Biol. Cybern. 89, 71–79 (2003) DOI 10.1007/s00422-003-0403-0 Ó Springer-Verlag 2003

Stabilizing function of antagonistic neuromusculoskeletal systems: an analytical investigation Heiko Wagner, Reinhard Blickhan Institute of Sports Science, Department of Science of Motion, Friedrich-Schiller-University Jena, 07749 Jena, Germany Received: 24 March 1999 / Accepted: 20 February 2003 / Published online: 22 May 2003

Abstract. Under normal conditions human walking or running consists of stable cyclic movements. Minor perturbances such as a stone or a pothole do not disrupt the cycle, and the system returns to its prescribed trajectory. We investigated whether a pair of antagonistic muscles is able to stabilize the movement without neuronal feedback. The human is represented by a model consisting of a massless two-segment linkage system (leg) topped by a point mass. Both the extensor and flexor muscles are described by a Hill-type muscle model. Conditions for stability are calculated analytically based on the Ljapunov Theory and the results are illustrated by numerical examples. The activation functions of both the extensor and flexor muscles can be calculated for a prescribed trajectory to maintain the self-stabilizing ability of such a system. Experimental evidence supports the prediction. Our investigation shows that a moving center of rotation of the kneejoint, a biarticular flexor muscle group, the force-velocity relation, and the ascending limb of the force-length relation improves the self-stabilizing ability of human movement.

1 Introduction Human locomotion is dynamically stable. A human walker or runner moves along a certain envisioned trajectory because of the stabilizing action of the central nervous system, which pushes the system back to its prescribed path after a perturbation. In dynamic situations, it is important that the system return to its prescribed condition within an acceptable time period (Ting et al. 1994). To execute very fast movements, biological systems need feedback loops with little or no delay preferably. Correspondence to: H. Wagner (e-mail: [email protected], Tel.:+49-3641-945706)

Van Soest and Bobbert (1993) found that the sensitivity of a musculoskeletal system to perturbations is less if muscle dynamics is included than if the force generation were simply controlled. Loeb (1995) has coined the term ‘‘preflex’’ for a zerodelay response to a perturbance. Taga (1998) investigated a model composed of a neural rhythm generator, a discrete movement generator and a musculoskeletal system with 8 segments and 20 muscles. Movements generated with this model emerge as a stable limit cycle and are stable against perturbations such as changes in the velocity and modulations of the step length. In order to avoid obstacles, a modification of the trajectory of the swing limbs is possible. The study, however, does not give information about the contribution of different features of the system to stability. We have shown previously that some properties of musculature do have the ability to stabilize cyclic movements of musculoskeletal systems. If the muscle architecture and the parallel elastic elements within a muscle are tuned to each other, the leg can have ‘‘preflexive’’ self-stabilizing properties (Wagner and Blickhan 1999). The term self-stability should indicate that the musculoskeletal properties themselves are able to provide stability without the need of a sensory feedback system. A series elastic element and a parallel damper element do not facilitate a preflex-like behavior. A cyclic movement calculated in a forward simulation using a simple Hill-type muscle model is highly unstable. A nonlinear feedback system is able to stabilize the trajectory of such a system. However, a neuronal feedback system cannot react with zero delay. A linear feedback system combined with a carefully adjusted geometry and mass distribution can provide stable operation of a torque-controlled runner or walker (Beletsky 1995; McGeer 1992; McGeer 1993). In order to demonstrate that an antagonistic musculoskeletal system can self-stabilize, it is necessary to select a sufficiently simple yet general example. Therefore we used a simple model with all the essentials of a dynamic musculoskeletal system. In the present study, we used an antagonistic flexor-extensor musculoskeletal

