using the functional reach test for probing the static stability of bipedal standing in humanoid...

Hindawi Publishing CorporationJournal of RoboticsVolume 2013 Article ID 126570 8 pageshttpdxdoiorg1011552013126570

Research ArticleUsing the Functional Reach Test for Probingthe Static Stability of Bipedal Standing in Humanoid RobotsBased on the Passive Motion Paradigm

Jacopo Zenzeri1 Dalia De Santis1 Vishwanathan Mohan1

Maura Casadio2 and Pietro Morasso12

1 RBCS Department Istituto Italiano di Tecnologia Via Morego 30 16163 Genoa Italy2 DIBRIS Department University of Genoa Viale Causa 13 16145 Genoa Italy

Correspondence should be addressed to Jacopo Zenzeri jacopozenzeriiitit

Received 24 December 2012 Revised 21 March 2013 Accepted 22 March 2013

Academic Editor G Muscato

Copyright copy 2013 Jacopo Zenzeri et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The goal of this paper is to analyze the static stability of a computational architecture based on the Passive Motion Paradigm forcoordinating the redundant degrees of freedom of a humanoid robot during whole-body reaching movements in bipedal standingThe analysis is based on a simulation study that implements the Functional Reach Test originally developed for assessing thedanger of falling in elderly peopleThe study is carried out in the YARP environment that allows realistic simulations with the iCubhumanoid robot

1 Introduction

In humans the ability to stand up on two legs is a necessaryprerequisite for bipedal walking Moreover there is ampleneurophysiological evidence that standing and walking arerather independent control mechanisms Therefore we sug-gest that also humanoid robots should be trained first tomaster the unstable standing posture in a generality of situa-tions and then learn to walk

We shall address this issue in relation with the humanoidrobot iCub [1] which has the size of a three-year-old child(height is 105 cm and weight is 142 Kg) and has 53 degrees offreedom (DoF) 7DoFs for each arm 9 for each hand 6 forthe head 3 for the trunk and spine and 6 for each leg iCub isstill unable to stand orwalk but only to crawl as baby toddlersof the same age Therefore the goal of this paper is to carryout a preliminary study of the computational processes thatmay allow iCub to achieve the sensorimotor competence thatis necessary for bipedal standing

The study builds upon what has already been achievedin the bimanual coordination of iCubrsquos movements [2 3]using the Passive Motion Paradigm which is a biomimetic

force-field based computational model based on the equi-librium point hypothesis The model has been evaluatedand validated both in a simulated environment and in realmovements However the present study is limited to thesimulation stage for ldquodevelopmental constraintsrdquo because thesensorimotor system of iCub has not matured enough toachieve the features that are necessary for standing (posturalcontrol system) and walking (bipedal locomotion system) ina biomimetic compliant way Compliant motion is currentlyoperating on the proximal joints of the upper and lower limbsBiomimetic postural control requires a compliant ankle andthis development will become available in the near future

As a matter of fact the whole body postural control sys-tem has two basic components static (P1) and dynamic (P2)P1 requires that for each time instant the projection of theCenter ofMass (CoM) on the ground remains inside the sup-port base (the convex hull of the points of contact of thebody with the ground) P1 is a problem of constrained co-ordination among the highly redundant DoFs of the humanhumanoid body but satisfying such coordination constraintsis not sufficient formaintaining balance unless the underlyingdynamic controller operates with very high levels of stiffness

2 Journal of Robotics

of the joints with particular reference to the ankle jointsThisis not what happens in the human case at least for healthysubjects who are characterized by a low level of stiffnesssmaller than the rate of growth of the toppling torque dueto gravity [4 5] thus inducing persistent sway movementsduring standing In contrast high-stiffness upright posturecharacterizes pathological conditions such as Parkinsonrsquosdisease [6] this entails apparently enhanced stability (the sizeof the sway movements is smaller in PD patients than incontrols) but also higher sensitivity to unexpected perturba-tions and thus higher danger of falling In humans robustdynamic stabilization with low levels of stiffness is achievedby intermittent control mechanisms [7ndash9] which generatesmall stabilizing control bursts on top of the body posturesdetermined by whole-body synergies

The focus of this paper is on P1 because the current imple-mentation constraints of the robot donot allow a low-stiffnessdynamic stabilization of the standing posture but self-adjusting static stabilization is the necessary prerequisite forachieving a general purpose mastery of the standing postureAs already mentioned we intend to address this problem byusing a biomimetic force-field based computational modelwhich takes inspiration from the Passive Motion Paradigm(PMP) [10] extended to include terminal attractor properties[11] We already used this approach for the coordinationof bimanual movements of the humanoid robot iCub andfor modeling whole-body reaching (WBR) movements inhumans [12] Here we investigate the feasibility of applyingthis model to the coordination of WBR movements in iCubwith particular emphasis on a specific form of WBR namelythe Functional Reach Test (FRT) FRT has been inventedas a dynamic clinical measure of balance [13] it measuresthe distance between the length of the arm and the maxi-mal forward reach in the standing position while maintain-ing a fixed base of support FRThas been tested for both valid-ity and reliability and is used in patients with diagnoses asdifferent as stroke Parkinsonrsquos disease vestibular hypofunc-tion multiple sclerosis and hip fractures FRT has also beenassociated with an increased risk of fall and frailty in elderlypeople who are unable to reach out at least 15 cm during bipe-dal standing

