Learning Force and Position Constraints in Human-robot Cooperative Transportation

Transcription

1 Learning Force and osiion Consrains in Human-robo Cooperaive Transporaion Leonel Rozo 1 Sylvain Calinon 12 and Darwin G. Caldwell 1 Absrac hysical ineracion beween humans and robos arises a large se of challenging problems involving hardware safey conrol and cogniive aspecs among ohers. In his conex he cooperaive (wo or more people/robos) ransporaion of bulky loads in manufacuring plans is a pracical example where hese challenges are eviden. In his paper we address he problem of eaching a robo collaboraive behaviors from human demonsraions. Specifically we presen an approach ha combines: probabilisic learning and dynamical sysems o encode he robo s moion along he ask. Our mehod allows us o learn no only a desired pah o ake he objec hrough bu also he force he robo needs o apply o he load during he ineracion. Moreover he robo is able o learn and reproduce he ask wih varying iniial and final locaions of he objec. The proposed approach can be used in scenarios where no only he pah o be followed by he ranspored objec maers bu also he force applied o i. Tess were successfully carried ou in a scenario where a 7 DOFs backdrivable manipulaor learns o cooperae wih a human o ranspor an objec while saisfying he posiion and force consrains of he ask. I. INTRODUCTION Robos are ofen envisaged as human-like machines ha can inerac wih people in a naural and safe way. Such human-robo ineracion (HRI) implies ha he robo is able o communicae wih he person undersand his/her needs and behave accordingly. This impression is paricularly imporan in siuaions where a human needs he help of anoher person o perform a given ask successfully. For example he ransporaion of bulky objecs demands a leas wo persons o carry he load cooperaively. This ask may become difficul when he load has o pass hrough narrow spaces and even more laborious if he objec is fragile enough so ha he ransporers mus concern abou he force hey apply o i. In his scenario one of he humans may be replaced by a roboic agen where he reasoning and adapaion abiliies of he human can be combined wih he robo s srengh and precision. Such funcionaliy may be achieved by programming he robo from demonsraions of he ask. Once he collaboraive behavior has been learned he robo can auonomously perform he ask expanding is own skills. rogramming by demonsraion (bd) [1] has been successfully applied o seings where a robo reproduces a learned skill in a sandalone fashion [2] [3]. However his approach has rarely been used in human-robo collaboraion 1 Deparmen of Advanced Roboics Isiuo Ialiano di Tecnologia (IIT) Via Morego Genova Ialy. name.surname@ii.i 2 Idiap Research Insiue Rue Marconi 19 O Box 592 CH-192 Marigny Swizerland. sylvain.calinon@idiap.ch This work was suppored by he STIFF-FLO European projec (F7- ICT ) and by he SAHARI European projec (F7-ICT ). Demonsraion Reproducion Fig. 1: Experimenal seing: demonsraion and reproducion phases. (HRC) scenarios where conrol-based soluions have dominaed (see Secion II). Ye mos conrol mehods require a model of he ask which becomes complex when a human is in he loop. In such insances bd emerges as a promising alernaive soluion allowing he naural ransfer of human knowledge abou he ask o he robo. In his conex a human eacher may for insance demonsrae o he robo is role in he ask [4] a rajecory o follow [5] or even how complian i should be [6]. We herefore propose o use bd o each a robo o simulaneously handle posiion and force consrains arising when a human and a robo cooperaively manipulae/ranspor an objec (see Fig. 1). Specifically our approach defines a se of virual dynamical sysems represening he consrains of he ask and driving he robo moion. Such sysems can ac on differen frames of reference for insance on coordinae sysems represening he robo s base he ranspored objec ec. To deal wih his problem we use a ask-paramerized formulaion of a Gaussian mixure model ha allows us no only o encode he human demonsraions bu also o exrac auomaically he imporance of he consrains acing a differen coordinae sysems along he ask [7]. Moreover our approach provides boh open-loop and feedback componens. This is specially imporan in HRC due o he fac ha pure open-loop sysems canno reac o perurbaions and pure feedback frameworks will impede he speed and fluency of

