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Advances in the Modelling and Control of Elastic Parallel Manipulators

Krzysztof Stachera Institute of Technology of Machines

and Automation of Production,Czestochowa University of Technology, Poland

Frank Schreiber, Walter Schumacher Institute of Control Engineering,

Technical University of Braunschweig, Germany

1 Introduction

Modern industrial processes require high performance manipulators, especially regarding the obtainable speedand precision. Parallel manipulators have the potential to meet some of those goals. As opposed to their om-nipresent serial counterparts, the drives of parallel mechanisms are mounted to the base or at least close to thebase allowing for a signicant reduction of the moved structural masses and at the same time increasing the ob-tainable precision of the parallel manipulators. Since the heavy drives do not have to be moved, the links and joints can consequently be designed lighter. The resulting light-weight manipulators are able to realize higheraccelerations and velocities, but tend to suffer from a reduced mechanical damping which may lead to unwantedvibrations during and after the execution of motions. Since the following steps of a handling and assembly pro-cess require the decay of the end-effector oscillations below a certain level of desired accuracy, these oscillationsincrease the necessary duration of each working cycle. To allow for the settling of the end-effector some of thetime gained by performing faster motions is unavoidably lost. Furthermore, the reduced stiffness of the structurecauses static deformations of the robot structure under the inuence of gravity, which decrease the manipulator’sprecision and which have to be compensated by the control strategies.

Among other means, the vibrations in a manipulator structure can be suppressed, using integrated adap-tronic components. These, along with the elasticities of the manipulator structure, introduce additional degreesof freedom ( Keimer & Sinapius, 2011 ). In comparison to rigid structures, these degrees of freedom increase themathematical and computational effort for the description and calculation of the manipulator’s kinematics anddynamics. In order to use the advantages of the light-weight constructions, efcient algorithms for the derivationof the manipulator models and their real time calculation for the motion control, as well as the structural controlhave to be proposed. The current lack of appropriate methods is caused by the common assumption, that theparallel manipulators are rigid structures, which is true for most, but not for all of them.

In order to solve the problems mentioned above and to ensure the high performance of the elastic paral-lel manipulators, computationally efcient algorithms for the calculation of the direct dynamics, based on the

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(Gabbert et al., 1997 ; Gabbert et al., 2002; Rose, 2005 ). The separate control of the elastic structure using piezo-electric patches and of the robot motion using conventional drives has been studied for several serial kinematics(Hermle, 2001 ). It was shown, that the control algorithms for the robot motion and for the structure’s vibrationscan be arbitrarily combined and laid out independently. First research on the separate control strategies hasbeen carried out by applying patches as well as stacks of piezoactive materials. These works were also able todemonstrate the effectiveness of the adaptronics in this area ( Wilson et al., 2000; Algermissen et al., 2004 ; Wa& Mills, 2004b ; Breitbach et al., 2005; Algermissen et al., 2008) . The nonlinear model-based control conceptsfor elastic manipulators have not yet been studied as thoroughly as in the eld of serial robots. This provided themotivation for the research in this area, described in this chapter.

3 Dynamics of elastic parallel manipulators

The elasticity of a manipulator’s links and of its joints can be considered as the main sources of structural vi-brations. Signicant progress was made to make the compliance of the manipulator joints accessible for con-trol ( Pavlovic et al., 2011 ). So, without loss of generality, in the presented approach the links of an elastic parallelmanipulator will be regarded as a main source of vibrations in the robot structure, since they are mainly built aslong and slender elements ( Piedboeuf, 2001; ?). The analysis of the dynamics of elastic parallel manipulators willbe started with a brief discussion of the methods for the description of the elastic robot arms. The derivation of the equations of motion of such a parallel manipulator will be based upon the Lagrange-D’Alembert formalismenhanced by elements to represent the links’ elasticities ( Nakamura & Ghodoussi, 1989 ; Park et al., 1999 ). addition, a new method for the derivation of the Jacobians of elastic parallel manipulators ( Stachera et al., 2007is introduced. Afterwards a new method for the simultaneous/distributed calculation of their direct dynamics(SCDD) will be presented and veried (Stachera & Schumacher, 2008 ).

