Reduced-OrderObserversforNonlinearStateEstimationin ... · observability and controllability [20]). Several approaches to model reduction have been presented in the literature to

Research ArticleReduced-Order Observers for Nonlinear State Estimation inFlexible Multibody Systems

Ilaria Palomba ,1 Dario Richiedei ,2 and Alberto Trevisani 2

1Faculty of Science and Technology, Free University of Bolzano-Bozen, Piazza Università 5, Bolzano 39100, Italy2Department of Management and Engineering (DTG), Università degli Studi di Padova, Stradella San Nicola 3,Vicenza 36100, Italy

Correspondence should be addressed to Alberto Trevisani; [email protected]

Received 10 August 2018; Accepted 8 October 2018; Published 1 November 2018

Academic Editor: Chao Tao

Copyright © 2018 Ilaria Palomba et al.%is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Modern control schemes adopted in multibody systems take advantage of the knowledge of a large set of measurements of themost important state variables to improve system performances. In the case of flexible-linkmultibody systems, however, the directmeasurement of these state variables is not usually possible or convenient. Hence, it is necessary to estimate them through accuratemodels and a reduced set of measurements ensuring observability. In order to cope with the large dimension of models adoptedfor flexible multibody systems, this paper exploits model reduction for synthesizing reduced-order nonlinear state observers.Model reduction is done through a modified Craig-Bampton strategy that handles effectively nonlinearities due to large dis-placements of the mechanism and through a wise selection of the most important coordinates to be retained in the model. Startingfrom such a reduced nonlinear model, a nonlinear state observer is developed through the extended Kalman filter (EKF). %emethod is applied to the numerical test case of a six-bar planar mechanism. %e smaller size of the model, compared with theoriginal one, preserves accuracy of the estimates while reducing the computational effort.

1. Introduction

Flexible-link multibody (FLMB) systems are highly prom-ising from an economical and sustainability point of viewbecause of the use of less material and the need for smalleractuators and less power consumption. However, flexibilityoften results in unwanted vibrations that limit motion ac-curacy and imposes advanced control schemes accountingfor the flexible dynamics. High-performance controlschemes are typically model-based [1–5] and often requirethe knowledge of a large set of state variables [3–5]. Forexample, a common application requiring the whole statevector (i.e., position and speed of all the state variables) isactive control through state feedback to assign the desiredmodal characteristics [6, 7]. %e knowledge of the tip po-sition of open-chain robots is also needed for precise controlof the end-effector trajectory [8]. In contrast, the directmeasurement of all the state variables is rarely feasible orconvenient [9], so the unmeasured state variables should beestimated by means of state observers that exploit a smaller

set of measurements, ensuring the system observability andan accurate system model. Most of the contributions in theliterature have been focused on the state estimation in rigid-body multibody systems [10–12], in flexible structures [13]or in single-link flexible mechanisms [2, 14–17]. Complexityis significantly exacerbated in the case of multilink flexiblesystems where state estimation is, to date, still a challengingtask, mainly because of the complex and large dimensionalmodels used to represent these systems that make cum-bersome the synthesis of observers.

In the past, due to limitations of the computationalpower of computers, the use of linearized [18] or linearizedreduced-order models has been proved to be a good trade-off between accuracy and computational complexity [18–20]. Indeed, computational effort should be reduced bykeeping model size as small as possible to perform real-timeestimations. Hence, model reduction is a powerful tool toboost the use of state observers, by representing through themodel just the state variables that are of interest for esti-mation and control purposes (i.e., those having higher

HindawiShock and VibrationVolume 2018, Article ID 6538737, 12 pageshttps://doi.org/10.1155/2018/6538737

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observability and controllability [20]). Several approaches tomodel reduction have been presented in the literature to datefor linear mechanical systems [21]. For example, in the fieldof structural dynamics, one of the most widespread re-duction techniques is the Craig-Bampton (CB) method [22]because of its effectiveness and ease of implementation.

In the very recent years, the use of nonlinear state ob-servers has been proved to be effective for improving ac-curacy of state estimation in multibody systems, comparedto linearized observers [2]. Indeed, the use of nonlinearmodels and variable gains of the observer handles moreeffectively the nonlinear behavior of these systems. %isincrease of the computational complexity of these observersstill calls for the use of reduced-order models.

