Abstract

Preparing simulation-driven surrogates for a coupled mechanical system can be challenging because the associated mechanical and actuator dynamics demand high-fidelity numerical solutions. Proposed here is a universal hydraulic surrogate (UHS), which can provide solutions to high-fidelity mechanical systems with a universal actuator in a surrogate-assisted monolithic approach. The UHS acts as an alternative to the standard lumped fluid theory by eliminating the hydraulic pressures differential equations. A surrogate-assisted universal actuator uses an approximated model to define hydraulic force in high-fidelity mechanical systems. The approximated force model was developed through training against the dynamics of a one-dimensional (1D) hydraulic cylinder and spring-damper. A covariance matrix adaption evolutionary strategy (CMA-ES) was used as an optimization algorithm to minimize differences between the standard dynamics and UHS approaches at the position and velocity levels. The robustness of resulting UHS was validated to predict the behaviors of the simple four-bar mechanism and the forestry crane. The focus was on numerical accuracy and computational efficiency. The maximum percent normalized root mean square error (PN-RMSE) between the states of the approximated force model and lumped fluid theory were approximately 2.04% and 6.95%, respectively. The proposed method was approximately 52 times faster than the standard lumped fluid theory method. By providing accurate predictions outside the training data, the simulation-driven UHS promises better computational performance leading to real-time simulation solutions for the coupled mechanical systems. The UHS can be applied in simulation, optimization, control, state and parameter estimation, and Artificial Intelligence (AI) implementations for coupled mechanical systems.

1 Introduction

Numerically efficient simulations (e.g., real-time simulations) of a coupled mechanical system [1] play a pivotal role in diverse applications including biomechanics [2], hydraulics [3], automotive and railway system engineering [47], control [810], and robotics [11,12]. Using real-time simulation methods, a human operator can interact with simulated machinery via simulators or hardware to replicate real machine operations [13]. However, preparing simulation-driven surrogates for a coupled mechanical system can be challenging because the associated mechanical and actuator dynamics demand high-fidelity numerical solutions, and the systems are complex, which leads to nonlinear solution methods with high computational burden.

Computer simulation of coupled mechanical systems often includes the modeling of mechanical and interconnected or actuator-driven subsystems [14]. The dynamic models of mechanical and actuator subsystems can be combined in monolithic [14] or cosimulation [3] approaches. High-fidelity modeling approaches, such as the finite element method [15], to mechanical subsystem simulation demand extensive computational power [16], which makes the real-time simulation of complex mechanical subsystems problematic. To achieve real-time performance, surrogate solutions are often used to reduce the model order of the high-fidelity models [1619].

Preparing a surrogate involves constructing a simplified approximation of a more complex model to reduce the computational cost of the simulation process [20]. In coupled mechanical systems, surrogate solutions use data-driven modeling approaches to learn from the patterns of system data [2126]. Data-driven surrogates are modeled using statistical tools such as regression analysis or machine learning techniques, i.e., neural networks, random forests, or Gaussian processes [27,28]. In general, these methods are time-consuming and demand significant computational power and high-cost hardware solutions [21]. Furthermore, data-driven surrogates offer accurate predictions within the scope of their training data [20,29]. However, the accuracy of these predictions declines when extrapolated beyond the boundaries of the training data [20,29].

Simulation-driven surrogates overcome these challenges by learning directly from the behaviors of simulation models [3032]. They preserve the physics of a system and provide more accurate solutions within the simulation model range [33]. However, the simulation-driven surrogates described in the literature [3234] make use of detailed high-fidelity system models, and solving high-fidelity models with computational efficiency while implementing simulation-driven surrogates can be challenging.

A coupled multibody-hydraulics system, where lumped fluid theory [35] describes the dynamics of the hydraulics [3638], is an example of such a system. Lumped fluid theory computes hydraulic pressure derivatives within a hydraulic volume by dividing the effective bulk modulus into small hydraulic control volumes. The introduction of these pressure derivatives leads to numerical stiffness in the simulation of the coupled multibody-hydraulics systems, which negatively affects computational efficiency.

To eliminate these deficiencies with simulation-driven surrogates, this study proposes a novel UHS [39] for hydraulically actuated mechanical systems. As mentioned in Ref. [39], the performance of a high-fidelity coupled mechanical system can be optimized using a low-fidelity actuator subsystem model, which can also be referred to as a universal actuator surrogate [39]. In a hydraulically actuated system, the UHS of a coupled mechanical system is prepared using a hydraulic cylinder model, a low-fidelity model, and an optimization algorithm.

To train surrogate models, the commonly used algorithms of the optimization problem are Bayesian optimization [40,41], surrogate-assisted optimization [42], gradient-based methods [43], evolutionary algorithms [4446], and hybrid methods [47]. Among these methods, evolutionary algorithms, referred to as CMA-ES, have recently been used in many engineering applications, because they require fewer generations, have good search properties, and offer easy-to-use parallelization methods [48].

A low-fidelity UHS model of a coupled mechanical-hydraulic subsystem can facilitate decision-making throughout a product lifecycle. In Ref. [39], a universal hydraulic solution was proposed to solve the hydraulics dynamics in a four-bar mechanism. However, the robustness of this approach has yet to be validated for an industrial application. This study serves to validate the universal hydraulic surrogate solution as a viable alternative to lumped fluid theory as demonstrated through its application to the four-bar mechanism. Its robustness is further validated by using it to predict the states of a forestry crane.

The objective of this study is to introduce a surrogate-assisted universal actuator approach to the modeling and simulation of coupled mechanical systems. To achieve this, a surrogate-assisted UHS is proposed as an alternative to lumped fluid theory in the framework of a hydraulically actuated and coupled mechanical system. The UHS was prepared by approximating the behaviors of lumped fluid theory using a 1D cylinder model with an approximated force model at the position and velocity levels. The CMA-ES algorithm was used during the training process to minimize differences between the results of the two approaches. The UHS enables real-time simulation solutions by reducing the number of differential equations for coupled hydraulically actuated and coupled systems in a surrogate-assisted monolithic approach.

The robustness of the UHS approach was validated in terms of numerical accuracy and computational efficiency by modeling first the four-bar mechanism and then the forestry crane. Achieving a numerical solution using the new method was approximately 52 times faster than when using the standard lumped fluid theory method, which makes application of the UHS suitable for real-time simulation, optimization, control, state and parameter estimation, and Artificial Intelligence (AI) implementations for coupled mechanical systems.

