Equivalent linear damping characterization in linear and nonlinear force–stiffness muscle models

Abstract

In the current research, the muscle equivalent linear damping coefficient which is introduced as the force–velocity relation in a muscle model and the corresponding time constant are investigated. In order to reach this goal, a 1D skeletal muscle model was used. Two characterizations of this model using a linear force–stiffness relationship (Hill-type model) and a nonlinear one have been implemented. The OpenSim platform was used for verification of the model. The isometric activation has been used for the simulation. The equivalent linear damping and the time constant of each model were extracted by using the results obtained from the simulation. The results provide a better insight into the characteristics of each model. It is found that the nonlinear models had a response rate closer to the reality compared to the Hill-type models.

1 Introduction

Many muscle models have been proposed in order to give a better insight into the muscle behavior toward activations coming from the central nervous system. The most known model in achieving this goal is the Hill’s model (Hill 1938). The phenomenological modeling approach based on Hill’s model dominates in musculoskeletal systems modeling for simplicity and low computational cost (Siebert et al. 2008).

On the basis of Hill’s muscle model, several Hill-type models have been emerged. These models consist mainly of three elements: contractile element (CE), series elastic element (SE), and parallel elastic element (PE) in diverse configurations (Zajac 1988; van Soest and Bobbert 1993; Günther and Ruder 2003; Siebert et al. 2008; Haeufle et al. 2012; Winters et al. 2012). The contraction dynamics of muscle has been recorded and described in numerous muscle experiments. CE describes the force–length and the force–velocity dependency of muscle fibers phenomenologically. The muscle fiber velocity depends on the CE force in a hyperbolic relation mostly known as the Hill’s relation (Zajac 1988). In a real muscle, the elasticity of CE and the surrounding passive tissues can be represented as elastic elements. These passive structures are usually divided into components thought to act purely in series (SE) and purely in parallel (PE) to the contractile element (CE) in classical Hill-type models. Therefore, two principal Hill-type models have emerged: one in which the parallel elastic element (PE) is arranged in parallel with the CE and the SE; the other, in which the PE is only in parallel with CE (Fig. 1). Both models have been used in the past to represent muscle contraction (Ettema and Meijer 2000; Haenen et al. 2003; Siebert et al. 2008). One of the deficiencies of these models is the lack of inclusion of mass and inertia of the muscle. The output is a one-dimensional force, which is applied to the models between the origin and the insertion points, or sometimes as moments by means of (constant) lever arms (Haeufle et al. 2014). The models inputs are neural muscle stimulation which acts as a force-generating term in CE, and the resulting force modulates through PE and SE.

The study of motor control gives a better insight into the effects the central nervous system has on the functional, goal-directed movement (Cheng et al. 2000). In the mid-sixties, a unique approach for development of a theory for human motor control was proposed by Asatryan and Feldman (1965a). As a result of a number of experiments, a model called the A model emerged. This model was based on the equilibrium point (EP) hypothesis that suggests the active movements to be a result of shifts in the equilibrium state of the motor system. In the mentioned model, the shifts are associated with setting the threshold (A) of the stretch reflex in its formulation (Asatryan and Feldman 1965a, b; Feldman 1996). Later this theory was extended and refurbished with more details (Feldman 1974a, b, 1976, 1986). Foisy and Feldman (2006) tested the two major theories of motor control (internal models and threshold control). The first model behaves as position-dependent EMG control which explains that the system counteracts the opposing forces by increasing the muscle activity in proportion to the distance from the initial posture. In contrast, the second model theory fully excludes these opposing forces by shifting thresholds as muscle activation parameter and thus resetting the equilibrium positions to a new posture. In general, two types of functional motor or muscle models provided in the literature by Hill and Feldman could be considered as models with nonlinear springs. The Hill’s model is characterized as models with adjustable stiffness due to excluding the reflex effect. On the other hand, including the reflex effect would result in adjustable starting length models that Feldman’s model could be considered one of them (Feldman 1966; Shadmehr and Arbib 1992). It was later investigated by Shadmehr and Arbib (1992) that while taking into account the muscle reflex effects, the same level of joint stiffness can be produced by much smaller muscle forces because of the nonlinear stiffness–force relationship.

