bandwidth maximizing design for hydraulically actuated...

Bandwidth Maximizing Design for HydraulicallyActuated Excavators

SEUNGJIN YOOCHEOL-GYU PARKDoosan Infracore Institute of Technology, Korea

BOKMAN LIMKYO IL LEEF. C. PARKSchool of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea([email protected])

(Received 28 December 2007� accepted 10 August 2009)

Abstract: In this paper we address the optimal design of a multi-degree-of-freedom hydraulically actuatedexcavator for reducing vibration and improving dynamic bandwidth. We specifically consider the problemof maximizing the minimum fundamental frequency of a hydraulic excavator with respect to various actuatorand structural design parameters, i.e. the piston areas of the hydraulic cylinders, and the masses and inertias ofthe mechanical links. An analytic gradient-based optimization algorithm, together with the design sensitivityequations, are derived. Detailed case studies involving a prototype hydraulic excavator are presented.

Keywords: Bandwidth, excavator, hydraulic, natural frequency, optimal design, vibration.

1. INTRODUCTION

Commercial excavators are almost exclusively hydraulically actuated, as hydraulic actuatorsare generally the most effective means of delivering the force-to-weight ratios required oftypical excavation tasks. While the compressibility of hydraulic oil is often regarded as neg-ligible, particularly under high operating pressures, in practice current hydraulic excavatorscontinue to suffer from low-frequency hydraulic oscillations� Ding et al. (2000) trace theseoscillations to oil compressibility, observing that the oscillations are further exacerbated bythe large masses and inertias of the structural links typical of most excavators. It has alsobeen observed that the low natural frequencies characteristic of hydraulic systems limit themaximum open loop position control gain, and more generally the overall dynamic band-width of the system (Merritt, 1967).

Several control strategies have been proposed for suppressing and reducing vibration inexcavators and other hydraulically actuated industrial machinery, e.g., Singh (1997), Ding

Journal of Vibration and Control, 16(14): 2109–2130, 2010 DOI: 10.1177/1077546309348855

��2010 SAGE Publications Los Angeles, London, New Delhi, Singapore

Figures 1, 4, 5 appear in color online: http://jvc.sagepub.com

2110 S. YOO ET AL.

et al. (2000), Park and Chang (2003), Honma et al. (1994), Kawaguchi et al. (1992),and Singhose et al. (1997). This paper takes an alternative perspective, and examines thequestion of how to design a hydraulically actuated excavator so as to improve its overalldynamic performance, particularly with respect to minimizing vibration.

Motivated by the fact that the minimum natural frequency imposes an upper bound on theposition control loop gain for single-degree-of-freedom (dof) hydraulic servomechanisms,we consider the generalized problem of maximizing the minimum fundamental frequencyof a multi-dof excavator with respect to the primary actuator and structural parameters, i.e.the piston areas of the hydraulic cylinders, and the masses and inertias of the mechanicallinks. We do not include kinematic parameters such as link lengths or offset angles as part ofthe design parameter set, since these are in most cases determined entirely from workspaceconsiderations. Constraints on the cylinder design parameters include velocity and forcerequirements of the hydraulic cylinders, the cylinder size (which can reasonably be regardedas being proportional to its cost), and the minimum rod diameter required to avoid buckling.

The optimal design problem addressed here shares a number of similarities with theapproach described by Lim and Park (2009), who maximize the minimum fundamental fre-quency of a rigid-link parallel mechanism with respect to link inertias and joint stiffnesses.Lim and Park’s main contribution, which is also motivated by a desire to minimize vibrations,is to prove that the problem as formulated reduces to a convex optimization problem involv-ing affine constraints on the design parameters—as is well known, for convex problems alllocal minimizers are global minimizers, and efficient and reliable interior point algorithmsare also readily available.

Lim and Park’s results are derived for the special case of parallel mechanisms, withjoint compliances modeled as one-dimensional linear or rotary springs, and input forces andtorques assumed directly applied to the joints. This paper extends the analysis to the muchlarger and far more ubiquitous class of hydraulically actuated industrial machinery such asexcavators. In this regard our paper makes the following specific contributions:

� A precise characterization is presented of the relationship between the hydraulic naturalfrequency and the fundamental frequency of a multi-dof hydraulic excavator. This char-acterization is based on an elasto-dynamic model that is amenable to design optimizationwhile capturing the primary sources of compliance in typical hydraulic excavators.

� Analytic gradients for the fundamental frequency with respect to the design parametersare derived. These are essential not only for developing robust gradient-based optimiza-tion algorithms that converge under a wide range of boundary conditions and design con-straints, but also for a sensitivity analysis to determine the design parameters that have themost influence on the performance criterion.

� Detailed formulations of several versions of the hydraulic excavator optimal design prob-lem are presented. In particular, we describe a number of simplifying but realistic as-sumptions on the excavator system such that the resulting design optimization problem isconvex.

� Detailed case studies are presented of optimal designs obtained for realistic hydraulicexcavators, and an assessment is made of the actual improvements that can be attainedthrough our proposed optimal design procedure.