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system and a knee joint based on the theory of Menschik (Menschik 1987), which includes the moving center of rotation (MCR) and the patella. To guarantee generality with respect to stability, it is necessary to follow concepts that can be used for analytical predictions like the Ljapunov Theory (Ljapunov 1892). In this study, we addressed the question of whether an antagonistic musculoskeletal system can stabilize simple cyclic vertical movements of humans if a specified activation function for both muscles is established. We compared the predicted activation functions with the normalized EMG of the human subject executing the prescribed trajectory. The difference between this investigation and others is that the stability of a human movement will be studied based not only on simulations but on analytical calculations as well. The musculoskeletal properties necessary for calculating the simulations will be determined individually through simple noninvasive measurements. We undertook this study as a first step toward estimating the influence of different properties of the musculoskeletal system on the self-stabilization of human movement. Specifically, we addressed the following research questions: do the (a) coactivation of antagonistic muscles, (b) knee-joint geometry, (c) biarticular arrangement of antagonistic flexor muscles, and (d) force-length relation of skeletal muscles influence the self-stabilization of human movement? 2 Materials and methods In order to investigate the self-stabilizing properties of the human leg, we considered cyclic vertical knee bends as a simple movement task. As a first approximation it is assumed that the knee bends have no horizontal movements that facilitate the mathematical description.

normalize the submaximal EMG-data of the measured movements. We recorded the kinematics of the knee bends from the lateral left side with a digital high-speed video system (Camsys 500, Mikromak). The hip joint and the ankle joint were marked with markers (Fig. 3). The sample frequency was 500 frames per second at a frame size of 256  256 pixels. The coordinates of the hip and ankle markers at each frame were digitized using a video analysis system (WinAnalyze). Vertical and two independent horizontal components of the ground reaction force were measured for the cyclic knee bends of the subject using a Kistler force plate at a sample rate of 1 kHz. At the leg press we measured one horizontal and the vertical component of the forces beneath each foot separately as well as the leg extension with a sample frequency of 1 kHz. 2.2 The model The human is represented by a model consisting of a massless two-segment linkage system (leg) topped by a point mass. We assumed that the knee bends were performed only in the vertical direction with no horizontal hip movement and with vertical alignment of the trunk. An antagonistic pair of knee-joint muscles has been included, and both the extensor and flexor muscles are described by a Hill-type muscle model. The muscle model consists of a force-velocity relation and a force-length relation, while series elasticity was neglected in this study. In general, the geometric transformation between the ground reaction force FG and the muscle forces for the extensor fme and flexor muscles fmf for vertical movements can be described by: FG ðt; X ; V Þ ¼ Ge ðX Þ  Ee ðtÞ  fme ðX ; V Þ  Gf ðX Þ  Ef ðtÞ  fmf ðX ; V Þ

2.1 The experimental procedure Five subjects were asked to execute several cyclic knee bends (Fig. 3b) at their preferred amplitude (about 0.2 m) and frequency (about 1.5 Hz). The subjects were instructed to keep their feet flat on the ground. Subsequently the subjects exerted maximum voluntary concentric contractions on a leg press (Fig. 3a) pushing against two different masses (m ¼ 25 and 85 kg) as well as isometric contractions. We measured the bipolar EMG (Biovision) of the following muscles: vastus lateralis, vastus medialis, rectus femoris, and biceps femoris. Surface skin electrode pairs were spaced about 2.5 cm apart on the skin along the longitudinal axis of the muscles. We used a sample frequency of 1 kHz and a gain of 1000. To normalize the EMG’s relative to maximum voluntary contractions (%MVC), we measured several maximum-effort isometric contractions for the extensor and the flexor muscles. From these data we selected an interval of MVC and integrated the rectified data over this interval. Hence, we calculated a mean amplitude for MVC to

ð1Þ

The functions Ee and Ef describe the activation function of the extensor and the flexor muscle group respectively, G – geometric function, fm – muscle force. 2.3 The geometric transformation The geometric function GðX Þ is defined as the quotient of the muscle moment arm and the effective moment arm of the ground reaction force FG according to the center of rotation of the knee joint. The muscle moment arms of the extensor he and flexor hf muscles are different. Therefore, it is necessary to calculate two different geometric transformations, Ge ðX Þ and Gf ðX Þ, for these two muscle groups. The transformation of velocity can be described for each muscle group: vme ¼ Ge ðX Þ  V

and vmf ¼ Gf ðX Þ  V

ð2Þ

with V being the shortening velocity of the leg, vme ¼ l_me , vmf ¼ l_mf the shortening velocity of the