Since the current state of the iCubrsquos competence for thestanding posture has still some ldquopathologicalrdquo aspects asregards the danger of falling the improvements coming frombetter design and better control could be appropriately evalu-ated with a test similar to FRT used with humans

2 FRT Network for iCub

Thenetwork architecturewhich has been applied for allowingiCub to carry out the Functional Reach Test is an extensionof the architecture developed for modeling whole-bodyreaching movements in humans The architecture is com-posed of four parts (1) task subnetwork (2) focal subnetwork(3) postural subnetwork and (4) temporal coordination unit(see Figure 1)

The three subnetworks are stable dynamical systemswith terminal attractor characteristics which are providedby the temporal coordination unit This unit generates a

Figure 1 PMP network for the iCub humanoid robot that carriesout the Functional Reach Test (FRT) 119909

119865and 119909

119879are the 3-dimen-

sional position vectors of the final and moving targets respectively119909119867is the position of the hand 119909

119862is the position of the CoM 119902 is

the 5-dimensional joint rotation vector (ankle knee hip shoulderelbow) 119869

119865is the 3times 5 Jacobian matrix of the whole kinematic chain

used by the focal sub-network 119869119875is the 3times 3 Jacobian matrix of the

partial kinematic chain (from ankle to hip) which is used by thepostural subnetwork 119865119865foc 119865119865pos are the force field generators ofthe two subnetworks 119865

119865 119865119875are the corresponding force fields 119879

119865

119879119875are the related torque fields 119860 is the 5times 5 diagonal admittance

matrix Γ(119905) is the temporal coordination function

time-varying gain Γ(119905) which is transmitted to the threesubnetworks and allows them to reach final equilibrium at thesame time

Γ (119905) =

120585

(1 minus 120585)

120585 (119905) = 6(

119905 minus 1199050

120591

)

5

minus 15(

119905 minus 1199050

120591

)

4

+ 10(

119905 minus 1199050

120591

)

3

(1)

Here 1199050is the initiation time and 120591 is the duration of the

coordinated forward-reaching movement 120585(119905) is a minimumjerk time base generator The general rationale of PMP as asynergy formation mechanism for highly redundant articu-lated systems is to express the goal of an action as a forcefield applied to the end effector and task-specific constraintsas additional force fields applied to task-related body parts Inour case the task-specific constraint is to keep the projectionof the center of mass on the ground inside the support basedetermined by the two feet and the associated force field isapplied to the hip according to a hip-balancing strategy Thetask and focal subnetworks are responsible of the former forcefield and the postural subnetwork takes care of the latter

21 Task Subnetwork It generates amoving target119909119879(119905) in 3D

space which attracts both hands of iCub (represented by thetime-varying vector 119909

119867(119905)) with a force field induced by the

focal sub-network 119909119879evolves from the initial position of the

hands which are supposed to be jointed to a final position119909119865 In order to maximize the forward reach in agreement

with the nature of the FRT the final position should be placedsomehow beyond the reachable target area In any case apeculiar feature of the PMP approach to synergy formationis that it always implies well-formed transformations thatdo not collapse in the vicinity of singular configurations inparticular if a chosen target is unreachable given the values

Journal of Robotics 3

of robometric parameters and task-related constraints themodel guides the end effector to the final state which is closestto the target

The investigated protocol corresponds to the most com-mon form of FRT that is the bimanual test We might alsoimplement in the same framework a unimanual paradigmin which one hand is attracted by a forward-moving targetand the other is either fixed to the body or is used as a furthercounterbalancing tool The support base is a function of theposition of the feet and in FRT they are supposed to be paralleland symmetric with respect to the body In FRT simulationswe positioned 119909

119865just outside the reachable workspace in the

anterior-posterior direction However the exact position isnot critical emphasizing the robustness of the computationalmodel

22 Focal Sub-Network It generates an attractive force field119865foc of elastic typewhich is applied to both hands implement-ing the focal part of the task whose goal is to allow the handsto reach or at least to approach as much as possible the target119865foc = 119870foc(119909119879 minus 119909119867) The force field 119865foc is mapped into atorque field 119879foc from the task space to the joint space byusing the following transformation 119879foc = 119869

119879

119865119865foc where 119869119865

is the Jacobian matrix of the overall kinematic chain (fromfeet to hands)

The admittance matrix119860 transforms the torque field intoa movement vector 119902 of the overall kinematic chain (notethat this is not a full admittance matrix but only its viscouscomponent) 119860 expresses the degree of participation of eachindividualDoFof the redundant kinematic chain of thewholebody to the common synergy Therefore by modulating therelative values of this matrix it is possible to implementdifferent equivalent synergies which may enhance the rangeof motion of a DoF with respect to the othersThemovementvector 119902 is thenmapped from the joint space to the task spaceby the same Jacobian matrix generating a prediction of thehand trajectory 119909

