Biomechanics and motor control

Biomechanics and Motor Control Defining Central Concepts Mark L Latash and Vladimir M Zatsiorsky Department of Kinesiology, The Pennsylvania State University, PA, USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright © 2016 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers may always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN: 978-0-12-800384-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Academic Press publications visit our website at http://store.elsevier.com/ Publisher: Nikki Levy Acquisition Editor: Nikki Levy Editorial Project Manager: Barbara Makinster Production Project Manager: Caroline Johnson Designer: Matthew Limbert Typeset by TNQ Books and Journals www.tnq.co.in Printed and bound in the United States of America Dedication To our wives, children, and grandchildren – the main source of happiness in life Preface Biomechanics of human motion and motor control are young fields of science While early works in biomechanics can be traced back to the middle of the nineteenth century (or even earlier, to studies of Borelli in the seventeenth century!), the first journal on biomechanics, the Journal of Biomechanics, has been published since 1968, the first international research seminar took place in 1969, and the International Society of Biomechanics was founded in 1973 at the Third International Conference of Biomechanics (with approximately 100 participants) Motor control as an established field of science is even younger While many consider Nikolai Bernstein (1896–1966) the father of the field of motor control, the journal Motor Control started only in 1997, the first conference—Progress in Motor Control—was held at about the same time (1996), and the International Society of Motor Control was established only in 2001 Both biomechanics and motor control have developed rapidly Currently, these fields are represented in many conferences, and many universities worldwide offer undergraduate and graduate programs in biomechanics and motor control This rapid growth is showing the importance of studies of biological movements for progress in such established fields of science as biology, psychology, and physics, as well as in applied fields such as medicine, physical therapy, robotics, and engineering New scientific fields explore new topics and work with new concepts Scientists are compelled to name them When the field is not completely mature, terms are often used with imprecise or varying meanings It is also tempting to adapt terms from more established fields of science (e.g., from physics and mathematics) and apply them to new objects of study, frequently with no appreciation for the fact that those terms have been defined only for a limited, well-defined set of objects or phenomena As a result, these established terms lose their initial meaning and become part of jargon This is currently the case in the biomechanics and motor control literature Lack of exactness and broad use of jargon are slowing down progress in these fields Inventing new terms, that is, renaming the same phenomena or processes without bringing a new well-defined meaning, can make the situation even worse The main purpose of this book is to try to clarify the meaning of some of the most frequently used terms in biomechanics and motor control The present situation can barely be called acceptable Consider, as an example, the title of a (nonexistent) paper: “The contribution of reflexes to muscle tone, joint stiffness, and joint torque in postural tasks.” As the reader will see in the ensuing chapters, all the main words in this title are “hints”: they are either undefined or defined differently by different researchers xii Preface There are two contrasting views on the importance of establishing precise terminology in new fields of science One of the leading mathematicians of the twentieth century, Israel M Gelfand (1913–2009; a winner of all the major prizes in mathematics and a member of numerous national academies) was seriously interested in motor control Israel Gelfand once said: “The worst method to describe a complex problem is to this with hints.” A contrasting quotation (from one of the prominent scientists in the field—we will not name him): “We should stop arguing about terms; this is a waste of time We should work.” The authors of this book consider themselves students of Israel Gelfand and share his opinion—arguing about terms is one of the very important steps in the development of science Using undefined or ambiguously defined terms (jargon) is worse than a waste of time; it leads to misunderstanding and sometimes creates factions in the scientific community where it becomes more important to use the “correct words” than to understand what they mean A cavalier attitude to terminology may lead to major confusion Consider, as an example, published data on muscle viscosity In the literature review on muscle viscosity (Zatsiorsky 1997) it was found that this term had been used with at least 11 different meanings, 10 of which disagree with the definition of viscosity in the International System of Units (SI) Diverse experimental approaches applied in similar situations resulted in sharply dissimilar viscosity values (the difference was, sometimes, thousand-fold) Even the units of measurement were different This is an appalling situation Biomechanics mainly operate with terms borrowed from classical (Newtonian) mechanics By themselves, these terms are precisely defined and impeccable However, their use in biomechanics needs caution In some cases, new definitions are necessary For instance, such a common term as joint moment does not exist in classical mechanics It is essentially jargon Skeptics are encouraged to peruse the mechanics textbooks; you will not find this term there In other cases, application of notions from classical mechanics needs some refinement For instance, the classical mechanical concept of stiffness cannot be (and should not be!) applied to the joints within the human body The term stiffness describes resistance of deformable bodies to imposed deformation; however, the joints are not bodies and joint angles can be changed without external forces In other words, if for deformable bodies, for instance, linear springs, there exist one-to-one relations between the applied force and the spring length, there is no such a relation between the joint angle and joint moment, or muscle length and muscle force In the mechanics of deformable bodies, stiffness is represented by the derivative of the force–deformation relation However, to call any joint moment–joint angle derivative, as some do, joint stiffness, would make this term a misnomer In some situations adding an adjective to the main term, for example, using the term apparent stiffness, can be an acceptable solution As compared to