%e typical implementation of the aforementioned CBmethod for reducing FLMB systems exploits the componentmode synthesis strategy [21]. Basically, the linear FE modelsof all the links are generated and reduced and, then, they areembedded in a moving frame [23] or in a corotational frame[24], to represent large displacements of the mechanism.Such a component-level reduction strategy might negativelyaffect the reduced model correctness since it does not fullyaccount for modal characteristic changes due to the motionof the multibody system. To overcome this issue, somestudies have recently suggested the use of model reduction atthe system-level, i.e., of the model of the whole system. Onthe one hand, this choice represents more effectively thechange of the modal characteristics of the system [25]. Onthe other hand, system-level reduction ensures a more ef-fective selection of the state variables to be retained in thereduced-order model, thus providing a better trade-offbetween model size and accuracy [26].

By taking advantage of the idea of using reduced-ordernonlinear models in the design of nonlinear observers, thiswork proposes a novel and comprehensive approach forefficient and accurate state estimation in FLMB systems. %emethod exploits the modified nonlinear CB reductionsuitable for flexible-link mechanisms based on the equiva-lent rigid-link system (ERLS) and formulated through or-dinary differential equations (and hence independentcoordinates) outlined in [25] and a wise selection of the mostimportant coordinates, as proposed in [26]. Such a model,that ensures that the dynamics with the highest observabilityand controllability are modeled in the system, is used for thesynthesis of an extended Kalman filter (EKF) [27] to deliveraccurate estimates of both the large motion and of the elasticvibrations of a FLMB system by means of a small set ofsensed signals and with a reduced computational effort.

%e method is validated numerically, by investigatingsensitivity to model uncertainty and measurement noise, bymeans of a planar 6-link FLMB system.

2. Modeling

2.1. Full-Order Dynamic Model: ERLS Approach. In thiswork, the equations of motion for a FLMB system undergoinglarge rigid-body motion are modeled through the equivalentrigid-link system (ERLS) formulation. FLMB systems withscleronomous and holonomic constraints are assumed. %e

motion of the system is notionally separated into the largemotion of a rigid-link moving reference configuration (ERLS)and the small elastic deformation of the flexible links withrespect to the ERLS itself. %erefore, the configuration ofa FLMB system is defined through a set of generalized rigid-body coordinates q and elastic coordinates u that are the full-order model independent coordinates:

x �q

u , (1)

where the dimension of x, denoted as n, is the number ofdegrees of freedom (dofs) of the full-order model.

%e system model is given by the following set ofnonlinear ODEs [23]:

ST(q)M(q)S(q) ST(q)M(q)

M(q)S(q) M(q)⎡⎣ ⎤⎦

√√√√√√√√√√√√√√√√√√√√√√√√√√√√M(q)

€q

€u

+ST(q)M(q) _S(q, _q) ST(q) 2MG(q) + C(q)(

M(q) _S(q, _q) 2MG(q) + C(q)

√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√C(q)

_q_u

+0 00 K(q)

√√√√√√√√K(q)

qu �

ST(q) ST(q)M(q)I M(q)

f(q)g(q)

√√√√√√√√√√√√√√√√√√√√√√√√√√F(q)

.

(2)

Finite elements are used in Equation (2) to get the mass,stiffness, damping, and centrifugal and Coriolis matrices(denoted M, K, C, and MG, respectively) by assembling theconsistent matrices of each finite element. In Equation (2), gand f denote the gravity acceleration and the external forcevector, respectively. S is the ERLS sensitivity coefficientmatrix for all the nodes of the finite elements and _S is its timederivative. %e mass, damping, and stiffness matrices of thefull-order model have been introduced in Equation (2) anddenoted as M, C, and K, respectively. F is the full-ordervector of the external forces.

2.2. Reduced-Order Dynamic Model: Modified Craig-Bampton Method. A dynamic model reduced through theCB method is represented through a hybrid set of co-ordinates, including some physical coordinates xm (namedthe master dofs) and some nonphysical coordinates η(named interior modal coordinates).

%e full-order set of physical coordinates x of the originalmodel is approximated through a reduced set of hybridcoordinates p by means of the CB transformation matrix H:

x ≈ H · p,

x ∈ Rn,

p ∈ Rr, r≪ n.