3.1 Description of the elasticities in the manipulator’s structure

The deformation of a continuum can be described by a partial differential equation:

µi ẅi( x, t ) + EI iwi( x, t ) + α i µi ẇi( x, t ) + βi EI i ẇi ( x, t ) = 0,

with µi( x) = ρi · Ai( x) the mass distribution, ρ i the mass density, A the cross sectional area perpendicular to theneutral ber of the beam, EI i( x) the bending stiffness, I i( x) the geometrical moment of inertia and wi( x, t ) deformation vector of the ith arm (see Figure 1). The factors α i and βi have to be determined from experi-ment ( Magnus & Popp, 1997 ). This type of description is only possible for geometrically simple shapes and onlyin cases in which forces are exclusively applied to the end points of the element ( Stachera, 2009 ). The samlimitations hold for the Ritz method with the global trial functions and its approximative solution

M i ¨ qie(t ) + D i ˙ qie(t ) + K i qie(t ) = 0

in the form of ordinary differential equations (ODE), with M i ∈R n× n representing a symmetrical positive denitemass or inertia matrix, K i ∈R n× n a symmetrical positive denite or semidenite stiffness matrix and qie(t ) ∈R nthe vector of the elastic modes. D i = α iM i + βiK i denotes the attenuation matrix according to a specic Rayleighansatz (Magnus & Popp, 1997 ).

The nite element method (FEM) allows an approximative description of more complicated continua ( Cchos, 1989; Gross et al., 2007; Schwertassek & Wallrapp, 1999) . The solution obtained with this method is alsoformulated using ordinary differential equations (ODE), in a form corresponding to eq. ( 2). One disadvantage othis approach is the high order of the obtained matrices, which might make a real time calculation of the equations

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y izsi+1 xsi+1

y si+1

y ei+1

zei+1x ei+1

x

A

x i

z0

y0

x0

zi

r ,

O0

sO

eO

iO

ci

wi

0ir Ti ,

Figure 1: Deformation of a elastic robot link

of motion unfeasible. One signicant possibility to reduce the order of the model is the method of concentratedparameters which also provides solutions of the form ( 2) (Benedict & Tesar, 1978; Isermann, 2002 ; Khalil &Gautier, 2000 ). A reduction of the obtainable precision has to be taken into account, though. Depending uponthe precision demands and the goal of the modelling, one of the two at last mentioned methods is applied. Theresulting models obtained by these discretization methods share the same structure, but differ with regard to theinterpretation of the coordinates and the precision of the solution.

3.2 Derivation of the equations of motion

The Lagrange-D’Alembert formulation can be used for the derivation of the dynamic equations of elastic parallelmanipulators ( Nakamura & Ghodoussi, 1989 ; Nakamura, 1991; Park et al., 1999; Yiu et al., 2001) . The modelforms a system in which the coordinates of the active, passive and elastic degrees of freedom are interdependentdue to the system’s constraints. In order to derive the dynamic equations, this system is transformed into a repre-

sentation in which the independent generalized coordinates are dened in such a way that the motion constraintsare implicitly fullled. The algorithm to derive the equations of motion in these coordinates is detailed below.

3.2.1 Inverse Dynamics

The procedure for the derivation of the inverse dynamics corresponds to methods, which are known from serialmanipulators and consists of the following three steps ( Nakamura & Ghodoussi, 1989 ):

1. Transformation of the System : Each closed kinematic loop of the parallel manipulator is separated at apassive joint, end-effector or link. The result is a reduced system which possesses a tree structure. Conse-quently, only serial kinematic chains are found in this system. Furthermore it is assumed that all remainingpassive joints possess virtual actuators.

2. Calculation of the Torques : The torques and forces of the real and virtual actuators are computed foreach kinematic chain. These torques and forces cause a movement in every chain, and these movementscorrespond to the movement of the original closed-link structure.

3. Transformation of the Torques : The torques and forces of the original parallel manipulator’s actuators arecalculated from the forces and torques of the tree structure by considering the additional closed kinematicloop constraints.

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It is assumed that the manipulator consists of l closed kinematic loops and possesses a total of e + a + p =one DOF joints, with e and p of them being discrete elastic DOF qe and passive DOF q p, respectively. Thcoordinates of the active joints qa and elastic DOF qe form a set of non-redundant coordinates. The controllabilityof the manipulator in absence of elasticity is assumed. According to the rst step, this system is divided into atree structure. The dimensions of qa and qe remain identical to those of the original structure ( na = ( n − p− e) ane = e respectively). The number of passive joint coordinates q p amounts to n p = ( p − l). Thus the coordinateof the tree structure are:

qt = qt ( qa , q p, qe),

where qa ∈ R na × 1 , q p ∈ R n p× 1 , qe ∈ R ne× 1 and the dimension of qt ∈ R nt × 1 , with nt = na + n p + ne . The redundacoordinates of the passive joints q p depend upon the coordinates of the active joints qa and the elastic DOF qe

q p = q p( qae ),

where qae ∈ R nae × 1 and nae = na + ne . Generally, the relation represented in ( 4) may not always exist analytically,but a solution to determine the relationship between the velocities and accelerations of the active and passive joints can always be found (Merlet, 2000 ; Stachera, 2005; Yiu et al., 2001 ). For this purpose the closed kinematicloop constraints h of the parallel manipulator are introduced:

h( qt ) = h( qae , q p) = 0.

