Hybrid Force/Position Control of a Very Flexible Parallel Robot

Manipulator in Contact with an Environment

Fatemeh Ansarieshlaghi

and Peter Eberhard

Institute of Engineering and Computational Mechanics, University of Stuttgart,

Pfaffenwaldring 9, 70569 Stuttgart, Germany

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 interaction

with environments and humans. This paper investigates a novel strategy to control a very ﬂexible parallel

manipulator interacting with an environment. The controller is complicated when the used robot manipulator

is a ﬂexible multibody dynamics system and the ﬂexibility shall be taken into account in the modeling and

controlling 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 hybrid

force/position control strategy is utilized. Therefore, two controllers based on the rigid and ﬂexible models

of the robot are designed and implemented to interact with a surface. The simulation results show that the

controllers based on the ﬂexible model have a better performance than those based on the rigid model and

successfully track a trajectory and interact with an environment.

1 INTRODUCTION

Light-weight manipulators attract a lot of research

interest 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 systems

and their ﬂexibility must be taken into account in the

controller design. The ﬂexible multibody system with

signiﬁcant deformations complicates the control de-

sign because there are more generalized coordinates

than control inputs. This complicated controller has

a more complex design when the robot interacts with

an environment. The controller shall be able to regu-

late the interaction force and move along the surface

at the same time. In order to obtain a high perfor-

mance in the end-effector’s contact part, an accurate

and efﬁcient model as well as a nonlinear controller is

necessary.

In this research, to investigate the controller a ﬂex-

ible robot available in hardware in our laboratory is

https://orcid.org/0000-0003-2693-0882

https://orcid.org/0000-0003-1809-4407

used. This robot has a λ-shape and its links are

very ﬂexible. In previous researches on this robot,

only its modeling and position control were inves-

tigated. The modeling is described in (Burkhardt

et al., 2013a). Also, an experimental study on the

modeling and the modeling improvement were done

in (Burkhardt et al., 2014) and (Ansarieshlaghi and

Eberhard, 2017). The position control of this system

can be done by a model-based or non-model-based

feedback controller. Non-model-based feedback con-

trollers in joint space were investigated in (Burkhardt

et al., 2014) and (Morlock et al., 2016) and the ﬂexi-

ble model based nonlinear feedback controller in joint

space was investigated in (Ansarieshlaghi and Eber-

hard, 2018b). An experimental study of the rigid

model-based controller on the lambda robot in the

workspace 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 space is

desired. Also, a part as a contact force regulator is

added to the position controller in order to control

the interaction with an environment. For interacting

with an environment, control of the contact force in

the interactive part is required. There are two meth-

ods to control this force, i.e., direct and indirect con-

tact force control methods. In the direct approach, the

controller regulates the force with a control loop and

Ansarieshlaghi, F. and Eberhard, P.

Hybrid Force/Position Control of a Very Flexible Parallel Robot Manipulator in Contact with an Environment.

DOI: 10.5220/0007833400590067

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 59-67

ISBN: 978-989-758-380-3

using feedback of the force measurement and desired

value of the force. For this method, an explicit model

of the interaction environment is required, see (Luh

et al., 1983). In the indirect force control methods,

the force is controlled via position controller as an

impedance (Hogan, 1985).

The contact force control of the light weight

robots is very important task and there are many ap-

plications on this topic, e.g., industrial (Schindlbeck

and Haddadin, 2015; Suarez et al., 2018), surgical (Li

et al., 2018; Sandoval et al., 2018; Kamikawa et al.,

2018), or soft robotic application (Vogel et al., 2015).

Also, the contact force control of a ﬂexible robot is in-

vestigated with one ﬂexible link in (Endo et al., 2017;

Feliu-Talegon et al., 2019) and for multi ﬂexible links

in (Suarez et al., 2018).

In our research, a hybrid force/position con-

trol is designed using the impedance controller ap-

proach (Siciliano and Khatib, 2016; Jung et al., 2004)

for a very ﬂexible robot manipulator.