biomechanics, the motor control terminology is at a disadvantage— in contrast to biomechanics, it cannot be based on strictly defined concepts and terms of classical mechanics Some of the terms used in motor control, if not invented specifically for this field, are borrowed from such fields as medicine and physiology Not all of them are precisely defined and understood by all users in the same way Examples are such basic and commonly used terms as reflex, synergy, and muscle tone Preface xiii Motor control, as a field of science, aims at discovery of laws of nature describing the interactions between the central nervous system, the body, and the environment during the production of voluntary and involuntary movements This definition makes motor control a subfield of natural science or, simply put, physics At the contemporary level of science, relevant neural processes cannot be directly recorded In fact, the situation is even worse Even if one had an opportunity to get information about activity of all neurons within the human body, it is not at all obvious what to with these hypothetical recordings of such activity The logic of the functioning of the central nervous system cannot be deduced from knowledge about functioning of all its elements; this was well understood by Nikolai Bernstein and his students This makes motor control something like “physics of unobservables”—laws of nature are expected to exist, but relevant variables are not directly accessible for measurement To overcome these obstacles, scientists introduce various models and hypotheses that can be only in part experimentally confirmed (or disproved) As a result, the motor control scientists work with unknown variables, and these unknowns should be somehow named It is a challenging task to find a proper term for something that we not know A delicate balance should be maintained; the term should be as precise as possible, and, at the same time, it should not induce a false impression that we really know what is happening within the brain and the body The target audience of this book is researchers and students at all levels We believe that using exact terminology has to start from the undergraduate level; hence, we tried to make the contents of the book accessible to students with only minimal background knowledge While the book is not a textbook, it can be used as additional reading in such courses as Biomechanics, Motor Control, Neuroscience, Physiology, Physical Therapy, etc Individual chapters in the book were selected based on personal views and preferences of the authors We tried to cover a broad range, from relatively clear concepts (such as joint torque) to very vague ones (such as motor program and synergy) There are many other concepts that deserve dedicated chapters But some of the frequently used concepts are covered in the existing chapters (e.g., internal models are covered in the chapter on motor programs, similarly to how these notions are presented in Wikipedia); others have been covered in recent reviews (such as normal movement, Latash and Anson 1996, 2006); and with respect to others, the authors not feel competent enough (e.g., complexity) We hope that our colleagues will join this enterprise and write comprehensive reviews or books covering important notions that are not covered in this book The book consists of four parts Part covers biomechanical concepts It includes the chapters on Joint torques, Stiffness and stiffness-like measures, Viscosity, damping and impedance, and Mechanical work and energy Part deals with basic neurophysiological concepts used in the field of motor control such as Muscle tone, Reflex, Preprogrammed reactions, Efferent copy, and Central pattern generator Part concentrates on some of the central motor control concepts, which are specific to the field and have been used and discussed extensively in the recent motor control literature They include Redundancy and abundance, Synergy, Equilibrium-point hypothesis, and Motor program Part includes two chapters xiv Preface with examples from the field of motor behavior, Posture and Prehension Only two behaviors have been selected based on the personal experience of the authors; they cover two ends of the spectrum of human movements, from whole-body actions to precise manipulations The book ends with the detailed Glossary, in which all the important terms are defined Mark L Latash Vladimir M Zatsiorsky Acknowledgments The book reflects the personal views of the authors developed over decades of work in the fields of biomechanics and motor control During that time, we have been strongly influenced by many of our colleagues and students We would like to thank all our colleagues/friends (too many to be named!) who helped us develop our views, participated in numerous exciting research projects, and provided frank critique of our own mistakes Our graduate and postdoctoral students have played a very important role not only by performing studies cited in the book but also by asking questions and engaging in discussions that forced us to select words carefully to achieve maximal exactness and avoid embarrassment Many of our colleagues are unaware of the importance of their influence on our current understanding of the fields of biomechanics and motor control We are very grateful to all researchers who performed first-class studies (many of which are cited in the book) leading to the current state of the field of movement science Joint Torque Concept of joint torques—or joint moments as many prefer to call them—is one of the fundamental concepts in the biomechanics of human motion and motor control A computer search in Google Scholar for the expression joint torques yielded 194,000 research papers Even if we discard the returns that are due to the possible “search noise,” the number of publications in which the above concepts were used or mentioned is huge The authors themselves were surprised with these enormous figures Such popularity should suggest that the term is well and uniformly understood and its use does not involve any ambiguity It is not the case, however In classical mechanics the concept of joint torques (moments) is not defined and is not used Peruse university textbooks on mechanics You will not find these terms there One of the authors vividly remembers a conference on mechanics attended mainly by the university professors of mechanics where a biomechanist presented his data He was soon interrupted with a question: “Colleague, you are using the term ‘joint moment’ which is unknown to us Please explain what exactly you have in mind.” 1.1 Elements of history An idea that muscles generate moments of force at joints was understood already by G Borelli (1681) Joints were represented as levers with a fulcrum at the joint center and two forces, a muscle force and external force acting on a