(3)

In the standard formulation of the CB method, usuallyadopted for structures and linear systems, the transformationmatrix in Equation (3) is constant. Conversely, in the case of

2 Shock and Vibration

mechanisms performing large displacements, it should bereformulated as a configuration-dependent matrix to copewith the nonlinearities of the FLMB models [25]:

x ≈ H q, q∗( p,

x �xmxs

∈ Rn�m+s,

p �xmη

∈ Rr�m+p, p≪ s.

(4)

In Equation (4), xTm � qT uTm ∈ Rm is the m-di-

mensional set of the master dofs, which comprises all theERLS coordinates q and some meaningful elastic co-ordinates um (e.g., those where external loads are applied).%e s-dimensional vector xTs � u

Ts ∈ R

s (m + s � n) in-cludes the remaining elastic coordinates that are trans-formed into interior modal coordinates through thetransformation matrix H. %e configuration-dependent CBreduction matrix in Equation (4) is defined as follows:

H q, q∗( �I 0

B(q) Ψ q, q∗( . (5)

In Equation (5), I and 0 are the identity and null ma-trices, respectively, B is the position-dependent Guyan’smatrix and Ψ is the configuration-dependent interior modematrix. %e definition of matrix Ψ � Ψ(q, q∗) is a key pointof the method and is obtained by the following three steps:

(1) Computation of the Interior Mode Matrix Φ at anEquilibrium Configuration q∗. %e so-called interiormode matrix Φ of the subsystem made by the slavedofs is computed through local modal analysis. Sincethe model is nonlinear, modal analysis is done byplacing the ERLS at a given equilibrium configurationq∗ (obtained by setting _q � €q � 0) and linearizing thesystem with the master dofs constrained. %e equi-librium point can be conveniently chosen as anyarbitrary point belonging to the desired trajectory.

(2) Selection of the Interior Modes. A suitable number ofcolumns ofΦ is removed to get the truncated interiormode matrix Φ. Such a rectangular matrix approxi-mates the full-order set of slave dofs with a smaller setof meaningful interior coordinates (Equation (4)). Akey point for getting accurate reduced models is theproper selection of the interior modes to be retained[26, 28, 29]. In Section 4.1, the strategy followed in thispaper will be briefly explained, even if any arbitraryselection method can be adopted.

(3) Definition of the Configuration-Dependent InteriorMode Matrix Ψ. %e truncated interior mode matrixΦ, computed for q � q∗, should be transformed tofollow the motion of the ERLS. Under the reasonablehypothesis of small elastic deformations with respectto the ERLS, on which the ERLS theory is based,eigenmodes change slowly if expressed with respect tothe reference system of the finite elements. Such anidea is schematically represented in Figure 1, wherefor sake of clarity a planar single flexible link has been

represented. In the ERLS model, the components ofeach eigenvector φ are expressed with respect to theERLS configurations (Figures 1(a) and 1(b)), and it isevident that these components are different. However,if a local reference frame {x,y} is adopted to expressthe eigenvectors, the coordinates are almost the samealso for different ERLS configurations (Figures 1(c) and1(d)). %is hypothesis has been assessed in [25].%erefore, it is possible to extend the validity of theinterior mode matrix Φ, computed at q∗, in a widesubset of the workspace, by projecting it onto the localreference frame of each finite elements and thenprojecting it again onto the actual instantaneous ERSLq through a suitable transformation matrix R �R(q, q∗) (see [25] for a detailed description):

Ψ q, q∗( � R q, q∗( Φ q∗( . (6)

Once H is defined, the reduced model is obtained bypremultiplying Equation (2) by HT:

HT q, q∗( M(q)H q, q∗( √√√√√√√√√√√√√√√√√√√√

M

€xm€η

+ HT q, q∗( C(q, _q)H q, q∗( √√√√√√√√√√√√√√√√√√√√√√

C

_xm_η

+ HT q, q∗( K(q)H q, q∗( √√√√√√√√√√√√√√√√√√√√

K

xmη

� HT q, q∗( F(q)√√√√√√√√√√√√

F

.

(7)

%emass, damping, and stiffness matrices of the reducedmodel have been introduced in Equation (7) and denoted asM, C, and K respectively. F is the reduced-order vector of theexternal forces.

3. State Estimation

3.1. Model Formulation. %e synthesis of a state observerrequires the dynamicmodel in Equation (7) to be formulatedin a state-space (first-order) form, as follows:

_z � f(z, F),

y � g(z, F),

⎧⎨

⎩ (8)

where z � pT _pT Tis the reduced-order state vector, _z is its

derivative, and y is the vector of the measured outputs (oftendenoted as the observations). Both the system equation f andthe measurement equation g in Equation (8) are usuallynonlinear functions of the state and of the input F. %e directuse of ODEs for the formulation of the multibody model inEquation (2) makes straightforward the achievement of thestate-space form required for state estimation.