BANDWIDTH MAXIMIZING DESIGN 2111

This paper is organized as follows. Section 2 presents a general elasto-dynamic modelfor a hydraulically actuated excavator, together with the fundamental frequency and its an-alytic gradient. Section 3 presents a formulation of the design problem, followed by thedetailed case study in Section 4. Section 5 concludes with a summary and some suggesteddirections for further inquiry.

2. ELASTO-DYNAMIC MODELING

In what follows we do not consider compliance of the ground, which would require complexmodels of soil–track interaction. We assume that all links are rigid, and regard compliance inthe hydraulically actuated joints to be the primary source of oscillation. This is not to implythat elastic deformations of the links (particularly the base link) are irrelevant� indeed, finiteelement analysis of the overall system is clearly essential at a later stage. Rather, our opti-mal design formulation, which as we show leads to a computationally tractable optimizationproblem without sacrificing physical relevance, is an early stage design optimization proce-dure, and intended to precede any subsequent finite element-based structural analysis thatwould follow at a later stage in the design process.

Our analysis is based on the following elasto-dynamic equations for a hydraulic excava-tor. With only a minimal loss of generality we consider a three-dof planar excavator modelof the type shown in Figure 1, consisting of the boom, arm, and bucket, together with a basichydraulic cylinder model. The rigid-body dynamic equations for this articulated structurecan be written in the form

M�q� �q � C�q� �q�� G�q� � �� (1)

where q � �3 denotes the generalized (joint) coordinates of the structure, M�q� � �33

is the generalized mass matrix, C�q� �q� � �3 denotes the Coriolis and centrifugal terms,G�q� � �3 denote the gravity terms, and � � �3 denotes the generalized input forces at theprismatic joints. For our hydraulic excavator the elements of the joint vector q � �3 consistof the hydraulic cylinder displacements �lb� la� lk� and the elements of the joint force vector� � �3 consist of hydraulic cylinder forces � fb� fa� fk� where the subscripts respectivelydenote “boom”, “arm”, and “bucket”.

Figure 2 portrays a schematic diagram of a hydraulic cylinder. The actuator dynamicsof the hydraulic cylinder can be derived via application of the continuity equations for eachchamber:

�Ph � �

Ah�l0 � l��Qh Ah �l�� (2)

�Pr � �

Ar�lc t l0 l��Ar �l � Qr�� (3)

where Ph and Pr denote the pressure in the head and rod side chamber of the hydraulic cylin-der, � is the bulk modulus of the hydraulic oil (assumed to be uniform over the chambers),

2112 S. YOO ET AL.

Figure 1. (a) Actual hydraulic excavator. (b) Kinematic structure.

Ah and Ar are the cylinder areas for the head and rod, respectively, Qh and Qr are the con-trolled flow rates into each chamber, l0 is the minimum stroke position, lc is the length ofthe cylinder barrel, l is the displacement of the rod from the minimum stroke position, andt denotes the thickness of the piston. The discharged oil from the pumps flows into the hy-draulic cylinder through the control valve that varies the metering area and the direction offlow according to the operator’s joystick command. For example, if an operator controls thejoystick to push the rod, the discharged oil from the pump flows into the piston head with rate


Figure 2. Schematic of a hydraulic cylinder.

Qh , while the oil in the rod side chamber returns to the reservoir with rate Qr . If an operatordoes not manipulate the control joystick, the hydraulic oil is trapped in each chamber suchthat the flowrate Qh and Qr are both zero.

In order to compute the fundamental frequency of the excavator (which is also a goodindicator of speed of dynamic response), it is enough to consider the situation when thehydraulic control system is unactuated such that the controlled flow rate Qh and Qr are bothzero in the actuator dynamic equations (2) and (3). The equivalent hydraulic spring constantdue to the oil trapped in each chamber then becomes

�Ph � �

Ah�l0 � l�Ah�l� (4)

�Pr � �

Ar �lc t l0 l�Ar�l� (5)

where �Ph and �Pr denote changes in the pressure in each chamber and �l denotes the defor-mation of hydraulic spring where the hydraulic spring force is represented by Equation (6):

f � Ah Ph Ar Pr � (6)

Under the assumptions described above, the resulting elasto-dynamic equations for thecombined excavator–actuator system can be summarized as follows:

0 � M�q� �q � C�q� �q�� G�q�� K �q��q� (7)

K �q� � diag

��

�Ai�h

�l0�i � li�� Ai�r

�lc�i ti l0�i li�

��i�b�a�k

� (8)

Note that the resulting equations of motion are valid when the hydraulic cylinders are un-forced or unactuated. Like the generalized mass matrix, the stiffness matrix K �q� is alsoconfiguration dependent.

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2.1. Hydraulic Natural Frequency Versus System Fundamental Frequency

By regarding the hydraulic cylinders other than those of interest as rigidly structured links,the hydraulic natural frequency for a single-dof hydraulic servomechanism is defined byMerritt (1967) as follows:

fh�i � 1

2�

�kii

mii� (9)

where fh�i denotes the natural frequency (in Hertz) corresponding to subsystem i , mii denotesthe i th diagonal entry of the mass matrix M , and kii the corresponding i th diagonal entry ofthe stiffness matrix K .

