques are net values.
Therefore any antagonist or agonist co-contraction effects cannot be determined or teased
apart. Finally, there are not good models to obtain muscle activation from muscle force.
Because force tends to be a low-passed version of activation, it is easier to go from
activation to force than vice versa.
Muscle Activation Dynamics
Muscle Activation Dynamics (Twitch Response)
for different values of g1, g2
1.0
-0.9, -0.9
-0.5, -0.9
-0.2, -0.9
-0.05, -0.9
-0.5, -0.5
-0.2, -0.5
-0.2, -0.2
u(k) / umax
0.8
0.6
0.4
0.2
0.0
0
10
20
TIME (samples)
30
Chart above is from muscle_activation_dynamics.xlsx.
How to convert EMG to muscle force? First, convert raw EMG to smoothed EMG by
rectify and low-pass filter (“linear envelope” filter). Then normalize by EMGmax,
measured during MVIC. This should give e(t) in range (0, +1). This signal is converted
to muscle activation a(t) by a nonlinear dynamic process. We do this by using a pure
delay cascaded with a linear dynamic system cascaded with a static, or instantaneous
(means the same thing) nonlinearity. The delay converts e(t) to e(t-d), where d is
magnitude of the delay. The linear dynamic system converts e(t) to u(t). (Like e(t), u(t)
will be in the range (0,1).) The linear system has unity gain at DC.1 Then an
“instantaneous” (meaning it has no dynamics, no smoothing, etc) nonlinearity is used to
convert u(t) to a(t), where a(t) is also in the range (0, +1). The nonlinearity is introduced
because if one plots steady isotonic force vs. EMG for various levels of force, one finds a
nonlinear relation, especially near the low end. (See Fig. 4A in paper.) This results in the
muscle activation signal a(t).
Converting Muscle Activation to Force
Muscle activation is converted to muscle force by scaling, with some more complicated
effects added as well. First the simple part: active muscle force = FA(t) = Fmax*a(t),
where Fmax= peak muscle force. We cannot stop here for two reasons: force-velocity
relation and force-length relation. In general, a muscle that is shortening produces less
force than an isometric muscle. We need to take this into account. Furthermore, a
muscle has a length, or range of lengths, at which it produces peak active force, and at
shorter or longer lengths it produces less force.
Effects of length
A muscle generates some force due to its passive elastic properties. The active and
passive components of the muscle are considered to be in parallel, which means the
forces generated by them are additive. Thus muscle force is now
F(t,L) = FA(t) + FP(L)
where FA(t)=Fmax*a(t)*fA(L) and FP(L)=Fmax*fP(L). fA(L) varies from 0 to 1 and and
fP(L) varies from 0 to infinity. Both are actually functions of normalized muscle length,
i.e. length divided by “optimum length”, where optimum length = L0 = length at which
active force is maximal. fA(L) captures the length-dependency of muscle force. This
length dependence appears to be instantaneous in the model used here. It can be
understood as a consequence of the number of overlapping crossbridges among the
sliding actin and myosin filaments, although the real story is more complicated. The
optimum length is somewhat activation-dependent (this is an experimental finding): L0
gets somewhat shorter (about 15% or so in some muscles) as activation a(t) increases
from 0 to 1. A muscle that is <50% or > 150% of optimum length generates no force, i.e.
fA(L/L0) is 0 if L/L0 <0.5 or if L/L0 > 1.5.
1
The linear dynamic system used is a second order system, analogous to a mass-spring-dashpot system. It
can be modeled like other second order lowpass filters. The particular form chosen by the authors is a
second order model of the form
u(t) + b1*u(t-1) + b2*u(t-2) = a*e(t), i.e. u(t)=a*e(t) - b1*u(t-1) - b2*u(t-2).
The requirement of unity gain means that a=1+b1+b2. Therefore there are only 2 adjustable parameters: b1
and b2. We can convert b1, b2 to an equivalent pair of variables g1 and g2 which are equally good at
characterizing the dynamic system. The advantage of using g1 and g2 is that the allowed values for g1 and
g2 that guarantee filter stability (i.e. non-infinite values) are simple: |g1|<1 and |g2|<1. The relation
between b1,b2 and g1,g2 is that 1+b1*x + b2*x2 = (1 + g1*x)(1+g2*x), i.e. b1=g1+g2 and b2=g1*g2.
A graph of the allowed values of g1,g2 and the corresponding allowed values of b1,b2 is in spreadsheet
muscle_activation_dynamics.xlsx. The same spreadsheet shows impulse response of the system as g1, g2
are varied. It turns out that if either g1 or g2 or both are >0, the system’s response to an impulse (i.e. the
force of a twitch) is oscillatory, which is non-physiologic. If either g1 or g2=0 the response is a simple
decaying exponential which is also non-physiologic. If both g1 and g2 are <0 (but > -1 for stability) then
physiological twitch forces are generated. See muscle_activation_dynamics.xlsx.
Effects of shortening or lengthening velocity
The Hill equation is used to adjust FA(t,L):
FA(t,L) = Fmax*a(t)*fA(L)
is replaced with
FA(t,L) = [(Fmax*b - a*v)/(v+b)]*a(t)*fA(L)
where a and b are constants that describe the shape of the force-velocity relationship.
Modeling the tendon
A smooth and slightly nonlinear relation is used. Force in muscle and force in tendon are
considered to be always and instantaneously equal since they are in series.
Pennation angle
Component of muscle force along long axis, in series with the tendon is
Flong(t,L) = F(t,L)*cos(φ(t)) where φ=pennation angle. φ=0 means muscle fibers are
along the long axis of muscle. Pennation angle changes as muscle length changes. The
effect is considered to be instantaneous and is modeled by eq. 33 which simplifies to
L(t)*sin(φ(t)) = L0*sin(φ0), where φ0 = pennation angle at optimum muscle fiber length
L0. This can be rewritten
sin(φ(t))/ sin(φ0) = L0/ L(t). (Reminiscent of Snell’s law for refraction.) It keeps the
width of the muscle constant, since the width is L(t)*sin(φ(t)).
Adjusting Parameters
Adjustable parameters that will not change with time include maximum force Fmax,
optimum fiber length L0, pennation angle at optimum length φ0, and tendon slack length.
These are determined separately for each muscle. Maximal shortening velocity vmax
(related to Hill force-velocity eqn) is considered constant, from the literature, same in all
muscles modeled.
Running the Model
A second order linear differential equation relates e(t) and u(t) for each muscle. The
many other nonlinearities present are also accounted for to get a nonlinear system of
differential equations. This can be written as a system of first order (ordinary, not partial)
differential equations. The solution can be obtained with a Runge-Kutta algorithm with
adaptive stepsize. Runge-Kutta-Fehlberg is one such algorithm, Matlab’s ode45() is a
very similar Runge-Kutta routine. Syntax for using ode45 is
[T,Y] = ode45(odefun,[Tstart Tstop],y0)
where ode45() puts values into the vector T and the array Y. T contains the times at
which solution is calculated; the columns of Y are the different variables Y1(t), Y2(t),
etc. odefun is a handle with the name of a function defining the first derivatives of the
Yi’s; the vector y0 has the initial conditions for the Yi’s.