By taking advantage of Equation (7), the nonlinearcontinuous-time system equations in Equation (8) take thefollowing form:

_z �_p€p

�0 I

− M−1(q)K(q) − M−1(q)C(q, _q)

p_p

+0

M−1(q) F.

(9)

Shock and Vibration 3

3.2. �e Extended Kalman Filter (EKF). �e estimation al-gorithm employed in this work is the EKF [25] that is thesimplest and most widespread extension of the well-knownKalman �lter to nonlinear state estimation. Such a �lterrequires a discrete-time representation of the system modelin Equation (8):

zk � fd zk−1, F̃k−1, vk−1( ),yk � g zk,wk( ),

(10)

fd is the discrete-time state equation, obtained by dis-cretizing f through any arbitrary scheme and k refers to thekth time sample. In Equation (10), vk−1 and wk are theprocess noise and measurement noise that are assumed inthe Kalman theory to be zero-mean uncorrelated Gaussiannoise with covariance matrices Q and R, respectively.

�e EKF is a discrete-time recursive algorithm thatoperates in two stages: the prediction and the correction.Only a brief explanation is provided here. �e readers couldrefer to the quoted literature for a more detailed description.

In the �rst stage, at each time step k, the �lter predicts thestate and the observation by means of the nonlinear modeland measurement equations, respectively:

ẑk|k−1 � fd ẑk−1|, F̃k−1|, vk−1|( ),

ŷk|k−1 � g ẑk|k−1,wk( ).(11)

Additionally, it propagates the state and observationerror covariance matrices (denoted as P̂zzk|k− 1, P̂

yyk|k− 1, and

P̂zyk|k− 1) by means of the Jacobians of the system and mea-surement equations (Âk−1 and D̂k|k−1 respectively), com-puted about the estimated state trajectory:

Âk−1 �zfzz

∣∣∣∣∣∣∣̂zk−1 ,̃Fk−1,

P̂zzk|k−1 � Âk−1P̂zzk−1Â

Tk−1 +Q,

(12)

D̂k|k−1 �zgzz

∣∣∣∣∣∣∣̂zk|k−1,

P̂yyk|k−1 � D̂k|k−1P̂zzk|k−1D̂

Tk|k−1 + R,

P̂zyk|k−1 � P̂zzk|k−1D̂

Tk|k−1.

(13)

Once the next measurements yk are available, the pre-dicted state is corrected through the error of output esti-mation (yk − ŷk|k−1) weighed through the �lter gain Kk:

ẑk � ẑk|k−1 + Kk yk − ŷk|k−1( ),

Kk � P̂zyk|k−1 P̂

yyk|k−1[ ]−1.

(14)

Finally, the state error covariance matrix is updated:

XYq∗

φiφi,Y

φi,X

i th ele

ment

r(q∗)

FLMBERLS

XYq

φi

φi,Yφi,Xi

th ele

ment

r(q)FLMBERLS

y ix i

XYq∗

φiφi,y

φi,x

i th ele

ment

r(q∗)

FLMBERLS

y i x i

XYq

φiφi,y

φi,xi th el

emen

t

r(q)FLMBERLS

Glo

bal r

efer

ence

fram

e{X

; Y}

Loca

l ref

eren

ce fr

ame

{xi; y

i}

q∗ q

(a) (b)

(c) (d)

Figure 1: Example of eigenvector components in a planar exible single link for two ERLS con�gurations: q∗ (a–c) and q (b–d), with respectto the global reference frame (a-b) and the local reference frame (c-d).


Pzzk � Pzzk|k− 1 −Kk P

yyk|k− 1K

Tk . (15)

Equations (12) and (13) reveal that gain Kk of the filtercorrection depends on the covariance matrices Q and R. Asuccessful estimation relies both on an accurate model and ona good tuning of these matrices: while R can be measured, Qshould be usually tuned to give the proper weights to modelpredictions and noisy measurements in state estimates.

