06690238

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014 1929

A General Actuator Model Based onNonlinear Equivalent Networks

Neville Hogan

Abstract—To facilitate interaction control, variable-impedanceactuators have been developed; however, they are fundamentallynonlinear. A general nonlinear actuator model which could be iden-tified unambiguously from observations with the actuator in situwould be useful. Equivalent circuits seem promising because theyseparate forward path dynamics from interactive dynamics. Canthis linear circuit concept be applied to nonlinear systems? In thispaper, equivalent circuits are first generalized to comprise two non-linear parts joined by a linear connector. Modeling actuators in thisway reveals an important insight: a Thevenin-type network doesnot permit unambiguous identification of the forward path dy-namics, whereas a Norton-type network does. That is because theNorton source is a zero of the interaction-port operator, whereasthe Thevenin source is not. With that insight, an equivalent circuitis further generalized to a network of two nonlinear parts joinednonlinearly, with the restriction that the source/forward path is azero of the interaction port operator. A Norton-type network inthat nonlinear form is unambiguously identifiable. It appears toprovide a versatile representation of nonlinear actuators.

Index Terms—Actuator, admittance, equivalent, impedance,modeling.

I. INTRODUCTION

R ECENT advances in sensing, communication, and controlhave yielded impressive robotic applications but to realize

their full potential requires advances in contact robotics: closephysical contact and cooperation between robots, humans, andother robots. Contact robotics requires reliable, robust controlof physical interaction, but despite substantial recent progress,that remains a significant challenge [1], [2]. Variable-impedanceactuators have been developed to facilitate interaction control[3]–[13]. However, they are fundamentally nonlinear becauseany linear model must exhibit constant mechanical impedance.A general model to describe and compare nonlinear actuatorswould prove useful, especially if the details of that model couldbe identified from observations with the actuator in situ. Thispaper explores whether a generalization of linear equivalentcircuits to nonlinear systems may serve as a general model ofactuators.

Manuscript received December 1, 2012; revised June 30, 2013 and Octo-ber 8, 2013; accepted November 16, 2013. Date of publication December 20,2013; date of current version June 13, 2014. Recommended by Technical EditorM. O’Malley. This work was supported in part by the Sun Jae Professorship, inpart by the Eric P. and Evelyn E. Newman Fund, and in part by a grant from theDefense Advanced Research Projects Agency under the Warrior Web ProgramBAA-11–72.

The author is with the Mechanical Engineering Department and the Brain andCognitive Science Department, Massachusetts Institute of Technology, Cam-bridge, MA 02139, USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMECH.2013.2294096

A. Information and Energy

Most systems and control theory is permeated by aninformation-processing or “signals” perspective: each systemcomponent is described by a mathematical operator that unilat-erally determines its output as a function of its input—but notvice versa. Describing the action of multiple components is astraightforward composition of their operators. To achieve thismodularity, each component is implemented so that its behavioris substantially unaffected by its assembly into a system (e.g.,using buffer amplifiers). However, the interactions due to phys-ical contact are fundamentally bi-lateral—each system affectsthe other, an observation first made in Newton’s third law [14].When a manipulator physically interacts with an object, the re-sulting system combines the manipulator’s dynamics with thoseof the contacted object. The combination is not well-modeledby a simple composition of operators and the signals perspectivedoes not work well.

Physical system dynamics may be described as “energy pro-cessing”; storage, transmission, and dissipation of energy arefundamental to physical system dynamics. Information process-ing is quite different. The components of a silicon-based com-puter or the neurons of a carbon-based mammalian brain are, ofcourse, subject to physical laws, including those of thermody-namics, but that is largely irrelevant to their function. Physicallaws may determine limits on processing speed but not on pro-cessing operations.

The distinction between information-processing and energy-processing is important because the former is subject to far fewerconstraints. The only constraints on signal-processing appearto be temporal causality (no output before the correspondinginput) and boundedness. In contrast, physical system dynamicsare subject to numerous additional constraints, such as energyconservation, entropy production, and so forth.

Actuators are the interfaces between these two disparate do-mains. A competent description of actuators should have two un-ambiguously distinct parts, describing a “forward path” throughwhich signals influence physical events; and interactive dynam-ics through which physical events evoke a physical response.A biological archetype is mammalian muscle and its associ-ated neural circuitry. Neuro–muscular dynamics is notoriouslynonlinear, complicated and difficult to quantify. The interplayof biochemistry, electrophysiology, and kinematics that con-stitute muscle behavior varies widely between muscles and isprohibitively difficult to measure in vivo. A summary represen-tation with experimentally-identifiable parts would significantlyfacilitate analysis.

One remarkably effective engineering analysis tool is theconcept of an equivalent circuit. Linear circuits of arbitrary

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1930 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014

complexity behave as though they comprised only two partswith a simple connection; and those parts may be identified un-ambiguously by simple experiments. This paper explores howthe equivalent circuit concept might be re-purposed to providea general description of nonlinear actuators.

II. EQUIVALENT NETWORKS

A fascinating history of the equivalent circuit concept is pre-sented by Johnson [15], [16]. Originally Helmholtz and later(independently) Thevenin showed that any circuit containingelectromotive forces (voltage sources) and resistances could bereplaced at any pair of terminals by a single voltage source inseries with a single resistance [17], [18]. Subsequently Mayerand Norton1 formulated an equivalent circuit comprised of acurrent source in parallel with a resistance [19], [20]. The con-cept of impedance introduced by Heaviside and its dual, ad-mittance, allowed equivalent circuits to be extended to includedynamic behavior (e.g., capacitance and inductance) [21], [22].Circuits comprising arbitrarily complicated assemblies of volt-age sources, current sources, resistances, capacitances, and in-ductances may be represented by Thevenin or Norton equiva-lent circuits. This prodigious simplification is one reason whyequivalent circuits remain one of the core conceptual tools ofengineering analysis.

The “terminal pair” of an electric circuit may be general-ized to an interaction port through which power flows into orout of a system. An interaction port is not confined to a sin-gle pair of variables that define power flow (voltage and cur-rent, force and velocity, etc.) but may be extended to multiplepairs of variables which need not be associated with the samephysical location provided their inner product defines powerflow. An important detail is that the two variable types trans-form differently between coordinate frames. If we consider therelation between the configuration variables of a manipulator(e.g., joint angles, link lengths) and the coordinates of its end-effector, the transformations associated with motion (position,displacement, velocity, etc.) are uniquely defined from configu-ration variables to end-effector variables, even when the formerhave many more dimensions. Conversely, the transformationsof momentum, force, torque, etc. are uniquely defined from end-effector to configuration variables. For notational convenience,force, momentum, etc. will generically be termed “exertion”variables; velocity, displacement, etc. will generically be termed“motion” variables.

Equivalent networks2 may be defined in any physical do-main: if the elements of a network model are linear, an equiv-alent network may be defined. Johnson proposed to describethe two alternative types as the “voltage-source” and “current-source” equivalents [15]. However, a broader repertoire of inputor “source” components is required. Using the analogy intro-duced by Maxwell, electrical voltage corresponds to mechanical

1By an astonishing coincidence, both Mayer and Norton published their workindependently in the same month of 1926.

2The term “network” is used because the connections in general physicalsystem models do not necessarily form closed circuits.

force and current corresponds to velocity3 [24]. However, it isnecessary—especially in robotic applications—to consider vari-ables other than force or velocity as inputs. For example, po-sition or configuration plays a key role in mechanical systemsand it will often be useful to define a “position source” as wellas a “velocity source”; similarly, we may define a “momentumsource” as well as a “force source”. This paper refers to thetwo alternative types as “exertion-source” and “motion-source”equivalent networks.