2 Dynamics of Coupled Systems

A coupled mechanical system comprises multiple interconnected components or subsystems. It can be modeled using monolithic [14,37] or cosimulation [3,49] approaches in terms of differential-algebraic equations (DAEs) or ordinary differential equations (ODEs). As an example, Fig. 1 shows a hydraulically actuated and coupled mechanical system.

Fig. 1
A hydraulically actuated and coupled mechanical system—the hydraulic cylinder actuates the mechanical subsystem of bodies 1 to 6. The corresponding mechanical subsystem is represented by the angles z1, z2, z3, z4, z5, and z6. The hydraulic pressures p1 and p2 describe the hydraulic subsystem.
Fig. 1
A hydraulically actuated and coupled mechanical system—the hydraulic cylinder actuates the mechanical subsystem of bodies 1 to 6. The corresponding mechanical subsystem is represented by the angles z1, z2, z3, z4, z5, and z6. The hydraulic pressures p1 and p2 describe the hydraulic subsystem.
Close modal

The dynamics of the mechanical subsystem in Fig. 1 can be modeled using classical multibody system dynamics formulations [50], whereas lumped fluid theory [35] describes the performance of the hydraulic subsystem. The dynamic modeling of mechanical and hydraulic subsystems in a coupled system is described briefly below.

2.1 Modeling the Mechanics.

Coupled-system mechanics can be modeled using multibody system dynamics. In a multibody system, multiple bodies are interconnected via joints. A body i can be defined using Cartesian velocities Z˙i and accelerations Z¨i. In this study, semirecursive formulations based on relative coordinates describe the system bodies [50]. Using the principle of virtual power, the inertial and external forces acting on a body can be described as Ref. [50]
i=1NbZ˙i*T(M¯iZ¨iQ¯i)=O
(1)
where M¯i6×6 represents the mass matrix, Q¯i6×1 is the external force vector acting on the body, and * denotes the virtual velocities. In Eq. (1), Z¨=TRdz¨+d(TRd)dtz˙ is introduced to formulate the equation of motion for an open loop system as [51,52]
RdTTTM¯TRdz¨=RdTTT(Q¯M¯d(TRd)dtz˙)
(2)

where z˙Nj×jf and z¨Nj×jf are the vectors of relative joint velocities and accelerations, T6Nb×6Nb denotes the constant path matrix, and Rd6Nb×Nj corresponds to the block-diagonal velocity transformation matrix. Here, jf is the degree-of-freedom of joints [50]. Nb and Nj are the numbers of bodies and joints in a multibody system, respectively. Equation (2) defines a system in terms of the relative coordinates z˙ and z¨. Note that this study focuses on the application of the formulation to revolute joints where jf = 1. However, this formulation can also be extended to other kinds of joints [53].

For a closed-loop system, as shown in Fig. 1, the cut-joint technique is needed with a set of Nm cut-joint constraint equations Φ(z)=0. The constraint equations can be accounted for in the equations of motion using the coordinate partitioning method [51]. Φ(z) are assumed to be holonomic and scleronomic in this method [51]. z˙ and z¨ can be mapped onto a set of an independent relative joint velocity vector z˙iNf and an acceleration vector z¨iNf with a velocity transformation matrix RzNb×Nf as
z˙=Rzz˙i,z¨=Rzz¨i+R˙zz˙i
(3)
where R˙z is the time derivative of Rz and Nf is the degree-of-freedom in system. Rz is computed as Rz=[(Φzd)1ΦziINf]. Here, ΦziNm×Nf and ΦzdNm×Nm are the independent and dependent columns of the Jacobian matrix Φz, where Nm represents the number of constraint equations. Substituting Eq. (3) into Eq. (2) yields the equation of motion for a closed-loop system
RzRdT(TTM¯T)RdRzz¨i=RzRdTTT(Q¯M¯D)
(4)

where D=TRd[(Φzd)1(Φ˙zz˙)0]+TR˙dz˙ include the absolute accelerations, when z¨i is set to zero. Further details on D can be found in Ref. [54].

2.2 Modeling the Actuators: Lumped Fluid Theory.

A hydraulic cylinder couples the hydraulic and mechanical subsystems. Figure 2 gives a representation of the hydraulic cylinder.

Fig. 2
A double-acting hydraulic cylinder—pa and pb are the hydraulic pressures in the areas Aa and Ab, respectively
Fig. 2
A double-acting hydraulic cylinder—pa and pb are the hydraulic pressures in the areas Aa and Ab, respectively
Close modal
In the coupling process, the hydraulic force produced by the cylinder Fh=paAapbAbFμ is applied to the mechanical subsystem. Friction force in the cylinder, Fμ, is ignored in this study. Hydraulic circuit pressures and effective bulk modulus Beh can be modeled using lumped fluid theory [35] as
{p˙h=BehVh(dVhdt+Qs),Beh=(1Bo+c=1nhVcVhBc)1
(5)

where p˙h is the first derivative of the hydraulic pressure, Vh is the hydraulic volume, and Qs is the sum of the incoming and outgoing flows. In Eq. (5), Bo defines the bulk modulus of the hydraulic fluid, Vc is the subvolume, and Bc is the bulk modulus of the subvolume. A semi-empirical method can be used to compute the flowrate in Eq. (5) [35].

Flow rate Qd through a directional control valve can be expressed as Qd=CvUsgn(Δp)|Δp|. Cv is the semi-empirical flowrate coefficient, U is the relative position of the spool, sgn(·) is the signum function to define the direction of flowrate, and Δp is the pressure difference over the valve ports. The voltage signal U can be expressed as U˙=UrefUτ. Here, Uref defines the reference voltage signal, and τ is the time constant describing the valve dynamics.

2.3 Monolithic Approach.

In a monolithic approach, the dynamics of mechanical and hydraulics subsystems can be combined to formulate the equations of motion as follows,
{MΣ(z)z¨i=QΣ(z,z˙,p,U),p˙=u0(z,z˙,p,U)
(6)

where MΣ(z)=RzRdT(TTM¯T)RdRz, and QΣ(z,z˙,p,U)=RzRdTTT(Q¯M¯D). Here, p and p˙ are the respective vectors of hydraulic pressure and pressure build-up, and u0 are the pressure variation equations. Equation (6) demonstrates the equations of motion for a hydraulically actuated and coupled mechanical system. The state vector of these equations can be represented as x=[(zi)T(z˙i)TpT]T.

