Hybrid Force/Position Control of a Very Flexible Parallel RobotManipulator in Contact with an Environment

Fatemeh Ansarieshlaghi a and Peter Eberhard b

Institute of Engineering and Computational Mechanics, University of Stuttgart,Pfaffenwaldring 9, 70569 Stuttgart, Germany

{fatemeh.ansari, peter.eberhard}@itm.uni-stuttgart.de

Keywords: Flexible Parallel Robot, Position Control, Force Control, Impedance Control

Abstract: There are many applications for robot manipulators and these tasks are complicated when they have interactionwith environments and humans. This paper investigates a novel strategy to control a very flexible parallelmanipulator interacting with an environment. The controller is complicated when the used robot manipulatoris a flexible multibody dynamics system and the flexibility shall be taken into account in the modeling andcontrolling process. Also, interaction with an unknown environment is another challenge of our research.Hence, a sophisticated controller is designed to overcome the respective challenges. To this end, a hybridforce/position control strategy is utilized. Therefore, two controllers based on the rigid and flexible modelsof the robot are designed and implemented to interact with a surface. The simulation results show that thecontrollers based on the flexible model have a better performance than those based on the rigid model andsuccessfully track a trajectory and interact with an environment.

1 INTRODUCTION

Light-weight manipulators attract a lot of researchinterest because of their complementing advantages.The advantages of light-weight robots include low en-ergy usage, less mass, and often high working speeds.However, due to the light-weight design, the bod-ies have a significant flexibility which yields unde-sired deformations and vibrations. These manipula-tors can be modeled as flexible multibody systemsand their flexibility must be taken into account in thecontroller design. The flexible multibody system withsignificant deformations complicates the control de-sign because there are more generalized coordinatesthan control inputs. This complicated controller hasa more complex design when the robot interacts withan environment. The controller shall be able to regu-late the interaction force and move along the surfaceat the same time. In order to obtain a high perfor-mance in the end-effector’s contact part, an accurateand efficient model as well as a nonlinear controller isnecessary.

In this research, to investigate the controller a flex-ible robot available in hardware in our laboratory is

a https://orcid.org/0000-0003-2693-0882b https://orcid.org/0000-0003-1809-4407

used. This robot has a λ-shape and its links arevery flexible. In previous researches on this robot,only its modeling and position control were inves-tigated. The modeling is described in (Burkhardtet al., 2013a). Also, an experimental study on themodeling and the modeling improvement were donein (Burkhardt et al., 2014) and (Ansarieshlaghi andEberhard, 2017). The position control of this systemcan be done by a model-based or non-model-basedfeedback controller. Non-model-based feedback con-trollers in joint space were investigated in (Burkhardtet al., 2014) and (Morlock et al., 2016) and the flexi-ble model based nonlinear feedback controller in jointspace was investigated in (Ansarieshlaghi and Eber-hard, 2018b). An experimental study of the rigidmodel-based controller on the lambda robot in theworkspace was also done in (Morlock et al., 2017).

In contrast, in the previous works, here, a feed-back controller of the lambda robot in work spaceis desired. Also, a part as a contact force regulatoris added to the position controller in order to con-trol the interaction with an environment. For inter-acting with an environment, control of the contactforce in the interactive part is required. There aretwo methods to control this force, i.e., direct and in-direct contact force control methods. In the direct ap-proach, the controller regulates the force with a con-

trol loop and using feedback of the force measure-ment and desired value of the force. For this method,an explicit model of the interaction environment is re-quired, see (Luh et al., 1983). In the indirect forcecontrol methods, the force is controlled via positioncontroller as an impedance (Hogan, 1985). In our re-search, a hybrid force/position control is designed us-ing the impedance controller approach (Siciliano andKhatib, 2016; Jung et al., 2004) for a very flexiblerobot manipulator.

The novelty of this work is that a nonlinear flex-ible model based position controller of the lambdarobot in workspace is designed. Then, this posi-tion controller is combined with an impedance partin order to have a controller architecture that can beused for controlling the position and acting force ofthe end-effector’s contact part during its interactionwith an environment. The impedance part computesa new end-effector position based on the measured orestimated force and the position of the end-effector.Therefore, the impedance part generates a new end-effector position for tracking by the position con-troller part.

