The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS13.020

Unified Contact Force Control Approach for Cable-driven Parallel Robots usingan Impedance/Admittance Control Strategy

C. Reichert∗ T. Bruckmann†

University of Duisburg-Essen University of Duisburg-EssenDuisburg, Germany Duisburg, Germany

Abstract— In this paper a unified contact force controlapproach for cable-driven parallel robots (CDPR) is pro-posed. The contact force controller is governed by two con-trol loops: 1) An inner impedance control loop enforcing adynamical relationship between the internal forces and theend-effector velocities to maintain a desired tension levelin the cable system. 2) An outer admittance control loopconsidering the contact wrench by altering the desired end-effector trajectory. For the parametrization of the admit-tance controller the desired stiffness of the virtual manip-ulator system is defined by the mechanical stiffness of theend-effector. Further, the influences of changing the stiff-ness of the virtual system are discussed. To reconstruct thecontact wrench a disturbance observer based on the gener-alized momentum approach is incorporated into both con-trol loops. Experiments with a 6-DOF CDPR with indus-trial electric synchronous machines are presented validat-ing the proposed contact force control approach.

Keywords: cable-driven parallel robots, actuation redundancy,contact force control, impedance and admittance control loops, dis-turbance observer.

I. Introduction

It is known that cable-driven parallel robots (CDPR) havesome advantages in comparison to serial kinematic manip-ulators (SKM) in terms of high end-effector (EE) acceler-ations over a wide workspace [1]. The platform is guidedalong a predefined path by a system of cables in a parallelconfiguration which are conventionally wound up by winchdrives attached to the base. Due to the unilateral propertiesof the cables – cables can only pull, but never push – a de-sired tension level in the cable system must be guaranteed.For that purpose usually actuation redundancy is required tocompletely restrain the platform [1]. Actuation redundancycan be achieved by additional winch drives without chang-ing the degree-of-freedom (DOF) of the underlying mecha-nism. That means, the degree-of-actuation (DOA) exceedsthe DOF of the platform. This redundancy allows an en-ergy optimal distribution of the cable forces guaranteeing adesired tension level in the cable system and thus, the me-chanical stiffness of the EE can be directly adjusted. More-over, it increases and homogenizes the force capabilities of

∗reichert@imech.de†bruckmann@imech.de

a redundantly actuated CDPR in terms of acceleration andpayload – contributing to the reduction of the overall powerconsumption – despite the additionally actuated cables [2].Exactly these properties must be supported by the controllerto take full advantage from the capabilities of cable-drivenparallel robots.

Based on their versatile positive properties CDPRs pos-sess a great potential for industrial applications and rangefrom tasks, where on the one hand SKMs cannot cover therequired workspace and, on the other hand, the masses tobe handled further exclude SKMs. These properties are ex-ploited within the EU project “CableBOT” for the devel-opment of a new generation of modular and reconfigurableCDPRs which are capable for many different tasks, e.g. inthe area of maintenance or logistics of large components.One example of an application is the painting and sand-blasting of an airplaine as shown in Fig. 1:

Fig. 1. Modular and reconfigurable cable-driven parallel robot inthe field of maintenance and logistic of large structures. Source:http://www.cablebot.eu

This work is related to the development of a unified con-tact force control approach for CDPRs. Especially tworequirements create challenging tasks: First, establishingand, second, maintaining an accurate contact between themanipulator and its environment. Eppinger and Seeringprovide in [3] an analytical overview on the involved dy-namics and clarify possible influences which can lead tobandwidth limitations in robot force control. Particularlynon-linearities in form of discontinuities during the con-tact phase can decrease the performance of the contact forcecontroller.

To overcome these limitations, the well-known impedancecontrol approach [4] or the hybrid position/force control ap-

proach [5] represent elegant solutions. In the hybrid posi-tion/force control, contact forces are applied in a predefineddirection decoupled from the motion controlled directions.Kraus et al. applied the hybrid position/force control ap-proach to CDPRs in [6] and show relevant experimental re-sults verifying the presented approach. Especially the in-corporated state-machine defining how the manipulator in-teracts with its environment provides an interesting point ofview with respect to an industrial application. Nevertheless,the drawback of the hybrid position/force control approachis that an EE contact wrench can only be regulated in pre-defined directions. Therefore, an EE contact wrench in amotion controlled direction can damage the manipulator orthe object to be handled.

The idea behind the impedance control approach is to usethe EE velocities in addition to an EE contact wrench feed-back to achieve a desired response of the manipulator bycreating a virtual manipulator system with specific prop-erties. Hence, no knowledge about the directions of theEE contact wrench is required. Caccavale et al. presentin [7] a general impedance control scheme for cooperatingmanipualtors. In more detail, a general impedance controlapproach is presented with a centralized impedance con-trol strategy [8] creating a compliant behaviour at the objectlevel and a decentralized impedance control strategy [9] en-forced for each manipualtor to avoid large internal forcesonto the object.