limb, respectively The concept of levers in the analysis of muscle action was also used by W and E Weber (1836) Only static tasks have been considered Determining joint moments during human movements is a sophisticated task (usually called the inverse problem of dynamics) It requires: Knowledge of the mass-inertial characteristics of the human body segments, such as their mass, location of the centers of mass, and moments of inertia (German scientists Harless (1860) and Braune and Fisher (1892) were the first to perform such measurements on cadavers) Recording the movements with high precision that allows computing the linear and angular accelerations of the body links This was firstly achieved by Braune and Fisher, and the study was published in several volumes in 1895e1904 It took the authors almost 10 years to digitize and analyze by hand the obtained stroboscopic photographs of two steps of free walking and one step of walking with a load Later, in 1920, N A Bernstein (English edition, 1967) improved the method, both the filming and the digitizing techniques It took then “only” about month to analyze one walking step With contemporary techniques it can be done in seconds Biomechanics and Motor Control http://dx.doi.org/10.1016/B978-0-12-800384-8.00001-6 Copyright © 2016 Elsevier Inc All rights reserved Biomechanics and Motor Control Solving the inverse problem mathematically and performing all the computations For simple two-link planar cases (such as a human leg moving in a plane), this was first done by Elftman (1939, 1940) The computations were done by hand With the development of modern computers the opportunity arose to study more complex (but still planar) movements (Plagenhoef, 1971) For the entire body moving in three dimensions the first successful attempts of computing the joint torques during walking and running in main human joints in 3D were reported only in the mid-1970s (Zatsiorsky and Aleshinsky, 1976; Aleshinsky and Zatsiorsky, 1978) Existence of the interactive forces and torques, i.e., the joint torques and forces induced by motion in other joints, and their importance, was well recognized by N A Bernstein (1967) for whom it was one of the main motivations for developing his ideas on the motor control 1.2 What are the joint torques/moments? Consider first what classical mechanics tell us This is really an elementary material 1.2.1 Return to basics: moment of a force and moments of a couple In mechanics two basic concepts are introduced, moment of force and moment of couple Moment of force According to Newton’s second law (F ¼ ma) a force acting on an unconstrained body induces a linear acceleration of the body in the direction of the force Also, any force that does not intersect a certain point generates a turning effect about this point, a moment of force The moment acts on a body to which the force is applied The moment of force MO about a point O is defined as a cross product of vectors r and F, where r is the position vector from O to the point of force application and F is the force vector MO ¼ r  F (1.1) The line of action of MO is perpendicular to the plane containing vectors r and F The line is along the axis about which the body tends to rotate at O when subjected to the force F The magnitude of moment is Mo ¼ F(rsinq) ¼ Fd, where q is the angle between the vectors r and F and d is a shortest distance from O to the line of action of F, the moment arm The moment arm is in the plane containing O and F The direction of the moment vector MO follows the right-hand rule in rotating from r to F: when the fingers curl in the direction of the induced rotation the vector is pointing in the direction of the thumb Oftentimes the object of interest is not a moment of force about a point MO but the moment about an axis OeO, for instance a flexioneextension axis at a joint Such a moment MOO equals a component (or projection) of the moment MO along 394 Glossary Specific tension See Specific force Spectral analysis A method of signal processing that represents a signal as a superposition of sine waves at different frequencies and with different amplitudes Speed The time rate of covering distance Speed is a scalar Speed-accuracy trade-off When a person moves at different speeds to a visual target, movement time varies as a linear function of the ratio of distance to the standard deviation of the final position Speed-difficulty trade-off (Fitts’s Law) Under the instruction to be both fast and accurate, movement time shows a logarithmic dependence on the ratio of movement distance to target size Spinal cord The elongated structure within the central nervous system, from the medulla to the lumbar spinal vertebrae, which plays a major role in the sensorimotor function It contains pathways that deliver sensory information to and neural commands from the brain, and interneurons that mediate reflexes and play a role in the control of certain movements Roots Sites of entrance and exit of axons Dorsal Sites of entrance of axons that carry sensory information into the spinal structures Ventral Sites of exit of axons that carry signals from the spinal cord to peripheral structures such as muscles Segments Each segment receives sensory information from a particular area of the body and innervates muscles in more or less the same area of the body There are cervical segments, 12 thoracic segments, lumbar segments, and sacral segments Spinal ganglia Groups of neurons outside the spinal cord but in close proximity to it Spinal ganglia contain bodies of proprioceptive neurons Spring constant See Stiffness Spring-like actions (of muscles and extremities) Elastic energy storage and recoil Stability A physical property of a system reflecting its ability to keep the current state or trajectory when subjected to a transient perturbation Dynamic An ability to move toward the original trajectory after a transient perturbation Postural An ability to maintain posture despite possible external perturbations Static An ability to return to the same steady state after a transient perturbation Stages of motor learning Three stages suggested by N Bernstein: freezing redundant degrees of freedom, releasing those degrees of freedom, and utilizing external forces to one’s advantage Static analysis (of kinematic chains) Direct static analysis Computing the end-effector force from known joint torques Inverse static analysis Computing the joint torques from known external forces Static optimization Optimization that neglects time Stiffness The amount of force per unit of deformation Stiffness is the inverse of compliance Application of this term to “active objects,” such as muscles or human extremities, is questionable because their resistance to deformation, for example, to stretch, is time dependent and is under neural control See also Apparent stiffness Apparent stiffness A