AN ANALYSIS OF EQUIVALENT IMPEDANCE ...bsys.hiroshima-u.ac.jp/pub/pdf/C/C_166.pdfMULTIBODY DYNAMICS 2007, ECCOMAS Thematic Conference C.L. Bottasso, P. Masarati, L. Trainelli (eds.)

MULTIBODY DYNAMICS 2007, ECCOMAS Thematic ConferenceC.L. Bottasso, P. Masarati, L. Trainelli (eds.)

Milano, Italy, 25–28 June 2007

AN ANALYSIS OF EQUIVALENT IMPEDANCE CHARACTERISTICSBY MODELING THE HUMAN MUSCULOSKELETAL STRUCTURE

AS A MULTIBODY SYSTEM

Masatoshi Hada?, Daisuke Yamada?, and Toshio Tsuji†

? TOYOTA Central Research & Development Laboratories, IncorporatedNagakute, Aichi, 480–1192, Japan

e-mails:[email protected] , [email protected] ,web page:http://www.tytlabs.co.jp

† Graduate School of Engineering, Hiroshima University1–4–1 Kagamiyama, Higashi-Hiroshima, 739–8527, Japan

e-mail: [email protected] ,web page:http://www.bsys.hiroshima-u.ac.jp

Keywords: Equivalent impedance, Human-machine system, Multibody dynamics, Muscu-loskeletal system, Muscle-tendon complex

Abstract.This paper deals with an analysis of equivalent impedance characteristics of human-machine

systems taking contact and constraint conditions into account. The analysis consists of fourphases: modeling human-machine system, solving the equation of motion, deriving equivalentimpedance characteristics, and visualizing the results. First, each muscle tendon complex ofthe human is modeled as a Hill-type model and the muscle path as a series of line segmentswith viapoints using a wrapping technique. Then differential-algebraic equations (DAEs) of thehuman-machine system are formulated by modeling both of the human musculoskeletal struc-ture and the object as the unified multibody system. Next, the joint torque of the human and thegeneralized forces of the object are obtained from inverse dynamics using motion and externalforce data. Muscle forces are estimated using a sequential quadratic programming with main-taining a balance against the joint torque. In the analysis phase, muscle stiffness and viscosityare calculated from the Hill-type model. According to coordinate transformation of tensor,human muscle impedance is transformed into reference points defined on the object, and theyare synthesized with the object impedance so that equivalent impedance of the human-machinesystem then be obtained. The results are visualized three-dimensionally to enhance usefulnessof the analysis. An upper extremities motion in forklift steering operation was analyzed as anapplication example. Equivalent inertia and stiffness at the human hand and around a steeringcolumn are calculated. The results show an effectiveness of the equivalent impedance analysisto investigate driver’s strategy in a steering maneuver.

1

[email protected]

[email protected]

http://www.tytlabs.co.jp

[email protected]

http://www.bsys.hiroshima-u.ac.jp

Masatoshi Hada, Daisuke Yamada and Toshio Tsuji

1 INTRODUCTION

When driving a vehicle on a bumpy road, we sometimes grasp the steering wheel of the vehi-cle more strongly than on a flat road. Grasping is orthogonal to the steering torque that is a con-trol input for traveling direction of the vehicle. It is believed that human beings subconsciouslyperform other task than turning the steering wheel, with applying this unnecessary-like effort.It is well known that muscles, only actuator for movements of human beings, provide variousviscoelastic characteristics from environmental inputs [1]. Considering mechanical impedancecharacteristics of both the human musculoskeletal structures and the object he/she is manipu-lating together, it might be possible to clarify the physical significance of the skillful strategiesof human beings.

In the researches that analyze dynamics of the human musculoskeletal system, there are sev-eral software systems have been proposed. Delp et al.[2] firstly developed a software packagecalled SIMM that enables users to develop, alter, and evaluate three dimensional musculoskele-tal structures. Eberhard el al.[3] has also investigated dynamical analysis of human motion bycombining musculoskeletal structure modeling and optimal control techniques. Rasmussen etal.[4] has proposed the software system called AnyBody and Nakamura et al.[5] has reportedsame sort of software. The users of these system specify the surfaces of bones, the kinematicsand passive torque characteristics of the joints, the muscle path and force generating parame-ters of the muscles. Then the softwares estimate muscle forces during movement by solvinginverse dynamics and optimization, or generate human motion by integrating forward dynamicsfrom controlled input forces. These softwares can be effective in the case of human motionanalysis. However, they have not yet led to analyze equivalent impedance characteristics ofhuman-machine systems.

Some studies have been undertaken for simulating the impedance characteristics of humanmusculoskeletal systems. Takeda et al.[6] expressed the viscoelastic properties of muscles bypolynomial formula in a joint space using experimental data, and proposed a method to expressthe impedance characteristics of a human hand in a task space. Furthermore, Stroeve [7] cal-culated the equivalent impedance characteristics of an upper extremities taking the dynamics,from the motor-command to the muscle contraction, into consideration. However, these studieswere not extended to include both the viscoelastic properties of the object and the constraintconditions. In almost all of the operations, both the human and the objects are constrained in avariety of ways. Therefore, it is difficult to directly apply these studies to a practical situationwhere a human operates on an object.

