Adapting to Contacts: Energy Tanks and Task Energy forPassivity-Based Dynamic Movement Primitives

Erfan Shahriari1, Aljaz Kramberger2, Andrej Gams2, Ales Ude2 and Sami Haddadin1

Abstract— In this paper, we develop a framework to en-code demonstrated trajectories as periodic dynamic motionprimitives (DMP) for an impedance-controlled robot and theirmodification to fulfil the task objective, i. e. to adapt basedon the force feedback and encoded desired wrench profilevia an admittance controller. This behavior by itself canviolate stability. Therefore, a passivity analysis for the wholesystem is presented, and based on input power ports and thedemonstrated reference power, a passivity observer (PO) isdesigned. Subsequently, a DMP phase altering law is introducedaccording to the passivity criterion in order to adjust the phasebased on the passivity criterion. However, since this does notnecessarily guarantee passivity, a suitable virtual energy tankis used. Experimental results on a Kuka LWR-4 robot polishingan unknown surface underline the real world applicability thesuggested controller.

I. INTRODUCTION

Transfer of knowledge from a human to a robot has beenthoroughly investigated in robotics [1]. However, transfer ofmotion to achieve a task is only meaningful with matchingcorrespondences – if we demonstrate squatting a robot,it should perform squatting, not move legs while fallingover [2]. Transfer of knowledge has been applied to manydifferent tasks humans perform in various environments,including households. Wiping of surfaces, e.g., is one of themost common household tasks, showing semantic similaritywith numerous actions [3]. Consequentially, it is no surprisethat wiping – rubbing on a surface with a cloth or sponge –has been extensively explored in robotics literature. As statedin [4], wiping demands rich and complex actions, which canresult in different effects depending on the concrete execution- it may be plain wiping, or even polishing, painting orremoving of material (grinding). Our experimental humanoidrobotic polishing setup is depicted in Fig. 1.

Just as many actions can be semantically similar to wiping,many different approaches have been described for wipingin the robotics literature, ranging from kinematic [5] anddynamic approaches [6], to high-level of abstraction [4], [7]and dealing with deformable objects [8]. Within learning bydemonstration (LbD), methods combining learning and forcecontrol within the scope of adaptive LbD have been used toacquire and maintain contact on arbitrary surfaces [9].

While many papers deal with the execution of wipingtasks as the appropriate use-case scenario, only few dealwith energy transfer between robot and environment in thescope of maintaining system stability and optimality for bothrobot and environment, respectively. Ensuring stability is

1The authors are members of the Institute of AutomaticControl at Leibniz Universitat Hannover, Hannover, Germany,lastname@irt.uni-hannover.de

2The authors are with the Humanoid and Cognitive Robotics Lab,Dept. of Automatics, Biocybernetics and Robotics, Jozef Stefan Institute,Ljubljana, Slovenia, aljaz.kramberger@ijs.si

Fig. 1: Robot polishing a table. The task was trained by ahuman operator, the challenge is to maintain a desired normalforce to the table and remain passive although the initialmovements may alter due to environmental changes.

tightly related with the concept of passivity, which essentiallyrefers to a system’s property not to produce more energythan it receives. A useful feature of the concept is itsmodularity [10], which makes it possible to divide a systeminto subsystems with properly chosen interconnections andafter analyzing the passivity of each subsystem, concludeoverall system passivity.

In this paper, we embed adaptive LbD methods into theconcept of passivity-based control in order to design andprove overall system stability during complex tasks such astactile wiping. The considered problem statement and relatedwork are introduced next, leading to the overall contributionsmade.

A. Problem StatementLet us consider a system that is able to execute motions

demonstrated by a human operator or that were generated byassociated planning algorithms, while adapting these basedon force feedback to maintain desired contact behavior withenvironment. Our overall goal is that the system is able to• adapt its speed at high performance based on an ob-

server that monitors system passivity in real-time.• ensure overall system stability based on the continuous

transfer of energy between the involved subsystems.

B. Related workRelated work can be separated into two main topics:

1) stability in learning by imitation and 2) passivity-basedcontrol.

