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Appendix

Muscle Formulation

LifeMOD™ muscles follow a series of physiologically-determined equations in order to produce the necessary forces that replicate the desired motion of the body, while staying within each muscle's physiological limit. The assumption is that if enough muscles are included, the calculated muscle forces will be very close the the physical muscle force values for the same activity. To decrease the problem of redundancy -- cases where there are several muscles contributing to the motion of a given joint -- the user may condition the output of any muscle from 0 to 500%. This is also defined as the muscle "tone."

Aside from the passive recording muscle -- which functions based on a user-tunable spring damper that records motion during a training simulation -- the muscles in LifeMOD all stem from two basic control algorithms and fit into one of two strength-conditioning (or force-limiting) muscle types, as shown in Figure 1.

Figure 1: Muscle Matrix outlining the control algorithms and strength_conditioning muscle factors.

The Hill-type muscle formulation is the traditional combination of a contractile element (CE) and a parallel elastic element (PE) describing the passive force. The contractile element contains an muscle activation state which controls the active muscle force capability for the muscle. Data from an EMG test may be used as activation curves for the contractile element.

Various muscle parameters may be adjusted. See Parameters to tune the model for simulation including also see Choosing Model Parameters for data sources and information on how to select the parameters mentioned in this section.

Open Loop Simple Muscle Formula

Open loop muscles fire via a user-defined activation curve, A(t), between 0 and 100% with no contraints or target values. The simple muscle limits the force admitted by the A(t) through the formula:

Where the terms meet the following conditions:

A(t) = activation curve with values between 0 and 1
F_max = product of the physiological cross sectional area and maximum isometric muscle stress (σ_max)
Tone = muscle force output filter value between 0 and 2
Preload = user-defined constant force value

Open Loop Hill Muscle Formula

The open Hill muscle formulas combine the A(t) curve and the physiological characteristics of the Hill-based muscles [Hill, '38], which operate on the traditional combination of active contractile elements (CE) and parallel passive elements (PE) with force-length and force-velocity contraints.

Equation 2 shows the formula used to place the strength-conditioning limits on open loop Hill muscles where F_CE can be found using the formula shown in Equation 3.

Where:

A(t) = activation state (normalized between 0 and 1)
F_max= product of the physiological cross sectional area and maximum isometric muscle stress (σ_max)
f_H= the normalized active force-velocity relation (Hill-curve)
f_L= the normalized active force-length relation
v_r= dimensionless lengthening velocity
l_r= dimensionless muscle length

For more detail on the definitions and terms in this equation, see the formula in the detailed Hill Muscle Formulation section.

Closed Loop Muscle Formulas

Closed loop muscles contain proportional-integral-differential (PID) controllers. The PID controller algorithm uses a target length/time curve to generate the muscle activation and the muscles follow this curve. Because of this, they require an inverse dynamics simulation using passive recording muscles prior to simulation with closed loop muscles. The closed loop algorithm is governed by the following formula:

Where

And

D_error= first derivative of P_error
I_error = time integral of P_error

The maximum force generated by a closed loop muscle is limited by the muscle strength conditioning formulas, listed in the open loop section, with A(t)=1. If the PID calculations result in a larger value than this, the force is limited and the controller will not force the model to exceed this physiological limit.

To apply maximum force limits to closed loop muscle groups, set A(t)=1 and compare the open loop strength conditioning formulas against the closed loop formula. In open loop muscles, A(t) is always an input into the formulations. For closed loop muscles, however, it serves as a calculated output value.

See Choosing Model Parameters for data sources and information on how to select the parameters mentioned in this section.

In-depth Hill-Based Muscle Formulation

The Hill-type muscle model is developed from the material behavior of the muscle model adapted from the original work by [Hill, '38] which results in a popular state equation applicable to skeletal muscle that has been stimulated to show tetanus. Reviews of this model and extensions can be found in [Winters, '88] and [Zajac, '89]. The Hill-type muscle model (Figure 1), consists of a contractile element (CE) which is in series with a series elastic element (SEE) both of which are in parallel with a passive element (PE). The SEE, shown in grey in figure 1 is often neglected when a series tendon is added. The main assumptions of the Hill model are that the contractile element is entirely stress free and freely distensible in the resting state, and is described by Hill's equation. When the muscle is activated, the series and paralell elements are elastic, and the whole muscle is a simple combination of identical sarcomeres in series and parallel.