Mörl et al. (2012) investigated the extent that Hill-type muscle models are able to explain the electromechanical delay (EMD). The EMD is the lag between the changes in the simulation and the force produced in the muscle. In addition, Millard et al. (2013) compared the computational speed and biological accuracy of three musculotendon models: two with an elastic tendon (an equilibrium model and a damped equilibrium model) and one with a rigid tendon. Stearne et al. (2012) also used the muscles force–length–velocity relationship driven from an individually scaled musculoskeletal OpenSim’s model in determination of the triceps surae muscles to Achilles tendon force using a Hill-type model. The nonlinear force–velocity also plays a significant role in stabilizing a muscle model and results in lower energy cost. This effect was investigated on computational model of an anguilliform swimmer by  Hamlet et al. (2015).

The purpose of this study is to simulate the Hill’s and Feldman’s muscle models, estimating the equivalent damping and time constant of each, and comparing the results with those obtained by the known models of OpenSim simulation pack. First, Hill’s and Feldman’s models and their properties for the cat soleus muscle are described. Afterward, the Hill’s model isometric contraction with linear passive properties is simulated. Then the equivalent damping and the time constant of each model is evaluated. Analysis of these results allows us to reach a better understanding of the speed, accuracy and stabilization of each muscle model. Due to nonlinear passive properties of muscles, the procedure explained above is once again implemented considering nonlinear passive properties, and the results are compared. We also simulate Hill’s muscle using the models in OpenSim. This gives a better insight into their responses.

2 Methods

2.1 Adjustable stiffness model (Hill-type model)

The classical Hill-type model has usually two representations called the [CE + PE] model and [CE + SE] model as illustrated in Fig. 1. The former consists of a contractile component in parallel to an elastic component named as PE and an elastic component in series with the whole model named as SE, whereas in the latter the SE component is in series with the CE part and the combination of these two would be in parallel to the PE component. When modeling solely the active properties of the muscles, there would be no significant difference in these two models; however, the difference would rise when the passive properties of the muscle would come into action.

Fig. 1

Hill-type model representations, where (left) is the [CE + SE] model and (right) is the [CE] model, and both models consist of a series element (SE), a contractile element (CE) and a parallel element (PE)

 Siebert et al. (2008) investigated that which representations of Hill’s model best describe the muscle response. Comparing the force–velocity relationship, the stiffness and the muscle’s yield strains, they concluded that the [CE + PE] model better captures the muscle dynamics. As a result, in order to simulate the muscle contraction, [CE + PE] model was chosen. The force generated in contractile element is a function of the level of activation known as a signal coming from the nervous system to simulate the muscle ($F_\mathrm{A}$), the maximum active isometric force ($F_\mathrm{max}$), the force–length ($F_\mathrm{L}$) and the force–velocity characteristics $(F_\mathrm{v}$):

$$\begin{aligned} F_\mathrm{CE}=F_\mathrm{L} F_\mathrm{v} F_\mathrm{A} F_\mathrm{max} \end{aligned}$$

(1)

The level of activation $F_\mathrm{A}$ was chosen to be modeled as a ramp starting from 0 at time 0 and reaching the maximum level of activation (1) in 0.1 s ($t_\mathrm{max}$) and maintaining this value for the remaining time of the simulation (0.9 s). The activation applied to the muscle is:

$$\begin{aligned} F_\mathrm{A} = \left\{ \begin{array}{ll} \frac{t}{t_\mathrm{max}} &{}\quad \text{ if } t < t_\mathrm{max} \\ 1 &{}\quad \text{ if } t \ge t_\mathrm{max} \end{array} \right. \end{aligned}$$

(2)

The force–velocity relationship, which is illustrated in Fig. 2, is the hyperbola proposed by Hill (1938). The hyperbola is an evident of the damping characteristic of the muscle for concentric contractions explaining that the faster the muscle contracts, the less the force generated (Van Leeuwen and Kier 1997).

$$\begin{aligned} F_\mathrm{v}= \begin{array}{ll} \frac{v_\mathrm{CEmax}-v_\mathrm{CE}}{v_\mathrm{CEmax}+(v_\mathrm{CE}.\mathrm{curv})}&\quad \hbox {if } v_\mathrm{CEmax}<0 \end{array} \end{aligned}$$

(3)

where $v_\mathrm{CEmax} < 0$ is the maximal CE shortening velocity and curve is a curvature parameter  (Siebert et al. 2008).