2 he collaboraion. We successfully es our approach in a realworld scenario where a 7 DOFs roboic manipulaor learns o perform a cooperaive ask requiring differen force and posiion consrains o be saisfied. A brief review on works dealing wih similar HRC problems is given in Secion II. Deails abou our approach can be found in Secions III and IV while resuls for he cooperaive ransporaion experimen are shown in Secion V. Conclusions and fuure work are presened in Secion VI. A. Conrol-based approaches II. RELATED WORK Human-robo collaboraion has been invesigaed from he early nineies when purely conrol-based approaches were dominan. Kosuge e al. [8] proposed an impedance conrol based on he apparen mechanical impedance of an objec manipulaed by muliple robos and a human. The force applied by he human was ransferred o he robo conrollers so ha he human could command he moion of he objec while he robos behave as followers. The proposed conroller was compared agains differen classic conrol scheme where is low damping version performed he bes [9]. Al-Jarrah and Zheng inroduced a wo-levels conrol schema where an admiance conroller is driven by a higher level force conrol. The idea here was o rigger a reflex conroller when he robo aced as a load for he human by seing a force-based hreshold ha governs he moion of he manipulaor [1]. Duchaine and Gosselin [11] considered ha he human inenion in cooperaive asks is ypically based on he direcion and magniude of he force measured a he robo s end-effecor. They proposed o add he rae of change of he sensed force o an impedance conroller [12] while varying is damping as funcion of he changes of magniude of he force. Dumora e al. decomposed a collaboraive ask ino a sequence of non-holonomic robo moions [13]. Every moion primiive was represened by a predefined virual mechanism coupled o he robo s impedance conroller. Hence he whole ask could be carried ou by sequencing he differen primiives according o he user s inenion [14]. Noe ha he key feaure in all hese works has been he need for a model of he ask linked o an analysis of he possible robo movemens so ha boh he parameers and he srucure of he conroller can be designed accordingly. Unforunaely mos of hese frameworks are no flexible in he sense ha if a new ask is required or if an addiional consrain needs o be considered he conrollers need o be redesigned. B. Human performance-based approaches Several works rely on human-human collaboraion sudies o assis in design of he robo conrollers. Ikeura e al. proposed o approximae human cooperaion using variable impedance conrol (wih zero siffness). From daa colleced when wo people carried an objec he damping parameer was esimaed according o he precision required by he ask eiher from leas squares [12] or by minimizing a cos funcion ha penalized high raes of change [15]. The approach was hen improved by inroducing siffness ino he conroller [16]. Here he parameers were esimaed from force and posiion daa colleced when a single human compleed he ask following minimum jerk robo movemens (i.e. he robo aced as he leader while he human was he follower). Noe ha he minimum jerk model [17] has also been an inspiraion for Maeda e al. [18]. They proposed o use such a model o esimae he human hand posiion in a human-robo carrying ask. This esimaion was used as he reference for he robo impedance conroller. Tsumugiwa e al. [19] used an impedance conroller where he damping varied according o he esimae of he human arm siffness. Their approach assumed ha a low velociy cooperaive sysem remains sable if he robo s damping proporionally varies as he human siffness. arker e al. [2] unlike former works proposed o ieraively une he parameers of he robo s admiance conroller from he user s preferences wih regards o he roboic parner. Noice ha he idea behind all hese approaches is mainly o emulae he way humans ac in a collaboraive scenario. This aim has been achieved eiher by shaping he parameers of a predefined conroller using moion/force paerns sensed while a human-human pair carries ou he ask or by uning he conroller based on users feedback. The success of hese mehods mainly relies on how well he robo conroller fis he human collaboraive behavior. C. Learning-based approaches Evrard e al. [4] proposed a probabilisic framework based on Gaussian mixure models (GMM) and Gaussian mixure regression (GMR) o respecively encode and reproduce robo collaboraive behaviors. The main idea was o demonsrae by eleoperaion he leader/follower roles during a cooperaive lifing ask. GMM encapsulaed he robo moion and he sensed forces whils GMR generaed he reference inpus corresponding o a given sensed force during reproducion. On he oher hand Medina e al. [5] endowed heir robo wih a cogniive sysem which provided segmenaion encoding and clusering capabiliies for demonsraions of collaboraive behavioral primiives. These were represened by a primiive graph and a primiive ree using hidden Markov models (HMM) ha were incremenally updaed during reproducion [21]. One of he main differences wih respec o [4] was ha here he robo sared behaving as a follower bu is role became more proacive as i acquired more knowledge abou he ask. Gribovskaya e al. [22] proposed a hybrid srucure based on bd and adapive conrol ha drives he robo using an adapive impedance conroller. Firs a model of he ask was learned from demonsraions encoded by a GMM o generae feedforward conrol signals. Then he impedance parameers were adaped as funcion of he kinemaic and force errors generaed during he execuion of he ask. In conras o he work presened above where he rajecory o be followed maered we proposed in [6] o each differen compliance levels o a robo by kinesheic eaching.