Equivalent networks are usually defined so that positive powerflows out of the network, though that is entirely a matter of con-vention. The remaining substantial choice is the operationalform4 of the network equation: which port variable (exertion ormotion) is chosen as input and which as output. A voltage-sourceequivalent circuit may be expressed as e = eT − Zi where e isterminal voltage, eT is Thevenin voltage, Z is the equivalentimpedance, and i is terminal current. It may alternatively beexpressed as i = Y (eT − e), where Y = Z−1 is the equivalentadmittance. Similarly, a current-source equivalent network maybe expressed as i = iN − Y e, where iN is Norton current or al-ternatively as e = Z (iN − i). These alternatives will be referredto as impedance forms and admittance forms, respectively.

III. NONLINEAR EQUIVALENT NETWORKS

Can the concept of an equivalent circuit be applied to nonlin-ear systems? A complete correspondence should not be expectedas superposition is not a general property of nonlinear systems.However, network models of physical systems are highly struc-tured. If the connections between subsystems are (a) power-continuous, neither supplying nor dissipating power, nor stor-ing energy and (b) symmetric in the sense of invariance underpermutation of the interaction ports—any two ports of the con-nector may be exchanged without affecting its behavior—thenthere are only two possible connectors and they are both linear5.This remarkable fact is true even if the subsystems connectedare nonlinear [25]. This fundamental linearity of the connec-tions in network models of physical systems justifies guardedoptimism.

1) Nonlinear Parts, Linear Connection: Nonlinear systemswith linear connections can be formulated [26] and, indeed, arefamiliar. Consider a reasonably general model of the inertialmechanics of a manipulator

I(θ) θ + C(θ, θ)θ = τacc

where θ is an array of configuration variables, I (θ) is iner-tia, C(θ, θ) includes Coriolis and centrifugal terms and τacc isaccelerating torque. In state-determined form,

3The dual analogy—current with force, voltage with velocity—may be usedwithout affecting the argument. See [23] for a detailed discussion.

4The term causality may also be encountered but it risks confusion withthe more familiar meaning of causality as a time-ordering with outputs notpreceding inputs.

5These are the familiar series (common current) and parallel (common volt-age) connections of electric circuits or the common force and common motionconnections of mechanical systems.

HOGAN: GENERAL ACTUATOR MODEL BASED ON NONLINEAR EQUIVALENT NETWORKS 1931

Fig. 1. Block diagram of nonlinear manipulator mechanics in the admit-tance operational form of an exertion-source equivalent network with a linearconnection.

θ = ω

ω = I (θ)−1 [−C (θ, ω) ω + τacc ] .

Inertia fundamentally embodies a connection. To explain,even though its state equations are nonlinear, inertia respondsto the signed linear sum of applied torques

τacc =∑

k

σkτk

where k indexes the distinct torques, σk = 1 if positive torqueacts to accelerate the inertia and σk = −1 if positive torqueacts to decelerate the inertia. Further, all of the subsystems thatgenerate those applied torques experience the same velocity

ωj = ωk ∀j, k.

We may include frictional losses and gravitational loads thatcommonly influence manipulator motion

τfriction = ZF (θ, ω) ω

τgravity = ZG (θ)

as well as actuator torques, τact , and exogenous torques, τexo ,due to interaction with external objects. The net acceleratingtorque is the signed sum of these torques

τacc = τact − τfriction − τgravity − τexo .

The important point is that any or all of these phenomena—friction, gravity, etc.—may be described by nonlinear equationsbut they all share a common motion and superimpose theirtorques linearly. For the external interaction port defined byτexo and ω, this fundamental property of inertia correspondsto representation as the admittance form of an exertion-sourceequivalent network comprising nonlinear parts combined witha linear connection (see Fig. 1).

A linear equivalent network has two possible operationalforms; not so in the nonlinear case. Inertial dynamics may notbe well-formulated in impedance form. Forces and/or torquesmay always be exerted on a robot and it will always determinea resulting motion, but if kinematic constraints are included,sometimes that motion may be zero; arbitrary motion may not beimposed. The important point is that the operational form is notarbitrary. An admittance form is always possible; an impedanceform is not.

IV. EQUIVALENT NETWORK MODELS OF ACTUATOR

DYNAMICS

Which operational form is appropriate for nonlinear actua-tors? Because they interact with mechanism inertia, most roboticactuators are best described with exertion (momentum, force,torque etc.) as output, in the operational form of mechanicalimpedance. With a strict analogy to electrical impedance, me-chanical impedance could be defined to map velocity onto force.However, that definition would not reflect the profound role ofgeometry in all aspects of mechanical dynamics. Herein theterm “mechanical impedance” refers generically to interactivedynamics with motion variables as input and exertion variablesas output.

A. Exertion-Source Equivalent Network

As a typical robotic actuator generates force or torque, it mayseem reasonable to represent it as an exertion source modifiedby interactive dynamics in the operational form of mechani-cal impedance, i.e., as generalization of Thevenin’s equivalentnetwork. However, a requirement that its parts should be identi-fiable unambiguously from observations at the interaction portargues against using this generalized exertion-source equivalentnetwork.

Assuming a linear connection, a state-determined represen-tation of an exertion-source equivalent network in impedanceoperational form is as follows. The source describes the forward-path dynamics

w = Tstate (w, c)

fT = Toutput (w, c)

where w is an array of source state variables, Tstate (·) andToutput (·) denote algebraic functions, c is an array of controlinputs, and fT is the exertion variable—in this case, force—determined by the control inputs. The interactive dynamics are

z = Zstate (z, x, π)

fZ = Zoutput (z, x, π)

where z is an array of state variables, Zstate (·) and Zoutput (·)denote algebraic functions, x is the interaction port displace-ment, π is an array of control inputs that influence actuatorforce indirectly by modulating actuator impedance, and fZ isthe “impedance force” evoked by displacement x. The interac-tion port force, f is the difference between fT and fZ

f = fT − fZ .

This first generalization of a linear Thevenin equivalent circuitto a nonlinear mechanical system admits nonlinear dynamicsbut preserves the linearity of the connection (see Fig. 2); seealso [26].

1) Identifying an Exertion-Source Equivalent Network: Oneapproach to identifying the exertion source is to enforce zeropower exchange, i.e., set x = 0 or x = constant = xc and ob-serve the interaction force required to do so

z = Zstate (z, xc , π) .


Fig. 2. Block diagram of an actuator model in the impedance operationalform of a nonlinear exertion-source (Thevenin) equivalent network with a linearconnection.

With this approach, source exertion should at least be identifi-able when the interactive dynamics are stable and have reachedsteady state. If so,

z = 0 ⇒ z = zss (xc, π)

and the corresponding impedance force is

fZ,ss (xc, π) = Zoutput (zss (xc) , xc , π) .

The steady-state interaction force observed when x = xc andz = 0 is

fss = fT − fZ,ss .

If the steady-state impedance force is nonzero, fZ,ss �= 0, theinteraction force observed when the interaction port is immobi-lized is not that of the exertion source: fss �= fT . Identificationof the exertion source in this way would be compromised.

This is not merely a mathematical nuance of no practicalimportance. Nonzero steady-state impedance force is common-place and may be introduced deliberately. The act of pushingon a tool induces a static instability that must be offset bya steady-state relation between force and displacement6. Thatrelation may be an intrinsic property “built-into” the actua-tor, as with series-elastic actuators or variable-stiffness actu-ators [4]–[13], [28], [29]. It may also be imposed or augmentedby the actuator’s local control system.

