Learning and Generalization of Motor Skillsby Learning from Demonstration

Peter Pastor, Heiko Hoffmann, Tamim Asfour, and Stefan Schaal

Abstract— We provide a general approach for learningrobotic motor skills from human demonstration. To representan observed movement, a non-linear differential equation islearned such that it reproduces this movement. Based on thisrepresentation, we build a library of movements by labelingeach recorded movement according to task and context (e.g.,grasping, placing, and releasing). Our differential equation isformulated such that generalization can be achieved simply byadapting a start and a goal parameter in the equation to thedesired position values of a movement. For object manipulation,we present how our framework extends to the control of gripperorientation and finger position. The feasibility of our approachis demonstrated in simulation as well as on a real robot. Therobot learned a pick-and-place operation and a serving-watertask and could generalize these tasks to novel situations.

I. INTRODUCTION

Humanoid robots assisting humans can becomewidespread only if the humanoids are easy to program.Easy programming might be achieved through learning fromdemonstration [1], [2]. A human movement is recordedand later reproduced by a robot. Three challenges needto be mastered for this imitation: the correspondenceproblem, generalization, and robustness against perturbation.The correspondence problem means that links and jointsbetween human and robot may not match. Generalizationis required because we cannot demonstrate every singlemovement that the robot is supposed to make. Learning bydemonstration is feasible only if a demonstrated movementcan be generalized to other contexts, like different goalpositions. Finally, we need robustness against perturbation.Replaying exactly an observed movement is unrealistic ina dynamic environment, in which obstacles may appearsuddenly.

To address these issues, we present a model that is basedon the dynamic movement primitive (DMP) framework (see[3], [4], [5], [6]). In this framework, any recorded movementcan be represented with a set of differential equations.Representing a movement with a differential equation has theadvantage that a perturbance can be automatically correctedfor by the dynamics of the system; this behavior addressesthe above mentioned flexibility. Furthermore, the equationsare formulated in a way that adaptation to a new goalis achieved by simply changing a goal parameter. Thischaracteristic allows generalization. Here, we will present

H. H. is supported by DFG, T.A. is supported by the University ofKarlsruhe, and S. S. is supported by DARPA, NASA, NSF.

Peter Pastor, Heiko Hoffmann, and Stefan Schaal are with the Uni-versity of Southern California, Los Angeles, USA. Tamim Asfour iswith the University of Karlsruhe, Germany. Email: pastorsa@usc.edu,heiko@clmc.usc.edu, asfour@ira.uka.de, sschaal@usc.edu

a new version of the dynamic equations that overcomesnumerical problems with changing the goal parameter thatoccurred in the previous formulation [3], [4], [5], [6].

We will use the dynamic movement primitives to representa movement trajectory in end-effector space; thus, we addressthe above-mentioned correspondence problem. For objectmanipulation – here, grasping and placing – besides the end-effector position, we also need to control the orientationof the gripper and the position of the fingers. The DMPframework allows to combine the end-effector motion withany further degree-of-freedom (DOF); thus, adding gripperorientation in quaternion notation and finger position isstraight-forward. In our robot demonstration, we use standardresolved motion rate inverse kinematics to map end-effectorposition and gripper orientation onto the appropriate jointangles [7].

To deal with complex motion, the above framework canbe used to build a library of movements primitives out ofwhich complex motion can be composed by sequencing.For example, the library may contain a grasping, placing,and releasing motion. Each of these movements, which wasrecorded from human demonstration, is represented by adifferential equation, and labeled accordingly. For movingan object on a table, a grasping-placing-releasing sequence isrequired, and the corresponding primitives are recalled fromthe library. Due to the generalization ability of each dynamicmovement primitive, an object may be placed between twoarbitrary positions on the table based solely on the threedemonstrated movements.

In the remainder of this article, we explain in Section II thedynamic movement primitive framework and present the newmodified form. In Section III we emphasize the generationof a library of movements. In Section IV we present anapplication of the framework on a simulated as well as on areal robot arm. In Section V we conclude this approach andprovide an outlook for future work.

II. DYNAMIC SYSTEMS FOR MOVEMENTGENERATION

This section briefly describes the dynamic movementprimitive framework and presents a modification to allowadaptation to a new goal position in a more robust andhuman-like way.