119867and thus closing the loop

The dynamics induced in the network by the temporalcoordination unit allows the hand to reach the final positionat the same time in which the final target is reached by themoving target but there is no guarantee that in the pro-cess the CoM remains within the support base Thus if onlydriven by this mechanism iCub would reach the final posi-tion but fall forward immediately after The postural subnet-work is intended to prevent such unfortunate event

23 Postural Subnetwork It modifies the torque field 119879focgenerated by the focal sub-network by adding a ldquoposturalrdquocomponent 119879pos that takes into account the position ofthe CoM on the support base The two torque fields arethen superimposed thus generating a total torque fields thatcombines the focal drive and the postural stabilization 119879tot =119879foc + 119879pos 119879pos is computed by projecting from the taskspace to the joint space the postural field 119865pos which isdefined in the vein of the so-called ldquohip strategyrdquo as a forceapplied backward to the hip in order to counteract the for-ward shift of the CoM induced by the focal field This forcewas implemented in the model by a nonlinear functionthat diverges to high values when the CoM position 119909

119862

approaches the forward limit 119909max of the support base 119865pos =minus119870pos 119909119862(119909max minus 119909119862) The motion 119909

119862of the CoM is derived

from the motion of the whole kinematic chain by using adifferent Jacobian matrix 119869

119875that takes into account only the

ankle and knee joints The activation of the postural fieldis meant to induce the following effects (1) a smaller for-ward shift of the CoM (2) a backward shift of the hip (3)a forward tilt of the trunk associated with the lowering of theCoM It is worth noticing that this complex control patternis not explicitly programmed but is implicitly coded by thedynamics of the networkThe postural field 119865pos is thenmap-ped from the task space to the joint space by the followingtransformation 119879pos = 119869

119879

119875119865pos The focal and postural fields

are superimposed generating the combined motion throughthe common admittance matrix

In summary the integrated dynamics of the interactingsubnetworks are characterized by the following equationswhich achieve a balance between the forward pull applied tothe hand and the backward pull applied to the hip

119879= Γ (119905) (119909119865

minus 119909119879)

119867= Γ (119905) 119869119865

119860 119879tot

119862= Γ (119905) 119869119875

119860 119879tot

119879tot = 119879foc + 119879pos

119879foc = 119869119879

119865119870foc (119909119879 minus 119909119867)

119879pos = minus119869119879

119875119870pos

119909119862

119909max minus 119909119862

(2)

The simulations of the iCub Functional Reach Test whichcorrespond to the integration of the equations above usedthe iCubrsquos simulator [14] implemented under the YARP open-source middleware [15]

3 Simulation Experiments with FRT Network

Thecomputational architecture described in the previous sec-tion was tested by using the iCub simulatorThe ldquorobometricrdquoparameters of iCub (length mass) are summarized here

(i) leg (213 cm 095 Kg)(ii) thigh (224 cm 15 Kg)(iii) trunk (127 cm 4Kg including head)(iv) humerus (152 cm 115 Kg)(v) forearm + hand (0137 cm 05 Kg)

As suggested by the FRT protocol the following set ofangles defines the initial posture of the test

(i) ankle 85∘(ii) knee 92∘(iii) hip 85∘(iv) shoulder 330∘(v) elbow 0∘


These angular values are absolute referring to the horizontalline in the sagittal planeWith this posture the initial positionof the hand has a distance of 2905 cm with respect to thevertical line passing through the ankle joint and the CoM isshifted 352 cm forward The final position 119909

119865of the target

was set 50 cm forward which is slightly beyond the maxi-mum reachable forward distance and the limit for the CoMdisplacement (119909max) was set equal to 13 cm considering thatthe length of iCubrsquos foot is 15 cm

The control parameters of the FRTnetwork and the valuesused in the simulations are as follows

(i) gain of the focal field 119870foc = 700Nm(ii) gain of the postural field119870pos = 2N(iii) admittance matrix of the whole kinematic chain119860 In

the reported experiments the matrix is 5times 5 becausethe following five joints are involved ankle knee hipshoulder elbow For simplicity we chose the matrixto be diagonal also because this allows us to chooseits values in a rational way What is important fromthe point of view of synergy formation are the relativevalues of the matrix diagonal For example if they areall equal it means that all the joints have the sameweight in the participation to the common action ifone is much smaller than the others then the cor-responding joint will change very little its angularvalue with respect to the other joints if one admit-tance value is much greater than the others thenthe corresponding joint will be the one that willmove the most The relative values of the matrixdiagonal can be scaled up and down with little effecton the overall synergy because the nonlinear gatingprovided by the Γ function tends to normalize theoverall gain In the simulation example we set thefive elements to the following values 119860

1(ankle) =

002 radNms 1198602(knee) = 001 radNms 119860

3(hip) =

03 radNms1198604(shoulder) = 01 radNms119860

5(elbow)

= 007 radNms The hip admittance has the highestvalue in order to facilitate the counterbalancing of theforward shift of the CoM induced by the arm-reach-ing movement with a suitable backward shift of thepelvis For the same reason the admittance of the kneeand elbow is relatively smaller But we can changethe pattern of admittance values in a large range as afunction of specific task or physical constraints of therobot

