# A closed loop musculoskeletal model of postural coordination dynamics

TL;DR: This work proposes to identify spindle reflex equivalent feedback gains for several target's frequency values in motor-control litterature, where spindle feedback gains are scaled by the central nervous system to adapt muscle stiffness depending on the postural task.

Abstract: A closed-loop model with actuator dynamics and sensory feedback has been developed to capture the complex postural behaviors observed in a human head tracking task. In motor-control litterature, spindle feedback gains are scaled by the central nervous system to adapt muscle stiffness depending on the postural task. We propose to identify spindle reflex equivalent feedback gains for several target's frequency values. Comparison with experimental results on human shows the relevance of this modeling, since our musculoskeletal model is able to exhibit reasonably well the behavioral invariants observed in human postures.

## Summary (3 min read)

### Introduction

- This simple task allowed the observation of several self-organized properties of the postural system, such as phase transition, multistability, critical fluctuations, hysteresis, and critical slowing down.
- An in-phase mode for low frequencies and an anti-phase mode for high frequencies.
- The authors will report human experimentations first, in order to describe the postural coordination concept, and then they will describe the modelling and the identification of model parameters in subsequent sections.

### A. Methods

- Following previous studies [11], [12], the experiment consists in tracking a moving target with the head while standing.
- Participants stood on a force platform in front of a physical target moved by a linear motor in antero-posterior direction, with the knees locked and the soles constantly in contact with the ground (Fig. 1-2-3).
- The experiment was performed on 11 healthy male subjects, with mean age 25, mean weight 75kg and mean size 1.79m.
- Target motion was sinusoidal with 10cm as amplitude, the frequency increases from 0.1Hz to 0.65Hz by 0.05Hz steps and during 10 periods.
- To capture the joint positions, a motion capture system (VICON NEXUS) was used, with 8 cameras (MX13) tracking 15 makers on the right side of the subject.

### B. Experimental results

- Fig.4 shows typical results for a representative subject (weight 75kg, size 1.80m).
- A transition is observed from in-phase to anti-phase mode around 0.4Hz.
- Each point is the mean value of the maximum (or minimum) joint position reached during the 10 oscillation periods performed at each frequency step.
- At the transition frequency, the ankle amplitude become very small (Fig. 10), and the relative phase between ankle and hip is difficult to estimate.
- Fig. 4(c) depicts mean values for torque amplitude estimation at each frequency step.

### A. Biomechanical model

- Barin [15] shows the relevance of an inverted pendulum structure in the case of a human sagittal plane task.
- Balance is described by the position of the CoP within the BoS, which can be expressed as a function of the dynamic parameters (eq.1).
- Euler’s equations were used for the calculation of the ground reaction forces as proposed by Cahouet et al. [16].
- The muscle length evolutions, function of the joint positions, are computed by using a double pulley model at each joint (see Fig. 5).

### B. Joint actuator dynamics

- Musculotendon contraction dynamics is often modeled with Hill type models.
- Hill type models are often linearized [18], but in their application, the authors will keep the non-linear dynamic phenomenon.
- Furthermore many musculoskeletal models as [19] consider the tendon completely stiff.
- The muscle model is composed of an active contractile element (CE) a passive parallel element (PE), which represents the intrinsic viscoelastic muscle properties, and a serial linear tendon (SE).
- The contractile element force is the product of the force-length Fl and the force-velocity Fv relationships and the activation act, as follows: Fce = act.

### C. Muscle redundancy and torque feedback

- Therefore muscle coactivation needs to be addressed.
- Previous human experimentation on sinusoidal standing perturbations [21] have shown a very small coactivation between the agonist and antagonist muscle group.
- Therefore, the coactivation will be neglected in their modeling.
- In human, a muscle force feedback is given by the Golgi Tendon Organs (GTO), since the GTO is in serial with the tendon, the gain KGTO is known to be constant.
- The muscle model activation must be positive and since (QT )+ is constant, the selection of the pulling muscle is based on the sign of the torque error.