After the differentiation of ( 5) a Pfafan form of the constraints is obtained:

∂h∂ qT ae

˙ qae + ∂h∂ qT p

˙ q p = 0.

In order to parameterize the conguration space of the manipulator, a Jacobian of the kinematic constraintsparameterized by the non-redundant coordinates has to be derived. For this purpose it will be assumed that therobot is normally actuated and away from actuator singularity. Therefore, the matrix ∂h∂ qT p from ( 6) is square aninvertible. The conguration space of the manipulator can be smoothly parameterized by the coordinates of the

active joints and the elastic DOF qae :

˙ q p = − ∂h∂ qT p

− 1 ∂h∂ qT ae

˙ qae =∂ q p∂ qT ae

˙ qae .

Based on ( 7) the sought Jacobian of the parameterization matrix can be written as follows:

W = ∂ qt ∂ qT ae

.

In accordance with the second step, the torques/forces of the tree-structure have to be calculated. They can bederived from the equations of motion in matrix form:

M t ( qt ) ¨ qt + C t ( ˙ qt , qt ) ˙ qt + η t ( qt ) + K t qt + D t ˙ qt = τt ,

where M t ( qt ), C t ( ˙ qt , qt )∈R nt × nt are the inertia matrix and the Coriolis matrix of the tree structure. These matricessatisfy the following structural properties:

1. M t ( qt ) is a symmetric and positive denite matrix,

2. Ṁ t ( qt ) − 2C t ( ˙ qt , qt ) is a skew-symmetric matrix.

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τt ∈ R nt × 1 represents all torques/forces of the real and virtual drives of the reduced system and η t ( qt ) ∈ R nt × 1 isthe vector of the gravitational force projected into joint space. K t ∈ R nt × nt and D t ∈ R nt × nt represent the diagonalmatrices of the discrete elasticities and discrete damping elements in joint space. By using the matrix W from(8) the third step of this method is performed to transform the equations of the dynamics of the tree structure ( 9)into the equations of the closed-link mechanism. In the following, they are expressed only as a function of thecoordinates of the active joints qa and the elastic DOF qe:

M c( qt ) ¨ qae + C c( ˙ qt , qt ) ˙ qae + η c( qt ) + K c qae + Dc ˙ qae = τc, (10)

with:

M c = WT M t W ∈ R nae × nae , (11)C c = W T M t Ẇ + W T C t W ∈ R nae × nae , (12)

η c = W T η t ∈ R nae × 1 , (13)K c = W T K t W ∈ R nae × nae , (14)Dc = W T D t W ∈ R nae × nae , (15)

τc = WT τt ∈ R nae × 1 , (16)

Proofs of these transformations and their derivations have been published in (Nakamura & Ghodoussi, 1989;Nakamura, 1991 ). The redundant coordinates of the passive joints, which are necessary for the computation of the compact matrices, result from the closed kinematic loop constraints of the parallel manipulator ( 4) as well asfrom their rst ( 7) and second derivatives.

Using this method, the equations of the dynamics of the tree structure ( 9) are transformed into the compactequations of the closed-link mechanism ( 10) and parameterized by the non-redundant coordinates qae (11 )–(16).Consequently the drive torques can be calculated.

3.3 Derivation of the Jacobian matrix of the parallel manipulator

In conventional methods the Jacobian of the parallel manipulator is derived using the velocity vector-loop method(Tsai, 1999 ) or by analyzing of the parallel manipulator’s statics ( Chakarov, 1999; Merlet, 2000 ; Kock, 2001 ).The L-D’A formulation allows to systematically convert between the single models of serial kinematic chains andthe model of the compact parallel manipulator ( Nakamura, 1991 ; Park et al., 1999 ; Yiu et al., 2001 ). However, itwas not yet shown how the Jacobians of the serial kinematic chains of the tree structure J i can be transformed intothe Jacobian of the parallel manipulator G . An algorithm has been developed to perform this transformation. Forthis purpose, the W matrix, representing the parameterization of the conguration space from ( 8) and the staticmatrix S of the parallel manipulator are used. The static matrix S describes the relationship between the forces inthe arms of the parallel manipulators f b ∈ R na × 1 and the force f x on its end-effector “e”, shown in the Figure 2.These forces in the branches f b can be calculated using the following relationship:

f b = s1 . . . sna− 1 f x = S− 1 f x. (17)

If the matrix S is not square then S− 1 is the pseudo-inverse S+ . The elements of the S-matrix si(qai , q pi, qei)comprise the vector that relates f bi , the branch force of the ith chain, and f xi, the Cartesian force. That relationcan be written:

f xi = si f bi . (18)

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f bi

f b1 f bn a

f x

e

Figure 2: Distribution of the Cartesian force on the end effector f x into the branch forces f bi of theparallel manipulator’s arms

and in the matrix form with matrix U this relation takes the form:

f bx = s1 . . . 0...