With these parameter values we could obtain the sim-ulation results illustrated in Figures 2 and 3 In particularFigure 2 shows the initial and final postures of the FRT Theend-effectors of iCub reach forward at a distance of 4675 cmwhich is shorter than the target distance (50 cm) because thebackward pull of the postural force field allows the forwardshift of the CoM (1195 cm) to remain inside the plannedlimit of 13 cm Thus static stability is preserved and the robotbody is stretched forward as much as possible The increaseof the hand forward reach is 177 cm with respect to the initialposture Incidentally this value is greater than the thresholdof 15 cm which is considered clinically relevant in the sense

that people who are unable to exceed such forward reach inthe test have a significant risk of falling

Figure 3 shows the time profile of the relevant variablesFigure 3(a) displays the intensities of the focal and posturalforce fields respectively together with the profile of the tem-poral coordination unit Γ(119905) (dashed) which provides term-inal attractor characteristics to overall model Figure 3(b)shows the rotations patterns of the five joints (ankle kneehip shoulder elbow) from the initial to the final posturePlease note that some angles evolve monotonously from ini-tial to termination time whereas others do not In particularthe elbow joint angle remains equal to 0 throughout thewholemovement for two reasons (1) it was set to 0 initially in agree-ment with the FRT protocol and (2) it remained 0 becauseboth force fields were directed horizontally (the focal fieldforward and the postural field backward resp) Figure 3(c)plots the forward displacements of the hand and the CoMthese curves evolve monotonously as should do to the finalshift values that must be compared to the final position of thetarget and the maximum admitted forward shift of the CoMrespectively It turns out that the hand stops about 5 cmbeforethe target because the latter is outside the workspace of therobot the CoM stops a few millimeters before the prefixedlimit of stability Finally Figure 3(d) displays the speedprofiles of the hand and the CoM respectively they appear tobe bell shaped and synchronized in agreement with the basicfindings of the research in WBR in general [16ndash18] whichshows indeed that the two parts of the WBR strategy are notindependent but strictly coupled by a common action genera-tion mechanism Figures 3(a) 3(b) and 3(c) also display thetime course of the Γ function emphasizing its role in theordered coordination and synchronization of so many differ-ent variables

How robust is the proposed synergy formation model fortesting the static stability of the standing posture in humanoidrobots The question can be analyzed from two points ofview (1) how effective is the model for inducing an optimalor quasi optimal forward reach and (2) how capable is it inmaintaining static stability although with a narrow marginIn order to answer such questions we carried out a sensitivityanalysis of the model with respect to the main parameters ofthe model (1) the final position of the target 119909

119865 (2) the gain

factors of the two force fields namely 119870foc and 119870pos (3) therelative values of the admittance matrix 119860

Figure 4 shows how the modulation of 119909119865in a large

range influences the variation of the final reach of the hand(119909119867) and the forward shift of the CoM (119909

119862) For values of

119909119865which are inside the range of reachable positions up to

about 45 cm there is a proportionality between 119909119865 on one

side and 119909119867

or 119909119862 on the other When 119909

119865exceeds 55 cm

there is a saturation for both 119909119867 at about 48 cm and 119909

119862

which approaches the threshold of 13 cm without ever cross-ing it (the maximum value is 1289 cm) Therefore in theFunctional Reach Test iCub cannot stretch the hand beyond48 cm and such performance is weakly dependent on theposition of the target provided that it is greater than 55 cmMoreover with the nominal values of the parameters used forthe simulations illustrated in Figures 2 and 3 static stability ispreserved for any final position of the target


00

01

02

03

04

05

06

(m)

HandTarget

CoM

01 02 03 04 05 06

(m)

119909max

Figure 2 Initial and final poses of iCub in the Functional Reach Test

Of the two gain parameters (119870foc and 119870pos) the formerone may have some influence on performance (how far iCubcan reach forward given a final position of the target) and thelatter can affect in principle the static stability of the synergyHowever the influence is verymild In particular if we doublethe value of 119870foc (from 700 to 1400Nm) the forward reachis increased by less than 1 cm (from 4675 cm to 4729 cm) if itis halved to 350Nm the forward reach is decreased by about6 cm Therefore 119870foc = 700Nm seems to be a lower boundon the gain of the focal field Stability is preserved in all casesin the sense that 119909

119862remains always behind the threshold

As regards the gain of the postural field the nominal value119870pos = 2N pushes the CoM quite close to the limit (1195 cmvs 13 cm) but preserves static stability Without such fieldthat is by setting 119870pos = 0 the stability limit would be over-come by several centimeters By reducing 119870pos from thenominal value the CoM will be pushed closer and closer tothe limit but it is necessary to go as low as 119870pos = 0005Nbefore losing static stability

The admittance matrix 119860 has an effect on the final post-ure of the body when equilibrium is reached however theinfluence on the task-related variables is very mild Forexample if all the elements of the diagonal are set all equal to

01 Nmsrad the forward reach of the hand is changed from4675 cm to 4680 cm and the forward shift of the CoM isslightly reduced from 1195 cm to 1193 cm Similar results areobtained by changing the elements of the matrix by 50 ormore in different combinations