3 Fig. 2: Illusraive example of he robo moion driven by a virual spring-damper aracor and consrained o exernal ineracion forces. The gray line represens he demonsraed pah of he end-effecor. The red line depics he rajecory of he aracor y. The core idea was o virually connec he robo s endeffecor o a se of virual springs driving he robo behavior. A ask-paramerized GMM [7] encoding he demonsraions defined he equilibrium poins of his sysem. The model was hen augmened by including siffness marices esimaed from he raining daa. Ye no resricions regarding he forces applied o he load were given neiher was a specific pah o follow. These specificaions become paricularly relevan in cooperaive ransporaion. Consequenly we address here he problem of learning force and posiion consrains in human-robo cooperaive ransporaion where he sar and arge locaions of he load vary. III. ROBLEM FORMULATION The problem ackled in his paper is ha of a human and a robo cooperaively ransporing an objec from a sar locaion o a arge posiion. Moreover we consider he case in which he given objec should be manipulaed wih some desired forces i.e. he forces applied o he load should allow he objec o be ranspored by pressing i on he sides wihou breaking i. In his conex he robo needs no only o follow a specific rajecory in is workspace bu also o physically inerac wih he user hrough he ranspored load under some force and posiion consrains. To formalize he problem le us consider he operaional space dynamics model of he robo under ineracion wih he environmen as Λ(x)ẍ+µ(xẋ)+p(x) = F F e (1) where Λ(x) µ(xẋ) and p(x) are he ineria marix he vecor of cenrifugal and Coriolis forces and he graviy componens respecively. The pose of he robo is denoed by x (i.e. posiion and orienaion of he end-effecor) F is he generalized forces vecor and F e is he vecor of conac forces exered by he end-effecor on he environmen. We assume a perfec nonlinear dynamic coupling which means he robo s end-effecor can be reaed as equivalen o a single uni mass moving in he Caresian space [23]. This allows us o formulae our problem as he case of finding he generalized force vecor F o aain he desired Fig. 3: Three differen demonsraions of he collaboraive ransporaion ask. The solid and dashed lines respecively depic he end-effecor and aracor rajecories. The sar and he end of robo moion are represened by colored dos and crosses. The dark and ligh boxes show he saring and arge locaions of he ranspored objec. ask dynamics. To achieve his aim we propose ha he robo behavior during he ineracion is driven by a virual aracor represened as a spring-damper sysem as shown in Fig. 2. Specifically he desired robo s moion during ineracion is given by ẍ = K (y x) K V ẋ F e (2) where K K V and y are he siffness marix he damping and he pah of he virual aracor respecively. The learning problem is herefore formulaed as esimaing he pah of y ha will induce he end-effecor o follow he cooperaive behaviors demonsraed by he eacher. Noice ha x and is firs and second ime derivaives are direcly obained from demonsraions. Also he conac forces F e are provided by a force sensor mouned a he robo s end-effecor. I is worh highlighing ha he sensor readings depend on he forces applied by he human and he robo while moving he load in oher words he sensed forces conain informaion abou he desired force o be applied o he objec. IV. LEARNING AND RERODUCTION We propose o probabilisically encode he se of demonsraions hrough a ask-paramerized version of he Gaussian mixure model [7]. Such a model allows us o consider he ask consrains given a differen frames of reference (i.e. he parameers of he ask). Formally he ask parameers are represened as coordinae sysems defined a ime sep n by {b nj A nj } represening respecively he origin of he frame and a se of basis vecors {e 1 e 2...} forming a ransformaion marix A=[e 1 e 2 ]. A movemen is observed from hese differen viewpoins forming a hird order ensor daase X R D N composed of rajecory samples X R D N. Every X corresponds o a marix composed of D-dimensional observaions a N ime seps. In our applicaion 1 D = 4 1 For sake of simpliciy he end-effecor orienaion was no considered in he experimens.