The novelty of this work is that a nonlinear ﬂex-

ible model based position controller of the lambda

robot in workspace is designed. Then, this posi-

tion controller is combined with an impedance part

in order to have a controller architecture that can be

used for controlling the position and acting force of

the end-effector’s contact part during its interaction

with an environment. The impedance part computes

a new end-effector position based on the measured or

estimated 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 in

Cartesian space is simulated and compared with the

rigid model based one. To test the designed pure

position controller, the end-effector tracks a trajec-

tory with low and high speed. Comparison results

show that the ﬂexible model based controller can

track a trajectory with much higher accuracy than the

rigid 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 a

ﬂat surface in two scenarios, interaction with a known

and an unknown environment. Simulation results on

the lambda robot show that the ﬂexible based con-

troller has better performance than the rigid based for

both scenarios and the robot using this ﬂexible based

impedance 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 electrical

parts. 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 position

controller and impedance controller. In Section 5, the

proposed 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 of

the long link using rigid bodies. This connection cre-

ates a closed loop kinematics constraint that causes

the parallel conﬁguration of the robot. This robot has

two prismatic actuators connecting the links to the

ground. The links are connected using passive rev-

olute joints to the linear actuators. Another revolute

joint is used to connect the short link and the middle

of the long link. An additional rigid body is attached

to the free end of the long link as an end-effector. The

drive positions and velocities are measured with two

optical encoders. Three full Wheatstone bridge strain

gauge sets are attached on the long ﬂexible link to

measure 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 some

power 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 with

a Speedgoat performance real-time target machine

running a Mathworks xPCtarget kernel, which is

called Simulink Real-Time since Matlab R2014a.

Also, to observe the controller progress a graphical

user interface is available for the input, output, safety

logic of the lambda robot and their communication.

3 MODELING OF THE

FLEXIBLE LAMBDA ROBOT

The modeling process of the ﬂexible manipulator with

λ conﬁguration can be separated into three major

steps. First, the ﬂexible components of the system

are modeled with the linear ﬁnite element method in

the commercial ﬁnite element code ANSYS. Next, in

order to control the λ robot, the degrees of freedom

of the ﬂexible bodies are reduced. Therefore, model

order reduction is utilized. Then, all the rigid and ﬂex-

ible parts are modeled as a ﬂexible multibody system

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

Figure 1: Mechanical setup of the ﬂexible parallel lambda

robot.

with a kinematic loop using the generalized coordi-

nates q ∈ R

M (q) ¨q = f(q, ˙q) + B(q)u + C

(q)λ , (1a)

c(q) = 0 . (1b)

The symmetric, positive deﬁnite mass matrix

M ∈ R

5×5

depends on the joint angles and the elastic

coordinates. The vector f ∈ R

contains the gener-

alized centrifugal, Coriolis and Euler forces, and the

vector of applied forces and inner forces due to the

body elasticity. The input matrix B ∈ R

5×2

maps the

input vector u ∈ R

to the system. The constraint

equations are deﬁned by c ∈ R

. The Jacobian ma-

trix of the constraint C = ∂c(q)/∂q ∈ R

2×5

maps the

reaction force λ ∈ R

due to the kinematic loop.

The ﬂexible lambda robot is modeled as a ﬂexi-

ble multibody system in Neweul-M

(Burkhardt et al.,

2013b). The simulated model of the lambda robot is

shown in Figure 2, see also (Burkhardt et al., 2013a;

Ansarieshlaghi and Eberhard, 2018a).

While the system has a kinematics loop as a con-

straint in Equation (1b), the system accelerations ¨q

can be divided into dependent ¨q

∈ R

and indepen-

dent ¨q

∈ R

coordinates and the system constraints

can be written as

¨c = C ¨q + c

= C

¨q

+ C

¨q

+ c

(2)

where q

= [α

,α

]

, q

= [s

]

, C

and C

are

dependent and independent parts of the Jacobian ma-

trix of the constraint and c

presents local acceler-

ations due to the constraints. The parameter q

de-

scribes elastic coordinates of the ﬂexible long link.