A. Dynamic Movement Primitives

Dynamic movement primitives can be used to generatediscrete and rhythmic movements. Here, we focus on discretemovements. A one dimensional movement is generated by

integrating the following set of differential equations1, whichcan be interpreted as a linear spring system perturbed by anexternal forcing term:

τ v = K(g − x) −Dv + (g − x0)f (1)

τx = v , (2)

wherex and v are position and velocity of the system;x0

andg are the start and goal position;τ is a temporal scalingfactor;K acts like a spring constant; the damping termDis chosen such that the system is critically damped, andfis a non-linear function which can be learned to allow thegeneration of arbitrary complex movements. This first setof equations is referred to as a transformation system. Thenon-linear function is defined as

f(θ) =

∑

iwiψi(θ)θ

∑

iψi(θ)

, (3)

whereψi(θ) = exp(−hi(θ − ci)2) are Gaussian basis func-

tions, with centerci and width hi, andwi are adjustableweights. The functionf does not directly depend on time;instead, it depends on a phase variableθ, which monotoni-cally changes from1 towards0 during a movement and isobtained by the equation

τ θ = −α θ , (4)

where α is a pre-defined constant. This last differentialequation is referred to as canonical system. These sets ofequations have some favorable characteristics:

• Convergence to the goalg is guaranteed (for boundedweights) sincef(θ) vanishes at the end of a movement.

• The weightswi can be learned to generate any desiredsmooth trajectory.

• The equations are spatial and temporal invariant, i.e.,movements are self-similar for a change in goal, startpoint, and temporal scaling without a need to change theweightswi.

• The formulation generates movements which are robustagainst perturbation due to the inherent attractor dynam-ics of the equations.

To learn a movement from demonstration, first, a move-ment y(t) is recorded and its derivativesy(t) and y(t) arecomputed for each time stept = 0, . . . , T . Second, thecanonical system is integrated, i.e.,θ(t) is computed foran appropriately adjusted temporal scaling tau. Using thesearrays,ftarget(θ) is computed based on (1) according to

ftarget(θ) =−K(g − y) +Dy + τ y

g − x0, (5)

where x0 and g are set toy(0) and y(T ), respectively.Thus, finding the weightswi in (3) that minimizes the errorcriterion J =

∑

θ

(

ftarget(θ) − f(θ))2

is a linear regressionproblem, which can be solved efficiently.

A movement plan is generated by reusing the weightswi,specifying a desired startx0 and goalg, setting θ = 1,

1We use a different notation as in [3], [4] to high-light the spring-likecharacter of these equations.

and integrating the canonical system, i.e. evaluatingθ(t).As illustrated in Fig. 1, the obtained phase variable thendrives the non-linear functionf which in turn perturbs thelinear spring-damper system to compute the desired attractorlandscape.

Fig. 1. Sketch of a one dimensional DMP: the canonical system drivesthe nonlinear functionf which perturbs the transformation system.

B. Generalization to New Goals

In this section, we describe how to adapt the movementto a new goal position by changing the goal parameterg.The original DMP formulation has three drawbacks: first,if start and goal position,x0 andg, of a movement are thesame, then the non-linear term in (1) cannot drive the systemaway from its initial state; thus, the system will remain atx0. Second, the scaling off with g − x0 is problematic ifg−x0 is close to zero; here, a small change ing may lead tohuge accelerations, which can break the limits of the robot.Third, whenever a movement adapts to a new goalgnew suchthat (gnew− x0) changes its sign compared to (goriginal − x0)the resulting generalization is mirrored. As an example fromour experiments, a placing movement on a table has start andgoal positions with about the same height; thus, the originalDMP formulation is not suitable for this kind of movementadaptation.

Y1

Y2

startgoal

Y2

Y1

start goal

Y1

Y2

goalstart

online adaptationafter 300ms

Y1

Y2

online adaptationafter 300ms

start goal

Fig. 2. Comparison of goal-changing results between old (Left) andnew (Right) DMP formulation in operational space(Y1, Y2) with onetransformation system for each dimension. The same original movement(solid line) and goals are used for both formulations. The dashed lines showthe result of changing the goal before movement onset (Top) and during themovement (Bottom).

Here, we present a modified form of the DMPs thatcures these problems (see Fig. 2), while keeping the same

favorable properties, as mentioned above. We replace thetransformation system by the following equations [8]:

τ v = K(g − x) −Dv −K(g − x0)θ +Kf(θ) (6)

τx = v , (7)

where the non-linear functionf(θ) is defined as before. Weuse the same canonical system as in (4). An important dif-ference to (1) is that the non-linear function is not multipliedanymore by(g−x0). The third termK(g−x0)θ is requiredto avoid jumps at the beginning of a movement. Learning andpropagating DMPs is achieved with the same procedure asbefore, except that the target functionftarget(θ) is computedaccording to

ftarget(θ) =τ y −D y

K− (g − y) + (g − x0) θ . (8)

The trajectories generated by this new formulation fordifferent g values are shown in Fig. 2. In our simulationand robot experiments we use this new formulation.