The nonlinear flexible model based controller inCartesian space is simulated and compared with therigid model based one. To test the designed pureposition controller, the end-effector tracks a trajec-tory with low and high speed. Comparison resultsshow that the flexible model based controller cantrack a trajectory with much higher accuracy than therigid based and it shows very high tracking perfor-mance. To investigate the hybrid force/position con-troller, the end-effector’s contact part interacts with aflat surface in two scenarios, interaction with a knownand an unknown environment. Simulation results onthe lambda robot show that the flexible based con-troller has better performance than the rigid based forboth scenarios and the robot using this flexible basedimpedance controller can apply the desired force dur-ing moving along the surface with high accuracy.

The paper is organized as follows: Section 2 de-scribes the robot, i.e., its mechanical and electricalparts. Section 3 includes the modeling of the flexi-ble parallel lambda robot. Section 4 explains the ar-chitecture of the nonlinear controller, i.e., the positioncontroller and impedance controller. In Section 5, theproposed nonlinear controller is simulated and the re-sults are discussed.

2 FLEXIBLE LAMBDA ROBOT

The used lambda robot is a simple parallel robot ma-nipulator. This robot has highly flexible links. The

end of the short link is connected in the middle ofthe long link using rigid bodies. This connection cre-ates a closed loop kinematics constraint that causesthe parallel configuration of the robot. This robot hastwo prismatic actuators connecting the links to theground. The links are connected using passive rev-olute joints to the linear actuators. Another revolutejoint is used to connect the short link and the middleof the long link. An additional rigid body is attachedto the free end of the long link as an end-effector. Thedrive positions and velocities are measured with twooptical encoders. Three full Wheatstone bridge straingauge sets are attached on the long flexible link tomeasure its deformation. The lambda robot configu-ration shown in Figure 1 has been built in hardware,see (Burkhardt et al., 2014) at the Institute of Engi-neering and Computational Mechanics of the Univer-sity of Stuttgart.

The electrical part of the hardware includes somepower supplies for motors, strain gauge’s ampli-fiers, digital/analog input-outputs boards, one Speed-goat target, a host computer, etc. For control-ling the robot, the online control is done witha Speedgoat performance real-time target machinerunning a Mathworks xPCtarget kernel, which iscalled Simulink Real-Time since Matlab R2014a.Also, to observe the controller progress a graphicaluser interface is available for the input, output, safetylogic of the lambda robot and their communication.

Figure 1: Mechanical setup of the flexible parallel lambdarobot

3 MODELING OF THEFLEXIBLE LAMBDA ROBOT

The modeling process of the flexible manipulator withλ configuration can be separated into three majorsteps. First, the flexible components of the systemare modeled with the linear finite element method inthe commercial finite element code ANSYS. Next, inorder to control the λ robot, the degrees of freedom

of the flexible bodies are reduced. Therefore, modelorder reduction is utilized. Then, all the rigid and flex-ible parts are modeled as a flexible multibody systemwith a kinematic loop using the generalized coordi-nates q ∈ R5 as

M(q)q = f(q, q)+B(q)u+CT(q)λ , (1a)c(q) = 0 . (1b)

The symmetric, positive definite mass matrixM ∈R5×5 depends on the joint angles and the elasticcoordinates. The vector f ∈ R5 contains the gener-alized centrifugal, Coriolis and Euler forces, and thevector of applied forces and inner forces due to thebody elasticity. The input matrix B ∈ R5×2 maps theinput vector u ∈ R2 to the system. The constraintequations are defined by c ∈ R2. The Jacobian ma-trix of the constraintC = ∂c(q)/∂q ∈R2×5 maps thereaction force λ ∈ R2 due to the kinematic loop.

The flexible lambda robot is modeled as a flexi-ble multibody system in Neweul-M2 (Burkhardt et al.,2013b). The simulated model of the lambda robot isshown in Figure 2, see also (Burkhardt et al., 2013a;Ansarieshlaghi and Eberhard, 2018a).

s1

s2

α2

α1

end-effector

Figure 2: Mechanical model of flexible parallel robot

While the system has a kinematics loop as a con-straint in Equation (1b), the system accelerations qcan be divided into dependent qd ∈ R2 and indepen-dent qi ∈ R3 coordinates and the system constraintscan be written as

c=Cq+c′′ =Ciqi +Cdqd +c′′ (2)

where qd = [α1,α2]T , qi = [s1,s2,qe]

T ,Cd andCi aredependent and independent parts of the Jacobian ma-trix of the constraint and c′′ presents local acceler-ations due to the constraints. The parameter qe de-scribes elastic coordinates of the flexible long link.Based on Equation (2), the system’s generalized ac-celeration can be formulated as

q=

[qiqd

]=

[I3×3−C−1

d Ci

]qi+

[03

−C−1d c′′

]=Jcqi+b

′′.