For later industrial applications a SKM with a wrist-mounted 6-DOF wrench sensor to measure the EE contactwrench can be attached onto the platform of a CDPR. How-ever, in the following work, the force sensors are only lo-cated at winch drives to measure the cable forces. Throughthis, the EE contact wrench must be reconstructed using themeasured cable forces. One suitable approach is the dis-turbance observer presented in [1] inspired by the failuredetection and isolation algorithm introduced in [10]. It isshown that the incorporated disturbance observer featuressome passive properties. This fact allows to incorporatethem into the controller design [1] based on the presentedpassivity analysis.

In this work, two separated control loops similar to [7]are used. Using this control approach, some modificationsare required: An inner internal force-based impedance con-troller with the aim to enforce a dynamical relationship be-tween the EE velocities and the internal forces onto the EEbeing manipulated. This approach can guarantee a desiredtension level in the cable system for given EE movementsbased on the work [1]. In general the internal force-basedimpedance controller is governed by a computed-torquecontrol (CTC) structure with a shaping of the internal forcesresulting from the chosen mass matrix. An outer admittancecontrol loop inspired by the work [11] which is used to reg-ulate the EE contact wrench by the generation of a virtualmanipulator system to alter the desired trajectory of the EE.By manipulating the properties of the virtual manipulator

system the dynamical behavior of the EE can be adjustedregarding to the application. When the EE goes into contactwith its environment, energy will be released by the virtualmanipulator system to get a stable contact. Beyond that,the shape of the free-form surface will be reconstructed bythe outer admittance control loop as an input for the innerimpedance control loop. Therefore, an accurate trajectorytracking behaviour of the EE to follow a path over the free-form surface can be guaranteed. Hence, a unified controlapproach considering both the regulation of internal forcesguaranteeing a desired tension level in the cable system andan EE contact wrench to establish a safe contact with theenvironment is designed.

The paper is organized as follows: First, the dynamics ofa CDPR formulated in EE coordinates are shown in section2. Furthermore, a force distribution algorithm is describedto solve the inverse dynamics problem. In section 3, a dis-turbance observer for redundantly actuated CDPRs is de-scribed for the reconstruction of an EE contact wrench. Inthe following section 4, the derivation of the unified contactforce control approach is introduced based on two separatedimpedance and admittance control loops, respectively. Fur-ther a control strategy is presented to handle the appliedEE contact wrench. In section 5, experiments on a 6-DOFCDPR with industrial electric synchronous machines arepresented. The paper is summarized in section 6.

II. Manipulator Dynamics

A. Formulation in End-Effector Coordinates

A CDPR can be modeled as a multi-body system (MBS)consisting of a working platform (end-effector) with δ-DOFs, constrained by m flexible cables in a parallel con-figuration. The EE is considered as a rigid body driven byconstraint forces and task forces (environment). As shownin [1], the equations of motion for CDPRs can be formu-lated by Lagrange’s equation of the first kind, with n gener-alized joint coordinates q ∈ Vn separated into active (cablelength l) and passive (tilt angles α and β) joint coordinatesdescribing the motion of the mechanism and δ ∈ N EEcoordinates x =

[o φ

]T(task-space) according to the po-

sition o and orientation φ with respect to a local coordinatesystem P . To describe the orientation of the EE with re-spect to an inertial coordinate system B, the rotation matrixR based on Bryant angles ψ, ϑ and ϕ is used.

These equations can be obtained by cutting each kine-matic chain of the mechanism at the EE and introducingredundant geometric and kinematic closure conditions [1]:

0 = h(q,o, φ) , (1)0 = J(q) q+ JX (φ) x , (2)

where h(q,o,φ) defines the forward kinematics of eachkinematic chain between the cable exit point and the EE’scenter of mass. Assuming a pulley-based cable guidance,

the cable length li can be computed as follows:

li = bi + di − pi , i = 1, . . . ,m . (3)

The vector bi describes the locations of the pulleys, whichare normally fixed in space. The vector Ppi represents theplatform attachment points and can be transformed into theinertial coordinate system B by the help of the rotation ma-trix R :

pi = o+R Ppi , i = 1, . . . ,m . (4)

Depending on the tilt angels α and β, the cable exit pointsdi defined in the local reference system G are given by:

dG =

ρi cosαi (1 + sinβi)ρi sinαi (1 + sinβi)

−ρi cosβi

, i = 1, . . . ,m . (5)

The cuts which have to be made to get the tree structurefor the given mechanism are shown in Fig. 2. The numberof fundamental loops can be constituted in the topologicalgraph and corresponds to the number of the winch drivesfor a redundantly actuated CDPR:

P

B

G

EE

Fig. 2. Tree structure of a cable-driven parallel robot after opening thekinematic chains at the end-effector

The m external control forces u generated by the winchdrives directly guide the mechanism along a predefined tra-jectory x and the equations of motion become[

Dq+QMXx+KXx+QX

]=

[u0δ

]+

[JT

JTX

]λ , (6)

where D is the inertia matrix related to each winch driveand Q includes all remaining potential forces, especiallyfriction forces. The subscript X denotes elements for theparticular linear EE equations of motion. Accordingly, thegeneralized mass matrix is MX, the generalized Coriolisand centrifugal matrices are summarized in KX. In Qx allgeneralized EE disturbances and the gravity forces are in-cluded.