C. Obstacle Avoidance

A major feature of using dynamic systems for movementrepresentation is robustness against perturbation [3]. Here,we exploit this property for obstacle avoidance [9], [10] byadding a coupling termp(x,v) to the differential equationsof motion

τ v = K(g−x)−Dv−K(g−x0)θ+Kf(θ)+p(x,v) . (9)

We describe obstacle avoidance in 3D end-effector space,therefore the scalarx, v, v turn into vectorsx,v, v and thescalarsK,D became positive definite matricesK,D. Forthe experiment in this paper we used the coupling term

p(x,v) = γRvϕexp(−cϕ) , (10)

whereR is a rotational matrix with axisr = (x−o)×v andangle of rotation ofπ/2; the vectoro is the position of theobstacle,γ and c are constant, andϕ is the angle betweenthe direction of the end-effector towards the obstacle and theend-effector’s velocity vectorv relative to the obstacle. Theexpression (10) is derived from [11] and empirically matcheshuman obstacle avoidance.

III. BUILDING A LIBRARY OF MOVEMENTS

This section briefly motivates the concept of a libraryof movements and their application in object manipulationtasks.

A. Motion Library Generation

Learning DMPs only requires the user to demonstratecharacteristic movements. These DMPs form a set of basicunits of action [1]. For movement reproduction only a simplehigh level command - to choose a primitive (or a sequenceof them) and set its task specific parameters - is required.Moreover, adaption to new situations is accomplished by ad-justing the startx0, the goalg, and the movement durationτ .Thus, a collection of primitives referred to asmotion libraryenables a system to generate a wide range of movements.

Fig. 3. Conceptual sketch of an imitation learning system (adapted from[1]). The components of perception (yellow) transform visual informationinto spatial and object information. The components of action (red) generatemotor output. Interaction between them is achieved using a common motionlibrary (blue). Learning (green) improves the mapping between perceivedactions and primitives contained in the motion library for movementrecognition and selection of the most appropriate primitive for movementgeneration.

On the other side, such a motion library can be employed tofacilitate movement recognition in that observed movementscan be compared to the pre-learned ones [3]. If no existingprimitive is a good match for the demonstrated behavior,a new one is created (learned) and added to the system’smovement repertoire (see Fig. 3). This makes the presentedformulation suited for imitation learning.

B. Attaching Semantic

As for imitation learning with DMPs a low-level approach,namely imitation of trajectories [2], was chosen, additionalinformation is needed by the system to successfully performobject manipulation tasks. For a pick-and-place operationfor example the system has to select a proper sequence ofmovement primitives, that is, first a grasping, then a placingand finally a releasing primitive. Therefore, it is necessaryto attach additional information to each primitive movementwhich facilitates this selection. Moreover, once a library ofmovement primitives is acquired, it is desirable to have thesystem able to search sequences of primitive movements thataccomplish further tasks. Traditional artificial intelligenceplanning algorithms tackle this problem by formalizing thedomain scenario. In particular, they define a set of operatorswith pre- and post-conditions and search for a sequenceof them which transfers the world from its initial state

Fig. 4. Objects are defined through actions that can be performed on them(Left), e.g. a cup is represented as a thing which can be used to drink waterfrom. On the other side, actions are defined through objects (Right), e.g. theway of grasping an object depends on the object - a can requires a differentgrip than a pen.

to the goal state. The post-conditions provides informationabout the change in the world, whereas the preconditionsensure that the plan is executable. Thus, such algorithmsare based on discrete symbolic representations of object andaction, rather than the low-level continuous details of actionexecution.

A link between the low-level continuous control repre-sentation (as typical in robotic applications) and high-levelformal description of actions and their impact on objects (asnecessary for planning) has been, for example, formalized byObject-Action Complexes [12], [13]. This concept proposesthat objects and actions are inseparably intertwined (seeFig. 4).

C. Combination of Movement Primitives

The ability to combine movement primitives to generatemore complex movements is a prerequisite for the conceptof a motion library. Here, we show how the presentedframework provides this ability.