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muscle, and lme and lmf the length of the extensor and flexor muscles, respectively. In this investigation, we compare two different types of knee models. The first model describes the knee joint as a simple hinge joint (Sust et al. 1997) and the second, more realistic, model includes the moving center of rotation and the patella based on the theory of Menschik (Menschik 1987). We described the flexor muscle group as a monoarticular or biarticular muscle. An analysis of the literature (Gu¨nther 1997) shows that for the biarticular flexor muscle group described in this study the moment arm at the hip joint is about twice the size of the moment arm at the knee joint. In this case, for the vertical knee bends the contraction velocity of the flexor muscle is approximately zero and the partial derivative with respect to X and V of the muscle force is zero, too. Hinge-joint model Assuming the knee as a simple hingejoint (Sust et al. 1997), the explicit formulation for the extensor model is given by Ge ðX Þ ¼

r  sin b X lo  lu sin a

Gf ðX Þ ¼

kof  kuf  X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 þ k 2  kof kuf l2 þ l2  X 2 lo  lu kof o u uf lo lu

ð3Þ

 while a ¼ 2b þ arcsin kro sin b þ arcsin kru sin b and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 X ¼ lo þ lu  2lo lu cos a. The explicit formulation for the flexor model is

Fig. 1. The hinge-joint model and the geometric function for the flexor and extensor muscles Gf ðX Þ and Ge ðX Þ: Both functions increase directly as a function of X . For the model, lo;u : length of the thigh and the shank; ko;u : length of the extensor muscle from attachment to the middle of the patella and length of the patella tendon from mid-patella to its insertion; r: distance between center of rotation and mid-patella; kof ;uf : attachments of the flexor muscle at the femur and the tibia, respectively; X : distance between the ankle joint and the hip joint

ð4Þ

We would like to point out that for a human knee joint the functions Ge ðX Þ and Gf ðX Þ increase monotonously (Fig. 1). Hence, the partial derivative with respect to X is positive for both muscles: GeX ðX Þ ¼

oGe ðX Þ > 0; oX

GfX ðX Þ ¼

oGf ðX Þ >0 oX

ð5Þ

The advantage of this simple knee model is that it allows one to obtain individual musculoskeletal properties through simple noninvasive measurements. We used the prescription according to Sust et al. (1997) shown in Fig. 1. Moving-center-of-rotation model Based on the theory of Menschik (Menschik 1987) we developed a knee model that involves the moving center of rotation (MCR). In this model, the cruciate ligaments and their connecting lines between the attachment points can be described as a planar four-bar linkage. We included a patella in this model in order to calculate the moment arms of the flexor and extensor muscles depending on the knee flexion angle or the leg extension. In this simple model, the momentary center of rotation is represented as the point at which the cruciate ligaments crossed (Fig. 2). The moment arm of a muscle can be defined as the perpendicular distance between the line of action of the muscle and the center of rotation of the joint.

Fig. 2. The knee joint based on the theory of Menschik (1987) and the geometric functions for the monoarticular flexor (dash-dotted) and the extensor (solid) muscles. The anterior ACL and posterior PCL cruciate ligaments are described as a planar four-bar linkage system. The center of rotation is represented as the point at which the cruciate ligaments are crossed. The shortest distance between the tendon and the center of rotation represents the moment arm of the described muscle. The geometric functions increase directly as a function of the leg extension X

With this model it is possible to calculate the shape of the femoral condylus, depending on the location of the cruciate ligaments, and the shape of the tibial-articulating surface (Muller 1993a,b). There are two ways to determine the individual parameters of this model. First, we can use x-ray or