4 .4 y3s y2s y3s y1s y1s.4.6 y2s y3t.2 y3t y2t y1t y1t y2t Fig. 4: robabilisic encoding of he demonsraions a he differen candidaes frames of he ask. The firs row shows he model in he sar frame S while he second row displays he GMM in he arge frame T. The gray lines depic he aracor rajecories observed from he corresponding candidae frame. The ellipsoids represen he componens of he model. corresponding o he aggregaion of he ime variable and he Caresian of he aracor y herefore h... posiion i 1 N X = y... y. The parameers of he model wih K 1 N E-sep: Q γni = K k=1 πk N X n µ i Σi Q. N X n µk Σk πi = Σi = n=1 γni N µi = n=1 γni X n N n=1 γni Σni 1 1 (Anj Σi A nj ) X (4) (Anj Σi A nj ) 1 (Anj µi +bnj). The model parameers are iniialized wih a k-means procedure redefined using a similar process o ha used for he modified EM algorihm. By using he emporary GMM parameers compued in Eq. (4) for a given se of ask parameers we resor o Gaussian mixure regression o rerieve a each ime sep he aracor posiion during reproducion. Specifically GMR relies on he join disribuion ( y) learned by he ask-paramerized GMM. The condiional probabiliy (y n n ) is esimaed as y an oupu disribuion N (µ yn Σ n ) ha is also Gaussian wih X 1 hni (n ) µyni + Σy (n µni ) µ yn = ni (Σni ) = X 1 y h2ni (n ) Σyni Σy Σni ni (Σni ) (5) and acivaion weighs hni (n ) defined as (3) The learned model can be used o reproduce movemens in new siuaions (for new posiions and orienaions of candidae frames). The model firs rerieves a each ime sep n a GMM by compuing a produc of linearly ransformed Gaussians Y N Anj µi +bnj Anj Σi A nj N (µni Σni ) = i n=1 γni (X n µi )(X n µi ). N n=1 γni µni y Σ n N N X i M-sep: N = K componens are defined by {πi {µi Σi } }i=1 where πi are he mixing coefficiens µi and Σi are he mode-j cener and covariance marix of he i-h Gaussian componen. Learning of he parameers is achieved by seing he consrained problem of maximizing he log-likelihood under he consrains ha he daa in he differen frames are generaed from he same source resuling in an EM process o ieraively updae he model parameers unil convergence. πi Σni πi N (n µi Σi ) hni (n ) = K. k πk N (n µk Σk ) V. TRANSORTATION EXERIMENT We es he performance of our approach in an experimen where a human-robo dyad ranspors an objec from a sar locaion o a desired arge. The deailed descripion abou he seing he demonsraion and reproducion phases as well as he obained resuls are given below.

5 A. Descripion of he ask A he beginning of he ransporaion ask wo paricipans simulaneously reach for he objec. Once hey make conac wih he load hey sar joinly ransporing he objec along a given pah o reach he arge locaion. When he objec ges o he final posiion he human-human pair releases i and moves away from he objec. Noe ha boh he saring and goal objec posiion/orienaion may vary across repeiions. As saed in Secion I he aim is o inroduce a robo ino such a ask by replacing one of he human paricipans by a robo. Specifically we used a orque-conrolled 7 DOFs WAM robo fied wih a 6-axis force/orque sensor. In he demonsraion phase he graviy-compensaed robo is kinesheically guided by he eacher while cooperaively achieving he ask wih he oher human parner as shown in Figure 1. Five examples of collaboraive behavior are given o he robo. The eacher shows he robo boh he pah o be followed and he force paern i should use while ransporing he load (see Figure 3). The demonsraions are hen used for raining a five-saes ask-paramerized GMM (K = 5 seleced empirically) wih wo candidae coordinae sysems ( = 2) namely he frames represening he sar and arge locaions of he objec. They are respecively defined as 2 and [ 1 A n1 = R S [ 1 A n2 = R T ] b n1 = [ x S o ] ] [ ] b n2 = x T. o Here R S and R T respecively represen he sar and final orienaion of he objec wih roaion marices while x S o and x T o define is corresponding Caresian posiions. 3 Finally he aracor s rajecory is compued using Eq. (2) wih predefined values for he marices K = 5 I and K V = 6 I. During reproducion of he ask he sar and arge frames are given o he model in order o obain he emporary GMM parameers using Eq. (4). Then he robo and is human parner ranspor he objec owards he arge locaion. Here for each ime sep n he robo obains a new aracor locaion from Eq. (5) (as explained in Secion III) ha generaes a new desired acceleraion in he operaional space of he robo. For sake of simpliciy in he experimens he orienaion of he robo s end-effecor is fixed. B. Resuls Fig. 4 shows he resuling encoding of he aracor rajecories compued from Eq. (2) and observed from he wo differen candidae frames. Noice how he muliple demonsraions are locally consisen when he robo approaches 2 Noe ha he duraion of he ask is no modulaed by he ask parameers. 