3 Surrogate-Assisted Universal Actuator

A surrogate-assisted hydraulic actuator optimizes the performance of a hydraulic actuator using a simple force model FS [55]. The simple force model can be represented as,
FS=[a1a2a3a4a0][UsSs˙Ss0S1]
(7)

where χ=[a1a2a3a4a0] represents the vector of force parameters, and Y=[UsSs˙Ss0S1]T includes variables in the simple force model. In Eq. (7), a1 is the surrogate gain parameter a2. Parameters a3, a4, and a0 are the surrogate bias, sS is the surrogate actuator position, s˙S is the surrogate actuator velocity, and s0S is the initial surrogate actuator position. The surrogate actuator is activated with the voltage signal U. The simple force model in Eq. (7) is independent of the pressures and flow rates.

In Eq. (7), FS is expressed in terms of linear sS, s˙S and U, as Fh is also the function of these variables in linear form Ref. [56]. Here, U is computed from the valve spool differential equation, representing the valve dynamics. The simple force model was employed in its linear form. However, the nonlinear terms, including the frictional forces in the hydraulic cylinder, have not been considered in the hydraulic force model. This aspect presents an opportunity for further investigation in future studies. CMA-ES optimization minimizes the fitness function f:L,yf(y). The vector y includes the errors in the positions and velocities of the hydraulics and surrogate actuators. L is the length of vector y. The performance of the algorithm is determined by the number of evaluations taken by the CMA-ES algorithm to minimize the function f.

As a result of successful training, the vector of force parameters in Eq. (7) is also optimized to χmin. The solution χmin is the surrogate solution. As a consequence of force optimization, the cylinder and parameters of the hydraulic actuators are optimized in terms of χmin. Therefore, χmin with the simple force model in Eq. (7) can be used as an alternative to conventional lumped fluid theory in the dynamic simulation of any coupled mechanical system. Therefore, χmin is also referred to as the UHS.

This approach requires less number of tuning parameters as compared to the data-driven surrogates. Typically, in data-driven studies, the selection of appropriate parameters is one of the key concerns [57]. Furthermore, the data-driven surrogates use the black-box approach [57], where the procedure of generating a surrogate is based solely on the availability of large amounts of data. However, this approach employs the physics of model as a gray-box in preparing the simulation-driven surrogate.

3.1 Surrogate-Assisted Monolithic Approach.

As can be seen in Eq. (7), the simple model does not depend upon hydraulic pressures and cylinder parameters. In the surrogate-assisted monolithic approach, the dynamics of the coupled system can be described as
M¯Σ(z)z¨i=Q¯Σ(z,z˙,U)
(8)

Equation (8) demonstrates the equations of motion for a UHS-assisted coupled mechanical system. The state vector of these equations can be represented as xS=[(zi)T(z˙i)T]T.

4 Application of Universal Hydraulic Surrogate in Academic and Industrial Applications

The robustness of the surrogate-assisted monolithic approach was verified by implementing it in the modeling of a coupled mechanical system. It was applied to a single cylinder four-bar mechanism and to a forestry crane. As reference solutions, these applications were also solved taking a monolithic approach using standard lumped fluid theory. The preparation and implementation of each UHS for these applications is discussed in the following paragraphs.

4.1 Hydraulic Four-Bar Mechanism.

Figure 3 shows the application of the universal surrogate solution for the four-bar mechanism. The hydraulics of the four-bar mechanism including cylinder, hydraulic volumes, 4/3 directional control valve, pP, and pT are illustrated in Fig. 3(a). Lumped fluid theory was used to compute the dynamics of this mechanism for comparison with the simple force model.

Fig. 3
Hydraulically actuated and surrogate-assisted four-bar mechanisms: (a) lumped-theory-based four-bar mechanism and (b) surrogate model based four-bar mechanism. The surrogate force model Fs is converted into F using the position vector.
Fig. 3
Hydraulically actuated and surrogate-assisted four-bar mechanisms: (a) lumped-theory-based four-bar mechanism and (b) surrogate model based four-bar mechanism. The surrogate force model Fs is converted into F using the position vector.
Close modal

Detailed modeling of the hydraulically actuated four-bar mechanism with lumped fluid theory can be found in the reference study [39]. The physical parameters of the hydraulically actuated four-bar mechanism and its UHS are presented in Table 1. Figure 4 shows the UHS-actuated four-bar mechanism.

Fig. 4
Lumped fluid theory model of universal hydraulic surrogate
Fig. 4
Lumped fluid theory model of universal hydraulic surrogate
Close modal
Table 1

Physical parameters of the hydraulically actuated four-bar mechanism and its lumped fluid theory model of universal hydraulic surrogate

ParameterValue
(a) Cylinder and hydraulic parameters used in lumped fluid theory
Cylinder80/35×535 mm
Stroke1020 mm
V16.35×105 m3
V26.35×105 m3
Bo1.6×109 N/m2
Bc2.1×1011 N/m2
Bh2.3×1010 N/m2
Cv2.138×108m3/sPa
pP140 bar
pT1 bar
(b) Physical properties of the four-bar mechanism and UHS
L1(OA)2 m
L2(AB)8 m
L3(BC)5 m
OC10 m
m12 kg
m28 kg
m35 kg
m100 kg
k10 N/m
c0.85 Ns/m
ParameterValue
(a) Cylinder and hydraulic parameters used in lumped fluid theory
Cylinder80/35×535 mm
Stroke1020 mm
V16.35×105 m3
V26.35×105 m3
Bo1.6×109 N/m2
Bc2.1×1011 N/m2
Bh2.3×1010 N/m2
Cv2.138×108m3/sPa
pP140 bar
pT1 bar
(b) Physical properties of the four-bar mechanism and UHS
L1(OA)2 m
L2(AB)8 m
L3(BC)5 m
OC10 m
m12 kg
m28 kg
m35 kg
m100 kg
k10 N/m
c0.85 Ns/m

4.1.1 Designing the Universal Hydraulic Surrogate.

Figure 4 shows the schematic diagram of the UHS for the four-bar mechanism. It comprises the mechanism hydraulics, the spring-damper system, and mass m.

Following lumped fluid theory, the dynamics of this model can be represented as
{s˙=s˙,s¨=(Fh+cs˙k(sx0))m,p˙1=Be1V1(Qd1A1s˙),p˙2=Be2V2(A2s˙Qd2)
(9)

where s˙ is actuator velocity, s¨ is actuator acceleration, p˙1 and p˙2 are the first-time derivatives of the pressures p1 and p2, and Fh is hydraulic force. Qd1 and Qd2 are the flow rates in volumes V1 and V2, respectively. Be1 and Be2 are the effective bulk moduli, and A1 and A2 are the areas of the piston side and piston-rod side of the hydraulic cylinder, respectively. The hydraulic force Fh is computed as Fh=p1A1p2A2.