### D. Spindle reflex feedback

- The restraining reflex joint torques are considered in the model in order to limit the joint motion and to improve the fit with human joint positions.
- It’s well known that spindle feedback reflexes (SFR) are important in postural muscles, especially in terms of stiffness control and disturbance rejection.
- Then CNS adapt the SFR gains, in order keep a good measure sensitivity.
- This theory is the target of a vivid debate but there is converging evidence.

### E. Closed-loop modeling

- The CNS needs to manage redundant sets of actuators and sensors to perform the tracking task.
- By analogy with inverse kinematics, the pseudoinverse matrix used in the control scheme depicted on Fig. 8 minimizes the norm of the supraspinal vector ||Uss(t)||.
- This equilibrium constraint is managed by the SFR loops which constrain muscle lengths (i.e. joint positions) close to rest positions.
- They correspond to the same typical subject described in section II.
- In addition, the operational space gains are taken as P = 800 and D = 1000, in order to let the closed-loop model follow the desired head position with a good accuracy for all frequencies.

### B. Comparison of experimental data with identified model results

- In order to evaluate the quality of their model, hence to give indications about the validity of their modelling and identification results, the authors will compare here the identified model simulation results and actual data.
- Fig. 9, 10 and 11 show the actual human movement of the head, ankle and hip joints, the CoP location and the ones obtained by the identified model parameters.
- The identified model simulation joint angles results for in-phase and anti-phase modes and during the phase transition are similar to the ones measured experimentally on human being, as it obviously should be since the criterion 4 minimizes the quadratic norm of modeldata joint angles discrepancy.
- Moreover, it is worth noting that the CoP location obtained by their identified model is very close to actual one.
- In addition, since simulated CoP remains within the base of support, the equilibrium constraint is satisfied.

### C. Hysteresis phenomenon

- A further model validation can be assessed by analysing the abilities of their model to exhibit the hysteresis phenomenon, hallmark of non-linear systems, which has been observed in human experiments [11].
- Note also, that this hysteresis phenomenon has never been modeled in the postural coordination framework.
- When the target frequency is up-chirped and then down-chirped, their closed loop model, with SFR gains taken as the mean values of the previous identified gains KII1 = 1500, KIa1 = 300, KII2 = 150, KIa2 = 10, exhibits the hysteresis phenomenon.
- Note that the gain values of the controller and the dynamics of the reference target influence the hysteresis region.
- Current work now examines the energetic cost for different types of reference dynamics around the transition frequency in order to better understand the hysteresis phenomenon.

### V. CONCLUSIONS

- The musculoskeletal closed-loop model the authors have developed provides realistic predictions of postural sway movements during head tracking task.
- Nevertheless, the authors believe that their model of postural coordination is promising in capturing behavioral invariants observed in human postures.
- More detailed sensory dynamics, as non-linear spindles, in addition to the time delay are under development to be considered in the model.
- Finally, to obtain a more realistic model, the authors will add more muscle actuators.
- But then, sensory information fusion and redundancy resolution will become an issue.

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### Cites background from "A closed loop musculoskeletal model..."

...The hip amplitude is larger than the ankle one.[2] ....

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...Coordinative in-phase small displacement of the ankle and the hip.[2] ....

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...3 [2] Typical human experimental results....

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...The hip amplitude is larger than the ankle one.[2]...

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...Coordinative in-phase small displacement of the ankle and the hip.[2]...

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3 citations

### Cites background from "A closed loop musculoskeletal model..."

...Similarly, to (Bonnet et al., 2009) the pulling muscle is determined based on the error in the system....

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...In this study, SFR gains identification is addressed in the stochastic framework, where the maximum-likelihood approach makes it possible to derive a criterion to be optimized to estimate these gains and an asymptotic uncertainty domain associated with the estimated gains [26]....

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...The asymptotic standard deviation associated with the estimated parameters are taken as the square root of the diagonal entry of the inverse of Fisher information matrix [26]....

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