The linearized dynamic model for an excavator at a given posture, neglecting damping,is of the form

M �q � Kq � �� (10)

Under “free vibration,” i.e. under a motion of the system (10) caused by nonzero initialconditions and a zero excitation� , the foregoing equation can be solved for �q:

�q � Dq� D � M1 K (11)

with the matrix D commonly referred to as the dynamic matrix in the literature. This ma-trix determines the behavior of the system at hand, as its eigenvalues are the fundamentalfrequencies of the system and its eigenvectors the modal vectors.

The system fundamental frequencies of the excavator are obtained as solutions to thefollowing eigenvalue problem: �

K 2i M�� i � 0� (12)

where fi � i�2� is the i th fundamental frequency, � i � �n is the i th natural mode, andM and K (the arguments have been suppressed for convenience) are the mass and stiffnessmatrices from the previous dynamics formulation, respectively. The 2

i can be identifiedwith the eigenvalues of M1 K , which in turn coincide with the eigenvalues of the symmet-ric matrix M1�2 K M1�2, where M1�2 is chosen to be the unique positive-definite inversesquare root of M (note that M and K are always symmetric positive-definite, thus ensuringthat the eigenvalues 2

i always remain positive). Alternatively, the eigenvalues of M1 K alsocoincide with the eigenvalues of the symmetric matrix K 1�2 M1 K 1�2.

We now relate the system fundamental frequencies with the hydraulic natural frequenciesas given in Equation (9). From Rayleigh’s inequality, note that the ratio kii�mii is boundedfrom above and below by

min

�M1 K

� � kii

mii� max

�M1 K

�� (13)


where min and max are the minimum and maximum eigenvalues of the corresponding ma-trix, respectively. It follows easily that the hydraulic natural frequencies fh�i are boundedfrom above and below by the respective maximum and minimum system fundamental fre-quencies fmax and fmin.

For our design optimization we consider subsets of the following collection of designparameters, a common feature being that none of these parameters have an influence on thekinematic structure:

p � [ Ab�h� Ab�r � Aa�h� Aa�r � Ak�h� Ak�r � Ib�z� Ia�z� Ik�z�

mb�ma�mk� rb�x � rb�y� ra�x � ra�y� rk�x � rk�y ]� (14)

Here A denotes the areas of the respective hydraulic cylinders, Iz denotes the rotationalinertia of the corresponding link about its mass center, m denotes the corresponding linkmass, and r denotes the coordinates of the mass center with respect to the correspondingbody-fixed frame for each link.

2.2. Sensitivity Analysis

The sensitivity of the system fundamental frequency with respect to arbitrary design para-meters can be analytically determined via the following result (see, e.g., Condren and Gedra(2002), Aryana and Bahai (2003), and Lee et al. (1996)): given a real symmetric matrixA with eigenvalue–eigenvector pair � � x�, and p some arbitrary scalar argument of A, it isknown that

� �A�

�p� xT��A��p�x

xTx� (15)

For our objective function f � � �M1�2 K M1�2��2� , the above derivative formula can

be applied to obtain the following:

� f

�p� f

p

��log f �

��log p�(16)

��log f �

��log p�� 1

2p

��T

�K1�2 �K

�pK1�2

�� T

�M1�2 �M

�pM1�2

��

�� (17)

where � denotes the eigenvector of unit magnitude for M1�2 K M1�2 corresponding to ,and � is the normalized unit length version of the vector K 1�2 M1�2�. Proof of this formu-lation can be found in Appendix II. Note that the resulting sensitivity equation is invariantunder uniform scaling of the fundamental frequency f and design parameter p. Therefore,the relative sensitivity ��log f ��log p� is robust to uncertainties in the oil’s bulk modulus,which is assumed to be uniform over the chambers and independent of the design parameterunits.

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To evaluate Equation (17), the derivatives of both the stiffness and mass matrices withrespect to the joint variables and other parameters of interest are required. The nonzero termsof the derivative for the stiffness matrix are as follows:

�Ki

�Ai�h� �

�l0�i � li�� (18)

�Ki

�Ai�r� �

�lc�i ti l0�i li�� (19)

Recursive algorithms for obtaining the derivatives of the mass matrix M with respect toarbitrary design parameters can be found in Park et al. (1995, 1999). In the formulation ofM for our excavator, it is assumed that the mass of each cylinder and rod are proportionalto its size� further assuming a standard slender rod geometry, the cylinder inertias are thenparameterized in a linear fashion with respect to the piston areas.

3. OPTIMAL DESIGN FORMULATION

We now consider the problem of determining the optimal sizes of the cylinders, and also thelink mass and inertia distributions, so as to maximize the minimum fundamental frequencyof the hydraulic excavator system. Note that the mass matrix M�q� is configuration depen-dent� because of this feature our design process is necessarily iterative. The design is thusconducted for a strawman task, by choosing a set of representative excavator configurations,which in turn lead to a set of fixed mass matrix values for each configuration. The frequencyspectrum for all of these configurations can be made, by design, to lie above the frequencyrange of the strawman task. Since the excavator will be commanded to execute tasks that aremore general than the strawman task, simulation of alternative and varied tasks is essentialto ensure that the design is safe from a resonance viewpoint.

