Meccanica (2011) 46: 171–182DOI 10.1007/s1101201094078
PARALLEL MANIPULATORS
A ﬂuidicmuscle driven forcecontrolled parallel platformfor physical simulation of virtual spatial forcedisplacementlaws
Mahendra Dhanu Singh
·
Kusnadi Liem
·
Vladimirs Leontjievs
·
Andrés Kecskeméthy
Received: 30 September 2009 / Accepted: 6 December 2010 / Published online: 4 January 2011© Springer Science+Business Media B.V. 2010
Abstract
Described in this paper is a sixleggedStewartGough parallel platform driven by a relativelynew type of ﬂuidic muscles. The advantage of the platform is that it is virtually free of stickslip effects.Thus, the device is wellsuited for ﬁnetuned forcecontrol and for physical simulation of virtual forcedisplacement laws. The legs of the platform are of type RRPS and are equipped with a coaxial coil springand a ﬂuidic muscle providing push and pull forces.Each leg is equipped with a force sensor, a pressuresensor, and a magnetostrictive position encoder. Thecontrol for the platform consists of six control loopsfor the six operated actuators with modelbased forcecontrol comprising individual gas models as well asthe rubber nonlinearities for each leg. The control lawalso includes the gas ﬂow in the proportional directional control valve in 3/3way function. The presentpaper describes the basic architecture of the platform,the dynamic models, as well as testbed results for theexisting ﬂuidicmuscle parallel platform DynaHex. Itis shown that the presented control scheme leads to astableforcecontroloftheplatformforquasistaticmotion. As an application, the device will be employed inﬁelds of biomechanics, as well as in general environments requiring physical simulation.
M. Dhanu Singh
·
K. Liem
·
V. Leontjievs
·
A. Kecskeméthy (
)Lehrstuhl für Mechanik und Robotik, UniversitätDuisburgEssen, 47057 Duisburg, Germanyemail: andres.kecskemethy@unidue.de
Keywords
Fluidic muscle
·
Force control
·
Parallelplatform
·
Physical simulator
1 Introduction
Parallel platforms are wellestablished today and areavailable in multiple architectures and technologies[1]. In this setting, most parallel platforms are positioncontrolled [2]. A special kind of task results when theparallel manipulator is targeted to provide force control. In these cases, one seeks to control the actuatorssuch that a given force at the endeffector is achieved.Current methodologies for this task are based on hybrid force/position control schemes, where the actuators are velocitycontrolled with some assumptions onthe compliance of the environment [3]. For example,the forceimpedance control scheme proposed in [4]uses a robot system that consists of an industrial serialsixaxis robot and a small six DOF parallel manipulator ﬁxed at its ﬂange. By this approach, the merits of the large workspace of the serial robot are combinedwiththehighbandwidthoftheparallelplatformdrivenbyelectricmotors.Thisconceptisalsousede.g.forindustrial forcefeedback control [5]. However, in someapplications, such as very stiff environments, or in environments with unknown or discontinuous stiffnessproperties, direct force control would be more desirable. An example of such an application is a testingdevice for cervical pairs [6], where sudden facet jointcontact would lead to probe damage due to errors in
172 Meccanica (2011) 46: 171–182
position control. An approach to circumvent this problem is the use of ﬂuidic muscles, which have the advantage of avoidance of slipstick effects, and whichalso have a very good transmission behavior betweenpressure and force. Additional advantages of ﬂuidicmuscles are their small size with respect to achievable forces, as well as their long durability. Pneumaticmuscles are already used to control the pose of parallel platforms [7]. By active force control, such ﬂuidicmuscles can produce any desired forcedisplacementlaw. In this paper, we analyze the use of pneumaticmuscles for directly controlling the force at the platform. Although ﬂuidic muscles are more difﬁcult tocontrol, we make use of the compliance and directforce controllability to improve the endeffector behavior when sudden contact arises. The optimal designof the platform has to take into account the problemsof singularity avoidance [8], optimization of manipulability index [9] and dexterity index [10], direct kine
matics computation [11], as well as general technological issues [1].This design task is described in the ﬁrst part of thepaper, the modeling and control of the pneumatic muscles is described in the second part and ﬁnally, experimental results obtained with the developed force controller are presented in the third part.