Figure 1: Discrete model for muscle contraction dynamics based on a Hill-type representation. The total force F_MUSCLE is the sum of a passive force F_PE and an active force F_CE. The passive element (PE) is a function of the instantaneous muscle length, l_curr. The contractile element (CE) is a function of the instantaneous muscle length, l_curr, instantaneous muscle shortening velocity v_curr and the time dependent activation state A(t).

When ignoring the SEE, The total muscle force calculated from the Hill formulation comes from the sum of a passive element force F_PE with an active element force F_CE. F_MUSCLE is the sum of both forces, thus:

F_MUSCLE = F_CE + F_PE

Passive Element F_PE

The passive element (PE) is a function of the instantaneous muscle length, l_curr.The muscle input parameters for the passive element are displayed in Figure 2.

Figure 2: Section of the Hill muscle dialog box with descriptions of the input parameters for the passive element.

The passive element force F_PE is modeled with a passive muscle stress, σ, value multiplied by the physiological cross sectional area, pCSA, of the particular muscle.

F_PE = σ · pCSA , where

σ = passive muscle stress
pCSA = physiological cross sectional area

Passive muscle stress, σ, modeled by the nonlinear stress-strain relationship [Deng, '87]

σ = (k · ε)/(1-ε/asym), where

ε = strain defined as the elongation relative to the resting length of the muscle,
k = passive muscle stiffness
asym = strain asymptote

The strain is defined by:

ε = (l_curr - l_free)/ l_free , where

l_curr = current (instantaneious) length of the muscle
l_free = free length of the muscle at rest when it is removed from the body

The l_free results in a smaller free length than the muscle length in its initial position in the model. The initial position in the body is define by l_{rest .}Assuming a linear relationship between the sarcomere length s [Magid, '85, Meyers, '98, Rack, '69], and the muscle length, the free length of the muscle can be calculated as:

l_free = l_rest· (S_free )/ (S_rest ),

where the muscle reference length l_ref is based on [Brelin-Fornari, '98]

l_ref = l_rest· (S_ref )/ (S_rest ), where

S_free = average sarcomere length of "free" muscle
S_rest = length of sarcomere at rest
S_ref = length of sarcomere at muscle optimal length

Active Element F_CE

The active or contractile element (CE) is a function of the instantaneous muscle length (l_curr) instantaneous muscle shortening velocity (v_curr) and the time dependent activation state A(t). The muscle input parameters for the contractile element are displayed in figure 4.

Figure 3: Section of the Hill muscle dialog box with descriptions of the input parameters for the contractile element.

Active muscle behavior (contractile element) is modeled with a normalized activation state and a maximum muscle force at activation.

F_CE = A(t) · F_max · f_H (v_r) · f_L(l_r) , where

A(t) = activation state (normalized betwee 0 - resting to 1 - maximum activation)
F_max= muscle force at maximum activation isometric conditions
f_H= the normalized active force-velocity relation (Hill-curve)
f_L= the normalized active force-length relation
v_r= dimensionless lengthening velocity
l_r= dimensionless muscle length

The muscle force at maximum activation is calculated by:

F_max = σ_max · pCSA, where

σ_max = is the maximum isometric muscle stress
pCSA = physiological cross sectional area

The function f_H is the normalized active force-velocity relation (Hill-curve). The default value used in LifeMOD was developed by [Delp, 1990]. Separate functions are defined for lengthening and shortening of the CE-element.

Where,

v_r = v_curr/V_max
v_curr = the instantaneous lengthening velocity V_max = the maximum shortening velocity of the muscle
CE_sh= shape force-velocity curve (shortening)
CE_shl= shape force-velocity curve (lengthening)
CE_ml= maximum relative force (lengthening)

The shape is determined by the parameters CE_sh and CE_shl, where CE_ml defines the maximum force the muscle can generate during lengthning relative to the maximum isometric force F_max.

Figure 4: Normalized muscle force as a function of normalized muscle length (l_r) for both active, f_l(l_r), (blue curve) and passive, F_PE, (other) element curves.

Figure 5: Standard force-velocity curve f_h for the conditions CE_sh= .5, CE_shl= .075, C_ml= 1.5

The function f_lis the normalized active fore-length relation.

f_l(l_r) = e -((l_r-1)/S_k)**2 , where

l_r = l_curr/l_ref
l_curr = the instantaneous muscle length
l_ref = optimum reference length at which the active force is generate most efficiently
S_k = determines the shape of the curve (figure 2)

Muscle activation, A(t), is described by a user-defined data spline. The data spline uses time as the independent variable and the normalized activation A as the dependent variable. A library of EMG data is avaliable via the XChange function in the main LifeMOD panel.