Adaptive Attitude Control of Redundant Time-Varying Complex Model of Human Body in The Nursing Activity

Dong, H., Luo, Z., and Nagano, A.

Paper:

Adaptive Attitude Control of Redundant Time-Varying ComplexModel of Human Body in The Nursing Activity

Haiwei Dong, Zhiwei Luo, and Akinori NaganoDepartment of Computational Science, Graduate School of System Informatics, Kobe University

1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, JapanE-mail: [email protected]

[Received December 14, 2009; accepted April 13, 2010]

With the development of human society, there aremore and more elderly people need to be taken care of.However, there is not enough labor force to take thenursing jobs. Nowadays robots play more and moreimportant roles in our daily life, especially in nursingactivities. In this paper, we illustrate a new attitudecontrol approach to lift human regardless of the indi-vidual differences, such as height, weight, and so on.In detail, considering our daily experience that onlyvery few joints are critical for accomplishing the liftingup task, we treats the human body as a redundant sys-tem. We use robust adaptive control to eliminate theeffects from the “uninterested joints” and identify thehuman parameters in real time. In addition, the con-vergence analysis, including tracking time and trackerror, is also given. The approach is simulated by lift-ing a human skeleton with two robot arms, which ver-ifies the efficiency and effectiveness of our strategy.

Keywords: human body, model reduction, adaptive atti-tude control, human parameters identification

1. Introduction

As more and more countries step into the aging soci-ety, much attention is drawn to the aging of populationthan ever before. The fact is that increasing number ofelderly people need to be looked after; however, there isnot enough labor force to take the nursing jobs. How todeal with the aging problem has drawn attention of allthe countries in the world. Apart from many social so-lutions, as the advanced robotics technology is becomingmature, we believe that the mentioned aging problem canbe partially solved by using robots in the nursing activi-ties, which inspires our research [1].

Specifically, this paper focuses on how to carry up thehuman body to a desired position and posture, which canbe utilized in the nursing activities in hospitals. However,few efforts have been made in the literatures to changethe attitude of human body [2, 3]. From a theoreticalviewpoint, the most relevant approach is whole arms co-operation control. However, many preceding researchesconsider simple objects with small Degrees of Freedom

(DOF) whereas the human body is much more complexwith considerable DOF [4–6]. From a practical stand-point, the most successful work on lifting human is the de-velopment of a robot named RI-MAN which was selectedas one of the best inventions by the TIME Magazine in2006 [7–9]. RI-MAN has many perceptive functions suchas sight, smell, hearing, touch, etc. Our research is basedon some of the dynamics simulations of RI-MAN.

It has been thought that the human body has about 206bones and numerous joints connecting adjacent bones.Based on the physiological structure of human joints, thejoints can be mainly divided into five types as hinge (1-DOF), pivot (1 DOF), saddle (2 DOF), gliding (2 DOF),ball socket (3 DOF). While in dynamic equations, eachDOF is expressed as one differential equation. Hence, it ispredicted that the overall set of equations of human bodydynamics is very complex to handle. Hence, the difficultyarises to compute such a big dynamic model.

Actually, the human body can be considered as a free-floating multi-link rigid object with passive moments.The objective is to change the attitude of the mentionedobject by external forces. Hence, two difficulties comeout: the first one is about free-floating multi-link rigidobject. The previous studies on free-floating object aremainly in controlling spacecraft. In the literatures, thespacecrafts were accurately modeled. Based on the pre-cise models, various methods, e.g., generalized Jacobianmethods, were proposed to complete the attitude controltask [10, 11]. However, in our case, the model of humanbody cannot be modeled accurately. That is not only be-cause there are some human parameters which can not bemeasured, but also because human bodies have individ-ual differences. The second one is about external forces.As the human body is such a complex model with veryhigh dimension, application of external forces on the hu-man body is also very complicated. The calculation needsconsiderable time and real-time performance is impossi-ble. In addition, the process of lifting human must be ab-solutely safe. If we cannot make sure that the computationis done in real-time, the safety cannot be guaranteed.

In consideration of the two difficulties above, the ba-sic idea for solution comes from our daily experience.When human lift a person we do not care about the an-gle of ankle, the position of hands and so on. What wedo have to care about are the position of the head, the ver-

418 Journal of Robotics and Mechatronics Vol.22 No.4, 2010

Adaptive Attitude Control of Time-Varying Model of Human Body

tical deflection of upper limb and the angle of hip. Herewe call them “interested states.” From the viewpoint ofsystem theorem, we treat the human body as a large re-dundant system whose dimension is reduced by divertingthe effects of other “uninterested joints” to the ones of“interested joints.” The newly constructed reduced humanmodel has very few DOF while having unmodeled uncer-tainty, unfortunately. In order to eliminate the unmod-eled uncertainty, human attitude controller is designed.Furthermore, the human parameter estimator is also de-signed.Thus, the whole attitude control approach for lift-ing up human body is capable to overcome the individualdifferences, such as height, weight, and so on.

This paper is organized as follows. The second sec-tion models the lifting problem and reduces the humanmodel. The third section illustrates the detailed derivationof the controller and the estimator. The fourth sectionanalyzes the convergence of the proposed approach, in-cluding tracking error and tracking time. The fifth sectiontakes a normal human body for example in simulation totest the effectiveness of the proposed approach. The sixthsection concludes the whole paper.

2. Adaptive Attitude Control of Human Body

2.1. Human Model Simplification

If we consider human body as a rigid multi-link object,each bone of the human body corresponds to a link andeach joint of human body corresponds to a human jointconnecting adjacent links. Moreover, the joints of humanbody has passive torques corresponding to the constric-tion forces and moments developed by ligaments, jointcapsules and other soft tissues. Hence, we write the dy-namics of human body model in the form

H(q)q+C(q, q)q+G(q) = τpass, . . . . . (1)

where qn×1 is generalized states of human body, whichinclude the position of head and the angles of all thejoints. H(q)n×n is inertia matrix. It is a symmetric andpositive semi-definite matrix, which contains informa-tion with regard to the instantaneous mass distributionof the human body. C(q, q)n×n is centripetal and cori-olis torques. The terms of C(q, q) contain products ofangular speeds. When the degrees of freedom are rota-tional, the terms of C(q, q) represent the moments of cen-trifugal forces. G(q)n×1 is gravitational torques. BecauseG(q) changes as the posture configuration of the humanbody model, the terms of it are functions of the general-ized states. τpass n×1 is passive joint torques. It containsthe torques and moments arising from muscular activa-tions and passive elastic structures surrounding the humanjoints. τrob n×1 is the torques exerted by the robot armswhich is controllable.