4 Discussion

Theproposed synergy formationmodel is just an example of alarge class of complex whole-body tasks that can be capturedby the Passive Motion Paradigm In a recent paper [19]PMP was proposed as an alternative to optimal control formotor cognition in human and humanoid neuroscience Thenext step will be to integrate in the formalism the dynamicsderived from physical interaction between the body and theenvironment

One of themost remarkable features of themodel is that itis ldquoself-adaptiverdquo with respect to the articulation of the under-lying body schema for the main following reasons degrees offreedom can be added or deleted for example for incorpo-rating in the body schema manipulated tools with internaldynamics or for taking into account the reduced mobi-lity due to some kind of impairment moreover the synergy


0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)

Figure 3 (a) Time course of the forces generated by the focal and the postural subnetworks respectively (continuous lines) time-basegenerator (dashed line) (b) Time course of the joint rotation angles after subtracting the mean value 119902

1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902

5(elbow) = 0 radThe angular values are absolute relative to the horizontal line (c) Forward

shift of the hand (Functional Reach) related to the forward position of the target (50 cm with respect to the ankle) and forward shift of theCoM related to the maximum stable position on the support base (119909max = 13 cm) (d) Velocity profiles of the hand and the CoM Panel (a)also displays the time course of the Γ function (dashed line)

formation capabilities of the model remain intact providedthat the modifications of the Jacobian matrices are learnedthrough an appropriate training see the appendix for a poss-ible simple procedure of approximation and learning

Another aspect of such computational robustness is thatthere is no need to compute the timing and the specific velo-city profiles of all the joints because they are implicit con-sequence of the internal model simulation process In thissense the curse of dimensionality that in most cases affectsthe efficiency of planningcontrol methods in highly redun-dant robots does not apply to the proposed model

Clearly there is a link between WBR tasks like functionalreaching and APAs (Anticipatory Postural Adjustments)which have been studied by many authors [20ndash22] In fact

any voluntary movement of the upper arms is intrinsicallya source of disturbance to the posture and the stability ofthe whole body It has been demonstrated that APAs arenot reflexes but coordinative structures superimposed on thepostural stabilization processesThis is also the case forWBRin general and for the FRT task in particular However thereis a difference in WBR and FRT there is a strong couplingbetween the movements of the lower and upper extremitiesbecause they are both directly involved in both concurrenttasks namely the focal and postural task In contrast in APAexperiments there are no targets to be reached and simple armraises or trunk flexionsextensions are performed in order toperturb the standing posture as a consequence the posturalsystem can operate independently of the arm system and this


045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max

Figure 4 Top panel final position of the hand in the forwarddirection (119909

119867) as a function of the final position of the target (119909

119865)

Bottom panel final position of the CoM (119909119862) as a function of the

final position of the target (119909119865) stability requires that 119909

119862is smaller

than a critical threshold 119909max (dashed line)

lowers the degree of correlation between the two kinds ofmovements

Finally let us consider the possible impact of biomedicalrobotics and biomechatronics technology on healthcareThere are many reasons for assuming that humanoid roboticscan give a significant contribution to this theme in a verygeneral sense There is indeed a widely shared vision that anew generation of robotic technologiesmdashrobot companionsfor citizensmdashcan help our society to come to terms withthe special needs of an ageing population in such a way toremain creative productive autonomous and independentFor example this is the vision of the FET Flagship InitiativesRoboCom which envisages a new generation of soft sentientmachines that will help and assist humans in activitiesof daily living We believe that the computational modelpresented in this paper goes in that direction because itallows a humanoid robot to acquire a degree of competencein focalpostural activities which is a prerequisite for safesoft friendly interaction between a robot companion and aneeding human

Appendix

Jacobian Matrix of the Whole-Body andBabbling Movements for Learning It

The whole body model has five degrees of freedom 119902 = [1199021

1199022 1199023 1199024 1199025] which identify the following joints ankle knee

hip shoulder elbow A related vector 119871 = [1198711 1198712 1198713 1198714 1198715]

stores the corresponding link lengths The position of theend effector is identified by a three-dimensional vector 119901 =[1199011 1199012 1199013] where 119901

1is the coordinate in the mediolateral

direction 1199012in the anteroposterior direction and 119901

3in the

vertical direction For the movements considered in thispaper the kinematic function 119901 = 119891(119902) can be written asfollows

1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902

1 and so forth From this we can

immediately derive 119869119865

119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0

minus11987111199041minus11987121199042minus11987131199043minus11987141199044minus11987151199045

+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)

119869119875is similar it includes only the first three columns of 119869

119865 If

the precise robometric variable is not known the kinematicfunction 119901 = 119891(119902) and the corresponding Jacobian matrixcan be approximated by means of a neural network trainedby means of babbling movements

Given a generic kinematic chain which maps the jointvector 119902 into the position 119901

119890119890= 119891(119902) of the end-effector the

nonlinear function 119901119890119890= 119891(119902) can be approximated by a neu-

ral network and the network parameter can be learned withthe aid of a training set namely a large representative set ofinput-output patterns 119901