The difficulty might be addressed by re-defining the exertionsource and impedance. Specifically, add and subtract the steady-state impedance force from the net force equation and re-groupterms

f = (fT − fZ,ss) − (fZ − fZ,ss) = fT − fZ .

As a result, in steady state fZ ,ss = 0 so that fxc ,ss = fT asrequired. But fZ,ss depends on xc , therefore this re-defined forcesource fT will also depend on the interaction port displacement,xc . That would compromise the separation of forward-path andinteractive dynamics. One of the signal benefits of an equivalentnetwork would be lost.

B. Motion-Source Equivalent Network

Assuming a linear connection, a state-determined representa-tion of a motion-source network in impedance operational form

6In fact, the limits of human force production appear to be determined by therequirement to produce a stabilizing static impedance rather than the requirementto produce force [27].

Fig. 3. Block diagram of an actuator model in the impedance operationalform of a nonlinear motion-source (Norton) equivalent network with a linearconnection.

is as follows. The motion source describes the forward-pathdynamics

w = Nstate (w, c)

xN = Noutput (w, c)

where w is an array of source state variables, Nstate (·) andNoutput (·) denote algebraic functions, c is a control input, andxN is the motion variable—in this case position—determinedby the control input.

The common-exertion connection requires a linear superpo-sition of motion variables

Δx = xN − x

where Δ denotes deviation, in this case deviation of actual mo-tion from source motion. The interactive dynamics in impedanceform are as follows:

z = Zstate (z,Δx, π)

f = Zoutput (z,Δx, π)

with the same notation as before. In this case, the impedanceforce is the interaction port force. As above, this first generaliza-tion of a linear Norton equivalent circuit to nonlinear mechanicalsystems admits nonlinear dynamics but preserves the linearityof the connector (see Fig. 3).

1) Identifying a Motion-Source Equivalent Network: Toidentify the motion source, zero power flow may be imposedby setting interaction port force to zero, f ≡ 0

0 = Zoutput (z,Δx, π) .

This defines the zero dynamics of the impedance [30]. Thesource motion should at least be identifiable when the impedancezero dynamics are stable and have reached steady state. In gen-eral, the steady-state relation between force and displacementdoes not guarantee that zero force requires zero displacement7

but that is often a reasonable assumption, especially for series-elastic or variable-stiffness actuators [3]–[13], [28], [29]. Inthose cases, zero steady-state interaction force implies zero dis-placement from the motion defined by the forward-path dynam-ics, i.e., Δx = 0 hence xN = x. The port motion for whichthe output force is identically zero identifies the motion sourceunambiguously, with dynamics given by w = Nstate (w, c) andxN = Noutput (w, c).

7For example, a system with backlash may exhibit zero force for a range ofdisplacements.


The requirement to have interactive zero dynamics in steadystate may seem restrictive but it is not prohibitive (at least inprinciple) and has been applied in practice as reviewed below.If the actuator begins with all state variables (w and z) in steadystate, the time history of port motion x (t) that maintains f ≡ 0for a specified input c (t) is the source motion xN (t). The non-linear interactive dynamics (impedance) may be identified byimposing motions x (t) that deviate from the source-definedmotion xN (t) and observing the forces evoked. Of course, non-linear system identification may be extremely challenging butthere is no fundamental reason it cannot be accomplished [31].

C. Example: Inertial Mechanics

Consider again a model of inertial mechanics, including fric-tional losses and gravitational loads

θ = ω

ω = I (θ)−1 [−C (θ, ω) ω + τacc ]

τfriction = ZF (θ, ω) ω

τgravity = ZG (θ)

τacc = τact − τfriction − τgravity − τexo .

It may seem reasonable to describe the actuator as a nonlinearexertion-source equivalent network as defined above but thatwill prove to compromise identification. For clarity, consideronly static (steady-state) actuator behavior

τact = τT (c) − ZA (θ, π)

where τT (·) and ZA (·) denote algebraic functions and othernotation is as before.

First consider an external port defined by τexo and θ, i.e.,where mechanical work done by the manipulator is dW =dθtτexo . To identify the exertion source seen at that interac-tion port, we might immobilize the port and observe the ex-ertion required to do so. Assume the system is stable andreaches steady state with ω = ω = 0. As a result, C (θ, ω) ω =0, ZF (θ, ω) ω = 0, and θ = constant = θc , hence,

τexo (c, θc , π) = τT (c) − ZA (θc , π) − ZG (θc) .

The observed torque τexo depends on configuration θc anddoes not reveal an exertion source independent of port motion.The separation of forward path behavior from interactive be-havior has been compromised.

An alternative model describes the actuator as a nonlinearmotion-source equivalent network as defined above. Again, con-sider only static (steady-state) behavior

τact = ZA (Δθ, π)

Δθ = θN (c) − θ

where θN (·) denotes an algebraic function and other notationis as before. To identify the motion source seen at the in-teraction port, we may observe the motion that results wheninteraction torque is identically zero, τexo = 0. Again, as-sume that the system is stable and reaches steady state, sothat ω = ω = 0, C (θ, ω) ω = 0, ZF (θ, ω) ω = 0. As a result,

θ = constant = θc (c). That θc (c) is the motion source seen atthe external port.

For clarity in what follows, re-label it as θc (c) = θN (c). Thatmotion source, θN (c), may be related to the actuator motionsource, θN (c), as follows. In steady state,

τacc = 0 = ZA ((θN − θN ), π) − ZG (θN ).

In any region of configuration space where the inverse ofZA (·) exists, we may solve for θN

θN = θN + Z−1A (ZG (θN ), π).

With nonzero gravitational torque, θN and θN differ. That isbecause they relate to two different interaction ports−θN forthe external port through which the manipulator interacts withthe world, θN for the actuator port through which the actuatorinteracts with the manipulator. The relation between θN and θN

is analogous to the familiar circuit-theory current-divider, albeitgeneralized to a class of nonlinear static impedance operators.

1) Nonlinear Parts, Nonlinear Connection: This exampleshows that the linear connection is not always preserved. Theexternal port static impedance is the relation between τexo andθ

τexo = ZA (θN − θ) − ZG (θ)

where the dependence on control inputs π has been omitted forclarity. Express this in terms of the external port motion source

τexo = ZA (θN + Z−1A (ZG (θN )) − θ) − ZG (θ)

= ZP (θN , θ)

where ZP (·) denotes the static impedance seen at the exter-nal port defined by τexo and θ. The actuator static impedancedepends only on the difference between θN and θ, that is,ZA (θN , θ) = ZA (Δθ). The external static impedance cannot,in general, be expressed as a function only of the deviation ofθN from θ. Nevertheless, in both cases, the motion source is azero of the impedance operator. For the actuator, substitutingθ = θN , τact = ZA (θN , θN ) = 0. For the external port, substi-tuting θ = θN ,

τexo = ZA (θN + Z−1A (ZG (θN )) − θN ) − ZG (θN )

= ZA (Z−1A (ZG (θN ))) − ZG (θN ) = 0.

In both cases, the impedance operator determines the exertionevoked by deviation of port motion from source motion. In themore general case, that impedance operator may vary with portmotion; ZP (θN , θ). This is a modest departure from the casewith a linear connector, akin to the variation of manipulator in-ertia with configuration. In fact, if the impedance is controllable(as in a variable-impedance actuator) it must necessarily dependon more than the deviation of port motion from source motion(e.g., on the control variables π).