. . . ...

0 . . . sna

f b = U f b. (1

Now, the Jacobian J t of the tree-structure is introduced:

J t =J 1 . . . 0...

. . . ...

0 . . . J n a

, (2

where J i( qai , q pi, qei ) are the na - Jacobians of the serial kinematic chains. In order to eliminate the dependenciesof the coordinates of the passive joint q p for the calculation of the Jacobian G of parallel manipulator, the matrixJ t is parametrized with the matrix W . After this parametrization, the new matrix does not yet represent themapping between the joint and Cartesian velocities nor that between the joint torques/forces and end-effector

torques/forces of the parallel manipulator. In order to obtain this mapping, the matrices S and U have to introduced. Applying the transformations ( 17) and (19) , the Jacobian of the parallel manipulator can be derivedfrom the following relation:

G + T = W T J Tt US −1 . (2

This pseudo-inverse Jacobian G + T represents the necessary mapping between the Cartesian force f x on the eneffector of parallel manipulator and the forces/torques τae ∈ Rnae × 1 in the manipulator’s structure in the jointspace:

τae = G + T f x (2

The presented method has the signicant advantage that the derivation of the serial Jacobians is less complex incomparison to the direct derivation of the compact Jacobian using standard methods.

3.4 Direct Dynamics

The equations of the direct dynamics are obtained from the compact equations ( 10) of the manipulator’s inversedynamics:

¨ qae = M c( qt )− 1( τc − C c( ˙ qt , qt ) ˙ qae − η c( qt ) − K c qae − Dc ˙ qae ), (2

This method therefore produces compact equations of the direct dynamics of the elastic parallel manipulator. Thedisadvantage is that the computations cannot be executed in parallel. If more parameters need to be considered

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for the modelling, e.g. Finite-Element-Method ( Wang & Mills, 2004a ), the matrices of the compact system mayno longer be feasibly calculated in real time due to their size and complexity.

3.4.1 Simultaneous calculation of the direct dynamics

The compact form of the direct dynamics equations, as mentioned above, can lead to complications in the real

time calculation, especially in control systems with short cycle times. In order to overcome this limitation, anew method for the simultaneous/distributed calculation of the direct dynamics of the elastic parallel manipulator- SCDD is suggested ( Stachera, 2006a ; Stachera et al., 2007; Stachera & Schumacher, 2008) . In this method,the manipulator is handled as a reduced system similar to the approach for the inverse dynamics of the L-D’Aformulation. However, in this system the closed kinematic loop constraints of the elastic parallel structure arerepresented by forces and torques, just like in the case of the Lagrangian equations of the rst type ( Miller &Clavel, 1992 ; Tsai, 1999 ).

The equations of the tree-structure have the known form shown in (9). These equations will now befactored into the equations of motion of the individual serial kinematic chains:

M ti ( qti ) ¨ qti + C ti ( ˙ qti , qti ) ˙ qti + η ti ( qti ) + J Ti f xi + K ti qti + D ti ˙ qti = τti , (24)

for i = 1 . . .na , where i denotes one kinematic chain with the associated variables qT ti = qai qT pi qT ei and torques τT ti = τai τT pi τT ei of the active τai and passive τ pi joints and additional structure torques τei . f xi represents an

external force acting on the end of the ith -branches of the tree-structure. The equations of the direct dynamics of each such chain can then be formulated:

¨ qti = M ti ( qti )− 1 τti − C ti ( ˙ qti , qti ) ˙ qti − η ti ( qti ) − K ti qti − D ti ˙ qti − J Ti f xi , (25)

for i = 1 . . .na . Thus, the direct dynamics of each serial kinematic chain can be calculated. The inputs of eachof these equations are the external forces acting upon the end of the particular serial kinematic chain f bxi andthe input torques τai and τ pi. The torques of the elastic DOF τei result from the material properties like stiffnessand damping. The input of the tree-structure ( 9) and of the compact parallel manipulator ( 10) is the torque/forcevector τc . According to the D’Alembert principle the performed virtual work for both systems, the reduced andthe original one, has to be equal ( Jarzebowska & Jarzebowski, 2000 ; Nakamura & Ghodoussi, 1989 ). Moreover,the working conditions of both systems have to be equal. It implies following conditions ( Stachera, 2009 ):

τa = τc, τ p = 0.

(26)

In order to fulll these conditions, the new torques/forces of the closed-loop constraints acting on the end of each branch, must be calculated. Along with the drive torques they are applied to the partial equations of directdynamics ( 25) .