119890119890(119905119896) 119902(119905119896) 119896 = 1 sdot sdot sdot 119899 obtained

experimentally via ldquobabbling movementsrdquo For example wecan use a three-layered artificial neural network (ANN)

119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)

where 119899 is the number of neurons of the hidden layer ℎ119895is

an intermediate variable 119911119895is the output of the hidden layer

119892(sdot) is a sigmoid nonlinearity 119908119894119895 119908119895119896

are the connectionweights from input to hidden and from hidden to outputlayers respectively 119902

119894and 119901

119896are the inputs and outputs

of the ANN After training by means of the standard backpropagationmethod we can extract the Jacobianmatrix fromthe trained neural network in the following way

119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments

The research leading to these results has received fundingfrom the European Communityrsquos Seventh Framework Programme (FP72007-2013) projects DARWIN (httpwwwdarwin-projecteu Grant no FP7-270138) EFAA (httpefaaupfedu Grant no FP7-270490) and ROBOCOM(httpwwwrobotcompanionseu Grant no FP7-284951)This research is also supported by IIT (Istituto Italiano diTecnologia RBCS Dipartimento)


References

[1] GMetta L Natale F Nori et al ldquoThe iCub humanoid robot anopen-systems platform for research in cognitive developmentrdquoNeural Networks vol 23 no 8-9 pp 1125ndash1134 2010

[2] VMohan PMorasso GMetta and G Sandini ldquoA biomimeticforce-field based computational model for motion planningand bimanual coordination in humanoid robotsrdquo AutonomousRobots vol 27 no 3 pp 291ndash307 2009

[3] V Mohan P Morasso J Zenzeri G Metta V S Chakravarthyand G Sandini ldquoTeaching a humanoid robot to draw lsquoShapesrsquordquoAuton Robots vol 31 pp 21ndash53 2011

[4] I D Loram and M Lakie ldquoDirect measurement of humanankle stiffness during quiet standing the intrinsic mechanicalstiffness is insufficient for stabilityrdquo Journal of Physiology vol545 no 3 pp 1041ndash1053 2002

[5] M Casadio P G Morasso and V Sanguineti ldquoDirect measure-ment of ankle stiffness during quiet standing implications forcontrol modelling and clinical applicationrdquo Gait and Posturevol 21 no 4 pp 410ndash424 2005

[6] M G Carpenter J H J Allum F Honegger A L Adkin andB R Bloem ldquoPostural abnormalities to multidirectional stanceperturbations in Parkinsonrsquos diseaserdquo Journal of NeurologyNeurosurgery and Psychiatry vol 75 no 9 pp 1245ndash1254 2004

[7] I D Loram C N Maganaris and M Lakie ldquoHuman posturalsway results from frequent ballistic bias impulses by soleus andgastrocnemiusrdquo Journal of Physiology vol 564 no 1 pp 295ndash311 2005

[8] A Bottaro Y Yasutake T NomuraM Casadio and PMorassoldquoBounded stability of the quiet standing posture an intermittentcontrol modelrdquo Human Movement Science vol 27 no 3 pp473ndash495 2008

[9] Y Asai Y Tasaka K Nomura T Nomura M Casadio and PMorasso ldquoA model of postural control in quiet standing robustcompensation of delay-induced instability using intermittentactivation of feedback controlrdquo PLoS ONE vol 4 no 7 ArticleID e6169 2009

[10] F A M Ivaldi P Morasso and R Zaccaria ldquoKinematic net-worksmdasha distributed model for representing and regularizingmotor redundancyrdquo Biological Cybernetics vol 60 no 1 pp 1ndash16 1988

[11] M Zak ldquoTerminal attractors for addressable memory in neuralnetworksrdquo Physics Letters A vol 133 no 1-2 pp 18ndash22 1988

[12] P Morasso M Casadio V Mohan and J Zenzeri ldquoA neuralmechanismof synergy formation forwhole body reachingrdquoBio-logical Cybernetics vol 102 no 1 pp 45ndash55 2010

[13] P W Duncan D K Weiner J Chandler and S StudenskildquoFunctional reach a new clinical measure of balancerdquo Journalsof Gerontology vol 45 no 6 pp M192ndashM197 1990

[14] V Tikhanoff A Cangelosi P Fitzpatrick G Metta L Nataleand F Nori ldquoAn open-source simulator for cognitive roboticsresearchrdquo Cogprints 6238 2008

[15] GMetta P Fitzpatrick and L Natale ldquoYARP yet another robotplatformrdquo International Journal of Advanced Robotic Systemsvol 3 no 1 pp 43ndash48 2006

[16] T R Kaminski ldquoThe coupling between upper and lowerextremity synergies during whole body reachingrdquo Gait andPosture vol 26 no 2 pp 256ndash262 2007

[17] T Pozzo P J Stapley and C Papaxanthis ldquoCoordinationbetween equilibrium and hand trajectories during whole bodypointingmovementsrdquoExperimental Brain Research vol 144 no3 pp 343ndash350 2002

[18] P J Stapley T Pozzo G Cheron and A Grishin ldquoDoes thecoordination between posture and movement during humanwhole-body reaching ensure center of mass stabilizationrdquoExperimental Brain Research vol 129 no 1 pp 134ndash146 1999