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Fig. 3. a The subject sitting on the leg press. The horizontal and vertical forces beneath the feet are measured independently for each foot with a sample frequency of 1 kHz. b The time course of leg extension X while performing vertical cyclic movements can be determined using the markers at the hip joint and the ankle joint. The ground reaction force can be measured using a Kistler force plate. The bipolar EMGs of the following muscles of the right leg are recorded: vastus lateralis, vastus medialis, rectus femoris, and biceps femoris

MRI-pictures of a lateral view of the knee joint to make a nonlinear regression between the digitized shape of the femoral condylus and the calculated condylus. The result of this regression is the length and location of the cruciate ligaments. The size of the patella can be determined from the images, or it can be estimated from a simple measurement at the skin. The other parameters, such as the length of the thigh and the shank, necessary to calculate the model can be measured noninvasively according to the prescription shown in Fig. 1. If an x-ray or MRI picture of the subject is lacking, a second method of determining the model parameters can be used. We calculated an averaged model of the knee using data of ten regressions of the first method. It is possible to scale the knee model to the depth of the knee, which can be measured with an anthropometric caliper (Wank et al. 2000). In the present study, we used the second method.

2.4 The muscle model The force-velocity relation of the flexor and extensor muscles can be described as a Hill-type muscle model for concentric contractions (Sust et al. 1997): c  a8vm 0 ð6Þ fH ðX ; V Þ ¼ vm ðX ; V Þ þ b For eccentric contractions we likewise apply a Hill-type structure just being reflected on the ordinate and on the abscissa (Van Leeuwen 1992): fH ðX ; V Þ ¼

C þ A8vm < 0 vm ðX ; V Þ  B

ð7Þ

Equations 6 and 7 are continuously decreasing. Assuming a continuous force-velocity relationship close to vm ¼ 0 it is possible to reduce the number of free

parameters to four (Wagner and Blickhan 1999). The remaining parameters can be estimated individually applying a nonlinear regression method established by Sust et al. (1997). The subjects were asked to perform maximum voluntary contractions, first concentrically against two different external loads followed by isometric contractions. The parameter values of the muscle model are varied by a mathematical algorithm until the simulated forces are in agreement with the measures. The muscle properties a, b, and c can be expressed with the more physiological properties fiso (maximum isometric force), vmax (maximum contraction velocity), and pmax (maximum power output), and vice versa. The force-length relation representing the myosin-actin overlap can be described as follows: 8 0 8 lm < lmin > > > lm lmin > > 8 lmin lm < l > < l lmin fl ¼ 1 ð8Þ 8 l lm lþ > > lm lmin > > 8 lþ < lm lmax > > : l lmin 0 8 lm > lmax with l ¼ lopt  Dlopt 2 , lopt being the optimum muscle length, Dlopt the depth of the plateau, and lmin (lmax Þ the minimum (maximum) length at which the muscle can generate a force. It is generally assumed that a summation of force-length relations of many fibers results in a smooth force-length relation for a whole muscle, resulting in a nonconstant slope. In the present study, we investigate the influence of different slopes of the forcelength relation on the stability of the system in general. Therefore, we decided to use this simplified model of the force-length relation. The maximum voluntary force production of the muscle fm results as the product of fH and fl : fm ¼ fl  fH :

ð9Þ

We would like to point out that an index e and f denoting an extensor and a flexor muscle, respectively, e.g., fme , defines the maximum voluntary force production of an extensor muscle. 2.5 The Ljapunovian stability analysis The equation of motion of the antagonistic model can be described as follows: 

 V X_ ð10Þ ¼ 1 1 V_ m Ge  Ee  fme  m Gf  Ef  fmf  g The Jacobian of the equation of motion (Eq. 10) is:

 0 1 0 ~ ð11Þ h ðY Þ ¼ a2 a1   o 1 while a2 ¼ oX mGe ðX Þ  Ee ðtÞ  fme ðX ; V Þ  o 1  oX m Gf ðX Þ  Ef ðtÞ  fmf ðX ; V Þ

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  and a1 ¼ oVo m1 Ge ðX Þ  Ee ðtÞ  fme ðX ; V Þ  oVo m1 Gf ðX Þ  Ef ðtÞ  fmf ðX ; V ÞÞ

Ef ¼

From this the eigenvalues rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 a1 þ a2 k1=2 ¼ 2 4

If it is possible to calculate a solution for both activation functions that takes values within the interval [0,1], the minimum condition for a self-stabilized trajectory can be realized with this antagonistic musculoskeletal-system.