stiffness-like measure obtained from an active object, for example, an active muscle The word apparent underscores that this measure differs from the analogous parameters of passive objects Joint stiffness A misnomer trying to represent joint mechanical behavior as that of a spring “Apparent joint stiffness” is recommended Limb stiffness A misnomer trying to represent limb mechanical behavior as that of a spring Glossary 395 Short-range stiffness Muscle resistance to stretch during the initial phase of muscle elongation Stiffness ellipse (at the endpoint) The ellipse obtained by connecting the tips of the restoring forces in response to a unit deflection in all directions Ellipse orientation The angle between the major axis of the ellipse and the X-axis of a fixed reference system Ellipse shape (Eccentricity) The ratio of the major to minor axes of an ellipse Ellipse size The ellipse area Stiffness matrix A matrix relating the differential force or torque increments with the differential deflection or deformation Strain A relative elongation Dl/l where l is the initial length of the object and Dl is its elongation Apparent strain (of a tendon) Amount of elongation per unit of tendon length Eulerian strain The ratio of the change in length of a line segment to its final length after the deformation Lagrangian strain The ratio of the change in length of a line segment to its initial length before the deformation Macroscopic strain (of a tendon) See Apparent strain Material strain (of a tendon) The tendon fiber length change per unit of fiber length Microscopic strain See Material strain Transverse strain Strain normal to the applied load Stress The amount of force per unit of area Normal stress Stress in a plane orthogonal to force direction Shear stress Stress in a plane parallel to the force direction Stress relaxation Decrease of stress over time under constant deformation Stretch reflex Change in muscle activation induced by muscle stretch Stretch-shortening cycle The sequence of muscle stretch and shortening Synapse A structure consisting of a presynaptic membrane, a synaptic cleft, and a postsynaptic membrane that uses physical and chemical processes to enable transmission of excitation and inhibition between excitable cells Excitatory A synapse that causes depolarization of the postsynaptic membrane, that is, brings its membrane potential closer to the threshold for action potential generation Inhibitory A synapse that causes hyperpolarization of the postsynaptic membrane, that is, brings its membrane potential further away from the threshold for action potential generation Neuromuscular A synapse between a terminal branch of the axon of a motoneuron and a muscle fiber Non-obligatory A synapse that, when it acts alone, is unable to bring the postsynaptic membrane to the threshold for action potential generation Neuro-neural synapses are typically nonobligatory Obligatory A synapse that always brings the postsynaptic membrane to the threshold for action potential generation Neuromuscular synapses are typically obligatory Synergy (a) Stereotypical patterns of activation of major flexor or extensor muscles typical of certain movement disorders, for example, those seen after stroke; (b) A set of variables that show correlated changes in time or with changes in task parameters; or (c) A neural organization that shares performance among a redundant set of variables and ensures stability of performance variables by using flexible combinations of elemental variables Covariation A feature of a synergy characterized by covariation of elemental variables that helps maintain a desired value of a performance variable 396 Glossary Linear torque synergy Linear scaling of joint torques during voluntary movement of a multi-joint chain Multi-muscle (a) A group of muscles that show correlated changes in activation levels over time or with changes in task parameters; (b) A neural organization ensuring covaried adjustments in muscle activation levels that keep a desired value or a desired time profile of a performance variable Sharing A feature of a synergy characterized by a pattern of relative involvement of elemental variables T Tangent modulus of elasticity See Young’s modulus Tendon A band of fibrous tissue that connects a muscle to a bone Tendon action (of muscles) Muscles’ behavior as nonextensible struts Tetanic Pertaining to Tetanus Tetanus A sustained muscle contraction caused by high-frequency firing of motoneurons Thalamus A brain structure that consists of a number of nuclei; it is part of several major loops that link the cortex of the large hemispheres, the basal ganglia, and the cerebellum, participating in the production of movement Thixotropy Property of a substance to decrease its viscosity when it is shaken or stirred Three-element muscle model See Hill’s model Toe region A region of the stress–deformation curve of a tendon at low levels of the stress Toe-region modulus The average slope of the stress–deformation curve in the toe region Tonic stretch reflex A reflex mechanism with an unknown neural path that links muscle length to active muscle force production Torque See Moment of a couple Active joint torques Joint torques generated by torque actuators Joint torque Two moments of force about the joint rotation axis acting on the adjacent segments The moments are equal in magnitude and opposite in direction Interaction torques (forces) The joint torques and forces induced by motion in other joints Net joint torque The total torque produced at a joint; the sum of the active and passive torques Passive joint torques Joint torque resisted by passive structures, in particular by the skeleton and ligaments Reactive torques The torques arising due to Newton’s third law; see also Interaction torques Torque agonists (in grasping) The fingers resisting external torque and hence helping to keep rotational equilibrium Torque antagonists (in grasping) The fingers assisting external torque Trajectory In kinematics, a path of the moving point Generally, a continuous time function of a physical variable Control The time function of a control variable Equilibrium The time function of variables characterizing equilibrium states of the system Virtual An imprecise term, used sometimes as a synonym of equilibrium trajectory and sometimes as a trajectory that the system would have shown if the external load were zero Transcranial magnetic stimulation A method of noninvasive stimulation of neural structures using eddy currents produced by a quickly changing magnetic field Translation (in kinematics) Movement of a body so that any line fixed with the body remains parallel to its original position Glossary 397 Trembling (in the rambling-trembling hypothesis) Oscillation of the body around the rambling trajectory while standing Tremor Involuntary cyclic movement associated