The mass matrix MX described above – defined in theplatform-fixed coordinate system P – is given as follows:

MX :=

[mpI −mp [c]

mp [c] J−mp [c] [c]

].

Here, mp is the mass of the EE and J is the inertia of theEE with respect to the platform fixed coordinate system P .A shift of the center point of mass c according to a variablepayload is considered by a skew-symmetric matrix [c].

On this occasion, the Lagrange multipliers λ charac-terize the constraint forces. The set of all valid con-figurations of a CDPR, according to the constraints, de-fines the so-called configuration space (c-space) V :={q ∈ Vn | h (q,o,φ) = 0} of the CDPR [1]. Based on theconstraint Jacobian J – which has locally full rank – and itsinverse, denoted by J−1, the generalized velocities can beexpressed as [

qx

]= Fx, F =

[−J−1JX

Iδ

](7)

to parametrize all admissible configurations q ∈ Vn. Inthe next step, the so-called orthogonal complement F of[J JX

]is defined, fulfilling

[J JX

]F ≡ 0. The defined

orthogonal complement is used to eliminate the unknownconstraint forces λ by the projection of the equations ofmotion (6) onto the c-space V . Since the vector of externalcontrol forces u only comprises non-zero entries for the mactuated joints with generalized coordinates, a submatrix Aof the orthogonal complement F can be identified so that

FT[u0δ

]= ATc ,

where c ≡ (c1, . . . , cm) is a vector of generalized controlforces corresponding to the actuator coordinates in form ofthe cable forces f . With the help of the parametrization inEq. (7) and using its time derivative

q = Ax+ Ax , (8)

the equations of motion for CDPRs formulated in EE coor-dinates can be described by:

M(q) x+K (q, q) x+Q(q, q) = AT(q) f , (9)

where

M := FT [diag {D,MX}]F ,

K := FT[diag {0m,KX}F + diag {D,MX} F

],

Q := FT[QT QTX

]T.

In Eq. (9), the matrix AT is the so-called structure matrixhaving full rank δ. It describes the influence of the cableforces f onto the EE. Furthermore, it allows to classify theDOA α = rank (A) . Hence, the DOA determines howmany generalized accelerations are affected by the con-troller [1]. The degree-of-redundancy (DOR) is definedas follows ρ := m − α. To fulfill the requirements of afully constrained CDPR, guaranteeing that the cables canbe tensed within the wrench-feasible workspace, the DORneeds to be ρ ≥ 1.

B. Force Distribution Algorithm

Due to the unilateral properties of cables, the number ofcables of a redundantly actuated CDPR exceeds its DOF byρ. In this case (ρ > 0), the structure matrix AT is not squareand its kernel (null-space) is of dimension ρ. ThereforeEq. (9) results in an under-determined system of equationswith an infinite number of possible solutions and cannot besolved unambiguously for the cable forces f ∈ Vm. Hence,it is possible to generate cable forces lying in the kernel ofAT ∈ Vn×m, which have no effect on the motion but canbe used to generate a tension level in the cable system.

The inverse dynamics problem (9) consists in determin-ing the required cable forces fd for a given trajectory x:

fd =(AT)+

W

(Mx+Kx+Q

)+NAT,Wf0 . (10)

Therewith(AT)+W

:= W−1A(ATW−1A

)−1is a

weighted pseudo inverse, where W is a positive definiteweighting matrix, with respect to the drive capabilities. Thenull-space projector NAT,W ∈ Vn×r generates the tensionlevel in the cable system close to the desired value of f0 [1].

Moreover, the cable force distribution f must satisfy thefollowing inequality constraint

0 ≤ fmin ≤ f ≤ fmax , (11)

where fmin defines a lower bound guaranteeing a minimumdesired tension level in the cable system and fmax definesan upper bound considering the breaking load of the cablesor the maximum torque generation of the winch drives, re-spectively.

For example, the defined inequality constraint (11) canbe illustrated for the case of a CDPR withm = 3 and r = 2as a cube C shown in Fig. 3:

f1

f2

f3

SC

F

fp

fc

f

fm

Fig. 3. Cable force distribution of a CDPR with three cables and a two-dimensional actuation redundancy

To compute valid cable forces f , the so-called Puncture-Method described in [12] is used. In contrast to conven-tional methods where optimization algorithms are prepared

to solve Eq. (9) for the cable forces f considering Eq. (11),here, a geometrically inspired method is introduced.