Based on Equation (2), the system’s generalized ac-

celeration can be formulated as

¨q =



¨q





3×3

−C

−1



¨q



−C

−1



= J

¨q

(3)

end-effector

Figure 2: Mechanical model of ﬂexible parallel robot.

Here, J

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 J

and replacing ¨q by Equation (3), Equa-

tion (1a) can be written as

M (J

¨q

+ b

) = J

(f + Bu + C

(q)λ) (4)

while the multiplication matrix of the rewritten Jaco-

bian matrix and the Jacobian matrix of the constraint

is J

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

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

M ¨q

f +

Bu. (5)

In Equation (5), the inertia matrix

M ∈ R

3×3

, the input matrix

B ∈ R

3×2

is J

B, and the

vector

f ∈ R

is J

(Mb

+ f ) from Equation (4).

The independent generalized acceleration can be ob-

tained by

¨q

−1

(

f +

Bu). (6)

Equation (6) will be used for the workspace de-

scription of the system dynamics and designing the

nonlinear feedback controller.

Joint Space to Work Space

Transformation

The system kinematics is formulated as

x = s(q

), (7)

where x ∈ R

is the end-effector position and s ∈ R

is a nonlinear function of the independent system co-

ordinates q

. The velocity and acceleration of the end-

effector in the Cartesian space are calculated by the

time derivative of the end-effector position function

in Equation (7) by

˙x = J ˙q

, (8a)

¨x = J ¨q

J ˙q

, (8b)

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

In Equation (8b), J and

J are the Jacobian ma-

trix of the robot kinematics and its derivative. Finally,

by replacing ¨q

from Equation (6) to Equation (8b),

the dynamics description can be found in workspace

coordinates with

¨x = J(

−1

(

f +

Bu)) +

J ˙q

. (9)

Equation (9) is used to control the position of the

end-effector as well as the contact force during the

interaction with an environment.

4 CONTROL OF THE FLEXIBLE

LAMBDA 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 on

the robot work space, the joint space, the kinemat-

ics, and the dynamics constraints of the robot. The

kinematics constraints include the maximum velocity,

position of the robot joints, and the closed loop kine-

matics. The maximum current, the maximum acting

force, and the maximum ﬂexible coordinates oscilla-

tion amplitude are deﬁned as dynamics constraints,

see (Seifried et al., 2011). The feedback control part

computes the lambda robot inputs based on the feed-

back linearization method (Khalil, 2002) using the

nonlinear 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 kinematics

or 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 next

case, the feedback controller includes a position con-

troller and an observer when the joint space controller

of the robot is desired, see (Ansarieshlaghi and Eber-

hard, 2018b). Also, for controlling the end-effector

position in the workspace, the position controller and

its combination with the forward kinematics is re-

quired. In the most complicated case, the feedback

controller is a combination of the position controller,

the impedance controller, and the forward kinematics

part. This combination is used when the contact part

of the robot end-effector has an interaction with an en-

vironment in the Cartesian space. Therefore, the joint

space to the workspace transformation of the system

dynamics is required and computed by Equation (9).

Based on the robot work space, the position con-

troller shall generate the robot’s inputs in Cartesian

space. Therefore, the control law is obtained as

u = −

−1

(

f +

−1

(

J ˙q

−K

e − K

˙e)). (10)

forward

kinematics

lambda

˙q

˙x

robot

controller

position

controller

feedforward

kinematics

dynamics

constraint

Figure 3: Nonlinear position control structure.

The error and its dynamics are calculated by e =

x − x

and ˙e = ˙x − ˙x

where x

is the desired value

of x, consequently ˙x

is the desired value for ˙x. The

desired trajectories of the end-effector can be com-

puted based on the robot workspace as x

and ˙x

The matrices K

and K

correspond to the weight-

ing of feedback errors and can be designed via the

LQR method or tuned by hand. Also, they should sat-

isfy the stability conditions for nonautonomous sys-

tems as a uniform stability, based on the Lyapunov

theorem.