4. Numerical Validation

Numerical simulations have been performed throughMatlab to validate the nonlinear reduced-order observer.Such a validation allows for an effective assessment of theobserver outcomes since it is possible to evaluate andcompare any state variable, including the elastic ones whichare difficult to measure in an experimental setup. Addi-tionally, it is possible to evaluate the impact of the modelreduction on the model accuracy and its effect on the ob-server stability. Indeed, a wrong model reduction causesrelevant estimation errors and might lead to instabilitybecause of observation spillover phenomena [20].

4.1. System Description. %e planar mechanism shown inFigure 1 is used as the test case for the validation. %emechanism lies on the horizontal plane and is supposedto be driven by three motors: two motors drive theabsolute rotation of links 1 and 2, while the third one drivesthe relative rotation between links 4 and 5. All the linkshave circular cross section and dimensions as shown inFigure 2(a) and are supposed to be made of aluminium withelastic modulus 69GPa and mass density 2700 kg/m3. Two-node, six-dof beam elements have been employed to modelthe links, leading to the finite element model shown inFigure 2(a). %e joints, the brakes, and the rotors of themotors have been modeled as lumped masses and nodalinertias, whose values are stated in Table 1.

%e resulting full-order model has 30 dofs, includingthree ERLS coordinates (defined as the absolute rotations oflinks 1, 2, and 5) and 27 elastic displacements. %e full-ordermodel has been reduced by applying the reduction strategydiscussed in Section 2.2 and by selecting the interior modesto be retained by means of the interior mode ranking (IMR)method [25]. Such a method ranks the interior modesbased on their contributions to the dynamics of the full-order system at the frequencies of interest, i.e., to representthe vibrational modes of the full system that have highercontrollability and observability and hence mainly affectthe system dynamics. By means of a correct representationof these vibrational modes, it is possible to synthesizea reduced-order model, and hence an observer, that ac-curately represents the system dynamics that are excitedand observed. In the test case under investigation, a re-duced model accurately matching the full system dy-namics in the frequency range 0–180Hz has been obtainedthat is a reasonable value if compared with the typicalbandwidths of motion controllers. Indeed, vibrationalmodes outside the controller bandwidth cannot be neither

excited nor controlled by the control loop. To reach sucha goal, the interior modes have been ranked trough theIMR method, and then the number of interior modes to beretained has been evaluated through accuracy indices,comparing all the eigenstructure of the full-order model inthe frequency range 0–180Hz with the ones of thereduced-order model synthesized at the equilibriumconfiguration q∗ � 1.0821 2.6354 4.7123 T. Two indicesare used. %e first one is normalized cross orthogonality(NCO), which provides a measure of the mass-matrix-orthogonality between the vibrational mode of interestrepresented through the full-order model φ and thereduced-order one φ:

NCO �φTMHφ

φTMφ( φTHTMHφ( . (16)

%e second index is the relative percentage error on thenatural frequencies:

εf �|f− f|

f· 100. (17)

In Equation (17), f and f are the frequency of the modeof interest in the full and in the reduced-order model, re-spectively. %e target value for the NCO is one, while it iszero for εf. By assuming reasonable thresholds on theseindices, the number of interior modes to be retained isdetermined and hence the truncated interior mode matrix Φis determined.

%e set of the reduced coordinates obtained in thisexample has just 13 dofs and contains 8 physical coordinatesxm (i.e., the master dofs, shown in red in Figure 2(b)) and 5interior modal coordinates η:

xm � q1 q2 q3 x14 y15 θ24 x25 y26 T,

η ∈ R5x1.(18)

Table 2 shows the values of the NCO and the εf for all thevibrational modes of the system in the frequency range ofinterest at the equilibrium configuration q∗. It is evident thatthe system dynamic is correctly represented. Similar resultshave been obtained by evaluating the accuracy of sucha reduced model in a neighborhood of the equilibriumconfiguration (see [25] for a detailed description).

Having reduced to 13 the dimensions of the model, thefirst-order formulation of the reduced model has a 26-di-mensional state vector versus the 60 state variables of thefull-order model. %us, a great reduction has been obtained.

%e measured inputs of the state observer are the threetorques exerted by the three actuators (T1, T2, and T3 inFigure 2(b)):

F � T1 T2 T3 0 0 −T3 0 0 0 0 0 0 0 T.