As long as the EE remains in the workspace, an intersec-tion set F between the hypercube C and the solution spaceS exists. This intersection set F contains all possible ca-ble force distributions f respecting the force equilibriumof Eq. (9) and Eq. (11). Here, the objective is formulatedto find the cable force distribution delivering nearly min-imum cable forces f . It can be computed as follows (seealso Fig. 3):

• Compute the projection of the center point fm of the hy-bercube C onto the solution space S. This valid point isdenoted by fc.

• Compute the projection of the origin of the coordinatesystem of the cable forces f onto the solution space S.This step provides a second point fp lying outside of thehypercube C.

• Connect fp and fc. The resulting line penetrates one sideof the hypercube C. This point delivers nearly minimumcable forces f since a minimum energy consumption ofthe winch drive system should may be reasonable, de-pending on the application.

The introduced Puncture Method is particularly character-ized by two properties: It computes a nearly minimumcable force distribution f and has a constant computationtime. Thus, it can be implemented on a real-time capablecontrol system. In addition, this method provides valid so-lutions for CDPRs (r > 2) of higher redundancies.

III. Disturbance Observer

A. Motivation

The proposed disturbance observer is inspired by themomentum-based collision detection algorithm introducedin [10]. The idea behind the disturbance observer is to con-sider the EE contact wrench as an error within the actuatedwinch drive system. One advantage of the proposed distur-bance observer is that it can be seamlessly integrated intothe control scheme. This statement can be emphasized bythe passivity analysis shown in [1].

B. Derivation of the Observer Law

In the following it is exploited that – based on the projec-tion method shown in Eq. (9) – CDPRs take (up on the c-space V) the form of an affine control system of the secondorder. Using this fact, the proposed disturbance observercan be applied to CDPRs.

Including all disturbances in form of an EE contactwrench η containing forces and torques, the equations ofmotion (9) can be described by

M(q) x+K (q, q) x+Q(q, q) = AT(q) f − η . (12)

One drawback of the proposed disturbance observer is thatthe crucial source of disturbances η is not limited to an

EE contact wrench. The disturbance observer will recon-struct any uncertainties in the equations of motion for theunderlying mechanism. Therefore, a parameter estimationalgorithm e.g. based on a recursive least-squares algorithm(RLS) is highly preferable to eliminate all model uncertain-ties like a variable mass matrix.

As proposed in [10] for the design of a disturbance ob-server the generalized momentum p is used:

p = M(q) x . (13)

The time derivative of the momentum equation is

p = M(q) x+ M(q) x . (14)

Furthermore, the accelerations in Eq. (14) can be replacedby the equations of motion in Eq. (9) to get an expressionfor p. In combination with the measured momentum p,the following description of the disturbance observer canbe yielded:

p = AT(q) f − ζ (q, q) +KR(p− p

), (15)

η = KR(p− p

), (16)

with

ζ (q, q) = −M(q) x+K (q, q) x+Q(q, q) . (17)

By the positive definite gain matrix KR the observer errorr = p − p is fed back and an asymptotic convergence ofthe observer error can be ensured [1].

By using Eq. (15) following residual vector can be givento estimate the required EE contact wrench η:

η = KR

t∫0

(AT(q) f − ζ (q, q)− η

)dt− p

. (18)

For the implemantion of Eq. (18), the measured EE veloci-ties x and forces f are required. No EE accelerations x norinversion of the mass matrix M(q) are needed.

The residual dynamics satisfy the following form

˙η(t) = −KRη +KRη , (19)

which can be identified as a linear exponentially stable sys-tem driven by the fault η [10]. Furthermore, the disturbanceobserver delivers decoupled EE contact wrenches for eachdirection of movement. This fact can be shown by follow-ing transfer function [1]:

ηi

ηi=

KRi

s+ KRi

, i = 1, . . . , δ . (20)

From this it is obvious that for

KR →∞ ⇒ η ≈ η, (21)

and in the case that all model uncertainties are eliminated,the disturbance observer yields the EE contact wrench η onits output.

IV. Contact-Force Controller

A. Motivation

Consider the CDPR in Fig. 2 as a multi-manipulator sys-tem handling an object (EE), where the object is going toestablish a contact with its environment. Especially two re-quirements are challenging tasks: First, establishing and,second, maintaining an accurate contact between the ma-nipulator and its environment. Moreover, in the case ofredundantly actuated CDPRs, the used control approachmust feature the characteristics of giving each manipulatorknowledge of the others to form a cooperating interactionof the manipulators. Otherwise, sagging effects degeneratethe performance of the CDPR and cause an undesired ten-sion level in the cable system. For this reason, force controlstrategies have to be integrated, taking a significant influ-ence on the dynamical behavior of the CDPR. Summariz-ing, the following desirable properties should be providedwithin the design of a controller where the EE is going toestablish a contact with its environment:

• Each manipulator features a compliance to eliminate un-desired interaction forces.

• Platform dynamics take no influence on tracking orsteady state position errors.

• An accurate and safe contact between the EE and its en-vironment must be maintained.

All these properties can be considered in the design of a uni-fied contact force controller using an impedance/admittancecontrol strategy as presented in the following.