3.1. Optimal Cylinder Sizing

We first consider the problem of optimal cylinder sizing only, assuming that all of the linkmasses and inertias are fixed. The design parameter vector we consider consists of the fol-lowing cylinder areas:

x � [ Ab�h Ab�r Aa�h Aa�r Ak�h Ak�r ]T� (20)

The problem formulated here proceeds on the assumption that the variation in the massmatrix elements with respect to cylinder areas will not be significant relative to the largeinertias of the links. In this case the corresponding optimization problem reduces to a convexone of the form described in Lim and Park (2009), whose main results we summarize inAppendix I. In our later case studies we examine the validity of this assumption.

We now list the various physical constraints imposed on the design optimization. Con-straints on the pump pressure and hydraulic driving force, and also on the pump flow rate, ofthe following form need to be considered:


fi � Ppump Ai�h� (21)

�fi � Ppump Ai�r � (22)

�li � Qpump

Ai�h� (23)

��li � Qpump

Ai�r� (24)

where Ppump and Qpump are the maximum pump pressure and flow rate, respectively, and fi ��fi and

�li ��li denote the user-specified specifications on the hydraulic force and veloc-

ity prescribed at the i th hydraulic cylinder. The above constraints impose lower and upperbounds on the piston areas. Another constraint involves the prevention of buckling of thecylinder rod, which requires that the minimum rod diameter be kept above a certain limitDi�m as follows:

Ai�h Ai�r � �

4D2

i�m� (25)

Modeling the cylinder rod as a clamped-hinged column, the Euler buckling criterion can beapplied to obtain the minimum rod diameter:

Pcr � 2�05�2

l2E I� (26)

where E is the Young’s modulus and I is the second moment of area of the piston rod.Including a safety factor �i for the minimum rod diameter of each hydraulic cylinder, thefollowing equation can be solved to obtain the minimum rod area under a specified maximumexternal load Pcr :

Pcr � 2�05� 2 E

�i l2i�r

�64

D4i�m

� �

4D2

i�m � 2li�r

��i Pcr

2�05�E� (27)

The resulting constraint to avoid buckling of the cylinder rod can thus be expressed as alinear inequality constraint on the piston areas:

Ai�h Ai�r � 2li�r

��i Pcr

2�05�E� (28)

Finally, the cost of the cylinder is also an important factor in selecting the hydraulic cylinder.We assume that the cost of the cylinder is proportional to its size or volume, leading to thefollowing linear equality constraint:�

i

�Ai�hli�c �

�Ai�h Ai�r

�li�r

� � c� (29)

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where c represents the total cost.Given any physically meaningful combination of the above constraints, we now seek to

maximize the minimum natural frequency with respect to the cylinder areas, i.e.

maxx min

�M�q�� x�1 K �q�� x�

�� (30)

where x denotes the vector of cylinder areas, and q� is a fixed given configuration. In theevent that there are multiple given configurations, one can choose any number of stan-dard optimization formulations, e.g., to optimize with respect to x over the worst caseconfiguration, or a weighted linear combination of the objective function values over thegiven configurations.

3.2. Optimal Cylinder Sizing and Inertia Parameter Distribution

We now consider the problem of simultaneously determining both the cylinder sizes and thelink mass and inertia distributions so as to maximize the minimum fundamental frequency.In this case the design parameter x consists of the cylinder areas listed in Equation (20) to-gether with all of the link masses, inertias, and mass centers. The constraints in this probleminclude those on the cylinder area presented in the previous section, and constraints on theinertia parameters of the boom, arm, and bucket links. First, we arbitrarily allow each inertiaparameter to vary within �30% of the current initial value:

0�7mi�o � mi � 1�3mi�o� (31)

0�7Ii�z�o � Ii�z � 1�3Ii�z�o� (32)

The total system mass is also assumed to be conserved such that the following holds:�mi �

�mi�o� (33)

The mass centers of each link are also allowed to vary within �30% of the current value.Taking all of the constraints into account leads to the following optimization formulation:

maxx

min

�M�x�1�2 K �x�M�x�1�2

�(34)

Subject to

fi

Ppump� Ai�h�

�fi

Ppump� Ai�r (35)

Ai�h � Qpump �li

� Ai�r � Qpump��li

(36)

Ai�h � Ai�r � 2li�r

��i Pcr

2�05�E� 0 (37)


�i

�Ai�h

�li�c � li�r

� Ai�r li�r

� � c (38)

0�7mi�o � mi � 1�3mi�o (39)

0�7Ii�z�o � Ii�z � 1�3Ii�z�o (40)�mi �

�mi�o (41)

0�7ri�x0 � ri�x � 1�3ri�x0 (42)

0�7ri�y0 � ri�y � 1�3ri�y0� (43)

Again, multiple configurations can be treated as described in the previous section.Note that this general formulation is not convex, because of the explicit dependence of

the inertia matrix on the choice of cylinder sizes and link inertias. However, the restrictedcase involving the approximated objective function for cylinder sizing, as well as the problemthat considers only the mass and inertia parameters of the links as the design parameters (andassumes that the cylinder sizes are fixed), lead to convex problems of the type described byLim and Park (2009). In our case studies described below, we find that given practical boundson the design parameters, in nearly all of the cases the optimization converges to a globalminimizer.