(19)

Six sensed outputs are available to compute the observerinnovation: the angular positions of the three-actuated links(q1, q2, and q3) and the curvatures (strains) at the midpointsof links 1, 2, and 4 (Figure 2(b)), denoted as c1, c2, and c4:


y � q1 q2 q3 c1 c2 c4{ }T. (20)

It should be noted that c1, c2, and c4 are nonlinearcombination of the state variables. �is choice of sensedoutputs ensures that system is observable, as corroborated bythe rank analysis of the observability matrix obtained bylinearizing the model around a set of equilibrium con�g-urations in the whole manipulator workspace.

4.2. Estimation Results. �e measurements have been gen-erated through the full-ordermodel, which can be thought ofas the “real mechanism.” In contrast, the observer is based

on the nonlinear reduced-order model and uses noisymeasurements since all the simulated measured signals thatare fed to observer are supposed to be corrupted by Gaussianwhite noise with realistic values for the sensors. Such a noisehas been generated as normally distributed random num-bers (by the “randn” function in MATLAB) having zeromean and the standard deviation (denoted as σ) stated inTable 3 for all the sensed measurements. �en, all themeasured signals have been digitized trough a 24-bit analog-to-digital converter, with input range ±10V.

A multirate EKF has been implemented: the continuous-time equations of motion of the real mechanism in Equation(9) have been discretized through the fourth-orderRunge–Kutta method with an integration step of 0.1ms,while the measured signals have been updated at 500Hz.

A motion lasting 3 seconds has been tested. �e manip-ulator end-eector is required to track a path describinga semicircumference of radius 0.15m (shown in Figure 3).�e motion of the mechanism is open-loop controlled by thecomputed torques shown in Figure 4 (computed through therigid-body model). No closed-loop control of the trajectory isperformed, since it is out of the scope of the paper. Addi-tionally, no vibration control is performed in order to excite thevibrational modes and hence to clearly evaluate if the reduced-order model and the observer are able to represent such dy-namics.�e noisy sensed outputs (Equation (18)) employed forcomputing the observer innovation are shown in Figure 5.

�e estimates of the angular positions and velocities ofthe three-actuated links are plotted in Figures 6 and 7, re-spectively, and are compared with those delivered by the realmechanism (i.e., those without noise). �e same �gures alsoshow the estimation error, i.e., the dierence between thetwo signals. �ese �gures clearly show that the observer isstable, and it delivers accurate estimates of the manipulatorgross motion, despite the presence of noise aecting mea-surements. Indeed, the amplitude of the estimation error issmaller than the measurement noise, as corroborated by thecomparison of Figures 4 and 5.

0.02m

0.282

mLI

NK

5

Node 14

Node 11

Node 4

Nodes 2, 6

Nodes 5, 10

Nodes 1, 3

Node 7

Nodes 8, 9

Nodes 12, 13

1.2m

0.5m

LINK 4

LINK 10.5m

0.03m0.

03m 0.01

6m

0.85

mLI

NK

2

0.85

mLI

NK

3

0.024m

(a)

Master dofsInputsOutputs

x25y26

x14

y15

q3

q2q1

T3

T2T1

γ4

γ2

γ1

θ24

(b)

Figure 2: Studied mechanism: �nite element model (a) variables involved in the estimation process (b).

Table 1: Lumped masses and nodal inertia.

Node Mass (kg) Inertia (10−2 kgm2)1 7.644 1.302 0.392 0.003 9.517 2.294 0.400 0.005 0.648 0.006 0.671 0.008 0.383 0.009 0.658 0.0010 0.308 0.0012 1.537 0.0013 0.095 0.1514 0.046 0.00

Table 2: Evaluation of the accuracy of the reduced model.

Mode frequency (Hz) NCO (–) εf (%)13.28 1.0000 0.011043.29 1.0000 0.001264.63 1.0000 0.0699124.25 0.9937 1.7157143.21 0.9983 0.5725159.36 1.0000 0.0112


Similar considerations can be obtained from Figures 8and 9, which show the linear elastic displacements andvelocities of the manipulator tip that might be of interest forimplementing trajectory control schemes. Again, there isa good agreement between the estimated variables and theactual ones, both in terms of amplitude and frequencycontent of the time-histories.

As a consequence of these precise estimates, the absolutedisplacement of the manipulator tip is very accurately es-timated, and its trajectory in the Cartesian space is almostoverlapped to the actual one, as shown in Figure 2.