Fig. 2

Hill-type force–velocity curve; negative velocity shows the concentric motion and positive velocity shows the eccentric motion

In an eccentric contraction, the applied force value would grow when the velocity of lengthening increases. Behavior in extension follows another curve rather than the Hill’s hyperbola  (Katz 1939).

$$\begin{aligned} F_\mathrm{v} = \begin{array}{ll} p-(p-1) \frac{v_\mathrm{CEmax}+v}{v_\mathrm{CEmax}-\mathrm{curv}.k_\mathrm{ec}.v} &{}\quad \text{ if } v_\mathrm{CEmax}\ge 0 \\ \end{array} \end{aligned}$$

(4)

where $k_\mathrm{ec}$ is an eccentric muscle constant, which can be computed from the difference between the slopes of the force velocity curves on both sides of the zero velocity. If the slope in eccentric section is n times the slope in concentric section, $k_\mathrm{ec}$ becomes  (Nazari 2011):

$$\begin{aligned} {k_\mathrm{ec}}=\frac{n\left( \frac{1}{\hbox {curv}}+1\right) }{p-1}-\frac{1}{\hbox {curv}} \end{aligned}$$

(5)

The p parameter, indicating the threshold of the muscle to withstand the elongation, value is 1.8 times of the maximum isometric force. As the muscle contraction is simulated, the eccentric part of the force–velocity function (positive velocities) was ignored.

Fig. 3

The force–length relationship was normalized relative to $F_\mathrm{max}$ and considered as a linear model . $l_4$ and $l_1$ determining the width of the force–length relationship. $l_3$ is the length at which the ascending limb changes slope, $f_\mathrm{c}$ is the corresponding normalized force, and $l_2$ is equal to the width of the plateau

The CE force–length relation was assumed to be linear as shown in Fig. 3:

$$\begin{aligned} F_\mathrm{l} = \left\{ \begin{array}{ll} 0 &{}\quad \text{ if }\;\; l_\mathrm{CE}\le l_4 \\ \frac{f_\mathrm{c}}{(l_3-l_4) } (l_\mathrm{CE}-l_4 ) &{}\quad \text{ if }\;\; l_4\le l_\mathrm{CE}\le l_3\\ \frac{(f_\mathrm{c}-1)}{l_3}(l_\mathrm{CE}-l_3)+1 &{}\quad \text{ if }\;\; l_3\le l_\mathrm{CE}\le 0\\ 1 &{}\quad \text{ if }\;\; 0\le l_\mathrm{CE}\le l_2\\ \frac{(l_\mathrm{CE}-l_2)}{(l_2-l_1)}+1 &{}\quad \text{ if }\;\; l_2\le l_\mathrm{CE}\le l_1 \\ 0 &{}\quad \text{ if }\;\; l_1\le l_\mathrm{CE} \end{array} \right. \end{aligned}$$

(6)

where $l_\mathrm{CE}$ is equal to $l-l_\mathrm{opt}$, $f_\mathrm{c}$ is the force at which the ascending limb changes slope and $l_1$, $l_2$, $l_3$ and $l_4$ are crucial lengths that the force–length relation slope changes  (Siebert et al. 2008).