In this paper, we propose a method for analyzing equivalent impedance characteristics ofhuman-machine systems taking both contact and constraint conditions into account. As shownin Fig.1, this analysis consists of four phases: modeling human-machine system, solving theequation of motion, deriving equivalent impedance characteristics, and visualizing the results.First, each muscle tendon complex of the human is modeled as a Hill-type model and the musclepath as a series of line segments with viapoints using a wrapping technique. Then differential-algebraic equations (DAEs) of the human-machine system are formulated by modeling both ofthe human musculoskeletal structure and the object as the unified multibody system. Next, thejoint torque of the human and the generalized forces of the object are obtained from inversedynamics using motion and external force data. Muscle forces are estimated using a sequen-tial quadratic programming with maintaining a balance against the joint torque. In the analysisphase, muscle stiffness and viscosity are calculated from the Hill-type model. According tocoordinate transformation of tensor, human muscle impedance is transformed into reference

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S o l v i n g D A E si n v e r s e d y n a m i c s

O p t i m i z i n g m u s c l e f o r c e d i s t r i b u t i o n

E q u i v a l e n t i m p e d a n c ec h a r a c t e r i s t i c s

M u s c l e - t e n d o n i m p e d a n c e

M o d e l i n g

H u m a n O b j e c t

M u s c u l e - t e n d o n c o m p l e xP E

S EC E

S o l v i n g

A n a l y s i s

V i s u a l i z a t i o n

Figure 1:Overview of equivalent impedance characteristics analysis.

points defined on the object, and they are synthesized with the object impedance so that equiv-alent impedance of the human-machine system then be obtained. In this derivation, contact andconstraint conditions are taken into account by using orthogonal complementary projections tothe null space of the contacts and constraints. The results are visualized three-dimensionally toenhance usefulness of the analysis. An upper extremities motion in forklift steering operationwas analyzed as an application example. Equivalent inertia and stiffness at the human handand around steering column are calculated. The results show an effectiveness of the equivalentimpedance analysis to investigate driver’s strategy in a steering maneuver.

In the following discussion, muscle refers to the muscle-tendon complex unless otherwisestated. When the muscle and tendon are compared, muscle refers to the muscle of the muscle-tendon complex.

2 MODELING HUMAN MUSCULOSKELETAL STRUCTURE

2.1 Modeling Muscle Tendon Complex

As introduced in Ref.[8], there are several ways to express muscle tendon complex behaviors.In this study, we deal a three-component model that is composed of contractile element (CE)describing muscle belly, serial elastic element (SE) describing tension of the tendon, and parallelelastic element (PE) describing passive force. Using this model, a muscle force is obtained fromthe following equation;

fu = afmaxfLfV cosα, (1)

= fmaxfSE, (2)

wherefL is tension-length relationship,fV is force-velocity relationship,fSE is serial elasticforce property as respectively shown in Fig.2 (a), (b) and (c).a is the muscle activity level(0 ≤ a ≤ 1) which represents the input signal from central nervous system.α is the musclepennation angle.

The tension-length relationshipfL is modeled as follows [9][10]:

fL = e−(lm−1)2/SL , (3)

lm = lm/lm0. (4)

3


( d ) t e n s i o n l e n g t h r e l a t i o n s h i p i n t h e p a r a l l e l e l e m e n t

f P E

0 . 8 1 . 0 1 . 50

1

N o r m a l i z e d m u s c l e l e n g t h ( - )No

rmalize

d musc

le forc

e (-)

0 . 5 1 . 0 1 . 50

0 . 5

1 . 0

N o r m a l i z e d m u s c l e f i b e r l e n g t h ( - )

Norma

lized m

uscle f

orce (-

)a = 1 . 0a = 0 . 8a = 0 . 6a = 0 . 4a = 0 . 2

a f L

( a ) t e n s i o n l e n g t h r e l a t i o n s h i p i n t h e c o n t r a c t i l e e l e m e n t ( b ) f o r c e v e l o c i t y r e l a t i o n s h i p i n t h e c o n t r a c t i l e e l e m e n t

- 1 . 0 0 1 . 00

1

N o r m a l i z e d m u s c l e f i b e r v e l o c i t y ( - )

Norma

lized m

uscle f

orce (-

)

f V

( c ) f o r c e s t r a i n r e l a t i o n s h i p i n t h e s e r i e s e l e m e n t

0 0 . 0 2 40

0 . 3 3

1 . 0

N o r m a l i z e d t e n d o n s t r a i n ( - )

Norma

lized te

ndon fo

rce (-)

f S E

Figure 2:A model of muscle tendon complex and characteristics of the muscle force

lm is the normalized length based on optimal muscle lengthlm0.The force-velocity relationshipfV is as follows [9]:

fV =

0 (vm ≤ −1),1+vm

1−vm/Af(−1<vm≤0),

(Bf−1)+vm(2+2/Af )Bf

(Bf−1)+vm(2+2/Af )(vm > 0),

(5)

vm = lm/vmax, (6)

wherevm is normalized velocity based on maximum contract velocityvmax.The following equations are used to describe the serial elastic elementfSE [9],

fSE =

0 (lt ≤ 0),ftoe(ekselt/εtoe−1)

ekse−1(0<lt≤εtoe),

klin

(lt − εtoe

)+ ftoe (εtoe < lt),

(7)

lt = (lt − lt0)/lt0, (8)

wherelt is the normalized length based on slack length of the tendonlt0.As shown in Fig.2 (d), the parallel elastic elementfPE is modeled as

fPE =ekpe(l−1)/ε0− 1

ekpe − 1, (9)

l = l/(lm0cosα + lt0). (10)