Further evidences of the observer eectiveness comefrom the fast Fourier transforms (FFT) of the tip elasticdisplacements, as shown in Figure 10: the spectra of theestimated ones are almost perfectly overlapped to those ofthe real mechanism. Such a result further con�rms thecapability of the reduced-order model to represent correctlythe dynamics of the manipulator in the frequency range0–180Hz, where the reduced model has been tuned.

4.3. Sensitivity Analysis. A sensitivity analysis has beencarried out to assess observer robustness to model

uncertainties. In particular, the payload mass carried at thetip of the real mechanism has been modi�ed by increasing itof the 25%, 50%, 75%, and 100% of its original value. Incontrast, the payload mass of the reduced-order modeladopted in the observer has been kept constant and equal to thenominal value. Randommeasurement noise, as in the previoustest, has been adopted to corrupt the measured signals.

Simulation outcomes have been summarized throughthe mean μ and the standard deviation σ of the estimationerrors averaged over 100 simulations for each value of thea payload mass:

μτ �1100

∑100

j�11/ns∑

ns

k�1τk,j − τ̂k,j∣∣∣∣∣

∣∣∣∣∣︸��︷︷��︸

μj

,

στ �1100

∑100

j�1

��1

ns − 1∑ns

k�1τk,j − τ̂k,j∣∣∣∣∣

∣∣∣∣∣− μj( )2

√√

.

(21)

In Equation (21), τ denotes a generic estimated variable;τk,j and τ̂k,j are the estimated and the actual values of τ,

t (s)

–10

–5

0

5

T 1 (N

m)

0 1 2 3t (s)

–20

–10

0

10

T 2 (N

m)

0 1 2 3 0 1 2 3t (s)

–0.05

0

0.05T 3

(Nm

)

Figure 4: Actuator forces.

–1.4 –1.35 –1.3 –1.25 –1.2 –1.15 –1.1x end effector (m)

–0.42

–0.36

–0.3

–0.24–0.26–0.28

–0.32–0.34

–0.38–0.4

y en

d ef

fect

or (m

)

ObserverSimulator

Figure 3: End-eector trajectory: actual (red) and estimated (blue).

Table 3: Standard deviations assumed for each measurement noise.

σT1(Nm)

σT2(Nm)

σT3(Nm)

σq1(rad)

σq2(rad)

σq3(rad)

σc1(1/m)

σc2[1/m]

σc4(1/m)

2 · 10−1 4 · 10−1 1 · 10−3 3.1 · 10−3 3.5 · 10−3 1 · 10−2 5 · 10−5 8 · 10−5 1.5 · 10−4


t (s)

0.65

0.7

0.75

0.8

0.85q 1

(rad

)

0 1 2 3t (s)

2.5

2.55

2.6

2.65

2.7

q 2 (r

ad)

0 1 2 3t (s)

4.2

4.4

4.6

4.8

5

5.2

5.4

q 3 (r

ad)

0 1 2 3

(a)

t (s)

–1

–0.5

0

0.5

1

1.5

γ 1 (1

/m)

×10–3

0 1 2 3

γ 2 (1

/m)

×10–3

t (s)

–2

0

2

4

0 1 2 3

γ 4 (1

/m)

×10–3

t (s)

–4

–2

0

2

4

6

0 1 2 3

(b)

Figure 5: Measured outputs: angular positions of the actuated links (a); curvatures of links 1, 2, and 4 (b).

ObserverSimulator

ObserverSimulator

ObserverSimulator

t (s)

0.65

0.7

0.75

0.8

0.85

q 1 (r

ad)

0 1 2 3t (s)

2.5

2.55

2.6

2.65

2.7

q 2 (r

ad)

0 1 2 3t (s)

4.2

4.4

4.6

4.8

5

5.2

5.4q 3

(rad

)

0 1 2 3

(a)

ObserverSimulator

ObserverSimulator

ObserverSimulator

×10–4

t (s)

–4

–2

0

2

4

q 1 er

ror (

rad)

0 1 2 3

×10–4

t (s)

–4

–2

0

2

4

q 2 er

ror (

rad)

0 1 2 3

×10–3

0 1 2 3t (s)

–1

0

1

q 2 er

ror (

rad)

(b)

Figure 6: Estimated angular positions of the actuated links (a) and estimation errors (b).