The task assigned to robot is to lift up the human body,and control the position and posture of the body to thedesired states. In this process, the human body is exertedby external forces. Hence, the dynamics of human model

Fig. 1. RI-MAN lifts up human skeleton. The robot RI-MAN was designed for taking care of elderly in the nursingactivity. Hence, he has many sensors for perception, includ-ing sight, smell, hearing, and touch.

can be modified into the following form

H(q)q+C(q, q)q+G(q) = τpass + τrob, . . . (2)

where τrob n×1 is the torques exerted by the robot arms,which is controllable.

Actually, the robot considered in this paper is RI-MANwhich has two manipulation arms (Fig. 1). In this case,the location of force application is the back and the kneeof human body, which is shown as F1 and F2. The forcesF1 and F2 have relation with τrob as

τrob = JT1 F1 + JT

2 F2, . . . . . . . . . . . (3)

where J1 and J2 are Jacobian matrix of the human model.After introducing the external forces in the human dy-

namics, the case we have to deal with becomes attitudecontrol of a free-floating multi-link rigid object. The ba-sic idea of our approach is to reduce the human model intoa small one with less DOF, which includes the followingthree steps:

(a) Choose “interested states.” These states include thefundamental performance indexes of the lifting task. Inother words, based on these states, we can easily checkwhether the task is completed or not. Let us define

q1 = [q1, . . . ,qm]Tm×1 . . . . . . . . . . (4)

as the “interested states” and

q2 = [qm+1, . . . ,qn]T(n−m)×1 . . . . . . . . (5)

as the “uninterested states” consisting of the other states.We can easily obtain that q = [q1 q2]

T .

(b) Arrange the dynamic equation set. Based on the divi-sion of “interested states” in (a), we change the positionsof elements of H, C, G, τpass and τrob extract the dynam-ics of the “interested states”. Then we divide the dynamicequation set into two parts where H, C, G, τpass and τrobare written into the form of block matrix (or block vector).

Journal of Robotics and Mechatronics Vol.22 No.4, 2010 419

owner



The expanded form of new dynamics equation set is[H11 H12H21 H22

][¨q1¨q2

]+

[C11 C12C21 C22

][˙q1˙q2

]+

[G1G2

]

=

[τpass,1τpass,2

]+

[τrob,1τrob,2

], . . (6)

where the dimensions of sub block matrices of H11, H12,H21, H22 are m×m, m× (n−m), (n−m)×m, (n−m)×(n−m), respectively. And the dimensions of sub blockmatrices of C11, C12, C21, C22 are m ×m, m × (n−m),(n−m)×m, (n−m)× (n−m), respectively. The dimen-sions of vectors G1, τpass,1, τrob,1 are m × 1, and G2,τpass,2, τrob,2 are (n−m)×1.(c) Construct the reduced human dynamics. First of all,we define the generalized states of the reduced humanmodel as

qs = q1. . . . . . . . . . . . . . . (7)

Extracting the parts of the human dynamics which we areinterested in, we get

[H11 H12

][ ¨q1¨q2

]+

[C11 C12

][ ˙q1˙q2

]+ G1

= τpass,1 + τrob,1. . . . (8)

Considering that the dynamic model is time-varying, afterarranging Eq. (8), we obtain

H11(t) ¨q1 + C11(t) ˙q1

+(G1(t)+ H12(t) ¨q2 +C12(t) ˙q2 − τpass,1(t)

)= τrob,1(t). . . . . . . . . . . . (9)

By defining the inertia matrix, centripetal matrix, gravi-tational matrix and torque vector of the reduced humanmodel as

Hs(t) = H11(t)Cs(t) = C11(t)Gs(t) = G1(t)+ H12(t) ¨q2 +C12(t) ˙q2 − τpass,1(t)τs(t) = τrob,1(t)

(10)

we obtain the general mechanical form of the reduced hu-man model

Hs(t)qs +Cs(t)qs+Gs(t) = τs(t) . . . . . (11)

where the subscript s denotes the small system. Actu-ally, our basic idea is to consider the influences from the“uninterested human joints” (in this case from state q2) asperturbations and then change the attitude of human adap-tively by estimating the human parameters of Hs, Cs andGs in real time. The detailed estimation meanings are

• Estimating Hs and Cs – make the system adap-tively adjusts itself to various people with differentweights;

• Estimating Gs – eliminate the perturbations from the“uninterested joints.”

Considering the basic idea above, the approach to be pro-posed in this paper should be able to identify and controlthe dynamics of the reduced human model at the same

time. Assuming that the human model is totally unknownin advance, for the safety in the nursing activity, the iden-tification process needs to be performed in real-time. Onthe other hand, the weights and heights etc. of the humanbodies are different between individuals. Hence, the strat-egy also has to be able to tolerate these individual differ-ences.

2.2. Human Attitude Control and Parameter Iden-tification

First of all, we assume that we do not have any pri-ori knowledge before lifting human, i.e., the initial valueof Hs, Cs, Gs are set as zero matrices (or zero vectors).The benefit of such assumption is that the proposed trajec-tory is much more robust and can be adaptive to variouspeople with different heights and weights. Whereas, theshortcoming is also obvious, i.e., generating unmodeleddynamics. To overcome the above disadvantage, we userobust controller to change the human attitude. Moreover,online human parameter identification is also done so asto estimate the human body in real time.