4. DESIGN CASE STUDY

We now consider a design case study involving a prototype excavator with specifications asgiven in Table 1� these specifications are based on the actual excavator (Doosan DX300LC)shown in Figure 1. The overload relief valves installed in the hydraulic circuits limit thepressure in each cylinder chamber, protecting it from failure when an external impact load isapplied to the excavator tip. Therefore, the critical load in Equation (26) is obtained from thecracking pressure multiplied by the initial piston area of each hydraulic cylinder. The safetyfactor �i in (27) is assumed to be 3.5 and the effective bulk modulus � is assumed to be12,000 bar. Design specifications on the velocity and force on each hydraulic cylinder are setto be half the initial capability, such that the piston areas that meet the design specificationsin (21), (22), (23), and (24) have values between half and twice the initial piston areas. Weuse Matlab to perform the dynamic analysis, and the corresponding Matlab OptimizationToolbox to solve the optimization problem on a Pentium 4 (2.6 GHz) personal computer.

4.1. Excavator Fundamental Frequency

Figure 3 shows the fundamental frequencies over a range of postures of the excavator. Theminimum fundamental frequency of the target excavator is obtained to be 2.3705 Hz at theposition �1�152� 0� 0�46�, with the numbers corresponding to the displacements of the boom,arm, and bucket, respectively.

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Figure 3. Fundamental frequency distribution of an excavator.

Table 1. Excavator model parameters.

Boom Arm BucketHydraulic Rod length li�r [m] 1.8200 2.1725 1.5350cylinders Cylinder length li�c [m] 1.8000 2.1000 1.5000

Initial piston area (m2) Ab�h Ab�r Aa�h Aa�r Ak�h Ak�r

0.0308 0.0166 0.0177 0.0090 0.0154 0.0090Maximum pump pressurePpump (bar) 310Cracking pressure PO R (bar) 350Young’s modulus E (Pa) 2�1 1011 (Carbon steel)Maximum pump flow-rateQpump (l/min1) 247 2Oil’s bulk modulus � (bar) 12000

The driving frequency of the excavator, anticipated to be between 0 and 3 Hz from thefast Fourier transform (FFT) of the hydraulic cylinder acceleration during a cycle of severalexcavation experiments, is also expected to cause resonant hydraulic oscillations. Therefore,it is observed that the target excavator may suffer from low-frequency oscillations near theoperator sensitive frequency bands. By increasing the first fundamental frequency of theexcavator at the minimum fundamental frequency position �1�152� 0� 0�46�, it would appearto be possible to avoid operator sensitive, resonant hydraulic oscillations. Moreover, theincreased minimum fundamental frequency will increase the bandwidth of the excavator suchthat the worst position control performance of the potential automated excavator would beimproved. At the same time, by decreasing the maximum first fundamental frequency whileincreasing the minimum second fundamental frequency, it will be possible to avoid operatorsensitive, vertical vibrations of the excavator. Numerical results show that the maximum


Figure 4. Semi-log sensitivity ��2�q2� 1�q1��log p�.

first fundamental frequency is 5.7979 Hz at configuration �0� 1�5795� 0�, while the minimumsecond fundamental frequency is 5.1148 Hz at configuration �1�1520� 1�5795� 0�6900�.

4.2. Fundamental Frequency Sensitivity

The relative sensitivity equation (17) also provides gradient information for the optimizationproblems formulated in Section 3, as well as insights into different design strategies, e.g.,avoiding operator sensitive vertical vibrations. Numerical results however indicate that therelative sensitivity of the first fundamental frequency at q1 � �0� 1�5795� 0� does not differsignificantly from that of the second fundamental frequency at q2 � �1�1520� 1�5795� 0�6900�.These findings suggest that the design strategy of increasing the minimum second fundamen-tal frequency while decreasing the maximum first fundamental frequency will be difficult torealize with the set of design parameters considered in this paper.

Figure 4 shows the semi-log sensitivity of the difference between the first and secondfundamental frequencies at the positions q1 and q2, respectively, which are obtained fromthe following relative sensitivity equation (the vectors �i and � i are associated with thefundamental frequency i ):

��2 1�

��log p�� p

22

��T

2

�K1�2 �K

�pK1�2

��2 �T

2

�M1�2 �M

�pM1�2

��2

�

p

21

��T

1

�K1�2 �K

�pK1�2

��1 �T

1

�M1�2 �M

�pM1�2

��1

�� (44)

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Figure 5. Relative sensitivity ��log1��log p��q�q� .

Table 2. Solution to the optimal cylinder sizing problem.

Frequency Cylinder area (m2)(Hz) Ab�h Ab�r Aa�h Aa�r Ak�h Ak�r

Initial 2.3705 0.0308 0.0166 0.0177 0.0090 0.0154 0.0090Optimal(approximation) 3.1668 0.0467 0.0332 0.0144 0.0089 0.0093 0.0057Optimal (exact) 3.1683 0.0476 0.0332 0.0137 0.0082 0.0081 0.0045Error (%) 0.05 1.86 0 5.11 8.53 14.26 25.76

From the above observation, it seems that there are no alternative design strategies foravoiding operator sensitive oscillatory response other than maximizing the minimum funda-mental frequency.