with alternating activation bursts in the agonist and antagonist muscles Parkinsonian tremor Is typically postural; it may be alleviated by voluntary movement Cerebellar tremor Has postural, kinetic (increases during movement), and intentional (increases when the extremity approaches the target) components Triphasic activation pattern A muscle activation pattern consisting of an initial agonist burst accompanied by a low level of antagonist co-contraction, followed by an antagonist burst (during which the agonist is relatively quiescent), and ending with one more agonist burst Twitch A brief muscle contraction in response to a single stimulus Two-joint muscles Muscles that cross and serve two joints U Ultrasound Sound pressure with a frequency exceeding the upper limit of human hearing, approximately 20 kHz Uncontrolled manifold A subspace in the space of elemental variables corresponding to a fixed value of a performance variable; the controller does not have to exert control as long as the elemental variables stay within the uncontrolled manifold Hypothesis A hypothesis that the central nervous system organizes an uncontrolled manifold (UCM) corresponding to a desired value of a potentially important performance variable and then acts to keep the elemental variables within the UCM V Variability (of human motion) A feature of motor performance reflecting different solutions for the same motor problems that are seen over time or during repetitive trials Variance A measure of variability equal to standard deviation squared “Good” variance A component of variance that does not affect a specific performance variable “Bad” variance A component of variance that affects a specific performance variable Vector A physical or mathematical variable having magnitude and direction; a unidimensional array of numbers Binormal vector Vector perpendicular to the tangent and normal vectors Couple vector A free vector that represents a force couple The couple vector is normal to the plane of the couple Curvature vector See Normal vector Normal vector Vector in the direction of normal acceleration, also called the Curvature vector Tangent vector Vector in the direction of the derivative at the point of interest on a curve If the curve were representing a trajectory of a point in time, the tangent vector would be a unit vector in the direction of the instantaneous velocity of the point Vector product See Cross product Velocity The time rate of change of position Velocity is a vector Angular velocity The rate of movement in rotation Joint velocity Angular velocity at a joint Segment velocity Velocity of a body segment as viewed by an external observer, in an absolute reference frame Segment velocity can be both linear and angular 398 Glossary Velocity-dependent response (of human muscles to stretch with constant moderate speed) The second phase of the response, during which force continues to increase but with progressively slower rate until the force reaches its peak at the end of stretch Vestibular apparatus Structures in the inner ear that contain vestibular receptors; these receptors convert linear and angular accelerations into sequences of action potentials Vestibular nuclei Four paired nuclei located in the brain stem: the superior, lateral, medial, and inferior nuclei They receive projections from the vestibular receptors and from other structures including the cerebellum Vestibulospinal tracts The lateral vestibulospinal tract innervates antigravity muscles; the medial vestibulospinal tract innervates neck muscles that are responsible for stabilizing the head Vibration-induced falling (VIF) Violations of vertical posture produced by high-frequency, low-amplitude vibration applied to postural muscles (VIFs can also be induced by vibration applied to some of the other muscles) Virtual displacement A hypothetical small displacement of a body or a system from an equilibrium position Virtual finger An imaginary digit with the mechanical action equal to that of a set of fingers (commonly, the four fingers of a hand) combined Viscoelasticity Combination of viscous and elastic properties in materials Viscosity Internal friction between molecules Viscosity hypothesis Under standard stimulation a muscle generates similar internal forces but these forces are not manifested externally due to the viscous resistance within the muscle The hypothesis was abandoned after A.V Hill (1938) showed that the heat production is sharply different in concentric and eccentric muscle actions W Weakest-link principle Force development rate is determined by the slowest muscle fibers Weight (of an object) The gravity force exerted on the object Work (of a force) Scalar product of the vectors of force and displacement External work (in biomechanics) Work done by the performer on the environment Internal work (in biomechanics) Work done by the performer to change the mechanical energy of own body or body segments Negative muscle work Work done by a muscle to resist its elongation The absolute value of this work equals to the value of work done on the muscle by an external force to extend the muscle Positive work (of a force or a moment of force) Work done over a linear or angular displacement in the direction of the force or moment Quasi-mechanical work An imagined work that would be done on a body segment by the resultant force and couple if there were no transformation between the gravitational potential energy and kinetic energy Virtual work The work done by a force over a virtual displacement Work-energy principle (for a rigid body) The total work done on the body by several forces equals the change in the body’s mechanical energy Working envelope The set of boundary points that can be reached by the end effector Wrench A force and a couple with the vectors along the same line X X-rays Electromagnetic radiation with the frequencies outside the visible range, 1016–1019 Hz Glossary 399 Y Yield effect Sudden reduction of muscle force during a fast stretch Young’s modulus The ratio of the uniaxial stress over the uniaxial strain in the range of the linear stress–strain relation Z Z-membrane A membrane separating two sarcomeres Zero-work paradox For movements beginning and ending at rest at the same location the energy is expended but the total work is zero Index Note: Page numbers followed by “f” indicate figures and “t” indicate tables A Abundance coordination, 178–179 definition, 177 in movements, 195–197, 196f principle of, 185–186 Alpha-motoneurons (a-MN), 108–109, 108f Anticipatory postural adjustments (APAs), 322–323 Anticipatory synergy adjustments (ASAs), 228–229, 229f, 324–326, 325f–326f Apparent stiffness leg stiffness, 44, 45f one-degree-of-freedom joints, 40–41 planar kinematic chains cross-coupling terms, 43 direct terms, 43 endpoint apparent stiffness, 43 endpoint deflection