This generalized equivalent network has a different connec-tion structure from that of the actuator model (common exertionand a linear superposition of motions), but it nevertheless sepa-rates forward path dynamics from interactive dynamics. More-over, because it is a zero of the external port impedance operator,


the external port motion source θN may be observed unambigu-ously simply by setting the external torque τexo to zero—withoutrequiring knowledge of the external port impedance.

To observe the motion source θN at the actuator port, it issufficient to impose zero actuator torque even for nonzero con-trol input c. This requires removing torques due to interveningdynamics, e.g., inertia; a method to do this is reviewed be-low. In this way, the actuator motion source may be identifiedunambiguously without knowledge of the actuator impedanceoperator, the only requirement being that the latter has a uniquezero within some well-defined range of its motion argument.

2) A One-Dimensional Example: A highly-simplified one-dimensional example may be informative. Assume an actuatorcharacterized by

τact = (π − θ)2 c

where 0 ≤ θ ≤ π and c ≥ 0 is a control input with units ofNm/rad. This is qualitatively similar to the characteristics of“pneumatic muscle actuators” [32]. Usually these actuators aredeployed in antagonistic groups, but for simplicity assume asingle actuator is opposed by a linear spring as in [33] and actsto drive a single degree-of-freedom inertial load

I..

θ = τacc = (π − θ)2 c − Kθ − τexo .

If we attempt to represent this actuator as a nonlinear exertion-source (Thevenin) equivalent network, to identify the exertionsource τT , we might immobilize the system θ = θ = 0, θ = θc

and observe the exertion required to do so

τexo (c, θc) = (π − θc)2 c − Kθc.

This observed torque depends on configuration θc as well ascontrol input c and does not reveal an exertion source indepen-dent of port motion.

If instead, we represent this actuator as a nonlinear motion-source (Norton) equivalent network, to identify the motionsource, we may impose zero exertion at the interaction portof interest. If that is the external port defined by τexo and θ, anordinary differential equation (ODE) relates θN to c

I¨θN + KθN = (π − θN )2c.

This ODE defines the forward-path dynamics relating θN (t)to the corresponding c (t) with no dependence on τexo .

If the interaction port of interest is between the spring-loadedactuator and inertia, the corresponding motion source θN maybe identified by imposing zero accelerating torque. If c mayvary, this may be achieved by setting τexo = −Iθ. In this simpleexample, it is sufficient to assume θ = 0 and τexo = 0, resultingin

0 = (π − θN )2 c − KθN .

The Norton motion source θN (c) may be identified with noa priori knowledge of the impedance operator by tabulatingsteady state θ at zero external torque for a suitable range ofc. The impedance operator may subsequently be identified foreach value of c by tabulating the relation between steady-statetorque and angle.

In this example, the control input may be expressed as afunction of the motion source

c = KθN / (π − θ)2

and the impedance operator may be written as

τacc = K

[θN

(π − θ)2

(π − θN )2 − θ

].

This nonlinear relation cannot be expressed solely as a func-tion of Δθ = θN − θ, but it has a well-defined zero at θN = θ.

In this simple example, the net accelerating torque due to theactuator and spring could be determined from external torquemeasurements by assuming the system was at equilibrium insteady state. Usually that is not possible due to the presence ofstatic friction, which is considered next.

D. Example: Electromagnetic Actuator with Friction

Direct-current permanent-magnet torque motors have manydesirable properties for high-performance control, but they com-monly exhibit nonideal behavior due to friction. Consider thefollowing model with linear electrical dynamics and frictionbased on a nonlinear Dahl model, a simplification of the LuGremodel [34], [35]

di

dt= Kaβec − βi

dϕ

dt= ω − σ0

τc|ω|ϕ

τDahl = σ0ϕ + σ1dϕ

dt

dω

dt= I−1

m (Km i − τact − τDahl (ω, ϕ) − Bω)

where i is motor current, Ka is a trans-conductance amplifiergain, ec is the voltage command into the amplifier, β is amplifierbandwidth, ω is motor shaft angular velocity measured withrespect to the motor frame8, Im is motor armature inertia, Km

is motor torque sensitivity, τact is motor output torque, τDahlis friction torque, B is an angular damping coefficient, ϕ is theDahl model state variable with units of angular displacement,σ0 is a constant with units of angular stiffness, σ1 is a constantwith units of angular damping, and τc is a constant with unitsof torque, the magnitude of sliding friction. If ω is a nonzeroconstant and the frictional dynamics are in steady state, dϕ/dt =0, then

ϕss =τc

σ0

ω

|ω|τDahl = τcsgn (ω) .

A trans-conductance amplifier imposes motor current pro-portional to its input voltage (at least over the amplifier band-width) which determines the electromagnetic torque exerted bythe motor. It may seem natural to represent this actuator as anexertion-source equivalent network.

8For clarity, in this example, the motor frame is assumed to be at rest in aninertial frame. The more general case does not affect the argument.


The product of motor torque, τact and angular velocity, ωdefine the actuator port. To identify the exertion source seen atthat interaction port, we might immobilize the motor shaft andobserve the torque required to do so. Substituting

dϕ

dt

∣∣∣ω≡0

= 0;ϕ = constant = ϕc

τDahl

∣∣∣ω≡0

= σ0ϕc

τact

∣∣∣ω≡0

= Km i − σ0ϕc

di

dt= Kaβec − βi.

These equations define the forward-path dynamics from volt-age input ec to exertion source output. Assume ec is constantand the system comes to steady state

di/dt = 0; iss = Kaec

τact,ss = Km Kaec − σ0ϕc = τem − σ0ϕc

where subscript ss denotes steady-state and τem denotes theelectromagnetic torque. The term σ0ϕc is the steady-state fric-tional torque at zero velocity, which may lie anywhere in therange −τc ≤ σ0ϕc ≤ τc . Within that range, friction exhibits theproperty of accommodation, balancing the net applied torque toprevent movement.

Due to friction, the observed value of τact,ss is not the elec-tromagnetic torque τem . The relation between τact,ss and ec isambiguous with an uncertainty of ±τc . Moreover, the precisevalue of the steady-state frictional torque depends on the sys-tem’s motion history. An unambiguous separation of forward-path behavior from interactive behavior has not been achieved.

Instead, consider describing this actuator as a motion-sourceequivalent network. To identify the interaction port motionsource, we may observe the motion that results when interactiontorque is identically zero, τact ≡ 0

di

dt= Kaβec − βi

dϕ

dt= ωN − σ0

τc|ωN |ϕ


dt

dωN

dt=

1Im

(Km i − τDahl (ωN , ϕ) − BωN )

where the notation ωN is introduced to distinguish this motionfrom the more general motion when τact �= 0. These equationsdescribe the forward-path dynamics, providing an unambiguousrelation between voltage input ec and motion source ωN (“free”motion, ω when τact ≡ 0).

Of course, the frictional nonlinearity is still present. Assumeec is constant and the system comes to steady state.

di/dt = 0; iss = Kaec

dϕ/dt = 0;ϕ = constant = ϕc

dω/dt = 0;ω = constant = ωN,ss ;Km Kaec

= τDahl (ωN,ss , ϕc) + BωN,ss

ωN,ss =

⎧⎨

⎩

(Km Kaec − τc)/B if Km Kaec > τc

0 if |Km Kaec | ≤ τc .(Km Kaec + τc)/B if Km Kaec < −τc

This relation between the steady-state motion source ωN,ss

and the command voltage ec is piecewise-linear and noninvert-ible. Nevertheless, it is unambiguous.