For the convenience of mathematical derivation, we de-fine the actual human parameter vector

P =[PT

H PTC PT

G]T

, . . . . . . . . . . (12)

where

PH = [Hs,11 Hs,12 . . . Hs,1n . . . Hs,n1 Hs,n2 Hs,nn]T

PC = [Cs,11 Cs,12 . . . Cs,1n . . . Cs,n1 Cs,n2 Cs,nn]T

PG = [Gs,1 Gs,2 . . . Gs,n]T

. (13)

and estimated human parameter vector as

P =[PT

H PTC PT

G]T

, . . . . . . . . . . (14)

where

PH =[Hs,11 Hs,12 . . . Hs,1n . . . Hs,n1 Hs,n2 Hs,nn

]T

PC =[Cs,11 Cs,12 . . . Cs,1n . . . Cs,n1 Cs,n2 Cs,nn

]T

PG =[Gs,1 Gs,2 . . . Gs,n

]T(15)

then the estimation error matrix can be defined as

P = P−P. . . . . . . . . . . . . . (16)

In fact, not any combination of H, C and G corresponds toa physical system. Therefore, the first step is to prove thatthe reduced human model represents a physical system. Itis easy to prove that by verifying that Hs −2Cs is a skew-symmetric matrix, i.e., the reduced human model satisfiesconservation of energy (the detailed derivation is in theproof of Theorem 1).

We proposed a theorem for changing the “interestedstates” of the large complex human body as in Theorem 1.Theorem 1 is composed of a human attitude control lawand a human parameter update law. In fact, the two pro-cesses of control and identification run at the same time.In the proof of Theorem 1, the global stability is shownby proving that the derivative of Lyapunov function can-didate is less than zero.



Theorem 1Consider a time-varying system

Hs(t)qs +Cs(t)qs+Gs(t) = τs(t) . . . . . (17)

without any pre-knowledge about Hs, Cs and Gs. The vec-tor qs,d means the desired states. Define a sliding term sas

s = ˙qs +Λqs = (qs−qs,d)′+Λ(qs −qs,d) . . (18)

where Λ is a positive diagonal matrix. From the concep-tual view of velocity, we define the reference velocity qs,rand reference acceleration qs,r as

qs,r = qs− sqs,r = qs− s . . . . . . . . . . . . . (19)

If we choose the human attitude control law

τs = Hs(t)qs,r +Cs(t)qs,r + Gs(t)− k · sgn(s) . (20)

and human parameter update law

˙P=−Γ−1 [s1qTs,r . . . snqT

s,r s1qTs,r . . . snqT

s,r s1 . . . sn](21)

under the assumption of∥∥k · sT sgn(s)∥∥>

∥∥PT ΓP∥∥ . . . . . . . . (22)

where k and Γ are positive diagonal matrices, sgn(·) is asignal function, then the whole system tracks the desiredtrajectory and the parameter matrices Hs, Cs and Gs con-verge to actual values globally.

Proof:Define a Lyapunov function candidate

V (t) =12

sT Hss+14

PT (2Γ+ I) P . . . . . (23)

where Γ is a positive diagonal matrix. The first part ofV (t) can be written as(

12

sT Hss)′

= sT (Hsqs −Hsqs,r)+12

sT Hss. (24)

From Eq. (17), Hsqs = τs −Csqs −Gs, then( 12 sT Hss

)′= sT (τs −Cs(s+ qs,r)−Gs −Hsqs,r)+

12 sT Hss

= sT (τs −Hsqs,r −Csqs,r −Gs)+12 sT (Hs −2Cs)s

(25)

According to the previous research on mechanical system,the system in the form of Eq. (2) satisfies

qT (H −2C)

q = 0 . . . . . . . . . . . (26)

i.e., H − 2C is a skew-symmetric matrix. Hence, the fol-lowing relation satisfies

Hi j −2Ci j =

{0 if i = j−(

H ji −2Cji)

otherwise . (27)

Without loss of generality, we choose q1 as the new statevector which we are “interested in” (Eq. (4)) and followthe same system reduction procedures in Eqs. (4)-(11).According to the relation in Eq. (27), the reduced system

satisfies˙H11(iu,iv)−2C11(iu,iv)

=

{0 if iu = iv−(

˙H11(iv,iu)−2C11(iv,iu)

)otherwise (28)

where H11 and C11 are defined in Eq. (6). Hence, ˙H11 −2C11 is a skew-symmetric matrix. Based on the definitionsof Hs and Cs in Eq. (10), Hs − 2Cs is a skew-symmetricmatrix, hence(

12

sT Hss)′

= sT (τs −Hsqs,r −Csqs,r −Gs) . (29)

Therefore, V (t) can be simplified as

V(t) =(

12

sT Hss)′

+

(12

PT ΓP)′

= sT (τs −Hsqs,r −Csqs,r −Gs)+( ˙P− P)T ΓP (30)

Applying the human attitude control law Eq. (20)

τs=Hs(t)qs,r +Cs(t)qs,r + Gs(t)− ksgn(s)

=

qTs,r 0 · · · 0 qT

s,r 0 · · · 0 1 0 · · · 00 qT

s,r · · · 0 0 qTs,r · · · 0 0 1 · · · 0

.... . .

......

. . ....

.... . .