Figure 5 shows the relative sensitivity of the first fundamental frequency at the positionq� � �1�1520� 0� 0�460�, at which the minimum fundamental frequency is attained.

It is observed that the boom cylinder’s piston areas are the most effective parametersfor increasing the minimum fundamental frequency. The greater effectiveness of the rodside piston area owes to the boom cylinder displacement in q�, where the stiffness is variedaccording to (8), giving more weight to the rod side piston area in the stiffness of hydraulicoil spring.

4.3. Optimal Cylinder Sizing

At the excavator position achieving the minimum first fundamental frequency, the results ofthe cylinder sizing problem are shown in Table 2 and Figure 6. The optimization variables


Figure 6. Optimal cylinder area distribution.

are the cylinder areas. The underlying bars in the figure denote the upper and lower boundsof each piston area of the cylinder, and the resulting cross-sectional area of the cylinderrod.

The minimum fundamental frequency is maximized to 3.1683 and 3.1668 Hz under theexact and approximated versions of the problem, respectively. The total optimization timesare on average about 5 seconds. By approximately solving the fundamental frequency of theexcavator neglecting the cylinder’s inertia variations during the optimization, the problem isconvex as mentioned in Section 3.

While the cylinder area distribution is slightly different under the two optimization prob-lem, the resulting fundamental frequencies are almost coincident. The reason for this isshown in the frequency sensitivity evaluated in Figure 5, where the “arm” and “bucket” pis-ton areas are shown to have only a minor effect on the fundamental frequency.

We now increase the total cost up to twice the initial cost, and solve the optimal cylindersizing problems repeatedly. The results of the optimization are shown in Figure 7. As theinertia effects of the hydraulic cylinder may lower the fundamental frequency by increasingthe size or cost of the hydraulic cylinder, the equality constraint (29) is relaxed to be aninequality constraint. The results, indicated by the gray line denoting the optimization resultby setting the total cost to be an equality constraint, testify to this observation. The dottedline indicates the approximated optimization result, which does not consider variations of thecylinder mass fixing it to its initial value regardless of the cylinder area variation. It is shownthat the optimization error increases as the constraint on the total size of the cylinder is furtherrelaxed, at the cost of any simplifications to make the problem convex. In fact, it is shownthat the constraint on total cost of the cylinder is always active in the approximated problem,

2124 S. YOO ET AL.

Figure 7. Maximized first fundamental frequency according to the cost relaxation.

because the piston areas only affect the stiffness matrix. It is also observed experimentallythat the exact problem, although it cannot be ensured to be convex in the general case, asin the approximated problem appears to find the global optimum within a few steps. Thisconclusion is based on our observation that, starting from a number of different initial designparameters lying on a prescribed bounded admissible region, the optimization procedurealways converges to the same identical solution.

The upper bound on each cylinder area is linear in the maximum pump flow-rates asin (23) and (24). It is observed that the upper bound on the boom cylinder’s rod side pistonarea is always activated. Therefore, the minimum fundamental frequency can be furtherincreased if the maximum flow capacity of the pump is increased. Maintaining the total costas the initial value, the upper bound on each piston is relaxed up to twice the initial value, byincreasing the maximum pump flow rate. Figure 8 shows the optimization results.

In this case, the optimization error is not significant because both optimization problemsattain the maximum boom rod side piston area under the total initial cost of the cylinder.Therefore, the approximated convex optimization problem shows good accuracy due to themoderate inertia effects of the cylinder.


Figure 8. Maximized first fundamental frequency according to the enlarged maximum pump flow.

4.4. Optimal Cylinder Sizing and Inertia Parameter Distribution

Figure 9 shows the optimization results for the problem formulated in Section 3.2. Theoptimization variables are the cylinder areas, link masses, inertias, and mass centers. Theminimum fundamental frequency is 3.8382 Hz, which is increased by 63�6% from the initial2.3705 Hz from the optimization. The total optimization time is about 6 seconds. It is shownthat the inertia parameters other than mass all settle to the corresponding lower bound. Inparticular, the mass centers are located close to the origin of each body fixed frame attachedat the joint axis. This in turn minimizes the rotational inertias of the links about the origin ofeach body fixed frame. As in the results of Section 4.3, the global optimum is also determinedwithin a few iterations.

5. CONCLUSION

This paper has addressed the problem of selecting the actuator and structural parameters ofa hydraulically actuated excavator in an optimal fashion, so as to minimize vibrations andto increase the dynamic control bandwidth. In our approach we consider the generalizedproblem of maximizing the minimum fundamental frequency with respect to the cylinder

2126 S. YOO ET AL.

Figure 9. Optimal cylinder area and inertia parameter distribution.

areas, and the masses and inertias of the mechanical links, subject to various velocity, force,and buckling constraints on the cylinder design parameters, as well as manufacturing cost.