Learning by imitation has been tackled in different man-ners, one of which is with the use of dynamic movementprimitives (DMPs) [11], [12]. DMPs per se have been shownstable [11], however, this does not imply the stability of in-contact robotic operations. For example, stability of learning

2017 IEEE-RAS 17th International Conference onHumanoid Robotics (Humanoids)Birmingham, UK, November 15-17, 2017

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for maintaining environmental contact through DMP cou-pling terms has been separately investigated in [13]. DMPsare of course not the only means of encoding trajectories.For example, Gaussian Mixture Models (GMMs) can be used[14], [15]. GMMs were used to estimate the entire attractorlandscape of a movement skill from several demonstrations.While the authors showed how to maintain the stability ofthe dynamical system, they did not specifically tackle thestability of in-contact behavior of the system. More recently,variable stiffness was applied as one of possible approachesin combination with motion imitation to ensure safety ofinteraction [16].

In our current work we show how the overall systemstability of in-contact robot behaviors can be maintainedthrough observing the passivity of the system.

Passivity-based control in humanoid robots was mostlyconsidered for gait stabilization and posture control [17]–[19]. For impedance controlled robots passivity was investi-gated for contact free scenarios [20] and the case of regula-tion [21]. When considering following desired force profileswith impedance controlled robots, the recent approach ofunified force/impedance control proposed in [22] allows foraccurate force tracking and simultaneous impedance control.In the current paper, an admittance controller is used toadapt the previously learned trajectory to respect the desiredforce profile. Several works have proposed the combinationof admittance and motion control [23]–[25], among thosemost are for teleoperation applications and followed differ-ent goals. Moreover, although some have applied passivityanalysis and specifically the concept of reference energy [26],[27], none has modeled the overall system from an energyand passivity point of view such that the encoded power canbe interpreted as a system with its own storage function.Finally, the augmentation of virtual energy tanks has drawnsignificant attention for stability analysis in robotics [22],[28], [29], and is used in this paper for passivity proof aswell as robustness in task performance.

C. Contributions

The contributions of the present work are as follows.1) Passivity analysis for desired trajectory and wrench en-

coding based on the concept of demonstrated referencepower.

2) Design and validation of a passivity observer (PO) foronline phase altering of encoded trajectory and wrenchprofile.

3) Design and validation of a passivity enforcing virtualenergy tank for task robustness.

The remainder of the paper is organized as follows. InSection II the basic models and notations for CartesianImpedance controlled robots with DMP trajectory generationare reviewed. Section III explains the trajectory modificationscheme based on admittance control. Passivity analysis of theproposed control subsystems follows in Section IV where itis shown that trajectory adaptation based on force feedbackvia admittance control is not always passive. Section Vexplains how system passivity can be observed and accord-ingly reacted to via reference velocity adaptation in case ofviolation. Finally, a virtual tank is introduced to guaranteethe overall passivity. Experimental evaluation of the proposed

passive interaction based on a surface polishing task ispresented in Section VI, with conclusions given in SectionVII.

II. PRELIMINARIES

In this section, a Cartesian Impedance controlled robot forwhich the desired trajectory is encoded via the well knownDMP formulation is modeled1.

A. Rigid-body Robot ModelConsidering the robot pose x = (pT ,ϕT )T ∈ R6 where

p ∈ R3 is the translational part and ϕ ∈ R3 is the Eulerangles representing the rotational part, dynamics of rigidbody robots in Cartesian space with n degrees of freedommay be written as

MK(q)x+CK(q, q)x+ F g(q) = Fm + F ext, (1)

where q ∈ Rn denotes the link position and q ∈ Rn thelink velocity. F ext ∈ R6 is the external wrench appliedto the robot end-effector and obtained from wrist sensing.It consists of translational forces f ext ∈ R3 and rotationalmoments mext ∈ R3. The controller input τm ∈ Rn and therobot gravitational vector g(q) ∈ Rn relate to the Cartesianspace wrenches Fm,F g(q) ∈ R6 by

τm = JT (q)Fm (2)

g(q) = JT (q)F g(q), (3)

where J(q) is the robot Jacobian matrix. MK(q) ∈ Rn×nand CK(q, q) ∈ Rn×n are the inertia matrix and the Coriolisand centrifugal matrix of the robot in Cartesian space, whichare related to the equivalent joint quantities M(q) ∈ Rn×nand C(q, q) ∈ Rn×n via [31]

MK(q) = J#T

(q)M(q)J#(q) (4)

CK(q, q) =(J#T

C(q, q)−MK(q)J(q))J#(q). (5)

Here, the right-hand Moore-Penrose pseudo inverse2

J#(q) = JT (q)(J(q)JT (q))−1 is used for inverting theJacobian matrix.