This control approach represents a class of control al-gorithms where the controller directly gives each manipu-lator the necessary compliance. It is based on the workspresented in [9] and [11]. Two separated impedance andadmittance control loops are implemented, respectively.This remarkable feature imparts each manipulator the nec-essary robustness guaranteeing a desired tension level inthe cable system and establishing an accurate and safecontact between the EE and its environment. Thus, theimpedance/admittance control strategy becomes an attrac-tive control approach for controlling CDPRs when the EEis going into contact with its environment.

B. Derivation of the Impedance Control Law

The inner impedance control loop can be classified as adecentralized control loop based on [9]. It is enforced foreach winch drive, avoiding large internal forces onto theEE. For that purpose, the equation of motion for each winchdrive can be described as follows [1]:

ui = Mi (q) qi +Qi (qi, qi) + fi , (22)

where M := D +(AT)+

MX (A)+ represents the joint

space mass matrix. The compensation of disturbances Qensures that the external control forces u are generated by

the winch drives exactly. To maintain the desired tensionlevel in the cable system, the measured cable forces f haveto be replaced by the inverse dynamics solution fd.

When multiple manipulators are connected to a platform,the forces f generated by each manipulator expressed in theCartesian space can be decomposed into motion-inducingforces fM and internal forces fI, with

f = fM + fI . (23)

Internal forces fI produce no net forces on the EE andthus they must be chosen to lie in the range of the null-space projector NAT . This means that internal forces fI arenot affected by motion-inducing forces fM and can be reg-ulated simultaneously [1]. Using this fact, the followingdecomposition method can be used to compute the internalforces for each manipulator

fIi = AThi

(Im −

(AT)#

h ATh

)f , (24)

where(AT)#

h is a generalized inverse of ATh in the ho-

mogenized form as shown in Sec. VI-A. A valid solutionis(AT)#

h = ATh

(AhA

Th

)-1which is the Moore-Penrose

pseudo inverse. The decomposition method defined in Eq.(24) results in zero interaction forces when fI = 0 holds.

As described in [9], impedance control in the applicationof cooperating manipulators must enforce a relationship be-tween the EE velocities and the internal forces on the ma-nipulated EE. Otherwise, if the total forces imposed by theenvironment on the manipulator will be incorporated in theimpedance relationship, platform dynamics will contributeto tracking and steady-state position errors.

Each manipulator is equipped with the followingimpedance describing a linear second-order function ex-pressed in the Cartesian space as shown in [1]:

GIδx+BIδx+CIδx = KPIδfIi , (25)

where δx = xd − x = pose errors of the platform,δfIi = fIi − fIid = internal force errors,GI, BI, CI = desired mass-, damping- and stiff-

ness matrices.

The force controller with the diagonal gain matrix KPI

is incorporated into the impedance function in Eq. (25),achieving a well-defined force tracking behaviour of the in-ternal forces fI. Here, the trajectory of the EE will be ma-nipulated in a way so that the errors in the internal forcesδfI are going to zero [13].

One property of the choosen impedance is that separateforce and position control loops are not required. Therefore,the impedance parameters can be chosen almost freely incontrast to the mentioned bandwidth limitations shown by[1]. Additionally, internal forces produce no net forces onthe EE and therefore, the platform dynamics have no influ-ence on tracking or steady state position errors.

To meet the stability constraints shown in [9], theimpedance mass G should be chosen equally to the pro-jected mass matrix M := MX +ATDA.

Each joint acceleration is related to the EE accelerationby its structure matrix

qi = Aixi + Aixi . (26)

Solving Eq. (25) for x, substituting into Eq. (26), and in-corporating into each winch drive dynamic equation (22),yields the following control law for each manipulator:

ui = Mi

{Ai

(xd +G

−1I . . .

. . .[BIδx+CIδx− δfIi

])+ Aixi

}+Qi + fi .

(27)

To achieve an accurate trajectory tracking behaviour,model-based controllers are required to compensate the dy-namics of the platform. In general, the internal force-basedimpedance controller is governed by a CTC structure withthe shaping of the internal forces resulting from the chosenmass matrix [1]. Here, the compensation of generalizedEE disturbances including effects as a variable mass matrixM are handled by the incorporated RLS algorithm. Thecompensation is executed considering the inverse dynam-ics solution. Therefore, an energy optimal distribution ofthe generalized EE disturbances to the m winch drives canbe performed. The block diagram of the proposed internalforce-based impedance controller is shown in Fig. 4.

CDPR

Fig. 4. Block diagram of the internal force-based impedance controller

The computational effort of the internal force-basedimpedance controller is quite low. Based on a desired tra-jectory xd and fId, measuring the pose x and solving theinverse dynamics problem shown in Eq. (10) to get an ex-pression for the force distribution fd guaranteeing a desiredtension level in the cable system, the required external con-trol forces u can be computed. The internal forces fI arecomputed using Eq. (24) from the measured cable forces ffor all winch drives.