1) Nonlinear Parts, Nonlinear Connection: The corre-sponding impedance operator might be identified by applyingcommand voltages, ec (t), imposing motions, ω (t), and ob-serving the evoked interaction torques, τact (t). Repeating thatprocess for all pairs of command voltages and motions wouldidentify the operator, τact (t) = Z (ωN (t) , ω (t)), that maps ac-tuator motion onto actuator exertion. As in the previous example,it may be expressed as an operation on ωN (t) and ω(t) but notexclusively on their difference.

A simplified form is obtained when the command voltagesare constants, corresponding to a steady-state motion source,ωN,ss . An imposed motion, ω (t), has no effect on the amplifierdynamics relating command voltage to motor current. Assum-ing the amplifier is in steady state, iss = Kaec , the simplifiedimpedance operator is as follows:

dϕ

dt= ω − σ0

τc|ω|ϕ


dt

τact = Km Kaec − τDahl (ω, ϕ) − Bω − Imdω

dt.

Because the relation between constant ec and ωN,ss cannotbe inverted, a simple substitution of ec with a function of ωN,ss

is not available. Instead, re-write the first equation

τact = Km Kaec + τc − τc − τDahl (ω, ϕ) − Bω − Imdω

dt.

If Km Kaec > τc , then ωN,ss > 0 and Km Kaec − τc =BωN,ss . The impedance operator becomes

dϕ

dt= ω − σ0

τc|ω|ϕ


dt

τact = B (ωN,ss − ω) + τc − τDahl (ω, ϕ) − Imdω

dt.

If a positive constant angular velocity, ω > 0, is im-posed and dϕ/dt = 0, then τDahl = τc . In that case, τact =B (ωN,ss − ω). Similarly, if Km Kaec < −τc , then ωN,ss < 0and τact = B (ωN,ss − ω). Thus, for a constant nonzero angularvelocity, the impedance operator converges to a linear functionof the deviation between port motion and source motion.

The exertion evoked by a more general motion cannot beexpressed in this simple form. First, if dω/dt �= 0, the nonzeroinertial torque Im dω/dt must be defined in terms of motion with


respect to an inertial frame; it cannot, in general, be defined interms of motion relative to the source motion ωN . Second, if theactuator port is immobilized (ω ≡ 0), then the value of τDahl de-pends on the trajectory taken to come to rest. For example, if restis slowly approached from positive values, i.e., ω (t) ≥ 0 suchthat over some interval dϕ/dt = 0 when ω ≡ 0, then τDahl = τc

and τact = Km Kaec + τc − 2τc . If Km Kaec ≥ τc , then substi-tuting Km Kaec − τc = BωN,ss yields τact = BωN,ss . How-ever, if Km Kaec ≤ −τc , then substituting Km Kaec + τc =BωN,ss yields τact = BωN,ss − 2τc . This was the problem en-countered above with the exertion-source equivalent networkmodel: friction introduces an ambiguity at ω ≡ 0 which de-pends on the motion history. It is an impedance which cannotbe expressed solely in terms of deviation from source motion.

Nevertheless, the motion source is a zero of the impedanceoperator. From above, if ωN,ss �= 0, then τact = B (ωN,ss − ω).If ω = ωN,ss �= 0, then τact = 0 as required. If ωN,ss = 0, thenKm Kaec = τDahl (ωN,ss , ϕc). Substitute

τact = Km Kaec − τDahl (ω, ϕ) − Bω − Imdω

dt

and assume steady state with ω = ωN,ss = 0

τact = Km Kaec − τDahl (ωN,ss , ϕc) = 0

also as required.Despite the ambiguity of the torque required to achieve

“blocked” motion, a motion-source equivalent network canbe defined that separates forward path dynamics from in-teractive dynamics. The extreme nonlinearity of the frictionmodel requires an impedance operational form. The motionsource is a zero of the impedance operator. It can be identi-fied unambiguously—even trivially—as the “free” motion ofthe interaction port.

Summarizing, a nonlinear equivalent-network representationof actuators is unambiguously identifiable if it satisfies the fol-lowing conditions.

1) It is in the operational form of mechanical impedance(motion in, exertion out).

2) The impedance operator has a unique zero within somewell-defined range of its motion argument.

3) The zero dynamics of the impedance operator are stable.

V. APPLICATION: HUMAN INTERACTIVE DYNAMICS

The challenge of identifying human interactive dynamics isparticularly acute. First, the biological actuator (muscle) hasnotoriously complicated and highly nonlinear dynamics [36].Second, there is as yet no ethical way to “open the box” andreliably observe variables internal to the human neural controlsystem. The value of an equivalent network representation thatcould be identified unambiguously from external measurementsseems unarguable.

Can an equivalent network representation be justified? It isreadily apparent that humans can bring the upper limbs to equi-librium at any configuration within their workspace. Interactivedynamics may then be described as an operator that determinesthe forces/torques evoked by displacement from that equilib-rium. This is consistent with a motion-source (Norton) equiv-

alent network as described above, for which identically zeroforce identifies the source motion.

Can the motion source be identified during movement, i.e.,when commands from the central nervous system are changing?Several attempts have been made by assuming a reasonable formfor interactive dynamics (e.g., time-varying mass-spring-and-damper behavior), identifying parameters of that model, andextrapolating from the results. Unfortunately, the outcome isexquisitely sensitive to the assumed form of the model—see [37]but compare with [38]. In fact, even the order of interactivedynamics is not reliably known. Though it is reasonable toassume that high-frequency dynamic behavior is dominated byskeletal inertia, in fact there is evidence of anti-resonance dueto muscle mass interacting with skeletal inertia [39], [40].

However, an equivalent network motion source can, in princi-ple, be identified without any knowledge of the neuro–muscularactuator impedance. A workable method to do so during armmovements was presented in [41]. The essence of the methodwas to estimate multivariable skeletal inertia, then generate ex-ogenous forces with a robotic manipulandum so that the neuro–muscular forces were nominally zero throughout a discretereaching movement. The resulting trajectory was the motionsource of an equivalent network model of the neuro–muscularactuator. Iteration over several nominally-identical movementswas required to improve the estimate and the exogenous forceswere presented on randomly selected movements to preclude hu-man adaptation to the stimulus. Passing over the details, whichare presented in [41], [42], for present purposes the main resultwas that, despite the intrinsic variability of human motor control,the method converged rapidly. The result was a reliable estimateof the motion source output, independent of any assumptionsabout neuro–muscular mechanical impedance.

VI. DISCUSSION

Intimate physical contact and interaction between mecha-tronic systems and objects in their environment—includinghumans—is becoming commonplace. Examples are found inrecent developments in amputation prostheses [43]–[45]. Thosecited examples feature variable mechanical impedance as anessential aspect of their design and control. The need for vari-able mechanical impedance has motivated extensive and ongo-ing research on variable-impedance actuators [3]–[13]. Variableimpedance actuators are fundamentally nonlinear.

A. Equivalent Networks as Actuator Models

Actuators occupy the interface between the information-processing of control systems and the energy-processing ofmechanical physics. Equivalent circuits provide a compact rep-resentation of complex linear electrical systems, separating bi-lateral interactive dynamic behavior from unilateral (forwardpath) dynamic behavior, and summarizing those behaviors intwo distinct parts. This paper explored to what extent equiva-lent circuit concepts might be re-purposed to provide a generalmodel of nonlinear actuators.