ObserverSimulator

t (s)

–0.4

–0.2

0

0.2

dq1/

dt (r

ad/s

)

0 1 2 3

ObserverSimulator

t (s)

–0.2

0

0.2

0.4

dq2/

dt (r

ad/s

)

0 1 2 3

ObserverSimulator

t (s)

–0.2

0

0.2

0.4

0.6

dq3/

dt (r

ad/s

)

0 1 2 3

(a)

ObserverSimulator

t (s)

–0.01

0

0.01

dq1/

dt er

ror (

rad/

s)

0 1 2 3

ObserverSimulator

t (s)

–0.01

0

0.01dq

2/dt

erro

r (ra

d/s)

0 1 2 3

ObserverSimulator

0 1 2 3t (s)

–0.06

–0.04

–0.02

0

0.02

0.04

dq3/

dt er

ror (

rad/

s)(b)

Figure 7: Estimated angular velocities of the actuated links (a) and estimation errors (b).

ObserverSimulator

t (s)

–1

–0.5

0

0.5

1

1.5

x 25 (

m)

×10–3

0 1 2 3

ObserverSimulator

t (s)

–3

–2

–1

0

1

2

y 26 (

m)

×10–3

0 1 2 3

(a)

ObserverSimulator

t (s)

–4

–2

0

2

4

x 25 e

rror

(m)

×10–5

0 1 2 3

ObserverSimulator

t (s)

–5

0

5

y 26 e

rror

(m)

×10–5

0 1 2 3

(b)

Figure 8: Estimated elastic displacements of the elastic coordinates of the end-eector (a) and estimation error (b).


respectively, at the kth simulation step of the jth run; and nsis the number of sample of each run.

�e means (markers) and the standard deviations(lines) of the estimation errors of the linear displacementsand velocities of the tip coordinates are shown in Figure 11.Clearly, the case of no added mass corresponds to theresults shown in Figures 5–9 for random noise. Althoughmismodeling aects negatively the estimation by in-creasing both the mean value and the standard deviation

of the errors, the estimation errors are small even in thepresence of a payload mass twice the expected one. �isresult corroborates the correctness of both the modelreduction approach and the estimation scheme proposed.

5. Conclusions

A nonlinear state observer based on the extended Kalman�lter and on a reduced dynamic model of a exible-link

ObserverSimulator

t (s)

–0.05

0

0.05

0.1

dx25

/dt (

m/s

)

0 1 2 3

ObserverSimulator

t (s)

–0.1

–0.05

0

0.05

0.1

dy26

/dt (

m/s

)

0 1 2 3

(a)

ObserverSimulator

t (s)

–0.02

–0.01

0

0.01

0.02

dx25

/dt e

rror

(m/s

)

0 1 2 3

ObserverSimulator

0 1 2 3t (s)

–0.01

0

0.01

dy26

/dt e

rror

(m/s

)

(b)

Figure 9: Estimated elastic velocities of the elastic coordinates of the end-eector (a) and estimation error (b).

50 100 150

2

4

6

8

10

12

14

16

18

|X25

|

×10–5

ObserverSimulator

ObserverSimulator

50 100 150Frequency (Hz)Frequency (Hz)

0.5

1

1.5

2

|Y26

|

×10–4

Figure 10: FFT of the elastic displacements of the end-eector.


manipulator has been successfully synthesized. Model re-duction has been performed through a modi�ed Craig-Bampton strategy that handles nonlinearities by properlymodifying the transformation matrix and that trades-obetween model size and accuracy through a wise selectionof the most important interior modes to be retained.

�e numerical results prove that such an observer deliversaccurate estimates of both the rigid-body motion and theelastic displacements. Observer robustness to measurement

noise and model uncertainties has been also proved by in-cluding noise and parameter variation.

�e great reduction of the computational eort due tothe reduction of the model size makes the proposed observerpromising for getting e§cient state estimates in the controlof exible-link multibody systems.

Data Availability

�e data used to support the �ndings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

�e authors declare that they have no conicts of interest.

References

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Percentage of mass added to the end-effector

1

2

3

μ x25

, σx2

5 (m

)

×10–5

0 25 50 75 100

μ y26

, σy2

6 (m

)

Percentage of mass added to the end-effector

0

1

2

3

4

0 25 50 75 100

×10–5

(a)

0 25 50 70 100Percentage of mass added to the end-effector

2

4

6

8

μ dx2

5/dt

, σdx

25/dt (

m/s

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×10–3

μ dy2

6/dt

, σdy

26/dt (

m/s

)

0 25 50 75 100Percentage of mass added to the end-effector

2

4

6

×10–3

(b)

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