B. Cartesian Impedance controlIn order to have a closed loop Cartesian impedance control

with the desired stiffness Kx ∈ R6×6, damping Dx ∈R6×6 and inertia identical to the robot inertia MK(q), theimpedance control law becomes [30]:

τm = −JT (q)[Kxx+Dx

˙x−MK(q)xd −CK(q, q)xd]

+ g(q), (6)

where x := x− xd and xd ∈ R6 is the desired robot pose.As a result, considering (1) and (6), the closed loop dynamicsbecomes

MK(q)¨x+ (CK(q, q) +Dx) ˙x+Kxx = F ext. (7)

At this point it shall be mentioned that due to the modularityof the passivity concept one may choose any other controllerthat together with the rigid-body robot results in a passivesystem w.r.t. the relative ports explained in Sec. IV-A.

1Please note that in this work, only rigid-body robots are considered. Formodeling flexible-joint robots see e.g. [30].

2If the left-hand pseudo inverse of J#(q) = (JT (q)J(q))−1JT (q)

is used, J#T(q) should be replaced by JT#

(q).

137

C. Dynamic Movement PrimitivesDynamic movement primitives are an elegant approach of

encoding trajectories, typically used in learning by demon-stration. In the following we provide a brief recap of periodicDMPs, which are used in our control scheme. Further detailsabout DMPs can be found [11].

A periodic DMP, which is in our case one of the task-spacevariables3 as introduced in Section II-A, is given by

z = Ω(αz(βz(g − xDMP)− z) + f(φ)), (8)xDMP = Ωz. (9)

The output of the DMP is xDMP, with g being the anchor ofoscillations. The periodic phase φ is the output of the phaseoscillator

φ = Ω, (10)

where Ω is the frequency of oscillations and f(φ) denotesthe periodic forcing term.

Encoding of the periodic forcing term uses a linear com-bination of N periodic basis functions Γ

f(φ) =

∑Ni=1wiΓi(φ)∑Ni=1 Γi(φ)

r, (11)

Γi(φ) = exp(hi(cos(φ− ci)− 1)), (12)

where r is the amplitude of the oscillator and hi > 0.Throughout this work, we use N = 25 and hi = 2.5N .The movement frequency Ω can be automatically determinedfrom the data [12], [33].

III. TRAJECTORY MODIFICATION VIA ADMITTANCECONTROL

While the Cartesian impedance controller (6) is responsi-ble for empowering the closed loop behavior (7), the higherlevel goal would be to adapt the trajectory reference xd withrespect to environmental contacts. A common example is torespect a desired end-effector wrench profile F d ∈ R6 whileperforming a desired motion. Obviously, if the desired posetrajectory is encoded as a DMP, the desired wrench repre-sentation should also share the same phase function as theDMP. Here, we propose that the according modification maybe done via the following standard admittance controller.

F d + F ext = Maxa +Daxa +Kaxa (13)

Its dynamics are governed by a mass-spring-damper behaviorwith Ma,Da,Ka being the desired inertia, damping andstiffness, respectively. Overall, the system has the inputF d + F ext and output xa ∈ R6, which encodes the DMPadaptation.

xd = xDMP + xa (14)

Figure 2 depicts the overall control architecture. It shouldbe mentioned that while any other trajectory generator couldbe used, here DMP’s are exploited due to their conveniencein reference motion encoding on the one hand, and on theother hand the intuitive way of reference velocity adaptationbased on the passivity criterion introduced in section V.

3Cartesian DMP formulation using quaternion notation of rotations wasintroduced in [32], which results in modified formulation of orientationDMPs. Since we omit orientations in this paper for sake of clarity we staywith the original notation.

Fig. 2: Block diagram of the overall system, consisting ofDMP and desired wrench which share the same phase func-tion, the admittance controller which modifies the referencetrajectory, the Cartesian impedance controller, the rigid bodyrobot and the environment. Moreover the effect of the energytank on the system is depicted as dotted lines.

IV. PRELIMINARY PASSIVITY ANALYSIS

In order to monitor stability of the overall system whileinteracting with environment, we rely on the concept ofpassivity-based control. As noted above one of the importantproperties of passivity analysis is that the proper interconnec-tion of passive subsystems leads to an overall passive system.After the following definition of passivity each subsystem isanalyzed and the overall system passivity is investigated inthe end.