Winters et al. (2012) proposed the $F_\mathrm{SE} (\bigtriangleup {l_\mathrm{SE}})$ relationship as:

$$\begin{aligned} F_\mathrm{SE} (\bigtriangleup {l_\mathrm{SE}} )= \left\{ \begin{array}{ll} \frac{F_1}{(e^{k_\mathrm{sh} }-1)}.\left( e^{\frac{(k_\mathrm{sh}.\bigtriangleup {l_\mathrm{SE}})}{(\bigtriangleup {l_\mathrm{SE1}})}}-1\right) &{}\quad \text{ if } 0<\bigtriangleup {l_\mathrm{SE}}<\bigtriangleup {l_\mathrm{SE1}} \\ F_1+k.(\bigtriangleup {l_\mathrm{SE}}-\bigtriangleup {l_\mathrm{SE1}}) &{}\quad \text{ if } \bigtriangleup {l_\mathrm{SE1}}<\bigtriangleup {l_\mathrm{SE1}} \\ \end{array} \right. \end{aligned}$$

(7)

where $\bigtriangleup {l_\mathrm{SE1}}$ and $F_1$ are the elongation and the force at which the force–elongation relation changes from exponential to linear, respectively. k was calculated from $\bigtriangleup {l_\mathrm{SEC1}}$, $F_1$, the dimensionless shape parameter $k_\mathrm{sh}$ and the constraint of equal stiffness at the place where $\bigtriangleup {l_\mathrm{SEC}}$ equals $\bigtriangleup {l_\mathrm{SEC1}}$ (Siebert et al. 2008). In model [CE + PE], the PE influences the intrinsic contraction dynamics. A PE force–elongation relationship represented as $F_\mathrm{PE} (\bigtriangleup {l_\mathrm{PE}})$ depending on $k_1$ and $k_2$ was taken from (Brown et al. 1996)

$$\begin{aligned} F_\mathrm{PE} ( \bigtriangleup {l_\mathrm{PE}} )= \begin{array}{ll} k_1.(e^{k_2. \bigtriangleup {l_\mathrm{PE}}} -1) &{} \text{ if } \bigtriangleup {l_\mathrm{PE}}>0 \\ \end{array} \end{aligned}$$

(8)

Two characterizations of the passive elements have been implemented. First they are linearized at starting position (linear model) and then the nonlinear characteristics are considered (nonlinear model). In the linear model, the slope at the starting position is considered as linear stiffness of passive elements. For SE the stiffness becomes:

$$\begin{aligned} k_\mathrm{SE}= \left\{ \begin{array}{ll} \frac{F_1.k_\mathrm{sh}}{(e^{k_\mathrm{sh} }-1).\bigtriangleup {l_\mathrm{SE1}} } &{}\quad \text{ if } 0<\bigtriangleup {l_\mathrm{SE}}<\bigtriangleup {l_\mathrm{SE1}} \\ k&{}\quad \text{ if } \bigtriangleup {l_\mathrm{SE1}}\le \bigtriangleup {l_\mathrm{SE}} \end{array} \right. \end{aligned}$$

(9)

The passive properties of the muscle and the Hill’s activation parameters are shown in Table 1.

2.2 Adjustable starting length model (Feldman’s model)

The Feldman and Hill-type muscle models have the similar passive properties. Their difference emerges in the contractile element function (active properties).

Table 1 Single muscle parameters of model [CE + PE] based on the corresponding ramp experiments (Siebert et al. 2008)

Feldman (1986) proposed a model considering the force generated in contractile component as a nonlinear function of activation level and the velocity:

$$\begin{aligned} F_\mathrm{CE}=f(L,v,l_\mathrm{threshold}) \end{aligned}$$

(10)

In Feldman’s model, the activation level depends on the difference between the muscle length and the zero-force length ($l_\mathrm{thereshold}$), which is the control parameter  (Feldman 1986). This model is considered as an adjustable starting length mode, where muscle stiffness changes as a nonlinear function of force for different activation levels. Feldman (2011) described that for a given activation command, muscle contractile force depends on combination of the stretch–reflex mechanism and the force–length characteristic, which is an exponential curve called the invariant characteristics (IC):

$$\begin{aligned}&F_{\mathrm{CE-Feldman}}=F_\mathrm{max} \nonumber \\&\quad \times \left( \hbox {exp}{\frac{[l(t-d_\mathrm{e})-l_\mathrm{threshold}+ \mu (v(t-d_\mathrm{e}))]^+}{l_\mathrm{c} }}-1\right) \nonumber \\ \end{aligned}$$

(11)

where $F_\mathrm{max}$ is the maximum force generation capacity of a muscle and is a function of the physical cross-sectional area of it, $l_\mathrm{threshold}$ is zero-force length (threshold length), $l_\mathrm{c}$ is the characteristic length, v is muscle velocity and $\mu$ is a damping coefficient. Both muscle length and velocity in this equation are delayed values at time $t-d_\mathrm{e}$. $[]^+$ means that the force is equal to zero if the expression within $[]^+$ is negative.