3 The posiion and orienaions of he objec were predefined in he experimen bu hese can alernaively be obained using a vision or opical racking sysem. Informaion regarding he moion of he human parner was no considered here. x x 2.5 (a) Reproducions wih varying sar and arge locaions of he objec. x x 2 (b) Nearly consan force applied by he human parner o he objec along he whole reproducion. x x 2.5 (c) Human varies he applied force along he whole reproducion. The robo adaps accordingly. Fig. 5: The solid lines represen he robo s rajecory while he dashed lines depic he rajecory of he aracor y obained from GMR. The green area display he sensed force a he robo s end-effecor. The dos and crosses respecively display he sar and end of he reproducion. he sar locaion of he objec (i.e. frame S) and when he manipulaor moves away once he load has been placed a is arge posiion (i.e. frame T ). This is refleced by he small ellipsoids in hese pars of he ask. Afer learning he obained model was used o es he reproducion of he ask on he real plaform. Three differen ypes of ess were carried ou o evaluae he performance of he robo. Firs he human and robo cooperaively ranspored he load as demonsraed i.e. he force applied o he load was similar o hose given previously. Fig. 5a shows hree successful reproducions under he aforemenioned condiion where boh he saring and arge locaion varied. Fig. 5b shows one of hese reproducions where i is clearly observed ha he sensed force profile remains nearly consan hroughou he whole reproducion. I is worh highlighing ha he observed offse beween he end-effecor posiion and he aracor pah allows he robo o apply he desired force o he load while ransporing i. The second es consised of applying a varying force o evaluae how he robo reaced o force perurbaions no observed during learning. The human operaor sared he ask pushing he objec wih a force higher han hose augh during he demonsraions. Then he applied force was significanly reduced and finally i reached values similar o he demonsraions as shown in Fig. 5c. I is observed ha he robo could successfully adap o hese force variaions.

6 When he force is high he robo behaves complianly allowing small deviaions from he pah sill ensuring ha he posiion consrain remains wihin a feasible range deermined by he observed variabiliy in he demonsraions and he conroller gains. In conras when he force is very low (i.e. he human may be losing he conac wih he load) he robo moves o apply more force and preven he objec from being dropped. Noe ha despie he force variaions he robo was able o ranspor he objec along a similar pah in he oher dimensions by showing a collaboraive behavior ha is an appropriae compromise beween force and posiion consrains auomaically exraced from he saisical represenaion of he demonsraions. The las es concerned he siuaion in which he human compleely releases he objec. In his case he robo s endeffecor moved forward bu only wihin a boundary defined by he variaions of he demonsraed possible pahs. In oher words he robo does no push indefiniely bu i insead moves in an appropriae manner according o he unexpeced circumsances and is prior knowledge of he ask learned from he demonsraions. A video of he experimen and he ask-parameerized GMM sourcecode are available a hp://programming-by-demonsraion.org/roman214/. VI. CONCLUSIONS AND FUTURE WORK We presened a bd approach for eaching a robo o cooperaively ranspor objecs. Our mehod explois he advanages of a dynamical sysem formulaion for modelling boh he moion and he ineracion of he robo wih he user and he environmen along he ask. The dynamics of he sysem is learned from a se of demonsraions ha is encapsulaed in a ask-paramerized GMM. Noe ha in conras o previous works he robo exracs he desired pah o follow and he needed force o be applied o he load from he examples provided by a human user. Therefore he approach does no depend on a specific model of he ask bu i auomaically exracs he differen consrains of he problem. The resuls showed ha he robo successfully