...0 0 · · · qT

s,r 0 0 · · · qTs,r 0 0 · · · 1

m×2m2

× PH

PCPG

2m2×1

− k

sgn(s1)sgn(s2)...sgn(sm)

m×1

. . . . (31)

into V (t), which leads to

V (t) = sT (Hs(t)qs,r +Cs(t)qs,r + Gs(t)−Hs(t)qs,r

−Cs(t)qs,r −Gs(t))− k · sT sgn(s)+ PTΓ( ˙P− P)

= sT (Hs(t)qs,r +Cs(t)qs,r + Gs(t))

−k · sT sgn(s)+ PTΓ( ˙P− P)

= [s1qTs,r . . . snqT

s,r]PH +[s1qTs,r . . . snqT

s,r]PC

+[s1 . . . sn]PG − sT sgn(s)+ PTΓ( ˙P− P) (32)

where

qs,r =[qs

r,1 qsr,2 . . . qs

r,m

]T

m×1

qs,r =[qs

r,1 qsr,2 . . . qs

r,m

]T

m×1

We obtain

V (t) = PT1×2m2

×[s1qT

s,r . . . snqTs,r s1qT

s,r . . . snqTs,r s1 . . . sn

]T2m2×1

−k · sT sgn(s)+ PTΓ( ˙P− P). . . . . . (33)

Taking the human parameter update law of Eq. (21),finally we obtain

V (t) =−PT ΓP− k · sT sgn(s). . . . . . . (34)



Fig. 2. Scheme of the proposed attitude control. The input signals are the desired trajectories of the “interested states” of the humanmodel. The output is the actual motions of the human model. The proposed approach both controls the position and posture of thehuman model and identifies the human parameters online at the same time.

According to the assumption of Eq. (22), V (t) < 0.Hence, the tracking error and parameter estimation errorconverge to zero asymptotically.

It is noted that the signals required in Eqs. (20) and (21)are s, qs,r, qs,r. According to the definitions of s, qs,r, qs,rin Eqs. (18) and (19) , the basic signals required are qs,qs, qs. Actually, the “interested states” qs, qs representthe attitude position and acceleration of the human body.Hence, they are obvious by binocular vision technology.However, qs is hard to measure. To avoid the accelera-tion term, we use filtering technology. Specifically, letw(t) be the impulse response of a stable, proper filter. Forexample, for the first-order filter ε/(p+ ε) where ε > 0,p= d/dt, the impulse response is e−εt . Then using partialintegration, qs can be integrated as∫ t

0w(t − r)qrdr = w(t − r)qs|t0 −

∫ t

0

dwdr

qsdr

= w(0)qs−w(t)qs(0)

−∫ t

0[w(t − r)qs − w(t − r)qs]dr (35)

which means qs = f (qs,w), i.e., the acceleration signalcan be obtained from velocity signal.

To illustrate the strategy more clearly, the scheme il-lustration is shown in Fig. 2. The input of the proposedapproach is the desired attitude of human body and theoutput is the actual attitude. There are four blocks in thescheme figure as signal transform unit, attitude controller,model parameter estimator and human body model. De-tailed explanations are as follows.

• Signal transform unit. This unit has two input sig-nals, including desired attitude and actual attitude.Actually, we can get many kinds of error signals,such as position error signal, angular velocity er-ror signal and angular acceleration signal and so on.Meanwhile, the combination of the above error sig-nals can also be obtained. Specifically, in the pro-posed approach, we create the sliding signal s, ve-locity reference error signal qs,r and acceleration ref-erence error signal qs,r.

• Attitude controller. The controller is a robust con-troller which can tolerate the uncertainty of reducedhuman model. In fact, the controller has its own es-timated human model in the form of

H(q)q+C(q, q)q+ G(q) = τpass + τrob. (36)

Each time, the controller gives orders to robot armsto lift human body up according to its own estimatedhuman model.

• Model parameter estimator. Although there is an in-ternal estimated human model in the controller, theuncertainty of human model estimation has a strongimpact on the control performance. In other words,if we can get a more accurate approximation modelof human body, the performance is better. Based onthe above consideration, model parameter estimatoruses system identification technology to refine theestimated human model. In the proposed approach,the calculation of the derivative of P revises the esti-mated human model.

• Human body model. The control orders are trans-ferred to control the manipulators of the robot forexerting external forces.

3. Convergence Analysis

First of all, we assume that the position and velocitycannot “jump,” so that any desired trajectory feasible fromtime t = 0 necessarily starts with the same position andvelocity as those of plant. Let qs = qs − qs,d be the track-ing error in the variable qs, i.e.,

qs =

[qs −qs,dqs − qs,d

]. . . . . . . . . . . (37)

Furthermore, define a time-varying surface in the state-space Rn by the scalar equation s(qs; t) = 0, where

s = ˙qs+λ qs. . . . . . . . . . . . . (38)



Given initial assumption qs,d(0) = qs(0), the problem oftracking qs ≡ qs,d is equivalent to that of remaining qs onthe surface S(t) for all t > 0; indeed s ≡ 0 represents a lin-ear differential equation whose unique solution is qs ≡ 0,given initial condition qs,d(0) = qs(0). Thus, the problemof tracking the n-dimensional vector qs,d can be reducedto that of keeping the scalar quantity s at zero. More pre-cisely, the problem of tracking the n-dimensional vectorqs,d can in effect be replaced by a 1st-order stabilizationproblem in s. Actually, the stabilization process in s canbe divided into two phases. The fist phase is to make s ap-proach and finally hit the manifold B(t) which is definedas

B(t) = {qs|s(qs; t)≤ φ} . . . . . . . . . (39)

where φ > 0 denotes the boundary layer thickness. Whilethe second phase is to make s converge to zero asymptot-ically in the boundary layer (Fig. 3).3.1. Tracking Time Analysis

In the proof of Theorem 1, we define the Lyapunovfunction as

V (t) =12

sT Hss+14

PT (2Γ+ I) P =V1(t)+V2(t) (40)

After taking the control law Eq. (20) and parameter updatelaw Eq. (21), finally the derivative of V (t) can be writtenas

V (t) =−k · sT sgn(s)− PT ΓP = V1(t)+ V2(t) . (41)

Extracting parts of the elements in V1(t) asn∑

k=1s2

kHs,kk and

differentiating the ordinary element, we obtain(12

s2kHs,kk

)′

≤ −kksksgn(sk) =−kk |sk| . . (42)