2 Platform design
2.1 Basic structureFor the realization of the targeted platform, a sixlegged
RRPS
type parallel platform [1] was chosen,each leg being hinged by a universal joint (RR) at thebase and a spherical joint (S) at the platform, witha prismatic joint along its leg axis as input stroke(Fig. 1).In order to avoid slipstick effects, the actuators arechosen as ﬂuidic (pneumatic) muscles (MAS 20, FestoAG & Co. KG. [12]), which operate by contracting inaxial direction when inﬂated.2.2 ActuatorsAs aﬂuidicmusclecanonlyproducepullingforces,anadditional force element is required in order to providealsoapushforce.Thisisrealizedinthepresentcontextby embedding the ﬂuidic muscle into a coaxial coil
Fig. 1
Concept of the developed parallel manipulator
Fig. 2
Single actuator with force, position and pressure sensors
spring whose stiffness is determined such that the required maximal pushing forces can be obtained withinthe stroke of the ﬂuidic muscle (Fig. 2). Each actuator is equipped with sensors measuring force, pressureand stroke, as illustrated in Fig. 2. The stroke of thehybrid actuator depends on the speciﬁc type of the ﬂuidic muscle and its length. For the chosen ﬂuidic muscle MAS 20, the stroke is
ℓ
=
40 mm.The relationship between air pressure and contractionforceisnonlinearanddependsontheactuallengthof the muscles (Fig. 3).The maximal and minimal actuator forces
F
min
,
F
max
depend on the current length
ℓ
of the actuatordue to the nonlinear characteristic of the ﬂuidic muscle: at a constant pressure the muscle force decreasesif the contraction increases. The corresponding maximal and minimal forces in Newton [N] at the attachment point of the platform (“
+
”
=
pull, “
−
”
=
push)are plotted for the current actuators in Fig. 4. The corresponding relationships can be approximated by the
Meccanica (2011) 46: 171–182 173
Fig. 3
Forcecontraction curves of ﬂuidic muscle
Fig. 4
Maximal and minimal actuator forces as function of actuator length
equations
F
min
=
7Nmm
ℓ
−
4
,
768
.
5 N
F
max
=
0
.
304Nmm
2
ℓ
2
−
361
.
3Nmm
ℓ
+
107
,
095
.
7 N2.3 Platform parametersAs basic design concept for the platform, a
symmetric simpliﬁed manipulator
[13] was chosen, where theattachment points at the base and the platform are located along a circle in three symmetrically distributedpairs of attachment points, respectively (Fig. 5). Thusone needs to consider as design parameters only therelative angle offsets
α
A
and
α
B
and the circle radii
r
a
and
r
b
at base and platform. This has many advantagesin manufacturing.As an additional design parameter, the height
h
EE
of the endeffector location with respect to the planepassing through the platform spherical joints (Fig. 6A)was considered. As will be seen, this parameter hassigniﬁcant inﬂuence on the required stroke of the actuators for completing a given platform motion.A further important issue was the manufacturing of the spherical joints, which needed to feature tilt anglesof
±
25
◦
in order to allow for the desired large roll andpitch angles of the platform. For this purpose, a customized version was built consisting of a twopiecesocket manufactured from brass and held together byspring screws (Fig. 6B).For the placement of the hinge points at base andplatform, a theoretical investigation of the best placement points was conducted in [6].The design consisted of the phases: (1) ﬁnding asingularityfree basic conﬁguration, and (2) verifying by interval analysis that the platform will achievethe expected results (stiffness, collisionavoidance,limited stroke, expected forces) within the targetedworkspace.
Fig. 5
Attachment points of the joint centers
174 Meccanica (2011) 46: 171–182
Fig. 6
(
A
) Distance of the end effector from the platform plane. (
B
) Limit of spherical joint
In order to achieve a singularityfree design of theplatform, the determinant of the inverse of the endeffector Jacobian
J
−
1
EE
should greater than zero, whichcorresponds to a positive “manipulability index” [9].The endeffector Jacobian
J
EE
hereby maps inﬁnitesimal length changes
δ
ℓ
of the legs to correspondinginﬁnitesimal variations
δt
EE
of the platform pose atthe end effector:
δt
EE
=
J
EE
δ
ℓ
.