(3)

Here, Jc and b′′ are Jacobian matrix and local ac-celerations due to rewriting the generalized accelera-tion based on the system constraints. Finally, by left-side multiplication with the transposed of the Jaco-bian matrix Jc and replacing q by Equation (3), Equa-tion (1a) can be written as

JTc M(Jcqi +b

′′) = JTc (f +Bu+CT(q)λ) (4)

while the multiplication matrix of the rewritten Jaco-bian matrix and the Jacobian matrix of the constraintis JT

c CT = 0, the equation of motion (1a) can be de-

termined, see (Burkhardt et al., 2013a), by

Mqi = f + Bu. (5)

In Equation (5), the inertia matrix M ∈ R3×3 isJT

c MJc, the input matrix B ∈R3×2 is JTc B, and the

vector f ∈ R3 is JTc (Mb′′+ f) from Equation (4).

The independent generalized acceleration can be ob-tained by

qi = M−1(f + Bu). (6)

Equation (6) will be used for the workspace descrip-tion of the system dynamics and designing the non-linear feedback controller.

3.1 Joint Space to Work SpaceTransformation

The system kinematics is formulated as

x= s(qi), (7)

where x ∈ R2 is the end-effector position and s ∈ R2

is a nonlinear function of the independent system co-ordinates qi. The velocity and acceleration of the end-effector in the Cartesian space are calculated by thetime derivative of the end-effector position functionin Equation (7) by

x= Jqi, (8a)

x= Jqi + J qi, (8b)

where J and J are the Jacobian matrix of the robotkinematics and its derivative. Finally, by replacing qifrom Equation (6) to Equation (8b), the dynamics de-scription can be found in workspace coordinates with

x= J(M−1(f + Bu))+ J qi. (9)

Equation (9) is used to control the position of the end-effector as well as the contact force during the inter-action with an environment.

4 CONTROL OF THE FLEXIBLELAMBDA ROBOT

The lambda robot controller is separated into feedfor-ward and feedback controller parts. In the feedfor-ward part, the desired signals are calculated based onthe robot work space, the joint space, the kinemat-ics, and the dynamics constraints of the robot. Thekinematics constraints include the maximum velocity,position of the robot joints, and the closed loop kine-matics. The maximum current, the maximum actingforce, and the maximum flexible coordinates oscilla-tion amplitude are defined as dynamics constraints,see (Seifried et al., 2011). The feedback control partcomputes the lambda robot inputs based on the feed-back linearization method (Khalil, 2002) using thenonlinear dynamics of the robot, the measured states,and the computed end-effector position.

The feedback controller part based on the appli-cation can have three parts, i.e., a position controller,an impedance controller, and the forward kinematicsor an observer. In the simplest case, the controller in-cludes only the position controller in the joint space,see (Eberhard and Ansarieshlaghi, 2019). As a nextcase, the feedback controller includes a position con-troller and an observer when the joint space controllerof the robot is desired, see (Ansarieshlaghi and Eber-hard, 2018b). Also, for controlling the end-effectorposition in the workspace, the position controller andits combination with the forward kinematics is re-quired. In the most complicated case, the feedbackcontroller is a combination of the position controller,the impedance controller, and the forward kinematicspart. This combination is used when the contact partof the robot end-effector has an interaction with an en-vironment in the Cartesian space. Therefore, the jointspace to the workspace transformation of the systemdynamics is required and computed by Equation (9).