The lambda robot is an underactuated robot and

the input matrix

B is therefore not of full row rank.

Thus, the inverse of the input matrix is not so straight-

forward to calculate. Therefore, the existing left-

inverse is used as a pseudo-inverse to yield

−1

I. The vector u presents the control input of the robot

manipulator and that is the output of the designed po-

sition controller based on the feedback linearization

method. Figure (3) shows the control structure for

controlling the end-effector position in order to track

a trajectory in Cartesian space.

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 and

hybrid force/position controller is the desired input

signal to the controller. This structure is extended to

control the contact force to interact with an environ-

ment. However, the interactive environment can be a

known or an unknown surface.

The robot has only two actuated joints and can just

move in the horizontal plane. Hence, the end-effector

can only interact with an environment in its move-

ment directions in the horizontal plane. That means

that the end-effector shall have contact in one or both

directions that the end-effector moves. The position

and acting force are dependent and shall be controlled

dependently. So the contact force is controlled by us-

ing indirect control methods. One of the approaches

that is used for indirect force control is the impedance

controller method, see (Siciliano and Villani, 1999;

Siciliano and Khatib, 2016).

Based on the feedback controller structure pre-

sented in Figure 3, the new feedback controller struc-

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

forward

kinematics

lambda

robot

controller

position

controller

impedance

controller

feedforward

˙q

˙x

Figure 4: Force/position control structure for interacting

with an environment.

ture is developed to control the force and position at

the same time. The developed new feedback con-

troller is shown in Figure 4.

The new feedback controller is composed of the

position controller, force controller, and the forward

kinematics part as a hybrid force/position controller.

In Figure 4, the feedforward controller computes the

desired values y

based on the interactive environ-

ment that can be a known or an unknown surface. For

a known scenario, the desired values y

are the po-

sition of the surface x

, the moving velocity on the

surface ˙x

, and the magnitude of the acting force on

the surface f

in Cartesian space. The desired values

for the unknown case are the magnitude of the moving

velocity v

and the acting force f

on the surface.

The impedance part of the feedback controller

computes a new desired position and velocity of the

robot’s end-effector based on the desired values by

the feedforward controller, the measured states q

and

˙q

, the contact force f , and the computed or observed

position of the contact part by the forward kinematics.

The position controller acts as the last part in Figure 3

based on its input signals and the dynamics model of

the robot.

In order to interact with an environment, a tool

that is shown in Figure 5 is designed and installed on

the end-effector in Figure 2.

The basic idea of the designed tool is to control the

contact force during moving on the surface. There-

fore, based on the robot conﬁguration and its degrees

of freedom, the contact part’s force should be deﬁned

as a function of the end-effector position. Thus, the

tool is designed to be a transfer function, to convert

the interaction force of the contact part to the end-

effector position displacement. This transfer tool is

designed such that the acting force accurately and re-

liably is transferred to the force sensor. The tool in-

cludes a body for interacting as a contact part, a mov-

able body which is connected to a spring for trans-

ferring the contact force, a force sensor to measure

the interacting force, and a structure to constrain the

spring in the force direction and to avoid its bending

and deformation. Also, for the movable body, a safety

stop is considered to limit the measured force and pro-

tect the force sensor.

contact

surface

contact

body

force

sensor

safety stop

spring

motion constraint

to avoid bending

end-effector

Figure 5: Designed tool for interacting with an environ-

ment.

Impedance Control

Impedance control is a method used for indirect force

regulation. 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 that

the impedance controller transfers the contact force as

a function of the contact part’s position and then, the

position controller regulates it.