[19] V Mohan and P Morasso ldquoPassive motion paradigm analternative to optimal controlrdquo Frontiers in Neurorobotics vol5 no 4 pp 1ndash28 2011

[20] A S Aruin ldquoThe organization of anticipatory postural adjust-mentsrdquo Journal of Automatic Control vol 12 pp 31ndash37 2002

[21] S Bouisset and M Zattara ldquoBiomechanical study of the pro-gramming of anticipatory postural adjustments associated withvoluntary movementrdquo Journal of Biomechanics vol 20 no 8pp 735ndash742 1987

[22] W A Lee T S Buchanan and M W Rogers ldquoEffects of armacceleration and behavioral conditions on the organization ofpostural adjustments during arm flexionrdquo Experimental BrainResearch vol 66 no 2 pp 257ndash270 1987

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



of the joints with particular reference to the ankle jointsThisis not what happens in the human case at least for healthysubjects who are characterized by a low level of stiffnesssmaller than the rate of growth of the toppling torque dueto gravity [4 5] thus inducing persistent sway movementsduring standing In contrast high-stiffness upright posturecharacterizes pathological conditions such as Parkinsonrsquosdisease [6] this entails apparently enhanced stability (the sizeof the sway movements is smaller in PD patients than incontrols) but also higher sensitivity to unexpected perturba-tions and thus higher danger of falling In humans robustdynamic stabilization with low levels of stiffness is achievedby intermittent control mechanisms [7ndash9] which generatesmall stabilizing control bursts on top of the body posturesdetermined by whole-body synergies

The focus of this paper is on P1 because the current imple-mentation constraints of the robot donot allow a low-stiffnessdynamic stabilization of the standing posture but self-adjusting static stabilization is the necessary prerequisite forachieving a general purpose mastery of the standing postureAs already mentioned we intend to address this problem byusing a biomimetic force-field based computational modelwhich takes inspiration from the Passive Motion Paradigm(PMP) [10] extended to include terminal attractor properties[11] We already used this approach for the coordinationof bimanual movements of the humanoid robot iCub andfor modeling whole-body reaching (WBR) movements inhumans [12] Here we investigate the feasibility of applyingthis model to the coordination of WBR movements in iCubwith particular emphasis on a specific form of WBR namelythe Functional Reach Test (FRT) FRT has been inventedas a dynamic clinical measure of balance [13] it measuresthe distance between the length of the arm and the maxi-mal forward reach in the standing position while maintain-ing a fixed base of support FRThas been tested for both valid-ity and reliability and is used in patients with diagnoses asdifferent as stroke Parkinsonrsquos disease vestibular hypofunc-tion multiple sclerosis and hip fractures FRT has also beenassociated with an increased risk of fall and frailty in elderlypeople who are unable to reach out at least 15 cm during bipe-dal standing

Since the current state of the iCubrsquos competence for thestanding posture has still some ldquopathologicalrdquo aspects asregards the danger of falling the improvements coming frombetter design and better control could be appropriately evalu-ated with a test similar to FRT used with humans

2 FRT Network for iCub

Thenetwork architecturewhich has been applied for allowingiCub to carry out the Functional Reach Test is an extensionof the architecture developed for modeling whole-bodyreaching movements in humans The architecture is com-posed of four parts (1) task subnetwork (2) focal subnetwork(3) postural subnetwork and (4) temporal coordination unit(see Figure 1)

The three subnetworks are stable dynamical systemswith terminal attractor characteristics which are providedby the temporal coordination unit This unit generates a

Figure 1 PMP network for the iCub humanoid robot that carriesout the Functional Reach Test (FRT) 119909

119865and 119909

119879are the 3-dimen-

sional position vectors of the final and moving targets respectively119909119867is the position of the hand 119909

119862is the position of the CoM 119902 is

the 5-dimensional joint rotation vector (ankle knee hip shoulderelbow) 119869

119865is the 3times 5 Jacobian matrix of the whole kinematic chain

used by the focal sub-network 119869119875is the 3times 3 Jacobian matrix of the

partial kinematic chain (from ankle to hip) which is used by thepostural subnetwork 119865119865foc 119865119865pos are the force field generators ofthe two subnetworks 119865

119865 119865119875are the corresponding force fields 119879

119865

119879119875are the related torque fields 119860 is the 5times 5 diagonal admittance

matrix Γ(119905) is the temporal coordination function

time-varying gain Γ(119905) which is transmitted to the threesubnetworks and allows them to reach final equilibrium at thesame time

Γ (119905) =

120585

(1 minus 120585)

120585 (119905) = 6(

119905 minus 1199050

120591

)

5

minus 15(

119905 minus 1199050

120591

)

4

+ 10(

119905 minus 1199050

120591

)

3

(1)

Here 1199050is the initiation time and 120591 is the duration of the

coordinated forward-reaching movement 120585(119905) is a minimumjerk time base generator The general rationale of PMP as asynergy formation mechanism for highly redundant articu-lated systems is to express the goal of an action as a forcefield applied to the end effector and task-specific constraintsas additional force fields applied to task-related body parts Inour case the task-specific constraint is to keep the projectionof the center of mass on the ground inside the support basedetermined by the two feet and the associated force field isapplied to the hip according to a hip-balancing strategy Thetask and focal subnetworks are responsible of the former forcefield and the postural subnetwork takes care of the latter