Based on the robot work space, the position con-troller shall generate the robot’s inputs in Cartesianspace. Therefore, the control law is obtained as

u=−B−1(f+MJ−1(J qi−KPe−KDe)). (10)

The error and its dynamics are calculated by e =x−xd and e= x− xd where xd is the desired valueof x, consequently xd is the desired value for x. Thedesired trajectories of the end-effector can be com-puted based on the robot workspace as xd and xd.The matrices KP and KD correspond to the weight-ing of feedback errors and can be designed via theLQR method or tuned by hand. Also, they should sat-isfy the stability conditions for nonautonomous sys-tems as a uniform stability, based on the Lyapunovtheorem. The inverse of the input matrix B is not

so straightforward to calculate, since it is not of fullrow rank. Therefore, the existing left-inverse is usedas a pseudo-inverse to yield BB−1 = I . The vectoru presents the control input of the robot manipulatorand that is the output of the designed position con-troller based on the feedback linearization method.Figure (3) shows the control structure for controllingthe end-effector position in order to track a trajectoryin Cartesian space.

forwardkinematics

lambda

qi

qi

xx

xd

xdu

robotcontrollerposition

controllerfeedforward

kinematics

dynamics

constraint

constraintFigure 3: Nonlinear position control structure

The position control for all scenarios as the po-sition or the force/position controller is same as Fig-ure 3. The only difference between the position andhybrid force/position controller is the desired inputsignal to the controller. This structure is extended tocontrol the contact force to interact with an environ-ment. However, the interactive environment can be aknown or an unknown surface.

The robot has only two actuated joints and can justmove in the horizontal plane. Hence, the end-effectorcan only interact with an environment in its move-ment directions in the horizontal plane. That meansthat the end-effector shall have contact in one or bothdirections that the end-effector moves. The positionand acting force are dependent and shall be controlleddependently. So the contact force is controlled by us-ing indirect control methods. One of the approachesthat is used for indirect force control is the impedancecontroller method, see (Siciliano and Villani, 1999;Siciliano and Khatib, 2016). Based on the feedbackcontroller structure presented in Figure 3, the newfeedback controller structure is developed to controlthe force and position at the same time. The devel-oped new feedback controller is shown in Figure 4.

The new feedback controller is composed of theposition controller, force controller, and the forwardkinematics part as a hybrid force/position controller.In Figure 4, the feedforward controller computes thedesired values yd based on the interactive environ-ment that can be a known or an unknown surface. Fora known scenario, the desired values yd are the po-sition of the surface xd, the moving velocity on thesurface xd, and the magnitude of the acting force onthe surface fd in Cartesian space. The desired valuesfor the unknown case are the magnitude of the mov-ing velocity vd and the acting force fd on the surface.

forwardkinematics

lambdarobotcontroller

positioncontrollerimpedance

controllerfeedforward

qi

qi

xx

x′dx′d

u

f

yd

Figure 4: Force/position control structure for interactingwith an environment

The impedance part of the feedback controllercomputes a new desired position and velocity of therobot’s end-effector based on the desired values bythe feedforward controller, the measured states qi andqi, the contact force f , and the computed or observedposition of the contact part by the forward kinematics.The position controller acts as the last part in Figure 3based on its input signals and the dynamics model ofthe robot.

In order to interact with an environment, a toolthat is shown in Figure 5 is designed and installed onthe end-effector in Figure 2. The basic idea of thedesigned tool is to control the contact force duringmoving on the surface. Therefore, based on the robotconfiguration and its degrees of freedom, the contactpart’s force should be defined as a function of the end-effector position. Thus, the tool is designed to be atransfer function, to convert the interaction force ofthe contact part to the end-effector position displace-ment. This transfer tool is designed such that the act-ing force accurately and reliably is transferred to theforce sensor. The tool includes a body for interactingas a contact part, a movable body which is connectedto a spring for transferring the contact force, a forcesensor to measure the interacting force, and a struc-ture to constrain the spring in the force direction andto avoid its bending and deformation. Also, for themovable body, a safety stop is considered to limit themeasured force and protect the force sensor.

4.1 Impedance Control

Impedance control is a method used for indirect forceregulation. The impedance refers the dynamic rela-tion of the motion and contact force of the manipula-tor during an interaction with an object. It means thatthe impedance controller transfers the contact force asa function of the contact part’s position and then, theposition controller regulates it.