Definition: A system with the state space model of χ =f(χ,u) with initial state of χ(0) = χ0 ∈ Rm, input vectoru ∈ Rl and output y = h(χ,u) is said to be passive, ifthere exists a positive semidefinite function S : Rm → R+,called storage function, such that [10]

S(χ(σ))− S(χ0) ≤∫ σ

0

uT (t)y(t)dt (15)

for all input signals u : [0, σ] → Rl, initial states χ0 ∈ Rmand σ > 0. Thus proving passivity of a system is equivalentto finding an appropriate storage function S(χ) such that

S ≤ uTy ∀(χ,u). (16)

A. Subsystem passivitya) Environment: In passivity-based analysis there is

no need for an exact system model, since it is commonlyassumed that the environment is passive w.r.t. the pair〈x,−F ext〉. In other words, there exists a storage functionSenv such that

Senv ≤ −xTF ext. (17)

b) Cartesian impedance controlled robot: Considering(7), the proposed storage function for the super-system ofCartesian impedance controller and rigid body robot is

Sm =1

2xTKxx+

1

2˙xTMK(q) ˙x. (18)

Its time derivative is

Sm = ˙xTKxx+1

2˙xTMK(q) ˙x+ ˙xTMK(q)¨x

= ˙xTKxx+1

2˙xTMK(q) ˙x

+ ˙xT(−CK(q, q) ˙x−Dx

˙x−Kxx+ F ext

)= ˙xTF ext − ˙xTDx

˙x︸ ︷︷ ︸≥0

+1

2˙xT(MK(q)− 2CK(q, q)

)˙x︸ ︷︷ ︸

=0

=⇒ Sm ≤ ˙xTF ext = xTF ext − xTd F ext, (19)

138

where the skew-symmetry of the matrix MK(q) −2CK(q, q) is taken into account. As a result, the Cartesianimpedance controlled robot is passive w.r.t. the pair 〈(x −xd),F ext〉. In case of variable stiffness (19) can not beachieved, and the time-derivative of Kx has to be considered[34].

c) Admittance control: The storage function for theadmittance controller (13) may be defined as

Sa =1

2xTaMaxa +

1

2xaKaxa. (20)

The time derivative of this function is

Sa =xTa (Maxa +Kaxa)

=xa(F d + F ext)− xTaDaaa︸ ︷︷ ︸≥0

=⇒ Sa ≤ xTaF d + xTaF ext. (21)

Consequently, the admittance control law system is passivew.r.t. the pair 〈xa, (F d + F ext)〉.

B. Passivity of overall systemBy defining the overall storage function as

Stot = Senv + Sm + Sa, (22)

passivity of the overall system as an integration of theaforementioned subsystems can be investigated by analyzingthe time derivative of Stot. Considering (17), (19) and (21),and replacing xd with its equivalence via (14) it can beconcluded that

Stot = Senv + Sm + Sa

≤ xTF ext − xTDMPF ext − xTaF ext + xTF ext

+ xTaF d + xTaF ext

≤ xTaF d − xTDMPF ext. (23)

Hence, the overall system is passive w.r.t. the pairs 〈xa,F d〉and 〈xDMP,−F ext〉, see Fig. 3.

The deduced passivity does not necessarily imply thestability of the system, as one should also pay attention tothe power being injected to the system through the ports〈xa,F d〉 and 〈xDMP,−F ext〉. We propose that this powershould be compared with a reference power trajectory thatis computed from the desired velocity and force commandsas shown next.

C. Reference energyConsidering the desired encoded motion x†DMP with its

desired task frequency Ω† and the desired wrench profile F †d,both being defined by the user, the desired power trajectorybecomes

P †in := xT†DMPF†d. (24)

P †in can be considered as power coming from a passivesystem with the storage function S† such that according to(16)

S† ≤ −P †in. (25)

Considering (23) and (25), it is straight-forward to see that

Stot + S† ≤ xTaF d − xTDMPF ext − P †in. (26)

Fig. 3: Port-based modelling of the subsystems and theirrelative power variables.

Thus, for the overall passivity it is enough to show that

xTaF d − xTDMPF ext − P †in ≤ 0. (27)

F d is assigned in the beginning, F ext and consequentlyxa are not modifiable4. P †in is always known after definingthe desired motion via (24). Therefore, the only possibilityto keep (27) valid5 is to modify the norm of xDMP throughits phase function by changing the task period.