Pilon and Feldman (2006) have also proposed a threshold model for the torque applied to human elbow as an exponential function of the adjustable length as below:

$$\begin{aligned} M=a\left( \hbox {exp}\left( \frac{ A}{l_\mathrm{c}}\right) -1\right) \end{aligned}$$

(12)

where a acts as the maximum moment that the joint is able to insert and A is considered as:

$$\begin{aligned} A=\frac{1}{1+r}(l(t-d_\mathrm{e})-l_\mathrm{threshold}+\mu v(t-d_\mathrm{e})) \end{aligned}$$

(13)

where $d_\mathrm{e}$, $\mu$ and $l_\mathrm{threshold}$ have the similar definitions as our representation of Feldman’s model and r is a weight coefficient with a value of 0.05. The effect of r in this research can be simulated by replacing $l_\mathrm{c}$ with $l_\mathrm{c}(1+r)$. As a result, the two models are simillar exept for this difference.

A passive force should be added to this active force to take into account the passive mechanical property of the muscle (Nazari et al. 2013). Muscle invariant characteristics (IC) at a given velocity is shown in Fig. 4, where the gray path shows an example of a voluntary contraction of muscle when motor commands (starting lengths) are changing.

Fig. 4

Feldman muscle invariant characteristics (IC) at a given velocity (Nazari et al. 2013)

3 Results

The simulation was performed under the assumption of isometric contraction. In modeling the linear passive properties, linear springs were used, and while simulating the nonlinear model, the nonlinear passive functions replaced the linear springs.

The muscle damping property is defined through its force–velocity characteristic. In order to evaluate the damping characteristic of Hill’s model, the force–velocity function was removed and replaced by a linear damper with the coefficient of B.

The Feldman’s $l_\mathrm{threshold}$ was chosen to be a ramp function starting from $l_{\mathrm{th-initial}}$ that was chosen to be equal to $l_\mathrm{opt}$ (the optimum muscle length), reaching $l_\mathrm{th-final}$ in 0.1 s. The value of $l_\mathrm{th-final}$ was chosen in such a manner to provide the same maximum force reached by Hill’s model at the same muscle length.

In the literature the characteristic length $l_\mathrm{c}$ varies between 9 mm (Laboissiere et al. 1996) and 25 mm (Buchaillard et al. 2006) with a role that is to stabilize the model. The value of 8.93 mm was chosen for $l_\mathrm{c}$ which is close to the range mentioned above and obtained by Asatryan and Feldman (1965b) from torque–angle curves for the human elbow (Laboissiere et al. 1996).

The parameter $\mu$ is called the damping coefficient. In the physiological literature, damping is associated with regulation of muscle force as a function of velocity. In contrast, the damping coefficient, $\mu$, characterizes the dependence of muscle’s threshold length on velocity. Both proprioceptive feedback and muscle intrinsic properties are velocity dependent and thus contribute to the force-related damping in a nonlinear way. The component of force-related damping dependent on proprioceptive feedback is, to a first approximation, proportional to the product of muscle stiffness and the coefficient $\mu$ (St-Onge et al. 1993). Thus, even though $\mu$ is considered, for simplicity, the same for all muscles and constant for a given movement (0.05 s), the force-related damping may still vary as a function of stiffness both for a single muscle and between muscles (Laboissiere et al. 1996).

Considering the fact that the Hill’s force–length curve uses the difference between current muscle length and optimal length as its characteristic length, the Feldman’s model uses the current muscle length. Having the current muscle length, the following relation can be used to compute the threshold length:

$$\begin{aligned} l_\mathrm{threshold}=l-l_c{\hbox {ln}}\left( \frac{F_\mathrm{in}}{F_\mathrm{max}}+1\right) \end{aligned}$$

(14)

where $F_\mathrm{in}$ and l are the current force and length of the muscle and $l_\mathrm{c}$ and $F_\mathrm{max}$ have the values defined earlier in the text.