4


x

y

z

pi (xi , yi , zi)

po (xo, yo, zo)pq (xq, yq, zq)

p

p

ab

c

0

pt

pt

Ellipsoid E

Muscle path

Control pointOrigin

Insertion

i

o

o

i

e

e

e

Figure 3:Muscle path wrapping an ellipsoid.

pq

po pi

pto

pto

pto

pt i

1

2

3

1

pq2

pq3

Figure 4:Viapoints along ellipsoid surface.

2.2 Modeling Muscle Path

In modeling a musculoskeletal system of a human body, muscle paths are usually expressedby line segments connecting viapoints between the origin and insertion[2, 3, 4, 5, 6, 7]. How-ever, because the viapoints are fixed to bones that are modeled as rigid bodies, there is a limit todescribe smooth muscle paths. Garner et al.[11] and Charlton et al.[12] used a wrapping methodto connect between the origin and insertion along with the surface of an ellipsoid, cylinder orcone. These methods, however, need an excessive computational load as convergent calculationis performed on each muscle. In this study, a new wrapping algorithm that allows muscles toconform to an ellipsoid with no convergent calculations are introduced.

As shown in Fig.3, origin and insertion of a muscle are defined aspo = (xo, yo, zo)T and

pi = (xi, yi, zi)T respectively. They are both defined in the frame of a wrapped ellipsoidsE,

radius of each axis isae, be, andce. FramesΣpo andΣpiare then defined on these two points.

Suppose thatΣpo move towards the same direction asΣpiin the case thatΣpo is rotated along

the axisu ∈ <3, |u| = 1 by φ (0 ≤ φ ≤ π). Then, a control pointpq = (xq, yq, zq)T is defined

as followspq = po + ηpio + A(u, ηφ)v, (11)

wherepio ∈ <3 is the vector from pointpo to pointpi, A(u, φ) ∈ <3×3 is the rotation matrix,v ∈ <3 is an arbitrary constant vector and0 ≤ η ≤ 1.

Considering the tangential line of the ellipsoidE that passes through the pointpo and is in the

5


plane containing the pointspo, pi andpq. The tangent pointpto = (xt, yt, zt)T can be obtained

by solving the following system of equations.

xt2

ae2

+yt

2

be2 +

zt2

ce2

= 1, (12)

xoxt

ae2

+yoyt

be2 +

zozt

ce2

= 1, (13)∣∣∣∣∣∣∣∣∣

xt yt zt 1xo yo zo 1xq yq zq 1xi yi zi 1

∣∣∣∣∣∣∣∣∣= 0. (14)

Although there are two solutions forpto, the one with a shorter distance topq is selected. Ro-tating the vectorpio in small steps untilpto coincidepti, the above calculation is repeatedlyperformed with the obtained point being inserted as a viapoint.

Because origin and insertion of muscle are usually on different bones, pointpq moves smoothlydepending on movement of the bone. Therefore, ifΣpi

, Σpo，η andv are properly defined, itis possible to enhance biofidelity of the muscle path. The algorithm was implemented so as towork well even in the case that multiple ellipsoids are wrapped. Muscles with larger attachmentarea, such as the broadest muscle of the back and the cowl muscle, are modeled as multiplemuscle paths sharing a muscle tendon complex model.

2.3 Formulating Equation of Motion

H u m a n c o n t a c tp o i n t c o o d i n a t e

e

C o n t a c t t r a n s m i s s i o n v i r t u a l c o o r d i n a t e

t r H u m a n g e n e r a l i z e d

c o o r d i n a t e

h

O b j e c t c o n t a c tp o i n t c o o r d i n a t e

c

O b j e c t g e n e r a l i z e d c o o r d i n a t e

m

O b j e c t I n t e r a c t i o nR e f e r e n c e p o i n t c o o r d i n a t e

r H u m a n - m a c h i n e s y s t e m H u m a n

H u m a n

O b j e c t

C o n t a c t p o i n t

r R e f e r e n c e p o i n t

H u m a n m u s c l ec o o r d i n a t e

u

Figure 5:Frames of a human-machine system.

The frames in a human-machine system is illustrated in Fig.5. Σu is the frame for the lengthof the musclelh ∈ <nu , nu is the number of muscles.Σh is the frame that is composed ofthe generalized coordinatesqh ∈ <nh, and that expresses the movement of the human, whileΣe is the frame for the human’s contact with the object and is composed of the contact pointcoordinateXe ∈ <ch. Similarly,Σm is the frame that expresses the movement of the object andis composed of the generalized coordinatesqm ∈ <nm, while Σc is the frame for the object’scontact with the human and is composed of the contact point coordinatesXc ∈ <cm. Σtr isthe frame for thentr dimension that is used to express the transmission of force at the contact

6


point between the human and the object[13, 14]. In the state where the human and the objectare in contact, the three framesΣe, Σtr and Σc coincide, with their shared z axis oriented in thedirection of a line normal to the plane of contact, and assuming that no shifting occurs in thecontact point.Σr is the frame with its origin atr, the reference point for equivalent impedancecharacteristics, which can be defined at anywhere on the object.

