Optimal Workloop Energetics of Muscle-Actuated Systems: An Impedance Matching View

Citation: Farahat WA, Herr HM ( Optimal Workloop Energetics of Muscle-Actuated Systems: An Impedance Matching View Waleed A. Farahat 0 Hugh M. Herr 0 Konrad P. Ko rding, Northwestern University, United States of America 0 1 Department of Mechanical Engineering, Massachusetts Institute of Technology , Cambridge, Massachusetts , United States of America, 2 The Media Laboratory and MIT - Harvard Division of Health Sciences and Technology, Massachusetts Institute of Technology , Cambridge, Massachusetts , United States of America Integrative approaches to studying the coupled dynamics of skeletal muscles with their loads while under neural control have focused largely on questions pertaining to the postural and dynamical stability of animals and humans. Prior studies have focused on how the central nervous system actively modulates muscle mechanical impedance to generate and stabilize motion and posture. However, the question of whether muscle impedance properties can be neurally modulated to create favorable mechanical energetics, particularly in the context of periodic tasks, remains open. Through muscle stiffness tuning, we hypothesize that a pair of antagonist muscles acting against a common load may produce significantly more power synergistically than individually when impedance matching conditions are met between muscle and load. Since neurally modulated muscle stiffness contributes to the coupled muscle-load stiffness, we further anticipate that power-optimal oscillation frequencies will occur at frequencies greater than the natural frequency of the load. These hypotheses were evaluated computationally by applying optimal control methods to a bilinear muscle model, and also evaluated through in vitro measurements on frog Plantaris longus muscles acting individually and in pairs upon a massspring-damper load. We find a 7-fold increase in mechanical power when antagonist muscles act synergistically compared to individually at a frequency higher than the load natural frequency. These observed behaviors are interpreted in the context of resonance tuning and the engineering notion of impedance matching. These findings suggest that the central nervous system can adopt strategies to harness inherent muscle impedance in relation to external loads to attain favorable mechanical energetics. - Funding: This work was supported by the DARPA Biomolecular Motors Program (award N66001-02-C-8034) and the MIT Media Lab. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. The capability of skeletal muscles to deliver mechanical power is key in determining the neuromechanical performance envelope of organisms. How fast and how far animals run, fly, swim, or jump is clearly limited by the mechanical power delivered by the muscletendon units to skeletal and environmental loads. Therefore, estimating the mechanical energetics of muscles (henceforth simply called energetics) has been of interest in diverse fields such as organismal biomechanics, biomimetic robotics and prosthetics [13]. Many factors influence the neuromechanical performance of organisms, including i) the dynamics and mechanical properties of muscle actuators, ii) skeletal mechanics, iii) neural control and iv) influence of loads external to the organism. Integrative approaches have been proposed to capture the interaction of all, or subsets of these factors. For example, the connection between muscle impedance (particularly stiffness) and neural control has been studied in depth with respect to postural and dynamic stability [4,5], locomotory functions [69], manipulation [10,11], and other biomechanical tasks [12]. In this work, we adhere to the definition of muscle mechanical impedance as the static and dynamic relation between muscle force and imposed stretch [4]. Muscle impedance encompasses muscle stiffness, which is the static relation between muscle force stretch only. In the context of muscle energetics, most investigations focused on experimentally measuring the power output of individual muscles at a range of frequencies, phases and electrical stimulation parameters, and finding maximal power generating capability of muscles under prescribed motion trajectories. However, the role of muscle-load interaction on output energetics has not been formalized. The central premise of this work is that the mechanical energetics of a muscle-actuated system cannot be determined in a meaningful manner without considering the coupling of muscle properties, load dynamics and neural activation. By considering this coupling explicitly, we arrive at phenomena that cannot be captured using standard workloop testing methodologies, including the opportunity to harness muscle-load interaction in an energetically advantageous manner. Muscle energetics have been characterized under dynamic conditions, both in vitro [13] and in vivo [9,14,15]. In vitro measurements relied almost invariably on the workloop technique [16]. In this approach, isolated muscles are subjected to predetermined periodic length variations in time (typically sinusoidal, but not always [17]) by means of an external motion source. At a given phase of the imposed oscillation, an electrical stimulus is delivered synchronously, resulting in periodic muscle contractions. A plot of muscle contractile force versus displacement results in a cyclic workloop, with the integrated area within Movement in organisms is a result of the interplay between biomechanics, neural control, and the influence of external environmental loads. Understanding the interaction between these factors is important not only for scientific reasons but also for engineering robotic systems and prostheses that strive to match biological performance. Muscle mechanical impedance is key in defining the mechanical interaction between muscles and their loads. It is well known that neural activation modulates muscle impedance, particularly stiffness, and that such modulation can be used advantageously to stabilize the posture and motion in organisms. Here, we show computationally and experimentally that stiffness modulation can also be used to enhance the capability of muscle to generate mechanical power, which is key in determining how fast animals can run, fly, swim, or jump. When muscles are activated optimally in relation to their external loads, they can create resonance conditions at optimal frequencies that significantly enhance their mechanical energetics by up to 7-fold. These findings can be interpreted in the context of the engineering notions of impedance matching and resonance tuning, which are commonly used as guiding principles in the design of diverse power optimal systems, such as communication circuits and robotic systems. the loop being a measure of the net muscle work done. These and similar measurements have been reproduced (...truncated)