It is straight forward to start executing a DMP after thepreceding DMP has been executed completely, since theboundary conditions of any DMP are zero velocity andacceleration. However, DMPs can also be sequenced suchthat complete stops of the movement system are avoided(see Fig. 5). This is achieved by starting the execution of thesuccessive DMP before the preceding DMP has finished. Inthis case, the velocities and accelerations of the movementsystem between two successive DMPs are not zero. Jumpsin the acceleration signal are avoided by properly initializingthe succeeding DMP with the velocities and positions of itspredecessor (vpred→ vsucc andxpred → xsucc).

Y1

Y2

(b)

(a)

Fig. 5. Chaining of a single minimum jerk movement primitive generalizedto four goals (black dots) resulting in a square like movement (a). Themovement generated by the DMPs are drawn alternating with blue solid andred dashed lines to indicate the transition between two successive DMPs.The movement direction is indicated by the arcs. The remaining movements(b) result from using different switching times (lighter color indicates earlierswitching time).

IV. EXPERIMENT

The following Section describes how we applied thepresented framework of DMPs on the Sarcos Slave arm toaccomplish object manipulation tasks, such as grasping andplacing.

A. Robot Setup

As experimental platform we used a seven DOF anthropo-morphic robot arm (see Fig. 6) equipped with a three DOFend-effector.

Fig. 6. Sketch of the Sarcos Slave arm, a seven DOF anthropomorphicrobot arm with a three DOF end-effector.

B. Learning DMPs from Demonstration

Learning DMPs from demonstration is achieved by regard-ing each DOF separately and employing for each of them anindividual transformation system. Thus, each DMP is setupwith a total of ten transformation systems to encode eachkinematic variable. In particular, the involved variables arethe end-effector’s position (x, y, z) in Cartesian space, theend-effector’s orientation (q0, q1, q2, q3) in quaternion space,and finger position(θTL, θTV , θFAA) in joint space. Each ofthem serve as separated learning signal, regardless of theunderlying physical interpretation. However, to ensure theunit length of the quaternionq, a post-normalization stepis incorporated. The setup is illustrated in Fig. 7, note, a

Fig. 7. Sketch of the 10 dimensional DMP used to generate movementplans for the Sarcos Slave arm.

single DMP encodes movements in three different coordinateframes simultaneously.

To record a set of movements, we used a 10 DOFexoskeleton robot arm, as shown in Fig. 8. Visual observa-tion and appropriate processing to obtain the task variableswould be possible, too, but was avoided as this perceptualcomponent is currently not the focus of our research.

Fig. 8. Sarcos Master arm used to record a human trajectory in end-effector space. Here, the subject demonstrates a pouring movement whichafter learning the DMP enabled a robot to pour water into several cups (seeFig. 12).

The end-effector position and orientation are recorded at480 Hz. The corresponding trajectories for the finger move-ments are generated afterwards accordingly: for a graspingmovement, for example, a trajectory was composed out oftwo minimum jerk movements for opening and closing thegripper. The corresponding velocities and accelerations forall DOF were computed numerically by differentiating theposition signal.

These signals served as input into the supervised learningprocedure described in II-A. For each demonstrated move-ment a separate DMP was learned and added to the motionlibrary.

C. Movement Generation

To generate a movement plan, a DMP is setup with the taskspecific parameters, i.e., the startx0 and the goalg. In ourDMP setup (see Fig. 7), this are the end-effector position,end-effector orientation, and the finger joint configuration.The startx0 of a movement is set to the current state ofthe robot arm. The goalg is set according to the context ofthe movement. For a grasping movement, the goal position(x, y, z) is set to the position of the grasped object and thegrasping width is set according to the object’s size. However,finding appropriate goal orientation is not straight forward,as the end-effector orientation needs to be adapted to thecharacteristic approach curve of the movement. Approachingthe object from the front results in a different final posture asapproaching it from the side. In case of a grasping movement,we developed a method to automatically determine the finalorientation by propagating the DMP to generate the Cartesianspace trajectory and averaging over the velocity vectors tocompute the approach direction at the end of the movement.For other movements, like placing and releasing, we setthe end-effector orientation to the orientation recorded fromhuman demonstration. Finally, we useτ to determine theduration of a each movement.

In simulation we demonstrate the reproduction and gen-eralization of the demonstrated movements. Our simulatedrobot arm has the same kinematic and dynamic propertiesas the Sarcos Slave arm. The reproduction of grasping and

Fig. 9. Snapshots of theSL Simulator showing a simulation of the SarcosSlave arm performing a grasping (Top) and a placing movement (Bottom).