Notions of impedance or admittance underpin much com-mon engineering insight. The gear ratios of mechanical


transmissions may be chosen to maximize power transfer to aload by “impedance matching” (at least under nominal operatingconditions). However, mechanical transmissions are notoriouslynonlinear, especially when their main cause of inefficiency—friction—is taken into account. This paper is an attempt to testwhether these widely-used intuitive ideas might be placed on arigorous basis.

B. Choice of Operational Form

Equivalent networks come in two types (Thevenin or Norton)and two operational forms (impedance or admittance). For linearsystems, these types and forms are interchangeable but that is notgenerally true for nonlinear systems. For example, the nonlinearforce-length behavior of active muscle is such that, for a givenlevel of activation, multiple lengths may be associated with thesame force; however, for a given level of activation, each lengthis associated with a unique force. Similarly, the omni-presentfriction that accompanies almost all actuators is well-describedas mechanical impedance, with input motion and output force,but not vice versa.

C. Choice of Network Type

The choice of equivalent network type also requires care. Ingeneral, the connection of an equivalent network may not beassociated with any physically identifiable location; thereforethe two parts of a network should be identifiable from externalmeasurements—i.e., without “opening the box”. This is espe-cially important in biological applications but is also relevantfor mechatronic systems.

In the foregoing, it was shown that if an actuator is describedin impedance operational form as an exertion-source equivalentnetwork (e.g., a controllable force source modified by mechani-cal impedance), then its parts cannot always be identified unam-biguously. In contrast, if an actuator is described in impedanceoperational form as a motion-source equivalent network (e.g., acontrollable motion source modified by mechanical impedance),then identifying its parts is substantially simpler. Imposing zeropower exchange at an interaction port is sufficient to identify themotion source unambiguously. Moreover, detailed knowledge ofthe interactive dynamic behavior (impedance or admittance) isnot required to identify the motion source.

D. Variable Impedance Actuators

These may seem like abstract considerations but they are es-sentially practical. Linear models are commonplace and can beeffective; but linear models of actuators have limitations. Actua-tors with variable stiffness or variable mechanical impedance arereceiving growing attention [3]–[13]. Any linear model must ex-hibit constant mechanical impedance and cannot describe theirdistinctive property, variable mechanical impedance. It is impor-tant to understand to what extent concepts derived from linearequivalent circuits can be applied to these fundamentally non-linear actuators.

The need to identify an actuator model from external mea-surements is also essentially practical. It is clearly important fora biological actuator like muscle. Studies of muscle in vitro con-

tinue to be an active area of research but they are not sufficient toinform the behavior of the neuro–muscular actuator in vivo andstudies of animal muscle cannot reliably be extrapolated to hu-mans [46]. For that reason, several methods have been proposedto identify neuro–muscular mechanical impedance in awakebehaving humans [47]–[54]. However, as reviewed above, iden-tification of an equivalent network motion source has provenchallenging [37], [38] though possible [41].

1) Mechatronic Applications: The need to identify actuatordynamics from external measurements also applies to mecha-tronic systems. An actuator’s dynamic behavior when assembledinto a machine may differ substantially from its isolated behav-ior. Complex internal forces can accentuate behavior that maybe insignificant in an isolated actuator—for example, side-loadsmay exacerbate friction. Sophisticated methods to calibratemodels of fully assembled robots have been developed [55].Those authors demonstrate methods to identify kinematics, in-ertial and frictional properties and approximate resonant struc-tural modes but they do not include actuator dynamics. Yet,actuator characteristics may be the principal determinants ofnonideal behavior [1]. The considerations presented here mayfacilitate including actuator dynamics—both forward-path andinteractive—in these procedures.

The main conclusion here—that a motion-source equivalentnetwork is the better model of an actuator—may seem counter-intuitive. Mammalian muscle is commonly perceived as a con-trollable force generator and a robot actuator is typically viewedas a controllable force or torque generator. However, the require-ment to identify forward path and interactive dynamics fromexternal measurement imposes a constraint. The motion-sourceequivalent network still describes the actuator with force ortorque as its output variable (i.e., in the operational form of me-chanical impedance)—consistent with common intuition—butdefines mechanical impedance to determine that output forceor torque in response to deviation of actual motion from thenominal motion determined by the source element. In fact,that is a good description of the now-popular series elasticactuators [28], [29]. The series elastic element determines anoutput force in response to the difference between actual mo-tion and that determined by a servo-motor, which is typicallynonback-drivable and hence well-described as a controllablemotion source. The same model is also fully consistent withrecent developments of variable-stiffness actuators [3], [4].

E. Implications

Though it may be an obvious point, interactive dynamics can-not be identified unambiguously without knowing the sourcemotion. Because it resides at the interface between informa-tional and physical dynamics, an actuator with a single output—exertion for an actuator in impedance form—must have a leasttwo inputs: one to account for physical interaction and one to ac-count for forward-path commands. If only variables associatedwith an interaction port are observed (e.g., joint torques and an-gular displacements) the relation between exertion and motion isindeterminate. Remarkably, this fact is frequently ignored in es-timates of lower-limb impedance during locomotion [56]–[59].


Because it is a zero of the impedance operator, the forward-path dynamics between control input and motion source outputat an external port may be identified without detailed knowl-edge of the impedance operator. A complete characterizationof forward-path dynamics would require an observation of thecontrol input, which is challenging in biological applications,but the consequences of the forward-path response to controlinputs are summarized in the equivalent network source, andfor the motion-source network, that may readily be observed ifthe interaction port exertion (e.g., force or torque) is zero [41].

To apply a similar approach to identify a robotic actuator (i.e.,independent of any inertial load it drives) would be equivalentto disconnecting the actuator from its load. One method is toestimate the load and, using a high-performance robotic manip-ulator, apply that load so that the net actuator-generated forceis zero. This method, which has been applied successfully toinvestigate the control of the human upper-extremity [41], [42]may prove to be applicable to mechatronic systems. Given areliable estimate of the kinematic, inertial, and frictional prop-erties of a robot such as [55] provide may enable a methodsuch as that presented by [41] to identify a robot’s actuator dy-namics in situ. The practicality of this approach remains to bedemonstrated.

Is such a complicated method really necessary? Are not “clas-sical” methods, e.g., based on “first-principles” modeling of me-chanical physics adequate? To address this question, considerthe results presented in a recent paper [4] which described an iso-lated variable stiffness actuator, not subject to the complex loadsdue to assembly into a functioning robot. Even under these idealconditions, a model based on “classical” first-principle methodsand adjusted to fit experimental observations was demonstra-bly inaccurate (see Fig. 10 of [4]). If “close enough” is not“good enough”—and often it is not—then empirical identifica-tion based on a representation such as the motion-source (Nor-ton) equivalent network presented here is necessary.

F. Limitations

In the attempt to generalize from linear electrical circuits toan important class of nonlinear systems, some features of equiv-alent networks have been retained but some have not. The con-cept has been generalized to any domain subject to the laws ofthermodynamics and the Norton current source has been gener-alized to include velocity, displacement, position—any motionvariable that has a well-defined transformation in the same direc-tion as a generalized velocity (e.g., from joint angle to end-pointposition). However, much has not been retained.

For general linear systems, the existence of equivalent net-works is guaranteed, no matter how complex the underlyingsystem, and extension to a broader class of sources (e.g., po-sition or momentum rather than velocity or force) simply addsintegrators or differentiators to the source element. For nonlin-ear systems, the author is unaware of any comparable proof. Inparticular, the linear connection of an equivalent circuit has notbeen retained. A shown above, in general a nonlinear relationbetween the source motion and the impedance operator may berequired.