By developing an elasto-dynamic model that is amenable to design optimization whilecapturing the primary sources of compliance in typical hydraulic excavators, we have pre-cisely characterized the relationship between an excavator’s hydraulic natural frequency andthe fundamental frequency. Analytic gradients for the fundamental frequency with respectto the design parameters have also been derived, and used for both sensitivity analysis and ina robust gradient-based optimization algorithm. Detailed formulations of several versions ofthe hydraulic excavator optimal design problem have presented, with a focus on simplifyingbut realistic assumptions on the excavator system that render the resulting design optimiza-tion problem convex.

Detailed optimal design case studies involving an actual hydraulic excavator have beenpresented to assess the actual improvements that can be attained through our optimal de-


sign procedure. Even for problems in which convexity cannot be guaranteed in general, ourresults suggest that under practical assumptions on the bounded nature of the admissibledesign parameter space, the optimization will in practice converge to a global minimizer.Our sensitivity analysis results further suggest that our design strategy allows one to avoidoperator-sensitive hydraulic resonant oscillations in the horizontal direction, as well as toimprove position control performance in excavator automated control. The minimum fun-damental frequency of the target excavator is shown to be increased from 33�7% to 63�6%depending on the choice of optimization design parameters.

Acknowledgements. This work represents the results of principal investigator S. Yoo’s Ph.D. dissertation, undertakenat SNU. This research was partially supported by Doosan Infracore, CBMS-KRF-2007-412-J03001, KIST-CIR, theIntelligent Autonomous Manipulation Research Center, ROSAEC, and IAMD-SNU.

APPENDIX I: CONVEX DESIGN FORMULATION

Lim and Park (2009) considers the minimum vibration design of multi-link mechanisms withnegligible link deformations, so that the mechanism’s structural elasticity can be attributedprimarily to joint torsional flexibility. The dynamic equations in this case are of the form

M�q� p� �q � C�q� �q� p� �q � N �q� p�� K �q q0� � 0�

S�q� �q0 � K �q0 q� � � � (45)

Here q0 � �n represents the vector of motor positions, q � �n the joint positions, p � �m

the vector of mass and inertial parameters, M�q� p� � �nn is the mass matrix, C�q� �q� p� ��nn is the Coriolis matrix, N �q� p� � �n reflects gravity forces, S�q� � �nn is the diago-nal matrix of rotor inertias, and K � diag �k1� � � � � kn�, ki � 0, is the diagonal joint stiffnessmatrix. The mass matrix M�q� p� furthermore admits the factorization (Park et al., 1995)

M�q� p� � ATLT�q�D�p�L�q�A� (46)

where A � �6nn is a constant matrix constructed entirely from the kinematic parameters,L�q� � �6n6n is a lower triangular matrix dependent on the joint values q, and D�p� ��6n6n is a block-diagonal link inertia matrix that is linearly dependent on the inertial andmass parameters p � �m . From this latter property it readily follows that M�q� p� dependslinearly on p. Methods for deriving the rigid-body dynamics of exactly actuated closedchains in the above form can be found in, e.g., Park et al. (1999).

Except where noted otherwise, for the remainder of this appendix we take the designparameters to be either the elements k � �k1� � � � � kn� � �n , ki � 0 of the diagonal stiffnessmatrix K , or the mass and inertial parameters p � �m in the mass matrix M . First assum-ing fixed p, maximizing the minimum natural frequency with respect to the joint stiffnessparameters k leads to the following optimization problem:

maxk��n

min�M�p�1�2 K �k�M�p�1�2�

subject to k � �� (47)

2128 S. YOO ET AL.

where min�� denotes the minimum eigenvalue, and � denotes a convex set in �n corre-sponding to the search space (e.g. a hypercube in �n). Alternatively, noting that the eigen-values of a given invertible matrix A1 are simply the inverses of the eigenvalues of A, andintroducing the compliance parameters ci � 1�ki , i � 1� � � � � n, the above maximization canbe equivalently formulated as the following minimization problem:

minc��n

max�M�p�1�2C�c�M�p�1�2�

subject to c � �� (48)

where C�c� � diag �c1� � � � � cn� and � is the corresponding convex set in c-space.Now assuming fixed k (or c), maximizing the minimum natural frequency with respect to

the mass and inertial parameters p can be formulated as the following optimization problem:

maxp��m

min�K �k�1�2 M1�p�K �k�1�2�

subject to p � �� (49)

for some convex set � in �m . The equivalent minimization formulation is given by

minp��m

max�C�c�1�2 M�p�C�c�1�2�

subject to p � � � (50)

In Lim and Park (2009) the two optimization problems of Equations (48) and (50) areshown to be convex. While optimizing min�M�p�1�2 K �k�M�p�1�2� with respect to pwhile keeping k fixed, or alternatively, optimizing the same objective function with respectto k while keeping p fixed, are both convex problems, it turns out that simultaneously op-timizing with respect to both p and k is not in general convex. A simple counterexampleis provided by a single-dof chain, in which M�p� and K �k� are both scalar. Case studiesdescribed by Lim and Park (2009) suggest, however, that for bounds on p and k that are notoverly large, many problems encountered in practical settings do indeed produce a uniqueglobal optimizer.