The calculation of these torques/forces starts with the computation of the interdependent torques/forcesof the tree-structure resulting from the input torque vector τc . The relation between them is established in (16 ).All the torques/forces of the passive joints of the tree-structure have to be calculated. To perform this task, thosepartial matrices and vectors of the equations of the reduced system ( 24) are used, which are associated with thevirtual torques of the passive joints:

τ pi = M pij ( qti ) ¨ qti + C pij ( ˙ qti , qti ) ˙ qti + η pi j( qti ) + K pij qti + Dpij ˙ qti + J Tij f bxi , (27)

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for j = 1 . . .n p, i = 1 . . .na . In accordance with the conditions ( 26) and regarding relation ( 16) the differencbetween acting torques/forces of the tree-structure and the compact parallel manipulator can be calculated:

∆ τa = τc − τa =∂ q p∂ qT a

T

τ p. (2

Since the inuence of the torques/forces of the elastic DOF ( τe) upon the manipulator’s movement has alreadybeen taken into account during the calculation of the virtual torques τ p, only the virtual torques (27) of tpassive joints are used for the calculation of these differences ∆ τa . Now, the new constraints torques/forces canbe calculated:

ˆ f xi = J − Ti [∆ τai − τ pi], (2

for i = 1 . . .na . The distribution of this force (29 ) in the manipulator’s structure implies the compliance of thecondition (26) . Now, the external forces acting on the end-effector of the compact parallel manipulator have tobe distributed between all the separate serial kinematic chains. The relations (17)–(19) can be combined:

f bx = US + f x, (3

with f T bx = f T

x1 . . . f T xi . . . f T bn a . These forces have to be complemented with the resulting constraint forces ( 2

and the force takes the end form: f xi = f xi + ˆ f xi, (3

for i = 1 . . .na . The forces from ( 31) in combination with the torques/forces from ( 26) cause the same movemenof the tree-structure as the movement of the original parallel manipulator as well as the same force and torquesdistribution in both structures.

3.5 Verication of the new calculation method

In the new method - the Simultaneous Calculation of the Direct Dynamics (SCDD), the system is divided intoa tree structure in order to accelerate the inversion of the inertia matrix. The most frequently used method, theLU-Gaussian elimination, has the complexity O(n3); more sophisticated algorithms can reduce the complexity of

the inversion to O(ne

). For the computation of the direct dynamics in the joint space of large scale systems, it istherefore better to process several small matrices rather than a single large one. Additionally, the computationsof the direct dynamics with this decomposition can be performed in parallel. In the Table 1 the complexity of thmatrix inversion indicated by the number of the necessary arithmetical operations is calculated for three differentmanipulators of the Collaborative Research Center SFB562 (Hesselbach et al., 2005 ). These calculations wercarried out with the assumption that one elastic DOF nek exists in each kinematic chain.

na n pk nek SCDD L-D’A Reduction

F IVE -B AR 2 1 1 54 64 16%

H EX A 6 2 1 384 1728 78%

TRIGLIDE

3 2 1 162 729 78%Table 1: Complexity of the direct dynamics calculation

The results indicate a considerable reduction of the calculation complexity by using the proposed algo-rithm. Therefore each kinematic chain can be modeled with more parameters and accordingly higher accuracy(Stachera & Schumacher, 2008 ).

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The new SCDD method has been compared with the L-D’A method in the simulation. In order to do this, amodel of the elastic planar parallel manipulator F IV E -B AR was derived ( Stachera, 2009 ). The discrete elasticitiesk 14 = k 24 = 5.464 × 106 [ N / m] were considered in all upper arms of the manipulator, shown in Figure 3. Q11 andQ21 represent the motors. The other parameters of this model are not relevant for this comparison.

−0.4 0 0.4 0.8

0

0.2

0.4

0.6

0.8

pA

pE

x in m

y i n

m pE

Q 11 Q 21

k 14k 24

Figure 3: Trajectory and workspace of elastic planar parallel manipulator - F IV E -B AR

A straight line trajectory between two points p A and p E has been chosen. The models have then beencontrolled by torques, which were created by a rigid body model without control. The black line represents thereference trajectory. The dark gray line is a result of the L-D’A model and the light gray line from the SCDDmodel. These results show that the models both followed the trajectory with comparable accuracy. The existingdifferences between the paths traveled by these two elastic models can be accounted for by the numerical precision(Stachera & Schumacher, 2008 ). Figure 4 shows the distance between the endpoints of both kinematic chains.The light gray line in Figure 4(a) shows the numerical error of the SCDD model and the black line of the L-D’Amethod. For better investigation, the error of the standard L-D’A is additionally presented in Figure 4(b). Even forthis standard method, a small numerical error, which increases slowly with time, is identiable. The error in theSCCD method results from the parallel computation and rounding errors and depends upon the sample intervalof the simulation: the smaller the interval, the smaller the error. This error does not constitute any encumbranceto implement this new method, because in the eld of numerical methods algorithms are known that deal with thestabilization of the numerical calculation and increasing of the computation accuracy ( Baumgarte, 1972) .