In this work, for modeling the impedance partonly a spring with stiffness k is used. The freelength of the designed tool is l0. For controlling thecontact part’s position and its acting force, the end-effector position shall be controlled. For this goal,based on the desired input yd, measurement signalsy = [qi, qi, f ]T , and the feedforward kinematics, the

contactsurface

contactbody

forcesensor

safety stop

spring

motion constraintto avoid bending

end-effector

Figure 5: Designed tool for interacting with an environment

new desired end-effector position and velocity are ob-tained as

x′d = g1(yd,y,x,k, l0), (11a)

x′d = g2(yd,y, x). (11b)

The important point in this work is that the environ-ment model is unknown and we do not have any infor-mation about its shape. The environment can be flex-ible, deformable, movable, or compressible. The onlydifference between a known and an unknown environ-ment for this work is, when the position and movingvelocity on the surface is known, the environment isnamed known. For an unknown environment, onlythe magnitude of moving velocity is given and basedon the measured force, the impedance part can distin-guish wherever that the contact body is in contact ornot. Based on this idea, it computes the new desiredvalues (x′d, x′d).

After introducing the environments that the robotcan interact with, now parameters that are used forupdating the new desired values are defined. The pa-rameters are determined based on the inputs of thefunctions g1 and g2 of Equation (11) in Figure 6.

Figure 6 shows a schematic view of the lambdarobot and its designed tool in interacting with an en-vironment. The designed contact tool’s length canchange based on the acting force. The angles β1 andβ2 are between the y-direction of the global coordi-nate system and the robot end-effector in two config-urations. It is calculated by

β = g4(qi) =π

2− (α1 + rqe). (12)

The constant r is the gain for the elastic rotation anddisplacement of the long flexible link. The angle θ

is computed based on the position of the robot end-effector at the current sample time t and the previoussample time t−4t

θ =g3(xt,xt−4t) (13)

=arctan(xxt

xyt

)− arctan(xxt−4t

xyt−4t

).

s1

s2

β2

l2

β1

f > fd

l1

f < fd

x

y

θ

Figure 6: Schematic view of the lambda robot in contactwith an environment

In Equation (14a), xxt is the end-effector positionat the current time in the x-direction and xyt is in the y-direction. For previous sample time, the end-effectorposition in the x-direction and y-direction are xxt−4tand xyt−4t .

The new desired position and velocity of the end-effector for a known case using Equation (11) can berewritten using the new defined parameters in Equa-tions (12) and (13) for the next sample time t+4tas

x′d(t+4t) = g1(β, fd, f ,xd,k, l0) (14a)

x′d(t+4t) = g2(θ, x, xd). (14b)

In this case, the new desired position x′d and thefunction g1 are computed using Equation (14a) as

x′d = g1 = xd−[

l0 sin(β)l0 cos(β)

]+Pj

k

[fe sin(β)fe cos(β)

]. (15)

In Equation (15), fe is the regulation force error asfe = fd− f and the matrixPj presents the design gain.The subscript j is related to the different situations areshown in Figure 6 and is a function of the force regu-lation’s amplitude error fe and its sign. Using differ-ence gain matrices for difference conditions in a gainscheduling method improves the position tracking andforce regulation performances and adapts the robot tothe new interaction situation.

Also, to increase the accuracy of the positiontracking and improve the force regulation tasks, thefunction g2 in Equation (14b) is defined by

x′d =g2 (16)

=

[cos(θ) sin(θ)sin(θ) cos(θ)

]xd +An

[cos(θ) sin(θ)sin(θ) cos(θ)

]e.

The matrixAn is the design gain. The subscript n de-pendents on the magnitude of the velocity error ||e||and adapts the robot end-effector velocities for differ-ent contact configuration.

For an unknown interaction, the new desired val-ues are computed as

x′d = x+Pj

k

[fe sin(β)fe cos(β)

]. (17)

For next sample time, the new velocity of the end-effector is calculated

x′d =

[vd sin(θ)vd cos(θ)

]+An

[cos(θ) sin(θ)sin(θ) cos(θ)

]ev, (18)

where vd is the magnitude of the moving velocity onthe surface and it can be a constant or function of timeor the end-effector position. The velocity error vectorev is the difference of the measured and desired ve-locity and is computed using x= [vx,vy]

T as

ev =

[vd sin(θ)vd cos(θ)

]−[

vxvy

]. (19)

The impedance controller obtains the next posi-tion and velocity of the end-effector based on the in-teracted environment and the defined goal for eachinteraction. It means that for the known interactiontask, controlling the position of the acting force has agreat importance and regulating the acting force in thespecified positions is desired. In contrast, for the un-known case, controlling the specified desired actingforce during moving along the surface is desired.