Each segment of the human body and each part of the object are treated as rigid bodies, wherethe mass, center of gravity and moment of inertia of each of these rigid bodies are defined. Oneach body, frames called marker are then defined to describe joints, muscle paths, positions ofcontact and constraint. In accordance with above definitions and assumptions, an equation ofmotion for the human-machine system are formulated as the following differential algebraicequations (DAEs),

[M GT

G 0

] [qλ

]=

[Q− g − h− JT

e Fe

γ

], (15)

whereM = diag. [Mh,Mm] ∈ <n×n, n = nh + nm is the inertia tensor of the human-machinesystem consisting of the human inertiaMh ∈ <nh×nh and the object inertiaMm ∈ <nm×nm.

q =[qh

T , qmT]T ∈ <n is the vector of the generalized coordinate consisting ofqh ∈ <nh and

qm ∈ <nm. G = diag. [Gh, Gm] ∈ <c×n, c = ch + cm is the constraint Jacobian.Gh ∈<ch×nh , rank(Gh) = ch < nh is the Jacobian in terms of the human constraintΦh. Gm ∈<cm×nm , rank(Gm) = cm < nm is the Jacobian in terms of the object constraintΦm and

joint constraint of the objectΦj. λ =[F T

φh, λT

m

]T ∈ <ch+cm is the constraint force and theLagrange multiplier. In this study, only the holonomic constraint expressed asGhδqh = 0 isconsidered.Q =

[τTh , QT

m

]∈ <n is the generalized force.g =

[gT

h , gTm

]∈ <n is the gravity

term. h =[hT

h , hTm

]∈ <n is the centrifugal and Coliolis force.γ ∈ <c is the term related to

derivative of the constraints.Je = [Jeh, Jem ] ∈ <nc×n is the Jacobian to contact points between

the human and the object.Fe ∈ <nc is the contact force. In this study, relative coordinates areused to describe the equation of motion of the human body [15], and the absolute coordinatesto describe the equation of motion for the object [16]. Therefore,nh is the total number of thedegrees of joint freedom of the human body, andnm is six times as large as the number of rigidbodies comprising the object model.

3 SOLVING HUMAN-MACHINE SYSTEM

3.1 Solving Inverse Dynamics

Giving motionsq, q, q and forcesFe into the formulated equation of motion Eq.(??), gener-alized forceQ is calculated by using a projection method [17] as

PMq = P (Q− g − h− JTe Fe), (16)

whereP ∈ <n×n is the matrix that describes the projection ofG onto the null space. Inthis method, it is unnecessary to calculate the constraint force and the Lagrange multiplierλ.Therefore, the joint torque of the humanτh can be obtained with the minimum set of input. Inthis computation, the muscle length vectorLu ∈ <nu and the muscle contraction velocityVu ∈<nu are also obtained by using muscle paths defined in the human musculoskeletal model. Also,the muscle JacobianJu ∈ <nh×nu , which indicates the muscle moment arm characteristics, is

7


Muscle jf j

k

d jk Muscle path

jk

(a) Before conditioning

Rigid body

Joint k

djk

f j

k

(b) After conditioning

Figure 6:Preconditioning muscle Jacobian

calculated from principle of virtual work.

Ju =∂Lu

∂qh

. (17)

3.2 Preconditioning Muscle Jacobian

Conventional musculoskeletal models assume that the direction of each muscle force is al-ways only on the muscle path line. This assumption might be reasonable if the joint movementis to a limited extent, but in the case of large displacements, it does not hold true because actualmuscles have volume and line of action can vary as reported in Ref.[18]. So we process a pre-conditioning for the muscle JacobianJu to give more effectively equilibrium with the humanjoint torque. Fig.6 illustrates the joint torqueτk and the moment arm vectordjk before and afterthe preconditioning. The outline of the algorithm is introduced here, the detail and effectivenessis described in Ref.[19].

Let τj ∈ <3 define the torque vector ofj th joint of the human with three rotational degreesof freedom as

τj = [τi, τi+1, τi+2]T , (18)

whereτi is i th element ofτh. Moment arm vectordjk ∈ <3 of musclek around jointj is alsodefined as follows,

djk = − [Jk,i, Jk,i+1, Jk,i+2]T , (19)

whereJk,i is (k, i) element ofJu. Then the magnitude of muscle force of musclek is definedasfuk

∈ <1, these provide the following equation:

τj =nu∑

i=1

djkfui. (20)

This indicates that|djk| represents a magnitude of a joint torque of jointj which is generatedby musclek . The direction ofdjk matches that of joint torque of jointj which is generated bymusclek. Therefore, the closer the direction ofdjk to that ofτj, the more efficiently musclekproduce a component ofτj.

In accordance with the following rules, the moment arm vectordjk are conditioned intodjk.

djk = Rjk(κjk, ψjk)djk, (21)

8


κjk =djk × τj

|djk||τj| , (22)

ψjk =exp(βj − ξjk)

1 + exp(βj − ξjk)ξjk, (23)

ξjk = asin

( |djk × τj||djk||τj|

), (24)

whereRjk ∈ <3×3 is the rotation matrix, which rotational axis isκjk ∈ <3 and rotation angle isψjk ∈ <1. ξjk ∈ <1 represents the angle betweendjk andτj. β is the conditioning factor whichdetermines a degree of change in direction ofdjk.

