Nabipour, Mahdi

Sartori, M.

English

Hill-type muscle-tendon unit (MTU) models, are widely used in a variety of applications while maintaining an appropriate balance between model complexity and accuracy. Throughout the literature there are models of the MTU that use Splines and other conditional statements for modeling the hill-type muscle. Finding a closed-form equation for this system enables to implement different controllers for controlling the MTU force in closed-loop fashion

University of Twente Research Information

 28th Congress of the European Society of Biomechanics, July 9-12, 2023, Maastricht, the Netherlands  CLOSED-FORM MODELING OF THE SOLEUS MUSCULOTENDON UNIT  Mahdi Nabipour* (1), Massimo Sartori (2) *m.nabipour@utwente.nl  1 and 2. Faculty of Biomechanical Engineering, Engineering Technology, University of Twente, Netherlands   Introduction Hill-type muscle-tendon unit (MTU) models, Fig. 1, are widely used in a variety of applications while maintaining an appropriate balance between model complexity and accuracy. Throughout the literature there are models of the MTU that use Splines and other conditional statements for modeling the hill-type muscle [1]. Finding a closed-form equation for this system enables to implement different controllers for controlling the MTU force in closed-loop fashion.  Figure 1: The pennate hill-type muscle model that is modified using a parallel damping element (in red) Methods There were some attempts in the literature to obtain the closed-form equation for the MTU [2]. The tendon force equation is derived by the following equation:  ( )cos sin .T T MT M MF k L L L  = − +  (1) where, 𝐹𝑇and 𝑘𝑇 are the tendon force and stiffness, respectively. The MTU length, 𝐿𝑀𝑇 , and the pennation angle, 𝛼, are derived by the limb kinematics but the muscle fibre length, 𝐿𝑀 is derived by the integral of the following equation’s inverse:  ( )( ) ( )cosTPEM MV M M Mo LFFF La t F F L−=  (2) In this equation, 𝐹𝑂𝑀 𝐹𝐿𝑀, 𝐹𝑉𝑀, and 𝐹𝑃𝐸represent the optimal muscle force, muscle force-length, force-velocity, and the parallel element force, respectively. Since the muscle activation, 𝑎(𝑡), is in the denominator of equation (2), the Clay Anderson’s muscle model with optimal parameters becomes unstable when 𝑎(𝑡) ≅ 0:  Figure 2: Tibialis Anterior tendon force during plantar/dorsiflexion using a dynamometer Two strategies were used to solve this issue. First, different ?̇?𝑀s were tested using optimization and regression. Since in this strategy the optimization should be done for every muscle and gait separately, a second strategy was developed. In this strategy, equation (2) was simplified to a linear model, ?̃?𝑉𝑀 = 𝑎?̇̃?𝑀 + 𝑏, and damping was added to the hill-type muscle model (Fig. 1). The linearized model assumption is valid while |?̇?𝑀| ≤ 0.5 which is almost always the case for walking and running gaits [3]. Results Using the new formulation, the model conforms to the results obtained by the neuromusculoskeletal modeling toolbox, CEINMS [4], for different gaits and muscles.  Figure 3: Comparison of the tendon forces obtained by the proposed MTU modeling technique with CEINMS toolbox for: a/b) Soleus and latGas using dynamometer, c/d) Soleus in gaits with various speeds As a result, equation (1) can now be written in closed-form, ideal for designing different closed-loop controllers for lower-limb muscles (e.g., Soleus). References 1. M. Sartori, et al, "Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines," Journal of Biomechanics, vol. 45, no. 3, pp. 595-601, 2012. 2. C. Anderson, "Equations for Modeling the Forces Generated by Muscles and Tendons," presented at the BioE215 Physics-based Simulation of Biological Structures, 2007.  3. E.M., Arnold, et al, "How muscle fiber lengths and velocities affect muscle force generation as humans walk and run at different speeds," Journal of Experimental Biology, 216(11), pp.2150-2160. 4. C. Pizzolato, et al, "CEINMS: A toolbox to investigate the influence of different neural control solutions on the prediction of muscle excitation and joint moments during dynamic motor tasks," Journal of biomechanics, 48(14), pp.3929-3936. a b c d 

Closed-form Modeling of the Soleus Musculotendon Unit

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Closed-form Modeling of the Soleus Musculotendon Unit

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