carried ou he ask and ha i was able o adap o force perurbaions no observed during he learning phase. In fuure work similarly as in [24] we plan o exend his research owards he esimaion of he siffness and damping marices of he virual aracor. In conras o [6] where only he siffness gains were esimaed hrough a wo-sep process we wan o learn he ask and esimae siffness and damping in a one-sho fashion. This would allow he robo o shape is compliance level along he ask according o he provided demonsraions. We also plan o explore he variabiliy of he demonsraions encapsulaed in he covariance marices of he model which could be exploied o deec if he robo reaches an unexpeced siuaion ha is oo far from he demonsraions (e.g. in case of failures) requiring he user o provide new demonsraions. [2] B. Akgun M. Cakmak J. Yoo and A. Thomaz Trajecories and keyframes for kinesheic eaching: A human-robo ineracion perspecive in HRI pp [3] L. Rozo. Jiménez and C. Torras A robo learning from demonsraion framework o perform force-based manipulaion asks Journal of Inelligen Service Roboics Special Issue on Arificial Inelligence Techniques for Roboics: Sensing Represenaion and Acion ar 2 vol. 6 no. 1 pp [4] S. Calinon. Evrard E. Gribovskaya A. Billard and A. Kheddar Learning collaboraive manipulaion asks by demonsraion using a hapic inerface in ICAR pp [5] J. Medina M. Lawizky A. Morl D. Lee and S. Hirche An experience-driven roboic assisan acquiring human knowledge o improve hapic cooperaion in IROS pp [6] L. Rozo S. Calinon D. G. Caldwell. Jiménez and C. Torras Learning collaboraive impedance-based robo behaviors in AAAI Conf. on Arificial Inelligence pp [7] S. Calinon Z. Li T. Alizadeh N. Tsagarakis and D. Caldwell Saisical dynamical sysems for skills acquisiion in humanoids in Humanoids pp [8] K. Kosuge H. Yoshida and T. Fukuda Dynamic conrol for robohuman collaboraion in RO-MAN pp [9] K. Kosuge and N. Kazamura Conrol of a robo handling an objec in cooperaion wih a human in RO-MAN pp [1] O. Al-Jarrah and Y. Zheng Arm-manipulaor coordinaion for load sharing using reflexive moion conrol in ICRA pp [11] V. Duchaine and C. Gosselin General model of human-robo cooperaion using a novel velociy based variable impedance conrol in EuroHapics pp [12] R. Ikeura and H. Inooka Variable impedance conrol of a robo for cooperaion wih a human in ICRA pp [13] J. Dumora F. Geffard C. Bidard and. Fraisse Towards a roboic parner for collaboraive manipulaion in HRI - Workshop on Collaboraive Manipulaion pp [14] J. Dumora F. Geffard C. Bidard T. Brouille and. Fraisse Experimenal sudy on hapic communicaion of a human in a shared human-robo collaboraive ask in IROS pp [15] R. Ikeura T. Moriguchi and K. Mizuani Opimal variable impedance conrol for a robo and is applicaion o lifing an objec wih a human in RO-MAN pp [16] M. Rahman R. Ikeura and K. Mizuani Invesigaing he impedance characerisic of human arm for developmen of robos o cooperae wih human operaors in IEEE Inl. Conf. on Sysems Man and Cyberneics pp [17] T. Flash and N. Hogan The coordinaion of arm movemens: An experimenally confirmed mahemaical model Journal of Neuroscience vol. 5 no. 7 pp [18] Y. Maeda T. Hara and T. Arai Human-robo cooperaive manipulaion wih moion esimaion in IEEE/RSJ Inl. Conf. on Inelligen Robos and Sysems (IROS) pp [19] T. Tsumugiwa R. Yokogawa and K. Hara Variable impedance conrol based on esimaion of human arm siffness for human-robo cooperaive calligraphic ask in ICRA pp [2] C. arker and E. Crof Design and personalizaion of a cooperaive carrying robo conroller in ICRA pp [21] D. Kulić W. Takano and Y. Nakamura Incremenal learning clusering and hierarchy formaion of whole body moion paerns using adapive hidden Markov chains IJRR vol. 27 no. 7 pp [22] E. Gribovskaya A. Kheddar and A. Billard Moion learning and adapive impedance for robo conrol during physical ineracion wih humans in ICRA pp [23] O. Khaib A unified approach for moion and force conrol of robo manipulaors: The operaional space formulaion IEEE Journal on Roboics and Auomaion vol. 3 no. 1 pp [24] S. Calinon D. Bruno and D. G. Caldwell A ask-parameerized probabilisic model wih minimal inervenion conrol in ICRA (Hong Kong China) May-June 214. REFERENCES [1] A. Billard S. Calinon R. Dillmann and S. Schaal Robo programming by demonsraion in Springer Handbook of Roboics (B. Siciliano and O. Khaib eds.) pp Springer 28.