Substituting the conditioned moment arm vectordjk to corresponding elements ofJu, we canobtain the conditioned muscle Jacobian matrixJu ∈ <nh×nu . Consequently, the equilibriumbetween the joint torques and the muscle forces becomes

τh = −JuT (qh, τh, β)fu. (25)

Note thatJu becomes a function ofqh, τh andβ, so this technique is effective only in inversedynamics analysis.

3.3 Optimizing Muscle Force Distribution

The muscle forcefu ∈ <nu can be obtained from the joint torqueτh according to the follow-ing steps. Firstly, the applying joint torque of the humanτh ∈ <nh is calculated eliminating thepassive force of the muscle fromτh,

τh = τh + JuT FPE, (26)

whereFPE ∈ <nu is the passive muscle force vector composing of the parallel elastic elementof each musclefPE as describe in Eq.(9).

Therefore, the muscle forcefu can be obtained by solving the constrained optimization prob-lem with

minimize J (fu) = fTu W T Wfu, (27)

subject to τh = −JuT fu, (28)

0 ≤ fu ≤ fmax, (29)

whereJ (fu) is an objective function offu. Eq.(28) is an equality constraint condition of equi-librium between joint torques and muscle forces. Eq.(29) is an inequality constraint conditionregarding range offu. fu must be set greater than or equal to0 because muscle can act onlycontraction.fmax is a maximum muscle force vector.W ∈ <nu×nu is a weighting factor matrixand defined as following equation.

W = diag.1/kρig, (30)

ρi is physiological cross sectional area (PCSA) of musclei and determined refereed to Ref.[20].k is a coefficient which represents muscle force per unit PCSA and set as 5.5 [21]. g is grav-itational acceleration. An et al [22] showed that the muscle force obtained from the objectivefunction using Eq.(30) is consistent with the muscle activity distribution obtained by measur-ing electromyography (EMG). In this study, the above equation is solved by using sequentialquadratic programming [23].

9


d X rK r

d X eH

H T H T

H J r

J rT

K e

TJ e m

K c

J e md X t rK t r

d X cJ e h

J e hT F e F t r F c F r

K h

h

d q h

K h- 1


d q m

K m

m

K m- 1

I n t e r a c t i o n

H u m a n g e n e r a l i z e d f r a m e h

H u m a n c o n t a c tp o i n t f r a m e e

C o n t a c t t r a n s m i s s i o n v i r t u a l f r a m e t r

O b j e c t c o n t a c tp o i n t f r a m e c

O b j e c t g e n e r a l i z e d f r a m e m

R e f e r e n c e p o i n t f r a m e r

J u

J uT

K u

u

d l

K u- 1

H u m a n m u s c l ef r a m e h

K e- 1 K t r

- 1 K c- 1 K r

- 1

Figure 7:Force/displacement relationships of a human-machine system.

4 ANALYZING EQUIVALENT IMPEDANCE CHARACTERISTICS

In this section, we describe a method for obtaining the muscle stiffnessKu ∈ <nu×nu andmuscle viscosityBu ∈ <nu×nu from the muscle forcefu. Then the human muscle impedance istransformed into reference pointsr, according to both coordinate transformation of tensor andan orthogonal complementary projection technique [17].

4.1 Muscle Stiffness and Viscosity

By using Eq.(1), activity level of the musclea ∈ <nu is calculated as

a =fu

fmaxfLfV cosα. (31)

The muscle stiffnesskm and tendon stiffnesskt are calculated by partial differentiation of Eq.(1)and (2) with respect to the muscle length as follows;

km = afmax∂fL

∂lmfV cosα, (32)

kt = fmax∂fSE

∂lt. (33)

Further, the muscle-tendon complex stiffnesskuiis expressed by:

ku =kmkt

km + kt

, (34)

and the stiffness of the parallel elastic elementkp is expressed by:

kp = fmax∂fPE

∂lucosα. (35)

Making upku ∈ <nu , kt ∈ <nu andkp ∈ <nu from ku, kt andkp of each muscle, The musclestiffness matrices becomes

Ku = diag.ku, (36)

Kut = diag.kt, (37)

Kup = diag.kp. (38)

10


Ku ∈ <nu×nu expresses the stiffness of the muscle-tendon complex,Kut ∈ <nu×nu is thestiffness of the tendon, andKup ∈ <nu×nu is the passive stiffness of the parallel elastic element.In the same manner as described above, the muscle viscosity matrices are obtained by partiallydifferentiating Eq.(1) with respect to the muscle velocity.