5 SIMULATION RESULTS

To validate the designed nonlinear controller,some tasks in the Cartesian space are specified asthe trajectory tracking by the position controller ofthe robot end-effector and interacting with a surfacevia the hybrid force/position controller. The nonlin-ear controllers are designed based on the flexible andrigid model of the lambda robot and tested on the sim-ulated model.

For pure position control, the controllers track aline trajectory with low and high speed. The resultsare shown in Figure 7. Based on the simulation resultsof two controllers, the flexible model based controllertracks the trajectory with higher accuracy and less os-cillation amplitude than the rigid model based, seeFigure 7. The difference of controllers performancesis increased when the end-effector tracks a trajectorywith high speed in Figure 7b. Also, the results showthe end-effector position controller based on the flex-ible model can track a trajectory with less than 1%normalized root mean square error.

To validate the hybrid force/position controller, ina first scenario, the robot interacts with a known flatsurface during high speed moving. Figure 8 shows

00.10.20.30.40.50.60.7

e[m

m]

rmsfrmsroscfoscr

maxfmaxr

(a) Tracking the line trajectory with low speed

0

0.5

1

1.5

22.5

3

e[m

m]

(b) Tracking the line trajectory with high speed

Figure 7: Comparison of the end-effector trajectory track-ing results based on the rigid (r) and flexible (f) controllers.In these comparisons, the root-mean-squared (rms), maxi-mum tracking error (max), and maximum oscillation am-plitude (osc) are used as benchmarks.

interaction results. For interacting with a known flatenvironment, the maximum normalized root meansquare error of the flexible based impedance con-troller for moving in the desired position and regulat-ing acting force on the surface are 1% and 5%, respec-tively, see Figures 8b and 8a. These results show thatthe hybrid force/position controller is able to applythe force at the desired position with high accuracy.

As the next scenario, a more difficult and chal-lenging task is investigated where the robot is inter-acting with an unknown environment. Figure 9 showsthe robot manipulation results. For interacting withan unknown environment, the error for moving withdesired velocity and regulating the acting force on thesurface are investigated. The results show the flexi-ble model based controller achieved drastically betterperformance than the rigid model based. Also, the in-teraction results show that the flexible controller canmove on an unknown surface during force regulatingwith more than 99% accuracy and at minimum 88%for tracking moving velocity.

The performance of the impedance controllerin each interaction scenario shows that the flexiblemodel based controller reach their desired goal suc-

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5time [s]

f act

[N]

0

0.5

1

1.5

2

2.53

desiredflexiblerigid

no acting force

(a) Acting force on a known surface in fast movement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5time [s]

eec

[m]

-.60

-.55

-.50

-.45

-.40

-.35

-.30

x-directionno contact

y-direction

(b) Contact part position on a known surface in fast move-ment

Figure 8: Comparison of the robot’s interaction with aknown surface. Flexible presents nonlinear controller basedon the flexible model and rigid shows the result of the con-troller based on the rigid model.

cessfully and overcome to the challenges.

6 CONCLUSIONS

In this paper, a high performance end-effectorposition controller and its combination with animpedance controller are presented for a flexible par-allel robot manipulator to interact with an environ-ment. The designed position controller based on theindependent system coordinates computes the robot’sinput using the measurable states of the system, theobtained, and the desired position and velocity ofthe end-effector. The nonlinear feedback controlleris extended with an impedance part in order to regu-late the contact force, too. The simulation results onthe lambda robot show that the nonlinear controllersbased on the flexible model have drastically betterperformance than the rigid model based controllers.Also, the hybrid force/position controller did all com-plicated tasks with high accuracy and successfullyovercame to all challenges, too. For future work, the

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

time [s]

f act

[N]

desiredflexiblerigid

(a) Acting force on an unknown surface in fast movement

0 1 2 3 4 5 60

2

4

6

8

10

12

time [s]

v[c

m/s

]

(b) End-effector tracking velocity on an unknown surface infast moving

Figure 9: Comparison the robot interaction with an un-known surface. Flexible presents nonlinear controller basedon the flexible model and rigid show result of the controllerbased on the rigid model.

designed controllers, i.e., the position and impedancecontroller will be tested on the real robot and theirperformance will be investigated.

ACKNOWLEDGEMENTS

This research presented in the Cluster of Excellencein Simulation Technology SimTech at the Universityof Stuttgart and is partially funded by the Landes-graduierten kolleg Baden-Wurttembergs. The authorsappreciate these discussions.

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