It is also important to recognize that an equivalent network isnot a minimal representation. The parameters of the same circuitcomponents appear in both parts of an equivalent circuit. Someduplication may occur because the same physical phenomenamay contribute to the forward-path response and to interactivebehavior (though perhaps in different ways). That is the price ofseparating source dynamics from interactive dynamics.

Despite these limitations, a representation in which the sourcemotion is a zero of the impedance operator appears to be ver-satile. Any system, robotic or human, which is capable of es-tablishing equilibrium at different locations in its workspaceappears to be compatible with a motion-source (Norton) equiv-alent network description. That is why the source, the output ofthe forward-path dynamics, may be identified without detailedknowledge of the interactive dynamics. A proof (or disproof) ofthe generality of the motion-source equivalent network remainsfor future work.

REFERENCES

[1] S. H. Kang, M. Jin, and P. H. Chang, “A solution to the accu-racy/robustness dilemma in impedance control,” IEEE /ASME Trans.Mechatronics, vol. 14, no. 3, pp. 282–294, Jun. 2009.

[2] C. Ott, R. Mukherjee, and Y. Nakamura, “Unified Impedance and Admit-tance Control,” presented at the IEEE Int. Conf. Robot. Autom., Anchor-age, AK, USA, 2010.

[3] A. Jafari, N. G. Tsagarakis, and D. G. Caldwell, “A novel intrinsicallyenergy efficient actuator with adjustable stiffness (AwAS),” IEEE/ASMETrans. Mechatronics, vol. 18, no. 1, pp. 355–365, Feb. 2013.

[4] S. S. Groothuis, G. Rusticelli, A. Zucchelli, S. Stramigioli, and R. Carloni,“The variable stiffness actuator vsaUT-II: Mechanical design, modeling,and identification,” IEEE/ASME Trans. Mechatronics, to be published,DOI: 10.1109/TMECH.2013.2251894.

[5] R. Carloni, L. C. Visser, and S. Stramigioli, “Variable stiffness actuators: Aport-based power-flow analysis,” IEEE Trans. Robot., vol. to be published,DOI: 10.1109/TMECH.2013.2251894,.

[6] M. Howard, D. J. Braun, and S. Vijayakumar, “Constraint-based equilib-rium and stiffness control of variable stiffness actuators,” presented at theInt. Conf. Robot. Autom., Shanghai, China, 2011.

[7] M. Lauria, M.-A. Legault, M.-A. Lavoie, and F. Michaud, “High perfor-mance differential elastic actuator for robotic interaction tasks,” presentedat the AAAI Spring Symp. Series, Stanford, CA, USA, 2007.

[8] J. W. Hurst, J. E. Chestnutt, and A. A. Rizzi, “An actuator with physicallyvariable stiffness for highly dynamic legged locomotion,” presented at theInt. Conf. Robot. Autom., New Orleans, LA, USA, 2004.

[9] C. E. English and D. Russell, “Implementation of variable joint stiffnessthrough antagonistic actuation using rolamite springs,” Mechanism Mach.Theory, vol. 34, pp. 27–40, 1999.

[10] C. E. English and D. Russell, “Mechanics and stiffness limitations of avariable stiffness actuator for use in prosthetic limbs,” Mechanism Mach.Theory, vol. 34, pp. 7–25, 1999.

[11] T. Morita and S. Sugano, “Development and evaluation of seven DOFMIA ARM,” presented at the Int. Conf. Robot. Autom., Albuquerque,NM, USA, vol. 1, pp. 462–467, 1997.

[12] T. Morita and S. Sugano, “Development of four-D.O.F. manipulator us-ing mechanical impedance adjuster,” presented at the Int. Conf. Robot.Autom., Minneapolis, MN, USA, vol. 4, pp. 2902–2907, 1996.

[13] T. Morita and S. Sugano, “Development of One-D.O.F. robot arm equippedwith mechanical impedance adjuster,” presented at the Int. Conf. Intell.Robots Syst., Pittsburgh, PA, USA, vol. 1, 1995.

[14] I. Newton, Philosophiæ Naturalis Principia Mathematica, 1687.[15] D. H. Johnson, “Origins of the equivalent circuit concept: The voltage-

source equivalent,” Proc. IEEE, vol. 91, no. 4, pp. 636–640, Apr.2003.

[16] D. H. Johnson, “Origins of the equivalent circuit concept: The current-source equivalent,” Proc. IEEE, vol. 91, no. 5, pp. 817–821, May2003.

[17] H. V. Helmholtz, “II. Uber einige gesetze der vertheilung elek-trischer strome in korperlichen leitern mit anwendung auf die


thierisch-elektrischen versuche [some laws concerning the distributionof electrical currents in conductors with applications to experiments onanimal electricity],” Ann. Phys. Chem., vol. 89, pp. 211–233, 1853.

[18] L. C. Thevenin, “Sur un nouveau theoreme d’electricite dynamique[On a new theorem of dynamic electricity],” Comptes Rendus Seancesl’Academie Sci., vol. 97, pp. 159–161, 1883.

[19] H. F. Mayer, “Ueber das ersatzschema der verstarkerrohre [on equivalentcircuits for electronic amplifiers],” Telegraphen- und Fernsprech-Technik,vol. 15, pp. 335–337, 1926.

[20] E. L. Norton, “Design of finite networks for uniform frequency character-istic,” Western Electric Company, Inc., New York, NY, USA, Tech. Memo.TM26-0-1860, Nov. 1926.

[21] O. Heaviside, Electrical Papers, vol. 1. Boston, MA, USA: Macmillan,1925.

[22] O. Heaviside, Electrical Papers, vol. 2. Boston, MA, USA: Macmillan,1925.

[23] N. Hogan and P. C. Breedveld, “The physical basis of analogies in phys-ical system models,” in The Mechatronics Handbook, R. H. Bishop, Ed.Boca Raton, FL, USA: CRC Press, 2002.

[24] J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford, U.K.:Clarendon, vol. 1, 1873.

[25] N. Hogan, “Modularity and causality in physical system modeling,”Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 109, pp. 384–391, Dec.1987.

[26] M. F. Moad, “On Thevenin’s and Norton’s equivalent circuits,” IEEETrans. Edu., vol. 25, no. 3, pp. 99–102, Aug. 1982.

[27] D. Rancourt and N. Hogan, “The biomechanics of force production,” inProgress in Motor Control: A Multidisciplinary Perspective, D. Sternad,Ed. New York, NY: Springer-Verlag, 2009, pp. 645–661.

[28] M. M. Williamson, “Series elastic actuators,” M.S. dissertation, Dept.Elect. Eng. Comput. Sci., Massachusetts Inst. Technol., Cambridge, MA,USA, 1995.

[29] G. A. Pratt and M. M. Williamson, “Series elastic actuators,” presented atthe Int. Conf. Intell. Robots Syst., Pittsburgh, PA, vol. 1, 1995.

[30] A. Isidori, Nonlinear Control Systems, 2nd ed. Berlin, Germany:Springer-Verlag, 1989.

[31] L. Ljung, “Perspectives on system identification,” presented at the Int.Fed. Autom. Control Congr., Seoul, Korea, 2008.

[32] F. Daerden and D. Lefeber, “Pneumatic artificial muscles: Actuators forrobotics and automation,” Eur. J. Mech. Environ. Eng., vol. 47, pp. 10–21,2002.

[33] K. Bharadwaj, K. W. Hollander, C. A. Mathis, and T. G. Sugar, “Springover muscle (SOM) actuator for rehabilitation devices,” presented at the26th Annu. Int. Conf. IEEE Eng. Med. Biol. Soc., San Francisco, CA,USA, 2004.