In the next section, we introduce a new scheme to de-termine the phase function of xDMP such that the systemintends to improve performance subject to remaining passive.

V. SYSTEM PASSIFICATION

A. Passivity observer (PO)Based on (27), a passivity criterion can be found, by which

system passivity can be observed at all times. For this, thepower Pacv is defined as

Pacv(t) = xTa (t)F d(φ)− xTDMP(φ)F ext(t)− P †in(φ), (28)

where φ is obtained via (10). Based on the sign of Pacv(t)the proposed passivity observer becomes

Passive Pacv(t) ≤ 0

Non-passive Pacv(t) > 0.(29)

Note that ideally one can assume F ext = −F d, leading toxa = 0. Consequently, Pacv(t) = 0 holds ideally.

B. DMP phase-function assignmentThe output velocity of the DMP can be changed according

to (29) such that if passivity is being violated, xDMP isreduced to let the system tend back to show passive behavior.Moreover, if the system is already passive with some margin,xDMP may be increased. Such performance may be realizedby changing the phase function (10) through assigning a taskfrequency Ω via

Ω = Ω† −KPacv, (30)

where Ω† denotes the original task frequency. Ω is theprimary assigned task frequency and K is a positive-definitescalar quantity. As shown in Sec. VI, using this approachwill bring Pacv towards zero. However, this policy is notnecessarily sufficient to ensure overall system passivity. Notethat instead of DMP other trajectory generators could be usedif their velocity can be accessed similarly.

4In this paper we do not consider variable-gain admittance control.5as long as F ext is not in the null-space of xDMP.

139

Fig. 4: The super-system with storage function SΣ consistingof energy tank, reference energy and overall system of admit-tance control + impedance controlled robot + environment.

C. Energy tank augmentationChanging DMP output velocity via (30) does not necessar-

ily ensure passivity of system. In order to guarantee passivitya virtual energy tank is introduced that takes care of potentialpassivity violating powers in (28). The tank is designed suchthat it provides the system with potentially demanded extrapower (i.e. Pacv(t) > 0) up to a limited amount of energy.This way, not only passivity is preserved, but there existsan extra robustness margin for system in case a reasonableamount of additional energy is required to perform the task.Finally, if this energy is not enough, the velocity decreasesuntil system stop.

The tank energy E associated to its system state xt isdefined as

Et =1

2x2t . (31)

The tank dynamics are defined as

xt =αtxt

(γt − 1)Pacv −βtxtγtPacv, (32)

where γt is defined as

γt =

1 if Pacv ≤ 0

0 else.(33)

βt is responsible to keep the energy tank below the upperlimit Eup and is defined as

βt =

1 if Et < Eup

0 else.(34)

αt keeps the energy tank above the lower limit Elow and isdefined as

αt =

1 if Elow + δE ≤ Et12

[1− cos

(Et−Elow

δEπ)]

if Elow ≤ Et < Elow + δE

0 else,(35)

where δE is the threshold above the lower limit, in whichthe possible energy that can be taken from the tank reducesto zero when Et = Elow.

By setting the initial tank energy Et,0 = Et(t = 0) thetank scales the task frequency as follows.

Ω = αtΩ (36)

Ω is the final DMP task frequency. Moreover, as dynamics ofthe tank also depend on the output velocity of the admittance

Fig. 5: Block diagram of the phase function assignment algo-rithm including the passivity observer (PO), virtual energytank, its upper and lower limit variables, and primary andfinal frequency assignment

control xa, the tank energy level will also have an effect onthe admittance controller via

˙xa = αtxa, (37)

where xa is the admittance control output obtained from(13), and xa is the final value before sending to the systemvia (14). Hence, if there is not enough energy in the tank, αtwill decrease to zero, and consequently the output velocityof the DMP and admittance controller will decay to zero,eventually stopping the whole system.

Considering (32) to (37), as long as βt 6= 0, it is straight-forward to see that

Et = −Pacv. (38)

Thus, a super-system consisting of the overall system storagefunction Stot, the reference energy S† and tank energy Etcan eventually be defined, see Fig. 4:

SΣ = Stot + S† + Et (39)

Considering (23), (25), (28) and (38) one gets

SΣ = Stot + S† + Et

≤ xTaF d − xTDMPF ext − P †in︸ ︷︷ ︸Pacv

−Pacv = 0. (40)

Moreover, if βt = 0 the tank power Et becomes

Et =

0 if Pacv ≤ 0

−Pacv else(41)

and (40) turns intoSΣ ≤ Pacv if Pacv ≤ 0

SΣ ≤ 0 else=⇒ SΣ ≤ 0 (42)

Thus, the super-system is passive. The phase modificationapproach and the tank effect are depicted in Fig. 5.