Eq. (42) states that the “distance” to the surface, as mea-sured by s2

k , decrease along all system trajectories. Thus,it constrains trajectories to point towards the manifoldB(t), as illustrated in Fig. 3. In detail, let treach

k be the re-quired time of the k-th generalized coordinate qs

k to reachthe surface sk = 0. Integrating the left side of Eq. (42)between t = 0 and t = treach

k leads to

∫ treachk

0

12

ddt

s2kHs,kkdt =

12

Hskks2

k

∣∣∣∣t=treach

k

t=0

= −12

Hs,kksk(t = 0)2 . (43)

while the integration of the right side between t = 0 andt = treach

k can be written as

−∫ treach

k

0kk |sk|dt ≤ −

∫ treachk

0kk |sk(t = 0)|dt

=−kk |sk(t = 0)|treachk . (44)

Applying the inequality relation in Eq. (42), we get the ac-quired time for any generalized coordinate qs

k to get sk = 0

treachk ≤ Hs,kk

2kk|sk(t = 0)| . . . . . . . . . (45)

Fig. 3. Two phases for transferring interested state to the de-sired state. The first phase is to drive the initial state qs,d,k(0)to reach the boundary layer; the second phase is to make thestate qs,k(t) converge to the desired state qs,d,k(t f ) asymptot-ically.

Fig. 4. Position error of the “interested states” in the secondphase. The error between the desired state and actual stateof the human joint decreases to zero exponentially.

Furthermore, manifold definition of B(t) implies that onceon the surface, s(t) = 0 , i.e.,

˙qs +λ qs = 0. . . . . . . . . . . . . (46)

The solution to the Eq. (46) is

qs = e−λ t . . . . . . . . . . . . . . (47)

Plugging the definition in Eq. (37), we can obtain qs −qs,d = e−λ t . Hence, the tracking error qs − qs,d tends tozero with a time constant λ exponentially (Fig. 4). Inother words, the proposed approach is exponentially sta-ble.

3.2. Static Tracking Error AnalysisAt the second phase, bounds on s can be directly trans-

lated into bounds on the tracking error vector qs, andtherefore the scalar s represents a true measure of tracking



Fig. 5. Relation between qs,k, ˙qs,k and sk (1 ≤ k ≤ n) inLaplace space. From the knowledge of Laplace transform, itis possible to get the upper bound of joint angle by integra-tion.

performance. Indeed, by definition Eq. (38), the tackingerror qs is obtained from s through a first-order lowpassfilters (Fig. 5), where p = d/dt is the Laplace operator.

From the definition of s in Eq. (38), for the first term ofs, we obtain

s1 = ˙qs,1 +λ1qs,1 . . . . . . . . . . . . (48)

i.e.,

qs,1 = (p+λ1)s1 . . . . . . . . . . . . (49)

where p = d/dt. According to the signal processingknowledge, we have

L(e−λ1t) =

∫ ∞

0e−λ1t e−ptdt =

1p+λ

. . . . (50)

where L(·) is Laplace operator. By applying the convolu-tion theorem, we obtain

qs,1 =

∫ t

0e−λ1(t−T )s1(T )dT . . . . . . . (51)

According to the assumption qs(0) = 0 and |s(t)| < φ ,|sk(t)| < φk (1 ≤ k ≤ n) the upper bound of qs,1(t) canbe obtained

|qs,1(t)| ≤ φ1

∫ t

0e−λ1(t−T )dT =

φ1

λ1e−λ1T−λ1t

∣∣∣∣T=t

T=0

=φ1

λ1

(1− e−λ1t

)≤ φ1

λ1. . . . . (52)

The derivation is the same of qs,k, where 1 ≤ k ≤ n. In all,we obtain∣∣qs,k(t)

∣∣≤ φk

λk, (1 ≤ k ≤ n). . . . . . . (53)

For the purpose of making use of the above derivation, werewrite the lowpass filter unit as

pp+λ

= 1− λp+λ

. . . . . . . . . . . (54)

Then the upper bounds of derivatives of qs,1 can be ob-

tained as∣∣ ˙qs,1(t)∣∣ ≤ |s1(t)|

(1−

∫ t

0λ1eλ1T−λ1tdT

)

= |s1(t)|(

1+1λ1

λ1eλ1T−λ1t∣∣∣T=t

T=0

)

= φ1

(1+

λ1

λ1− e−λ1t

)≤ 2φ1. . (55)

The same derivation to the other generalized joints, weobtain∣∣ ˙qs,k(t)

∣∣≤ 2φk, 1 ≤ k ≤ n. . . . . . . . (56)

4. Simulation and Analysis

In the simulation, at first we used AUTOLEV to con-struct the human model and then exported the model as aMATLAB code. After that, we added the proposed strat-egy codes, including human attitude control and humanparameter update, into the MATLAB code. By runningthe code, we obtained all the information about the po-sitions, velocities and accelerations of the human model.The animation was done based on these data with VOR-TEX where the skeleton model was constructed by con-necting the bones composed of polygon points. It is notedthat in the simulation, we assume the robot realize perfectforce control.

4.1. Human Model ConfigurationIn the simulation, we take a normal human body into

account which is composed of 16 parts, including head,chest, mid-trunk, lower-trunk, upper arms (left and right),lower arms (left and right), hands (left and right), upperlegs (left and right), lower legs (left and right) and feet(left and right). Each two parts (or two links) are con-nected by one joint. According to the physiological struc-ture of the human body, the joints vary from 1 DOF to3 DOF. In all, the human body model we considered has35 DOF with 1.7142 m height and 72.81 kg weight. Thedetailed parameters of the human body model are shownin Table 1 [12, 13].

As we all know that the human joint can not rotatefrom 0◦ to 360◦. For example, the neck can only rotate inthe interval [−π/2 π/2] when we turn around our head.Moreover, there are passive joint moments correspondingto the constriction forces and moments developed by lig-aments, joint capsules and other soft tissues around thejoints. Based on the previous researches by Anderson etal. and Yamaguchi, we used the passive moment τpass inthe simulation as

τpass = α+eβ+(q−q+) +α−eβ−(q−q−) . . . . (57)

where q+ and q− are the threshold angle beyond whichthe passive moment takes effect. The passive moment issmall in the interval q− ≤ q ≤ q+ and it becomes largevery quickly if q > q+ or q < q− as shown in Fig. 6.