C. Derivation of the Admittance Control Law

As described in the introduction, the regulation of an EEcontact wrench is a challenging task. The object of theouter admittance control loop in form of a centralized con-trol loop is featuring the characteristics of generating a safecontact between the EE and its environment by manipulat-ing the dynamical properties of the manipulator system.

During the contact phase the residual vector η (outputof the disturbance observer) will rapidly increase. When adesired threshold of the EE contact wrench is exceeded, theouter admittance control loop described by a linear secondorder function will be activated as shown in [11]:

GAδx+BAδx+CAδx = KPAδη , (28)

where δx = xd − x = pose errors of the platform,δη = η − ηd = EE contact wrench errors,GA, BA, CA = desired mass-, damping- and stiff-

ness matrices.

The measured EE contact wrench η will be directly fedback by the outer admittance controller altering the desiredtrajectory xd by integrating twice

δx = G−1A

(−BAδx−CAδx+KPAδη

), (29)

while enforcing a desired EE contact wrench ηd. Thus, areflex reaction of the EE is enforced to establish a safe con-tact between the EE and its environment.

The idea behind this strategy is to enforce a virtual ma-nipulator system featuring the property that the manipula-tor will counteract against the external EE contact wrenchalong the same resulting direction. After the contact estab-lishment and in a first approximation, by combining Eq. (9),Eq. (28), futher considering the inverse dynamics solutionin Eq. (10) and assuming the limit case of η (for largeKR), the manipulator dynamics in the corresponding con-tact force direction become:

M(xd +G

−1A

[BAδx+CAδx . . .

. . . −KPAδη])

= ATf .(30)

The EE contact wrench errors δη are shaped by a factordepending on the chosen admittance mass matrix GA. Infact, by changing the virtual manipulator system the effec-tive mass of the EE can be lowered [10] and the applied EEcontact wrench η can be regulated faster. Nevertheless, theproposed control approach defines an active controller dueto the fact that additional energy is fed into the system af-ter the contact establishment. That means, there is a upperlimit where the controller loses its stability.

D. Control Strategy

In the following, the control strategy is proposed to bringthe EE in contact with its environment. For that purpose,

a schematic representation of the environment is shown inFig. 5 and the corresponding coordinate system T of thetask and the applied EE contact wrench η are given.

P

B

T

ηz

1©

2©

3©EE

path

Fig. 5. Schematic representation of the environment

The presented control strategy considers all three phaseswithin free motion of the EE, establishing and maintainga contact between the EE and its environment:

1 Free motion of the EE

During the free motion of the EE it is highly preferableto use the proposed internal force-based impedance con-troller. Based on its properties it provides an accuratetrajectory tracking while showing a passive feature like aspring-damper system. Through this, an unexpected be-haviour after establishing a contact between the EE and itsenvironment can be prevented.

2 Establishing a contact

When the EE is going to establish a contact with its en-vironment, the dynamical properties of the system will bechanged suddenly. The simplest reaction strategy would beto stop the trajectory. However, this implies that the EEwould still have a contact with its environment. Therefore,in this phase when an EE contact wrench η is detected bythe presented disturbance observer, contact energy will bereleased. Hereafter, a desired EE contact wrench ηd is ap-plied using the outer admittance control loop by the genera-tion of a virtual manipulator system with defined propertieslike a smooth reflex reaction according to the intensity ofthe EE contact wrench η.

3 Maintaining a contact

In this phase, the EE moves over the free-form surface.Here, the outer admittance control loop must manipulate thetrajectory which is the input for the inner impedance controlloop in a way that the EE keeps the contact with its envi-ronment guaranteeing the desired EE contact wrench ηd.When the contact is lost, the impedance controller bringsthe EE again into contact with its environment.

After finishing the task, the desired EE contact wrenchwill be set to zero, so that the EE can release the contactand cross over to the free motion.

V. Experiments

A. Prototype

The presented unified contact force control approach wasimplemented on the 6-DOF SEGESTA prototype shown inFig. 6. This system is operated by an industrial controlsystem based on Beckhoff TwinCAT 3 at a sampling fre-quency of 2 kHz. For the actuation of the CDPR, ten elec-tric synchronous machines (Beckhoff AM8031) are usedconnected to the winches with a nominal torque generationof cnom = 1.2 Nm. Thus m = 10, i.e. the CDPR hasa fourfold actuation redundancy. The winches are placedat the vertices of a nearly symmetric cuboid with an over-all dimension of 1.2 m x 1.4 m x 1.1 m (l x L x h).Two pulleys can be reconfigured to enhance the mechan-ical stiffness of the EE located on the opposite sides ofthe cuboid. The variable pulleys are actuated by linear ac-tuators (IGUS ZLW-1040-03-S-150-L-1100) with steppermotors type NEMA 34. The mass of the EE is approxi-mately 0.8 kg. Strain-gauge beam arrangements (MegatronKM302) are integrated into the winch drives and used tomeasure the cable forces.