The UHS of the model in Fig. 4 replaces Fh with the simple force model described in Eq. (7). Designing a UHS solution for the mechanism requires finding the initial hydraulic force Fh0, the hydraulics and parameters k, c and m. The initial hydraulic force Fh0 is calculated using the static equilibrium of the four-bar mechanism at the angle z0=60deg. In this configuration, the initial pressure p20=3.97bar is used to calculate initial pressure p10 using the expression p10=(Fs0+p20A2)/A1. Furthermore, at static equilibrium, k and m can be found to ensure that the surrogate force expression covers the complete range of the forces generated by the actuator. The initial spring length is x0=0mm. However, parameters c is tuned until minimum error is achieved between the lumped fluid theory and UHS model at the force level.

4.1.2 Optimization Algorithm.

The damping constant c and the CMA-ES parameters are selected by trial and error. The aim is to minimize the PN-RMSE between the position, velocity, and acceleration states of the surrogate model and lumped fluid theory. The initial dataset for training the surrogate was considered to be a1=[1000900]T,a2=[100100]T,a3=[200200]T,a0=1, a4 = a2, and τ=0.0045. Here, τ is the time constant defining the valve dynamics.

The training of χ=[a1a2a3a4a0] with the behaviors of lumped fluid theory was carried out using the CMA-ES. The objective of the CMA-ES algorithm is to minimize the fitness function f:L,yf(y). f can be described in the UHS case for the four-bar mechanism as
f=(|ssSs0S|)N+(|s˙s˙S|)N
(10)

where s,sS,s˙, and s˙S are the actuator positions and velocities from lumped fluid theory and the surrogate models, respectively. Initial actuator position sS0 corresponds to the initial position of the surrogate model. This training yields χmin described in Table 10 to two decimal places, which is used in the simple force model to solve the four-bar mechanism shown in Fig. 3(b).

Table 10

The UHS solutions for the four-bar mechanism and forestry crane

χminχlminχtmin
1.56×1031.35×1051.28×105
2.19×1031.19×1051.29×105
3.30×1011.24×1026.03×102
1.22×1022.36×1041.20×104
1.43×1021.67×1041.10×104
1.18×1059.45×1051.14×105
6.71×1031.43×1061.23×104
2.67×1022.96×1023.43×102
χminχlminχtmin
1.56×1031.35×1051.28×105
2.19×1031.19×1051.29×105
3.30×1011.24×1026.03×102
1.22×1022.36×1041.20×104
1.43×1021.67×1041.10×104
1.18×1059.45×1051.14×105
6.71×1031.43×1061.23×104
2.67×1022.96×1023.43×102

4.2 Forestry Crane.

The application of the UHS solution for the forestry crane is shown in Fig. 5. Figure 5(a) reveals that the forestry crane is a two-degree-of-freedom coupled mechanical system. The bodies in the mechanical subsystem are the pillar, lift boom, bracket 1, bracket 2, and tilt boom. The mechanical subsystem also includes a multibody closed-loop system. The geometric, mass, and inertial properties of the crane bodies are shown in Tables 2 and 3, respectively.

Fig. 5
Application of Universal Hydraulic Surrogate in a forestry crane, available at the Laboratory of Intelligent Machines, LUT University: (a) experimental setup of forestry crane and (b) surrogate models based forestry crane. The surrogate force models Fsl and Fst are converted into the vector form Fl and Ft using the position vectors.
Fig. 5
Application of Universal Hydraulic Surrogate in a forestry crane, available at the Laboratory of Intelligent Machines, LUT University: (a) experimental setup of forestry crane and (b) surrogate models based forestry crane. The surrogate force models Fsl and Fst are converted into the vector form Fl and Ft using the position vectors.
Close modal
Table 2

Geometric properties of the bodies

BodyLengthxy
Lift boom2.87 m1.229 m0.055 m
Bracket 10.45 m0.257 m0.004 m
Bracket 20.48 m0.213 m0.000 m
Tilt boom2.77 m0.660 m0.251 m
BodyLengthxy
Lift boom2.87 m1.229 m0.055 m
Bracket 10.45 m0.257 m0.004 m
Bracket 20.48 m0.213 m0.000 m
Tilt boom2.77 m0.660 m0.251 m
Table 3

Inertial properties of the bodies

BodyMassInertia
Lift boom143.66 kg110.45 kgm2
Bracket 111.52 kg0.27 kgm2
Bracket 27.90 kg0.22 kgm2
Tilt boom157.88 kg67.05 kgm2
BodyMassInertia
Lift boom143.66 kg110.45 kgm2
Bracket 111.52 kg0.27 kgm2
Bracket 27.90 kg0.22 kgm2
Tilt boom157.88 kg67.05 kgm2

The forestry crane is actuated by lift and tilt hydraulic cylinders and the hydraulics. Following lumped fluid theory, the pressures inside the lift cylinder can be represented by p1 and p2, whereas the pressures inside the tilt cylinder can be described as p3 and p4, respectively. The detailed modeling of the forestry crane and its physical parameters can be found in Ref. [56]. The cylinder and valve parameters for the forestry crane are listed in Table 4.

Table 4

Hydraulic parameters

SymbolsValue
Lift cylinder100/56×535mm
Tilt cylinder100/56×780mm
Lift stroke820 mm
Tilt stroke1050 mm
V11.9002×104 m3
V21.9002×104 m3
V32.5335×104 m3
V42.5335×104 m3
Cv12.138×108m3/sPa
Cv22.138×108m3/sPa
pP100 bar
pT1 bar
SymbolsValue
Lift cylinder100/56×535mm
Tilt cylinder100/56×780mm
Lift stroke820 mm
Tilt stroke1050 mm
V11.9002×104 m3
V21.9002×104 m3
V32.5335×104 m3
V42.5335×104 m3
Cv12.138×108m3/sPa
Cv22.138×108m3/sPa
pP100 bar
pT1 bar

In the surrogate-assisted monolithic approach, the lift cylinder, tilt cylinders, and their hydraulics are replaced by the lift and tilt cylinder surrogates. The design and implementation of these UHSs are explained in the following paragraphs.