4.2 Stiffness Transformation

Fig.7 illustrates relationships between force and displacement of a human-machine system.According to tensor coordinate transformation, the muscle stiffnessKu andKup in the muscleframeΣu are transformed into the human stiffnessKh ∈ <nh×nh expressed inΣh

Kh = JuT KuJu + Ju

T KupJu. (39)

Here, by usingPh ∈ <nh×nh, orthogonal complementary projection ofGh, PhGhT = 0, the

constrained stiffnessK ′h ∈ <nh×nh and the constrained JacobianJe ∈ <ne×nh can be obtained

as follows:

K ′h = Kh + PhKh − (PhKh)

T , (40)

Jeh= Jeh

Ph. (41)

Furthermore, considering contact constraint matrixH [13, 14] and the internal force effect ofthe contact forceFc, the constrained equivalent stiffness of the human inΣm, hK ′

m ∈ <nm×nm,becomes

hK ′m = hK ′

mj +hK ′mf , (42)

hK ′mj = JT

emHT (H K ′

e−1HT )−1HJem , (43)

K ′e = (Jeh

K ′h−1J

Teh

)−1, (44)

hK ′mf =

∂JTem

Fc

∂qm

, (45)

wherehK ′mj ∈ <nm×nm is the equivalent stiffness transmitted from the muscle stiffnessKu, and

hK ′mf ∈ <nm×nm the stiffness due to the internal force effect byFc.Now, by adding the object stiffnessKm ∈ <nm×nm in Σm, and considering the constraint on

the objectΦm in the same manner as be done with Eq. (40) and (41), we have

hmK ′′m = hmK ′

m + PmhmK ′

m − (PmhmK ′

m)T , (46)hmK ′

m = hK ′m + Km, (47)

Jr = JrPm, (48)

wherePm ∈ <nm×nm is the orthogonal complementary projection ofGm, PmGmT = 0, Jr ∈

<nr×nm is the constrained Jacobian matrix on the referencer of the object.Therefore, the equivalent stiffness of the human-machine systemhmK ′′

m ∈ <nr×nr taking theconstraints on both the human and the object into consideration can be expressed by

hmK ′′r = (Jr

hmK ′′m−1JT

r )−1. (49)

Meanwhile the equivalent viscosityhmB′′m ∈ <nr×nr can be obtained by defining the muscle

viscosityBu ∈ <nu×nu instead of the muscle stiffnessKu, and performing similar transforma-tion without the second term of Eq.(42).

11


Proof of Eq. (40)

If the generalized forceQ can be expressed by stiffnessK ∈ <n×n and infinitesimal dis-placementδq ∈ <n as follows:

Q = Kδq, (50)

The equation of motion for a multibody system that is constrained by the environment is de-scribed as

Mq + g + h + GT λ = Kδq. (51)

Considering the static condition,q = q = 0. Then Eq.(51) becomes

g + GT λ = Kδq. (52)

While the constraint can be expressed by the Jacobian matrixG and the infinitesimal dis-placementδq as follows:

Gδq = −c, (53)

wherec = ∂Φ/∂t ∈ <ch+cm. Multiplying both sides of Eq.(52) by the matrixP ,

Pg = PKδq (54)

is obtained. Eq.(53) can be expressed using the pseudo inverse matrix ofG, G+ [17];

(I − P )δq = −G+c. (55)

Premultiplying both sides of Eq. (55) by K, adding Eq.(54) to the result, and rearranging theequation and taking into considerationP T = P , KT = K,

Pg −KG+c =K + PK − (PK)T

δq (56)

is obtained.The second term on the left hand side of Eq.(56) becomes0 if the displacement constraint is

not included; therefore, Eq.(56) can be expressed by

Pg =K + PK − (PK)T

δq. (57)

Comparing Eq.(57) with Eq.(50), the right hand side of Eq.(57) satisfies the constraint and isequal to elastic force due to the infinitesimal displacementsδq. Therefore,K + PK − (PK)T

can be regard as the stiffness taking the constraintΦ into consideration.

4.3 Inertia Transformation [ 24]

Fig.8 illustrates relationships between force and acceleration in a human-machine system. Inthe same manner as in the stiffness transformation, the constrained inertia matrixM ′

h ∈ <nh×nh

is

M ′h = Mh + PhMh − (PhMh)

T . (58)

The detail of derivation is in Ref.[17]. Then the equivalent inertiahM ′m ∈ <nm×nm in Σm can

be obtained as

hM ′m = JT

emHT (H hM ′

e−1HT )−1HJem , (59)

hM ′e = (Jeh

M ′h−1J

Teh

)−1. (60)

12


X rM r

M r- 1

. .X eH

H T H T

H J r

J rT

M e

M e - 1M c

M c - 1

X t r

M t r- 1

M t r

X c. . . . . .

F e F t r F c F r

M h

M h - 1

h

q h. .


q m

M mM m

- 1

. .

m

I n t e r a c t i o n

H u m a n g e n e r a l i z e d f r a m e h

H u m a n c o n t a c tp o i n t f r a m e e

C o n t a c t t r a n s m i s s i o n v i r t u a l f r a m e t r

O b j e c t c o n t a c tp o i n t f r a m e c

O b j e c t g e n e r a l i z e d f r a m e m

R e f e r e n c e p o i n t f r a m e r

J e m

TJ e mJ e hT

J e h

Figure 8:Force/acceleration relationships of a human-machine system.