In this work, for modeling the impedance part

only a spring with stiffness k is used. The free

length of the designed tool is l

. For controlling the

contact part’s position and its acting force, the end-

effector position shall be controlled. For this goal,

based on the desired input y

, measurement signals

y = [q

, ˙q

, f ]

, and the feedforward kinematics, the

new desired end-effector position and velocity are ob-

tained as

= g

,y,x, k, l

), (11a)

˙x

= g

,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 only

difference between a known and an unknown environ-

ment for this work is, when the position and moving

velocity on the surface is known, the environment is

named known. For an unknown environment, only

the magnitude of moving velocity is given and based

on the measured force, the impedance part can distin-

guish wherever that the contact body is in contact or

not. Based on this idea, it computes the new desired

values (x

, ˙x

After introducing the environments that the robot

can interact with, now parameters that are used for

updating the new desired values are deﬁned. The pa-

rameters are determined based on the inputs of the

functions g

and g

of Equation (11) in Figure 6.

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

f > f

f < f

Figure 6: Schematic view of the lambda robot in contact

with an environment.

Figure 6 shows a schematic view of the lambda

robot and its designed tool in interacting with an en-

vironment. The designed contact tool’s length can

change based on the acting force. The angles β

and

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 = g

) =

− (α

+ rq

). (12)

The constant r is the gain for the elastic rotation

and displacement 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 previous

sample time t − 4t

θ =g

t−4t

) (13)

=arctan(

) − arctan(

t−4t

In Equation (14a), x

is the end-effector position

at the current time in the x-direction and x

is in the y-

direction. For previous sample time, the end-effector

position in the x-direction and y-direction are x

t−4t

and x

t−4t

The new desired position and velocity of the end-

effector for a known case using Equation (11) can be

rewritten using the new deﬁned parameters in Equa-

tions (12) and (13) for the next sample time t + 4t

(t + 4t) = g

(β, f

, f ,x

,k, l

) (14a)

˙x

(t + 4t) = g

(θ, ˙x, ˙x

). (14b)

In this case, the new desired position x

and the

function g

are computed using Equation (14a) as

= g

= x

−



sin(β)

cos(β)





sin(β)

cos(β)



. (15)

In Equation (15), f

is the regulation force error as

= f

− f and the matrix P

presents the design gain.

The subscript j is related to the different situations are

shown in Figure 6 and is a function of the force regu-

lation’s amplitude error f

and its sign. Using differ-

ence gain matrices for difference conditions in a gain

scheduling method improves the position tracking and

force regulation performances and adapts the robot to

the new interaction situation.

Also, to increase the accuracy of the position

tracking and improve the force regulation tasks, the

function g

in Equation (14b) is deﬁned by

˙x

(16)



cos(θ) sin(θ)

sin(θ) cos(θ)



˙x

+ A



cos(θ) sin(θ)

sin(θ) cos(θ)



˙e.

The matrix A

is the design gain. The subscript n

dependents 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 +



sin(β)

cos(β)



. (17)

For next sample time, the new velocity of the end-

effector is calculated

˙x



sin(θ)

cos(θ)



+ A



cos(θ) sin(θ)

sin(θ) cos(θ)



, (18)

where v

is the magnitude of the moving velocity on

the surface and it can be a constant or function of time

or the end-effector position. The velocity error vector

is the difference of the measured and desired ve-

locity and is computed using ˙x = [v

]



sin(θ)

cos(θ)



−





. (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 each

interaction. It means that for the known interaction

task, controlling the position of the acting force has a

great importance and regulating the acting force in the

speciﬁed positions is desired. In contrast, for the un-

known case, controlling the speciﬁed desired acting

force during moving along the surface is desired.

5 SIMULATION RESULTS

To validate the designed nonlinear controller, some

tasks in the Cartesian space are speciﬁed as the tra-

jectory tracking by the position controller of the robot

end-effector and interacting with a surface via the hy-

brid force/position controller.

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

The nonlinear controllers are designed based on

the ﬂexible model with ﬁve degrees of freedom and

the rigid model with four degrees of freedom of the

lambda robot. The controllers are implemented on the

robot model with eleven degrees of freedom because

the previous researches and testing on the lambda

robot in (Ansarieshlaghi and Eberhard, 2018b; Eber-

hard and Ansarieshlaghi, 2019) show that the real sys-

tem behavior is very close to this model.

For pure position control, the controllers track a

line trajectory with low and high speed. The results

are shown in Figure 7.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

e [mm]

rms

osc

max

(a) Tracking the line trajectory with low speed.