0.36

0.51

0.65

0.24

0.39

0.54

0.05

0.1

z[m

]

x [m]

y [m]0.0

0.25

0.5

0.4

0.5

0.6

0.2

0.1

x [m]

y [m]

z[m

]

Fig. 10. The desired trajectories (blue lines) from the movements shownin Fig. 9 adapted to new goals (red lines) indicated by the grid.

placing are show in Fig. 9. The generalization of thesemovements to new targets is shown in Fig. 10.

D. Task Space Control

To execute DMPs on the simulated and on the realrobot we used a velocity based inverse kinematics controlleras described in [14], [7]. Thus, the task space referencevelocitiesxr are transformed into the reference joint spacevelocities θr (see Fig. 11). The reference joint positionθr

and accelerationθr are obtained by numerical integrationand differentiation of the reference joint velocitiesxr. Thedesired orientation, given by the DMP as unit quaternions,is controlled using quaternion feedback as described in [15],[7].

Fig. 11. DMP control diagram: the desired task space positions andvelocities arexd , xd , the reference task space velocity command isxr ,the reference joint positions, joint velocities and joint accelerations areθr,θr, and θr .

The reference joint position, velocities and accelerationare transformed into appropriate torque commandsu usinga feed-forward and a feedback component. The feed-forwardcomponent estimates the corresponding nominal torques tocompensate for all interactions between the joints, while thefeedback component realizes a PD controller.

Fig. 12. Movement reproduction and generalization to new goal with the Sarcos Slave Arm. The top row shows the reproduction of a demonstratedpouring movement in Fig. 8, and the bottom row shows the result of changing the goal variable.

E. Robot Experiment

We demonstrate the utility of our framework in a robotdemonstration of serving water (see Fig. 12). First, a humandemonstrator performed a grasping, pouring, retreating bot-tle, and releasing movement as illustrated in Fig. 8. Second,the robot learned these movements and added them to themotion library. Third, a bottle of water and three cups wereplaced on the table. Fourth, an appropriate sequence ofmovement primitives were chosen. Fifth, each DMP weresetup with corresponding goalg. Finally, the robot executedthe sequence of movements and generalized to different cupposition simply through changing the goalg of the pouringmovement.

To demonstrate the framework’s ability to online adaptto new goals as well as avoiding obstacles, we extendedthe experimental setup with a stereo camera system. Weused a color based vision system to visually extract the goalposition as well as the position of the obstacle. The task wasto grasp a red cup and place it on a green coaster, whichchanges its position after movement onset, while avoidinga blue ball-like obstacle (see Fig. 13). To accomplish thistask a similar procedure was used as before. Except, thistime, the Cartesian goal of the grasping movement wasset to the position of the red cup and the goal of theplacing movement was set to the green coaster. The goalorientation for the grasping movement were set automaticallyas described in Section IV-C, whereas the orientation ofthe placing and releasing were adopted from demonstration.This setup allows us to demonstrate the framework’s abilityto generalize the grasping movement by placing the redcup on different initial positions. Our robot could adaptmovements to goals which change their position during therobot’s movement. Additionally, movement trajectories wereautomatically adapted to avoid moving obstacles (see Fig. 13and video supplement).

V. CONCLUSIONS AND FUTURE WORK

A. Conclusions

This paper extended the framework of dynamic movementprimitives to action sequences that allow object manipu-lation. We suggested several improvements of the originalmovement primitive framework, and added semantic infor-mation to movement primitives, such that they can codeobject oriented action. We demonstrated the feasibility ofour approach in an imitation learning setting, where a robotlearned a serving water and a pick-and-place task fromhuman demonstration, and could generalize this task to novelsituations.

The approach is not restricted to the presented exper-imental platform. Any type of motion capturing systemthat is capable of extracting the end-effector’s position andorientation can substitute the Sarcos Master arm and anymanipulator that is able to track a reference trajectory intask space can substitute the Sarcos Slave arm.

B. Future Work

Future work will significantly extend the movement librarysuch that a rich movement repertoire can be represented. Fur-thermore, work will focus on associating objects with actions(similar to [13]) to enable planning of action sequences. Weintend to use reinforcement learning to learn the high levelparameters, such as switching time between two successivemovement primitives, to generate smooth transitions whichstill accomplish a given task. Finally, we will apply thisextended framework on a humanoid robot.