4.2.1 Designing Lift and Tilt Universal Hydraulics Surrogates.

In Fig. 5(b), the lift cylinder and tilt cylinder surrogates demonstrate the UHS applications for the forestry crane. Table 5 describes the spring-damper properties of the lift and tilt cylinder surrogates. Hydraulic parameters are not required in the simulation of a simple forces-based model of the forestry crane. As shown by Fig. 5(a), the lift cylinder pushes and the tilt cylinder pulls on the bodies. The lift and tilt cylinder forces are positive and negative, respectively.

Table 5

Physical parameters of the lumped fluid theory models of lift and tilt universal hydraulic surrogates

ParameterValue
(a) Spring-damper in lift cylinder surrogate
m12100 kg
k15024 N/m
c18350 Ns/m
(b) Spring-damper in tilt cylinder surrogate
m2100 kg
k22286 N/m
c26000 Ns/m
ParameterValue
(a) Spring-damper in lift cylinder surrogate
m12100 kg
k15024 N/m
c18350 Ns/m
(b) Spring-damper in tilt cylinder surrogate
m2100 kg
k22286 N/m
c26000 Ns/m

Figure 6 illustrates the universal hydraulic surrogate solutions for the hydraulics in the forestry crane. At static equilibrium, Fl0=2.45×104N, and Ft0=2.76×103N. These are the initial forces in the lift and tilt cylinders, respectively. Therefore, to produce high-magnitude positive and negative forces, the universal surrogate solutions must be configured as presented in Fig. 6.

Fig. 6
Universal hydraulic surrogate solutions to the hydraulics of forestry crane application (a) lumped fluid theory model lift cylinder hydraulics surrogate and (b) lumped fluid theory of tilt cylinder hydraulics surrogate
Fig. 6
Universal hydraulic surrogate solutions to the hydraulics of forestry crane application (a) lumped fluid theory model lift cylinder hydraulics surrogate and (b) lumped fluid theory of tilt cylinder hydraulics surrogate
Close modal
Using lumped fluid theory, the dynamics of the lift cylinder surrogate can be represented by the following expressions:
{s˙1=s˙1,s¨1=(Fl+cs˙1k1(s1xl0)m1g)m1g,p˙1=Be1V1(Qd1A1s˙1),p˙2=Be2V2(A2s˙1Qd2)
(11)
Similarly, the dynamics of the tilt cylinder surrogate are expressed as
{s˙2=s˙2,s¨2=(Ftc2s˙2+k2(s2xt0))m2,p˙3=Be3V3(Qd3A3s˙2),p˙4=Be4V4(A4s˙2Qd4)
(12)

s1, s2, s˙1, and s˙2 are the lift and tilt cylinder positions and velocities in Eqs. (11) and (12). m1 and m2 are the masses, k1 and k2 are the spring constants, c1 and c2 are the damping constants, xl0 and xt0 are the initial spring lengths, and Fl and Ft are the actuator forces. Be1,Be2,Be3, and Be4 are the effective bulk modulus in V1, V2, V3, and V4 hydraulic volumes. p˙1,p˙2,p˙3, and p˙4 are the pressure derivatives, and Qd1,Qd2,Qd3, and Qd4 are the flow rates in the lift and hydraulic cylinder UHSs, respectively.

At static equilibrium, k1, k2, m1, and m2 are found to ensure UHS solutions cover the full range of hydraulic actuator force. The initial spring lengths are considered xl0=250mm and xt0=250mm in the surrogates models. Parameters c1 and c2 are tuned until a satisfactory level of accuracy is achieved between lumped fluid theory and the surrogate models at the force levels.

4.2.2 Optimization Algorithm.

The initial CMA-ES training parameters are chosen through trial and error until a satisfactory level of accuracy in both states and forces is achieved between lumped fluid theory and the surrogate models. The simple force models in the lift and tilt UHS can be expressed as
{Fsl=al1U1+al2s1+al3s˙1+al4s10+al0,Fst=at1U2+at2s2+at3s˙2+at4s20+at0
(13)

where χl=[al1al2al3al4al0] and χt=[at1at2at3at4at0] are vectors of force parameters in the UHS of the lift and tilt cylinders. s10 and s20 are the initial actuator positions of the lift and tilt cylinders. The initial parameters of χl and χt are listed in Table 9.

Table 9

Initial parameters of the simple force models in lift cylinder and tilt cylinder surrogates

ParameterValue
(a) Simple force in the lift surrogate
al1[18002000]T
al2[200200]T
al3[30003000]T
al4[200200]T
al01
(b) Simple force in the tilt surrogate
at1[1000900]T
at2[100100]T
at3[45004500]T
at4[100100]T
at01
ParameterValue
(a) Simple force in the lift surrogate
al1[18002000]T
al2[200200]T
al3[30003000]T
al4[200200]T
al01
(b) Simple force in the tilt surrogate
at1[1000900]T
at2[100100]T
at3[45004500]T
at4[100100]T
at01
The CMA-ES algorithm minimizes the fitness functions f1:L1,y1f(y1) and f2:L2,y2f(y2) using the initial parameters of χl and χt. The parameters χl and χt are trained against lumped fluid theory using the following expressions:
{f1=(|s1s1Ss10S|)N1+(|s˙1s˙1S|)N1,f2=(|s2s2Ss20S|)N2+(|s˙2s˙2S|)N2
(14)

where s1,s1S,s2, and s2S are the actuator positions of the lift and tilt hydraulic cylinders using lumped fluid theory and the simple model, respectively. s10S and s20S are the initial actuator positions in each lift and tilt UHS. s˙1,s˙1S,s˙2, and s˙2S are the corresponding actuator velocities according to lumped fluid theory and the simple model, respectively.

Table 10 describes the universal surrogate result training solutions χlmin and χtmin. χlmin and χtmin are coupled with the multibody system dynamics using the simple force expressions described in Eq. (13) instead of lumped fluid theory.

5 Results and Discussion

The preparation and implementation of each UHS for the four-bar mechanism and forestry crane were carried out in the matlab environment. The universal surrogate was trained on a computer with an 11th generation Intel® Core™ i5-11500H CPU running at 2.92 GHz with 32 GB RAM. The operating system was 64-bit Windows 11. In the figures, lumped fluid theory solutions are indicated using () and referred to as lumped theory. The surrogate solutions, solved using the simple force model, are illustrated using (). The performance of each UHS was evaluated based on training, numerical accuracy, and computational efficiency for the four-bar mechanism and the forestry crane applications.

5.1 Universal Hydraulic Surrogate Training.

The accuracy of UHS solutions in any application depends upon their prediction accuracy of states and forces with respect to lumped fluid theory. To this end, surrogate model states such as position, velocity, acceleration, and forces must be compared to states from lumped fluid theory. Figure 7 describes the accuracy of predicted states for the UHS solutions for the four-bar mechanisms and forestry crane with respect to lumped theory, respectively.