0.5

1.5

2.5

e [mm]

(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.

Based on the simulation results of two controllers,

the ﬂexible model based controller tracks the trajec-

tory with higher accuracy and less oscillation ampli-

tude than the rigid model based, see Figure 7 for both

tracking velocities. The difference of controllers per-

formances is increased when the end-effector tracks a

trajectory with high speed in Figure 7b. Also, the re-

sults show the end-effector position controller based

on the ﬂexible model can track a trajectory with less

than 1% normalized root mean square error.

To validate the hybrid force/position controller,

in a ﬁrst scenario, the robot interacts with a known

ﬂat surface during high speed moving. The spring

stiffness for this scenario, in Equations (15) is k =

100 N/m. Figure 8 shows interaction results, e.g.,

tracking the surface and the acting force on the sur-

face by the robot contact tool.

For interacting with a known ﬂat environment, the

maximum normalized root mean square error of the

ﬂexible based impedance controller for moving in the

desired position and regulating acting force on the

surface are used as benchmarks and these are 1% and

5%, respectively, see Figures 8b and 8a.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

time [s]

act

[N]

0.5

1.5

2.5

desired

ﬂexible

rigid

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

4.5

time [s]

[m]

-.60

-.55

-.50

-.45

-.40

-.35

-.30

x-direction

no contact

y-direction

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

ment.

Figure 8: Comparison of the robot’s interaction with a

known surface. Flexible presents nonlinear controller based

on the ﬂexible model and rigid shows the result of the con-

troller based on the rigid model.

These results show that the hybrid force/position

controller is able to apply the force at the desired po-

sition with high accuracy.

As the next scenario, a more difﬁcult and chal-

lenging task is investigated where the robot is in-

teracting with an unknown environment. The used

spring for impedance modeling for the contact tool, in

Equations (17) has same value as in the known case.

Figure 9 shows the robot manipulation results. In con-

trast of the known case, the controller goal is related

to the acting force regulation during moving on a ﬂat

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

0 1 2 3 4 5 6

0.5

1.5

2.5

time [s]

act

[N]

desired

ﬂexible

rigid

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

0 1 2 3 4

time [s]

v [cm/s]

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

in fast moving.

Figure 9: Comparison the robot interaction with an un-

known surface. Flexible presents nonlinear controller based

on the ﬂexible model and rigid show result of the controller

based on the rigid model.

surface.

For interacting with an unknown environment, the

moving with desired velocity and regulating the act-

ing force on the surface are investigated. The results

show the ﬂexible model based controller achieved

drastically better performance than the rigid model

based. Also, the interaction results show that the ﬂex-

ible controller can move on an unknown surface dur-

ing force regulating with more than 99% accuracy and

at minimum 88% for tracking moving velocity.

The performance of the force/position controller

in each interaction scenario shows that the ﬂexible

model based controller reaches their desired goals

successfully and overcome to the challenges.

6 CONCLUSIONS

In this paper, a high performance end-effector posi-

tion controller and its combination with an impedance

controller are presented for a ﬂexible parallel robot

manipulator to interact with an environment. The de-

signed position controller based on the independent

system coordinates computes the robot’s input using

the measurable states of the system, the obtained, and

the desired position and velocity of the end-effector.

The nonlinear feedback controller is extended with

an impedance part in order to regulate the contact

force, too. The simulation results on the lambda robot

model show that the nonlinear controllers based on

the ﬂexible model have drastically better performance

than the rigid model based controllers. Also, the

hybrid force/position controller did all complicated

tasks with high accuracy and successfully overcame

to all challenges, too.

For future work, the designed controllers, i.e., the

position and impedance controller will be tested on

the real robot and their performance will be investi-

gated.

ACKNOWLEDGEMENTS

This research presented in the Cluster of Excellence

in Simulation Technology SimTech at the University

of Stuttgart and is partially funded by the Landes-

graduierten kolleg Baden-W

urttembergs. The authors

appreciate these discussions.

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