(1)For the given architecture, it is easier to computethe inverse
J
−
1
EE
of the Jacobian. This can be donee.g. using the concept of kinematical differentials [14],which is not reproduced here as the correspondingequations are trivial.A possible, plausible criterion for the design of theplatformistominimizetherequiredstroketomovetheplatform, such as to have large platform motions withrestricted ﬂuidic muscle strokes. To this end, ﬁrst, thestrokeasafunctionofangle
α
A
wasregarded,showingthat the minimal stroke is attained for
α
A
=
α
B
, andin particular for
α
A
=
0
◦
. However, the manipulabilityindex at this point always has a minimum equal to zerofor
α
A
=
α
B
, meaning that a conﬁguration
α
A
=
α
B
isalways singular. Hence, this optimizing criterion is notsuitable for the platform design.As a second possibility, the dependency of the legstroke on the endeffector offset was regarded. Thisis shown in Fig. 7, where the stroke is plotted overthe upper platform radius for different values of theplatform offset. As can be seen, an operation point below of the platform always leads to higher requiredstroke, while positive offset values reduce the amountof the required stroke. Taking
h
EE
=
125 mm, onecan obtain a stroke of 40 mm at the prescribed upper radius of 100 mm, which is a realistic stroke for
Fig. 7
Strokeasfunctionof
h
EE
and
r
B
(
α
A
= −
20
◦
,α
B
=
5
◦
)
the chosen ﬂuidic muscles. This shows that choosinga suitable operation point is also relevant for an appropriate design of the platform. For the chosen angle values of
α
A
= −
20
◦
and
α
B
=
5
◦
, the platformwas proven to be singularityfree in the whole targetedworkspace [6].In addition to singularity avoidance, the rangeof motion and the target forces of the endeffectorwere analyzed in [6] using the intervalarithmeticlibrary ALIAS [15]. Hereby, the functionality wasveriﬁed throughout the complete workspace of theplatform deﬁned by the maximal stroke interval
ℓ
min
=
625
.
5 mm and
ℓ
max
=
665
.
5 mm.
3 Modeling and control of ﬂuidic muscles
3.1 Model a ﬂuidic muscleFor developing a modelbased control, the characteristics of each ﬂuidic muscle were identiﬁed by individ
Meccanica (2011) 46: 171–182 175
ualmeasurementsduetopossibledifferencesinthesixactuators. At ﬁrst, the relationship between force
F
onthe one side, and stroke
s
and pressure
p
on the other(as depicted by Fig. 4) was approximated by a ninthorder polynomial:
F
=
g(s,p)
≈
i
+
j
≤
9
i,j
=
0
a
ij
s
i
p
j
(2)The coefﬁcients were obtained from experimentsanddataﬁttingusingleastsquares[16].Thehighpolynomial order was chosen such as to be able to collectpossible higherorder nonlinearities in the model. Although lowerorder approximations might also proveto be sufﬁcient, such an analysis was out of the scopeof the present work.For the dynamic gas model, we regard the ideal gaslaw and the polytropic equation
pV
=
mRT,
(3)
pρ
−
γ
=
const.
,
(4)where
ρ
,
R
,
T
,
V
and
m
denote the density, gas constant, temperature, volume of the ﬂuidic muscle andgas mass, respectively, and
γ
is the polytropic exponent, taken here as
γ
=
1
.
4. After taking the timederivative of (4) and backsubstituting (4), one obtains
˙
p
=
γ RT V
˙
m
−
γ pV
˙
V.
(5)Note that in this derivation all thermodynamical variables, including temperature, are assumed to changewith time.For the performed actuations it can be assumedthat the volume of the ﬂuidic muscle only depends onthe stroke. We approximate this volumestroke dependency displayed in Fig. 8 by a third order polynomial
V(s)
=
3
i
=
0
b
i
s
i
(6)leading to the time derivative
˙
V
=
d
V(s)
d
t
=
d
V(s)
d
s
˙
s
=
3
i
=
1
ib
i
s
i
−
1
˙
s.
(7)The mass ﬂow
˙
m
with which the muscle is inﬂatedhas a nonlinear characteristic and is a function of pres
Fig. 8
Volume of pneumatic muscle over stroke
sure
p
inside the muscle and the input voltage
u
of thevalve
˙
m
=
φ(p,u),
(8)where the supply pressure is assumed to be constantand the function
φ(p,u)
is still unknown.By substituting (8) and (7) into (5), one obtains the
representation for
˙
p
˙
p
=
γ RT V(s)φ(p,u)
f
1
(s,p,u)
−
γ pV(s)
d
V(s)
d
s
˙
s
f
2
(s,p,
˙
s)
(9)which can be viewed as a superposition of two terms
f
1
(s,p,u)
and
f
2
(s,p,
˙
s)
.In (9), the second term can be readily computedoncetheapproximation
V(s)
isknown(frommeasurements) and the pressure is given.The difﬁcult term in (9) is the ﬁrst term, as the nonlinear valve function
φ(p,u)
is problematic to measure. In the sequel, we propose an alternative way of determining this term, based on an interpretation thatmakes its measurement easier.Assume the position of the actuator to be constant.Then, for the velocity it holds
˙
s
=
0, and the secondterm of the right side of the equation vanishes. Thusthe ﬁrst term in (9) is just the pressure rate for ﬁxed
actuator length and given voltage and pressure values
ˆ˙
p
s
(p,u)
= ˙
p(s,p,u)

s
=
const
.
.
(10)In order to determine this twoparametric function, the actuator is ﬁxed in a rigid frame at differ