In this section we present a simulation of the model described above. The simulation was run using ode45 solver that has a variable time step in order to achieve the tolerance specified. The tolerance was set to 0.001 ms. The assumption of linear passive elements was made followed by the implementation of nonlinear passive elements. As can be seen in Fig. 5, the maximum force generated taking into account nonlinearity is 20.47 N which is greater than the force with linear assumption 17.68 N.

Fig. 5

Hill’s muscle model force–time plot considering linear passive properties (black line) and nonlinear passive properties (gray dashed line)

In addition, there exists a little offset between the force obtained during nonlinear simulation and $F_\mathrm{max}$. This offset could be due to the errors existing in consideration of the mean value of several experiments. The Hill’s model with nonlinear passive properties was also implemented in the OpenSim software (Delp et al. 2007). OpenSim is considered as a software that provides its users with the ability to observe and analyze the forces generated during movement in different muscles of the body.

In order to simulate the Hill’s response in OpenSim, the same activation as in Hill’s model (Eq. 2) is applied on a model of an isometric muscle. The muscle modeling used for this purpose is based on the modeling proposed by Millard et al. (2013).

The force applied to the muscle in the Millard’s model is:

$$\begin{aligned} f^{M}=f_o^M\left( \alpha , f^L (\hat{l}^M), f^V (\hat{v}^M)\right) + f^\mathrm{PE} (\hat{l}^M) \end{aligned}$$

(15)

where the first term ($f_o^M$) is the Hill’s active force that is a function of $\alpha$ (activation), $f _{L}$ (active force–length relationship), $f_{v}$ (active force–velocity relationship), and the second term ($f^\mathrm{PE}$) is the passive force which is a function of the current muscle length ($\hat{l}^M$). Changing the active force–length, active force–velocity, the maximum isometric force, and the optimal muscle length to be the same as the Hill’s model, the plots shown in Fig. 6 are obtained.

As it can be seen Fig. 6, both models have reached their desired maximum isometric force; however, there exists a small difference between the results in the transient part. This could be due to the different passive and active properties considered for each modeling. As for the active force–length relation, the OpenSim ignores the 0 to $l_2$ part in Fig. 3 where the normalized force has the constant value of one. The more these two obey the same formulation, the more similar the results would become.

Fig. 6

The Hill’s model (gray line) and the OpenSim’s model (black line)

Using Eq. 14 and the maximum force of 20.47 N achieved at active length of 44.3 mm, the $l_\mathrm{th-final}$ was set to 38.11 mm. Therefore, the $l_\mathrm{th}$ starts from 53 mm ($l_\mathrm{opt}$) to 44.3 mm, $l_\mathrm{th-final}$ during the first 0.1 ms.

In the literature, different values of the reflex delay have been reported. For example, Luschei and Goldberg (2011) used 10 ms delay for mastication in monkeys and Buchaillard et al. (2009) used 17 ms in tongue muscle modeling.  Pilon and Feldman (2006) have studied the effect of reflex delay on the stability of the inserted torque to the elbow movements and illustrated that their model is stable from 30 to 100 ms. For this purpose we have studied the effect of changing the delay values on output response. It is shown in Fig. 7 that the proposed model in this study remains stable for reflex delays up to to 100 ms and the response becomes unstable as the reflex delay becomes higher than 100 ms. As it can be seen, the delay values less than 38 ms generates oscillations for the start of activation and the values more than 38 ms generates overshooting in force values at the beginning of response. Hence we picked up the delay value as 38 ms in our simulations.