4 Control of the structure’s vibration of a parallel manipulator

One of the principal requirements for the structural, as well as the motion control is a constant control qualityall over the workspace. This demand can not be fullled using gain-scheduling-control ( Wang & Mills, 2004b;Algermissen et al., 2005; Algermissen & Sinapius, 2011 ). As the properties of the manipulator are pose depen-dent, this implies the necessity of the use of model-based control algorithms ( Asada & Slotine, 1986; Moallemet al., 2000 ). Furthermore the control algorithms have to be robust enough to be able to tolerate uncertainties of the robot model, which can only be determined with a limited precision. According to the Singular-Perturbation-method, the control law for vibration suppression of robot structure can be designed separately from the motioncontrol of the manipulator ( Siciliano & Book, 1988; Levine, 1996 ; Hermle, 2001) .

In this chapter a model-based control algorithm is described in Section 4.1 along with the update strategy

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0 0.05 0.1 0.15 0.2 0.25

0

2

4

6

8x 10

−10

∆ s

i n m

t in s

((a)) SCDD Method

0 0.05 0.1 0.15 0.2 0.25

0

2

4

6

8x 10

−14

∆

s i n m

t in s

((b)) L-D’A Method

Figure 4: Numerical error of the SCDD and LDA method

in Section 4.2 to adjust the controller parameters to the varying nonlinear plant. The approach is validated inSects. 4.3 and 4.4 using the experimental results obtained with the F IVE -B AR manipulator.

4.1 Linear robust controllerThe choice of the control structure was motivated by methods for passive vibration suppression, which are com-monly applied to linear systems (Fischer & Stephan, 1993 ). These methods use additional spring-mass-damper-systems as dynamic vibration absorbers, which are installed with poorly damped systems and tuned in order tosuppress its natural frequency. In the selected approach the actuator is chosen to behave like a spring resulting ina controller with a D2T 2 transfer function ( Stachera & Schumacher, 2003 ):

Ga (s) = L { f a }L { f s}

= L {mmod ¨ xa }

L { f s} =

mmod va s2

mmod s2 + d a s + k a, (3

with d a denoting the virtual attenuation factor, k a the virtual stiffness, mmod the effective vibrating mass, xa actuator displacement and va the controller gain. The transfer function Ga (s) describes the relationship betweenthe force measured by the sensor f s (controller input) and the actuator forces caused by the control action(controller output). Therefore, the effect of the controller upon the system mainly equals the product of theaccelerations produced by the actuator ¨ xa and the effective vibrating mass mmod . Figure 5 depicts the basic idefor the controller design.

Each vibration mode possesses its own controller. The controller coefcients for each mode are calculatedwith the assumption, that the structure exhibits a very low damping and its natural damping factors can be ne-glected ( d mod ≈ 0). The parameters of the virtual spring-mass-damper-system formed by the actuator and each

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y0x0

k VS

k mod dmod

f S

xa

xr

actuator

sensor

f emmod

((a)) One-mode-model of the vibration

Σ

Controller

f 12

f 13

f 13

f 12

1-D T2 2

n-D T2 2

((b)) Superposition of the modal controllers

Figure 5: Basic ideas for the controller design

one-mode-system have to be chosen in a way that allows the closed-loop control system to return to its restingpoint as quickly as possible without oscillating. Therefore the closed-loop system has to show a behavior of acritically damped second order system. By comparing the coefcients of the closed-loop system with the desiredtransfer function the parameters k a and d a can be determined:

k a = 4

π2 − 4k mod = βmod k mod , (33)

with βmod the relation factor between both stiffnesses and the virtual attenuation factor, which has to ensure verygood damping of the system:

d a = 2

k a mmod . (34)

The parameters k mod and mmod represent the stiffness and the effective vibrating mass of a single structural modeprojected into the direction of the actuator for a complex systems. Any of those modal controllers is designedfor each of the controlled structural modes of the manipulator and for each of the used actuators. The actuatorresponse is computed from superposition of the modal controllers associated with him, as shown in Figure 5(b) .Further information regarding the used formula and the proof of robustness and stability are detailed in ( Stachera& Schumacher, 2003 ; Stachera, 2009 ). Since most manipulators belong to the family of nonlinear systems, thefrequencies of their respective modes change during motion. Consequently the parameters k mod and mmod have tobe updated according to the end-effector position.