EE PP

drive��

cable��

pulleySS

surface PP

Fig. 6. 6-DOF SEGESTA prototype

To measure the EE pose, a forward kinematics com-putation based on the Levenberg-Marquardt-Fletcher algo-rithm is used. Additionally, the cable stiffness is incor-porated in the forward kinematics to get more exact cablelengths. The motors have a high stiction of about 0.12 Nmwhich makes it necessary to use an additional friction ob-server introduced in [14] to support the internal force-basedimpedance controller. In addition, the compensation of fur-ther unmodeled effects in the position-controlled direction,considered as a variable mass matrix M, are handled bythe proposed disturbance observer. The disturbance ob-server is parametrized with KR = 150. Further, a recursiveleast-squares estimator is used to support the disturbanceobserver. To compute a desired tension level in the cablesystem, the introduced Puncture-Method is used due to itsproperty to deliver nearly minimal cable forces with fminequal to 10 N.

B. Results

For the experiment, each manipulator impedance de-scribed in Eq. (25) was chosen using the guidelines in-troduced in [9] examining the projected mass matrix M.The impedance stiffness matrix CI was chosen such that animpedance bandwidth of approximately 3 Hz is achievedand BI was chosen to achieve critical damping: CI =1000M and BI = 63M. The given parametrization wasalso chosen according to [1], supporting a well behaviourof the EE. By lowering the desired impedance mass matrixGI, the internal forces errors can be minimized. However,as described in [9], there is a lower limit where the con-troller looses its stability. Additionally, the admittance con-trol loop was parametrized with the following parameters:The admittance stiffness matrix CA was chosen equally tothe mechanical stiffness of the EE described in App. VI-B.Based on the assumption that the environmental stiffnessgoes to infinity, this choice of CA leads to a natural posi-tion tracking behaviour of the EE over the free-form sur-face. In the same way, the admittance damping matrix CAwas chosen to achieve critical damping, respectively. Theadmittance mass matrix GI results from the chosen stiff-ness matrix CA. The control gains KPI and KPA for theincorporated force controllers in the impedance and admit-tance control loops were manually adjusted considering theguidelines defined in [13].

The following scenario is used to demonstrate the per-formance of the proposed contact force control approach(see Fig. 5): At the beginning ( 1© 7→ 2©) the EE moves inthe z-direction with a constant velocity of 1 m/s, establish-ing a contact with the free-form surface. After establishingthe contact ( 2© 7→ 3©) with the free-form surface, the EEmoves in the y-direction along the shown path with a max-imum EE velocity of 1 m/s. The start- and end-point on thefree-form surface are defined by the length of dc = 0.6 mwhich corresponds to the given path. Within the experi-ment, the desired EE contact wrench in the z-direction wasset to ηd =10 N. Experimental results are shown when theEE goes into contact with the free-form surface and whenthe EE moves over the free-form surface. The evolutionof the EE contact wrench errors δη and the internal forceerrors δfI expressed in the joint space during the run areshown in Fig. 7 and Fig. 8, respectively. The EE pose er-rors during the run are shown in Fig. 9.

At low speeds the presented contact force control ap-proach has the ability to regulate the desired EE contactwrench ηd during the establishment of the contact quitefast. After a short time period, the contact could be estab-lished with a small overshoot. Nevertheless, both criterias– overshoot and settling time – depend on the chosen EEvelocities. The EE retracts itself and moves rapidly awayfrom the free-form surface while guaranteeing the desiredEE contact wrench ηd. The performance of the control ap-proach can be further adjusted by KR due to the fact that

0 2.5 5 7.5−4.5

−3

−1.5

0

1.5

3

4.5

time / s

forc

es /

N

ηx

ηy

ηz

1© 7→ 2©

2© 7→ 3© released

JJ

Fig. 7. EE contact wrench errors when moving the end-effector along thegiven path over the free-form surface

0 2.5 5 7.5−4.5

−3

−1.5

0

1.5

3

4.5

time / s

forc

es /

N

1© 7→ 2©

2© 7→ 3© released

JJ

Fig. 8. Internal force errors when moving the end-effector along the givenpath over the free-form surface

this gain matrix defines the low-pass character of the dis-turbance observer. Additionally, the results shown in Fig. 9furthermore demonstrate that the control approach featuresa well-defined force tracking behaviour while regulating theEE on the given path despite of the high EE velocities. Dur-ing the run there are small oscillations in the measured EEcontact wrench and also peaks in the EE pose errors (about1 mm) can be observed due to stick-slip effects. Besides,the results shown in Fig. 9 and Fig. 7 demonstrate the im-portance of the disturbance observer, due to the elimina-tion of steady state errors at the rest positions. Beyond that,the disturbance observer delivers a decoupled EE contactwrench η for each direction of movement and, thereby, setsan improved position and force tracking behaviour of theEE. In addition, the internal force errors δfI shown in Fig. 8are going to zero at steady state, due to the compensationof joint disturbances like friction forces. Moreover, the pre-sented Puncture Method delivers a valid cable force distri-

0 2.5 5 7.5−4.5

−3

−1.5

0

1.5

3

4.5

time / s

pose

/ m

x 10−3

ex

ey

ez

1© 7→ 2©

2© 7→ 3© released

JJ

Fig. 9. Pose errors when moving the end-effector along the given pathover the free-form surface

bution f within the run, so that at all times a desired tensionlevel in the cable system can be guaranteed.