Fig. 7
Performance of the states in surrogate models and lumped fluid theory during the training process: (a) four-bar mechanism UHS position, (b) four-bar mechanism UHS velocity, (c) lift hydraulics UHS position, (d) lift hydraulics UHS velocity, (e) tilt hydraulics UHS position, and (f) tilt hydraulics UHS velocity
Fig. 7
Performance of the states in surrogate models and lumped fluid theory during the training process: (a) four-bar mechanism UHS position, (b) four-bar mechanism UHS velocity, (c) lift hydraulics UHS position, (d) lift hydraulics UHS velocity, (e) tilt hydraulics UHS position, and (f) tilt hydraulics UHS velocity
Close modal

As can be seen in Figs. 7(a), 7(b), 7(e), and 7(f), each UHS was actuated for 10 s in the training phase. However, in case of lift cylinder, 5 s simulation is considered enough for preparing lift UHS. The surrogate models were moved from the initial position to the minimum actuator length and then brought back to the initial actuator length. Figures 7(a)–7(f) reveal that the actuator positions and velocities of the surrogate solutions very closely follow the behaviors of the lumped fluid theory models. Further, in Table 6, the PN-RMSE values indicate the high accuracy of the surrogate model states with respect to lumped fluid theory.

Table 6

Performance of CMA-ES in UHS training for the four-bar mechanism and forestry crane applications

CasesxPN-RMSEEvaluationsfminTraining time
Four-bar UHSs, s˙0.01%, 0.87%23129.8×104161 s
Lift UHSs1, s˙10.01%, 3.04%19824.00×103158 s
Tilt UHSs2, s˙20.01%, 2.57%36634.20×103238 s
CasesxPN-RMSEEvaluationsfminTraining time
Four-bar UHSs, s˙0.01%, 0.87%23129.8×104161 s
Lift UHSs1, s˙10.01%, 3.04%19824.00×103158 s
Tilt UHSs2, s˙20.01%, 2.57%36634.20×103238 s

In Table 6, the low PN-RMSE of 0.01% in s, s1, and s2 demonstrates the high accuracy of the surrogate models compared to lumped fluid theory at the position level. The PN-RMSE of 3.04% in s˙,s˙1, and s˙2, which is relatively higher than at the position level, is attributed to the higher stiffness values used in the preparation of the UHS model compared to lumped fluid theory. However, as indicated in Fig. 7, the UHS follows the behaviors of lumped theory at the velocity levels.

The training time in Table 6 is shorter than the conventional data-driven surrogates. The selection of vectors χ,χl, and χt is not based on the model. Instead, these vectors were identified through a procedure that is both simple and reliable. Consequently, this approach facilitated the quick and appropriate identification of the surrogate model parameters. Further, Table 11 of Appendix  A also compares the PN-RMSE values in the accelerations and forces of the UHS models with respect to lumped fluid theory. This table shows that accuracy in the accelerations and forces is required in preparing the UHS solution for an application.

Table 11

Percent normalized root mean square error of accelerations and forces of UHS

CasesStatesPN-RMSE
Four-bar UHSs¨, F1.82%, 0.95%
Lift UHSs¨1, Fl7%, 7%
Tilt UHSs¨2, Ft1.96%, 1.54%
CasesStatesPN-RMSE
Four-bar UHSs¨, F1.82%, 0.95%
Lift UHSs¨1, Fl7%, 7%
Tilt UHSs¨2, Ft1.96%, 1.54%

No UHS replicates the high frequencies of lumped fluid theory at the acceleration and forces levels. This demonstrates that UHS solutions do not represent the compressible nature of hydraulic oil. However, compared to the lumped theory, each UHS provides reliable and computationally efficient solutions by solving less stiff numerical differential equations. Note that simple force expression is inspired from Ref. [55] and terms representing compressible oil might be missing in Eq. (7). Further, this study ignores the friction force Fμ in hydraulic cylinders, which reduces the frequency of these oscillations. This behavior can be observed in the reference studies [14,56], where similar systems were studied.

5.2 Validating Universal Hydraulic Surrogate Performance in Applications.

χmin,χlmin, and χtmin are the computed results of the UHS training. They are coupled with the surrogate-assisted dynamics of the four-bar mechanism and the forestry crane via the simple force models. The solutions for the hydraulics in these applications with the simple force models are explained in terms of numerical accuracy and computational efficiency in the following paragraphs. These results are also compared with lumped fluid theory.

5.2.1 Numerical Accuracy.

The χmin-based simple force model solution for the hydraulics of four-bar mechanism is shown in Figs. 8(a) and 8(b) for a 22 s simulation. The simulation time in these figures is more than the simulation time in UHS training. The four-bar mechanism was actuated using a different input signal and simulation time than for its UHS. See Fig. 7. Angle and angular velocity for the simple model follow patterns similar to lumped theory.

Fig. 8
Comparing the accuracy of dynamic simulation solutions for the simple model with respect to lumped theory: (a) angle of z the four-bar mechanism, (b) angular velocity z˙ of the four-bar mechanism, (c) lift boom angle z1, (d) lift boom angular velocity z˙1, (e) tilt boom angle z4, and (f) tilt boom angular velocity z˙4
Fig. 8
Comparing the accuracy of dynamic simulation solutions for the simple model with respect to lumped theory: (a) angle of z the four-bar mechanism, (b) angular velocity z˙ of the four-bar mechanism, (c) lift boom angle z1, (d) lift boom angular velocity z˙1, (e) tilt boom angle z4, and (f) tilt boom angular velocity z˙4
Close modal

The PN-RMSE in z of the simple mode with respect to lumped fluid theory is 2.06% whereas this difference is 0.75% for z˙. Further, this difference occurs at the static position of the hydraulic cylinder during 2–4, 8–12, 14–18, and 20–22 s of the simulation. However, the difference between z and z˙ is quite small. Similar mass and inertial properties of bodies have been used for the simple model and lumped fluid theory.

The dynamic simulation solutions of the forestry crane using the simple model closely resemble the solutions from lumped fluid theory. Figures 8(c)8(f) reveal that z1, z4, z˙1, and z˙4 computed from the simple model follow a similar pattern despite the differences in input signals from the UHS. The PN-RMSE in z1 and z4 are 1.29% and 6.72%, respectively. In contrast, this difference is 2.80% and 4.26% for z4 and z˙4.