The new D2T 2-Controller ( 32) combined with the proposed parameters (33) and (34) was compared inthe simulation with LQ-Controller in order to investigate its performance. The simulation’s results for bothcontrollers are presented in Figure 6. The black line shows the behavior of the system under control, the lightgray line the undamped vibration without control, and the dark gray line the behavior of the actuator. The answerof the controlled closed-loop system shows clearly, that the new controller possesses an efciency comparable tothe reference control law derived for that specic point and at the same time requires a lower computational effortthan the LQ-Controller. This feature is very important for complex non-linear systems, where the controllers’parameters have to be calculated for a high number of controllers at every time step of the working digital controlsystem. Additionally, as can be seen by the symmetrical actuator action, the new control algorithm allows a betterusage of the actuators’ range than the reference control law.

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0 5 10 15 20 25−10

0

10

20

30

40

f i n

N

t in ms

× 10

((a)) D2T 2-Controller

0 5 10 15 20 25−10

0

10

20

30

40

f i n

N

t in ms

× 10

((b)) LQ-Controller

Figure 6: Comparison of the control strategies

4.2 Application of the manipulator characteristics for the adjustment of the controller parameter

A linear controller was proposed for the control of the structural vibrations of an elastic parallel manipulator. Inorder to adjust the modal controller to the varying nonlinear plant, the features of the force and inertia ellipsoidswere utilized ( Tsai, 1999 ; Nakamura & Ghodoussi, 1989; Gosselin, 1990 ; Chakarov, 1999 ; Ceccarelli & Carbone2005 ).

The force ellipsoids allow the determination of the robot properties with regard to the generation of forcesand the resistance against forces and, therefore, also the calculation of the structural stiffness in an arbitrarydirection in space. The stiffness can be derived as follows:

k mod = k DIR xy = f T x NTx Nx f x

− 1

(3

with Nx = K − 1x denoting the compliance matrix (the inverse of the stiffness matrix in the task coordinate frame),the unit vector f x = 1 describing the spatial direction. In some cases the generation of the elds for every modeis advisable to be able to analyze them separately.

The inertial ellipsoid describes the ability of the manipulator to generate a unit force in an arbitrary spatialdirection, therefore it determines the inertia of the robot structure into a given direction. The effective vibratingmass differs from the projected inertia by a factor λ and can be calculated from:

mmod = m DIR xy = λ ¨ xT x M Tx M x ¨ x x (3

with the unit vector ¨ x x = 1 determining the spatial direction and M Tx describing the inertia matrix of the ma-nipulator in the task frame ( Stachera, 2009 ). Here again, the analysis of separated modes can require the separatecalculation of each eld.

The compliance matrix N x and the inertia matrix M x are pose-dependent and have to be calculated in eachcontrol step using the non-linear model of the manipulators dynamics, as derived in Sec. 3. Thus the contralgorithm consists of the transfer function ( 32), the parameters of the controller ( 33) and (34) and the adaptioformulas (35) and ( 36) .

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4.3 Verication of the proposed control law

The proposed control algorithm was examined using the elastic planar parallel manipulator F IVE -B AR (see Figure7(a) ). In the robot model each upper arms’ elasticities are modeled by the lumped parameters k 14 and k 24 , asdepicted in Figure 7(b). The parameters have the following values: i = 1, 2, li1 = 0.3 [m], mi1 = 0.523 [kg] - stiff

((a)) Elastic planar parallel manipulatorF IV E -B AR

z1

y1

x1 x 5

y5

z 5

O2

O3

Q11

O4

e pa

Q21

l12l22

O5O1

l13

l14

l23

l24

l11

l21

m , k 12 12

m , k 13 13

m , k 14 14

m 11

m e

m , k 24 24

m , k 23 23m , k 22 22

m 21

((b)) Model parameters of the F IVE -B ARmanipulator

Figure 7: Elastic planar parallel manipulator F IV E -B AR

crank , li2 = 0.016 [m], k i2 = 4.22 × 109 [ N / m], mi2 = 0.063 [kg], d i2 = 819 .0 - piezo sensors, li3 = 0.08 [m], k i3 ,mi3 = 0.125 [kg] - piezo actuators, li4 = 0.404 [m], k i4 = 5.464 × 106 [ N / m], mi4 = 0.186 [kg], d i4 = 10.0 - elasticrobot arms, me = 1.0 [kg] - additional mass at the end-effector, I drive = 0.089 [kgm2] - moment of inertia at thedrives.