direction, 41, 41f force–displacement relations, 41 stiffness ellipse, 42–43, 42f B Babinski reflex, 118 Back-coupling, 236–237 Bernstein’s engrams, 278–279 Brain stem reflexes, 107 C Center of mass of a body (CoM), 12 Central neural drive (CND), 348 Central pattern generators (CPGs), 289 afferent signals, 170 alpha-motoneurons, activation patterns, 170 cyclic vs discrete actions, 168–170, 169f definition, 160 descending control, 165–166 half-center model, 162–163, 162f–163f history “deafferented” persons, 160 fictive locomotion, 158–159 flexion and crossed-extension reflexes, 157–158 locomotor strip, 158–159 locust ganglia, 158 mesencephalic locomotor region (MLR), 158–159, 158f neural activity pattern, 159 peripheral receptors, 159 rhythmic activity, 159 kinetic/kinematic variables, 157 neuronal mechanisms, 164 noradrenergic system, 164–165 persistent inward currents (PICs), 160 postural activity, 161 postural control system, 170–171 rhythmic patterns, hierarchy, 166, 167f scratching and stepping rhythmic movements, 164, 164f sensory effects corrective stumbling reaction, 167–168, 168f discomfort-induced therapy, 168 rhythmic locomotor-like patterns, 166–167 stance and swing phases, 166 triphasic pattern, 164 voltage–current characteristics, 160, 161f Centripetal coupling coefficients, 21, 23 Conditioned reflexes, 107 Contact force, 13, 22 Contemporary motion analysis systems, 179 Contractile component (CC), 36 Controlled-release method, 127 Coupling inertia coefficients, 21, 23 402 CPGs See Central pattern generators (CPGs) Crossed extensor reflex, 109–110 D Discomfort-induced therapy, 168 Dynamic force enhancement, 109 E Early postural adjustments (EPAs), 323–324, 324f Efferent copy afferent signals (reafference), 142–143, 142f, 336 Bayesian probability, 141–142 exafference, 140 history, 139–140 kinesthetic perception alpha- and gamma-motoneurons, 144–146, 145f Golgi tendon organs, 146, 146f intrafusal muscle fibers, 144, 145f muscle stimulation, 148–149, 148f muscle vibration, 147 neural signals, 143 posture and locomotion, vibrationinduced effects, 147–148 power-producing extrafusal muscle fibers, 144, 145f proprioceptor neurons, 143, 144f sensory endings, types, 143 motor command generation, 140, 141f motor control forces and muscle activations, 149–150 methods, 149 referent configuration hypothesis See Referent configuration hypothesis posture-stabilizing mechanisms, 142 preflexes, 142 Elftman’s approach, 64 End effector force, 13, 22 Energy saving mechanisms, 64 energy expenditure, walking, 69 energy recuperation/compensation, 80–81 in human motion, 70 mechanical energy transfer adjacent segments, 73–75, 74f–75f fraction approach, 75–76 Jumping Jack, 76–78, 78f Index kinematic chain, two-joint muscle, 75, 76f mechanical energy expenditure (MEE), 79 nonintercompensated sources, 79 source approach, 75–76 source intercompensation, 79 two-joint muscles, 80 variants of, 76, 77t pendulum-like motions energy conservation, 70 energy conservation coefficient, 73, 74f ideal pendulum, 70, 71f quasi-mechanical work, 71–72 swing phases, walking, 70–71 time course, fractions, 70–71, 72f Enslaving, 214–215, 230–231 Equifinality, 264–266, 265f Equilibrium-point (EP) hypothesis, 115, 150–151, 151f, 308, 315 alpha-model, 252, 252f definition, 251 equifinality, 264–266, 265f history alpha-model, 250 dynamic systems and ecological psychology approach, 247–248 mass-spring system, 249–250 muscle force-length characteristics, 248, 249f servo-control hypothesis, 249, 250f voluntary movements, 247–248 invariant/tonic stretch reflex characteristics, 251–252 joint control agonist and antagonist, 259–260, 259f co-activation command, 260 joint compliant characteristic (JCC), 259–260 measuring control and equilibrium trajectories, 260–262, 261f–262f reciprocal command, 260, 260f l-model, 251–252 motor redundancy, 247 muscle length values, 251, 251f nonzero activation, 253 Index referent configuration hypothesis, 247, 262–264, 263f single muscle external load characteristic, 256–257 factors, 258, 259f passive and active movement, 255–256, 256f three basic trajectories, 257–258, 257f synergies elemental variables, trials adjustments, 267–268 kinetic, kinematic/electromyographic groups, 267 referent configurations, 269–270, 269f spasticity, 266, 267f stereotypical motor pattern, 266 tonic stretch reflex, 268–269, 268f threshold control hypothesis, 247 threshold elements cartoon neuron, 253–254, 254f excitatory sensory feedback, 255 membrane potential, 254 refractory period, 253–254 stretch reflex, 255, 255f subthreshold depolarization, 254–255 Equivalent joint torque, 22 F Fenn’s approach, 64 Finger enslaving inter-finger connection matrix, 349–350 matrix equation, 349–350 normal force decomposition, 350, 351f three-layer neural network, 348–349, 349f torque antagonists, 350 Flexor reflex, 109–110 Force-couple system, 18 Free moments, 5, 5f Functional stretch reflex, 123 G Gamma-motoneurons (g-MN), 108–109, 108f Generalized external force, 13, 22 Goal-equivalent manifold (GEM), 220 Golgi tendon organs (GTO), 105, 106f Grasping antagonist fingers, 347, 348f “central” fingers, 347 403 closures, classifications, 335 finger forces, 347 form and force closures, 335 grasp force collinear forces, 343 hand dynamometer, 342–343 load force, 343 movement kinematics, 344–345, 345f static, statodynamic and dynamic fractions, 343, 344f three-digit grasps, 343–344, 344f grasp moment, 345–346 grasp pattern selection, 335 hand and digit actions finger enslaving See Finger enslaving force deficit, 347–348 grasp force and moment enslaving, 350–352, 352f history hand dexterity, 335, 335f “inverted-T ” handle/beam apparatus, 336, 336f pinch grasps, 335–336 internal force, 341–342 manipulation force, 341–342 multidigit grasps, 346 multifinger precision grasps, 339–340 object manipulation, 335 orthonormal basis vectors, 342 “peripheral” fingers, 347 power and precision grips, 335 prehension synergies chain effects, 354–355, 355f–356f local effects, 353, 353f multi finger grasping tasks, 352–353 superposition principle, 354–358, 357f synergic effects, 353, 353f uncontrolled manifold (UCM) hypothesis, 354 VF and thumb tangential forces, 354, 354f reach-to-grasp movement, 335 robotic manipulators, 342 single-finger contacts apparent tangential stiffness, 337 digit–object interaction, 337 fingertip deformation, 337 hard-finger contact, 337 point contact, 337 404 Grasping (Continued) six-component force sensors, 337 slip prevention, 338–339, 338f soft-finger contact, 337 tangential forces, 338 torque agonists and antagonists, 346–347, 346f vector-by-vector analysis, 342 virtual finger and hierarchical control, 340–341, 341f H Heterogenic reflex, 106 Hypothetical hierarchical system, 281–282, 281f I Impaired motor programs, 293–294, 294f Inborn reflexes, 107 J Joint compliant characteristics (JCC), 