From the viewpoint of geometrics, α+ and α− denote



Table 1. Anthropological parameter values. The human body model used in the simulation is a normal body with 1.7142 m heightand 72.81 kg weight which totally has 35 DOF. Specifically, the anthropological items include length, mass center position andinertia coefficients

Segment Length (m) Mass (kg) MCS pos (m) I11 (kg.m2) I22 (kg.m2) I33 (kg.m2)Head 0.2429 5.07 0.1215 0.027 0.020 0.030Chest 0.2421 11.65 0.1226 0.174 0.148 0.070Mid-Trunk 0.2155 11.92 0.0970 0.129 0.121 0.081Lower-Trunk 0.1457 8.15 0.0891 0.065 0.060 0.053Upper Arm 0.2817 1.98 0.1626 0.013 0.004 0.011Lower Arm 0.2689 1.18 0.1230 0.007 0.001 0.006Hand 0.0862 0.45 0.0681 0.001 0.001 0.001Upper Leg 0.3960 10.34 0.1622 0.175 0.036 0.175Lower Leg 0.4300 3.16 0.1890 0.037 0.006 0.035Foot (antero posterior) 0.1788 0.90 0.0652 0.001 0.004 0.004Foot (vertical) 0.0420 — 0.0210 — — —Hip Width 0.0835

−0.5 0 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

5

q (rad)

Pas

sive

mom

ent (

kg.m

2 )

q+

q−−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−6

−4

−2

0

2

4

6

q (rad)

Pas

sive

mom

ent (

kg.m

2 )

(a) (b)

Fig. 6. Passive moment term. The parameters are set as q+ = 0.175, q− = −0.175, α+ = 0.085, α− = 0.085, β+ = 30.72,β− = −30.72. It is easy to see that the value of passive moment in the interval of [q−,q+] is very small while increasing rapidlyoutside that interval.

how sensitive the passive moment is and β+ and β− de-note what magnitude level the passive moment is. InFig. 6, it is shown that α+ and α− decide the steepness ofthe curve and β+ and β− determine the vertical extensionof the curve. In the simulation, the passive moments wereimplemented in the joints of chest-midtrunk, midtrunk-lowertrunk, lowertrunk-upperleg, upperleg-lowerleg andlowerleg-foot where the parameters of passive momentare shown in Table 2 [14].

For the other joints, there were no passive momentmodels available in the literature. However, we have tomake sure the angles of these joints are in the reasonablerange. Hence, we add passive angle joint moment

τang cont = α+eβ+(q−q+) +α−eβ−(q−q−)− γ q . (58)

into the human model where the passive term −γ q acts asa damping component.

4.2. Simulation Results

The simulation was implemented by coordination ofthree software packages, including AUTOLEV, MAT-LAB and VORTEX. The detailed cooperation relationsare explained as follows. AUTOLEV is used to constructthe dynamic model of human body for further computa-tion [15]. As MATLAB is very powerful in computing,we choose it to do the main computation as a solution toolfor ordinary differential equations; although VORTEX isable to do physical simulation, the programming grammaris a bit complex. Hence, we just use its stereoscopic pre-sentation function to make animations. More specifically,the simulation process is explained as the following threephases:

• Construct the dynamics of human body model. Weimported the human body parameters, including set-tings of link, joint and passive moment, into AU-TOLEV. For efficient computation, AUTOLEV auto-



Table 2. Passive moment parameters of human joints. The values of α+, α−, β+, β−, q+ and q− are based on the research ofhuman lower extremity model.

Joint Joint Joint Axes α+ β+ α− β− q+ q−Type

Chest-MidTrunk 3 DOF1 0 0 −0.250 20.360 0.250 −20.360 0.085 −0.0850 1 0 −0.250 20.360 0.250 −20.360 0.085 −0.0850 0 1 −0.350 30.720 0.250 −20.360 0.085 −0.085

MidTrunk-LowerTrunk 3 DOF1 0 0 −0.250 20.360 0.250 −20.360 0.085 −0.0850 1 0 −0.250 20.360 0.250 −20.360 0.085 −0.0850 0 1 −0.350 30.720 0.250 −20.360 0.085 −0.175

LowerTrunk-UpperLeg 3 DOF1 0 0 −0.030 14.940 0.030 −14.940 0.500 −0.5000 1 0 −0.030 14.940 0.030 −14.940 0.920 −0.9200 0 1 −0.244 5.050 1.510 −21.880 1.810 −0.470

UpperLeg-LowerLeg 1 DOF 0 0 1 −6.090 33.940 11.030 −11.330 0.130 −2.400LowerLeg-Foot 1 DOF 0 0 1 −2.030 38.110 0.180 −42.120 0.520 −0.740

Fig. 7. Flow diagram of simulation program. The sim-ulation uses three software packages including AUTOLEV,MATLAB and VORTEX. After adding control module andidentification module, the MATLAB source code generatedby AUTOLEV is executed. Such computation results (in-cluding how the human body model moves) are used formaking animations by VORTEX.

matically generated 6773 intermediate variables andconstructed 35 dynamic equations. After compiling,the final dynamic equations were exported as exe-cutable source code of MATLAB (or C) for furthercomputation.

• Compute the motion of human body. The MATLABsource code generated by AUTOLEV does not con-tain any attitude control strategy. Hence, we haveto add the codes of human attitude control and hu-man parameter update into the basic MATLAB code.More specifically, one execution cycle can be ex-plained as follows (Fig. 7). The function ReadUser-Input() initiates all the coefficients and parametersfor simulation, such as the mass, inertia coefficient,and initial posture of human body model. Then Ope-nOutputFilesAndWriteHeadings() makes text files

(from human.1 to human.45) for storing final motionresults. The new mdlDerivatives() with adaptive con-trol function calculates and returns the derivativesof the continuous “interested states” of the humanmodel. After that, the ordinary differential equationsrepresenting the dynamics of human body under con-trol and identification is solved by solver Ode45()and the final results are outputted to VORTEX foranimation.