VI. ACKNOWLEDGMENTS

This research was supported in part by National Sci-ence Foundation grants ECS-0325383, IIS-0312802, IIS-0082995, ECS-0326095, ANI-0224419, the DARPA programon Learning Locomotion, a NASA grant AC98-516, an

Fig. 13. Sarcos Slave arm placing a red cup on a green coaster. The first row shows the placing movement on a fixed goal. The second row shows theresulting movement as the goal changes (white dashed arc) after movement onset. The third row shows the resulting movement as a blue ball-like obstacleinterferes the placing movement.

AFOSR grant on Intelligent Control, the ERATO KawatoDynamic Brain Project funded by the Japanese Science andTechnology Agency, the ATR Computational NeuroscienceLaboratories, DFG grant HO-3887/1-1, and partially by theEU Cognitive Systems project PACO-PLUS (FP6-027657).

REFERENCES

[1] S. Schaal, “Is imitation learning the route to humanoid robots?” inTrends in Cognitive Sciences, 1999.

[2] A. Billard and R. Siegwart, “Robot Learning from Demonstration,”Robotics and Autonomous Systems, vol. 47, pp. 65–67, 2004.

[3] A. J. Ijspeert, J. Nakanishi, and S. Schaal, “Movement Imitation withNonlinear Dynamical Systems in Humanoid Robots,” inProceedingsof the IEEE International Conference on Robotics and Automation(ICRA), 2002.

[4] ——, “Learning Attractor Landscapes for Learning Motor Primitives,”in Advances in Neural Information Processing Systems 15 (NIPS),2003.

[5] S. Schaal, J. Peters, J. Nakanishi, and A. J. Ijspeert, “Control, Planning,Learning, and Imitation with Dynamic Movement Primitives,” inWorkshop on Bilateral Paradigms on Humans and Humanoids, IEEEInternational Conference on Intelligent Robots and Systems (IROS),2003.

[6] S. Schaal, A. J. Ijspeert, and A. Billard, “Computational Approachesto Motor Learning by Imitation,” inPhilosophical Transaction of theRoyal Society of London: Series B, Biological Sciences, 2003.

[7] J. Nakanishi, R. Cory, M. Mistry, J. Peters, and S. Schaal, “Operationalspace control: a theoretical and emprical comparison,”InternationalJournal of Robotics Research, vol. 27, pp. 737–757, 2008.

[8] H. Hoffmann, P. Pastor, and S. Schaal, “Dynamic movement primitivesfor movement generation motivated by convergent force fields in frog,”in Fourth International Symposium on Adaptive Motion of Animals andMachines, R. Ritzmann and R. Quinn, Eds., Case Western ReserveUniversity, Cleveland, OH, 2008.

[9] D. Park, H. Hoffmann, and S. Schaal, “Combining dynamic movementprimitives and potential fields for online obstacle avoidance,” inFourth International Symposium on Adaptive Motion of Animals andMachines, R. Ritzmann and R. Quinn, Eds., Case Western ReserveUniversity, Cleveland, OH, 2008.

[10] D. Park, H. Hoffmann, P. P., and S. Schaal, “Movement reproductionand obstacle avoidance with dynamic movement primitives and poten-tial fields,” in IEEE International Conference on Humanoid Robotics,J. H. Oh, Ed., Daejeon, Korea, submitted.

[11] B. R. Fajen and W. H. Warren, “Behavioral dynamics of steering,obstacle avoidance, and route selection,”Journal of ExperimentalPsychology: Human Perception and Performance, vol. 29, no. 2, pp.343–362, 2003.

[12] T. Asfour, K. Welke, A. Ude, P. Azad, J. Hoeft, and R. Dillmann,“Perceiving Objects and Movements to Generate Actions on a Hu-manoid Robot.” inIEEE International Conference on Robotics andAutomation (ICRA), 2007.

[13] C. Geib, K. Mourao, R. Petrick, N. Pugeault, M. Steedman,N. Krueger, and F. Woergoetter, “Object Action Complexes as anInterface for Planning and Robot Control,” inProceedings of the IEEE-RAS International Conference on Humanoid Robots (Humanoids),2006.

[14] J. Nakanishi, M. Mistry, J. Peters, and S. Schaal, “Experimentalevaluation of task space position/orientation control towards compliantcontrol for humanoid robots,” inProceedings of the IEEE InternationalConference on Intelligent Robotics Systems, 2007, pp. 1–8.

[15] J. S. Yuan, “Closed-loop manipulator control using quaternion feed-back,” IEEE Journal of Robotics and Automation, vol. 4, no. 4, pp.434–440, 1988.