By using orthogonal complementary projection ofGm, we have

hmM ′′m = hm M ′

m + hmM′m, (61)

hmM′m = Pm

hmM ′m − (Pm

hmM ′m)T , (62)

hmM ′m = hM ′

m + Mm. (63)

Therefore, the equivalent inertia of the human-machine systemhmM ′′r ∈ <nr×nr taking the

constraints on both the human and the object into consideration can be expressed by

hmM ′′r = (Jr

hmM ′′m−1Jr

T)−1. (64)

The above gives the detailed method for calculating the equivalent impedance characteristicsof a human-machine system under constrained environment. In the next section, using thismethod, we configure a three dimensional model for a driver-steering system and analyze theequivalent impedance characteristics for a steering maneuver.

5 APPLICATION TO UPPER EXTREMITIES MOTION

In order to demonstrate the effectiveness of the proposed analysis described above, a forklifttruck steering operation were measured measured and analyzed using a musculoskeletal modelof upper extremities. The subject was a male, skilled in operating industrial vehicles, 36 yearsold, 1.83 m tall, and weighing 80.0 kg.

5.1 Experimental Setups

Fig. 9 depicts the experimental setup consisting of an driving position adjustable mockup,a steering column with a DC motor, a steering control computer, a steering unit for measuringoperation force of the subject, a liquid crystal display monitor, an electromyographies monitor,and a motion capture system. Longitudinal and vertical position of the steering column andthe seat, inclination angle of the steering unit can be changed to correspond to various type ofvehicles. In this study, the layout parameters listed in the Fig.9 were set.

As shown in Fig.10, a round bar with a counter weight was attached to the head of thecolumn; the operation force that the subject exerts on it was measured by a force/torque trans-ducer(ATI NANO25, load rating:125N/3Nm). A bearing was fitted between the knob and the

13


M o t o r

M o t i o n c a p t u r e

O p e r a t i o n f o r c e

R e a c t i o n t o r q u e

S t e e r i n g u n i tD i s p l a y

y0x

S t e e r i n g a n g l eS t e e r i n g t o r q u e

U p p e r b o d y p o s t u r e

C o m p u t e r sH i p p o i n t( - 0 . 3 4 0 , 0 . 4 5 1 )

( 0 . 0 9 9 , 0 . 6 9 6 )I n c l i n a t i o n a n g l e = 0 . 9 7 8

Figure 9:Experimental setups.

F o r c e / t o r q u e t r a n s d u c e r

yx

z

r a d i u s = 0 . 1 5 3 m

s t e e r i n g a n g l e

S t e e r i n g k n o b

C o u n t e r w e i g h t

Figure 10:Steering unit.

C o u r s e

C u r s o r

Figure 11:Example of display.

supporting part, so that the torque around thez axis in Fig.10 was not transmitted. HenceH inEq. (43) and (59) can be expressed by:

H =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

. (65)

As a subject operate the steering unit, the reaction torque was given to the subject gener-ated in DC motor (rated load torque: 2.25Nm, gear ratio: 14.67). A rotary encoder (Danaher,5000pulse/rotation) was attached to the motor, and a torque sensor (Kistler***) was placed atthe head of the column. In the steering control unit, the reaction torque was kept at the value of1.2 Nm by using angular velocity and torque feedback.

Fig.11 shows an illustration of the task. A target course was continually shown from theupper part of the display. The cursor in the display was moved from side to side depending onthe angle of the steering unit.

14


Figure 12:A musculoskeletal model of upper extremities.

SL vmax Af Bf ftoe εtoe kse klin kpe ε0

0.45 10lm0 0.25 1.4 0.33 0.0241 3.0 42.8 5.00 0.60

Table 1:Physiological parameters for the muscle model [9].

5.2 Musculoskeletal Model of Upper Extremities

A musculoskeletal model of the upper extremities was scaled to the subject as shown inFig.12. The model consists of 29 muscles and 9 rigid bodies including the upper arms, forearms and hands. The origins and insertions, optimal length, maximum force of the muscleswere refereed to such as Ref.[20] and so on. The muscle moment arm characteristics werealso adjusted according to published literature such as [25] and so on. Furthermore, the massand moment of inertia of each rigid body were set after referring to Ref.[26]. The physiolog-ical parameters for each muscle are shown in Table1. The pennation angleα is treated as 0,precondition factorβ asπ/18 for all muscles.

5.3 Experimental Method

The subject sat on the driving position-adjustable bench, and gripped the steering knob withhis left hand. The subject was instructed to follow the course by turning the steering wheelcounterclockwise six times at a speed of 2π rad/sec, stopping it at the 9 o’clock position, holdingit there for about one second, turning it clockwise six times, and then stopping it again atthe 9 o’clock position. The posture of the upper body while operating was measured usingmotion capture (Motionanalysis EAGLE). In this study, 30 markers was attached and the threedimensional positions of them were measured using 10 cameras. the operating force of the handwas measured using the force/torque transducer, and measured values were used as inputs forthe following analysis. Electromyograph of four different muscles around the joint of the leftshoulder were also measured for verification.

15


- 1 001 0

Fx (N)

1 0

0

Fy (N)

0 1 2 3 4 5 - 2 0- 1 00

Fz (N)

t i m e ( s e c )Figure 13:Measured operation force during clockwise rotation.

5.4 Equivalent impedance of the Hand

Fig.13 shows the measured operation force for clockwise rotation. The tangential force,Fy,required to rotate the steering wheel was held constant at around 8.0N. On the other hand, thepatterns of the normal forceFx and the pressing forceFz vary depending on the position of theknob. The operation force for clockwise rotation and the operational posture data were analyzedusing the musculoskeletal model.