Fig. 7

The effect of reflex delay on the stability of Feldman’s muscle model + Hill’s force–velocity term

In order to evaluate the damping characteristic of each model, the Feldman’s muscle model was simulated. The resulting oscillations were not damped during the simulation time, which could be an evidence of the model’s very low damping characteristic. According to this observation, the Hill’s force–velocity term was added to the Feldman’s model in order to damp the oscillations. The effect of the Feldman velocity term was also a matter of interest. As a result, two different models including and excluding the Feldman’s velocity term were considered in Fig. 8. The results show that there is a little difference between the two proposed models, compared to the difference they have with Hill’s model. Therefore, the Feldman’s damping characteristic is not significant compared to the Hill’s force–velocity term. The damping coefficient (B) introduced via the model’s velocity term was calculated by omitting the force–velocity terms and replacing them with a damper that provides the exact maximum force which had been achieved while using those terms. The muscle mass was needed in this part which was assumed to be 3.4 g  (Scott et al. 1996). This is the mean value of several experiments conducted on cat’s soleus muscle. It can be seen in Fig. 9 that the damping coefficient of Hill’s model is 0.98 $\frac{\mathrm{N}\,\mathrm{s}}{\mathrm{mm}}$, whereas the proposed Feldman’s combined model with Hill’s force–velocity term is 2 $\frac{\mathrm{N}\,\mathrm{s}}{\mathrm{mm}}$ as shown in Fig. 10. In addition, Fig. 11 shows the effect of omitting the Feldman’s velocity term on the model mentioned earlier resulting in damping of 1.5 $\frac{\mathrm{N}\,\mathrm{s}}{\mathrm{mm}}$. This is an evidence that the Feldman’s model velocity term would enhance the damping ability of the model. On the other hand, it is possible to conclude the fact that combined Feldman’s model has a better damping characteristic than the model proposed by Hill.

Fig. 8

Hill’s muscle model force–time plot (black dashed line), Feldman’s muscle model combined with Hill’s force–velocity term (black line) and Feldman’s muscle model without its velocity term combined with Hill’s force–velocity term (gray line)

Fig. 9

The equivalent damping characteristic of Hill’s muscle model

Fig. 10

The equivalent damping characteristic of Feldman’s muscle model + Hill’s force–velocity term

Fig. 11

The equivalent damping characteristic of Feldman’s muscle model (excluding its own velocity term) + Hill’s force–velocity term

Table 2 Time constant and equivalent damping of three nonlinear models

The time constant was chosen to be the time the muscle force reaches 63.2 % of its final value. It can be seen in Fig. 8 that the time constant of Feldman’s combined model with Hill’s force–velocity term was 0.09 s, while excluding the Feldman’s velocity term resulted in the time constant of 0.12 s. The time constant of the Hill’s model was 0.21 s. This could be an evidence of the fast response of Feldman’s model to the activation, and the effect of Feldman’s velocity term in enhancing the time constant. The results are shown in Table 2.

It is shown in Fig. 12 that the linear passive properties produce less force than the nonlinear ones. As expected, the damping characteristic for the linear Feldman’s muscle is 1.2 $\frac{\mathrm{N}\,\mathrm{s}}{\mathrm{mm}}$ which is less than 2 $\frac{\mathrm{N}\,\mathrm{s}}{\mathrm{mm}}$ that is achieved by the nonlinear one. In addition, Fig. 12 implies that nonlinearity would enhance the time constant by 0.03 s, helping the muscle in reaching its desired maximum force in less time.

Fig. 12

The effect of nonlinearity on damping and time constant of Feldman’s muscle model + Hill’s force–velocity term

4 Conclusion

We have simulated Hill’s model and proposed a new muscle model combining the Feldman’s stretch reflex force with Hill’s force-velocity term during isometric contraction. The modeling, numerical analysis and some comparison with the OpenSim software which provides its users with a vast variety of muscle simulations were presented in this article. The equivalent damping characteristic and the time constant for three different models were computed. As far as nonlinear passive properties are considered instead of linear properties, the ability of the model to damp the oscillations and show a closer response to the stimuli coming from the central nervous system would improve. A new model using Feldman’s formulation and taking advantage of the Hill’s velocity term while taking into account the nonlinear passive properties was proposed. The model has displayed an excellent behavior toward isometric simulation, and has a time constant closer to reality and a higher damping characteristic than other models.

The present study can be extended in many directions. For example, different activation schemes such as isotonic activation will shed light on other features of the proposed model; modeling the Feldman’s muscle model in OpenSim enables a better comparison on a unique platform.