• Animate the whole dynamics. At first, the poly-gons of human body bones were built by the soft-ware PRO/ENGINEER. Then the polygons of boneswere imported into VORTEX and presented in stereodisplay. After that, the bones were connected accord-ing to the physiological annexation. Finally, the mo-tion data resolved by MATLAB was used to drivethe joints. With the movement of skeleton model,the animation was presented.

In the simulation, we chose the position of head (denotedas Ph,x, Ph,y, Ph,z), the angle drift off the horizontal lineof lower-trunk (denoted as θ1,x, θ1,y, θ1,z), the angle oflower-trunk and upper-leg (denoted as θ2,x, θ2,y, θ2,z) toconstitute the “interested states” (Fig. 8), i.e.,

qs =[Ph,x,Ph,y,Ph,z,θ1,x,θ1,y,θ1,z,θ2,x,θ2,y,θ2,z

]T . (59)

The initial velocity and acceleration are set as qs(0) =qs(0) = 0 and the desired states is set as

qs,d =

[0.2,0.8,0.01,0.01,0.01,−0.7854,0.01,0.01,1.5708]T

qs,d = qs,d = [0, 0, 0, 0, 0, 0, 0, 0, 0]T .. . . . . . . . . . . . . . . . . . (60)

Applying the control law in Eq. (20) and adaptation lawin Eq. (21), we obtain

τs =[Fh,x, Fh,y, Fh,z, τ1,x, τ1,y, τ1,z, τ2,x, τ2,y, τ2,z

]T (61)



Fig. 8. Choice of the “interested states” in lifting up hu-man body. When lifting up human body, not all the motionsare essential to measure and control. Based on the proposedapproach, the “interested states” are chosen as follows: po-sition of the head, angle drift off the horizontal line of lower-trunk and angle of lower-trunk and upper-leg.

Then we choose

F1 =

Fh,x

Fh,yFh,z

F2 ≈

(τ1,zl1,z sinθ1,z + τ2,zl2,z sinθ2,z)+(τ1,yl1,y cosθ1,y + τ2,yl2,y cosθ2,y)

(τ1,zl1,z cosθ1,z + τ2,zl2,z cosθ2,z)+(τ1,xl1,x sinθ1,x + τ2,xl2,x sinθ2,x)

(τ1,xl1,x cosθ1,x + τ2,xl2,x cosθ2,x)+(τ1,yl1,y sinθ1,y + τ2,yl2,y sinθ2,y)

(62)

where [l1,x, l1,y, l1,z]T denotes the distance between the

head position and the buttock. [l2,x, l2,y, l2,z]T denotes

the distance between the buttock and the application pointof F2. By applying F1 and F2 in the human model, attitudecontrol is achieved.

The animation of lifting up human body in our ap-proach is shown in Fig. 9. At the beginning of the simu-lation, we assume that we do not have any pre-knowledgeabout the human body. Hence, the initial values of Hs, Csand Gs are set to zero matrices (or zero vectors). As theestimation of the human parameters goes on, Hs, Cs andGs converge to their true values of Hs, Cs and Gs.

The energy, position and angle changes are shown inFig. 10. It is easy to see that it takes about 1 sec to changethe attitude of human body. There is a peak of kinematicsenergy at the time of about 0.2 sec which means at thattime, the attitude changes very quickly (Fig. 10(a)). Thatis because we assume no pre-knowledge of the humanbody at the beginning of the simulation. Moreover, as theidentification of human body needs only about 0.2 sec, theidentification process is fast enough for safe nursing.

We set the desired position of the head as (0.2 m, 0.8 m,0.01 m). Compared with other joints rotating in x or y di-rection, the joints rotating in z direction turn significantly.Thus, the angle changes of these joints affect the head po-sition in x direction greater (Fig. 10(b)).

(a)

(c)

(e)

(b)

(d)

Fig. 9. Snapshots of the lifting process. The five snapshotslabeled from (a) to (e) are taken in the equivalent time in-terval, which shows the whole process of lifting up humanbody.

In the proof of Theorem 1, it was shown that Hs−2Cs isa skew-symmetric matrix which indicates that parts of thestates (or their linear combination) can be controlled asa whole. In the simulation, we constructed a new statewhich is the angle sum of head, chest, mid-trunk, andlower-trunk. The angle drift off the horizontal line of thenew state changes to −0.7854 rad (i.e., −45◦) as shown inFig. 10(c). The angle between lower-trunk and upper-legchanges to 1.5708 rad (i.e., 90◦) at the time about 1 sec.(Fig. 10(d)).

5. Conclusion

In this paper, a new reduced model adaptive force con-trol strategy for lifting up human body was proposed.Compared with previous researches, there are two signif-icant advantages in the proposed attitude control. First isthat it is not necessary to measure human body, like heightand weight, in advance because the proposed approachcan automatically identify the human parameters online.Second is that the human attitude control law guaranteesthe accuracy. Moreover, the robust controller which weused also tolerates the unmodeled uncertainty of the re-duced human model. The proposed approach was ana-lyzed completely from the view point of algorithm con-vergence. From the derivation of tracking time and track-ing error, the approach is reliable. The simulation verifiedthe proposed approach by lifting up a normal human body



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

Kin

emat

ic e

nerg

y (J

)

Energy of the simplified human model

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−500

0

500

Pot

entia

l ene

rgy

(J)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−500

0

500

Time (s)

Tot

al e

nerg

y

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

X d

irect

ion

(m)

Position of the head

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

Y d

irect

ion

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

Z d

irect

ion

(m)

Time (s)

(b)

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

Ang

le (

rad)

Angle and volicity off the hirizontal line of lower−trunk

0 0.5 1 1.5 2−6

−4

−2

0

2

Time (s)

Vol

icity

(ra

d/s)

(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Ang

le (

rad)

Angle and velocity of the joint of lowertrunk−upperleg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

1.5

2

2.5

time

Vel

ocity

(ra

d/s)

(d)

Fig. 10. Energy and angle change with time when the hu-man body is lifted up. (a) The change of energy. (b) Theposition of the head. (c) The angle and velocity drift off thehorizontal line of the lowertrunk. (d) The angle and velocityof the joint of lowertrunk-upperleg.