VI. Conclusions

In this paper, the design of a unified contact force con-trol approach based on two separated impedance and ad-mittance control loops was proposed. In a first step, theequations of motion for CDPRs in terms of EE coordinatesare presented using a projection method. On this base, adisturbance observer reconstructing the applied EE con-tact wrench was derived. Inspired by the idea of consid-ering CDPRs as a multi-manipulator system establishinga contact with its environment, the presented unified con-tact force control approach was introduced. Both the inter-nal forces to guarantee a desired tension level in the cablesystem and an EE contact wrench to establish an accurateas well as a safe contact between the EE and its environ-ment can be considered. One advantage of the inner internalforce-based impedance controller is that no explicit positionand force control loops are necessary and the impedance pa-rameters can be freely chosen. The advantages of the outeradmittance controller are: During the contact phase energywill be released to get a safe contact with the free-form sur-face and during the motion of the EE over the free-formsurface the desired trajectory will be altered defined by thevirtual manipulator system keeping the EE into contact.Additionally, the necessary guidelines to parametrize theused admittance controller based on the mechanical stiff-ness of the end-effector are shown. The implementation isdiscussed and the feasibility is shown. Experimental resultsare reported for the 6-DOF SEGESTA prototype, showingthat an accurate force tracking behavior and a desired ten-sion level in the cable system can be guaranteed during theEE motion over the free-form surface. Future works are fo-cusing on the connection of a serial kinematic manipulatorwith the working platform of a cable-driven parallel robot.This may result in tasks in which the serial kinematic ma-nipulator goes into contact with its environment, e.g. thesandblasting of a large workpiece as shown in Fig. 1.

Appendix

A. Homogenization of the Structure Matrix AT

To avoid different physical units within the computationof the internal forces fI in case of redundantly actuated CD-PRs the structure matrix AT must be homogenized (see also[15]). In accordance with this, the homogenized structurematrix AT

h contains consistent physical units.The homogenization of the structure matrix AT will be

executed using the kinematic closure conditions (2) andthey are defined in homogenized form as follows [2]:

0 = J(q) q+ JXh (φ) diag (jx) x . (31)

On this occasion, only the EE-to-manipulator Jacobian JXis of importance. jx contains several characteristic lengths

serving as divisors for individual matrix elements of the EE-to-manipulator Jacobian JX in the form of

jx =(1, 1, 1, jx, jy, jz

)T. (32)

The vector jx is chosen in a form that the homogenizedEE-velocities xh = diag (jx) x only contain elements withlength units. By the help of the platform attachment pointsPpi, the characteristic lengths (middle platform radiuses) jxcan be computed as follos:

jx =1

m

m∑i=1

|Ppxi| , (33)

jy =1

m

m∑i=1

|Ppyi| , (34)

jz =1

m

m∑i=1

|Ppzi| . (35)

Using Eq. (31), the generalized joint velocities q dependingon the homogenized EE-coordinates xh are defined by:

q = −J−1JXh xh . (36)

From this, the homogenized structure matrix ATh can be ex-

pressed as shown in Sec. II-A.

B. Derivation of the Mechanical Stiffness K

The stiffness matrix K(x) describes a linear relationship

δw = K(x) δx (37)

between an infinitesimal wrench δw and the resulting dis-placements δx of the EE according to [2]. In the static equi-librium of Eq. (9) the following description can be given:

δw = −∂AT

∂xf δx−AT δf . (38)

On this occasion, the cable forces

δf = Kl δl (39)

cause virtual infinitesimal displacements δl of the cablelengths. The matrix Kl = diag

(kl,1, · · · , kl,m

)defines the

stiffness matrix expressed in the joint space, wherein thediagonal elements kl,i are significantly defined by the com-pliance of the winch drives, which in turn depend on theused control approach.

Equally, infinitesimal displacements δx cause an oppo-site change δl in the cable length:

δl = −A δx . (40)

In summary, the wrench can be computed by:

δw = −∂AT

∂xf δx+AT Kl A δx . (41)

From the resulting Cartesian stiffness matrix

K(x) = −∂AT

∂xf +AT Kl A (42)

two different influences are obvious. The first term Ka isnamed the active stiffness and describes the changes in thestructure matrix A by a displacement of the EE. It is also af-fected by the cable forces f . The second term Kp indicatesthe passive stiffness, which depends mainly on the controlapproach.

Acknowledgments

The research leading to these results has received fund-ing from the European Community’s Seventh FrameworkProgramme under grant agreement No. NMP2-SL-2011-285404 (CableBOT).

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