APPENDIX II: DERIVATION OF SENSITIVITY EQUATIONS

The derivative of the fundamental frequency f � � �M1�2 K M1�2��2� with respect to

the design parameters is given by

� f

�p� 1

4� �

M1�2 K M1�2� �

�M1�2 K M1�2

��p

� (51)

From the eigenvalue sensitivity equation for the symmetric matrix (Lee et al., 1996), it fol-lows that the identity (52) holds:


� �

M1�2 K M1�2�

�p� �T �

�M1�2 K M1�2

��p

�� (52)

where � is the corresponding eigenvector.Using the fact that �M1�2��p � M1�2��M1�2��p�M1�2, Equation (52) can be

rewritten as follows:

� �

M1�2 K M1�2�

�p� �T

��M1�2

�pK M1�2 � M1�2 K

�M1�2

�p� M1�2 �K

�pM1�2

��

� �T

� M1�2 �M1�2

�pM1�2 K M1�2

M1�2 K M1�2 �M1�2

�pM1�2 � M1�2 �K

�pM1�2

�� (53)

From the definition of � and utilizing the identity M1�2��M1�2��p�M1�2 � ��M1�2��p�M1�2 � M1�2��M1�2��p�, Equation (54) can be derived:

� �

M1�2 K M1�2�

�p� �M1�2 K M1�2

��T

�M1�2 �M

�pM1�2

��

� �T M1�2 K 1�2

�K1�2 �K

�pK1�2

�K 1�2 M1�2�� (54)

Defining the vector � � K 1�2 M1�2��K 1�2 M1�2��, Equation (54) is rewritten as follows:

� �

M1�2 K M1�2�

�p� �M1�2 K M1�2

��T

�M1�2 �M

�pM1�2

��

� �

M1�2 K M1�2��T

�K1�2 �K

�pK1�2

�� (55)

As a result, Equations (56) and (57) follow from (51) and (55):

� f

�p� f

2

��T

�K1�2 �K

�pK1�2

�� T

�M1�2 �M

�pM1�2

��

�(56)

��log f �

��log p�� p

f

� f

�p� p

2

��T

�K1�2 �K

�pK1�2

�� T

�M1�2 �M

�pM1�2

��

�� (57)

For the scalar case where the fundamental frequency is given as f � 1�2��

k�m, it is notedthat Equation (57) reduces to the following:

� f

�p� 1

2�

�k

m

�1

2k

�k

�p 1

2m

�m

�p

�� f

2

�1

k

�k

�p 1

m

�m

�p

�� (58)

2130 S. YOO ET AL.

REFERENCES

Aryana, F. and Bahai, H., 2003, “Sensitivity analysis and modification of structural dynamic characteristic usingsecond order approximation,” Engineering Structures 25(10), 1279–1287.

Condren, J. E. and Gedra, T. W., 2002, “Eigenvalue and eigenvector sensitivity applied to power system steady-state operating point,” in Proceedings of the 45th Midwest Symposium on Circuits and Systems, Tulsa, OK,August 4–7, pp. 683–686.

Ding, G., Qian, Z., and Pan, S., 2000, “Active vibration control of excavator working equipment with ADAMS,” in2000 International ADAMS User Conference, Orlando, FL, June 19–21.

Honma, K., Nakajima, K., Izumi, E., Aoyagi, Y., and Watanabe, H., 1994, “Improvement of damping characteristicsof hydrostatically driven hydraulic excavator,” Transactions of the Japan Society of Mechanical EngineersPart C 60(580), 4175–4182.

Kawaguchi, M., Maekawa, A., Kobayashi, N., Suzuki, K., and Yoshimura, T., 1992, “Study on active vibrationcontrol of arm for construction machinery,” Journal of the Japan Society of Mechanical Engineers Series C58(549), 1417–1423.

Lee, H. B., Lee, S. W., Jung, H. K., Hahn, S. Y., Cheon, C., and Kim, H. S., 1996, “An optimum design method foreigenvalue problems,” IEEE Transactions on Magnetics 32(3), 1246–1249.

Lim, B. and Park, F. C., 2009, “Minimum vibration mechanism design via convex programming,” ASME Journal ofMechanical Design 131(1), 011009.

Merritt, H. E., 1967, Hydraulic Control Systems, Wiley, New York.Park, F. C., Bobrow, J. E., and Ploen, S. R., 1995, “A Lie group formulation of robot dynamics,” The International

Journal of Robotics Research 14(6), 609–618.Park, F. C., Choi, J. H., and Ploen, S. R., 1999, “Symbolic formulation of closed chain dynamics in independent

coordinates,” Mechanism and Machine Theory 34(5), 731–751.Park, J. Y. and Chang, P. H., 2003, “Vibration control of a telescopic handler using time delay control and com-

mandless input shaping technique,” Control Engineering Practice 12(6), 769–780.Singh, S., 1997, “The state of the art in automation of earthmoving,” ASCE Journal of Aerospace Engineering 10(4),

179–188.Singhose, W. E., Porter, L. J., and Seering, W. P., 1997, “Input shaped control of a planar gantry crane with hoisting,”

in Proceedings of the 1997 American Control Conference, Albuquerque, NM, June 4–6, pp. 97–100.

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