VI. EXPERIMENTAL RESULTS

The proposed method is tested on a 7 DOF Kuka LWR-4 impedance controlled robot, see Fig. 8. The consideredchallenge is to polish a surface, wich location may alterduring process execution. External wrenches F ext are mea-sured with an additional ATI force/torque sensor mountedon the wrist of the robot arm. In this experiment the taskis to maintain the normal contact force Fd = 10 N with

140

the table. Therefore, only forces in global z−direction arerelevant. The upper and lower energy limit of the tank werechosen to be Eup = 0.6 J and Elow = 0 J.

0 20 40 60 80 1000

0.5

1

Rob

otpos.[m

]

Sec.1 Sec.2 Sec.3 Sec.4

Robotpos x Robotpos y Robotpos z

0 20 40 60 80 100

-0.2

0

0.2

Ref.DMP

vel.[m

/s]

DMPvel x DMPvel y DMPvel z

0 20 40 60 80 100-0.05

0

0.05

0.1

Ref.Addmitan

cevel.[m

/s]

Admvel x Admvel y Admvel z

0 20 40 60 80 100Time[s]

5

10

15

20

Forces[N

] Fext z Fd z

Fig. 6: Motion and force data for the wiping task experiment

In the first phase of the experiment the desired polishingmotion was demonstrated to the robot according to [9],encoded with periodic DMPs, and executed, see Fig. 8a.For better understanding the experimental results are dividedinto sections (Sec.) shown in Figs. 6, 7, 8, and in theaccompanying video. Section 1 of the experiment shows theinitial, stable, periodic polishing motion. Pacv remains 0,the system passivity is preserved and no energy is drainedfrom the tank. In Section 2 one corner of the table is lifted,introducing a change to the environment. Passivity (28) isviolated and the DMP is slowed down. This gives time tothe admittance controller to adapt to the change. We can seefrom the bottom plot in Fig. 7 that the tank energy is partiallydrained and filled up again in the same period of the motion,preserving the overall tank energy.

Section 3 depicts a stable motion. The other corner ofthe table is lifted, introducing a different change to theenvironment, see Fig. 8c. Passivity is preserved. In theadaptation process some energy from the tank was injected into the system. Therefore, the overall tank energy has reduced.

Section 4 shows a strong passivity violation, where onecorner of the table is rapidly lifted. The tank energy isimmediately fully drained and the system stops, preventingdamage from the robot and the environment, respectively.

VII. CONCLUSIONOverall, the presented method gives a robot the ability to

follow desired wrench profiles in rather unknown environ-ments by provably stable trajectory adaptation. The methodconsists of following parts:

0 20 40 60 80 1000

1

2

Omega[rad

/s]

Sec.1 Sec.2 Sec.3 Sec.4

0 20 40 60 80 100-0.5

0

0.5

Passivitysign

al

0 20 40 60 80 100Time[s]

0

0.2

0.4

0.6

Tan

kenergy

[J]

Tanke EnergyMax tank energy limitLow tank energy limit

Fig. 7: Phase and energy data for the wiping task experiment.

Based on a new passivity criterion an observer is designedthat monitors the overall system passivity. This scheme isused to design a DMP phase control law, altering the execu-tion speed and thus allowing system performance improve-ment in case passivity is ensured. Finally, this requirementis enforced by a virtual energy tank by modification of thetask frequency of the DMP and the admittance controlleroutput. The successful experiments underline the validity ofthe scheme.

ACKNOWLEDGMENT

We greatly acknowledge the funding of this work by theAlfried Krupp von Bohlen und Halbach Foundation and FoF-RIA Project AUTOWARE (No. 723909).

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(a) Sec. 1: Robot executesdemonstrated wiping motion.

(b) Sec. 2: Environment haschanged in comparison todemonstrated motion. Therobot has to adapt to thechange.

(c) Sec. 3: Robot adapts to thechange, tank energy and passiv-ity ensured.

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Fig. 8: Humanoid robot polishing a previously unknown environment.

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