259–260 Joint torque center of mass of a body (CoM), 12 centripetal coupling coefficients, 21, 23 coupling inertia coefficients, 21, 23 definition, delimitations actual joint torques and mechanical energy, 11 equivalent torques, 12 flexion moment, 9–10 movement analysis, slow horizontal arm extension, 10–11, 11f two one-joint muscles, three-link system, 10, 10f elbow flexion movement, 18–19, 19f equivalent joint torque, 22 external contact force control force-couple system, 18 scalar method (Jacobian method), 15–16 vector method (geometric method), 16–17, 17f–18f inverse problem solution, joint flexion, joint moment, 21 linear actuators, Index linear and angular accelerations, body links, mass-inertial characteristics, moment of couple, 5–6, 5f moment of force, 4–5 muscle and external force, muscle-tendon complex, patellar force/quadriceps force, single-joint muscles, statics Jacobian matrix, 13 kinematic chain, 13 motor redundancy, 13 overdetermined task, 13 skeletal system, 13 tools, 12 transpose Jacobian, 14–15, 14f torque actuator, two-joint muscles actual joint torques, adjacent body segments, biarticular muscle, 7–8 equivalent torques, three-segment system, 7, 8f two-link planar chain, 20, 20f wrist/hand behavior, 19 K Kinematic synergies, 221–222 Kinetic synergies force-related and moment-related components, 224 multi-element parallel chains, 222 multi-joint kinetic synergies, 225–226 parallel force/moment generating elements, 222 three-finger grasp, 224–225, 225f two-finger pressing task, 223, 223f two-level hierarchical scheme, 224, 224f L Linear and angular accelerations, body links, Long-loop reflex, 124 M Marginal redundancy, 184–185, 184f Mechanical energy expenditure (MEE), 79 Index Mechanical work and energy energy saving mechanisms See Energy saving mechanisms external work, 63 history, 64–65 in human movements, 66–67 fraction approach, 67–68 source approach, 68–69 internal work, 63 movement economy, 63 negative muscle work, 66 scalar product, 65 static forces, 65 Minimum-torque-change model, 190 Monosynaptic reflexes, 103–104, 104f Monosynaptic stretch reflexes, 34–35 Moore–Penrose pseudoinverse, 186 Motor program action plan, 278 agonist activation, 279, 280f Bernstein’s engrams, 278–279 brain clock vs emergent timing, 282 contemporary sources, 275–276 cortical neuron activity, 291, 292f definition, 278 dynamical neural field theory, 281–282 history, 276–278 hypothetical hierarchical system, 281–282, 281f impaired motor programs, 293–294, 294f internal models, 276 action potential transmission velocity, 284–285, 285f classical mechanics and control theory, 283 force field, 286, 287f multijoint limb, 283, 284f physical/physiological approach, 286 signal transmission, 284–285 threshold elements, 283–284 transfer effects, 286 motor learning, 295–297, 296f muscle activation patterns, 279 neuronal population vectors, 291–293 referent coordinates, 288–289 relative phase and standard deviation, 279–280, 280f spinal cord, 289–290, 290f 405 symbolic movements, 275 time series, 279, 280f two-element tasks, 279–280 Motor synergy anticipatory synergy adjustments (ASAs), 228–229, 229f atypical development, 230 back-coupling, 236–237 definition, 205 enslaving, 230–231 error compensation, 205 history, 206–207 matrix factorization techniques, 239 neurological disorders, 231, 232f optimal feedback control, 235–236 with practice agonist–antagonist muscles, 232 intertrial data distributions and variance components, 232–233, 233f multi-finger accurate force production task, 233–234, 234f synergy index, 233–234 transcranial magnetic stimuli (TMS), 235 uncontrolled manifold hypothesis, 232 referent configurations, 237–239, 237f requirements, 205 synergy-A, 207–208, 208f synergy-B advantages and disadvantages, 209 nonnegative matrix factorization (NNMF), 208–212, 210f principal component analysis (PCA), 208–212, 210f–211f synergy-C See Synergy-C MT See Muscle tone (MT) MTU See Muscle–tendon units (MTU) Muscle components fiber-reinforced composites, 35–36 Hill-type models, 36, 36f lumped-parameter model, 36 muscle behavior, 36–37, 37f quick-release and controlled-release methods, 37, 38f SEC compliance vs muscle tension, 37, 39f Muscle synergies, 226–228, 227f Muscle–tendon units (MTU), 33, 38–40 Muscle tone (MT) apparent stiffness, 88 Bernstein definition, 88 406 Muscle tone (MT) (Continued) complete relaxation, 87 Down syndrome (DS), 86–87 high muscle tone, 94–96, 95f history, 85 low muscle tone, 92–94, 94f muscle–tendon complex, 86–87 operational definitions, 86 relaxed muscle, 91–92, 92f tonic stretch reflex active muscle force, 88, 89f characteristics, 90–91, 91f deafferentation, 88 equilibrium-point hypothesis, 89–90 Muscle viscosity theory, 58 Fenn effect, 51–53, 52f history, 50 muscle contraction, 51, 52f N Neuromotor system, 310 Neuromuscular synapses, 103 Nonlinear time series analysis, 311–312 O Oligosynaptic reflex, 104–105 P Parallel elastic components (PEC), 36 Phasic reflex, 106 Phasic stretch reflex, 108–109, 108f Phrenology, 290–291 Polysynaptic reflex, 105, 106f Posture anticipatory postural adjustments (APAs), 322–323 anticipatory synergy adjustments (ASAs), 324–326, 325f–326f definition, 307 early postural adjustments (EPAs), 323–324, 324f equilibrium-point hypothesis, 308, 315 feed-forward adjustments, 322 history, 306–307 muscle force–length characteristics, 315–316, 316f postural control, 305 posture-movement paradox, 308 Index posture-stabilizing mechanisms, 316, 317f See also Posture-stabilizing mechanisms predictive control, 321 standing posture, 308 steady-state process center of mass (COM) projection, 309, 309f center of pressure (COP), 310, 311f inverted pendulum model, 312–313 movement disorders, 311 neuromotor system, 310 noise hypothesis, 310–311 nonlinear time series analysis, 311–312 rambling-trembling (Rm-Tr) decomposition, 313–315, 313f–314f stability conditions, 310–311 sway analysis, 311–312 tabes dorsalis, 309–310 vestibular system, 310 synergies, 308–309 agonist–antagonist pair, 320 co-contraction M-modes, 319 muscle activation, triphasic pattern, 319–320, 320f primitives, 318–319 three joint configurations, 318, 318f two-joint system, equilibrium trajectories, 320, 321f two-joint wrist-elbow system, 319–320, 320f uncontrolled manifold (UCM) hypothesis, 317–318 whole-body elementary motions, 319 vertical posture, 305, 308 Posture-stabilizing mechanisms, 142, 316, 317f long-latency responses, 327 preprogrammed reactions, 328 reflexes, 328 short-latency responses, 327 zero-delay mechanisms, 327–328 Preprogrammed reaction agonist muscle, 128 atypical preprogrammed reactions, 133, 134f characteristics, 122–123, 123f complex nature, 126 controlled-release method, 127 Index corrective stumbling reaction, 130–131, 131f–132f functional stretch reflex, 123 grip reaction, 128–129, 129f history, 121–122, 122f joint