Comparatively, higher PN-RMSE occurs in z4 than z1 due to the drifting of the tilt cylinder surrogate at the static position from 5 to 5.5 s of the simulation. The disparities in z˙1 and z˙4 are the consequence of the corresponding UHS differences at the velocity levels shown in Figs. 7(d) and 7(f). Valve dynamics are not used for the simple model. The lift and tilt cylinder surrogates are kept at static equilibrium positions for a shorter time than in the four-bar mechanism.

5.2.2 Computational Efficiency.

The performance of UHS is further evaluated in terms of computational efficiency. This is accomplished by solving the dynamics of the used cases using both stiff and nonstiff integration schemes, with an initial time-step of 1×103 s. The stiff integration schemes included matlab-integrated solvers such as ode15 s, ode23 s, ode23t, and ode23tb [58]. Nonstiff solvers are ode45, ode23, ode78, ode89, ode113, and RK4 [58]. During these simulation experiments, the variable-step solver parameters were used as the relative tolerance 1×1010, absolute tolerance 1×1010, maximum step size 1×102 and maximum order 2 for matlab ODE solvers. However, fixed-step solver was used with RK4 method. Note the numerical accuracy of UHS solutions remains approximately same with all above stiff and nonstiff integration schemes.

Figure 9 shows the comparison of computational time taken by the simple model and lumped fluid theory in solving the case studies. The vertical axis represents the computational time required by the modeling method, displayed on a logarithmic scale with a base of 10. The horizontal axis shows the different integration schemes used in the simulation experiment. In Fig. 9, the blue bars are significantly higher than those of the simple model, indicating that the lumped fluid approach requires more computational time.

Fig. 9
Comparing the computational time of simulation solutions on the logarithmic scale for solving the dynamics of four-bar mechanism and forestry crane: (a) computational time in the four-bar mechanism and (b) computational time in the forestry crane (color version online)
Fig. 9
Comparing the computational time of simulation solutions on the logarithmic scale for solving the dynamics of four-bar mechanism and forestry crane: (a) computational time in the four-bar mechanism and (b) computational time in the forestry crane (color version online)
Close modal

Tables 7 and 8 further summarize the computational efficiency of these two approaches with respect to the different integration schemes. In these tables, Method 1 represents lumped fluid theory, and the simple model is indicated by Method 2. The ratio column corresponds to the efficiency of the simple model with respect to lumped fluid theory. It is obtained by dividing lumped fluid theory's computational time with that of the simple model. It demonstrates how much faster the simulation solutions get by using the simple model as compared to lumped fluid theory.

Table 7

Comparison of the computational time using the simple model and lumped fluid in solving the four-bar mechanism

ode45ode23ode78ode89ode113ode15 sode23 sode23tode23tbRK4
Method 149379.5019.8523.889.07357.351444.95219.14440.446.86
Method 28.1212.588.6416.595.0310.2927.996.1912.718.53
Ratio6.0430.162.301.441.8034.7251.6235.3834.640.80
ode45ode23ode78ode89ode113ode15 sode23 sode23tode23tbRK4
Method 149379.5019.8523.889.07357.351444.95219.14440.446.86
Method 28.1212.588.6416.595.0310.2927.996.1912.718.53
Ratio6.0430.162.301.441.8034.7251.6235.3834.640.80
Table 8

Comparison of the computational time using the simple model and lumped fluid in solving the forestry crane

ode45ode23ode78ode89ode113ode15 sode23 sode23tode23tbRK4
Method 129.70206.1019.30277.71471311.80129636111.24
Method 28.0132.807.2011.982.2536.80155.1028469.50
Ratio3.706.282.672.253.413.998.454.507.851.20
ode45ode23ode78ode89ode113ode15 sode23 sode23tode23tbRK4
Method 129.70206.1019.30277.71471311.80129636111.24
Method 28.0132.807.2011.982.2536.80155.1028469.50
Ratio3.706.282.672.253.413.998.454.507.851.20

Table 7 shows that maximum computational advantage achieved from using the simple force model in the case of a four-bar mechanism with ode23 s results in achieving a solution 52 times faster than with lumped fluid theory. However, in the case of a forestry crane, the simple model is approximately 8.5 times faster under the given solver settings. Computational efficiency, of course, depends on the case study, the implementation details, and the operating system used [59]. The differences in the computational efficiency occur due to less number of differential equations in simple model. The lumped fluid theory introduces additional stiff pressure differential equations, resulting in reduced computational efficiency compared to the simple model. However, in the case of RK4, the lumped fluid theory model takes approximately the same computation time as the simple model in the four-bar mechanism and forestry crane applications. This is due to less stiffness in differential equations. Note, in general, RK4 is not recommended for the stiff differential equations due to requirement of small integration time-step for stability [60].

The performance of ode23 s solver at variable time-step is further explained in Tables 12 and 13 of Appendix  C. In both applications, lumped fluid theory takes higher number of successful steps, function evaluations, LU decompositions and linear system solutions [61]. This indicates prior computational speed of simple model as compared to lumped fluid theory due to removing the pressure differential equations. However, it can be observed in Tables 12 and 13, that number of failed steps occur more in case of simple model. This suggests the better stability of lumped fluid theory [61].

Table 12

Comparison of the computational time using the simple model and lumped fluid in solving the forestry crane

Successful stepsFailed stepsEvaluationsLU decompositionLinear systems
Method 11.53×1061871.23×1071.53×1064.61×106
Method 24.58×1048492.77×1054.67×1041.40×105
Successful stepsFailed stepsEvaluationsLU decompositionLinear systems
Method 11.53×1061871.23×1071.53×1064.61×106
Method 24.58×1048492.77×1054.67×1041.40×105

6 Conclusion

This study presents a surrogate-assisted universal actuator approach for high-fidelity coupled mechanical systems within the domain of simulation-driven surrogates. It can provide more accurate predictions outside the training data for a hydraulically actuated and coupled mechanical system compared to data-driven surrogates. As an example, in the framework of hydraulic coupled systems, a UHS as an alternative to lumped fluid theory has been proposed by using a simple force model. The dynamics of UHS has been described using lumped fluid theory in a 1D cylinder and mass spring-damper system.

The behaviors of the simple force model were approximated against lumped fluid theory at the position and velocity levels using the CMA-ES optimization algorithm during the process to minimize differences between the standard and UHS approaches. As demonstrated in Table 6, the maximum training time taken by CMA-ES training was 238 s for the forestry crane application. This training time is considerably shorter than that of the data-driven surrogates [2426] and conventional simulation-driven surrogates [3234].