[34] P. R. Dahl, “Solid friction damping of mechanical vibrations,” AIAA J.,vol. 14, pp. 1675–1682, 1976.

[35] K. J. Astrom and C. Canudas-de-Wit, “Revisiting the LuGre frictionmodel,” IEEE Control Syst. Mag., vol. 28, no. 6, pp. 101–114, Dec. 2008.

[36] G. I. Zahalak, “Modeling muscle mechanics (and energetics),” inMultiple Muscle Systems: Biomechanics and Movement Organization,J. M. Winters and S. L.-Y. Woo, Eds. New York, NY, USA: Springer-Verlag, 1990, pp. 1–23.

[37] H. Gomi and M. Kawato, “Equilibrium-point control hypothesis examinedby measured arm stiffness during multijoint movement,” Science, vol. 272,pp. 117–120, Apr. 1996.

[38] P. Gribble, D. J. Ostry, V. Sanguinetti, and R. Laboissiere, “Are complexcontrol signals required for human arm movement?” J. Neurophysiol.,vol. 79, pp. 1409–1424, 1998.

[39] J. M. Wakeling and B. M. Nigg, “Modification of soft tissue vibrationsin the leg by muscular activity,” J. Appl. Physiol., vol. 90, pp. 412–420,2001.

[40] J. M. Wakeling, A.-M. Liphardt, and B. M. Nigg, “Muscle activity reducessoft-tissue resonance at heel-strike during walking,” J. Biomechanics,vol. 36, pp. 1761–1769, 2003.

[41] A. J. Hodgson and N. Hogan, “A model-independent definition of attractorbehavior applicable to interactive tasks,” IEEE Trans. Syst., Man, Cybern.C, Appl. Rev., vol. 30, no. 1, pp. 105–118, Feb. 2000.

[42] A. J. Hodgson, “Inferring central motor plans from attractor trajectorymeasurements,” Ph.D. dissertation, Dept. Mech. Eng., Massachusetts Inst.Technol., Cambridge, MA, USA, 1994.

[43] C. D. Hoover, G. D. Fulk, and K. B. Fite, “Stair ascent with a poweredtransfemoral prosthesis under direct myoelectric control,” IEEE/ASMETrans. Mechatronics, vol. 18, no. 3, pp. 1191–1200, Jun. 2013.

[44] F. Sup, H. A. Varol, J. Mitchell, T. J. Withrow, and M. Goldfarb, “Pre-liminary evaluations of a self-contained anthropomorphic transfemoralprosthesis,” IEEE/ASME Trans. Mechatronics, vol. 14, no. 6, pp. 667–676, Dec. 2009.

[45] S. K. Au, J. Weber, and H. Herr, “Powered ankle–foot prosthesis improveswalking metabolic economy,” IEEE Trans. Robot., vol. 25, no. 1, pp. 51–66, Feb. 2009.

[46] W. A. Farahat and H. M. Herr, “Optimal workloop energetics of muscle-actuated systems: An impedance matching view,” PLoS Comput. Biol.,vol. 6, p. e1000795, 2010.

[47] F. A. Mussa-Ivaldi, N. Hogan, and E. Bizzi, “Neural, mechanical, andgeometric factors subserving arm posture in humans,” J. Neurosci., vol. 5,pp. 2732–2743, 1985.

[48] J. M. Dolan, M. B. Friedman, and M. L. Nagurka, “Dynamic and loadedimpedance components in the maintenance of human arm posture,” IEEETrans. Syst., Man, Cybern., vol. 23, no. 3, pp. 698–709, May/Jun. 1993.

[49] T. Tsuji, P. G. Morasso, K. Goto, and K. Ito, “Human hand impedancecharacteristics during maintained posture,” Biol. Cybern., vol. 72, pp. 475–485, 1995.

[50] H. Gomi and M. Kawato, “Human arm stiffness and equilibrium-pointtrajectory during multi-joint movement,” Biol. Cybern., vol. 76, pp. 163–171, 1997.

[51] E. Burdet, R. Osu, D. W. Franklin, T. Yoshioka, T. E. Milner, andM. Kawato, “A method for measuring end-point stiffness during multi-joint arm movements,” J. Biomechanics, vol. 33, pp. 1705–1709, 2000.

[52] E. J. Perreault, R. F. Kirsch, and P. E. Crago, “Effects of voluntary forcegeneration on the elastic components of endpoint stiffness,” Exp. BrainRes., vol. 141, pp. 312–323, 2001.

[53] J. J. Palazzolo, M. Ferraro, H. I. Krebs, D. Lynch, B. T. Volpe, andN. Hogan, “Stochastic estimation of arm mechanical impedance duringrobotic stroke rehabilitation.,” IEEE Trans. Neural Syst. Rehabil. Eng.,vol. 15, no. 1, pp. 94–103, Mar. 2007.

[54] P. H. Chang, K. Park, S. H. Kang, H. I. Krebs, and N. Hogan, “Stochas-tic estimation of human arm impedance with robots having nonlinearfrictions: An experimental validation,” IEEE/ASME Trans. Mechatronics,vol. 18, no. 2, pp. 775–786, Apr. 2012.

[55] D. Kostic, B. de Jager, and M. Steinbuch, “Modeling and identificationfor robot motion control,” in Robotics and Automation Handbook, T. R.Kurfess, Ed. New York, NY, USA: CRC Press, 2005, pp. 14-1–14-26.

[56] C. T. Farley and O. Gonzalez, “Leg stiffness and stride frequency in humanrunning,” J. Biomechanics, vol. 29, pp. 181–186, 1996.

[57] C. T. Farley, H. H. P. Houdijk, C. V. Strien, and M. Louie, “Mechanism ofleg stiffness adjustment for hopping on surfaces of different stiffnesses,”J. Appl. Physiol., vol. 85, pp. 1044–1055, 1998.

[58] D. P. Ferris, K. Liang, and C. T. Farley, “Runners adjust leg stiffnessfor their first step on a new running surface,” J. Biomechanics, vol. 32,pp. 787–794, 1999.

[59] C. T. Moritz and C. T. Farley, “Human hopping on damped surfaces:Strategies for adjusting leg mechanics,” in Proc. R. Soc. Lond. B, vol. 270,pp. 1741–1746, 2003.

Neville Hogan received the Dip. Eng. degree(with distinction) from Dublin Institute of Technol-ogy, Dublin, Ireland, and the M.S., M.E., and Ph.D.degrees from Massachusetts Institute of Technology(MIT), Cambridge, MA, USA.

He is currently Sun Jae Professor of MechanicalEngineering and a Professor of Brain and CognitiveSciences at the Massachusetts Institute of Technol-ogy (MIT). He is also Director of the Newman Lab-oratory for Biomechanics and Human Rehabilitationand founder and Director of Interactive Motion Tech-

nologies, Inc. His research includes robotics, motor neuroscience, and rehabil-itation engineering, emphasizing the control of physical contact and dynamicinteraction.

Dr. Hogan has received numerous awards, including Honorary Doctoratesfrom Delft University of Technology, Delft, The Netherlands, and Dublin Insti-tute of Technology, Ireland; the Silver Medal of the Royal Academy of Medicinein Ireland; and the Henry M. Paynter Outstanding Investigator Award and theRufus T. Oldenburger Medal from the Dynamic Systems and Control Divisionof the American Society of Mechanical Engineers(ASME). He has been a mem-ber of the ASME since 1983.

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