ElasticParallel

ManipulatorAdjustment of the controllers

Control of the structure

Control of therigid body

Trajectory

f 23f 13

f 22f 12

t 11t 21

x,y

P a r a m e t e r s

S t a t e s

P a r a m e t e r s

P a r a m e t e r s

Figure 8: Control Architecture implemented for the F IV E -B AR elastic planar parallel manipulator

The architecture of the proposed controller, as implemented to control the structural vibrations of theelastic planar parallel manipulator F IVE -B AR is shown in the Figure 8. τ11 and τ21 are the torques of the left andright drive respectively, f 12 and f 22 represent the sensors’ forces measured in the left and right upper link. f 13 and f 23 are the actuators’ forces of the piezo-actuators integrated into the upper left and right link. The controllers’parameters are calculated in the block “Adjustment of the controllers” based on the measured or observed statesof the elastic manipulator at each time step. Consequently, the new parameters are transmitted to the controllers,

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which compute the new output of the structural control.

0 0.05 0.1 0.15 0.2 0.25−250

−125

0

125

250

f 1 2

i n

N

0 0.05 0.1 0.15 0.2 0.25−250

−125

0

125

250

t in s

f 2 2

i n

N

Figure 9: Vibration suppression on the elastic parallel manipulator F IV E -B AR

The simulations were performed in the following way: The controllers were activated during a motionalong a straight line trajectory, as shown in Figure 3. Every actuator was driven by a one single-mode controller.The force measured at the respective piezo sensor served as an input for the controller. Figure 9 depicts tforces f 12 and f 22 at the sensors during the motion along the given trajectory with (black) and without (light gray)structural control. The vibrations of the elastic modes were excited by the stepwise change of the accelerationscaused by the drive torques. Without the use of the controller, distinct vibrations are visible, whereas they areefciently attenuated after the activation of the control, during the motion as well as after the completion of thetrajectory. The magnications in Figure 9 provide further details for two points on the trajectory. The boxes showmagnications by 1 .5 and 2 .5, respectively. The effort of the actuators, as well as the control quality, remainsnearly constant in each point of the trajectory. This meets the demands placed upon the control algorithm.

4.4 Experimental results

The presented experiments were carried out for one pose of the manipulator ( Q11 = 135 ◦ and Q12 = 45 ◦ ) and wian additional payload at the end-effector me = 2 kg. These experiments have shown that the main sources of thevibrations are located in the passive joints of the manipulator. The dominant oscillations were perpendicular tothe working plane and therefore a 3-axial acceleration sensor attached to the end-effector was used for the controlstrategy. The results of the controller action are shown in Figure 10(a) for the piezo-sensors and in Figure 10for the 3-axial acceleration sensor. The attenuation of the z-components of the oscillations at the end-effector was7.3 dB .

These experimental results conrm the applicability of the new control algorithm and the chosen designstrategy, although the experiments were performed only for static poses and the performance of the control wasnot good enough. Responsible for this poor performance was a phase shift from the signal processing, whichwas expected but not considered and modeled in this rst implementation of the new control strategy 1. Theimplementation difculties, e.g. the phase shift due to the digital signal processing, still need to be addressed in

1A detailed discussion of the experimental and simulative results can be found in ( Stachera, 2009 )

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2 2.5 3 3.5 4−120

−80

−40

0

40

80

f 1 2

i n

N

2 2.5 3 3.5 4−120

−80

−40

0

40

80

f 2 2

i n

N

t in s

((a)) Measurements usingpiezosensors

2 2.5 3 3.5 4−50

0

50

a 1 2

i n

m / s 2

2 2.5 3 3.5 4−50

0

50

a 2 2

i n

m / s 2

2 2.5 3 3.5 4−5.0

0

5.0

a z i n

m / s 2

t in s

((b)) Measurements using a 3-axialaccelerometer

Figure 10: Measurements

future work in order to apply the proposed controller scheme to a wider range of mechanisms.

5 Conclusions

In this chapter the modelling and control of structural vibrations of elastic parallel manipulators has been dis-cussed. For the modelling of the manipulators’ dynamics, an enhancement of the Lagrange-D’Alembert formu-lation has been proposed using a new algorithm for the derivation of the Jacobian of parallel manipulators. Basedupon these results, a method has been presented to simultaneously calculate the direct dynamics of parallel andfurthermore elastic parallel manipulators. It allows a signicant reduction of the complexity of the calculations,where this calculation time can be a critical point for the implementation of the system’s observers or sophisticatedmodel-based control strategies. This procedure has been demonstrated for the F IV E -B AR manipulator.

The knowledge of manipulator’s dynamics has allowed the design of a new model-based control law forvibration suppression in the manipulator’s structure. A robust linear controller combined with a pose-dependentadjustment law has been conceptualized and presented. The performance and potential of the new algorithm hasbeen conrmed by simulative and experimental examinations. Future work will be directed towards applying the

presented methods to parallel kinematic machines offering more degrees of freedom.

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