21 Task Subnetwork It generates amoving target119909119879(119905) in 3D

space which attracts both hands of iCub (represented by thetime-varying vector 119909

119867(119905)) with a force field induced by the

focal sub-network 119909119879evolves from the initial position of the

hands which are supposed to be jointed to a final position119909119865 In order to maximize the forward reach in agreement

with the nature of the FRT the final position should be placedsomehow beyond the reachable target area In any case apeculiar feature of the PMP approach to synergy formationis that it always implies well-formed transformations thatdo not collapse in the vicinity of singular configurations inparticular if a chosen target is unreachable given the values







119879

119865119865foc where 119869119865






119862






119879




119879= Γ (119905) (119909119865

minus 119909119879)

119867= Γ (119905) 119869119865

119860 119879tot

119862= Γ (119905) 119869119875

119860 119879tot

119879tot = 119879foc + 119879pos

119879foc = 119869119879

119865119870foc (119909119879 minus 119909119867)

119879pos = minus119869119879

119875119870pos

119909119862

119909max minus 119909119862

(2)









119865of the target





1(ankle) =


3(hip) =


5(elbow)






119865 (2) the gain







side and 119909119867


119865exceeds 55 cm


119862



00

01

02

03

04

05

06

(m)

HandTarget

CoM

01 02 03 04 05 06

(m)

119909max







4 Discussion




0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)


1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902








045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119879

119865119865foc where 119869119865






119862






119879




119879= Γ (119905) (119909119865

minus 119909119879)

119867= Γ (119905) 119869119865

119860 119879tot

119862= Γ (119905) 119869119875

119860 119879tot

119879tot = 119879foc + 119879pos

119879foc = 119869119879

119865119870foc (119909119879 minus 119909119867)

119879pos = minus119869119879

119875119870pos

119909119862

119909max minus 119909119862

(2)









119865of the target





1(ankle) =


3(hip) =


5(elbow)






119865 (2) the gain







side and 119909119867


119865exceeds 55 cm


119862



00

01

02

03

04

05

06

(m)

HandTarget

CoM

01 02 03 04 05 06

(m)

119909max







4 Discussion




0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)


1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902








045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119865of the target





1(ankle) =


3(hip) =


5(elbow)






119865 (2) the gain







side and 119909119867


119865exceeds 55 cm


119862



00

01

02

03

04

05

06

(m)

HandTarget

CoM

01 02 03 04 05 06

(m)

119909max







4 Discussion




0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)


1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902








045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

01

02

03

04

05

06

(m)

HandTarget

CoM

01 02 03 04 05 06

(m)

119909max







4 Discussion




0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)


1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902








045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

Time (ms)

Postural

Focal

Forc

e fiel

d in

tens

ity (N

)

Γ

(a)

0 500 1000 1500 2000 2500

0

02

04

06

Join

t ang

les (

rad)

minus1

minus08

minus06

minus04

minus02

Time (ms)

1199024

11990251199022

1199021

1199023

(b)

0 500 1000 1500 2000 25000

01

02

03

04

05

CoM

Hand

Forw

ard

disp

lace

men

t (m

)

Target

Time (ms)

119909max

(c)

0 500 1000 1500 2000 25000

005

01

015

02

025

03

035

04

045

CoM

Hand

Spee

d (m

s)

Time (ms)

(d)


1(ankle) = 138 rad 119902

2(knee) = 160

rad 1199023(hip) = 083 rad 119902

4(shoulder) = 595 rad 119902








045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










045 05 055 06 065 07 075 080404204404604805

045 05 055 06 065 07 075 0800801012014

(m)

(m)

119909119867

119909119862

119909119865 (m)

119909max



119865)



119862is smaller




Appendix








3in the


1199011= 0

1199012= 11987111198881+ 11987121198882+ 11987131198883+ 11987141198884+ 11987151198885

1199013= 11987111199041+ 11987121199042+ 11987131199043+ 11987141199044+ 11987151199045

(A1)

with 1198881= cos 119902

1 1199041= sin 119902



119869119865=

120597119901

120597119902

=[

[

0 0 0 0 0


+11987111199041+11987121199042+11987131199043+11987141199044+11987151199045

]

]

(A2)


119865 If








119901 = 119891 (119902)

997904rArr

ℎ119895= sum119894119908119894119895119902119894

119911119895= 119892 (ℎ

119895) 119894 = 1 5 119895 = 1 119899 119896 = 1 3

119901119896= sum119895119908119895119896119911119895

(A3)





119894and 119901



119869119896119894=

120597119901119896

120597119902119894

= sum

119895

120597119901119896

120597119911119895

120597119911119895

120597ℎ119895

120597ℎ119895

120597119902119894

= sum

119895

1199081198951198961198921015840(ℎ119895)119908119894119895 (A4)

Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









using the functional reach test for probing the static stability of bipedal standing in humanoid...

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