perturbation, agonist–antagonist pair, 127–128, 127f long-loop reflex, 124 in movements, 132–133 neurophysiological mechanism, 124 perturbation magnitude, 125 postural reactions, 129–130 sensory source, 125–126 transcortical reflex, 124 triggered reactions, 124 Presynaptic inhibition, 108–109, 108f R Reciprocal inhibition, 104–105, 105f Redundancy alpha-motoneurons, 179–180, 180f Bernstein problem, 179 coordination, 178–179 cost/objective function, 180 definition, 177 degree-of-freedom, 181 elemental variables, 181 finger motion/force production, 181–182 joint torque, 182–183 kinematic redundancy, human arm, 177–178, 178f marginal redundancy, 184–185, 184f multimuscle systems, 182 neuromotor system, 179–180 optimal feedback control, 193–195, 194f optimization analytical inverse optimization, 192–193 complex cost functions, 192 cost functions, 186 kinematic cost functions, 188–189, 188f kinetic cost functions, 189–190, 189f minimum-norm approach, 187 Moore–Penrose pseudoinverse, 186 physiological cost functions, 190–191 psychological cost functions, 191–192 range-space motion and self-motion, 187 two-finger accurate force production task, 186–187, 187f 407 physiological minimization, 180–181 single-joint level, 182–183, 182f state redundancy, 184 task-specific/intention-specific structural units, 180–181 time-varying factors, 181 trajectory redundancy, 183–184, 183f voluntary movements anticipatory synergy adjustments, 198 impaired motor function, 197 multielement time-varying system, 197 task-specific control, 199 task-specific covariation, 199–200 three-finger accurate force production task, 198, 199f uncontrolled manifold (UCM) hypothesis, 198 Referent configuration hypothesis, 153f, 247, 262–264, 263f components, 151–152, 152f equilibrium-point hypothesis, 151, 151f kinesthetic perception, 150–151 muscle activation, 151–152 Reflexes Babinski reflex, 118 classification animal’s experience, 107–108 neurophysiological structures, 107 number of synapses, 103–105, 104f response location, 106 time pattern, 106 crossed extensor reflex, 109–110 definitions, 100, 102–103 flexor reflex, 109–110 F-response, 111–112 history, 99–100 motor actions, 102 motor response, 102 neural transmission (synapse), 101 preprogrammed reactions, 110–111, 111f reflex-like reactions, 101–102 role of vs central pattern generators, 112–113, 114f force-related reflex effects, 115 mechanical characteristics, 113–114, 115f 408 Reflexes (Continued) reflex reversals, 117 tonic stretch reflex, 115–116, 116f velocity-sensitive reflexes, 114 spasticity, 117 spinal central pattern generator, 102 stimulus link, 100–101 stretch reflex, 108–109, 108f time delay, 101 tonic vibration reflex, 110 vs voluntary movements, 100, 101f Relaxed muscles force–length properties, 31–32, 32f Hill’s three-component muscle model, 33 muscle–tendon units (MTU), 33 structural elements, 32 viscous stiffness, 33 Residual force enhancement, 109 S Scalar method (Jacobian method), 15–16 Serial elastic components (SEC), 36 Servo-control hypothesis, 249, 250f Stiffness and stiffness-like measures active biological objects, 45–46 agonist–antagonist muscle pairs, 29 apparent stiffness, 29 See also Apparent stiffness compliance, 27 definition, 26–27 elastic deformable bodies, 44–45 elastic properties constant velocity stretch, 34, 35f dynamic force enhancement, 33–34 frog muscle fiber, tetanic stimulation, 33, 34f monosynaptic stretch reflexes, 34–35 muscle components See Muscle components muscle force/deformation, 33 muscle–tendon units, 38–40 relaxed muscles See Relaxed muscles residual force enhancement, 34 short-range stiffness, 33–34 tendons, 30–31, 31f Hooke’s law, 25 incremental stiffness, 28–29 Index mass-spring system, 26 passive joints, 26 passive vs active object properties, 28 stiff muscles, 27 stress and strain, 27 three-component lumped-parameters model, 25 velocity and acceleration effects, dynamic stiffness, 29–30 Stretch reflex, 108–109, 108f, 255, 255f Synergy-A, stereotypical muscle activation patterns, 207–208, 208f Synergy-B, variables with parallel scaling advantages and disadvantages, 209 nonnegative matrix factorization (NNMF), 208–212, 210f principal component analysis (PCA), 208–212, 210f–211f Synergy-C basketball throw task, 220–221, 221f covariation, 220 definition, 214 elemental variables identification finger mode, 216, 216f muscle modes, 217 two-finger pressing task, 214–215, 215f factors, 220 goal-equivalent manifold (GEM), 220 hammer trajectory, 213 intertrial and intratrial variance, 218–219, 219f–220f Jacobian matrix, 217–218 kinematic synergies, 221–222 kinetic synergies force-related and moment-related components, 224 multi-element parallel chains, 222 multi-joint kinetic synergies, 225–226 parallel force/moment generating elements, 222 three-finger grasp, 224–225, 225f two-finger pressing task, 223, 223f two-level hierarchical scheme, 224, 224f multi-element system, 213 muscle synergies, 226–228, 227f Index task-specific stability, 212–213 TNC method, 221 tolerance, 220 two-dimensional pointing task, 212–213, 213f UCM-based analysis, 220 T Tendon reflex/T-reflex, 104 Threshold control hypothesis, 247 Tonic reflex, 106 Tonic stretch reflex, 108–109, 108f, 115–116, 116f active muscle force, 88, 89f characteristics, 90–91, 91f deafferentation, 88 equilibrium-point hypothesis, 89–90 muscle movement, 89–90, 90f Tonic vibration reflex, 110 Transcortical reflexes, 107, 124 Transcranial magnetic stimuli (TMS), 235 U Uncontrolled manifold (UCM) hypothesis, 198, 232, 317–318, 354 covariation, 198, 214, 220 elemental variables, 221, 238–239 Jacobian, 221 relation to synergies, 230, 317–318 409 V Vector method (geometric method), 16–17, 17f–18f Velocity-dependent resistance active objects active muscles, 55, 56f joints, 56–57 kinematic chains, 57–58 damping coefficient, 53 hysteresis loop, 50, 50f mechanical impedance, 58–59 motor control, 53 muscle viscosity theory, 58 Fenn effect, 51–53, 52f history, 50 muscle contraction, 51, 52f passive objects passive muscles and joints, 54–55 synovial fluid, 54 tendons, 54 spasticity, 60 viscosity, definition, 49, 49f Velocity-sensitive reflexes, 114 Vibration-induced fallings (VIFs), 147 W Work of a muscle negative muscle work, 66 scalar product, 65 static forces, 65 ... of motor control, the journal Motor Control started only in 1997, the first conference—Progress in Motor Control was held at about the same time (1996), and the International Society of Motor Control. .. center; C and S stand for a cosine and sine function, correspondingly; subscripts 1, 2, and 12 refer to the a1 and a2, and a1ỵ2, correspondingly; a_ stand for angular velocity and acceleration; and. .. (1.11) 16 Biomechanics and Motor Control For a three-link planar chain, for instance for an arm model that includes an upper arm, forearm, and hand and describes a human arm grasping a handle, the