Fig.14 shows the estimated muscle force of the anterior and posterior of the deltoideus overfour times of clockwise turning. Rectified and filtered EMG data are also shown in the Fig.14.The horizontal axis is normalized based on the knob position such that the 9 o’clock positionis 0. Fig.15 (a) shows the equivalent inertia ellipsoids of the hand at four different positions.Contraction patterns of both muscles are in good agreement with measurement results. The size,shape and direction of the inertia ellipsoid closely approximate the results of hand impedanceestimated when the posture is maintained [27]. Considering these two results, it is thought thatboth the estimated muscle force and the equivalent impedance calculated by this analysis areappropriate.

On the other hand, Fig.15(b) shows a stiffness ellipsoid drawn by hand stiffnesshK ′mj based

on the stiffnessKu of muscles in motion. The size of the ellipsoid is considerably smaller thanthat estimated by Tsuji et al.[27]. This is thought to be attributed to the equivalent stiffnessidentified by Tsuji et al. containing the effects of reflection, whereas, in this method, the equiv-alent stiffness was calculated without taking into account the effects of reflection, skin, tissues,etc.

5.5 Equivalent Impedance around the Steering Column

The equivalent inertiahmM ′′r ∈ <1 and the equivalent stiffnesshmK ′′

r ∈ <1 around the steer-ing column axis was calculated. Thinking of the previous result on the equivalent stiffness,hmK ′′

r was also decomposed of the equivalent stiffness due to the musclehmK ′′rj ∈ <1 the equiv-

alent stiffness due to the internal force effecthmK ′′rf ∈ <1. In the calculation, the inertia and

16


00 . 51 . 0

Norma

lized m

uscle f

orce (-

) M u s c l e f o r c eD e l t o i d a n t e r i o r

IEMG (

-)

0

00 . 10 . 20 . 3

0

K n o b p o s i t i o n ( r a d )

I E M G

D e l t o i d p o s t e r i o r

0

Figure 14:Comparison of muscle force with IDEM during clockwise rotation.

12

34

1 0 . 0 N / m

0 . 1 0 m

( b ) S t i f f n e s s1 . 0 k g0 . 1 0 m

12

34

( a ) I n e r t i aFigure 15:Simulated stiffness and inertia ellipsoids at different postures during clockwise rotation.

stiffness of the steering unit was assumed to be 0, being necessary to clarify the effects of thehuman. The results are plotted in Fig.16, lateral axis is normalized same as in Fig.14.

As for the equivalent inertiahmM ′′r , there is a region where it contributes greatly to the steer-

ing column axis, while there is another region where it hardly contributes at all. Comparingwith Fig.15(a), it is found that the equivalent inertiahmM ′′

r increases in a region where a longeraxis of the hand inertia ellipsoid approaches the tangential direction of the steering, while itdecreases in a region where a shorter axis of the hand inertia ellipsoid approaches the tangentialdirection of the steering.

It should be noted that as the equivalent stiffnesshmK ′′r is affected by internal force, it takes

on a negative value when the knob is at a position where the hand is almost fully extended.This is because the subject turns the steering wheel at an angular speed of 2 rad/sec, causing theoperator to exert an operation force inwards in the direction of a normal line. According to theresults of a crank turning experiment conducted by Ota, et al.[28], it is pointed out that the handforce works inwards in the direction of a normal line. Equivalent stiffness may attain a positivevalue in the actual steering operation. However, the steering operation of this human-machine

17


0

0 . 1

0

1

2

- 20

2

K n o b p o s i t i o n ( r a d )0

2 ] [kg

.mhmM r''

] [N

m/rad

hmK rj''

] [N

m/rad

hmK r''

Figure 16:Equivalent inertia and stiffness around steering column axis during clockwise rotation.

system may be made more stable if the effects of internal force and those of muscle stiffnessare properly combined. Also, if a method of allowing equivalent stiffness to become positive inall regions can be developed by controlling the operation system configuration.

6 CONCLUSIONS

In this paper, we propose a method for analyzing equivalent impedance characteristics ofhuman-machine systems considering not only contact between the human and the object butalso constraint on the human and on the object. The proposed analysis was applied to the upperextremities motion in forklift steering operation. The results reveal that this analysis is an usefultool in examining the physical significance of human-machine systems and can be effective indesigning a human-machine system.

The proposed method can easily apply to other type of motions, and objective functions inthe muscle force estimation. However, we think that the musculoskeletal model will need fur-ther improvements in accuracy. More physiological experimental data and moment arm char-acteristics in complex joint movements will be required in order to define accurate parametersof the muscle model and muscle path. More studies are needed concerning validation of theimpedance characteristics during steering and other behaviors. Furthermore, a new solutionwould be necessary if one taking effects of muscle activation dynamics and reflective feedbackinto consideration.

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AN ANALYSIS OF EQUIVALENT IMPEDANCE ...bsys.hiroshima-u.ac.jp/pub/pdf/C/C_166.pdfMULTIBODY DYNAMICS 2007, ECCOMAS Thematic Conference C.L. Bottasso, P. Masarati, L. Trainelli (eds.)

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