These surrogate solutions were implemented in the four-bar mechanism and forestry crane applications using a surrogate-assisted monolithic approach. Note that the inertial properties and lengths of bodies in case examples were not provided in the training phase. However, masses and angles are needed to compute the forces at the static equilibrium. These forces are needed to design the mass spring-damper properties in the surrogate models to ensure that the surrogate force expression covers the complete range of the forces generated by the actuator. The UHS solutions can actuate the booms in case examples beyond the training data due to the physics of the simulation models.

With fewer differential equations, the UHS offers significant advantages in real-time simulation for coupled mechanical systems. Tables 12 and 13 demonstrate that, in both applications, simulating the coupled mechanical system with the simple force model consumes substantially less computational power compared to lumped fluid theory. This difference is observed in terms of successful steps, function evaluations, LU decompositions and linear system solutions. The lumped fluid theory yields more stable solutions but requires greater computational resources, making it less efficient compared to the simple model. The maximum computational benefits achieved by this new approach in these applications resulted in achieving solutions approximately 52 times faster with respect to lumped fluid theory. Regarding accuracy, the PN-RMSE between the states of the approximated force model and lumped fluid theory is approximately 2.04% and 6.95%, respectively.

This new approach offers opportunities for businesses and research in real-time simulation, simulation-driven surrogates, optimization, control, state and parameter estimation, and AI applications for coupled mechanical systems. By reducing the number of differential equations, it provides advantages in linearizing nonlinear dynamic systems. For future studies, a simple force expression should be derived analytically that may account for the compressible nature of hydraulic oil. The nonlinear simulation-driven surrogates can also be studied for a more in-depth investigation of the problems in question. This approach will allow for further investigation into the sensitivity of this method. In these studies, the results of UHS, lumped fluid theory and analytically derived expression should be compared to provide valuable insights for the readers. This exploration could pave the way for researchers to optimize and simulate complicated high-fidelity problems such as contacts, friction, biomechanics, computational fluids, electric, and hybrid electric systems, etc., in real-time.

Acknowledgment

This work was supported by the Business Finland (SANTTU-Oulu 8896/31/2021 and SANTTU-LUT 8859/31/2021). Authors would also like to acknowledge Dr. Scott Semken from the Laboratory of Machine Design, Department of Mechanical Engineering, LUT University for providing proof reading for this article.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

Aa, Ab =

hydraulic cylinder area (m2)

Bc,Beh,Bo =

bulk modulus (Pa)

c =

damping constant (Ns/m)

D =

matrix

Fh, FS =

force (N)

Fμ =

frictional force (N)

k =

Spring constant (N/m)

M¯ =

mass matrix

ph =

hydraulic pressure (Pa)

p =

hydraulic pressure vector

p˙ =

time derivative of hydraulic pressure vector

Qd =

flow rate (m3s1)

Q¯ =

force vector

r˙ =

translational velocity vector

r¨ =

translational acceleration vector

s =

actuator position (m)

s˙ =

actuator velocity (ms1)

Rd =

block-diagonal velocity transformation matrix

Rz =

second velocity transformation matrix

T =

constant path matrix

U =

control signal (V)

Vh =

hydraulic volumes (m3)

zi =

angle of body

z =

relative position vector

Z˙ =

Cartesian velocity vector

Z¨ =

Cartesian acceleration vector

Greek Symbols
Φ =

vector of kinematic constraint equations

Φz =

Jacobian matrix of the kinematic constraint equations

Φzd =

dependent columns of Jacobian matrix

Φzi =

independent columns of Jacobian matrix

χ,χl,χt =

vector of surrogate force parameters

Y =

variables in simple force model

Dimensionless Groups
a0,a1,a2,a3,a4 =

surrogate force parameters

jf =

degree of freedom in joints

Nb =

number of bodies

Nf =

degree of freedom in system

Nj =

number of joints

Nm =

number of the constraint equations

Superscripts and Subscripts
h =

hydraulics

d =

dependent coordinates

i =

independent coordinates

i =

number of body

UHS Solutions for the Applications

The initial parameters in χl and χt of simple force models are described in Table 9. The parameter vectors χl and χt are needed to compute the simple force models for the lift and tilt forces in Eq. (13), respectively. The parameters in χl and χt are adjusted through trial and error until satisfactory accuracy between the lumped fluid theory and UHS is achieved.

Table 10 presents the optimized values of these parameters in simple force models for the four-bar mechanism and forestry cranes. These parameters are obtained in result of surrogate training against the lumped theory using CMA-ES. The optimized χmin, χlmin, and χtmin described in this table are used as universal actuators in the surrogate-assisted monolithic approach.

Accelerations and Forces of Surrogate Models

The PN-RMSE values in accelerations and forces of UHS models with respect to lumped fluid theory are shown in Table 11. These values represent both the four-bar mechanism and forestry crane. The maximum PN-RMSE values occurred in case of lift UHS. This difference can also be observed in Fig. 7(c) and 7(d), where lumped theory shows more oscillations in comparison to surrogate models. This table demonstrates that accuracy in the acceleration and forces is required in preparing the UHS solution for an application.

Statistics of ode23 s

Table 12 presents a comparison of the computational time taken by the simple model and the lumped fluid model. The comparison is based on solving the four-bar mechanism using the ode23 s solver. Method 1, indicating lumped fluid theory, takes more successful steps, evaluations, LU decomposition and linear systems as compared to Method 2.

This aligns to the superior computational speed of simple model, represented by Method 2, in solving the four-bar mechanism. However, Method 1 has less failed steps. This indicates more stability of lumped fluid theory compared to the simple model.

Table 13 further verifies this trend in solving the forest crane example with ode23 s solver. As observed, the lumped fluid theory requires a greater number of successful steps, evaluations, LU decompositions, and linear system solutions, indicating lower efficiency in comparison to the alternative method. However, the comparison of failed steps in Table 13 corresponds to the prior stability lumped fluid theory.

Table 13

Comparison of the computational time using the simple model and lumped fluid in solving the forestry crane

Successful stepsFailed stepsEvaluationsLU decompositionLinear systems
Method 12.63×1053063.42×1062.63×1057.91×105
Method 26.22×10411315.62×1056.33×1041.90×105
Successful stepsFailed stepsEvaluationsLU decompositionLinear systems
Method 12.63×1053063.42×1062.63×1057.91×105
Method 26.22×10411315.62×1056.33×1041.90×105

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