(35 DOF) with passive moments. It is novel that the ap-proach proposed in this paper is not only designed for thespecific case of lifting human, but also can be used muchmore widely for physically interacting with human. Ourfuture research will concern on how to change the lyingposture of human body by robot.

References:[1] P. Padetsky, “The Man Who Mastered Motion,” Science, Vol.7,

pp. 52-60, 1986.[2] T. Yamamoto, K. Terada, and Y. Kuniyoshi, “Lifting Techniques for

the Humanoid Robots: Insight from Human Movements,” Proc. ofIEEE-RAS Inter. Conf. on Humanoid Robotics, pp. 251-258, 2008.

[3] R. Takeda, K. Nakadai, T. Takahashi, K. Komatani, T. Ogata, andH. G. Okuno, “Automatic Estimation of Reverberation Time withRobot Speech to Improve ICA-based Robot Audition,” Proc. ofIEEE-RAS Int. Conf. on Humanoid Robotics, 2009.

[4] Z. W. Luo et al., “On Cooperative Manipulation of Dynamic Ob-ject.,” Advanced Robotics, Vol.10, pp. 621-636, 1996.

[5] F. Asano et al. “Dynamic Modeling and Control for Whole BodyManipulation,” Proc. of IEEE Int. Conf. of Robotics and Automa-tion, 2003.

[6] P. Song, M. Yashima, and V. Kuma, “Dyanmics and Control ofWhole Arm Graps,” Proc. of Int. Conf. on Robotics and Automa-tion, 2001.

[7] T. Mukai et al., “Development of The Tactile Sensor System of AHuman Interactive Robot ‘RI-MAN,’” IEEE Trans. on Robotics,Vol.24, pp. 502-512, 2008.

[8] M. Onishi et al. “Generation of Human Care Behaviors by Human-interactive Robot RI-MAN, ” Proc. of IEEE Int. Conf. on Roboticsand Automation, 2007.

[9] T. Odashima et al. “A Soft Human-interactive Robot RI-MAN,”Video Proc. of Inter. Conf. on Intelligent Robots and Systems, 2006.

[10] Y. Umetani and K. Yoshida, “Resolved Motion Rate Controlof Space Manipulators with Generalized Jacobian Matrix,” IEEETrans. on Robotics and Automation, Vol.5, pp. 303-314, 1989.

[11] K. Yoshida and Y. Umetani, “Control of Space Manipulators withGeneralized Jacobian Mtrix,” Space Robotics: Dynamics and Con-trol, pp. 165-204, 1992.

[12] A. Nagano, R. Himeno, and S. Fukashiro, “An Introduction toThree-dimensional Rigid Body Dynamics: Vol.4 Simulation Us-ing An Assisting Software Package,” Japanse J. of Biomechanicsin Sports and Exercise, Vol.8, pp. 209-223, 2004.

[13] A. Nagano et al., “A Three-dimensional Linked Segment Model ofThe Whole Human Body,” Int. J. of Sport and Health Science, Vol.3,pp. 311-325, 2005.

[14] F. C. Anderson and M. G. Pandy, “A Dynamic Optimization Solu-tion for Vertical Jumping in Three Dimensions,” Computer Methodsin Biomechanics and Biomechanical Engineering, Vol.2, pp. 201-231, 1999.

[15] T. R. Kane and D. A. Levinson, “Dynamics: Theory and Applica-tions,” McGraw-Hill, 1985.



Name:Haiwei Dong

Affiliation:Department of Computational Science, GraduateSchool of System Informatics, Kobe University

Address:1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, JapanBrief Biographical History:2005 Received B.E. from Nanjing University of Science and Technology2008 Received M.E. from Shanghai Jiaotong University2008- Working towards Ph.D. degree in Kobe UniversityMain Works:• H. Dong et al., “Novel Information Matrix Sparsification Approach forPractical Implementation of Simultanesous Localization and Mapping,”Advanced Robotics, Vol.24, pp.819-838, 2010.

Name:Zhiwei Luo

Affiliation:Professor, Kobe University

Address:1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, JapanBrief Biographical History:1992- Assistant Professor, Toyohashi University of Technology1994- Frontier Researcher, The Institute of Physical and ChemicalResearch1999- Associate Professor, Yamagata University2001- Team Leader, The Institute of Physical and Chemical Research2006- Professor, Kobe UniversityMain Works:• Development of a human-interactive robot RI-MAN• Immersion-type dynamic simulation of human-robot interaction• Innovation of health engineeringMembership in Academic Societies:• The Society of Instrument and Control Engineers (SICE)• The Robotics Society of Japan (RSJ)• Japanese Neural Network Society (JNNS)• The Institute of Electrical and Electronics Engineers (IEEE)

Name:Akinori Nagao

Affiliation:Department of Computational Science, GraduateSchool of System Informatics, Kobe University

Address:1-1 Rokkodai, Nada, Kobe, HyogoBrief Biographical History:1996 B.L.A., University of Tokyo,1998 M.A., University of Tokyo,2001 Ph.D., Arizona State University2002-2003 Boston University & Harvard Medical School2003-2006 RIKEN2006-2007 University of Aberdeen2007- Kobe UniversityMain Works:• Nagano et al., “An analysis of directional changes in the center ofpressure trajectory during stance,” Gait and Posture.• Nagano et al., “Neuromusculoskeletal computer modeling and simulationof upright, straight-legged, bipedal locomotion of Australopithecusafarensis (A.L. 288-1),” American J. of Physical Anthropology.• Nagano and Komura, “Longer moment arm results in smaller jointmoment development, power and work outputs in fast motions,” J. ofBiomechanics.Membership in Academic Societies:• International Society of Biomechanics (ISB)• American Society of Biomechanics (ASB)• Japanese Society of Biomechanics (JSB)• Japanese Society of Physical Education