ð12Þ

can be calculated. For the described movement the eigenvalues are time dependent. Therefore, we are using a stability measure that guarantees local stability in such a way that the real parts of the eigenvalues k1=2 must be negative at any given moment. Hence, based on the Routh-Hurwitz criterion the following conditions must be fulfilled: a1 < 0;

ð13Þ

a2 < 0

Consideration of the first condition a1 < 0 The geometric transformation Ge and Gf for the extensor and the flexor as well as the mass m are positive. Therefore, the condition defined by Eq. 12 is true when the partial derivative of fHe according to V is negative and one fHf is positive. This demand is true for both functions in general. Consideration of the second condition a2 < 0 To simplify the notation, we use the following abbreviations: o ðGe ðX Þ  fme ðX ; V ÞÞ ; S ¼ oX   o B ¼ oX Gf ðX Þ  fmf ðX ; V Þ

Multiplying a2 by m and rewriting we can generate the following relation, which must be fulfilled: Ee ðtÞ  S  Ef ðtÞ  B < 0

ð14Þ

In order to calculate the activation function for the extensor and flexor muscles to generate a stable movement, we must distinguish the four different cases listed in Table 1. Only the first two cases result in workable solutions. We can use the first two cases of Table 1 together with Eq. 1 to determine both activation functions for the extensor and the flexor muscles Ee and Ef as the minimum solution for self-stabilization. ( FG 8S > 0; B > 0 S ð15Þ Ee ¼ GeFfGme Gf fmf B 8S < 0; B > 0 Ge fme

Table 1. Four different cases to calculate the activation function for the extensor muscle

Ee  BS 8S > 0; 0 8S < 0;

B>0 B>0

ð16Þ

3 Results In order to determine the muscle properties of the five subjects individually, we used a software package (JOP Kinematics) to calculate a nonlinear regression of the muscle properties (fiso , vmax , pmax , and P Þ to six different data sets measured at the leg press (see Sect. 2.2) simultaneously (Fig. 4). The ability to calculate a fit for up to eight trials with different boundary conditions simultaneously improves the quality of the fit significantly. In order to quantify the normalized maximum eccentric force fecc according to fiso , we used a value of fecc = 1.5. With that value and the constants a, b, and c it is possible to calculate the constants A, B, and C to describe the eccentric part of the Hill-type muscle model (Eq. 7). The muscle properties of the five subjects are shown in Table 2. There are several properties contributing to the stabilization, but mostly they are not sufficient to stabilize the movement alone. An interaction of several stabilizing properties seems to be necessary. For one trial of the prescribed vertical movement we have combined the measured data of the leg extension X , its velocity V and the ground reaction force FG (Fig. 5b). In order to generate a self-stabilized movement, we calculated the activation functions of both the extensor (Eq. 15) and the flexor (Eq. 16) muscles using kinematic and dynamic data and the previously determined individual musculoskeletal properties of the subjects. We calculated the eigenvalues of the measured knee bends applying four different models. A simple hingejoint model (HJ) and the model including the moving center of rotation of the knee joint (MCR). For both models we further compared a monoarticular with a biarticular flexor muscle arrangement. Only for the MCR-model with a biarticular flexor muscle was it possible to calculate negative eigenvalues for the whole knee-bend cycle (Fig. 6). For all the other models parts of the cycle had positive eigenvalues.

Case

S

B

Ef

remarks

1

S>0

B>0

1 Ef > Ee  BS 0

2

S0

1 Ef 0 > Ee  BS

3

S>0

B